Abstract

We present a three-dimensional model based on the finite-element method for solving the time-harmonic Maxwell equation in optics. It applies to isotropic or anisotropic dielectrics and metals and to many configurations such as an isolated scatterer in a multilayer, bi-gratings, and crystals. We discuss the application of the model to near-field optical recording.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  5. V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
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  6. N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
    [CrossRef]
  7. J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
    [CrossRef]
  8. W. Chew, C. Lu, and Y. Wang, "Efficient computation of three-dimensional scattering of vector electromagnetic waves," J. Opt. Soc. Am. A 11, 1528-1537 (1994).
    [CrossRef]
  9. K. Yee, "Numerical solutions of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
    [CrossRef]
  10. K. Shlager and J. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas Propag. Mag. 37, 39-56 (1995).
    [CrossRef]
  11. A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, 1st ed. (Artech House, 1998).
  12. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  13. J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).
    [CrossRef]
  14. W.-C. Liu and D. P. Tsai, "Optical tunneling effect of surface plasmon polaritons and localized surface plasmon resonance," Phys. Rev. B 65, 155423 (2002).
    [CrossRef]
  15. G. Bao, "Finite element approximation of time harmonic waves in periodic structures," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 32, 1155-1169 (1995).
    [CrossRef]
  16. G. Bao and H. Yang, "A least-squares finite element analysis for diffraction problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 37, 665-682 (2000).
  17. J. Nédélec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
    [CrossRef]
  18. J. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
    [CrossRef]
  19. G. Mur and A. de Hoop, "A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media," IEEE Trans. Magn. MAG-21, 2188-2191 (1985).
    [CrossRef]
  20. P. Monk, Finite Element Methods for Maxwell's Equations, 1st ed. (Oxford U. Press, 2003).
    [CrossRef]
  21. G. C. Cohen, High-Order Numerical Methods for Transient Wave Equations, 1st ed. (Springer, 2001).
  22. P. Monk and L. Demkowicz, "Discrete compactness and the approximation of Maxwell's equations in R3," Math. Comput. 70, 507-523 (2001).
    [CrossRef]
  23. W. Rachowicz and L. Demkowicz, "An hp-adaptive finite element method for electromagnetics. Part II: A 3D implementation," Int. J. Numer. Methods Eng. 53, 147-180 (2002).
    [CrossRef]
  24. C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
    [CrossRef]
  25. C. Geuzaine, "High order hybrid finite element schemes for Maxwell's equations taking thin structures and global quantities into account," Ph.D. thesis (Université de Liège, Liège, Belgium, 2001).
  26. T. Eibert and V. Hansen, "3-D FEM/BEM-hybrid approach based on a general formulation of Huygen's principle for planar layered media," IEEE Trans. Microwave Theory Tech. 45, 1105-1112 (1997).
    [CrossRef]
  27. W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
    [CrossRef]
  28. M. Benzi, "Preconditioning techniques for large linear systems: a survey," J. Comput. Phys. 182, 418-477 (2002).
    [CrossRef]
  29. Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
    [CrossRef]
  30. G. Mur and A. de Hoop, "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. Magn. 28, 1130-1133 (1992).
    [CrossRef]
  31. J. Gozani, A. Nachshon, and E. Trukel, "Conjugate gradient coupled with multigrid for an indefinite problem," in Advances in Computer Methods for Partial Differential Equations V (Springfield, 1984), pp. 425-427.
  32. H. Elman, O. Ernst, and D. O'Leary, "A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations," SIAM J. Sci. Comput. (USA) 23, 1291-1315 (2002).
    [CrossRef]
  33. S. Kim, "Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media," SIAM J. Sci. Comput. (USA) 24, 684-701 (2003).
    [CrossRef]
  34. Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, "A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation," SIAM J. Sci. Comput. (USA) 27, 1471-1492 (2006).
    [CrossRef]
  35. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (Society for Industrial and Applied Mathematics, 2003).
    [CrossRef]
  36. A. Kononov, X. Wei, and H. Urbach, "An efficient preconditioner for the finite element method applied to the time-harmonic Maxwell equations," J. Comput. Phys. (to be published).
  37. F. Zijp, M. van der Mark, J. Lee, and C. Verschuren, "Near-field read-out of a 50-GB first-surface disk with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system," in Proc. SPIE 5380, 209-223 (2004).
  38. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]

