Axometrics, Inc. 103 Quality Circle, Suite 215 Huntsville, AL 35806 Phone: (256) 704-3332 Fax: (256) 704-6002 E-Mail: info@axometrics.com Website: http://www.axometrics.com EllipsoView TM USER S MANUAL Elipsometry Software Version 2.x 2013 Axometrics, Inc. All rights reserved. AxoScan, AxoView, Axometrics and the Axometrics logo are trademarks of Axometrics, Inc. 1
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CONTENTS 1 Introduction... 2 1.1 Basic Operation... 2 1.1.1 Toolbar... 2 1.1.2 Sample Layers... 3 1.1.3 Data Table... 4 1.1.4 Graphic Display... 6 1.2 Basic Operation (AxoScan)... 8 2 Measuring and Loading Axoscan Data... 9 2.1 Performing a Measurement Using LCDView (AxoScan Only)... 9 2.1.1 Measure Spectrum:... 10 2.1.2 Arbitrary Axis Measurement... 11 2.1.3 Rotation Measurement... 12 2.2 Loading A Dataset Measured Using AxoView... 13 2.2.1 Spectral Dataset Files... 14 2.2.2 Two-Axis Tilt dataset files... 15 2.2.3 Arbitrary Tip-Tilt Datasets... 15 2.3 Removing or Adjusting a Dataset... 16 3 Configuring Layers... 18 4 Coordinate systems... 24 4.1 VRTF-APM coordinate system... 24 5 Data Fitting Details... 25 5.1 Merit Functions... 26 5.2 Special notes regarding non-linear fitting... 27 5.3 Basic Theory and Partial Coherence... 27 5.3.1 Thin Film Simulations... 28 5.3.2 Basics of Coherence... 28 5.3.3 Partial Coherence Simulation... 29 5.3.4 Region Definition... 29 5.3.5 Complete Calculation... 31 6 Program Options... 33 6.1 Display Options... 33 6.1.1 Toolbar... 33 6.1.2 Poincare Sphere... 33 6.1.3 Data Plot... 34 6.1.4 Dispersion Plot... 35 6.2 Analysis... 36 6.2.1 Settings... 36 6.2.2 Data Fitting:... 38 6.2.3 Partial Coherence... 39 6.2.4 Materials Library... 40 6.2.5 Hardware Settings... 41 1
1 INTRODUCTION EllipsoView is a powerful data analysis and measurement software package that is designed to measurement the optical properties of thin film samples. The Axometrics EllipsoView Analysis package is secured using a hardware key. This hardware key must be attached to the USB port of your computer to run EllipsoView. IMPORTANT: DO NOT LOSE THE HARDWARE KEY. The replacement cost for the USB hardware key is the complete purchase price of the LCDView software package. 1.1 Basic Operation Tool Bar Sample Layers Graphs Data Files 1.1.1 TOOLBAR The Toolbar controls on the front panel are: 2
New Project: Pressing this button will clear out all of the current measured data loaded into EllipsoView. Load Measured Data: Press this button to load measured data into EllipsoView Undo (Ctrl-Z) Undo the last change to the Ellipsometry layer Parameters. Import Layers File: Loads Ellipsometry Layers from a file (File>Import Recipe Ctrl I) Export Layers File: Saves the Current LCD Parameter Recipe. (File>Export Recipe) Add Layer: Adds a new layer to the current layer structure. Edit Layer: Edit the properties of the current layer. Remove Layer: Removes the selected layer from the structure Open Program Options: Opens the Program Options window (identical to Tools>Program Options from the pull-down menus) Measure Sample with AxoScan: (AxoScan systems only) If the AxoScan Engine is running pressing this button is used to perform a measurement. If the AxoScan Engine is not running, pressing this button will intialize the AxoScan Engine. Optimize Data Fit: Fits the Ellipsometry parameters to the current imported data Perform Cell Mapping (AxoStep Only): This performs ellipsometry mapping of the Imported AxoStep Data. 1.1.2 SAMPLE LAYERS The Sample Layers table is shown below 3
Each Layer in the structure is defined by two parameters Material Name and Layer Geometry. These are described below. Material Name: Each layer with the same material name will have the same optical constants (N,K, Dispersion etc). Therefore if the optical constants of a layer are changed, all layers with the same name will change in the same way. The first copy of the material in the list will be written in black (or green) font. Additional layers with that name will have their optical constants written in blue and the value in the dispersion column will indicate a pick up and the layer number, indicates which layer it is copying the dispersion parameters from. Layer Geometry: Layer Geometry refers to the physical dimensions and orientations of the layer, which are independent in each layer. These include the Thickness, and Euler angles (alpha, beta and gamma) Several Options are available by right clicking on a layer in the layer table These options allow the user to quickly change parameters in a layer without needing to open the Edit Layer window (see Section 3) 1.1.3 DATA TABLE An example of the data table is shown below. There are two separate rotation measurements (*.rtmx) that have been imported. The table shows Wavelength, Polar Angle and Azimuth Angle. In any given scan, one of these parameters is variable and two are constant. In these data scans, the Azimuth angle is the variable, and the wavelength and polar angle are constant. 4
The Plots for the data are shown on the Data tab on the right side of the main window. Each scan that is checked in the data table will be shown in the data graphs. The table also shows whether the measurement was made in reflection or transmission mode, and if the user moves the horizontal slider to the right, the merit function used for fitting that scan becomes visible. For More information regarding importing data sets, see section 2. 5
1.1.4 GRAPHIC DISPLAY The Graphic Displays are shown on the right side of the front window and are controlled using the tabs at the bottom of the screen. Dispersion: The dispersion graph shows the value of refractive index (or permitivity) of the material that is currently selected in the Layers window. Right clicking on either of the graphs allows the user to change their appearance as necessary. The dispersion can be displayed in any one of several units commonly used in ellipsometry. For more information on the display options see Section 6.1.4 6
Data: This graph shows the data and the simulations (fit) graphs for each data set that is checked in the Data Table In addition to the measured data and the fit, this tab also shows the current RMS Fit value and the Merit Function being used. Parameter to Plot: This pull down menu is used to select any of 32 different parameters to display. Unwrap Delta: When this box is checked the values of Delta that cross 180 degrees will be properly unwrapped (Not this only works for the Delta setting and not for Retardance Magnitude. For More information on the settings for data plots see Section 6.1.3 Poincare Sphere: The measured and simulated data can also be displayed on the Poincare Sphere 7
The View pull-down menu allows the user to select between the Retardance, Diattenuation or Polarizance states. QC Report: See Section?? for information on the QC report. 1.2 Basic Operation (AxoScan) The basic procedure analyzing a Sample Using AxoScan is as follows 1. Perform a Measurement: Perform a two-axis measurement, or a spectral measurement of the sample using an AxoScan polarimeter system. If the computer is attached to an AxoScan polarimeter, this can be done by pressing the button on the toolbar. 2. Load Axoscan data into LCD view: The software will read in all of the measurement parameters, and use this measurement geometry for all data fitting and simulation. If the measurement was performed directly in EllipsoView the data is automatically imported into the program. 3. Create a Layer Structure: See Section 3 for detailed instruction on creating a layer structure. 4. Optimize the fitting: If the a manual fit was used press the Optimize button to optimize the fit to the data 8
2 MEASURING AND LOADING AXOSCAN DATA 2.1 Performing a Measurement Using LCDView (AxoScan Only) If LCDView is operating on a computer that is directly connected to an AxoScan polarimeter (or connected to a remote polarimeter, through TCP/IP) a measurement can be performed by pushing the button on the toolbar. If the software does not detect either AxoView, or the AxoScan Engine running on the host computer, this button will be grayed out. If the user starts AxoView (or the AxoScan Engine) after starting LCD view, or the user wishes to connect to a remotely running version of AxoView, the user can select Tools>Connect to AxoScan from the pull-down menus on the front panel. This will attempt to connect to the AxoScan engine that is configured in the Hardware category Program Options (Section 6.2.5). If no AxoScan Engine is detected, the software will try to initialize the AxoScan Engine (provided that a path location for the AxoScan engine is provided) Once LCDView has been connected to the AxoScan Engine, the the Measure Button on the data import controls will be enabled. button on the toolbar and Pushing the button on the toolbar, or by pushing the Measure button on the data import controls brings up the following control, allowing the user to select between Measure Spectrum and a Tip-Tilt Measurement. NOTE: The Measurement Fixture control must correctly represent the measurement fixture that the data was taken on, since the coordinate systems are different for different measurement fixtures If the polarimeter is connected to an XY table, the Control XY Table button will be enabled allowing the user to launch the XY Stage control utility to correctly position the sample in the polarimeter 9
Once the stage is correctly positioned, the user has three measurement options: Measure Spectrum, Two-Axis measurement and Arbitrary Axis Measurement. 2.1.1 MEASURE SPECTRUM: If the user selects Measure Spectrum, the following dialog box appears, allowing the user to configure the spectral measurement. The control is designed to operate the same as the Measure Spectrum control in AxoView. Once the user presses the measure button, a signal will be sent to AxoView (or the AxoScan Engine) and the measurement will be performed. Once the measurement is complete, a file dialog box will appear allowing the user to save the data. Note that if the user presses cancel on the file dialog box the data is not saved but it will be imported into LCDView. Once the data is 10
