Abstract

An expression is derived for the effects of a uniformly moving diffuser plate on the far-zone spectrum of a polychromatic plane wave that is incident normally upon the plate. The expression clearly shows effects due to both the motion of the diffuser plate and the statistical properties of the plate. With a strong diffuser plate the spectrum of the scattered light is found to be essentially a Doppler-shifted version of the spectrum of the incident field, independent of the form of the correlation function of the plate. In the weak-diffuser limit it is found to depend on the fourth power of the frequency, on the correlation function of the diffuser plate, and on the speed with which the plate is moving.

© 1991 Optical Society of America

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References

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  1. For a review of some of this work see, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232;P. W. Milonni, S. Singh, “Source correlations and optical spectra,” in “Some recent developments in the fundamental theory of light,” contribution in Advances in Atomic and Molecular Physics, D. Bates, R. Bedenson, eds. (Academic, San Diego, Calif., 1991), Vol. 28, Chap. VIII, pp. 127–137.
    [CrossRef]
  2. E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989);J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
    [CrossRef] [PubMed]
  3. D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990);D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
    [CrossRef]
  4. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
    [CrossRef]
  5. A. Lagendijk, “Terrestrial redshift from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
    [CrossRef]
  6. W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [CrossRef]
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
    [CrossRef]
  8. J. H. Churnside, “Speckle from a rotating diffuse object,” J. Opt. Soc. Am. 72, 1464–1469 (1982);J. C. Leader, “An analysis of the frequency spectrum of laser light scattered from moving rough objects,” J. Opt. Soc. Am. 67, 1091–1098 (1977).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  10. The general Galilean transformation would also lead to a frequency change in the argument of the integrand in Eq. (2.2). There is no such change in the present case because the wave vector of the incident field is perpendicular to the direction of motion of the plate.
  11. Except for a trivial change in the notation this formula readily follows from Eq. (5.4) of E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978) and the relation Jω′(u′)/r2= S(∞)(ω′; ru′) between the radiant intensity, Jω′(u′), and the far-zone spectrum, S(∞)(ω ′; ru′).
    [CrossRef]
  12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  13. R. L. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. 64, 592–598 (1974).
    [CrossRef]
  14. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  15. H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
    [CrossRef]

1990

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990);D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[CrossRef]

A. Lagendijk, “Terrestrial redshift from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[CrossRef]

1989

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

1982

1978

1976

H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
[CrossRef]

1974

1964

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Asakura, T.

H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Churnside, J. H.

Fante, R. L.

Foley, J. T.

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989);J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[CrossRef] [PubMed]

Fujii, H.

H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[CrossRef]

Gori, F.

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989);J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[CrossRef] [PubMed]

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

James, D. F. V.

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990);D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Lagendijk, A.

A. Lagendijk, “Terrestrial redshift from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[CrossRef]

Martienssen, W.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Spiller, E.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

Uozumi, J.

H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
[CrossRef]

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Wolf, E.

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990);D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[CrossRef]

Except for a trivial change in the notation this formula readily follows from Eq. (5.4) of E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978) and the relation Jω′(u′)/r2= S(∞)(ω′; ru′) between the radiant intensity, Jω′(u′), and the far-zone spectrum, S(∞)(ω ′; ru′).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989);J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[CrossRef] [PubMed]

For a review of some of this work see, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232;P. W. Milonni, S. Singh, “Source correlations and optical spectra,” in “Some recent developments in the fundamental theory of light,” contribution in Advances in Atomic and Molecular Physics, D. Bates, R. Bedenson, eds. (Academic, San Diego, Calif., 1991), Vol. 28, Chap. VIII, pp. 127–137.
[CrossRef]

Am. J. Phys.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[CrossRef]

J. Opt. Am.

H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to surface profile,” J. Opt. Am. 66, 1222–1236 (1976).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Opt. Commun.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73, 169–172 (1989).
[CrossRef]

Phys. Lett. A

A. Lagendijk, “Terrestrial redshift from a diffuse light source,” Phys. Lett. A 147, 389–392 (1990).
[CrossRef]

D. F. V. James, E. Wolf, “Doppler-like frequency shifts generated by dynamic scattering,” Phys. Lett. A 146, 167–171 (1990);D. F. V. James, M. P. Savedoff, E. Wolf, “Shifts of spectral lines caused by scattering from fluctuating random media,” Astrophys. J. 359, 67–71 (1990).
[CrossRef]

Other

For a review of some of this work see, for example, E. Wolf, “Influence of source-correlations on spectra of radiated fields,” in International Trends in Optics, J. W. Goodman, ed. (Academic, New York, 1991), pp. 221–232;P. W. Milonni, S. Singh, “Source correlations and optical spectra,” in “Some recent developments in the fundamental theory of light,” contribution in Advances in Atomic and Molecular Physics, D. Bates, R. Bedenson, eds. (Academic, San Diego, Calif., 1991), Vol. 28, Chap. VIII, pp. 127–137.
[CrossRef]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6, 1142–1149 (1989);J. T. Foley, E. Wolf, “Frequency shifts of spectral lines generated by scattering from space–time fluctuations,” Phys. Rev. A 40, 588–598 (1989).
[CrossRef] [PubMed]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

The general Galilean transformation would also lead to a frequency change in the argument of the integrand in Eq. (2.2). There is no such change in the present case because the wave vector of the incident field is perpendicular to the direction of motion of the plate.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (6)

Fig. 1
Fig. 1

Moving diffuser plate in front of an aperture illuminated by a normally incident polychromatic plane wave.

