Abstract

An accurate global model is proposed for a two-dimensional probe–sample system of photon scanning tunneling microscopy in near-field optics. A coupling of a finite-element method in the inhomogeneous sample and a boundary integral method on the artificial boundary of the truncated domain is developed. Numerical experiments are included to demonstrate the effectiveness of the proposed method and to show the features of wave propagation in photon scanning tunneling microscopy.

© 2008 Optical Society of America

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  1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Vol. 93of the Applied Mathematical Sciences Series (Springer-Verlag, 1998).
  2. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, 2003).
  3. C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657-699 (1996).
    [CrossRef]
  4. E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189-195 (1992).
    [CrossRef] [PubMed]
  5. D. Courjon and C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989-1028 (1994).
    [CrossRef]
  6. P. Carney and J. Schotland, “Near-field tomography,” in Inside Out: Inverse Problems and Applications, G.Uhlmann, ed. (Cambridge U. Press, 2003), pp. 133-168.
  7. G. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726-735 (1982).
    [CrossRef]
  8. D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
    [CrossRef]
  9. P. Carney and J. Schotland, “Determination of three-dimensional structure in photon scanning tunneling mircoscopy,” J. Opt. A, Pure Appl. Opt. 4, S140-S144 (2002).
    [CrossRef]
  10. P. A. Temple, “Total internal reflection microscopy: A surface inspection technique,” Appl. Opt. 20, 2656-2664 (1981).
    [CrossRef] [PubMed]
  11. A. Cvitkovic, N. Ocelic, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 15, 8550-8565.
    [CrossRef] [PubMed]
  12. C. Girard, “Theoretical atomic-force-microscopy study of a stepped surface: Nonlocal effects in the probe,” Phys. Rev. B 43, 8822-8828 (1991).
    [CrossRef]
  13. N. Gregersen, B. Tromborg, and S. Bozhevolnyi, “Vectorial modeling of near-field imaging with uncoated fiber probes: Transfer function and resolving power,” Appl. Opt. 45, 8739-8747 (2006).
    [CrossRef] [PubMed]
  14. S. Wang, “Analysis of probe-sample interaction in near-field optical image of dielectric structure,” Microsc. Microanal. 5, 290-295 (1999).
    [CrossRef] [PubMed]
  15. J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
    [CrossRef]
  16. D. Van Labeke and D. Barchiesi, “Probes for scanning tunneling optical microscopy: A theoretical comparison,” J. Opt. Soc. Am. A 10, 2193-2201 (1993).
    [CrossRef]
  17. S. Bozhevolnyi, S. Berntsen, and E. Bozhevolnaya, “Extension of the macroscopic model for reflection near-field microscopy: Regularization and image formulation,” J. Opt. Soc. Am. A 11, 609-617 (1994).
    [CrossRef]
  18. J.-J. Greffet and R. Carminati, “Image formulation in near-field optics,” Prog. Surf. Sci. 56, 133-237 (1997).
    [CrossRef]
  19. A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
    [CrossRef]
  20. J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
    [CrossRef]
  21. L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
    [CrossRef]
  22. H. Furukawa and S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: Effects of multiple scattering,” Opt. Commun. 132, 170-178 (1996).
    [CrossRef]
  23. K. Tanaka, M. Tanaka, and T. Omoya, “Boundary integral equation for a two-dimensional simulator of a photon scanning tunneling microscopy,” J. Opt. Soc. Am. A 15, 1918-1931 (1998).
    [CrossRef]
  24. S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
    [CrossRef]
  25. S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
    [CrossRef]
  26. F. Brezzi and C. Johnson, “On the coupling of boundary integral and finite element methods,” Calcolo 16, 189-201 (1979).
    [CrossRef]
  27. G. N. Gatica and W. L. Wendland, “Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems,” Appl. Anal. 63, 39-75 (1996).
    [CrossRef]
  28. G.-C. Hsiao, “The coupling of BEM and FEM--a brief review,” in Boundary Element X, (Springer, 1988), Vol. 1, pp. 431-445.
  29. C. Johnson and J.-C. Nédélec, “On the coupling of boundary integral and finite element methods,” Math. Comput. 35, 1063-1079 (1980).
    [CrossRef]
  30. T. Tran, “The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: Local estimates,” Math. Comput. 64, 501-513 (1995).
  31. P. Persson and G. Strang, “A simple mesh generator in Matlab,” SIAM Rev. 46, 329-345 (2004).
    [CrossRef]
  32. J. Jin, The Finite Element Method in Electromagnetics, (Wiley, 1993).
  33. J. Coyle, “Locating the support of objects contained in a two-layered background medium in two dimensions,” Inverse Probl. 16, 275-292 (2000).
    [CrossRef]

