Atmospheric optical communication with a Gaussian Schell beam

856 J. Opt. Soc. Am. A/Vol. 20, No. 5 /May 2003 J. C. Ricklin and F. M. Davidson

Atmospheric optical communication with aGaussian Schell beam

Jennifer C. Ricklin

U.S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20723

Frederic M. Davidson

Johns Hopkins University, Department of Electrical and Computer Engineering, 3400 North Charles Street,Baltimore, Maryland 21218-2686

Received July 24, 2002; revised manuscript received November 8, 2002; accepted December 18, 2002

We consider a wireless optical communication link in which the laser source is a Gaussian Schell beam. Theeffects of atmospheric turbulence strength and degree of source spatial coherence on aperture averaging andaverage bit error rate are examined. To accomplish this, we have derived analytic expressions for the spatialcovariance of irradiance fluctuations and log-intensity variance for a Gaussian beam of any degree of coherencein the weak fluctuation regime. When spatial coherence of the transmitted source beam is reduced, intensityfluctuations (scintillations) decrease, leading to a significant reduction in the bit error rate of the optical com-munication link. We have also identified an enhanced aperture-averaging effect that occurs in tightly focusedcoherent Gaussian beams and in collimated and slightly divergent partially coherent beams. The expressionsderived provide a useful design tool for selecting the optimal transmitter beam size, receiver aperture size,beam spatial coherence, transmitter focusing, etc., for the anticipated atmospheric channel conditions.© 2003 Optical Society of America

OCIS codes: 060.4510, 030.1640, 010.7060, 010.1300.

1. INTRODUCTIONFree-space laser communication offers an attractive alter-native for transferring high-bandwidth data when fiber-optic cable is neither practical nor feasible. However,random fluctuations in the atmosphere’s refractive indexcan severely degrade the wave front of a signal-carryinglaser beam, causing intensity fading at the receiver. Thisresults in increased system bit error rates (BERs) espe-cially along horizontal propagation paths. For manycases of practical interest, the limiting factor in free-spacelaser communication system performance is the presenceof clear-air atmospheric turbulence in the optical channel.

Here we consider the performance of an atmosphericlaser communication link where the transmitter beam ispartially (spatially) coherent. The Gaussian Schellbeam1 provides a theoretical framework for describingspatial partial coherence effects in a transmitted laserbeam. One way to create a Gaussian Schell beam is topass an initially spatially coherent laser beam through aphase diffuser such that the transmitted signal beam be-comes partially coherent.2 A Gaussian Schell beam stillretains its beamlike (i.e., highly directional) properties,although with an increased divergence angle.

An analytic expression valid in the weak fluctuation re-gime is derived here that approximates the spatial cova-riance of irradiance fluctuations in the receiver plane thatare due to atmospheric turbulence in the optical channel.This expression includes the spatial coherence and focus-ing characteristics of the source beam. An expression forthe log-intensity variance of the received beam is also ob-tained that incorporates both the aperture-averaging ef-

1084-7529/2003/050856-11$15.00 ©

fects of the lens at the receiver and the spatial coherenceproperties of the beam. With knowledge of the log-intensity variance, it is possible to calculate the resultingaverage BER of a direct-detection optical communicationlink that employs on–off signaling.

The results obtained here show that the average BERcan be significantly reduced by the use of a partially co-herent source beam. These results are in agreementwith an earlier theoretical treatment where it was foundthat, as the initial field coherence is decreased, intensityfluctuations of the observed radiation also decrease.3

Such behavior was previously noticed in several prelimi-nary experiments.4,5

In essence, using a partially coherent source beam withan appropriate focal length allows one to use the same la-ser power level to achieve a prescribed BER, thus to someextent compensating for the presence of atmospheric tur-bulence in the channel. The approach described in thispaper provides an important practical result for the de-sign of a free-space optical communication system–i.e., amethod for calculating the optimal transmitted beam size,receiver aperture size, beam spatial coherence, etc., forthe anticipated atmospheric channel conditions.

We have also identified an enhanced aperture-averaging effect that occurs in tightly focused coherentGaussian beams and in collimated and slightly divergentpartially coherent beams. It has long been recognizedthat there is a significant reduction in scintillation at thefocal point of a tightly focused beam. However, exploit-ing this scintillation reduction would entail unrealisti-cally precise focusing and pointing. The use of a par-

2003 Optical Society of America

J. C. Ricklin and F. M. Davidson Vol. 20, No. 5 /May 2003/J. Opt. Soc. Am. A 857

tially coherent source beam has the effect of shifting thisenhanced reduction in irradiance fluctuations into the re-gime of collimated and divergent Gaussian beams whereprecise focusing and pointing is not as critical.

This paper is organized as follows. Section 2 brieflydescribes mathematical analysis of the optical communi-cation receiver and calculation of the bit error probability(bit error rate). The reduction in intensity fluctuationsthat is due to lens aperture averaging is discussed in Sec-tion 3. An analytic expression for the spatial covarianceof irradiance fluctuations valid in the weak fluctuation re-gime is derived in Section 4 for a Gaussian beam of anydegree of spatial coherence. Section 5 compares theaperture-averaging factor based on the spatial covarianceof irradiance fluctuations derived here with actual valuesexperimentally measured by Churnside.6 The reductionin scintillations due to aperture averaging when thesource beam is partially coherent is investigated in Sec-tion 6. An expression for the log-intensity variance of aGaussian Schell beam is developed in Section 7. Finally,the combined effects of source beam coherence and atmo-spheric turbulence strength on receiver BER performanceare examined in Section 8.

