Abstract

We show that, at multiple scattering, the scattered field amplitude at a plane close to the surface of a perfectly reflecting grating, and thus the image of the grating, does not at all resemble the grating profile. We demonstrate the performance of an inverse-scattering procedure that applies within the range of validity of the Rayleigh hypothesis and that permits the attainment of superresolution in the reconstructed surface profile. This is the first solution, to our knowledge, for cases in which the near-field optics methods established so far do not work.

© 1993 Optical Society of America

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  1. D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
    [CrossRef]
  2. G. A. Massey, Appl. Opt. 23, 658 (1984).
    [CrossRef] [PubMed]
  3. E. Betzig, A. Harootunian, A. Lewis, M. Isaacson, Appl. Opt. 25, 1890 (1986); E. Betzig, J. K. Trautman, J. S. Weiner, T. D. Harris, R. Wolfe, Appl. Opt. 31, 4563 (1992); E. Betzig, J. K. Trautman, Science 257, 189 (1992).
    [CrossRef] [PubMed]
  4. J. M. Vigoureux, C. Girard, D. Courjon, Opt. Lett. 14, 1039 (1989).
    [CrossRef] [PubMed]
  5. D. Courjon, K. Sarayeddine, M. Spajer, Opt. Commun. 71, 23 (1989).
    [CrossRef]
  6. D. W. Pohl, in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1990), pp. 243–312.
  7. C. Girard, D. Courjon, Phys. Rev. B 42, 9340 (1990).
    [CrossRef]
  8. B. Labani, C. Girard, D. Courjon, D. Van Labeke, J. Opt. Soc. Am. B 7, 936 (1990).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 30.
  10. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 7, p. 232.
  11. J. L. Urestky, Ann. Phys. (N.Y.) 33, 400 (1965).
    [CrossRef]
  12. R. Petit, M. Cadilhac, C. R. Acad. Sci. B 262, 468 (1966).
  13. R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
    [CrossRef]
  14. E. Wolf, M. Nieto-Vesperinas, J. Opt. Soc. Am. A 2, 886 (1985).
    [CrossRef]
  15. N. Garcia, N. Cabrera, Phys. Rev. B 18, 576 (1978).
    [CrossRef]
  16. M. Nieto-Vesperinas, J. M. Soto-Crespo, Phys. Rev. B 38, 7250 (1988).
    [CrossRef]
  17. M. Nieto-Vesperinas, N. Garcia, Opt. Acta 28, 1651 (1981).
    [CrossRef]
  18. K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
    [CrossRef]

1990

1989

J. M. Vigoureux, C. Girard, D. Courjon, Opt. Lett. 14, 1039 (1989).
[CrossRef] [PubMed]

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

1988

M. Nieto-Vesperinas, J. M. Soto-Crespo, Phys. Rev. B 38, 7250 (1988).
[CrossRef]

1986

1985

1984

D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

G. A. Massey, Appl. Opt. 23, 658 (1984).
[CrossRef] [PubMed]

1981

M. Nieto-Vesperinas, N. Garcia, Opt. Acta 28, 1651 (1981).
[CrossRef]

K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
[CrossRef]

1978

N. Garcia, N. Cabrera, Phys. Rev. B 18, 576 (1978).
[CrossRef]

1969

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

1966

R. Petit, M. Cadilhac, C. R. Acad. Sci. B 262, 468 (1966).

1965

J. L. Urestky, Ann. Phys. (N.Y.) 33, 400 (1965).
[CrossRef]

Betzig, E.

Cabrera, N.

N. Garcia, N. Cabrera, Phys. Rev. B 18, 576 (1978).
[CrossRef]

Cadilhac, M.

R. Petit, M. Cadilhac, C. R. Acad. Sci. B 262, 468 (1966).

Celli, V.

K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
[CrossRef]

Courjon, D.

Denk, W.

D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Garcia, N.

M. Nieto-Vesperinas, N. Garcia, Opt. Acta 28, 1651 (1981).
[CrossRef]

K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
[CrossRef]

N. Garcia, N. Cabrera, Phys. Rev. B 18, 576 (1978).
[CrossRef]

Girard, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 30.

Harootunian, A.

Isaacson, M.