imported, it cannot be saved from LCD view. It is recommended that the user always save the data when the measurement is complete. 2.1.2 ARBITRARY AXIS MEASUREMENT If the user selects the Arbitrary Axis measurement, the following control appears allowing the user to set up polar angle scans along arbitrary rotation angles. This control is similar to the Arbitrary Axis measurement control in AxoView, and can be used to set up a fixed number of equally spaces rotation axis. This control also allows the user to manually input arbitrary rotation angles. Max Tilt Angle ( ): This control determines the Maximum Tilt Angle for the scan Increment ( ): This control determines the step size to be taken for the scan Scans angles are generated as follows: Transmission Measurements: Scans are created using the maximum number of increments than can be generated between the -1*Max Tilt Angle and +1*Max Tilt Angle, always containing the value of 0. For Example if Max Tilt Angle is 45 and the Increment is 10 the steps will be (- 40,-30,-20,-10, 0, 10, 20, 30, 40 ). However if the increment is set to 5 then the steps will be (-45,-40,...,0,...,40,45 ). Reflection Measurements: Scans are created by starting at the minimum possible angle (configured in the AxoStep Engine) and continue until the Maximum Tilt Angle is exceeded. For Example if Max Tilt Angle is 45 and the Increment is 5 and the minimum tilt angle (from the AxoStep Engine is 31 ) the steps will be (31,36,41 ). Number of Rotation Angles: The software will create this number of evenly spaced tilt scans at rotation angles between the value in Initial Rotation Angle and 180 +Initial Rotation Angle. Initial Rotation Angle: This is the rotation angle used for the first tilt scans. Number of Measurements to Average: This control sets the number of measurements to average for each scan location. 11
Retardance Order: This control is grayed out because retardance order is not used in Ellipsoview calculations. Wavelength: is the wavelength of the measurement. Manually Select Rotation Angles: Check this box to manually enter a list of desired rotation angles into the table on the right of the control window. Once the user presses the Accept button, a signal will be sent to the AxoScan Engine and the measurement will be performed. Once the measurement is complete, a file dialog box will appear allowing the user to save the data. Note that if the user presses cancel on the file dialog box the data is not saved but it will be imported into EllipsoView. 2.1.3 ROTATION MEASUREMENT If the user selects the Rotation Measurement, the following dialog appears allowing the user to set a rotation angle scan along an arbitrary Tilt angle. Number to Average: This control sets the number of measurements to average for each scan location. Tilt Angle: This is the tilt angle to perform the measurement at. Number of Rotation Angles. This control sets the number of evenly spaced rotation angles starting at Initial Rotation Angle and ending at 360 + Initial Rotation Angle. Initial Rotation Angle: This is the rotation angle used for the first measurement. Measurement Wavelength: Is the wavelength of the measurement 12
Reflection: Check this box if you want to perform a Reflection Measurement. The box is only visible if the system is capable of making a reflection measurement. The box is grayed out if the system can ONLY perform reflection measurements (ex. VVRM). List of Rotation Angles: This indicator shows the list of rotation angles that will be measured for the Scan. Once the user presses the Scan button, a signal will be sent to the AxoScan Engine and the measurement will be performed. Once the measurement is complete, a file dialog box will appear allowing the user to save the data. Note that if the user presses cancel on the file dialog box the data is not saved but it will be imported into EllipsoView. 2.2 Loading A Dataset Measured Using AxoView The following instructions describe how to load a previously saved Mueller matrix dataset. 1. Select File>Open from the pull down menus, or click on the button on the toolbar. 2. A dialog box will appear prompting you to select the type of data set to load 3. The software can import any of the 4 basic AxoScan data types. NOTE: It is important that the Measurement Fixture control, correctly represent the measurement fixture that the data was taken on, since the coordinate systems are different for different measurement fixtures. See Chapter Error! Reference source not found. for more information on Coordinate systems. 4. After selecting the type of dataset to load, a standard Windows dialog box will appear prompting you to select the appropriate file. For Spectra Datasets (*.mmsp) the file contains only a single scan of data, which will automatically be imported into the analysis program. 13
The Two-Axis Tilt dataset files (*.tamm) and the Arbitrary Tip-Tilt Dataset files (*.opmm) contain multiple scans of data. When these dataset types are selected, the user will be prompted to select which scan to be imported into the analysis program. Once the data is imported it becomes visible in the analysis section of the front panel You can View the data on the Data tab to the right. 2.2.1 SPECTRAL DATASET FILES The original AxoScan spectral data format (*.mmsp) does not contain any information about tilt and Azimuth angle. Therefore, this information must be manually entered when loading spectral data that is in the original AxoScan data format. 14
2.2.2 TWO-AXIS TILT DATASET FILES Although LCDView can still import two-axis(*.tamm) data files it is not recommended. 1. When the Two-Axis Tilt Dataset file is selected the program opens the following window, allowing the user to select between the scan taken about the slow axis or the scan taken about the fast axis. 2. Select either Tilt About Slow-Axis, Tilt About Fast Axis, or Both 3. Click Import this Scan to import the data into the Analysis program, or click Cancel. 2.2.3 ARBITRARY TIP-TILT DATASETS 1. When the Arbitrary Tip-Tilt dataset is selected the program opens the following window, allowing the user to select any of the scans contained in the dataset. The user selects one of the rotation angles. The diagram shows a polar plot of the points taken in the scan. 15