2007

J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

2006

2005

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

2004

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

P. Persson and G. Strang, “A simple mesh generator in Matlab,” SIAM Rev. 46, 329-345 (2004).
[CrossRef]

2003

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, 2003).

P. Carney and J. Schotland, “Near-field tomography,” in Inside Out: Inverse Problems and Applications, G.Uhlmann, ed. (Cambridge U. Press, 2003), pp. 133-168.

2002

P. Carney and J. Schotland, “Determination of three-dimensional structure in photon scanning tunneling mircoscopy,” J. Opt. A, Pure Appl. Opt. 4, S140-S144 (2002).
[CrossRef]

2000

J. Coyle, “Locating the support of objects contained in a two-layered background medium in two dimensions,” Inverse Probl. 16, 275-292 (2000).
[CrossRef]

1999

S. Wang, “Analysis of probe-sample interaction in near-field optical image of dielectric structure,” Microsc. Microanal. 5, 290-295 (1999).
[CrossRef] [PubMed]

1998

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Vol. 93of the Applied Mathematical Sciences Series (Springer-Verlag, 1998).

K. Tanaka, M. Tanaka, and T. Omoya, “Boundary integral equation for a two-dimensional simulator of a photon scanning tunneling microscopy,” J. Opt. Soc. Am. A 15, 1918-1931 (1998).
[CrossRef]

A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
[CrossRef]

1997

J.-J. Greffet and R. Carminati, “Image formulation in near-field optics,” Prog. Surf. Sci. 56, 133-237 (1997).
[CrossRef]

1996

J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
[CrossRef]

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657-699 (1996).
[CrossRef]

H. Furukawa and S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: Effects of multiple scattering,” Opt. Commun. 132, 170-178 (1996).
[CrossRef]

G. N. Gatica and W. L. Wendland, “Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems,” Appl. Anal. 63, 39-75 (1996).
[CrossRef]

1995

T. Tran, “The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: Local estimates,” Math. Comput. 64, 501-513 (1995).

L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
[CrossRef]

1994

1993

1992

E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189-195 (1992).
[CrossRef] [PubMed]

1991

C. Girard, “Theoretical atomic-force-microscopy study of a stepped surface: Nonlocal effects in the probe,” Phys. Rev. B 43, 8822-8828 (1991).
[CrossRef]

1989

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
[CrossRef]

1988

G.-C. Hsiao, “The coupling of BEM and FEM--a brief review,” in Boundary Element X, (Springer, 1988), Vol. 1, pp. 431-445.

1982

G. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726-735 (1982).
[CrossRef]

1981

1980

C. Johnson and J.-C. Nédélec, “On the coupling of boundary integral and finite element methods,” Math. Comput. 35, 1063-1079 (1980).
[CrossRef]

1979

F. Brezzi and C. Johnson, “On the coupling of boundary integral and finite element methods,” Calcolo 16, 189-201 (1979).
[CrossRef]

Bainier, C.

D. Courjon and C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989-1028 (1994).
[CrossRef]

Barchiesi, D.

