Abstract

A compressed sensing scheme for near-field imaging of corrugations of relative sparse Fourier components is proposed. The scheme employs random sparse measurement of near field to recover the angular spectrum of the scattered field. Surprisingly, it can be shown heuristically and numerically that under the Rayleigh hypothesis the angular spectrum is compressible and amenable to compressed sensing techniques. Iteration schemes are developed for recovering the surface profile from the angular spectrum. The proposed nonlinear least squares in the Fourier basis produces accurate reconstructions even when the Rayleigh hypothesis is known to be false.

© 2012 Optical Society of America

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  1. P. Beckmann, “Scattering of light by rough surfaces.” Prog. Opt. 6, 53–69 (1967).
    [CrossRef]
  2. F. B. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).
  3. O. Ivanyshyn and T. Johanson, “Nonlinear integral equation methods for the reconstruction of an acoustically sound soft obstacle,” J. Integral Equ. Appl. 19, 289–308 (2007).
    [CrossRef]
  4. A. Schatzberg and A. J. Devaney, “Rough surface inverse scattering within the Rytov approximation,” J. Opt. Soc. Am. A 10, 942–950 (1993).
    [CrossRef]
  5. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [CrossRef]
  6. R. J. Wombell and J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
    [CrossRef]
  7. R. J. Wombell and J. A. DeSanto, “Reconstruction of rough-surface profiles with Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
    [CrossRef]
  8. A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
    [CrossRef]
  9. E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
    [CrossRef]
  10. B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
    [CrossRef]
  11. A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
    [CrossRef]
  12. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
    [CrossRef]
  13. B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
    [CrossRef]
  14. N. Garcia and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
    [CrossRef]
  15. M. Nieto-Vesperinas and N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  16. K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
    [CrossRef]
  17. A. Fannjiang, “Compressive imaging of subwavelength structures,” J. Imaging Sci. 2, 1277–1291 (2009).
    [CrossRef]
  18. B. Deutsch, R. Hillenbrand, and L. Novotny, “Near-field amplitude and phase recovery using phase-shifting interferometry,” Opt. Express 16, 494–501 (2008).
    [CrossRef]
  19. E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
    [CrossRef]
  20. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  21. A. Kirsch, “Diffraction by periodic structures,” in Inverse Problems in Mathematical Physics (Springer-Verlag, 1993), Vol. 422, pp. 87–102.
  22. T. Arens and T. Hohage, “On radiation conditions for rough surface scattering problems,” IMA J. Appl. Math. 70, 839–847 (2005).
    [CrossRef]
  23. G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
    [CrossRef]
  24. R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Math. Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
    [CrossRef]
  25. R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Math. Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [CrossRef]
  26. J. B Keller, “Singularities and Rayleigh’s hypothesis for diffraction gratings,” J. Opt. Soc. Am. A 17, 456–457 (2000).
    [CrossRef]
  27. T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).
  28. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
    [CrossRef]
  29. H. Rauhut, “Stability results for random sampling of sparse trigonometric polynomials,” IEEE Trans. Inf. Theory 54, 5661–5670 (2008).
    [CrossRef]
  30. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).
  31. J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
    [CrossRef]
  32. A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
    [CrossRef]
  33. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).
  34. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer, 1998).
  35. J. Yang and Y. Zhang, “Alternating direction algorithms for L1 problems in compressive sensing,” TR09-37 (CAAM, Rice University , 2010).
  36. J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

2009 (1)

A. Fannjiang, “Compressive imaging of subwavelength structures,” J. Imaging Sci. 2, 1277–1291 (2009).
[CrossRef]

2008 (3)

H. Rauhut, “Stability results for random sampling of sparse trigonometric polynomials,” IEEE Trans. Inf. Theory 54, 5661–5670 (2008).
[CrossRef]

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).

B. Deutsch, R. Hillenbrand, and L. Novotny, “Near-field amplitude and phase recovery using phase-shifting interferometry,” Opt. Express 16, 494–501 (2008).
[CrossRef]

2007 (2)

J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

O. Ivanyshyn and T. Johanson, “Nonlinear integral equation methods for the reconstruction of an acoustically sound soft obstacle,” J. Integral Equ. Appl. 19, 289–308 (2007).
[CrossRef]

2006 (5)

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
[CrossRef]

T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).

