Effect of atmospheric turbulenceon heterodyne lidar performance

Mikhail S. Belen'kii

The effect of atmospheric turbulence on heterodyne lidar performance is studied by use of scatteringtheory. A theoretical analysis is carried out for both bistatic and monostatic lidar systems withindependently variable transmitter and receiver parameters in regimes of weak and strong intensityfluctuations. The conditions of validity of a diffuse target model for description of the optical wavescattering by aerosols in a turbulent atmosphere are presented. The equations for signal powerdegradation and the conditions under which the time-averaged output of a heterodyne lidar does notdepend on either turbulent conditions of propagation along the path or the transmitter parameters,including transmitter coherence length, are obtained. A physical interpretation of these results is given,and a comparison with the data of previous theories is made.

Key words: Atmospheric turbulence, laser beam propagation, optical wave scattering, heterodynelidar performance.

Introduction

The effect of atmospheric turbulence on heterodynelidar detection efficiency has been studied theoretical-lyl 2 and experimentally.3 4 However, as it has beennoted in Ref. 3: "In general, there are no publishedtheoretical analyses on the detection efficiency of abistatic or monostatic heterodyne lidar which allowsthe practical freedom to vary independently the sizeof the transmitter, receiver and detector. What isrequired for lidar system design is a theory whichpredicts the turbulent field coherence length effect onheterodyne detection efficiency."

In a recently published paper- such a theory hasbeen developed. However, a simplified diffuse targetmodel, which did not take into account a randomfinite number of the particles and their randompositions in the range-spread scattering volume, wasused in this paper for description of wave scatteringin a turbulent atmosphere. The conditions of valid-ity of this model for the description of real wave-scattering processes by discrete independent scatter-ers are unknown. Besides Ref. 5 there is no detailedanalysis of the conditions under which the time-

When this research was performed the author was with Elec-tronic Power Devices, Inc., 3482 Oakcliff Road, Atlanta, Georgia30340. He is now with the Georgia Tech Research Institute,Georgia Institue of Technology, Atlanta, Georgia 30332.

Received 16 March 1992.

0003-6935/93/275368-05$06.00/0.© 1993 Optical Society of America.

averaged output of the heterodyne laser radar doesnot depend on the turbulent conditions of propaga-tion along the path or the transmitter parameters,which are important for the design of practical sys-tems and the analysis of theoretical and experimentalresults.

This paper is devoted to solving the above problems.The theory of optical wave scattering by discreteindependent scatterers in a turbulent atmosphere isused to study the influence of atmospheric turbulenceon the detection efficiency of heterodyne lidar sys-tems and to determine the conditions of validity of thediffuse target model.

Coherent Properties of Scattered Radiation

Let a space-limited laser beam be propagated in aturbulent atmosphere and scattered by the ensembleof discrete independent scatterers with the origin of aCartesian coordinate system placed in the center ofscattering volume. In a single-scattering (Born) ap-proximation6 and wave zone of an individual particle(L >> a2/x, where L is the distance from the center ofthe scattering volume to the lidar, and & is theaverage size of the particles) the scattered wave fieldin the transmitter plane is written in the form

N

Us(R) = A.G(R, Rm)P(Rmij)U(Rm),m=1

(1)

where Ui(Rm) = | d2RG(Rm, Ro)Uo(Ro) is the inci-dent wave field, G(Rm, Ro) is the Green's function of

5368 APPLIED OPTICS / Vol. 32, No. 27 / 20 September 1993

the turbulent medium, Uo(Ro) is the source field, Amand Rm are the backscatter amplitude and the radiusvector of the mth particle, N is the number of theparticles in the scattering volume, and P(Rmii) =exp(-Rm112/1,12) is the spatial weighting function of thescattering volume. The longitudinal size of the spa-tial weighting function l! = [(7r)1/2 cTi/2] is determinedby the pulse duration, Ti, and the speed of light, c.

