Abstract

We show theoretically and experimentally that the spectrum of coherently scattered light from a randomly rough interface in reflection and transmission is redshifted with a shrinkage in spectral width. In reflection mode the amounts of the redshift and the shrinkage depend on interface roughness, incident angle, and the spectral width of the illuminating light. In transmission mode they also depend on the refractive indices of the surrounding media. The redshift and width shrinkage increase with decrease of the coherently scattered light intensity. This study shows that the spectrum of the diffusely scattered light is blueshifted in the specular direction and in directions with small scattering angles only in situations with appreciable intensity of the coherently scattered light. With decrease of the latter intensity the blueshift reduces and turns into redshift. Also, the redshift and blueshift decay with increase of the scattering angle. An experimental investigation has been carried out, on sheet glasses with different roughness on one side, in reflection and transmission modes. The experimental results and theoretical predictions are quite consistent.

© 2009 Optical Society of America

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  1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
    [CrossRef] [PubMed]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Sec. 5.8.
  3. E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220-2223 (1989).
    [CrossRef] [PubMed]
  4. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
    [CrossRef]
  5. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef] [PubMed]
  6. G. Popescu and A. Dogariu, “Spectral anomalies at wave-front dislocations,” Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef] [PubMed]
  7. M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
    [CrossRef]
  8. M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored castastrophes,” Opt. Lett. 33, 1863-1865 (2008).
    [CrossRef] [PubMed]
  9. M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
    [CrossRef]
  10. O. V. Angelsky, P. V. Polyanskii, and S. G. Hanson, “Singular-optical coloring of regularly scattered white light,” Opt. Express 14, 7579-7586 (2006).
    [CrossRef] [PubMed]
  11. V. K. Polyanskii, O. V. Angelsky, and P. V. Polyanskii, “Scattering-induced spectral changes as a singular optical effect,” Opt. Appl. 32, 843-848 (2002).
  12. O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
    [CrossRef]
  13. M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397-2405 (2008).
    [CrossRef]
  14. M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
    [CrossRef]

2008 (4)

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397-2405 (2008).
[CrossRef]

M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored castastrophes,” Opt. Lett. 33, 1863-1865 (2008).
[CrossRef] [PubMed]

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
[CrossRef]

2006 (1)

2005 (1)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

2004 (1)

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

2002 (3)

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

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior 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]

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

1989 (1)

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220-2223 (1989).
[CrossRef] [PubMed]

1986 (1)

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

Amiri, M.

M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored castastrophes,” Opt. Lett. 33, 1863-1865 (2008).
[CrossRef] [PubMed]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

O. V. Angelsky, P. V. Polyanskii, and S. G. Hanson, “Singular-optical coloring of regularly scattered white light,” Opt. Express 14, 7579-7586 (2006).
[CrossRef] [PubMed]

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

Dashtdar, M.

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397-2405 (2008).
[CrossRef]

Dogariu, A.

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

Ebadi, Z.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Gbur, G.

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

Hanson, S. G.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

O. V. Angelsky, P. V. Polyanskii, and S. G. Hanson, “Singular-optical coloring of regularly scattered white light,” Opt. Express 14, 7579-7586 (2006).
[CrossRef] [PubMed]

James, D. F. V.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Karimi, E.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

Khalesifard, H. R.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

Maksimyak, A. P.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

Maksimyak, P. P.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Sec. 5.8.

Nahal, A.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Negrych, A. L.

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

Polyanskii, P. V.

O. V. Angelsky, P. V. Polyanskii, and S. G. Hanson, “Singular-optical coloring of regularly scattered white light,” Opt. Express 14, 7579-7586 (2006).
[CrossRef] [PubMed]

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

Popescu, G.

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

Tavassoly, M. T.

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397-2405 (2008).
[CrossRef]

M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored castastrophes,” Opt. Lett. 33, 1863-1865 (2008).
[CrossRef] [PubMed]

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
[CrossRef]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Visser, T. D.

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

Wolf, E.

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

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220-2223 (1989).
[CrossRef] [PubMed]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Sec. 5.8.

