Abstract

The pseudospectral time-domain (PSTD) method greatly extends the physical volume of biological tissue in which light scattering can be calculated, relative to the finite-difference time-domain (FDTD) method. We have developed an analogue of the total-field scattered-field source condition, as employed in FDTD, for introducing focussed illuminations into PSTD simulations. This new source condition requires knowledge of the incident field, and applies update equations, at a single plane in the PSTD grid. Numerical artifacts, usually associated with compact PSTD source conditions, are minimized by using a staggered grid. This source condition’s similarity with that used by the FDTD suggests a way in which existing FDTD codes can be easily adapted to PSTD codes.

© 2014 Optical Society of America

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  1. Q. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Techn. Lett. 15, 158–165 (1997).
    [CrossRef]
  2. Q. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE T. Geosci. Remote 37, 917–926 (1999).
    [CrossRef]
  3. D. Kosloff, E. Baysal, “Forward modeling by a Fourier method,” Geophysics 47, 1402–1412 (1982).
    [CrossRef]
  4. J. Virieux, S. Operto, “An overview of full-waveform inversion in exploration geophysics,” Geophysics 74, WCC127 (2009).
    [CrossRef]
  5. M. Ding, K. Chen, “Staggered-grid PSTD on local Fourier basis and its applications to surface tissue modeling,” Opt. Express 18, 9236–9250 (2010).
    [CrossRef] [PubMed]
  6. S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, J. T. Walsh, “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express 13, 3666–3672 (2005).
    [CrossRef] [PubMed]
  7. G. Chen, P. Yang, G. Kattawar, “Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles,” J. Opt. Soc. Am. A 25, 785–789 (2008).
    [CrossRef]
  8. S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, J. T. Walsh, “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium,” Opt. Lett. 29, 1393–1395 (2004).
    [CrossRef] [PubMed]
  9. T. -W. Lee, S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenn. Wirel. Pr. 3, 253–256 (2004).
    [CrossRef]
  10. X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
    [CrossRef]
  11. D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
    [CrossRef]
  12. J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
    [CrossRef]
  13. P. Petre, T. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE T. Antenn. Propag. 40, 1348–1356 (1992).
    [CrossRef]
  14. C. Balanis, Advanced Engineering Electromagnetics (John Wiley and Sons, 1989).
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    [CrossRef]
  16. R. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
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    [CrossRef]
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    [CrossRef]
  22. Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
    [CrossRef]
  23. A. Oskooi, S. G. Johnson, “Electromagnetic wave source conditions,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, S. G. Johnson, eds. (Artech House, 2013), pp. 65–100.
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    [CrossRef]
  26. H. -W. Chen, “Staggered-grid pseudospectral viscoacoustic wave field simulation in two-dimensional media,” J. Acoust. Soc. Am. 100, 120–131 (1996).
    [CrossRef]
  27. C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
    [CrossRef]
  28. G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
    [CrossRef]
  29. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems ii. Structure of the image field in an aplanatic system,” P. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  30. S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, J. T. Walsh, “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium: addendum,” Opt. Lett. 30, 56–57 (2005).
    [CrossRef] [PubMed]

2012 (2)

C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
[CrossRef]

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

2010 (1)

2009 (2)

Z. Lin, L. Thylen, “An analytical derivation of the optimum source patterns for the pseudospectral time-domain method,” J. Comput. Phys. 228, 7375–7387 (2009).
[CrossRef]

J. Virieux, S. Operto, “An overview of full-waveform inversion in exploration geophysics,” Geophysics 74, WCC127 (2009).
[CrossRef]

2008 (2)

2005 (2)

2004 (3)

S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, J. T. Walsh, “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium,” Opt. Lett. 29, 1393–1395 (2004).
[CrossRef] [PubMed]

T. -W. Lee, S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenn. Wirel. Pr. 3, 253–256 (2004).
[CrossRef]

X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
[CrossRef]

2002 (1)

Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
[CrossRef]

1999 (1)

Q. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE T. Geosci. Remote 37, 917–926 (1999).
[CrossRef]

1997 (1)

Q. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Techn. Lett. 15, 158–165 (1997).
[CrossRef]

1996 (2)

T. Özdenvar, G. A. McMechan, “Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation,” Geophy. J. Int. 126, 819–828 (1996).
[CrossRef]