2006 (1)

Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, "A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation," SIAM J. Sci. Comput. (USA) 27, 1471-1492 (2006).
[CrossRef]

2004 (1)

Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
[CrossRef]

2003 (1)

S. Kim, "Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media," SIAM J. Sci. Comput. (USA) 24, 684-701 (2003).
[CrossRef]

2002 (4)

M. Benzi, "Preconditioning techniques for large linear systems: a survey," J. Comput. Phys. 182, 418-477 (2002).
[CrossRef]

H. Elman, O. Ernst, and D. O'Leary, "A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations," SIAM J. Sci. Comput. (USA) 23, 1291-1315 (2002).
[CrossRef]

W. Rachowicz and L. Demkowicz, "An hp-adaptive finite element method for electromagnetics. Part II: A 3D implementation," Int. J. Numer. Methods Eng. 53, 147-180 (2002).
[CrossRef]

W.-C. Liu and D. P. Tsai, "Optical tunneling effect of surface plasmon polaritons and localized surface plasmon resonance," Phys. Rev. B 65, 155423 (2002).
[CrossRef]

2001 (1)

P. Monk and L. Demkowicz, "Discrete compactness and the approximation of Maxwell's equations in R3," Math. Comput. 70, 507-523 (2001).
[CrossRef]

2000 (1)

G. Bao and H. Yang, "A least-squares finite element analysis for diffraction problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 37, 665-682 (2000).

1999 (1)

C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
[CrossRef]

1997 (2)

T. Eibert and V. Hansen, "3-D FEM/BEM-hybrid approach based on a general formulation of Huygen's principle for planar layered media," IEEE Trans. Microwave Theory Tech. 45, 1105-1112 (1997).
[CrossRef]

J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
[CrossRef]

1996 (1)

1995 (3)

G. Bao, "Finite element approximation of time harmonic waves in periodic structures," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 32, 1155-1169 (1995).
[CrossRef]

K. Shlager and J. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas Propag. Mag. 37, 39-56 (1995).
[CrossRef]

J. Judkins and R. Ziolkowski, "Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings," J. Opt. Soc. Am. A 12, 1974-1983 (1995).
[CrossRef]

1994 (3)

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

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

W. Chew, C. Lu, and Y. Wang, "Efficient computation of three-dimensional scattering of vector electromagnetic waves," J. Opt. Soc. Am. A 11, 1528-1537 (1994).
[CrossRef]

1992 (2)

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

G. Mur and A. de Hoop, "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. Magn. 28, 1130-1133 (1992).
[CrossRef]

1990 (1)

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
[CrossRef]

1986 (1)

J. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

1985 (1)

G. Mur and A. de Hoop, "A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media," IEEE Trans. Magn. MAG-21, 2188-2191 (1985).
[CrossRef]

1983 (1)

1982 (1)

1980 (1)

J. Nédélec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

1978 (1)

1966 (1)

K. Yee, "Numerical solutions of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Bao, G.

G. Bao and H. Yang, "A least-squares finite element analysis for diffraction problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 37, 665-682 (2000).

G. Bao, "Finite element approximation of time harmonic waves in periodic structures," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 32, 1155-1169 (1995).
[CrossRef]

Benzi, M.

M. Benzi, "Preconditioning techniques for large linear systems: a survey," J. Comput. Phys. 182, 418-477 (2002).
[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]

Chew, W.

J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
[CrossRef]

W. Chew, C. Lu, and Y. Wang, "Efficient computation of three-dimensional scattering of vector electromagnetic waves," J. Opt. Soc. Am. A 11, 1528-1537 (1994).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Cohen, G. C.

G. C. Cohen, High-Order Numerical Methods for Transient Wave Equations, 1st ed. (Springer, 2001).

de Hoop, A.

G. Mur and A. de Hoop, "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. Magn. 28, 1130-1133 (1992).
[CrossRef]

G. Mur and A. de Hoop, "A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media," IEEE Trans. Magn. MAG-21, 2188-2191 (1985).
[CrossRef]

Demkowicz, L.

