Abstract

Optical diffraction tomography is an imaging technique that permits retrieval of the map of permittivity of an object from its scattered far field. Most reconstruction procedures assume that single scattering is dominant so that the scattered far field is linearly linked to the permittivity. In this work, we present a nonlinear inversion method and apply it to complex three-dimensional samples. We show that multiple scattering permits one to obtain a power of resolution beyond the classical limit imposed by the use of propagative incident and diffracted waves. Moreover, we stress that our imaging method is robust with respect to correlated and uncorrelated noise.

© 2006 Optical Society of America

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  1. 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]
  2. N. Destouches, C. A. Guérin, M. Lequime, and 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]
  3. E. Wolf, 'Three-dimensional structure determination of semi-transparent objects from holographic data,' Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  4. S. Kawata, O. Nakamura, and S. Minami, 'Optical microscope tomography. I. Support constraint,' J. Opt. Soc. Am. A 4, 292-297 (1987).
    [CrossRef]
  5. P. S. Carney and J. C. Schotland, 'Three-dimensional total-internal reflection microscopy,' Opt. Lett. 26, 1072-1074 (2001).
    [CrossRef]
  6. P. Chaumet, K. Belkebir, and A. Sentenac, 'Three-dimensional subwavelength optical imaging using the coupled dipole method,' Phys. Rev. B 69, 245405 (2004).
    [CrossRef]
  7. E. M. Purcell and C. R. Pennypacker, 'Scattering and absorption of light by nonspherical dielectric grains,' Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).
  9. 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]
  10. P. C. Chaumet and M. Nieto-Vesperinas, 'Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,' Phys. Rev. B 61, 14119-14127 (2000).
    [CrossRef]
  11. P. C. Chaumet and M. Nieto-Vesperinas, 'Electromagnetic force on a metallic particle in the presence of a dielectric surface,' Phys. Rev. B 62, 11185-11191 (2000).
    [CrossRef]
  12. P. C. Chaumet and M. Nieto-Vesperinas, 'Time-averaged total force on a dipolar sphere in an electromagnetic field,' Opt. Lett. 25, 1065-1067 (2000).
    [CrossRef]
  13. B. T. Draine, 'The discrete dipole approximation and its application to interstellar graphite grains,' Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  14. W. C. Chew and Y. M. Wang, 'Reconstruction of two-dimensional permittivity distribution using the distorted wave Born iterative method,' IEEE Trans. Med. Imaging 9, 218-235 (1990).
    [CrossRef] [PubMed]
  15. N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
    [CrossRef]
  16. A. G. Tijhuis, 'Born-type reconstruction of material parameters of an inhomogeneous, lossy dielectric slab from reflected-field data,' Wave Motion 11, 151-173 (1989).
    [CrossRef]
  17. A. G. Tijhuis, K. Belkebir, A. C. S. Litman, and B. P. de Hon, 'Theoretical and computational aspects of 2-D inverse profiling,' IEEE Trans. Geosci. Remote Sens. GE-39, 1316-1330 (2001).
    [CrossRef]
  18. R. E. Kleinman and P. M. van den Berg, 'A modified gradient method for two-dimensional problems in tomography,' J. Comput. Appl. Math. 42, 17-35 (1992).
    [CrossRef]
  19. R. E. Kleinman and P. M. van den Berg, 'An extended range-modified gradient technique for profile inversion,' Radio Sci. 28, 877-884 (1993).
    [CrossRef]
  20. K. Belkebir and 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]
  21. K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, and M. Saillard, 'Validation of 2D inverse scattering algorithms from multi-frequency experimental data,' J. Electromagn. Waves Appl. 14, 1637-1667 (2000).
    [CrossRef]
  22. K. Belkebir and A. Sentenac, 'High resolution optical diffraction microscopy,' J. Opt. Soc. Am. A 20, 1223-1229 (2003).
    [CrossRef]
  23. P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
    [CrossRef] [PubMed]
  24. P. M. van den Berg and R. E. Kleinman, 'A contrast source inversion method,' Inverse Probl. 13, 1607-1620 (1997).
    [CrossRef]
  25. A. Abubakar, P. M. van den Berg, and B. J. Kooij, 'A conjugate gradient contrast source technique for 3D profile inversion,' IEICE Trans. Electron. E83-C, 1864-1874 (2000).
  26. A. Abubakar and P. M. van den Berg, 'The contrast source inversion method for location and shape reconstructions,' Inverse Probl. 18, 495-510 (2002).
    [CrossRef]
  27. W. H. Press, B. P. Flannery, S. A. Teukolski, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).
  28. K. Belkebir, R. E. Kleinman, and C. Pichot, 'Microwave imaging--Location and shape reconstruction from multifrequency scattering data,' IEEE Trans. Microwave Theory Tech. 45, 469-476 (1997).
    [CrossRef]
  29. L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
    [CrossRef]
  30. R. E. Kleinman and P. M. van den Berg, 'Two-dimensional location and shape reconstruction,' Radio Sci. 29, 1157-1169 (1994).
    [CrossRef]
  31. J. Daillant and A. Gibaud, X-Ray and Neutron ReflectivityLecture Notes in Physics (Springer-Verlag, 1999), p. 130.
  32. C.-A. Guérin and A. Sentenac, 'Second-order perturbation theory for scattering from heterogeneous rough surfaces,' J. Opt. Soc. Am. A 21, 1251-1260 (2004).
    [CrossRef]
  33. T. M. Habashy, R. W. Groom, and 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]
  34. P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
    [CrossRef]
  35. A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
    [CrossRef]
  36. P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
    [CrossRef]