2005 (1)

T. Arens and T. Hohage, “On radiation conditions for rough surface scattering problems,” IMA J. Appl. Math. 70, 839–847 (2005).
[CrossRef]

2001 (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

2000 (3)

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

J. B Keller, “Singularities and Rayleigh’s hypothesis for diffraction gratings,” J. Opt. Soc. Am. A 17, 456–457 (2000).
[CrossRef]

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

1998 (1)

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

1993 (2)

1991 (3)

R. J. Wombell and J. A. DeSanto, “Reconstruction of rough-surface profiles with Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

R. J. Wombell and J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

1984 (2)

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

1981 (2)

M. Nieto-Vesperinas and N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
[CrossRef]

1972 (1)

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef]

1971 (1)

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Math. Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

1969 (1)

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Math. Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

1967 (1)

P. Beckmann, “Scattering of light by rough surfaces.” Prog. Opt. 6, 53–69 (1967).
[CrossRef]

1965 (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Akamine, S.

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Akduman, I.

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

Arens, T.

T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).

T. Arens and T. Hohage, “On radiation conditions for rough surface scattering problems,” IMA J. Appl. Math. 70, 839–847 (2005).
[CrossRef]

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

Ash, E. A.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef]

Bass, F. B.

F. B. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

Beckmann, P.

P. Beckmann, “Scattering of light by rough surfaces.” Prog. Opt. 6, 53–69 (1967).
[CrossRef]

Candès, E. J.

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Carney, P. S.

J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

Celli, V.

K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
[CrossRef]

Chandler-Wilde, S. N.

T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer, 1998).

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

Deckert, V.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Denk, W.

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Derveaux, G.

G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
[CrossRef]

DeSanto, J.

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

DeSanto, J. A.

T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).

R. J. Wombell and J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

R. J. Wombell and J. A. DeSanto, “Reconstruction of rough-surface profiles with Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

Deutsch, B.

Devaney, A. J.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Erdmann, G.

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

Fannjiang, A.

A. Fannjiang, “Compressive imaging of subwavelength structures,” J. Imaging Sci. 2, 1277–1291 (2009).
[CrossRef]

Fuks, I. M.

F. B. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
[CrossRef]

K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
[CrossRef]

M. Nieto-Vesperinas and N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Hadimioglu, B.

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Harootunian, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Hecht, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Hereman, W.

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

Hillenbrand, R.

Hohage, T.

T. Arens and T. Hohage, “On radiation conditions for rough surface scattering problems,” IMA J. Appl. Math. 70, 839–847 (2005).
[CrossRef]

Isaacson, M.

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Ivanyshyn, O.

O. Ivanyshyn and T. Johanson, “Nonlinear integral equation methods for the reconstruction of an acoustically sound soft obstacle,” J. Integral Equ. Appl. 19, 289–308 (2007).
[CrossRef]

Johanson, T.

O. Ivanyshyn and T. Johanson, “Nonlinear integral equation methods for the reconstruction of an acoustically sound soft obstacle,” J. Integral Equ. Appl. 19, 289–308 (2007).
[CrossRef]

Keller, J. B

Khuri-Yakub, B. T.

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Kirsch, A.

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

A. Kirsch, “Diffraction by periodic structures,” in Inverse Problems in Mathematical Physics (Springer-Verlag, 1993), Vol. 422, pp. 87–102.

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer, 1998).

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

Lanz, M.

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Lewis, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Martin, O. J. F.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Meier, A.

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

Millar, R. F

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Math. Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Math. Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

Misra, M.

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

Murray, A.

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Nicholls, G.

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, “Near-field optics inverse-scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
[CrossRef]

M. Nieto-Vesperinas and N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Novotny, L.

Ozdemir, O.

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

Papanicolaou, G.

G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
[CrossRef]

Pohl, D. W.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Quate, C. F.

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Rauhut, H.

H. Rauhut, “Stability results for random sampling of sparse trigonometric polynomials,” IEEE Trans. Inf. Theory 54, 5661–5670 (2008).
[CrossRef]

Riedera, K. H.

K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
[CrossRef]

Sahinturk, H.

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Schatzberg, A.

Schotland, J. C.

J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

Sick, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Sun, J.

J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

Tao, T.

