Time-delayed statistics for a bistatic coherent lidaroperating in atmospheric turbulence

Farzin Amzajerdian and J. Fred Holmes

Analytical expressions are derived for the time-delayed statistics and the heterodyne signal power as afunction of the transmitter-receiver spacing for the general case of a bistatic coherent lidar. The effect of theturbulence-induced correlation between the outgoing and return paths was included and the 5/3 law form ofthe wave structure function was used.

Introduction

The study of speckle propagation in the turbulentatmosphere is of considerable interest because of wide-spread use of coherent lidars and imaging systems.However, early work on the statistics of the returnedintensityl 2 included the assumption of independenteffects of turbulence on outgoing and return paths.

The effect of the correlation between the incidentand reflected waves has been considered by Cliffordand Wandzura3 in analyzing the performance of amonostatic heterodyne lidar where the paths are coaxi-al. They predicted that the signal degradation causedby the presence of atmospheric turbulence is less se-vere for the monostatic lidar than previously predictedby the independent path assumption. Later work byAksenov et al. 4 takes the dependence effect of theturbulence on the two paths into account for the twoasymptotic cases of spherical and plane waves reflect-ed from a diffuse target. Their work showed an in-crease in intensity fluctuations as the separation be-tween the source and the receiver decreased.

In this paper the work of Clifford and Wandzura isextended to the general bistatic case, which in the twolimits of large and small transmitter and receiver sepa-ration approaches the independent path and the mon-ostatic path cases. Analytic expressions are obtaineddescribing the time-delayed statistics, the covariance

Farzin Amzajerdian is with Litton Aero Products, Moorpark,California 93021-2699. J. Fred Holmes is with the Department ofApplied Physics and Electrical Engineering, Oregon Graduate Insti-tute, 19600 N.W. von Neumann Drive, Beaverton, Oregon 97006-1999.

Received 29, November 1989.0003-6935/91/213029-05$05.00/0.© 1991 Optical Society of America.

and the heterodyne signal power, and the signal-to-noise ratio reduction factor.

Mutual Intensity Function

The extended Huygens-Fresnel principle is utilized toobtain the effect of refractive turbulence on the outgo-ing and return paths. The field at the target is thenwritten as

U(p) = k exp(ikL) drU(r) exp ikp - r 2 +,Pj(pr)2wriL f L2L vI Ij (1)

where V11(p,r) is the complex phase factor describingthe effect of turbulence on a spherical wave travelingfrom point r in the transmitter plane to point p in thetarget plane, L is the path length, and U(r) is thesource amplitude distribution, which for a single-modelaser beam is given by

U(r) = U0 exp(-r 2 /2a02

- ikr2 /2F,), (2)

and where ao, Ft, and k are the transmitter beam radi-us, focus distance, and wave number, respectively.With the turbulence effect from the target to the re-ceiver denoted by 0 2(P,P), the field at the receiver is

U() = k exp(lkL)I dpT(p)U(p) ex p p-P12 + p,P], (3)

where T(p) is the complex reflectivity function of thetarget and p is the location of the field point in thereceiver plane. If the transverse coherence length ofthe incident wave is much greater than the surfacecorrelation distanced then

(T(pi)T*(p 2)) = 4i T 26(po - P2),

0(4)

where ( ) denotes an ensemble average over the targetstatistics and T0

2 is the target reflectivity coefficient.The mutual intensity function is then given by

20 July 1991 / Vol. 30, No. 21 / APPLIED OPTICS 3029

r(p 1,p 2 ) = (U(p 1 U*(P2 ))

=(2 L) JJdpjdp2 (T(pi)T*( 2) (U(PlU (12)

X ex [k (IP1 -p12 - IP2 - P212) + V12(P1,Pl) + 2*(P2,P2) (5)