2004 (2)

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

C.-A. Guérin and A. Sentenac, 'Second-order perturbation theory for scattering from heterogeneous rough surfaces,' J. Opt. Soc. Am. A 21, 1251-1260 (2004).
[CrossRef]

2003 (2)

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

P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
[CrossRef]

2002 (2)

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

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]

2001 (6)

N. Destouches, C. A. Guérin, M. Lequime, and 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]

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

K. Belkebir and 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]

P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
[CrossRef] [PubMed]

A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
[CrossRef]

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

2000 (5)

P. C. Chaumet and 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, and 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, and M. Saillard, 'Validation of 2D inverse scattering algorithms from multi-frequency experimental data,' J. Electromagn. Waves Appl. 14, 1637-1667 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, 'Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,' Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, 'Electromagnetic force on a metallic particle in the presence of a dielectric surface,' Phys. Rev. B 62, 11185-11191 (2000).
[CrossRef]

1998 (1)

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
[CrossRef]

1997 (2)

K. Belkebir, R. E. Kleinman, and C. Pichot, 'Microwave imaging--Location and shape reconstruction from multifrequency scattering data,' IEEE Trans. Microwave Theory Tech. 45, 469-476 (1997).
[CrossRef]

P. M. van den Berg and R. E. Kleinman, 'A contrast source inversion method,' Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

1996 (1)

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

1994 (1)

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

1993 (2)

T. M. Habashy, R. W. Groom, and 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]

R. E. Kleinman and P. M. van den Berg, 'An extended range-modified gradient technique for profile inversion,' Radio Sci. 28, 877-884 (1993).
[CrossRef]

1992 (2)

R. E. Kleinman and 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 (1)

N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
[CrossRef]

1990 (1)

W. C. Chew and Y. M. Wang, 'Reconstruction of two-dimensional permittivity distribution using the distorted wave Born iterative method,' IEEE Trans. Med. Imaging 9, 218-235 (1990).
[CrossRef] [PubMed]

1989 (1)

A. G. Tijhuis, 'Born-type reconstruction of material parameters of an inhomogeneous, lossy dielectric slab from reflected-field data,' Wave Motion 11, 151-173 (1989).
[CrossRef]

1988 (1)

B. T. Draine, 'The discrete dipole approximation and its application to interstellar graphite grains,' Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1987 (1)

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]

1969 (1)

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

Abubakar, A.

A. Abubakar and 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, and B. J. Kooij, 'A conjugate gradient contrast source technique for 3D profile inversion,' IEICE Trans. Electron. E83-C, 1864-1874 (2000).

Belkebir, K.

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

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

K. Belkebir and 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, and B. P. de Hon, 'Theoretical and computational aspects of 2-D inverse profiling,' IEEE Trans. Geosci. Remote Sens. GE-39, 1316-1330 (2001).
[CrossRef]

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

Bryant, G. W.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Carney, P. S.

P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
[CrossRef] [PubMed]

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

Chaumet, P.

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

Chaumet, P. C.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
[CrossRef]

A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, 'Electromagnetic force on a metallic particle in the presence of a dielectric surface,' Phys. Rev. B 62, 11185-11191 (2000).
[CrossRef]

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

P. C. Chaumet and M. Nieto-Vesperinas, 'Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,' Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
[CrossRef]

Chew, W. C.

W. C. Chew and Y. M. Wang, 'Reconstruction of two-dimensional permittivity distribution using the distorted wave Born iterative method,' IEEE Trans. Med. Imaging 9, 218-235 (1990).
[CrossRef] [PubMed]

Daillant, J.

