Wave Optics Module. Model Library Manual VERSION 4.4

Transcription

1 Wave Optics Module Model Library Manual VERSION 4.4

2 Wave Optics Module Model Library Manual COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; 7,623,991; and 8,457,932. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement ( and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see Version: November 2013 COMSOL 4.4 Contact Information Visit the Contact COMSOL page at to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at Other useful links include: Support Center: Product Download: Product Updates: COMSOL Community: Events: COMSOL Video Center: Support Knowledge Base: Part number: CM023503

3 Beam Splitter Solved with COMSOL Multiphysics 4.4 Introduction A beam splitter is used for splitting a beam of light in two. One way of making a splitter is to deposit a thin layer of metal between two glass prisms. The beam is slightly attenuated within the layer and then split into two paths. This example models the thin metal layer using a transition boundary condition, which reduces the memory requirements. Losses in the metal layer are also computed. Output Thin metal layer Input Output Figure 1: A beam splitter composed of two prisms with a thin layer of metal between them. Model Definition Model the beam splitter in the 2D plane, as shown in Figure 1, under the assumption that the electric field is polarized perpendicular to the plane. A Gaussian beam of wavelength 700 nm propagates in the x direction through the glass prism of refractive index n 1.5. A 13 nm thin layer of silver sandwiched between the two prisms splits the beams. The model geometry is a square region around the region where the Gaussian beam crosses the silver layer. The focus of the beam is at the left boundary, so the expression for the beam intensity at the focal plane can be used as the excitation. The expression for the relative electric field intensity at the focal plane of a Gaussian is 1 BEAM SPLITTER

4 Ey y = exp --- w 2 (1) where w = 3500 nm is the beam waist, and the y = 0 line is the centerline of the beam. Use this expression in a Port boundary condition on the left side to model the incident beam. Model all the other domain boundaries using Scattering Boundary Conditions. These conditions are appropriate when they are placed several wavelengths away from any scattering objects and the wave is known to be traveling at normal or almost normal incidence. The thin silver layer is modeled using a Transition Boundary Condition. At a free-space wavelength of 700 nm, the dielectric of silver is about r i, where the imaginary part accounts for the losses. Thus, you can set the conductivity of the metal to zero. This boundary condition allows for a discontinuity in the fields across the interface by splitting the mesh at the boundary. It can introduce both losses and a phase shift across the interface. It does not require a mesh of the thickness of the domain, and thus saves significant memory. Mesh the two domains with triangular elements, with the maximum size set such that there are six elements per wavelength in the glass. Results and Discussion Figure 2 shows the electric field intensity in the modeling domain. The beam is split into two beams, one propagating in the x direction and the other one in the y direction. The splitting can be evaluated by computing the flux crossing the incoming boundary and the two outgoing boundaries. Figure 3 plots the power crossing these boundaries as well as the losses at the mirror. 2 BEAM SPLITTER

5 Figure 2: The electric field intensity shows that the incoming beam is split into two beams of approximately equal intensity. Figure 3: The power flux crossing the input boundary and the two output boundaries as well as the losses at the silver surface. 3 BEAM SPLITTER

6 Model Library path: Wave_Optics_Module/Optical_Scattering/ beam_splitter Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description lda0 700[nm] 7.000E-7 m Wavelength f0 c_const/lda E14 1/s Frequency h_max 0.2*lda E-7 m Maximum mesh size eps_ag *i i Relative dielectric constant, Silver Here, c_const is a predefined COMSOL constant for the speed of light in vacuum. 4 BEAM SPLITTER

7 GEOMETRY 1 1 In the Model Builder window, under Component 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose µm. Create a triangle using Polygon for one prism. Polygon 1 1 Right-click Component 1>Geometry 1 and choose Polygon. 2 In the Polygon settings window, locate the Coordinates section. 3 In the x edit field, type In the y edit field, type Click the Build Selected button. Rotate the triangle to create the other prism. Rotate 1 1 On the Geometry toolbar, click Rotate. 2 Select the object pol1 only. 3 In the Rotate settings window, locate the Input section. 4 Select the Keep input objects check box. 5 Locate the Rotation Angle section. In the Rotation edit field, type BEAM SPLITTER

8 6 Click the Build All Objects button. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN (EWFD) Now set up the physics. Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 6 BEAM SPLITTER

9 2 Select Boundaries 2, 4, and 5 only. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 1 only. 3 In the Port settings window, locate the Port Properties section. 7 BEAM SPLITTER

10 4 From the Wave excitation at this port list, choose On. 5 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x 0 y exp(-(y/3500[nm])^2) 6 In the edit field, type ewfd.k. Transition Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Transition Boundary Condition. 2 Select Boundary 3 only. z 3 In the Transition Boundary Condition settings window, locate the Transition Boundary Condition section. 4 From the Electric displacement field model list, choose Relative permittivity. 5 From the r list, choose User defined. In the associated edit field, type eps_ag. 6 From the r list, choose User defined. Leave the default value of 1. 7 From the list, choose User defined. Leave the default value of 0. 8 In the d edit field, type 13[nm]. 8 BEAM SPLITTER

11 MATERIALS Next, assign material properties. Use Glass (quartz) for all domains. 1 On the Home toolbar, click Add Material. ADD MATERIAL 1 Go to the Add Material window. 2 In the tree, select Built-In>Glass (quartz). 3 In the Add material window, click Add to Component. 4 Close the Add material window. MESH 1 Choose the maximum mesh size in the air domain smaller than 0.2 wavelengths using the parameter h_max that you defined earlier. Scale the mesh size by the inverse of the refractive index. Size 1 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type h_max/1.5. Free Triangular 1 1 In the Model Builder window, right-click Mesh 1 and choose Free Triangular. 2 Right-click Free Triangular 1 and choose Build All. STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 On the Home toolbar, click Compute. 9 BEAM SPLITTER

12 RESULTS Electric Field (ewfd) The default plot shows the E-field norm. Compare the plot with Figure 2. Follow the steps below to reproduce the plot in Figure 3. 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 On the 1D Plot Group 2 toolbar, click Line Graph. 3 Select Boundary 1 only. 4 In the Line Graph settings window, locate the y-axis Data section. 5 In the Expression edit field, type -ewfd.npoav. 6 Locate the x-axis Data section. From the Parameter list, choose Expression. 7 In the Expression edit field, type y. 8 On the 1D Plot Group 2 toolbar, click Line Graph. 9 Select Boundary 5 only. 10 In the Line Graph settings window, locate the y-axis Data section. 11 In the Expression edit field, type ewfd.npoav. 12 Locate the x-axis Data section. From the Parameter list, choose Expression. 13 In the Expression edit field, type y. 14 On the 1D Plot Group 2 toolbar, click Line Graph. 15 Select Boundary 4 only. 16 In the Line Graph settings window, locate the y-axis Data section. 17 In the Expression edit field, type ewfd.npoav. 18 Locate the x-axis Data section. From the Parameter list, choose Expression. 19 In the Expression edit field, type x. 20 On the 1D plot group toolbar, click Line Graph. 21 Select Boundary 3 only. 22 In the Line Graph settings window, locate the y-axis Data section. 23 In the Expression edit field, type ewfd.qsrh. 24 Locate the x-axis Data section. From the Parameter list, choose Expression. 25 In the Expression edit field, type x. 10 BEAM SPLITTER

13 26 On the 1D Plot Group 2 toolbar, click Plot. The plot describes the power flux crossing the input boundary and the two output boundaries together with the losses at the silver surface. Compare with Figure BEAM SPLITTER

14 12 BEAM SPLITTER Solved with COMSOL Multiphysics 4.4

15 Dielectric Slab Waveguide Introduction A planar dielectric slab waveguide demonstrates the principles behind any kind of dielectric waveguide such as a ridge waveguide or a step index fiber, and has a known analytic solution. This model solves for the effective index of a dielectric slab waveguide as well as for the fields, and compares to analytic results. Figure 1: The guided modes in a dielectric slab waveguide have a known analytic solution. Model Definition A dielectric slab of thickness h slab = 1 m and refractive index n core = 1.5 forms the core of the waveguide, and sits in free space with n cladding = 1. Light polarized out of the plane of propagation, of wavelength = 1550 nm, is perfectly guided along the axis of the waveguide structure, as shown in Figure 1. Here, only the TE 0 mode can propagate. The structure varies only in the y direction, and it is infinite and invariant in the other two directions. 1 DIELECTRIC SLAB WAVEGUIDE

16 The analytic solution is found by assuming that the electric field along the direction of propagation varies as E z E(y)exp(-ik x x), where E(y) C 1 cos(k y y) inside the dielectric slab, and E(y) C 0 exp( y h slab 2 in the cladding. Because the electric and magnetic fields must be continuous at the interface, the guidance condition is where k y and satisfy k y 2 t slab = k tan y k y = 2 k core 2 k cladding 2 with k core = 2 n core and k cladding = 2 n cladding. It is possible to find the solution to the above two equations via the Newton-Raphson method, which is used whenever COMSOL Multiphysics detects a system of nonlinear equations, the only requirement being that of an adequate initial guess. This model considers a section of a dielectric slab waveguide that is finite in the x and y directions. Because the fields drop off exponentially outside the waveguide, the fields can be assumed to be zero at some distance away. This is convenient as it makes the boundary conditions in the y direction irrelevant, assuming that they are imposed sufficiently far away. Use Numerical Port boundary conditions in the x direction to model the guided wave propagating in the positive x direction. These boundary conditions require first solving an eigenvalue problem that solves for the fields and propagation constants at the boundaries. Results and Discussion Figure 2 shows the results. The numerical port boundary condition at the left side excites a mode that propagates in the x direction and is perfectly absorbed by the numerical port on the right side. The analytic and numerically computed propagation constants agree. 2 DIELECTRIC SLAB WAVEGUIDE

17 Figure 2: The electric field in a dielectric slab waveguide. Model Library path: Wave_Optics_Module/Verification_Models/ dielectric_slab_waveguide Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 3 DIELECTRIC SLAB WAVEGUIDE

18 4 Click the Study button. 5 In the tree, select Custom Studies>Empty Study. 6 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description lambda0 1550[nm] 1.550E-6 m Wavelength n_core Refractive index, core n_cladding Refractive index, cladding h_core 1[um] 1.000E-6 m Thickness, core h_cladding 7[um] 7.000E-6 m Thickness, cladding w_slab 5[um] 5.000E-6 m Slab width k_core k_cladding 2*pi[rad]*n_core/ lambda0 2*pi[rad]*n_cladding/ lambda E6 rad/m Wave number, core 4.054E6 rad/m Wave number, cladding f0 c_const/lambda E14 1/s Frequency GEOMETRY 1 1 In the Model Builder window, under Component 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose µm. Rectangle 1 1 Right-click Component 1>Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type w_slab. 4 In the Height edit field, type h_core. 5 Locate the Position section. From the Base list, choose Center. 4 DIELECTRIC SLAB WAVEGUIDE

19 6 Click the Build Selected button. Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type w_slab. 4 In the Height edit field, type h_cladding. 5 Locate the Position section. From the Base list, choose Center. 6 Click the Build All Objects button. 7 Click the Zoom Extents button on the Graphics toolbar. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN The wave is excited at the port on the left side. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundaries 1, 3, and 5 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Numeric. 5 From the Wave excitation at this port list, choose On. Now, add the exit port. Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundaries 8 10 only (the boundaries on the right side). 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Numeric. MATERIALS Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 In the Material settings window, locate the Material Contents section. 5 DIELECTRIC SLAB WAVEGUIDE

20 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_cladding 1 Refractive index 4 Right-click Component 1>Materials>Material 1 and choose Rename. 5 Go to the Rename Material dialog box and type Cladding in the New name edit field. 6 Click OK. By default, the first material you add applies on all domains. Add a core material. Material 2 1 Right-click Materials and choose New Material. 2 Select Domain 2 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_core 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type Core in the New name edit field. 7 Click OK. MESH 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type lambda0/n_cladding/8. Size 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 6 DIELECTRIC SLAB WAVEGUIDE

21 4 Select Domain 2 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type lambda0/n_core/8. 8 Click the Build All button. STUDY 1 Step 1: Boundary Mode Analysis 1 On the Study toolbar, click Study Steps and choose Other>Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Search for modes around edit field, type n_core. This value should be in the vicinity of the value that you expect the fundamental mode to have. 4 In the Mode analysis frequency edit field, type f0. Add another boundary mode analysis, for the second port. Step 2: Boundary Mode Analysis 2 1 On the Study toolbar, click Study Steps and choose Other>Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Search for modes around edit field, type n_core. 7 DIELECTRIC SLAB WAVEGUIDE

22 4 In the Port name edit field, type 2. 5 In the Mode analysis frequency edit field, type f0. Finally, add the study step for the propagating wave in the waveguide. Step 3: Frequency Domain 1 On the Study toolbar, click Study Steps and choose Frequency Domain>Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 On the Study toolbar, click Compute. RESULTS Electric Field (ewfd) The default plot shows the norm of the electric field. Modify the plot to shows the z-component (compare with Figure 2). 1 In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1. 2 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>electric field, z component (ewfd.ez). 3 Locate the Coloring and Style section. From the Color table list, choose WaveLight. 4 On the Electric Field (ewfd) toolbar, click Plot. Finish by comparing the simulation results to the analytic solution. To compute the latter, add a Global ODEs and DAEs interface and then set up and solve the relevant equations. COMPONENT 1 On the Home toolbar, click Add Physics. ADD PHYSICS 1 Go to the Add Physics window. 2 In the Add physics tree, select Mathematics>ODE and DAE Interfaces>Global ODEs and DAEs (ge). 8 DIELECTRIC SLAB WAVEGUIDE

23 3 Find the Physics in study subsection. In the table, enter the following settings: Studies Solve Study 1 4 In the Add physics window, click Add to Component. 5 Close the Add physics window. ROOT On the Home toolbar, click Add Study. ADD STUDY 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Custom Studies>Preset Studies for Some Physics>Stationary. 3 Find the Physics in study subsection. In the table, enter the following settings: Physics Solve Electromagnetic Waves, Frequency Domain (ewfd) 4 In the Add study window, click Add Study. 5 Close the Add study window. GLOBAL ODES AND DAES Global Equations 1 1 In the Model Builder window, under Component 1>Global ODEs and DAEs click Global Equations 1. 2 In the Global Equations settings window, locate the Global Equations section. 3 In the table, enter the following settings: Name f(u,ut,utt,t) (1) Initial value (u_0) (1) alpha k_y alpha-k_y*tan(k_y*h_core /2) k_y^2-(k_core^2-k_claddi ng^2-alpha^2) k_core/2 0 k_core/2 0 Initial value (u_t0) (1/s) Description STUDY 2 On the Study toolbar, click Compute. 9 DIELECTRIC SLAB WAVEGUIDE

24 RESULTS Derived Values Finally, compare analytical and computed propagation constants. 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Ports>Propagation constant (ewfd.beta_1). 3 Locate the Expression section. Select the Description check box. 4 In the associated edit field, type Propagation constant,beta_1. 5 Click the Evaluate button. 6 On the Results toolbar, click Global Evaluation. 7 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Ports>Propagation constant (ewfd.beta_2). 8 Locate the Expression section. Select the Description check box. 9 In the associated edit field, type Propagation constant, beta_2. 10 Right-click Results>Derived Values>Global Evaluation 4 and choose Evaluate>Table 1 - Global Evaluation 3 (ewfd.beta_1). 11 On the Results toolbar, click Global Evaluation. 12 In the Global Evaluation settings window, locate the Data section. 13 From the Data set list, choose Solution Locate the Expression section. In the Expression edit field, type sqrt(k_core^2-k_y^2). 15 Select the Description check box. 16 In the associated edit field, type Propagation constant, computed. 17 Click the Evaluate button. 10 DIELECTRIC SLAB WAVEGUIDE

25 Directional Coupler Introduction Directional couplers are used for coupling a light wave from one waveguide to another waveguide. By controlling the refractive index in the two waveguides, for instance by heating or current injection, it is possible to control the amount of coupling between the waveguides. Port 1 and Port 2 Port 3 and Port 4 Cores Cladding Figure 1: Schematic drawing of the waveguide structure. The structure consists of the two waveguide cores and the surrounding cladding. Port 1 and 2 are used for exciting the waveguides and Port 3 and 4 absorb the waves. Light that propagates through a dielectric waveguide has most of the power concentrated within the central core of the waveguide. Outside the waveguide core, in the cladding, the electric field decays exponentially with the distance from the core. However, if you put another waveguide core close to the first waveguide (see Figure 1), that second waveguide will perturb the mode of the first waveguide (and vice versa). Thus, instead of having two modes with the same effective index, one localized in the first waveguide and the second mode in the second waveguide, the modes and their respective effective indexes split and you get a symmetric supermode (see Figure 2 and Figure 4 below), with an effective index that is slightly larger than 1 DIRECTIONAL COUPLER

26 the effective index of the unperturbed waveguide mode, and an antisymmetric supermode (see Figure 3 and Figure 5), with an effective index that is slightly lower than the effective index of the unperturbed waveguide mode. Since the supermodes are the solution to the wave equation, if you excite one of them, it will propagate unperturbed through the waveguide. However, if you excite both the symmetric and the antisymmetric mode, that have different propagation constants, there will be a beating between these two waves. Thus, you will see that the power fluctuates back and forth between the two waveguides, as the waves propagate through the waveguide structure. You can adjust the length of the waveguide structure to get coupling from one waveguide to the other waveguide. By adjusting the phase difference between the fields of the two supermodes, you can decide which waveguide that will initially be excited. Model Definition The directional coupler, as shown in Figure 1, consists of two waveguide cores embedded in a cladding material. The cladding material is GaAs, with ion-implanted GaAs for the waveguide cores. The structure is modeled after Ref. 1. The core cross-section is square, with a side length of 3 µm. The two waveguides are separated 3 µm. The length of the waveguide structure is 2 mm. Thus, given the tiny cross-section, compared to the length, it is advantageous to use a view that doesn t preserve the aspect ratio for the geometry. For this kind of problem, where the propagation length is much longer than the wavelength, the Electromagnetic Waves, Beam Envelopes interface is particularly suitable, as the mesh does not need to resolve the wave on a wavelength scale, but rather the beating between the two waves. The model is setup to factor out the fast phase variation that occurs in synchronism with the first mode. Mathematically, we write the total electric field as the sum of the electric fields of the two modes, Er = E 1 exp j 1 x + E 2 exp j 2 x = E 1 + E 2 exp j 2 1 x exp j 1 x The expression within the square parentheses is what will be solved for. It will have a beat length L defined by 2 1 L = 2 2 DIRECTIONAL COUPLER

27 or 2 L = In the simulation, this beat length must be well resolved. Since the waveguide length is half of the beat length and the waveguide length is discretized into 20 subdivisions, the beat length will be very well resolved in the model. The model uses two numeric ports per input and exit boundary (see Figure 1). The two ports define the lowest symmetric and antisymmetric modes of the waveguide structure. Results and Discussion Figure 2 to Figure 5 shows the results of the initial boundary mode analysis. The first two modes (those with the largest effective mode index) are both symmetric. Figure 2 shows the first mode. This mode has the transverse polarization component along the z-direction. The second mode, shown in Figure 4, has transverse polarization along the y-direction. Notice that your plots may look different from the plots below, as the plots show the real part of the boundary mode electric fields. The computed complex electric fields can have different phase factors than for the plots below. Thus, the color legends can have different scales and the fields can either show minima (a blue color) or maxima (a red color) at the locations for the waveguide cores. However, for a symmetric mode, 3 DIRECTIONAL COUPLER

28 it will have the same field value for both waveguide cores and for an antisymmetric mode, it will have opposite field values for the two waveguide cores. Figure 2: The symmetric mode for z-polarization. Notice that the returned solution can also show the electric field as positive values in the peaks at the cores. Figure 3: The antisymmetric mode for z-polarization. 4 DIRECTIONAL COUPLER

29 Figure 3 and Figure 5 show the antisymmetric modes. Those have effective indexes that are slightly smaller than those of the symmetric modes. Figure 3 shows the mode for z-polarization and Figure 5 shows the mode for y-polarization. Figure 4: The symmetric mode for y-polarization. Notice that the returned solution can also show the electric field as positive values in the peaks at the cores. Figure 5: The antisymmetric mode for y-polarization. 5 DIRECTIONAL COUPLER

30 Figure 6 shows how the electric field increases in the receiving waveguide and decreases in the exciting waveguide. In a longer waveguide, the waves would oscillate back and forth between the waveguides. Figure 6: Excitation of the symmetric and the antisymmetric modes. The wave couples from the input waveguide to the output waveguide. Notice your result may show that the wave is excited in the other waveguide core, if your mode fields have different signs than what is displayed in Figure 2 to Figure 5. Figure 7 shows the result, when there is a phase difference between the fields of the exciting ports. In this case, the superposition of the two modes results in excitation of the other waveguides (as compared to the case in Figure 6). 6 DIRECTIONAL COUPLER

31 Figure 7: The same excitation conditions as in Figure 6, except that there is a phase difference between the two ports of radians. Notice your result may show that the wave is excited in the other waveguide core, if your mode fields have different signs than what is displayed in Figure 2 to Figure 5. Reference 1. S. Somekh, E. Garmire, A. Yariv, H.L. Garvin, and R.G. Hunsperger, Channel Optical Waveguides and Directional Couplers in GaAs-lmbedded and Ridged, Applied Optics, vol. 13, no. 2, pp , Model Library path: Wave_Optics_Module/Waveguides_and_Couplers/ directional_coupler Modeling Instructions From the File menu, choose New. 7 DIRECTIONAL COUPLER

32 NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Beam Envelopes (ewbe). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Boundary Mode Analysis. 6 Click the Done button. GLOBAL DEFINITIONS First, define a set of parameters for creating the geometry and defining the material parameters. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description wl 1.15[um] 1.150E-6 m Wavelength f0 c_const/wl 2.607E14 1/s Frequency a 3[um] 3.000E-6 m Side of waveguide cross-section d 3[um] 3.000E-6 m Distance between the waveguides len 2.1[mm] m Waveguide length width 6*a 1.800E-5 m Width of calculation domain height 4*a 1.200E-5 m Height of calculation domain ncl Refractive index of GaAs dn Refractive index increase in waveguide core nco ncl+dn Refractive index in waveguide core GEOMETRY 1 Create the calculation domain. 8 DIRECTIONAL COUPLER

33 Block 1 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type len. 4 In the Depth edit field, type width. 5 In the Height edit field, type height. 6 Locate the Position section. From the Base list, choose Center. Now add the first embedded waveguide. Block 2 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type len. 4 In the Depth edit field, type a. 5 In the Height edit field, type a. 6 Locate the Position section. From the Base list, choose Center. 7 In the y edit field, type -d. Add the second waveguide, by duplicating the first waveguide and modifying the position. Block 3 1 Right-click Component 1>Geometry 1>Block 2 and choose Duplicate. 2 In the Block settings window, locate the Position section. 3 In the y edit field, type d. 4 Click the Build All Objects button. DEFINITIONS Since the geometry is so long and narrow, don't preserve the aspect ratio in the view. 1 In the Model Builder window, expand the Component 1>Definitions node. Camera 1 In the Model Builder window, expand the Component 1>Definitions>View 1 node, then click Camera. 2 In the Camera settings window, locate the Camera section. 3 Clear the Preserve aspect ratio check box. 9 DIRECTIONAL COUPLER

34 4 Click the Apply button. 5 Click the Zoom Extents button on the Graphics toolbar. MATERIALS Now, add materials for the cladding and the core of the waveguides. Material 1 1 In the Home toolbar, click New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n ncl 1 Refractive index 4 Right-click Component 1>Materials>Material 1 and choose Rename. 5 Go to the Rename Material dialog box and type GaAs cladding in the New name edit field. 6 Click OK. Material 2 1 In the Home toolbar, click New Material. 10 DIRECTIONAL COUPLER

35 2 Select Domains 2 and 3 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n nco 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type Implanted GaAs core in the New name edit field. 7 Click OK. ELECTROMAGNETIC WAVES, BEAM ENVELOPES Since there will be no reflected waves in this model, it is best to select unidirectional propagation. 1 In the Electromagnetic Waves, Beam Envelopes settings window, locate the Wave Vectors section. 2 From the Number of directions list, choose Unidirectional. 3 Specify the k 1 vector as ewbe.beta_1 x 0 y 0 z This sets the wave vector to be that of the lowest waveguide mode. Add two numeric ports per port boundary. The first two ports excite the waveguides. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundaries 1, 5, and 10 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Numeric. 5 From the Wave excitation at this port list, choose On. Now duplicate the first port and rename it. 11 DIRECTIONAL COUPLER

36 Port 2 1 Right-click Component 1>Electromagnetic Waves, Beam Envelopes>Port 1 and choose Duplicate. 2 In the Port settings window, locate the Port Properties section. 3 In the Port name edit field, type 2. Next create the ports at the other end of the waveguides. Port 3 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundaries only. 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Numeric. Duplicate this port and give it a new unique name. Port 4 1 Right-click Component 1>Electromagnetic Waves, Beam Envelopes>Port 3 and choose Duplicate. 2 In the Port settings window, locate the Port Properties section. 3 In the Port name edit field, type 4. MESH 1 Define a triangular mesh on the input boundary and then sweep that mesh along the waveguides. Free Triangular 1 1 In the Mesh toolbar, click Boundary and choose Free Triangular. 2 Select Boundaries 1, 5, and 10 only. Size 1 1 Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size. Set the maximum mesh element size to be one wavelength, which will be enough to resolve the modes. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 12 DIRECTIONAL COUPLER

37 5 In the associated edit field, type wl. 6 Select the Minimum element size check box. 7 In the associated edit field, type wl/2. Sweep the mesh along the waveguides. Twenty elements along the waveguide will be sufficient to resolve the mode-coupling that will occur. Swept 1 In the Model Builder window, right-click Mesh 1 and choose Swept. 2 In the Model Builder window, click Mesh I>Size. 3 In the Size settings window, locate the Element Size section. 4 Click the Custom button. 5 Locate the Element Size Parameters section. In the Maximum element size edit field, type len/20. 6 Click the Build All button. STUDY 1 Don't generate the default plots. 1 In the Model Builder window, click Study 1. 2 In the Study settings window, locate the Study Settings section. 13 DIRECTIONAL COUPLER

38 3 Clear the Generate default plots check box. Step 1: Boundary Mode Analysis Now analyze the four lowest modes. The first two modes will be symmetric. Since the waveguide cross-section is square, there will be one mode polarized in the z-direction and one mode polarized in the y-direction. Mode three and four will be antisymmetric, one polarized in the z-direction and the other in the y-direction. 1 In the Model Builder window, under Study 1 click Step 1: Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Desired number of modes edit field, type 4. Search for the modes with effective index close to that of the waveguide cores. 4 In the Search for modes around edit field, type nco. 5 In the Mode analysis frequency edit field, type f0. Compute only the boundary mode analysis step. 6 Right-click Study 1>Step 1: Boundary Mode Analysis and choose Compute Selected Step. RESULTS Create a 3D surface plot to view the different modes. 3D Plot Group 1 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 On the 3D Plot Group 1 toolbar, click Surface. First look at the modes polarized in the z-direction. 3 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Boundary mode analysis>tangential boundary mode electric field>tangential boundary mode electric field, z component (ewbe.tebm1z), by double-clicking it or selecting it and pressing Enter. 4 In the Model Builder window, click 3D Plot Group 1. 5 In the 3D Plot Group settings window, locate the Data section. 6 From the Effective mode index list, choose the largest effective index. 7 On the 3D plot group toolbar, click Plot. This plot shows the symmetric mode polarized in the z-direction. Compare with Figure 2. 8 From the Effective mode index list, choose the third largest effective index. 14 DIRECTIONAL COUPLER

39 9 On the 3D plot group toolbar, click Plot. This plot shows the anti-symmetric mode polarized in the z-direction. Compare with Figure In the Model Builder window, under Results>3D Plot Group 1 click Surface In the Surface settings window, locate the Expression section. 12 In the Expression edit field, type ewbe.tebm1y. 13 In the Model Builder window, click 3D Plot Group In the 3D Plot Group settings window, locate the Data section. 15 From the Effective mode index list, choose the second largest effective index. 16 On the 3D plot group toolbar, click Plot. This plot shows the symmetric mode polarized in the y-direction. Compare with Figure From the Effective mode index list, choose the smallest effective index. 18 On the 3D plot group toolbar, click Plot. This plot shows the anti-symmetric mode polarized in the y-direction. Compare with Figure 5. Derived Values You will need to copy the effective indexes for the different modes and use them in the boundary mode analyses for the different ports. 1 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Ports>Propagation constant (ewbe.beta_1), by double-clicking it or selecting it and pressing Enter. 2 Click the Evaluate button. TABLE Copy all information in the table to the clipboard. Then paste that information in a text editor, so you easily can enter the values later in the boundary mode analysis steps. 1 In the Table window, click Full Precision, then click Copy Table and Headers to Clipboard. STUDY 1 Step 1: Boundary Mode Analysis 1 In the Model Builder window, under Study 1 click Step 1: Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Desired number of modes edit field, type DIRECTIONAL COUPLER