W. Rachowicz and L. Demkowicz, "An hp-adaptive finite element method for electromagnetics. Part II: A 3D implementation," Int. J. Numer. Methods Eng. 53, 147-180 (2002).
[CrossRef]

P. Monk and L. Demkowicz, "Discrete compactness and the approximation of Maxwell's equations in R3," Math. Comput. 70, 507-523 (2001).
[CrossRef]

Dular, P.

C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
[CrossRef]

Eibert, T.

T. Eibert and V. Hansen, "3-D FEM/BEM-hybrid approach based on a general formulation of Huygen's principle for planar layered media," IEEE Trans. Microwave Theory Tech. 45, 1105-1112 (1997).
[CrossRef]

Elman, H.

H. Elman, O. Ernst, and D. O'Leary, "A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations," SIAM J. Sci. Comput. (USA) 23, 1291-1315 (2002).
[CrossRef]

Enghetta, N.

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

Erlangga, Y. A.

Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, "A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation," SIAM J. Sci. Comput. (USA) 27, 1471-1492 (2006).
[CrossRef]

Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
[CrossRef]

Ernst, O.

H. Elman, O. Ernst, and D. O'Leary, "A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations," SIAM J. Sci. Comput. (USA) 23, 1291-1315 (2002).
[CrossRef]

Gaylord, T. K.

Geuzaine, C.

C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
[CrossRef]

C. Geuzaine, "High order hybrid finite element schemes for Maxwell's equations taking thin structures and global quantities into account," Ph.D. thesis (Université de Liège, Liège, Belgium, 2001).

Gozani, J.

J. Gozani, A. Nachshon, and E. Trukel, "Conjugate gradient coupled with multigrid for an indefinite problem," in Advances in Computer Methods for Partial Differential Equations V (Springfield, 1984), pp. 425-427.

Hansen, V.

T. Eibert and V. Hansen, "3-D FEM/BEM-hybrid approach based on a general formulation of Huygen's principle for planar layered media," IEEE Trans. Microwave Theory Tech. 45, 1105-1112 (1997).
[CrossRef]

Judkins, J.

Kim, S.

S. Kim, "Multigrid simulation for high-frequency solutions of the Helmholtz problem in heterogeneous media," SIAM J. Sci. Comput. (USA) 24, 684-701 (2003).
[CrossRef]

Knop, K.

Kononov, A.

A. Kononov, X. Wei, and H. Urbach, "An efficient preconditioner for the finite element method applied to the time-harmonic Maxwell equations," J. Comput. Phys. (to be published).

Lee, J.

F. Zijp, M. van der Mark, J. Lee, and C. Verschuren, "Near-field read-out of a 50-GB first-surface disk with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system," in Proc. SPIE 5380, 209-223 (2004).

Legros, W.

C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
[CrossRef]

Li, L.

Liu, W.-C.

W.-C. Liu and D. P. Tsai, "Optical tunneling effect of surface plasmon polaritons and localized surface plasmon resonance," Phys. Rev. B 65, 155423 (2002).
[CrossRef]

Lu, C.

J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
[CrossRef]

W. Chew, C. Lu, and Y. Wang, "Efficient computation of three-dimensional scattering of vector electromagnetic waves," J. Opt. Soc. Am. A 11, 1528-1537 (1994).
[CrossRef]

Meys, B.

C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn. 35, 1442-1445 (1999).
[CrossRef]

Moharam, M. G.

Monk, P.

P. Monk and L. Demkowicz, "Discrete compactness and the approximation of Maxwell's equations in R3," Math. Comput. 70, 507-523 (2001).
[CrossRef]

P. Monk, Finite Element Methods for Maxwell's Equations, 1st ed. (Oxford U. Press, 2003).
[CrossRef]

Mur, G.

G. Mur and A. de Hoop, "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. Magn. 28, 1130-1133 (1992).
[CrossRef]

G. Mur and A. de Hoop, "A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media," IEEE Trans. Magn. MAG-21, 2188-2191 (1985).
[CrossRef]

Murphy, W.

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

Nachshon, A.

J. Gozani, A. Nachshon, and E. Trukel, "Conjugate gradient coupled with multigrid for an indefinite problem," in Advances in Computer Methods for Partial Differential Equations V (Springfield, 1984), pp. 425-427.

