Abstract

We simulate a total internal reflection tomography experiment in which an unknown object is illuminated by evanescent waves and the scattered field is detected along several directions. We propose a full-vectorial three-dimensional nonlinear inversion scheme to retrieve the map of the permittivity of the object from the scattered far-field data. We study the role of the solid angle of illumination, the incident polarization, and the position of the prism interface on the resolution of the images. We compare our algorithm with a linear inversion scheme based on the renormalized Born approximation and stress the importance of multiple scattering in this particular configuration. We analyze the sensitivity to noise and point out that using incident propagative waves together with evanescent waves improves the robustness of the reconstruction.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
  2. N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
    [CrossRef]
  15. A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetics fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
    [CrossRef]
  16. P. C. Chaumet, A. Sentenac, A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
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    [CrossRef]
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  21. K. Belkebir, A. G. Tijhuis, “Modified2 gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
    [CrossRef]
  22. P. C. Chaumet, K. Belkebir, A. Sentenac, “Three-dimensional subwavelength optical imaging using the coupled dipole method,” Phys. Rev. B 69, 245405 (2004).
    [CrossRef]
  23. A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
    [CrossRef]
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2004

P. C. Chaumet, K. Belkebir, A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

P. C. Chaumet, A. Sentenac, A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

P. C. Chaumet, K. Belkebir, A. Sentenac, “Three-dimensional subwavelength optical imaging using the coupled dipole method,” Phys. Rev. B 69, 245405 (2004).
[CrossRef]

2003

2002

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

A. Abubakar, P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495–510 (2002).
[CrossRef]

2001

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

K. Belkebir, A. G. Tijhuis, “Modified2 gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[CrossRef]

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

P. So, H. Kwon, C. Dong, “Resolution enhancement in standing-wave total-internal reflection microscopy: a point spread function engineering approach,” J. Opt. Soc. Am. A 18, 2833–2845 (2001).
[CrossRef]

P. S. Carney, J. C. Schotland, “Three-dimensional total-internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
[CrossRef]

2000

G. Cragg, P. So, “Standing wave total-internal reflection microscopy,” Opt. Lett. 25, 46–48 (2000).
[CrossRef]

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

D. Fischer, “Subwavelength depth resolution in near-field microscopy,” Opt. Lett. 25, 1529–1531 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

1997

K. Belkebir, R. E. Kleinman, C. Pichot, “Microwave imaging—location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech. 45, 469–476 (1997).
[CrossRef]

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

1995

1994

F. Pincemin, A. Sentenac, J.-J. Greffet, “Near-field scattered by a dielectric rod below a metallic surface,” J. Opt. Soc. Am. A 11, 1117–1127 (1994).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

1993

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations—a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. [Solid Earth] 98, 1759–1775 (1993).
[CrossRef]

1992

R. E. Kleinman, P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[CrossRef]

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetics fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

1991

1987

1973

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

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Abubakar, A.

A. Abubakar, P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495–510 (2002).
[CrossRef]

A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Belkebir, K.

P. C. Chaumet, K. Belkebir, A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

P. C. Chaumet, K. Belkebir, A. Sentenac, “Three-dimensional subwavelength optical imaging using the coupled dipole method,” Phys. Rev. B 69, 245405 (2004).
[CrossRef]

K. Belkebir, A. Sentenac, “High resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20, 1223–1229 (2003).
[CrossRef]

K. Belkebir, A. G. Tijhuis, “Modified2 gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[CrossRef]

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

K. Belkebir, R. E. Kleinman, C. Pichot, “Microwave imaging—location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech. 45, 469–476 (1997).
[CrossRef]

Bonnard, S.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1959).

Carney, P. S.

Chaumet, P. C.

P. C. Chaumet, K. Belkebir, A. Sentenac, “Three-dimensional subwavelength optical imaging using the coupled dipole method,” Phys. Rev. B 69, 245405 (2004).
[CrossRef]

P. C. Chaumet, A. Sentenac, A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

P. C. Chaumet, K. Belkebir, A. Sentenac, “Superresolution of three-dimensional optical imaging by use of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).
[CrossRef] [PubMed]

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

Cragg, G.

de Fornel, F.

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

F. de Fornel, Evanescent Waves, Vol. 73 of Springer Series in Optical Sciences (Springer Verlag, 2001).
[CrossRef]

de Hon, B. P.