Using Eqs. (1), (2), and (4), we can reduce the expres-sion above to

ToUo /k 2 r.k (p2 2] r(P1lP 2) = ,L (21__) exp~[ 2 (p12

-P 2 JJ dpdr 1 dr 2

x exJ- r12+ 2 + i 2k (1- L/F)(r 2 -r 2

2)

ik [p (r1 + r2) + P (-P 2)J} ( exp[, 1(prI)

+ ,1*(p,r 2 ) + P2(P 1,P) + t2*(P 2,P)]) (6)

where the ensemble averages over the atmosphericturbulence and target statistics are taken indepen-dently. Making use of the principle of reciprocity[I2(P,P) = 1&(P,P)], one can write the atmospheric tur-bulence ensemble average above as

( ) = exp{- 1/2 [DO(ri - r2) + DO(p,-P2)

+ D1(r, - P2) + D1(r 2-) -(r- ) -D(r 2-P2 ), (7)

where D,(x) = 2(x/po)5/3 is the wave structure func-tion,5

Po = [0.546Lk2J Cn2(W)dwJ

is the turbulence coherence length, and C 2 is thestructure constant of the index of refraction (Kolmo-gorov spectrum). The log-amplitude covariance func-tion C has not been included in Eq. (7), since itseffects are negligiblel2 in comparison to the wavestructure function at low-to-intermediate turbulenceand longer wavelengths (log-amplitude variance<0.05). Making the change of variables

r=ri-r 2 , 2R=ri+r2 ,

P = P1 -P 2 , 2q = P + P2 ,

Eq. (6) can be rewritten as

T02U0

2 k 2 /k \r(P1P2)= °2° - L expli-p-q l drdRH(rpRq)

X ex{-4 2- 2 + i -(1- L/F)r. RJ

X J dp exl{-i k p . (r + p)

where H is defined as

H(r,p,R,q) = exp{- [D,(r) + D,(p) + D,(R + - q + 2

+ D(R -- q - P- D,(R + r q -

(8)

- D, R ~--q + 2 )]} (9)

The integration over p and r can be performed bymaking use of the following identity:

dp ex-i L-P (r + p) = -kL (r + p).

Thus the mutual intensity function simplifies to

r(P1 ,P2 ) = T0 U2 exp(i L P q)exp(_p2/4.2) dR

XexJ-- - i-(1-L/Fp*R)lexp[- (2p5 +21R-qLao 2 LJL P0

5 35/3

-|R-q -p5/ - IR-q + 5/3)] (10)

It can be seen from the expression above that, as thetransmitter-receiver spacing increases, the correlationbetween the two paths becomes less significant com-pared with the correlation between any two pointsalong each path. Consequently, Eq. (10) reduces tothe solution for the independent path assumptionwhen q is much greater than ao and p, regardless of theturbulence coherence length po.

Time-Delayed Statistics

The time-delayed mutual intensity function is definedby

r(p 1,p 2,r) = (U(P11t)U(P 2,t 2)), (11)

where = tl - t2 . With the time delay r included inEq. (6), the last term must be replaced by

(exp[,Pj(p,r,t) + Pl*(,r2,t2) + 2(P1,Pt) + P2*(P2,Pt2)I)-

(12)

It can be seen from expression (12) and Fig. 1 that anadditional time-delayed correlation exists as the atmo-spheric eddies along one path drift through the otherpath with the crosswind. Using the principle of reci-procity, one can write expression (12) as

expl- /2 [D#(ri - r2,0,Vr) + D#(pi - P 2,0,VT)

+ D#(r, -p 2,0,Vr) + D,-(r2 - P1 0 _Vr)- D (rj -P,0,0) - D(r 2 - 2,0,0]I. (13)

The time-delayed wave structure function is given by5

Target Plane

ReceiverAperture-- --

- - - - Turbulence

TransmitterAperture

Transmitter - Receiver Plane

Fig. 1. Turbulent eddies drift with the wind through both outgoingand return paths.

