Abstract

The formulas of the energy and energy flux density of partially coherent electromagnetic beams in atmospheric turbulence are derived by using Maxwell’s equations. Expressions expressed by elements of electric cross spectral density matrixes of the magnetic and the mutual cross spectral density matrix are obtained for the partially coherent electromagnetic beams. Taken the partially coherent Cosh-Gaussian (ChG) electromagnetic beam as a typical example, the spatial distributions of the energy and energy flux density in atmospheric turbulence are numerically calculated. It is found that the turbulence shows a broadening effect on the spatial distributions of the energy and energy flux density. Some interesting results are obtained and explained with regard to their physical nature.

© 2009 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media (SPIE Press, 1998)
  2. J. Wu, "Propagation of a Gaussian-Schell beams through turbulent media," J. Mod. Opt. 37, 671 - 684 (1990).
    [CrossRef]
  3. J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355 - 1363 (1991).
    [CrossRef]
  4. X. L. Ji and X. Q. Li, "Directionality of Gaussian array beams propagating in atmospheric turbulence," J. Opt. Soc. Am. A 26, 236 - 243 (2009).
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  7. A. L. Yang, E. T. Zhang, X. L. Ji and B. D. Lü, "Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence," Opt. Express 16, 8366 - 8380 (2008).
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  8. H. T. Eyyuboglu, Y. Baykal and C. J. Yang, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891 - 2901 (2007).
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  11. J. C Leader, "Atmospheric propagating of partially coherent radiation," J. Opt. Soc. A.  68, 175-185 (1978.).
    [CrossRef]
  12. J. C. Leader, "Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence," J. Opt. Soc. A. 69, 73-84 (1979)
    [CrossRef]
  13. J. C. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. A. 71, 542-558 (1981).
    [CrossRef]
  14. Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
    [CrossRef]
  15. Y. Baykal, M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. A. 2, 2124-2132 (1985).
    [CrossRef]
  16. R. Conan, "Mean-square residual error of a wave front after propagation through atmospheric turbulence and after correction with Zernike polynomials," J. Opt. Soc. Am. A 25, 526 - 536 (2008).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st. (Cambridge University Press, New York, 1995)
  18. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  19. A. A. Tovar and L. W. Casperson, "Production and propagation of Hermite-sinusoidal-Gaussian laser beams," J. Opt. Soc. Am. A 15, 2425 - 2432 (1998)
    [CrossRef]
  20. M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gauss beams," Opt. Commun. 70, 361-364 (1989).
    [CrossRef]
  21. H. T. Yura, "Mutual coherence function of a finite cross-section optical beam propagating in a turbulent medium," Appl. Opt. 11,1399 - 406 (1972).
    [CrossRef] [PubMed]

2009

2008

2007

2006

2004

1998

1991

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

1990

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

1989

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gauss beams," Opt. Commun. 70, 361-364 (1989).
[CrossRef]

1985

Y. Baykal, M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. A. 2, 2124-2132 (1985).
[CrossRef]

1983

Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

1981

J. C. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. A. 71, 542-558 (1981).
[CrossRef]

1979

J. C. Leader, "Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence," J. Opt. Soc. A. 69, 73-84 (1979)
[CrossRef]

1972

Baykal, Y.

H. T. Eyyuboglu, Y. Baykal and C. J. Yang, "Complex degree of coherence for partially coherent general beams in atmospheric turbulence," J. Opt. Soc. Am. A 24, 2891 - 2901 (2007).
[CrossRef]

Y. Baykal, M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. A. 2, 2124-2132 (1985).
[CrossRef]

Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Boardman, A. D.

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

Casperson, L. W.

Conan, R.

Dogariu, A.

Eyyuboglu, H. T.

Gbur, G.

Ji, X. L.

Leader, J. C.

J. C. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. A. 71, 542-558 (1981).
[CrossRef]

J. C. Leader, "Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence," J. Opt. Soc. A. 69, 73-84 (1979)
[CrossRef]

Li, X. Q.

Lü, B. D.

Plonus, M. A.