40 4 In the Search for modes around edit field, type , by selecting the value in you text editor and then copying and pasting it here. This should be the largest effective index. The last figures could be different from what is written here. Step 3: Boundary Mode Analysis 1 1 Right-click Study 1>Step 1: Boundary Mode Analysis and choose Duplicate. 2 In the Model Builder window, under Study 1 click Step 3: Boundary Mode Analysis 1. 3 In the Boundary Mode Analysis settings window, locate the Study Settings section. 4 In the Search for modes around edit field, type , by selecting the value in you text editor and then copying and pasting it here. This should be the third largest effective index. The last figures could be different from what is written here. 5 In the Port name edit field, type 2. Step 4: Boundary Mode Analysis 2 1 Select the two boundary mode analyses, Step 1: Boundary Mode Analysis and Step 3: Boundary Mode Analysis 1. 2 In the Model Builder window, right-click Step 1: Boundary Mode Analysis and choose Duplicate. 3 In the Model Builder window, under Study 1 click Step 4: Boundary Mode Analysis 2. 4 In the Boundary Mode Analysis settings window, locate the Study Settings section. 5 In the Port name edit field, type 3. Step 5: Boundary Mode Analysis 3 1 In the Model Builder window, under Study 1 click Step 5: Boundary Mode Analysis 3. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Port name edit field, type 4. Step 2: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 2: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. Finally, move Step2: Frequency Domain to be the last study step. 4 Right-click Study 1>Step 2: Frequency Domain and choose Move Down. Repeat this command twice. 5 On the Home toolbar, click Compute. 16 DIRECTIONAL COUPLER

41 RESULTS 3D Plot Group 1 Remove the surface plot and replace it with a slice plot of the norm of the electric field. 1 In the Model Builder window, under Results>3D Plot Group 1 right-click Surface 1 and choose Delete. Click Yes to confirm. 2 Right-click 3D Plot Group 1 and choose Slice. 3 In the Slice settings window, locate the Plane Data section. 4 From the Plane list, choose xy-planes. 5 In the Planes edit field, type 1. 6 Right-click Results>3D Plot Group 1>Slice 1 and choose Deformation. 7 In the Deformation settings window, locate the Expression section. 8 In the z component edit field, type ewbe.norme. 9 On the 3D plot group toolbar, click Plot. 10 Click the Go to View 1 button on the Graphics toolbar. 11 Click the Zoom Extents button on the Graphics toolbar. The plot shows how the light couples from the excited waveguide to the unexcited one; compare with Figure 6. ELECTROMAGNETIC WAVES, BEAM ENVELOPES Port 2 To excite the other waveguide, set the phase difference between the exciting ports to. 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Beam Envelopes click Port 2. 2 In the Port settings window, locate the Port Properties section. 3 In the in edit field, type pi. STUDY 1 On the Home toolbar, click Compute. RESULTS 3D Plot Group 1 Now the other waveguide is excited and the coupling occurs in reverse direction, compared to the previous case. Compare your result with that in Figure DIRECTIONAL COUPLER

42 18 DIRECTIONAL COUPLER Solved with COMSOL Multiphysics 4.4

43 Fabry-Perot Cavity Solved with COMSOL Multiphysics 4.4 Introduction A Fabry-Perot cavity is a slab of material of higher refractive index than its surroundings, as shown in Figure 1. Such a structure can act as a resonator at certain frequencies. Although such solutions can be found analytically, this model demonstrates how to find the resonant frequencies and the Q-factor. L T 2 R 1 T 1 x n 2 n 1 Figure 1: A Fabry-Perot cavity. An electromagnetic wave, traveling at normal incidence, will be partially reflected and transmitted at each interface between differing dielectrics. When the length, L, is an integer fraction of the wavelength, this will act as a resonator. Model Definition The geometry is a slab of a material with refractive index higher than the surrounding medium. It is assumed that the mode of interest is polarized with the electric field out of the plane, and that the wave vector of the mode of interest is parallel to the x-axis. 1 FABRY-PEROT CAVITY

44 Because the mode of interest propagates in the x direction, the model s y-dimension is arbitrary. The model space is composed of three types of domains: a central domain of unit width and refractive index n 4 domains of n 1 on both sides of the central domain two outer perfectly-matched-layer (PML) domains The PMLs absorb without reflection any incoming evanescent or propagating wave. The boundary condition on the top and bottom edges is perfect magnetic conductor (PMC), which implies that the solution will be mirror symmetric about those planes. A scattering boundary condition (SBC) applies at the left and right sides. This boundary condition is only perfectly transparent to an incoming plane wave, and will partially reflect any other component. Using a PML backed by an SBC reduces any artificial reflections due to the boundary conditions. EIGENFREQUENCY MODEL First, solve the model as an eigenvalue problem, which requires that you specify the number of eigenfrequencies to solve for and the frequency range around which to search. The PML and the SBC make this problem nonlinear, by introducing a damping term that depends upon the frequency. This, in turn, requires that you specify an eigenvalue transform point, which only needs to be within an order of magnitude or so of the expected resonant frequency. FREQUENCY-DOMAIN MODEL The approach described above has several drawbacks. First, the results must be manually examined to identify the spurious, nonphysical, modes. Second, it requires solving a nonlinear eigenvalue problem using a memory-intensive direct solver. For a 2D model, this is not a computational hurdle, but for structurally complex 3D cases, where far more mesh elements are required, it can be a concern. The convergence rate and solution time of the eigenvalue solver also depend on the choice of starting guess at the resonant frequency, the number of modes requested as output, and the spacing between these modes. An alternative approach to determining the resonant frequency and Q-factor is to recast this as a frequency-domain model, and to excite the structure over a range of frequencies. The excitation should be as isotropic as possible, so that it can excite all possible modes. The present example uses a line current condition applied to a point. The model is run in the frequency domain over a range covering the expected resonances. 2 FABRY-PEROT CAVITY

45 Results and Discussion EIGENFREQUENCY MODEL The results of the eigenvalue analysis are plotted in Figure 2, and the Q-factor is reported in Table 1. Some of these results are clearly nonphysical, and in fact represent numerical modes that is, solutions to the numerical eigenvalue problem that have no physically meaningful interpretation. These nonphysical eigenmodes can be identified in two ways: A visual examination of the field solutions can reveal that some modes exist purely in the PML regions. This is, however, a manual task, and it is not always obvious that a mode is indeed physical. Alternatively, it is possible to examine the Q-factor for each mode. A nonphysical mode has a Q-factor less than 1 2. Figure 2: The electric field across the entire modeling domain for various solutions to the eigenvalue problem. Only physical modes are visualized. 3 FABRY-PEROT CAVITY

46 TABLE 1: COMPUTED RESONANT FREQUENCY AND Q-FACTORS, PHYSICAL MODES ARE HIGHLIGHTED Resonant frequency (MHz) Q-factor Note ~0 ~0 Nonphysical mode FREQUENCY-DOMAIN MODEL Figure 3 plots the results of the frequency-domain analysis. The total energy density is monitored at a point inside the cavity region. The peaks in this graph correspond to the resonant frequencies, f 0, and the Q-factor can be computed as Q f 0 f, where f is the full width at half maximum. This method is an alternative approach to finding the resonant frequencies and Q-factors that requires less memory, an important concern for 3D models. This approach entirely avoids the problem of finding and eliminating spurious modes. The only limitations are that some care must be taken to ensure that the desired modes are indeed excited and that evaluation of the Q-factor requires manual postprocessing of the data. 4 FABRY-PEROT CAVITY

47 Figure 3: Plot of energy density within the cavity over a range of frequencies. This plot can be used to find the resonant frequencies and Q-factors. Model Library path: Wave_Optics_Module/Verification_Models/ fabry_perot Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 5 FABRY-PEROT CAVITY

48 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Eigenfrequency. 6 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description f_min 20[MHz] 2.000E7 Hz Minimum frequency in sweep f_max 100[MHz] 1.000E8 Hz Maximum frequency in sweep GEOMETRY 1 Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Height edit field, type Locate the Position section. From the Base list, choose Center. 5 In the x edit field, type Click the Build Selected button. Array 1 1 On the Geometry toolbar, click Array. 2 Select the object r1 only. 3 In the Array settings window, locate the Size section. 4 From the Array type list, choose Linear. 5 In the Size edit field, type 5. 6 Locate the Displacement section. In the x edit field, type 1. 7 Click the Build All Objects button. 6 FABRY-PEROT CAVITY

49 8 Click the Zoom Extents button on the Graphics toolbar. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Now set up the physics. The model is based on differences in refractive indices and you solve for the E-field's out-of-plane component. 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain. 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Components section. 3 From the Electric field components solved for list, choose Out-of-plane vector. Assign a PMC condition on the top and bottom edges. Perfect Magnetic Conductor 1 1 On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor. 7 FABRY-PEROT CAVITY

50 2 Select Boundaries 2, 3, 5, 6, 8, 9, 11, 12, 14, and 15 only. DEFINITIONS Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 Select Domains 1 and 5 only. 8 FABRY-PEROT CAVITY

51 ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 Select Boundaries 1 and 16 only. Line Current (Out-of-Plane) 1 1 On the Physics toolbar, click Points and choose Line Current (Out-of-Plane). 9 FABRY-PEROT CAVITY

52 2 Select Point 5 only. 3 In the Line Current (Out-of-Plane) settings window, locate the Line Current (Out-of-Plane) section. 4 In the I 0 edit field, type 1. MATERIALS Now, specify the material properties. First, define the medium surrounding the slab. Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 10 FABRY-PEROT CAVITY

53 2 Select Domains 1, 2, 4, and 5 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n 1 1 Refractive index 5 Right-click Component 1>Materials>Material 1 and choose Rename. 6 Go to the Rename Material dialog box and type n=1 in the New name edit field. 7 Click OK. The refractive index of the slab is 4. Material 2 1 Right-click Materials and choose New Material. 11 FABRY-PEROT CAVITY

54 2 Select Domain 3 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n 4 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type n=4 in the New name edit field. 7 Click OK. MESH 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extremely fine. 12 FABRY-PEROT CAVITY

55 4 Click the Build All button. STUDY 1 Step 1: Eigenfrequency 1 In the Model Builder window, under Study 1 click Step 1: Eigenfrequency. 2 In the Eigenfrequency settings window, locate the Study Settings section. 3 In the Desired number of eigenfrequencies edit field, type 8. 4 In the Search for eigenfrequencies around edit field, type 1e7. Because of the PMLs and SBCs, this is a nonlinear eigenvalue problem and an additional setting is required to handle the associated damping term. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 1 node, then click Eigenvalue Solver 1. 3 In the Eigenvalue Solver settings window, locate the Values of Linearization Point section. 4 Find the Value of eigenvalue linearization point subsection. In the Point edit field, type 1e7. 5 In the Model Builder window, click Study FABRY-PEROT CAVITY

56 6 In the Study settings window, locate the Study Settings section. 7 Clear the Generate default plots check box. 8 On the Home toolbar, click Compute. RESULTS Derived Values 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Global>Quality factor (ewfd.qfactor), by double-clicking it or selecting it and pressing Enter. 3 Click the Evaluate button. TABLE Review the evaluated eigenfrequencies and Q-factors; Q-factors less than 0.5 correspond to nonphysical eigenmodes. RESULTS 1D Plot Group 1 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, click to expand the Legend section. 3 From the Position list, choose Lower right. 4 Locate the Data section. From the Eigenfrequency selection list, choose Manual. 5 In the Eigenfrequency indices (1-5) edit field, type On the 1D Plot Group 1 toolbar, click Line Graph. 7 In the Line Graph settings window, locate the Selection section. 8 Click Paste Selection. 9 Go to the Paste Selection dialog box. 10 In the Selection edit field, type 3, 6, 9, 12, FABRY-PEROT CAVITY

57 11 Click the OK button. 12 In the Line Graph settings window, locate the y-axis Data section. 13 In the Expression edit field, type Ez. 14 Click to expand the Legends section. Select the Show legends check box. 15 From the Legends list, choose Manual. 16 In the table, enter the following settings: Legends 37.5 MHz 74.9 MHz MHz MHz 17 On the 1D plot group toolbar, click Plot. Compare the plot with that shown in Figure 2. ROOT Add a new study as an alternative approach to examine the Q-factor for each mode. 1 On the Home toolbar, click Add Study. 15 FABRY-PEROT CAVITY

58 ADD STUDY 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Frequency Domain. 3 In the Add study window, click Add Study. STUDY 2 Step 1: Frequency Domain 1 In the Model Builder window, under Study 2 click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type range(f_min,0.25[mhz],f_max). 4 On the Home toolbar, click Compute. RESULTS Electric Field (ewfd) 1 In the 2D Plot Group settings window, locate the Data section. 2 From the Parameter value (freq) list, choose 3.75e7. 3 On the Electric Field (ewfd) toolbar, click Plot. 16 FABRY-PEROT CAVITY

59 4 Click the Zoom Extents button on the Graphics toolbar. This is the E-field norm at 37.5 MHz. Finally, reproduce the plot in Figure 3. 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 On the 1D Plot Group 3 toolbar, click Global. 5 In the Global settings window, locate the y-axis Data section. 6 In the table, enter the following settings: Expression Unit Description ewfd.intwe+ewfd.intwm J Total energy 7 Click to expand the Legends section. Clear the Show legends check box. 8 On the 1D plot group toolbar, click Plot. Using the definition Q = f0/ f you can use this plot to evaluate the Q-factor at each resonance. 17 FABRY-PEROT CAVITY

60 18 FABRY-PEROT CAVITY Solved with COMSOL Multiphysics 4.4

61 Fresnel Equations Solved with COMSOL Multiphysics 4.4 Introduction A plane electromagnetic wave propagating through free space is incident at an angle upon an infinite dielectric medium. This model computes the reflection and transmission coefficients and compares the results to the Fresnel equations. Model Definition A plane wave propagating through free space (n 1) as shown in Figure 1 is incident upon an infinite dielectric medium (n 1.5) and is partially reflected and partially transmitted. If the electric field is p-polarized that is, if the electric field vector is in the same plane as the Poynting vector and the surface normal then there will be no reflections at an incident angle of roughly 56, known as the Brewster angle. Reflected Unit cell Transmitted Incident n 1 n 2 Figure 1: A plane wave propagating through free space incident upon an infinite dielectric medium. Although, by assumption, space extends to infinity in all directions, it is sufficient to model a small unit cell, as shown in Figure 1; a Floquet-periodic boundary condition applies on the top and bottom unit-cell boundaries because the solution is periodic along the interface. This model uses a 3D unit cell, and applies perfect electric conductor and perfect magnetic conductor boundary conditions as appropriate to 1 FRESNEL EQUATIONS

62 model out-of-plane symmetry. The angle of incidence ranges between 0 90 for both polarizations. For comparison, Ref. 1 and Ref. 2 provide analytic expressions for the reflectance and transmittance. Reflection and transmission coefficients for s-polarization and p-polarization are defined respectively as n 1 cos incident n 2 cos transmitted r s = n 1 cos incident + n 2 cos transmitted 2n 1 cos incident t s = n 1 cos incident + n 2 cos transmitted n 2 cos incident n 1 cos transmitted r p = n 1 cos transmitted + n 2 cos incident 2n 1 cos incident t p = n 1 cos transmitted + n 2 cos incident (1) (2) (3) (4) Reflectance and transmittance are defined as R = r 2 (5) T = cos t cos n 2 n 1 transmitted incident 2 (6) The Brewster angle at which r p 0 is defined as n 2 B = atan----- n 1 (7) 2 FRESNEL EQUATIONS

63 Results and Discussion Figure 2 is a combined plot of the y component of the electric-field distribution and the power flow visualized as an arrow plot for the TE case. Figure 2: Electric field, E y (slice) and power flow (arrows) for TE incidence at 70 inside the unit cell. 3 FRESNEL EQUATIONS

64 For the TM case, Figure 3 visualizes the y component of the magnetic-field distribution instead, again in combination with the power flow. Figure 3: Magnetic field, H y (slice) and power flow (arrows) for TM incidence at 70 inside the unit cell. Note that the sum of reflectance and transmittance in Figure 4 and Figure 5 equals 1, showing conservation of power. Figure 5 also shows that the reflectance around 56 the Brewster angle in the TM case is close to zero. 4 FRESNEL EQUATIONS

65 Figure 4: The reflectance and transmittance for TE incidence agree well with the analytic solutions. Figure 5: The reflectance and transmittance for TM incidence agree well with the analytic solutions. The Brewster angle is also observed at the expected location. 5 FRESNEL EQUATIONS

66 References 1. C.A. Balanis, Advanced Engineering Electromagnetics, Wiley, B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, Wiley, Model Library path: Wave_Optics_Module/Verification_Models/ fresnel_equations Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Define some parameters that are useful when setting up the mesh and the study. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 6 FRESNEL EQUATIONS

67 3 In the table, enter the following settings: Name Expression Value Description n_air Refractive index, air n_slab Refractive index, slab lda0 1[m] m Wavelength f0 c_const/lda E8 1/s Frequency alpha 70[deg] rad Angle of incidence beta asin(n_air*sin(alpha)/ n_slab) rad Refraction angle h_max lda0/ m Maximum element size, air alpha_brewster atan(n_slab/n_air) rad Brewster angle, TM only r_s r_p t_s t_p (n_air*cos(alpha)-n_slab* cos(beta))/ (n_air*cos(alpha)+n_slab* cos(beta)) (n_slab*cos(alpha)-n_air* cos(beta))/ (n_air*cos(beta)+n_slab*c os(alpha)) (2*n_air*cos(alpha))/ (n_air*cos(alpha)+n_slab* cos(beta)) (2*n_air*cos(alpha))/ (n_air*cos(beta)+n_slab*c os(alpha)) Reflection coefficient, TE Reflection coefficient, TM Transmission coefficient, TE Transmission coefficient, TM The angle of incidence is updated while running the parametric sweep. The refraction (transmitted) angle is defined by Snell's law with the updated angle of incidence. The Brewster angle exists only for TM incidence, p-polarization, and parallel polarization. DEFINITIONS Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 7 FRESNEL EQUATIONS

68 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description ka ewfd.k0 rad/m Propagation constant, air kax ka*sin(alpha) rad/m kx for incident wave kay 0 ky for incident wave kaz ka*cos(alpha) rad/m kz for incident wave kb n_slab*ewfd.k0 rad/m Propagation constant, slab kbx kb*sin(beta) rad/m kx for refracted wave kby 0 ky for refracted wave kbz kb*cos(beta) rad/m kz for refracted wave GEOMETRY 1 First, create a block composed of two domains. Use layers to split the block. Block 1 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type In the Depth edit field, type In the Height edit field, type Click to expand the Layers section. Find the Layer position subsection. In the table, enter the following settings: Layer name Thickness (m) Layer Click the Build All Objects button. 8 FRESNEL EQUATIONS

69 8 Click the Zoom Extents button on the Graphics toolbar. Choose wireframe rendering to get a better view of each boundary. 9 Click the Wireframe Rendering button on the Graphics toolbar. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Set up the physics based on the direction of propagation and the E-field polarization. First, assume a TE-polarized wave which is equivalent to s-polarization and perpendicular polarization. E x and E z are zero while E y is dominant. The wave is excited from the port on the top. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 7 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Wave excitation at this port list, choose On. 5 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x exp(-i*kax*x)[v/m] y 0 z 6 In the edit field, type abs(kaz). 9 FRESNEL EQUATIONS

70 Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 3 only. 3 In the Port settings window, locate the Port Mode Settings section. 4 Specify the E 0 vector as 0 x exp(-i*kbx*x)[v/m] y 0 z 5 In the edit field, type abs(kbz). The bottom surface is an observation port. The S21-parameter from Port 1 and Port 2 provides the transmission characteristics. The E-field polarization has E y only and the boundaries are always either parallel or perpendicular to the E-field polarization. Apply periodic boundary conditions on the boundaries parallel to the E-field except those you already assigned to the ports. Periodic Condition 1 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 2 Select Boundaries 1, 4, 10, and 11 only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. 10 FRESNEL EQUATIONS

71 5 Specify the k F vector as kax x 0 y 0 z Apply a perfect electric conductor condition on the boundaries perpendicular to the E-field. This condition creates a virtually infinite modeling space. Perfect Electric Conductor 2 1 On the Physics toolbar, click Boundaries and choose Perfect Electric Conductor. 11 FRESNEL EQUATIONS

72 2 Select Boundaries 2, 5, 8, and 9 only. MATERIALS Now set up the material properties based on refractive index. The top half is filled with air. Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 Select Domain 2 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_air 1 Refractive index 5 Right-click Component 1>Materials>Material 1 and choose Rename. 6 Go to the Rename Material dialog box and type Air in the New name edit field. 7 Click OK. The bottom half is glass. 12 FRESNEL EQUATIONS

73 Material 2 1 Right-click Materials and choose New Material. 2 Select Domain 1 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_slab 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type Glass in the New name edit field. 7 Click OK. MESH 1 The periodic boundary condition performs better if the mesh is identical on the periodicity boundaries. This is especially important when dealing with vector degrees of freedom, as will be the case in the TM version of this model. The maximum element size is smaller than 0.2 times the wavelength. The bottom half domain is scaled inversely by the refractive index of the material. Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type h_max. Size 1 1 In the Model Builder window, under Component 1>Mesh 1 click Size 1. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 1 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type h_max/n_slab. 13 FRESNEL EQUATIONS

74 Free Triangular 1 1 In the Model Builder window, right-click Mesh 1 and choose Free Triangular. 2 Select Boundaries 1 and 4 only. Copy Face 1 1 Right-click Mesh 1 and choose Copy Face. 2 Select Boundaries 1 and 4 only. 3 In the Copy Face settings window, locate the Destination Boundaries section. 4 Select the Destination group focus toggle button. 5 Select Boundaries 10 and 11 only. Free Tetrahedral 1 1 Right-click Mesh 1 and choose Free Tetrahedral. 2 Right-click Mesh 1 and choose Build All. STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 14 FRESNEL EQUATIONS

75 Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names alpha Parameter value list range(0,2[deg],90[deg]) Use a direct solver instead of an iterative one for faster convergence. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Stationary Solver 1 right-click Direct and choose Enable. 3 Right-click Study 1>Solver Configurations>Solver 1>Stationary Solver 1>Direct and choose Compute. RESULTS Electric Field (ewfd) The default plot is the E-field norm for the last solution, which corresponds to tangential incidence. Replace the expression with E y, add an arrow plot of the power flow (Poynting vector), and choose a more interesting angle of incidence for the plot. 1 In the Model Builder window, under Results>Electric Field (ewfd) click Multislice 1. 2 In the Multislice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>electric field, y component (ewfd.ey). 3 Locate the Multiplane Data section. Find the x-planes settings and in the Planes edit field, type 0. 4 Find the z-planes settings and in the Planes edit field, type 0. 5 Locate the Coloring and Style section. From the Color table list, choose Wave. 6 In the Model Builder window, right-click Electric Field (ewfd) and choose Arrow Volume. 7 In the Arrow Volume settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, 15 FRESNEL EQUATIONS

76 Frequency Domain>Energy and power>power flow, time average (ewfd.poavx,...,ewfd.poavz). 8 Locate the Arrow Positioning section. Find the y-grid points and in the Points edit field, type 1. 9 Locate the Coloring and Style section. From the Color list, choose Green. 10 In the Model Builder window, click Electric Field (ewfd). 11 In the 3D Plot Group settings window, locate the Data section. 12 From the Parameter value (alpha) list, choose On the 3D plot group toolbar, click Plot. 14 Click the Zoom Extents button on the Graphics toolbar. The plot should look like that in Figure 2. Add a 1D plot to see the reflection and transmission versus the angle of incidence. 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Plot Settings section. 3 Select the x-axis label check box. 4 In the associated edit field, type Angle of Incidence. 5 Select the y-axis label check box. 6 In the associated edit field, type Reflectance and Transmittance. 7 Click to expand the Legend section. From the Position list, choose Upper left. 8 On the 1D plot group toolbar, click Global. 9 In the Global settings window, locate the y-axis Data section. 10 In the table, enter the following settings: Expression Unit Description abs(ewfd.s11)^2 1 Reflectance abs(ewfd.s21)^2 1 Transmittance 11 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Line list, choose None. 12 From the Marker list, choose Cycle. 13 From the Line list, choose None. 14 From the Marker list, choose Cycle. 16 FRESNEL EQUATIONS

77 15 On the 1D plot group toolbar, click Global. 16 In the Global settings window, locate the y-axis Data section. 17 In the table, enter the following settings: Expression Unit Description abs(r_s)^2 Reflectance, analytic n_slab*cos(beta)/ (n_air*cos(alpha))*abs(t_s)^2 Transmittance, analytic 18 On the 1D plot group toolbar, click Plot. 19 In the Model Builder window, right-click 1D Plot Group 2 and choose Rename. 20 Go to the Rename 1D Plot Group dialog box and type Reflection and Transmission in the New name edit field. 21 Click OK. Compare the resulting plots with Figure 4. The remaining instructions are for the case of TM-polarized wave, p-polarization, and parallel polarization. In this case, E y is zero while E x and E z characterize the wave. In other words, H y is dominant while H x and H z are effectless. Thus, the H-field is perpendicular to the plane of incidence and it is convenient to solve the model for the H-field. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Port 1 1 In the Port settings window, locate the Port Mode Settings section. 2 From the Input quantity list, choose Magnetic field. 3 Specify the H 0 vector as 0 x exp(-i*kax*x)[a/m] 0 z Port 2 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain click Port 2. 2 In the Port settings window, locate the Port Mode Settings section. 3 From the Input quantity list, choose Magnetic field. y 17 FRESNEL EQUATIONS

78 4 Specify the H 0 vector as 0 x exp(-i*kbx*x)[a/m] y 0 z Perfect Electric Conductor 2 The model utilizes the H-field for the TM case and the remaining boundaries need to be perfect magnetic conductors. 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain right-click Perfect Electric Conductor 2 and choose Disable. Perfect Magnetic Conductor 1 1 On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor. 2 Select Boundaries 2, 5, 8, and 9 only. To keep the solution and plots for the TE case, do as follows: STUDY 1 Solver 1 1 In the Model Builder window, under Study 1>Solver Configurations right-click Solver 1 and choose Solution>Copy. RESULTS Electric Field (ewfd) 1 In the Model Builder window, under Results Ctrl-click to select both Results>Electric Field (ewfd) and Results>Reflection and Transmission, then right-click and choose Duplicate. Electric Field (ewfd) 1 In the Model Builder window, under Results click Electric Field (ewfd). 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. Reflection and Transmission 1 In the Model Builder window, under Results click Reflection and Transmission. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution FRESNEL EQUATIONS

79 STUDY 1 On the Home toolbar, click Compute. RESULTS Electric Field (ewfd) 1 1 In the Multislice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Magnetic>Magnetic field>magnetic field, y component (ewfd.hy). 2 On the 3D plot group toolbar, click Plot. This reproduces Figure 3. Reflection and Transmission 1 1 In the Model Builder window, expand the Results>Reflection and Transmission 1 node, then click Global 2. 2 In the Global settings window, locate the y-axis Data section. 3 In the table, enter the following settings: Expression Unit Description abs(r_p)^2 n_slab*cos(beta)/ (n_air*cos(alpha))*abs(t_p)^2 Reflectance, analytic Transmittance, analytic 4 On the 1D plot group toolbar, click Plot. The plot should look like Figure 5. The Brewster angle is observed around 56 degrees, which is close to the analytic value. 19 FRESNEL EQUATIONS

80 20 FRESNEL EQUATIONS Solved with COMSOL Multiphysics 4.4

81 Mach-Zehnder Modulator Introduction Optical modulators are used for electrically controlling the output amplitude or the phase of the light wave passing through the device. To reduce the device size and the driving voltage, waveguide-based modulators are used for communication applications. To control the optical properties with an external electric signal, the electro-optic effect, or Pockels effect, is used, where the birefringence of the crystal changes proportionally to the applied electric field. A refractive index change results in a change of the phase of the wave passing through the crystal. If you combine two waves with different phase change, you can interferometrically get an amplitude modulation. The device in Figure 1 is a Mach-Zehnder modulator. The input wave is launched into a directional coupler. The power of the input is split equally into the two output waveguides of the first directional coupler. Those two waveguides form the two arms of a Mach-Zehnder interferometer. On one of the arms, you can apply an electric field to modify the refractive index in the material and, thus, modify the phase for the wave propagating through that arm. The two waves are then combined into another 50/50 directional coupler. By changing the applied voltage you can continuously control the amount of light exiting from the two output waveguides. V 0 In Out 1 Out 2 50/50 Directional Coupler Mach- Zehnder Interferometer 50/50 Directional Coupler Figure 1: Schematic drawing of the Mach-Zehnder modulator. A common material for fabricating waveguide modulators is lithium niobate, LiNbO 3. Lithium niobate is a ferroelectric crystal that exhibits uniaxial birefringence. 1 MACH-ZEHNDER MODULATOR

82 Waveguide structures can be fabricated by either indiffusion of Ti into the core regions or by annealed proton exchange, where lithium ions are exchanged with protons from an acid bath. Model Definition This model shows how the Electromagnetic Waves, Beam Envelopes user interface can be combined with the Electrostatics user interface to perform simulations of the properties of an optical waveguide modulator. The model is implemented in a 2D geometry, but could be extended to a full 3D simulation. The Electromagnetic Waves, Beam Envelopes interface is formulated assuming that the electric field is defined as the product of a slowly varying envelope function and a rapidly varying phase function E = E 1 exp jk r, where E 1 is the envelope function, k is a wave vector and r is the position. If k is properly selected for the problem, the envelope function E 1 will have a spatial variation occurring on a length scale much larger than the wavelength. A good assumption, for this model, is that the wave is well approximated in the straight domains using the wave vector for the incident mode, However, in the waveguide bends the wave vector can be written as where k 0 n eff is the propagation constant for the mode, k 0 is the vacuum wave number, n eff is the effective index of the waveguide mode, is the angle from the x-axis, and x and y are the unit vectors in the x- and y-directions, respectively. The wave vector difference is thus 2 2 = cos x + sin y, = cos 1 x + sin y. It is the wave vector difference that determines the phase variation for the envelope field. Thus, we must make sure that the phase variation is well resolved by the mesh. For instance, 2 r where N is a suitably large number, for instance 6. From the relations above, we get that the maximum mesh element sizes in the x- and y-direction should be 2 N, 2 MACH-ZEHNDER MODULATOR

83 h x max = Nn eff 1 cos and h y max = Nn eff sin. Results and Discussion The first part of the model is to define a minimum bend radius that provides low loss. Figure 2 shows the power transmission for an S-shaped bend. As seen, a bend radius of 2.5 mm will give a transmission of approximately 98% of the power. We accept the 2% loss and fix the bend radius to be 2.5 mm. Figure 2: The transmission through an S-bent waveguide versus the radius of curvature for the bend. 3 MACH-ZEHNDER MODULATOR

84 Figure 3 shows the electric field norm for the wave propagating in the S-shaped bend, for a bend radius of 2.5 mm. As seen, the wave follows the waveguide in the bend, as expected. Figure 3: The electric field norm for the wave in the S-bent waveguide for a radius of curvature of 2.5 mm. We want the directional coupler structures to operate as 50/50 couplers. That is, half of the incident power should exit from each of the two output arms. To find the coupler length where this condition is met, we monitor the power difference in the two arms of the Mach-Zehnder interferometer and sweep the length of the directional coupler. Figure 4 shows the result of the parameter sweep. A coupler length of 380 m 4 MACH-ZEHNDER MODULATOR

85 gives zero power difference between the two arms. That is, the power is the same in the two arms. Figure 4: The absolute value of the power difference between the two waveguide arms in the Mach-Zehnder interferometer versus the length of the directional coupler. 5 MACH-ZEHNDER MODULATOR

86 Figure 5 shows that the electric field norms for the two arms indeed seem to be the same. Figure 5: The electric field norm in the two waveguide arms of the Mach-Zehnder interferometer. As shown, the fields are almost the same for a directional coupler length of 380 m. Finally, a voltage is applied across the waveguide in one of the arms. The voltage modifies the refractive index in the arm and, thus, there will be a phase difference between the wave propagating through the two Mach-Zehnder interferometer arms. As expected, Figure 6 shows that the wave can be switched between the two output waveguides by tuning the applied voltage. Thus, if all input and output ports are connected to other waveguides or fibers, you can use the device as a spatial switch. 6 MACH-ZEHNDER MODULATOR

87 However, if only one input port and one output port are active, the device operates as an amplitude modulator. Figure 6: The transmission to the upper (S21) and the lower (S41) output waveguide versus the applied voltage, V0. Model Library path: Wave_Optics_Module/Waveguides_and_Couplers/ mach_zehnder_modulator Modeling Instructions The parameterized geometry for the Mach-Zehnder modulator is quite complicated to set up. To get straight to the physics modeling, start by importing the MPH-file. In the imported MPH-file, the parameters for the geometry and the materials are already defined, as well as the default physics settings. The study node includes a default boundary mode analysis study sequence. 7 MACH-ZEHNDER MODULATOR

88 If you want to build the geometry manually, follow the steps in Appendix: Geometry Modeling Instructions, starting on page 31, and then return here, but skip the following two steps to import the MPH-file. 1 From the File menu, choose Open. 2 Browse to the model s Model Library folder and double-click the file mach_zehnder_modulator_geom_sequence.mph. Now, start by defining the materials for the waveguide structure. MATERIALS Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_clad 1 Refractive index 4 Right-click Component 1>Materials>Material 1 and choose Rename. 5 Go to the Rename Material dialog box and type Cladding in the New name edit field. 6 Click OK. Material 2 1 Right-click Materials and choose New Material. 2 In the Material settings window, locate the Geometric Entity Selection section. 8 MACH-ZEHNDER MODULATOR

89 3 From the Selection list, choose Core. 4 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_core 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type Core in the New name edit field. 7 Click OK. ELECTROMAGNETIC WAVES, BEAM ENVELOPES Setup the interface for unidirectional propagation, using the wave number calculated in the boundary mode analysis. 1 In the Model Builder, under Component 1, click Electromagnetic Waves, Beam Envelopes node. 2 Locate the Components section. From the Electric field components solved for list, choose Out-of-plane vector. 3 Locate the Wave Vectors section. From the Number of directions list, choose Unidirectional. 9 MACH-ZEHNDER MODULATOR

90 4 Specify the k 1 vector as ewbe.beta_1 x 0 y Now define the input and the output ports. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 In the Port settings window, locate the Boundary Selection section. 3 From the Selection list, choose Port 1. 4 Locate the Port Properties section. From the Type of port list, choose Numeric. 5 From the Wave excitation at this port list, choose On. Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 In the Port settings window, locate the Boundary Selection section. 10 MACH-ZEHNDER MODULATOR

91 3 From the Selection list, choose Port 2. 4 Locate the Port Properties section. From the Type of port list, choose Numeric. Port 3 1 On the Physics toolbar, click Boundaries and choose Port. 2 In the Port settings window, locate the Boundary Selection section. 11 MACH-ZEHNDER MODULATOR

92 3 From the Selection list, choose Port 3. 4 Locate the Port Properties section. From the Type of port list, choose Numeric. Port 4 1 On the Physics toolbar, click Boundaries and choose Port. 2 In the Port settings window, locate the Boundary Selection section. 12 MACH-ZEHNDER MODULATOR

93 3 From the Selection list, choose Port 4. 4 Locate the Port Properties section. From the Type of port list, choose Numeric. Use the scattering boundary condition to absorb some of the light that is not guided by the waveguide. The scattering boundary condition is only absorbing light propagating close to the normal direction to the boundary, so it will not absorb non-guided light propagating with large angles of incidence. Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 In the Scattering Boundary Condition settings window, locate the Boundary Selection section. 13 MACH-ZEHNDER MODULATOR

94 3 From the Selection list, choose Scattering boundary condition. MESH 1 Define a mesh on the edge and then map it over the whole domain. 1 In the Model Builder window, under Component 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Sequence type list, choose User-controlled mesh. Free Triangular 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Triangular 1 and choose Delete. Click Yes to confirm. Edge 1 1 Right-click Mesh 1 and choose More Operations>Edge. 14 MACH-ZEHNDER MODULATOR

95 2 Select Boundaries 1, 3, 5, 8, 10, and 12 only. Size 1 1 Right-click Component 1>Mesh 1>Edge 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type hy. Size 2 1 Right-click Edge 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 Click Clear Selection. 4 Select Boundaries 3 and 10 only, that correspond to the cores of the waveguides. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 15 MACH-ZEHNDER MODULATOR

96 7 In the associated edit field, type min(hy,w/4). Mapped 1 1 In the Model Builder window, right-click Mesh 1 and choose Mapped. 2 In the Mapped settings window, click to expand the Advanced settings section. 3 Locate the Advanced Settings section. Select the Adjust evenly distributed edge mesh check box. Size 1 1 Right-click Component 1>Mesh 1>Mapped 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type hx. 6 Click the Build All button. 16 MACH-ZEHNDER MODULATOR

97 STUDY 1 Step 1: Boundary Mode Analysis Now define the boundary mode analysis study steps for the numeric ports and the frequency domain study for finding the domain solution. 1 In the Model Builder window, under Study 1 click Step 1: Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Study Settings section. 3 In the Search for modes around edit field, type n_core. 4 In the Mode analysis frequency edit field, type f0. Step 3: Boundary Mode Analysis 1 1 Right-click Study 1>Step 1: Boundary Mode Analysis and choose Duplicate. 2 In the Model Builder window, under Study 1 right-click Step 3: Boundary Mode Analysis 1 and choose Move Up. 3 In the Boundary Mode Analysis settings window, locate the Study Settings section. 4 In the Port name edit field, type 2. Step 4: Boundary Mode Analysis 2 1 Right-click Study 1>Step 3: Boundary Mode Analysis 1 and choose Duplicate. 2 In the Model Builder window, under Study 1 right-click Step 4: Boundary Mode Analysis 2 and choose Move Up. 3 In the Boundary Mode Analysis settings window, locate the Study Settings section. 4 In the Port name edit field, type 3. Step 5: Boundary Mode Analysis 3 1 Right-click Study 1>Step 4: Boundary Mode Analysis 2 and choose Duplicate. 2 In the Model Builder window, under Study 1 right-click Step 5: Boundary Mode Analysis 3 and choose Move Up. 3 In the Boundary Mode Analysis settings window, locate the Study Settings section. 4 In the Port name edit field, type 4. Step 5: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 5: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 On the Home toolbar, click Compute. 17 MACH-ZEHNDER MODULATOR

98 DEFINITIONS Input S-bend Click the Zoom Selected button on the Graphics toolbar. As seen from the result graph, the wave is not bound to the core when the bend radius is so small. To make the wave follow the waveguide core, the bend radius must be increased. Thus, we should make a parametric sweep of the bend radius, to find the smallest radius that gives a sufficient transmission. STUDY 1 Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click the Add button. 4 In the table, enter the following settings: Parameter names r0 Parameter value list 18 MACH-ZEHNDER MODULATOR

99 5 Click Range. 6 Go to the Range dialog box. 7 In the Start edit field, type 100[um]. 8 In the Step edit field, type 400[um]. 9 In the Stop edit field, type 2500[um]. 10 Click the Replace button. 11 On the Study toolbar, click Compute. RESULTS 1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 6. 4 On the 1D Plot Group 1 toolbar, click Global. 5 In the Global settings window, locate the y-axis Data section. 6 In the table, enter the following settings: Expression Unit Description sqrt(abs(ewbe.s21)^2+abs(ewbe.s41)^2) 1 7 Locate the x-axis Data section. From the Axis source data list, choose r0. 8 In the Model Builder window, click 1D Plot Group 3. 9 Locate the Plot Settings section. Select the x-axis label check box. 10 In the associated edit field, type Bend radius of curvature (m). 11 Select the y-axis label check box. 12 In the associated edit field, type Total modal transmission. 13 Click to expand the Title section. From the Title type list, choose None. 14 Click to expand the Axis section. Select the Manual axis limits check box. 15 In the y minimum edit field, type In the y maximum edit field, type In the Model Builder window, under Results>1D Plot Group 3 click Global In the Global settings window, click to expand the Legends section. 19 Clear the Show legends check box. 19 MACH-ZEHNDER MODULATOR

100 20 On the 1D plot group toolbar, click Plot. Your graph should look the same as the graph in Figure 2. A loss of approximately 2% seems reasonable, as we get for a bend radius of 2.5 mm. Electric Field (ewbe) 1 1 Zoom-in on a part of the waveguide bend. 2 On the 2D plot group toolbar, click Plot. Compare your graph to Figure 3. As you see, for a 2.5 mm bend radius, the wave is bound to the waveguide core. Thus, now set the bend radius parameter to 2.5 mm. GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description r0 2.5[mm] m Bend radius Now make sure that the directional coupler splits power of the incoming wave equally much into its output ports. To do this, you compare the power in the two waveguide arms of the Mach-Zehnder interferometer. DEFINITIONS Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 From the Selection list, choose End of upper Mach-Zehnder waveguide. 20 MACH-ZEHNDER MODULATOR

101 5 Click Zoom Selected. Integration 2 1 Right-click Component 1>Definitions>Integration 1 and choose Duplicate. 2 In the Integration settings window, locate the Source Selection section. 3 From the Selection list, choose End of lower Mach-Zehnder waveguide. 21 MACH-ZEHNDER MODULATOR

102 4 Click Zoom Selected. Variables 1a 1 In the Model Builder window, right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description P1 intop1(ewbe.npoav) W/m Power in upper waveguide P2 intop2(ewbe.npoav) W/m Power in lower waveguide STUDY 1 Parametric Sweep Modify the parametric sweep for a sweep of the directional coupler length. 1 In the Model Builder window, expand the Study 1 node, then click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 In the table, select the parameter d_dc. 4 Click Range. 22 MACH-ZEHNDER MODULATOR

103 5 Go to the Range dialog box. 6 In the Start edit field, type 80[um]. 7 In the Step edit field, type 50[um]. 8 In the Stop edit field, type 430[um]. 9 Click the Replace button. 10 On the Home toolbar, click Compute. RESULTS 1D Plot Group 3 1 In the Model Builder window, under Results>1D Plot Group 3 click Global 1. 2 In the Global settings window, locate the y-axis Data section. 3 In the table, enter the following settings: Expression Unit Description abs(p2-p1) W/m 4 In the Model Builder window, click 1D Plot Group 3. 5 In the 1D Plot Group settings window, locate the Plot Settings section. 6 In the x-axis label edit field, type Directional coupler length (m). 7 In the y-axis label edit field, type Absolute power difference (W/m). 8 Locate the Axis section. In the x minimum edit field, type 8e-5. 9 In the x maximum edit field, type 4.3e In the y minimum edit field, type Locate the Grid section. Select the Manual spacing check box. 12 In the x spacing edit field, type 5e In the y spacing edit field, type On the 1D Plot Group 3 toolbar, click Plot. Your graph should now look like Figure 4. Electric Field (ewbe) 1 1 In the Model Builder window, under Results click Electric Field (ewbe) 1. 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Parameter value (d_dc) list, choose 3.8e-4. 4 On the Electric Field (ewbe) 1 toolbar, click Plot. 23 MACH-ZEHNDER MODULATOR

104 5 Click the Zoom Extents button on the Graphics toolbar. 6 Click the Zoom In button on the Graphics toolbar four times. Your graph should now look like Figure 5. As shown in Figure 4 and Figure 5, the power in the two waveguides is almost the same when the directional coupler waveguides are 380 m long. Thus, set the parameter d_dc to 380 m. GLOBAL DEFINITIONS Parameters 1 In the Model Builder window, expand the Global Definitions node, then click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description d_dc 380[um] 3.800E-4 m Length of directional coupler waveguides The final geometry parameter to fix is the Mach-Zehnder waveguide length. Set that length to 2 cm. 4 In the table, enter the following settings: Name Expression Value Description d_mz 2[cm] m Length of Mach-Zehnder waveguides Finally, add an Electrostatics user interface to apply an electric field across the waveguide in one of the arms of the interferometer. COMPONENT 1 On the Home toolbar, click Add Physics. ADD PHYSICS 1 Go to the Add Physics window. 2 In the Add physics tree, select AC/DC>Electrostatics (es). 3 In the Add physics window, click Add to Component. 4 Close the Add physics window. 24 MACH-ZEHNDER MODULATOR

105 MATERIALS Cladding 1 In the Model Builder window, expand the Component 1>Materials node, then click Cladding. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity epsilonr epsr 1 Basic Core 1 In the Model Builder window, under Component 1>Materials click Core. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity epsilonr epsr 1 Basic GEOMETRY 1 Add two lines for the terminals - one for the ground and one for the applied voltage. Polygon 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Polygon. 2 In the Polygon settings window, locate the Coordinates section. 3 In the x edit field, type d0+2*dx_bend+d_dc d0+2*dx_bend+d_dc+d_mz. 4 In the y edit field, type w_tot/2-w w_tot/2-w. Polygon 2 1 Right-click Component 1>Geometry 1>Polygon 1 and choose Duplicate. 2 In the Polygon settings window, locate the Coordinates section. 3 In the y edit field, type w_tot/2+w w_tot/2+w. 25 MACH-ZEHNDER MODULATOR

106 4 Click the Build All Objects button. Now, add a voltage terminal and a ground. ELECTROSTATICS Electric Potential 1 1 On the Physics toolbar, click Boundaries and choose Electric Potential. 26 MACH-ZEHNDER MODULATOR

107 2 Select Boundary 72 only. 3 In the Electric Potential settings window, locate the Electric Potential section. 4 In the V 0 edit field, type V0. Ground 1 1 On the Physics toolbar, click Boundaries and choose Ground. 27 MACH-ZEHNDER MODULATOR

108 2 Select Boundary 66 only. MATERIALS Cladding Also make sure that the refractive index is changed by the applied static electric field. 1 In the Model Builder window, under Component 1>Materials click Cladding. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_clad-0.5*n_clad^3*r13*es.e y 1 Refractive index Core 1 In the Model Builder window, under Component 1>Materials click Core. 2 In the Material settings window, locate the Material Contents section. 28 MACH-ZEHNDER MODULATOR

109 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n_core-0.5*n_core^3*r13*es.e y 1 Refractive index STUDY 1 Parametric Sweep 1 In the Model Builder window, under Study 1 click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 In the table, choose the parameter V0. 4 Click Range. 5 Go to the Range dialog box. 6 In the Start edit field, type 0[V]. 7 In the Step edit field, type 10[V]. 8 In the Stop edit field, type 80[V]. 9 Click the Replace button. Step 6: Stationary 1 On the Study toolbar, click Study Steps and choose Stationary>Stationary. 2 In the Model Builder window, under Study 1 right-click Step 6: Stationary and choose Move Up. 3 In the Stationary settings window, locate the Physics and Variables Selection section. 4 In the table, click on the cell in the Solve for column corresponding to Electromagnetic Waves, Beam Envelopes to deactivate the physics in this study step. Step 1: Boundary Mode Analysis 1 In the Model Builder window, under Study 1 click Step 1: Boundary Mode Analysis. 2 In the Boundary Mode Analysis settings window, locate the Physics and Variables Selection section. 3 In the table, click on the cell in the Solve for column corresponding to Electrostatics to deactivate the physics in this study step 4 Repeat steps 1-3 for the other Boundary Mode Analysis and Frequency Domain steps. 5 On the Home toolbar, click Compute. 29 MACH-ZEHNDER MODULATOR

110 RESULTS 1D Plot Group 3 1 In the Model Builder window, under Results click 1D Plot Group 3. 2 In the 1D Plot Group settings window, locate the Plot Settings section. 3 Clear the x-axis label check box. 4 Clear the y-axis label check box. 5 Locate the Axis section. Clear the Manual axis limits check box. 6 Locate the Grid section. Clear the Manual spacing check box. 7 In the Model Builder window, under Results>1D Plot Group 3 click Global 1. 8 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Ports>S-parameter, db>s-parameter, db, 21 component (ewbe.s21db). 9 Click Add Expression in the upper-right corner of the y-axis data section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Ports>S-parameter, db>s-parameter, db, 41 component (ewbe.s41db). 10 Locate the Legends section. Select the Show legends check box. 11 From the Legends list, choose Manual. 12 In the table, enter the following settings: Legends S-parameter, db, 21 component S-parameter, db, 41 component 13 On the 1D plot group toolbar, click Plot. Compare your graph with Figure MACH-ZEHNDER MODULATOR

111 Appendix: Geometry Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Beam Envelopes (ewbe). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Boundary Mode Analysis. 6 Click the Done button. GLOBAL DEFINITIONS Start by adding parameters for the geometry and the material properties. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 31 MACH-ZEHNDER MODULATOR

112 3 In the table, enter the following settings: Name Expression Value Description wl 1.55[um] 1.550E-6 m Wavelength f0 c_const/wl 1.934E14 1/s Frequency w 2[um] 2.000E-6 m Waveguide width w_tot 30[um] 3.000E-5 m Total waveguide width n_clad Cladding refractive index n_core Core refractive index d0 2*wl 3.100E-6 m Initial straight waveguide dy_bend 0.6*w_tot 1.800E-5 m Total displacement in y-direction at S-bend r0 100[um] 1.000E-4 m Bend radius alpha acos((r0-dy_be rad Bend angle nd/2)/r0) dx_bend 2*r0*sin(alpha ) 8.292E-5 m Total length in the x-direction for S-bend d_dc 90[um] 9.000E-5 m Length of directional coupler waveguides d_mz 100[um] 1.000E-4 m Length of Mach-Zehnder waveguides The length of the Mach-Zehnder arms has purposely been set to a small value, to make it easier to build the geometry. You will later change the length to a realistic value. 4 In the table, enter the following settings: Name Expression Value Description hx wl/(6*n_core)/ 1.293E-6 m Maximum element size in (1-cos(alpha)) x-direction hy wl/(6*n_core)/ sin(alpha) 2.807E-7 m Maximum element size in y-direction dy_wg 3[um] 3.000E-6 m Distance between directional coupler waveguides r13 30[pm/V] 3.000E-11 C/N Electro-optic coefficient 32 MACH-ZEHNDER MODULATOR

113 Name Expression Value Description V0 100[V] V Applied voltage epsr Low-frequency relative permittivity GEOMETRY 1 Start defining the first S-shaped bend that is part of the first directional coupler. You will later study the radius of curvature for these bends, to find a value that is a good compromise between insertion loss and device size. Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type d0. 4 In the Height edit field, type w_tot. 5 Click to expand the Layers section. In the table, enter the following settings: Layer name Layer 1 Thickness (m) (w_tot-w)/2 6 Select the Layers on top check box. Circle 1 1 In the Model Builder window, right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type r0+w_tot/2. 4 Locate the Position section. In the x edit field, type d0. 5 In the y edit field, type w_tot/2-r0. 6 Locate the Size and Shape section. In the Sector angle edit field, type alpha. 7 Locate the Rotation Angle section. In the Rotation edit field, type 90-alpha. Circle 2 1 Right-click Component 1>Geometry 1>Circle 1 and choose Duplicate. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type r0+w/2. 33 MACH-ZEHNDER MODULATOR

114 Circle 3 1 Right-click Component 1>Geometry 1>Circle 2 and choose Duplicate. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type r0-w/2. Circle 4 1 In the Model Builder window, under Component 1>Geometry 1 right-click Circle 1 and choose Duplicate. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type r0-w_tot/2. 4 On the Home toolbar, click Build All. 5 Click the Zoom Extents button on the Graphics toolbar. Difference 1 1 On the Geometry toolbar, click Difference. 2 Select the objects c2, c3, and c1 only. 3 In the Difference settings window, locate the Difference section. 4 Select the Objects to subtract toggle button. 34 MACH-ZEHNDER MODULATOR

115 5 Select the object c4 only. 6 Click the Build All Objects button. 7 Click the Zoom Extents button on the Graphics toolbar. Rotate 1 1 On the Geometry toolbar, click Rotate. 35 MACH-ZEHNDER MODULATOR

116 2 Select the object dif1 only. 3 In the Rotate settings window, locate the Input section. 4 Select the Keep input objects check box. 5 Locate the Rotation Angle section. In the Rotation edit field, type Locate the Center of Rotation section. In the x edit field, type d0+dx_bend/2. 7 In the y edit field, type w_tot/2-dy_bend/2. 8 Click the Build All Objects button. 36 MACH-ZEHNDER MODULATOR

117 9 Click the Zoom Extents button on the Graphics toolbar. Rectangle 2 1 In the Model Builder window, under Component 1>Geometry 1 right-click Rectangle 1 and choose Duplicate. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type d_dc. 4 Locate the Position section. In the x edit field, type d0+dx_bend. 5 In the y edit field, type -dy_bend. 6 Click the Build All Objects button. 37 MACH-ZEHNDER MODULATOR

118 7 Click the Zoom Extents button on the Graphics toolbar. Mirror 1 1 On the Geometry toolbar, click Mirror. 38 MACH-ZEHNDER MODULATOR

119 2 Select the objects dif1 and rot1 only. 3 In the Mirror settings window, locate the Input section. 4 Select the Keep input objects check box. 5 Locate the Point on Line of Reflection section. In the x edit field, type d0+dx_bend+d_dc/2. 6 Click the Build All Objects button. 39 MACH-ZEHNDER MODULATOR

120 7 Click the Zoom Extents button on the Graphics toolbar. Rectangle 3 1 Right-click Rectangle 1 and choose Duplicate. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type d_mz. 4 Locate the Position section. In the x edit field, type d0+2*dx_bend+d_dc. 5 Click the Build All Objects button. 40 MACH-ZEHNDER MODULATOR

121 6 Click the Zoom Extents button on the Graphics toolbar. Mirror 2 1 On the Geometry toolbar, click Mirror. 2 Select the objects dif1, r2, mir1(1), rot1, r1, and mir1(2) only. 3 In the Mirror settings window, locate the Input section. 4 Select the Keep input objects check box. 5 Locate the Point on Line of Reflection section. In the x edit field, type d0+2*dx_bend+d_dc+d_mz/2. 41 MACH-ZEHNDER MODULATOR

122 6 Locate the Input section. Under Input objects, click Zoom Selected. Rectangle 4 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type 2*(d0+d_dc)+d_mz+4*dx_bend. 4 In the Height edit field, type (w_tot-w-dy_wg)/2. 5 Locate the Position section. In the y edit field, type -dy_bend. 42 MACH-ZEHNDER MODULATOR

123 6 Click the Build All Objects button. Difference 2 1 On the Geometry toolbar, click Difference. 43 MACH-ZEHNDER MODULATOR

124 2 Select the objects dif1, r2, mir2(1), mir2(4), mir1(1), r1, mir1(2), mir2(2), mir2(6), mir2(3), rot1, mir2(5), and r3 only. 3 In the Difference settings window, locate the Difference section. 4 Select the Objects to subtract toggle button. 44 MACH-ZEHNDER MODULATOR

125 5 Select the object r4 only. 6 Click the Build All Objects button. Mirror 3 1 On the Geometry toolbar, click Mirror. 45 MACH-ZEHNDER MODULATOR

126 2 Select the object dif2 only. 3 In the Mirror settings window, locate the Input section. 4 Select the Keep input objects check box. 5 Locate the Point on Line of Reflection section. In the y edit field, type (w_tot-w-dy_wg)/2-dy_bend. 6 Locate the Normal Vector to Line of Reflection section. In the x edit field, type 0. 7 In the y edit field, type 1. 8 Click the Build All Objects button. 46 MACH-ZEHNDER MODULATOR

127 9 Click the Zoom Extents button on the Graphics toolbar. DEFINITIONS Finally, add some selections that will be useful when defining the model. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 Select Domains 1 3, 7 9, 13, 15, and 17 only. 3 Right-click Component 1>Definitions>Explicit 1 and choose Rename. 4 Go to the Rename Explicit dialog box and type Input S-bend in the New name edit field. 5 Click OK. 47 MACH-ZEHNDER MODULATOR

128 6 Click the Zoom Selected button on the Graphics toolbar. 7 Click the Zoom Extents button on the Graphics toolbar. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 Select Domains 2, 5, 8, 11, 15, 16, 20, 23, 26, 29, 33, 34, 38, 41, 44, 47, 51, 52, 56, 59, 62, 65, 69, 70, 74, and 77 only. 3 Right-click Component 1>Definitions>Explicit 2 and choose Rename. 4 Go to the Rename Explicit dialog box and type Core in the New name edit field. 48 MACH-ZEHNDER MODULATOR

129 5 Click OK. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 Select the All domains check box. 4 Locate the Output Entities section. From the Output entities list, choose Adjacent boundaries. 5 Right-click Component 1>Definitions>Explicit 3 and choose Rename. 6 Go to the Rename Explicit dialog box and type Exterior boundaries in the New name edit field. 49 MACH-ZEHNDER MODULATOR

130 7 Click OK. Explicit 4 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 1, 3, and 5 only. 5 Right-click Component 1>Definitions>Explicit 4 and choose Rename. 6 Go to the Rename Explicit dialog box and type Port 1 in the New name edit field. 7 Click OK. Make sure your selection match the one shown below. 50 MACH-ZEHNDER MODULATOR

131 8 Click the Zoom Selected button on the Graphics toolbar. 9 Click the Zoom Extents button on the Graphics toolbar. Explicit 5 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries only. 5 Right-click Component 1>Definitions>Explicit 5 and choose Rename. 6 Go to the Rename Explicit dialog box and type Port 2 in the New name edit field. 7 Click OK. Make sure your selection match the one shown below. 51 MACH-ZEHNDER MODULATOR

132 8 Click the Zoom Selected button on the Graphics toolbar. 9 Click the Zoom Extents button on the Graphics toolbar. Explicit 6 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries only. 5 Right-click Component 1>Definitions>Explicit 6 and choose Rename. 6 Go to the Rename Explicit dialog box and type Port 4 in the New name edit field. 7 Click OK. Your selection should match the one shown below. 52 MACH-ZEHNDER MODULATOR

133 8 Click the Zoom Selected button on the Graphics toolbar. 9 Click the Zoom Extents button on the Graphics toolbar. Explicit 7 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 8, 10, and 12 only. 5 Right-click Component 1>Definitions>Explicit 7 and choose Rename. 6 Go to the Rename Explicit dialog box and type Port 3 in the New name edit field. 7 Click OK. Your selection should match the one shown below. 53 MACH-ZEHNDER MODULATOR

