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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 20746–20758
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Spatially partially coherent beam parameter optimization for free space optical communications

Deva K. Borah and David G. Voelz  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 20746-20758 (2010)
http://dx.doi.org/10.1364/OE.18.020746


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Abstract

The problem of coherence length optimization in a spatially partially coherent beam for free space optical communication is investigated. The weak turbulence regime is considered. An expression for the scintillation index in a series form is derived and conditions for obtaining improvement in outage probability through optimization in the coherence length of the beam are described. A numerical test for confirming performance improvement due to coherence length optimization is proposed. The effects of different parameters, including the phase front radius of curvature, transmission distance, wavelength and beamwidth are studied. The results show that, for smaller distances and larger beamwidths, improvements in outage probability of several orders of magnitude can be achieved by using partially coherent beams.

© 2010 Optical Society of America

1. Introduction

It is known that the use of a spatially partially coherent beam (PCB) can improve the performance of free space laser communications through atmospheric turbulence [1

1. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43(2), 330–341 (2004). [CrossRef]

], [2

2. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef]

]. One of the reasons for this improvement is a reduction in coherent interference, which lowers the intensity fluctuation (scintillation) at the receiver. The reduction in intensity fluctuations of Gaussian beams with decreasing source spatial coherence was observed and reported as early as 1980s [3

3. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk 54, 1054–1059 (1983).

]. However, the divergence of a PCB is greater than an analogous coherent beam, causing a reduction of the signal level that degrades the performance of an optical communication system. Therefore, for designing a laser communication system, an important issue is the trade-off between reduction in scintillation and reduction in the mean signal intensity at the receiver and how this trade-off can be achieved through optimization of the PCB parameters.

There is also a large body of literature that focuses on the SI of different beams propagating through the atmospheric turbulence channels. More recently, the works in [6

6. H.T. Eyyuboglu, Y. Baykal, E. Sermtlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009). [CrossRef]

] and [7

7. H. T. Eyyuboglu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Applied Physics B - Laser and Optics 98(4), 857–863 (2010). [CrossRef]

] consider Bessel-Gaussian and Laguerre Gaussian beams and demonstrate that lower scintillations at on-axis and off-axis positions can be obtained using certain beam orders. Other recent works include multibeam investigations of scintillation [8

8. Y. Baykal, H. T. Eyyuboglu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Applied Optics 48(10), 1943–1954 (2009). [CrossRef] [PubMed]

], and propagation properties of a stochastic GSM beam [9

9. S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Optics Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

]. However, since the performance of a communication system, in terms of bit error rates or outage probabilities, typically depends nonlinearly on beam parameters, including the SI, the exact trade-offs for these beams for communication purposes are not obvious.

The paper is organized as follows. Section 2 presents the GSM beam model. In Section 3, the communication performance metric and the simplified cost function are described. Section 4 presents the derivation of the SI for PCBs in a series form. The coherence length optimization condition is discussed in detail, and a numerical test to confirm performance gains due to PCBs is given. This section also discusses the effects of various parameters, such as the phase front radius of curvature and the beamwidth, on coherence length optimization. Finally, Section 5 concludes our study and outlines directions for future research.

2. Beam model

We consider a Gaussian Schell model (GSM) beam [10

10. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

], [11

11. S. R. Seshadri, “Partially coherent Gaussian schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16(6), 1373–1380 (1999). [CrossRef]

]. After propagation through the atmospheric turbulence channel over a distance z, the mean intensity profile, Ī(ρ), on the receiver plane is modeled as [12

12. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

],

I¯(ρ)=KnW02W2exp(2ρ2W2)
(1)

where ρ is the radial distance from the mean beam center on the transverse plane, Kn=Wr2/W02 is a normalization constant so that the total power is (1/2)πWr2 , Wr is the reference beam radius or beamwidth, W 0 is the effective beam radius at the transmitter, W is the receiving beam size given by