Labani, B.

Lanz, M.

D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Lewis, A.

Massey, G. A.

Millar, R. F.

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, J. M. Soto-Crespo, Phys. Rev. B 38, 7250 (1988).
[CrossRef]

E. Wolf, M. Nieto-Vesperinas, J. Opt. Soc. Am. A 2, 886 (1985).
[CrossRef]

M. Nieto-Vesperinas, N. Garcia, Opt. Acta 28, 1651 (1981).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 7, p. 232.

Petit, R.

R. Petit, M. Cadilhac, C. R. Acad. Sci. B 262, 468 (1966).

Pohl, D. W.

D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

D. W. Pohl, in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1990), pp. 243–312.

Rieder, K. H.

K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
[CrossRef]

Sarayeddine, K.

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

Soto-Crespo, J. M.

M. Nieto-Vesperinas, J. M. Soto-Crespo, Phys. Rev. B 38, 7250 (1988).
[CrossRef]

Spajer, M.

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

Urestky, J. L.

J. L. Urestky, Ann. Phys. (N.Y.) 33, 400 (1965).
[CrossRef]

Van Labeke, D.

Vigoureux, J. M.

Wolf, E.

Ann. Phys.

J. L. Urestky, Ann. Phys. (N.Y.) 33, 400 (1965).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. W. Pohl, W. Denk, M. Lanz, Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

C. R. Acad. Sci. B

R. Petit, M. Cadilhac, C. R. Acad. Sci. B 262, 468 (1966).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

M. Nieto-Vesperinas, N. Garcia, Opt. Acta 28, 1651 (1981).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Phys. Rev. B

C. Girard, D. Courjon, Phys. Rev. B 42, 9340 (1990).
[CrossRef]

N. Garcia, N. Cabrera, Phys. Rev. B 18, 576 (1978).
[CrossRef]

M. Nieto-Vesperinas, J. M. Soto-Crespo, Phys. Rev. B 38, 7250 (1988).
[CrossRef]

Proc. Camb. Philos. Soc.

R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

Surf. Sci.

K. H. Rieder, N. Garcia, V. Celli, Surf. Sci. 108, 169 (1981).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 30.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 7, p. 232.

D. W. Pohl, in Advances in Optical and Electron Microscopy, C. J. R. Sheppard, T. Mulvey, eds. (Academic, New York, 1990), pp. 243–312.

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Figures (3)

Fig. 1
Fig. 1

NFO scattering geometry.

Fig. 2
Fig. 2

(a) The solid curve is test profile D(x) = h[cos(2πx/a)] for a = 3.18λ, h/a = 0.08. The dashed curve coincident on the test profile is the reconstructed profile from the scattered field at z0 = h and the upper dashed curve is the scattered field at z0 = h. (b) Same as (a) for a = 0.16λ, h/a = 0.08. Lower dashed curves from top to bottom are the reconstructed profiles from z0 = 0, h, 3h, and 10h, respectively. The upper dashed curve is the scattered field at z0 = h.

Fig. 3
Fig. 3

Same as Fig. 2(a) for D(x) = h[sin(2πz/a) + cos(6πx/a)]. The lower dashed curves are reconstructions from the scattered field at z0 = h for a = 3.18λ, h = 0.095λ (curve 1) and a = 1.1λ, h = 0.03λ (curve 2). The upper dashed curves are scattered fields at z0 = h for a = 3.18λ, h = 0.095λ (curve 1) and a = 1.1λ, h = 0.03λ (curve 2). Notice the loss of contrast in the reconstruction with the smaller a.

Equations (5)

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U ( R , z ) = A ( K ) exp [ i ( K R + k z z ) ] d K ,
A G = ( 1 / k G z ) a / 2 a / 2 J ( R ) exp [ i ( K G R + k G z z ) ] d R ,
exp [ i k i z D ( R ) ] + G A G exp { i [ G R + k G z D ( R ) ] } = 0 .
D n ( R ) = ( 1 / 2 k i z ) ln | G A G exp ( i G R ) × exp { i [ k G z i k i z D n 1 ( R ) ] } | ,
D 0 ( R ) = ( 1 / 2 k i z ) ln | G A G exp ( i G R ) | .

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