134 8 Click the Zoom Selected button on the Graphics toolbar. Difference 1 1 On the Definitions toolbar, click Difference. 2 In the Difference settings window, locate the Geometric Entity Level section. 3 From the Level list, choose Boundary. 4 Locate the Input Entities section. Under Selections to add, click Add. 5 Go to the Add dialog box. 6 In the Selections to add list, select Exterior boundaries. 7 Click the OK button. 8 In the Difference settings window, locate the Input Entities section. 9 Under Selections to subtract, click Add. 10 Go to the Add dialog box. 11 In the Selections to subtract list, choose Port 1, Port 2, Port 4, and Port Click the OK button. 13 Right-click Component 1>Definitions>Difference 1 and choose Rename. 54 MACH-ZEHNDER MODULATOR

135 14 Go to the Rename Difference dialog box and type Scattering boundary condition in the New name edit field. 15 Click OK. Your selection should match the one shown below. 16 Click the Zoom Extents button on the Graphics toolbar. Explicit 8 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries only. 5 Right-click Component 1>Definitions>Explicit 8 and choose Rename. 6 Go to the Rename Explicit dialog box and type End of lower Mach-Zehnder waveguide in the New name edit field. 7 Click OK. Verify your selection with the one shown below. 55 MACH-ZEHNDER MODULATOR

136 8 Click the Zoom Selected button on the Graphics toolbar. 9 Click the Zoom Out button on the Graphics toolbar. 10 Click the Zoom Out button on the Graphics toolbar. Explicit 9 1 On the Definitions toolbar, click Explicit. 2 In the Explicit settings window, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries only. 5 Right-click Component 1>Definitions>Explicit 9 and choose Rename. 6 Go to the Rename Explicit dialog box and type End of upper Mach-Zehnder waveguide in the New name edit field. 7 Click OK. Verify your selection with the one shown below. 56 MACH-ZEHNDER MODULATOR

137 8 Click the Zoom Selected button on the Graphics toolbar. Union 1 1 On the Definitions toolbar, click Union. 2 In the Union settings window, locate the Geometric Entity Level section. 3 From the Level list, choose Boundary. 4 Locate the Input Entities section. Under Selections to add, click Add. 5 Go to the Add dialog box. 6 In the Selections to add list, choose End of lower Mach-Zehnder waveguide and End of upper Mach-Zehnder waveguide. 7 Click the OK button. 8 Right-click Component 1>Definitions>Union 1 and choose Rename. 9 Go to the Rename Union dialog box and type End of Mach-Zehnder waveguides in the New name edit field. 57 MACH-ZEHNDER MODULATOR

138 10 Click OK. Finally, check your last selection with the one shown below. 58 MACH-ZEHNDER MODULATOR

139 Defining a Mapped Dielectric Distribution of a Metamaterial Lens Introduction This example demonstrates how to set up a spatially varying dielectric distribution, such as might be engineered with a metamaterial. Here, a convex lens shape is defined via a known deformation of a rectangular domain. The dielectric distribution is defined on the undeformed, original rectangular domain and is mapped onto the deformed shape of the lens. Although the lens shape defined here is convex, the dielectric distribution causes the incident beam to diverge. r (X g, Y g ) (x, y) = F(X g, Y g ) Figure 1: A convex metamaterial lens. Both the shape and the dielectric distribution are defined on a rectangular domain, and mapped into the deformed state. 1 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

140 Model Definition Consider a 2D model geometry as shown in Figure 2. A square air domain, bounded by a perfectly matched layer (PML) on all sides, encloses a rectangular region in which the metamaterial lens is defined. PML Figure 2: The modeling domain consists of the metamaterial lens in an air domain, and a surrounding PML. A Gaussian beam is incident from the left. Model a Gaussian beam entering the domain from the left side, via a surface current excitation at an interior boundary. The surface current, J s0,can also be thought of as a displacement current excitation. The waist of the beam is at the boundary, so the excitation at this boundary can be specified as J s y = exp 2 where w 0 is the waist size. The excitation is at the boundary between a domain of free space and the PML, and excites a wave propagating in both directions into the PML and into the modeling domain. The wave propagating into the PML is completely absorbed, and the wave propagating into the domain is diffracted by the lens. w 0 2 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

141 Both the shape and the dielectric distribution of the metamaterial lens are defined with respect to the original Cartesian coordinate system, as shown in Figure 1. The true shape of the lens is described by the relationship x y F x X g Y g = = F y X g Y g X 2 g 2 Y g Y g x2 where X g, Y g are the Cartesian coordinates of the undeformed frame. The dielectric distribution is defined on the original Cartesian domain as: r Y 2 = + 2 g The above expression introduces a variation in the dielectric in the y-coordinate of the undeformed lens. On the deformed lens, the dielectric varies in both directions. The Deformed Geometry functionality is used to define the mapping of the dielectric from the initially rectangular domain onto the desired shape. The deformation and the dielectric distribution within the lens domain is completely specified via the above functions. Results and Discussion The model is solved for the out-of-plane electric field. Figure 3 plots the electric field norm, showing a Gaussian beam with minimal divergence incident upon the lens from the left. The beam is diffracted by the convex lens and spreads out. Figure 4 displays the dielectric distribution, and shows variation in both directions defined via the mapping described above. 3 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

142 Figure 3: The norm of the electric field shows the Gaussian beam diffracted by the metamaterial lens. Figure 4: Contour plot of the dielectric distribution. 4 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

143 Model Library path: Wave_Optics_Module/Gratings_and_Metamaterials/ mapped_metamaterial_distribution Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Mathematics>Deformed Mesh>Deformed Geometry (dg). 3 Click the Add button. 4 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 5 Click the Add button. 6 Click the Study button. 7 In the tree, select Custom Studies>Preset Studies for Some Physics>Stationary. 8 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description f0 3[GHz] 3.000E9 Hz Operating frequency lda0 c_const/f m Free space wavelength w0 lda0* m Gaussian beam waist size Here, c_const is a predefined COMSOL constant for the speed of light in vacuum. 5 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

144 GEOMETRY 1 First, create a square for the entire model domain. Add a layer on each side of the square. Square 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Square. 2 In the Square settings window, locate the Size section. 3 In the Side length edit field, type 3. 4 Locate the Position section. From the Base list, choose Center. 5 Click to expand the Layers section. In the table, enter the following settings: Layer name Layer 1 Thickness (m) lda0 6 Select the Layers to the left check box. 7 Select the Layers to the right check box. 8 Select the Layers on top check box. 9 Click the Build Selected button. Add a rectangle for the lens. Rectangle 1 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Height edit field, type 2. 4 Locate the Position section. From the Base list, choose Center. 6 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

145 5 Click the Build All Objects button. DEFINITIONS Add a selection for the lens domain which will be recalled frequently while setting up the model properties. Explicit 1 1 On the Definitions toolbar, click Explicit. 7 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

146 2 Select Domain 7 only. 3 Right-click Component 1>Definitions>Explicit 1 and choose Rename. 4 Go to the Rename Explicit dialog box and type Lens in the New name edit field. 5 Click OK. Next, add a set of variables for the shape and the dielectric distribution of the metamaterial lens. Variables 1a 1 On the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Lens. 8 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

147 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description xp 0.5[m]*Xg[1/m]*(2-(Yg[1/m])^2) m Mapping of Xg -> x yp Yg*(1+(0.5*(xp[1/m])^2)) m Mapping of Yg -> y erp (1+0.5*(Yg[1/m])^2)^2 Dielectric distribution Here, Xg and Yg are predefined Deformed Geometry physics variables representing the Cartesian coordinates of the undeformed frame. Add a perfectly matched layer (PML). Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 Select Domains 1 4, 6, and 8 10 only. DEFORMED GEOMETRY Set up Deformed Geometry. You need to specify Free Deformation, Prescribed Mesh Displacement and Prescribed Deformation. 1 In the Model Builder window, under Component 1 click Deformed Geometry. 2 In the Deformed Geometry settings window, locate the Frame Settings section. 9 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

148 3 From the Geometry shape order list, choose 1. Free Deformation 1 1 On the Physics toolbar, click Domains and choose Free Deformation. 2 Select Domain 5 only. Prescribed Mesh Displacement 2 1 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement. 10 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

149 2 Select Boundaries only. 3 In the Prescribed Mesh Displacement settings window, locate the Prescribed Mesh Displacement section. 4 In the d x edit field, type xp-xg. 5 In the d y edit field, type yp-yg. Prescribed Deformation 1 1 On the Physics toolbar, click Domains and choose Prescribed Deformation. 2 In the Prescribed Deformation settings window, locate the Domain Selection section. 3 From the Selection list, choose Lens. 4 Locate the Prescribed Mesh Displacement section. In the d x edit-field array, type xp-xg on the first row. 5 In the d y edit-field array, type yp-yg on the 2nd row. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN In Electromagnetic Waves, Frequency Domain, the dielectric distribution is configured via the user-defined variable erp and the Gaussian beam is modeled as entering the domain from the left side, via a surface current excitation. 11 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

150 Wave Equation, Electric 2 1 From the Select Physics list, now showing Deformed Geometry, select Electromagnetic Waves, Frequency Domain. 2 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 3 In the Wave Equation, Electric settings window, locate the Domain Selection section. 4 From the Selection list, choose Lens. 5 Locate the Electric Displacement Field section. From the Electric displacement field model list, choose Relative permittivity. 6 From the r list, choose User defined. In the associated edit field, type erp. 7 Locate the Magnetic Field section. From the r list, choose User defined. Leave the default value 1. 8 Locate the Conduction Current section. From the list, choose User defined. Leave the default value 0. Surface Current 1 1 On the Physics toolbar, click Boundaries and choose Surface Current. 2 Select Boundary 10 only. 3 In the Surface Current settings window, locate the Surface Current section. 12 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

151 4 Specify the J s0 vector as 0 x 0 y exp(-(y/w0)^2) z MATERIALS Set all domain with vacuum. The lens domain material properties are explicitly configured by Wave Equation, Electric 2 in Electromagnetic Waves, Frequency Domain. 1 On the Home toolbar, click Add Material. ADD MATERIAL 1 Go to the Add Material window. 2 In the tree, select Built-In>Air. 3 In the Add Material window, click Add to Component. 4 Close to the Add Material window. MATERIALS MESH 1 Size 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Free Triangular. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type lda0/10. 5 In the Minimum element size edit field, type In the Model Builder window, right-click Mesh 1 and choose Build All. You may zoom in a few times to check the quality of the mesh. STUDY 1 The model is analyzed with two study steps. First, make sure that Stationary study step is solved only for Deformed Geometry. 13 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

152 Step 1: Stationary 1 In the Model Builder window, under Study 1 click Step 1: Stationary. 2 In the Stationary settings window, locate the Physics and Variables Selection section. 3 In the table, enter the following settings: Physics Solve for Discretization Electromagnetic Waves, Frequency Domain physics Add a Frequency Domain study step and set as solved only for Electromagnetic Waves, Frequency Domain. Step 2: Frequency Domain 1 On the Study toolbar, click Study Steps and choose Frequency Domain>Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 Locate the Physics and Variables Selection section. In the table, enter the following settings: Physics Solve for Discretization Deformed Geometry physics 5 On the Study toolbar, click Compute. RESULTS Electric Field (ewfd) The default plot shows the magnitude of electric fields. Change the default color pattern and add a contour plot for the magnitude. 1 In the Model Builder window, under Results>Electric Field (ewfd) click Surface 1. 2 In the Surface settings window, locate the Coloring and Style section. 3 From the Color table list, choose RainbowLight. 4 In the Model Builder window, right-click Electric Field (ewfd) and choose Contour. 5 In the Contour settings window, locate the Levels section. 6 In the Total levels edit field, type Locate the Coloring and Style section. From the Coloring list, choose Uniform. 8 From the Color list, choose Black. 14 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

153 9 Clear the Color legend check box. See Figure 3 to compare the reproduced plot. Add a filled contour plot describing the dielectric distribution over the lens. 2D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 2D Plot Group. 2 In the Model Builder window, under Results right-click 2D Plot Group 2 and choose Contour. 3 In the Contour settings window, locate the Expression section. 4 In the Contour settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Material properties>relative permittivity, average (ewfd.epsrav), by double-clicking it or selecting it and pressing Enter. 5 Locate the Levels section. In the Total levels edit field, type Locate the Coloring and Style section. From the Contour type list, choose Filled. 7 From the Color table list, choose GrayScale. 8 Select the Reverse color table check box. The plot for the dielectric distribution is shown in Figure DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

154 16 DEFINING A MAPPED DIELECTRIC DISTRIBUTION OF A METAMATERIAL LENS

155 Nanorods Introduction A Gaussian electromagnetic wave is incident on a dense array of very thin wires (or rods). The distance between the rods and, thus, also the rod diameter is much smaller than the wavelength. Under these circumstances, the rod array does not function as a diffraction grating (see the Plasmonic Wire Grating model). Instead, the rod array behaves as if it was a continuous metal sheet for light polarized along the rods, whereas for light polarized perpendicular to the rods, the array is almost transparent to the electromagnetic wave. However, for the latter case dipole coupling between the rods occurs, thereby coupling electromagnetic excitation between the rods also outside of the illuminated region. Model Definition Figure 1 shows the array of rods and the incident Gaussian beam. The beam propagates in negative y-direction. The wavelength is 750 nm and the spot radius of the beam equals the wavelength. For a more detailed discussion about Gaussian beams, see the model Second Harmonic Generation of a Gaussian Beam. Notice that the closed-form expression for the electric field of a Gaussian beam is only an approximation to the solution of Helmholtz equation. For tightly focused beams, as in this model, you also need to include an electric field component in the propagation direction, for in-plane polarization, as described in Ref NANORODS

156 y x Figure 1: The modeled array of nanorods and the incident Gaussian beam. The model solves for the electric field in a 2D plane. Thus, the beam and rod properties are constant in the out-of-plane dimension. The rods radius is 20 nm and the separation between the rods is 150 nm. Figure 1 shows that half of the beam illuminates the first part of the array of rods. Thus, most of the 40 rods in the model have a very low illumination. The rods are metallic, with a dispersion formula for the relative permittivity given by 2 1 P = (1) where the angular frequency is defined by = 2 c (2) and p is the plasma frequency. The plasma frequency is set to negative relative permittivity similar to that of gold. 21, resulting in a Results and Discussion Figure 2 shows the norm of the electric field for the Gaussian beam. In this configuration, the light beam is polarized in the x-direction, that is, in the direction of 2 NANORODS

157 the grating vector. As seen from the surface plot, the nanorod array illuminates half of the Gaussian light beam. However, for this polarization there is no noticeable reflection nor diffraction from the nanorod array. Figure 2: Surface plot of the electric field norm. The electric field of the Gaussian light beam is polarized in the x direction (along the grating vector). Figure 3 shows a zoom-in of the left most part of the nanorod array and the center of the light beam. The plot shows that there is a dipolar coupling between the rods. Figure 4 shows that the coupling between the nanorods extends much longer than the intensity distribution of the exciting Gaussian light beam. 3 NANORODS

158 Figure 3: Zoom-in on the center of the Gaussian beam and the left-most nanorods. As for Figure 2, the polarization of the beam is in the x direction. 4 NANORODS

159 Figure 4: Line plot of the x component of the electric field (blue line) and the Gaussian beam background field (green line). The beam is polarized in the x direction. When the light beam is polarized along the rods, that is, in the z direction (the out-of-plane direction), the interaction between the beam and the nanorod array is much stronger. As a comparison to Figure 3, Figure 5 shows a zoom-in of the center of the Gaussian beam and the left-most nanorods. When the beam is polarized in the out-of-plane direction, the nanorod array appears for the beam almost as an opaque metal sheet. Thus, the beam is reflected and there is strong edge diffraction. 5 NANORODS

160 Figure 5: Zoom-in of the electric field norm for the center of the Gaussian beam and the left-most nanorods. The Gaussian light beam is polarized along the rods (out-of-plane in the z direction). The effects of reflection and edge diffraction is very obvious from Figure 6. Above the array there is a standing wave, formed by the incident beam and the beam reflected from the array, and below the array edge diffraction is evident. 6 NANORODS

161 Figure 6: Surface plot of the full Gaussian beam and all the nanorods. The beam is polarized in the out-of-plane (z) direction. Notes About the COMSOL Implementation This model implementation demonstrates the use of perfectly matched layers for absorbing a propagating wave. Furthermore, when defining the relative permittivity, the Drude-Lorentz dispersion model is used. This setting is found in the wave equation feature. Reference 1. M. Lax, W.H. Louisell, and W.B. McKnight, From Maxwell to paraxial wave optics, Physical Review A, vol. 11, no. 4, pp , Model Library path: Wave_Optics_Module/Optical_Scattering/nanorods 7 NANORODS

162 Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Add parameters for the Gaussian light beam, the geometric domains, and the critical material parameter for the nanorod. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description wl 750[nm] 7.500E-7 m Wavelength f c_const/wl 3.997E14 1/s Frequency w0 wl 7.500E-7 m Spot radius z0 pi*w0^2/wl 2.356E-6 m Rayleigh range k 2*pi/wl 8.378E6 1/m Propagation constant E0 1[V/m] V/m Electric field amplitude r_np 20[nm] 2.000E-8 m Radius of nanorods N_NP Number of nanorods dx_np 150[nm] 1.500E-7 m Separation between nanorods 8 NANORODS

163 Name Expression Value Description omega_p sqrt(21)*2*pi* f 1.151E16 1/s Plasma frequency for nanorod material w_air_lef t 5*w E-6 m Width of air domain for x < 0 w_air_rig ht max(5*w0,1.2*( N_NP-1)*dx_NP) 7.020E-6 m Width of air domain for x > 0 w_air w_air_left+w_a ir_right 1.077E-5 m Width of air domain h_air 4*wl 3.000E-6 m Height of air domain d_pml wl 7.500E-7 m Thickness of PML domains Now add functions used for describing the input Gaussian beam. Define the radius of curvature function, R, as a piecewise function, since it is infinite when its argument is zero. Analytic 1 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type R_finite. 4 Locate the Definition section. In the Expression edit field, type y*(1+(z0/y)^2). 5 In the Arguments edit field, type y. 6 Locate the Units section. In the Arguments edit field, type m. 7 In the Function edit field, type m. Piecewise 1 1 On the Home toolbar, click Functions and choose Global>Piecewise. 2 In the Piecewise settings window, locate the Function Name section. 3 In the Function name edit field, type R. 4 Locate the Definition section. Find the Intervals subsection. In the Argument edit field, type y. 5 In the table, enter the following settings: Start End Function -1e10*z0-1e-8*z0 R_finite(y) -1e-8*z0 0 R_finite(-1e-8*z0) 9 NANORODS

164 Start End Function 0 1e-8*z0 R_finite(1e-8*z0) 1e-8*z0 1e10*z0 R_finite(y) 6 Locate the Units section. In the Arguments edit field, type m. 7 In the Function edit field, type m. Analytic 2 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type w. 4 Locate the Definition section. In the Expression edit field, type w0*sqrt(1+(y/ z0)^2). 5 In the Arguments edit field, type y. 6 Locate the Units section. In the Arguments edit field, type m. 7 In the Function edit field, type m. Analytic 3 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type E. 4 Locate the Definition section. In the Expression edit field, type E0*sqrt(w0/ w(y))*exp(-(x/w(y))^2)*exp(j*k*x^2/ (2*R(y)))*exp(j*(k*y-0.5*atan(y/z0))). 5 In the Arguments edit field, type x, y. 6 Locate the Units section. In the Arguments edit field, type m,m. 7 In the Function edit field, type V/m. 8 Click to expand the Advanced section. Select the May produce complex output for real arguments check box. GEOMETRY 1 Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 10 NANORODS

165 3 In the Width edit field, type w_air+2*d_pml. 4 In the Height edit field, type h_air+2*d_pml. 5 Locate the Position section. In the x edit field, type -w_air_left-d_pml. 6 In the y edit field, type -h_air/2-d_pml. Now create layers surrounding the rectangle for the perfectly matched layers (PMLs). 7 Click to expand the Layers section. In the table, enter the following settings: Layer name Layer 1 Thickness (m) d_pml 8 Select the Layers to the left check box. 9 Select the Layers to the right check box. 10 Select the Layers on top check box. Create the nanorods by first defining a circle, and then create a linear array of those circles. Circle 1 1 In the Model Builder window, right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type r_np. Array 1 1 On the Geometry toolbar, click Array. The circle, representing a nanorod, is very small. Thus, you probably need to zoom in to select it. 2 Click the Zoom Box button on the Graphics toolbar. 3 Select the object c1 only. 4 Click the Zoom Extents button on the Graphics toolbar. 5 In the Array settings window, locate the Size section. 6 From the Array type list, choose Linear. 7 In the Size edit field, type N_NP. 8 Locate the Displacement section. In the x edit field, type dx_np. 9 Locate the Selections of Resulting Entities section. Select the Create selections check box to simplify later selection of the nanorods. 11 NANORODS

166 10 Click the Build Selected button. Rename the array to make it easier to later select the nanorods. 11 Right-click Component 1>Geometry 1>Array 1 and choose Rename. 12 Go to the Rename Array dialog box and type Nanorods in the New name edit field. 13 Click OK. 14 Right-click Geometry 1 and choose Build All Objects. DEFINITIONS Create a view to be used later for zooming-in on the plots. View 2 1 In the Definitions toolbar, click View. 2 In the View settings window, locate the View section. 3 Select the Lock axis check box. Axis 1 In the Model Builder window, expand the View 2 node, then click Axis. 2 In the Axis settings window, locate the Axis section. 3 In the x minimum edit field, type -0.5e-6. 4 In the x maximum edit field, type 0.5e-6. 5 In the y minimum edit field, type -0.5e-6. 6 In the y maximum edit field, type 0.5e-6. Switch back to the regular view now. 7 Right-click Component 1>Definitions>View 2>Axis and choose Go to View 1. Define the layers surrounding the central rectangular domain as PML regions. Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 Select Domains 1 4 and 6 9 only. MATERIALS Let the default material be air. The material properties of the nanorods will later be defined in a wave equation feature. 1 On the Home toolbar, click Add Material. 12 NANORODS

167 ADD MATERIAL 1 Go to the Add Material window. 2 In the tree, select Built-In>Air. 3 In the Add material window, click Add to Component. 4 Close the Add material window. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Calculate the first solution for light polarized along the out-of-plane z-direction. Solve for the scattered field and set the background field to a propagating 2D Gaussian beam. 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain. 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Settings section. 3 From the Solve for list, choose Scattered field. 4 Specify the E b vector as 0 x 0 y E(x,y) z Because there are no in-plane field components in this case, it suffices to compute the solution for the out-of-plane component. 5 Locate the Components section. From the Electric field components solved for list, choose Out-of-plane vector. Create a separate wave equation feature for the nanorod domains. Wave Equation, Electric 2 1 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 2 In the Wave Equation, Electric settings window, locate the Domain Selection section. 3 From the Selection list, choose Nanorods. Select the Drude-Lorentz dispersion model. The material parameters will result in a negative real part of the relative permittivity. This is normal for many metals, for frequencies below the plasma frequency. 4 Locate the Electric Displacement Field section. From the Electric displacement field model list, choose Drude-Lorentz dispersion model. 13 NANORODS

168 5 In the P edit field, type omega_p. 6 In the table, enter the following settings: Oscillator strength (1) Resonance frequency (rad/s) Damping in time (Hz) MESH 1 Create a swept mesh for the PML regions. Mapped 1 1 In the Mesh toolbar, click Mapped. 2 In the Mapped settings window, locate the Domain Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domains 1 4 and 6 9 only (the outer layers). Distribution 1 1 Right-click Component 1>Mesh 1>Mapped 1 and choose Distribution. 2 Select Boundaries 6, 12, 15, and 18 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 8. Size Set the maximum mesh size to be less than one-eighth of the wavelength. 1 In the Model Builder window, under Component 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type wl/8. Free Triangular 1 1 In the Mesh toolbar, click Free Triangular, then click Build Mesh. STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f. 14 NANORODS

169 4 On the Study toolbar, click Compute. RESULTS Electric Field (ewfd) Compare the results with that in Figure 6. The rod array appears for the field almost as a metal sheet. Thus, there are effects of edge diffraction and the reflection leads to standing wave effects in the region above the nanorod array. Select the previously defined view, to zoom-in on the center of the beam and the leftmost rods. 1 In the 2D Plot Group settings window, locate the Plot Settings section. 2 From the View list, choose View 2. 3 On the Electric Field (ewfd) toolbar, click Plot. Compare the result with that in Figure 5. The zoom-in shows that light is blocked by the nanorod array and that it is diffracted by the edge of the array. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Now set the field to be polarized mainly in the x direction, which means that the field is polarized along the array of nanorods (along the grating vector). Notice, however, that the tightly focused beam also have an electric field component in the propagation direction. 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain. 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Settings section. 3 Specify the E b vector as E(x,y) x j*d(e(x,y),x)/ewfd.k0 y 0 z Since there is no field component in the out-of-plane direction, the solution can be performed for only the in-plane components. 4 Locate the Components section. From the Electric field components solved for list, choose In-plane vector. 5 On the Study toolbar, click Compute. 15 NANORODS

170 RESULTS Electric Field (ewfd) Compare the result with the graph in Figure 3. The zoomed-in view shows that the nanorods act as dipoles that enhance the field strength between the rods. Click the Zoom Extents button on the Graphics toolbar. Compare the graph with Figure 2. In the zoomed-out view the Gaussian beam seems to be almost unperturbed by the nanorod array that cuts into half of the beam. Data Sets Make a cut line through the centers of the rods, to make a detailed line graph of the field distribution along the line. 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Line Data section. 3 In row Point 1, set x to -w_air_left. 4 In row Point 2, set x to w_air_right. 1D Plot Group 2 1 On the Results toolbar, click 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D Plot Group 2 toolbar, click Line Graph. Plot the magnitude of the electric field, polarized in the x-direction, and compare with the background electric field. 5 In the Line Graph settings window, locate the y-axis Data section. 6 In the Expression edit field, type abs(ewfd.ex). 7 Locate the x-axis Data section. From the Parameter list, choose Expression. 8 In the Expression edit field, type x. 9 Right-click Results>1D Plot Group 2>Line Graph 1 and choose Duplicate. 10 In the Line Graph settings window, locate the y-axis Data section. 11 In the Expression edit field, type abs(ewfd.ebx). 12 On the 1D Plot Group 2 toolbar, click Plot. Set the axis limits to get a detailed view of the field distribution. Notice the field enhancement along the array of the rods. 13 In the Model Builder window, click 1D Plot Group NANORODS

171 14 In the 1D Plot Group settings window, click to expand the Axis section. 15 Select the Manual axis limits check box. 16 In the x minimum edit field, type In the x maximum edit field, type 6E In the y minimum edit field, type In the y maximum edit field, type 5e On the 1D plot group toolbar, click Plot and compare the graph with Figure NANORODS

172 18 NANORODS Solved with COMSOL Multiphysics 4.4

173 Modeling a Negative Refractive Index Introduction It is possible to engineer the structure of materials such that both the permittivity and permeability are negative. Such materials are realized by engineering a periodic structure with features comparable in scale to the wavelength. It is possible to model both the individual unit cells of such a material, as well as to model to properties of a bulk negative index material. This example demonstrates the correct way to model a metamaterial domain with bulk negative permittivity and permeability. Reflected Unit cell Transmitted Incident 1, 1 2, 2 Figure 1: A plane wave of light incident upon an infinite half-space of material with negative permittivity and permeability. Model Definition A plane wave ( 1 m) traveling through vacuum ( r 1, r 1) is incident at an angle of 30 upon an infinite half-space of metamaterial with bulk negative permittivity and permeability ( r 1, r 1) as shown in Figure 1. The objective of the analysis is to observe the fields in the metamaterial. Since the model space is infinite and invariant along the interface, it is possible to model a finite-sized unit cell around the interface, and use Floquet-periodic boundary 1 MODELING A NEGATIVE REFRACTIVE INDEX

174 conditions. The incident wave is modeled using a Port boundary condition, which will both launch the incident wave, as well as absorb the reflected wave. Two different ways are demonstrated for modeling boundary to the infinite metamaterial domain. In the first case, a Port boundary condition is used. This boundary condition requires computing the wavevector in the metamaterial, and manually adjusting the propagation constant at the boundary to account for the negative index. In the other case, a Perfectly Matched Layer (PML) is used to truncate the domain. The PML acts as an absorbing medium for all energy incident upon it, but must also be adjusted to account for the negative index. It is simpler to use, but increases the model size. The transition between the two materials requires some special care. The natural boundary condition between domains of different material properties does not account for the change in direction of the flux. An additional degree of freedom has to be added to the model. This can be done by using the Transition Boundary Condition (TBC), which allows for a change in flux across the boundary. The TBC takes as input the material properties on one side of the domain, it does not matter which side. The thickness of the TBC should be approximately of the wavelength. If it is too small, it can introduce numerical difficulties. If it is too big, it will alter the results significantly. The TBC only needs to be used in this way if the effective refractive index is similar in magnitude, but opposite in sign. 2 MODELING A NEGATIVE REFRACTIVE INDEX

