Abstract

An algorithm for the numerical solution of the inhomogeneous Maxwell’s equations is presented. The algorithm solves the inhomogeneous vector wave equation of the electric field by writing the solution as a convergent Born series. Compared to two dimensional finite difference time domain calculations, solutions showing the same accuracy can be calculated more than three orders of magnitude faster.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Fast semi-analytical solution of Maxwell’s equations in Born approximation for periodic structures

Maxim Pisarenco, Richard Quintanilha, Mark G. M. M. van Kraaij, and Wim M. J. Coene
J. Opt. Soc. Am. A 33(4) 610-617 (2016)

Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium

Snow H. Tseng, Jethro H. Greene, Allen Taflove, Duncan Maitland, Vadim Backman, and Joseph T. Walsh
Opt. Lett. 29(12) 1393-1395 (2004)

Power-series solutions to the optical Maxwell–Bloch equations

Ljubomir Matulic and Christopher Palmer
J. Opt. Soc. Am. B 6(3) 356-363 (1989)

References

  • View by:
  • |
  • |
  • |

  1. W. Drexler and J. G. Fujimoto, eds., Optical Coherence Tomography (Springer, 2008).
    [Crossref]
  2. D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
    [Crossref]
  3. R. L. Gordon and C. A. Mack, “Lithography simulation employing rigorous solutions to Maxwell’s equations,” Proc. SPIE 3334, 176–196 (1998).
    [Crossref]
  4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
    [Crossref] [PubMed]
  5. A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
    [Crossref]
  6. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [Crossref]
  7. J. Schäfer and A. Kienle, “Scattering of light by multiple dielectric cylinders: comparison of radiative transfer and Maxwell theory,” Opt. Lett. 33, 2413–2415 (2008).
    [Crossref] [PubMed]
  8. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
    [Crossref]
  9. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [Crossref]
  10. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method (Artech House, 1995).
  11. Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15, 158–165 (1997).
    [Crossref]
  12. Q. H. Liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Modell. Electron. Networks Devices Fields 17, 299–323 (2004).
    [Crossref]
  13. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  14. M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
    [Crossref]
  15. F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
    [Crossref]
  16. G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
    [Crossref]
  17. Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. and Stat. Comput. 7, 856–869 (1986).
    [Crossref]
  18. J.-P. Schäfer, “ http://www.mathworks.com/matlabcentral/fileexchange/36831-matscat ,” Last accessed October 13, 2014, 3:47 pm.
  19. J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
    [Crossref]
  20. S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
    [Crossref]
  21. K. Moriya, “Light scattering from defects in crystals: Scattering by dislocations,” Philos. Mag. B 64, 425–445 (1991).
    [Crossref]
  22. J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
    [Crossref] [PubMed]

2016 (2)

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

2012 (2)

J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
[Crossref]

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

2008 (1)

2007 (1)

M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[Crossref]

2006 (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

2004 (1)

Q. H. Liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Modell. Electron. Networks Devices Fields 17, 299–323 (2004).
[Crossref]

1998 (1)

R. L. Gordon and C. A. Mack, “Lithography simulation employing rigorous solutions to Maxwell’s equations,” Proc. SPIE 3334, 176–196 (1998).
[Crossref]

1997 (1)

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

1996 (1)

1993 (1)

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

1992 (1)

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[Crossref]

1991 (1)

K. Moriya, “Light scattering from defects in crystals: Scattering by dislocations,” Philos. Mag. B 64, 425–445 (1991).
[Crossref]

1986 (1)

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. and Stat. Comput. 7, 856–869 (1986).
[Crossref]

1985 (1)

D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

Assous, F.

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
[Crossref]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Degond, P.

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

Gordon, R. L.

R. L. Gordon and C. A. Mack, “Lithography simulation employing rigorous solutions to Maxwell’s equations,” Proc. SPIE 3334, 176–196 (1998).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method (Artech House, 1995).

Heintze, E.

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

Heitmann, D.

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Hoekstra, A.

M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Kienle, A.