Determining Am in Eq. (1) in the form Am=2T'1/2a1/2(am, 0)k- 1, where is the scattering crosssection of a particle with radius am, 0 = 2 Tr - I R l/L isthe scattering angle and k is the wave number, wecalculate the mutual coherence function of the scat-tered wave F2s = (U(R1 )U*(R2)). We make indepen-dent averagings over the longitudinal variable ofrandom particle positions in the scattering volumeRml (the particles are considered to be distributedhomogeneously), the number of the particles N(whichhas a Poisson distribution and where N >> 1), andparticle sizes and refractive-index fluctuations alongthe path.

After averaging we obtain

F2S(Rl, R2)

4Tr2IPalllCI

= k2 ||| d2Rod2Rotd2Rm± Uo(Ro)Uo*(Ro')

X r 40 (Rm±, R1, R2) X r 4 t(R 1 , R2; RO, Ro'), (2)

where

r4O(Rml; R1, R 2 )

- (2-L exp{2 [(Ro2 - Ro 2) + (R12 - R22)

+ 2Rm(Ro' - Ro + R2 - R)]}

F4t = ((Ro, Rmj)G(Rm., Rj)

x G*(Ro', Rm±)G*(Rm±, R2));

P3a is the volume aerosol backscatter coefficient (m-sr-'), P = (N/V)f f(am)dam, f(am) is a function ofparticle size distribution, and N is the average num-ber of particles. The angle brackets in Eq. (3) indi-cate averaging over the refractive index fluctuationsalong the path, and G is a random part of Green'sfunction.

It should be noted that according to scatteringtheory,6 Eq. (1) is fulfilled under the condition ofsmall optical thickness of the atmosphere ( < 1), andif the optical thickness is not small (T 1), additionalterms that describe the effects of multiple scatteringin the scattering volume will appear in Eq. (1). Inaveraging over the longitudinal variable of randomparticle positions (Rm11) the two following assumptionshave been used: the longitudinal intensity correla-tion scale of a sounding beam (liii = L) is larger thanthe scattering volume length, L >> CTi/2 and thescattering volume length is larger then the wave

length cTi/2 >> . Under the last condition thesecond term is Eq. (4), which describes the interfer-ence of the waves scattered by different scatterers(m # k),

N N N Nl

r2S = (Us(Ri)Us*(K3)) ( I + : 1 \m=km=k mkmfkI Rml

(4)

becomes negligible.Equation (2), except for the constant before the

integrals, coincides with the corresponding expres-sion for the mutual coherence function of a wavescattered by a diffuse target5 7 (Lambert surface);thus the conditions of validity of the diffuse targetmodel for description of the optical wave scattering bydiscrete scatterers in the range-spread volume in aturbulent atmosphere have the form

r < 1, L >> a 2 /A, L >> cri/2 >> . (5)

Furthermore, we consider a regime of weak (q =L/kpo 2 << 1), and strong (q >> 1) intensity fluctua-tions. Here po is the atmospheric turbulent coher-ence length, which is given by

Po = [1.45 k2 f c 2()(1 - /L)5/3dg] 3'5

where c 2 is the refractive index structure characteris-tic. In a regime of weak intensity fluctuation (q <<1) we use the following representation of the randompart of Green's function G = exp(iS), where S is arandom phase of a spherical wave, calculated by ageometrical optics approach and distributed accord-ing to the Gaussian law. After averaging Eq. (3) overthe refractive index fluctuations, we have

F4t = exp{-/2F(RO, Ro'; R1, R2)1,

where

+ D(RI - Ro') - D(Rj - Ro') - D(R 2 - Ro'). (6)

The two first structure functions D in Eq. (6)describe the phase fluctuations of the sounding andscattered waves, respectively. The other four func-tions describe the mutual correlation of phase fluctu-ations of the sounding and scattering waves. Conse-quently for a bistatic scheme R/po > 1, R = R ,1 +R21/2, the function F takes the form

F(R Ro'; R1, R2 ) = D(Ro - Ro') + D(R1 - R2 ). (7)

We take the initial field distribution at the transmit-ter aperture to be of the form

Uo(Ro) = UO exp(-R 02 /2a0

2 - ikR02/2Fj), (8)

where U0 is the amplitude of the transmitter field, aois the effective transmitter aperture radius, and F is

20 September 1993 / Vol. 32, No. 27 / APPLIED OPTICS 5369

(3)

the initial radius of the wave-front curvature of thebeam.

Exact results for the coherence function 12s mightbe obtained by performing the integration in Eq. (2)with the use of Eqs. (6)-(8) and the exact representa-tion for the structure function D(R): D(R) =2(Rl/po)5/3. Unfortunately, Eq. (2) in this case be-comes analytically intractable. The approximate re-sults might be obtained with the use of the quadraticapproximation D(R) 2(R/po)2. Such approxima-tion has been used in many cases, including, thecalculation of a mutual coherence function of asounding beam8 and a reflected wave.9 Rigorousasymptotic and numerical estimations of the error inthis approximation were obtained in Ref. 10. Afterthe integration of Eq. (2) we have

F2s(RP = Ba l2 exp[-(l/pca2 + /po2)p2 + iRpI

where the total transmitter intensity flux Ioy =r, ao2 U 02.

Then the degree of coherence is

1F2 (R, p) I= (I(R + p/2)I(R - p/2))

= exp[-(1/pa 2 + 1/po2)p2], (9)

where Pca = 2L/kaef is the aerosol coherence lengthdetermined by the VanCittert-Zernice theorem,"'aef2 = ad2 + a2 is the effective size of the soundingbeam on the distance L from the transmitter, ad2 =

a02(1 - L/F) 2 + (L/kao)2 is the diffractive size of the

beam, and at = 2L/kpo is the turbulent addition, p =I Pi - P2 1 Note that Eq. (9) above coincides with Eq.(2) in Ref. 12 and with Eqs. (38) and (40) in Ref. 9.

From Eq. (9) it is easy to see that the smallest scalefrom Pca and PO determines the coherence length ofthe scattered wave Pc Indeed, if we determine thecoherence length, Pc, from the equation y(Pc) =exp(- 1), we have

k [ ( L2 ]l-/2PC = 4L [fF1 + f)A1 + 8q }, (10)

where = kao2/L is the Fresnel parameter of thetransmitter aperture. In a homogeneous medium(q = 0) the coherence length Pc = Pco = 2L/kaddetermines the speckle structure scale of the scat-tered wave caused by the decorrelation of the scatter-ers. In a turbulent atmosphere under the conditionsq << 1, ql << 1, l < 1, or qQ-l << 1, l > 1 for acollimated beam (L/Fi = 0) and condition ql << 1 fora focused beam (L/F = 1), when Pco << po thecoherence length PC = Pco and does not depend on theturbulent conditions of propagation along the path.

On the other hand, in a regime of strong intensityfluctuations (q >> 1) under the conditions ql >> 1 orqQ-l >> 1 for L/Fi = 0 and condition ql >> 1 forL/Fj = 1, when Pco >> po, the scattered wave coher-

ence length Pc = po/21/2 and does not depend on thetransmitter parameters fl and L/Fi.

For a monostatic system (R << p), from Eqs.(2)-(8) it follows that

-y(p) = exp[-(1/pca 2 - /po2)p2 ]. (11)

The coherence length, Pc, as is seen from Eqs. (9) and(11), depends on the distance R. According to Eq.(11) the coherence length is equal to the correspond-ing value in a homogeneous medium Pc = P =2L/(kad). This is connected to a tilt-compensationmechanism2 under the double passage of the wavesthrough the same inhomogeneities of a medium.However, as Eq. (11) is approximate, as a conse-quence of the use of a quadratic approximation forthe structure function D(R), the last results for y(p)are true only under the following conditions: L/F, =0, ql < 1 for l < 1 or qf- << 1 for l > 1 andL/Fj = 1, qQ << 1, when pco << p and coherencelength does not depend on the turbulent conditionson the path.

In a regime of strong intensity fluctuation (q >> 1)function F4t is represented as the sum of the productsof the second moments of the random functions Gaccording to the Gaussian statistics laws'3 (zero-orderfourth-moment solution) and a quadratic approxima-tion is used for the structure functions D(R). Asimple calculation shows

F2S(R, P) = r2(l)(R, p) + r2(2)(Rp), (12)

where

F2(')(R, p) = I exp[-(1/pCa2 + 1/po2 )p2 - (ik/L)Rp]

(13)

P2(2(R Ra [ 2 p 2- p

F2 (2 ) (R,p) =I 2 exp ------ R pl2a 0 a0

2 4a02 L

(14)

where

I = BalI0L2.

