Efficient computation of Maxwell eigenmodes in axisymmetric cavities using hierarchical vector finite elements

摘要

Computation of Maxwell eigenmodes in an axisymmetric cavity using hierarchical vector finite elements is presented. The use of curl conforming vector basis functions, which span the null space of the curl operator, leads to the appearance of spurious modes with zero eigenvalues. Such spurious modes lead to electric flux solution with non-zero divergence. Constraining the solution space in the variational statement for the eigenvalue problem by weakly enforcing the flux to be divergence-free leads to the elimination of such modes. Discrete equivalent of such a constraint equation is developed for axisymmetric problems solved using hierarchical vector and scalar basis functions of orders complete to p = 2. The discrete constraint equation, developed individually for Fourier modes m = 0 and m ≥ 1, is efficiently integrated with a subspace iteration-based eigenvalue solution technique such as the Lanczos/Arnoldi method. The resulting solution technique is free of spurious modes added with an advantage of seeking a solution of a positive definite matrix during each iteration of the eigenvalue solver. Convergence in solution is demonstrated for orders up to p = 2, while the proposed technique can be extended to basis functions of arbitrary order.