Page 1

Spatially partially coherent beam

parameter optimization for free space

optical communications

Deva K. Borah and David G. Voelz

Klipsch School of Electrical and Computer Engineering, New Mexico State University,

Las Cruces, NM 88003, USA

dborah@nmsu.edu

Abstract:

partially coherent beam for free space optical communication is investi-

gated. 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 perfor-

mance 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.

The problem of coherence length optimization in a spatially

© 2010 Optical Society of America

OCIS codes: (030.7060) Turbulence; (200.2605) Free-space optical communication;

(140.3295) Laser beam characterization; (030.1640) Coherence.

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).

2. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: impli-

cations for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).

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).

5. D. G. Voelz and X. Xiao, “Metric for optimizing spatially partially coherent beams for propagation through

turbulence,” Opt. Eng. 48(3), (2009).

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).

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).

8. Y. Baykal, H. T. Eyyuboglu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbu-

lence,” Applied Optics 48(10), 1943-1954 (2009).

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).

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).

11. S. R. Seshadri, “Partially coherent Gaussian schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16(6),

1373–1380 (1999).

12. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd Ed., The Society of

Photo-Optical Instrumentation Engineers, 2005.

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1.Introduction

It is known that the use of a spatially partially coherent beam (PCB) can improve the perfor-

mance of free space laser communications through atmospheric turbulence [1], [2]. One of the

reasons for this improvement is a reduction in coherent interference, which lowers the inten-

sity 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]. 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 sys-

tem. 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.

In previous work [4], Schulz showed that a coherent beam maximizes the expected intensity

whereas a PCB minimizes the scintillation index (SI). He provided an optimizing criterion for

the modes of the PCB for minimizing the SI. Voelz and Xiao described a heuristic metric for

determining near-optimal link performance that incorporates both the reduction in scintillation

and the increase in beam spread [5]. Their work considered a collimated Gaussian Schell-model

(GSM) beam propagating through turbulence, and investigated optimization of the transverse

coherence length (lc) as a function of beam size, wavelength, turbulence strength, and propa-

gation distance under the heuristic metric criterion. For a more detailed discussion on PCBs,

please see the references listed in [1], [5].

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] and [7] 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], and propagation properties of a stochastic

GSM beam [9]. 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.

Our work in this paper focuses on scalar PCBs for optical communications. Although PCBs

generally lower the SI, this alone does not guarantee performance improvement in communi-

cation systems. In fact, in certain communications scenarios, the use of a PCB can degrade

performance, and a coherent beam is rather the optimal beam configuration. The choice of a

PCB over a coherent beam is influenced by several parameters including wavelength, phase

front radius of curvature, and beamwidth. To our best knowledge, studies to date have not pro-

vided an analytical framework that leads to simple intuitive conclusions regarding the effects

of these parameters on coherence length optimization of PCBs. In general, previous work in-

volves extensive evaluations of a cost function or a non-linear optimization to merely confirm

the potential benefits of using a PCB for a given parameter set. Detailed interpretation of PCB

performance results is generally difficult using the available mathematical expressions.

In our work, the problem of optimizing the transverse coherence length of a PCB for min-

imal outage probability in a communications link is considered. This optimization approach

and metric are more specific to the optical communications problem than have been considered

previously. The study concerns the GSM beam and performance optimization as a function of

various link parameters. We are not aware of other work in the literature that considers op-

timization of beam parameters for PCBs except the studies reported in [4], [5]. A significant

extension of previous work is the inclusion of beam focus (phase front radius of curvature) and

beamwidth as study parameters. A numerical test is proposed for confirming possible improve-

ment with a PCB. We demonstrate that large initial beam size, small propagation distance and

short wavelength are favorable conditions for performance enhancement using a PCB.

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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], [11]. After propagation through the at-

mospheric turbulence channel over a distance z, the mean intensity profile,¯I(ρ), on the receiver

plane is modeled as [12],

¯I(ρ) = KnW2

W2exp

where ρ is the radial distance from the mean beam center on the transverse plane, Kn=W2

is a normalization constant so that the total power is (1/2)πW2

or beamwidth, W0is the effective beam radius at the transmitter, W is the receiving beam size

given by

?

r0= 1−z/F0is a focusing parameter, F0is the phase-front radius of curvature at the trans-

mitter, z0= 2z/(kW2

length, ξs= 1+2W2

ρ0= (0.55C2

expressed in units of m−2/3. Noting that the Rytov variance is σ2

on or close to the beam center, including beam wander effects for an untracked beam, is given

by (p. 274 in [12])

0

?

