Lecture 6 Dielectric Waveguides and Optical Fibers Slab Waveguide, Modes, V-Number Modal, Material, and Waveguide Dispersions Step-Index Fiber, Multimode and Single Mode Fibers Numerical Aperture, Coupling Loss Bit-Rate, dispersion and optical bandwidth Graded-index fibers Absorption and Scattering Fiber Manufacture
y Step-Index Fiber y Cladding Core φ r z Fiber axis n 2 n 1 n The step index optical fiber. The central region, the core, has greater refractive index than the outer region, the cladding. The fiber has cylindrical symmetry. We use the coordinates r, φ, z to represent any point in the fiber. Cladding is normally much thicker than shown. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Rays and Modes Along the fiber 1 Meridional ray Fiber axis 3 1, 3 (a) A meridiona ray always crosses the fibe axis. 2 2 1 2 Fiber axis 3 Skew ray 4 5 5 4 1 2 3 (b) A skew ray does not have to cross the fiber axis. It zigzags around the fiber axis. Ray path along the fiber Ray path projected on to a plane normal to fiber axis Illustration of the difference between a meridional ray and a skew ray. Numbers represent reflections of the ray. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fiber Modes (a) The electric field of the fundamental mode (b) The intensity in the fundamental mode LP 01 (c) The intensity in LP 11 (d) The intensity in LP 21 Core Cladding E E 01 r The electric field distribution of the fundamental mod in the transverse plane to the fiber axis z. The light intensity is greatest at the center of the fiber. Intensity patterns in LP 01, LP 11 and LP 21 modes. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Normalized Parameters 1 b 0.8 0.6 0.4 LP 01 LP 11 LP 21 LP 02 0.2 0 0 1 2 3 4 5 6 2.405 V Normalized propagation constant b vs. V-number for a step index fiber for various LP modes. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Numerical Aperture α < α m ax A B α > α m ax n 2 n 0 n 1 Lost B θ < θ c θ > θ c Fiber axis Cladding Propagates A Core Maximum acceptance angle α max is that which just gives total internal reflection at the core-cladding interface, i.e. when α = α max then θ = θ c. Rays with α > α max (e.g. ray B) become refracted and penetrate the cladding and ar eventually lost. 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
EE 230: Optical Fiber Communication Fibers from the view of Geometrical Optics From the movie Warriors of the Net
Principal Types of Optical Fiber Types of Fibers Single mode/multi-mode Step Index/Graded Index Dispersion Shifted/Non-dispersion shifted Silica/fluoride/Other materials Major Performance Concerns for Fibers Wavelength range Maximum Propagation Distance Maximum bitrate Crosstalk Understanding Fiber Optics-Hecht
Total Internal Reflection Snells Law : n sin θ = n sin θ 1 1 2 2 Re flection Condition θ = θ 1 3 When n > n and as θ increases eventually θ 1 2 1 2 goes to 90 deg rees and n n θ = θ = 2 1sin c n2 or sin c n 1 θc is called the Critical angle For θ > θ there is no pro pagating refracted ray 1 c
Reflection as a function of angle The reflectivities of waves polarized parallel and perpendicular to the plane of incidence as given by the Fresnel equations This additional Phase Shift is not accounted for in geometrical wave approach Fiber Optics Communication Technology-Mynbaev & Scheiner
To help protect your privacy, PowerPoint prevented this external picture from being automatically downloaded. To download and display this picture, click Options in the Message Bar, and then click Enable external content. Basic Step index Fiber Structure Fiber Optics Communication Technology-Mynbaev & Scheiner
Ray Trajectories in Step Index fiber Meridional Rays Skew Rays
Rays and Their E-field Distribution
Coupling Light into an Optical Fiber Fiber Optics Communication Technology-Mynbaev & Scheiner
Acceptance Angle The acceptance angle (θ i ) is the largest incident angle ray that can be coupled into a guided ray within the fiber The Numerical Aperature (NA) is the sin(θ i ) this is defined analagously to that for a lens NA = sin θ = n 1 n n 1 i 2 = n 2 2 1 n2 n1 2 f # f f º = D FullAccep tanceangle = 1 2 NA Optics-Hecht & Zajac
Geometrical View of Modes Ray approximation valid in the limit that λ goes to zero Geometrical Optics does not predict the existance of discrete modes Maxwells Equations and dielectric boundary conditions give rise to the requirement that the fields and phase reproduce themselves each cycle Fiber Optics Communication Technology-Mynbaev & Scheiner
Intermodal Dispersion T = L n n 2 1 2 c BL (bit/s m) < n n 2 2 1 c Fiber Optics Communication Technology-Mynbaev & Scheiner
Graded Index Fiber α ρ n( ρ) = n1 1 for ρ<a a n( ρ) = n 1 =n for ρ>a [ ] 1 2 for α = 2 a "parabolic profile" Fiber Optic Communication Systems-Agarwal 1 ρ NA=n ( ) 2 1 2 1 which varies with ρ a 2 t = GI Ln1 8c 2 Fiber Optic Communications-Palais
