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

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References

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  1. D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
  2. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, Mineola, New York, 2001).
  3. 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.
  4. 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.
  5. 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]
  6. 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.
  7. Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999).
    [CrossRef]
  8. G. X. Fan, Q. H. Liu, “FDTD and PSTD simulations for plasma applications,” IEEE Trans. Plasma Sci. 29, 341–348 (2001).
    [CrossRef]
  9. B. Tian, Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithms,” Prog. Electromagn. Res. 28, 253–273 (2000).
    [CrossRef]
  10. 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]
  11. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  12. Q. H. Liu, “PML and PSTD algorithm for arbitrary lossy anisotropic media,” IEEE Microwave Guid. Wave Lett. 9, 48–50 (1999).
    [CrossRef]
  13. Q. H. Liu, “A frequency-dependent PSTD algorithm for general dispersive media,” IEEE Microwave Guid. Wave Lett. 9, 51–53 (1999).
    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. 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).
  18. W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001).
    [CrossRef]
  19. J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  20. A. Dutt, V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
    [CrossRef]
  21. N. Nguyen, Q. H. Liu, “The regular Fourier matrices and non-uniform fast Fourier transforms,” SIAM J. Sci. Comput. 21, 283–293 (1999).
    [CrossRef]
  22. Q. H. Liu, “An accurate algorithm for nonuniform fast Fourier transforms,” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
    [CrossRef]
  23. C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).
  24. A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992).
    [CrossRef]
  25. A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).
  26. X. Gao, “Design, fabrication and characterization of small diffractive optical elements,” M.S. thesis (University of Delaware, Newark, Delaware, 2000).

2001

G. X. Fan, Q. H. Liu, “FDTD and PSTD simulations for plasma applications,” IEEE Trans. Plasma Sci. 29, 341–348 (2001).
[CrossRef]

W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001).
[CrossRef]

2000

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]

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]

B. Tian, Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithms,” Prog. Electromagn. Res. 28, 253–273 (2000).
[CrossRef]

1999

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]

Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999).
[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]

N. Nguyen, Q. H. Liu, “The regular Fourier matrices and non-uniform fast Fourier transforms,” SIAM J. Sci. Comput. 21, 283–293 (1999).
[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]

1998

Q. H. Liu, “An accurate algorithm for nonuniform fast Fourier transforms,” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

1997

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]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993

A. Dutt, V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

1992

A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992).
[CrossRef]

1965

J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Bayliss, A.

A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, Mineola, New York, 2001).

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Chan, C. H.

Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999).
[CrossRef]

Chen, Y. C.

W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001).
[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]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Dinesen, P. G.

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]

Dutt, A.

A. Dutt, V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

Fan, G. X.

G. X. Fan, Q. H. Liu, “FDTD and PSTD simulations for plasma applications,” IEEE Trans. Plasma Sci. 29, 341–348 (2001).
[CrossRef]

Gao, X.

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).

X. Gao, “Design, fabrication and characterization of small diffractive optical elements,” M.S. thesis (University of Delaware, Newark, Delaware, 2000).

Ge, D.

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]

Gottlieb, D.

D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).

Hesthaven, J. S.

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]

Hussaini, M. Y.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Leung, W. K.

W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001).
[CrossRef]

Leung, Y. F.

Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999).
[CrossRef]

Li, Q. L.

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]

Liu, Q. H.

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, 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]

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]

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]

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]

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]

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.

Lynov, J. P.

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]

Mirotznik, M. S.

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).

Nguyen, N.

N. Nguyen, Q. H. Liu, “The regular Fourier matrices and non-uniform fast Fourier transforms,” SIAM J. Sci. Comput. 21, 283–293 (1999).
[CrossRef]

Orszag, S. A.

D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

Prather, D. W.

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).

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Rokhlin, V.

A. Dutt, V. Rokhlin, “Fast Fourier transforms for nonequi-spaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

Taflove, A.

A. Taflove, S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, Norwood, Mass., 2000).

Tian, B.

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]

B. Tian, Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithms,” Prog. Electromagn. Res. 28, 253–273 (2000).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Turkel, E.

A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992).
[CrossRef]

Xu, X. M.

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]

Zang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

Zhang, Z. Q.

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]

IEEE Microwave Guid. Wave Lett.

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. H. Liu, “An accurate algorithm for nonuniform fast Fourier transforms,” IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

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]

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]

IEEE Trans. Plasma Sci.

G. X. Fan, Q. H. Liu, “FDTD and PSTD simulations for plasma applications,” IEEE Trans. Plasma Sci. 29, 341–348 (2001).
[CrossRef]

J. Comput. Phys.

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]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

A. Bayliss, E. Turkel, “Mappings and accuracy for Chebyshev pseudo-spectral methods,” J. Comput. Phys. 101, 342–359 (1992).
[CrossRef]

Math. Comput.

J. W. Cooley, J. W. Tukey, “Algorithm for the machine computation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Microwave Opt. Technol. Lett.