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
[Crossref]

J. Schäfer and A. Kienle, “Scattering of light by multiple dielectric cylinders: comparison of radiative transfer and Maxwell theory,” Opt. Lett. 33, 2413–2415 (2008).
[Crossref] [PubMed]

Kieß, S.

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

Krüger, B.

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Lee, S.-C.

J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
[Crossref]

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[Crossref]

Leedumrongwatthanakun, S.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

Liu, Q. H.

Q. H. Liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Modell. Electron. Networks Devices Fields 17, 299–323 (2004).
[Crossref]

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

Lynch, D. R.

D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
[Crossref]

Mack, C. A.

R. L. Gordon and C. A. Mack, “Lithography simulation employing rigorous solutions to Maxwell’s equations,” Proc. SPIE 3334, 176–196 (1998).
[Crossref]

Mackowski, D. W.

Mendach, S.

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Mishchenko, M. I.

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Moriya, K.

K. Moriya, “Light scattering from defects in crystals: Scattering by dislocations,” Philos. Mag. B 64, 425–445 (1991).
[Crossref]

Osnabrugge, G.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

Paulsen, K. D.

D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
[Crossref]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Pfannkuche, D.

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Raviart, P.

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

Rothe, T.

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

Rottler, A.

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. and Stat. Comput. 7, 856–869 (1986).
[Crossref]

Schäfer, J.

J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
[Crossref]

J. Schäfer and A. Kienle, “Scattering of light by multiple dielectric cylinders: comparison of radiative transfer and Maxwell theory,” Opt. Lett. 33, 2413–2415 (2008).
[Crossref] [PubMed]

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. and Stat. Comput. 7, 856–869 (1986).
[Crossref]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Segre, J.

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

Simon, S.

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Stark, J.

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

Strohbehn, J. W.

D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method (Artech House, 1995).

Vellekoop, I. M.

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

Yurkin, M.

M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[Crossref]

Zhao, G.

Q. H. Liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Modell. Electron. Networks Devices Fields 17, 299–323 (2004).
[Crossref]

Astrophys. J. (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[Crossref]

Int. J. Numer. Modell. Electron. Networks Devices Fields (1)

Q. H. Liu and G. Zhao, “Review of PSTD methods for transient electromagnetics,” Int. J. Numer. Modell. Electron. Networks Devices Fields 17, 299–323 (2004).
[Crossref]

J. Comput. Phys. (3)

F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segre, “On a finite-element method for solving the three-dimensional Maxwell equations,” J. Comput. Phys. 109, 222–237 (1993).
[Crossref]

G. Osnabrugge, S. Leedumrongwatthanakun, and I. M. Vellekoop, “A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media,” J. Comput. Phys. 322, 113–124 (2016).
[Crossref]

D. R. Lynch, K. D. Paulsen, and J. W. Strohbehn, “Finite element solution of Maxwell’s equations for hyperthermia treatment planning,” J. Comput. Phys. 58, 246–269 (1985).
[Crossref]

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

J. Quant. Spectrosc. Radiat. Transf. (3)

M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[Crossref]

J. Schäfer, S.-C. Lee, and A. Kienle, “Calculation of the near fields for the scattering of electromagnetic waves by multiple infinite cylinders at perpendicular incidence,” J. Quant. Spectrosc. Radiat. Transf. 113, 2113–2123 (2012).
[Crossref]

S.-C. Lee, “Scattering by closely-spaced radially-stratified parallel cylinders,” J. Quant. Spectrosc. Radiat. Transf. 48, 119–130 (1992).
[Crossref]

Microw. Opt. Technol. Lett. (1)

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

Opt. Lett. (1)

Philos. Mag. B (1)

K. Moriya, “Light scattering from defects in crystals: Scattering by dislocations,” Philos. Mag. B 64, 425–445 (1991).
[Crossref]

Phys. Med. Biol. (1)

J. Stark, T. Rothe, S. Kieß, S. Simon, and A. Kienle, “Light scattering microscopy measurements of single nuclei compared with GPU-accelerated FDTD simulations,” Phys. Med. Biol. 61, 2749–2761 (2016).
[Crossref] [PubMed]

