Pseudospectral Methods Sahar Sargheini Laboratory of Electromagnetic Fields and Microwave Electronics (IFH) ETHZ 7th Workshop on Numerical Methods for Optical Nano Structures, ETH Zurich, July 4-6, 2011

Pseudospectral Methods Numerical methods for solving PDEs Approximate the differential operatore Approximate the solution Finite difference Spectral methods

Pseudospectral Methods Weighted residues Pseudospectral or Collocation method of selected points Galerkin method Least square

Pseudospectral Methods Finite Elements Method: Finite Difference Method: Pseudospectral Methods N point method

Pseudospectral Methods Created by Kreiss and Oliger in 1972. Were first introduced to the electromagnetic community around 1996 by Liu. Error Memory usage and time consumption will be reduced significantly Infinite order / Exponential convergence

Pseudospectral Methods Basis functions Periodic functions Non periodic functions Trigonometric Chebyshev or Legendre Semi-Infinite functions Infinite functions Laguerre Hermite

Fourier PSFD

Fourier PSFD Liu extended the pseudospectral methods to the frequency domain (2002). All proposed PSFD methods used Chebyshev basis functions. However for periodic structures, trigonometric basis functions will be much more suitable. In addition,using trigonometric functions, we can benefit from characteristics of Fourier series, and that is why we call this method Fourier PSFD Conventional single-domain PSFD methods suffer from staircasing error. This error will not be reduced unless the number of discretization points increases. To overcome this difficulty in a multidomain method, curved geometries should be divided into several subdomains whereas this method is complicated and time consuming to some extend. We used a new technique to overcome the staircasing error in a single-domain PSFD method. We formulate the constitutive relations with the help of a convolution in the spatial frequency domain.

Fourier PSFD Constitutive relation Bloch-Floquet: Periodic functions Conventional PSFD method Conventional PSFD method C-PSFD method C-PSFD method

Fourier PSFD Photonic crystals TMz mods TEz mods C-PSFD: 10×10 Error: Second band at the M point of the first Brillouin zone Conventional PSFD: 6×6

Fourier PSFD Photonic crystals TMz modes Error: Second band at the M point of the first Brillouin zone C-PSFD: 8×8

Fourier PSFD Photonic crystals TMz modes C-PSFD: 12×12 Error: seventeenth band at

Fourier PSFD Left-handed binary grating

Fourier PSFD Left-handed binary grating

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