Abstract

Some published computational work has suggested that partially coherent beams may be less susceptible to distortions caused by propagation through random media than fully coherent beams. In this paper this suggestion is studied quantitatively by examining the mean squared width of partially coherent beams in such media as a function of the propagation distance. The analysis indicates under what conditions, and to what extent, partially coherent beams are less affected by the medium.

© 2002 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.
  2. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Vol. 22, pp. 341–398.
  3. L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).
  4. M. J. Beran, “Propagation of the mutual coherence function through random media,” J. Opt. Soc. Am. 56, 1475–1480 (1966).
    [CrossRef]
  5. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
    [CrossRef]
  6. J. Wu, A. D. Boardman, “Coherence length of a Gauss-ian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
    [CrossRef]
  7. Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radio Quantum Electron. 10, 33–35 (1967).
    [CrossRef]
  8. M. Santarsiero, F. Gori, R. Borghi, G. Cincotti, P. Vahimaa, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
    [CrossRef]
  9. G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
    [CrossRef]
  10. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  11. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [CrossRef]

2001 (1)

G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
[CrossRef]

1999 (1)

1991 (1)

J. Wu, A. D. Boardman, “Coherence length of a Gauss-ian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

1990 (1)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

1978 (1)

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

1967 (1)

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radio Quantum Electron. 10, 33–35 (1967).
[CrossRef]

1966 (1)

Andrews, L. C.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).

Beran, M. J.

Boardman, A. D.

J. Wu, A. D. Boardman, “Coherence length of a Gauss-ian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

Borghi, R.

Cincotti, G.

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Vol. 22, pp. 341–398.

Feizulin, Z. I.

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radio Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Foley, J. T.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Gbur, G.

G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
[CrossRef]

Gori, F.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.

Kravtsov, Yu. A.

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radio Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Mandel, L.

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

Phillips, R. L.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).

Santarsiero, M.

Vahimaa, P.

Wolf, E.

G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
[CrossRef]

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

Wu, J.

J. Wu, A. D. Boardman, “Coherence length of a Gauss-ian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

J. Mod. Opt. (2)

J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

J. Wu, A. D. Boardman, “Coherence length of a Gauss-ian Schell-model beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
[CrossRef]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[CrossRef]

Radio Quantum Electron. (1)

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radio Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vols. 1 and 2.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 1985), Vol. 22, pp. 341–398.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).

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

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Figures (4)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Illustration of the condition for the validity of approximation (17).

Fig. 3
Fig. 3

Spreading of Gaussian Schell-model beams in free space (FS) and in turbulent media (T) for various values of the width of the spectral degree of coherence in the waist plane, σμ. The turbulence distance zT is indicated by a dashed line. Case (a) represents a beam produced by a spatially fully coherent source, while case (d) represents a beam produced by a spatially incoherent source. In all cases w0=0.01 m and RT1.

Fig. 4
Fig. 4

Spreading of Gaussian Schell-model beams in free space (FS) and in turbulent media (T) for two values of the width of the spectral degree of coherence, σμ. In both cases zT=2220 m, w0=0.1 m, and RT1.

Equations (54)

