A Fast Algorithm Of 3-D Photonic Crystal Energy Band In Oblique Coordinate System

Photonic crystals(PhCs)are new type of artificial bandgap material proposed in the 1980s,which are called "optical semiconductors" or "future semiconductors" in the scientific and industrial circles.Numerical calculation of the band structure of PhCs has always been an important and challenging topic from both application and scientific research.Mathematically,the calculation of the energy band structure is finally represented as solving several minimum positive eigenvalues and corresponding eigenvectors of a series of large-scale generalized eigenvalue problems.This paper mainly studies the calculation of the energy band structure of threedimensional(3D)PhCs,and develops a fast numerical algorithm FAMEobl based on the modified Yee’s finite difference discretization scheme in the oblique coordinate system,which has verified the effectiveness by a large number of numerical experiments.The research of this paper mainly includes the following three parts.First,we construct a set of oblique coordinate system based on the 3D lattice translation vectors a1,a2,a3,and we combine Yee’s scheme to discretize the Maxwell eigenvalue problem and constitutive relations of 3D PhCs.The advantage is that general nonorthogonal Bravais lattices and orthorhombic Bravais lattices,as well as anisotropic and isotropic cases of permittivity or permeability,can be treated equally in a unified manner,and the second-order accuracy of its differential format is preserved.After discretizing the model,the Bloch quasi-periodic boundary conditions can be simply incorporated into the discrete single curl operator,resulting in a discrete Maxwell eigenvalue problem model.Second,we perform an eigendecomposition on the coefficient matrix of the discrete Maxwell eigenvalue problem model.The discrete single curl operator is written as a 3D discrete Fourier transform matrix multiplied by a sparse matrix,and the standard eigenvalue problem without null space is obtained by using the null space free method.A fast algorithm FAMEobl with a computational complexity of O(n log n)is proposed for the above model.Finally,we conduct relevant numerical comparison experiments for the proposed algorithm FAMEobl for 3D anisotropic and isotropic photonic crystals,respectively.The results of the numerical experiments verify the effectiveness and efficiency of our algorithm,and a summary and outlook are given at the end of this paper.