Statistical model for fading return signals in coherent lidars

Statistical model for fading return signalsin coherent lidars

Aniceto BelmonteDepartment of Signal Theory and Communications, Technical University of Catalonia,

BarcelonaTech, 08034 Barcelona, Spain ([email protected])

Received 21 July 2010; accepted 24 October 2010;posted 27 October 2010 (Doc. ID 131960); published 6 December 2010

A statistical model for the return signal in a coherent lidar is derived from the fundamental principles ofatmospheric scattering and turbulent propagation. The model results in a three-parameter probabilitydistribution for the coherent signal-to-noise ratio in the presence of atmospheric turbulence and affectedby target speckle. We consider the effects of amplitude and phase fluctuations, in addition to local os-cillator shot noise, for both passive receivers and those employing active modal compensation of wave-front phase distortion. We obtain exact expressions for statistical moments for lidar fading and evaluatethe impact of various parameters, including the ratio of receiver aperture diameter to the wavefrontcoherence diameter, the speckle effective area, and the number of modes compensated. © 2010 OpticalSociety of AmericaOCIS codes: 010.3640, 010.1330, 010.1290.

1. Introduction

During thepast fewdecades, coherent lidar has estab-lished itself as a unique instrument in atmosphericremote sensing and its application area has broa-dened considerably (see, e.g., [1–10]) along with thetechnologies contributing to it [11]. Coherent lidarsare currently employed in a large variety of atmo-spheric applications in fields as diverse as laser vibro-metry and target identification [1,2], meteorologicalobservation and atmospheric wind determination[3,4], aerosol and atmospheric constituent concentra-tion measurements [5–7], and tracking and control ofpollutants atmospheric fluxes [8–10].

Many coherent lidar applications impose stringentpower constraints while requiring high levels of sen-sitivity and accuracy. Therefore, to optimize the lidarsystem parameters and ensure full utilization of lim-ited resources, it is of paramount importance to havea clear understanding of all the possible sources ofexternal disturbances, i.e., target speckle and atmo-spheric refractive turbulence, affecting lidar mea-surements. It is, then, natural that a problem of

continuing interest among researchers studying co-herent lidar concerns the definition of a model forthe physical mechanisms involved in the lidar pro-blem and the statistical form of the lidar signal fluc-tuations (see, e.g., [12–18]).

Evaluating the performance of a heterodyne orhomodyne lidar receiver in the presence of atmo-spheric turbulence and target speckle is generallydifficult because of the complex ways turbulence andspeckle affect the coherence of the received signalthat is to be mixed with the local oscillator. Thedownconverted heterodyne or homodyne power ismaximized when the spatial field of the received sig-nal matches that of the local oscillator [12]. Any mis-match of the amplitudes and phases of the two fieldswill result in a loss in downconverted power, i.e., fad-ing. Here we study, in a unified framework, the ef-fects of both wavefront distortion and amplitudescintillation, in addition to local oscillator shot noise,for both passive receivers and those employing activemodal compensation of wavefront phase distortion.Adaptive compensation of atmospheric wavefrontphase distortion to improve the performance of atmo-spheric systems has been an important field of studyfor many years. In particular, the modal compensa-tion method involves correction of several modes of

0003-6935/10/356737-12$15.00/0© 2010 Optical Society of America

10 December 2010 / Vol. 49, No. 35 / APPLIED OPTICS 6737

an expansion of the total phase distortion in a set ofbasis functions.

In lidar systems, signal fluctuations could resultfrom physical mechanisms other than target speckle,primarily refractive turbulence, so thesemechanismsmust also be the focus of this analysis. In general, fluc-tuations induced by turbulence are not as intense asthose due to speckle in optical remote sensing systemsbut, although their normalized variance is smaller,they still need to be considered to properly describethe performance of any practical coherent lidar.Muchwork has been published on coherent lidar theory.Early analytical work [12–17] on the problem concen-trated on the reduction of heterodyne system effi-ciency caused by beam distortion under limitedconditions that provide a reliable basis for prelimin-ary assessment of lidar performance. These mostlyheuristic analyses statistically quantified turbu-lence-induced fading through its mean and variance,although they alone are not adequate to fully charac-terize system performance. Later analyses used inheterodyne lidar [18] attempted to overcome theselimitations and fully characterize the statistics of het-erodyne optical systems by defining theoretical ex-pressions based on the path-integrated technique(see, e.g., [19]), one of the theoretical asymptoticmeth-ods for estimating the higher moments of the fieldspropagated through random media. This techniquepredicts the leading-order effects of refractive turbu-lence with minimal approximations, providing aframework valid for any typical path-integrated at-mospheric refractive turbulence. However, theoreti-cal calculation of beam propagation and the highermoments of the field is still difficult and, conse-quently, no simple analytical solutions are knownbeyond those obtained for simplified beam configura-tions and unrealistic atmospheric characterization.More recently, full-wave simulation of beam propaga-tion has been used to examine the uncertainty inher-ent to the process of lidar heterodyne optical powermeasurement because of the presence of refractiveturbulence for both passive receivers [20] and thoseusing modal compensation of wavefront phase distor-tion [21].With the two-beammodel, the lidar return isexpressed in terms of the overlap integral of the trans-mitter and virtual (backpropagated) local oscillatorbeams at the target, reducing the problem to one ofcomputing irradiance along the two propagationpaths. Although the simulation of beam propagationpermits the full statistical examination of the signaldegradation in a heterodyne receiver caused by re-fractive turbulence under general atmospheric condi-tions and at arbitrary transmitter and receiverconfigurations, the technique is computationallydemanding.

In this work, our formulation circumvents the needfor a detailed description of the compound speckleand turbulence problem. Such a specification is dif-ficult because of the inherent complexity of the pro-pagation, as well as the random distribution of theatmospheric scatters. Instead, our phenomenological

model first replaces the turbulence-distorted wave-front by an equivalent speckle representation of ran-domly dispersed coherent contributions. It thencompounds the resulting turbulence fading statisticswith target speckle statistics. The generalized Kdistribution proposed in this study to describe fadingreturn signals in heterodyne lidar receivers wasderived under the assumption that the average co-herent signal is a random variable driven by atmo-spheric turbulence. In this situation, the mean of thetarget speckle signal is smeared by turbulencespeckle fluctuations, and we need to define multiplystochastic (compound) statistics to describe the re-turn signals in a coherent lidar.