J. Daillant and A. Gibaud, X-Ray and Neutron ReflectivityLecture Notes in Physics (Springer-Verlag, 1999), p. 130.

de Fornel, F.

A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
[CrossRef]

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
[CrossRef]

de Hon, B. P.

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

Destouches, N.

N. Destouches, C. A. Guérin, M. Lequime, and 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]

Draine, B. T.

B. T. Draine, 'The discrete dipole approximation and its application to interstellar graphite grains,' Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Duchêne, B.

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

Dufour, J.-P.

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
[CrossRef]

Flannery, B. P.

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

Gibaud, A.

J. Daillant and A. Gibaud, X-Ray and Neutron ReflectivityLecture Notes in Physics (Springer-Verlag, 1999), p. 130.

Giovannini, H.

N. Destouches, C. A. Guérin, M. Lequime, and 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]

Groom, R. W.

T. M. Habashy, R. W. Groom, and 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, and 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]

Guérin, C.-A.

Habashy, T. M.

T. M. Habashy, R. W. Groom, and 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]

Hugonin, J.-P.

N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
[CrossRef]

Jackson, J. D.

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

Joachimowicz, N.

N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
[CrossRef]

Kawata, S.

Kleinman, R. E.

K. Belkebir, R. E. Kleinman, and C. Pichot, 'Microwave imaging--Location and shape reconstruction from multifrequency scattering data,' IEEE Trans. Microwave Theory Tech. 45, 469-476 (1997).
[CrossRef]

P. M. van den Berg and R. E. Kleinman, 'A contrast source inversion method,' Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

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

R. E. Kleinman and P. M. van den Berg, 'An extended range-modified gradient technique for profile inversion,' Radio Sci. 28, 877-884 (1993).
[CrossRef]

R. E. Kleinman and 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, and B. J. Kooij, 'A conjugate gradient contrast source technique for 3D profile inversion,' IEICE Trans. Electron. E83-C, 1864-1874 (2000).

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, and 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]

Lesselier, D.

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

Litman, A. C. S.

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

Markel, V. A.

P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
[CrossRef] [PubMed]

Minami, S.

Nakamura, O.

Nieto-Vesperinas, M.

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

P. C. Chaumet and M. Nieto-Vesperinas, 'Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,' Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, 'Electromagnetic force on a metallic particle in the presence of a dielectric surface,' Phys. Rev. B 62, 11185-11191 (2000).
[CrossRef]

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]

Pezin, F.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, and 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, and C. Pichot, 'Microwave imaging--Location and shape reconstruction from multifrequency scattering data,' IEEE Trans. Microwave Theory Tech. 45, 469-476 (1997).
[CrossRef]

N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
[CrossRef]

Press, W. H.

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

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]

Rahmani, A.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
[CrossRef]

A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
[CrossRef]

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
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Sabouroux, P.

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, and 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, and 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.

P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
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P. S. Carney and J. C. Schotland, 'Three-dimensional total-internal reflection microscopy,' Opt. Lett. 26, 1072-1074 (2001).
[CrossRef]

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Souriau, L.

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

Spies, B. R.

T. M. Habashy, R. W. Groom, and 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]

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

Tijhuis, A. G.

K. Belkebir and 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, and B. P. de Hon, 'Theoretical and computational aspects of 2-D inverse profiling,' IEEE Trans. Geosci. Remote Sens. GE-39, 1316-1330 (2001).
[CrossRef]

A. G. Tijhuis, 'Born-type reconstruction of material parameters of an inhomogeneous, lossy dielectric slab from reflected-field data,' Wave Motion 11, 151-173 (1989).
[CrossRef]

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A. Abubakar and 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, and B. J. Kooij, 'A conjugate gradient contrast source technique for 3D profile inversion,' IEICE Trans. Electron. E83-C, 1864-1874 (2000).

P. M. van den Berg and R. E. Kleinman, 'A contrast source inversion method,' Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

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

R. E. Kleinman and P. M. van den Berg, 'An extended range-modified gradient technique for profile inversion,' Radio Sci. 28, 877-884 (1993).
[CrossRef]

R. E. Kleinman and 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, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1986).

Wang, Y. M.