Phys. Rev. B (1)

A. Rottler, B. Krüger, D. Heitmann, D. Pfannkuche, and S. Mendach, “Route towards cylindrical cloaking at visible frequencies using an optimization algorithm,” Phys. Rev. B 86, 245120 (2012).
[Crossref]

Proc. SPIE (1)

R. L. Gordon and C. A. Mack, “Lithography simulation employing rigorous solutions to Maxwell’s equations,” Proc. SPIE 3334, 176–196 (1998).
[Crossref]

Science (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[Crossref] [PubMed]

SIAM J. Sci. and Stat. Comput. (1)

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. and Stat. Comput. 7, 856–869 (1986).
[Crossref]

Other (4)

J.-P. Schäfer, “ http://www.mathworks.com/matlabcentral/fileexchange/36831-matscat ,” Last accessed October 13, 2014, 3:47 pm.

W. Drexler and J. G. Fujimoto, eds., Optical Coherence Tomography (Springer, 2008).
[Crossref]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
[Crossref]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-Domain Method (Artech House, 1995).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 (left) Scheme of the anti-aliasing algorithm to assign a refractive index to the simulation cells. The scatterer is a cylinder whose center is not necessarily aligned with the rectangular grid. Its circumference is denoted by the blue line. The grid cells (solid black lines) are divided into subcells (dashed black lines). For each subcell a black dot at its center denotes that the subcell is considered to be inside the cylinder. The effective refractive index of the grid cell (denoted by the gray scale) depends on the number of subcells that are considered to be inside the cylinder. (right) Our test system consists of five cylinders with a radius of 5 µm and a refractive index n2 that are placed in a homogeneous medium with a refractive index of n1. The near field is calculated in a square that contains all cylinders. From the calculated near field the far field is calculated by a near to far field transformation on the boundary of this square.
Fig. 2
Fig. 2 Near field calculated by the Born series with 10 cells per µm. The subfigures (a)–(c) and (g)–(i) show the amplitude of the electric near field for the first and second set of refractive indices, respectively. Shown are the x ((a) and (g)), the y ((b) and (h)), and the z component ((c) and (i)). Subfigures (d)–(f) and (j)–(l) show the absolute values of the respective differences to the near field calculated analytically. All values are normalized to the amplitude of the incident wave E0. For the sake of illustration the lower end of the colormap has been cut even though there are values that are smaller than the range shown in the color bar.
Fig. 3
Fig. 3 Near field calculated by FDTD with 50 cells per µm. The subfigures (a)–(c) and (g)–(i) show the amplitude of the electric near field for the first and second set of refractive indices, respectively. Shown are the x ((a) and (g)), the y ((b) and (h)), and the z component ((c) and (i)). Subfigures (d)–(f) and (j)–(l) show the absolute values of the respective differences to the near field calculated analytically. All values are normalized to the amplitude of the incident wave E0. For the sake of illustration the lower end of the colormap has been cut even though there are values that are smaller than the range shown in the color bar.
Fig. 4
Fig. 4 Calculated far field for the five cylinders shown in Fig. 1 with a refractive index n1 = 1.0 of the medium and a refractive index n2 = 1.2 of the cylinders. The figures show the components parallel ((a) and (b)) and perpendicular ((c) and (d)) to the scattering plane. A magnification of the region of backward scattering is shown in (e) and (f). The figures compare the results from the analytical theory in (a), (c), and (e) with the present algorithm using a cell size of 100 nm and in (b), (d), and (f) with the FDTD calculation using a cell size of 20 nm.
Fig. 5
Fig. 5 Similar data as in Fig. 4 but for a refractive index n1 = 1.36 of the medium and a refractive index n2 = 1.39 of the cylinders. The figures compare the results from the analytical theory in (a), (c), and (e) with the present algorithm using a cell size of 100 nm and in (b), (d), and (f) with the FDTD calculation using a cell size of 12.5 nm.
Fig. 6
Fig. 6 Run time of the present algorithm and the FDTD method for different cell sizes. The value on the abscissa is the accuracy that is defined in Eq. (76) and is determined from a comparison with the respective analytical results. The results from the Born series and the FDTD are fitted with a power law for both sets of refractive indices. A comparison of these fits shows that for an accuracy of 0.1 the Born series is up to a factor of 7300 faster, depending on the simulated system.