2. RECEIVER PERFORMANCEFigure 1 shows the direct-detection binary optical com-munication link considered here consisting of a lasertransmitter, atmospheric channel containing clear-airturbulence, and a maximum-likelihood receiver. Themaximum-likelihood receiver consists of an avalanchephotodiode (APD) photodetector, a matched filter imple-mented in the form of a moving integrator, and a clockedcomparator. The clocked comparator performs a thresh-old test at bit interval boundaries. Descriptions of re-ceiver performance similar to the approach used herehave been discussed by other authors.7–9

The photodetector is an APD that uses current gain toreduce the effects of thermal (Boltzmann) noise. Therandomness of the avalanche current gain correspond-ingly increases the noise of the output photocurrent abovethe Poisson shot-noise limit. For an average currentgain G, this excess noise is characterized by the factorF10:

F 5 keff G 1 ~1 2 keff!~2 2 1/G !, (1)

where keff is the ratio of the ionization coefficient for holesto electrons. If the APD were instead a p-i-n photodiode,the value of both the current gain G and the excess noisefactor F would be unity.

When more than a few hundred photons are absorbedduring a single bit interval, the APD output photocurrent

Fig. 1. Direct-detection optical communication link consisting ofa laser transmitter, log-normal atmospheric channel, andmaximum-likelihood receiver.

can be modeled as a Gaussian stochastic process with anonzero mean and a variance proportional to the mean.7

The probability density function (pdf) for the output of themoving integrator at t 5 Tb is described by the Gaussianpdf:

p~ yTb! 5

1

A2psyTb

2expF2~ yTb

2 ^ yTb&!2

2syTb

2 G . (2)

In Eq. (2) yTbis a Gaussian-distributed random variable

that represents the output of the integrating filter at theend of a bit interval. The integrator integrates the APDoutput only for the previous Tb s. In the absence of at-mospheric fading, the mean ^ yTb

& and variance syTb

2 in

Eq. (2) are given by7

^ yTb& 5

eh

hfGPopt 5 eGnf ,

syTb

2 5 2BS e2h

hfFG2Popt 1

2KBT

RLD , (3)

where nf 5 hPopt /hf is photon flux absorbed by the de-tector in photons per second, 2B 5 1/Tb is the effectivebandwidth (data rate) of the moving integrator, h is quan-tum efficiency, e is electric charge in coulombs, hf is thephoton energy, and Popt is the instantaneous opticalpower incident on the receiver in watts. The quantity2KBT/RL represents thermal noise of the load resistorseen by the APD, where KB 5 1.38 3 10223J/K isBoltzmann’s constant, T is the effective noise tempera-ture in Kelvin and RL is the load resistor input to the am-plifier in ohms. For simplicity, background light andphotodetector dark current are neglected.

The effects of atmospheric turbulence on the receivedoptical power can be expressed by the dimensionless ran-dom variable Z11,12:

Z [Popt

^Popt&, (4)

which describes atmospheric turbulence-induced fadingin the received signal. In Eq. (4) ^Popt& is the received op-tical power with atmospheric turbulence fluctuations av-eraged out so that the average value ^Z& 5 1.

Both experimental and theoretical evidence haveshown that in the weak fluctuation regime the randomvariable Z describing atmospheric turbulence-induced op-tical power fluctuations can be considered log normal. Inthe strong fluctuation regime, however, a log-normallymodulated exponential distribution of amplitude fluctua-tions has been found to agree better withobservations.13–16 An expression defining the boundarybetween the weak and the strong fluctuation regimes isprovided in expressions (32) below. Here we restrict con-sideration to the weak fluctuation regime by modeling op-tical power fluctuations as log normal with a pdf givenby17,18

p~Z !dZ 51

A2ps ln Z2

expF2~ ln Z 112 s ln Z

2 !2

2s ln Z2 G 1

ZdZ.

(5)


In Eq. (5) the average of ln Z is ^ln Z& 5 212sln Z

2 , wherethe quantity s ln Z

2 represents fluctuations (scintillations)in the optical power as measured by a point receiver. Itis assumed that Z remains constant over the time interval@t, t 1 Tb# but varies slowly over many bit times Tb .

For on–off signaling, the optical power levels at the re-ceiver can be represented under the two signaling hypoth-eses as either a digital 1 (ON state) given by

Popt1 5 Z^Popt& (6)

or a digital 0 (OFF state) given by

Popt0 5 eZ^Popt&, (7)

where e represents residual laser light in the OFF statethat is due to the source modulator having an imperfectextinction ratio. For a given Z, the mean and variance ofthe moving integrator output representing the GaussianAPD photocurrent under H1 (ON state) are

^ yTb

1 & 5 eGZh

hf^Popt& 5 eGZnf ,

s12 [ var~ yTb

1 !

5 2BS e2h

hfFG2Z^Popt& 1

2KBT

RLD , (8)

and under H0 (OFF state) they are

^ yTb

0 & 5 eGeZh

hf^Popt& 5 eGeZnf ,

s02 [ var~ yTb

0 !

5 2BS e2h

hfFG2eZ^Popt& 1

2KBT

RLD . (9)

In Eqs. (8) and (9) the variance considers the effects ofboth excess shot noise and thermal noise from the APDand load resistor.

The maximum-likelihood receiver8 consists of a circuitthat implements the decision rule

p~ yTbuH1 !

Decide H1.

,

Decide H0

p~ yTbuH0 !. (10)

This yields the following quadratic equation for the deci-sion threshold value yTh :

S s12

s02 2 1 D yTh

2 1 2S ^ yTb

1 & 2s1

2

s02 ^ yTb

0 & D yTh 2 s12 lnS s1

2

s02D

1s1

2

s02 ^ yTb

0 &2 2 ^ yTb

1 &2 5 0. (11)

The decision threshold value given by Eq. (11) depends onthe instantaneous value of the fading parameter Zthrough the mean and variance of the moving integratoroutput. Here we consider only a fixed threshold receiverwhere Z is replaced by its average value ^Z& 5 1 in Eqs.(8) and (9) so that the decision threshold is independent ofZ.

In the absence of fading, the receiver probability of er-ror (BER) is given by the well-known expression:

Pr~erroruno fading! 51

2QS yTh 2 ^ yTb

0 &

2s0D

11

2QS yTh 2 ^ yTb

1 &

2s1D , (12)

where Q(x) is the Gaussian tail integral defined as

Q~x ! 51

A2pE

x

`

expS 2t2

2 D dt 51

2erfcS x

A2D . (13)

The effects of atmospheric turbulence-induced log-normalfading on receiver BER are included by averaging Eq. (12)over the fading variable Z, while keeping in mind that allthe quantities of the Q(x) functions (except yTh) dependon Z. An adaptive receiver would include the depen-dence of yTh on Z as well. Adaptive receivers cannot beimplemented in practice because it is not possible to de-termine an exact value of Z from the photodetector outputphotocurrent.

