Turbulence-induced fading probability in coherent opticalcommunication through the atmosphere

N. PerlotInstitute of Communication and Navigation, DLR Oberpfaffenhofen, 82234 Wessling, Germany

nicolas.perlot@dlr.de

Received 2 April 2007; revised 6 August 2007; accepted 10 August 2007;posted 15 August 2007 (Doc. ID 81719); published 5 October 2007

To assess the coherent detection of an optical signal perturbed by atmospheric turbulence, the loss in themean signal-to-noise ratio (SNR) is usually invoked although it constitutes a limited description of thesignal fluctuations. To produce statistical distributions of the SNR, we generate random optical fields. A5�3-power law for the phase structure function is considered. The benefit of a wavefront tilt correction isassessed. Based on the 1%-probability fade, an optimum receiver size is found. For phase fluctuationsonly, a similarity between the signal distribution and the beta distribution is observed. Phase andamplitude are assumed independent, and the influence of amplitude perturbations is assessed with ascintillation index of 2. Turbulence impairments are compared for a coherent receiver and a direct-detection receiver. © 2007 Optical Society of America

OCIS codes: 010.1300, 010.1330, 060.1660, 060.4510.

1. Introduction

When an optical signal is transmitted, there existseveral reasons for using a coherent detection. Com-pared to its incoherent counterpart, a coherent re-ceiver is generally more sensitive to the signal andless sensitive to background light. However, whenthe signal is transmitted through the atmosphere,optical turbulence randomly distorts the phase andthe amplitude of the wave. Signal fades associatedwith optical turbulence strongly depend on the detec-tion type. Unlike direct detection, coherent detectionis impaired by phase distortions, which set a limit onthe effective receiver size.

Much work has been done on the characterizationof scintillation and on its impact on direct-detectionsystems [1,2]. For coherent systems, the performanceevaluation is generally based on the mean signal-to-noise ratio (SNR) characterized by Fried in 1967[1,3,4]. This long-term SNR does not, however, allowpredictions regarding signal fading probabilities. Theinvestigation of the heterodyne signal distributionstarted more than three decades ago. Churnside andMcIntyre derived analytically an approximate distri-

bution for a limited range of the receiving aperturesize [5,6]. The analytical derivation of the coherent-signal distribution is difficult, even when the statis-tics of the optical field are known. The reason is thespatial averaging operated by the receiving apertureon the optical field, which generally makes expres-sions of the signal statistics intractable. Distributionscomputed from the generation of correlated phasesamples in the receiving aperture were also reported[7,8]. There is still, however, no accurate descriptionof the signal distribution as a function of the aperturesize. Increasing the aperture size reduces scintilla-tion but also the heterodyne efficiency. Recently,there has been a regain of interest in optical free-space coherent communications with the fabricationof terminals implementing a homodyne binary phaseshift keying (BPSK) [9,10]. To use this technologythrough the atmosphere, a precise prediction of theturbulence influence is required.

Here, we also resort to simulations. Spatial real-izations of the optical field in the receiving-apertureplane are produced. Associated SNR statistics areanalyzed as a function of the aperture size. As apossible improvement through adaptive optics, weconsider the tilt correction. The paper is structured asfollows. First, we define the detected signal photocur-rent and express it in terms of the perturbed optical

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7218 APPLIED OPTICS � Vol. 46, No. 29 � 10 October 2007

field. In Section 3, the SNR distribution is studiedunder phase perturbations only. Section 4 deals withthe additional effect of amplitude perturbations (scin-tillation). Under scintillation, a brief comparison ofimpairments is made with direct detection. The paperconcludes with Section 5.

2. Signal Definition

A coherent detection is obtained by adding a localoscillator (LO) to the received field on the photode-tector. Because optical turbulence has a negligibledepolarization effect [11], we assume a polarizationmatch of the two fields on the detector and considerscalar fields. The analysis of the photocurrent result-ing from the detection can be done by considering thecombined fields either in the focal plane or in theaperture plane [3]. In the aperture plane, the pho-tocurrent term i containing the interference betweenthe scalar LO field ELO and the scalar signal field Es

is given by [3]:

i � Re��W�r�Es�r�ELO*�r�dr�, (1)

where Re denotes the real part, r is a two-dimensional vector in the aperture plane, and W�r� isthe aperture window function. Depending on the def-inition of the irradiance, Eq. (1) may vary by a con-stant factor. To simplify the analysis, we assume thatthe LO as well as the unperturbed signal field aremonochromatic plane waves over the aperture andthat the LO phase maximizes the current i at anytime. This amounts to considering a receiver with aperfect optical phase-locked loop assuring a homo-dyne detection. Temporal spectra of turbulence-induced wavefront perturbations can be found in [12].Setting ELO�r� � 1, Eq. (1) yields

i ���W�r�Es�r�dr�� ��W�r�A�r�exp�j��r�dr�, (2)

where A�r� and ��r�, respectively, are the perturbedamplitude and phase of the optical signal, andj � �1. We assume A�r� and ��r� are statisticallyhomogeneous over the aperture and we normalize theamplitude according to

