Abstract

We deal with the inverse-scattering problem for a dielectric slab embedded in a three-layered medium starting from multifrequency scattered field data under the framework of the Born approximation. This allows us to state the problem as a linear inverse one, and the singular-value decomposition (SVD) of the relevant operator makes it possible to investigate and to solve it. In particular, the SVD tool allows an analysis of the reconstruction capabilities of the algorithm in terms of spatial variability of the unknowns that can be retrieved. The new contribution consists in an analysis of the role of the discontinuity of the dielectric properties between the second and the third medium. This analysis is performed with regard both to the class of retrievable dielectric profiles and to the model error deriving from the Born approximation and shows, finally, that this discontinuity can be troublesome.

© 2004 Optical Society of America

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References

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  1. J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).
  2. E. Nyfors, “Industrial microwave sensors,” Subsurface Sens. Technol. Appl. 1, 23–43 (2000).
    [CrossRef]
  3. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  4. A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  5. R. Persico, F. Soldovieri, R. Pierri, “On the convergence properties of a quadratic approach to the inverse scattering problem,” J. Opt. Soc. Am. A 19, 2424–2428 (2002).
    [CrossRef]
  6. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).
  7. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
  8. J. Xia, A. K. Jordan, J. A. Kong, “Electromagnetic inverse-scattering theory for inhomogeneous dielectrics: the local reflection model,” J. Opt. Soc. Am. A 11, 1081–1086 (1994).
    [CrossRef]
  9. D. B. Ge, “An iterative technique in one dimensional profile inversion,” Inverse Probl. 3, 399–406 (1987).
    [CrossRef]
  10. T. Uno, S. Adachi, “Inverse scattering method for one dimensional inhomogeneous layered media,” IEEE Trans. Antennas Propag. AP-35, 1456–1466 (1987).
    [CrossRef]
  11. I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
    [CrossRef]
  12. C. J. Trantanella, D. G. Dudley, K. A. Nabulsi, “Beyond the Born approximation in one-dimensional profile reconstruction,” J. Opt. Soc. Am. A 12, 1469–1478 (1995).
    [CrossRef]
  13. A. G. Tijhuis, “Born-type reconstruction of material parameters in an inhomogeneous lossy dielectric slab from reflected-field data,” Wave Motion 11, 151–173 (1989).
    [CrossRef]
  14. G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999);“Errata,” J. Opt. Soc. Am. A 16, 2310 (1999).
    [CrossRef]
  15. R. Pierri, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multifrequency, multiview, and multifrequency–multiview approach,” J. Opt. Soc. Am. A 16, 2392–2399 (1999).
    [CrossRef]
  16. R. Pierri, G. Leone, R. Persico, F. Soldovieri, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento 24, 245–261 (2001).
  17. A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
    [CrossRef]
  18. R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
    [CrossRef]
  19. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1990), pp. 1–120.
  20. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  21. R. W. Deming, A. J. Devaney, “Diffraction tomography for multi-monostatic ground penetrating radar imaging,” Inverse Probl. 13, 29–45 (1997).
    [CrossRef]
  22. O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
    [CrossRef]
  23. A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
    [CrossRef]
  24. O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
    [CrossRef]
  25. A complex contrast function also accounts for possible losses in the investigation domain. This entails that the contrast function also depend on the work frequency. However, in the present analysis we consider this dependence negligible.
  26. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  27. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).
  28. G. D. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
    [CrossRef]
  29. A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).
  30. L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, Oxford, UK, 1982).

2002 (2)

R. Persico, F. Soldovieri, R. Pierri, “On the convergence properties of a quadratic approach to the inverse scattering problem,” J. Opt. Soc. Am. A 19, 2424–2428 (2002).
[CrossRef]

R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
[CrossRef]

2001 (1)

R. Pierri, G. Leone, R. Persico, F. Soldovieri, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento 24, 245–261 (2001).

2000 (2)

E. Nyfors, “Industrial microwave sensors,” Subsurface Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

1999 (3)

1998 (2)

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

1997 (1)

R. W. Deming, A. J. Devaney, “Diffraction tomography for multi-monostatic ground penetrating radar imaging,” Inverse Probl. 13, 29–45 (1997).
[CrossRef]

1995 (3)

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

C. J. Trantanella, D. G. Dudley, K. A. Nabulsi, “Beyond the Born approximation in one-dimensional profile reconstruction,” J. Opt. Soc. Am. A 12, 1469–1478 (1995).
[CrossRef]

1994 (1)

1989 (1)

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

1987 (2)

D. B. Ge, “An iterative technique in one dimensional profile inversion,” Inverse Probl. 3, 399–406 (1987).
[CrossRef]

T. Uno, S. Adachi, “Inverse scattering method for one dimensional inhomogeneous layered media,” IEEE Trans. Antennas Propag. AP-35, 1456–1466 (1987).
[CrossRef]

1984 (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Adachi, S.