In principle, the form of the maximum-likelihood re-ceiver can also be determined by first averaging theGaussian pdf [Eq. (2)] over the fading process Z to obtain

p~ yTbuHi! 5 E

0

` 1

A2psyTb

2 ~z !

3 expH 2@ yTb2 ^ yTb

i ~z !&#2

2s i2~z !

J p~z !dz,

(14)

where p(z) is obtained from Eq. (5) and i 5 1, 0. How-ever, this cannot be accomplished in closed form, and con-sequently it is not possible to find the decision thresholdas a function of s ln Z

2 and ^Popt&. Nonetheless, some au-thors approximate p( yTb

uHi) by a Gaussian pdf with ap-propriate means and variances in order to approximatethe BER by equations similar to Eq. (12).

3. APERTURE-AVERAGING FACTORThe random variable Z defined in Eq. (4) can be expressedin an equivalent form as the ratio of optical irradiances:

Z 5I~r, z !

Î~r, z !&, (15)

where I(r, z) represents the instantaneous intensity in-cident on a single point of the receiver aperture, Î(r, z)&is the average intensity at the same point on the receiverwith fluctuations that are due to atmospheric turbulenceaveraged out, ^ • & denotes ensemble averaging, and r5 (x2 1 y2)1/2 is the transverse distance from the beamcenterline in the plane perpendicular to the z axis defin-ing the direction of propagation. The quantity s ln Z

2 canthen be described as the log-intensity variance for a singlepoint on the receiver aperture11:


s ln Z2 [ K H lnF I~r, z !

Î~r, z !&G J 2

2 K lnF I~r, z !

Î~r, z !&G L 2L .

(16)

In earlier experiments19–23 it was noted that when theaperture diameter of an optical receiver was increased,fluctuations in the received optical intensity correspond-ingly decreased. This phenomenon was called apertureaveraging. To account for the averaging effect of a non-point aperture, Tatarskii11 modified the definition of thelog-intensity variance by introducing the aperture-averaging factor A:

s ln Z82 5 As ln Z2 , (17)

where A is given by11,21

A 516

pE

0

1

xdxbI~r, z !@cos21~x ! 2 x~1 2 x2!1/2#.

(18)

In Eq. (18) x 5 r/D, D is the receiver aperture diameter,and bI(r, z) 5 BI(r, z)/BI(0, z) is the normalized spatialcovariance of irradiance fluctuations.

It is important to note that the aperture-averaging fac-tor, which varies between zero and one, is not a statisticalquantity describing the variance of the field intensity.Instead, the aperture-averaging factor provides the frac-tional decrease in the log-intensity variance that is due tohaving a nonpoint receiver aperture. To account for theaveraging effect of the receiver lens in Eq. (5), one shouldreplace the log-intensity variance s ln Z

2 with the quantityAs ln Z

2 .

4. SPATIAL COVARIANCE OF IRRADIANCEFLUCTUATIONSHere we develop an analytic expression that closely ap-proximates the spatial covariance of irradiance fluctua-tions bI(r, z) in the weak fluctuation regime for a field ofany degree of source coherence. This expression includesthe focusing characteristics of a Gaussian beam.

Expand the spatial covariance of irradiance fluctua-tions BI(r1 , r2 , z) as follows24:

BI~r1 , r2 , z ! 5 Î~r1 , z !I~r2 , z !& 2 Î~r1 , z !&

3 Î~r2 , z !&. (19)

Under conditions of weak turbulence, BI(r1 , r2 , z) canbe approximated as Io(r1 , z)Io(r2 , z)4^x(r1 , z)3 x(r2 , z)&, where x(r1 , z) is a Gaussian random pro-cess and Io(r,z) is the field intensity in the absence of tur-bulence. We can then estimate BI(r1 , r2 , z) throughthe use of

Î~r1 , z !I~r2 , z !& 5 Î~r1 , z !&Î~r2 , z !&

1 uG~r1 , r2 , z !u2, (20)

with the result that the spatial covariance of irradiancefluctuations can be expressed as

BI~r1 , r2 , z ! 5 uG~r1 , r2 , z !u2. (21)

The only place this approximation is used in this paper isthe calculation of the aperture-averaging factor in Eq.(26). Equation (20) is a statement only about the first

and second moments of the irradiance fluctuations; thepdf for these will continue to be considered log normal.

If the statistical properties of light in the source planeare independent of those in the atmospheric channel,for sufficiently narrowband source fields the mutualcoherence function G(r1 , r2 , z, t 5 0) [ Û* (r1 , z)3 U(r2 , z)& can be obtained from2,25,26

G~r1 , r2 , z ! > W~r1 , r2 , z !

51

~lz !2 EEEEd2r1d2r2Wo~r1 , r2 , 0!

3 êxp@C* ~r1 , r1! 1 C~r2 , r2!#&

3 expH jk

2z@~r1 2 r1!2 2 ~r2 2 r2!2#J .

(22)

In expression (22) r, r are two-dimensional vectors thatlie in the source and receiver planes, respectively, z is thedistance between the source and the receiver planes, andW(x1 , x2 , z) is the cross-spectral density function de-fined as W(x1 , x2 , z) [ Û* (x1 , z)U(x2 , z)&.

The random part of the complex phase of a sphericalwave that is due to propagating through homogeneousturbulence can be approximated by27

êxp@C* ~r1 , r1! 1 C~r2 , r2!#&

> expF21

ro2 ~rd

2 1 rd • rd 1 rd2 !G , (23)

where ro(z) 5 (0.55Cn2k2z)23/5 is the coherence length of

a spherical wave propagating in atmospheric turbulencecharacterized by the refractive-index structure parameterCn

2 and k 5 2p/l is the optical wave number. The vec-tors rd , rd are defined as rd 5 r1 2 r2 and rd 5 r12 r2 .