H. -W. Chen, “Staggered-grid pseudospectral viscoacoustic wave field simulation in two-dimensional media,” J. Acoust. Soc. Am. 100, 120–131 (1996).
[CrossRef]

1992 (1)

P. Petre, T. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE T. Antenn. Propag. 40, 1348–1356 (1992).
[CrossRef]

1982 (1)

D. Kosloff, E. Baysal, “Forward modeling by a Fourier method,” Geophysics 47, 1402–1412 (1982).
[CrossRef]

1980 (1)

D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
[CrossRef]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems ii. Structure of the image field in an aplanatic system,” P. R. Soc. A 253, 358–379 (1959).
[CrossRef]

1950 (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pur. Appl. Math. 3, 355–391 (1950).
[CrossRef]

1939 (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Backman, V.

Balanis, C.

C. Balanis, Advanced Engineering Electromagnetics (John Wiley and Sons, 1989).

Baysal, E.

D. Kosloff, E. Baysal, “Forward modeling by a Fourier method,” Geophysics 47, 1402–1412 (1982).
[CrossRef]

Carbotte, S.

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

Chen, G.

Chen, H. -W.

H. -W. Chen, “Staggered-grid pseudospectral viscoacoustic wave field simulation in two-dimensional media,” J. Acoust. Soc. Am. 100, 120–131 (1996).
[CrossRef]

Chen, K.

Chen, Y.

Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
[CrossRef]

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Cížek, V.

V. Čížek, Discrete Fourier Transforms and Their Applications (Adam Hilger, 1986).

Corrêa, G. J. P.

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

Ding, M.

Fisher, R.

D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
[CrossRef]

Gao, X.

X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
[CrossRef]

Greene, J. H.

Hagness, S.

A. Taflove, S. Hagness, Computational Electrodynamics, Third Edition (Artech House, 2005).

Hagness, S. C.

T. -W. Lee, S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenn. Wirel. Pr. 3, 253–256 (2004).
[CrossRef]

Harrington, R.

R. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

Johnson, S. G.

A. Oskooi, S. G. Johnson, “Electromagnetic wave source conditions,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, S. G. Johnson, eds. (Artech House, 2013), pp. 65–100.

Kattawar, G.

Kim, Y. L.

Kosloff, D.

D. Kosloff, E. Baysal, “Forward modeling by a Fourier method,” Geophysics 47, 1402–1412 (1982).
[CrossRef]

Kriezis, E. E.

Lee, T. -W.

T. -W. Lee, S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenn. Wirel. Pr. 3, 253–256 (2004).
[CrossRef]

Levine, H.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pur. Appl. Math. 3, 355–391 (1950).
[CrossRef]

Li, C.

Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
[CrossRef]

Li, Q.

Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
[CrossRef]

Lin, Z.

Z. Lin, L. Thylen, “An analytical derivation of the optimum source patterns for the pseudospectral time-domain method,” J. Comput. Phys. 228, 7375–7387 (2009).
[CrossRef]

Liu, C.

C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
[CrossRef]

Liu, Q.

Q. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE T. Geosci. Remote 37, 917–926 (1999).
[CrossRef]

Q. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Techn. Lett. 15, 158–165 (1997).
[CrossRef]

Maitland, D.

McMechan, G. A.

T. Özdenvar, G. A. McMechan, “Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation,” Geophy. J. Int. 126, 819–828 (1996).
[CrossRef]

Merewether, D.

D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
[CrossRef]

Mirotznik, M.

X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
[CrossRef]

Munro, P. R. T.

Mutter, J. C. C.

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

Operto, S.

J. Virieux, S. Operto, “An overview of full-waveform inversion in exploration geophysics,” Geophysics 74, WCC127 (2009).
[CrossRef]

Oskooi, A.

A. Oskooi, S. G. Johnson, “Electromagnetic wave source conditions,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, S. G. Johnson, eds. (Artech House, 2013), pp. 65–100.

Özdenvar, T.

T. Özdenvar, G. A. McMechan, “Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation,” Geophy. J. Int. 126, 819–828 (1996).
[CrossRef]

Panetta, R. L.

C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
[CrossRef]

Petre, P.

P. Petre, T. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE T. Antenn. Propag. 40, 1348–1356 (1992).
[CrossRef]

Prather, D.