3030 APPLIED OPTICS / Vol. 30, No. 21 / 20 July 1991

D,[(p,0,V) = 2.91LK2 Cn2(w)I(1 - w)p - V(w)r1513 dw,

where w is the normalized path length from the trans-mitter-receiver to the target. The first two terms inexpression (13) are the result of the individual pathsand the rest describe the dependence of the two paths.Following the same steps as before and integratingover dp and dr, the time-delayed mutual intensity canbe written as

r(P 1 ,P2 ,r) = T 2 expQ kp q) exp(-p2/4a 2) dR

X ex{- 2 i- (1- L/F,)p. R]

X exp 15/3 (q - R + p15/3 + q-R - p15 3)JPo5 3

2.91Lk 2 2

+ d1C(+ 2()[(1 -)p - V(W)-r1513

+ 11- W)p + V(W) 1513 + (1 - w)(R - q) - V(W)T 15/3

+ (1- w)(R - q) + V(W)r15'3]1}.

I .C

0.8t-AUT0c 0.60VARA 0.4NCE

0.2k

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Time-Delay NSEC

Fig. 2. Autocovariance for the independent path, bistatic, andmonostatic cases.

0.9

(14)

It has been shown in the past that, for most atmo-spheric conditions and operating at longer wave-lengths, the fields at the receiver can be assumed to bejointly Gaussian," 2' 6 for which the time-delayed co-variance of the intensity is given by

C(P1 ,P2 ,r) = Ir(p 1lp2",)1 2.

w

< 0.5

0U

(15)

Using Eq. (14) in Eq. (15) and performing the integra-tion numerically, we obtained the time-delayed covari-ance function. The normalized autocovariance versustime delay , and the covariance versus detector spac-ing P are shown in Figs. 2 and 3 for three cases:monostatic, bistatic, and with the independent pathassumption. For these results it has been assumedthat the transmitter beam is focused on the target andthat the wind and the turbulence are uniform.

It should be noted that the turbulence outer scalesize has not been considered here and is taken to beinfinite.5 Therefore, the statistics may approach theirasymptotic form of independent paths more rapidlywith q than is shown in Figs. 2 and 3 when q is largerthan the outer scale size.

Figure 2 shows the autocovariance falling faster withtime delay for smaller spacings. This reduction inautocovariance can be significant in some applicationssuch as coherent differential absorption lidar measure-ments7 and remote sensing of atmospheric turbulenceand crosswinds,8'9 where the measurements are aver-aged in time and consequently the amount of correla-tion between samples effects the overall error. It isinteresting to see from Eq. (14) that the effect of theturbulence on the time-delayed covariance functionapproaches additive or multiplicative behavior in as-ymptotic cases of uncorrelated and correlated paths asnoted by Aksenov et al. 4 This can be seen by setting q

0.0 0.0 25.0

DETECTOR SPACING IN MILLIMETERS

50.0

Fig. 3. Covariance function for the independent path, bistatic, andmonostatic cases.

>> ao, p, and Vr for the independent path case and bysetting q = 0 and Vr >> p and ao for correlated paths.

Heterodyne Signal Power

The turbulence-induced correlation between the twopaths can affect the performance of a heterodyne de-tection system. The heterodyne signal power is nowdeveloped as a function of transmitter-receiver spac-ing.

The heterodyne signal power is given by

(i2) = (If U(p)ULo(p)dp12 )

= f f dpldp 2 ULO(pl)ULo*(P2 )r(Pl P2 ), (16)

where ULO(p) is the local oscillator field distributiongiven by

ULOWp) = UA exp[ (p- 12

1+ -ikp ] (17)

which has been assumed to be an untruncated Gauss-ian beam of radius dlo centered at a with a phase front

20 July 1991 / Vol. 30, No. 21 / APPLIED OPTICS 3031

a,,Independent> . Pi~~~ths

'.," ,..,qo15 cry......

q 0ss "a

aO'2.5 c.

pow 25cm

V 5 /sec

angle specified by the aiming vector s. Vector a there-fore defines the location of the center of the receivingaperture, which is also the distance between the trans-mitter and receiver, and FLO is the effective local oscil-lator focus distance backpropagated through the re-ceiver optics. On substituting Eqs. (10) and (17) intoEq. (16), the signal power can be expressed as

(i2 L-O2 fJ dpdq exp(-Iq - a2#2

• ex i ~ .) p q] exp(i-P p-a - ikp .s)

i[ L ( FLO) ] FLO)

X exp(-p 2 /4#02 -2 /4a0

2 ) f dR exp(-R 2 /a02 )

[ exp[iL (1T)p R] exp[ -

I- L Ft ~~[Po 5/3

X (2p5 /3 + 2IR - q15/3 - R - q - p1/3- IR - q + P153)] (18)

where the integration variables are changed to p and q.To account for any aiming error or misalignment

between the signal and local oscillator wave fronts,aiming vector s was used in Eq. (18). However, for thewave fronts of the signal and local oscillator to besuperimposed, vector s must line up with the vectorconnecting point a to the origin of the coordinate sys-tem in the target plane, thereby making the phasefactor exp(-ikp . s) equal to exp[-i(k/L)p . a]. Invok-ing this condition and once again making the change ofvariables = R-q, 2 = R + q, and setting a0 = ogives

(i 2) TO2UAUo J dp exp(-p2 /2aO2 ) exp(i a p a)

X| dtd7 ex -i ,Y + 2LP)1.. 2aO 2

X exp -i 2LO) p-] exP(-217 _ -ll /aO2 - I + a12/2ao2)

X exp[ 1 (2p5/3 + 2' - t --p15/3 - + P5/3)], (19)

where

ka = k2 ( -L

PLO -2 ( L)

Now the integration over t can be performed, result-ing in

Ti,2 ) 2ALO2o p 2 21-i2 TO UA U° 2O 2 dp exp(-p 2 /2a0

2 ) ex.-i 8i 2

L p

X ex|-i. t Lop a1 | dt exp(-t + a12 /2aO2)2a02 p f

X ex i LO) P ] exp o51 3 (2p5/3 + 2t5/3

I t4 - p15/3 - I + P153). (20)

1.

SNR

RE 0.DUC

I0N

FA a.CT0R

0.001 "-0.1 1.0 10.0

RATIO OF RECEIVER APERTURE RADIUS TO RHO SUB-ZERO

Fig. 4. Heterodyne SNR reduction factor vs the normalized receiv-ing aperture radius for the independent path, bistatic, and monosta-tic cases.

Considering the usual case for which Ut = QLO andexpressing the integrand in terms of new variables p' =p/ao and (' = V/ao, we can write the resultant signalpower in a more convenient form as

2ir2 T02UA2U 2aO6

T2PLOPO (°2) FD(aO,r,,Q), (21)

where PLO and P0 are the local oscillator and the trans-mitted power, respectively, ao = a/ao, r = ao/po, Q= 2 t = QLO, and F0 is the signal-to-noise ratio reductionfactor3 caused by atmospheric turbulence, given by

FO(aoroQ) = 1I dp' exp(-p 2/2) exp(-iQp' ao) dt'

X exp(-Ip + ao12/2) exp(-igp' * ')

X exp[-ro5/ 3 (2p'5 /3 + 2t'5/3 - it' - p'l5/3 - I' + p'I5/3)]. (22)

The two complex phase terms in Eq. (22) can be ne-glected when the transmitter and receiver are focusedon the target or when the far-field condition holds, i.e.,kao2 << L. For the case of independent paths, thesignal reduction factor is given by

Fo=J dp'ex{[P4 (1+ °) exp[- QUt2 + °2°LO)]

X exp[-2(r~p') 5 /3 ], (23)

where 2 LO is redefined as

LO kB2 (1 -L )L FLO/

For comparison with Eq. (22), 800 will be set equal to aoand QLO equal to Qt, which reduces Eq. (23) to