2. Select the rotation angle for the imported dataset. 3. To import multiple scans the user has 3 options. a. Press the Select All button to select all the scans in the data file. b. Holding down the <Control> Key allows the user to click on individual scans in the list. c. Holding down the <Shift> Key allows the user to select a range of scans 4. Click Import this Scan to import the data into the Analysis program, or click CANCEL. 2.3 Removing or Adjusting a Dataset The user can clear all currently loaded data by clicking on the New Project toolbar. button on the To remove individual scans, right click on the data table. and choose Remove Unchecked Data from the pull-down menu. To adjust parameter settings, you can right click on a data scan and select Adjust Parameters. This will bring up the following dialog box, 16
allowing the user to change the settings for a given data scan. The following controls are available on this dialog control. Tilt Angle: This specifies the PSG Tilt Angle that the measurement was performed at. The dialog will automatically import the tilt value recorded in the data file. It can be changed here if necessary. Azimuth Angle (actually rotation angle): This specifies the PSG Rotation Angle that the measurement was performed at. The dialog will automatically import the rotation value recorded in the data file. It can be changed here if necessary. Beam configuration: This control can be used to change whether the data was taken in Reflection or Transmission Mode. Some older Axometrics software applications do no record the beam configuration when saving measurement data, and Reflected data might be imported as Transmitted data. Use this control to change that attribute if necessary. Merit Function: Since EllipsoView allows the user to select different merit functions for each scan, the user can change the merit function for this particular scan using this control. For more information about merit functions, see Section 5.1. 17
3 CONFIGURING LAYERS The layer structure is shown on the front panel. An empty layer structure consists of an incident medium and a substrate medium. The default for these materials is air (n=1.0, k=0.0). The user can change the optical properties of the substrate, just like any other layer. However the incident medium refractive index must be changed in the program options. To insert a new layer right click on any layer in the structure And choose New Layer (Above... or Below...). You can also press the Add Layer button on the toolbar. This will bring up a dialog asking if you want to insert the layer Above or Below the current layer. The Layer Editor is automatically opened when adding a new layer 18
The Geometry tab (on the left), shows the thickness and orientation of the current layers. It also shows the current material of that layer. The Parameters tab (on the right) shows the properties of the material for the current layer. 19
The Materials Pull down menu shows the current materials in the project that can be selected from for this layer. By default the only material is air. Use the buttons on the right to add and remove materials from the project. : Create New Material. This brings up the following dialog box Material Name: You must give the material a name. 20
Material Type: Choose from Isotropic, Uniaxial or Biaxial. Dispersion Model: This is the model that controls how the refractive index properties change with wavelength. There are currently 6 dispersion models to choose from: Constant: define a constant (same for all wavelengths) n and k value for the material. Cauchy: a refractive index model that assumes k=0 (used for transparent materials) Sellmeier: a more complex refractive index model that also assumes k=0 (used for transparent materials) Drude: a dispersion model that assumes a perfect conductor (used for metals) Lorentz: a general purpose dispersion model for absorbing and conducting materials (used for color filters, ITO and IZO). Table: use tabulated data for the refractive index. after clicking OK, the material appears in the Materials list, and the parameters for that material are shown in the Parameters table: To determine the definition of any of the parameters, you can right click 21
and select What s This from the pull down menu. This will bring up a window showing you information about the selected parameter Check the boxes for any parameters you wish to optimize. Import material from library: Press this button to import a material from the current library of available materials. After pressing OK, the material is loaded into the list of Materials. 22
Press OK to add the layer to the project (make sure you set the thickness parameter) Continue adding layers until you have configure the structure you with to measure. 23
4 COORDINATE SYSTEMS 4.1 VRTF-APM coordinate system When measurements are made on the VRTF or the Axometrics Panel Mapper (APM) the x-axis is rotated clockwise 90 from the OPMF coordinate system. This is shown in Figure 2. Looking down β is in this plane Inhomogeneous tilts are in this plane too Sample +φ X Nx Azimuth Y Doors Figure 1: APM coordinate system 24