X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems ii. Structure of the image field in an aplanatic system,” P. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Sarkar, T.

P. Petre, T. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE T. Antenn. Propag. 40, 1348–1356 (1992).
[CrossRef]

Schwinger, J.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pur. Appl. Math. 3, 355–391 (1950).
[CrossRef]

Smith, F.

D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
[CrossRef]

Spiegelman, M.

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Taflove, A.

Thylen, L.

Z. Lin, L. Thylen, “An analytical derivation of the optimum source patterns for the pseudospectral time-domain method,” J. Comput. Phys. 228, 7375–7387 (2009).
[CrossRef]

Török, P.

Tseng, S. H.

Virieux, J.

J. Virieux, S. Operto, “An overview of full-waveform inversion in exploration geophysics,” Geophysics 74, WCC127 (2009).
[CrossRef]

Walsh, J. T.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems ii. Structure of the image field in an aplanatic system,” P. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Yang, P.

C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
[CrossRef]

G. Chen, P. Yang, G. Kattawar, “Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles,” J. Opt. Soc. Am. A 25, 785–789 (2008).
[CrossRef]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[CrossRef]

Commun. Pur. Appl. Math. (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pur. Appl. Math. 3, 355–391 (1950).
[CrossRef]

Geophy. J. Int. (1)

T. Özdenvar, G. A. McMechan, “Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation,” Geophy. J. Int. 126, 819–828 (1996).
[CrossRef]

Geophysics (3)

G. J. P. Corrêa, M. Spiegelman, S. Carbotte, J. C. C. Mutter, “Centered and staggered Fourier derivatives and Hilbert transforms,” Geophysics 67, 1558–1563 (2012).
[CrossRef]

D. Kosloff, E. Baysal, “Forward modeling by a Fourier method,” Geophysics 47, 1402–1412 (1982).
[CrossRef]

J. Virieux, S. Operto, “An overview of full-waveform inversion in exploration geophysics,” Geophysics 74, WCC127 (2009).
[CrossRef]

IEEE Antenn. Wirel. Pr. (1)

T. -W. Lee, S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antenn. Wirel. Pr. 3, 253–256 (2004).
[CrossRef]

IEEE T. Antenn. Propag. (3)

X. Gao, M. Mirotznik, D. Prather, “A method for introducing soft sources in the PSTD algorithm,” IEEE T. Antenn. Propag. 52, 1665–1671 (2004).
[CrossRef]

P. Petre, T. Sarkar, “Planar near-field to far-field transformation using an equivalent magnetic current approach,” IEEE T. Antenn. Propag. 40, 1348–1356 (1992).
[CrossRef]

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302–307 (1966).
[CrossRef]

IEEE T. Geosci. Remote (1)

Q. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE T. Geosci. Remote 37, 917–926 (1999).
[CrossRef]

IEEE T. Nucl. Sci. (1)

D. Merewether, R. Fisher, F. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE T. Nucl. Sci. 27, 1829–1833 (1980).
[CrossRef]

J. Acoust. Soc. Am. (1)

H. -W. Chen, “Staggered-grid pseudospectral viscoacoustic wave field simulation in two-dimensional media,” J. Acoust. Soc. Am. 100, 120–131 (1996).
[CrossRef]

J. Comput. Phys. (1)

Z. Lin, L. Thylen, “An analytical derivation of the optimum source patterns for the pseudospectral time-domain method,” J. Comput. Phys. 228, 7375–7387 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Ra. (1)

C. Liu, R. L. Panetta, P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Ra. 113, 1728–1740 (2012).
[CrossRef]

Microw. Opt. Techn. Lett. (2)

Q. Li, Y. Chen, C. Li, “Hybrid PSTD-FDTD technique for scattering analysis,” Microw. Opt. Techn. Lett. 34, 19–24 (2002).
[CrossRef]

Q. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Techn. Lett. 15, 158–165 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

P. R. Soc. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems ii. Structure of the image field in an aplanatic system,” P. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton, L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[CrossRef]

Other (6)

R. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

A. Taflove, S. Hagness, Computational Electrodynamics, Third Edition (Artech House, 2005).

A. Taflove, A. Oskooi, S. Johnson, eds., Advances in FDTD Computational Electrodynamics. Photonics and Nanotechnology (Artech House, 2013).

