Abstract

In an optical diffraction microscopy experiment, one measures the phase and amplitude of the field diffracted by the sample and uses an inversion algorithm to reconstruct its map of permittivity. We show that with an iterative procedure accounting for multiple scattering, it is possible to visualize details smaller than λ/4 with relatively few illumination and observation angles. The roles of incident evanescent waves and noise are also investigated.

© 2003 Optical Society of America

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References

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  1. P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
    [CrossRef]
  2. O. Haeberlé, A. Dieterlen, S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. 26, 1684–1686 (2001).
    [CrossRef]
  3. J. Enderlein, “Theoretical study of detection of a dipole emitter through an objective with high numerical aperture,” Opt. Lett. 25, 634–636 (2000).
    [CrossRef]
  4. J.-J. Greffet, R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
    [CrossRef]
  5. S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
    [CrossRef]
  6. P. S. Carney, J. C. Schotland, “Three-dimensional total internal reflection microscopy,” Opt. Lett. 26, 1072–1074 (2001).
    [CrossRef]
  7. C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
    [CrossRef]
  8. M. Lambert, D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563–576 (2000).
    [CrossRef]
  9. V. Lauer, “New approach to optical diffraction tomographyyielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205, 165–176 (2002).
    [CrossRef] [PubMed]
  10. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  11. S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
    [CrossRef]
  12. 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]
  13. W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
    [CrossRef]
  14. J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
    [CrossRef]
  15. N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
    [CrossRef]
  16. 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]
  17. R. E. Kleinman, P. M. van den Berg, “An extended range-modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
    [CrossRef]
  18. 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]
  19. L. Souriau, B. Duchêne, D. Lesselier, R. E. Kleinman, “Modified gradient approach to inverse scattering for binary objects in stratified media,” Inverse Probl. 12, 463–481 (1996).
    [CrossRef]
  20. R. E. Kleinman, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
    [CrossRef]
  21. 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]
  22. W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 1986).

2002 (1)

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

2001 (5)

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]

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
[CrossRef]

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

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

O. Haeberlé, A. Dieterlen, S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. 26, 1684–1686 (2001).
[CrossRef]

2000 (4)

M. Lambert, D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563–576 (2000).
[CrossRef]

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[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]

J. Enderlein, “Theoretical study of detection of a dipole emitter through an objective with high numerical aperture,” Opt. Lett. 25, 634–636 (2000).
[CrossRef]

1997 (2)

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]

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

1996 (1)

L. Souriau, B. Duchêne, D. Lesselier, 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, P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci. 29, 1157–1169 (1994).
[CrossRef]

1993 (1)

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

1992 (1)

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]

1991 (1)

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[CrossRef]

1990 (1)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

1989 (1)

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

1987 (1)

1969 (1)

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

Belkebir, K.

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]

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]

Bewersdorf, J.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[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]

Carminati, R.

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

Carney, P. S.

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

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

Chew, W. C.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Dieterlen, A.

Duchêne, B.

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

Enderlein, J.

Flannery, B. P.

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

Goldberg, B. B.

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
[CrossRef]

Greffet, J.-J.

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

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

Haeberlé, O.

Hell, S. W.

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

Hugonin, J.-P.

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[CrossRef]

Ippolito, S. B.

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
[CrossRef]

Jacquey, S.

Joachimowicz, N.

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[CrossRef]

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]

L. Souriau, B. Duchêne, D. Lesselier, 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, 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, “An extended range-modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[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]

Lambert, M.

M. Lambert, D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563–576 (2000).
[CrossRef]

Lauer, V.

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

Lesselier, D.

M. Lambert, D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563–576 (2000).
[CrossRef]

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

Minami, S.

Nakamura, O.

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]

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[CrossRef]

Press, W. H.

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

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.

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

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

Souriau, L.

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

Teukolski, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 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]

Ünlü, M. S.

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
[CrossRef]

van den Berg, P. M.

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, “An extended range-modified gradient technique for profile inversion,” Radio Sci. 28, 877–884 (1993).
[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 University, Cambridge, UK, 1986).

Wang, Y. M.

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

Wolf, E.