In Section 2, after discussing the general proper-ties of speckle and turbulence statistics, we proceedto develop the double stochastic representation of thecoherent lidar return signals leading to the general-ized K distribution. In Section 3, we calculate andpresent the performance for both passive receiversand those employing active modal compensation ofwavefront phase distortion. We consider the effectsof amplitude and phase fluctuations. The conclusionis provided in Section 4.

2. Statistical Model for Coherent FadingReturn Signals

For a heterodyne receiver, with average power con-straint P and noise power spectral density N0=2,γ0P=N0B is the signal-to-noise ratio (SNR) per unitbandwidthB. The SNR γ0 for a quantum or shot-noiselimited signal canbe interpreted as the detected num-ber of photons (photocounts) per pulsewhen1=B is thepulse period. Coherently detected signals are mod-eled as narrowband rf signals with additive whiteGaussian noise (AWGN). For a heterodyne lidarsystem, in the presence of target speckle and atmo-spheric turbulence, we must consider fading signals,which are signals also affected by multiplicativenoise. In the fading AWGN channel, we let α2 denotethe atmospheric channel power fading and ðP=N0BÞα2 ¼ γ0α2 denote the instantaneous received SNRper pulse.For a shot-noise-limited coherent optical re-ceiver, the SNR of the envelope detector can be takenas the number of signal photons detected on the recei-ver aperture γ0multiplied byaheterodynepowermix-ing efficiency α2: In addition to the effective delivery ofthe signal to the detector, the performance of the op-tical link also depends on the receiver sensitivity,measured in terms of received photons. For systemswith perfect spatial mode matching, the mixing effi-ciency is equal to 1. When the spatial modes arenot properly matched, the contribution to the currentsignal from different parts of the receiver aperturecan interfere destructively and result in reduced in-stantaneous heterodyne mixing and consequent fad-ing. Note that, conditional on a realization of theatmospheric channel describedbyα2, this is anAWGNchannel with instantaneous received SNR γ ¼ γ0α2.This quantity is a function of the random channelpower fading α2 and is, therefore, random. The

6738 APPLIED OPTICS / Vol. 49, No. 35 / 10 December 2010

statistical properties of the atmospheric randomchannel fade α, with mean-square value Ω ¼ a2 andprobability density function (PDF) PαðαÞ, provide astatistical characterization of the SNR γ ¼ γ0α2. Inthis study, we define a statistical model for the fadingamplitude α (i.e., SNR γ) of the received signal scat-tered by the atmospheric target after propagationthrough the atmosphere.

When the spatial field of the received signal doesnot match that of the local oscillator, referred to thereceiving aperture and assumed to be uniform there,the random channel fading

α ¼ 4

πD2

ZdrWðrÞESðrÞ ð1Þ

depends on amplitude and phase mismatches of thetwo fields incident on the receiving aperture throughthe random fluctuations ESðrÞ of a complex field pat-tern. ESðrÞ represents the amplitude fluctuationsand phase distortions introduced by atmospheric tur-bulence and target speckle in the return signal. Ingeneral, fading is a complex magnitude α ¼ αrþjαi, where αr and αi represent integrals over the col-lecting aperture of the real and imaginary parts, re-spectively, of the normalized optical field reachingthe receiver. These real and imaginary parts canbe considered as the components of a complex ran-dom phasor. The circular receiving aperture of diam-eter D is defined by the aperture function WðrÞ,which equals unity for jrj ≤ D=2, and equals zerofor jrj > D=2. Note that α has been normalized tobe unity when there is a perfect match between boththe received signal and the local oscillator, i.e.,ESðrÞ ¼ 1 for all r. An explicit expression for themean-square value Ω ¼ a2,

Ω ¼ αα� ¼�

4

πD2

�2Z

drWðrÞESðrÞZ

dr0Wðr0ÞE�Sðr0Þ;

ð2Þcan be found by writing the two integrals as a doubleintegral and bringing the averaging operator insidethe double integral:

Ω ¼�

4

πD2

�2Z Z

drdr0WðrÞWðr0ÞESðrÞE�Sðr0Þ: ð3Þ

By changing the variables of integration from r, r0 toρ ¼ r − r0 and R ¼ ðrþ r0Þ=2, and recognizing μðρÞ ¼ESðRþ ρ=2ÞE�

SðR − ρ=2Þ as the spatial coherencefunction describing the average spatial correlationlength of field fluctuations, we get

Ω ¼ 4

πD2

ZdρKDðρÞμðρÞ: ð4Þ

Here, KDðρÞ is the receiver circular aperture func-tion, i.e., the fractional area of two overlapping cir-cles of diameter D whose centers are displaced adistance ρ:

KDðρÞ¼4

πD2

ZdRW

�Rþ1

2ρ

�W

�R−

12ρ

�

¼2π

�a cos

�ρD

�−

�ρD

��1−

�ρD

�12��

; ρ≤D: ð5Þ

Note that the single integral in Eq. (4) can be char-acterized as a coherent effective area Aeff of fieldfluctuations, in which case the value of Ω,

Ω ¼ 4

πD2 Aeff ; ð6Þ

decreases with the ratio of the receiver measurementarea to Aeff . Note that the fading amplitude mean-square value Ω corresponds to the so-called apertureefficiency, a parameter describing the effective por-tion of the aperture area πD2=4 collecting heterodyneoptical power [22]. Thus the inverse 1=Ω can be inter-preted as the average number of field coherent areas(spatial modes of the field pattern) influencing themeasurement area. Consistent with this interpreta-tion of 1=Ω, the average number of photons detectedon the receiver aperture from a single pulse, i.e., theaverage SNR per pulse �γ ¼ γ0a2 ¼ γ0Ω, is the effec-tive photocount coherently detected from a singlepulse in a correlation area of the field pattern onthe aperture plane. The exact number of spatialmodes 1=Ω in the field pattern affecting the measure-ment area depends on the field spatial coherence andthe area of integration of the aperture. Exact expres-sions for Ω can only be found when the correlationfunction μðρÞ is specified. That being said, in the caseof a receiver area that is very large compared withthe average correlation length of the field fluctua-tions, the receiver aperture function KDðρÞ is muchwider than the autocorrelation function μðρÞ, andwe can factor out KDðρ ¼ 0Þ ¼ 1 from the integralin Eq. (4):

Ω ≈4

πD2

ZdρμðρÞ; ð7Þ

where ρ ≤ D. On the other hand, when the field cor-relation area is much wider than the aperture area,in Eq. (4) we can simplify μðρÞ ¼ 1 and, using the in-tegral of KD equal to the aperture area πD2=4, theresult is

Ω ≈4

πD2

ZdρKDðρÞ ¼ 1: ð8Þ

As we would have expected, in this limit, the numberof field coherent areas affecting the measurementarea approaches unity.

A. Coherent Detection in Atmospheric Turbulence

The presence of atmospheric turbulence needs to beconsidered into the statistical description of coherentlidar return signals. Turning attention to fading α,we study how amplitude and phase turbulence-induced fluctuations of the optical field define the


statistics of the fading intensity α2 ¼ α2r þ α2i . We notethat the two random magnitudes αr and αi are ex-pressed in Eq. (1) as integrals over the apertureand, hence, are the sums of contributions from eachpoint in the aperture. In order to proceed with theanalysis, we could consider a statistical model inwhich these continuous integrals are expressed as fi-nite sums over N statistically independent cells inthe aperture. Under the assumption that the numberof independent coherent regions N is large enough,we can consider that αr and αi asymptotically ap-proach jointly normal random variables. Then, theprobability density function of the length fadingamplitude α can be well approximated by a Rayleighdistribution.

Just as in a speckle pattern, the Rayleigh distribu-tion for the turbulence amplitude fading length is aconsequence of the central-limit theorem. However,under conditions of weak turbulence in which thenumber of coherent terms is small, the fadingmay ac-tually be the result of summing a small number ofterms. In this case, the fading α is not likely to beRayleigh. Rather than assuming that α is alwaysRayleigh distributed for all conditions of turbulence,it is more realistic to assume that α satisfied ageneralized Rayleigh distribution that becomesRayleigh only in the limit as the number of coherentterms N becomes large. Such a distribution is theNakagami-m distribution [23], which in essence is acentral chi-square distribution described by

pαðαÞ ¼ 2ðmNÞm α2m−1

ΓðmÞ expð−mNα2Þ: ð9Þ

Here, Γ is the complete gamma function. The pa-rameter m characterizes the amount of turbulentfading. When m → 1, the number of contribution co-herent areas N is very large and the m distributionreduces to Rayleigh. Note that the Nakagami-m dis-tribution closely approximates the Rice distribution[23], which we previously used to model the impactof atmospheric turbulence-induced fading on free-space optical communication links using coherentdetection [24]. Applying the Jacobian of the transfor-mation α2 ¼ γ=γ0, the corresponding SNR γ distribu-tion can be described according to a gammadistribution given by

pγðγÞ ¼�mNγ0

�m γm−1

ΓðmÞ exp�−mNγ0

γ�: ð10Þ

When m → 1, the gamma distribution reduces to ex-ponential distribution. The Nakagami-m parameterm and fading parameter N are measures of turbu-lence effects. Here, N is the inverse of the fadingmean-square value

N ¼ 1=Ω ¼ 1=a2 ð11Þ

and, according to Eq. (6), it describes the number offield coherent areas affecting the fading measure-ment. Also, it can be shown that the distributionmoments are given by

�γk ¼ Γðmþ kÞmkΓðmÞ �γ

k; ð12Þ

which yields, by using the average SNR �γ ¼ γ0Ω ¼γ0a2, an expression for m

1m

¼ σ2γ�γ2 ¼ a4

a22− 1: ð13Þ

Since αr and αi can be considered jointly normalrandom variables, it is possible to relate high-ordermoments with the lower-order moments and replacesthe fourth-order moment in Eq. (13) by a4 ¼3a22

− 2a4. It results in

1m

¼ 2 − 2

�a2

a2

�2: ð14Þ

Although a Nakagami-m distribution can be reducedto a Rayleigh distribution, the parameter m, by char-acterizing the amount of fading through the normal-ized SNR γ variance, gives more control over theextent of the turbulence fading. When m → 1 andthe number of contribution coherent areas is large,the normalized variance is one, as expected forRayleigh distributions. When m grows m → ∞, anda very small number of contribution terms combine,the normalized variance decreases. Here, the densityfunction becomes highly peaked around the meanvalue �γ ¼ γ0=N and there is only a small fading tobe considered.

To produce a measure of turbulence effects, itwould be necessary to develop procedures to estimatem and N. Equivalently, as Eqs. (11) and (14) describefading parameters m and N in terms of fading firstand second moments, we need to establish closed ex-pressions for �α and a2. In order to assess the impactof turbulence, both log-amplitude and phase fluctua-tions should be considered. As a consequence, thecomplex field fluctuation pattern in the pupil planeis expressed as

ESðrÞ ¼ exp½χðrÞ − jϕðrÞ�; ð15Þwhere χðrÞ and ϕðrÞ represent the log-amplitude fluc-tuations (scintillation) and phase variations (aberra-tions), respectively, introduced by atmosphericturbulence. Bringing the averaging operator intoEq. (1), we find the mean fading amplitude

�α ¼ 4

πD2

ZdrWðrÞESðrÞ

¼ 4

πD2

ZdrWðrÞexp½χðrÞ − jϕðrÞ�: ð16Þ


The mean log amplitude can be extracted out of theintegral, and assuming independence of χ and ϕ,yields

�α ¼ 4

πD2 expð�χÞZ

drWðrÞexp½χðrÞ − �χ�exp½−jϕðrÞ�:ð17Þ

We note that, because α results from atmosphericturbulence, we can consider phases ϕ that obeyhomogeneous, isotropic, zero-mean Gaussian statis-tics. Subject to this assumption, and the expressionsfor the mean of exponential functions of Gaussianvariables, we can write

expð−jϕÞ ¼ exp�−12σ2ϕ

�: ð18Þ

The phase variance σ2ϕ frequently is used to charac-terize the statistics of phase aberrations caused byatmospheric turbulence in considering a Kolmogorovspectrum of turbulence [25]:

σ2ϕ ¼ 1:0299

�Dr0

�5=3

: ð19Þ

The coefficient 1.0299 in the phase variance σ2ϕassumes that no terms are corrected by a receiveremploying active modal phase compensation. InEq. (19), the receiver aperture diameter D is normal-ized by the wavefront coherence diameter r0, whichdescribes the spatial correlation of phase fluctua-tions in the receiver plane [12]. If it is also assumedthat the log amplitudes χ are normal random vari-ables [26], we can use energy conservation, and themean of exponential functions of normal variables, toobtain classical results for the log-amplitude and am-plitude means:

�χ ¼ −σ2χ ; expðχ − �χÞ ¼ exp�12σ2χ�: ð20Þ

The mean of the irradiance β ≡ exp 2ðχk − �χÞ is givenby expð2σ2χ Þ. The log-amplitude variance σ2χ is oftenexpressed as a scintillation index σ2β ¼ expð4σ2χ Þ − 1.Replacing Eqs. (18) and (20) into Eq. (17), and notingthat the integral of the aperture functionWðrÞ is sim-ply the aperture area πD2=4, yields

�α ¼ exp�−12σ2χ�exp

�−12σ2ϕ

�: ð21Þ

Equation (4) describes the fading mean-square valuea2 in terms of the field fluctuation coherence functionμðρÞ. In atmospheric turbulence, based on the factthat the statistics of phase and log amplitude arehomogeneous, isotropic, and Gaussian, we have [12]

μðρÞ ¼ expf½χðrÞ þ χðr0Þ� − j½ϕðrÞ − ϕðr0Þ�g

¼ exp�−12DWðρÞ

�; ð22Þ

where ρ ¼ jr − r0j. Here, based on the Kolmogorov the-ory of turbulence, the wave structure function DWðρÞdescribes the statistics of optical field variation in theatmosphere in terms of the coherence diameter r0 as

DWðρÞ ¼ 6:88

� ρr0

�53

: ð23Þ

By replacing Eqs. (22) and (23) into Eq. (4), we obtainan explicit expression for the fading a2

a2 ¼ 4

πD2 2πZD=2

0

ρdρKDðρÞ exp�−126:88

� ρr0

�53�; ð24Þ

where we note that the integrand is an isotropic func-tion of ρ, so that the angular part of the ρ integrationin Eq. (4) can be performed with minimal effort. Inthe case of a receiver area that is large comparedwith the average correlation length of the field fluc-tuations, we can factor out KDðρ ¼ 0Þ ¼ 1 from theintegral and solve it in a closed form to obtain

a2 ¼ 1:09

�r0D

�2Γ�65; 1:08

�Dr0

�5=3

�: ð25Þ

Here, Γða; xÞ is the lower incomplete gamma func-tion. Physical insight into this result may be ob-tained by considering the limiting case in which thereceiver aperture is much greater than the coherencediameter r0, i.e., D ≫ r0. In this case, the lower in-complete gamma function in Eq. (25) can be replacedby the gamma function Γð6=5Þ to obtain a2 ¼1:007ðr0=DÞ2. To a good approximation, being the in-verse of the fading mean-square value of the numberof field coherent areas affecting the fading measure-ment N [Eq. (11)], the aperture can be considered toconsist of ðD=r0Þ2 independent cells, each ofdiameter r0.

By using Eqs. (21) and (24) in Eqs. (11) and (14), itis simple to express the proposed Nakagami-m para-meters N and m in terms of three well-known mag-nitudes in atmospheric turbulence studies: thelog-amplitude variance σ2χ , often used to characterizethe statistics of amplitude fluctuations; the phasevariance σ2ϕ, needed to depict the statistics of phaseaberrations; and the wave structure function DWðρÞwhich, depending on the aperture diameter D nor-malized by the wavefront coherence diameter r0, de-scribes the spatial correlation of phase fluctuationsin the receiver plane.

B. Speckle Coherent Detection Statistics

One principal performance limitation encounteredby lidar receivers is produced by target speckle


processes that appear because the backscattered sig-nal is composed of a multitude of independentphased additive complex components. When only tar-get speckle is considered, Eq. (1) represents the inte-gral of the complex field of a speckle pattern that, bythe Central Limit Theorem—the number of randomphasors contributing from a rough target to thespeckle pattern is large—it is a circular complexGaussian random process over space. The integralrepresents a linear transformation of that processand, consequently, must also obey circular complexGaussian statistics [27]. It follows that the channelfading amplitude α must obey Rayleigh statistics

pαðαÞ ¼ 2Mα expð−Mα2Þ; ð26Þ

where the inverse of the fading mean-square valueM ¼ 1=Ω represents, consistent with the previous in-terpretation, the average number of speckles influen-cing the coherent measurement. Hence, using theaverage SNR �γ ¼ γ0Ω ¼ γ0=M and the Jacobian of thetransformation α2 ¼ γ=γ0, the instantaneous SNRper pulse γ is distributed according to an exponentialdistribution given by

pγðγÞ ¼Mγ0

exp�−Mγ0

γ�: ð27Þ

In a more general case of speckle, where the returnsignal is defined not by a single speckle pattern butrather by the sum of n independent speckle patternsgenerated when n identical laser shots are averaged,the exponential distribution in Eq. (27) becomes agamma density function of order n [27]:

pγðγÞ ¼�nMγ0

�n γn−1ΓðnÞ exp

�−nM

γγ0

�: ð28Þ

The parameter M ¼ 1=Ω determines the statistics ofthe speckled SNR γ and, consequently, we need to con-sider it closely. The number of spatial speckle modesM depends on the correlation function of speckle μSand the receiving aperture function KD. In the caseof a Gaussian-shaped field pattern on the scatteringtarget, and by virtue of the van Cittert–Zernike the-orem described by the Fourier integral in the spatialcoherence function μS that propagates from the spa-tially incoherent target plane ν, we get