Berntsen, S.

Betzig, E.

E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189-195 (1992).
[CrossRef] [PubMed]

Binnig, G.

G. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726-735 (1982).
[CrossRef]

Bouju, X.

A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
[CrossRef]

Bozhevolnaya, E.

Bozhevolnyi, S.

Brezzi, F.

F. Brezzi and C. Johnson, “On the coupling of boundary integral and finite element methods,” Calcolo 16, 189-201 (1979).
[CrossRef]

Carminati, R.

J.-J. Greffet and R. Carminati, “Image formulation in near-field optics,” Prog. Surf. Sci. 56, 133-237 (1997).
[CrossRef]

Carney, P.

J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

P. Carney and J. Schotland, “Near-field tomography,” in Inside Out: Inverse Problems and Applications, G.Uhlmann, ed. (Cambridge U. Press, 2003), pp. 133-168.

P. Carney and J. Schotland, “Determination of three-dimensional structure in photon scanning tunneling mircoscopy,” J. Opt. A, Pure Appl. Opt. 4, S140-S144 (2002).
[CrossRef]

Castiaux, A.

A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Vol. 93of the Applied Mathematical Sciences Series (Springer-Verlag, 1998).

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, 2003).

D. Courjon and C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989-1028 (1994).
[CrossRef]

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
[CrossRef]

Coyle, J.

J. Coyle, “Locating the support of objects contained in a two-layered background medium in two dimensions,” Inverse Probl. 16, 275-292 (2000).
[CrossRef]

Cvitkovic, A.

Danzebrink, H.

A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
[CrossRef]

de Fornel, F.

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
[CrossRef]

Dereux, A.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657-699 (1996).
[CrossRef]

Dufour, J.

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

Furukawa, H.

H. Furukawa and S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: Effects of multiple scattering,” Opt. Commun. 132, 170-178 (1996).
[CrossRef]

Gatica, G. N.

G. N. Gatica and W. L. Wendland, “Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems,” Appl. Anal. 63, 39-75 (1996).
[CrossRef]

Girard, C.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657-699 (1996).
[CrossRef]

C. Girard, “Theoretical atomic-force-microscopy study of a stepped surface: Nonlocal effects in the probe,” Phys. Rev. B 43, 8822-8828 (1991).
[CrossRef]

Goudonnet, J.-P.

J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
[CrossRef]

Goumri-Said, S.

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

Greffet, J.-J.

J.-J. Greffet and R. Carminati, “Image formulation in near-field optics,” Prog. Surf. Sci. 56, 133-237 (1997).
[CrossRef]

Gregersen, N.

Hillenbrand, R.

Hsiao, G.-C.

G.-C. Hsiao, “The coupling of BEM and FEM--a brief review,” in Boundary Element X, (Springer, 1988), Vol. 1, pp. 431-445.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, (Wiley, 1993).

Johnson, C.

C. Johnson and J.-C. Nédélec, “On the coupling of boundary integral and finite element methods,” Math. Comput. 35, 1063-1079 (1980).
[CrossRef]

F. Brezzi and C. Johnson, “On the coupling of boundary integral and finite element methods,” Calcolo 16, 189-201 (1979).
[CrossRef]

Kawata, S.

H. Furukawa and S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: Effects of multiple scattering,” Opt. Commun. 132, 170-178 (1996).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Vol. 93of the Applied Mathematical Sciences Series (Springer-Verlag, 1998).

Nédélec, J.-C.

C. Johnson and J.-C. Nédélec, “On the coupling of boundary integral and finite element methods,” Math. Comput. 35, 1063-1079 (1980).
[CrossRef]

Novotny, L.

L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
[CrossRef]

Ocelic, N.

Omoya, T.

Persson, P.

P. Persson and G. Strang, “A simple mesh generator in Matlab,” SIAM Rev. 46, 329-345 (2004).
[CrossRef]

Pohl, D.