Nédélec, J.

J. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

J. Nédélec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

O'Leary, D.

H. Elman, O. Ernst, and D. O'Leary, "A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations," SIAM J. Sci. Comput. (USA) 23, 1291-1315 (2002).
[CrossRef]

Oosterlee, C. W.

Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, "A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation," SIAM J. Sci. Comput. (USA) 27, 1471-1492 (2006).
[CrossRef]

Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
[CrossRef]

Rachowicz, W.

W. Rachowicz and L. Demkowicz, "An hp-adaptive finite element method for electromagnetics. Part II: A 3D implementation," Int. J. Numer. Methods Eng. 53, 147-180 (2002).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rokhlin, V.

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

V. Rokhlin, "Rapid solution of integral equations of scattering theory in two dimensions," J. Comput. Phys. 36, 414-439 (1990).
[CrossRef]

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (Society for Industrial and Applied Mathematics, 2003).
[CrossRef]

Schneider, J.

K. Shlager and J. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas Propag. Mag. 37, 39-56 (1995).
[CrossRef]

Shlager, K.

K. Shlager and J. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas Propag. Mag. 37, 39-56 (1995).
[CrossRef]

Song, J.

J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
[CrossRef]

Taflove, A.

A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, 1st ed. (Artech House, 1998).

Trukel, E.

J. Gozani, A. Nachshon, and E. Trukel, "Conjugate gradient coupled with multigrid for an indefinite problem," in Advances in Computer Methods for Partial Differential Equations V (Springfield, 1984), pp. 425-427.

Tsai, D. P.

W.-C. Liu and D. P. Tsai, "Optical tunneling effect of surface plasmon polaritons and localized surface plasmon resonance," Phys. Rev. B 65, 155423 (2002).
[CrossRef]

Urbach, H.

A. Kononov, X. Wei, and H. Urbach, "An efficient preconditioner for the finite element method applied to the time-harmonic Maxwell equations," J. Comput. Phys. (to be published).

van der Mark, M.

F. Zijp, M. van der Mark, J. Lee, and C. Verschuren, "Near-field read-out of a 50-GB first-surface disk with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system," in Proc. SPIE 5380, 209-223 (2004).

Vassiliou, M.

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

Verschuren, C.

F. Zijp, M. van der Mark, J. Lee, and C. Verschuren, "Near-field read-out of a 50-GB first-surface disk with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system," in Proc. SPIE 5380, 209-223 (2004).

Vuik, C.

Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, "A novel multigrid-based preconditioner for the heterogeneous Helmholtz equation," SIAM J. Sci. Comput. (USA) 27, 1471-1492 (2006).
[CrossRef]

Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
[CrossRef]

Wang, Y.

Weedon, W. H.

W. C. Chew and W. H. Weedon, "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates," Microwave Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Wei, X.

A. Kononov, X. Wei, and H. Urbach, "An efficient preconditioner for the finite element method applied to the time-harmonic Maxwell equations," J. Comput. Phys. (to be published).

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Yang, H.

G. Bao and H. Yang, "A least-squares finite element analysis for diffraction problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 37, 665-682 (2000).

Yee, K.

K. Yee, "Numerical solutions of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
[CrossRef]

Zijp, F.

F. Zijp, M. van der Mark, J. Lee, and C. Verschuren, "Near-field read-out of a 50-GB first-surface disk with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system," in Proc. SPIE 5380, 209-223 (2004).

Ziolkowski, R.

Appl. Numer. Math. (1)

Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, "On a class of preconditioners for the study of the Helmholtz equation," Appl. Numer. Math. 50, 405-425 (2004).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

K. Shlager and J. Schneider, "A selective survey of the finite-difference time-domain literature," IEEE Antennas Propag. Mag. 37, 39-56 (1995).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

N. Enghetta, W. Murphy, V. Rokhlin, and M. Vassiliou, "The fast multipole method (FMM) for electromagnetic scattering problems," IEEE Trans. Antennas Propag. 40, 634-642 (1992).
[CrossRef]

J. Song, C. Lu, and W. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propag. 45, 1488-1493 (1997).
[CrossRef]

K. Yee, "Numerical solutions of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
[CrossRef]

IEEE Trans. Magn. (3)

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Figures (14)

Fig. 1
Fig. 1

Example of a geometry of the CD Ω for a nonperiodic configuration.