W. C. Chew and Y. M. Wang, 'Reconstruction of two-dimensional permittivity distribution using the distorted wave Born iterative method,' IEEE Trans. Med. Imaging 9, 218-235 (1990).
[CrossRef] [PubMed]

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

Astrophys. J. (2)

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

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[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Joachimowicz, C. Pichot, and J.-P. Hugonin, 'Inverse scattering: An iterative numerical method for electromagnetic imaging,' IEEE Trans. Antennas Propag. 39, 1742-1751 (1991).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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

IEEE Trans. Med. Imaging (1)

W. C. Chew and Y. M. Wang, 'Reconstruction of two-dimensional permittivity distribution using the distorted wave Born iterative method,' IEEE Trans. Med. Imaging 9, 218-235 (1990).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory Tech. (1)

K. Belkebir, R. E. Kleinman, and 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. (1)

A. Abubakar, P. M. van den Berg, and 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]

Inverse Probl. (4)

K. Belkebir and 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. Abubakar and P. M. van den Berg, 'The contrast source inversion method for location and shape reconstructions,' Inverse Probl. 18, 495-510 (2002).
[CrossRef]

L. Souriau, B. Duchêne, D. Lesselier, and R. E. Kleinman, 'Modified gradient approach to inverse scattering for binary objects in stratified media,' Inverse Probl. 12, 463-481 (1996).
[CrossRef]

P. M. van den Berg and R. E. Kleinman, 'A contrast source inversion method,' Inverse Probl. 13, 1607-1620 (1997).
[CrossRef]

J. Comput. Appl. Math. (1)

R. E. Kleinman and 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. (1)

K. Belkebir, S. Bonnard, F. Pezin, P. Sabouroux, and 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] (1)

T. M. Habashy, R. W. Groom, and 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]

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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]

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N. Destouches, C. A. Guérin, M. Lequime, and 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. (2)

Phys. Rev. A (1)

A. Rahmani, P. C. Chaumet, and F. de Fornel, 'Environment-induced modification of spontaneous emission: Single-molecule near-field probe,' Phys. Rev. A 63, 023819 (2001).
[CrossRef]

Phys. Rev. B (5)

P. C. Chaumet, A. Rahmani, and G. W. Bryant, 'Generalization of the coupled dipole method to periodic structure,' Phys. Rev. B 67, 165404 (2003).
[CrossRef]

P. C. Chaumet, A. Rahmani, F. de Fornel, and J.-P. Dufour, 'Evanescent light scattering: The validity of the dipole approximation,' Phys. Rev. B 58, 2310-2315 (1998).
[CrossRef]

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

P. C. Chaumet and M. Nieto-Vesperinas, 'Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,' Phys. Rev. B 61, 14119-14127 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, 'Electromagnetic force on a metallic particle in the presence of a dielectric surface,' Phys. Rev. B 62, 11185-11191 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

P. S. Carney, V. A. Markel, and J. C. Schotland, 'Near-field tomography without phase retrieval,' Phys. Rev. Lett. 86, 5874-5877 (2001).
[CrossRef] [PubMed]

Radio Sci. (2)

R. E. Kleinman and P. M. van den Berg, 'An extended range-modified gradient technique for profile inversion,' Radio Sci. 28, 877-884 (1993).
[CrossRef]

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

Wave Motion (1)

A. G. Tijhuis, 'Born-type reconstruction of material parameters of an inhomogeneous, lossy dielectric slab from reflected-field data,' Wave Motion 11, 151-173 (1989).
[CrossRef]

Other (3)

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

J. Daillant and A. Gibaud, X-Ray and Neutron ReflectivityLecture Notes in Physics (Springer-Verlag, 1999), p. 130.

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

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Figures (10)

Fig. 1
Fig. 1

Sketch of the illumination and detection configuration of the ODT 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, which denote a plane wave propagating toward positive z. For the ODT experiments, the authors took as illumination 16 plane waves in both planes ( x , z ) and ( y , z ) . The angle between the incident wave vector and the z axis ranges over 80 ° to 80°. See text for more detail.

Fig. 2
Fig. 2

Reconstructed permittivity of a single cube of permittivity ε = 2.25 of widths λ 20 (dashed curve) and λ 4 (solid curve). Upper figure, plot of the relative permittivity along the x axis; lower figure, plot along the z axis. The legend on the left is for the solid curve (large cube), on the right for the dashed curve (small cube).

Fig. 3
Fig. 3

(a) Map in the y = 0 plane of the reconstructed permittivity of two dipoles (cubes of width λ 20 , ε = 2.25 ) separated by 0.6 λ along the z axis. (b) Normalized reconstructed permittivity contrast [ ( ε 1 ) max ( ε 1 ) ] versus z λ for x = y = 0 : dashed curve, two dipoles separated by 0.6 λ along the z axis; solid curve, two cubes of width λ 4 , ε = 2.25 , whose centers are separated by the same distance as the dipoles, 0.6 λ , along the z axis.

