Abstract

The spreading of partially coherent beams propagating through atmospheric turbulence is studied by use of the coherent-mode representation of the beams. Specifically, we consider partially coherent Gaussian Schell-model beams entering the atmosphere, and we examine the spreading of each coherent mode, represented by a Hermite–Gaussian function, on propagation. We find that in atmospheric turbulence the relative spreading of higher-order modes is smaller than that of lower-order modes, whereas the relative spreading of all order modes is the same as in free space. This modal behavior successfully explains why under certain circumstances partially coherent beams are less affected by atmospheric turbulence than are fully spatially coherent laser beams.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).
  2. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Chap. VI.
  3. L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, 1998).
  4. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
    [CrossRef]
  5. J. Wu, A. D. Boardman, “Coherence length of a Gaussian–Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991).
    [CrossRef]
  6. G. Gbur, E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).Equation (29) of this paper contains a small error. It should read F2=4π23∫0∞κ3Φn(κ)dκ.
    [CrossRef]
  7. S. A. Ponomarenko, J.-J. Greffett, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  8. A. Dogariu, S. Amarande, “Propagation of partially co-herent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [CrossRef] [PubMed]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  10. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  11. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  12. J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
    [CrossRef] [PubMed]
  13. T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
    [CrossRef]
  14. G. Gbur, E. Wolf, “The Rayleigh range of partially coherent beams,” Opt. Commun. 199, 295–304 (2001).
    [CrossRef]
  15. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), Sec. 70.
  16. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, Calif., 1994).

2003 (1)

2002 (2)

2001 (1)

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

1997 (1)

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

1991 (1)

J. Wu, A. D. Boardman, “Coherence length of a Gaussian–Schell 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]

1989 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982 (1)

Amarande, S.

Andrews, L. C.

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

Boardman, A. D.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).

Dogariu, A.

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Chap. VI.

Friberg, A. T.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

Gbur, G.

Gori, F.

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, Calif., 1994).

Greffett, J.-J.

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

Habashy, T.

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

Mandel, L.

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

Phillips, R. L.

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

Ponomarenko, S. A.

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, Calif., 1994).

Starikov, A.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), Sec. 70.

Tervonen, E.

Turunen, J.

Wolf, E.

G. Gbur, E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).Equation (29) of this paper contains a small error. It should read F2=4π23∫0∞κ3Φn(κ)dκ.
[CrossRef]

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

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

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

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

Wu, J.

J. Wu, A. D. Boardman, “Coherence length of a Gaussian–Schell 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]

Inverse Probl. (1)

T. Habashy, A. T. Friberg, E. Wolf, “Application of the coherent-mode representation to a class of inverse source problems,” Inverse Probl. 13, 47–61 (1997).
[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 Gaussian–Schell 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. (3)

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

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

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

Opt. Lett. (2)

Other (7)

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

J. W. Strohbehn, ed., Laser Beam Propagation in the Atmosphere (Springer-Verlag, New York, 1978).

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Chap. VI.

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

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971), Sec. 70.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic Press, San Diego, Calif., 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Illustration of the notation relating to beam propagation through atmospheric turbulence.

Fig. 2
Fig. 2

Normalized rms width of the coherent modes propagating through atmospheric turbulence for (a) m=n=0, (b) m=n=1, (c) m=n=2, (d) m=n=4, and (e) the normalized rms width of the modes (for any m and n) in free space. These curves are plotted under the conditions that k=107 m-1, Cn2=10-14 (Cn2=0 for free-space propagation) m-2/3, l0=0.01 m, and w0=0.01 m.

Fig. 3
Fig. 3

Normalized intensity distribution of (a) a beam generated by a Gaussian Schell-model source with the degree of global coherence lg=0.4 and (b) a fully coherent Gaussian beam under the conditions that k=107 m-1, Cn2=10-14 (Cn2=0 for free-space propagation) m-2/3, l0=0.01 m, σS=5 mm (i.e., the width of the beam in the source plane is the same), and the propagation distance z=5000 m. With these parameters, the strong-fluctuation condition used to simplify Eq. (33) is satisfied.

