Abstract

The correlation function between two speckle patterns produced by different incident waves is studied by means of stochastic approaches. The angular-frequency memory effect, which includes the angular memory effect as a special case, is discussed, and the concept of the conjugate memory effect is introduced, which implies the interference of a complex-conjugate pair of scattering processes in the intensity.

© 1999 Optical Society of America

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References

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  1. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [CrossRef]
  2. H. M. Pedersen, “Second-order statics of light diffracted from gaussian, rough surfaces with application to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
    [CrossRef]
  3. G. Zhang, L. Tsang, “Angular correlation function of wave scattering by a random rough surface and discrete scatterers and its application in the detection of a buried object,” Waves Random Media 7, 467–478 (1997).
    [CrossRef]
  4. S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
    [CrossRef] [PubMed]
  5. T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
    [CrossRef]
  6. M. E. Knotts, T. R. Michel, K. A. O’Donnell, “Angular correlation functions of polarized intensities scattered from a one-dimensionally rough surface,” J. Opt. Soc. Am. A 9, 1822–1831 (1992).
    [CrossRef]
  7. V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
    [CrossRef]
  8. V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
    [CrossRef] [PubMed]
  9. M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
    [CrossRef]
  10. D. Léger, J. C. Perrin, “Real-time measurement of surface roughness by correlation of speckle patterns,” J. Opt. Soc. Am. 66, 1210–1217 (1976).
    [CrossRef]
  11. M. Nieto-Vesperinas, A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
    [CrossRef]
  12. M. Nieto-Vesperinas, A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
    [CrossRef]

1997 (4)

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

G. Zhang, L. Tsang, “Angular correlation function of wave scattering by a random rough surface and discrete scatterers and its application in the detection of a buried object,” Waves Random Media 7, 467–478 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

1993 (1)

1992 (3)

1988 (1)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

1976 (2)

1975 (1)

H. M. Pedersen, “Second-order statics of light diffracted from gaussian, rough surfaces with application to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Feng, S.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Goodman, J. W.

Kane, C.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Knotts, M. E.

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Léger, D.

Leskova, T. A.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

Malyshkin, V.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

Maradudin, A. A.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

McGurn, A. R.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

Michel, T. R.

Nieto-Vesperinas, M.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from a weakly rough one-dimensional random metal surface,” Opt. Lett. 22, 946–948 (1997).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
[CrossRef]

M. Nieto-Vesperinas, A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[CrossRef]

O’Donnell, K. A.

Pedersen, H. M.

H. M. Pedersen, “Second-order statics of light diffracted from gaussian, rough surfaces with application to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Perrin, J. C.

Sanchez-Gil, A.

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

M. Nieto-Vesperinas, A. Sanchez-Gil, “Intensity angular correlations of light multiply scattered from random rough surfaces,” J. Opt. Soc. Am. A 10, 150–157 (1993).
[CrossRef]

M. Nieto-Vesperinas, A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[CrossRef]

Shchegrov, A. V.

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Tsang, L.

G. Zhang, L. Tsang, “Angular correlation function of wave scattering by a random rough surface and discrete scatterers and its application in the detection of a buried object,” Waves Random Media 7, 467–478 (1997).
[CrossRef]

Zhang, G.

G. Zhang, L. Tsang, “Angular correlation function of wave scattering by a random rough surface and discrete scatterers and its application in the detection of a buried object,” Waves Random Media 7, 467–478 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

H. M. Pedersen, “Second-order statics of light diffracted from gaussian, rough surfaces with application to the roughness dependence of speckles,” Opt. Acta 22, 523–535 (1975).
[CrossRef]

Opt. Commun. (1)

M. Nieto-Vesperinas, A. A. Maradudin, A. V. Shchegrov, A. Sanchez-Gil, “Speckle patterns produced by weakly rough random surfaces: existence of a memory effect twin peak in the angular correlations and its consequences,” Opt. Commun. 142, 1–6 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

M. Nieto-Vesperinas, A. Sanchez-Gil, “Enhanced long-range correlations of coherent waves reflected from disordered media,” Phys. Rev. B 46, 3112–3115 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

Waves Random Media (2)

G. Zhang, L. Tsang, “Angular correlation function of wave scattering by a random rough surface and discrete scatterers and its application in the detection of a buried object,” Waves Random Media 7, 467–478 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Memory effect.

Fig. 2
Fig. 2

Scattering structure.

Fig. 3
Fig. 3

Optical path difference.

Fig. 4
Fig. 4

Scattered waves that are expressed by a complex-conjugate pair of random-walk processes on a complex plane.

Fig. 5
Fig. 5

Symmetry with respect to the xy plane.

Fig. 6
Fig. 6

Incident and scattering directions for space-reversal symmetry.

Fig. 7
Fig. 7

Wave vectors that satisfy the condition for the ordinary memory effect.

Fig. 8
Fig. 8

Wave vectors that satisfy the condition for the conjugate memory effect.

Fig. 9
Fig. 9

Model of multiple scattering.

Fig. 10
Fig. 10

Intensity correlation on the memory line, where θi1=30°, θs1=50°, 2π/k1=2π/k2=1, Lx=1000, D=0.1, 0.3, 1, and 10, N=1000, and the number of samples=1000.