T. Uno, S. Adachi, “Inverse scattering method for one dimensional inhomogeneous layered media,” IEEE Trans. Antennas Propag. AP-35, 1456–1466 (1987).
[CrossRef]

Akilov, G. P.

L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, Oxford, UK, 1982).

Akudman, I.

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

Arsenine, V. Y.

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bernini, R.

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1990), pp. 1–120.

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

Brancaccio, A.

R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
[CrossRef]

A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

Bucci, O. M.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Crocco, L.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

Daniels, J.

J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).

de Villiers, G. D.

G. D. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Deming, R. W.

R. W. Deming, A. J. Devaney, “Diffraction tomography for multi-monostatic ground penetrating radar imaging,” Inverse Probl. 13, 29–45 (1997).
[CrossRef]

Devaney, A. J.

R. W. Deming, A. J. Devaney, “Diffraction tomography for multi-monostatic ground penetrating radar imaging,” Inverse Probl. 13, 29–45 (1997).
[CrossRef]

Dudley, D. G.

Ebbini, E. S.

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Ge, D. B.

D. B. Ge, “An iterative technique in one dimensional profile inversion,” Inverse Probl. 3, 399–406 (1987).
[CrossRef]

Haddadin, O. S.

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Idemen, M.

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

Isernia, T.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

Jordan, A. K.

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Kantorovic, L. V.

L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, Oxford, UK, 1982).

Kong, J. A.

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Leone, G.

R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
[CrossRef]

R. Pierri, G. Leone, R. Persico, F. Soldovieri, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento 24, 245–261 (2001).

G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999);“Errata,” J. Opt. Soc. Am. A 16, 2310 (1999).
[CrossRef]

A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

McNally, B.

G. D. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Nabulsi, K. A.

Nyfors, E.

E. Nyfors, “Industrial microwave sensors,” Subsurface Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

Pascazio, V.

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

Persico, R.

Pierri, R.

Pike, E. R.

G. D. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Soldovieri, F.

R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
[CrossRef]

R. Persico, F. Soldovieri, R. Pierri, “On the convergence properties of a quadratic approach to the inverse scattering problem,” J. Opt. Soc. Am. A 19, 2424–2428 (2002).
[CrossRef]

R. Pierri, G. Leone, R. Persico, F. Soldovieri, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento 24, 245–261 (2001).

Tichonov, A. N.

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Tijhuis, A. G.

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

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).

Trantanella, C. J.

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).

Uno, T.

T. Uno, S. Adachi, “Inverse scattering method for one dimensional inhomogeneous layered media,” IEEE Trans. Antennas Propag. AP-35, 1456–1466 (1987).
[CrossRef]

Xia, J.

AEU Int. J. Electron. Commun. (1)

R. Pierri, A. Brancaccio, G. Leone, F. Soldovieri, “Electromagnetic prospection via homogeneous and inhomogeneous plane waves: the case of an embedded slab,” AEU Int. J. Electron. Commun. 56, 11–18 (2002).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. Uno, S. Adachi, “Inverse scattering method for one dimensional inhomogeneous layered media,” IEEE Trans. Antennas Propag. AP-35, 1456–1466 (1987).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

O. M. Bucci, L. Crocco, T. Isernia, V. Pascazio, “Inverse scattering problems with multifrequency data: reconstruction capabilities and solution strategies,” IEEE Trans. Geosci. Remote Sens. 38, 1749–1756 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998).
[CrossRef]

Inverse Probl. (4)

R. W. Deming, A. J. Devaney, “Diffraction tomography for multi-monostatic ground penetrating radar imaging,” Inverse Probl. 13, 29–45 (1997).
[CrossRef]

G. D. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

I. Akudman, M. Idemen, “On the use of Gaussian beams in one-dimensional profile inversion connected with lossy dielectric slabs,” Inverse Probl. 11, 315–328 (1995).
[CrossRef]

D. B. Ge, “An iterative technique in one dimensional profile inversion,” Inverse Probl. 3, 399–406 (1987).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995).
[CrossRef]

J. Opt. Soc. Am. A (6)

Nuovo Cimento (1)

R. Pierri, G. Leone, R. Persico, F. Soldovieri, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento 24, 245–261 (2001).

Subsurface Sens. Technol. Appl. (1)

E. Nyfors, “Industrial microwave sensors,” Subsurface Sens. Technol. Appl. 1, 23–43 (2000).
[CrossRef]

Wave Motion (1)

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

Other (10)

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

A. N. Tichonov, V. Y. Arsenine, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

J. Daniels, Surface Penetrating Radar (Institution of Electrical Engineers, London, 1996).

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics (Academic, New York, 1990), pp. 1–120.

