Derive the time-independent wave equation

Assignment Help Physics
Reference no: EM131503313

Report

The term report is like a "small research project" to allow you to apply the physical principles and techniques taught in class to solve some real photonic device problems. Here you will be asked to solve the same problem in several different approaches, to develop physical insight about photonic devices.

To enhance your physical understanding of the problem, you are NOT supposed to use COMSOL Physics or any EM solver to numerically solve the problem. You need to mathematically formulate the problem step by step, and then solve the problem analytically (if possible) or numerically with the assistance of Matlab.

1. We have a 1-D periodic structure made of alternating layers with their index of refraction being n1 and n2 (n1 = 2.7 and n2 = 3.5).

2112_Figure.png

(a) Derive the time-independent wave equation as:

(b) Find the eigen values and eigen functions using the Bloch theorem.

The above time-independent wave equation with periodic index is similar to the 1D Schrodinger equation. Let us try to solve the problem in a similar approach as we solve the Schrodinger equation in a Kronig-Penny model (i.e. having a square wave-like periodic potential).

Write the general solution of the E-field in region 1 (with index n1) and region 2 (with index n2) as:

E(z) = Aejβ_1z + BAe-jβ_1z           0 ≤ z ≤ d1                        β1 = kon1

E(z) = Cejβ_2z + De-jβ_2z            d1 ≤ z ≤ d1 + d2 = Λ         β2 = kon2

Use the boundary conditions at z = d1, we obtain two equations due to the continuity of tangential E and tangential H at the interface z = d1.

Because the solution can be written as a "Bloch wave" as the electron wave-function in a periodic potential, the Bloch theorem in semiconductor physics tells us

E(z + Λ) = E(z)ejβΛ

This gives us two additional boundary conditions at the position z = d1 + d2 = Λ.

Write down the 4 boundary conditions in the following matrix form:

2085_Figure1.png

Remember ω = koc. Therefore, by solving the equation: det[V] = 0, we obtain ω as a function of β. This is the "photonic band structure" for this 1D photonic crystal.

Use Matlab program to solve the above problem and plot ω(β). Also plot the ω(β) diagram in the "reduced band structure" in the "1st Brillouin zone (i.e. limit the β value to the "photonic first Brillouin zone: - (π/Λ) ≤ β ≤ (π/Λ)).

Identify the "photonic bandgaps" in your band structure.

(c) Find the coefficients A, B, C, D (i.e. B, C, D in terms of A, which can be an arbitrary number). Show that E(z) is indeed a Bloch wave in the following form:

E(z) = Uβ(z)jβz where Uβ(z) = Uβ(z + NΛ) is a periodic function.

2. For the same 1D photonic crystal in (1), you will analyze the problem in a different approach (s-matrix) as described below:

(a) Consider the layer 0 ≤ z ≤ d1. Treat the layer as a "two-port system" with incident and reflected waves at the input and output ports.

For a single layer 0 ≤ z ≤ d1, we can assume A(0), B(d1) to be incident waves at ?? = 0 and d1.

Write the S-matrix as

1431_Figure2.png

where β1 = kon1, r21 = n1-n2/n1+n2 is the field-reflectivity for a plane wave incident from an n2 layer and reflected at the n2/n1 interface.

From the above relation, reorganize the terms to prove that

579_Figure3.png

Note that now the transfer matrix can be cascaded.

(b) Following the similar approach in (a), find the transfer matrix T2 in the following expression:

2258_Figure4.png

(c) Assuming the photonic crystal has a total of N period, find the transfer matrix, T, of the entire crystal:

1735_Figure5.png

(d) Calculate the reflectivity of the N-period photonics crystal (Bragg reflector). Also show the expression for the special case when n1d1o = n2λ2o = ¼.

r = B(0)/A(0)|B(NΛ) = 0

(e) Compare your results in (d) with the result from the coupled-mode theory discussed in class. Under what conditions the two results become close to each other (note that the coupled-mode theory results are approximations and here the analysis is "exact"). Can you find the expression of the "coupling coefficient" by comparing the two results?

