Analytical Analysis of Slab Waveguide

Presented by

Dr Mohamed Farhat

Professor, Senior Member IEEE, OSA

Nanotechnology and Nanoelectronics Engineering Program

Zewail City of Science and Technology

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.

What is an optical mode?

• Modes are the possible solutions of the Helmholtz equations.
• A mode (in general) is a time-harmonic solution to Maxwell’s equations (∝ e^iωt)

What is an optical mode?

• 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

Modal Analysis of TE Dielectric Slab WG

Analytical method:

Modal Analysis of TE Dielectric Slab WG

Starting from Maxwell's’ equations

Helmholtz Wave Equation

Modal Analysis of TE Dielectric Slab WG

Starting from Maxwell's’ equations

Helmholtz Wave Equation

Assuming:

Modal Analysis of TE Dielectric Slab WG

From Maxwell's’ equations

Modal Analysis of TE Dielectric Slab WG

From Maxwell's’ equations

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

Taking into consideration the field characteristics in each layer, we have three different constants:

Modal Analysis of TE Dielectric Slab WG

0

w

x

Modal Analysis of TE Dielectric Slab WG

0

w

x

Modal Analysis of TE Dielectric Slab WG

Applying Boundary Condition :

0

w

x

Modal Analysis of TE Dielectric Slab WG

0

w

x

Modal Analysis of TE Dielectric Slab WG

0

w

x

Boundary conditions: The tangent electric filed is continuous at the boundaries

Will be continuous

Modal Analysis of TE Dielectric Slab WG

0

w

x

At x=0:

Will be continuous

So from the boundary conditions:

Modal Analysis of TE Dielectric Slab WG

At x=0:

Is continuous

Modal Analysis of TE Dielectric Slab WG

0

w

x

At x=w:

Modal Analysis of TE Dielectric Slab WG

At x=w:

Is continuous

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

Modal Analysis of TE Dielectric Slab WG

From the above equations, we can get:

Modal Analysis of TE Dielectric Slab WG

Solving The Transcendental Equation:

Substituting by these equations in the Transcendental Equation:

Modal Analysis of TE Dielectric Slab WG

Solution of the Transcendental Equation

• For example: the MATALB sentence “x = fzero(@func, xo)” will return the zero of the function func which takes one input x using the initial guess xo

• There are many techniques to solve for the zeros of the characteristic equation.

• The good news are, Matlab provides built-in zero finding function called fzero.

• The way fzero works, it takes an identifier for the function (written as the function name preceded by the symbol “@”) and the initial value or the range where the zero is predicted to exist.

Solution Strategy

Solution Strategy for the modal analysis of Slab Waveguide

Solve the transcendental equation

Get the fields amplitudes in each layer

Generate the field in each layer

Numerical Solution For Transcendental Equation

Numerical Solution For Transcendental Equation

Fundamental Mode M=0

Numerical Solution For Transcendental Equation

First Order Mode M=1

Graphical Solution

Graphical Solution

Graphical Solution

Modal Analysis of TE Dielectric Slab WG

Important Notes on Slab Waveguide Modes:

Mode Orthogonality

Mode Orthogonality

4- All modes should be Orthogonal

Proof

Starting from the wave equation:

It can be rewritten as:

Is the Eigen value of the above equation

Mode Orthogonality

Mode Orthogonality

Mode Orthogonality

Mode Orthogonality

Then, there are two possibilities:

Mode Orthogonality

Mode Orthogonality

Mode Orthogonality