L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
[CrossRef]

Regli, P.

L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
[CrossRef]

Rohrer, H.

G. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726-735 (1982).
[CrossRef]

Salomon, L.

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

Sarayeddine, K.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
[CrossRef]

Schotland, J.

J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

P. Carney and J. Schotland, “Near-field tomography,” in Inside Out: Inverse Problems and Applications, G.Uhlmann, ed. (Cambridge U. Press, 2003), pp. 133-168.

P. Carney and J. Schotland, “Determination of three-dimensional structure in photon scanning tunneling mircoscopy,” J. Opt. A, Pure Appl. Opt. 4, S140-S144 (2002).
[CrossRef]

Spajer, M.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
[CrossRef]

Strang, G.

P. Persson and G. Strang, “A simple mesh generator in Matlab,” SIAM Rev. 46, 329-345 (2004).
[CrossRef]

Sun, J.

J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

Tanaka, K.

Tanaka, M.

Temple, P. A.

Tran, T.

T. Tran, “The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: Local estimates,” Math. Comput. 64, 501-513 (1995).

Trautman, J. K.

E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189-195 (1992).
[CrossRef] [PubMed]

Tromborg, B.

Van Labeke, D.

Wang, S.

S. Wang, “Analysis of probe-sample interaction in near-field optical image of dielectric structure,” Microsc. Microanal. 5, 290-295 (1999).
[CrossRef] [PubMed]

Weeber, J.-C.

J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
[CrossRef]

Wendland, W. L.

G. N. Gatica and W. L. Wendland, “Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems,” Appl. Anal. 63, 39-75 (1996).
[CrossRef]

Zayats, A.

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

Appl. Anal.

G. N. Gatica and W. L. Wendland, “Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems,” Appl. Anal. 63, 39-75 (1996).
[CrossRef]

Appl. Opt.

Calcolo

F. Brezzi and C. Johnson, “On the coupling of boundary integral and finite element methods,” Calcolo 16, 189-201 (1979).
[CrossRef]

Helv. Phys. Acta

G. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726-735 (1982).
[CrossRef]

Inverse Probl.

J. Coyle, “Locating the support of objects contained in a two-layered background medium in two dimensions,” Inverse Probl. 16, 275-292 (2000).
[CrossRef]

J. Appl. Phys.

A. Castiaux, H. Danzebrink, and X. Bouju, “Glass and silicon probes: A comparative theoretical study for near-field optical microscopy,” J. Appl. Phys. 84, 52-57 (1998).
[CrossRef]

J. Sun, P. Carney, and J. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103103 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

P. Carney and J. Schotland, “Determination of three-dimensional structure in photon scanning tunneling mircoscopy,” J. Opt. A, Pure Appl. Opt. 4, S140-S144 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Math. Comput.

C. Johnson and J.-C. Nédélec, “On the coupling of boundary integral and finite element methods,” Math. Comput. 35, 1063-1079 (1980).
[CrossRef]

T. Tran, “The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: Local estimates,” Math. Comput. 64, 501-513 (1995).

Microsc. Microanal.

S. Wang, “Analysis of probe-sample interaction in near-field optical image of dielectric structure,” Microsc. Microanal. 5, 290-295 (1999).
[CrossRef] [PubMed]

Opt. Commun.

J.-C. Weeber, F. de Fornel, and J.-P. Goudonnet, “Numerical study of the tip-sample interaction in the photon scanning tunneling microscope,” Opt. Commun. 126, 285-292 (1996).
[CrossRef]

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23-28 (1989).
[CrossRef]

H. Furukawa and S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: Effects of multiple scattering,” Opt. Commun. 132, 170-178 (1996).
[CrossRef]

S. Goumri-Said, L. Salomon, J. Dufour, F. de Fornel, and A. Zayats, “Numerical simulations of photon scanning tunneling microscopy: Role of probe tip geometry in image formation,” Opt. Commun. 224, 245-258 (2005).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

S. Goumri-Said, L. Salomon, J. Dufour, and F. de Fornel, “Two-dimensional numerical simulations of photon scanning tunneling microscopy: Fourier modal method and R-matrix algorithm,” Opt. Quantum Electron. 36, 787-806 (2004).
[CrossRef]

Phys. Rev. B

C. Girard, “Theoretical atomic-force-microscopy study of a stepped surface: Nonlocal effects in the probe,” Phys. Rev. B 43, 8822-8828 (1991).
[CrossRef]

Prog. Surf. Sci.