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n(r)=1+n1(r),
{2+k2[1+2n1(r)]}U(r)=0,
U(r)=2ikzU0(ρ)exp[ψ(ρ, r)]G0(ρ, r)d2ρ,
G0(ρ, r)=expikz+ik(x-x)2+(y-y)22z,
I(ρ, z)|U(ρ, z)|2=4k2z2U0*(ρ1)U0(ρ2)exp[ψ*(ρ1, ρ, z)+ψ(ρ2, ρ, z)]G0*(ρ1, ρ, z)×G0(ρ2, ρ, z)d2ρ1d2ρ2.
I(ρ, z)=4k2z2W0(ρ1, ρ2)Cψ(ρ1, ρ2; ρ, ρ, z)G0*(ρ1, r)G0(ρ2, r)d2ρ1d2ρ2,
W0(ρ1, ρ2)=U0*(ρ1)U0(ρ2)
Cψ(ρ1, ρ2; ρ1, ρ2, z)=exp[ψ*(ρ1, ρ1, z)+ψ(ρ2, ρ2, z)].
Cψ(ρ1, ρ2, ρ1, ρ2, z)=exp[2E1(z)+E2(ρ2-ρ1, ρ2-ρ1, z)],
E1(z)=-2π2k2z0κΦn(κ)dκ,
E2(ρ2-ρ1, ρ2-ρ1, z)=4π2k2z010κΦn(κ)×J0[κ|(1-ξ)(ρ2-ρ1)+ξ(ρ2-ρ1)|]dκdξ.
Cψ(ρ1, ρ2, ρ1, ρ2, z)Cψ(ρ2-ρ1, ρ2-ρ1, z).
W0(ρ1, ρ2)=I0(ρ1)I0(ρ2)μ0(ρ2-ρ1),
I(ρ, z)=4k2z2I0(ρ1)I0(ρ2)μ0(ρ2-ρ1)×Cψ(ρ2-ρ1, 0, z)G0*(ρ1, ρ, z)×G0(ρ2, ρ, z)d2ρ1d2ρ2.
I1(ρ)=I0(ρ),
μ1(ρ)=μ0(ρ)Cψ(ρ, 0, z).
I(ρ, z)Cψ(0, 0, z)4k2z2×I0(ρ1)I0(ρ2)μ0(ρ2-ρ1)×G0*(ρ1, ρ, z)G0(ρ2, ρ, z)d2ρ1d2ρ2.
ρ2(z)¯ρ2I(ρ, z)d2ρI(ρ, z)d2ρ.
ρ2(z)¯=a2+c2(z)z2,
a2=I1(ρ)ρ2d2ρI1(ρ)d2ρ,
c2(z)=W˜1(-ks, ks)s2d2sW˜1(-ks, ks)d2s,
W˜1(K1, K2)=1(2π)4W1(ρ1, ρ2)×exp[-i(K1·ρ1+K2·ρ2)]d2ρ1d2ρ2
a2=σI2,
σI2=I0(ρ)ρ2d2ρI0(ρ)d2ρ,
c2(z)=σJ2-1k2ρ2Cψ(ρ, 0, z)|ρ=0,
σJ2=J(s)s2d2sJ(s)d2s,
J(s)=(2πk)2W˜0(-ks, ks)
-1k2ρ2Cψ(ρ, 0, z)|ρ=0=zF2,
F2=2π230κ3Φn(κ)dκ.
ρ2(z)¯=σI21+σJ2σI2z2+F2σI2z3.
zR=σI/σJ
ρ2(z)¯=σI21+z2zR2+F2σI2z3.
ρ2(zT)¯turb-ρ2(zT)¯freeρ2(zT)¯turb=0.1.
ρ2(z)¯free=σI2+σJ2z2.
ρ2(z)¯freeσJ2z2,
ρ2(z)¯turbσJ2z2+F2z3.
zT19σJ2F2.
σJ2=[σJ2]coh-1k2ρ2μ0(ρ)|ρ=0,
[σJ2]coh=W˜coh(-ks, ks)s2d2sW˜coh(-ks, ks)d2s,
Wcoh(ρ1, ρ2)=I0(ρ1)I0(ρ2),
σJ3σIF21.
ρ2(z)¯freeσI2,
ρ2(zT)¯turbσI2+F2z3.
zT19σI2F23.
σJ3σIF231.
RTσJ3σIF2.
F2zR3σI2=F2σIσJ3=1RT.
I0(ρ)=A exp[-2ρ2/w02],
μ0(ρ)=exp[-ρ2/2σμ2].
σI2=w022,
σJ2=2k21w02+1σμ2,
zR=kw021w02+1σμ2-1/2.
Φn(κ)=0.033Cn2κ-11/3 exp[-κ2/κm2],
F2=1.095Cn2l0-1/3.

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