Tables (1)

Tables Icon

Table 1 Positions of the five cylinders in the calculated system relative to the center of the simulation area. The system is depicted in Fig. 1.

Equations (76)

Equations on this page are rendered with MathJax. Learn more.

( ϵ E ) = 0 ,
( μ H ) = 0 ,
× E = i ω μ H , and
× H = i ω ϵ E .
× ( × E ) = ω 2 μ ϵ E
( E ) 2 E = k 2 E Helmholtz equation
2 ψ + k 2 ψ = s
( E ln ( ϵ ) ) = 0
E = 1 ϵ ( ϵ E ) ( ϵ E ) ϵ ϵ 2 = E ln ( ϵ )
2 ψ + ( k 0 2 + i δ ) ψ = ( k 2 k 0 2 i δ ) ψ s = v ψ s ,
2 g ( r ) + ( k 0 2 + i δ ) g ( r ) = δ D ( r ) ,
( k x 2 + k y 2 + k z 2 ) g ^ ( k ) + ( k 0 2 + i δ ) g ^ ( k ) = 1 ,
g ^ ( k ) = 1 k x 2 + k y 2 + k z 2 k 0 2 i δ .
Ψ = G V Ψ + G S
Ψ = ( 𝟙 G V ) 1 G S
Ψ = n = 0 ( G V ) n G S .
γ = i δ V .
i δ ( k 2 k 0 2 i δ ) .
γ Ψ = γ G V Ψ + γ G S .
Ψ = [ 𝟙 ( γ G V γ + 𝟙 ) ] 1 γ G S
Ψ = n = 0 ( γ G V γ + 𝟙 ) n γ G S .
Ψ n + 1 = ( γ G V γ + 𝟙 ) Ψ n + γ G S
δ max r | k 2 ( k ) k 0 2 |
U = 𝟙 + 2 i δ G
Re ( k 2 k 0 2 ) = 0
Im ( k 2 k 0 2 ) = δ
δ > max r | k 2 ( r ) k 0 2 |
2 E ( E ) + ( k 0 2 + i δ ) E = v E s .
2 g ( r ) ( g ( r ) ) + ( k 0 2 + i δ ) g ( r ) = δ D ( r ) 𝟙 ,
( k y 2 k z 2 k x k y k x k z k y k x k x 2 k z 2 k y k z k z k x k z k y k x 2 k y 2 ) g ^ ( k ) + ( k 0 2 + i δ ) g ^ ( k ) = 𝟙 ,
( k y 2 k x k y 0 k y k x k x 2 0 0 0 k x 2 k y 2 ) g ^ ( k ) + ( k 0 2 + i δ ) g ^ ( k ) = 𝟙
g ^ m , n ( k ) = δ m , n k 2 k 0 2 i δ + k m k n ( k 2 k 0 2 i δ ) ( k 0 2 i δ ) ( 1 δ m , z ) ( 1 δ n , z ) ,
2 D 1 [ 1 k 2 + a 2 ] = K 0 ( a r ) 2 π .
2 D 1 [ k m k n ( k 2 + a 2 ) a 2 ] = r m r n K 0 ( a x 2 + y 2 ) 2 π a 2 ,
g m , n ( r ) = K 0 ( a r ) 2 π δ m , n + K 1 ( a r ) 2 π a r δ m , n ( 1 δ m , z ) K 2 ( a r ) r m r n 2 π r 2 ( 1 δ m , n ) ( 1 δ n , z )
g ^ m , n ( k ) = δ m , n k 2 k 0 2 i δ + k m k n ( k 2 k 0 2 i δ ) ( k 0 2 i δ ) .
3 D 1 [ 1 k 2 + a 2 ] = e a r 4 π r ,
3 D 1 [ k m k n ( k 2 + a 2 ) a 2 ] = r m r n e a x 2 + y 2 + z 2 4 π a 2 x 2 + y 2 + z 2 .
