Scool of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 007 Homework 11 Due on Nov. 9, 007 by 5:00 PM Reading Assignments: i) Review te lecture notes. ii) Review sections 7.1, 7., 7.4, and te entire capter of te paperback book Electromagnetic Waves. Problem 11.1: (Dielectric optical waveguides) Consider te following integrated dielectric optical waveguide on a Silicon cip: Silicon Dioide (n = 1.5) Silicon (n 1 = 3.5) Silicon Dioide (n = 1.5) Te dielectric waveguide is designed to carry ligt wose free-space wavelengt is 1.55 µm (you can find te corresponding frequency ω from tis information). A side note: 1.55 µm is te also wavelengt wic is used for fiber-optic communications. Tis wavelengt is not in te visible part of te spectrum and so cannot be seen by te eye. Propagation vectors: a) Assuming tat = 1.0 µm, find te values of k for te first tree TE modes of te waveguide for te frequency ω indicated above. Hint: You will need to solve a transcendental equation grapically to find te value of k for eac TE mode, and ten using te value of k find te corresponding value of k. Tose of you wo are tinking of using routines in Matlab or Matamatica to solve transcendental equations need to be careful of te fact tat many suc routines may not give all te solutions to an equation if multiple solutions eist. Effective indices: b) Usually te value k is not specified directly (because it can be a long cumbersome number). Instead te effective inde n eff defined by te relation, 1
k neff = ω c is used to specify k. Te effective inde is te inde of a ypotetical medium in wic a plane wave of frequency ω would ave te same wavevector k as tat of te waveguide mode. Using your results from part(a), find te effective indices of te first tree TE modes. You need to give numerical values as answers. Single-mode waveguides: c) In many cases, one wants to ave optical waveguides tat can propagate only one mode for a given frequency ω. Te way to acieve tis in practice is to srink te dimensions of te waveguide core until only te lowest mode can propagate at te given frequency ω and te cut-off frequencies of all te iger modes are larger tan ω (note tat te cut-off frequencies of modes increase as te waveguide core dimensions are reduced). Assuming te frequency indicated above for ligt of free-space wavelengt 1.55 µm, find ow small te eigt of te waveguide ougt to be suc tat only te lowest TE mode (i.e. TE 1 mode) can propagate in te waveguide and all te iger TE modes cannot propagate. Give a numerical answer. Problem 11. (Test your concepts) Consider a dielectric slab waveguide sown below: n = 1.0 θ n 1 = 4.0 n = 1.0 In te figure above, te guided wave angle of incidence θ is also sown. a) Consider guided TE mode for two different frequencies ω 1 and ω were 1 ω ω >. Will te guided θ at wave angle of incidence θ 1 at frequency ω 1 be larger or smaller tan te angle of incidence frequency ω? Eplain your answer. No points will be awarded for wrong or inadequate eplanation. b) Consider guided TE 1 mode. Wat is te group velocity v g = dω dk of te guided TE 1 mode in te limit wen te frequency ω approaces 0. Wat is te group velocity v g = dω dk of te guided TE 1 mode in te limit wen te frequency ω approaces. Eplain your answer. No points will be awarded for wrong or inadequate eplanation. Your answer sould be in terms of te material and waveguide parameters. c) Consider guided TE and TM modes for te same frequency ω tat is larger tan te cut-off frequency of bot te modes. Will te angle of incidence θ TE of te TE mode be larger or smaller tan
te angle of incidence θ TM of te TM mode? Eplain your answer. No points will be awarded for wrong or inadequate eplanation. Problem 11.3: (Metal-dielectric ybrid waveguides) Consider te following integrated metal-dielectric ybrid waveguide on a Silicon cip: / Silicon (n 1 = 3.5) Silicon Dioide (n = 1.5) Te propagating mode is guided by reflection from te perfect-metal layer at te top, and by total internal reflection at te bottom interface. Te possible epressions for te E-field of te TE-modes are given below: r E r E j k (, ) = yˆ E sin( k ) e 0 α + j k (, ) = yˆ E e e 1 o a) Using te all te boundary conditions at your disposal - at all te interfaces - and te field epressions given above, find a transcendental equation relating k to te frequency ω. b) It turns out tat te transcendental equation for te TE-modes of te above waveguide can also be found in a way tat is smarter tan wat you did in part (a). Consider te waveguide structure of problem 11.1. If you reason carefully you ougt to reac a conclusion tat all te TE-modes of te above metaldielectric ybrid waveguide are in fact a subset of te TE-modes of te waveguide in problem 11.1. Can you specify wic TE-modes of te waveguide in problem 11.1 correspond to te TE-modes of te above metal-dielectric ybrid waveguide? Using tis metod of reasoning, find te transcendental equation tat relates k to te frequency ω for te above metal-dielectric ybrid waveguide and confirm tat it is te same as tat found in part (a) above. Problem 11.4: (Asymmetric dielectric optical waveguides) Consider te following integrated optical waveguide on a Silicon substrate: 3
Silicon Dioide W Silicon Silicon Substrate As long as te widt W of te waveguide is muc larger tan its eigt, te waveguide can be approimated as a slab waveguide sown below: Air (n 3 = 1.0) Silicon (n 1 = 3.5) Silicon Dioide (n = 1.5) Te essential feature about tis dielectric waveguide is tat it is asymmetric. Te cladding layers at top and bottom ave different indices. Ligt is guided in te Silicon core layer. Te following two problems will teac you tat a lot can be learned about dielectric waveguides witout calculating all te details. a) Find te epression for te cut-off frequency ω m of te TE m mode? Hint: Do not attempt to find te modes or do some elaborate mat to get te answer. If you ave understood te concepts beind waveguiding in dielectric waveguides you sould be able to get te answer in one or two lines witout even knowing te epressions for te modes or te transcendental equation tat describes te relationsip between k and ω for te asymmetric waveguide. b) Sketc te dispersion curves (k vs ω) for te first tree TE modes. Your sketces sould also include (wit dased lines) te tree dispersion curves for plane waves traveling in media wit indices n 1, n, and n 3, respectively. Indicate on te sketc te cut-off frequencies for te first tree modes as well as te asymptotic beavior of te dispersion curves for large frequencies. 4
Problem 11.5: (Two Hertian dipoles) Consider two Hertian dipoles sown below: y J 1 φ J Te current densities of te dipoles are specified by te pasors: r r ( ) = ˆ 3 r J1 r Id δ r + ˆ r r ( ) = ˆ j α 3 r J r Id e δ r ˆ a) Assuming α = π, = λ ( θ = π, φ ), find te radiation pattern ( θ = π, φ) p in te y- plane. Sketc p and indicate te angular location of all te nulls and te maima in te radiation pattern. Te matlab routine polar can be used for plotting. Problem 11.6: (Two more Hertian dipoles) Consider two Hertian dipoles sown below: J 1 θ J 5
Te current densities of te dipoles are specified by te pasors: r r ( ) = ˆ 3 r J1 r Id δ r ˆ r r ( ) = ˆ jα 3 r J r Id Ae δ r + ˆ a) Assuming A = 1, α = π, = λ ( θ, φ = 0) do not miss a ( θ ), find te radiation pattern ( θ, φ = 0) p in te - plane. Sketc p and indicate te angular location of all te nulls in te radiation pattern. Hint: Make sure you sin term tat also comes in due to te angular dependence of te radiation patterns of te individual dipoles. 6