5 DATA FITTING DETAILS The polarization metric that is used to determine the quality of the fitted data is the RMS difference in the retardance Mueller matrix. This metric compares the measured retardance Mueller matrix from the polarimeter with a Mueller matrix calculated using the parameter values currently in the LCD parameter control.. The RMS % Error on the front panel toolbar is a figure of merit that describes how well the data simulated using the optimized parameters fits the measured data. For a perfect fit, the RMS % Error would equal zero. We have found that an RMS % error less than 1.0 function generally represents a good fit. When you press the Optimize button the computer begins using the fitting algorithm to determine the optimum values of the LCD parameters selected by the user. The user selects which parameters to optimize by checking the optimize selector for each parameter in the LCD Properties Control. While the program is optimizing the appearance of the optimization window changes as shown. You can press the Stop button at any time and the program will stop the data fitting routine. Note that prematurely stopping the optimization can result in a poor fit. Pressing the Abort button will cancel the optimization. 25
5.1 Merit Functions The merit function is the value that is compared between the measured data and the simulated data to determine the quality of the fit. In general the merit function is a non linear vector function, used to compare multiple values for each data point. EllipsoView has 5 options for merit functions that the user can choose from: 1. Normalized Mueller matrix: Compares the measured Normalized Mueller matrix from the polarimeter with a Mueller matrix calculated using the parameter values that describe the optical properties of the layer structure. Note, the normalized Mueller matrix removes the absolute reflectivity or transmission values. This is usually not a problem when measuring in reflection or measuring samples whose transmission is approximately constant with wavelength (i.e. clear with no color) This is the best general purpose merit function. Use this merit function if you are unsure about which one to select. 2. Psi and Delta: These are the merit functions that are used in standard ellipsometry systems. They are defined as follows: where r p and r s are the reflectivity in the p and s planes of the measurement (corresponding to the x and y coordinates of the polarimeter head). In this case the value of Delta (δ) is exactly the same as the retardance magnitude, and the value of Psi (ψ) is closely related to the diattenuation magnitude. It is important to note that Psi and Delta contain no information about the ellipticity or orientation of either the retardance or the diattenuation. In most cases Psi and Delta should only be used on isotropic (not uniaxial or biaxial) samples. 3. Input States: Compares the measured input (diattenuation) polarization states with the values calculated using the parameter values in the layer structure. Use this setting to ignore any layers below the polarizer film. 26
Merit Function 4. Output States: Compares the measured output (polarizance) polarization states with the values calculated using the parameter values in the layer structure. Use this setting to ignore any layers above the polarizer film. 5. Full Mueller Matrix: This is the same as the Normalized Mueller Matrix, except that absolute transmittance and/or reflectance is included in the calculation. Usually this is only necessary when measuring samples in transmission whose absorption changes with wavelength (Mostly Color Filters, and maybe ITO/IZO) The RMS % Error in the Optimize window is a figure of merit that describes how well the data simulated using the optimized parameters fits the measured data. For a perfect fit, the RMS % Error would equal zero. 5.2 Special notes regarding non-linear fitting When performing a non-linear optimization, the computer algorithm assigns a merit function to determine how close a simulated result is to a measured result. The program starts with an initial estimate of the variables then adjusts them in an effort to reduce the merit function. To do this the program will try different values, and see which values will reduce the merit function. Values that reduce the merit function are used as guesses for the next step in the algorithm. By iteratively repeating this process the program will find a solution. A common drawback to using this approach in non-linear systems is illustrated below. Plotted is a hypothetical merit function plotted as a function of a parameter value. Notice that if the initial guess is close to one of the local minima, the solution found by the program will then to converge to one of those (incorrect) values. It is therefore necessary that the first initial guess be close to the True Solution for the algorithm to find the correct parameter value. Local Minima True solution Parameter value Where this situation occurs in the Multilayer software is when selecting the Azimuth angle parameters for each layer. Often times if the Azimuth parameter in a particular layer is off by 90 a local minima sometimes results, causing the software to give incorrect results. 5.3 Basic Theory and Partial Coherence 27