Y. Baykal, M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. A. 2, 2124-2132 (1985).
[CrossRef]

Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Qian, X. M.

Rao, R. Z.

Schwartz, C.

Tovar, A. A.

Tyson, R. K.

Wang, S. J.

Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Wen, Y.

Wu, J.

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

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

Yang, A. L.

Yang, C. J.

Yura, H. T.

Zahid, M.

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gauss beams," Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Zhang, E. T.

Zubairy, M. S.

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gauss beams," Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

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

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

J. Opt. Soc. A.

J. C. Leader, "Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence," J. Opt. Soc. A. 69, 73-84 (1979)
[CrossRef]

J. C. Leader, "Beam intensity fluctuations in atmospheric turbulence," J. Opt. Soc. A. 71, 542-558 (1981).
[CrossRef]

Y. Baykal, M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. A. 2, 2124-2132 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. Zahid and M. S. Zubairy, "Directionality of partially coherent Bessel-Gauss beams," Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Opt. Express

Radio Sci.

Y. Baykal, M. A. Plonus and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Other

J. C Leader, "Atmospheric propagating of partially coherent radiation," J. Opt. Soc. A.  68, 175-185 (1978.).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media (SPIE Press, 1998)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st. (Cambridge University Press, New York, 1995)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

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Figures (5)

Fig. 1.
Fig. 1.

The spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in free space. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour

Fig. 2.
Fig. 2.

The spatial distributions and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C 2 n =10-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

Fig. 3.
Fig. 3.

The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=10.0m in turbulent atmosphere with C 2 n =10-15. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

Fig. 4.
Fig. 4.

The spatial distribution and contour of the energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.6 and β=0.8 at the plane z=10.0m in turbulent atmosphere with C 2 n =10-16. (a) β=0.8 (b) β=0.6

Fig. 5.
Fig. 5.

The spatial distribution and contour of the energy and energy flux of partially coherent ChG electromagnetic beam with spatial correlation parameters β=0.8 at the plane z=5.0 m in turbulent atmosphere with C 2 n =10-16. (a) energy spatial distribution and its contour (b) energy flux spatial distribution and its contour map

Equations (57)