175 Results and Discussion The electric field is plotted in Figure 2. The fields within the PML are not plotted, since these have no physical meaning. The wave is observed to switch direction at the interface between vacuum and the metamaterial. Figure 2: The electric fields in vacuum and in the metamaterial. The waves are observed to switch direction. Model Library path: Wave_Optics_Module/Gratings_and_Metamaterials/ negative_refractive_index Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 3 MODELING A NEGATIVE REFRACTIVE INDEX

176 MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Define parameters that are useful when setting up the physics and the mesh. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Nam e Expression Value Description lda0 1[m] m Wavelength, vacuum f0 c_const/lda E8 1/s Frequency e_a Relative permittivity, vacuum mu_a Relative permeability, vacuum e_b Relative permittivity, metamaterial mu_b Relative permeability, metamaterial n_a sqrt(e_a*mu_a) Refractive index, vacuum n_b -sqrt(e_b*mu_b) Refractive index, metamaterial alph a 30[deg] rad Angle of incidence beta asin(n_a*sin(alpha)/ n_b) rad Refraction angle h_ma x 0.1*lda0/abs(n_b) m Maximum mesh element size Here, c_const is a predefined COMSOL constant for the speed of light in vacuum. 4 MODELING A NEGATIVE REFRACTIVE INDEX

177 DEFINITIONS Variables 1 1 In the Model Builder window, under Component 1 right-click Definitions and choose Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description ka ewfd.k0 rad/m Propagation constant, air kax ka*sin(alpha) rad/m kx for incident wave kay -ka*cos(alpha) rad/m ky for incident wave kb n_b*ewfd.k0 rad/m Propagation constant, slab kbx kb*sin(beta) rad/m kx for refracted wave kby -kb*cos(beta) rad/m ky for refracted wave The ewfd. prefix gives the correct physics-interface scope for the propagation constant. GEOMETRY 1 Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Height edit field, type 3. 4 Click to expand the Layers section. In the table, enter the following settings: Layer name Thickness (m) Layer Right-click Component 1>Geometry 1>Rectangle 1 and choose Rename. 6 Go to the Rename Rectangle dialog box and type Two port model in the New name edit field. 7 Click OK. The first rectangle consists of two rectangular domains representing the vacuum and meta material, respectively. Rectangle 2 1 Right-click Geometry 1 and choose Rectangle. 5 MODELING A NEGATIVE REFRACTIVE INDEX

178 2 In the Rectangle settings window, locate the Size section. 3 In the Height edit field, type 4. 4 Locate the Position section. In the x edit field, type In the y edit field, type Locate the Layers section. In the table, enter the following settings: Layer name Thickness (m) Layer 1 1 Layer Right-click Component 1>Geometry 1>Rectangle 2 and choose Rename. 8 Go to the Rename Rectangle dialog box and type PML model in the New name edit field. 9 Click OK. The second rectangle consists of three rectangular domains representing the vacuum, meta material, and PML. 10 In the Geometry toolbar, click Build All. 11 Click the Zoom Extents button on the Graphics toolbar. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN 1 In the Model Builder, under Component 1, click Electromagnetic Waves, Frequency Domain. 6 MODELING A NEGATIVE REFRACTIVE INDEX

179 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Components section. 3 From the Electric field components solved for list, choose Out-of-plane vector. Wave Equation, Electric 1 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain click Wave Equation, Electric 1. 2 In the Wave Equation, Electric settings window, locate the Electric Displacement Field section. 3 From the Electric displacement field model list, choose Relative permittivity. Next, set up the physics. Use two different ways to model the infinite metamaterial domain. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 5 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Wave excitation at this port list, choose On. 7 MODELING A NEGATIVE REFRACTIVE INDEX

180 5 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x 0 y exp(-i*kax*x) z 6 In the edit field, type abs(kay). Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 2 only. 3 In the Port settings window, locate the Port Mode Settings section. 4 Specify the E 0 vector as 0 x 0 y exp(-i*kbx*x) z 5 In the edit field, type -abs(kby). The first method uses two ports. Periodic Condition 1 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 8 MODELING A NEGATIVE REFRACTIVE INDEX

181 2 Select Boundaries 1, 3, 6, and 7 only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. 5 Specify the k F vector as kax x 0 y Port 3 1 On the Physics toolbar, click Boundaries and choose Port. 9 MODELING A NEGATIVE REFRACTIVE INDEX

182 2 Select Boundary 14 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Wave excitation at this port list, choose On. 5 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x 0 y exp(-i*kax*x) z 6 In the edit field, type abs(kay). Periodic Condition 2 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 10 MODELING A NEGATIVE REFRACTIVE INDEX

183 2 Select Boundaries 8, 10, 12, and only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. 5 Specify the k F vector as kax x 0 y DEFINITIONS Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 11 MODELING A NEGATIVE REFRACTIVE INDEX

184 2 Select Domain 3 only. 3 In the Perfectly Matched Layer settings window, locate the Scaling section. 4 In the PML scaling factor edit field, type -1. The second method uses one port and the PML. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Transition Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Transition Boundary Condition. 12 MODELING A NEGATIVE REFRACTIVE INDEX

185 2 Select Boundaries 4 and 13 only. 3 In the Transition Boundary Condition settings window, locate the Transition Boundary Condition section. 4 From the Electric displacement field model list, choose Relative permittivity. 5 From the r list, choose User defined. In the associated edit field, type mu_a. 6 From the r list, choose User defined. In the associated edit field, type e_a. 7 From the list, choose User defined. Leave the default value 0. 8 In the d edit field, type lda0/1000. MATERIALS Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 13 MODELING A NEGATIVE REFRACTIVE INDEX

186 2 Select Domains 2 and 5 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity epsilonr e_a 1 Basic Relative permeability mur mu_a 1 Basic Electrical conductivity sigma 0 S/m Basic Material 2 1 In the Model Builder window, right-click Materials and choose New Material. 14 MODELING A NEGATIVE REFRACTIVE INDEX

187 2 Select Domains 1, 3, and 4 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity epsilonr e_b 1 Basic Relative permeability mur mu_b 1 Basic Electrical conductivity sigma 0 S/m Basic MESH 1 For the periodic boundary conditions, the mesh needs to be identical on the periodic boundaries. Copy edges to generate the identical pairs. Edge 1 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose More Operations>Edge. 15 MODELING A NEGATIVE REFRACTIVE INDEX

188 2 Select Boundaries 1 and 3 only. Copy Edge 1 1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Copy Edge. 2 Select Boundaries 1 and 3 only. 3 In the Copy Edge settings window, locate the Destination Boundaries section. 4 Select the Destination Group Focus toggle button. 16 MODELING A NEGATIVE REFRACTIVE INDEX

189 5 Select Boundaries 6 and 7 only. Edge 2 1 Right-click Mesh 1 and choose More Operations>Edge. 2 Select Boundaries 8, 10, and 12 only. 17 MODELING A NEGATIVE REFRACTIVE INDEX

190 Copy Edge 2 1 Right-click Mesh 1 and choose More Operations>Copy Edge. 2 Select Boundaries 8, 10, and 12 only. 3 In the Copy Edge settings window, locate the Destination Boundaries section. 4 Select the Destination Group Focus toggle button. 5 Select Boundaries only. Size 1 Right-click Mesh 1 and choose Free Triangular. It is recommended to choose the maximum mesh size smaller than 0.2 wavelengths in each domain. Here, use 0.1 wavelengths of the metamaterial domain for the smooth result visualization using the parameter h_max that you defined earlier. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type h_max. 18 MODELING A NEGATIVE REFRACTIVE INDEX

191 5 In the Mesh toolbar, click the Build All button. STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 On the Home toolbar, click Compute. RESULTS Data Sets The PML is not of interest for the result visualization; use only the air and metamaterial domains for this purpose. 1 In the Model Builder window, expand the Results>Data Sets node. 2 Right-click Solution 1 and choose Add Selection. 3 In the Selection settings window, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Domain. 19 MODELING A NEGATIVE REFRACTIVE INDEX

192 5 Select Domains 1, 2, 4, and 5 only. Electric Field (ewfd) The default plot shows the norm of the electric field. Modify the plot to show the z-component of the electric field, then add a contour plot. 1 In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1. 2 In the Surface settings window, locate the Expression section. 3 In the Expression edit field, type Ez. 4 Locate the Coloring and Style section. From the Color table list, choose WaveLight. 5 In the Model Builder window, right-click Electric Field (ewfd) and choose Contour. 6 In the Contour settings window, locate the Expression section. 7 In the Expression edit field, type Ez. 8 Locate the Levels section. In the Total levels edit field, type Locate the Coloring and Style section. From the Color table list, choose GrayPrint. 10 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot with that shown in Figure MODELING A NEGATIVE REFRACTIVE INDEX

193 Photonic Crystal Solved with COMSOL Multiphysics 4.4 Photonic crystal devices are periodic structures of alternating layers of materials with different refractive indices. Waveguides that are confined inside of a photonic crystal can have very sharp low-loss bends, which may enable an increase in integration density of several orders of magnitude. Introduction This model describes the wave propagation in a photonic crystal that consists of GaAs pillars placed equidistant from each other. The distance between the pillars prevents light of certain wavelengths to propagate into the crystal structure. Depending on the distance between the pillars, waves within a specific frequency range are reflected instead of propagating through the crystal. This frequency range is called the photonic bandgap (Ref. 1). By removing some of the GaAs pillars in the crystal structure you can create a guide for the frequencies within the bandgap. Light can then propagate along the outlined guide geometry. Pillar (GaAs) Outgoing wave Incoming wave Air 1 PHOTONIC CRYSTAL

194 Model Definition The geometry is a square of air with an array of circular pillars of GaAs as described above. Some pillars are removed to make a waveguide with a 90 bend. The objective of the model is to study TE waves propagating through the crystal. To model these, use a scalar equation for the transverse electric field component E z, E z n 2 2 k 0Ez = 0 where n is the refractive index and k 0 is the free-space wave number. Because there are no physical boundaries, you can use the scattering boundary condition at all boundaries. Set the amplitude E z to 1 on the boundary of the incoming wave. Results and Discussion Figure 1 contains a plot of the z component of the electric field. It clearly shows the propagation of the wave through the guide. Figure 1: The z component of the electric field showing how the wave propagates along the path defined by the pillars. 2 PHOTONIC CRYSTAL

195 If the angular frequency of the incoming wave is less than the cutoff frequency of the waveguide, the wave does not propagate through the outlined guide geometry. In Figure 2 the wavelength has been increased by a factor of 1.2. Figure 2: A longer wavelength will not propagate through the guide. This picture shows the norm of the electric field. References 1. J.D. Joannopoulus, R.D. Meade, and J.N. Winn, Photonic Crystals (Modeling the Flow of Light), Princeton University Press, Chuang Shun Lien, Physics of Optoelectronic Devices, Wiley Series in Pure and Applied Optics, Wiley-Interscience, Model Library path: Wave_Optics_Module/Waveguides_and_Couplers/ photonic_crystal 3 PHOTONIC CRYSTAL

196 Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GEOMETRY 1 Import 1 1 On the Home toolbar, click Import. 2 In the Import settings window, locate the Import section. 3 Click the Browse button. 4 Browse to the model s Model Library folder and double-click the file photonic_crystal.mphbin. 5 Click the Import button. 6 Click the Zoom Extents button on the Graphics toolbar. MATERIALS The refractive index of GaAs depends on the frequency. The expression used in this model defines a linearized frequency dependency between the refractive index values corresponding to the free space wavelengths m and m, according to Ref. 2. Material 1 1 On the Home toolbar, click New Material. 2 Right-click Material 1 and choose Rename. 3 Go to the Rename Material dialog box and type GaAs in the New name edit field. 4 PHOTONIC CRYSTAL

197 4 Click OK. 5 Select Domains 1 and 3 86 only. This is most easily done by removing Domain 2 from the list once you have selected all domains. 6 In the Material settings window, click to expand the Material properties section. 7 In the Material properties tree, select Electromagnetic Models>Refractive Index>Refractive index (n). 8 Click Add to Material. GaAs 1 In the Model Builder, expand the Materials>GaAs node, then click Refractive index. 2 In the Property Group settings window, locate the Local Properties section. 3 In the Local properties table, enter the following settings: Property Expression Unit lambda0 c_const/freq m 4 Locate the Output Properties and Model Inputs section. Find the Output properties subsection. In the table, enter the following settings: Property Variable Expression Unit Size Refractive index n ; nii = n, nij = e5*lambda x3 Material 2 1 On the Home toolbar, click New Material. 2 Right-click Material 2 and choose Rename. 3 Go to the Rename Material dialog box and type Air in the New name edit field. 4 Click OK. 5 Select Domain 2 only. 6 In the Material settings window, click to expand the Material properties section. 7 Locate the Material Properties section. In the Material properties tree, select Electromagnetic Models>Refractive Index>Refractive index (n). 8 Click Add to Material. 9 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n 1 1 Refractive index 5 PHOTONIC CRYSTAL

198 ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 In the Scattering Boundary Condition settings window, locate the Boundary Selection section. 3 From the Selection list, choose All boundaries. Scattering Boundary Condition 2 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 Select Boundary 5 only. 3 In the Scattering Boundary Condition settings window, locate the Scattering Boundary Condition section. 4 From the Incident field list, choose Wave given by E field. 5 Specify the E 0 vector as 0 x 0 y 1 z MESH 1 The default mesh settings aim for a good resolution of all curved boundaries. In this model, they would lead to a high number of elements. By relaxing the growth rate and the curvature resolution requirement, you can get a decently accurate solution with fewer elements. Size 1 On the Mesh toolbar, click Free Triangular. 2 In the Model Builder, under Mesh 1, click on Size. 3 In the Size settings window, locate the Element Size section. 4 Click the Custom button. 5 Locate the Element Size Parameters section. In the Maximum element growth rate edit field, type In the Curvature factor edit field, type Click the Build All button. 6 PHOTONIC CRYSTAL

199 STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type 3e8/1e-6 3e8/1.2e-6. This will get you one solution for a free space wavelength of 1 m, and one for a free space wavelength of 1.2 m. 4 On the Study toolbar, click Compute. RESULTS Electric Field (ewfd) The default plot shows the distribution of the electric field norm for the lowest of the frequencies. Because this is below the cutoff frequency of the waveguide, the wave does not propagate through the outlined guide geometry. 1 In the Model Builder window, click Electric Field (ewfd). 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Parameter value (freq) list, choose 3e14. 4 On the Electric Field (ewfd) toolbar, click Plot. 300 THz, or a free space wavelength of 1 m, is within the bandgap. The wave propagates all the way through the geometry, losing only a little of its energy. Try visualizing the instantaneous value of the field. 5 In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1. 6 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>electric field, z component (ewfd.ez). 7 Locate the Coloring and Style section. From the Color table list, choose WaveLight. The WaveLight color table looks better if the range is symmetric around zero. 8 Select the Symmetrize color range check box. 9 On the Electric Field (ewfd) toolbar, click Plot. Finally, create a line plot comparing how the electric field magnitude falls off as the waves of the two frequencies under study enter the waveguide. 7 PHOTONIC CRYSTAL

200 Data Sets 1 On the Results toolbar, click Cut Line 2D. 2 In the Cut Line 2D settings window, locate the Line Data section. 3 In row Point 1, set y to 0.75e-6. 4 In row Point 2, set x to 2.5e-6. 5 In row Point 2, set y to 0.75e-6. 6 Click the Plot button. 1D Plot Group 2 1 On the Results toolbar, click 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Cut Line 2D 1. 4 On the 1D Plot Group 2 toolbar, click Line Graph, then click Plot. 8 PHOTONIC CRYSTAL

201 Plasmonic Wire Grating Introduction A plane electromagnetic wave is incident on a wire grating on a dielectric substrate. The model computes transmission and reflection coefficients for the refraction, specular reflection, and first order diffraction. Model Definition Figure 1 shows the considered grating, with a gold wire on a dielectric material with refractive index n. The grating constant, or the distance between the wires, is d. A plane-polarized wave traveling through a medium with refractive index n is incident on the grating, at an angle in a plane perpendicular to the grating. n Au d n Figure 1: The modeled grating. The model considers a unit cell of a slice through this geometry. The grating is assumed to consist of an infinite number of infinitely long wires. If the wavelengths involved in the model are sufficiently short compared to the grating constant, one or several diffraction orders can be present. The diagram in Figure 2 1 PLASMONIC WIRE GRATING

202 shows two transmissive paths taken by light incident on adjacent cells of the grating, exactly one grating constant apart. d sin( n n m d sin( m Figure 2: The geometric path lengths of two transmitted parallel beams. The optical path length is the geometric path length multiplied by the local refractive index. The criterion for positive interference is that the difference in optical path length along the two paths equals an integer number of vacuum wavelengths, or: m 0 = dn sin m n sin (1) with m = 0, ±1,± 2,..., 0 the vacuum wavelength, and m the transmitted diffracted beam of order m. For m = 0, this reduces to refraction, as described by Snell's law: sin 0 = n sin n Because the sine functions can only vary between -1 and 1, the existence of higher diffraction order requires that n + n m n d + n The model instructions cover only first order diffraction, and are hence only valid for under the condition 2 0 dn sin + n (2) Note that for the special cases of perpendicular and grazing incidence, the right-hand side of the inequality evaluates to dn and d n n respectively. 2 PLASMONIC WIRE GRATING

203 Figure 3 shows the corresponding paths of the reflected light. d sin( m m d sin( n n Figure 3: The geometric path lengths of two parallel reflected beams. For positive interference we get m 0 = dn sin m sin, (3) where m is the reflected beam of diffraction order m. Setting m = 0 in this equation renders sin 0 = sin, or specular reflection. The condition for no reflected diffracted beams of order 2 or greater being present is 2 0 dn a 1 + sin. (4) The model uses n 1 for air and n 1.2 for the dielectric substrate. Allowing for arbitrary angles of incidence and with a grating constant d = 400 nm, Equation 2 sets the validity limit to vacuum wavelengths greater than 440 nm. The model uses 0 = 441 nm. For the wire, a complex-valued permittivity of i approximates that of gold at the corresponding frequency. The performance of the grating depends on the polarization of the incident wave. Therefore both a transverse electric (TE) and a transverse magnetic (TM) case are considered. The TE wave has the electric field component in the z direction, out of the modeling xy-plane. For the TM wave, the electric field vector is pointing in the xy-plane and perpendicular to the direction of propagation, whereas the magnetic field 3 PLASMONIC WIRE GRATING

204 has only a component in the z direction. The angle of incidence is for both cases swept from 0 to /2, with a pitch of /40. Results and Discussion As an example of the output from the model, Figure 4 and Figure 5 show the electric field norm for an angle of incidence equal to /5, for the TE and TM case respectively. Figure 4: Electric field norm for TE incidence at /5. 4 PLASMONIC WIRE GRATING

205 Figure 5: Electric field norm for TM incidence at /5. All the computed transmission and reflection coefficients for TE incidence are plotted in Figure 6. R 0, the coefficient for specular reflection, increases rather steadily with the angle of incidence. This is both because of reflection in the material interface and because the wave sees the wire as increasingly wider at greater angles the same effect as achieved by a Venetian blind. T 0, the refracted but not diffracted transmission, decreases accordingly. For the considered wavelength to period length ratio, the transmitted diffracted beam T 1 is propagating only for nearly perpendicular incidence. The reflected diffraction order R 1 would need a shorter wavelength or a larger grating period to show up. Instead, the most prominent diffraction orders are R 1 and T 1. Note first that the sum of all coefficients is consistently less than 1. This is because of the dielectric losses in the wire. This is even more apparent for TM incidence, as Figure 7 shows. Here, approximately half of the wave is absorbed in the wire. Another important feature of the TM case is that there is very little specular reflection (R 0 ) around 60 degrees. 5 PLASMONIC WIRE GRATING

206 Figure 6: Transmission and reflection coefficients for TE incidence. Figure 7: Transmission and reflection coefficients for TM incidence. 6 PLASMONIC WIRE GRATING

207 Notes About the COMSOL Implementation The model is set up for one unit cell of the grating, flanked by Floquet boundary conditions describing the periodicity. As applied, this condition states that the solution on one side of the unit equals the solution on the other side multiplied by a complex-valued phase factor. The phase shift between the boundaries is evaluated from the perpendicular component of the wave vector. Because the periodicity boundaries are parallel with the y-axis, only the x-component is required. Note that due to the continuity of the field, the phase factor will be the same for the refracted and reflected beams as for the incident wave. Port conditions are used both for specifying the incident wave and for letting the resulting solution leave the model without any non-physical reflections. In order to achieve perfect transmission through the port boundaries, one port for each mode (m 0, m 1, m 1) in each direction must be present. This gives a total of 6 ports. The input to each periodic port is an electric or magnetic field amplitude vector and an angle of incidence. The angle of incidence is defined as k n = ksin z, where k is the propagation vector of the incident wave, n is the normalized normal vector, k is the wave number, is the angle of incidence, and z is the unit vector in the z direction. Note that this definition means that the angle of incidence on the opposite sides have opposite signs. To automatically create ports for the diffraction orders, you also provide the refractive index at the port boundary and the maximum frequency (which in this model is the single frequency that is used). The below table lists the parameters names used in the model. Internal means that the variable is not provided as an input parameter TABLE 4-1: PARAMETER NAMES MODEL DESCRIPTION MODEL DESCRIPTION n na Refractive index, air n nb Refractive index, dielectric alpha Angle of incidence 1 Internal Reflected diffraction angle, order 1-1 Internal Reflected diffraction angle, order -1 0 beta Refraction angle 7 PLASMONIC WIRE GRATING

208 TABLE 4-1: PARAMETER NAMES MODEL DESCRIPTION MODEL DESCRIPTION 1 Internal Refracted diffraction angle, order 1-1 Internal Refracted diffraction angle, order -1 Model Library path: Wave_Optics_Module/Gratings_and_Metamaterials/ plasmonic_wire_grating Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description na Refractive index, air nb Refractive index, dielectric d 400[nm] 4.000E-7 m Grating constant 8 PLASMONIC WIRE GRATING

209 Name Expression Value Description lam0 441[nm] 4.410E-7 m Vacuum wavelength f0 c_const/lam E14 1/s Frequency alpha 0 0 Angle of incidence beta asin(na*sin(alpha)/nb) 0 rad Refraction angle Although the angle of incidence will not remain constant at 0, it needs to be specified as a parameter to be accessible to the parametric solver. GEOMETRY 1 Create the geometry entirely in terms of the grating constant, for easy scalability. Rectangle 1 1 In the Model Builder window, under Component 1 right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type d. 4 In the Height edit field, type 3*d. 5 Click the Build Selected button. Rectangle 2 1 In the Model Builder window, right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type d. 4 In the Height edit field, type 3*d. 5 Locate the Position section. In the y edit field, type -3*d. 6 Click the Build Selected button. Circle 1 1 Right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type d/5. 4 Locate the Position section. In the x edit field, type d/2. 5 Click the Build Selected button. 9 PLASMONIC WIRE GRATING

210 6 Click the Zoom Extents button on the Graphics toolbar. The geometry now consists of two rectangular domains for the air and the dielectric, and a circle centered on their intersection. You can remove the line through the circle if you first create a union of the objects. Union 1 1 On the Geometry toolbar, click Union. 2 From the Edit menu, choose Select All. 3 Click the Build Selected button. Delete Entities 1 1 Right-click Geometry 1 and choose Delete Entities. 2 On the object uni1, select Boundary 6 only. This is the horizontal diameter of the circle in the center of the geometry. 3 Click the Build Selected button. Form Union 1 In the Model Builder window, under Component 1>Geometry 1 right-click Form Union and choose Build Selected. 10 PLASMONIC WIRE GRATING

211 ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Before setting up the materials, define which constitutive relations you want to use in the Electromagnetic Waves interface. Wave Equation, Electric 2 1 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 2 Select Domain 3 only. 3 In the Wave Equation, Electric settings window, locate the Electric Displacement Field section. 4 From the Electric displacement field model list, choose Dielectric loss. MATERIALS Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n na 1 Refractive index 4 Right-click Component 1>Materials>Material 1 and choose Rename. 5 Go to the Rename Material dialog box and type Air in the New name edit field. 6 Click OK. Material 2 1 Right-click Materials and choose New Material. 2 Select Domain 1 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n nb 1 Refractive index 5 Right-click Component 1>Materials>Material 2 and choose Rename. 6 Go to the Rename Material dialog box and type Dielectric in the New name edit field. 11 PLASMONIC WIRE GRATING

212 7 Click OK. Material 3 1 Right-click Materials and choose New Material. 2 Select Domain 3 only. 3 In the Material settings window, locate the Material Contents section. 4 In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity (imaginary part) epsilonbis Dielectric losses Relative permittivity (real part) epsilonprim Dielectric losses Relative permeability mur 1 1 Basic Electrical conductivity sigma 0 S/m Basic 5 Right-click Component 1>Materials>Material 3 and choose Rename. 6 Go to the Rename Material dialog box and type Gold in the New name edit field. 7 Click OK. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN In the first version of this model, you will assume a TE-polarized wave. This means that E x and E y will be zero throughout the geometry, and that you consequently only need to solve for E z. 1 In the Electromagnetic Waves, Frequency Domain settings window, locate the Components section. 2 From the Electric field components solved for list, choose Out-of-plane vector. Now define the excitation port. A periodic port assumes that the structure is periodic and simplifies the setup of ports for the diffraction orders. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 5 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Periodic. 5 From the Wave excitation at this port list, choose On. Notice that you define the electric field by only setting the amplitude. A phase factor should not be entered. 12 PLASMONIC WIRE GRATING

213 6 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x 0 y 1 z 7 In the edit field, type alpha. 8 In the n edit field, type na. 9 In the f max edit field, type f0. The order in which you set up the ports will determine how the S-parameters are labeled. You have just created Port 1 for the excitation. If you set up the next port for the transmission of the purely refracted beam, the S21-parameter will contain information on the zero order transmission. Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 2 only. 3 In the Port settings window, locate the Port Properties section. 4 From the Type of port list, choose Periodic. 5 Locate the Port Mode Settings section. Specify the E 0 vector as 0 x 0 y 1 z The angle of incidence on the exit side corresponds to the angle of incidence an incident wave on that side would have to provide the correct propagation angle in the material. Notice that this also means that the sign is opposite that on the entry side. 6 In the edit field, type -beta. 7 In the n edit field, type nb. 8 In the f max edit field, type f0. Port 1 Continue with the ports for the reflected diffraction orders. Since these diffraction orders are not propagating at normal incidence, you have to add the ports manually. 13 PLASMONIC WIRE GRATING

214 Diffraction Order 1 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain right-click Port 1 and choose Diffraction Order. 2 In the Diffraction Order settings window, locate the Port Mode Settings section. 3 From the Components list, choose Out-of-plane vector. 4 In the m edit field, type -1. Diffraction Order 2 1 Right-click Component 1>Electromagnetic Waves, Frequency Domain>Port 1>Diffraction Order 1 and choose Duplicate. 2 In the Diffraction Order settings window, locate the Port Properties section. 3 In the Port name edit field, type 4. 4 Locate the Port Mode Settings section. In the m edit field, type 1. The transmitted diffraction orders are propagating at normal incidence. Thus, you can create them automatically by clicking the Compute Diffraction Orders button. Port 2 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain click Port 2. 2 In the Port settings window, locate the Port Mode Settings section. 3 Click the Compute Diffraction Orders button. You now find the Diffraction Order ports as subfeatures to Port 2. Periodic Condition 1 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 2 Select Boundaries 1, 3, 7, and 8 only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. The wave vector in the direction for the periodicity is used by the periodic port. Thus, you can use that wave vector also for the Floquet periodic condition. 5 From the k-vector for Floquet periodicity list, choose From periodic port. MESH 1 The periodic boundary conditions perform better if the mesh is identical on the periodicity boundaries. This is especially important when dealing with vector degrees of freedom, as will be the case in the TM version of this model. 14 PLASMONIC WIRE GRATING

215 1 In the Model Builder window, under Component 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Sequence type list, choose User-controlled mesh. Free Triangular 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Triangular 1 and choose Delete. Click Yes to confirm. Size 1 In the Model Builder window, under Component 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 From the Predefined list, choose Extra fine. Edge 1 1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Edge. 2 Select Boundaries 1 and 3 only. Copy Edge 1 1 Right-click Mesh 1 and choose More Operations>Copy Edge. 2 Select Boundary 3 only. 3 In the Copy Edge settings window, locate the Destination Boundaries section. 4 Select the Destination Group Focus toggle button. 5 Select Boundary 8 only. Copy Edge 2 1 Right-click Mesh 1 and choose More Operations>Copy Edge. 2 Select Boundary 1 only. 3 In the Copy Edge settings window, locate the Destination Boundaries section. 4 Select the Destination Group Focus toggle button. 5 Select Boundary 7 only. Free Triangular 1 1 Right-click Mesh 1 and choose Free Triangular. 2 Right-click Mesh 1 and choose Build All. To set up the study to sweep for the angle of incidence, some modifications of the solver is required. 15 PLASMONIC WIRE GRATING

