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 (1)

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

2005 (1)

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

2004 (1)

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

2003 (1)

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

2002 (1)

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

2000 (1)

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

1998 (2)

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 (1)

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

1996 (1)

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

1995 (1)

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

1989 (1)

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. (1)

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

Inverse Probl. (4)

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. (1)

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

J. Opt. A, Pure Appl. Opt. (1)

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

Math. Models Meth. Appl. Sci. (1)

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

Opt. Commun. (1)

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

Rep. Prog. Phys. (1)

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

SIAM J. Appl. Math. (1)

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

SIAM J. Sci. Comput. (USA) (1)

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

Other (4)

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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