W=W0[r02+(ξs+2W02ρ02)z02]1/2,

r 0 = 1 − z/F 0 is a focusing parameter, F 0 is the phase-front radius of curvature at the transmitter, z0=2z/(kW02) is normalized distance, k = 2π/λ is the wave number, λ is the wavelength, ξs=1+2W02/lc2 is the source coherence parameter, lc is the spatial coherence length, ρ0=(0.55Cn2k2z)3/5 , and Cn2 is the index of refraction structure parameter of the atmosphere expressed in units of m−2/3. Noting that the Rytov variance is σR2=1.23Cn2k7/6z11/6 , the SI σI2 on or close to the beam center, including beam wander effects for an untracked beam, is given by (p. 274 in [12

12. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

])

σI2=4.42σR2ze5/6σpe2W02(r02+ξsz02)+3.86σR2{0.4[(1+2re)2+4ze2]5/12cos[56tan1(1+2re2ze)]1116ze5/6}
(2)

where

re=r0r02+ξsz02,ze=ξsz0r02+ξsz02

and

σpe2=ζ1(λz2W0)2(2W0Fp)5/3[1ζ2(Fp2Cr2W02+ζ3)1/6]

with ζ 1 = 0.54, ζ 2 = 8/9, and ζ 3 = 0.5 for a focused beam, while ζ 1 = 0.48, ζ 2 = 1 and ζ 3 = 1 for a collimated beam, and Fp=(0.16Cn2k2z)3/5 . Let us write σI2=σI,bw2+σI,nw2 , where σI,bw2=4.42σR2ze5/6σpe2W02(r02+ξsz02) is due to beam wander effects and essentially corresponds to the radial component of the SI evaluated at ρ = σpe, and σI,nw2 represents the remaining term in (2). Note that we consider the received signal at the beam center and, therefore, σI,bw2 does not contain ρ.

3. Performance metric

We consider the outage probability, Pout, which is the probability that the received signal-to-noise ratio (SNR) falls below a threshold SNR resulting in unacceptable error rates. For a given noise level, this occurs when the intensity I becomes less than a threshold intensity Ith. We will use this simplified approach in this work. Thus, we write

Pout=0Ithp(I)dI
(3)

where p(I) is the probability density function of I. Using the log-normal (LN) model for I [12

12. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

], we can write

Pout=0Ith1I2πσlnI2exp{[ln(I/I¯)+0.5σlnI2]22σlnI2}dI

where σlnI2=ln(1+σI2) . It can be easily shown that

Pout=Q(yth)

where

yth=lnIthlnI¯+0.5σlnI2σlnI,

and Q(x)=1/2πxexp(y2/2)dy is the Gaussian tail probability. For weak turbulence and small Pout, yth is a large negative number. Under these conditions, we need to minimize

ythlnIthlnI¯+0.5σlnI2σlnI1σlnIln(Ith/I¯)

For weak turbulence, σI2σlnI2 , and so we consider the following cost function for minimization,

φ(lc)=1σIln(IthI¯)
(4)

Note that for Ith < Ī, the value of the above cost function is a negative number, and hence zero is not the minimum value of the above cost function in general.

4. PCB Parameter selection

For certain beam configurations and turbulence scenarios, the optimal beam is simply the coherent beam. In other words, lc = ∞ will minimize (4), and PCB is not even necessary. However, this can be ascertained only after performing a non-linear optimization on (4) with respect to lc, or evaluating the cost function (4) or outage probability (3) for various values of lc. This becomes more difficult when one studies the effects of combinations of other parameters, such as F 0 and W 0, over lc. We show in this section that this can be done without conducting extensive optimization. Toward that end, we first provide an alternative series expansion for the SI and develop interpretations.

