Modes in infinite slab waveguides ELECE 3240 Photonics

Modes in infinite slab waveguides ELEC-E 3240 Photonics Spring 2016 Group 1: Dream Team Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License.

Lecture frame 1. Basics: 1. 2. Maxwell equations EM constants and variables 1. Waveguides 1. 2. 3. 4. 5. 6. Basics Dielectric slab waveguide Helmholtz equation What is an optical mode? Boundary conditions Cutoff frequency Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 2

1. 1 Basics: Maxwell equations • The mathematical basis of waveguide calculations if formed by the Maxwell equations: Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 3

1. 1 Basics: Maxwell equations • One can also derive the time-harmonic Maxwell equations in terms of phasors using the previous equations: Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 4

1. 2 EM constants and variables • Angular frequency ω • Measure of the rate at which a particular wave pulsates. • Wave vector k • Vector, whose direction is the same as the equivalent waves propagation direction, and whose magnitude is the same as the equivalent waves wavenumber. • Permittivity ε • Measure of a material's ability to enhance electric fields. • Permeability μ • Measure of a material's ability to enhance magnetic fields. Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 5

2. 1 Waveguides: Basics • Waveguide is a structure built to guide physical waves (electromagnetic, sound, etc. ) • The main idea behind a waveguide is to restrict the waves expansion to every possible direction, and guide it towards a desired direction with minimal losses. • There a number of different kinds of waveguides for different kinds of waves. Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 6

2. 2 Dielectric slab waveguide • A slab waveguide consists of three layers with different dielectric constants. • The wave is guided in the middle section called ”the core”. This happens only when total optical reflection occurs at the boundaries of the different layers. ECE 303 – Fall 2005 – Farhan Rana – Cornell University Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 7

2. 3 Helmholtz equation • The time-independent form of the general wave equation is called the Helmholtz equation: • Helmholtz equation is used when we want to find stationary solutions of the wave equation. Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 8

2. 4 What is an optical mode? • Modes are the possible solutions of the Helmholtz equations. • These solution depend on the boundary conditions of the given problem. • When the electric field is transverse to the direction of propagation -> TE mode • When the magnetic field is transverse to the direction of propagation -> TM mode Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 9

2. 4 Helmholtz equation and modes • Considering an infinite slab waveguide in (y, z) plane, to study TM mode(Hz=0) we need to solve the following equation: • The propagation constant γ determines how the amplitude and phase of an EM wave vary along the propagation direction. Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 10

2. 5 Boundary conditions ECE 303 – Fall 2005 – Farhan Rana – Cornell University ● Both even (left) and odd (right) modes exist ● Tangential components of both E and H vectors must be continuous across the boundary • Nonzero values makes the calculation difficult Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 11

2. 6 Cutoff frequency ● ● Frequencies below the cutoff frequency are not able to propagate in the waveguide o In a sense, a waveguide is a high-pass filter Mode with the lowest cutoff frequency is the dominant mode Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 12

Further information ● David K. Cheng – Fundamentals of Engineering Electromagnetics – Addison-Wesley Publishing Company Inc. – 1993 ● David K. Cheng – Field and wave electromagnetics – Addison. Wesley Publishing Company Inc. – 1985 Except otherwise noted, this work is licensed under a Creative Commons Attribution 4. 0 International License. 18. 4. 16 13