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

Destouches, N.

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Dong, C.

Draine, B. T.

Fischer, D.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Flatau, P. J.

Giovannini, H.

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Girard, C.

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

Goodman, J. J.

Greffet, J.-J.

Groom, R. W.

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations—a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. [Solid Earth] 98, 1759–1775 (1993).
[CrossRef]

Guérin, C. A.

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Gustafsson, M.

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

Habashy, T. M.

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations—a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. [Solid Earth] 98, 1759–1775 (1993).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Kawata, S.

Kleinman, R. E.

K. Belkebir, R. E. Kleinman, C. Pichot, “Microwave imaging—location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech. 45, 469–476 (1997).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[CrossRef]

Kooij, B. J.

A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Kwon, H.

Lakhtakia, A.

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetics fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

Lequime, M.

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

Litman, A. C. S.

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

Minami, S.

Nakamura, O.

Nieto-Vesperinas, M.

Pennypacker, C. R.

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

Pezin, F.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

Pichot, C.

K. Belkebir, R. E. Kleinman, C. Pichot, “Microwave imaging—location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech. 45, 469–476 (1997).
[CrossRef]

Pincemin, F.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Purcell, E. M.

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

Rahmani, A.

P. C. Chaumet, A. Sentenac, A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

Sabouroux, P.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

Saillard, M.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

Schotland, J. C.

Sentenac, A.

So, P.

Spies, B. R.

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations—a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. [Solid Earth] 98, 1759–1775 (1993).
[CrossRef]

Stammes, J.

Teukolski, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Tijhuis, A. G.

K. Belkebir, A. G. Tijhuis, “Modified2 gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[CrossRef]

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

van den Berg, P. M.

A. Abubakar, P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495–510 (2002).
[CrossRef]

A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

R. E. Kleinman, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

R. E. Kleinman, P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Wedberg, T.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, 1959).

Astrophys. J.

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

IEEE Trans. Geosci. Remote Sens.

A. G. Tijhuis, K. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of 2-D inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

K. Belkebir, R. E. Kleinman, C. Pichot, “Microwave imaging—location and shape reconstruction from multifrequency scattering data,” IEEE Trans. Microwave Theory Tech. 45, 469–476 (1997).
[CrossRef]

IEICE Trans. Electron.

A. Abubakar, P. M. van den Berg, B. J. Kooij, “A conjugate gradient contrast source technique for 3D profile inversion,” IEICE Trans. Electron. E83-C, 1864–1874 (2000).

Int. J. Mod. Phys. C

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetics fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

Inverse Probl.

A. Abubakar, P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495–510 (2002).
[CrossRef]

K. Belkebir, A. G. Tijhuis, “Modified2 gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[CrossRef]

J. Comput. Appl. Math.

R. E. Kleinman, P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math. 42, 17–35 (1992).
[CrossRef]

J. Electromagn. Waves Appl.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, M. Saillard, “Validation of 2D inverse scattering algorithms from multi-frequency experimental data,” J. Electromagn. Waves Appl. 14, 1637–1667 (2000).
[CrossRef]

J. Geophys. Res. [Solid Earth]

T. M. Habashy, R. W. Groom, B. R. Spies, “Beyond the Born and Rytov approximations—a nonlinear approach to electromagnetic scattering,” J. Geophys. Res. [Solid Earth] 98, 1759–1775 (1993).
[CrossRef]

J. Microsc.

M. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef] [PubMed]

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Commun.

N. Destouches, C. A. Guérin, M. Lequime, H. Giovannini, “Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles,” Opt. Commun. 198, 233–239 (2001).
[CrossRef]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Lett.

Phys. Rev. A

A. Rahmani, P. C. Chaumet, F. de Fornel, C. Girard, “Field propagator of a dressed junction: fluorescence lifetime calculations in a confined geometry,” Phys. Rev. A 56, 3245–3254 (1997).
[CrossRef]

Phys. Rev. B

P. C. Chaumet, K. Belkebir, A. Sentenac, “Three-dimensional subwavelength optical imaging using the coupled dipole method,” Phys. Rev. B 69, 245405 (2004).
[CrossRef]

Phys. Rev. E

P. C. Chaumet, A. Sentenac, A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Radio Sci.

R. E. Kleinman, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

F. de Fornel, Evanescent Waves, Vol. 73 of Springer Series in Optical Sciences (Springer Verlag, 2001).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, 1959).