216 STUDY 1 Step 1: Frequency Domain 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names alpha Parameter value list 5 Click Range. 6 Go to the Range dialog box. 7 In the Start edit field, type 0. 8 In the Stop edit field, type pi/2-pi/40. 9 In the Step edit field, type pi/ Click the Replace button. 11 On the Home toolbar, click Compute. RESULTS Electric Field (ewfd) The default plot shows the electric field norm for the last solution, almost tangential incidence. Look at a more interesting angle of incidence. 1 In the 2D Plot Group settings window, locate the Data section. 2 From the Parameter value (alpha) list, choose On the Electric Field (ewfd) toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 4. Rename the plot group to make it clear that it shows the TE solution. 16 PLASMONIC WIRE GRATING

217 5 Right-click Results>Electric Field (ewfd) and choose Rename. 6 Go to the Rename 2D Plot Group dialog box and type 2D Plot Group TE in the New name edit field. 7 Click OK. Add a 1D plot to look at the various orders of reflection and transmission versus the angle of incidence. 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the 1D Plot Group settings window, click to expand the Title section. 3 From the Title type list, choose Manual. 4 In the Title text area, type Reflection and Transmission of TE Wave. 5 Locate the Plot Settings section. Select the x-axis label check box. 6 In the associated edit field, type Angle of incidence (degrees). 7 Select the y-axis label check box. 8 In the associated edit field, type Reflection and transmission coefficients. 9 On the 1D Plot Group 2 toolbar, click Global. Since the diffraction orders are not propagating for all angles, use a logic expression to set the S-parameter to zero when the wave is evanescent and to multiply with one when it is propagating. 10 In the Global settings window, locate the y-axis Data section. 11 In the table, enter the following settings: Expression Unit Description abs(ewfd.s11)^2 1 abs(ewfd.s21)^2 1 abs(ewfd.s31)^2*(imag(ewfd.beta_3)==0) 1 abs(ewfd.s41)^2*(imag(ewfd.beta_4)==0) 1 abs(ewfd.s51)^2*(imag(ewfd.beta_5)==0) 1 abs(ewfd.s61)^2*(imag(ewfd.beta_6)==0) 1 12 Locate the x-axis Data section. From the Parameter list, choose Expression. 13 In the Expression edit field, type alpha*180/pi. 14 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Cycle. 17 PLASMONIC WIRE GRATING

218 15 Click to expand the Legends section. From the Legends list, choose Manual. 16 In the table, enter the following settings: Legends R₀ T₀ R_-1 R₁ T_-1 T₁ 17 On the 1D Plot Group 2 toolbar, click Plot. 18 In the Model Builder window, right-click 1D Plot Group 2 and choose Rename. 19 Go to the Rename 1D Plot Group dialog box and type 1D Plot Group TE in the New name edit field. 20 Click OK. The plot should now look like Figure 6. The remaining instructions show to alter the physics so that you solve for an incident TM wave. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN 1 In the Electromagnetic Waves, Frequency Domain settings window, locate the Components section. 2 From the Electric field components solved for list, choose In-plane vector. You will now solve for E x and E y instead of E z ; for a TM wave, E z is zero. Port 1 The easiest way to specify a TM wave is to define the magnetic field, since only the z component is used. 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain click Port 1. 2 In the Port settings window, locate the Port Mode Settings section. 3 From the Input quantity list, choose Magnetic field. 4 Specify the H 0 vector as 0 x 18 PLASMONIC WIRE GRATING

219 0 y 1 z Diffraction Order 1 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain>Port 1 click Diffraction Order 1. 2 In the Diffraction Order settings window, locate the Port Mode Settings section. 3 From the Components list, choose In-plane vector. Diffraction Order 2 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain>Port 1 click Diffraction Order 2. 2 In the Diffraction Order settings window, locate the Port Mode Settings section. 3 From the Components list, choose In-plane vector. Port 2 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain click Port 2. 2 In the Port settings window, locate the Port Mode Settings section. 3 From the Input quantity list, choose Magnetic field. 4 Specify the H 0 vector as 0 x 0 y 1 z 5 Click the Compute Diffraction Orders button to change components for the diffraction orders that are propagating at normal incidence. ROOT Add a new study in order not to overwrite the TE solution. 1 On the Home toolbar, click Add Study. ADD STUDY 1 Go to the Add Study window. 2 Find the Studies subsection. In the tree, select Preset Studies>Frequency Domain. 3 In the Add study window, click Add Study. 19 PLASMONIC WIRE GRATING

220 4 Close the Add Study window. STUDY 2 Step 1: Frequency Domain 1 In the Model Builder window, expand the Study 2 node, then click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names alpha Parameter value list 5 Click Range. 6 Go to the Range dialog box. 7 In the Start edit field, type 0. 8 In the Stop edit field, type pi/2-pi/40. 9 In the Step edit field, type pi/ Click the Replace button. 11 On the Study toolbar, click Compute. RESULTS Electric Field (ewfd) 1 In the 2D Plot Group settings window, locate the Data section. 2 From the Parameter value (alpha) list, choose On the Electric Field (ewfd) toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. 5 Right-click Results>Electric Field (ewfd) and choose Rename. 6 Go to the Rename 2D Plot Group dialog box and type 2D Plot Group TM in the New name edit field. 20 PLASMONIC WIRE GRATING

221 7 Click OK. You have now reproduced Figure 5. For the transmission and the reflection of the TM waves, copy and reuse the 1D plot for the TE waves. 1D Plot Group TE 1 1 In the Model Builder window, under Results right-click 1D Plot Group TE and choose Duplicate. 2 In the 1D Plot Group settings window, locate the Title section. 3 In the Title text area, type Reflection and Transmission of TM Wave. 4 Locate the Data section. From the Data set list, choose Solution 2. 5 On the 1D Plot Group TE 1 toolbar, click Plot. 6 Right-click Results>1D Plot Group TE 1 and choose Rename. 7 Go to the Rename 1D Plot Group dialog box and type 1D Plot Group TM in the New name edit field. 8 Click OK. Compare the resulting plot with that in Figure PLASMONIC WIRE GRATING

222 22 PLASMONIC WIRE GRATING Solved with COMSOL Multiphysics 4.4

223 Scatterer on Substrate Introduction A plane TE-polarized electromagnetic wave is incident on a gold nanoparticle on a dielectric substrate. The absorption and scattering cross sections of the particle are computed for a few different polar and azimuthal angles of incidence. Model Definition Figure 1 shows the geometry, with the substrate considered to occupy the entire z<0 half-space. A plane 600 THz (500 nm wavelength) electromagnetic wave is incident at a polar angle and an azimuthal angle. The wave is plane-polarized with the electric field vector tangential to the surface of the substrate. n a Au n b Figure 1: The modeled geometry. The gray boundary represents the surface of the dielectric. The electric field vector of the incident wave points in the direction, orthogonal to the plane of incidence. The model uses n a 1 for air and n b 1.5 for the dielectric substrate. The scattering nanoparticle has a complex-valued permittivity of i, approximating that of gold at 600 THz (500 nm wavelength). 1 SCATTERER ON SUBSTRATE

224 The model computes the scattering, absorption, and extinction cross-sections of the particle on the substrate. The scattering cross-section is defined as 1 sc = ---- n S sc ds I 0 Here, n is the normal vector pointing outwards from the nanodot, S sc is the scattered intensity (Poynting) vector, and I 0 is the incident intensity. The integral is taken over the closed surface of the scatterer. The absorption cross section equals 1 abs = ---- QdV, where Q is the power loss density in the particle and the integral is taken over its volume. The extinction cross section is simply the sum of the two others: I 0 ext = sc + abs.. Results and Discussion As explained in Notes About the COMSOL Implementation, the model first computes a background field from the plane wave incident on the substrate, and then uses that to arrive at the total field with the nanoparticle present. Figure 2 and Figure 3 show the y-component and the norm of the electric background field, not yet affected by the nanoparticle, for the = 4, = 6 solution. In the air, this field is a superposition of the incident and reflected plane waves. In the substrate, only a transmitted plane wave exists. 2 SCATTERER ON SUBSTRATE

225 Figure 2: Background electric field, y-component for 4, 6, on three slices parallel with the yz-plane. Figure 3: Background electric field norm, for = 4, 6. 3 SCATTERER ON SUBSTRATE

226 Figure 4 and Figure 5 show the norm of the total electric field for the same angles of incidence, after it has been influenced both by the material interface and by the nanoparticle. Figure 4: Slice plot of the y-component of the total electric field for 4, 6. Figure 5: Slice plot of the total electric field norm for 4, 6. 4 SCATTERER ON SUBSTRATE

227 In Figure 6, the power loss density is shown in a horizontal slice through the nanoparticle. No apparent resonance is present and most of the losses take place near the surface of the particle. Figure 6: Power loss density in a slice through the nanoparticle. Table 1 shows the computed cross sections for the set of angles of incidence. TABLE 1: CROSS SECTIONS. abs (m 2 ) ext (m 2 ) / /6 / /4 / For this small sample of the angular space, both cross sections indicate a strong dependence on the polar angle but little variation with the azimuthal angle. For a comparison, the nanoparticle covers a geometric area of m 2 of the substrate. 5 SCATTERER ON SUBSTRATE

228 Notes About the COMSOL Implementation The Electromagnetic Waves, Frequency Domain interface features an option to solve for the scattered field, a perturbation to the total field caused by a local scatterer. The incident wave is then entered as a background electric field. This field should be a solution to the wave equation without the presence of the scatterer. If the scatterer is suspended in free space or any other homogeneous medium, the background field is simply what you are sending in, for example a Gaussian or a plane wave. With the scatterer placed on a substrate, the analytical expression for the background field becomes more complicated. It needs to be the correct superposition of an incident and a reflected wave in the free space domain, and a transmitted wave in the substrate. A simple and general way to avoid deriving and entering the analytical background field is to use a full field solution of the problem without the scatterer. To achieve this full field solution, the simulation is set up with two Port conditions. One defines the incident plane wave and allows for specular reflection. The other absorbs the transmitted plane wave. The side boundaries have Floquet conditions, stating that the solution on one side of the geometry equals the solution on the other side multiplied by a complex-valued phase factor. This effectively turns the model into a section of a geometry that extends indefinitely in the xy-plane. The local wave vector and the direction of the incident electric field vector are input parameters for the ports and the Floquet conditions. Using the coordinate system in Figure 1, the incident wave vector is k a = k x k y k az = k a cos a sin a sin a sin a cos a, where k a is the wavenumber in the first medium, here vacuum, a and a the azimuthal and polar angles of incidence. The expression for the tangentially polarized electric field vector at the plane of incidence becomes E 0 = E 0 exp ik x x+ k y y sin a cos a 0. The Port condition lets you define a total input power from which the electric field amplitude E 0 is derived. The model uses the value A P = I , cos where I 0 = 1 MW/m 2 is the intensity of the incident field and A the area of the boundary where the port is set up. 6 SCATTERER ON SUBSTRATE

229 In the substrate, the wave vector is k b = k x k y k bz = k b cos b sin b sin b sin b cos b, with k b = n b k a, n a b = a, sin b = n a sin a. n b Notice that the x and y components for the wave vector are the same for the wave in the substrate and the incident wave, due to field continuity. The electric field vector at the output port is proportional to exp ik x x + k y y sin b cos b 0. A second Electromagnetic Waves, Frequency Domain interface introduces the gold nanoparticle as the scatterer and surrounds the geometry with PMLs. With the full field solution from the first interface as the background field, only the scattered field needs to be absorbed in the PMLs. Model Library path: Wave_Optics_Module/Optical_Scattering/ scatterer_on_substrate Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 7 SCATTERER ON SUBSTRATE

230 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button twice. You should now see two Electromagnetic Waves, Frequency Domain entries in the Selected physics field. 4 Click the Study button. 5 In the tree, select Custom Studies>Empty Study. You will add steps to the study before solving the model. 6 Click the Done button. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. Define the model parameters. The Description field is optional. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description w 750[nm] 7.500E-7 m Width of physical geometry t_pml 150[nm] 1.500E-7 m PML thickness h_air 400[nm] 4.000E-7 m Air domain height h_subs 250[nm] 2.500E-7 m Substrate domain height na Refractive index, air nb Refractive index, substrate epsr_gold *i i Relative permittivity, gold at 600 THz f0 600[THz] 6.000E14 Hz Frequency phi 0 0 Azimuthal angle of incidence in both media theta 0 0 Polar angle of incidence in air thetab asin(na/ nb*sin(theta)) 0 rad Polar angle in substrate 8 SCATTERER ON SUBSTRATE

231 Name Expression Value Description I0 1[MW/m^2] 1.000E6 W/m² Intensity of incident field P I0*w^2*cos(theta) 5.625E-7 W Port power The first four parameters will be used in defining the geometry. The azimuthal angle in the substrate remains the same as the angle of incidence. As the polar angle of incidence gets other values in the study, the polar angle in the substrate will automatically be recomputed. DEFINITIONS Define expressions for the wave vector in both media. Variables 1 1 In the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description ka ewfd.k0*na rad/m Wave number in air kx ka*cos(phi)*sin(theta) rad/m ky ka*sin(phi)*sin(theta) rad/m kaz -ka*cos(theta) rad/m kb ewfd.k0*nb rad/m Wave number in substrate kbz -kb*cos(thetab) rad/m GEOMETRY 1 Import the nanoparticle. Import 1 1 On the Geometry toolbar, click Import. 2 In the Import settings window, locate the Import section. 3 Click the Browse button. 4 Browse to the model s Model Library folder and double-click the file scatterer_on_substrate.mphbin. 5 Click the Import button. Draw the air and the substrate using your model parameters. 9 SCATTERER ON SUBSTRATE

232 Block 1 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type w+2*t_pml. 4 In the Depth edit field, type w+2*t_pml. 5 In the Height edit field, type h_air+t_pml. 6 Locate the Position section. From the Base list, choose Center. 7 In the z edit field, type (h_air+t_pml)/2. 8 Click to expand the Layers section. Find the Layer position subsection. In the table, enter the following settings: Layer name Layer 1 Thickness (m) t_pml 9 Select the Left, Right, Front, Back, and Top check boxes. 10 Clear the Bottom check box. Block 2 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type w+2*t_pml. 4 In the Depth edit field, type w+2*t_pml. 5 In the Height edit field, type h_subs+t_pml. 6 Locate the Position section. From the Base list, choose Center. 7 In the z edit field, type -(h_subs+t_pml)/2. 8 Locate the Layers section. Find the Layer position subsection. In the table, enter the following settings: Layer name Layer 1 Thickness (m) t_pml 9 Select the Left, Right, Front, Back, and Bottom check boxes. Leave the Top check box cleared. 10 Click the Build All Objects button. 11 Click the Zoom Extents button on the Graphics toolbar. 10 SCATTERER ON SUBSTRATE

233 12 Click the Wireframe Rendering button on the Graphics toolbar. DEFINITIONS Define selections to separate between the part of your model where you will compute physical results and the part that will constitute the PML. For convenience, add separate selections for the nanoparticle. Explicit 1 1 On the Definitions toolbar, click Explicit. 2 Select Domains 18, 19, and 25 only (the non-pml domains). 3 Right-click Component 1>Definitions>Explicit 1 and choose Rename. 4 Go to the Rename Explicit dialog box and type Physical Domains in the New name edit field. 5 Click OK. Complement 1 1 On the Definitions toolbar, click Complement. 2 In the Complement settings window, locate the Input Entities section. 3 Under Selections to invert, click Add. 11 SCATTERER ON SUBSTRATE

234 4 Go to the Add dialog box. 5 In the Selections to invert list, select Physical Domains. 6 Click the OK button. 7 Right-click Component 1>Definitions>Complement 1 and choose Rename. 8 Go to the Rename Complement dialog box and type PML Domains in the New name edit field. 9 Click OK. Explicit 2 1 On the Definitions toolbar, click Explicit. 2 Select Domain 25 only (the nanoparticle). 3 Right-click Component 1>Definitions>Explicit 2 and choose Rename. 4 Go to the Rename Explicit dialog box and type Nanoparticle in the New name edit field. 5 Click OK. Explicit 3 1 On the Definitions toolbar, click Explicit. 2 Select Domain 25 only (the nanoparticle). 3 In the Explicit settings window, locate the Output Entities section. 4 From the Output entities list, choose Adjacent boundaries. 5 Right-click Component 1>Definitions>Explicit 3 and choose Rename. 6 Go to the Rename Explicit dialog box and type Nanoparticle Surface in the New name edit field. 7 Click OK. Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 In the Perfectly Matched Layer settings window, locate the Domain Selection section. 3 From the Selection list, choose PML Domains. 4 Locate the Scaling section. From the Physics list, choose Electromagnetic Waves, Frequency Domain 2. 5 In the PML scaling factor edit field, type 0.5. Reducing the scaling factor makes the coordinate stretching in the PML less aggressive, which reduces interior reflections and improves the iterative solver 12 SCATTERER ON SUBSTRATE

235 convergence. The downside is a slightly increased systematic error from reflections on the boundaries outside the PML. Only the second interface will be active in the PML domains. As this interface will use the electric field components from the first interface, define them to be 0 in the PML domains. Variables 2 1 In the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose PML Domains. 5 Locate the Variables section. In the table, enter the following settings: Name Expression Unit Description ewfd.ex 0 ewfd.ey 0 ewfd.ez 0 MATERIALS Define materials for the air, the substrate, and the nanoparticle. Material 1 1 In the Home toolbar, click New Material. 2 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n na 1 Refractive index 3 Right-click Component 1>Materials>Material 1 and choose Rename. 4 Go to the Rename Material dialog box and type Air in the New name edit field. 5 Click OK. You will later set up the physics interfaces to use the refractive index instead of the listed missing properties (the relative permittivity, the relative permeability, and the electrical conductivity). Material 2 1 Right-click Materials and choose New Material. 2 Select Domains 1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 26, 27, 30, 31, 34, and 35 only (all domains in the substrate, including the PML). 13 SCATTERER ON SUBSTRATE

236 3 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n nb 1 Refractive index 4 Right-click Component 1>Materials>Material 2 and choose Rename. 5 Go to the Rename Material dialog box and type Substrate in the New name edit field. 6 Click OK. You are now ready to specify the physics. Start by setting up the first interface so that it computes the full wave solution to the plane wave falling in on the semi-infinite substrate. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN 1 In the Electromagnetic Waves, Frequency Domain settings window, locate the Domain Selection section. 2 From the Selection list, choose Physical Domains. Wave Equation, Electric 2 1 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 2 In the Wave Equation, Electric settings window, locate the Domain Selection section. 3 From the Selection list, choose Nanoparticle. 4 Locate the Electric Displacement Field section. From the n list, choose User defined. In the associated edit field, type na. 5 From the k list, choose User defined. This redefines the nanoparticle as air. Port 1 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 68 only (the top boundary of the physical domain). 3 In the Port settings window, locate the Port Properties section. 4 From the Wave excitation at this port list, choose On. 5 In the P in edit field, type P. 14 SCATTERER ON SUBSTRATE

237 6 Locate the Port Mode Settings section. Specify the E 0 vector as -sin(phi)*exp(-i*(kx*x+ky*y)) cos(phi)*exp(-i*(kx*x+ky*y)) x y 0 z 7 In the edit field, type abs(kaz). Port 2 1 On the Physics toolbar, click Boundaries and choose Port. 2 Select Boundary 62 only (the bottom boundary of the physical domain). 3 In the Port settings window, locate the Port Mode Settings section. 4 Specify the E 0 vector as -sin(phi)*exp(-i*(kx*x+ky*y)) cos(phi)*exp(-i*(kx*x+ky*y)) x y 0 z 5 In the edit field, type abs(kbz). Periodic Condition 1 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 2 Select Boundaries 60, 63, 113, and 116 only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. 5 Specify the k F vector as kx ky x y 0 z The z-component of the wave vector does not affect the periodicity and can be left out. Periodic Condition 2 1 On the Physics toolbar, click Boundaries and choose Periodic Condition. 2 Select Boundaries 61, 64, 74, and 77 only. 3 In the Periodic Condition settings window, locate the Periodicity Settings section. 15 SCATTERER ON SUBSTRATE

238 4 From the Type of periodicity list, choose Floquet periodicity. 5 Specify the k F vector as kx x ky y 0 z Set up the second interface to compute how the plane wave solution from the first interface is affected by the nanoparticle. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN 2 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain 2. 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Settings section. 3 From the Solve for list, choose Scattered field. 4 Specify the E b vector as ewfd.ex ewfd.ey ewfd.ez x y z Wave Equation, Electric 2 1 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 2 In the Wave Equation, Electric settings window, locate the Domain Selection section. 3 From the Selection list, choose Nanoparticle. 4 Locate the Electric Displacement Field section. From the Electric displacement field model list, choose Relative permittivity. This time, the properties of the nanoparticle will be taken from your material specification. Thus, define the properties for gold. MATERIALS Material 3 1 In the Home toolbar, click New Material. 2 In the Material settings window, locate the Geometric Entity Selection section. 3 From the Selection list, choose Nanoparticle. 16 SCATTERER ON SUBSTRATE

239 4 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Relative permittivity epsilonr epsr_gold 1 Basic Relative permeability mur 1 1 Basic Electrical conductivity sigma 0 S/m Basic 5 Right-click Component 1>Materials>Material 3 and choose Rename. 6 Go to the Rename Material dialog box and type Gold in the New name edit field. 7 Click OK. MESH 1 The default mesh settings are sufficient to resolve the waves in the air and the substrate. You will use a locally finer mesh to resolve the 43 nm skin depth in the nanoparticle. To avoid interpolation errors across the periodic boundaries, they should be meshed identically. PMLs should preferably use a swept mesh with at least five elements across. A convenient way to combine the two latter requirements is to start from a mapped surface mesh. Mapped 1 1 In the Mesh toolbar, click Boundary and choose Mapped. 2 Select the boundaries on the top surface of the geometry: 13, 26, 39, 56, 69, 82, 109, 122, and 135 only. Distribution 1 1 Right-click Component 1>Mesh 1>Mapped 1 and choose Distribution. 2 Select Edges 42, 92, 135, and 150 only. To do this, you can click the Paste Selection button near the selection list and type 42, 92, 135, Leave 5 as the Number of elements. Distribution 2 1 Right-click Mapped 1 and choose Distribution. 2 Select Edges 78 and 79 only. 3 In the Distribution settings window, locate the Distribution section. 4 In the Number of elements edit field, type 10. Swept 1 1 In the Mesh toolbar, click Swept. 2 In the Swept settings window, locate the Domain Selection section. 17 SCATTERER ON SUBSTRATE

240 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose PML Domains. Distribution 1 1 Right-click Component 1>Mesh 1>Swept 1 and choose Distribution. The default distribution of 5 elements per domain in each direction works well. Convert 1 1 In the Mesh toolbar, click Modify and choose Convert. 2 In the Convert settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Boundary. 4 Select the boundaries between the PML and the physical domain: Boundaries 60-64, 68, 74, 77, 113, and 116 only. Converting the mesh on these boundaries to triangles makes it possible to connect it with a free tetrahedral mesh in the physical domain. Free Tetrahedral 1 1 In the Mesh toolbar, click Free Tetrahedral. Size 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Tetrahedral 1 and choose Size. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Nanoparticle. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type 43[nm]. 18 SCATTERER ON SUBSTRATE

241 8 Click the Build All button. Before solving the model, set up component couplings and variables for extracting the cross sections. DEFINITIONS Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Operator Name section. 3 In the Operator name edit field, type intop_vol. 4 Locate the Source Selection section. From the Selection list, choose Nanoparticle. Integration 2 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Operator Name section. 3 In the Operator name edit field, type intop_surf. 4 Locate the Source Selection section. From the Geometric entity level list, choose Boundary. 5 From the Selection list, choose Nanoparticle Surface. 19 SCATTERER ON SUBSTRATE

242 Variables 3 1 In the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description nrelpoav nx*ewfd2.relpoavx+ny*ewfd2.relp oavy+nz*ewfd2.relpoavz W/m² Relative normal Poynting flux sigma_sc intop_surf(nrelpoav)/i0 m² Scattering cross section sigma_abs intop_vol(ewfd2.qh)/i0 m² Absorption cross section sigma_ext sigma_sc+sigma_abs m² Extinction cross section The relative normal Poynting vector is defined from the outwards-facing normal vector and the automatically defined coordinate components of the Poynting flux. STUDY 1 Set up the solver for a few different combinations of angles. Because the second physics interface depends on the first one but not vice versa, the model can be solved sequentially. 1 In the Model Builder window, click Study 1. 2 In the Study settings window, locate the Study Settings section. 3 Clear the Generate default plots check box. Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click the Add button. 4 In the table, enter the following settings: Parameter names theta Parameter value list 0 pi/6 pi/6 pi/4 5 Click the Add button. 20 SCATTERER ON SUBSTRATE

243 6 In the table, enter the following settings: Parameter names phi Parameter value list 0 0 pi/4 pi/4 Step 1: Frequency Domain 1 On the Study toolbar, click Study Steps and choose Frequency Domain>Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 Locate the Physics and Variables Selection section. In the table, deactivate the second Electromagnetic Waves physics by clicking on the cell corresponding to the Solve for column: Physics Solve for Discretization Electromagnetic Waves, Frequency Domain 2 physics Step 2: Frequency Domain 2 1 On the Study toolbar, click Study Steps and choose Frequency Domain>Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 4 Locate the Physics and Variables Selection section. In the table, deactivate the first Electromagnetic Waves physics by clicking on the cell corresponding to the Solve for column: Physics Solve for Discretization Electromagnetic Waves, Frequency Domain physics Solver 1 1 On the Study toolbar, click Show Default Solver. The periodic boundary conditions used in the first interface perform better with a direct solver. 2 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1>Stationary Solver 1 node. 3 Right-click Direct and choose Enable. 4 On the Study toolbar, click Compute. 21 SCATTERER ON SUBSTRATE

244 Before generating the plots, set up the data sets for easy display of the surfaces of the substrate and the nanoparticle. RESULTS Data Sets 1 In the Model Builder window, expand the Results>Data Sets node. 2 Right-click Solution 1 and choose Rename. 3 Go to the Rename Solution dialog box and type Substrate in the New name edit field. 4 Click OK. 5 Right-click Results>Data Sets>Substrate and choose Add Selection. 6 In the Selection settings window, locate the Geometric Entity Selection section. 7 From the Geometric entity level list, choose Boundary. 8 Select Boundaries 65 and 87 only (the air-substrate interface). 9 In the Model Builder window, under Results>Data Sets right-click Substrate and choose Duplicate. 10 Right-click Results>Data Sets>Substrate 1 and choose Rename. 11 Go to the Rename Solution dialog box and type Particle in the New name edit field. 12 Click OK. 13 In the Model Builder window, expand the Results>Data Sets>Particle node, then click Selection. 14 In the Selection settings window, locate the Geometric Entity Selection section. 15 From the Selection list, choose Nanoparticle Surface. 16 In the Model Builder window, under Results>Data Sets right-click Solution 2 and choose Add Selection. 17 In the Selection settings window, locate the Geometric Entity Selection section. 18 From the Geometric entity level list, choose Domain. 19 From the Selection list, choose Physical Domains. 20 Select the Propagate to lower dimensions check box. The selection you just made will make the fields show up only in the physical domain. If you want to see how the relative field is damped in the PML, you can delete this selection. 22 SCATTERER ON SUBSTRATE

245 You will create plots for the y-component and the norm of the background field and the total field. Begin with a plot of the background field, with the substrate but not the nanoparticle in place. 3D Plot Group 1 1 On the Results toolbar, click 3D Plot Group. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 Right-click Results>3D Plot Group 1 and choose Rename. 5 Go to the Rename 3D Plot Group dialog box and type Background Field, y in the New name edit field. 6 Click OK. Background Field, y 1 On the Results toolbar, click Slice. 2 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>electric field, y component (ewfd.ey). 3 Locate the Plane Data section. In the Planes edit field, type 3. 4 On the Background Field, y toolbar, click Plot. You have now plotted the y-component from the first interface, for the theta = phi = /4 solution. You can look at the different solutions using the Parameter Value list in the settings window of the Background Field, y node. 5 In the Model Builder window, click Background Field, y. 6 In the 3D Plot Group settings window, locate the Data section. 7 From the Parameter value (theta,phi) list, choose 3: theta= , phi= On the 3D plot group toolbar, click Plot. Color only the substrate surface to make it clear that you are looking at the field distribution without the nanoparticle. 9 On the Background Field, y toolbar, click Surface. 10 In the Surface settings window, locate the Data section. 11 From the Data set list, choose Substrate. 12 Locate the Expression section. In the Expression edit field, type Click to expand the Title section. From the Title type list, choose None. 23 SCATTERER ON SUBSTRATE