5.3.1 THIN FILM SIMULATIONS When performing ellipsometric measurements it is important to understand the calculations being performed to fit the measured data. The simulations are performed by calculating the electric field present in each layer of the sample. These electric fields inside the layers of the structures are then combined and boundary conditions are applied to determine the reflected and transmitted electric fields that correspond to the measured data. In a general birefringent material, each layer in the structure is defined by a dielectric tensor of the form. where ε i =(n i +ik i ) 2 and R is a rotation matrix about the axis α,β, and γ. The most general solution to the electric field in a layer defined by the dielectric tensor shown above is known as the Berreman method (ref. 1). This solution method is a fully rigorous solution to Maxwell s equations and results in 4 different electric field waves inside the material, two propagating downward (e 1 and e 3 ) and two propagating upward (e 2 and e 4 ). This is shown below. ( 1 ) e 2 e e 4 3 e 1 Figure 2 If any layer in the structure being simulated is birefringent then a full Berreman calculation is necessary. However in many cases, simplifying approximation can be made. Fresnel calculation: In cases where the material is isotopric εx=εy=εz, a complete Berreman calculation is not require, and the material only contains only two electric field waves (a single upward and single downward propagating wave) in the material. In this case the much simpler Fresnel based, or Admittance matrix based calculations can be used (ref. 2). Although these calculations are simpler, for the case of isotropic material, they are still fully rigorous solutions to Maxwell s equations. Extended Jones matrix: The Berreman and Fresnel calculations described above assume that the light source that is used to measure the sample is perfectly monochromatic (like a laser) and that all of the electric fields (4 wave or 2 wave) coherently interfere with each other. However, if the optical source being used has a finite bandwidth, then the forward and backward propagating electric fields will not perfectly interfere in any layers that are significantly thicker than the coherence length in that material (defined below in Eq. (2) ). In the case of isotropic materials, the back reflections from multiple surfaces are simply added together, and in birefringent materials, the Extended Jones Matrix method (ref 3.), which only calculates the forward propagating electric fields in the material, is used to calculated the electric fields independently in the forward and reverse directions to determine the reflectivity of the sample. 5.3.2 BASICS OF COHERENCE When using a non-monochromatic source to perform ellipsometric measurements of multilayered samples that contain thick layers, transparent substrates or cover glass, care must be taken to insure that the coherence properties of the simulation are modeled properly. The key value when evaluating the coherence properties of an optical measurement is the coherence length, defined (for a Gaussian spectrum) as (refs. 4 and 5): 28
This distance represents the length at which interference fringing decreases by 50%. For example a wavelength(λ) of 450nm with a bandwidth(δλ) of 5nm, and in a material with refractive index(n) 1.5 has a coherence length of approximately 12 μm. Generally all multilayer structures will fall into one of the following four categories. 1. For multilayer structures with layer thicknesses(t) that are significantly thinner than the coherence length (t<<l c ) a Fresnel thin film model (or Berreman model) is used to calculate the reflectance and transmittance functions. 2. For multilayer structure where the layer thicknesses are significantly greater than the coherence length (t>>l c ) an Extended Jones Matrix method should be used to calculate the reflectance and transmittance functions. 3. For multilayer structure where the layer thicknesses are approximately equal to the coherence length (say: 0.5l c < t < 3l c ) a partially coherent Fresnel thin film model (or Berreman model) is used to calculate the reflectance and transmittance functions. This calculation method is described below. 4. For multilayer structures that are a combination of the previous three categories, the calculation is divided into regions to perform the calculations. The results from these regions are then combined appropriately to calculate the final reflectance and transmittance functions. 5.3.3 PARTIAL COHERENCE SIMULATION A common method of simulating partial coherence is to perform multiple coherent calculations over a narrow bandwidth, each with a slightly different wavelength. That is, ( 2 ) where I i is the normalized light intensity at wavelength λ i. ( 3) 5.3.4 REGION DEFINITION Region definition is based on layer thickness and coherence length and is separated into two stages. Stage 1: Separate Incoherent regions from partially and fully coherent regions. Separate regions are defined based on layer thickness. All layers above and below a pre-defined thickness (say 3 times the coherence length) are treated incoherently. This is illustrated below: 29
region 1 t»l c region 2 t»l c t «l c region 3 Figure 3 Stage 2: Determine if regions can be simulated using fully coherent or partially coherent light source. If all of the layer thicknesses are significantly less than the coherence length within each layer, then the regions can be simulated coherently. For example, the layers shown above in region 3 are all less than the coherence length so layer structure in this region is treated coherently (using a single monochromatic wavelength) t «l c region 3 Figure 4 However if any one of the layers is within a predefined range where it is too thick to be treated coherently but is not thick enough to be treated incoherently, then all the layers in this region must be treated using the partial coherence method described above. 30