μSðρÞ ¼2

πW2T

Zdν exp

�−2ρ2W2

T

�exp

�−j

kzν · ρ

�: ð29Þ

After solving the Fourier integral, the result is

μSðρÞ ¼ exp�−12

�kWT

2zρ�

2�≡ exp

�−

� ρρS

�2�: ð30Þ

Here, ρS is defined as the speckle coherence radius de-scribing the spatial size of the field correlated areas

ρS ¼ffiffiffi2

p �2z

kWT

�; ð31Þ

where k ¼ 2π=λ is the free-space wavenumber and z isthe distance to the target. WT is a measure of thetransmitted laser beam width at the target plane.In atmospheric propagation,where thepresence of re-fractive turbulence must be taken into account, theintensity 1=e2 radiusWTðzÞ for an initially collimatedbeam can be approximated well as [28]

W2TðzÞ ¼ W2

0

�1þ

�zz0

�2�þ 2

�4zkr0

�2: ð32Þ

The first term inEq. (32) is the free-space beam inten-sity radius,W0 is the initial 1=e2 intensity radius, andz0 ¼ kW2

0=2 is the Rayleigh range of the beam. Theterm 4z=kr0 in Eq. (32) describes the additional beamspreading introduced by turbulence. The wavefrontcoherence diameter r0 describes turbulence for thepath where the point source is located at the scatter-ing target plane and evaluated at the lidar location(27). It is important to note that the expressions forμSðρÞ given by Eqs. (29) and (31) only consider atmo-spheric turbulence on the one-way path from the lidarto the target scattering region of interest. Because, inlidar applications, we are concernedwith a round-trippath, we also need to introduce turbulence effects onthe return path from the target plane to the lidar lo-cation. If we assume that atmospheric turbulence isstatistically independent of the fluctuations asso-ciated with the rough target, the field degree of coher-ence in Eq. (29) must be modified as

μSðρÞ ¼ exp�−12

�kWT

2zρ�

2�exp

�−126:88

� ρr0

�53�:

ð33ÞBased on the Kolmogorov theory of turbulence, weused Eqs. (22) and (23) to describe the loss of field co-herence due to phase and log-amplitude fluctuationsin atmospheric turbulence. Equation (33) is equiva-lent to Eq. (30) by rewriting the exponential argu-ments in terms of a modified speckle coherentradius. By noting that the argument of the turbulenceexponential in Eq. (33) can be approximated asð2ρ=r0Þ5=3, and replacing 5=3 by 2 in the exponent,we can obtain a good analytical expression for thespeckle coherence radius affected by round-trip tur-bulence:

1

ρ2S¼ 1

2

�kWT

2z

�2þ 4

r20: ð34Þ

Here, the coherence diameter r0 in the receiver planerefers to the round-trip atmospheric path and is a fac-tor of 4−3=5 less than the corresponding coherencelength for the one-way path used in Eq. (32) [13]. Inthe limit of weak atmospheric turbulence, the degreeof coherence of the backscattered field is not strongly


affected by atmospheric turbulence, the term 4=r20 canbe excluded, and Eq. (34) becomes the one-way pathspeckle coherence radius in Eq. (31). Also, in thislimit, the turbulence spreading in Eq. (32) can beexcluded and the beam radius can be approachedby WT ≈ 2z=kW0. In this case of weak turbulence,Eq. (31) leads to the result ρS ≈

ffiffiffi2

pW0. On the other

hand, in the limit of strong atmospheric turbulence,the first term inEq. (34) canbe excludedas turbulencedominates the speckle coherence, and we can ap-proach 2ρS ≈ r0.

In the case of a circular aperture, by replacing thespeckle correlation function μS in Eq. (29) into Eq. (4),we obtain an expression for the parameter Ω ¼ 1=Mdescribing the average number of speckle areas inthe receiving aperture D:

Ω ¼ 4

πD2

ZdρKDðρÞ exp

�−

� ρρS

�2�: ð35Þ

Here, the speckle coherence diameter ρS is given byEq. (34). By assuming a receiving area that is largecompared with the speckle diameter 2ρS, we can fac-tor out KDðρ ¼ 0Þ ¼ 1 from the integral, to obtain theclosed-form expression

1M

¼ 4

πD2 2πZD=2

0

ρdρ exp�−

� ρρS

�2�

¼�2ρSD

�2�1 − exp

�−

�D2ρS

�2��

: ð36Þ

Physical insight is obtained by considering the limit-ing case where D ≫ 2ρS. To a good approximation,the aperture can be considered to consist of M ¼ðD=2ρSÞ2 independent cells, each of diameter 2ρS.

C. Detection Statistics in Heterodyne Lidars

Detection of rough targets in a turbulent atmosphererequires that both the fluctuations of the target andthe fluctuations of turbulence be taken into account(see Fig. 1). For typical atmospheric situations, thetime scale of the fluctuations due to turbulence isseveral orders of magnitude larger than that ofspeckle-induced fluctuations (milliseconds ratherthan microseconds). The long time constant of the re-turn signal fluctuation due to turbulence means thatthese fluctuations are essentially correlated over theshort correlation time associatedwith speckle. Conse-quently, the mean of the target speckle signal issmeared by turbulence speckle fluctuations. Theremust be a compounding of the statistics for the signalaffected by speckle Rayleigh fading, conditioned onknowledge of the mean value as described by the Na-kagami-m distribution characterizing the signal af-fected by turbulence. Here, the speckle process isdriven by the turbulence randomprocess, and thepro-blem must be analyzed by the application of condi-tional speckle statistics.