C. Balanis, Advanced Engineering Electromagnetics (John Wiley and Sons, 1989).

A. Oskooi, S. G. Johnson, “Electromagnetic wave source conditions,” in Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology, A. Taflove, A. Oskooi, S. G. Johnson, eds. (Artech House, 2013), pp. 65–100.

V. Čížek, Discrete Fourier Transforms and Their Applications (Adam Hilger, 1986).

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Figures (8)

Fig. 1
Fig. 1

Diagrams illustrating the field equivalence principle. (a) A lens focuses light into a region bounded by fictitious surface S = S1S2S3S4. (b) The lens and light source are no longer present but the same field is produced within S due to equivalent sources on S.

Fig. 2
Fig. 2

Demonstration of how the TFSF source condition introduces a focused field into the “Total field” region bounded by S = S1S2S3S4 (left) and how the focused field can be introduced via a single planar interface S1 (center). Three field sample points from the FDTD grid, in the vicinity of the planar interface (right). PML refers to the perfectly matched layer which absorbs outgoing radiation, allowing unbounded simulations to be performed. The intensity map depicting the incident field represents the electric field at an instant in time.

Fig. 3
Fig. 3

(a) An example plot of Hy at an instant in time, along a line of constant x with an interface between total and scattered regions at z = 0. (b) The central differences approximation to ∂Hy/∂z. (c) ∂Hy/∂z evaluated using discrete Fourier transform on a collocated grid. (d) ∂Hy/∂z evaluated using the discrete Fourier transform on a staggered grid. In (b)–(d) the analytically evaluated ∂Hy/∂z is plotted.

Fig. 4
Fig. 4

The values of D n c (a) and D n s + (c) for the case N = 32 and their discrete Fourier transforms which are purely real ((b) and (d)). ℜ and ℑ refer to real and imaginary parts respectively.

Fig. 5
Fig. 5

a) Plots of an x-polarised beam with wavelength 1325nm focused by a lens with numerical aperture 0.056. The values have been normalized by the analytic in-focus value. The beam was introduced at its waist, just after the PML. (top) Plot of |Ex| in the plane y = 0, the position where the beam was introduced, coincides with the perturbation in the field near the left vertical axis of the plot. (lower left) Plot of |Ex| in the plane z = 1.66μm, just after the beam is introduced. (lower right) Plot of |Ex| in the plane z = 129μm just before the PML. b) Plot of |Ex|, normalized by the in-focus analytic value along a line which passes through the focus of the beam which is at z = 0. The plot demonstrates that ripples in the incident beam are restricted to the vicinity of where the beam is introduced.

Fig. 6
Fig. 6

The relative error in Ex evaluated at several planes starting from the focus (z = 0) and for several PSTD simulations each having a different value of Δt. Δtmax is the maximum value of Δt for which the PSTD simulation is stable as given by the Courant stability criterion [17].

Fig. 7
Fig. 7

Images of the magnitude and relative error of each field component for the analytic and PSTD cases on a plane 4λ beyond the focus. Each field magnitude is normalized by the analytic on-axis and in-focus value of Ex. The error is calculated according to | E { x , y , z } Analytic E { x , y , z } Analytic | / max { | E { x , y , z } Analytic | } , where the numerator is the maximum in order to avoid spurious results when the field has a small magnitude. Best viewed on screen.

Fig. 8
Fig. 8

a) The relative error in each field component, evaluated on a plane 4λ behind the focus as the grid spacing is varied; and b) The relative error in Ex evaluated at several planes starting from the focus (z = 0) and for several PSTD simulations each having a different lateral dimension.

Tables (1)

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Table 1 Summary of simulation parameters used in each test, where Δ is the grid spacing employed in all directions.