4.1. Expressions for the Scintillation Index

Let us first consider the SI given by (2). This index depends on lc through trigonometric expressions. As lc increases or decreases, it is difficult to interpret from (2) whether σI2 will also increase or decrease. We define x=(1+2re)/(2ze)=(r02+ξsz02+2r0)/(2ξsz0) , and then show in the Appendix how σI,nw2 can be expressed in a series form. We obtain

σI2=4.42σR2ze5/6[σpe2W02(r02+ξsz02)+0.8733g(x)]
(5)

where

g(x)=0.7127(1+572x24554!64x4+43225144×66x6)11/16
(6)

for |x| < 1, while

g(x)=0.7127(x2)512{cos(5π12)[1+572(1x)245524×64(1x)4]±sin(5π12)[56(1x)3564(1x)3+]}1116
(7)

for the case |x| > 1, with the positive or negative sign selection before the sine function corresponding to the sign of x. Note that, in (5), the non-wander part is σI,nw2=3.86σR2ze5/6g(x) . When |x| is close to unity, the series converges slowly, and more terms in the series expansion are required for accurate representation.

4.2. Coherence length optimization

In order to obtain the optimum lc, we differentiate the cost function φ(lc) with respect to lc and set it to zero to obtain

dσI2dlc=(1Wr2)[I¯σI2ln(I/Ith)]32z2k2lc3
(8)

where we have used

dI¯dlc=(W0Wr)2I¯2z024W02lc3

Equation (8) shows that when lc is optimal, the derivative of the SI equals a target value, T(lc)=(1/Wr2)32I¯σI2z2/[ln(I¯/Ith)k2lc3] , represented by the right-hand-side (RHS) of (8). The SI derivative, dσI2/dlc , is given by

ddlcσI2=3.6833σR2ze1/6(dzedlc)[σpe2W02(r02+ξsz02)+0.8733g(x)]+4.42σR2ze5/6[4σpe2z02(r02+ξsz02)2lc3+0.8733g(x)]
(9)

where

g(x)=0.7127(1072x4×4554!64x3+6×43225144×66x5+)(dxdlc)
(10)

for |x| < 1, and

g(x)=0.7127[(56)(x2)712x{cos(5π12)[1+572(1x)245524×64(1x)4]±sin(5π12)[56(1x)3564(1x)3+]}+(x2)512{cos(5π12)[1072(1x)3+4×45524×64(1x)5]±sin(5π12)[56(1x)2+3×3564(1x)4]}](dxdlc)
(11)

for |x| > 1. In (9), we have to use

dzedlc=8zr02klc3(r02+ξsz02)2,
(12)

and

dxdlc=π(34zF0+z2F02)2λzlc(1+lc22W02)2
(13)

is required in (10) and (11). We now present the following proposition.

Proposition 1

For a sufficiently small value of Ith so that Ith < Ī, if the derivative ddlcσI2 is larger than or equal to the target value T(lc) at a high lc value, then an optimum finite lc exists.

Proof

First observe that the target value T(lc) increases rapidly as lc → 0, while T(lc) → 0 as lc → ∞. The SI derivative dσI2/dlc, on the other hand, approaches zero when lc → 0. This can be seen by observing that (12) and (13) both approach zero, x → 0.5z 0, and ze → 1/z 0 as lc → 0. If the SI derivative becomes larger than the target value for a large lc, then it implies that the SI derivative has crossed the target curve at least at one lc value, where the condition (8) is satisfied. This gives an optimum lc value.

The above propostion provides a useful test to guarantee optimization benefit from lc. All we need to do is to calculate the SI derivative and the target value at a large lc value, say at lc = 100W 0. If the scintillation derivative is larger than the target value, then finite optimized lc exists.

Since Ī decreases with a decrease in lc, it may not be possible to guarantee Ith < Ī for smaller values of lc when the outage probability is larger. In that case, T(lc) becomes negative when Ith > Ī. This occurs when lc becomes smaller than a certain minimum value lc 0, which can be obtained from (1) in the form

lc0=[k2W028z2{Wr2W02Ith[r02+(1+2W02ρ02)z02]}]1

In that case, one also needs to evaluate the SI derivative and T(lc) at an lc value close to but greater than lc 0, making sure that T(lc) is higher than the SI derivative at that point. Then Proposition 1 is used to verify if the SI derivative is larger than or equal to T(lc) at a large lc value, in which case an optimum finite lc is guaranteed.