Equations (67)

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

S(0)(ρ, ω)=A exp-|ρ|22σS2,
μ(0)(ρ1, ρ2, ω)=exp-|ρ1-ρ2|22σμ2,
W(0)(ρ1, ρ2, ω)=S(0)(ρ1, ω)S(0)(ρ2, ω)×μ(0)(ρ1, ρ2, ω)=A exp-|ρ1|2+|ρ2|24σS2×exp-|ρ1-ρ2|22σμ2.
W(0)(ρ1, ρ2, ω)=nαn(ω)φn*(ρ1, ω)φn(ρ2, ω),
σW(0)(ρ1, ρ2, ω)φn(ρ1, ω)d2ρ1=αn(ω)φn(ρ2, ω)
σφn*(ρ, ω)φm(ρ, ω)d2ρ=δnm,
αn(ω)0.
W(0)(ρ1, ρ2, ω)=mnβmn(ω)ϕmn(0)*(ρ1, ω)×ϕmn(0)(ρ2, ω),
βmn(ω)=Aπa+b+cba+b+cm+n
a=14σS2,b=12σμ2,
c=a2+2ab.
ϕmn(0)(ρ, ω)ϕ(0)(ρx, ρy, ω)=BmnHm2w0 ρxHn2w0 ρyexp-ρx2+ρy2w02,
Bmn=1w0π2m+n-1m!n!,
w0=1c,
lgσμ/σS,
βmn(ω)β00(ω)=1lg2/2+1+lg(lg/2)2+1m+n.
U(ρ, z, ω)=-ik exp(ikz)2πzU0(ρ0, ω)×expik (ρ-ρ0)22zexp[ψ(ρ, ρ0, z)]d2ρ0.
I(ρ, z, ω)U*(ρ, z, ω)U(ρ, z, ω)M=k2πz2d2ρ0d2ρ0U0*(ρ0, ω)×U0(ρ0, ω)×exp-ik (ρ-ρ0)2-(ρ-ρ0)22z×exp[ψ*(ρ, ρ0, z)+ψ(ρ, ρ0, z)]M,
exp[ψ*(ρ, ρ0, z)+ψ(ρ, ρ0, z)]M
=exp[-12Dsp(|ρ0-ρ0|, z)],
Dsp(|ρ|, z)=8π2k2z010κΦ(κ)[1-J0(κξ|ρ|)]dκdξ.
u=ρ0+ρ02,v=ρ0-ρ0.
I(ρ, z, ω)=k2πz2d2ud2v×U0*u+12v, ωU0u-12v, ω×exp-i kzuvexpi kzvρ×exp-4π2k2z010κΦ(κ)×[1-J0(κξv)]dκdξ.
[U0(ρ, ω)]mnϕmn(0)(ρ, ω)=BmnHm2w0 ρxHn2w0 ρy×exp-ρx2+ρy2w02.
wmn(z)ρ2Imn(ρ, z,ω)d2ρImn(ρ, z, ω)d2ρ1/2,
Imn(ρ, z, ω)=k2πz2|Bmn|2d2ud2v×Hm2w0ux+12 vxHm2w0ux-12 vx×Hn2w0uy+12 vyHn2w0uy-12 vy×exp-2u2w02exp-v22w02exp-i kzuv×expi kzvρexp-4π2k2z×010κΦ(κ)[1-J0(κξv)]dκdξ.
Imn(ρ, z, ω)d2ρ=1.
wnm(z)=m+n+12w021+2kw022z2+43 π20κ3Φ(κ)dκz31/2.
wmnN(z)wmn(z)wmn(0)=1+2kw022z2+8π23(m+n+1)w020κ3Φ(κ)dκz31/2.
[wmnN(z)]FS=1+2kw022z21/2.
Φ(κ)=0.033Cn2κ-11/3exp-κ2κm2,
wmnN(z)=1+2kw022z2+4.373(m+n+1)w02 Cn2l0-1/3z31/2.