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Eα(ρs,Zs)=12π(z=0)Eα(ρs0,0)zs(exp(ikRs)Rs)exp[ψα(ρs,ρs0,Zs)]d2ρs0,(α=x,y)'
Ez(ρs,zs)=12π(z=0)[Ex(ρs0,zs)xs(exp(ikRs)Rs)exp[ψxρsρs0zs)]
+ Ey (ρs0,0)ys(exp(ikRs)Rs)exp[ψy(ρs,ρs0,zs)]d2ρs0'
Rs=(xsxs0)2+(ysys0)2+zz2,
k=2πλ (wavenumber) ,
E=12 [(Ex*(ρ1,t)+Ex(ρ2,t+τ))i+(Ey*(ρ1,t)+Ey(ρ2,t+τ))j+(Ez*(ρ11,t,ω)+Ez(ρ2,t+τ))k] .
w=E·D2+B·H2(energydensity) ,
S=E×H , (energyfluxdensity),
<M*(r1,t+τ)N(r2,t)>=limT 12T TT M* (r1,t+τ)N(r2,t)dt,
<we(ρ1,ρ2,ω)>=ε04 Σα=x,y,z Wvv(ee) (ρ1,ρ2,ω),
Wvv(ab)(ρ1,ρ2,ω)=< Mv* (ρ1,ω)Nv(ρ2,ω),>,(a,b=e,h;M,N=E,H) ,
<we(ρ1,ρ2)>=ε04 0 Σv=x,y,z Wvv(ee) (ρ1,ρ2,ω) d ω ,
< wh (ρ1,ρ2,ω) >=1ω2μ02[(2z1y22y1z2)Wyz(ee)(ρ1,ρ2,ω)2y1y2Wzz(ee)(ρ1,ρ2,ω)2z1z2Wyy(ee)(ρ1,ρ2,ω) +
(2z1x22x1z2) Wxz(ee) (ρ1,ρ2,ω) 2z1z2 Wxx(ee) (ρ1,ρ2,ω)2x1x2Wzz(ee)(ρ1,ρ2,ω)+
(2y1x22x1y2)Wxy(ee)(ρ1,ρ2,ω)2x1x2Wyy(ee)(ρ1,ρ2,ω)2y1y2Wxx(ee)(ρ1,ρ2,ω),
< wh(ρ1,ρ2)>=μ04<wh(ρ1,ρ2,ω)>dω.
<s(ρ1,ρ2,ω)>=12 [(Wyz(eh)(ρ1,ρ2,ω)Wyz(he)(ρ1,ρ2,ω)i+(Wzx(eh)(ρ1,ρ2,ω),
Wzx(he) (ρ1,ρ2,ω)j+(Wxy(eh)(ρ1,ρ2,ω)Wxy(he)(ρ1,ρ2,ω))k]
<S(ρ1,ρ2)>=120+[(Wyz(eh)(ρ1,ρ2,ω)Wyz(he)(ρ1,ρ2,ω))i+(Wzx(eh)(ρ1,ρ2,ω)Wzx(he)(ρ1,ρ2,ω))j
+ (Wxy(eh)(ρ1,ρ2,ω)Wxy(he)(ρ1,ρ2,ω))k] d ω .
Wyz(eh)(ρ1,ρ2,ω)=iωμ0 [Wyy(ee)(ρ1,ρ2,ω)x2Wyx(ee)(ρ1,ρ2,ω)y2] ,
Wyz(he)(ρ1,ρ2,ω)=iωμ0[Wxz(ee)(ρ1,ρ2,ω)z1Wzz(ee)(ρ1,ρ2,ω)x1] ,
Wzx(eh) (ρ1,ρ2,ω)=iωμ0 [Wzz(ee)(ρ1,ρ2,ω)y2Wzy(ee)(ρ1,ρ2,ω)z2],
Wzx(he) (ρ1,ρ2,ω)=iωμ0[Wyx(ee)(ρ1,ρ2,ω)x1Wxx(ee)(ρ1,ρ2,ω)y1],
Wxy(eh) (ρ1,ρ2,ω)=iωμ0[Wxx(ee)(ρ1,ρ2,ω)z2Wxz(ee)(ρ1,ρ2,ω)x2] ,
Wxy(he) (ρ1,ρ2,ω)=iωμ0[Wzy(ee)(ρ1,ρ2,ω)y1Wzy(ee)(ρ1,ρ2,ω)z1],
Ex (ρ0)=exp(ρ02w02)cosh(Ω0x0)cosh(Ω0y0)'
Ey (ρ0)=0,
[Ŵ(ee)(ρ10,ρ20)]=[Wxx(ee)(ρ10,ρ20)0Wxx(ee)(ρ10,ρ20)000Wxz(ee)(ρ10,ρ20)0Wzz(ee)(ρ10,ρ20)]× ,
Wxx(ee)(ρ10,ρ20)=z1z24π2(z=0)[Wxx(ee)(ρ01,ρ02)](0)exp[ik(R2R1)]R13R23(ikR21),
×(1ikR1)<exp[ψx(ρ01,ρ10,z)+ψx(ρ02,ρ20,z)]>d2ρ01d2ρ02
Wzz(ee) (ρ10,ρ20)=14π2(z=0)(x1x01)(x2x02)[Wxx(ee)(ρ01,ρ02,0)](0)exp[ik(R2R1]R12R22(ikR21) ,