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Figures (8)

Fig. 1
Fig. 1

Illumination and detection configuration of the TIRT experiment. The observation points are regularly placed on the half-sphere Γ (with a radius of 400 λ ). The illumination is as represented by the arrows, corresponding to a plane wave propagating toward the positive values of z. For the TIRT experiments, the authors took as illumination 16 plane waves in both the planes ( x , z ) and ( y , z ) , either in p or s polarization. The angle between the incident wave vector and the z axis ranges over 80 to 80 deg .

Fig. 2
Fig. 2

Left side: map of the relative permittivity in the plane ( x , y ) just above the substrate, i.e., z = λ 40 . Right side: map of the relative permittivity in the plane ( x , z ) for y = 0 . We have a = λ 4 , c = λ 10 , ε s = 2.25 , θ inc [ 80 , 80 ] deg , and p-polarized incident waves.

Fig. 3
Fig. 3

Influence of the interface in the inverse scattering problem: (a) map of the relative permittivity when the interaction between the objects and the substrate is not taken into account ( S d = S = 0 ) , (b) map of the relative permittivity when the substrate is taken into account only in the far-field zone ( S = 0 ) .

Fig. 4
Fig. 4

Influence of the illuminations: (a) map of the relative permittivity with only propagative wave illuminations ( θ inc [ 40 , 40 ] deg ) , (b) same as (a) but with evanescent wave illuminations ( θ l inc [ 80 , 43 ] [ 80 , 43 ] deg ) .

Fig. 5
Fig. 5

Influence of the position of the sample with respect to the interface. This figure is the same as Fig. 2, except that the centers of the cubes are located at z 0.6 λ from the interface.

Fig. 6
Fig. 6

Reconstruction of the permittivity using s-polarized wave illumination. The parameters are the same as those for Fig. 2.

Fig. 7
Fig. 7

Map of the relative permittivity in using the renormalized Born approximation: (a) with only propagative waves ( θ inc [ 40 , 40 ] deg ) , (b) with both propagative and evanescent waves ( θ inc [ 80 , 80 ] deg ) , (c) with only evanescent waves ( θ l inc [ 80 , 43 ] [ 80 , 43 ] deg ) .

Fig. 8
Fig. 8

Robustness of the inverse scattering algorithm with respect to uncorrelated noise: (a) map of the relative permittivity using only evanescent wave illuminations ( θ l inc [ 80 , 43 ] [ 80 , 43 ] deg ) , (b) same as (a) but with both evanescent and propagative wave illuminations ( θ inc [ 80 , 80 ] deg ) .

Equations (43)