Figure 1 shows outage probability versus coherence length in meters, obtained numerically from (3) for Cn2=1014m2/3 , W 0 = 0.05 m, Ith = 0.01, F 0 = ∞, and λ = 1.55μm at distances of 1000, 1500 and 2000 meters. The Rytov variance for these distances corresponds to 0.1991, 0.4187 and 0.7095 respectively. The reference beamwidth Wr is 0.025 m, which will be used in all our numerical examples. Observe that the optimal lc increases with z, and the benefit from optimizing lc decreases with increasing z. We show the corresponding SI derivative and target values given by (8) in Fig. 2. Note that the meeting points of the SI derivative and target values correspond to the optimal lc values. As z increases, both the SI derivative and target values increase but target values increase more than the SI derivative for any given increase in z. Observe that for low outage probability or sufficiently small Ith, the RHS of (8) is a positive quantity. Therefore, for optimal lc to exist, the derivative dσI2/dlc must be positive. Note that since (12) is negative, the first term in (9) is negative. Therefore, the second term in (9) must be positive and large for the SI derivative to cross the target values. Although it is difficult to make a general statement without considering all terms involved, roughly speaking, a large dx/dlc helps to create a situation where an optimal finite lc value exists. This implies a small z, a small λ, and a large W 0.

Fig. 1 Outage probability for Cn2=1014m2/3 , W 0 = 0.05 m, F 0 = ∞, and λ = 1.55μm at distances of 1000, 1500 and 2000 meters.
Fig. 2 The derivative of the scintillation index and target values for the same parameters given in Fig. 1

4.3. Effects of phase front radius of curvature

Differentiating the cost function with respect to F 0, and setting it to zero, we obtain the following condition to find optimal F 0

dσI2dF0=(W0Wr)24zσI2I¯ln(Ith/I¯)(1zF0)1F02
(14)

Note that from the physical behavior of focusing, for F 0 less than but close to z, the SI increases with an increase in F 0 as the beam wander effects begin to dominate significantly when F 0 approaches z. So, the left-hand-side (LHS) of (14) is a positive quantity in this region. The RHS is also a positive quantity in this region for low threshold values (Ith < Ī), and the optimality condition is satisfied in this region.

Fig. 3 Outage probability versus F 0 for Cn2=1014m2/3 , W 0 = 0.05 m and λ = 1μm.
Fig. 4 The SI derivative with respect to lc and target values for the parameters given in Fig. 3

4.4. Effects of beamwidth

The conditions for optimum beamwidth can be obtained by differentiating the cost function with respect to W 0 and setting it to zero. This produces

dσI2dW0=4σI2W0ln(Ith/I¯)[18z2I¯k2Wr2(1lc2+1ρ02+1W02)]
(15)

We show beamwidth effects in Fig 5 by computing (3) for four cases of (z, λ) pairs under Cn2=1014m2/3 , F 0 = ∞, lc = ∞, and Ith = 0.025. These cases correspond to (z, λ) values of (1500 m, 1 μm), (1500 m, 1.55 μm), (2000 m, 1 μm), and (2000 m, 1.55 μm). The optimized W 0 values for the cases are found to be 0.016 m, 0.02 m, 0.018 m, and 0.024 respectively. In Fig. 6, we plot the SI derivative with respect to W 0, and the target values given by the RHS of (15) for lc = 0.02 m, z = 1500 m, λ = 1μm, F 0 = ∞, and Ith = 0.025. The optimal value for W 0, obtained from calculating outage probability (3), is found to be 0.019 m. This agrees with Fig. 6, where the SI derivative and the target curve are found to meet around the optimal value. To gain insight on the effects of lc for a given value of W 0, consider two cases:

Fig. 5 Outage probability versus beamwidths for Cn2=1014m2/3 , collimated beam, and lc = ∞. Cases 1, 2, 3 and 4 refer to (z, λ) pairs of (1500 m, 1 μm), (1500 m, 1.55 μm), (2000 m, 1 μm), and (2000 m, 1.55 μm) respectively.
Fig. 6 SI derivatives and target values with respect to beamwidths for z = 1500 m, λ = 1μm, F 0 = ∞, Ith = 0.025. The optimal value for W 0 is found to be 0.019 m.

Case 1: Suppose

W044z2ξsk2
(16)

That is,

W024z2k2lc2+16z4k4lc4+4z2k2

In this case, using z0=2z/(kW02) , we can write

ze=2z/kW02ξs(1+4z2ξsk2W04)2zξskW02(14z2ξsk2W04)
(17)

and also

x=3kW024zξs(1+4z2ξs3k2W04)
(18)

so that

ze32x

To keep the discussion simple, consider a collimated beam (F 0 = ∞), and ignore beam wander effects. The SI can then be expressed as

σI23.86σR2ze5/6g(x)
(19)

Therefore, the SI becomes a function of x. Now, from (18), we can write

x=3k4z(1/W02+2/lc2)
(20)

so W 0 and lc affect SI via x in nearly similar ways. As lc increases, x also increases. From (6) and (19), we observe that when |x| < 1, the SI will also increase if |x| is close to unity, otherwise the SI will decrease. If |x| > 1, then we can see from (7) and (19) that the SI will increase with increasing |x|.

W044z2ξsk2
(21)

That is,

W024z2k2lc2+16z4k4lc4+4z2k2

For this condition, we can write

zekW022z(1k2W044ξsz2)
(22)

and

x=zkW02(1+3k2W044z2ξs)
(23)

We get

ze12xkW022z

It appears that in this region, x is not affected much by lc. Therefore, SI is relatively unaffected by lc. However, the mean intensity increases with lc. Therefore, not much benefit can be obtained by optimizing lc for W 0 selected in this region, and coherent beam tends to perform better.

To understand the benefits of optimizing lc for different values of W 0, we consider the SI derivative with respect to lc and the target values given by (8). Figure 7 shows the derivative and the target values for different values of W 0 under Cn2=1014m2/3 , λ = 1μm, F 0 = ∞ evaluated at a large value of lc = 0.5 m. For the z = 1500 m case, we see from Fig. 5 that the optimal W 0 is 0.016. For the same parameters, the target values just meet the derivative curve in Fig. 7 at about W 0 = 0.016. Since the target curve does not exceed the SI derivative curve for this W 0, performance improvement by optimizing lc is not guaranteed. In fact, for all the optimal W 0 values observed in Fig. 5, we could not obtain further improvement in performance by optimizing lc, i.e., lc = ∞ gives best performance. Observe from Fig. 7 that for very small values of W 0, benefits from optimizing lc is not guaranteed. For larger values of W 0, there is guaranteed benefit from optimizing lc. This agrees with our theory observations that for small W 0, the SI becomes nearly independent of lc with no potential benefits from small lc values. Also, for longer distance, the minimum W 0 required for benefit from lc optimization increases.

Fig. 7 SI derivatives with respect to lc and target values to determine benefits of PCBs at various beamwidths for the parameters λ = 1μm and F 0 = ∞.

5. Conclusion

Partially coherent beams can provide significant performance improvement in free space optical communications, but such improvements may not be realized under all beam configurations and channel conditions. In order to analyze the effects of a PCB, a more intuitive series expression for the scintillation index of atmospheric turbulence is derived. Using this expression, a numerical test for confirming the performance improvement due to coherence length optimization is developed. The effects of different parameters, including the phase front radius of curvature and beamwidth are studied. The results show improvements in outage probability by several orders of magnitude from the use of PCBs for smaller distances, lower wavelengths and larger beamwidths. An important direction for future research is to study the characteristics of coherence length optimization when the receiver employs aperture averaging. Since aperture averaging also reduces scintillation, we expect further overall performance improvement with the relative improvement due to the PCB requiring additional investigation.