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

Appl. Phys. Lett. (3)

S. B. Ippolito, B. B. Goldberg, M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
[CrossRef]

C. M. Blanca, J. Bewersdorf, S. W. Hell, “Single sharp spot in fluorescence microscopy of two opposing lenses,” Appl. Phys. Lett. 79, 2321–2323 (2001).
[CrossRef]

P. S. Carney, J. C. Schotland, “Inverse scattering for near-field microscopy,” Appl. Phys. Lett. 77, 2798–2800 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Joachimowicz, C. Pichot, J.-P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1753 (1991).
[CrossRef]

IEEE Trans. Med. Imaging (1)

W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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]

Inverse Probl. (3)

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

M. Lambert, D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563–576 (2000).
[CrossRef]

J. Comput. Appl. Math. (1)

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. (1)

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. Microsc. (1)

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

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

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

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

Opt. Lett. (3)

Prog. Surf. Sci. (1)

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

Radio Sci. (2)

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

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

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Scattered far-field modulus (a) and phase (b) versus angles of detection for normal incidence. Noiseless data, bold curve; 10% noise, dashed curve.

Fig. 3
Fig. 3

Reconstruction from a single illumination: L=1, θ1=0°, and 19 angles of detection ranging from -60° to 60°. Reconstructed refractive index map in the investigated domain Ω with 2λ width and λ/2 height, represented with gray level.

Fig. 4
Fig. 4

Same as in Fig. 3 with 19 angles of incidence ranging from -35° to 35°.

Fig. 5
Fig. 5

Same as in Fig. 3 with 18 angles of incidence, 9 of them ranging from -70° to -43° and the others ranging from 43° to 70°. All the incident plane waves are totally reflected at the glass–air interface.

Fig. 6
Fig. 6

(a), (b) same as in Fig. 5 with 19 angles of incidence ranging from -70° to 70°; (c) cross-sectional cut of the reconstructed refractive index obtained in (b) for two given elevations, z=λ/10 (dotted curve) and z=λ/4 (dashed curve).

Fig. 7
Fig. 7

Same as in Fig. 6(a) but the inversion procedure applies the Born approximation.

Equations (29)

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

ΔEl(r)+r(z)k02El(r)=Sl-k02χ(r)El(r),
ΔG(r, r)+(z)k02G(r, r)=-δ(r-r)
El(r)=Elref(r)+k02ΩG(r, r)χ(r)El(r)dr,
Elref(r)=t(κl)exp(iκlx+iγlz),
Elref(r)=exp(iκlx+iγlbz)+r(κl)exp(iκlx-iγlbz),
G(r, r)=i4πγ(κ) {exp(iγ(κ)|z-z|)-r(κ)exp[iγ(κ)(z+z)]}×exp[iκ(x-x)]dκ.
G(r, r)=(2π/k0r)1/2γ(κ)g(κ, r)exp(ik0r-iπ/4),
eld(r)=Eld(κ)exp(ik0r)/r.
Eld(κΓ)=Ωk02χ(r)El(r)K(κ, r)dr,
Eld=KχEl,
El=Elref+GχEl.
El,n=El,n-1+αl,nυl,n,
χn=χn-1+βndn,
Fn(χn, El,n)=WΩl=1Lhl,n(1)Ω2+WΓl=1Lhl,n(2)Γ2,
WΩ=1l=1LElrefΩ2,WΓ=1l=1LflΓ2.
hl,n(1)=Elref-El,n-1+GχnEl,n,
hl,n(2)=fl-KχnEl,n.
χn=ξn2.
ξn=ξn-1+βn;ξdn;ξ.
dn;ξ=gn;ξ+γn;ξdn-1;ξ,
γn;ξ=gn;ξ, gn;ξ-gn-1;ξΩgn-1;ξΩ2,
gn;ξ=2ξn-1ReWΩl=1LE¯l,n-1Ghl,n-1(1)-WΓl=1LE¯l,n-1Khl,n-1(2),
υl,n=gl,n;E+γl,n;Elυl,n-1,
γl,n;El=gl,n;El, gl,n;El-gl,n-1;ElΩgl,n-1;ElΩ2,
gl,n;El=WΩ[χ¯n-1Ghl,n-1(1)-hl,n-1(1)]-WΓχ¯n-1Khl,n-1(2).
El,n=El,n-1+αl,n;υυl,n+αl,n;wwl,n.
wl,n=E˜l,n-1-El,n-1,
E˜l,n-1=[1-Gχn-1]-1Elinc,
fl=Eld[1+uA exp(iϕ)],

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