1. 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

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

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

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

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

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

7. Fourier PSFD

8. 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.

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

10. 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

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

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

13. Fourier PSFD
Left-handed binary grating

14. Fourier PSFD
Left-handed binary grating

15. Thank you for your attention