The two components of the function 2S haveessentially different properties. The first, 1 F2 1 1,does not depend on the distance R and has a character-istic scale equal top, for the variablep. This compo-nent describes the coherence of the scattered radia-tion for a bistatic scheme (R >> pO, ao). The secondfunction, r2(2), depends on the distance R and isnonzero only in the bounds of the transmitter aper-ture (R < a). It coincides, except for the constant1p0

2 /2a 02, with the coherence function of the sound-

ing beam at the transmitter plane. This componenttakes into account the field fluctuation correlationbetween the direct and reflected waves and describeswhat has been called "the effect of the long-rangecorrelation."' 3 " 4 In the following section, usingEqs.(9)-(14), we calculate the signal power of a heterodynelidar detector.

5370 APPLIED OPTICS / Vol. 32, No. 27 / 20 September 1993

Heterodyne Lidar Detection

According to Ref. 2, the time-averaged signal powerof a focused heterodyne detector is given by

(j 2 ) = 2 ffd2p d2RF2s(R, P)F20(R, p)W(R, p), (15)

where 20 is the mutual coherence function of thelocal oscillator signal, 6 is a detector efficiency, and Wis the product of the transmission functions of thereceiving telescope. We take the functions W andF20 in the following form:

W(, ) =exp(-4-2 -R F p) (16)

r2 0(R, p) = exp(- Rp)X (17)

where aL is the receiving telescope aperture radius, FLis the telescope focal length. It should be noted thatEq. (16) corresponds to the following approximationof the transmission function:

W(p) = exp(-2p2/aL2 - ikp 2 /2FL),

which gives the intensity distribution in the focalplane coinciding with the Eiry picture1 with an errorof approximately 9%.

It follows that the expression for the signal powerof a heterodyne bistatic lidar (R >> PO, a) in aturbulent atmosphere is

(j2= = 2 rraL4 IF1(aL/pO, LIF), (18)

where

Fl(aL/Po, QF) = [ + aL2/p02(1 + lF 2)]1'

lF2 = 1 + q-l-1/4 + q-lQ(1 - L/F,) 2/4.

(19)

Equations (18) and (19) describe the reduction ofthe signal power that is due to turbulence and has thesame form in the regime of weak (q << 1) and strong(q 1) intensity fluctuations. This allows indepen-dent variation of the transmitter and receiver param-eters.

These equations have a clear physical meaning:

(a) The reduction of signal power is determinedby the ratio of the receiving telescope aperture radiusto the turbulent coherence length, aL/po, and theeffective receiving telescope radius is limited", 2 by themagnitude ofp0 .

(b) The influence of the transmitter parameterson a heterodyne lidar signal is determined by thefunction lF, and this influence decreases with in-creased turbulent intensity along the path (F - 1 forq>> 1).

Note that for a multimode beam,8 the function IF

has the form

IF2 = 1 + q-'l-1/4 + q-1i(1 - 4 + q qk/4,

(20)

where qk = L/kPk 2 , and Pk is the transmitter fieldcoherence length. From Eqs. (18)-(20) it followsthat for the fixed turbulent condition along the path(q = constant) the maximum signal power may beobtained with a single-mode (qk = 0) focused (L/F, =1) laser beam under the condition L ». 1. Thus themultimode structure of the sounding beam reducesthe signal power of a heterodyne lidar, ( 2 )B -

Pk 2/aL 2 , under the conditions q-lqk »> 1 and qkQ >i1, and signal power does not depend onpk if q-lqk <<

I or qkl << 1.Based on Eq. (18) we obtain the conditions when

the signal power of heterodyne lidar does not dependon the transmitter parameters L and L/F or turbu-lent condition of propagation along the path. Equa-tion (18) shows that under the limitations ql- << 1,L > 1 for L/Fj = 0 and Qq << 1 for L/F = 1 thesignal power coincides with the corresponding valuein a homogeneous medium (q = 0)

F, = F,0 = {1 + 4a2 L+ 2(1 - F)]} 1 (21)

and does not depend on the turbulent conditionsalong the path. I

On the other hand, under the conditions ql » 1 orqQ-' »> 1 for L/Fj = Oand lq 1 for L/F = 1 thereduction factor F, = (1 + 2aL2/p 2)-' p02/2aL2 anddoes not depend on the transmitter parameters.Note that the above conditions for the signal power ofa heterodyne lidar and the scattered wave coherencelengthp, coincide with each other.