g m , n ( r ) = δ m , n e a r 4 π r + 1 + a r a 2 r 2 δ m , n e a r 4 π r a 2 r 2 + 3 a r + 3 a 2 r 2 r m r n r 2 e a r 4 π r ,
E ( r ) = d r g ( r r ) ( v ( r ) E ( r ) + s ( r ) )
Ψ = G V Ψ + G S
γ Ψ = γ G V Ψ + γ G S
γ = i δ V
δ > max r | k 2 k 0 2 | .
Ψ = [ 𝟙 ( γ G V γ + 𝟙 ) ] 1 γ G S
Ψ = n = 0 ( γ G V γ + 𝟙 ) n γ G S .
Ψ n + 1 = ( γ G V γ + 𝟙 ) Ψ n + γ G S = Ψ n + γ ( G ( V Ψ n + S ) Ψ n )
Ψ n + 1 = Ψ n + γ ( 1 G ^ ( V Ψ n + S ) Ψ n ) ,
G ^ = G 1 .
U = 𝟙 + 2 i δ G
U ^ = U 1
u ^ ( k ) = 𝟙 + 2 i δ g ^ ( k )
R = ( cos ( θ ) cos ( ϕ ) cos ( θ ) sin ( ϕ ) sin ( θ ) sin ( ϕ ) cos ( ϕ ) 0 sin ( θ ) cos ( ϕ ) sin ( θ ) sin ( ϕ ) cos ( θ ) ) .
g ^ = R T ( 1 k 2 k 0 2 i δ 0 0 0 1 k 2 k 0 2 i δ 0 0 0 1 k 0 2 i δ ) R = i 2 δ R T 𝟙 R i 2 δ R T ( k 2 k 0 2 + i δ k 2 k 0 2 i δ 0 0 0 k 2 k 0 2 + i δ k 2 k 0 2 i δ 0 0 0 k 0 2 + i δ k 0 2 i δ ) R .
u ^ = R T ( k 2 k 0 2 + i δ k 2 k 0 2 i δ 0 0 0 k 2 k 0 2 + i δ k 2 k 0 2 i δ 0 0 0 k 0 2 + i δ k 0 2 i δ ) R .
r n = 1 G ^ ( V Ψ n + S ) Ψ n ,
( E s ) 2 ( E i + E s ) = k 2 ( E i + E s ) .
2 E i + k m 2 E i = 0
( E s ) 2 E s = k 2 E s + ( k 2 k m 2 ) E i .
( E s ) 2 E s + ( k 0 2 + i δ ) E s = v E s ( k m 2 k 2 ) E i .
s = ( k m 2 k 2 ) E i .
k 2 ( d ) = k m 2 + α 2 ( N α d + 2 i k m d ) ( α d ) N 1 P N ( d ) N !
P N ( d ) = n = 0 N ( α d ) n n !
O ( N D )
O ( N D t Δ t ) .
Δ t = S c Δ ,
t = n c D L 2 = n c D L .
O ( L D + 1 Δ D + 1 )
k n = 2 Δ sin 1 ( 1 S sin ( π S Δ λ ) ) k a + π 3 ( 1 S 2 ) Δ 2 3 λ 3
ϕ max ( k n k a ) L π 3 ( 1 S 2 ) Δ 2 L 3 λ 3 ,
Δ 3 λ 3 ϕ max π 3 ( 1 S 2 ) L
O ( L 1.5 ( D + 1 ) λ 1.5 ( D + 1 ) ) .
O ( N D log ( N ) ) .
O ( L D + 1 λ D + 1 log ( L λ ) ) .
T L 3 λ 3 log ( L λ ) .
A = 1 2 mediam j ( | | E num ( ϕ j ) | 2 | E analyt ( ϕ j ) | 2 | | E analyt ( ϕ j ) | 2 ) + 1 2 mediam j ( | | E num ( ϕ j ) | 2 | E analyt ( ϕ j ) | 2 | | E analyt ( ϕ j ) | 2 ) ,

Metrics