Bandwidth for Various Fiber Types 1 Bit Rate BR< 4 Dt BR SI = 1 = c 4D t 4LnD SI BR GI = 2 c n LD 1 2 BR BR GI SI 2c n LD 2 1 8 = c = D 4LnD No intermodal time shift for single Mode Fiber Fiber Optics Communication Technology-Mynbaev & Scheiner
Derivation of Ray Propagation Equation Graded Index Slab Uniform in X and Z Fundamentals of Photonics - Saleh and Teich
EE 230: Optical Fiber Communication Waveguide/Fiber Modes From the movie Warriors of the Net
Optical Waveguide mode Optical Waveguide mode patterns seen in the end faces of small diameter fibers patterns Optics-Hecht & Zajac Photo by Narinder Kapany
Planar Mirror Waveguide The planar mirror waveguide can be solved by starting with Maxwells Equations and the boundary condition that the parallel component of the E field vanish at the mirror or by considering that plane waves already satisfy Maxwell s equations and they can be combined at an angle so that the resulting wave duplicates itself Fundamentals of Photonics - Saleh and Teich
Mode Components Number and Fields Fundamentals of Photonics - Saleh and Teich
Mode Velocity and Polarization Degeneracy Group Velocity derived by considering the mode from the view of rays and geometrical optics TE and TM mode polarizations Fundamentals of Photonics - Saleh and Teich
Multimode Propagation In general many modes are excited in the guide resulting in complicated field and intensity patterns that evolve in a complex way as the light propagates down the guide Fundamentals of Photonics - Saleh and Teich
Planar Dielectric guide Geometry of Planar Dielectric Guide The β m all lie between that expected for a plane wave in the core and for a plane wave in the cladding Characteristic Equation and Self-Consistency Condition Number of modes vs frequency Propagation Constants For a sufficiently low frequency only 1 mode can propagate Fundamentals of Photonics - Saleh and Teich
Planar Dielectric Guide Field components have transverse variation across the guide, with more nodes for higher order modes. The changed boundry conditions for the dielectric interface result in some evanescent penetration into the cladding The ray model can be used for dielectric guides if the additional phase shift due to the evanecent wave is accounted for. Fundamentals of Photonics - Saleh and Teich
Two Dimensional Rectangular Planar Guide In two dimensions the transverse field depends on both k x and k y and the number of modes goes as the square of d/λ The number of modes is limited by the maximum angle that can propagate θ c Fundamentals of Photonics - Saleh and Teich
Step Index Cylindrical Guide 2 2 æ2 p ö V = k0a( n1 - n2 )» ç an1 2D çè l ø Fundamentals of Photonics - Saleh and Teich
Number of Modes Propagation constant of the lowest mode vs. V number a æ2π ö V=k a(n -n )=2 ç an 2 2 2 0 1 2 p NA» 1 λ ç 0 λ çè 0 ø Graphical Construction to estimate the total number of Modes Fundamentals of Photonics - Saleh and Teich
ω-β Mode Diagram Straight lines of dω/dβ correspond to the group velocity of the different modes The group velocities of the guided modes all lie between the phase velocities for plane waves in the core or cladding c/n 1 and c/n 2
High Order Fiber modes Fiber Optics Communication Technology-Mynbaev & Scheiner
High Order Fiber Modes 2 Fiber Optics Communication Technology-Mynbaev & Scheiner
Power Confinement vs V- Number This shows the fraction of the power that is propagating in the cladding vs the V number for different modes. V, for constant wavelength, and material indices of refraction is proportional to the core diameter a As the core diameter is dereased more and more of each mode propagates in the cladding. Eventually it all propagates in the cladding and the mode is no longer guided Note: misleading ordinate lable
Macrobending Loss One thing that the geometrical ray view point cannot calculate is the amount of bending loss encountered by low order modes. Loss goes approximately exponentially with decreasing radius untill a discontinuity is reached.when the fiber breaks!
Comparison of the number of modes 1-d Mirror Guide M = 2 d l 0 The V parameter characterizes the number of wavelengths that can fit across the core guiding region in a fiber. 1-d Dielectric Guide M» 2 d NA l 0 For the mirror guide the number of modes is just the number of ½ wavelengths that can fit. 2-d Mirror Guide 2-d Dielectric Guide 0 2 M p æ2 d ö» 4 l ç çè ø p æ2 d ö M» NA 4 ç l çè ø 0 2 For dielectric guides it is the number that can fit but now limited by the angular cutoff characterized by the NA of the guide 4 æ 16 d ö» = ç ø 2 M V NA 2 p çè l 0 2-d Cylindrical Dielectric Guide 0 2 a V=2p NA λ
Confinement Factor