J. Europ. Opt. Soc. Rap. Public. (1)

O. V. Angelsky, S. G. Hanson, P. P. Maksimyak, A. P. Maksimyak, and A. L. Negrych “Experimental demonstration of singular-optical colouring of regularly scattered white light,” J. Europ. Opt. Soc. Rap. Public. 3, 08029 (2008).
[CrossRef]

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

Opt. Appl. (1)

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

Opt. Commun. (3)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 281, 2397-2405 (2008).
[CrossRef]

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (4)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior 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]

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

E. Wolf, “Correlation-induced Doppler-like frequency shifts of spectral lines,” Phys. Rev. Lett. 63, 2220-2223 (1989).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771-818 (1996).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Sec. 5.8.

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Figures (9)

Fig. 1
Fig. 1

(a) NSI of a light source with central wavenumber k 0 = 1.14 × 10 5 cm 1 and spectral width σ k = 2 × 10 4 cm 1 . The calculated modified NSIs of the CSLs from rough interfaces associated with (b) σ f = 60 nm , (c) σ f = 150 nm , and (d) σ f = 250 nm .

Fig. 2
Fig. 2

Sketch of the setup used to measure the scattered light intensity in a desired direction in reflection mode. The characters S, L m , Sl, MC, PMT, and PC denote the light source, lenses, slit, monochromator, photomultiplier tube, and personal computer, respectively.

Fig. 3
Fig. 3

Plots (a) and (b) are the NSIs of the CSLs from glass surfaces of roughness σ = 0.42 μ m and σ = 0.08 μ m , respectively, at incident angle θ i = 85 ° . Plot (r) is the NSI reflected from the reference interface. Plots (c) and (d) are the NSIs of the DSLs from the same rough interfaces, respectively, at scattering angle θ s = 1 ° .

Fig. 4
Fig. 4

Plots (a) and (b) are the NSIs for the CSLs from a glass surface of roughness σ = 0.42 μ m at incident angles θ i = 83 ° and θ i = 85 ° , respectively. Plot (r) is the NSI of the light reflected from the reference interface. Plots (c) and (d) are the NSIs of the DSLs from the same rough interface and at the same incident angles, respectively, at scattering angle θ s = 1 ° .

Fig. 5
Fig. 5

Plot (a) is the NSI of the transmitted CSL from a glass interface of roughness σ = 0.92 μ m , immersed in chloroform, at normal incidence. Plot (r) is the NSI of the light transmitted from the reference interface. Plot (b) is the NSI of the transmitted DSL under the same conditions, but at scattering angle θ s = 1 ° .

Fig. 6
Fig. 6

NSIs of the DSLs from an interface of roughness σ = 0.08 μ m at different scattering angles and the light reflected from the reference interface. The incident angle for all cases was θ i = 85 ° .

Fig. 7
Fig. 7

NSIs of the DSLs transmitted from an interface of roughness σ = 0.42 μ m at different scattering angles and the NSI of the light transmitted from the reference interface. The incident angle for all cases is θ i = 50 ° .

Fig. 8
Fig. 8

Circles represent the experimental modifying factors for a surface of roughness σ = 0.42 μ m at incident angle 79°. The solid curve is a fitted Gaussian function.

Fig. 9
Fig. 9

Plot (a) is the NSI of the light source after the light has been filtered by an interference filter. Plot (b) is the NSI of the filtered light after it is coherently scattered from a glass interface of roughness σ = 0.42 μ m at incident angle θ i = 77 ° . The solid curve is obtained by multiplying the profile (a) by the theoretically calculated modifying function, Eq. (7) in the text.

Equations (11)

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E ( k ) = E 0 ( k ) R + P n ( h ) exp ( i k h f ) d h ,
f = 2 cos θ i
f = N 1 ( N 2 sin 2 θ i cos θ i )
S ( k , P n , f ) = S 0 ( k ) R 2 | + P n ( h ) exp ( i k h f ) d h | 2 ,
M ( k , P n , f ) = | + P n ( h ) exp ( i k h f ) d h | 2 .
P n ( h ) = 1 σ 2 π exp ( h 2 2 σ 2 ) .
M ( k , P n , f ) = exp ( k 2 σ 2 f 2 ) .
S 0 ( k ) = s 0 exp ( ( k k 0 ) 2 2 σ k 2 ) ,
S ( k , p n , f ) = S 0 exp ( k 0 2 σ 2 f 2 ) exp [ ( k k 0 1 + 2 σ k 2 σ 2 f 2 ) 2 2 σ k 2 1 + 2 σ k 2 σ 2 f 2 ] ,
δ k 0 = k 0 1 + 1 2 σ k 2 σ 2 f 2 .
σ M = σ k 1 + 2 σ k 2 σ 2 f 2 .

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