Fig. 2
Fig. 2

Example of a configuration that is periodic in the x 1 direction. The CD Ω has width along the x 1 direction equal to the period.

Fig. 3
Fig. 3

Example of a configuration that is periodic in two directions in the ( x 1 , x 2 ) plane. The vectors a 1 and a 2 span the unit cell A in the x 3 = 0 plane and Ω = A × ( x 3 s , x 3 l ) .

Fig. 4
Fig. 4

Optical system with a focusing lens and SIL (left) and part of the disc (right) seen from the top and in cross section (the SIL as shown is much too small because its radius is 0.5 mm .)

Fig. 5
Fig. 5

Amplitudes of the E x i (top left), E y i (top right), and E z i components (bottom) of the incident field in the focal plane of the objective lens. The effective numerical aperture is 1.9.

Fig. 6
Fig. 6

Amplitudes of the E x (left) and the E y components (right) of the total electric field in a plane inside the air gap and parallel to the surface of the disc at a distance of 5 nm from it. The Gaussian beam is TE polarized, and the incident spot is focused on the center of a pit.

Fig. 7
Fig. 7

Modulus of the electric field as a function of x 3 in a SIL–air–Si three-layer system, calculated analytically and numerically on different meshes with the indicated number of points per wavelength. The incident plane wave is oblique, and the angle between the wave vector and the x 3 axis is 30 deg . Both S polarization (left) P polarization (right) are shown.

Fig. 8
Fig. 8

Mean error in the modulus of the electric field shown in Fig. 7 is plotted as a function of the number of points per wavelength. Both polarizations are considered.

Fig. 9
Fig. 9

Cross section of the amplitude of the E x component of the total field for the case of Fig. 6 in the ( x , z ) plane (left) and the ( y , z ) plane (right). Both planes are through the center of the spot.

Fig. 10
Fig. 10

Amplitudes of the E x (left) and the E y components (right) of the total electric field in a plane inside the air gap and parallel to the surface of the disc at a distance of 5 nm from it. The Gaussian beam is TM polarized, and the incident spot is focused on the center of a pit.

Fig. 11
Fig. 11

Amplitudes of the E x (left) and the E y components (right) of the total electric field in a plane inside the air gap and parallel to the surface of the disc at a distance of 5 nm . The Gaussian beam is TE polarized, and the incident spot is focused in the middle between two pits.

Fig. 12
Fig. 12

Amplitudes of the E x (left) and the E y components (right) of the total electric field in a plane inside the air gap and parallel to the surface of the disc at a distance of 5 nm . The Gaussian beam is TM polarized, and the incident spot is focused in the middle between two pits.

Fig. 13
Fig. 13

Intensity patterns of the reflected field (i.e., E x 2 + E y 2 of the reflected far field) when the incident Gaussian beam is TE polarized and the spot is focused in the center of a pit (left) and in the middle between two pits (right). The normalized detected intensities are 0.4795 (left) and 0.3724 (right), respectively.

Fig. 14
Fig. 14

Intensity patterns of the reflected field when the incident Gaussian beam is TM polarized and the spot is focused in the center of a pit (left) or in the middle between two pits (right). The normalized detected intensities are 0.4862 (left) and 0.4797 (right).

Tables (1)

Tables Icon

Table 1 Values of α ͇ , β ͇ , F, and G When the Total or Scattered Electric or Magnetic Field Is Calculated

Equations (78)

