Abstract
This study focused on the two-dimensional frequency-dependent photonic crystal band structure and structured-preserving Arnoldi method. Research related most to band structure of the frequency-independent material, it's mostly correspond to a linear spectral problem. Combined with the traditional algorithms for discrete eigenvalue problem, such as IRA, Jacobi-Davidson etal., It will be able to obtain the approximate solution of band structure corresponding to crystal structure and the material. However, on the frequency-dependent material, since the dielectric constant of the material and the frequency is related, it corresponds to the eigenvalue problem mostly nonlinear eigenvalue problem. Therefore, the general algorithm for linear eigenvalue problem is dicult to apply in such problems. C. Engstrom and M. Richter recommends using dispersion relation to transform the non-linear problem into a linear spectrum spectral problem. The spectral parameter also transformed from the frequency to the wavelength of the wave vector. Discrete eigenvalue problem corresponding to the transformed spectral problem can be written as a gyroscopic quadratic eigenvalue problem. It can be solved approximately by the structured-preserving algorithm SHIRA, which will batter than the general eigensolver, such as IRA. However, this algorithm will encounter diculties of solving its invariant subspace. This paper is focus on extracting invariant subspace described above, and the gyroscopic quadratic eigenvalue problem can be obtained to a palindromic eigenvalue problem. The latter will be better than the former on the computation, and without diculty to solve the invariant subspace.