In expression (22) the degree of source coherence of thepropagating laser beam is characterized by the global co-herence parameter z 5 1 1 (wo

2/sg2) 1 (2wo

2/ro2), where

sg2 is the variance of the Gaussian Schell beam describing

the ensemble average of spatially dependent randomphases at the transmitter:1,2,26

Wo~r1 , r2 , 0! 5 Io expS 2ur1 2 r2u2

2sg2 D . (24)

Coherence of the laser beam at the transmitter is de-scribed by the source coherence parameter zS 5 11 (wo

2/sg2).

An expression for the cross-spectral density functionW(r1 , r2 , z) for a partially coherent Gaussian beam wasderived in Ref. 2. Approximating the mutual coherencefunction G(r1 , r2 , z) by the cross-spectral density func-tion W(r1 , r2 , z), using this expression in Eq. (21), andthen normalizing the spatial covariance of irradiance fluc-tuations BI(r1 , r2 , z) yields


b1~r, z ! 5BI~r, z !

BI~0, z !

5 expH 2r2

ro2 F2 1

ro2

wo2z2

2ro

2f 2

wz2~z !

G J 5 m~2r, z !.

(25)

In Eq. (25) m(r, z) is the complex degree of spatial coher-ence; r 5 ur1 2 r2u, wo is beam size (radius) at the trans-mitter; z 5 z/ zd is normalized distance ( zd 5 kwo

2/2 isdiffractive distance); wz(z) 5 wo( r2 1 z z2)1/2 is beam sizeat the receiver; r(z) 5 (Ro 2 z)/Ro is the normalizedtransmitter focusing parameter, where Ro is the phasefront radius of curvature at the transmitter; and f5 ( r/ z) 2 z(wo

2/ro2). In terms of the focusing parameter,

convergent beams are defined by r , 1, collimated beamsby r 5 1, and divergent beams by r . 1.

Note in Eq. (25) that the normalized spatial covarianceof irradiance fluctuations has an analytic solution interms of the complex degree of coherence m(r, z) evalu-ated at twice the radial separation distance. Using thisresult in Eq. (18) yields an easily evaluated expression forthe aperture-averaging factor:

A 516

pE

0

1

xdx expH 2D2x2

ro2 F2 1

ro2

wo2z2

2ro

2f 2

wz2~z !

G J3 @cos21~x ! 2 x~1 2 x2!1/2#, (26)

which can be represented in closed form as28

A 54

p2 H 1 2 expS 21

2p2D F IoS 1

2p2D 1 I1S 1

2p2D G J ,

(27)

where In( p) is the modified Bessel function of the firstkind of order n and

p2 [D2

ro2 F2 1

ro2

wo2z2

2ro

2f 2

wz2~z !

G . (28)

An earlier expression for the aperture-averaging factor ofa partially coherent beam wave in turbulence29 containsseveral compounding mathematical errors.

5. COMPARISON WITH CHURNSIDE DATABy assuming the signal-carrying laser beam to be com-pletely coherent, it is possible to compare the aperture-averaging factor given in Eq. (26) with aperture-averaging data experimentally obtained by Churnside.6

In his experiments Churnside used a 10-mW He–Ne laseroperated in single mode at a wavelength of 633 nm. Forthe 500-m path a negative lens was placed in front of thelaser so that the beam diameter at the receiver was ap-proximately 1 m. This provides sufficient information toestimate the focusing parameter r. Although the size ofthe beam at the transmitting aperture was not specified,the typical beam size for this type of laser is of the orderof 1.5 mm, and this is the value used in the comparison.Thus the normalized path length z 5 z/ zd for the 500-mpath is estimated as 44.8.

There are two principle sources of error in the Churn-side data. Churnside used an array of six receiving ap-ertures with diameters of 1, 2.25, 5, 10, 25, and 50 mm.He assumed that the smallest aperture of 1 mm could beconsidered a point receiver and used this value to normal-ize the irradiance flux variances calculated from thelarger lenses. While this is a reasonably good assump-tion, it undoubtedly did introduce some error into thedata. The second source of error is intrinsically more dif-ficult to control and to quantify. Atmospheric turbulencestrength can vary significantly both within the local areaof the experimental site and as a function of time. Allmeasurements of turbulence strength are, at best, aver-age values that approximate the actual values at a par-ticular place and instant in time. Churnside used a scin-tillometer to measure atmospheric turbulence strengthover a 250-m path and assumed that, because of the uni-formity of the terrain, these values were representative ofthe actual values over the propagation path.

For the weak fluctuation regime case shown in Fig. 2the aperture-averaging factor derived here compares rea-sonably well with the Churnside data. The aperture-averaging factor A is unity when no aperture averagingoccurs and decreases in value as fluctuations in the in-coming wave are averaged owing to increasing receiveraperture diameter size. The aperture-averaging factorderived here tends to overpredict the aperture-averagingeffect for larger values of the normalized receiver aper-ture diameter (kD2/4L)1/2.

6. EFFECTS OF SOURCE COHERENCE ONAPERTURE AVERAGINGNext we consider the aperture-averaging factor when thesource beam is partially coherent. Using a partially co-herent source beam decreases the coherence size of theoptical wave front at the receiver. This will create an ar-

Fig. 2. Comparison of the Churnside data6 for the aperture-averaging factor A with the aperture-averaging factor calculatedwith Eq. (26) (‘‘new theory’’), as a function of normalized receiveraperture diameter (kD2/4L)1/2 for a strongly divergent beam( r 5 330) in the weak fluctuation regime ( s1

2 5 0.232, s12zrec

5/6

5 3.4 3 1024, Cn2 5 1.46 6 1.18 3 10213 m22/3).


tificial aperture-averaging effect similar to that obtainedwhen the receiver aperture size is increased.