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Tsogka, C.

G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
[CrossRef]

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Wild, U. P.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Wombell, R. J.

R. J. Wombell and J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

R. J. Wombell and J. A. DeSanto, “Reconstruction of rough-surface profiles with Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

Yamada, H.

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Yang, J.

J. Yang and Y. Zhang, “Alternating direction algorithms for L1 problems in compressive sensing,” TR09-37 (CAAM, Rice University , 2010).

Yapar, A.

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

Zenobi, R.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Zhang, Y.

J. Yang and Y. Zhang, “Alternating direction algorithms for L1 problems in compressive sensing,” TR09-37 (CAAM, Rice University , 2010).

Ann. Phys. (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Appl. Phys. Lett. (1)

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

C. R. Acad. Sci. (1)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Acad. Sci. 346, 589–592 (2008).

Commun. Comput. Phys. (1)

T. Arens, S. N. Chandler-Wilde, and J. A. DeSanto, “On integral equation and least squares methods for scattering by diffraction gratings,” Commun. Comput. Phys. 1, 1010–1042 (2006).

IEEE Trans. Antennas Propag. (1)

A. Yapar, O. Ozdemir, H. Sahinturk, and I. Akduman, “A Newton method for the reconstruction of perfectly conducting slightly rough surface profiles,” IEEE Trans. Antennas Propag. 54, 275–279 (2006).
[CrossRef]

IEEE Trans. Inf. Theory (3)

E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

H. Rauhut, “Stability results for random sampling of sparse trigonometric polynomials,” IEEE Trans. Inf. Theory 54, 5661–5670 (2008).
[CrossRef]

IMA J. Appl. Math. (1)

T. Arens and T. Hohage, “On radiation conditions for rough surface scattering problems,” IMA J. Appl. Math. 70, 839–847 (2005).
[CrossRef]

Inverse Probl. (2)

G. Derveaux, G. Papanicolaou, and C. Tsogka, “Resolution and denoising in near-field imaging,” Inverse Probl. 22, 1437–1456 (2006).
[CrossRef]

R. J. Wombell and J. A. DeSanto, “The reconstruction of shallow rough-surface profiles from scattered field data,” Inverse Probl. 7, L7–L12 (1991).
[CrossRef]

J. Appl. Phys. (1)

J. Sun, P. S. Carney, and J. C. Schotland, “Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102, 103013 (2007).

J. Chem. Phys. (1)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

J. Imaging Sci. (1)

A. Fannjiang, “Compressive imaging of subwavelength structures,” J. Imaging Sci. 2, 1277–1291 (2009).
[CrossRef]

J. Integral Equ. Appl. (2)

O. Ivanyshyn and T. Johanson, “Nonlinear integral equation methods for the reconstruction of an acoustically sound soft obstacle,” J. Integral Equ. Appl. 19, 289–308 (2007).
[CrossRef]

A. Meier, T. Arens, S. N. Chandler-Wilde, and A. Kirsch, “A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces,” J. Integral Equ. Appl. 12, 281–321 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

Math. Proc. Cambridge Philos. Soc. (2)

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Math. Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

R. F Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Math. Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

Nature (1)

E. A. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature 237, 510–512 (1972).
[CrossRef]

Opt. Acta (1)

M. Nieto-Vesperinas and N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

B. T. Khuri-Yakub, S. Akamine, B. Hadimioglu, H. Yamada, and C. F. Quate, “Near field acoustic microscopy,” Proc. SPIE 1556, 30–39 (1991).
[CrossRef]

Prog. Opt. (1)

P. Beckmann, “Scattering of light by rough surfaces.” Prog. Opt. 6, 53–69 (1967).
[CrossRef]

SIAM Rev. (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43, 129–159 (2001).
[CrossRef]

Surf. Sci. (1)

K. H. Riedera, N. Garcia, and V. Celli, “An effective procedure to determine corrugation functions from atomic beam-diffraction intensities,” Surf. Sci. 108, 169–180 (1981).
[CrossRef]

Ultramicroscopy (1)

A. Lewis, M. Isaacson, A. Harootunian, and A. Murray, “Development of a 500 Å spatial resolution light microscope. I. Light is efficiently transmitted through λ/16 diameter apertures,” Ultramicroscopy 13, 227–231 (1984).
[CrossRef]

Waves Random Media (1)

J. DeSanto, G. Erdmann, W. Hereman, and M. Misra, “Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces,” Waves Random Media 8, 385–414 (1998).
[CrossRef]

Other (5)

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer, 1998).

J. Yang and Y. Zhang, “Alternating direction algorithms for L1 problems in compressive sensing,” TR09-37 (CAAM, Rice University , 2010).

F. B. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, 1980).

A. Kirsch, “Diffraction by periodic structures,” in Inverse Problems in Mathematical Physics (Springer-Verlag, 1993), Vol. 422, pp. 87–102.

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Figures (8)

Fig. 1.
Fig. 1.

Plane wave incidence upon a rough surface.

Fig. 2.
Fig. 2.

Real part of the scattered field induced by the profile h(x)=0.2454sin(x) (top) and the profile h(x)=0.01sin(12x)+0.007cos(7x) (bottom) with the normal incident plane wave.

Fig. 3.
Fig. 3.

The profile h(x)=0.01(p=04sin((1+3p)x)) and the reconstructions (bottom). The top panels show the real part (top left) and the imaginary part (right) of the angular spectrum un (blue crosses), the reconstructed angular spectrum u~n (red dots), and the theoretical prediction vn (green circles).

Fig. 4.
Fig. 4.

The profile h(x)=0.01sin(12x)+0.007cos(7x) and the reconstructions (bottom). The labeling is the same as in Fig. 3.

Fig. 5.
Fig. 5.

Periodized Gaussian h(x)=b(e(ax)2erf(aπ)ππ)·χ~[0.9π,0.9π](x), a=2, b=0.01 and the reconstructions where χ~ is a smoothed indicator function (bottom). The labeling is the same as in Fig. 3.

Fig. 6.
Fig. 6.

Double subwavelength peaks h(x)=b(ζ(a(x12))+ζ(a(x+12))), a=2.5, b=0.01, ζ(x)=exp(11x21)χ(1,1)(x)+c0 and the reconstructions (bottom). Here the constant c0 is chosen such that ζ^0=0. The labeling is the same as in Fig. 3.

Fig. 7.
Fig. 7.

The sinusoidal profile h(x)=0.491cos(x) and the reconstructions (bottom). The labeling is the same as in Fig. 3.

Fig. 8.
Fig. 8.

The sinusoidal profile h(x)=0.589cos(x) and the reconstructions (bottom). The labeling is the same as in Fig. 3.

Equations (73)

