Fast Simulation Algorithms On Three-dimensional Photonic Crystals

Photonic crystals are artificial band-gap materials that are periodically arranged by a variety of media with different refractive indices at the optical scale,of which flexible and tunable electromagnetic structures can be used to control the propagation of light,are an important class of optical metamaterials.The numerical simulation of photonic band structure is of great significance to the research of photonic crystals and the development of optical devices.The propagation of light in photonic crystals is usually described by the Maxwell’s equations and needs to satisfy the Bloch boundary condition,combined with the constitutive properties of the medium relationship,that is,the eigenvalue problem corresponding to the photonic crystal energy band structure problem is obtained,which is called the Maxwell’s eigenvalue problem,which is transformed into a generalized eigenvalue problem after discretization.Numerically,solving the photonic energy band structure problem essentially corresponds to calculating several minimum positive eigenvalues of a series of large-scale generalized eigenvalue problems.In this paper,we consider the problem of calculating the energy band structure of three-dimensional(3D)photonic crystals,from the perspective of numerical algebra,based on the the structural analysis of coefficient matrices of discrete eigenvalue problems,we design an efficient numerical simulation algorithm FAME,and verify the effectiveness of the algorithm by a large number of numerical experiments.Therefore,the research in this paper has important theoretical significance and application value from the perspective of scientific computing and engineering background.The specific content and innovation points of this paper are as follows.Firstly,this paper designs a fast algorithm for the calculation of the band structure of(bi)isotropic 3D photonic crystals.The electromagnetic parameters in the(bi)-isotropic medium are scalar functions,for any Bravais lattice structure,we use the Yee’s finite difference method to discretize the Maxwell’s eigenvalue problem and obtain an explicit representation of the generalized eigenvalue problem.Further more,by deriving the explicit singular value decomposition of the discrete matrix,we compress the original generalized eigenvalue problem to an null-space free eigenvalue problem,which removes the interference of null space when calculating the minimum positive eigenvalue.We use the inverse Lanczos method to calculate the target eigenvalue,and the matrix-vector multiplication operation in the iterative process can be greatly accelerated by the fast Fourier transform.Finally,we combine the relevant theoretical analysis results and matrix calculation skills to design fast simulation algorithms FAMEi and FAMEbi for the calculation of isotropic and bi-isotropic 3D photonic crystal band structures.Secondly,this paper designs a fast algorithm for the calculation of the band structure of the(bi)-anisotropic 3D photonic crystals and analyses the special numerical phenomenon of the eigenpairs of discrete problems.The electromagnetic parameters in the(bi)-anisotropic medium are 3 × 3 matrix-valued functions,We propose the bi-Lebedev grid to discretize the Maxwell eigenvalue problem to ensure the complete configuration of electromagnetic field components and electromagnetic parameters at the same grid point.We derive the explicit singular value decomposition of the coefficient matrix for the discrete eigenvalue problem and the corresponding null-space free eigenvalue problem,and develop fast algorithms FAMEa and FAMEba for calculating the anisotropic and bi-anisotropic 3D photonic crystal band structures.For the case that the electromagnetic parameter matrix is singular or indefinite,we give the corresponding theoretical analysis,and obtain the important physical phenomenon of the bifurcation of eigenvalues and the localization of the electromagnetic fields.Finally,this paper uses a large number of numerical experiments to demonstrate the efficiency and effectiveness of the fast algorithm FAME.We implement the fast algorithm FAME as an efficient simulation software package FAMEm and FAMEg based on MATLAB and CUDA C languages,which are capable of calculating photonic band structures with 14 Bravais lattices based on CPU and GPU computing technologies,and finish the whole procedure in tens of minutes or hours for(bi)-isotropic and(bi)-anisotropic media with grid numbers larger than one million.For the bifurcation properties of eigenvalues and localization properties of electromagnetic fields of bi-anisotropic photonic crystals in theoretical analysis,we use the package FAMEg to verify them in detail.The innovations of this paper are as follows:(1)overcome the structural complexity of 3D photonic crystals and the difficulty of large discrete scale and develop efficient numerical algorithms;(2)through the structural analysis of the coefficient matrix of the discrete eigenvalue problem,deduce its explicit compact singular value decomposition,compress the null space of the eigenvalue problem,and excavate the fast Fourier transform structure contained in the coefficient matrix and vector multiplication,then easily computing the smallest positive eigenvalues using inverse Lanczos method;(3)propose the bi-Lebedev grid,in order to ensure that the electromagnetic parameters in the tensor form and electromagnetic field components for bianisotropic media have complete configurations;(4)develop the simulation packages FAMEm and FAME mboxg based on CPU and GPU computing technologies and share them on the FAME website,and a large number of numerical experiments have verified the efficiency and effectiveness of the algorithm in this paper,and the special phenomenon of bi-anisotropic 3D photonic crystals are addressed in detail.