We can regard the speckle distribution Eq. (28) tobe a conditional density function, conditioned onknowledge of the SNR γ0, which we represent hereby the variable x:

pγjxðγjxÞ ¼�nMx

�N γn−1ΓðnÞ exp

�−nM

γx

�: ð37Þ

The unconditional probability density function of theSNR γ is found by averaging the above density func-tion with respect to the statistics of the conditionalSNR x:

pγðγÞ ¼Z∞

0

pγjxðγjxÞpxðxÞdx ¼ ðnMÞn γn−1ΓðnÞ

Z∞

0

�1x

�n

× exp�−nM

γx

�pxðxÞdx: ð38Þ

By assuming that the speckle is driven by turbu-lence, we indicate that pxðxÞ) obeys the gammadistribution in Eq. (10):

pxðxÞ ¼�mNγ0

�m xm−1

ΓðmÞ exp�−mNγ0

x

�; ð39Þ

and, after some simple algebra, the result of theintegration in Eq. (37) can be reduced to

Fig. 1. (Color online) For typical atmospheric situations, the timescale of the coherent lidar signal fluctuations due to turbulence isseveral orders of magnitude larger than that of speckle-inducedfluctuations (milliseconds rather than microseconds). The longtime constant of the return signal fluctuation due to turbulencemeans that these fluctuations are essentially correlated overthe short correlation time associated with speckle. Here, the meanof the target speckle signal is smeared by turbulence speckle fluc-tuations, and we need to define multiply stochastic (compound)statistics to describe the return signals in a coherent lidar.


pγðγÞ¼2

ΓðnÞΓðmÞ�nMmN

γ0γ�nþm

2 1γKm−n

�2

�nMmN

γ0γ�1

2�:

ð40Þ

Here, Kβ is a modified Bessel function of the secondkind, order β. This result is a generalization of thewell-known K distribution. Equation (40) is the mainresult of this modeling effort. The model results in athree-parameter probability distribution for thecoherent SNR in the presence of atmospheric turbu-lence and affected by target speckle. These param-eters are the average number of speckles influencingthe coherent measurement M, the number of turbu-lent coherent areas in the receiver affecting the fad-ing measurement N, and the amount of fadingintroduced by atmospheric turbulence in the lidarsignal 1=m. These three parameters, along withthe number of averaged shots n and the turbu-lence-free photocount budget γ0, completely charac-terize the statistics of return fading signals incoherent lidars.

3. Performance Analysis of Coherent Lidars

The most common and well-understood performancemeasures of a coherent lidar system in the presenceof fading are the average SNR �γ and the normalizedvariance σ2γ =�γ2. They describe the inherent statisticaluncertainty of the process of heterodyne SNR γ mea-surement because of the presence of refractive turbu-lence and speckle. In order to describe these twoperformance measures, we need to know at least thefirst statistical moments of the instantaneous SNR γ.As a function of the independent laser shots n aver-aged at the receiver, with some algebra, the momentsabout the origin of the atmospheric SNR γ can be cal-culated from Eq. (40) in terms of the turbulence-freeSNR γ0 (detected photocounts), the fading parameterm characterizing the strength of atmospheric turbu-lence, and the number of spatial modesM andN thatdescribe target speckle and atmospheric turbulence,respectively,

�γk ¼Z∞

0

dγγkpγðγÞ ¼ΓðkþmÞΓðmÞ

Γðkþ nÞΓðnÞ

� γ0nMmN

�k:

ð41Þ

From here, the mean SNR can be written as

�γ ¼ γ0MN

; ð42Þ

indicating that the strength of the detected signal isdetermined by the number of photons falling on theturbulence effective area per speckle correlationarea. Of special interest is the ratio of the SNR var-iance to its mean square that describes the uncer-tainty in the lidar measurement process:

σ2γ�γ2 ¼ ΓðmÞΓðmþ 2Þ

Γ2ðmþ 1ÞΓðnÞΓðnþ 2ÞΓ2ðnþ 1Þ − 1: ð43Þ

Withm being a positive real number, Eq. (43) simpli-fies to

σ2γ�γ2 ¼ ðmþ 1Þ

mðnþ 1Þ

n− 1: ð44Þ

Note that, because the heterodyne lidar SNR resultsfrom compounding two sources of fluctuation (atmo-spheric turbulence and target speckle), this magni-tude could be greater than unity.

A. Noncompensated Passive Receivers

Figure 2 shows the effect of atmospheric turbulenceand speckle on the performance of Doppler lidars forpassive receivers, those nonemploying active modalcompensation of wavefront phase distortion. Westudy the mean SNR �γ in Fig. 2(a) and the SNR nor-malized variance σ2γ =�γ2 in Fig. 2(b) as functions ofseveral parameters: the average turbulence-freeSNR γ0, the receiver aperture diameter D, the num-ber of speckle modes M, and the strength of atmo-spheric turbulence. Turbulence is quantified bytwo parameters: the phase coherence length r0 andthe scintillation index σ2β . The value of the scintilla-tion index σ2β ¼ 1 corresponds to strong scintillation,but still below the saturation regime. When we as-sume no scintillation, σ2β ¼ 0, the effect of turbulenceis simply to reduce the coherence length r0. For afixed coherent diameter r0, as aperture diameter Dis increased, the normalized aperture diameterD=r0 increases, and turbulence reduces the hetero-dyne downconversion efficiency.

In Fig. 2(a), the mean SNR �γ ¼ γ0=MN is plottedagainst the normalized aperture diameter D=r0 fordifferent values of the speckle coherent diameter2ρS. The SNR is expressed in decibels, referenced tothe smallest aperture considered in the figure. Thespeckle coherence diameter 2ρS is expressed in termsof the turbulence coherence diameter r0. We also con-sider the nonspeckle case by plotting �γ ¼ γ0=N inFig. 2(a). The mean SNR depends on the numberof spatial modes M and N describing target speckleand atmospheric turbulence, but it is independent ofboth the amount of turbulence fading m and thenumber of averaged shots n accumulated. The effectsof scintillation on the mean SNR �γ are very weak andthey have not been considered in Fig. 2(a). The no-turbulence, no-speckle SNR ρ0 is also presented inFig. 2(a). For any aperture diameter, the value ofγ0 is proportional to D2. As is expected when speckleis not considered [1], if D is less than r0, the meanSNR �γ increases as the square of the diameter. Whendiameter D is larger than r0, atmospheric turbulencelimits the effective receiving aperture to the dimen-sions of the coherence diameter r0. This effectiveaperture defines the maximum possible coherentmean SNR �γ. When target speckle is taken into


account, the behavior of the mean SNR �γ is markedlydifferent. Here, for a fixed coherence diameter r0,when the normalized aperture diameter D=r0 islarge, we increase the number of speckle coherentareas M affecting the collected SNR. As a conse-quence, for large apertures the mean SNR �γ goesdown very quickly. For instance, when a large nor-malized aperture D=r0 ¼ 10 is considered, decreas-ing the speckle coherence diameter 2ρS from 5r0 to10r0 penalizes the mean SNR by more than 15dB.

