Abstract

When propagating through particulate media, optical beams are degraded owing to scattering. We found that the ratio between the width of the distorted beam and the width of the initial beam decreases when the spatial coherence of the incident beam is reduced. These experimental observations are well described within the paraxial approximation of the transport theory.

© 2003 Optical Society of America

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References

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  1. G. Gbur, E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [CrossRef]
  2. S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  3. A. Dogariu, S. Amarande, “Propagation of partially coherent beams: the turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [CrossRef] [PubMed]
  4. R. F. Lutomirski, “Atmospheric degradation of electrooptical system performance,” Appl. Opt. 17, 3915–3921 (1978).
    [CrossRef] [PubMed]
  5. N. S. Kopeika, D. Sadot, I. Dror, “Aerosol light scatter vs turbulence effects in image blur,” in Optics Atmospheric Propagation and Adaptive Systems II, A. Kohnle, A. D. Devir, eds., Proc. SPIE3219, 44–51 (1998).
    [CrossRef]
  6. Y. Kuga, A. Ishimaru, “Modulation transfer function and image transmission through randomly distributed spherical particles,” J. Opt. Soc. Am. A 2, 2330–2336 (1985).
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978), Vol. 2.
  8. S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
    [CrossRef] [PubMed]
  9. A. Wax, J. E. Thomas, “Measurement of smoothed Wigner phase-space distributions for small-angle scattering in a turbid medium,” J. Opt. Soc. Am. A 15, 1896–1908 (1998).
    [CrossRef]
  10. C.-C. Cheng, M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
    [CrossRef]
  11. Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).
  12. H. T. Yura, L. Thrane, P. E. Andersen, “Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium,” J. Opt. Soc. Am. A 17, 2464–2474 (2000).
    [CrossRef]
  13. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  14. For a review, see M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  15. D. A. de Wolf, “Coherence of a light beam through an optically dense turbid layer,” Appl. Opt. 17, 1280–1285 (1978).
    [CrossRef] [PubMed]
  16. P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
    [CrossRef]

2003

2002

G. Gbur, E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

2000

1998

1996

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

1986

1985

1978

1969

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Amarande, S.

Andersen, P. E.

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Bastiaans, M. J.

Carney, P. S.

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

Cheng, C.-C.

C.-C. Cheng, M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[CrossRef]

de Wolf, D. A.

Dogariu, A.

Dror, I.

N. S. Kopeika, D. Sadot, I. Dror, “Aerosol light scatter vs turbulence effects in image blur,” in Optics Atmospheric Propagation and Adaptive Systems II, A. Kohnle, A. D. Devir, eds., Proc. SPIE3219, 44–51 (1998).
[CrossRef]

Gbur, G.

Greffet, J.-J.

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Ishimaru, A.

John, S.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Kopeika, N. S.

N. S. Kopeika, D. Sadot, I. Dror, “Aerosol light scatter vs turbulence effects in image blur,” in Optics Atmospheric Propagation and Adaptive Systems II, A. Kohnle, A. D. Devir, eds., Proc. SPIE3219, 44–51 (1998).
[CrossRef]

Kuga, Y.

Lutomirski, R. F.

Mandel, L.

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

Pang, G.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Ponomarenko, S. A.

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Raymer, M. G.

C.-C. Cheng, M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[CrossRef]

Sadot, D.

N. S. Kopeika, D. Sadot, I. Dror, “Aerosol light scatter vs turbulence effects in image blur,” in Optics Atmospheric Propagation and Adaptive Systems II, A. Kohnle, A. D. Devir, eds., Proc. SPIE3219, 44–51 (1998).
[CrossRef]

Thomas, J. E.

Thrane, L.

Wax, A.

Wolf, E.

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

G. Gbur, E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

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

Yang, Y.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

Yura, H. T.

Appl. Opt.

J. Biomed. Opt.

S. John, G. Pang, Y. Yang, “Optical coherence propagation and imaging in a multiple scattering medium,” J. Biomed. Opt. 1, 180–191 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Commun.

P. S. Carney, E. Wolf, “An energy theorem for scattering of partially coherent beams,” Opt. Commun. 155, 1–6 (1998).
[CrossRef]

S. A. Ponomarenko, J.-J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. A

C.-C. Cheng, M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[CrossRef]

Sov. Phys. JETP

Yu. N. Barabanenkov, “On the spectral theory of radiation transport equations,” Sov. Phys. JETP 29, 679–684 (1969).

Other

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

N. S. Kopeika, D. Sadot, I. Dror, “Aerosol light scatter vs turbulence effects in image blur,” in Optics Atmospheric Propagation and Adaptive Systems II, A. Kohnle, A. D. Devir, eds., Proc. SPIE3219, 44–51 (1998).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978), Vol. 2.

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Figures (6)

Fig. 1
Fig. 1

Illustration of the notation relating to the propagation of the beam.

Fig. 2
Fig. 2

Experimental setup used to study the scattering of PCBs: FA, field aperture; D, diffuser; SM, scattering medium; L, lens.

Fig. 3
Fig. 3

Three-dimensional representation of the angular intensity S(θ) recorded by the CCD detector.

Fig. 4
Fig. 4

Divergence for PCBs with different coherence parameters. Squares, the experimental data; solid curve, the results of calculations based on Eq. (4).

Fig. 5
Fig. 5

Normalized angular scattered intensity for an incident beam with a diameter of 9 mm and a coherence parameter for A of σμ=390 µm and for B of σμ=57 µm. Solid curves, the experimental results; dashed curves, the calculations based on Eq. (4).

Fig. 6
Fig. 6

rms angular spread of the beam after the particulate medium relative to its initial value. The coherence length σμ of the PCB ranges from 1.3 mm to 57 µm.

Equations (16)

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W(x, k, z)=1(2π)2exp(-is·k)×Wx+s2, x-s2, zd2s,
vz+vkk·xW(x, k, z)=d2kF˜(k-k)W(x, k, z),
F˜(k-k)=-vμTδ2(k-k)+vNk2dσ(k-k)dΩ.
S(k)=exp[-Γ(s, 0, L)]H(s, 0, L)×exp(-ik·s)d2s,
Γ(s, 0, L)=LvF(s)=μTL-LNk2×exp(is·Δk)dσ(Δk)dΩd2Δk,
H(s, 0, L)=1(2π)2Wx+s2, x-s2, 0d2x.
Δθ=1kk2S(k)d2kS(k)d2k1/2.
Δθ2=1k2Sout(H(0, 0, L)s2{exp[-Γ(s, 0, L)]}s=0+exp[-Γ(0, 0, L)]s2[H(s, 0, L)]s=0),
Δθ2=LNk4Δk2dσ(Δk)dΩd2Δk+1k2Sins2[H(s, 0, L)]s=0.
Δθ2=LNσSΔθS2+Δθ02,
Δθ02=1k2Sins2[H(s, 0, L)]s=0,
ΔθS2=Δk2dσ(Δk)dΩd2Δkk4dσ(Δk)dΩd2Δk=Δk2dσ(Δk)dΩd2Δkk4σS.
D=Δθ2Δθ021/2=LNσSΔθS2+Δθ02Δθ021/2=1+LNσSΔθS2Δθ021/2.
W(r1, r2)=exp-r12+r22w02exp-|r1-r2|22σμ2
Δθ02=2k21σμ2+1w02,
W(r1, r2)=Discr1w0Discr2w01/2 exp-|r1-r2|22σμ2,

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