A complex contrast function also accounts for possible losses in the investigation domain. This entails that the contrast function also depend on the work frequency. However, in the present analysis we consider this dependence negligible.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science, Utrecht, The Netherlands, 1987).

L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, Oxford, UK, 1982).

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Figures (13)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Model error versus permittivity of the third medium for a weak scatterer.

Fig. 3
Fig. 3

Model error versus permittivity of the third medium for a strong scatterer.

Fig. 4
Fig. 4

Pictorial representation of the physical relationship between the second interface and the nonlinearity.

Fig. 5
Fig. 5

Singular-value behavior in the complex case [ε3=9 (solid curve), ε3=6 (dotted–dashed curve), ε3=1 (dashed curve), perfect electric conductor (asterisks)].

Fig. 6
Fig. 6

Spatial spectral content in the complex case (p.e.c., perfect electric conductor).

Fig. 7
Fig. 7

Spectrum of the first singular function in the complex case.

Fig. 8
Fig. 8

Spatial spectral content in the complex case for a threshold below the second flat areas of the singular-value behavior [ε3=6 (solid curve), ε3=1 (dashed curve)].

Fig. 9
Fig. 9

Singular-value behavior in the real case [ε3=9 (solid curve), ε3=6 (dotted–dashed curve), ε3=1 (dashed curve), perfect electric conductor (asterisks)].

Fig. 10
Fig. 10

Spectrum of the first singular function in the real case.

Fig. 11
Fig. 11

Spatial spectral content in the real case.

Fig. 12
Fig. 12

Reconstruction of a weak real scatterer.

Fig. 13
Fig. 13

Reconstruction of a strong real scatterer.

Equations (25)

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

E(z, ω)=Einc(z, ω)+k220dGi(z, z, ω)E(z, ω)χ(z)dz,
zD,ωΩ,
Es(ω)=k220dGe(z, ω)E(z, ω)χ(z)dz,
ωΩ,
χ(z)=εr(z)ε2-1,zD0,elsewhere
Es(ω)=F(χ)=k220dGe(z, ω)E(z, ω; χ(·))χ(z)dz, ωΩ.
Esl(ω)=A(χ)=k220dGe(z, ω)Einc(z, ω)χ(z)dz, ωΩ.
Esl(ω)=A(χ)=0dχ(z)[exp(-jk2z)+R23 exp(-j2k2d)exp(+jk2z)]2dz, ωΩ.
err=m|Es|m-Esl|m|2m|Es|m|21/2=m|F(χ)|m-A(χ)|m|2m|F(χ)|m|21/2,
Esl(ω)=0dχ(z)exp(-j2k2z)dz+2R23 exp(-j2k2d)0dχ(z)dz+R232 exp(-j4k2d)0dχ(z)exp(+j2k2z)dz=χ˜(2k2)+2R23 exp(-j2k2d)χ˜(0)+R232 exp(-j4k2d)χ˜(-2k2),
Esl(ω)=0dχ(z)exp(-2jk2z)dz=χ˜(2k2).
N=d(k2 max-k2 min)π=dε2c0π(ωmax-ωmin).
SSC(k)=m=1M|u˜m(k)|.
Einc(z, ω)=T12M2E0[exp(-jk2z)+R23 exp(-j2k2d)×exp(+jk2z)],zD,ωΩ,
Ge(z, ω)=-j T212k2M2[exp(-jk2z)+R23 exp(-j2k2d)×exp(+jk2z)],zD,ωΩ,
Gi(z, z, ω)=exp(-jk2|z-z|)2jk2+R21n1 exp(-jk2z)2jk2g+R23n2 exp[-jk2(d-z)]2jk2g, z, zD,ωΩ,
n1=exp(-jk2z)+R23 exp[-jk2(2d-z)],
n2=exp[-jk2(d-z)]+R21 exp[-jk2(d+z)],
g=1-R21R23 exp(-2jk2d).
Esl(ω)=A(χ)=4 exp[jϕ(ω)]0dχ(z)cosk2z-k2d+α(ω)22dz,
0dχ(z)[A1+(E)]*dz=ΩA1(χ)E*(ω)dω=Ω exp[jϕ(ω)]A(χ)E*(ω)dω=ΩA(χ){exp[-jϕ(ω)]E(ω)}*dω=0dχ(z){A+(exp[-jϕ(ω)]E)}*dz χL2(0, d),EL2(Ω).
A1+(·)=A+(exp[-jϕ(ω)](·))
A1+(A1(χ))=A+(exp[-jϕ(ω)]{exp[jϕ(ω)]A(χ)})=A+(A(χ)).
vn=1σn A(un),
vn=1σn A1(un)=1σn exp[jϕ(ω)]A(un)=exp[jϕ(ω)]vn.

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