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x 3 ( 0 ) > x 3 ( 1 ) > > x 3 ( N 1 ) .
{ ϵ ̃ r ( x 3 ) = ϵ r ( j ) μ ̃ r ( x 3 ) = μ r ( j ) } for x 3 ( j ) < x 3 < x 3 ( j 1 ) , j = 0 , , N .
L n p = { = i = 1 n p i a i ; for integer i } .
A = { r = i = 1 n p y i a i ; 1 2 < y 1 , , y n p < 1 2 } .
m = i = 1 n p m i b i ,
b i a j = 2 π δ i j , for 1 i , j n p ,
b i e ̂ j = 0 , for 1 i n p and for n p + 1 j 3 ,
b 1 = ( 2 π p , 0 , 0 ) T ;
b 1 = 2 π a 2 × e ̂ 3 a 1 ( a 2 × e ̂ 3 ) ,
b 2 = 2 π a 1 × e ̂ 3 a 2 ( a 1 × e ̂ 3 ) ;
b 1 = 2 π a 2 × a 3 a 1 ( a 2 × a 3 ) ,
b 2 = 2 π a 1 × a 3 a 2 ( a 1 × a 3 ) ,
b 3 = 2 π a 1 × a 2 a 3 ( a 1 × a 2 ) .
B = { k = i = 1 n p η i b i ; 1 2 η 1 , , η n p < 1 2 } .
ϵ r ͇ ( r ) = Re [ ϵ r ͇ ( r ) ] + i Im [ ϵ r ͇ ( r ) ] ,
μ r ͇ ( r ) = Re [ μ r ͇ ( r ) ] + i Im [ μ r ͇ ( r ) ] ,
ϵ r ͇ ( r + ) = ϵ r ͇ ( r ) , μ r ͇ ( r + ) = μ r ͇ ( r ) .
ϵ r ͇ , μ r ͇ , ϵ r ͇ 1 , μ r ͇ 1 L ( R 3 ) .
r ( exp ( i b 1 r ) , , exp ( i b n p r ) ) ,
Ω T n p × i = n p + 1 3 ( x i s , x i l ) ,
R n p × i = n p + 1 3 ( x i s , x i l )
x 3 s x 3 ( N 1 ) , x 3 l x 3 ( 0 ) ;
Ω = T n p × [ i = n + p + 1 3 [ x i s , x i l ] \ i = n + p + 1 3 ( x i s , x i l ) ] .
Ω ̃ T n p × i = n p + 1 3 ( x i s δ i s , x i l + δ i l ) ,
k = ( k 1 , k 2 , k 3 ) T , with k i = 0 for all n p + 1 i 3 .
V k ( r ) = e i k r V ( r ) ,
V k ( r + l ) = e i k l V k ( r ) ,
F ( E i ) ( k , x n p + 1 , , x 3 ) = E i ( r ) e i k r d x 1 d x n p .
E i ( r ) = ( 1 2 π ) n p F ( E i ) ( k , x n p + 1 , , x 3 ) e i k r d k 1 d k n p = ( 1 2 π ) n p m B + m F ( E i ) ( k , x n p + 1 , , x 3 ) e i k r d k 1 d k n p = B E k i ( r ) d k 1 d k n p ,
E k i ( r ) = ( 1 2 π ) n p m F ( E i ) ( k + m , x n p + 1 , , x 3 ) e i ( k + m ) r .
J ( r ) = B J k ( r ) d k 1 d k n p ,
J k ( r ) = ( 1 2 π ) n p m F ( J ) ( k + m , x n p + 1 , , x 3 ) e i ( k + m ) r .
E ( r ) = B E k ( r ) d k 1 d k n p .
E k s = E k E k 0 , H k s = H k H k 0 .
× E k s = i ω μ 0 μ r ͇ H k s + i ω μ 0 ( μ r ͇ μ ̃ r ) H k 0 ,
× H k s = i ω ϵ 0 ϵ r ͇ E k s + i ω ϵ 0 ( ϵ r ͇ ϵ ̃ r ) E k 0 ,
E k s and H k s satisfy the outgoing radiation conditions .
E k and H k are in L loc 2 ( R 3 ) ,
ω 2 ϵ 0 μ 0 ϵ r ͇ E k × ( μ r ͇ 1 × E k ) = i ω μ 0 J k ,