I(ρ, z, ω)=k2πz2d2ud2v×U0*u+12v, ωU0u-12v ,ω×exp-i kzuvexpi kzvρ×exp-4π2k2z010κΦ(κ)×[1-J0(κξv)]dκdξ.
I(ρ, z, ω)=k2πz2Ad2ud2v ×exp-u22σS2exp-v22σΔ2×exp-i kzuvexpi kzvρ×exp-4π2k2z010κΦ(κ)×[1-J0(κξv)]dκdξ,
1σΔ214σS2+1σμ2.
J0(κξv)1-14(κξv)2.
I(ρ, z, ω)=k2πz2Ad2ud2v ×exp-u22σS2exp-v22σΔ2×exp-i kzuvexpi kzvρ×exp-13 π2k2zv20κ3Φ(κ)dκ.
I(ρ, z, ω)=AΔ2(z)exp-ρ22σS2Δ2(z),
Δ(z)=1+1(kσSσΔ)2 z2+1σS223 π20κ3Φ(κ)dκz31/2.
Δ(z)=1+1(kσS)214σS2+1σμ2z2+1.093Cn2l0-1/3σS2 z31/2.
IN(ρ, z, ω)=1+(z/2kσS2)2Δ2(z)exp-ρ22σS2Δ2(z).
Fρ2Imn(ρ, z, ω)d2ρ,
F=F1+F2,
F1=-zk2|Bmn|2d2ud2v×Hm2w0ux+12 vxHm2w0ux-12 vx×Hn2w0uy+12 vyHn2w0uy-12 vy×exp-2u2w02exp-v22w02exp-i kzuv×exp-4π2k2z010κΦ(κ)×[1-J0(κξv)]dκdξδ(vx)δ(vy),
F2=-zk2|Bmn|2d2ud2v×Hm2w0ux+12 vxHm2w0ux-12 vx×Hn2w0uy+12 vyHn2w0uy-12 vy×exp-2u2w02exp-v22w02exp-i kzuv×exp-4π2k2z010κΦ(κ)×[1-J0(κξv)]dκdξδ(vx)δ(vy).
 x2exp(-i2πxs)dx=-1(2π)2 δ(s).
F1=-zk22π12mm!w0duxdvx×Hm2w0ux+12 vxHm2w0ux-12 vx×exp-2ux2w02exp-vx22w02exp-i kz uxvx×exp-4π2k2z010κΦ(κ)×[1-J0(κξvx)]dκdξδ(vx).
exp(-x2)[Hn(x)]2dx=2nn!π.
exp(-x2)Hm(x+y)Hn(x+z)dx
=2nπm!zn-mLmn-m(-2yz),formn,
F1=-zk2dvxexp-121w02+14kz2w02vx2×Lm1w02+14kz2w02vx2×exp-4π2k2z010κΦ(κ)×[1-J0(κξvx)]dκdξδ(vx),
 f(x)δ(x)dx=f(0),
F1=2m+14 w021+2kw022z2+23 π20κ3Φ(κ)dκz3.
F2=2n+14 w021+2kw022z2+23 π20κ3Φ(κ)dκz3.
F=m+n+12 w021+2kw022z2+43 π20κ3Φ(κ)dκz3.
Gexp[-12 Dsp(v, z)],
Dsp(v, z)=8π2k2z010κΦ(κ)[1-J0(κξv)]dκdξ
Δγl0,
Δγ(Cn2k2z)-3/5,
Cn2k2zl05/31,
Dsp(l0, z)Cn2k2zl05/3,
Dsp(l0, z)1.
vl0.
κκl01/l0,
0<ξ<1
κξv1.
F2=4π230κ3Φn(κ)dκ.

Metrics