Abstract

The topology of a partially developed speckle field was studied by use of interference techniques through computer simulation. Amplitude and phase structures in the vicinity of caustics for a coherent radiation field scattered at a surface with large inhomogeneities were investigated. It was confirmed that the caustics are indispensible components of the procedure for the formation of networks of amplitude zeros for a coherent field scattered by a rough surface with large inhomogeneities. It is shown that the formation of interference forklets in the field gives evidence of changes in the field’s topology, as these forklets are a diagnostic sign of transition from a caustic to a three-dimensional pattern of a diffraction catastrophe.

© 2004 Optical Society of America

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References

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  1. J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).
  2. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  3. O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).
  4. J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  5. I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  6. I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. 50, 5164–5172 (1994).
    [CrossRef]
  7. N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).
  8. N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
    [CrossRef]
  9. H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
    [CrossRef]
  10. N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
    [CrossRef]
  11. M. Mujat, A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003).
    [CrossRef] [PubMed]
  12. G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  13. V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).
  14. J. F. Nye, “From Airy rings to the elliptic umbilic diffraction catastrophe,” J. Opt. A Pure Appl. Opt. 5, 503–510 (2003).
    [CrossRef]
  15. J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
    [CrossRef]
  16. J. F. Nye, “Evolution from a Fraunhofer to a Pearcey diffraction pattern,” J. Opt. A Pure Appl. Opt. 5, 495–502 (2003).
    [CrossRef]
  17. O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, S. G. Hanson, “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. 42, 4529–4540 (2003).
    [CrossRef] [PubMed]
  18. A. Apostol, A. Dogariu, “Coherence properties of optical near fields,” Opt. Photon. News 14(12), 22 (2003).
    [CrossRef]
  19. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
    [CrossRef]
  20. O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).
  21. O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
    [CrossRef]

2003 (7)

M. Mujat, A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003).
[CrossRef] [PubMed]

V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).

J. F. Nye, “From Airy rings to the elliptic umbilic diffraction catastrophe,” J. Opt. A Pure Appl. Opt. 5, 503–510 (2003).
[CrossRef]

J. F. Nye, “Evolution from a Fraunhofer to a Pearcey diffraction pattern,” J. Opt. A Pure Appl. Opt. 5, 495–502 (2003).
[CrossRef]

O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, S. G. Hanson, “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. 42, 4529–4540 (2003).
[CrossRef] [PubMed]

A. Apostol, A. Dogariu, “Coherence properties of optical near fields,” Opt. Photon. News 14(12), 22 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

2002 (1)

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

2001 (1)

1997 (1)

O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).

1996 (1)

N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

1994 (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. 50, 5164–5172 (1994).
[CrossRef]

1993 (1)

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

1992 (1)

1987 (1)

J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
[CrossRef]

1983 (1)

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

1981 (1)

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

1974 (1)

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Allen, J. L.

N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Angelsky, O.

O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).

Angelsky, O. V.

V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).

O. V. Angelsky, D. N. Burkovets, P. P. Maksimyak, S. G. Hanson, “Applicability of the singular-optics concept for diagnostics of random and fractal rough surfaces,” Appl. Opt. 42, 4529–4540 (2003).
[CrossRef] [PubMed]

O. V. Angelsky, P. P. Maksimyak, V. V. Ryukhtin, S. G. Hanson, “New feasibilities for characterizing rough surfaces by optical-correlation techniques,” Appl. Opt. 40, 5693–5707 (2001).
[CrossRef]

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

Apostol, A.

A. Apostol, A. Dogariu, “Coherence properties of optical near fields,” Opt. Photon. News 14(12), 22 (2003).
[CrossRef]

Baranova, N. B.

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

Berry, M.

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Besaha, R.

O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).

Blok, H.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

Burkovets, D. N.

Dogariu, A.

A. Apostol, A. Dogariu, “Coherence properties of optical near fields,” Opt. Photon. News 14(12), 22 (2003).
[CrossRef]

M. Mujat, A. Dogariu, “Polarimetric and spectral changes in random electromagnetic fields,” Opt. Lett. 28, 2153–2155 (2003).
[CrossRef] [PubMed]

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Freilikher, V.

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. 50, 5164–5172 (1994).
[CrossRef]

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Gbur, G.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

Hanson, S. G.

Haws, D. R.

J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
[CrossRef]

He, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Lenstra, D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

Maksimyak, P. P.

Mamayev, A. V.

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

McDuff, R.

Mokhun, I.

O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).

Mujat, M.

Nye, J.

J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

Nye, J. F.

J. F. Nye, “From Airy rings to the elliptic umbilic diffraction catastrophe,” J. Opt. A Pure Appl. Opt. 5, 503–510 (2003).
[CrossRef]

J. F. Nye, “Evolution from a Fraunhofer to a Pearcey diffraction pattern,” J. Opt. A Pure Appl. Opt. 5, 495–502 (2003).
[CrossRef]

J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
[CrossRef]

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Padgett, M. J.