Fig. 11
Fig. 11

Same as Fig. 10, but for the conjugate memory line.

Fig. 12
Fig. 12

Intensity correlation on the memory line (theory), where θi1=30°, θs1=50°, 2π/k1=2π/k2=1, and D=0.1, 0.3, 1, and 10.

Fig. 13
Fig. 13

Same as Fig. 12, but for the conjugate memory line (theory).

Fig. 14
Fig. 14

Contour map of intensity correlation for the ordinary memory effect, where θi1=θi2=30°, θs1=50°, 2π/k1=1, Lx=1000, D=0.1, N=1000, and the number of samples=1000.

Fig. 15
Fig. 15

Same as Fig. 14, but for the conjugate memory effect.

Fig. 16
Fig. 16

Speckle pattern, where θi1=10°, 2π/k1=1, Lx=100, D=0, and N=100.

Fig. 17
Fig. 17

Speckle pattern, where θi1=10°, 2π/k1=1, Lx=100, D=1, and N=100.

Equations (60)

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

sin θs1-sin θi1=sin θs2-sin θi2.
cos θi dθi=cos θs dθs,
sin θs1-sin θi1=-sin θs2+sin θi2.
2 sin θi=sin θs1+sin θs2,
kiα=ksαkα,
kiαqiα+ezβiα=(kiα sin θiα cos ϕiα, kiα sin θiα sin ϕiα,-kiα cos θiα),
ksαqsα+ezβsα=(ksα sin θsα cos ϕsα, ksα sin θsα sin ϕsα, ksα cos θsα),
pαksα-kiαqα+ezβα,
rx+ezz,
rnxn+ezz,
I(ksα|kiα)I(pα)=AgV exp(-ipα·r)dB(r)2
=AgVV exp[ipα·(r1-r2)]dB(r1)dB(r2),
dB(r)=0,
dB(r)dB(r)=δ(r-r)drdr,
J(pα)I(pα)-I(pα)
=AgVV exp[ipα·(r1-r2)][dB(r1)dB(r2)-dB(r1)dB(r2)].
J(p1)J(p2)=Ag2VV{exp[i(p1+p2)·(r1-r2)]+exp[i(p1-p2)·(r1-r2)]}dr1dr2
=Ag2V exp[i(p1+p2)·r]dr2+V exp[i(p1-p2)·r]dr2.
p·r=qxx+qyy+βz,dr=dxdydz,
Γ(p)V sinc(qxLx/2)sinc(qyLy/2)sinc(βD/2),
sinc Xsin XX,
J(p1)J(p2)=Ag2[|Γ(p1-p2)|2+|Γ(p1+p2)|2].
qxLxqyLyβD0,p=(qx, qy, β).
|(q1x-q2x)Lx|+|(q1y-q2y)Ly|+|(β1-β2)D|0
|(q1x+q2x)Lx|+|(q1y+q2y)Ly|+|(β1+β2)D|0.
I(pα)=Apn=1N exp(-irn·pα)2=ApV exp(-ir·pα)dD(r)
=ApVV exp[i(r1-r2)·pα]dD(r1)dD(r2),
dD(r)=jδ(r-rj)dr,
dD(r)=dr,
dD(r)dD(r)=δ(r-r)drdr+drdr,
J(p1)J(p2)
=Ap2{Γ(p1-p2)|2+|Γ(p1+p2)|2+|Γ(p1)|2+|Γ(p2)|2+2[Γ(p1-p2)+Γ(p1+p2)]Γ(p1)Γ(p2)+V}
Ap2[|Γ(p1-p2)|2+|Γ(p1+p2)|2+V],
q1=q2.
q1=-q2.
qs1+qs2=2qi,
ks2=qs1-qi1+qi2±ezβs2
ks2=-qs1+qi1+qi2±ezβs2.
kpq+ezβ,
kmq-ezβ,
βkα2-|q|2,
sin θs1-sin θi1=sin θs2-sin θi2,
k1 sin θs1-k2 sin θs2=(k1-k2)sin θi,
(sin θi-sin θs)dk=k cos θs dθs,
k1(sin θs1-sin θi1)=-k2(sin θs2+sin θi2).
ks1-ki1=ks2-ki2,
ks1-ki1=-ks2+ki2,
ki1=ki2,ks1=ks2
ki1=-ks2,ks1=-ki2,
ki1=-ki2,ks1=-ks2,
ki1=ks2,ks1=ki2,
q1x=q2x,
q1x=-q2x.
Ψfαr1·kiα-rn·ksα+ψ,
Ψrαrn·kiα-r1·ksα+ψ,
Ψf1=Ψr2,
Ψf1=-Ψf2+2ψ,
I(θsα|θiα, kα)=ANn=1N exp{ikα[xn(sin θiα-sin θsα)-zn(cos θiα+cos θsα)]}2,
d+k1(sin θs1-sin θi1)-k2(sin θs2-sin θi2)=0
d-k1(sin θs1-sin θi1)+k2(sin θs2-sin θi2)=0

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