Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

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Computational Photonics

Fourier Modal Method (FMM)

Seminar 07, 30 June 2014

• Learn how to implement a 1D version of the Fourier

Mode solver in TE polarization

• Extend the code to calculate the diffraction efficiencies

in reflection and transmission

• (voluntary) learn about stability issues of the transfer

matrix algorithm

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM fundamentals

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• aims:

• solving for the Bloch eigenmodes in the grating region

• solving for the diffraction efficiencies in reflection / transmission

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM: eigenmodes of the grating layer

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• 1D, TE polarization

• Maxwell’s equations in periodic domain Bloch theorem

• all quantities are expanded into Fourier series on the periodic basis

• all Fourier components are stored in a column vector

• this yields an algebraic eigenvalue problem for the propagation constant

• decoupling electric and magnetic field results in a smaller problem for the

square of the propagation constant:

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM: eigenmodes of the grating layer

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• is a diagonal matrix with the square of the transverse wavevector

components, is a so-called Toeplitz matrix with the coefficients .

• The eigenvectors are the Fourier components of the electric field of

forward modes. All components of all modes are stored in a matrix .

Upper index mode number, lower index Fourier component.

• The normalized H field is given from Maxwell’s equations:

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Task 1: Fourier eigenmodes

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• write a program that calculates the propagation constants for a grating:

• Test it for the following grating structure:

[beta] = FMM_1D_TE(epsilon, period, lambda, theta, refIndices, N)

%

% INPUT:

% <epsilon>: vector with permitivity distribution

% <period>: period of the grating in micron

% <lambda>: wavelength in micron

% <theta>: angle of incidence in degree

% <refIndices>: vector [n1 n2] with refr.indices of surrounding

% <N>: number of Fourier orders

%

% OUTPUT:

% <beta>: vector with propagation constants of grating modes

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Programming hints 1: eigenmodes

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• for Fourier expanding the permitivity, make use of the relation eps_m = fft(epsilon) / M

where M = length(epsilon)

• note that MATLAB stores the Fourier coefficients in an order that will

appear unusual to you: first positive, then negative frequencies:

• As long as M>2N+1, you do not have to deal with the somewhat delicate

task to get straight about this ordering, just pick up the right components

for the Toeplitz matrix (first (2N+1) and last (2N) elements). Resist to use fftshift(), fft(epsilon, N) or similar unless you know what

you are doing!

• Thus, choose a sufficiently large sampling for the permitivity, say M=1001

(odd M Fourier truncation converges better)

• If you are interested in FFT details, you can have a look into the

FFT primer in the supplementary materials

• for the construction of the matrix, you can use the MATLAB function toeplitz()

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM: calculating diffraction efficiencies

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• what remains is to calculate the reflection / transmission amplitudes of the

different Rayleigh modes (plane waves) in the surrounding media

• therefore, we follow this idea:

• these modal amplitudes can be linked by transfer matrices for

forward/backward modal amplitudes:

express fields in

homogeneous

regions as

Rayleigh mode

expansion

express fields in

grating region as

eigenmode

expansion in Fourier

basis

Boundary

conditions yield

system of

equations for

modal coefficients

Solve for

reflected and

transmitted

amplitudes

grating region medium 2 medium 1

incident wave

(only zero Fourier

order non-zero)

reflected Fourier

orders Rm

transmitted

Fourier orders Tm

vector with zeros

(no incident wave

from behind)

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM: calculating diffraction efficiencies

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• now express fields as modal superposition:

• medium 1:

• medium 2:

• grating region:

propagator:

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

FMM: calculating diffraction efficiencies

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• Maxwell boundary conditions demand continuity at the two interfaces

• this allows to eliminate all modal amplitudes in the grating region

• what remains is the transfer matrix of the grating that links only the

quantities of interest:

• we are now ready to solve for the unknown amplitudes of the different

diffraction orders in reflection / transmission:

• diffraction efficiencies:

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Task 2: diffraction efficiencies

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• modify your program, so that it also calculates the diffraction efficiencies:

• Test it for the grating structure given before. Plot the efficiencies in a bar plot using bar().

[beta, eta_R, eta_T] = FMM_1D_TE(epsilon, thickness, period, ...

lambda, theta, refIndices, N)

%

% INPUT:

% <epsilon>: vector with permitivity distribution

% <thickness>: thickness of the grating in micron

% <period>: period of the grating in micron

% <lambda>: wavelength in micron

% <theta>: angle of incidence in degree

% <refIndices>: vector [n1 n2] with refr.indices of surrounding

% <N>: number of Fourier orders

%

% OUTPUT:

% <beta>: vector with propagation constants of grating modes

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Programming hints 2: diffraction efficiencies

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• All matrices are (2N+1) by (2N+1) square matrices, the block matrices are

thus 2*(2N+1) by 2*(2N+1).

• Vectors are of size (2N+1), block vectors twice that size.

• You can create block matrices / vectors in MATLAB just as you create

ordinary ones.

• Matrices usually do not commute! Care about the order of multiplication!

• Remember what we already discussed regarding solving linear systems of

equations.

• Remember that the “generic” index for the Fourier orders runs from –N...N,

so the zero Fourier component is in the middle!

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Task 3*: stability

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• Increase the thickness of the grating to 2.5µm

• You get a warning: what is its meaning?

• What is the reason that this happens?

• Why does it help reducing N to a value below 13 in that case?

• super brain: how can you prevent that?

Computational Photonics, Summer Term 2014, Abbe School of Photonics, FSU Jena, Prof. Thomas Pertsch

Homework 6 (30 June 2014)

• Solve at least tasks I & II.

• Prepare a one page report about your solution with a figure of some

calculated example.

• Submit your m-files of your program together with your one page report

electronically to teaching-nanooptics@uni-jena.de by 11 July 2014.

• Please put everything together in one single email which contains your

name (FAMILY NAME, Given Name) and matriculation number.

• Late submissions will not be accepted!

• 12 July the solutions of the tasks will be available online at the lectures

homepage www.iap.uni-jena.de/teaching >>> Computational Photonics.

• You are expected to solve the task yourself and a declaration of

independent work must be signed by every student at the end of the

semester.