OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 29, Iss. 4 — Apr. 1, 2012
  • pp: 617–626

Compressive imaging of subwavelength structures: periodic rough surfaces

Albert Fannjiang and Hsiao-Chieh Tseng  »View Author Affiliations

JOSA A, Vol. 29, Issue 4, pp. 617-626 (2012)

View Full Text Article

Enhanced HTML    Acrobat PDF (741 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

ToC Category:
Diffraction and Gratings

Original Manuscript: September 1, 2011
Manuscript Accepted: November 12, 2011
Published: March 27, 2012

Albert Fannjiang and Hsiao-Chieh Tseng, "Compressive imaging of subwavelength structures: periodic rough surfaces," J. Opt. Soc. Am. A 29, 617-626 (2012)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited