Abstract

When the surface roughness is comparable with the wavelength of the probing radiation, the scattered field contains both the regular (forward-scattered) component of coherent nature and the diffusely scattered part. Coloring of the regular component of white light scattered by a colorless dielectric slab with a rough surface is considered as a peculiar effect of singular optics with zero (infinitely extended) interference fringes. To explain the observed alternation of colors with respect to the increasing depth of the surface roughness, we apply a model of transition layers associated with the surface roughness. By applying the chromascopic technique, it is shown that the modifications of the normalized spectrum of the forward-scattered white light can be interpreted as the effect of a quarter-wavelength (anti-reflecting) layer for some spectral component of a polychromatic probing beam.

© 2006 Optical Society of America

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References

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  1. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef] [PubMed]
  2. G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  3. V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata 32, 843–848 (2002).
  4. S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
    [CrossRef]
  5. M. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
    [CrossRef]
  6. M. Berry, “Exploring the colours of dark light,” New J. Phys. 4, 74.1–74.14 (2002).
    [CrossRef]
  7. J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003).
    [CrossRef]
  8. I. Freund, “Polychromatic polarization singularities,” Opt. Lett. 28, 2150–2152 (2003).
    [CrossRef] [PubMed]
  9. A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
    [CrossRef]
  10. M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
    [CrossRef]
  11. O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “On the feasibility for determining the amplitude zeroes in polychromatic fields,” Opt. Express 13, 4396–4405 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. V. Shvedov, W. Krolokowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Yu. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express 13, 7393–7398 (2005).
    [CrossRef] [PubMed]
  14. P. V. Polyanskii, “Some current views on singular optics,” in Sixth International Conference on Correlation Optics,O. V. Angelsky, ed., Proc. SPIE 5477, 31–40 (2004).
    [CrossRef]
  15. S. A. Ponomarenko, “A class of partially coherent vortex beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
    [CrossRef]
  16. G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
    [CrossRef] [PubMed]
  17. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [CrossRef] [PubMed]
  18. Ch. V. Felde, “Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams,” Ukr. J. Phys. 49, 473–480 (2004).
  19. G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
    [CrossRef]
  20. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.
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  24. F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).
  25. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  26. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  27. A. Sommerfeld, Optics (Academic, New York, 1954).
  28. F. S. Grawford, Waves: Berkley Physics Course (McGraw-Hill, New York, 1968) Vol 3.
  29. R. M. Evans, An Introduction to Color (Wiley, New York, 1959).
  30. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
  31. S. R. Wilk, “Once in a blue moon,” Opt. Photonics. News 17, 20–21 (2006).

2006 (1)

S. R. Wilk, “Once in a blue moon,” Opt. Photonics. News 17, 20–21 (2006).

2005 (3)

2004 (4)

M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
[CrossRef]

P. V. Polyanskii, “Some current views on singular optics,” in Sixth International Conference on Correlation Optics,O. V. Angelsky, ed., Proc. SPIE 5477, 31–40 (2004).
[CrossRef]

Ch. V. Felde, “Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams,” Ukr. J. Phys. 49, 473–480 (2004).

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
[CrossRef]

2003 (5)

2002 (6)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

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

V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata 32, 843–848 (2002).

M. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
[CrossRef]

M. Berry, “Exploring the colours of dark light,” New J. Phys. 4, 74.1–74.14 (2002).
[CrossRef]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
[CrossRef]

2001 (2)

M. S. Soskin and M. V. Vasnetsov, “Singular Optics” in Progress in OpticsE. Wolf, ed., (North-Holland, Amsterdam, 2001) Vol.  42, 219–276.

S. A. Ponomarenko, “A class of partially coherent vortex beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
[CrossRef]

Angelsky, O. V.

Arkhelyuk, O. O.

M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
[CrossRef]

Bass, F. G.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

Berry, M.

M. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
[CrossRef]

M. Berry, “Exploring the colours of dark light,” New J. Phys. 4, 74.1–74.14 (2002).
[CrossRef]

Bogatyryova, G. V.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999.)

Burckhardt, Ch. B.

R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971).

Collier, R. J.

R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971).

Desyatnikov, A. S.

V. Shvedov, W. Krolokowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Yu. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express 13, 7393–7398 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.

Dogariu, A.

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

Evans, R. M.

R. M. Evans, An Introduction to Color (Wiley, New York, 1959).

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Felde, C. V.

Felde, Ch. V.

Ch. V. Felde, “Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams,” Ukr. J. Phys. 49, 473–480 (2004).

Freund, I.

Fuks, I. M.

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Grawford, F. S.

F. S. Grawford, Waves: Berkley Physics Course (McGraw-Hill, New York, 1968) Vol 3.

Hanson, S. G.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kivshar, Yu. S.

V. Shvedov, W. Krolokowski, A. Volyar, D. N. Neshev, A. S. Desyatnikov, and Yu. S. Kivshar, “Focusing and correlation properties of white-light optical vortices,” Opt. Express 13, 7393–7398 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.

Krolokowski, W.

Leach, J.

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003).
[CrossRef]

Lin, L. H.

R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971).

Maksimyak, A. P.

Maksimyak, P. P.

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

Neshev, D. N.

Padgett, M. J.

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003).
[CrossRef]

Polyanskii, P. V.

P. V. Polyanskii, “Some current views on singular optics,” in Sixth International Conference on Correlation Optics,O. V. Angelsky, ed., Proc. SPIE 5477, 31–40 (2004).
[CrossRef]

M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
[CrossRef]

G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef] [PubMed]

V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata 32, 843–848 (2002).

Polyanskii, V. K.

V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata 32, 843–848 (2002).

Ponomarenko, S. A.

Popescu, G.

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

Schouten, H. F.

Shvedov, V.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954).

Soskin, M. S.

M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
[CrossRef]

G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef] [PubMed]

M. S. Soskin and M. V. Vasnetsov, “Singular Optics” in Progress in OpticsE. Wolf, ed., (North-Holland, Amsterdam, 2001) Vol.  42, 219–276.

Torner, L.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics” in Progress in OpticsE. Wolf, ed., (North-Holland, Amsterdam, 2001) Vol.  42, 219–276.

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Volyar, A.

Volyar, A. V.

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Wilk, S. R.

S. R. Wilk, “Once in a blue moon,” Opt. Photonics. News 17, 20–21 (2006).

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
[CrossRef]

G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef] [PubMed]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211–1213 (2002).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999.)

J. Opt. A (1)

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wave-fields,” J. Opt. A 6, 239–242 (2004).
[CrossRef]

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

New J. Phys. (4)

M. Berry, “Coloured phase singularities,” New J. Phys. 4, 66.1–66.14 (2002).
[CrossRef]

M. Berry, “Exploring the colours of dark light,” New J. Phys. 4, 74.1–74.14 (2002).
[CrossRef]

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154.1–154.7 (2003).
[CrossRef]

M. S. Soskin, P. V. Polyanskii, and O. O. Arkhelyuk, “Computer-synthesized hologram-based rainbow optical vortices,” New J. Phys. 6, 196.1–196-8 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Opt. Photonics. News (1)

S. R. Wilk, “Once in a blue moon,” Opt. Photonics. News 17, 20–21 (2006).

Opt. Spectrosc. (1)

A. V. Volyar and T. A. Fadeeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 235–244 (2003).
[CrossRef]

Optica Applicata (1)

V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular-optical effect,” Optica Applicata 32, 843–848 (2002).

Phys. Rev. Lett. (2)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behaviour of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

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

Proc. SPIE (1)

P. V. Polyanskii, “Some current views on singular optics,” in Sixth International Conference on Correlation Optics,O. V. Angelsky, ed., Proc. SPIE 5477, 31–40 (2004).
[CrossRef]

Ukr. J. Phys. (1)

Ch. V. Felde, “Young’s diagnostics of phase singularities of the spatial coherence function at partially coherent singular beams,” Ukr. J. Phys. 49, 473–480 (2004).

Other (11)

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Progress in optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999.)

M. S. Soskin and M. V. Vasnetsov, “Singular Optics” in Progress in OpticsE. Wolf, ed., (North-Holland, Amsterdam, 2001) Vol.  42, 219–276.

R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971).

F. G. Bass and I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

A. Sommerfeld, Optics (Academic, New York, 1954).

F. S. Grawford, Waves: Berkley Physics Course (McGraw-Hill, New York, 1968) Vol 3.

R. M. Evans, An Introduction to Color (Wiley, New York, 1959).

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

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Figures (2)

Fig. 1.
Fig. 1.

Chromascopic simulation of the spectral changes of the forward-scattered component of a white light induced by a colorless glass rough surface (n i=1.52, n 2=1.233, and n 3=1) following the model of the transition layer.

Fig. 2.
Fig. 2.

Photos of a natural Moon (a), a blue Moon (b), and a red Moon (c).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

T g = exp [ iq cos ( 2 π p x ) ] = l = i l J l ( q ) exp ( i l 2 π p x ) ,
T r = exp [ i m q m cos ( 2 π p m x ) ] m [ l = i l J l ( q m ) exp ( i l 2 π p m x ) ] ,
η 0 = m J 0 2 ( q m ) .
I r I i = 4 [ 1 n 1 1 + n 1 ] 2 sin 2 [ π 2 ( λ i λ 0 1 ) ] ,
I f I i = 1 I r I i .
( R G B ) ( R G B ) max R G B .

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