t «l c 0.5 l < t < 5l c c region 2 Figure 5 5.3.5 COMPLETE CALCULATION The complete calculation is performed in four steps. Step 1: Mueller Matrices for the reflectance and transmittance are calculated for each region, using the thick (incoherent) regions as incident and substrate material as necessary. Step 2: If any incoherent layers are birefringent, propagation Mueller matrices are calculated for each incoherent layer using the Extended Jones Matrix Method. Otherwise incoherent layer propagation is treated as just a multiplicative absorption. 31
Step 3: Mueller Matrices are combined to represent the complete optical path for the multiple incoherent reflections within the sample as illustrated in the example below. M 1 =M 1r M= 2 M1t-M 2r M1t+ M 1r M= 2 M M 2t- M 3r M 2t+ M 1t- 1t+ region 1 M 1t M 2r region 2 M 2t M 3r region 3 M 3t M t =M 3t M 2t M 1t Figure 6 Step 4: All of the reflection Mueller matrices, calculated in Step 3, are added together to obtain the total reflected Mueller matrix (including depolarization). References 1. D.W. Berreman: J. Opt. Soc. Am., v62 (1972) 502 2. M. Born and E. Wolfe, Principles of Optics (6 th Ed), Cambridge Univ. Press, (1980) (sec. 1.6). 3. C.-J. Chen, et. al.: J. Opt. Soc. Am. A., v14, n11 (1997) 3125 4. G. K. Ackermann, Holography: A Practical Approach. Wiley-VCH. (2007) 5. http://en.wikipedia.org/wiki/coherence_length 32
6 PROGRAM OPTIONS To change the basic program options, select SETUP>PROGRAM OPTIONS from the pull-down menus on the front panel. This brings up the following dialog box, allowing the user to edit the software configuration to meet their needs 6.1 Display Options The Options under this menu heading affect the general apperance of the software (Plot colors, Line thickness etc.). All of the settings under this menu heading are straightforward and only affect the appearance of the software, and not the functionality. 6.1.1 TOOLBAR The toolbar section is used to customize which buttons are visible on the toolbar in the main window. Simply check or uncheck which buttons you would like to appear on the toolbar. 6.1.2 POINCARE SPHERE This section allows the user to configure the appearance of the Poincare Sphere on the Front Panel. 33
The controls on the left, affect the appearance of the sphere, and the controls on the right affect the appearance of the Data points, and the Fitted values displayed on the sphere. 6.1.3 DATA PLOT The Data Plot window controls the appearance of the data plots. Plot Style, Point Style and Line width control the appearance of the Data Display and the Fit display (Note, the Fit will always be displayed as a solid line). Click on the picture in any of these controls to chance their appearance on the front panel. 34
Data Color Array and Fit Color Array: This control is used to set the color of the data plots in the order that they are loaded on the front panel. Any of the colors can be changed by clicking on the color inside the array. Match Fit Color to Data Color: When this box is checked, the Fit line, and the Data points/line will be the same color. In this case, the Fit Color Array is ignored and grayed out. Match Simulation Plot Points to Data Plot Points. When this box is checked, the fit line will be constructed using only the points that directly correspond to the data points measured. Edit Display Ranges: Pressing this button brings up the following window Match Measured Data is the same as Match Simulation Plot Points to Data Plot Points above. Use Range In Data, uses the Min and Max values in the data, and the increment set in this window. Double Click any entry to change its value. 6.1.4 DISPERSION PLOT Use this section to change the appearance of data on the dispersion plots on the front panel. 35
Dispersion View: The use can select to display the optical constants for the material as either Refractive Index (N and K) or as Permitivity (ϵ R and ϵ I ). Show N and K together: Displays the real and imaginary dispersion on a single graph. Refractive Index Plot Colors: The User can select the colors used for Nx, Ny and Nz (or equivalent real permativities) Absorption Plot Colors: The User can select the colors used for Kx, Ky and Kz (or equivalent imaginary permativities) Dispersion X Axis: The User can select the units for the X axis on the dispersion plot. 6.2 Analysis The Options under this Menu heading, affect the program operation, and must be configured properly to obtain accurate measurement results. 6.2.1 SETTINGS 36
Retardance Orientation Center for Plots: This value defines the middle value of retardance used when plotting Retardance orientation. The maximum retardance orientation value displayed in the on-screen plots will be this value plus 90, and the minimum retardance orientation shown in the plots will be this value minus 90. Force Mueller Matrix Elements to Physical Values: When this box is checked, the algorithm given in Section 8.4 of [Goldstein, D.H., Polarized Light, 3 rd ed., CRC Press, Boca Raton, FL, 2011], that removes non-physical noise from the Mueller Matrix elements. Import Old Data as Reflected: When this box is checked, data that is saved using old AxoScan data formats (before Ver 5.0.0) will be imported as reflected if no information regarding system configuration is included in the data file. Incident Refractive Index: This is the refractive index of the incident material. Biaxial Model: This box is used to select how the calculation is performed for thin film birefringent (uniaxial and biaxial) material. Berreman uses a 4 wave model that is fully rigorous (consistent with Maxwell s equations). Extended Jones uses Two Wave Model (ignoring back reflection). In most ellipsometry calculations, the user should use Berreman (4 wave). When measuring samples with moderately thick birefringent layers 2μm to 35μm in transmission, (things like liquid crystal or discotic layers) using the Extended Jones Matrix will often give better fitted data without needing to use partial coherence (which takes a long time to calculate). Use FMM for Orientation Fitting. When this box is checked the software will use the Fast Mueller Matrix (FMM) method to calculate the orientation of the birefringent layer chosen in the pull-down menu labeled Layer. When this box is checked, the following options are available to configure the FMM method. 37
See the user s manual for the FMM for an explanation of the parameters in this control. 6.2.2 DATA FITTING: The following options are available from the Data Fitting menu item. Max Iterations is the maximum number of iterations that the Optimization algorithm will use to fit the data Tolerance is a parameter used by the algorithm to determine when to stop the optimization procedure. The lower the number, the more stable the solution. The computer stops optimizing when Tolerance reaches a value less than 0.00005. When this happens the algorithm thinks it has found a solution. Although the algorithm has found a solution, it might not be the correct solution. A common pitfall in non-linear optimization methods like the one being used here is that they are susceptible to local minima. See section 6.2 for a description of local-minima. The ultimate determination of how good the fit is rests with the merit function value, not the Tolerance Value. 38
Independent Merit Functions: When this box is checked it allows the user to use an different merit function for each data scan that is imported. Merit Function: This is the value that is used to compare the data and the simulation to determine the quality of the fit see Section 5.1 for more information Mapping Step Size Reduction: When performing Ellipsometry mapping using AxoStep, this control specifies the step size in pixels for the mapping operation. For example, Step Size Reduction=1 would calculate every pixel, whereas Step Size Reduction=3 would only calculate every third pixel. 6.2.3 PARTIAL COHERENCE The settings for Partial Coherence are shown below. Note: The Theoretical description of partial coherence is given in Section 5.3 Spectral Line Bandwidth: This is the line width of the light coming from the light source (default = 10 nm) Simulate Partial Coherence: Check this box if you want EllipsoView to simulate the effects of partial coherence. Note: This will slow down calculation times by a significant amount. A good rule of thumb to determine whether partial coherence is needed is if any of the layers in the sample are thicker than half the Coherence Length in Air to about 5 times the Coherence Length in Air. Spectral Line Shape: This specifies the shape of the spectral line to be used in the simulation. There are three options 39
Gaussian: This is the standard bell curve and is the most common Lorentzian: This is a bell shaped curve similar to the Gaussian Curve only the spectral amplitudes to not drop off as fast. Rectangle: This function is a simple rectangle function. These three functions are shown in the graph below: 1.2 1 0.8 0.6 0.4 0.2 Gausian Lorentzian Rectangle 0 520 530 540 550 560 570 580 wavelength (nm) Number of Discrete Wavelength Points: This is the number of discrete points used to generate the spectral curve from Equation (3) in Section 5.3.3. Random Dithering Scale: Set this value to randomly dither the discrete wavelengths when simulating partial coherence. This is usually only necessary when using the rectangle function. Automatically Calculate Coherence Length: When this is selected, the software will the following rules to determine how to perform the calculation: Less than 0.5*Coherence Length: Use fully coherent simulation. Between 0.5 and 5 Times Coherence Length: Use fully coherent or partially coherent simulation Greater than 5*Coherence Length: Use Incoherent Simulation Manually Configure Coherence Length: Use Minimum Layer Thickness and Maximum Layer Thickness to determine when to use Fully coherent, partially coherent and incoherent simulations. 6.2.4 MATERIALS LIBRARY The Materials Library is used to view the available libraries and the contents of each library. Its settings are shown below. 40
Library Directory: This control sets the directory where libraries are stored. 6.2.5 HARDWARE SETTINGS For LCDView to properly interact with Axometrics Hardware (AxoScan, and AxoStep), the Hardware Settings must be properly configured. 41
System Type: This sets the measurement system type used for acquiring the LCD Data. AxoScan IP address: This is the Ethernet IP address of the AxoScan Engine (If LCDView and The AxoScan Engine are on the same PC, use the default value of 127.0.0.1) AxoScan Communication Port: This is the Ethernet Port that the AxoScan engine uses for communcation (Default Value = 5000) AxoScan Status Port: This is the Ethernet Port that the AxoScan Engine uses for status checking (Default Value = 5002) Test Connection: Press this button to determine if the AxoScan Ethernet settings are configured Properly. Fixture Type: Use this to set the AxoScan Fixture type. This sets the coordinate system used for AxoScan Data Calculations (this can be changed when importing data) 42