−2ρ2

W2

?

(1)

r/W2

0

r,Wris the reference beam radius

W =W0

r2

0+

?

ξs+2W2

0

ρ2

0

?

z2

0

?1/2

,

0) is normalized distance, k = 2π/λ is the wave number, λ is the wave-

0/l2

nk2z)−3/5, and C2

cis the source coherence parameter, lcis the spatial coherence length,

nis the index of refraction structure parameter of the atmosphere

R= 1.23C2

nk7/6z11/6, the SI σ2

I

σ2

I= 4.42σ2

Rz5/6

e

σ2

0+ξsz2

pe

W2

0(r2

0)+3.86σ2

R

?

0.4

?

(1+2re)2+4z2

e

?5/12cos

?5

6tan−1?1+2re

2ze

??

(2)

−11

16z5/6

e

?

where

re=

r0

r2

0+ξsz2

0

,

ze=

ξsz0

0+ξsz2

r2

0

and

σ2

pe= ζ1

?λz

2W0

?2?2W0

Fp

?5/3?

1−ζ2

?F2

p

C2

rW2

0

+ζ3

?−1/6?

with ζ1= 0.54, ζ2= 8/9, and ζ3= 0.5 for a focused beam, while ζ1= 0.48, ζ2= 1 and

ζ3= 1 for a collimated beam, and Fp= (0.16C2

where σ2

e

W2

to the radial component of the SI evaluated at ρ = σpe, and σ2

in (2). Note that we consider the received signal at the beam center and, therefore, σ2

not contain ρ.

nk2z)−3/5. Let us write σ2

I= σ2

I,bw+σ2

I,nw,

I,bw=4.42σ2

Rz5/6

σ2

0+ξsz2

pe

0(r2

0)is due to beam wander effects and essentially corresponds

I,nwrepresents the remaining term

I,bwdoes

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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=

?Ith

0

p(I)dI

(3)

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

we can write

?Ith

I

?

where σ2

Pout=

0

1

2πσ2

lnI

exp

?

−[ln(I/¯I)+0.5σ2

lnI]2

2σ2

lnI

?

dI

lnI= ln(1+σ2

I). It can be easily shown that

Pout= Q(−yth)

where

yth=lnIth−ln¯I+0.5σ2

lnI

σlnI

,

and Q(x) = 1/√2π?∞

xexp(−y2/2)dy is the Gaussian tail probability. For weak turbulence and

small Pout, ythis a large negative number. Under these conditions, we need to minimize

yth≈lnIth−ln¯I+0.5σ2

lnI

σlnI

≈

1

σlnI

ln(Ith/¯I)

For weak turbulence, σ2

tion,

I≈ σ2

lnI, and so we consider the following cost function for minimiza-

φ(lc) =1

σIln

?Ith

¯I

?

(4)

Note that for Ith<¯I, 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 co-

herent 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 be-

comes more difficult when one studies the effects of combinations of other parameters, such as

F0and W0, 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 lcthrough trigonometric ex-

pressions. As lcincreases or decreases, it is difficult to interpret from (2) whether σ2

Iwill also

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increase or decrease. We define x = (1+2re)/(2ze) = (r2

in the Appendix how σ2

0+ξsz2

0+2r0)/(2ξsz0), and then show

I,nwcan be expressed in a series form. We obtain

σ2

I= 4.42σ2

Rz5/6

e

?

σ2

0+ξsz2

pe

W2

0(r2

0)+0.8733g(x)

?

(5)

where

g(x) = 0.7127

?

1+5

72x2−455

4!64x4+

43225

144×66x6−···

?

−11/16(6)

for |x| < 1, while

g(x)=

0.7127(x2)

5

12

?

6

cos

?5π

?

12

−35

??

64

1+5

72

?1

x

?2−

??

455

24×64

−11

16

?1

x

?4−···

?

±sin

?5π

12

??5

?1

x

?1

x

?3+···

(7)

for the case |x| > 1, with the positive or negative sign selection before the sine function cor-

responding to the sign of x. Note that, in (5), the non-wander part is σ2

When |x| is close to unity, the series converges slowly, and more terms in the series expansion

are required for accurate representation.

I,nw= 3.86σ2

Rz5/6

e g(x).

4.2.

In order to obtain the optimum lc, we differentiate the cost function φ(lc) with respect to lcand

set it to zero to obtain

dσ2

I

dlc

W2

r

where we have used

d¯I

dlc

Wr

Equation (8) shows that when lc is optimal, the derivative of the SI equals a target value,

T(lc) = (1/W2

SI derivative, dσ2

Coherence length optimization

=

?1

??