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Δutot+k2utot=0inΩR2,k>0utot=0onΩ,
Ω={r=(x,z)R2z>h(x)},hC(R)L(R).
uinc(r)=eikd^·r=eik(xcosθzsinθ),d^=(cosθ,sinθ),0<θ<π.
u(x+L,h(x))=eik((x+L)cosθh(x)sinθ)=eikLcosθu(x,h(x)).
u(x+nL,z)=einLkcosθu(x,z)for all(x,z)Ω,n.
u(x,z)=nZun(z)ei(n+kcosθ)x=nZun(z)eikαnx
αn=nk+cosθ,
u..n+k2(1αn2)un=0.
u(x,z)=|αn|1aneik(αnxβnz)(incoming waves)+|αn|1bneik(αnx+βnz)(outgoing waves)+|αn|>1cneik(αnx+βnz)(evanescent waves),
βn={1αn2,|αn|1iαn21,|αn|>1.
u(x,z)=nZuneik(αnx+βnz),z>hmax.
ΩΓ{(x,z)Ω,x[π,π)}
Γ{(x,z)Ωx[π,π)}.
un=i4πkβnππeik(αnx+βnh(x))(utot(r)ν|rΓ)1+h.2(x)dx.
|cosθ+nk|1,nZ
u(xj,z0)eikcosθxj=nZeinxjuneikβnz0
Xn=uneikβnz0m,
Yj=u(xj,z0)eikxjcosθ,
A=[Aj,n]=1meinxj,
Y=AX+E
minX1subject toYAX2ϵE2,
(1δ)Z22AZ22(1+δ)Z22,δ(0,1),
mlnmCδ2sln2slnNln1η,η(0,1)
Anj=1me2πinξj,n=N/2,,N/21,
δ2s<21
X~X2C01sXsX1+C1ϵ,X~X1C0XsX1+C1ϵ
u~n=1meikβnz0X~n.
|αn|=|cosθ+nk|1.
k|βn|z0Ce
βn=cos2θ+2nkcosθ+n2k21|n|k1.
k(|n|k1)z0k|βn|z0Ce
|n|n0Cez0+k,
mlnm>C(21)2sln2slnNln1η
u~u2eCem(C01sXsX1+C1ϵ)
[Aj,n]=1mein2π(ξj12)=1me2πinξj(1)n.
X~X2C01sXsX1+C1ϵ
X~X22=n=n0n0|u~neikβnz0muneikβnz0m|2+nΛ|X~nXn|2,
X~X22m|eCe|2n=n0n0|u~nun|2+0=me2Ceu~u22,
u~u2eCem(C01sXsX1+C1ϵ)
uinc(x,h(x))=u(x,h(x))=nZuneik(αnx+βnh(x)),
eikh(x)sinθ=nZuneikβnh(x)einx.
nZuneinxe1kh(x)sinθe1kβ0h(x)=1+2ikh(x)β0+O(k2|h|2).
unυn{1n=02ikh^nβ0n0,
XXs1XZ1=n=N/2N/21m|eikβnz0||unυn|mϵu,
u=(KiηS)ψonΩΓ
u(x,z)=ππ(νΦ((x,z),(x,h(x)))iηΦ((x,z),(x,h(x))))ψ(x)·1+h.2(x)dx.
uinc(x,h(x))=12ψ(x)+ππ(νΦ((x,h(x)),(x,h(x)))iηΦ((x,h(x)),(x,h(x))))ψ(x)1+h.2(x)dx
Φ(r,r)=i4nZe2πinkcosθH0(1)(k|rr2πn(1,0)|),x,x[π,π)
νH0(1)(k|rr|)=kH1(1)(k|rr|)(rr)·ν(r)|rr|
H0(1)(k|r|)=1πeik(|z|β+xα)dαβ,
β={1α2,|α|<1iα21,|α|>1,
u(x,z)=nZeik(αnx+βnz)(14πππeik(αnx+βnh(x))gn(x)ψ(x)dx),z>hmax,
gn(x)=kkh.(x)αnβn+ηβn1+h.2(x).
un=14πππeik(αnx+βnh(x))gn(x)ψ(x)dx
uinc(x,h(m)(x))=12ψ(m)(x)+ππ(νΦ((x,h(m)(x)),(x,h(m)(x)))iηΦ((x,h(m)(x)),(x,h(m)(x))))ψ(m)(x)1+|h.(m)|2(x)dx
un=14πππeik(αnx+βnh(m)(x))gn(m+1)(x)ψ(m)(x)dx,nZ,
gn(m+1)(x)=kkh.(m+1)(x)αnβn+ηβn1+|h.(m)|2(x).
eikh(x)·nZuneikαnx=eik(xcosθh(x)sinθ).
h[0](x)=ln(nZuneikαnx)2ik,
h[n+1](x)=ln(nZuneik(αnx+(βn1)h[n](x)))2ik
e(h;x)=eikxjcosθeikhsinθ+nZuneikαnxjeikβnh,
h[i+1]=h[i]e(h[i];x)ddhe(h[i];x)
F(a)=e(nanϕn(·),·)2=j|e(nanϕn(xj),xj)|2,
minaF(a).
uS(r)=ΓΦ(r,r)ψS(r)dS(r),
uD(r)=ΓνΦ(r,r)ψD(r)dS(r)
limρ0uS(r+)=limρ0uS(r)=uS(r0).
uD(r+)=ψD(r0)ΓνΦ0(r+,r)dS(r)+v(r+),
uD(r)=ψD(r0)ΓνΦ0(r,r)dS(r)+v(r)
v(r±)=ΓνΦ(r±,r)(ψD(r)ψD(r0))dS(r)+ψD(r0)Γ(νΦ(r±,r)νΦ0(r±,r))dS(r),
limρ0(uD(r+)uD(r))=ψD(r0)
ΓνΦ0(r±,r)dS(r)=12Bρ(r0)νΦ0(r±,r)dS(r)
=14πρBρ(r0)±1dS(r)±12,ρ0

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