Abstract

The question is examined as to whether the spectrum of the far field produced on scattering of a polychromatic plane wave on a spatially random object can be the same in every direction of scattering. It is shown that within the accuracy of the first Born approximation a sufficiency condition for this to be so is that the two-point correlation function of the dielectric susceptibility of the scatterer obey a certain scaling law. It is further shown that if, in addition, the isotropic far-zone spectrum is the same as the spectrum of the incident field, the second moment of the dielectric susceptibility of the scatterer must have the form of a 1/f noise, well known in the theory of fractals.

© 1997 Optical Society of America

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References

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  1. E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatiallyrandom media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989); erratum, 7, 173 (1990).
    [CrossRef]
  2. W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
    [CrossRef]
  3. R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), pp. 14–15.
  4. E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [CrossRef] [PubMed]
  5. E. Wolf, “Two inverse problems in spectroscopy with partially coherent sourcesand the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
    [CrossRef]
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), p. 171.
  7. See, for example, G. Wornell, Signal Processing with Fractals (Prentice-Hall, Upper Saddle River, N.J., 1996), Sec. 3.2.

1992 (1)

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sourcesand the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[CrossRef]

1989 (1)

1988 (1)

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

1986 (1)

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

Carter, W. H.

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

Foley, J. T.

Gori, F.

James, R. W.

R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), pp. 14–15.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), p. 171.

Wolf, E.

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sourcesand the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[CrossRef]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatiallyrandom media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989); erratum, 7, 173 (1990).
[CrossRef]

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), p. 171.

J. Mod. Opt. (1)

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sourcesand the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[CrossRef]

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

Opt. Commun. (1)

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (3)

R. W. James, The Optical Principles of Diffraction of X-rays (Bell, London, 1948), pp. 14–15.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), p. 171.

See, for example, G. Wornell, Signal Processing with Fractals (Prentice-Hall, Upper Saddle River, N.J., 1996), Sec. 3.2.

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Figures (1)

Fig. 1
Fig. 1

Illustration of the notation.

Equations (42)

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Cη(r2-r1, ω)=η*(r1, ω)η(r2, ω)
S()(ru, ω)=Vr2 ωc4C˜η[k(u-u0), ω]S(i)(ω),
k=ωc,
C˜η(K, ω)=Cη(r, ω)exp(-iKr)d3r
s()(ru, ω)=S()(ru, ω)0S()(ru, ω)dω.
s()(ru, ω)=ω4C˜η[k(u-u0), ω]S(i)(ω)0ω4C˜η[k(u-u0), ω]S(i)(ω)dω.
C˜η[k(u-u0), ω]=F(ω)H˜(u-u0),
s()(ru, ω)=ω4F(ω)S(i)(ω)0ω4F(ω)S(i)(ω)dω,
C˜η[0, ω]=F(ω)H˜(0),
C˜η[k(u-u0), ω]=1H˜(0) C˜η[0, ω]H˜(u-u0).
K=k(u-u0).
C˜η[K,ω]=1H˜(0) C˜η[0, ω]H˜(K/k).
|K|2k.
C˜η(K, ω)0for|K|>2k,
Cη(r, ω)=1(2π)3H˜(0) C˜η(0, ω)×H˜(K/k)exp(iKr)d3K.
Cη(r, ω)=k3H˜(0) C˜η(0, ω)H(kr),
H˜(K)=H(r)exp(-iKr)d3r.
μη(r, ω)=Cη(r, ω)Cη(0, ω).
0|μη(r, ω)|1,
μη(r, ω)=h(kr)(k=ω/c),
h(kr)=H(kr)H(0).
krk(r2-r1)=2πλ (r2-r1)
2πλ (r2-r1)=2πλ (r2-r1),
μ(ρ, ω)=f(kρ)(k=ω/c),
s()(ru, ω)=ω4C˜η(0, ω)S(i)(ω)0ω4C˜η(0, ω)S(i)(ω)dω,
ω4C˜η(0, ω)=const(α,say),
s(i)(ω)=S(i)(ω)0S(i)(ω)dω.
C˜η(0, ω)=Cη(r, ω)d3r
C˜η(0, ω)=Cη(0, ω)μ(r, ω)d3r.
C˜η(0, ω)=Cη(0, ω)h(kr)d3r
C˜η(0, ω)=1k3 Cη(0, ω)h(R)d3R.
Cη(0, ω)=βω,
η*(r, ω)η(r, ω)constω.
Cη(r, ω)=A exp-(k0x)22σx2+(k0y)22σy2+(k0z)22σz2,
C˜η(K, ω)=(2π)3/2Aσxσyσzk03 exp-12k02 [(Kxσx)2+(Kyσy)2+(Kzσz)2],
s()(ru, ω)=N(u, ω)S(i)(ω)0N(u, ω)S(i)(ω)dω,
N(u, ω)=ω4 exp-12 kk02[(ux-u0x)2σx2+(uy-u0y)2σy2+(uz-u0z)2σz2],
Cη(r, ω)=A exp-(kx)22σx2+(ky)22σy2+(kz)22σz2,
C˜η(K, ω)=(2π)3/2Aσxσyσzk3 exp-12k2 [(Kxσx)2+(Kyσy)2+(Kzσz)2].
s()(ru, ω)=ωS(i)(ω)0ωS(i)(ω)dω.
μη(r, ω)=h(kr)(k=ω/c).
η*(r, ω)η(r, ω)=const.ω

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