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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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Received 28 Jul 2010; revised 24 Aug 2010; accepted 3 Sep 2010; published 15 Sep 2010

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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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Received 28 Jul 2010; revised 24 Aug 2010; accepted 3 Sep 2010; published 15 Sep 2010

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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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Received 28 Jul 2010; revised 24 Aug 2010; accepted 3 Sep 2010; published 15 Sep 2010

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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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Received 28 Jul 2010; revised 24 Aug 2010; accepted 3 Sep 2010; published 15 Sep 2010

27 September 2010 / Vol. 18, No. 20 / OPTICS EXPRESS 20750