�A2� � �I� � 1, (3)

where �·� denotes ensemble averaging and I is theoptical field intensity. Furthermore, we consider acircular aperture of diameter D and normalize theaperture window function over the aperture area:

W�r� � 4���D2�, if �r� � D�20, if �r� � D�2. (4)

The photocurrent i will thus be analyzed as a dimen-sionless quantity. An important point underlying thefollowing analysis is the separation of two types ofvariations: turbulence-induced signal variations andreceiver noise variations. Turbulence-induced fluctu-ations are on the order of kilohertz, whereas thereceiver noise bandwidth for a common optical trans-mission is on the order of gigahertz. Because turbu-lent perturbations are slow compared to a bitduration, we can consider a turbulence-induced prob-ability distribution for the SNR. This distribution isdirectly related to that of the signal photocurrent i.

3. Phase Perturbations

A. Mean Signal-to-Noise Ratio

We consider the phase as a Gaussian field character-ized by the following spatial structure function,

D���� � 6.88� �

r��5�3

, (5)

where � is the module of the variable vector � in theaperture plane and r� is a characteristic length. Be-cause it defines the phase structure function, thecharacteristic length is noted r� (r0, the so-called co-herence diameter, is generally used to define thewave structure function [3]). Equation (5) is an ap-proximation of the structure function derived fromthe Rytov theory: there exist more complex and moreprecise expressions of D���� [13]. It has been arguedthat phase statistics under weak fluctuations shouldalso apply under strong fluctuations [11,14]. Note,however, that by considering the phase as a contin-uous Gaussian field, phase dislocations, which mayoccur under strong fluctuations [15], are not takeninto account.

Without scintillation �A � 1�, i is equal to i� de-fined by

i� ���W�r�exp�j��r�dr�, (6)

and included between 0 and 1. The mean squareequals [3]:

�i�2� � 2��HW���R�����d�, (7)

where HW is the autocorrelation function of the aper-ture window:

HW��� ��W�r�W�r ��dr. (8)

10 October 2007 � Vol. 46, No. 29 � APPLIED OPTICS 7219

R���� is the spatial correlation function of exp�j��r�given by

R���� � exp��0.5D����. (9)

It is known [3] that, with a structure function asdefined by Eq. (5), we have �i�

2� → r�2�D2 for

D�r� → . A well-designed coherent receiver is shot-noise limited, and the SNR is proportional to theproduct of i2 with the aperture area. Because of itsknown asymptotic behavior, we consider the follow-ing SNR:

SNR0 � �D�r��2i2. (10)

Numerical realizations of the phase field ��r� weregenerated as described in [16] with a 4 � 4 interpo-lator. Figure 1 shows �SNR0� in decibels (dB) fori � i�. Although �SNR0� is plotted versus D�r�, oneshould consider the abscissa as a variable D withrespect to a constant r� (rather than a variable r� withrespect to a constant D). This is because SNR0 takesinto account the antenna gain, which is determinedby D2 but independent of r�. The asymptotes D2�r�

2

and 1 are plotted on the graph as dashed curves.Beside the case of an uncorrected wavefront, correc-tions of the so-called Z and C tilts are considered. TheZ tilt corresponds to the second and third Zernikepolynomials. The C tilt is calculated from the dis-placement of the centroid of the focal spot. In theabsence of scintillation, the C tilt is equivalent to theG tilt, which is the average gradient of the wavefrontacross the aperture. The difference between C and Gtilts is discussed in [17]. The difference between Zand G tilts is discussed in [18]. From simulations,�SNR0� was evaluated for the three types of wave-front. Analytically, Eq. (7) was used for the no-correction case and [19] provided expressions for theZ-tilt case. However, no analytical expression of

�SNR0� for the C-tilt correction could be found. Weobserve in Fig. 1 a good agreement between the exact(analytical) and the simulation results. In the re-maining part of the paper we drop the Z tilt and focuson the C-tilt correction, as it is usually the one im-plemented in reality.

