## Applying a mapped pseudospectral time-domain method in simulating diffractive optical elements

JOSA A, Vol. 21, Issue 5, pp. 777-785 (2004)

http://dx.doi.org/10.1364/JOSAA.21.000777

Enhanced HTML Acrobat PDF (718 KB)

### Abstract

A new technique for the analysis of two-dimensional diffractive optical elements, by use of the pseudospectral time-domain (PSTD) method, is presented. In particular, the method uses a nonuniform (NU) grid and a mapping technique to obtain very accurate spatial derivatives in an efficient manner. To this end, we present the formulation of the PSTD method by using a NU grid and compare its application to the analysis with that of the finite-difference time-domain (FDTD) method. Using only a fraction of the memory and a fraction of the computation time used by FDTD, the mapped PSTD was able to obtain very close results to FDTD.

© 2004 Optical Society of America

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(220.3620) Optical design and fabrication : Lens system design

(350.3950) Other areas of optics : Micro-optics

**History**

Original Manuscript: July 24, 2003

Revised Manuscript: January 6, 2004

Manuscript Accepted: January 6, 2004

Published: May 1, 2004

**Citation**

Xiang Gao, Mark S. Mirotznik, Shouyuan Shi, and Dennis W. Prather, "Applying a mapped pseudospectral time-domain method in simulating diffractive optical elements," J. Opt. Soc. Am. A **21**, 777-785 (2004)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-5-777

Sort: Year | Journal | Reset

### References

- D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
- J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, Mineola, New York, 2001).
- A. V. Kabakian, “A spectral algorithm for electromagnetic wave scattering in the time domain application to RCS computation,” in Proceedings of the 27th AIAA Plasmadynamics and Lasers Conference (American Institute of Aeronautics and Astronautics, www.aiaa.org , 1996), Paper 96-2334.
- Q. H. Liu, “A spectral-domain method with perfectly matched layers for time-domain solutions of Maxwell’s equations,” presented at the 1996 URSI Meeting, Baltimore, Md., July 1996.
- Q. H. Liu, “The PSTD algorithm: a time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997). [CrossRef]
- B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the ACES 13th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, http://aces.ee.olemiss.edu , 1997), pp. 926–933.
- Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999). [CrossRef]
- G. X. Fan, Q. H. Liu, “FDTD and PSTD simulations for plasma applications,” IEEE Trans. Plasma Sci. 29, 341–348 (2001). [CrossRef]
- B. Tian, Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithms,” Prog. Electromagn. Res. 28, 253–273 (2000). [CrossRef]
- Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Remote Sens. 37, 917–926 (1999). [CrossRef]
- J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
- Q. H. Liu, “PML and PSTD algorithm for arbitrary lossy anisotropic media,” IEEE Microwave Guid. Wave Lett. 9, 48–50 (1999). [CrossRef]
- Q. H. Liu, “A frequency-dependent PSTD algorithm for general dispersive media,” IEEE Microwave Guid. Wave Lett. 9, 51–53 (1999). [CrossRef]
- Q. L. Li, Y. C. Chen, D. Ge, “Comparison study of the PSTD and FDTD methods for scattering analysis,” Microwave Opt. Technol. Lett. 25, 220–226 (2000). [CrossRef]
- J. S. Hesthaven, P. G. Dinesen, J. P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999). [CrossRef]
- Q. H. Liu, X. M. Xu, B. Tian, Z. Q. Zhang, “Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations,” IEEE Trans. Geosci. Remote Sens. 38, 1551–1560 (2000). [CrossRef]
- X. Gao, D. W. Prather, M. S. Mirotznik, “A method for introducing soft sources in the PSTD algorithm,” IEEE Trans. Antenna Propag. (to be published).
- W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001). [CrossRef]
- J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965). [CrossRef]
- A. Dutt, V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993). [CrossRef]
- N. Nguyen, Q. H. Liu, “The regular Fourier matrices and non-uniform fast Fourier transforms,” SIAM J. Sci. Comput. 21, 283–293 (1999). [CrossRef]
- Q. H. Liu, “An accurate algorithm for nonuniform fast Fourier transforms,” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998). [CrossRef]
- C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).
- A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992). [CrossRef]
- A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).
- X. Gao, “Design, fabrication and characterization of small diffractive optical elements,” M.S. thesis (University of Delaware, Newark, Delaware, 2000).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.