¯Iσ2

I

ln(¯I/Ith)

?32z2

4W2

0

l3

c

k2l3

c

(8)

=

?W0

?2¯I2z2

0

r)32¯Iσ2

I/dlc, is given by

Iz2/[ln(¯I/Ith)k2l3

c], represented by the right-hand-side (RHS) of (8). The

d

dlcσ2

I

=

3.6833σ2

Rz−1/6

e

?dze

dlc

4σ2

(r2

??

pez2

σ2

0+ξsz2

pe

W2

0(r2

0)+0.8733g(x)

?

+4.42σ2

Rz5/6

e

?

0

0)2l3

0+ξsz2

c

+0.8733g′(x)

?

(9)

where

g′(x) = 0.7127

?10

72x−4×455

4!64

x3+6×43225

144×66x5+···

??dx

dlc

?

(10)

for |x| < 1, and

g′(x) = 0.7127

??5

?

?5π

6

?

64

??

(x2)−7

12x

?

cos

?5π

??

64

12

??

1+5

72

?1

?5π

????dx

x

?2−

12

455

24×64

−10

72

?

?1

x

x

?4−···

?3+4×455

?

±sin

?1

?5π

?5−···

12

?

?5

±sin

6

?1

x

−35

?1

−5

x

?3+···

6

x

+(x2)

5

12

?

?4−···

cos

???1

24×64

x

?

12

?1

?2+3×35

?1

x dlc

(11)

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

dze

dlc

= −

8zr2

0+ξsz2

0

kl3

c(r2

0)2,

(12)

and

dx

dlc

=

π

?

3−4z

F0+z2

2λz

F2

0

?

lc

?

1+

l2c

2W2

0

?2

(13)

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

Proposition 1: For a sufficiently small value of Ithso that Ith<¯I, if the derivative

than or equal to the target value T(lc) at a high lcvalue, then an optimum finite lcexists.

Proof: First observe that the target value T(lc) increases rapidly as lc→ 0, while T(lc) → 0

as lc→ ∞. The SI derivative dσ2

can be seen by observing that (12) and (13) both approach zero, x → 0.5z0, and ze→ 1/z0as

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 lcvalue, where the condition

(8) is satisfied. This gives an optimum lcvalue.

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 lcvalue, say at

lc= 100W0. If the scintillation derivative is larger than the target value, then finite optimized lc

exists.

Since¯I decreases with a decrease in lc, it may not be possible to guarantee Ith<¯I for smaller

values of lcwhen the outage probability is larger. In that case, T(lc) becomes negative when

Ith>¯I. This occurs when lcbecomes smaller than a certain minimum value lc0, which can be

obtained from (1) in the form

d

dlcσ2

Iis larger

I/dlc, on the other hand, approaches zero when lc→ 0. This

lc0=

??k2W2

0

8z2

?W2

r

W2

0Ith

−

?

r2

0+

?

1+2W2

0

ρ2

0

?

z2

0

???−1

In that case, one also needs to evaluate the SI derivative and T(lc) at an lcvalue close to but

greater than lc0, 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 lcis guaranteed.

Figure 1 shows outage probability versus coherence length in meters, obtained numerically

from (3) forC2

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 Wris 0.025 m, which will be used

in all our numerical examples. Observe that the optimal lcincreases with z, and the benefit

from optimizing lcdecreases 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 lcvalues. 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 lcto exist, the derivative dσ2

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 lcvalue exists. This implies a small z, a small

λ, and a largeW0.

n= 10−14m−2/3,W0= 0.05 m, Ith= 0.01, F0= ∞, and λ = 1.55µm at distances

I/dlcmust be positive. Note that

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00.02 0.040.06 0.080.10.12 0.140.160.18 0.2

10

−35

10

−30

10

−25

10

−20

10

−15

10

−10

10

−5

10

0

Coherence length

Outage probability

z=1000 m

z=1500 m

z=2000 m

Fig. 1. Outage probability for C2

at distances of 1000, 1500 and 2000 meters.

n= 10−14m−2/3, W0= 0.05 m, F0= ∞, and λ = 1.55µm

0 0.02 0.04 0.060.08 0.10.12 0.140.16 0.18 0.2

10

−3

10

−2

10

−1

10

0

10

1

10

2

Coherence length

Scintillation index derivative and target values

Derivative at z=1000 m

Target values at z=1000 m

Derivative at z=1500 m

Target values at z=1500 m

Derivative at z=2000 m

Target values at z=2000 m

Fig. 2. The derivative of the scintillation index and target values for the same parameters

given in Fig. 1

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0 5001000 1500

F0 (m)

2000 25003000

10

−10

10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Outage probability

z=1000 m

z=1250 m

z=1500 m

Fig. 3. Outage probability versus F0for C2

n= 10−14m−2/3,W0= 0.05 m and λ = 1µm.

4.3. Effects of phase front radius of curvature

Differentiating the cost function with respect to F0, and setting it to zero, we obtain the follow-

ing condition to find optimal F0

dσ2

dF0

I

=

?W0

Wr

?2 4zσ2

I¯I

ln(Ith/¯I)

?

1−z

F0

?1

F2

0

(14)

Note that from the physical behavior of focusing, for F0less than but close to z, the SI

increases with an increase in F0as the beam wander effects begin to dominate significantly

when F0approaches 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<¯I), and the

optimality condition is satisfied in this region.

Figure 3 shows outage probabilities versus F0for λ = 1µm, C2

m, lc= ∞, and Ith= 0.1. A larger Iththan the previous figures is selected so that Pout does

not become too low. The optimal F0values for the z values of 1000 m, 1250 m, and 1500 m

are 860 m, 1030 m and 1180 m respectively. These values are less than z. A natural question

with respect to this set of results would be: will there be further performance improvement by

optimizing over lc? To answer this question, we can run optimization over lcfor each possible

value of F0, requiring significant computations. Fortunately, we can simply use Proposition 1,

and produce the SI derivative and the target values given by (8) for different values of F0. The

results so obtained are shown in Fig. 4. We observe that the SI derivative values are larger

than the target values for all values of F0. This implies that further improvement in Poutcan be

achieved using lcoptimization for each of these F0values.

n= 10−14m−2/3, W0= 0.05

4.4. Effects of beamwidth

The conditions for optimum beamwidth can be obtained by differentiating the cost function

with respect toW0and setting it to zero. This produces

dσ2

dW0

I

=

4σ2

I

W0ln(Ith/¯I)

?

1−8z2¯I

k2W2

r

?1

l2

c

+1

ρ2

0

+

1

W2

0

??

(15)

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0 50010001500 20002500

F0 (m)

3000 350040004500 5000

10

−7

10

−6

10

−5

10

−4

Scintillation index derivative and target values

Derivative at z=1000 m

Target values at z=1000 m

Derivative at z=1250 m

Target values at z=1250 m

Derivative at z=1500 m

Target values at z=1500 m

Fig. 4. The SI derivative with respect to lcand target values for the parameters given in

Fig. 3

0 0.02 0.040.060.08 0.10.12

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10

0

Beamwidth (m)

Outage probability

Case 1

Case 2

Case 3

Case 4

Fig. 5. Outage probability versus beamwidths forC2

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.

n= 10−14m−2/3, collimated beam, and

We show beamwidth effects in Fig 5 by computing (3) for four cases of (z,λ) pairs under

C2

(1500 m, 1 µm), (1500 m, 1.55 µm), (2000 m, 1 µm), and (2000 m, 1.55 µm). The optimized

W0values 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 W0, and the target values given by the RHS of

(15) for lc= 0.02 m, z = 1500 m, λ = 1µm, F0= ∞, and Ith= 0.025. The optimal value for

W0, 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 lcfor a given value ofW0, consider two cases:

n= 10−14m−2/3, F0= ∞, lc= ∞, and Ith= 0.025. These cases correspond to (z,λ) values of

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0 0.020.04 0.060.08 0.10.12

−20

−15

−10

−5

0

5

10

15

20

25

30

Beamwidth (m)

Scintillation index derivative and target values

Derivative

Target values

Fig. 6. SI derivatives and target values with respect to beamwidths for z = 1500 m, λ =

1µm, F0= ∞, Ith= 0.025. The optimal value forW0is found to be 0.019 m.

Case 1: Suppose

W4

0≫4z2ξs

k2

(16)

That is,

W2

0≫4z2

k2l2

c

+

?

16z4

k4l4

c

+4z2

k2

In this case, using z0= 2z/(kW2

0), we can write

ze=

2z/k

1+4z2ξs

W2

ξs

0

?

k2W4

0

? ≈2zξs

kW2

0

?

1−4z2ξs

k2W4

0

?

(17)

and also

x =3kW2

0

4zξs

?

1+4z2ξs

3k2W4

0

?