J.-J. Greffet and R. Carminati, “Image formulation in near-field optics,” Prog. Surf. Sci. 56, 133-237 (1997).
[CrossRef]

Rep. Prog. Phys.

D. Courjon and C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989-1028 (1994).
[CrossRef]

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657-699 (1996).
[CrossRef]

Science

E. Betzig and J. K. Trautman, “Near-field optics: Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189-195 (1992).
[CrossRef] [PubMed]

SIAM Rev.

P. Persson and G. Strang, “A simple mesh generator in Matlab,” SIAM Rev. 46, 329-345 (2004).
[CrossRef]

Ultramicroscopy

L. Novotny, D. Pohl, and P. Regli, “Near-field, far-field and imaging properties of the 2D aperture SNOM,” Ultramicroscopy 57, 180-188 (1995).
[CrossRef]

Other

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Vol. 93of the Applied Mathematical Sciences Series (Springer-Verlag, 1998).

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College Press, 2003).

P. Carney and J. Schotland, “Near-field tomography,” in Inside Out: Inverse Problems and Applications, G.Uhlmann, ed. (Cambridge U. Press, 2003), pp. 133-168.

J. Jin, The Finite Element Method in Electromagnetics, (Wiley, 1993).

G.-C. Hsiao, “The coupling of BEM and FEM--a brief review,” in Boundary Element X, (Springer, 1988), Vol. 1, pp. 431-445.

Cited By

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Figures (9)

Fig. 1
Fig. 1

Geometry of the model: Photon scanning tunneling microscopy. The probe length is a 1 + a 2 , where a 1 is the length of a rectangular part, a 2 is the radius of a semi-disc-shaped taper, and w is the probe width. The probe-substrate distance is a 3 ; n 1 , n 2 , n 3 , and n 4 are the refractive indices of the respective media.

Fig. 2
Fig. 2

Geometry of the computational domain and a mesh.

Fig. 3
Fig. 3

Example 1: The log–log scale of the error u u h 1 versus the number of nodes.

Fig. 4
Fig. 4

Intensity of the total field in Example 1: (a) The surface plot; (b) the image view.

Fig. 5
Fig. 5

Intensity of the total field in Example 2: (a) The surface plot; (b) the image view.

Fig. 6
Fig. 6

Intensity of the total field in Example 3: (a) The surface plot; (b) the image view.

Fig. 7
Fig. 7

Intensity of the total field with displacement in Example 3: (a) d = 100 nm ; (b) d = 100 nm .

Fig. 8
Fig. 8

Intensity of the total field in Example 4: (a) The surface plot; (b) the image view.

Fig. 9
Fig. 9

Intensity of the total field with displacement in Example 4: (a) d = 100 nm ; (b) d = 100 nm .