The aperture-averaging factor for a convergent (fo-cused) beam is shown in Fig. 3 as a function of the turbu-lence strength Cn

2 for a coherent (solid curve) and a par-tially coherent (dashed curve) beam. Two differentreceiver optics diameters, 2 and 10 cm, are considered.As expected, irradiance fluctuations significantly decreasewhen the receiver optics diameter is increased from 2 to10 cm. However, for the convergent beam there is no dis-cernable advantage in using a partially coherent sourcebeam. This behavior can be understood by examiningthe normalized covariance of irradiance fluctuationsbI(r, z) given in Eq. (25). For the transmitter beam sizeof 2.5 cm assumed here, f < 1 when r < 1 (condition fora convergent beam). The effect of the third term(f 2 ro

2)/@wz2(z)# in Eq. (25) describing the contribution to

bI(r, z) from partial coherence is significantly diminishedbecause f 2 is small.

Fig. 3. Aperture-averaging factor as a function of atmosphericturbulence strength for a convergent (focused) beam ( r 5 0.1) forcoherent (zS 5 1) and almost incoherent (zS 5 1000) beams.Two receiver diameters D are considered.

Fig. 4. Same as Fig. 3 but for a collimated beam ( r 5 1) and10-cm diameter receiver. Three source coherence levels are con-sidered: coherent beam (zS 5 1), slight partial coherence(zS 5 3), and an almost incoherent beam (zS 5 1000).

The effects of partial coherence become more apparentin the collimated beam ( r 5 1) shown in Fig. 4. In weakturbulence (Cn

2 ; 10216 to 10215 m22/3) there is almost anorder of magnitude decrease in irradiance fluctuations be-tween the coherent and the partially coherent beams. Asturbulence strength increases, this advantage is lost, and,for Cn

2 . 10214 m22/3, irradiance fluctuations are reducedequally for all three beams, regardless of the source beamcoherence. This occurs because when sg /ro ! 1 the ef-fects of source coherence dominate the global coherenceparameter z. Conversely, when sg /ro @ 1, any advanta-geous effect due to having a partially coherent sourcebeam is overtaken by the loss in wave-front coherence in-duced by atmospheric turbulence.

A similar situation exists with the divergent beamshown in Fig. 5, where the transmitter focusing param-eter r has been increased to 5. With this divergent beam,more than an order of magnitude of improvement in re-ducing irradiance fluctuations is possible, and this im-provement continues in the presence of stronger turbu-lence conditions than with the collimated beam. For the

Fig. 5. Same as Fig. 4 but for a divergent beam ( r 5 5).

Fig. 6. Aperture-averaging factor as a function of the transmit-ter focusing parameter r. Three source coherence states areconsidered, from a coherent beam (zS 5 1) to an almost incoher-ent beam (zS 5 10,000).


divergent beam f . 1, so the contribution of the thirdterm (f 2 ro

2)/@wz2(z)# in Eq. (25) is emphasized.

The behavior observed in Figs. 3–5 is summarized inFig. 6. Note that all beams, including coherent beams,experience an enhanced aperture-averaging effect forsmall r (tightly focused beams defined by r ! 1). As thesize of r increases, this enhanced ability to average irra-diance fluctuations is lost for all beams, regardless ofsource coherence. Varying the receiver diameter D willshift the set of curves in Fig. 6 vertically up or down: upfor smaller aperture sizes D, where less averaging of irra-diance fluctuations occurs, and down for larger aperturesizes D, where more averaging of irradiance fluctuationsoccurs.

This enhanced ability to reduce irradiance fluctuations(scintillations) is clearly a property of the Gaussian beamwave itself rather than an effect due to partial coherenceand occurs naturally with tightly focused coherent beams,where r ! 1. Using a partially coherent source beamhas the effect of shifting this enhanced reduction in irra-diance fluctuations into the regime of collimated ( r 5 1)and divergent ( r . 1) Gaussian beam waves. On the ba-sis of experimental evidence, earlier researchers noticedthat increasing the transmitter aperture size of a focusedbeam (thus further decreasing r) had the effect of reduc-ing scintillations in the beam focal plane.30–32 Later itwas discovered that this reduction in scintillation de-pends critically on the focus adjustment and is not prac-tically realizable.33 Using a partially coherent source ex-ploits this previously observed enhanced reduction inscintillation by shifting this effect so that it occurs withlarger transmitter focusing parameters r (collimated anddivergent beams) and thus is not dependent on precisebeam focusing.

7. APERTURE-AVERAGED LOG-INTENSITYVARIANCEThe log-intensity variance at a point, s ln Z

2 , can be closelyapproximated from exact expressions for the log-amplitude variance given in Ref. 34 by assuming a Kol-mogorov spectrum:35

s ln Z2 > 4.42s1

2zrec5/6

r2

wz2~z !

1 3.86s12H 0.4@~1 1 2 rrec!

2

1 4 zrec2 #5/12 cosF5

6tan21S 1 1 2 rrec

2 zrecD G2

11

16zrec

5/6J ,

(29)

where s12 5 1.23Cn

2k7/6 z11/6 is the Rytov variance for aplane wave. Here we have replaced the receiver beamparameters for a coherent beam in free space with theequivalent free-space receiver parameters for a partiallycoherent beam:

rrec~z ! 5Rz~z ! 1 z

Rz~z !, zrec~z ! 5

z

0.5kwz2~z !

. (30)

In Eqs. (30) the free-space phase front radius of curvaturefor a partially coherent beam at the receiver is given by2

Rz~z ! 5z~ r2 1 zSz2!

f z 2 zSz2 2 r2, f [

r

z2 z

wo2

ro2 , (31)

and the free-space receiver beam size for a partially co-herent beam is given by wz(z) 5 wo( r2 1 zSz2)1/2. Be-cause the effects of atmospheric turbulence are accountedfor in expression (29) by the Rytov variance, only thesource coherence parameter zS 5 1 1 (wo

2/sg2) is required

here.For a Gaussian beam wave the weak fluctuation regime

is characterized by the following restrictions34,35:

s12 , 1, s1

2zrec5/6 , 1. (32)

The effects of source coherence, aperture averaging,beam focusing, and atmospheric turbulence strength onthe log-intensity variance s ln Z

2 and the aperture-averaged

Fig. 7. Log-intensity variance as a function of radial distance r

from the beam center for slightly divergent ( r 5 5) source beamsranging from coherent (zS 5 1) to almost incoherent (zS5 1000). The beam size at the receiver increases as the sourcebeam loses coherence: (1) zS 5 1, wz(z) 5 12.7 cm; (2) zS5 20, wz(z) 5 15.8 cm; (3) zS 5 50, wz(z) 5 19.2 cm; (4) zS5 1000, wz(z) 5 64.5 cm.