Appendix

Consider the second part of the SI (2). Define x = (1 + 2re)/2ze and let |x| < 1. Using

tan1x=12jln(1+jx1jx)

we can write

cos{56tan1(1+2re2ze)}=12[exp(j56tan1x)+exp(j56tan1x)]=12{exp[512ln(1+jx1jx)]+exp[512ln(1+jx1jx)]}=(1+jx)5/6+(1jx)5/62(1+x2)5/12
(24)

Using the expansion (1 + y)q = 1 + qy + (1/2!)q(q − 1)y2 + ⋯ in (24), and recalling (2), we get the following expression after a few simplification steps,

σI,nw2=3.86σR2ze5/6[0.7127(1+572x24554!64x4+43225144×66x6)11/16]
(25)

When |x| > 1, we substitute y = 1/x and proceed as follows. Observe that

tan1x=±π2+12jln(1jy1+jy)

where the positive or the negative sign is taken according to the sign of x. Next, using

cos[56tan1(1+2re2ze)]=12[e±j5π12(1jy1+jy)5/12+ej5π12(1+jy1jy)5/12]
(26)

and performing several steps for simplification, we obtain

σI,nw2=3.86σR2ze5/6[0.7127(x2)5/12{cos(5π12)[1+572(1x)245524×64(1x)4]±sin(5π12)[56(1x)3564(1x)3+]}1116]
(27)

Acknowledgments

The authors would like to thank the Air Force Office of Scientific Research (AFOSR) for providing funding support to conduct this research under the Transformational Communications Advanced Technology Study (TCATS) program.

References and links

1.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43(2), 330–341 (2004). [CrossRef]

2.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef]

3.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk 54, 1054–1059 (1983).

4.

T. J. Schulz, “Optimal beam for propagation through random media,” Opt. Lett. 30(10), 1093–1095 (2005). [CrossRef] [PubMed]

5.

D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through turbulence,” Opt. Eng. 48(3), (2009). [CrossRef]

6.

H.T. Eyyuboglu, Y. Baykal, E. Sermtlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26(2), 387–394 (2009). [CrossRef]

7.

H. T. Eyyuboglu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Applied Physics B - Laser and Optics 98(4), 857–863 (2010). [CrossRef]

8.

Y. Baykal, H. T. Eyyuboglu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Applied Optics 48(10), 1943–1954 (2009). [CrossRef] [PubMed]

9.

S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Optics Express 18(12), 12587–12598 (2010). [CrossRef] [PubMed]

10.

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9(5), 796–803 (1992). [CrossRef]

11.

S. R. Seshadri, “Partially coherent Gaussian schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16(6), 1373–1380 (1999). [CrossRef]

12.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of Photo-Optical Instrumentation Engineers, 2005. [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.7060) Coherence and statistical optics : Turbulence
(060.2605) Fiber optics and optical communications : Free-space optical communication
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: July 28, 2010
Revised Manuscript: August 24, 2010
Manuscript Accepted: September 3, 2010
Published: September 15, 2010

Citation
Deva K. Borah and David G. Voelz, "Spatially partially coherent beam parameter optimization for free space optical communications," Opt. Express 18, 20746-20758 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20746


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References

  1. O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. 43(2), 330-341 (2004). [CrossRef]
  2. J. C. Ricklin, and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19(9), 1794-1802 (2002). [CrossRef]
  3. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk 54, 1054-1059 (1983).
  4. T. J. Schulz, "Optimal beam for propagation through random media," Opt. Lett. 30(10), 1093-1095 (2005). [CrossRef] [PubMed]
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