× (1ikR1)<exp[ψz(ρ01,ρ10,z)+ψz(ρ02,ρ20,z)]>d2ρ01d2ρ02
Wxz (ρ10,ρ20)=z4π2(z=0)(x2x02)Wxx(0)(ρ01,ρ02,0)exp[ik(R2R1)]R12R22(ikR21),
×(1ikR1)<exp[ψx(ρ01,ρ10,z)+ψz(ρ02,ρ20,z)]>d2ρ01d2ρ02
Wzx (ρ10,ρ20)=Wxz*(ρ10,ρ20).
< exp [ψα(ρ1,ρ10,z1)+ψb*(ρ2,ρ20,z2)] >
exp {1ρ02(z1,z2)[(ρ20ρ10)2+(ρ2ρ1)2+(ρ1ρ2)(ρ10ρ20)]} ,
ρ02(z1,z2)=(0.545Cn2k2)35(z1z2)310,
Rsρsxsxs0+ysys0ρs ,
ρs=xs2+ys2+zs2 ,
exp(αx2+cx)dx= πα exp (c24α) ,
Wxx(ee)(ρ1,ρ2)=π2z1z2f016AA1 {exp[h124A+(B1Ω0)24A1]+exp[h124A+(B1+Ω0)24A1]+exp[h224A+(B2Ω0)24A1]+exp[h224A+(B2+Ω0)24A1]} ,
× {exp[h324A+(B3Ω0)24A1]+exp[h324A+(B3+Ω0)24A1]+exp[h424A+(B4Ω0)24A1]+exp[h424A+(B4+Ω0)24A1]}
Wzx(ee)(ρ1,ρ2)=x1z2Wxx(ee)(ρ1,ρ2)+z2π2f032AA12[(B5Ω0)exp[h524A+(B5Ω0)24A1]+(B6Ω0)exp[h624A+(B6Ω0)24A1]+(B5Ω0)exp[h524A+(B5Ω0)24A1]+
(B5Ω0)exp[[h624A+(B6+Ω0)24A1]]{exp[h324A+(B3Ω0)24A1]+exp[h324A+(B3+Ω0)24A1]+exp[h424A+(B4Ω0)24A1]+exp[h424A+(B4Ω0)24A1]},
Wzz(ee)(ρ1,ρ2)=x1x2wxx(ee)(ρ1,ρ2)π2f032AA12{x1[(B1Ω0)exp[h124A+(B1Ω0)24A1]+(B2Ω0)exp[h224A+(B2Ω0)24A1]+(B1Ω0)
×exp[h124A+(B1+Ω0)24A1] + (B2+Ω0)exp[h224A+(B2Ω0)24A1]]+x2 [(B5Ω0)exp[h524A+(B5Ω0)24A1]+(B6Ω0)×
exp [h624A+(B6+Ω0)24A1] + ( B5 + Ω0 ) exp [ h524A+(B6Ω0)24A1 ] +(B6Ω0)exp[h624A+(B6Ω0)24A1]][1A(B1Ω0)×
exp [h124A+(B1Ω0)24A1] + ( B1 + Ω0 ) exp [ h124A+(B1Ω0)24A1 ] ) + h12A ((B2Ω0)exp[h224A+(B2Ω0)24A1]+(B2Ω0)×
exp [h224A+(B2+Ω0)24A1] ) ]B4AA1[2A1+(B2Ω0)2)exp[h224A+(B2+Ω0)24A1]+(2A1+(B2+Ω0)2)exp[h224A+
(B2+Ω0)24A1] + (2A1+(B1Ω0)2)exp[h124A+(B1Ω0)24A1] + (2A1+(B1+Ω1)2) exp [h124A+(B1+Ω0)24A1]] } { exp [h324A+
(B3Ω0)24A1] + exp [h324A+(B3+Ω0)24A1]+exp[h424A+(B4Ω0)24A1]+exp[h424A+(B4+Ω0)24A1]}
wzx(ee)(ρ1,ρ2,z)=[wzx(ee)(ρ1,ρ2,z)]*,
A=1w02+1σ02,h1=c3Ω0,h2=c3+Ω0,h3=c4Ω0,h4=c4+Ω0,h5=c1Ω0,h6=c1+Ω0,c1=ikx2ρ2 ,
c2=iky2ρ2,c3=ikx1ρ2,c4=iky1ρ2,B=2σ,A1=AB24A,B1=c1+Bh12A,σ=p0σ0p02+σ02,B2=c1+Bh22A,
B3=c2+Bh32A,B4=c2+Bh42A,B5=c3+Bh52A,B6=c3Bh62A,f0=exp[ik(ρ2ρ1)]4π2ρ13ρ23(ikρ21)(1ikρ1),

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