In Fig. 2(b), we plot the normalized SNR varianceσ2γ =�γ2 against the normalized aperture diameterD=r0for different values of the number of averaged shots nused to define the lidar signal. Each shot producesγ0=n undisturbed photons, so that the total received

photocount in both the single and the multiple shotssituation is the same γ0. As shown by Eqs. (43) and(44), the magnitude σ2γ =�γ2 depends on the number ofaveraged shots n and the amount of turbulence fad-ing m, but it is independent of the number of spatialmodes M and N describing target speckle andatmospheric turbulence, respectively. In the limitof weak turbulence (small normalized aperture di-ameter D=r0, i.e., m → ∞), the normalized variancetrends asymptotically to 1=n. For a single shotn ¼ 1, the normalized variance is unity, as expectedfor the classical negative exponential speckle. How-ever, in the limit of strong turbulence (large normal-ized aperture diameter D=r0, i.e., m → 1), thenormalized variance becomes 1þ 2=n. When n ¼ 1,we reach a maximum value of 3. As we observe inFig. 2(b), the effects of scintillation are noticeablefor the small aperture diameters and must be prop-erly considered. For relatively small apertures, am-plitude scintillation is dominant, and normalizedvariance is virtually unaffected by wavefront phasedistortions. When the aperture is larger, phasedistortion becomes dominant and the normalizedvariance is substantially independent of the scintil-lation index σ2β .

B. Active Modal-Compensated Receivers

Adaptive compensation of atmospheric wavefrontphase distortion to improve the performance of atmo-spheric systems is an important field of study. Inparticular, modal compensation method involves cor-rection of several modes of an expansion of the totalphase distortion in a set of basis functions. We canconsider the effects of amplitude and phase fluctua-tions for receivers employing active modal compensa-tion of wavefront phase distortion. Typically, onlyphase variations are compensated in adaptive sys-tems, andwe need tomodify only the twomagnitudesused to describe the statistics of wavefront phasefluctuations in atmospheric turbulence, i.e., thephase variance σ2ϕ and the wave structure functionDWðρÞ, according to the degree of compensation ap-plied to the receiving system.

In [25], the statistics of phase aberrations causedby atmospheric turbulence were characterized con-sidering a Kolmogorov spectrum of turbulence. Inthat analysis, classical results for the phase varianceσ2ϕ were extended to consider modal compensation ofatmospheric phase distortion. In such modal com-pensation, Zernike polynomials are widely used asbasis functions because of their simple analytical ex-pressions and their correspondence to classical aber-rations [29]. It is known that the residual phasevariance following modal compensation of J Zerniketerms is given by

σ2ϕ ¼ CJ

�Dr0

�5=3

; ð45Þ

where the aperture diameter D is normalized bythe wavefront coherence diameter r0, which describes

Fig. 2. (Color online) Mean coherent SNR �γ (photocounts) andSNR normalized variance σ2γ =�γ2 as a function of the normalized re-ceiver aperture diameterD=r0. (a) Mean coherent SNR is shown fordifferent coherence diameters of spatial speckle 2ρS over the re-ceiving aperture. The SNR is expressed in decibels, referencedto the smallest aperture considered in the figure. The no-specklecase (dashed curve) and free-space limit γ0 are included. (b) SNRnormalized variance is studied when n equal-strength inde-pendent laser shots are averaged. Amplitude fluctuations areexcluded (solid curves) by assuming σ2β ¼ 0. When scintillationis considered (dashed curves), the scintillation index is fixed atσ2β ¼ 1.


the spatial correlation of phase fluctuations in thereceiver plane [12]. In Eq. (45), the coefficient CJdepends on J [25]. For example, aberrations up totip–tilt, astigmatism, and coma correspond to J ¼3, 6, and 10, respectively. These aberrations are onthe scale of the largest wavefront distortions. Ideally,it is desirable to set J large enough for the residualvariance in Eq. (45) to become negligible. As the dis-tribution on the size of the wavefront distortionsthat are due to turbulence approximately followsKolmogorov’s spectrum, this means that most of theturbulence power is on the previous largest scalesaberrations. Removal of the variance due to theselargest scales would increase the turbulence coher-ence diameter r0 to a great extent. For example, com-pensating wavefront aberrations up to astigmatism(J ¼ 6) reduces the residual phase variance by a fac-tor of 15 (CJ ¼ 0:0648), which is roughly equivalentto increasing the coherence diameter by a factor of 5.

Completion of the statistical description of cor-rected wavefronts requires consideration of the spa-tial correlation of phase fluctuations. To considerthe detection statistics of receivers employing activemodal atmospheric compensation, a modified form ofstructure functionDWðρÞwould be applied to the com-pensated wavefront. In a first approach, after modalcorrection of the first J modes, we may be tempted toconsider the standard structure function in Eq. (23)with, according to Eq. (45), a proportionally larger co-herence diameter (as noted, aberration compensationup to astigmatismwill produce a factor of 5 increase incoherence diameter). Certainly, a more accurate ap-proach to the problem has been described [30]. Formodal compensation, when the atmospheric wave-front phase is partially corrected, the residual struc-ture function can be written analogously to theatmospheric phase structure function as [30]

DWðρÞ ¼ 6:44

� ρr0

�53

−DJðρÞ: ð46Þ

The function DJðρÞ, expressed in terms of the modeand cross-mode shape functions of Zernike polyno-mials [29] in this analysis, describes the compensa-tion for the residual phase structure function aftermodal correction of the first J modes.

Figure 3 shows the effect of atmospheric turbu-lence and speckle on the performance of Dopplerlidars for receivers employing active modal compen-sation of wavefront phase distortion. We study themean SNR �γ in Fig. 3(a) and the normalized varianceσ2γ =�γ2 in Fig. 3(b) as a function of the receiver aper-ture diameterD, the number of speckle modesM, thestrength of atmospheric turbulence, and the numberof spatial modes J removed by the compensation sys-tem. The compensating phases are expansions up totilt–tilt (J ¼ 3), astigmatism (J ¼ 6), and fifth-orderaberrations (J ¼ 20). The no-turbulence case isalso shown.

In Fig. 3(a), the mean SNR �γ is considered. Thespeckle coherence diameter 2ρS is fixed and equal

to the coherence diameter r0. No shot averaging isconsidered; i.e., n ¼ 1. We consider the nonturbu-lence case by plotting �γ ¼ γ0=M. We have found thatfor most typical lidar apertures, wavefront phasefluctuations are the dominant impairment, and com-pensation of a modest number of modes can reduceperformance penalties by several decibels. For in-stance, when a large normalized aperture D=r0 ¼10 is considered, compensation of just J ¼ 20 modesincreases the mean SNR �γ by more than 15dB.Speckle effects are not diminished by the use ofphase-compensation techniques.

In Fig. 3(b), we separately quantified the effectsof amplitude fluctuations and wavefront phase

Fig. 3. (Color online) (a) Mean coherent SNR �γ and (b) SNR nor-malized variance σ2γ =�γ2 are shown for different values of the nor-malized receiver aperture diameterD=r0 and the number of modesJ removed by phase compensation. Turbulence is characterized bya fixed phase coherence diameter r0. Without loss of generality, inall cases considered in these plots, the correlation diameter of thespeckle field 2ρS is equal to r0. The compensating phases are ex-pansions up to tilt (J ¼ 3), astigmatism (J ¼ 6), and fifth-orderaberrations (J ¼ 20). The no-correction case (J ¼ 0) is also consid-ered. The dashed curve corresponds to the measurement asso-ciated with no-turbulence, one-pulse speckle. The free-spacelimit γ0 is included in (a). (b) Amplitude fluctuations are excluded(solid curves) by assuming σ2β ¼ 0. When scintillation is considered(dashed curves), the scintillation index is fixed at σ2β ¼ 1.


distortion on the amount of turbulence fading σ2γ =�γ2.The value of the scintillation index σ2β ¼ 1 is kept be-low the saturation regime. When the turbulencereaches the saturation regime, wavefront distortionbecomes so severe that it would be unrealistic to con-sider phase compensation.We have identified two dif-ferent regimes of turbulence based on the receiveraperture diameter that has been normalized to the co-herence diameter of the wavefront phase. When thenormalized aperture diameter is relatively small, am-plitude scintillation dominates and, as phase fluctua-tions have little impact, performance is virtuallyindependent of the number of modes compensated.For small normalized apertures, only when scintilla-tion is excluded are the effects of phase compensationappreciable. For the single shot n ¼ 1 considered inFig. 3(b), when no-turbulence is contemplated, thenormalized variance is unity. When the normalizedaperture is larger, amplitude fluctuations becomenegligible and phase fluctuations become dominant,so that high-order phase compensation may beneeded to improve performance to acceptable levels.In any case, when no-compensation J ¼ 0 is used,the amount of fading introduced by turbulence is im-portant even for apertures as small as the coherencediameter r0. However, when J ¼ 20 modes are com-pensated, the amount of fading introduced by turbu-lence is a small fraction of speckle fading; thus it canbe excludedeven for apertures as large as ten times r0.

4. Conclusions

We have focused on elucidating those implications ofthe atmospheric propagation problem that bear onthe design and reliability of lidar coherent systems.We present recent studies on the impact of phase andamplitude fluctuations on Doppler lidars using co-herent detection and consider, in a unified frame-work, the effects of wavefront distortion, amplitudescintillation, and diffuse target speckle on perfor-mance, for both passive receivers and those employ-ing active modal compensation of wavefront phasedistortion. As the effects ascribed to turbulence andspeckle are random and must be quantified, we de-fine amathematical model for the probability densityfunction of the received coherent signal after its pro-pagation through the atmosphere. In our model, theparameters describing the signal statistics dependon turbulence and target conditions and the degreeof modal compensation applied in the receiver. Weprovide analytical expressions for every key modelparameter.

By noting that the impact of atmospheric turbu-lence on coherent receivers can be compounded withtarget speckle statistics, we have shown that theprobability density function of the lidar return signalis an example from the family of K distributions. Ourformulation circumvents the need for a detailed de-scription of the compound speckle and turbulenceproblem. Such a specification is difficult because ofthe inherent complexity of the propagation as wellas the random distribution of the numerous target

scatters. Our statistical model is developed fromthe fundamental principles of scattering and turbu-lence. This model results in a three-parameter distri-bution for the return signal, thereby making itpossible to gain information on the physical proper-ties of target and atmospheric turbulence underlyingthe parameter definitions. In this analysis, we esti-mate the parameters using a heuristic theory of co-herent signals.

Wehaveprovided analytical expressions for the sig-nal statistical moments and have used them to studythe effect of various parameters on performance, in-cluding turbulence level, signal strength, receiveraperture size, speckle effective area, and the extentof compensation. We have separately quantified theeffects of amplitude fluctuations andwavefront phasedistortion on system performance and have identifieddifferent regimes of turbulence, depending on the re-ceiver aperture diameter normalized to the coherencediameter of the wavefront phase. When the normal-ized aperture is larger, amplitude fluctuations be-come negligible, and phase fluctuations becomedominant. We have found that, for most typical lidarsituations, compensation of a modest number ofmodes can reduce performance penalties by severaldecibels.

This study was partially funded by the SpanishMinistry of Science and Innovation (MICINN) grantTEC 2009-10025.

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Statistical model for fading return signals in coherent lidars

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