Fig. 8. Log-intensity variances given in Fig. 7 showing the ef-fects of aperture averaging over a 10-cm-diameter receiver aper-ture.


Fig. 9. Effects of aperture averaging the log-intensity varianceover 1-, 3-, and 10-cm-diameter receiver apertures for a slightlyincoherent divergent beam ( r 5 5). Each curve stops at the ra-dial size of the specified aperture.

Fig. 10. Aperture-averaged log-intensity variance as a functionof the focusing parameter r. Values given are for the log-intensity variance at the edge of the receiver aperture (r5 5 cm).

Fig. 11. Aperture-averaged log-intensity variance showing theeffect of atmospheric turbulence strength ( r 5 5). Values givenare for the log-intensity variance at the edge of the receiver ap-erture (r 5 5 cm).

log-intensity variance As ln Z2 are examined in Figs. 7–11.

The log-intensity variance s ln Z2 is shown as a function of

the radial distance r from the axial beam center (r 5 0)in Fig. 7. Having a partially coherent source beam hasthe effect of damping off-axis intensity fluctuations in thereceived beam [note that the optical power in the beamwill greatly decrease beyond r 5 wz(z)].

The results given in Fig. 7 do not consider the effects ofaperture size. Figure 8 shows the effects of aperture av-eraging the point log-amplitude variances given in Fig. 7over a 10-cm-diameter receiver aperture. Because of ap-erture averaging, the log-intensity variance As ln Z

2 is sig-nificantly reduced for both the coherent and the partiallycoherent beams. The coherent laser beam (zS 5 1) stillshows evidence of off-axis intensity fluctuations, but theseoff-axis fluctuations decrease as the source beam becomesless coherent.

The combined benefits of aperture averaging and par-tial coherence are even more apparent in Fig. 9, where re-ceiver aperture diameters of 1, 3, and 10 cm are consid-ered for a slightly incoherent laser beam (zS 5 10).Increasing the receiver aperture diameter from 1 to 10 cmreduces off-axis intensity fluctuations by approximatelyhalf an order of magnitude.

The effect of beam focusing on the aperture-averagedlog-intensity variance is examined in Fig. 10. Here weconsider the worst-case scenario at the beam edge @r5 wz(z)#, where off-axis intensity fluctuations are mostsevere. Coherent and almost coherent (zS < 2) conver-gent and collimated beams ( r < 1) have significant off-axis intensity fluctuations, with off-axis intensity fluctua-tions diminishing for divergent ( r . 1) beams. Fromthis figure we see that having even a slightly partially co-herent source beam significantly reduces off-axis inten-sity scintillations, especially for convergent and colli-mated beams.

In Fig. 11 we consider the effects of atmospheric turbu-lence strength on the aperture-averaged log-intensityvariance for a divergent beam. In weak-to-moderate at-mospheric turbulence (Cn

2 , 10214 m22/3), off-axis values@r 5 wz(z)# of the aperture-averaged log-intensity vari-ance significantly decrease as the source beam becomesless coherent. As turbulence strength increases beyondCn

2 5 10214 m22/3, the degree of source coherence has lessof an effect on As ln Z

2 , although there is still a slight ad-vantage in having a partially coherent source beam evenin strong turbulence (Cn

2 . 10214 m22/3). When sg /ro! 1, the effects of source coherence are dominant, and,when sg /ro @ 1, loss in coherence of the propagatingwave front is primarily determined by atmospheric turbu-lence rather than the coherence properties of the sourcebeam.

8. BIT ERROR RATE PERFORMANCEThe effects of atmospheric turbulence strength and sourcecoherence on the average probability of error (BER) in adirect-detection binary optical communication link areshown in Figs. 12 and 13. Equation (14) was evaluatedwith a MATLAB-based Gauss–Lobatto quadrature nu-merical integration routine.36 Because MATLAB has a


limit of 104 iterations for the Gauss–Lobatto integrationroutine, the curves do not always terminate at the sameBER.

In these calculations we assume a current gain ofG 5 100, keff 5 0.01, 2B 5 1/Tb 5 1 Mbits/s (data rate of106 bits/s), load resistance RL 5 50 V, T 5 300 K, quan-tum efficiency h 5 0.8, and a 100:1 modulator extinctionratio (e 5 0.01). The aperture-averaged log-intensityvariance As ln Z

2 is obtained from Eq. (26) and expression(29), where we have used the average value of As ln Z

2 @r5 wz(z)/2#. The average optical power at the receiverPopt is expressed in decibel milliwatts, with a 1-mW opti-cal power corresponding to 0 dBm. The optical power isobtained from the photon flux nf by using the relation-ship Popt 5 hfnf /h. We assume there are no pointingerrors or beam motion or wander and that thetransmitter–receiver system is perfectly aligned. Oncethe average optical power is determined, the opticalpower in decibel milliwatts can be obtained from

Popt dBm 5 10 log10~Popt mW!. (33)

When expression (29) is examined for the log-intensityvariance, it is apparent that the beam size at the receiverplays a significant role in determining off-axis intensityscintillations. In Fig. 12 the receiver beam size wz(z) iskept constant by varying the focusing parameter r in or-der to isolate the effect of source coherence on the BERfrom any effect due to increased beam size. Since thebeam footprint at the transmitter is 1 m across and thereceiver aperture diameter is 10 cm, the detector does notreceive all of the transmitted power, and the set of curvesin Fig. 12 is shifted to the right by scaling the receivedoptical power Popt by the ratio of the beam footprint to thedetector area.

