Eigenvalue Problems Of Electromagnetic Field And Numerical Mode Matching Methods

Cavity resonators are important devices in microwave and optical technologies.The res-onant frequency is one of the crucial parameters in a cavity.Meanwhile,the waveguides are also important devices in the electromagnetics field,because it has the characteristic of guiding electromagnetic wave.Three-dimensional(3D)layered media are commonly found in microwave,millimeter wave and optical integrated circuits,electronic packag-ing,electromagnetic exploration technology and other fields.Therefore,it is crucial to simulate the interaction of electromagnetic waves in such complex structures.First,based on the mixed finite element method and the accelerated two-grid method,a new two-grid vector discretization scheme is presented for computing the resonant cavity problem.In this scheme,a Lagrangian multiplier is introduced to impose the divergence-free constraint,and the edge basis functions are used to discretize the electric field so that it can eliminate all spurious eigenvalues.In addition,the scheme is based on the shifted-inverse power method of the elliptic eigenvalue problem,and the Rayleigh quotient is used to refine the numerical solutions,which lead to the scheme is highly accurate and efficient.This scheme successfully solves the 2D and 3D cavity problems with complex geometries and filled inhomogeneous anisotropic medium.In particular,it also performs well when solving the practical problems such as the quintuple-mode bandpass filter and the open cavity.Second,based on the Bloch periodic boundary conditions(BPBC)and the periodic bound-ary conditions(PEC),two mixed variational formulations are presented for the Bloch peri-odic waveguides,and they are proved to be equivalent.Meanwhile,the absorbing bound-ary conditions(ABC)are set as the cutoff boundary of the open waveguide(named ABC waveguide),and the mixed variational formulations of ABC waveguide are derived.In the above mixed variational formulations,Gauss' law is introduced to satisfy the divergence-free conditions.The BPBC waveguide,PBC waveguide and the open waveguide are solved by the mixed spectral element method(MSEM)and the mixed finite element method(M-FEM).In MFEM,the incomplete 2nd-order edge basis functions(LT/QN)are used to discretize the transverse vector electric field,and the classical quadratic node basis func-tions are applied to discretize the longitudinal electric field.Similarly,for the SNMM,the vector basis functions consist of the Gauss-Lobatto-Legendre(GLL)points are used to discretize the transverse electric field and the scalar basis functions consist of GLL points are employed to discretize the longitudinal electric field.Therefore,both MFEM and M-SEM are completely spurious mode free.In this paper,such as the fiber Bragg grating,graphene waveguide,patterned lithography waveguide and ABC waveguides are calculat-ed by using the MSEM and MFEM.Numerical experiments indicate that our methods are high accuracy and efficiency.Thirdly,combined the MFEM,MSEM with the numerical model matching(NMM)method,this work develops the 3D finite element numerical mode matching(FNMM)method and the 3D spectral numerical mode matching(SNMM)method,respectively.They are used to explore the physical properties of the single metasurfaces in a half-space,including their anomalous reflection and refaction,surface plasmon polaritons(SPPs),and absorptance.The numerical mode matching methods are the semi-analytical slovers that decompose an d(d=2,3)dimensional electromagnetic field problem into a series of d-1 dimensional eigenvalue problems in the horizontal direction and a analytical solu-tion in the longitudinal direction.Therefore,the numerical mode matching methods are used for dimensionality reduction,which can save a lot of computational costs.They are used to explore the properties and singular physical phenomena of the graphene surface,black phosphorus surface,gradient metasurface,metal metasurface,and microwave patch metasurface.Numerical results indicate that for the homogeneous isotropic/anisotropic,inhomogeneous isotropic,graded metasurface,they can be simulated by the numerical mode matching methods with high accuracy and high efficiency.Finally,based on the MSEM,MFEM and numerical mode matching method,this work proposes the spectral numerical mode matching(SNMM)method and the finite element numerical mode matching(FNMM)method for the three-dimensional(3D)layered multi-region structure.Similarly,the SNMM and FNMM of layered multi-region are also semi-analytical solvers.In this paper,the excitation vector generated by the excitation sources,the generalized reflection matrix and the recursion formulations of the magnitude field in each region are derived detailedly.The SNMM and FNMM methods inherit the excellent quality of the MSEM,MFEM and NMM methods,which not only can effectively obtain high-precision solutions,but also greatly reduce the computational costs.In numerical experiments,two metasurface problems,two lithography models,a half-space open space scattering problem,and an excitation source problem in the open system are simulated by using the SNMM and FNMM,respectively.The numerical results show that the spec-tral numerical mode matching method and the finite element numerical mode matching method are high efficiency and accuracy.