F0 = dp' exp- P- (1 + 2)] exp[-2(rop') 5/31. (24)

3032 APPLIED OPTICS / Vol. 30, No. 21 / 20 July 1991

The signal-to-noise ratio reduction factors given byEqs. (22) and (24) were numerically evaluated, and theresults are shown in Fig. 4 for the focused case where Q= 0 and a defocused case where Q2 = 4 and for severalnormalized spacings ao between the transmitter andreceiver. As a increases, the bistatic solution ap-proaches the independent path assumption solution,as it should. If Ot = QLO and ao = 0 are used in Eq. (20),it becomes equivalent to the Clifford-Wandzura re-sult.

Summary and Conclusions

The time-delayed mutual intensity function, autoco-variance function, and covariance function for a bista-tic coherent lidar have been formulated, utilizing theextended Huygens-Fresnel formulation and a 5/3 lawform of the wave structure function, taking into ac-count the statistical dependence of the turbulencefluctuation on the outgoing and return paths. Theautocovariance was found to drop off significantlymore slowly with time delay as the bistatic distancebetween the transmitter and receiver increased,whereas there is little change in the covariance func-tion. Using the formulation for the mutual intensityfunction, we determined the heterodyne signal-to-noise ratio reduction factor Fo as a function of thebistatic spacing, the Fresnel numbers of the transmit-ter-receiver and local oscillator, and the ratio (ro) ofthe transmitted beam radius to the transverse phasecoherence length. From Fig. 4 it can be seen that F0becomes smaller (worse signal-to-noise ratio) as thebistatic spacing increases. Also, for the case where theFresnel number Q = 4, the curve for the monostaticcase (ao = 0) actually increases slightly.(enhancement)before decreasing with increasing r. However, thebistatic cases do not exhibit this behavior. In the Clif-ford-Wandzura paper, there is a curve for the monos-tatic and independent path cases for = 16. Theenhancement is even greater for the monostatic casewith the larger Fresnel number and again there is noeffect for the independent path case. It appears then

that the heterodyne signal enhancement is maximumfor the monostatic case and increases with increasingFresnel number. The enhancement of F0 thereforebehaves qualitatively like backscattered intensity en-hancement 0 and may be caused by the same phenom-enon.

This research was sponsored in part by the U.S.Army Research Office. The authors thank the review-ers for several useful suggestions and for alerting us toa serious error in the original version of the paper.

References1. M. H. Lee, J. F. Holmes, and J. R. Kerr, "Statistics of the Speckle

Propagation Through the Turbulent Atmosphere," J. Opt. Soc.Am. 66, 1164 (1976).

2. J. F. Holmes, M. H. Lee, and J. R. Kerr, "Effect of the Log-Amplitude Covariance Function on the Statistics of SpecklePropagation Through the Atmosphere," J. Opt. Soc. Am. 70,355(1980).

3. S. F. Clifford and S. Wandzura, "Monostatic Heterodyne LidarPerformance: the Effect of the Turbulent Atmosphere," Appl.Opt. 20, 514 (1981).

4. V. P. Aksenov, V. A. Banakh, and V. L. Mironov, "Fluctuationsof Retroreflected Laser Radiation in a Turbulent Atmosphere,"J. Opt. Soc. Am. A 1, 263 (1984).

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7. D. K. Killinger and N. Menyuk, "Effect of Turbulence-InducedCorrelation on Laser Remote Sensing," Appl. Phys. Lett. 38,968(1981).

8. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, "Use of Scintilla-tions to Measure Average Wind Across a Light Beam," Appl.Opt. 11, 239 (1972).

9. J. F. Holmes, F. Amzajerdian, R. V. S. Gudimetla, and J. M.Hunt, "Remote Sensing of Atmsopheric Winds Using Speckle-Turbulence Interaction, a CO2 Laser, and Optical HeterodyneDetection," Appl. Opt. 27, 2531 (1988).

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