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E ( r i ) = E inc ( r i ) + j = 1 , j i N T ( r i , r j ) α ( r j ) E ( r j ) + j = 1 N S ( r i , r j ) α ( r j ) E ( r j ) ,
α ( r j ) = 3 d 3 4 π ε ( r j ) 1 ε ( r j ) + 2 ,
E d ( r ) = j = 1 N [ T d ( r , r j ) + S d ( r , r j ) ] α ( r j ) E ( r j ) .
E = E inc + A ̿ p ,
E = [ E x ( r 1 ) , E y ( r 1 ) , E z ( r 1 ) , , E z ( r N ) ] ,
E inc = [ E x inc ( r 1 ) , E y inc ( r 1 ) , E z inc ( r 1 ) , , E z inc ( r N ) ] ,
p = [ p x ( r 1 ) , p y ( r 1 ) , p z ( r 1 ) , , p z ( r N ) ] ,
E E inc .
E l d = B ̿ p l ,
α n = α n 1 + a n d n ,
h l , n = f l B ̿ α n E l , n ,
E n , l = [ I ̿ A ̿ α n 1 ] 1 E l inc ,
F n ( α n ) = l = 1 L h l , n Γ 2 l = 1 L f l Γ 2 = W Γ l = 1 L h l , n Γ 2 ,
F n ( a n ) = W Γ l = 1 L ( h l , n 1 Γ 2 + a n 2 B ̿ d n E l , n Γ 2 2 a n Re h l , n 1 B ̿ d n E l , n Γ ) .
a n = l = 1 L Re h l , n 1 B ̿ d n E l , n Γ l = 1 L B ̿ d n E l , n Γ 2 .
d n = g n ; α + γ n d n 1 ,
g n ; α = W Γ l = 1 L E l , n * B ̿ h l , n 1 ,
γ n = g n ; α g n ; α g n 1 ; α Γ g n 1 ; α Γ 2 .
Re [ f ̃ l ; v ( r k ) ] = Re [ f l ; v ( r k ) ] + u A r ξ l ; v ,
Im [ f ̃ l ; v ( r k ) ] = Im [ f l ; v ( r k ) ] + u A i η l ; v ,
E m ( r ) = E inc ( r ) + V T ( r , r ) χ ( r ) E m ( r ) d r ,
T ( r , r ) = exp ( i k 0 R ) [ ( 3 R R R 2 I ) ( 1 R 3 i k 0 R 2 ) + ( I R R R 2 ) k 0 2 R ] 4 π 3 I δ ( R ) ,
E m ( r i ) = E inc ( r i ) + j = 1 , j i N T ( r i , r j ) χ ( r j ) d 3 E m ( r j ) ε ( r i ) 1 3 E m ( r i ) .
E ( r i ) = E inc ( r i ) + j = 1 , j i N T ( r i , r j ) α ( r j ) E ( r j ) ,
ε + 2 3 E m ( r i ) = E ( r i ) .
E m ( r i ) 3 ε + 2 E inc ( r i ) .
a = [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 1 2 ,
sin φ = ( x x 0 ) a ,
cos φ = ( y y 0 ) a ,
Δ p = w 1 ε s w 0 w 1 + ε s w 0 , Δ s = w 1 w 0 w 1 + w 0 ,
S ( r , r 0 ) = [ I 1 + cos ( 2 φ ) I 2 sin ( 2 φ ) I 2 sin φ I 3 sin ( 2 φ ) I 2 I 1 cos ( 2 φ ) I 2 cos φ I 3 sin φ I 3 cos φ I 3 I 4 ] ,
I 1 = i 2 ( 0 k 0 + 0 i ) d w 0 J 0 ( a u ) exp [ i w 0 ( z + z 0 ) ] ( k 0 2 Δ s w 0 2 Δ p ) ,
I 2 = i 2 ( 0 k 0 + 0 i ) d w 0 J 0 ( a u ) exp [ i w 0 ( z + z 0 ) ] ( k 0 2 Δ s w 0 2 Δ p ) ,
I 3 = ( 0 k 0 + 0 i ) d w 0 J 1 ( a u ) exp [ i w 0 ( z + z 0 ) ] Δ p w 0 u ,
I 4 = i ( 0 k 0 + 0 i ) d w 0 J 0 ( a u ) exp [ i w 0 ( z + z 0 ) ] Δ p u 2 .
S d ( r , r 0 ) = k 0 2 r exp { i k 0 [ x ( x x 0 ) + y ( y y 0 ) + z ( z + z 0 ) ] r } × [ ( x z r ρ ) 2 Δ p y 2 ρ 2 Δ s x y ρ 2 ( z 2 r 2 Δ p + Δ s ) x z r 2 Δ p x y ρ 2 ( z 2 r 2 Δ p + Δ s ) ( y z r ρ ) 2 Δ p x 2 ρ 2 Δ s y z r 2 Δ p x z r 2 Δ p y z r 2 Δ p ρ 2 r 2 Δ p ] ,
Δ p = ( ε s r 2 ρ 2 ) 1 2 ε s z ( ε s r 2 ρ 2 ) 1 2 + ε s z , Δ s = ( ε s r 2 ρ 2 ) 1 2 z ( ε s r 2 ρ 2 ) 1 2 + z .
S d ( z , z 0 ) = k 0 2 z exp [ i k 0 ( z + z 0 ) ] [ Δ p 0 0 0 Δ p 0 0 0 0 ] .
p l init = γ l B ̿ f l ,
M ( γ l ) = f l B ̿ p l init Γ 2 = f l γ l B ̿ B ̿ f l Γ 2 .
γ l = B B ̿ f l f l Γ B B ̿ f l Γ 2 .
E l init = E l inc + A ̿ p l init .
α init ( r ) = Re [ l = 1 L p l init ( r ) E l init * ( r ) l = 1 L E l init ( r ) 2 ] .

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