For a monostatic lidar (R = 0) in a regime of weakintensity fluctuation (q << 1, qQ-' << 1 for L/F = 0and Lq << 1 for L/F. = 1), the signal power coincideswith the corresponding value in a homogeneousmedium (q = 0)

(i 2) = C2'raL 4IF1O. (22)

In the regime of strong fluctuation (q > 1) we have

(i2 ) = 2 (i )B. (23)

Thus the signal power of a monostatic heterodynelidar is larger by a factor of 2 than that for a bistaticone. The signal power is connected with the contri-bution of the component 2(2), which describes thelong-range correlation of the scattered wave. Notethat the above result was also obtained in Ref. 5.

The comparison of the results for the reductionfactor, F1, with the corresponding data from Ref. 5shows that under the conditibns when lidar signalpower does not depend on the transmitter parameters

20 September 1993 / Vol. 32, No. 27 / APPLIED OPTICS 5371

or on the turbulent conditions on the path, the resultscoincide with each other. For comparison with datafrom Ref. 2 we put, as has been done in Ref. 2, aL =

Do/2, where Do is the diameter of the transmitter andreceiver aperture. Then we see that for a mono-static lidar (L/F = 1, q >> 1) our result F, =4(po/Do)2 coincides with the data of Ref. 2, but for abistatic configuration [F, = 2(po/Do)2] it differs fromthe data of Ref. 2 by a factor of 2.

Discussion and Conclusion

According to Eqs. (9) and (10), there are two tenden-cies12 that change the scattered wave coherence lengthas one moves further from the scattering volume:on the one hand, the scale Pc, according to VanCittert-Zernike theoreml (Pco = 2L/kaef) increasesbecause of diffraction on the scale aef; on the otherhand, Pc decreases because of the turbulent distortionof the wave-phase front ( - po). The relationbetween these two scales, Pco and p0, determines theconditions under which the coherence length of thescattered wave, Pc, and the signal power of a hetero-dyne lidar (j2) do not depend on the turbulent condi-tions on the path or on transmitter parameters.

Indeed, for a collimated beam L/F, = 0 with > 1in a homogeneous medium (q = 0) the aerosol coher-ence lengthpco = 2L/kad. When this value is smallerthan the turbulent scale Po(pco2 /po2 = 4ql1 << 1),the scattered wave coherence length, Pc, and thesignal power (2) do not depend on the turbulentconditions on the path (Pc = Pco, F, = Flo). On thecontrary, under the condition (pco

2/po

2 = 4qQli >> 1)the valuepc = po/21/2 and F, = po 2 /2aL 2 do not dependon the transmitter parameters. Thus the physicalsense of the above conditions becomes clear.

Note that when the time-averaged output of theheterodyne laser radar does not depend on the trans-mitter parameters or the turbulent conditions ofpropagation along the path, the above conditions areimportant for analysis of theoretical and experimen-tal results. For example, in Ref. 5 it was shown that"the assumption of statistically independent path isvalid for a monostatic configuration if the irradiancefluctuations at the target are small." However, it iswell known6 that irradiance fluctuations of the focus-ing beam become small under the condition Qq << 1.As was shown above, under the same condition thesignal power of a heterodyne lidar does not depend onthe turbulent conditions of propagation along thepath. Therefore the above conclusion from Ref. 5 is

valid when the influence of turbulence on the lidarsystem is negligible.

In general, the above results may be used forheterodyne lidar system design, planning of experi-ments, and prediction of the effect of atmosphericturbulence on heterodyne lidar performance

The author thanks R. G. Roper for helpful discus-sions.

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