(18)

so that

ze≈3

2x

To keep the discussion simple, consider a collimated beam (F0= ∞), and ignore beam wander

effects. The SI can then be expressed as

σ2

I≈ 3.86σ2

Rz5/6

e g(x)

(19)

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

x =

3k

0+2/l2

4z(1/W2

c)

(20)

so W0and lcaffect SI via x in nearly similar ways. As lcincreases, x also increases. From

(6) and (19), we observe that when |x| < 1, the SI will also increase if |x| is close to unity,

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otherwise the SI will decrease. If |x| > 1, then we can see from (7) and (19) that the SI will

increase with increasing |x|.

Case 2: We next consider the case

W4

0≪4z2ξs

k2

(21)

That is,

W2

0≪4z2

k2l2

c

+

?

16z4

k4l4

c

+4z2

k2

For this condition, we can write

ze≈kW2

0

2z

?

1−k2W4

4ξsz2

0

?

(22)

and

x =

z

kW2

0

?

1+3k2W4

4z2ξs

0

?

(23)

We get

ze≈1

2x≈kW2

0

2z

It appears that in this region, x is not affected much by lc. Therefore, SI is relatively unaf-

fected by lc. However, the mean intensity increases with lc. Therefore, not much benefit can be

obtained by optimizing lcfor W0selected in this region, and coherent beam tends to perform

better.

To understand the benefits of optimizing lcfor different values of W0, we consider the SI

derivative with respect to lcand the target values given by (8). Figure 7 shows the derivative

and the target values for different values of W0under C2

evaluated at a large value of lc= 0.5 m. For the z = 1500 m case, we see from Fig. 5 that the

optimal W0is 0.016. For the same parameters, the target values just meet the derivative curve

in Fig. 7 at about W0= 0.016. Since the target curve does not exceed the SI derivative curve

for this W0, performance improvement by optimizing lcis not guaranteed. In fact, for all the

optimalW0values 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 W0, benefits from optimizing lcis not guaranteed. For larger values of W0, there is

guaranteed benefit from optimizing lc. This agrees with our theory observations that for small

W0, the SI becomes nearly independent of lcwith no potential benefits from small lcvalues.

Also, for longer distance, the minimumW0required for benefit from lcoptimization increases.

n= 10−14m−2/3, λ = 1µm, F0= ∞

5.Conclusion

Partially coherent beams can provide significant performance improvement in free space opti-

cal 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 ex-

pression for the scintillation index of atmospheric turbulence is derived. Using this expression,

a numerical test for confirming the performance improvement due to coherence length opti-

mization 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

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0 0.020.04 0.060.08 0.10.12

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Beamwidth (m)

Scintillation index derivative and target values

Derivative at z=1000 m

Target values at z=1000 m

Derivative at z=1500 m

Target values at z=1500 m

Fig. 7. SI derivatives with respect to lcand target values to determine benefits of PCBs at

various beamwidths for the parameters λ = 1µm and F0= ∞.

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)/2zeand let |x| < 1. Using

tan−1x =1

2jln

?1+ jx

1− jx

?

we can write

cos

?5

6tan−1?1+2re

2ze

??

=

1

2

1

2

(1+ jx)5/6+(1− jx)5/6

2(1+x2)5/12

?

?

exp

?

?5

j5

6tan−1x

?

+exp

?

− j5

6tan−1x

−5

??

?1+ jx

=

exp

12ln

?1+ jx

1− jx

??

+exp

?

12ln1− jx

???

=

(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,

σ2

I,nw= 3.86σ2

Rz5/6

e

?

0.7127

?

1+5

72x2−455

4!64x4+

43225

144×66x6−···

?

−11/16

?

(25)

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

tan−1x = ±π

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

2+1

2jln

?1− jy

1+ jy

?

cos

?5

6tan−1?1+2re

2ze

??

=1

2

?

e±j5π

12

?1− jy

1+ jy

?5/12+e∓j5π

12

?1+ jy

1− jy

?5/12?

(26)

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and performing several steps for simplification, we obtain

σ2

I,nw

=

3.86σ2

Rz5/6

e

?5π

?

0.7127(x2)5/12?

??5

cos

?5π

12

??

??

1+5

72

?

?1

x

?2−

455

24×64

?1

x

?4−···

?

(27)

±sin

126

?1

x

?

−35

64

?1

x

?3+···−11

16

Acknowledgments

The authors would like to thank the Air Force Office of Scientific Research (AFOSR) for pro-

viding funding support to conduct this research under the Transformational Communications

Advanced Technology Study (TCATS) program.

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