Numerical methods for solving photonic crystal slabs

Numerical methods for solving photonic crystal slabs

Photonic crystal slabs: periodic structures of finite thickness

2IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

Outline


Photonic crystal (PhC) slabs Layered media (special case of PhC slabs) Transfer and scattering matrices Numerical stability of T and S-matrices Fourier Modal Method (FMM) Factorization rules FMM examples Nonrectangular unit cell Scattering matrix for software combination C method

Layered stack (special case of PhC slabs)


Goal: reflection, transmission, absorption, field distribution, dispersion diagrams

Solution procedure:


1) Solve Maxwell’s equation in each layer separately2) Compute interface and layer transfer matrices3) Assemble the matrix (S-matrix) which links the input amplitudes with output

4x4

Maxwell’s equations in an individual layer:


Maxwell’s equations in CGS unitsMaxwell’s equations in CGS units

constitutive relations

ˆ ˆ;xx xy xz xx xy xz

yx yy yz yx yy yz

zx zy zz zx zy zz

ε, μ are generally tensors



Maxwell’s equations for curl operatorMaxwell’s equations for curl operator

1) All fields have time dependence 2) Look for a solution as a plane wave E,H~ , ky=0 3) Eliminate Ez, Hz-components

6 unknowns



eigenvalue problem for kz

4x4 matrix



4 eigenvalues

For isotropic medium (μ, ε) – scalars: K+1=K+

2=-K-3=-K-

4

1 4 1 4( , ) exp( ) exp( ),x x xE x z E iK z ik x E iK z ik x K K

4 eigenvectors

Solution of the Maxwell’s equations in a layer

one can solve 2x2 matrices

Transfer matrix: links the amplitudes at z1 and z2


tij are 2x2 block matrices

z1

z2


Interface and layer matrices:

layer transfer matrix through the layer dp

interface transfer matrix

total transfer matrix of the structure

Transfer matrix is numerically unstable:


Reason: loss of precision when adding/subtracting big and small numbersThe error grows with the increase of the layer thicknesses

S-matrix or R-matrix algorithms are numerically stable

S-matrix recursive algorithm:


Tikhodeev et al. 2002

If a layer of thickness L is added

initial condition

recursive formula

no mixing of small and large exponents


Transfer and Scattering matrices:

4 periods of slabs:

md

i

md

25.0

;06.024.1

1.0

;7.6

2

2

1

1

T-matrix fails with absorptive materialsT-matrix fails with absorptive materials

14


Example: dual band omnidirectional mirror

N=14 layers

Fourier modal method (FMM):


some layers have 1D or 2D periodicity ε(x,y), μ=1

1) Solve Maxwell’s equation in each layer separately (by Floquet-Fourier decomposition)2) Compute interface and layer transfer matrices3) Assemble the matrix (S-matrix) which links the input amplitudes with output

Maxwell’s equation in an individual layer


T. Weiss, Master's thesis, 4th Physics Institute University of Stuttgart (2008).

Periodic layer:


(r)u

periodic function

decomposition of permittivity function

Ng=(2gx,max+1)(2gy,max+1) – truncation order


Periodic layer:


2Ng eigenvalues

• Compute interface and layer transfer matrices• Assemble the matrix (S-matrix) which links the input amplitudes with output

Plane wave convention and space discretization

Single frequency analysis removes time from equations

Rectangular discretizationis proved to be better

Toeplitz matrix

For the sake of simplicity we assume a permeability to be constant

Fourier transform of functions multiplication

Function “g” will be represented by vector and function “f” by Toeplitz matrix

Derivatives are representedby diagonal matrices

Factorization rules

It often happens that functions “g” and “h” have complementary jumps at certain points

Diploma thesis: Thomas Weiss (University of Stuttgart) (2008)L. Li, J. Opt. Soc. Am. A 13, 1870 (1996)

Factorization rules

From a continuity of functions E_z, D_x and D_y one can derive:

Factorization rules

Factorization in x and y directions

FMM simulations

The accuracy depends on a number of modes and layers

FMM simulations

Simulation of nanowires array

Non-rectangular grid

For some cases non-rectangular unit cell is the one with a smallest area

L. Li, J. Opt. Soc. Am. A, Vol. 14, No. 10 (1997)

Fourier decomposition incorresponding coordinates


Co- and contravariant coordinates

Incident light


Modes in photonic crystal

Field equations


Matrix representation


Material-matrix assembly

Order the modes Modes normalization on a field magnitude Discretization is made for x and y components but modes are TE and TM For non-rectangular grid covariant components are used

Scattering matrix and software combinations

The same structure with FEM and FMMCST-studio suite FEM solver


Connection between software Order of modes should be the same Modes normalization on the same field magnitude Phase of modes depend on the normalization

FEM-FMM combination FEM is more suitable for complex geometries FEM is volume dependable FMM is good for rectangular geometries FMM does not depend on layers thickness



Ray tracing technique

Matrix with embedded photonic elements Coupling wave with geometrical optics Could be used for structures too big for wave analysis Embedded photonic structures could have various shapes


Numerical optimization

Evolutionary strategy

Mirror optimization

The optimizer found a configuration with 3 Bragg reflectors

Mittwoch, 19. April 2023

38Departement/Institut/Gruppe

Methods for solving diffraction gratings

C method: curved coordinates


Li et al. 1999 “Rigorous and efficient grating-analysis method made easy for optical engineers”

Multilayer approximation in FMM

Introduce new coordinates

No multilayer approximation, matching of boundary conditions is easy!





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Numerical methods for solving photonic crystal slabs