Fig. 2
Fig. 2

Illustrating the notation relating to the determination of the far-zone spectrum.

Fig. 3
Fig. 3

Scattering of light by a rotating diffuser plate.

Fig. 4
Fig. 4

The normalized spectrum, S(∞)(ω′; ru′)/N′, where N′ = Alc2σh2k04 cos2 θ/(2π)3r2, with Q = 108, for (a) β = 0, ξ = 1000; (b) β = 2 × 10−7, ξ = 1000; (c) β = 0, ξ = 1010; (d) β = 0, ξ = 1020.

Fig. 5
Fig. 5

Normalized spectrum S(∞)(ω′ ru′)/N″, where N″ = Maxω′. [S(∞)(ω′; ru′)], with Q = 100 and β = 0, for (a) ξ = 10; (b) ξ = 100, (c) ξ = 1000.

Fig. 6
Fig. 6

Schematic illustration of the contributions of the different factors in expression (4.15), with the Fourier transform of the correlation function of heights given by Eq. (4.20): (a) sinc2[ξ1(sin θ)ω′/ω0], (b) sinc2[ξ2(sinθ)ω′/ω0], (c) ω′4, (d) high-Q spectral line (e) low-Q spectral line.

Equations (60)

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V ( ) ( ρ , t ) = 1 2 π 0 a ( ω ) e i ω t d ω .
V ( + ) ( ρ , t ) = 1 2 π 0 a ( ω ) T ( ρ v t ; ω ) e i ω t d ω .
Γ ( ρ 1 , t 1 ; ρ 2 , t 2 ) = V ( + ) * ( ρ 1 , t 1 ) V ( + ) ( ρ 2 , t 2 ) V D
Γ ( ρ 1 , t 1 ; ρ 2 , t 2 ) = 1 2 π 2 0 0 a * ( ω ) a ( ω ) V T * ( ρ 1 υ t 1 ; ω ) × T ( ρ 2 υ t 2 ; ω ) D exp [ i ( ω t 1 ω t 2 ) d ω d ω ,
a * ( ω ) a ( ω ) υ = S ( i ) ( ω ) δ ( ω ω ) ,
T * ( ρ 1 ; ω ) T ( ρ 2 ; ω ) D = C D ( ρ 2 ρ 1 ) .
Γ ( ρ , τ ) = 1 ( 2 π ) 2 0 S ( i ) ( ω ) C D ( ω ; ρ υ τ ) e i ω τ d ω ,
ρ = ρ 2 ρ 1 , τ = t 2 t 1 .
W ( ρ , ω ) = Γ ( ρ , τ ) exp ( i ω τ ) d τ .
W ( ρ , ω ) = 1 ( 2 π ) 2 d τ exp ( i ω τ ) 0 S ( i ) ( ω ) C D ( ω ; ρ v τ ) × exp ( i ω τ ) d ω .
S ( ) ( ω ; r u ) = ( ω cos θ 2 π c r ) 2 B ( ρ 1 ) B ( ρ 2 ) W ( ρ 1 ρ 2 , ω ) × exp [ i k ( ω ) u ( ρ 2 ρ 1 ) ] d 2 ρ 1 d 2 ρ 2 ,
k ( ω ) = ω / c
B ( ρ ) = { 1 when ρ A 0 when ρ A .
ρ 1 = ρ ρ / 2 , ρ 2 = ρ + ρ / 2 ,
S ( ) ( ω ; r u ) = A ( ω cos θ 2 π c r ) 2 C A ( ρ ) W ( ρ , ω ) × exp [ i k ( ω ) u ρ ] d 2 ρ ,
C A ( ρ ) = B ( ρ + ρ / 2 ) B ( ρ ρ / 2 ) d 2 ρ B ( ρ ) B ( ρ ) d 2 ρ
A = B ( ρ ) B ( ρ ) d 2 ρ
S ( ) ( ω ; r u ) = A ( 2 π ) 2 ( ω cos θ 2 π c r ) 2 d τ 0 d ω × d 2 ρ C A ( ρ ) S ( i ) ( ω ) C D ( ω ; ρ v τ ) × exp [ i ( ω ω ) τ ] exp [ i ω u ρ / c ] .
C A ( ρ ) C D ( ω ; ρ v τ ) exp [ i ω u ρ / c ] d 2 ρ C A ( v τ ) C D ( ω ; ρ v τ ) exp [ i ω u ρ / c ] d 2 ρ = ( 2 π ) 2 C A ( v τ ) C D [ ω ; ( ω u ) / c ] exp [ i ( ω u v τ ) / c ] ,
C D ( ω ; f ) = 1 ( 2 π ) 2 C D ( ω ; ρ ) exp ( if ρ ) d 2 ρ
S ( ) ( ω ; r u ) = A ( 2 π ) 2 ( ω cos θ 2 π c r ) 2 d τ 0 d ω C A ( v τ ) S ( i ) ( ω ) × C D [ ω ; ( ω u / c ) ] exp [ i ( ω ω ) τ exp [ i ω u ρ / c ] ,
S ( ) ( ω ; r u ) = A ( ω cos θ 2 π c r ) 2 0 d ω C A [ ω ( 1 u β ) ω ; v ] × S ( i ) ( ω ) C D [ ω ; ( ω u / c ) ] .
C A ( Ω ; v ) = C A ( v τ ) e i Ω r d τ ,
β = v / c .
T tr = L / | v | .
T tr τ coh ,
T ( ρ ; ω ) T ( ρ ; ω ± 1 / T tr ) .
0 d ω C A [ ω ( 1 u β ) ω ; v ] S ( i ) ( ω ) C D [ ω ; ( ω u ) / c ] S ( i ) ( ω ) C D ( ω ; K ) 0 d ω C A [ ω ( 1 u β ) ω ; v ] = 2 π S ( i ) ( ω ) C D [ ω ; K ] ,
K = ω c u ,
ω = ω ( 1 β u ) .
S ( ) ( ω ; r u ) = A 2 π ( cos θ c r ) 2 ω 2 C D ( ω ; K ) S ( i ) ( ω ) .
T ( ρ ; ω ) = exp { i ω [ n ( ω ) 1 ] h ( ρ ) c } ,
σ h 2 = [ h ( ρ ) h ( ρ ) ] 2
g ( ρ 1 ρ 2 ) = h ( ρ 1 ) h ( ρ 2 ) h ( ρ 1 ) h ( ρ 2 ) σ h 2
g ( k ) = g ( ρ ) exp ( i k ρ ) d 2 ρ .
C D ( ω ; ρ ) = exp { η ( ω ) [ 1 g ( ρ ) ] } ,
η ( ω ) = ω 2 [ n ( ω ) 1 ] 2 σ h 2 c 2 .
η ( ω ) 1 .
C D ( ω ; ρ ) exp [ η ( ω ) ρ 2 2 l D 2 ] .
1 l D 2 = 1 4 π 0 g ( K ) K 3 d K .
S ( ) ( ω ; r u ) = N [ n ( ω ) 1 ] 2 S ( i ) ( ω ) × exp { ( l D sin θ ) 2 2 [ σ h ( n ( ω ) 1 ) ( 1 β u ) ] 2 } ,
N = A l D 2 cos 2 θ [ 2 π r σ h ( 1 β u ) ] 2 ,
η ( ω ) 1 .
C D ( ω ; ) = [ 1 η ( ω ) ] ,
δ C D ( ω ; ρ ) = η ( ω ) g ( ρ ) .
S D ( ω ; r u ) = M [ n ( ω ) 1 ] 2 ω 4 g ( ω u / c ) S ( i ) ( ω ) ,
M = A σ h 2 cos 2 θ ( 1 β u ) 2 2 π ( r c 2 ) 2 .
g ( ρ ) = Δ ( ρ x / l c ) Δ ( ρ y / l c ) ,
Δ ( ρ x / l c ) = { 1 x / l c 0 when x l c . when x l c .
S ( i ) ( ω ) = E Q 2 ( ω ω 0 1 ) 2 + 1 ,
g ( ω u / c ) = l c 2 × sin c 2 ( ω ω 0 ξ u x ) × sin c 2 ( ω ω 0 ξ u y ) ,
ξ = l c ω 0 / 2 c = π l c / λ 0 .
S ( ) ( ω ; r u ) = A 2 π ( cos θ c r ) 2 ω 2 1 u β C D ( ω ; K ) S ( i ) ( ω ) ,
ω = ω [ 1 β ( u u ) ] ,
K = ( ω u ω u ) / c ,
C D ( ω ; ρ ) = C D ( ω ; ) + δ C D ( ω ; ρ ) ,
S ( ) ( ω ; r u ) = S A ( ) ( ω ; r u ) + S D ( ) ( ω ; r u ) ,
S A ( ) ( ω ; r u ) = A ( 2 π ) 2 ( ω cos θ 2 π c r ) 2 × d τ 0 d ω d 2 ρ C A ( ρ ) S ( i ) ( ω ) C D ( ω ; ) × exp [ i ( ω ω ) τ ] exp ( i ω u ρ / c ) .
S A ( ) ( ω ; r u ) = A 2 π ( ω cos θ c r ) 2 C A ( ω u / c ) S ( i ) ( ω ) C D ( ω ; ) ,
C A ( f ) = 1 ( 2 π ) 2 C A ( r ) exp ( i f r ) d 2 r

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