Equations (26)

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× E i ω μ H = J * H = ρ * μ × H + i ω ε E = J E = ρ ε ,
E ( x , y , z ) = 1 4 π S [ i ω μ ( n × H ) G + ( n × E ) × G + ( n E ) G ] d a ,
J s = n × H , J s * = n × E , η = ε n E
H y t = 1 μ [ E z x E x z J y * ] E x t = 1 ε [ H y z J x ] E z t = 1 ε [ H y x J z ] ,
H y ( i , k + 1 ; n t + 1 ) = H y ( i , k + 1 ; n t ) + Δ t μ Δ x z [ E z ( i + 1 / 2 , k + 1 ; n t + 1 / 2 ) E z ( i 1 / 2 , k + 1 ; n t + 1 / 2 ) + E x ( i , k + 1 / 2 ; n t + 1 / 2 ) E x ( i , k + 3 / 2 ; n t + 1 / 2 ) Δ x z J y * ( i , k + 1 ; n t + 1 / 2 ) ]
E x ( i , k + 1 / 2 ; n t + 1 / 2 ) = E x ( i , k + 1 / 2 ; n t 1 / 2 ) + Δ t ε Δ x z [ H y ( i , k ; n t ) H y ( i , k + 1 ; n t ) Δ x z J x ( i , k + 1 / 2 ; n t ) ]
E z ( i 1 / 2 , k + 1 ; n t + 1 / 2 ) = E z ( i 1 / 2 , k + 1 ; n t 1 / 2 ) + Δ t ε Δ x z [ H y ( i , k + 1 ; n t ) H y ( i 1 , k + 1 ; n t ) Δ x z J z ( i 1 / 2 , k + 1 ; n t ) ] ,
H y ( i , k s ; n t ) = { H y ( i , k s ; n t ) } ( 5 ) + Δ t μ Δ x z E x , i ( i , k s + 1 / 2 ; n t + 1 / 2 ) .
X n = k = 0 N 1 x k exp ( i n k 2 π / N ) , n = 0 , 1 , , N 1
x k = 1 N n = 0 N 1 X n exp ( i n k 2 π / N ) , k = 0 , 1 , , N 1 ,
a n = { n , 0 n N 2 n N , N 2 < n N 1 .
S n ( Δ ) = { exp ( i a n 2 π Δ / N ) ) , n N / 2 cos ( a n 2 π Δ / N ) , n = N / 2 ,
x k + Δ = 1 N n = 0 N 1 S n ( Δ ) X n exp ( i n k 2 π / N ) , k = 0 , 1 , , N 1 = IDFT { S n ( Δ ) DFT { x k } } ,
x k = lim Δ 0 x k + Δ x k Δ
x k ± 1 / 2 = lim Δ 0 x k ± 1 / 2 + Δ x k ± 1 / 2 Δ
x k = IDFT { D n c DFT { x k } }
x k ± 1 / 2 = IDFT { D n s ± DFT { x k } } ,
D n c = { i a n 2 π / N , n N / 2 0 , n = N / 2
D n s ± = { exp ( ± i a n π / N ) i a n 2 π / N , n N / 2 π , n = N / 2 ,
J s = k ^ × H i , J s * = k ^ × E i , η = ε k ^ E i .
H y ( i , k + 1 ; n t + 1 ) = H y ( i , k + 1 ; n t ) + Δ t μ Δ x z [ IDFT i { D n s DFT i { E z ( i + 1 / 2 , k + 1 ; n t + 1 / 2 ) } IDFT k { D n s + DFT k { E x ( i , k + 1 / 2 ; n t + 1 / 2 ) } δ k s , k + 1 Δ x z J y * ( i , k + 1 ; n t + 1 / 2 ) ]
E x ( i , k + 1 / 2 ; n t + 1 / 2 ) = E x ( i , k + 1 / 2 ; n t 1 / 2 ) Δ t ε Δ x z [ IDFT k { D n s + DFT k { H y ( i , k ; n t ) } } ]
E z ( i 1 / 2 , k + 1 ; n t + 1 / 2 ) = E z ( i 1 / 2 , k + 1 ; n t 1 / 2 ) + Δ t ε Δ x z [ IDFT i { D n s DFT i { H y ( i , k + 1 ; n t ) } } ] ,
H y ( i , k s ; n t ) = { H y ( i , k s ; n t ) } ( 5 ) + Δ t μ Δ x z 2 E i ( i , k s ; n t + 1 / 2 ) i ^
ε { x , y , z } ( k ) = i j | E { x , y , z } P S T D ( i Δ , j Δ , k Δ ) E { x , y , z } Analytic ( i Δ , j Δ , k Δ ) | 2 / i j | E { x , y , z } Analytic ( i Δ , j Δ , k Δ ) | 2
E x t = 1 ε [ H z y H y z J x ] ; E y t = 1 ε [ H x z H z x J y ] ; E z t = 1 ε [ H y x H x y J z ] .

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