Eigendecomposition of the Discrete Double-Curl Operator with Application to Fast Eigensolver for Three-Dimensional Photonic Crystals

Abstract

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of 5.184 million dimension eigenvalue problems within 50 to 104 minutes on a workstation with two Intel Quad-Core Xeon X5687 3.6 GHz CPUs.

Matrix representation of the double-curl operator for simulating three dimensional photonic crystals

“…Numerical Methods. For the numerical simulation of photonic crystals, the common methods include: plane-wave expansion method (PWE) [10,15,17,28], multiple scattering method [7,32], transfer-matrix method [4], finite-difference time-domain method (FDTD) [14,18,30,36], and finite-difference frequency-domain method (FDFD) [6,12,13,33,35,37]. Each method can be applicable on different objects and has its own advantages and disadvantages.…”

“…The dimension of null space of this generalized eigenvalue problem accounts for 1 3 of all, and if the several smallest positive eigenvalues are the target that we want to find, any of the currently known eigensolver will be affected by null space and difficult to get the answer. Huang et al [12] calculated the eigen-decomposition of the matrix, and then reduced the generalized eigenvalue problem to a null space free standard eigenvalue problem by some matrix computation techniques. The price is the original sparse matrix becomes dense matrix, so the matrix-vector multiplication is expensive.…”

“…By Theorem 4.3 and Theorem 4.4, we can observe that the matrix representations of K 3 has 4 general forms for various lattice structures, but the eigen-decomposition of K 3 has only two situations, the difference depends on the value cos θ α − cos θ β cos θ γ . Since the matrices C 1 , C 2 , and C 3 are commute to each other, they can be diagonalized by the same matrix, now we use the same argument in [12] to this matrix. Let…”

“…In [12], Huang et al developed the FFT based scheme to compute the matrix-vector multiplications T * p and T q for any vectors p and q. Here, we provide the sketch and the Algorithms for computing T * p and T q, respectively.…”

“…Algorithm 5.1 FFT-based matrix-vector multiplication for T * p [12] Input: Any vector p ∈ C n . Output: The vector T * p. 1: for k = 1, · · · , n 3 do 2:…”