Abstract

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.

© 2007 Optical Society of America

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References

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  1. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).
  2. F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).
  3. C. Giard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996).
    [CrossRef]
  4. D. Courjon, Near-field Microscopy and Near-field Optics (Imperial College Press, 2003).
  5. D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
    [CrossRef]
  6. I. Akduman and A. Alkumru, Inverse Probl. 11, 1125 (1995).
    [CrossRef]
  7. P.-M. Cutzach and C. Hazard, Math. Models Meth. Appl. Sci. 21, 433 (1998).
  8. P. Carney and J. Schotland, J. Opt. A, Pure Appl. Opt. 4, s140 (2002).
    [CrossRef]
  9. G. Bao and P. Li, J. Comput. Math. 25, 10 (2007).
  10. Y. Chen, Inverse Probl. 13, 253 (1997).
    [CrossRef]
  11. G. Bao and P. Li, SIAM J. Appl. Math. 65, 2049 (2005).
    [CrossRef]
  12. G. Bao and J. Liu, SIAM J. Sci. Comput. (USA) 25, 1102 (2003).
    [CrossRef]
  13. J. Coyle, Inverse Probl. 16, 275 (2000).
    [CrossRef]
  14. F. Natterer, Inverse Probl. 20, 447 (2004).
    [CrossRef]
  15. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
    [CrossRef]
  16. E. Turkel and A. Yefet, Appl. Numer. Math. 27, 533 (1998).
    [CrossRef]

2007

G. Bao and P. Li, J. Comput. Math. 25, 10 (2007).

2005

G. Bao and P. Li, SIAM J. Appl. Math. 65, 2049 (2005).
[CrossRef]

2004

F. Natterer, Inverse Probl. 20, 447 (2004).
[CrossRef]

2003

G. Bao and J. Liu, SIAM J. Sci. Comput. (USA) 25, 1102 (2003).
[CrossRef]

2002

P. Carney and J. Schotland, J. Opt. A, Pure Appl. Opt. 4, s140 (2002).
[CrossRef]

2000

J. Coyle, Inverse Probl. 16, 275 (2000).
[CrossRef]

1998

P.-M. Cutzach and C. Hazard, Math. Models Meth. Appl. Sci. 21, 433 (1998).

E. Turkel and A. Yefet, Appl. Numer. Math. 27, 533 (1998).
[CrossRef]

1997

Y. Chen, Inverse Probl. 13, 253 (1997).
[CrossRef]

1996

C. Giard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996).
[CrossRef]

1995

I. Akduman and A. Alkumru, Inverse Probl. 11, 1125 (1995).
[CrossRef]

1989

D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
[CrossRef]

Akduman, I.

I. Akduman and A. Alkumru, Inverse Probl. 11, 1125 (1995).
[CrossRef]

Alkumru, A.

I. Akduman and A. Alkumru, Inverse Probl. 11, 1125 (1995).
[CrossRef]

Bao, G.

G. Bao and P. Li, J. Comput. Math. 25, 10 (2007).

G. Bao and P. Li, SIAM J. Appl. Math. 65, 2049 (2005).
[CrossRef]

G. Bao and J. Liu, SIAM J. Sci. Comput. (USA) 25, 1102 (2003).
[CrossRef]

Carney, P.

P. Carney and J. Schotland, J. Opt. A, Pure Appl. Opt. 4, s140 (2002).
[CrossRef]

Chen, Y.

Y. Chen, Inverse Probl. 13, 253 (1997).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

Courjon, D.

D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
[CrossRef]

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

Coyle, J.

J. Coyle, Inverse Probl. 16, 275 (2000).
[CrossRef]

Cutzach, P.-M.

P.-M. Cutzach and C. Hazard, Math. Models Meth. Appl. Sci. 21, 433 (1998).

Dereux, A.

C. Giard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996).
[CrossRef]

Engl, H.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Giard, C.

C. Giard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996).
[CrossRef]

Hanke, M.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Hazard, C.

P.-M. Cutzach and C. Hazard, Math. Models Meth. Appl. Sci. 21, 433 (1998).

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

Li, P.

G. Bao and P. Li, J. Comput. Math. 25, 10 (2007).

G. Bao and P. Li, SIAM J. Appl. Math. 65, 2049 (2005).
[CrossRef]

Liu, J.

G. Bao and J. Liu, SIAM J. Sci. Comput. (USA) 25, 1102 (2003).
[CrossRef]

Natterer, F.

