A fast method to compute dispersion diagrams of three-dimensional photonic crystals with rectangular geometry

Abstract

We propose a method and codes for fast computation of complex dispersion relations in three-dimensional photonic crystals (PCs) with rectangular geometry. The main idea of the method is to convert the eigenproblem to a nonlinear equation equivalent to the zero-determinant condition. This equation is then solved iteratively either by fixed-point iteration or by rational approximation method. Additional mathematical elements include fast-converging continued-fraction expansion to compute the interaction tensor (appearing in the above nonlinear equation) and efficient accounting for the rectangular geometry in matrix-vector multiplications, which are involved in computing the continued fraction coefficients. The method allows one to perform realistic three-dimensional computations on a typical laptop computer, including finding the Bloch wave vector in the band gaps and in evanescent mode bands. This paper is focused on the method and includes its detailed explanation and illustration with examples. The associated computational package contains a detailed user guide and a set of further demonstrations, which can be run with the help of provided scripts. Program summaryProgram title: RectDisp v.1-3CPC Library link to program files:https://doi.org/10.17632/k7nk86cfbs.1Code Ocean capsule:https://codeocean.com/capsule/7141738Licensing provisions: CC BY NC 3.0Programming language: Fortran 2003Nature of problem: The program computes the complex Bloch wave number q as a function of real frequency ω in photonic crystals (PCs) with rectangular geometry. The PC constituents are characterized by complex dispersive permittivities and are non-magnetic. Projection of q onto the XY-plane of the PC is fixed and can be specified by the user, and the projection onto the Z-axis is computed.Solution method: The method converts the eigenproblem of finding q to the nonlinear equation equivalent to the zero-determinant condition for, at most, a 3×3 matrix, and solves the latter by an iterative method (either fixed-point iteration or rational approximation or a combination of the two approaches). Additional features include continued-fraction expansion for computing the coefficients in the nonlinear equation and utilization of variable separability characteristic of the rectangular geometry for fast matrix-vector multiplication.Additional comments including restrictions and unusual features: The current version of the codes computes at most one value of q for each frequency. This excludes spurious modes arising due to the artificial band folding. Such modes are usually not coupled to incident radiation. However, some other modes can be missed. Under most circumstances, this can happen at higher frequencies, i.e., for ωh/c>π, where h is the PC period. Verification of mode coupling to external radiation and exhaustive search for all coupled modes are not currently implemented but we plan to add these features in future releases. If a solution is not found at some frequency, it can potentially be found by fine-tuning input parameters as described in the User Guide.