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
3IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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)
4IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Goal: reflection, transmission, absorption, field distribution, dispersion diagrams
Solution procedure:
5IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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:
6IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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
7IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Maxwell’s equations in an individual layer:
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
8IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Maxwell’s equations in an individual layer:
eigenvalue problem for kz
4x4 matrix
9IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Maxwell’s equations in an individual layer:
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
10IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
tij are 2x2 block matrices
z1
z2
11IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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:
12IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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:
13IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Tikhodeev et al. 2002
If a layer of thickness L is added
initial condition
recursive formula
no mixing of small and large exponents
14IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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
15IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Example: dual band omnidirectional mirror
N=14 layers
Fourier modal method (FMM):
16IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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
17IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
T. Weiss, Master's thesis, 4th Physics Institute University of Stuttgart (2008).
Periodic layer:
18IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
(r)u
periodic function
decomposition of permittivity function
Ng=(2gx,max+1)(2gy,max+1) – truncation order
19IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
Periodic layer:
19IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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
Non-rectangular grid
Co- and contravariant coordinates
Incident light
Non-rectangular grid
Modes in photonic crystal
Field equations
Non-rectangular grid
Matrix representation
Non-rectangular grid
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
Scattering matrix and software combinations
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
Scattering matrix and software combinations
Scattering matrix and software combinations
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
Scattering matrix and software combinations
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
39IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
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!
40IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
C method: curved coordinates
41IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography
C method: curved coordinates