Fig. 4
Fig. 4

Two cubes of side a = λ 4 separated by a distance c = λ 7 along the x axis. (a) and (b) show reconstructed maps of permittivities with a test domain of size ( 1.6 × 1.6 × 1.6 ) λ 3 ; the square in dashed line indicates the position of the actual cubes: (a) map of the relative permittivity in the plane ( x , z ) for y = 0 ; (b) map of the relative permittivity in the plane ( x , y ) for z = 0 . (c) Relative permittivity versus x for y = z = 0 (dashed curve) and the actual profile (solid curve).

Fig. 5
Fig. 5

Two cubes of size a = λ 4 separated by a distance c = λ 3 along the z axis. (a) and (b) show reconstructed maps of permittivities for a test domain Ω sized ( 1.6 × 1.6 × 1.6 ) λ 3 ; the square in dashed line indicates the position of the actual cubes: (a) map of the relative permittivity in the plane ( x , z ) for y = 0 ; (b) map of the relative permittivity in the plane ( x , y ) for z = 0 . (c) Comparison between the reconstructed relative permittivity (dashed curve) and the actual profile (solid curve) versus z for x = y = 0 .

Fig. 6
Fig. 6

Two cubes of side a = λ 4 separated by a distance c = λ 3 along the z axis for different permittivities. We display the permittivity versus z λ for x = y = 0 . In solid is the actual profile, in dashed curve with diamond symbols, the reconstruction obtained with the nonlinear inversion scheme. The actual permittivity of the two cubes is ε (a) 1.01, (b) 2.25, (c) 4.0.

Fig. 7
Fig. 7

The objects are the same as those in Fig. 4, and we have kept the same representation, but with uncorrelated noise on the scattered field. The upper figures (a), (b), (c) are obtained for a level of noise u = 5 % , the lower figures (d), (e), (f) for a stronger noise u = 15 % .

Fig. 8
Fig. 8

Same as Fig. 7 but the noise consists now of multiplying the scattered field by a phase factor of the form e i ψ as specified in Eq. (17). The term of Gaussian noise is of a standard deviation σ = 5 ° , and for the correlated phase ψ a we have chosen γ = 10 ° . For the upper figures γ = 10 ° , for the lower figures, γ = 10 ° .

Fig. 9
Fig. 9

Nine cubes of side a = λ 4 distributed in a test domain of volume 8 λ 3 (see Table 1 for their positions). (a), (b), (c), (d) are the reconstructed maps of the relative permittivities: (a) map in the ( x , z ) plane for y λ = 0.375 , (b) map in the ( x , z ) plane for y λ = 0.575 , (c) map in the ( x , y ) plane for z λ = 0.675 , (d) map in the ( x , y ) plane for z λ = 0.575 . (e) Relative permittivity versus x λ along the horizontal line plotted in (a). (f) Relative permittivity versus z λ along the vertical line plotted in (a).

Fig. 10
Fig. 10

Same as Fig. 9 but the cubes have different relative permittivities as detailed in Table 1.

Tables (1)

Tables Icon

Table 1 Coordinates of the Center of the Nine Cubes ( a = λ 4 ) Embedded in an Investigation Domain Ω of Volume 8 λ 3 a

Equations (23)

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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 ) ,
α ( r j ) = 3 d 3 4 π ε ( r j ) ε 0 ε ( r j ) + 2 ε 0 ,
E d ( r ) = j = 1 N T ( 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 l d = B ̿ p l ,
α n = α n 1 + a n d n ,
h l , n = f l B ̿ α n E l ,
E l = [ I ̿ A ̿ α ] 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 ,
E l E l , n 1 = [ I ̿ A ̿ α n 1 ] 1 E l inc ,
F n ( a n ) = W Γ l = 1 L ( h l , n 1 Γ 2 + a n 2 B ̿ d n E l , n 1 Γ 2 2 a n Re h l , n 1 B ̿ d n E l , n 1 Γ ) .
a n = l = 1 L Re h l , n 1 B ̿ d n E l , n 1 Γ l = 1 L B ̿ d n E l , n 1 Γ 2 .
d n = g n ; α + γ n d n 1 ,
g n ; α = W Γ l = 1 L E l , n 1 * 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 ,
A r = max { [ Re ( f l ; v ) ] min [ Re ( f l ; v ) ] } l = 1 , , L ; v ,
A i = max { [ Im ( f l ; v ) ] min [ Im ( f l ; v ) ] } l = 1 , , L ; v .
f ̃ l ; v ( r k ) = f l ; v ( r k ) e i ψ l ; v , ψ l ; v = ψ l ; v g + ψ l a ,

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