N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Pilipetsky, N. F.

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

Polyanskii, P. V.

V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).

Polyanskii, V. K.

V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).

Popescu, G.

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Ryukhtin, V. V.

Schouten, H. F.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

Shkunov, V. V.

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

Shvartsman, N.

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. 50, 5164–5172 (1994).
[CrossRef]

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Simpson, N. B.

N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

Smith, C. P.

Smith, R. A.

J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
[CrossRef]

Visser, T. D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

White, A. G.

Zeldovich, B. Ya.

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

Appl. Opt. (2)

J. Mod. Opt. (3)

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

N. B. Simpson, J. L. Allen, M. J. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43, 2485–2491 (1996).
[CrossRef]

J. F. Nye, D. R. Haws, R. A. Smith, “Use of diffraction gratings with curved lines to study the optical catastrophes D6+ and D6-,” J. Mod. Opt. 34, 407–427 (1987).
[CrossRef]

J. Opt. A Pure Appl. Opt. (2)

J. F. Nye, “Evolution from a Fraunhofer to a Pearcey diffraction pattern,” J. Opt. A Pure Appl. Opt. 5, 495–502 (2003).
[CrossRef]

J. F. Nye, “From Airy rings to the elliptic umbilic diffraction catastrophe,” J. Opt. A Pure Appl. Opt. 5, 503–510 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

N. B. Baranova, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zeldovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A 73, 525–528 (1983).
[CrossRef]

JETP (1)

N. B. Baranova, B. Ya. Zeldovich, A. V. Mamayev, N. F. Pilipetsky, V. V. Shkunov, “Dislocation of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP 33, 1789–1797 (1981).

Opt. Appl. (2)

O. Angelsky, R. Besaha, I. Mokhun, “Appearance of wave front dislocations under interference among beams with simple wave fronts,” Opt. Appl. 27, 273–278 (1997).

V. K. Polyanskii, O. V. Angelsky, P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843–848 (2003).

Opt. Commun. (1)

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Opt. Lett. (2)

Opt. Photon. News (2)

A. Apostol, A. Dogariu, “Coherence properties of optical near fields,” Opt. Photon. News 14(12), 22 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Blok, “Phase singularities and enhanced transmission at a subwavelength slit,” Opt. Photon. News 14(12), 23 (2003).
[CrossRef]

Phys. Rev. (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. 50, 5164–5172 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

G. Popescu, A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, M. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (2)

J. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, Use of Optical Correlation Techniques for Characterizing Scattering Objects and Media, Vol. PM71 of SPIE Press Monograph Series (SPIE Press, Bellingham, Wash., 1999).

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Figures (19)

Fig. 1
Fig. 1

(a) Simulated random rough surface exponentially smoothed over three pixels. (b) The corresponding height distribution function; R q and R a are, respectively, the root-mean-square deviation and the arithmetic mean deviation of irregularities from a mean surface line, S k is the asymmetry coefficient, and K u is the kurtosis coefficient.

Fig. 2
Fig. 2

Notation for computation of the radiation field scattered an arbitrary distance z from a rough surface.

Fig. 3
Fig. 3

(a), (c), (e), (g) Intensity distributions and (b), (d), (f), (h) the corresponding interferograms of the scattered coherent radiation at various distances z from a rough surface: (a), (b) z 1 = 5 μm; (c), (d) z 2 = 8 μm; (e), (f) z 3 = 15 μm; (g), (h) z 4 = 500 μm. The most interesting areas are outlined by rectangles and shown as adjacent enlargements. The evolution of a diffraction pattern can be observed (a), (b) from a caustic neck, (c), (d) to a certain extent, and (e)–(h) to a three-dimensional pattern. An experimentally obtained interferogram from a rough surface is shown in fragment (i).

Fig. 4
Fig. 4

(a), (c) Intensity distributions and (b), (d) the corresponding interferograms of the scattered coherent radiation at distances (a), (b) z 1 = 560 μm and (c), (d) z 2 = 780 μm from a rough surface. Loci of amplitude zeros are depicted by small circles. One can see that the number of speckles is equal, on average, to the number of amplitude zeros, which are in dark areas of the field. So, for (a), (b) = 44, = 44; and for (c), (d) = 32, = 32, where and are average numbers of speckles and amplitude zeros, respectively.

Fig. 5
Fig. 5

(a) Fragment of intensity distribution and (b)–(e) corresponding interferograms of the field obtained at distance z = 35 μm from a rough surface. One can observe the breaking of (a) a two-charged amplitude zero into (b) two one-charged amplitude zeros. The areas of interest are outlined by rectangles. Amplitude zeros are visualized by interference forklets (b)–(e).

Fig. 6
Fig. 6

Interference patterns obtained by coaxial superposition of the field with a coherent reference wave at distances (a) z 1 = 6 μm, (b) z 2 = 9 μm, and (c) z 3 = 11 μm from a rough surface. Phase saddles are outlined by rectangles. Enlarged images of the phase saddles are shown at the right in (a)–(c). Fragment (d) shows schematically a phase surface in the vicinity of saddle point A.

Fig. 7
Fig. 7

Interferograms of the field at the caustic zone obtained via (a), (c) on-axis and (b), (d) off-axis interference. Phase saddles and amplitude zeros are outlined by rectangles and by circles, respectively. One can see that the number of phase saddles is equal to the number of amplitude zeros.

Fig. 8
Fig. 8

Evolution of a pair of anisotropic dislocations. Interferograms are obtained at distances (a) z 1 = 700 μm, (b) z 2 = 710 μm, (c) z 3 = 720 μm, (d) z 4 = 750 μm, (e) z 5 = 770 μm, and (f) z 6 = 800 μm from a rough surface. Amplitude zeros are marked by circles.

Fig. 9
Fig. 9

Intensity distributions of the field at the caustic zone at distances (a) z 1 = 6 μm, (b) z 2 = 7 μm, (c) z 3 = 8 μm, (d) z 4 = 9 μm, (e) z 5 = 10 μm, (f) z 6 = 11 μm, and (g) z 7 = 12 μm from a rough surface. Amplitude zeros are outlined by rectangles. One can observe clustering of amplitude zeros at caustic zones. The structure of clusters corresponds to the structure of caustics.

Fig. 10
Fig. 10

Schematic of discrete analysis of the field that determines the behavior of the lines of zero amplitude. A-D, discrete planes of analysis of the field propagating along the z axis. Filled circles, amplitude zeros. 1, 1′ and 2, 2′ are pairs of amplitude zeros, which are crossings of the lines of zero amplitude with planes A-D.

Fig. 11
Fig. 11

Fragments of interferograms of the field at distances (a) z 1 = 7 μm, (b) z 2 = 8 μm, (c) z 3 = 9 μm, (d) z 4 = 10 μm, (e) z 5 = 11 μm, (f) z 6 = 12 μm, (g) z 7 = 13 μm, and (h) z 8 = 14 μm from a rough surface. Amplitude zeros are marked by circles. One can observe (c), (d) birth and (g) annihilation of dislocations (marked by arrows).

Fig. 12
Fig. 12

Fragments of interferograms of the field at distances (a) z 1 = 600 μm, (b) z 2 = 650 μm, (c) z 3 = 700 μm, (d) z 4 = 760 μm, (e) z 5 = 770 μm, (f) z 6 = 800 μm, and (g) z 7 = 810 μm from a rough surface. Amplitude zeros obtained by software are marked by circles. One can observe the birth of amplitude zeros [fragment (a)] and their motion along a complex trajectory, including rotation, as the plane of observation is moved from the point of birth of singularities. The pair of the actual (nucleating and evolving) amplitude zeros are marked by arrows.

Fig. 13
Fig. 13

(a), (b) Fragments of intensity distribution and (c), (d) the corresponding interferograms at the caustic zone at distance z = 5 μm from an object. One can see diffraction maxima as the lines accompanying the caustic lines, as well as relative half-period shift of interference fringes along these lines.

Fig. 14
Fig. 14

(a) Fragment of intensity distribution with the computed amplitude zeros (marked by circles) at distance z = 6 μm from an object and (b) the corresponding interferogram.

Fig. 15
Fig. 15

(a), (b) Fragments of intensity distribution with the computed amplitude zeros (marked by circles) at distance z = 6 μm from an object and (b), (c) the corresponding interferograms.

Fig. 16
Fig. 16

(a) Amplitude and (b) phase distributions of the field scattered by a rough surface with large inhomogeneities at distance z = 5 μm from the object. Fragment (c) shows the amplitude and phase distribution of the field that result from superimposition of the distributions shown in fragments (a) and (b).

Fig. 17
Fig. 17

(a) Joint distribution of the field’s amplitude and phase with imposed computed amplitude zeros (marked by circles) and (b) the associated interferogram of the field at distance z = 6 μm from the object. The areas of interest are outlined by rectangles.

Fig. 18
Fig. 18

(a) Joint distribution of the field’s amplitude and phase with imposed computed amplitude zeros (marked by circles) and (d) the associated interferogram of the field at distance z = 10 μm from the object. The areas of interest are outlined by rectangles b and c.

Fig. 19
Fig. 19

Explanation of the effect of motion of the amplitude zeros that constitute a dipole. Plane FF′ is the focal plane of partial focusing. Planes 1–4 are the secant planes (the planes of observation). Circles mark the points of crossing of the lines of zero amplitude with the secant planes, which are the amplitude zeros (1′, 1″, 2′, and 2″) at the planes of observation. A coherent wave propagates along the z axis. (a), (b) Two-dimensional and three-dimensional patterns, respectively. (b) Illustrates the rotating and through-and-back motion of the dipole.

Equations (2)

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ReAx, y=0, ImAx, y=0,
Uξ, η= ziλ  Ax, yR2x, y, z, ξ, η ×exp-ikRx, y, z, ξ, ηn-1hx, ydxdy,

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