246 14 Locate the Coloring and Style section. From the Coloring list, choose Uniform. 15 From the Color list, choose Gray. If you zoom in and rotate the plot you just created, it should look like Figure 2. The most convenient way to reproduce Figure 3 is to duplicate and modify the y-component plot. Background Field, y 1 1 Right-click Background Field, y and choose Duplicate. 2 Right-click the new Background Field, y 1 node and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Background Field, Norm in the New name edit field. 4 Click OK. Background Field, Norm 1 In the Model Builder window, expand the Results>Background Field, Norm node, then click Slice 1. 2 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field norm (ewfd.norme). 3 On the Background Field, Norm toolbar, click Plot. The electric field norm from the first interface confirms that you have a standing wave pattern in the air and a propagating plane wave in the substrate. In order to further confirm that the first interface was set up correctly, verify that the power reflection at the material interface agrees with the analytical result. Derived Values 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 Locate the Expression section. In the Expression edit field, type abs(ewfd.s11)^2. 5 Click the Evaluate button. 24 SCATTERER ON SUBSTRATE

247 TABLE The results agree reasonably well with the analytical solution, as indicated in Table 2. For more information, see (Fresnel Equations). TABLE 2: COMPUTED AND ANALYTICAL POWER REFLECTION COEFFICIENTS. abs(emw.s11)^2 R / /6 / /4 / To visualize the total field, start out with another copy of one of your background field plots. You will change the plot expression and add the particle. RESULTS Background Field, y 1 1 In the Model Builder window, under Results right-click Background Field, y and choose Duplicate. 2 Right-click Background Field, y 1 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Total Field, y in the New name edit field. 4 Click OK. Total Field, y 1 In the Model Builder window, expand the Results>Total Field, y node, then click Slice 1. 2 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain 2>Electric>Electric field>electric field, y component (ewfd2.ey). 3 On the Total Field, y toolbar, click Surface. 4 In the Surface settings window, locate the Data section. 5 From the Data set list, choose Particle. 6 Locate the Expression section. In the Expression edit field, type 1. 7 Locate the Title section. From the Title type list, choose None. 8 Locate the Coloring and Style section. From the Coloring list, choose Uniform. 9 From the Color list, choose Yellow. The plot should now look like Figure SCATTERER ON SUBSTRATE

248 Create a plot of the total field norm to reproduce Figure 5. Total Field, y 1 1 Right-click Total Field, y and choose Duplicate. 2 In the Model Builder window, under Results right-click Total Field, y 1 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Total Field, Norm in the New name edit field. 4 Click OK. Total Field, Norm 1 In the Model Builder window, expand the Results>Total Field, Norm node, then click Slice 1. 2 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain 2>Electric>Electric field norm (ewfd2.norme). 3 On the 3D plot group toolbar, click Plot. The cross section expressions that you defined are available for global evaluation. Derived Values 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 Locate the Expression section. In the Expression edit field, type sigma_abs. 5 Click the Evaluate button. 6 Repeat steps 4 and 5 for sigma_sc, and finally for sigma_ext. The results should resemble those in Table 1. The remaining instructions result in a plot of the power loss in the particle, reproducing Figure 6. Total Field, Norm 1 1 In the Model Builder window, under Results right-click Total Field, Norm and choose Duplicate. 2 Right-click Total Field, Norm 1 and choose Rename. 3 Go to the Rename 3D Plot Group dialog box and type Power Loss in the New name edit field. 26 SCATTERER ON SUBSTRATE

249 4 Click OK. Power Loss 1 In the Model Builder window, expand the Results>Power Loss node, then click Slice 1. 2 In the Slice settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain 2>Heating and losses>total power dissipation density (ewfd2.qh). 3 Locate the Plane Data section. From the Plane list, choose xy-planes. 4 From the Entry method list, choose Coordinates. 5 In the z-coordinates edit field, type 50[nm]. 6 In the Model Builder window, under Results>Power Loss right-click Surface 2 and choose Disable. 27 SCATTERER ON SUBSTRATE

250 28 SCATTERER ON SUBSTRATE Solved with COMSOL Multiphysics 4.4

251 Optical Scattering Off a Gold Nanosphere Introduction This model demonstrates the calculation of the scattering of a plane wave of light off of a gold nanosphere. The scattering is computed for the optical frequency range, over which gold can be modeled as a material with negative complex-valued permittivity. The far-field pattern and the losses are computed. PMC symmetry plane Gold sphere k E PEC symmetry plane Figure 1: A gold sphere illuminated by a plane wave. Due to symmetry, only one-quarter of the sphere has to be modeled. Model Definition A gold sphere of radius r 100 nm is illuminated by a plane wave, as shown in Figure 1. The optical frequency range, corresponding to a free space wavelength of 400 nm 700 nm, is simulated. At these frequencies, gold can be modeled as having a complex valued permittivity, with real and imaginary components. The complex 1 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

252 permittivity is expressed as r. The real and imaginary part of the complex permittivity are extracted using the complex refractive index in Ref. 1. Over the frequency range of interest, it is possible to compute the skin depth via 1 = Re k 0 r (1) where k 0 is the free space wavenumber, and r is the complex-valued relative permittivity. The skin depth is shown in Table 1, and ranges from 27 nm 44 nm. The skin depth is evaluated with assumption of plane wave incidence over flat surface, so it is not directly applicable on the gold sphere in the model. TABLE 1: COMPLEX DIELECTRIC CONSTANT AND SKIN DEPTH FOR GOLD Frequency (THz) Skin Depth (nm) Due to the symmetry of the problem, only one-quarter of the sphere is modeled. A region of air around the sphere is also modeled, of with equal to half the wavelength in free space. A perfectly matched layer (PML) domain is outside of the air domain and acts as an absorber of the scattered field. The PML should not be within the reactive near-field of the scatterer, placing it a half-wavelength away is usually sufficient. The far field radiation pattern and the heat losses are computed. 2 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

253 Results and Discussion The far-field radiation pattern is plotted in Figure 2. These show that, at short wavelengths, a single gold sphere will scatter light forward, in the direction of propagation of the incident light. At longer wavelengths, the scattered fields from the sphere look more as the radiation pattern of a dipole antenna. The heat losses, plotted in Figure 3, show that the particle preferentially absorbs the shorter wavelengths. The radius of the sphere can also be varied to see how the absorption depends upon the geometry. Figure 2: The far-field radiation pattern in the E-plane (blue) and H-plane (green) when wavelength is 700 nm. 3 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

254 Figure 3: The resistive heating losses in the gold sphere. Reference 1. P.B. Johnson and R.W. Christy, Optical Constants of the Noble Metals, Phys. Rev. B, vol. 6, pp , Model Library path: Wave_Optics_Module/Optical_Scattering/ scattering_nanosphere Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 4 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

255 MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Define some parameters that are useful for setting up the mesh and the study. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description r0 100[nm] 1.000E-7 m Sphere radius lda 400[nm] 4.000E-7 m Wavelength f0 c_const/lda 7.495E14 1/s Frequency t_air lda/ E-7 m Thickness of air around sphere t_pml lda/ E-7 m Thickness of PML h_max lda/ E-8 m Maximum element size, air Here, c_const is a predefined COMSOL constant for the speed of light. Add two interpolation functions for the real and imaginary parts, respectively, of the complex relative dielectric constant (permittivity) as functions of the frequency. Interpolation 1 1 On the Home toolbar, click Functions and choose Global>Interpolation. 2 In the Interpolation settings window, locate the Definition section. 3 From the Data source list, choose File. 4 Click the Browse button. 5 Browse to the model s Model Library folder and double-click the file scattering_nanosphere_er_interpolation.txt. 5 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

256 6 Click the Import button. 7 In the Function name edit field, type eps_real. 8 Locate the Units section. In the Arguments edit field, type Hz. 9 In the Function edit field, type 1. Interpolation 2 1 On the Home toolbar, click Functions and choose Global>Interpolation. 2 In the Interpolation settings window, locate the Definition section. 3 From the Data source list, choose File. 4 Click the Browse button. 5 Browse to the model s Model Library folder and double-click the file scattering_nanosphere_ei_interpolation.txt. 6 Click the Import button. 7 In the Function name edit field, type eps_imag. 8 Locate the Units section. In the Arguments edit field, type Hz. 9 In the Function edit field, type 1. GEOMETRY 1 Create a sphere with layers. The outermost layer represents the PMLs and the core represents the gold sphere. The middle layer is the air domain. Sphere 1 1 On the Geometry toolbar, click Sphere. 2 In the Sphere settings window, locate the Size section. 3 In the Radius edit field, type r0+t_air+t_pml. 4 Click to expand the Layers section. In the table, enter the following settings: Layer name Layer 1 Layer 2 Thickness (m) t_pml t_air 5 Click the Build Selected button. 6 Click the Wireframe Rendering button on the Graphics toolbar to get a better view of the interior parts. Then, add a block intersecting one-quarter of the sphere. 6 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

257 Block 1 1 On the Geometry toolbar, click Block. 2 In the Block settings window, locate the Size section. 3 In the Width edit field, type 2*(r0+t_air+t_pml). 4 In the Depth edit field, type 2*(r0+t_air+t_pml). 5 In the Height edit field, type 2*(r0+t_air+t_pml). 6 Locate the Position section. In the x edit field, type -(r0+t_air+t_pml). 7 Click the Build Selected button. Generate the quarter sphere by intersecting two objects. Intersection 1 1 On the Geometry toolbar, click Intersection. 2 Select the objects sph1 and blk1 only. 3 Click the Build All Objects button. 7 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

258 4 Click the Zoom Extents button on the Graphics toolbar. DEFINITIONS Add a variable for the total heat losses in the gold sphere computed as a volume integral of resistive losses. First, add an integration coupling operator for the volume integral of the gold sphere. Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Operator Name section. 3 In the Operator name edit field, type int_l. 4 Select the gold sphere (Domain 3) only. Variables 1 1 On the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Variables section. 3 In the table, enter the following settings: Name Expression Unit Description l_gold int_l(ewfd.qrh) W Heat losses Here, the ewfd. prefix gives the correct physics-interface scope for the resistive losses. 8 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

259 ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Now set up the physics. You solve the model for the scattered field, so it needs background electric field (E-field) information. The background plane wave is traveling in the positive x direction, with the electric field polarized along the z-axis. The default boundary condition is perfect electric conductor, which applies to all exterior boundaries including the boundaries perpendicular to the background E-field polarization. 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Frequency Domain. 2 In the Electromagnetic Waves, Frequency Domain settings window, locate the Settings section. 3 From the Solve for list, choose Scattered field. 4 Specify the E b vector as 0 x 0 y exp(-j*ewfd.k0*x) z Apply a user-defined relative dielectric constant on the gold sphere. The constant can be complex with the real and imaginary parts generated by interpolating with the given tables. Wave Equation, Electric 2 1 On the Physics toolbar, click Domains and choose Wave Equation, Electric. 2 In the Wave Equation, Electric settings window, locate the Electric Displacement Field section. 3 From the Electric displacement field model list, choose Relative permittivity. 9 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

260 4 Select Domain 3 only. 5 From the r list, choose User defined. In the associated edit field, type eps_real(ewfd.freq)-i*eps_imag(ewfd.freq). 6 Locate the Magnetic Field section. From the r list, choose User defined. Locate the Conduction Current section. From the list, choose User defined. Leave the default value 0. Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 10 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

261 2 Select Boundaries 3 and 16 only. DEFINITIONS The outermost domains from the center of the sphere are the PMLs. Perfectly Matched Layer 1 (pml1) 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 Select Domains 1 and 5 only. 3 In the Perfectly Matched Layer settings window, locate the Geometry section. 11 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

262 4 From the Type list, choose Spherical. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN Set PMC on the boundaries parallel to the background E-field polarization. Perfect Magnetic Conductor 1 1 On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor. 12 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

263 2 Select Boundaries 1, 4, 8, 11, and 14 only. Far-Field Domain 1 1 On the Physics toolbar, click Domains and choose Far-Field Domain. 2 Select Domains 2 and 4 only. 13 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

264 Far-Field Calculation 1 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Frequency Domain>Far-Field Domain 1 click Far-Field Calculation 1. 2 In the Far-Field Calculation settings window, locate the Boundary Selection section. 3 Click Clear Selection. 4 Select Boundaries 6 and 15 only. 5 Locate the Far-Field Calculation section. Select the Symmetry in the y=0 plane check box. 6 Select the Symmetry in the z=0 plane check box. 7 From the Symmetry type list, choose Symmetry in H (PEC). MATERIALS Assign air as the material for all domains. Because you set the properties of the gold sphere explicitly in the Electromagnetic Waves, Frequency Domain interface, they are not affected by this setting. 1 On the Home toolbar, click Add Material. ADD MATERIAL 1 Go to the Add Material window. 2 In the tree, select Built-In>Air. 14 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

265 3 In the Add material window, click Add to Component. 4 Close the Add material window. MESH 1 The maximum mesh size is at most 0.2 wavelengths in free space. To evaluate the gold sphere up to the accuracy level of the skin depth, set the maximum element size inside the sphere around the half of the minimum skin depth over the frequency sweep range. Size I 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Size.. 2 In the Size settings window, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domain 3 only. 5 Locate the Element Size section. Click the Custom button. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated edit field, type 13.5[nm]. Size 1 In the Model Builder window, under Component 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. Locate the Element Size Parameters section. In the Maximum element size edit field, type h_max Free Tetrahedral 1 1 On the Mesh toolbar, click Free Tetrahedral. 2 In the Free Tetrahedral settings window, locate the Domain Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domains 2 4 only. Finally, use a swept mesh for the PMLs. Swept 1 1 On the Mesh toolbar, click Swept. 2 In the Swept settings window, locate the Domain Selection section. 3 From the Geometric entity level list, choose Domain. 15 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

266 4 Select Domains 1 and 5 only. Distribution 1 1 Right-click Component 1>Mesh 1>Swept 1 and choose Distribution. Leave the default Number of elements, 5. 2 On the Mesh toolbar, click Build Mesh. STUDY 1 Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names lda Parameter value list range(400[nm],300[nm]/30,700[nm]) Step 1: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 16 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

267 3 In the Frequencies edit field, type f0. 4 In the Model Builder window, click Study 1. 5 In the Study settings window, locate the Study Settings section. 6 Clear the Generate default plots check box. 7 On the Study toolbar, click Compute. RESULTS Begin the results analysis and visualization by adding a selection to see the resistive losses only inside the gold sphere. Data Sets 1 In the Model Builder window, expand the Results>Data Sets node. 2 Right-click Solution 2 and choose Add Selection. 3 In the Selection settings window, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Domain. 5 Select Domain 3 only. 3D Plot Group 1 1 On the Results toolbar, click 3D Plot Group. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 On the 3D Plot Group 1 toolbar, click Volume. 5 In the Volume settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Heating and losses>resistive losses (ewfd.qrh). 6 On the 3D Plot Group 1 toolbar, click Plot. 17 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

268 7 Click the Zoom Extents button on the Graphics toolbar. The following instructions reproduce the polar plot of the far-field at the E-plane and H-plane shown in Figure 2. Polar Plot Group 2 1 On the Results toolbar, click Polar Plot Group. 2 In the Polar Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 From the Parameter selection (lda) list, choose Last. 5 On the Polar Plot Group 2 toolbar, click Line Graph. 6 Click the Zoom Extents button on the Graphics toolbar. 7 Select Edges 5 and 15 only. 8 In the Line Graph settings window, click Replace Expression in the upper-right corner of the r-axis data section. From the menu, choose Electromagnetic Waves, Frequency Domain>Far field>far-field norm (ewfd.normefar). 9 Locate the Angle Data section. From the Parameter list, choose Expression. 10 In the Expression edit field, type atan2(z,x). 11 Click to expand the Title section. From the Title type list, choose None. 18 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

269 12 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Blue. 13 In the Model Builder window, under Results>Polar Plot Group 2 right-click Line Graph 1 and choose Duplicate. 14 In the Line Graph settings window, locate the Angle Data section. 15 In the Expression edit field, type atan2(-z,x). 16 Right-click Line Graph 1 and choose Duplicate. 17 In the Line Graph settings window, locate the Selection section. 18 Select the Selection focus toggle button. 19 Click Clear Selection. 20 Select Edges 6 and 21 only. 21 Locate the Angle Data section. In the Expression edit field, type atan2(y,x). 22 Locate the Coloring and Style section. Find the Line style subsection. From the Color list, choose Green. 23 In the Model Builder window, under Results>Polar Plot Group 2 right-click Line Graph 3 and choose Duplicate. 24 In the Line Graph settings window, locate the Angle Data section. 25 In the Expression edit field, type atan2(-y,x). 26 On the Polar Plot Group 2 toolbar, click Plot. Finish by plotting the heat losses inside the gold sphere. 1D Plot Group 3 1 On the Results, click 1D Plot Group. 2 In the 1D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 On the 1D Plot Group 1 toolbar, click Global. 5 In the Global settings window, click Replace Expression in the upper-right corner of the y-axis data section. From the menu, choose Definitions>Heat losses (l_gold). 6 Locate the x-axis Data section. From the Axis source data list, choose Outer solutions. 7 Click to expand the Legends section. Clear the Show legends check box. 8 On the 1D Plot Group 1 toolbar, click Plot. Compare the resulting plot with Figure OPTICAL SCATTERING OFF A GOLD NANOSPHERE

270 20 OPTICAL SCATTERING OFF A GOLD NANOSPHERE

271 Second Harmonic Generation of a Gaussian Beam Introduction Laser systems are an important application area in modern electronics. There are several ways to generate a laser beam, but they all have one thing in common: The wavelength is determined by the stimulated emission, which depends on material parameters. It is especially difficult to find lasers that generate short wavelengths (for example, ultraviolet light). With nonlinear materials it is possible to generate harmonics with frequencies that are multiples of the frequency of the laser light. Coherent light with half the wavelength of the fundamental beam is generated with second-order nonlinear materials. This model shows how to set up second harmonic generation as a transient wave simulation using nonlinear material properties. A Nd:YAG ( 1.06 m) laser beam is focused on a nonlinear crystal, so that the waist of the beam is inside the crystal. Model Definition To simplify the problem and to save some calculation time this model is not a full 3D simulation, but rather a 2D model. The model uses COMSOL Multiphysics standard 2D coordinate system, assuming that the beam is propagating in the x direction, it has a transverse Gaussian intensity dependence in the y direction and that the electric field is polarized in the out-of-plane z direction. y w 0 x When a laser beam propagates, it travels as an approximate plane wave with a cross-section intensity of Gaussian shape. At the focal point, the laser beam has its 1 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

272 minimum width, w 0. The solution of the time-harmonic Maxwell s equations for a 2D geometry, gives the following electric field (z-component) 1 : w E xyz = E e y w x wx 2 ky 2 cos t kx x R x ez, where x wx = w x In these expressions, w 0 is the minimum waist, is the angular frequency, y is the in-plane transverse coordinate, and k is the wave number. The wave front of the beam is not exactly planar; it propagates like a spherical wave with radius R(x). However, close to the focal point the wave is almost plane. The laser beam is also modeled as a pulse in time, using a Gaussian envelope function. This produces a wave package with a Gaussian frequency spectrum. These expressions are used as the input boundary conditions. The nonlinear properties for second harmonic generation in a material can be defined with the following matrix, x x = atan x 0 x 0 Rx = x x 2. E x 2 P = d 11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 2 E y 2 E z, d 31 d 32 d 33 d 34 d 35 d 36 2E z E y 2E z E x 2E x E y where P is the polarization. The model only uses the d 33 parameter for simplicity. To keep the problem size small the nonlinear parameter is magnified by some orders of 1. Notice that the electric field is defined for a 2D Gaussian beam. In 2D, the amplitude is defined as the square root of the spot radius ratio and the Gouy phase shift (x) is defined with a factor 1/2. In 3D the amplitude is defined using just the spot radius ratio and there is no factor 1/2 for the Gouy phase shift. 2 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

273 magnitude. The crystal here has a value of F/V, when the values for most materials usually are in the F/V range. Without this magnification, to get a detectable second harmonics requires a much longer propagation distance, resulting in a large computational problem. Results and Discussion The main purpose of this simulation is to calculate the second harmonic generation when the pulse travels along the 20 m geometry. So you have to solve for the time it takes for the pulse to enter, pass, and disappear from the volume. The pulse has a characteristic time of 10 fs, and below you can see the pulse after it has traveled 61 fs. Figure 1: The pulse after 61 fs. 3 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

274 After 90 fs the pulse has reached the output boundary (see Figure 2). The simulation has to continue for another 30 fs until the pulse completely disappears. Figure 2: The pulse after 90 fs. It has now reached the output boundary. The simulation stores the times between 60 fs and 120 fs, which is when the pulse passes the output boundary. The electric field at this boundary has a second harmonic component that can be extracted using a frequency analysis. The result appears in Figure 3. The small peak on the right side of the large peak is the second harmonic generation. To the left of the large peak is also a smaller peak, due to difference frequency generation. This optical rectification effect is also a second-order nonlinear optical process. 4 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

275 Figure 3: The frequency spectrum of the beam at the output boundary. The small peak on the right side of the large peak is the second harmonic generation. The smaller peak close to zero frequency is due to difference frequency generation. Reference 1. A. Yariv, Quantum Electronics, 3rd Edition, John Wiley & Sons, Model Library path: Wave_Optics_Module/Nonlinear_Optics/ second_harmonic_generation Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. 5 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

276 MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Transient (ewt). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Time Dependent. 6 Click the Done button. GEOMETRY 1 1 In the Model Builder window, under Component 1 click Geometry 1. 2 In the Geometry settings window, locate the Units section. 3 From the Length unit list, choose µm. GLOBAL DEFINITIONS Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description w0 2[um] E-6 m Minimum spot radius of laser beam lambda0 1.06[um] E-6 m Wavelength of input laser beam E0 30[kV/m] V/m Peak electric field x0 pi*w0^2/lambda E-5 m Rayleigh range k0 2*pi/lambda E6 1/m Propagation constant omega0 k0*c_const E15 1/s Angular frequency t0 25[fs] E-14 s Pulse time delay dt 10[fs] E-14 s Pulse width d33 1e-17[F/V] E-17 s 7 A 3 / (m 4 kg 2 ) Matrix element for second harmonic generation 6 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

277 Analytic 1 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type w. 4 Locate the Definition section. In the Expression edit field, type w0*sqrt(1+(x/ x0)^2). Analytic 2 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type eta. 4 Locate the Definition section. In the Expression edit field, type atan2(x,x0)/2. Analytic 3 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type R. 4 Locate the Definition section. In the Expression edit field, type x*(1+(x0/x)^2). GEOMETRY 1 Create a rectangular calculation domain, where y = 0 is the symmetry plane for the laser beam. Rectangle 1 1 Right-click Geometry 1 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type In the Height edit field, type 6. 5 Locate the Position section. In the x edit field, type In the y edit field, type In the x edit field, type Click the Build All Objects button. 9 Click the Zoom Extents button on the Graphics toolbar. 7 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

278 MATERIALS Assume that the laser beam propagates through an artificial material with the properties of air except for an additional second-order optical nonlinearity. You will specify the nonlinearity later, when setting up the wave equation. 1 On the Home toolbar, click Add Material. ADD MATERIAL 1 Go to the Add Material window. 2 In the tree, select Built-In>Air. 3 In the Add material window, click Add to Component. 4 Close the Add Material window. ELECTROMAGNETIC WAVES, TRANSIENT Perfect Magnetic Conductor 1 1 On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor. 2 Select Boundary 3 only. Scattering Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 Select Boundary 1 only. 3 In the Scattering Boundary Condition settings window, locate the Scattering Boundary Condition section. 4 From the Incident field list, choose Wave given by E field. 5 Specify the E 0 vector as 0 x 0 y E0*sqrt(w0/w(x))*exp(-y^2/ w(x)^2)*cos(omega0*t-k0*x+eta(x)-k0*y^2/ (2*R(x)))*exp(-(t-t0)^2/dt^2) z Scattering Boundary Condition 2 1 On the Physics toolbar, click Boundaries and choose Scattering Boundary Condition. 2 Select Boundary 4 only. 8 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

279 Wave Equation, Electric 1 1 In the Model Builder window, under Component 1>Electromagnetic Waves, Transient click Wave Equation, Electric 1. 2 In the Wave Equation, Electric settings window, locate the Electric Displacement Field section. 3 From the Electric displacement field model list, choose Remanent electric displacement. 4 Specify the D r vector as 0 x 0 y d33*ewt.ez^2 z MESH 1 Mapped 1 In the Model Builder window, under Component 1 right-click Mesh 1 and choose Mapped. Distribution 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Mapped 1 and choose Distribution. 2 Select Boundaries 1 and 4 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Explicit distribution. 5 In the Explicit element distribution edit field, type sin(range(0,0.025*pi,0.5*pi)). This creates a denser mesh closer to the upper boundary. Distribution 2 1 Right-click Mapped 1 and choose Distribution. 2 Select Boundaries 2 and 3 only. 3 In the Distribution settings window, locate the Distribution section. 4 From the Distribution properties list, choose Explicit distribution. 5 In the Explicit element distribution edit field, type range(0,5e-8,20e-6). 6 Click the Build All button. 9 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

280 7 Click the Zoom Extents button on the Graphics toolbar. DEFINITIONS To perform an FFT analysis, the number of time steps that have to be saved is very large. To store all solutions of the A-field results in a huge model file. However, for the FFT, it is only interesting to look at the field at the output boundary. You can take advantage of this fact by defining a Domain Point Probe that stores and displays the on-axis electric field at the output boundary, whereas the A-field is only stored at the times defined in the study. 1 On the Definitions toolbar, click Probes and choose Domain Point Probe. 2 In the Domain Point Probe settings window, locate the Point Selection section. 3 In row Coordinates, set x to In the Model Builder window, expand the Component 1>Definitions>Domain Point Probe 1 node, then click Point Probe Expression 1. 5 In the Point Probe Expression settings window, locate the Probe Settings section. 6 In the Probe variable edit field, type Eout. 7 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Transient>Electric>Electric Field>Electric field, z component (ewt.ez). 10 SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

281 STUDY 1 Step 1: Time Dependent Now define the times for which to save the full field solutions. 1 In the Model Builder window, under Study 1 click Step 1: Time Dependent. 2 In the Time Dependent settings window, locate the Study Settings section. 3 In the Times edit field, type 0 61[fs] 90[fs] 120[fs]. 4 Click to expand the Results while solving section. Locate the Results While Solving section. Select the Plot check box. Solver 1 1 On the Study toolbar, click Show Default Solver. Set the solver to calculate the output field every 0.2 fs. 2 In the Model Builder window, expand the Solver 1 node, then click Time-Dependent Solver 1. 3 In the Time-Dependent Solver settings window, click to expand the Time stepping section. 4 Locate the Time Stepping section. From the Steps taken by solver list, choose Manual. 5 In the Time step edit field, type 0.2[fs]. 6 On the Study toolbar, click Compute. RESULTS Electric Field Now define the surface plots, as displayed in Figure 1 and Figure 2. 1 In the Model Builder window, expand the Electric Field node, then click Surface In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Transient>Electric>Electric Field>Electric field, z component (ewt.ez) in the upper-right corner of the section. 3 Right-click Results>Electric Field>Surface 1.1 and choose Height Expression. 4 In the 2D Plot Group settings window, locate the Data section. 5 From the Time (s) list, choose 6.1e On the Electric Field toolbar, click Plot. You should now see the plot in Figure 1. 7 From the Time (s) list, choose 9e SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

282 8 On the Electric Field toolbar, click Plot. You should now see the plot in Figure 2. Probe 1D Plot Group 2 Finally, make a spectral analysis of the on-axis electric field at the output boundary. 1 In the Model Builder window, expand the Results>Probe 1D Plot Group 2 node, then click Probe Table Graph 1. 2 In the Table Graph settings window, locate the Data section. 3 From the Transformation list, choose Frequency spectrum. 4 Select the Frequency range check box. 5 On the Probe 1D Plot Group 2 toolbar, click Plot. Now you should have a graph similar to the one in Figure SECOND HARMONIC GENERATION OF A GAUSSIAN BEAM

283 Self-Focusing Solved with COMSOL Multiphysics 4.4 Introduction For low intensities, the refractive index is independent of the intensity of the light in the material. However, when the intensity is large, so large that the electric field of the light field actually start to perturb the electron clouds around the nuclei, the refractive index start to depend on the intensity. Thus, for dielectrics, like glass, the refractive index increases with the intensity. When a Gaussian beam propagates through a medium with an intensity-dependent refractive index, the index will be highest at the center of the beam. Thus, the induced refractive index profile will act as a lens or a waveguide that counteracts the spreading of the beam due to diffraction. The effect that the beam itself induces this positive, focusing, lens in the material is called self-focusing. Self-focusing manifests itself both in terms of whole-beam focusing, where the beam s properties are changed by the induced index profile, and by small-scale focusing, where noise across the beam s cross-sectional intensity distribution is amplified and the beam can break up in to several self-focused filaments. The nonlinear refractive index is written as n = n 0 + I where n 0 is the constant (linear) part of the refractive index, is the nonlinear refractive index coefficient and I is the intensity. The nonlinearity is due to the optical Kerr effect, which is a nonlinear distortion of the electron clouds around the nuclei in the material. For the standard optical glass BK-7, the nonlinear coefficient is cm 2 /W. Thus, for an intensity of 2.5 GW/cm 2, the induced refractive index change is This is a small change, but, as you will see, it will have a significant effect on the beam parameters. Self-focusing is important from a laser engineering perspective, as the modification of the beam must be incorporated in the design. Furthermore, if the threshold for self-focusing is exceeded, the material will be damaged. Thus, it important to know the self-focusing threshold values for the materials used in the design. Self-focusing occurs in dielectrics, like optical glasses and laser rod materials, such as Nd:YAG. A first estimate of the threshold power for self-focusing is obtain by assuming that the beam has a circular cross-section with a uniform intensity. Within this beam, the, 1 SELF-FOCUSING