(f) The E-field at each interface between the nth interface of n1and n2 layers can be written as:

E(nΛ) = A(nΛ) + B(nΛ)

E(nΛ + d1) = A(nΛ + d1) + B(nΛ + d1)   n = 0,1,2 ... ..

Find the E-field at these positions: E(nΛ) and E(nΛ + d1).

Find E(z) for all z positions and plot |E(z)|2. The results should be the same as the result in problem 1(c).

3. Problems 1, 2 talk about 1D photonic crystal, now let us extend the analysis to 2D photonic crystals. The index of refraction is represented as

N(x, z) = no + [∑mnmeiGmx][∑nnneiGnz]            G = 2π/Λ

(a) Show that the time-independent wave equation can be approximately written as

1812_Figure6.png

(b) Assume E(x, z) = X(x)Z(z). Show that the above wave equation can be written as

X'' + ko2nomnmeiGmx + βx2 = 0

Z'' + ko2nonnneiGnz + [ko2no2 - βx2] = 0

βx is the projection of the wave vector in the x-axis.

(c) We will leave the problem to this stage. Hope that by now you can fully appreciate the similarity between the problem of semiconductor and photonic crystal and the subtle differences as well.

If you like and have time, you can go on and actually solve a 2D photonic crystal band structure.

Reference no: EM131503313

Questions Cloud

Ideas of overcoming confusion with peers : Feel free to share any visuals or resources you may find online to help support your ideas of overcoming confusion with your peers.
Artifacts useful in the improvement of instruction : How are data and artifacts useful in the improvement of instruction?
Describe the ethical dilemma the therapist is confronted : Describe the ethical dilemma the therapist is confronted with in terms of issues related to confidentiality breaching.
A program to improve the relationship between two groups : Create a program to improve the relationship between two groups.Describe the context of the conflict. What kind of community or organization is this?
Derive the time-independent wave equation : We have a 1-D periodic structure made of alternating layers with their index of refraction being n1 and n2. Derive the time-independent wave equation
Identify common influences or predictors of adolescent drug : Identify common influences or predictors of adolescent drug and alcohol use indicated by the research presented in textbook and other resources.
Evaluate the effectiveness of the treatment interventions : Evaluate the effectiveness of the treatment interventions implemented by Dr. Heston, supporting your statements with information from the case.
Create the final output or that require numeric input : Your goal is to refactor your Project two purchase calculator to use lists and functions. Spend some time examining the refactored sandwich builder project for ideas.
Research on the nervous system is controversial : Research on the nervous system is controversial. In this assignment, you will explore nervous system research and the controversies surrounding it.

Reviews

Write a Review

Physics Questions & Answers

  Find the magnitude of the resulting magnetic field

A sphere of radius R is uniformly charged to a total charge of Q. It is made to spin about an axis that passes through its center with an angular speed ω. Find the magnitude of the resulting magnetic field at the center of the sphere.

  Find the equivalent resistance

A resistor is in the shape of a cube, with each side of resistance  R . Find the equivalent resistance between any two of its adjacent corners.

  What is the electric field at the location

Question: Field and force with three charges? What is the electric field at the location of Q1, due to  Q 2 ?

  What is the maximum displacement of the bridge deck

What is the maximum displacement of the bridge deck?

  What is the magnitude of the current in the wire

What is the magnitude of the current in the wire as a function of time?

  Blackbody

Questions on blackbody, Infra-Red Detectors & Optic Lens and Digital Image.

  Gravity conveyor

Illustrate the cause of the components accelerating from rest down the conveyor.

  Calculate the dc voltage

Calculate the dc voltage applied to the circuit.

  Quadrupole moments in the shell model

Quadrupole moments in the shell model

  Determine the tension in each string

Determine the tension in each string

  Introductory mechanics: dynamics

Calculate the smallest coefficient of static friction necessary for mass A to remain stationary.

  Evaluate maximum altitude

Evaluate maximum altitude?

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd