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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

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pseudospectral methods

Pseudospectral Methods

SaharSargheini

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 methods1
Pseudospectral Methods

Numerical methods

for solving PDEs

Approximate the

differential operatore

Approximate the

solution

Finite difference

Spectral methods

pseudospectral methods2
Pseudospectral Methods

Weighted residues

Pseudospectral

or

Collocation

or

method of selected points

Galerkin method

Least square

pseudospectral methods3
Pseudospectral Methods
  • Finite Elements Method:
  • Finite Difference Method:
  • Pseudospectral Methods

N point method

pseudospectral methods4
Pseudospectral Methods
  • 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 methods5
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 psfd1
Fourier PSFD

Bloch-Floquet:

Periodic functions

  • Constitutive relation
  • Conventional PSFD method
  • Conventional PSFD method
  • C-PSFD method
  • C-PSFD method
fourier psfd2
Fourier PSFD
  • Photonic crystals

TMz mods

TEz mods

C-PSFD: 10×10

C-PSFD: 6×6

Error: Second band at

the M point of the first Brillouin zone

Conventional PSFD: 6×6

fourier psfd3
Fourier PSFD
  • Photonic crystals

TMz modes

Error: Second band at

the M point of the first Brillouin zone

C-PSFD: 8×8

fourier psfd4
Fourier PSFD
  • Photonic crystals

TMz modes

C-PSFD: 12×12

Error: seventeenth band at

fourier psfd5
Fourier PSFD
  • Left-handed binary grating
fourier psfd6
Fourier PSFD
  • Left-handed binary grating