Numerical methods for solving photonic crystal slabs

1. Numerical methods for solving photonic crystal slabs

2. Photonic crystal slabs: periodic structures of finite thickness IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

3. 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 IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

4. Layered stack (special case of PhC slabs) Goal: reflection, transmission, absorption, field distribution, dispersion diagrams IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

5. Solution procedure: 4x4 • Solve Maxwell’s equation in each layer separately • Compute interface and layer transfer matrices • Assemble the matrix (S-matrix) which links the input amplitudes with output IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

6. Maxwell’s equations in an individual layer: Maxwell’s equations in CGS units constitutive relations ε, μ are generally tensors IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

7. Maxwell’s equations in an individual layer: Maxwell’s equations for curl operator 6 unknowns • All fields have time dependence • Look for a solution as a plane wave E,H~ , ky=0 • Eliminate Ez, Hz-components IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

8. Maxwell’s equations in an individual layer: eigenvalue problem for kz 4x4 matrix IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

9. Maxwell’s equations in an individual layer: 4 eigenvalues 4 eigenvectors For isotropic medium (μ, ε) – scalars: K+1=K+2=-K-3=-K-4 one can solve 2x2 matrices Solution of the Maxwell’s equations in a layer IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

10. Transfer matrix: links the amplitudes at z1 and z2 z1 z2 tij are 2x2 block matrices IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

11. Interface and layer matrices: layer transfer matrix through the layer dp interface transfer matrix total transfer matrix of the structure IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

12. Transfer matrix is numerically unstable: Reason: loss of precision when adding/subtracting big and small numbers The error grows with the increase of the layer thicknesses S-matrix or R-matrix algorithms are numerically stable IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

13. S-matrix recursive algorithm: initial condition recursive formula If a layer of thickness L is added no mixing of small and large exponents Tikhodeev et al. 2002 IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

14. Transfer and Scattering matrices: T-matrix fails with absorptive materials 4 periods of slabs: IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography 14

15. Example: dual band omnidirectional mirror N=14 layers IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

16. Fourier modal method (FMM): some layers have 1D or 2D periodicity ε(x,y), μ=1 • Solve Maxwell’s equation in each layer separately (by Floquet-Fourier decomposition) • Compute interface and layer transfer matrices • Assemble the matrix (S-matrix) which links the input amplitudes with output IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

17. Maxwell’s equation in an individual layer T. Weiss, Master's thesis, 4th Physics Institute University of Stuttgart (2008). IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

18. Periodic layer: periodic function Ng=(2gx,max+1)(2gy,max+1) – truncation order decomposition of permittivity function IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

19. Periodic layer: 2Ng eigenvalues • Compute interface and layer transfer matrices • Assemble the matrix (S-matrix) which links the input amplitudes with output 19 IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

20. Plane wave convention and space discretization Single frequency analysis removes time from equations Rectangular discretization is proved to be better

21. 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 represented by diagonal matrices

22. Factorization rules Diploma thesis: Thomas Weiss (University of Stuttgart) (2008) L. Li, J. Opt. Soc. Am. A 13, 1870 (1996) It often happens that functions “g” and “h” have complementary jumps at certain points

23. Factorization rules From a continuity of functions E_z, D_x and D_y one can derive:

24. Factorization rules Factorization in x and y directions

25. FMM simulations The accuracy depends on a number of modes and layers

26. FMM simulations Simulation of nanowires array

27. Non-rectangular grid L. Li, J. Opt. Soc. Am. A, Vol. 14, No. 10 (1997) For some cases non-rectangular unit cell is the one with a smallest area Fourier decomposition in corresponding coordinates

28. Non-rectangular grid Co- and contravariant coordinates Incident light

29. Non-rectangular grid Modes in photonic crystal Field equations

30. Non-rectangular grid Matrix representation

31. 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

32. Scattering matrix and software combinations CST-studio suite FEM solver The same structure with FEM and FMM

33. 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

34. Scattering matrix and software combinations

35. 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

36. Scattering matrix and software combinations Numerical optimization Evolutionary strategy

37. Mirror optimization The optimizer found a configuration with 3 Bragg reflectors

38. Methods for solving diffraction gratings Departement/Institut/Gruppe

39. C method: curved coordinates Multilayer approximation in FMM Introduce new coordinates No multilayer approximation, matching of boundary conditions is easy! Li et al. 1999 “Rigorous and efficient grating-analysis method made easy for optical engineers” IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

40. C method: curved coordinates IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography

41. C method: curved coordinates IFH - Electromagnetic Field Theory Group and Laboratory of Crystallography