ω 2 ϵ 0 μ 0 ϵ r ͇ E k s × ( μ r ͇ 1 × E k s ) = ω 2 ϵ 0 μ 0 ( ϵ r ͇ ϵ ̃ r ) E k 0 i ω μ 0 × [ ( 1 μ r ͇ 1 μ ̃ r ) H k 0 ] .
k 0 2 α ͇ U k × ( β ͇ × U k ) = F k + × G k ,
α ͇ = Re ( α ͇ ) + i Im ( α ͇ ) ,
β ͇ = Re ( β ͇ ) + i Im ( β ͇ ) ,
U k ( r ) = e i k r U ( r ) , F k ( r ) = e i k r F ( r ) , G k ( r ) = e i k r G ( r ) ,
× U k = e i k r ( i k + ) × U = e i k r k × U ,
k = i k + .
k 0 2 α ͇ U k × ( β ͇ k × U ) = F + k × G .
U H loc ( curl , T n p × R 3 n p ) ,
ζ = ( ζ 1 x 1 , ζ 2 x 2 , ζ 3 x 3 ) ,
k ζ = i k + ζ .
D ͇ ( ζ ) = [ ζ 1 0 0 0 ζ 2 0 0 0 ζ 3 ] ;
k ζ × V = i k × V + 1 det ( D ͇ ( ζ ) ) D ͇ ( ζ ) × [ D ͇ ( ζ ) V ] .
k ζ × V L 2 ( Ω ̃ ) × D ͇ ( ζ ) V L 2 ( Ω ̃ ) .
k 0 2 α ͇ U s k ζ × ( β ͇ k ζ × U s ) = F s + k × G s , in Ω ̃ .
U s = D ͇ ( ζ ) U s .
k ζ × U s = 1 det ( D ͇ ( ζ ) ) D ͇ ( ζ ) ( k × U s ) .
k 0 2 α ͇ U s k × β ͇ k × U s = F s + k × G s , in Ω ̃ ,
α ͇ = det ( D ͇ ( ζ ) ) D ͇ ( ζ ) 1 α D ͇ ( ζ ) 1 , in Ω ̃ ,
β ͇ = det ( D ͇ ( ζ ) ) 1 D ͇ ( ζ ) β ͇ D ͇ ( ζ ) , in Ω ̃ .
lim r r n × U s = lim r r n × U s , for r Ω ,
lim r r n × [ β ͇ k × U s ] + lim r r G s = lim r r n × [ β ͇ k × U s + G s ] , for r Ω .
n × U s = 0 , on Ω ̃ ;
n × β ͇ k × U s = 0 , on Ω ̃ ;
k 0 2 α ͇ U k × ( β ͇ k × U ) = F + k × G , in Ω ,
k 0 2 α ͇ U s k × ( β ͇ k × U s ) = 0 , in Ω ̃ \ Ω ,
lim r r n × U ( r ) lim r r n × U s ( r ) = lim r r n × U 0 ( r ) , for all r Ω ,
lim r r n × β ͇ k × U lim r r n × [ β ͇ k × U s ] = lim r r n × [ β ͇ k × U 0 + G s ] .
a j × U ( r + l a j ) = a j × U ( r ) ,
a j × U s ( r + l a j ) = a j × U s ( r ) ,
Ω [ k 0 2 α ͇ U V β ͇ k × U k × V ] d x 1 d x 2 d x 3 + Ω ̃ \ Ω [ k 0 2 α ͇ U s V β ͇ k × U s k × V ] d x 1 d x 2 d x 3 = Ω F V d x 1 d x 2 d x 3 + Ω G k × V d x 1 d x 2 d x 3 + Ω lim r r n × [ β ͇ k × U 0 + G G s ] V d S .
E E h L 2 + H H h L 2 C h k ,
U = m U m ϕ m , in Ω ,
U s = m U m s ϕ m , in Ω ̃ \ Ω ,
A n m ( U m U m s ) = ( F n F n s ) ,
U k ( r 0 ) = e i k U k ( r 0 + ) .
G ͇ k ( r , r 0 ) = exp ( i k r ) G ͇ ( r , r 0 ) ,
k 0 2 α ̃ G ͇ p k × β ̃ k × G ͇ p = p δ ( r r 0 ) ,
U s ( r 0 ) p = R 3 \ Ω U s [ k 0 2 α ̃ G ͇ p k × β ̃ k × G ͇ p ] = Ω lim r r n × U s ( r ) β ̃ k × G ͇ ( r , r 0 ) p d S ( r ) Ω lim r r n × [ β ̃ k × U s ( r ) ] G ͇ ( r , r 0 ) p d S ( r ) ,

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