W. K. Leung, Y. C. Chen, “Transformed-spaced nonuniform pseudospectral time-domain algorithm,” Microwave Opt. Technol. Lett. 28, 391–396 (2001).
[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]

Y. F. Leung, C. H. Chan, “Pseudospectral time-domain (PSTD) method with unsplit-field PML,” Microwave Opt. Technol. Lett. 22, 278–283 (1999).
[CrossRef]

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]

Prog. Electromagn. Res.

B. Tian, Q. H. Liu, “Nonuniform fast cosine transform and Chebyshev PSTD algorithms,” Prog. Electromagn. Res. 28, 253–273 (2000).
[CrossRef]

SIAM J. Sci. Comput.

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]

Other

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1987).

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).

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.

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.

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).

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Figures (12)

Fig. 1
Fig. 1

Diagrams of (a) the x u points and the average position and (b) grid-position deviation x * .

Fig. 2
Fig. 2

Applying the CLPF technique to the x * u mapping curve. The resulting NU index is max ( Δ x ) / min ( Δ x ) = 3.2298 .

Fig. 3
Fig. 3

Applying the CLPF technique to the y * u mapping curve. The resulting NU index is max ( Δ y ) / min ( Δ y ) = 2.0754 .

Fig. 4
Fig. 4

Convergence of the CLPF technique. The two curves are the resulting staircase errors from the x * u and y * u curves shown in Figs. 2(a) and 3(a) while an iterative algorithm was used to reduce them. The wavelength shown on the y axis equals 1 × 10 - 6 and is used to normalize staircase errors.

Fig. 5
Fig. 5

(a) Close-up view of the NU grid obtained from the spline function and a low-pass filter without use of the constraining technique. (b) Close-up view of the NU grid obtained from the same filter function but with the CLPF technique applied. As can be seen, (a) shows visible staircase error, whereas in (b) the staircase error is much smaller.

Fig. 6
Fig. 6

Maximum relative errors in derivative calculations for the x * u and y * u curves shown in Figs. 2 and 3. Derivatives d u / d x and d u / d y are calculated with Eq. (5); d F / d u are calculated with the FFT–IFFT technique. Machine-limited accuracy at a high sampling rate is achieved for the x * u and y * u curves when both are filtered. Without filtering, the errors are quite large.

Fig. 7
Fig. 7

Diagram of a 2D NU grid and a DOE profile.

Fig. 8
Fig. 8

Diagram of the discretization of a multilevel DOE. The location of the characteristic points ( x m ,   h m ) , where the discretized profile coincides with the continuous one, uniquely determines the geometry of the discretized profile.

Fig. 9
Fig. 9

Diagram of weight function ζ. The region where ζ = 1 is the total field, and ζ = 0 corresponds to the scattered field. The connecting region is where ζ is between 0 and 1. The incident terms should be added to all the connecting regions. But if the incident wave is added only to the meshed region shown in this figure, a windowed soft source can be created for the region above the meshed region.

Fig. 10
Fig. 10

Steady-state result of | H z | obtained from the mapped PSTD algorithm. The image is on a NU rectangular grid.

Fig. 11
Fig. 11

Absolute difference of | H z | 2 , | E x | 2 , and | E y | 2 steady-state values from FDTD and the mapped PSTD. x and y coordinates are grid numbers. The scattered field region in the mapped PSTD is excluded. The DOE profile is also plotted.

Fig. 12
Fig. 12

x-direction line scan of steady-state | H z | 2 at several distances away from the DOE surface. The locations are randomly picked. To avoid overlapping, curves at different locations are shifted differently. But curves at the same y location are shifted the same amount for the mapped PSTD and FDTD. Negative y’s correspond to locations below the DOE surface.

Tables (1)

Tables Icon

Table 1 Performance Comparison of Mapped PSTD and FDTD

Equations (13)

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F x x = x j = d u d x x = x j F u u = u j = d u d x x = x j IFFT   [ ik   FFT ( F ) ] | u = u j ,
x ¯ j = ( x N + 1 - x 1 )   j - 1 N + x 1 .
x j * = x j - x ¯ j .
x n * = m = 1 N a m exp ( ik m u n ) .
d x d u u = j = d ( x ¯ + x * ) d u u = j = x N + 1 - x 1 N + d x * d u u = j = x N + 1 - x 1 N + IFFT   [ ik   FFT ( x * ) ] | u = j .
x B * , n + 1 = x B * , n - a ( x _ B * , n - x B * , 0 ) ,
h ( x ) = mod H 0 - 1 ( n - n 0 )   ( x 2 + f 2 - f ) ,   H 0 ,
E ^ tot = E ^ inc + E scat = ζ   E inc + E scat ,
H ^ tot = H ^ inc + H scat = ζ   H inc + H scat .
E ^ x t = c r H ^ z y - ζ y   H z _ inc - σ y r 0   E ^ x ,
E ^ y t = - c r H ^ z x - ζ x   H z _ inc - σ x r 0   E ^ y ,
H ^ zx t = - c μ r E ^ y x - ζ x   E y _ inc - σ x * μ r μ 0   H ^ zx ,
H ^ zy t = c μ r E ^ x y - ζ y   E x _ inc - σ y * μ r μ 0   H ^ zy ,

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