Array Integral Equation-Fast Fourier Transform Solver for Simulation of Supercells and Aperiodic Penetrable Metamaterials

Abstract

A hybrid surface integral equation solver based on the hierarchical matrix, the integral equation fast Fourier transform method and the equivalent surface method is developed for simulation of (pseudo-) periodic optical metamaterial problems. The solver is typically suitable for problems with (1) large periodic supercell or aperiodic cell arrays, (2) dominance of near field interactions, (3) presence of deep subwavelength features and (4) high geometry surface to volume ratio. An integral equation accelerator is introduced which uses a four to six level block Toeplitz matrix structure which naturally maps to the pseudo-periodic structure of metamaterial array problems, eliminates the need for the correction process encountered in fast Fourier transform-based methods and leads to substantial recycling of near field matrices. Novel metamaterial designs such as a supercell metamaterials, an aperiodic metasurfaces an a surface Luneburg lens are presented and solved, verifying the efficiency and accuracy of the presented model.

Simulating Wave Phenomena in Large Graded-Pattern Arrays With Random Perturbation

“…Thus, using iterative matrix solvers, the solution process boils down to calculating the action of [Z ij ] on a sequence of guess vectors, and hence, being able to efficiently calculate the action of [Z ij ] on a given guess vector will be critical. Using the A-IE-FFT [28] method, this is achieved extremely efficiently for an important class of array-like problems as it is evident from the examples presented in this work. A detailed discussion on the A-IE-FFT method will deviate the discussion from the main intent of this article, i.e., important application of the method in large array MM design.…”

“…For this purpose, an integral equation representation of the problem must be adopted. Since these examples involve wave scattering in penetrable media, a JMCFIE formulation is adopted for this purpose [28,29]. The formulation is briefly stated here for the sake of consistency.…”

“…Since this the focus of this article is more on the new applications of the array-IE-FFT (A-IE-FFT) method, we refrain from diving too deep into the method itself and refer the interested reader to [28]. However, with (5) at hand, similar to other IE solvers a surface discretization of the integral equation results in a matrix equation of the form…”

“…The following section, however, the cons and the pros of this method are briefly discussed. A detailed presentation on the A-IE-FFT method can be found in [28].…”

“…In this work, we present the application of a recently introduced method regarded as 'Array Integral Equation Fast Fourier Transform' (A-IE-FFT) method [28], to some representative examples that highlight the importance of the technique in the above-mentioned modeling applications. Moreover, we will provide proof-of-concept simulation examples demonstrating its feasibility and highlighting its remarkable computational efficiency in problems involving large arrays.…”