284 refractive index will be higher than outside the beam. Thus, the beam itself induces a waveguide structure. You can equal the acceptance angle of the waveguide with the diffraction angle from the circularly confined light. From this equality, you get the critical power to be 2 P cr = (1) 8 n 0 Model Definition The geometry for the model is simple - just a cylinder. The incident beam is approximated by a Gaussian beam, polarized in the z-direction. The electric field is given by w 0 E xyz E y 2 + z , wx w 2 jk y2 + z 2 = exp exp x 2R x exp jkx x z where E 0 is the electric field amplitude, w 0 is the spot radius at the waist, k is the wave number, defined by k = 2 n 0 /, and z is the unit vector in the z-direction. The function w(x) defines the spot size variation as a function of the distance from the beam waist, wx = w x , (2) where x 0 = n 0 w 0 2 / is the Rayleigh range. The radius of curvature is defined by and the phase change close to the beam waist, the so called Gouy shift, is defined by x Rx x 1 x 0 = x 2 x x = atan x 0 2 SELF-FOCUSING

285 Results and Discussion Figure 1 shows the Gaussian beam for a low input intensity. As expected, the beam is symmetric around the beam waist location. Figure 1: The Gaussian beam for a low peak intensity, I 0 = 1 kw/cm 2. Figure 2 shows the beam for a high input intensity, I 0 = 6 GW/cm 2. The nominal induced refractive index, I 0, is As noted from Figure 3 the actual induced 3 SELF-FOCUSING

286 refractive index is more than ten times the nominal value. This is of course an effect of the self-focusing of the beam. Figure 2: The Gaussian beam for a high peak intensity, I 0 = 6 GW/cm 2. To demonstrate the effect of self-focusing, Figure 4 shows the calculated spot radius at the boundary between the propagation domain and the PML domain. The spot radius is defined by w = 2 Iyz y 2 + z 2 dydz A , (3) Iyz dydz A where A is the integration area and I(y,z) is the cross-sectional intensity distribution of the beam. For low peak intensities, the calculated spot radius, using Equation 3, give similar results as the Gaussian beam expression in Equation 2. However, with 4 SELF-FOCUSING

287 increasing intensities, the beam start to deviate from a Gaussian beam and the spot radius is reduced linearly with intensity. Figure 3: The induced refractive index change, I, for a high peak intensity, I 0 = 6 GW/ cm 2. 5 SELF-FOCUSING

288 Figure 4: The spot radius at the end of the propagation domain versus the peak intensity. The final intensity used in the parametric sweep is 6 GW/cm 2. This corresponds to a power that is 22% of the critical power, provided in Equation 1. As discussed in Ref. 1, it is expected that the critical power for a Gaussian beam is reduced by a factor of approximately 4/(1.22 ) 2 = Thus, the power corresponding to the last peak intensity in the sweep is approximately a factor 0.22/0.27 = 0.81 from the critical power for self-focusing for a Gaussian beam, where the induced refractive index profile completely balances the diffractive spreading of the beam. To compute the field for even higher intensities, a much finer mesh would be needed, as the beam can break up into filaments. For a nonlinear problem like this, it is important to verify the results by, for instance, repeating the simulation with a finer mesh. Reference 1. W. Koechner, Solid-State Laser Engineering, Springer, chap. 4.6, SELF-FOCUSING

289 Model Library path: Wave_Optics_Module/Nonlinear_Optics/self_focusing Modeling Instructions From the File menu, choose New. NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 3D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Beam Envelopes (ewbe). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Frequency Domain. 6 Click the Done button. GLOBAL DEFINITIONS Start by adding some global model parameters. Parameters 1 On the Home toolbar, click Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name Expression Value Description wl 1.064[um] 1.064E-6 m Wavelength f0 c_const/wl 2.818E14 1/s Frequency w0 100*wl 1.064E-4 m Nominal spot radius n Refractive index of BK-7 glass x0 pi*n0*w0^2/wl m Rayleigh range k 2*pi*n0/wl 8.976E6 1/m Propagation constant 7 SELF-FOCUSING

290 Name Expression Value Description I0 2.5[GW/cm^2] 2.500E13 W/m² Nominal peak intensity E0 sqrt(2*i0/ n0*sqrt(mu0_const/ epsilon0_const)) 1.113E8 V/m Nominal peak electric field length 4*x m Length of computation domain radius 2.5*w0*sqrt(1+(length/ (2*x0))^2) 5.948E-4 m Radius of computation domain gamma 4e-16[cm^2/W] 4.000E-20 s³/kg Nonlinear refractive index coefficient delta_n gamma*i E-6 Nominal refractive index change P_cr (1.22*pi)^2*wl^2/ (8*pi*n0*gamma) 1.088E7 W Critical power Now add the functions that describe the incident Gaussian beam. Start with the function describing the spot radius versus distance from focus. Analytic 1 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type w. 4 Locate the Definition section. In the Expression edit field, type w0*sqrt(1+(x/ x0)^2). 5 Locate the Units section. In the Arguments edit field, type m. 6 In the Function edit field, type m. Next, add the radius of curvature function. Analytic 2 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type R. 4 Locate the Definition section. In the Expression edit field, type x*(1+(x0/x)^2). 5 Locate the Units section. In the Arguments edit field, type m. 6 In the Function edit field, type m. Finally, add the function describing the phase shift around the focus region. 8 SELF-FOCUSING

291 Analytic 3 1 On the Home toolbar, click Functions and choose Global>Analytic. 2 In the Analytic settings window, locate the Function Name section. 3 In the Function name edit field, type eta. 4 Locate the Definition section. In the Expression edit field, type atan2(x,x0). 5 Locate the Units section. In the Arguments edit field, type m. GEOMETRY 1 Cylinder 1 1 On the Geometry toolbar, click Cylinder. 2 In the Cylinder settings window, locate the Size and Shape section. 3 In the Radius edit field, type radius. 4 In the Height edit field, type length. 5 Locate the Position section. In the x edit field, type -length/2. 6 Locate the Axis section. From the Axis type list, choose x-axis. 7 On the Home toolbar, click Build All Objects. GLOBAL DEFINITIONS Since the geometry is so long and thin, modify the view setting to not preserve the aspect ratio. DEFINITIONS Camera 1 In the Model Builder window, expand the Component 1>Definitions>View 1 node, then click Camera. 2 In the Camera settings window, locate the Camera section. 3 Clear the Preserve aspect ratio check box. 4 Click the Apply button. 5 Click the Zoom Extents button on the Graphics toolbar. MATERIALS Now add the BK-7 glass used in the model. 9 SELF-FOCUSING

292 Material 1 1 In the Model Builder window, under Component 1 right-click Materials and choose New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n n0+gamma*ewbe.poavx 1 Refractive index The variable ewbe.poavx represents the intensity in the propagation direction. 4 Right-click Component 1>Materials>Material 1 and choose Rename. 5 Go to the Rename Material dialog box and type BK-7 glass in the New name edit field. 6 Click OK. DEFINITIONS Setup a boundary integration operator, for calculation of the output power and the output spot radius. Integration 1 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Integration settings window, locate the Operator Name section. 3 In the Operator name edit field, type intop_output_boundary. 4 Locate the Source Selection section. From the Geometric entity level list, choose Boundary. 5 Select Boundary 6 only. Add the expressions for the output power and the output spot radius. Variables 1 1 In the Definitions toolbar, click Local Variables. 2 In the Variables settings window, locate the Variables section. 10 SELF-FOCUSING

293 3 In the table, enter the following settings: Name Expression Unit Description P intop_output_boundary(ewbe.n Poav) W Output power w_t sqrt(2*intop_output_boundary (ewbe.npoav*(y^2+z^2))/p) m Spot radius on output boundary ELECTROMAGNETIC WAVES, BEAM ENVELOPES Set the interface to use unidirectional propagation and define the wave vector component in the x-direction. 1 In the Model Builder window, under Component 1 click Electromagnetic Waves, Beam Envelopes. 2 In the Electromagnetic Waves, Beam Envelopes settings window, locate the Wave Vectors section. 3 From the Number of directions list, choose Unidirectional. 4 Specify the k 1 vector as k x 0 y 0 z Use a matched boundary condition to launch an incident Gaussian beam polarized in the z-direction. Matched Boundary Condition 1 1 On the Physics toolbar, click Boundaries and choose Matched Boundary Condition. 2 Select Boundary 1 only. 3 In the Matched Boundary Condition settings window, locate the Matched Boundary Condition section. 4 From the Incident field list, choose Electric field. 5 Specify the E 0 vector as 0 x 0 y E0*w0/w(x)*exp(-(y^2+z^2)/ w(x)^2)*exp(-i*(k*x-eta(x)+k*(y^2+z^2)/(2*r(x)))) z 11 SELF-FOCUSING

294 Matched Boundary Condition 2 1 On the Physics toolbar, click Boundaries and choose Matched Boundary Condition. 2 Select Boundary 6 only. MESH 1 1 In the Model Builder window, under Component 1 click Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Sequence type list, choose User-controlled mesh. Set the main Size settings to represent the discretization along the propagation direction. Since the beam is expected to behave as a slightly distorted Gaussian beam, it is sufficient to set the maximum mesh element size to half the Rayleigh range. Size 1 In the Model Builder window, under Component 1>Mesh 1 click Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Maximum element size edit field, type x0/2. Add a free triangular mesh on the input surface. This mesh should resolve the beam's cross-sectional distribution. Free Triangular 1 1 In the Model Builder window, right-click Mesh 1 and choose More Operations>Free Triangular. 2 Select Boundary 1 only. Size 1 1 Right-click Component 1>Mesh 1>Free Triangular 1 and choose Size. 2 In the Size settings window, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. Select the Maximum element size check box. 5 In the associated edit field, type w0/2. 6 Select the Minimum element size check box. 7 In the associated edit field, type w0/4. 12 SELF-FOCUSING

295 Now remove the default Free Tetrahedral 1 node and replace it with a Swept mesh node. Free Tetrahedral 1 1 In the Model Builder window, under Component 1>Mesh 1 right-click Free Tetrahedral 1 and choose Delete. Click Yes to confirm. 2 Right-click Mesh 1 and choose Swept. 3 Right-click Mesh 1 and choose Build All. STUDY 1 Don't generate the default plots. 1 In the Model Builder window, click Study 1. 2 In the Study settings window, locate the Study Settings section. 3 Clear the Generate default plots check box. Step 1: Frequency Domain 1 In the Model Builder window, under Study 1 click Step 1: Frequency Domain. 2 In the Frequency Domain settings window, locate the Study Settings section. 3 In the Frequencies edit field, type f0. 13 SELF-FOCUSING

296 Setup a parametric sweep of the nominal peak intensity, from 1 kw/cm 2 (corresponding to linear propagation) to 6 GW/cm 2 that will show a significant self-focusing effect. Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click the Add button. 4 In the table, enter the following settings: Parameter names I0 Parameter value list 1e7, 1e13, 2e13, 3e13, 4e13, 5e13, 6e13 5 Click to expand the Study extensions section. Locate the Study Extensions section. From the Use parametric solver list, choose Off, to turn off the parametric solver that otherwise would perform calculations also for intermediate intensities. Since this is a nonlinear problem, it is better to split the complex electric field variable into its real and imaginary parts. This will produce a more accurate Jacobian for the problem, leading to a faster convergence. Solver 1 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solver 1 node, then click Compile Equations: Frequency Domain. 3 In the Compile Equations settings window, locate the Study and Step section. 4 Select the Split complex variables in real and imaginary parts check box. 5 On the Study toolbar, click Compute. RESULTS 3D Plot Group 1 1 On the Results toolbar, click 3D Plot Group. 2 In the 3D Plot Group settings window, locate the Data section. 3 From the Data set list, choose Solution 2. 4 On the 3D Plot Group 1 toolbar, click Slice. 5 In the Slice settings window, locate the Plane Data section. 6 From the Plane list, choose xy-planes. 14 SELF-FOCUSING

297 7 In the Planes edit field, type 1. 8 Right-click Results>3D Plot Group 1>Slice 1 and choose Deformation. 9 In the Deformation settings window, locate the Expression section. 10 In the z component edit field, type ewbe.norme. 11 Click the Go to Default 3D View button on the Graphics toolbar. Take a look at the field distributions for the different intensities. 12 In the Model Builder window, click 3D Plot Group In the 3D Plot Group settings window, locate the Data section. 14 From the Parameter value (I0) list, choose the desired intensity. 15 On the 3D plot group toolbar, click Plot. Compare the graph at the intensity 6e13 with that in Figure 2. To really see how the beam changes with intensity, you can also visualize this by running the 3D plots in the Player. The first step would be to normalize the field solution with the nominal peak electric field. 16 In the Model Builder window, under Results>3D Plot Group 1 click Slice In the Slice settings window, locate the Expression section. 18 In the Expression edit field, type ewbe.norme/e0. 19 In the Model Builder window, under Results>3D Plot Group 1>Slice 1 click Deformation In the Deformation settings window, locate the Expression section. 21 In the z component edit field, type ewbe.norme/e0. Export 1 On the Results toolbar, click Player. 2 In the Player settings window, locate the Animation Editing section. 3 From the Loop over list, choose I0. 4 Locate the Playing section. In the Display each frame for edit field, type 1. 5 Right-click Results>Export>Player 1 and choose Play. If you check the Repeat box in the Playing settings, you can have the Player repeat the sequence. Now create a new plot of the refractive index change induced by the beam. 3D Plot Group 2 1 On the Results toolbar, click 3D Plot Group. 15 SELF-FOCUSING

298 2 On the Results toolbar, click Slice. 3 In the Slice settings window, locate the Plane Data section. 4 From the Plane list, choose xy-planes. 5 In the Planes edit field, type 1. 6 Click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Beam Envelopes>Material properties>refractive index>refractive index, xx component (ewbe.nxx). 7 Locate the Expression section. In the Expression edit field, type ewbe.nxx-n0. 8 On the 3D Plot Group 2 toolbar, click Plot. 9 Click the Go to Default 3D View button on the Graphics toolbar. Compare your graph with Figure 3. Now, visualize how the spot radius decreases with intensity, using a table and a table plot. Derived Values 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, locate the Expression section. 3 In the Expression edit field, type w_t. 4 Locate the Data section. From the Data set list, choose Solution 2. 5 Click the Evaluate button. TABLE In the Table window, click Table Graph. RESULTS 1D Plot Group 3 1 In the Model Builder window, under Results>1D Plot Group 3 click Table Graph 1. 2 In the Table Graph settings window, locate the Data section. 3 From the x-axis data list, choose I0. 4 From the Plot columns list, choose Manual. 5 In the Columns list, select Spot radius on output boundary (m). 6 In the Model Builder window, click 1D Plot Group 3. 7 In the 1D Plot Group settings window, locate the Axis section. 8 Select the Manual axis limits check box. 16 SELF-FOCUSING

299 9 In the y minimum edit field, type 1.5e In the y maximum edit field, type 2.5e Locate the Grid section. Select the Manual spacing check box. 12 In the x spacing edit field, type 1e In the y spacing edit field, type 0.2e On the 1D Plot Group 3 toolbar, click Plot. Your result should look similar to that in Figure 4. Now, compare the result with the output spot radius of the ideal Gaussian beam. Derived Values 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, locate the Expression section. 3 In the Expression edit field, type w(length/2). 4 Click the Evaluate button. You should find that the spot radius for the low intensity cases are close to that of the nominal beam. RESULTS Derived Values Finally, compare the total power in the beam with the critical power for self-focusing, as defined in Equation 1. 1 On the Results toolbar, click Global Evaluation. 2 In the Global Evaluation settings window, locate the Expression section. 3 In the Expression edit field, type P/P_cr. 4 Locate the Data section. From the Data set list, choose Solution 2. 5 Right-click Results>Derived Values>Global Evaluation 2 and choose Evaluate>New Table. You should find that the power ratio is approximately 22%. 17 SELF-FOCUSING

300 18 SELF-FOCUSING Solved with COMSOL Multiphysics 4.4

301 Step-Index Fiber Bend Introduction The transmission speed of optical waveguides is superior to microwave waveguides because optical devices have a much higher operating frequency than microwaves, enabling a far higher bandwidth. Today the silica glass (SiO 2 ) fiber is forming the backbone of modern communication systems. Before 1970, optical fibers suffered from large transmission losses, making optical communication technology merely an academic issue. In 1970, researchers showed, for the first time, that low-loss optical fibers really could be manufactured. Earlier losses of 2000 db/km now went down to 20 db/km. Today s fibers have losses near the theoretical limit of 0.16 db/km at 1.55 m (infrared light). One of the winning devices has been the single-mode fiber, having a step-index profile with a higher refractive index in the center core and a lower index in the outer cladding. Numerical software plays an important role in the design of single-mode waveguides and fibers. For a fiber cross section, even the most simple shape is difficult and cumbersome to deal with analytically. A circular step-index waveguide is a basic shape where benchmark results are available (see Ref. 1). This example is a model of a single step-index waveguide made of silica glass. The inner core is made of pure silica glass with refractive index n 1 = and the cladding is doped, with a refractive index of n 2 = These values are valid for free-space wavelengths of 1.55 m. The radius of the cladding is chosen to be large enough so that the field of confined modes is zero at the exterior boundaries. For a confined mode there is no energy flow in the radial direction, thus the wave must be evanescent in the radial direction in the cladding. This is true only if n eff n 2 On the other hand, the wave cannot be radially evanescent in the core region. Thus n 2 n eff n 1 The waves are more confined when n eff is close to the upper limit in this interval. For a bent fiber the mode is no longer completely guided by the refractive index structure. This can be qualitatively explained by considering that for a straight 1 STEP-INDEX FIBER BEND

302 waveguide, the wavefronts (planes with a constant phase) are orthogonal to the fiber axis. For a circularly bent fiber, the wavefronts rotate around the center point of the circle with a constant angular velocity. As a result, the propagation constant varies with the distance from the circle center point. At some distance from the center point, the propagation constant is larger that the local wave number, defined by the vacuum wavelength and the refractive index of the cladding material. Beyond this radius, the wave cannot have a constant angular velocity and the wavefronts must bend, implying that the wave starts to radiate energy out from the fiber. For a more complete discussion about waves in bent waveguides, see Ref. 2. Model Definition The first mode analysis is made on a cross-section in the xy-plane of the fiber. The wave propagates in the z direction and has the form E xyzt = E xy e where is the angular frequency and the propagation constant. An eigenvalue equation for the electric field E is derived from Helmholtz equation which is solved for the eigenvalue = j. j t z 2 2 E k 0n E = 0 As boundary condition along the outside of the cladding the electric field is set to zero. Because the amplitude of the field decays rapidly as a function of the radius of the cladding this is a valid boundary condition. The second mode analysis is performed for a 2D axisymmetric geometry. In this case, the wave propagates in the direction and the electric field is expressed as E r zt E rz e j t r 0 = where r 0 is an average radius for the mode in the bent fiber. The radius r 0 is often slightly larger than the radius of curvature for the bent fiber. The eigenvalue solved for in this case is = j r 0. As a consequence of this eigenvalue definition, the effective indices you provide as input to the eigenvalue solver and the effective indices that the solver returns are all scaled with the radius r 0. The geometry is defined as a rectangle surrounding the circular core domain. To absorb the radiating mode, there is a Perfectly Matched Layer (PML) surrounding the, 2 STEP-INDEX FIBER BEND

303 rectangular cladding domain. The wavelength in the PML should correspond to the radial wave vector component for the wave. Results and Discussion When studying the characteristics of straight optical waveguides, the effective mode index of a confined mode, n eff = k 0 as a function of the frequency is an important characteristic. A common notion is the normalized frequency for a fiber. This is defined as 2 a V = n 1 n 2 = k 0 a n 1 0 where a is the radius of the core of the fiber. For this simulation, the effective mode index for the fundamental mode, corresponds to a normalized frequency of The electric and magnetic fields for this mode is shown in Figure 1 below. n STEP-INDEX FIBER BEND

304 Figure 1: The surface plot visualizes the z component of the electric field. This plot is for the effective mode index STEP-INDEX FIBER BEND

305 Figure 2 shows the result for the bend fiber, indicating that the mode is leaky and radiates some power in the radial direction. Figure 2: Surface plot of the z component of the electric field for the mode in the bent fiber. The contour plot shows the component (in the direction of propagation) of the magnetic field and the arrow plot shows the electric field polarization. Reference 1. A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall, Model Library path: Wave_Optics_Module/Waveguides_and_Couplers/ step_index_fiber_bend Modeling Instructions From the File menu, choose New. 5 STEP-INDEX FIBER BEND

306 NEW 1 In the New window, click the Model Wizard button. MODEL WIZARD 1 In the Model Wizard window, click the 2D button. 2 In the Select physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click the Add button. 4 Click the Study button. 5 In the tree, select Preset Studies>Mode Analysis. 6 Click the Done button. COMPONENT 1 1 In the Model Builder window, right-click Component 1 and choose Rename. 2 Go to the Rename Component dialog box and type Straight Fiber in the New name edit field. 3 Click OK. GEOMETRY 1 1 In the Geometry settings window, locate the Units section. 2 From the Length unit list, choose µm. Circle 1 1 In the Model Builder window, right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type Click the Build Selected button. Circle 2 1 Right-click Geometry 1 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type 8. 4 Click the Build Selected button. 6 STEP-INDEX FIBER BEND

307 MATERIALS Material 1 1 On the Home toolbar, click New Material. 2 Right-click Material 1 and choose Rename. 3 Go to the Rename Material dialog box and type Doped Silica Glass in the New name edit field. 4 Click OK. 5 Select Domain 2 only. 6 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n Refractive index Material 2 1 On the Home toolbar, click New Material. 2 Right-click Material 2 and choose Rename. 3 Go to the Rename Material dialog box and type Silica Glass in the New name edit field. 4 Click OK. 5 Select Domain 1 only. 6 Locate the Material Contents section. In the table, enter the following settings: Property Name Value Unit Property group Refractive index n Refractive index ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN MESH 1 1 In the Model Builder window, expand the Straight Fiber>Electromagnetic Waves, Frequency Domain node, then click Straight Fiber>Mesh 1. 2 In the Mesh settings window, locate the Mesh Settings section. 3 From the Element size list, choose Finer. 4 Click the Build All button. 7 STEP-INDEX FIBER BEND

308 STUDY 1 Step 1: Mode Analysis 1 In the Model Builder window, under Study 1 click Step 1: Mode Analysis. 2 In the Mode Analysis settings window, locate the Study Settings section. 3 In the Search for modes around edit field, type The modes of interest have an effective mode index somewhere between the refractive indices of the two materials. The fundamental mode has the highest index. Therefore, setting the mode index to search around to something just above the core index guarantees that the solver will find the fundamental mode. 4 In the Mode analysis frequency edit field, type c_const/1.55[um]. This frequency corresponds to a free space wavelength of 1.55 m. 5 In the Model Builder window, right-click Study 1 and choose Rename. 6 Go to the Rename Study dialog box and type Study 1 (Straight Fiber) in the New name edit field. 7 Click OK. 8 On the Home toolbar, click Compute. RESULTS Electric Field (ewfd) 1 Click the Zoom Extents button on the Graphics toolbar. 8 STEP-INDEX FIBER BEND

309 2 Click the Zoom In button on the Graphics toolbar. The default plot shows the distribution of the norm of the electric field for the highest of the 6 computed modes (the one with the lowest effective mode index). To study the fundamental mode, choose the highest mode index. Because the magnetic field is exactly 90 degrees out of phase with the electric field you can see both the magnetic and the electric field distributions by plotting the solution at a phase angle of 45 degrees. Data Sets 1 In the Model Builder window, expand the Results>Data Sets node, then click Solution 1. 2 In the Solution settings window, locate the Solution section. 3 In the Solution at angle (phase) edit field, type 45. Electric Field (ewfd) 1 In the Model Builder window, under Results click Electric Field (ewfd). 2 In the 2D Plot Group settings window, locate the Data section. 3 From the Effective mode index list, choose (2). 9 STEP-INDEX FIBER BEND

310 4 In the Model Builder window, expand the Electric Field (ewfd) node, then click Surface 1. 5 In the Surface settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Electric>Electric field>electric field, z component (ewfd.ez). 6 On the Electric Field (ewfd) toolbar, click Plot. Add a contour plot of the H-field. 7 In the Electric Field (ewfd) toolbar, click Contour. 8 In the Contour settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electromagnetic Waves, Frequency Domain>Magnetic>Magnetic field>magnetic field, z component (ewfd.hz). 9 On the Electric Field (ewfd) toolbar, click Plot. The distribution of the transversal E and H field components confirms that this is the HE 11 mode. Compare the resulting plot with that in Figure 1. Data Sets Rename the dataset and the plot group to refer to the Straight Fiber model component. 1 In the Model Builder window, under Results>Data Sets right-click Solution 1 and choose Rename. 2 Go to the Rename Solution dialog box and type Solution 1 (Straight Fiber) in the New name edit field. 3 Click OK. Electric Field (ewfd) 1 In the Model Builder window, under Results right-click Electric Field (ewfd) and choose Rename. 2 Go to the Rename 2D Plot Group dialog box and type Straight Fiber in the New name edit field. 3 Click OK. COMPONENT 2 Now add a 2D axisymmetric model component to model the bent fiber. 1 On the Home toolbar, click Add Component and choose 2D Axisymmetric. 2 In the Model Builder window, right-click Component 2 and choose Rename. 10 STEP-INDEX FIBER BEND

311 3 Go to the Rename Component dialog box and type Bent Fiber in the New name edit field. 4 Click OK. GEOMETRY 2 1 In the Geometry settings window, locate the Units section. 2 From the Length unit list, choose µm. Add a circle representing the fiber core. Circle 1 1 Right-click Bent Fiber>Geometry 2 and choose Circle. 2 In the Circle settings window, locate the Size and Shape section. 3 In the Radius edit field, type 8. 4 Locate the Position section. In the r edit field, type Add a square cladding region, representing the domain the mode essentially is propagating in. Square 1 1 In the Model Builder window, right-click Geometry 2 and choose Square. 2 In the Square settings window, locate the Size section. 3 In the Side length edit field, type Locate the Position section. From the Base list, choose Center. 5 In the r edit field, type Finally, add three rectangle domains, where we will define Perfectly Matched Layers (PMLs). Rectangle 1 1 Right-click Geometry 2 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type In the Height edit field, type Locate the Position section. In the r edit field, type In the z edit field, type Click the Zoom Extents button on the Graphics toolbar. 11 STEP-INDEX FIBER BEND

312 Rectangle 2 1 Right-click Geometry 2 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type In the Height edit field, type Locate the Position section. In the r edit field, type In the z edit field, type -50. Rectangle 3 1 Right-click Geometry 2 and choose Rectangle. 2 In the Rectangle settings window, locate the Size section. 3 In the Width edit field, type In the Height edit field, type Locate the Position section. In the r edit field, type In the z edit field, type Click the Build All Objects button. 12 STEP-INDEX FIBER BEND

313 ADD PHYSICS Now, add the an Electromagnetic Waves, Frequency Domain interface. 1 On the Home toolbar, click Add Physics. 2 Go to the Add Physics window. 3 In the Add physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 4 In the Add physics window, click Add to Component. 5 Close the Add Physics window. ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN 2 Add a Perfect Magnetic Conductor (PMC) exterior boundary condition. Perfect Magnetic Conductor 1 1 On the Physics toolbar, click Boundaries and choose Perfect Magnetic Conductor. 2 Select Boundaries 1 3, 5, 7, 9, and only (the external boundaries). MATERIALS Add the same cladding and core materials as for the straight fiber. Material 3 1 On the Home toolbar, click New Material. 2 In the Material settings window, locate the Material Contents section. 3 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n Refractive index 4 Right-click Bent Fiber>Materials>Material 3 and choose Rename. 5 Go to the Rename Material dialog box and type Silica Glass in the New name edit field. 6 Click OK. Material 4 (mat4) 1 On the Home toolbar, click New Material. 2 Select Domain 7 only. 3 In the Material settings window, locate the Material Contents section. 13 STEP-INDEX FIBER BEND

314 4 In the table, enter the following settings: Property Name Value Unit Property group Refractive index n Refractive index 5 Right-click Bent Fiber>Materials>Material 4 and choose Rename. 6 Go to the Rename Material dialog box and type Doped Silica Glass in the New name edit field. 7 Click OK. DEFINITIONS Now add the PML domains. Perfectly Matched Layer 1 1 On the Definitions toolbar, click Perfectly Matched Layer. 2 Select Domains 1 and 3 6 only. 3 In the Perfectly Matched Layer settings window, locate the Geometry section. 4 From the Type list, choose Cylindrical. 5 Locate the Scaling section. From the Coordinate stretching type list, choose Rational. 14 STEP-INDEX FIBER BEND