Comparing behavior of the coherent beam (zS 5 1) infree space and in atmospheric turbulence in Fig. 12, at

Fig. 12. BER as a function of received optical power in decibelmilliwatts (1 mW 5 0 dBm). The focusing parameter r is ad-justed so that, regardless of source coherence, the 5-cm diametertransmitted beam always has a receiver beam size of wz(z)5 50 cm (beam footprint of 1 m). Left to right: (1) coherentbeam in free space (best possible performance); (2) r 5 1, zS

5 2500, As ln Z2 5 0.013; (3) r 5 15.45, zS 5 1000, As ln Z

2

5 0.017; (4) r 5 20, zS 5 1, As ln Z2 5 0.021 (coherent source

beam).

235 dBm the BER is increased by almost 6 orders of mag-nitude because of the presence of atmospheric turbulence.As the source beam becomes less coherent, the laser com-munication system begins to approach its free-space per-formance. For example, at 230 dBm the partially coher-ent (zS 5 2500) beam has a BER of the order of 1029,while the BER for the coherent beam is of the order of1026. This corresponds to approximately 3 orders ofmagnitude reduction in the BER. Using a partially co-herent source beam with an appropriate focal length al-lows one to use the same laser power level to achieve aprescribed BER.

Figure 13 shows the potential for BER reduction as afunction of atmospheric turbulence strength for a par-tially coherent source beam (zS 5 10). Any loss in re-ceived power that is due to the additional spreading of thepartially coherent laser beam has been disregarded (afterpropagating 2 km, because of pointing errors and beammotions, actual differences in received power between a6.4-cm beam footprint and 13.8–15.8-cm beam footprintswill be minimal). In moderate turbulence at 252.5 dBm,use of a partially coherent beam reduces the BER by morethan 3 orders of magnitude. However, when atmosphericturbulence strength increases to Cn

2 5 1.23 10214 m22/3, the reduction in the BER between the co-

herent and the partially coherent beams is only 1 order ofmagnitude. This is because the laser beam’s global co-herence is determined more by atmospheric turbulencestrength than by the source beam coherence in the stron-ger turbulence conditions.

9. CONCLUDING REMARKSIn this paper we have shown that reducing the spatial co-herence of the laser source in a wireless optical commu-nication link can, under many circumstances, signifi-

Fig. 13. Effect of atmospheric turbulence strength on BER as afunction of received optical power for collimated ( r 5 1) coherent(zS 5 1) and partially coherent (zS 5 10) beams. Left to right:(1) coherent beam in free space (best possible performance); (2)zS 5 10, As ln Z

2 5 0.013, wz(z) 5 6.9 cm; (3) zS 5 10, As ln Z2

5 0.05, wz(z) 5 7.9 cm; (4) zS 5 1, As ln Z2 5 0.045, wz(z)

5 3.2 cm; (5) zS 5 1, As ln Z2 5 0.118, wz(z) 5 3.2 cm.


cantly reduce intensity fluctuations (scintillations) thatare due to atmospheric turbulence in the optical commu-nication channel. This in turn can reduce the BER thatis due to atmospheric turbulence by as much as severalorders of magnitude, dependent on beam focusing and at-mospheric turbulence strength. This reduction in BER isin part due to an enhanced aperture-averaging effectidentified here that occurs when the wave-front coherenceat the receiver is reduced, owing to either decreasedsource coherence or atmospheric-turbulence-inducedwave-front fluctuations. As the laser source becomes lesscoherent, the log-intensity variance and off-axis intensityfluctuations will also decrease. The combined effect of areduced log-intensity variance and an enhanced aperture-averaging factor can be mutually exploited to minimizethe BER that is due to atmospheric turbulence. Placinga phase diffuser in front of the laser transmitter apertureis one simple way to create a partially coherent lasersource.

We have also derived analytic expressions for the spa-tial covariance of irradiance fluctuations and log-intensityvariance for a Gaussian beam of any degree of coherencein the weak fluctuation regime. We found that the nor-malized spatial covariance of irradiance fluctuations isgiven by the complex degree of spatial coherence evalu-ated at twice the radial separation distance—in short,fluctuations in the wave-front intensity are directly re-lated to the wave-front coherence. The global coherenceparameter provides a tool to investigate the relative con-tribution to wave-front coherence from the coherence ofthe laser source and random atmospheric turbulence-induced fluctuations. When atmospheric turbulence isweak, global wave-front coherence is determined prima-rily by the coherence of the laser source. As turbulencestrength increases, the effect (and any advantage) in hav-ing a partially coherent source diminishes, since wave-front coherence will instead be dominated by atmosphericturbulence.

It is important to note that decreasing the source coher-ence results in a larger beam divergence angle and alarger beam footprint at the receiver. To a certain extent,this is desirable feature and will help to reduce pointingerrors. However, there is a trade-off: As the beam foot-print increases, power incident on the receiver decreases.These issues are addressed in Ref. 2, where expressionsare derived for the average intensity, beam size, and co-herence length of a partially coherent laser beam in tur-bulence. Together with the expressions derived here,these provide a useful design tool for selecting the optimaltransmitted beam size, receiver aperture size, beam spa-tial coherence, transmitter focusing characteristics, etc.,for the anticipated atmospheric channel conditions.

ACKNOWLEDGMENTSWe thank Stephan Bucaille for writing the MATLAB codeused to calculate the average probability of error curvesand Larry Andrews for providing the Churnside data.

Corresponding author J. C. Ricklin can be reached byphone, 301-394-2535; fax, 301-394-0225; or e-mail,[email protected].

REFERENCES1. A. C. Schell, ‘‘The multiple plate antenna,’’ Ph.D. disserta-

tion (Massachusetts Institute of Technology, Cambridge,Mass., 1961).

2. J. C. Ricklin and F. M. Davidson, ‘‘Atmospheric turbulenceeffects on a partially coherent Gaussian beam: implica-tions for free-space laser communication,’’ J. Opt. Soc. Am.A 19, 1794–1802 (2002).

3. V. A. Banach, V. M. Buldakov, and V. L. Mironov, ‘‘Inten-sity fluctuations of a partially coherent light beam in aturbulent atmosphere,’’ Opt. Spektrosk. 54, 1054–1059(1983).