The mean normalized SNR for the uncorrectedwave, noted �SNR0,u�, is generally approximated by

�SNR0,u� ��D�r��2

�1 �D�r��5�36�5. (11)

For the C-tilt correction, it was found that the meannormalized SNR could be approximated by

�SNR0,C� � �SNR0,u��1 �0.985D�r��1.84

1 �0.331D�r��3.18�. (12)

The maximum error made by Eq. (12) is �10% com-pared to the simulation values.

B. Signal-to-Noise Ratio Distribution

The normalized variance of i is defined by

�i,02 �

�i2�

�i�2 � 1. (13)

Based on simulation results, an approximation of �i,02

when i � i� can be put in the form:

�i,02 � 0.273� �aD�r��c

1 �bD�r��d �eD�r��f

1 �eD�r��f�. (14)

The parameters of Eq. (14) are listed in Table 1 forthe cases with and without tilt correction. Figure 2shows �i,0

2 as a function of D�r� together with thefitting curves of Eq. (14). We observe that the nor-malized variance of i� starts a sharp increase atD � 0.5r� when there is no correction and a sharpincrease at D � 2r� when the tilt is corrected. Thebehavior of �i,0

2 for D�r� → can be deduced from thecentral-limit theorem (CLT) [20]. The integral in Eq.(6) can indeed be viewed as the infinite sum of inde-pendent optical-field cells. It results that i tends tohave a Rayleigh distribution and we have

limD�r�→

�i,02 �

4�

� 1. (15)

Fig. 1. (Color online) Mean SNR0 when the wavefront is notcorrected and when C- and Z-tilt corrections are performed. Sim-ulation results are compared with exact values.

Table 1. Parameter Values for Fitting Curves of �i,02

Parameter No Correction C-Tilt Correction

a 0.743 0.188b 0.532 0.180c 2.59 4.95d 3.30 5.78e 9.56 0.520f 22.4 5.22

7220 APPLIED OPTICS � Vol. 46, No. 29 � 10 October 2007

The factor 4�� � 1 � 0.273 was introduced in Eq. (14)to comply with Eq. (15).

In an attempt to find a simple distribution modelfor i�, the computed distribution of i� was comparedwith the beta distribution. Note that the choice of thebeta distribution is based on a resemblance of theobserved curves, and not on any statistical reasoning.The beta cumulated density function (CDF), notedFbeta, is given by

Fbeta�x� �Bx� , ��B� , ��

, 0 � x � 1, (16)

where B is the beta function, Bx the incomplete betafunction [21], � and � the two distribution parame-ters. To assess the merit of the beta distribution, weconsider, over the values taken by x, the maximumdifference between Fbeta and the CDF Fi�

of i�. Themaximum of |Fi�

� Fbeta| is shown in Fig. 3 as a

function of D�r�. Mean and variance are identical forFi�

and Fbeta. Thus moments of higher orders are herecompared. At approximately D�r� � 8, with a maxi-mum CDF difference less than 10�2, the beta fit canbe considered good. For D�r� � 10, the fit worsensbecause the beta model does not tend to the Rayleighdistribution.

To study the strength of the SNR0 fades as a func-tion of D�r�, we consider the level under which SNR0lies with a probability of 1%. In Fig. 4 the CDF ofSNR0 is plotted for the particular case D�r� � 2 andthe 1%-probability fade level is indicated. Figure 5shows the 1%-probability fade of SNR0 as a functionof D�r�. We note that, at 1% probability, the SNRstarts to decrease when �i,0

2 starts a sharp increase(compare with Fig. 2). Interestingly, whereas D�r�

� 2 represents a maximum for the fade level of thetilt-corrected wave, it is a local minimum for that ofthe uncorrected wave. For D�r� → , the fade level

Fig. 2. (Color online) Normalized variances of i� estimated fromsimulations and plotted along with their fitting curves.

Fig. 3. (Color online) Comparison of the i� distribution with thebeta distribution. The beta fit improves as max�|Fi�

� Fbeta|�decreases.

Fig. 4. (Color online) Cumulative density function of SNR0 for thecase D�r� � 2.

Fig. 5. (Color online) Level of fades occurring with a probability of1% for i � i�.

10 October 2007 � Vol. 46, No. 29 � APPLIED OPTICS 7221

tends to �18.9 dB as a result of the CLT. The 1%-probability fade is here an expedient means of as-sessing a link. Because most bit errors occur during1%-probability fades, the variations of the mean bit-error rate (in particular, variations with respect tothe aperture size) inversely follow the variations ofthese fades. Also relevant is the depth of the fadeswith respect to the mean �SNR0�. These fades, whichare supposed to occur 1% of the time, are known to bemuch longer than a bit duration. Error bursts asso-ciated with deep fades are then difficultly mitigatedby channel coding.

We observe in Figs. 2 and 5 a transition regimewhere the fluctuations strongly increase. This tran-sition regime occurs at approximately D�r� � 1 for anuncorrected wave and at approximately D�r� � 5 fora tilt-corrected wave. Similar transition regimes havebeen found by Andrews et al. when evaluating scin-tillation for an uplink scenario [22]. In this case, theprinciple of reciprocity is applied with the parameterD denoting the diameter of the transmitting aper-ture. In Fig. 10 of [22] the transition regimes for thescintillation index can be observed at approximatelyD�r� � 1 and D�r� � 5 for an uncorrected beam anda corrected beam, respectively.

4. Phase and Amplitude Perturbations

A. Intensity and Amplitude Statistics

Under weak fluctuations, amplitude and intensityfields are generally lognormally distributed [13]. In-tensity, amplitude, and log-amplitude � are relatedby I � A2 � exp�2��. Knowing the spatial covariancefunction BI��� of the lognormally distributed inten-sity, the spatial correlation function RA��� of the am-plitude is readily found from

RA��� � �BI��� 1

�I2 1 �1�4

. (17)

RA��� is related to the log-amplitude structure func-tion D���� by

RA��� � exp��0.5D����. (18)

From Eq. (17) we deduce that the mean amplitude isgiven by

�A� � ��I2 1��1�8. (19)

We will consider a spatial power spectrum FI��� of theintensity based on that predicted by the Rytov theoryfor a plane wave propagating through turbulence ofconstant strength and characterized by a Kolmogorovspectrum. We thus have [11]:

FI��� � �1 ��I

2

�2 sin� �2

�I2����11�3, (20)

where �I is the characteristic spatial frequency of theintensity field. Defining the intensity correlationlength �I as the 1�e2 crossing point of the normalized

intensity covariance function [1], we have �I �1.4��I. The scaling of the spectrum FI��� is deter-mined by �I

2.

B. Phase-Amplitude Dependence

Because amplitude fluctuations arise from phasefluctuations, a dependence between the transversephase and amplitude fields must be present. Eventhough the mean square photocurrent �i2� is foundnot to be strongly affected by this dependence [3],other moments may be. It is, however, generally dif-ficult to characterize this dependence for propaga-tions through turbulent media. How often does anarea of almost constant amplitude ride an area ofalmost constant phase? The Rytov theory allows theevaluation of the correlation function ������ betweenphase and amplitude fluctuations (see [11], p. 235).This correlation function is defined by

������ �D�����

D����D����, (21)

with D����� being the cross-structure function of � and�. Tatarskii [11] shows that ����0� is significantlyless than 1 �����0� � 0.33� for the case L�k �� l0where k is the wavenumber, L is the propagationdistance, and l0 is the inner scale of turbulence, witha constant turbulence strength over the path. For themore important case L�k �� l0, it was found thatR���0� �� 1. So, for constant turbulence strength, thecorrelation decreases as the propagation distance in-creases. Assuming field statistics as given by the Ry-tov theory, numerical realizations of amplitude andphase fields containing a cross correlation can be gen-erated [23]. Investigating phase and amplitude cor-relation, other authors [24] have observed from anumerical propagation simulation that the averagephase gradient at the center of a speckle spot (i.e., aspot of high amplitude) was less than 1�3 the averagephase gradient of the whole amplitude field; unfortu-nately, the physical parameters of the simulated sce-nario are not provided.

In Subsections 4.C–4.E we look at a scintillationindex of 2. Considering the fact that amplitude fluc-tuations require a given propagation through the tur-bulent medium to develop and strengthen whereasphase fluctuations do not, we easily imagine propa-gation scenarios where phase and amplitude fluctu-ations are highly uncorrelated. In the following, weassume amplitude and phase independent in the ap-erture plane.

C. Photocurrent Approximation

To use the simulation results obtained for i�, we ap-proximate the statistics of i by the statistics of iAi�

where

iA ��W�r�A�r�dr. (22)

Considering iAi�, the C-tilt correction is performedon just i�. The behavior of i for limit cases is given in

7222 APPLIED OPTICS � Vol. 46, No. 29 � 10 October 2007

Table 2. We have i � iAi� when D�r� �� 1, D��I �� 1,or �I

2 � 0. Otherwise a difference is expected betweeni and iAi�, and also between their statistics. To quan-tify this difference, lognormal amplitude fields with�I

2 � 2 were generated. They were first generated asGaussian fields according to [25], then applied to anexponential transformation. The power spectrum ofI as given by Eq. (20) was rendered with two cases forthe intensity correlation length: r���I � 1 and r���I

� 3. Letting r���I vary from 1 to 3 covers many prac-tical scenarios [13]. The mean squares of i and iAi�

were computed, as well as their normalized vari-ances. The ratio of the mean squares �iA

2��i�2���i2� is

shown in Fig. 6. The ratio �A�,02��i,0

2 of the normal-ized variances is shown in Fig. 7. To analyze thecurves, we note first that

�iA� � �A�, (23)

�iA2� � 2��HW���RA����d�. (24)

For the case without tilt correction, we can write

�iA2��i�

2�

�i2��

2��HW���RA����d��HW���R�����d�

�HW���RA���R�����d�

.

(25)

As D�r� → , HW��� becomes constant for � values overwhich R���� takes significant values. We also haveRA��� � �A�2 over most � values and 2��HW����d�� 1. Thus

limD�r�→

�iA2��i�

2�

�i2�� �A�2

�R�����d�

�RA���R�����d�

, (26)

which is also true for the tilt-corrected signal. Equa-tion (26) explains why, at large D�r�, the mean-square ratio is more departed from 1 for the caser���I � 1 than for the case r���I � 3. In Fig. 7, weobserve

limD�r�→

�A�,02

�i,02 � 1, (27)

which is a consequence of the CLT. The error madefor the tilt-corrected signal is of a different naturecompared to the static signal. The quality of the sig-nal with C-tilt correction is overestimated becauseapplying a tilt correction to a wave with only phasedistortions is more efficient than to a wave combiningamplitude and phase distortions.

Neglecting the difference in the statistics of i andiAi�, and assuming the independence of A and �, theCDF Fi of i is given by [20]:

Table 2. Limit Cases of the Photocurrent i

D�r� �� 1 D�r� �� 1

Limit conditions �I2 → 0 D��I �� 1 D��I → D��I �� 1 D��I →

Behavior of i i → 1 i � A i → �A� i � Ai� i → Rayleigh variable

Fig. 6. (Color online) Ratio of the mean squares evaluated withand without the approximation i � iAi�.

Fig. 7. (Color online) Same as Fig. 6 but for the ratio of thenormalized variances.

10 October 2007 � Vol. 46, No. 29 � APPLIED OPTICS 7223

Fi�x� ��0

1

fi��y�FiA�xy�dy, (28)

where fi�is the probability density function of i� and

FiA is the CDF of iA. Like A, iA is taken as a lognormalvariable.

D. Signal-to-Noise Fading

Scintillation does not affect strongly the mean SNR.Assuming i � iAi�, �SNR0� is proportional to �iA

2�,which, as D increases, decreases from 1 and saturatesat �A�2. For �I

2 � 2, we have �A2 0.76, which isnegligible compared to fade depths. Figure 8 shows,in presence of scintillation, the level of the 1%-probability fade for SNR0. Equation (28) was used tocalculate the fade levels. Scintillation introducesdeep fades at low D�r� values, that is, where wave-front distortions may be negligible. Note also that theoptimum aperture size, as seen from Fig. 8, is close tothat without scintillation (see Fig. 5).

E. Comparison with Direct Detection

We now wish to compare the impact of turbulence ona coherent system and on a direct-detection (DD) sys-tem. Direct detection suffers from scintillation as thesignal photocurrent is proportional to the opticalpower detected. In a DD system, two noise types canbe distinguished: a noise constant and independent ofthe received power, such as thermal noise, and shotnoise, which increases with the received power. Thesignal-to-noise ratio SNRDD,TN of a DD system limitedto thermal noise is proportional to the square of thereceived power,

SNRDD,TN � P2, (29)

with

P ��W�r�A2�r�dr. (30)

When the receiver is limited by shot noise, we have:

SNRDD,SN � P. (31)

For different types of receiver, Fig. 9 shows theimpairment (loss) caused by turbulence on the SNRas compared to a vacuum propagation. The impair-ment is calculated as the 1%-probability fade. Thescintillation is defined here by �I

2 � 2 and r���I

� 1. Like iA, P is assumed lognormally distributed.We see from Fig. 9 that, in the presence of turbulence,a coherent system can compete with a DD systemonly under the two following conditions: (i) the aper-ture size is limited, (ii) the DD counterpart suffersfrom significant thermal noise (or another signal-independent noise). For a coherent system to reachthe loss curve of the DD shot-noise-limited system, anideal adaptive-optics system would be required.

Now one may regard the DD impairment curves ofFig. 9 as somewhat optimistic. Indeed, wavefront dis-tortions (equivalently, r�) are not taken into accountin the impairment calculation for the DD cases. Todetect all the spatial modes contained in a distortedwavefront, it should be assured that the photodetec-tor placed in the focal plane fully overlaps the ex-panded focal spot. Making the detector area larger toaccount for turbulence-induced focal spot expansionis detrimental in terms of collected background light,and detector noise and bandwidth. The choice of thedetector size and its impact on the system perfor-mance can be a complex problem and is beyond thescope of this paper. However, it is of interest to quan-tify the necessary increase of detector size as wave-front distortions worsen. The average size of the focalspot is given by the point spread function (PSF). An-drews and Phillips provide approximations of thelong-term PSF (no tilt correction) and short-term PSF(with tilt correction) [13]. Noting PSFLT and PSFST

the long- and short-term PSFs, respectively, we have

Fig. 8. (Color online) Same as Fig. 5 but with scintillation. Fig. 9. (Color online) Impairment caused by turbulence on theSNR. Results for coherent and direct detections are displayed (DD� direct detection).

7224 APPLIED OPTICS � Vol. 46, No. 29 � 10 October 2007

PSFLT�r� � exp��KD2

�1 D�r��5�3 r2�, (32)

PSFST�r� � exp��KD2

�1 0.28D�r��5�3 r2�. (33)

K is here a constant. In the absence of turbulence, thePSF approximation gives PSF0�r� � exp��KD2r2�. Let�det be the factor by which the detector diameter mustbe increased due to turbulence. We readily have

�det � �1 D�r��5�6, (34)

when no correction is applied, and

�det � �1 0.28D�r��5�6, (35)

when the tilt is corrected. To give an impression of theimpact of wavefront distortions on DD systems, Fig.10 shows the aperture-averaging factor and the de-tector diameter factor �det on the same graph as afunction of D�r�. The aperture-averaging factor,noted a, is defined as the ratio of the variance of P tothe scintillation index �I

2:

a ��P

2

�I2 . (36)

It is thus a measure for the reduction of scintillationfades. Curves of a were evaluated for r���I values of 1and 3. The factor �det by which the detector sizeshould be increased was evaluated for uncorrectedand tilt-corrected waves. One sees that scintillationreduction through increased aperture size reaches alimit set by the maximum acceptable detector size. Inthe context of detrimental background light and high

D�r� values, tilt correction is beneficial also for directdetection since it allows one to keep a smaller detec-tor and thus to reduce the field of view.

5. Conclusion

In optical coherent communications through turbu-lence, the analysis of fading probability helps inchoosing the correct receiving aperture size. Spatialfields based on Kolmogorov statistics of turbulencehave been numerically generated and have provideddistribution instances of the coherent signal. It wasfound that, without tilt correction, deleterious SNRfades are expected for D�r� � 0.5 (Fig. 5). When thetilt is corrected, such fades are expected for D�r�

� 2, which contrasts with the mean SNR climaxing atD�r� � 5 (Fig. 1).

Scintillation was assumed independent of thephase perturbations and was characterized accordingto the weak-fluctuation theory. This may not be ac-curate with respect to what can be measured in thesaturation regime [1,26,27]. An approximation wasalso introduced to study scintillation independentlyof wavefront distortions. Nevertheless, some usefulassertions can be deduced regarding coherent receiv-ers. Although scintillation can greatly impair thecommunication quality, it has little influence on theoptimum aperture size (Fig. 8). The latter is predom-inantly determined by the ratio D�r�. Aperture aver-aging of scintillation has, for an uncorrected wave, alimited positive impact because the associated fadereduction is perceivable only when �I � D � r�.

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Fig. 10. (Color online) Aperture averaging factor a and detectordiameter factor �det plotted on the same graph. As the receiverdiameter D increases less scintillation is experienced due to aper-ture averaging. But the size of the detector in the focal plane mustbe adjusted to account for growing wavefront distortions.

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