F. Natterer, Inverse Probl. 20, 447 (2004).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).

Neubauer, A.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

Sarayeddine, K.

D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
[CrossRef]

Schotland, J.

P. Carney and J. Schotland, J. Opt. A, Pure Appl. Opt. 4, s140 (2002).
[CrossRef]

Spajer, M.

D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
[CrossRef]

Turkel, E.

E. Turkel and A. Yefet, Appl. Numer. Math. 27, 533 (1998).
[CrossRef]

Yefet, A.

E. Turkel and A. Yefet, Appl. Numer. Math. 27, 533 (1998).
[CrossRef]

Appl. Numer. Math.

E. Turkel and A. Yefet, Appl. Numer. Math. 27, 533 (1998).
[CrossRef]

Inverse Probl.

Y. Chen, Inverse Probl. 13, 253 (1997).
[CrossRef]

J. Coyle, Inverse Probl. 16, 275 (2000).
[CrossRef]

F. Natterer, Inverse Probl. 20, 447 (2004).
[CrossRef]

I. Akduman and A. Alkumru, Inverse Probl. 11, 1125 (1995).
[CrossRef]

J. Comput. Math.

G. Bao and P. Li, J. Comput. Math. 25, 10 (2007).

J. Opt. A, Pure Appl. Opt.

P. Carney and J. Schotland, J. Opt. A, Pure Appl. Opt. 4, s140 (2002).
[CrossRef]

Math. Models Meth. Appl. Sci.

P.-M. Cutzach and C. Hazard, Math. Models Meth. Appl. Sci. 21, 433 (1998).

Opt. Commun.

D. Courjon, K. Sarayeddine, and M. Spajer, Opt. Commun. 71, 23 (1989).
[CrossRef]

Rep. Prog. Phys.

C. Giard and A. Dereux, Rep. Prog. Phys. 59, 657 (1996).
[CrossRef]

SIAM J. Appl. Math.

G. Bao and P. Li, SIAM J. Appl. Math. 65, 2049 (2005).
[CrossRef]

SIAM J. Sci. Comput. (USA)

G. Bao and J. Liu, SIAM J. Sci. Comput. (USA) 25, 1102 (2003).
[CrossRef]

Other

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
[CrossRef]

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

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, 1998).

F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).

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Figures (2)

Fig. 1
Fig. 1

Photon scanning tunneling microscopy.

Fig. 2
Fig. 2

Example 1: (a) true scatterer, (b) reconstructed scatterer.

Equations (21)

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

Δ u + n 2 k 2 ( 1 + q ) u = 0 ,
n ( x ) = { 1 for x 2 > 0 , n 0 for x 2 < 0 .
Δ u ref + n 2 k 2 u ref = 0 ,
u = u ref + u s .
Δ u s + n 2 k 2 ( 1 + q ) u s = k 2 q u ref .
Δ G ( x , y ) + n 2 ( x ) k 2 G ( x , y ) = δ ( x y ) ,
u s ( x ) = k 2 D G ( x , y ) q ( y ) ( u ref ( y ) + u s ( y ) ) d y .
u s ( x ) = k 2 D G ( x , y ) q ( y ) u ref ( y ) d y ,
δ q = q k q k ̃ .
Δ u ̃ i s + n 2 k 2 ( 1 + q k ̃ ) u ̃ i s = k 2 q k ̃ u i ref ,
Δ u i s + n 2 k 2 ( 1 + q k ) u i s = k 2 q k u i ref .
Δ δ u i s + n 2 k 2 ( 1 + q k ̃ ) δ u i s = k 2 δ q ( u i ref + u ̃ i s ) .
M u i s ( x ) = [ u i s ( x 1 ) , , u i s ( x m ) ] T .
S i ( q k ) = M u i s .
R i ( q k ̃ ) = M ( δ u i s ) .
S i ( q k ̃ ) δ q = R i ( q k ̃ ) .
δ q = α S i ( q k ̃ ) * R i ( q k ̃ ) ,
Δ ψ i + n 2 k 2 ( 1 + q k ̃ ) ψ i = k 2 j = 1 m ζ i j δ ( x x j )
S i ( q k ̃ ) * R i ( q k ̃ ) = ( u i ref ¯ + u ̃ i s ¯ ) ψ i ,
δ q = α ( u i ref ¯ + u ̃ i s ¯ ) ψ i .
u i s ( x j ) ( 1 + σ rand ) u i s ( x j ) , j = 0 , , 32 , i = 0 , , 8 .

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