4. J. C. Ricklin, F. D. Davidson, and T. Weyrauch, ‘‘Free-spacelaser communication using a partially coherent source,’’ inOptics in Atmospheric Propagation and Adaptive SystemsIV, A. Kohnle, J. D. Gonglewski, and T. J. Schmugge, eds.,Proc. SPIE 4538, 13–23 (2001).

5. V. I. Polejaev and J. C. Ricklin, ‘‘Controlled phase dif-fuser for a laser communication link,’’ in ArtificialTurbulence for Imaging and Wave Propagation, J. D. Gon-glewski and M. A. Vorontsov, eds., Proc. SPIE 3432, 103–107 (1998).

6. J. H. Churnside, ‘‘Aperture averaging of optical scintilla-tions in the turbulent atmosphere,’’ Appl. Opt. 30, 1982–1994 (1991).

7. F. M. Davidson and X. Sun, ‘‘Gaussian approximation ver-sus nearly exact performance analysis of optical communi-cation receivers,’’ IEEE Trans. Commun. 36, 1185–1192(1988).

8. D. L. Snyder, Random Point Processes (Wiley Interscience,New York, 1975).

9. R. M. Gagliardi and S. Karp, Optical Communications(Wiley Interscience, New York, 1995).

10. R. J. McIntyre, ‘‘The distribution of gains in uniformly mul-tiplying avalanche photodiodes,’’ IEEE Trans. Electron De-vices 19, 703–713 (1972).

11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium(McGraw-Hill, New York, 1961).

12. J. H. Churnside and C. M. McIntyre, ‘‘Averaged thresholdreceiver for direct detection of optical communicationsthrough the lognormal atmospheric channel,’’ Appl. Opt. 16,2669–2676 (1977).

13. J. H. Churnside and R. J. Hill, ‘‘Probability density of irra-diance scintillations for strong path-integrated refractiveturbulence,’’ J. Opt. Soc. Am. A 4, 727–733 (1987).

14. J. H. Churnside and S. F. Clifford, ‘‘Log-normal Ricianprobability-density function of optical scintillations in theturbulent atmosphere,’’ J. Opt. Soc. Am. A 4, 1923–1930(1987).

15. S. M. Flatte, C. Bracher, and G. Wang, ‘‘Probability densityfunctions of irradiance for waves in atmospheric turbulencecalculated by numerical simulation,’’ J. Opt. Soc. Am. A 11,2080–2092 (1994).

16. R. J. Hill and R. G. Frehlich, ‘‘Probability distribution of ir-radiance for the onset of strong scintillation,’’ J. Opt. Soc.Am. A 14, 1530–1540 (1997).

17. D. L. Fried, G. E. Meyers, and M. P. Keister, ‘‘Measure-ments of laser-beam scintillation in the atmosphere,’’ J.Opt. Soc. Am. 57, 787–797 (1967).

18. V. I. Tatarskii, The Effects of the Turbulent Atmosphere onWave Propagation (National Technical Information Service,Springfield, Va., 1971).

19. H. Mikesell, A. A. Hoag, and J. S. Hall, ‘‘The scintillation ofstarlight,’’ J. Opt. Soc. Am. 41, 689–695 (1951).

20. S. H. Reiger, ‘‘Starlight scintillation and atmospheric turbu-lence,’’ Astron. J. 68, 395–406 (1963).

21. D. L. Fried, ‘‘Aperture averaging of scintillation,’’ J. Opt.Soc. Am. 57, 169–175 (1967).

22. G. E. Homstad, J. W. Strohbehn, R. H. Berger, and J. M.Heneghan, ‘‘Aperture-averaging effects for weak scintilla-tions,’’ J. Opt. Soc. Am. 64, 162–165 (1974).

23. R. S. Iyer and J. L. Bufton, ‘‘Aperture averaging effects instellar scintillation,’’ Opt. Commun. 22, 377–381 (1977).

24. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978).


25. R. Lutomirski and H. T. Yura, ‘‘Propagation of a finite opti-cal beam in an inhomogeneous medium,’’ Appl. Opt. 10,1652–1658 (1971).

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

27. H. T. Yura, ‘‘Mutual coherence function of a finite cross sec-tion optical beam propagating in a turbulent medium,’’Appl. Opt. 11, 1399–1406 (1972).

28. R. I. Joseph, Department of Electrical and Computer Engi-neering, Johns Hopkins University, Baltimore, Maryland21218 (personal communication, 2001).

29. S. J. Wang, Y. Baykal, and M. A. Plonus, ‘‘Receiver-apertureaveraging effects for the intensity fluctuation of a beamwave in the turbulent atmosphere,’’ J. Opt. Soc. Am. 73,831–837 (1983).

30. D. L. Fried and J. B. Seidman, ‘‘Laser beam scintilla-tion in the atmosphere,’’ J. Opt. Soc. Am. 57, 181–185(1967).

31. A. Ishimaru, ‘‘Fluctuations of a beam wave propagating

through a locally homogeneous medium,’’ Radio Sci. 4, 295–305 (1969).

32. F. P. Carlson, ‘‘Application of optical scintillation measure-ments to turbulence diagnostics,’’ J. Opt. Soc. Am. 59,1343–1347 (1969).

33. J. R. Kerr and R. Eiss, ‘‘Transmitter-size and focus effectson scintillations,’’ J. Opt. Soc. Am. 62, 682–684 (1972).

34. W. B. Miller, J. C. Ricklin, and L. C. Andrews, ‘‘Log-amplitude variance and wave structure function: a newperspective for Gaussian beams,’’ J. Opt. Soc. Am. A 10,661–672 (1993).

35. L. C. Andrews and R. L. Phillips, Laser Beam Propagationthrough Random Media (SPIE Press, Bellingham, Wash.,1998).

36. W. Gander and W. Gautschi, ‘‘Adaptive quadrature—revisited,’’ Department Informatik Institut fur Wissen-schaftliches Rechnen, Eidgenussische Technische Hochs-chule Zurich. Report available via anonymous ftp fromftp.inf.ethz.ch as doc/tech-rteports/1998/306.ps.

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Atmospheric optical communication with a Gaussian Schell beam