Equations (41)

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Δ u + κ 2 n 2 u = 0 in R 2 ,
κ ( x ) = { κ 0 for x 2 > 0 n 1 κ 0 for x 2 < 0 } .
Δ u ref + κ 2 u ref = 0 in R 2 .
u ref = { u t for x 2 > 0 u i + u r for x 2 < 0 } ,
u t = t exp ( i α x 1 + i γ x 2 ) , u r = r exp ( i α x 1 i β x 2 ) ,
γ ( α ) = { κ 0 2 α 2 for κ 0 > α i α 2 κ 0 2 for κ 0 < α } .
u = u ref + u s .
lim ρ Σ ρ u s ν i κ u s 2 d s = 0 ,
Δ u + κ 2 n 2 u = 0 in Ω ,
u ν = λ on Γ .
( φ , ψ ) = Ω φ ψ ¯ d x ,
φ , ψ = Γ φ ψ ¯ d s ,
a ( u , v ) λ , v = 0 for all v H 1 ( Ω ) ,
a ( u , v ) = ( u , v ) ( κ 2 n 2 u , v ) .
Δ u s + κ 2 u s = 0 in Ω e .
1 2 u s ( x ) = Γ G ( x , y ) ν ( y ) u s ( y ) d s ( y ) Γ G ( x , y ) u s ( y ) ν ( y ) d s ( y ) x Γ ,
1 2 u ref ( x ) = Γ G ( x , y ) ν ( y ) u ref ( y ) d s ( y ) + Γ G ( x , y ) u ref ( y ) ν ( y ) d s ( y ) , x Γ .
u ( x ) = 2 Γ G ( x , y ) ν ( y ) u ( y ) d s ( y ) 2 Γ G ( x , y ) u ( y ) ν ( y ) d s ( y ) + 2 u ref ( x ) , x Γ .
( S φ ) ( x ) 2 Γ G ( x , y ) φ ( y ) d s ( y )
( D φ ) ( x ) 2 Γ G ( x , y ) ν ( y ) φ ( y ) d s ( y ) .
u D u + S λ = 2 u ref on Γ .
u , μ D u , μ + S λ , μ = 2 u ref , μ , μ H 1 2 ( Γ ) .
a ( u , v ) λ , v = 0 for all v H 1 ( Ω ) ,
u , μ D u , μ + S λ , μ = 2 u ref , μ for all μ H 1 2 ( Γ ) .
X h = { u H 1 ( Ω ) : u T is linear , T T h } .
a ( u h , v ) λ h , v = 0 for all v X h ,
u h , μ D u h , μ + S λ h , μ = 2 u ref , μ for all μ Y h .
Δ G ( x , y ) + κ 2 ( x ) G ( x , y ) = δ ( x y ) ,
G ( x , y ) x 2 = 0 + = G ( x , y ) x 2 = 0 ,
G ( x , y ) x 2 x 2 = 0 + = G ( x , y ) x 2 x 2 = 0 ,
κ ( x ) = { κ 1 for x 2 > 0 κ 2 for x 2 < 0 } .
β i = { κ i 2 ξ 2 for κ i > ξ i ξ 2 κ i 2 for κ i < ξ } .
G ( x , y ) = Φ 1 ( x , y ) + Ψ 1 ( x , y ) ,
G ( x , y ) = Φ 2 ( x , y ) + Ψ 2 ( x , y ) ,
G ( x , y ) = Ψ 3 ( x , y ) ,
G ( x , y ) = Ψ 4 ( x , y ) ,
Ψ 1 ( x , y ) = i 4 π 1 β 1 β 1 β 2 β 1 + β 2 e i β 1 ( x 2 + y 2 ) e i ξ ( x 1 y 1 ) d ξ ,
Ψ 2 ( x , y ) = i 4 π 1 β 2 β 2 β 1 β 1 + β 2 e i β 2 ( x 2 + y 2 ) e i ξ ( x 1 y 1 ) d ξ ,
Ψ 3 ( x , y ) = i 2 π e i ( β 1 x 2 β 2 y 2 ) β 1 + β 2 e i ξ ( x 1 y 1 ) d ξ ,
Ψ 4 ( x , y ) = i 2 π e i ( β 1 y 2 β 2 x 2 ) β 1 + β 2 e i ξ ( x 1 y 1 ) d ξ ,
Φ i ( x , y ) = i 4 H 0 ( 1 ) ( κ i x y ) , i = 1 , 2 .

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