Pseudospectral Methods Sahar Sargheini Laboratory of Electromagnetic Fields and Microwave Electronics (IFH) ETHZ 1 7th Workshop on Numerical Methods for Optical Nano Structures, ETH Zurich, July 4-6, 2011

Pseudospectral Methods 2 Numerical methods for solving PDEs Approximate the differential operatore Approximate the solution Finite differenceSpectral methods

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

Pseudospectral Methods  Finite Elements Method: 4 Finite Difference Method: Pseudospectral Methods N point method

Pseudospectral Methods  Pseudospectral methods  Created by Kreiss and Oliger in  Were first introduced to the electromagnetic community around 1996 by Liu. 5 Error Infinite order / Exponential convergence Memory usage and time consumption will be reduced significantly

Pseudospectral Methods  Basis functions 6 Periodic functions Trigonometric Non periodic functions Chebyshev or Legendre Semi-Infinite functions Laguerre Infinite functions Hermite

7 Fourier PSFD

8 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 9 C-PSFD method Conventional PSFD method Bloch-Floquet: Periodic functions C-PSFD method

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

Fourier PSFD 11 Photonic crystals TMz modes Error: Second band at the M point of the first Brillouin zone C-PSFD: 8×8

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

Fourier PSFD 13 Left-handed binary grating

Fourier PSFD 14 Left-handed binary grating

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