Until now I have found some analytical and purely numerical ways to find a dispersion diagram of a 1D photonic crystal...
However I want to calculate it for my experimental 1D structure, which is finite. Can you recommend a numerical solver or a comprehensively described analytical method?
PS: my structure can't be assumed to be infinite. Very few layers.
Hey Marcin!
You can try computing BG by using optiFDTD /Matlab. For reference you may read up a book by Prof. Igor Sukhoivanov. I hope it'll help you to bits.
You could make it by comsol, just define the k vector in one direction (kx,ky=kz=0 )which makes your wave propagating only in 1 dimension.
By using the "RSoft", you can easily obtain the photonic band diagram.
Also, see the following link:
https://uk.mathworks.com/matlabcentral/fileexchange/?term=authorid%3A31800
There are different programs which can solve the problem numerically from which you can find OptiFDTD as a freeware. Since for calculating the dispersion curve, you just need to define a unit cell, you don't need to worry about its dimension (i.e. finite or infinite).
However, if you really want to deal with the finite crystal, you can use PML (perfectly matched layer) boundary condition instead of PBC (periodic boundary condition) in the direction of your periodicity.
Note that for the case of 1D PhC which the structure has periodicity in one direction and it is uniform in other two directions, the boundary conditions for those uniform facets are PML.
Hi, Marcin! My group is busy with this problem. A method valid especially for finite size resonators and photonic crystals was developed: Standing Wave Expansion.
1. E.Ya. Glushko, Stepanyuk A.M. "Electromagnetic modes inside the island kind 2d photonic crystal resonator”. In the Springer Proceedings in Physics, vol.195: Nanophysics, Nanomaterials, Interface Studies, and Applications, p. 263 - 274 (2017) © Springer International Publishing AG 2017, O. Fesenko, L. Yatsenko (eds.), DOI 10.1007/978-3-319-56422-7_18 2. Glushko E.Ya. Island kind 2D photonic crystal resonator, Ukrainian Phys. Journal v.62 №11, pp. 939- 946 (2017).
Good luck!
1) Numerical packs for electromagnetic simulation: CST Microwave Studio, COMSOL Multiphysics.
2) MPB - MIT Photonics Bands (https://mpb.readthedocs.io/en/latest/).
I just recently found an article, that claims, that semi-infinite approximation holds its validity even for structures with very few periods. Let me quote:
"We confirm below that if the field is well confined to the surface a few periods is sufficient. Indeed, it has already been reported [5] that for an Si/SiO2 structure, good agreement with the dispersion relation evaluated
for a semi-infinite system and a finite multilayer can be obtained even for N=2+1/2. (With the notation N+1/2 periods we indicate a structure composed of N periods and the first layer of the unit cell, which refers to those systems whose unit cell is constituted of two layers.)"
"Analysis of Bloch-surface-wave assisted diffraction-based biosensors" by Marco Liscidini and J. E. Sipe
Hi!
You can use free FDTD package MEEP. Link for method describing: https://meep.readthedocs.io/en/latest/Scheme_Tutorials/Resonant_Modes_and_Transmission_in_a_Waveguide_Cavity/
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