# Antenna parameters for incoherent backscatter heterodyne lidar

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Antenna parameters for incoherent backscatterheterodyne lidar

B. J. Rye

The antenna and beam geometry of lidar systems employing heterodyne reception of incoherent backscattersignals are discussed. Particular emphasis is placed on systems where the target extends uniformly acrossthe transmitted beam using topographic targets or atmospheric backscatter. The geometry is assumed tobe circularly symmetrical, but otherwise arbitrary obscurations are permitted. The effects of atmosphericscintillation are neglected. Parameters are defined which characterize the system efficiency, and the condi-tions under which these parameters may be maximized are considered.

I. IntroductionOptical techniques employing incoherent backscatter

from atmospheric scattering centers or topographictargets are currently finding application in remotesensing of the chemical composition and the physicalproperties of the atmosphere. The accuracy or sensi-tivity of these techniques will depend in some way onthe strength of the return signal and on the corre-sponding SNR. It is of engineering interest thereforeto be able to predict the variation of the former quantitywith the geometrical properties of the system. In thispaper relevant geometrical parameters are specified forincoherent backscatter systems under ideal conditionsin which the photodetector is operated as the mixingelement in an optical heterodyne receiver.

As is well known, incoherent backscatter leads toformation of a speckle pattern in the plane of the re-ceiver aperture. In atmospheric backscatter lidar thespeckles fluctuate in time as the transmitted beam scansthe scattering volume that gives rise to the instanta-neous return signal. The return field is assumed hereto have a Gaussian distribution' and cross-spectralpurity, 2' 3 which implies that the correlation function ofthe field can be factorized into time and space depen-dent terms. Only the latter are relevant to the geo--metrical problem, and the temporal parameters, whichare also, of course, relevant to signal processing, are notdiscussed here. The scattering volume has a depth (inrange) equal to half of the spatial length of the trans-mitted laser pulse. It is assumed here that this depth

The author is with University of Hull, Department of AppliedPhysics, Hull, HU6 7RX, England.

Received 23 May 1978.0003-6935/79/091390-09$00.50/0. 1979 Optical Society of America.

is small compared with the range and that the densityof the scattering centers is at most only range depen-dent. Moreover, in much of the discussion below ontopographic backscatter, the transmitted beam profileis assumed smaller than the target area. In short, forboth types of system the scattering source will be re-garded as a surface with a uniform backscatteringcoefficient. As to the geometry of the optical systemit is assumed that this is circularly symmetric, but thatotherwise the antenna geometry and transmitted beamprofile are arbitrary. The receiving telescope optics isassumed free of transmission loss and aberrations, whileall diffraction formulas are given in the Fresnel ap-proximation. Depolarization of the backscattered lightis neglected, as are the effects of atmospheric scintilla-tion; early calculations by Fried4 suggest that the latteris more realistic in the ir than at shorter wavelengths.

The fundamental properties of lidar systems thatemploy direct detection of an incoherently backscat-tered signal have been considered by Goodman. Inparticular the contribution to the noise in a return en-ergy measurement due to speckle intensity fluctuationsis shown to dominate that due to quantum noise if therate at which photoelectrons (or photocarriers) aregenerated exceeds the rate at which "correlation cells"of the speckle pattern are received. Discussion of thereturn from such a (quasi-thermal) incoherent scat-tering source closely parallels the classical optical dis-cussion of the field obtained from thermal sources wherethe speckle is usually referred to as the spatial modepattern of the observed field; in the latter case attentionwas drawn to the intensity fluctuation term as a self-beating signal in the ac coupled output of the photo-detector by Forrester et al. 6 For heterodyne receptionthe signal is also in the ac coupled output of the photo-detector, and the significance of the spatial mode orspeckle structure of the partially coherent field at the

1390 APPLJED OPTICS / Vol. 18, No. 9 / 1 May 1979

receiver aperture is to limit the mean optical poweravailable to the receiver; a discussion of the physicalprinciples involved in this is given in the review byCummins and Swinney.7 The antenna theorem of rel-evance when a heterodyne receiver is illuminated by anextended source was discussed by Siegman8; in generalthis defines the acceptance pattern of the receiver, andin the present context it determines the efficiency withwhich the local oscillator is mixed with the return, whichis of special importance in lidar systems since it is afunction of range. Sonnenschein and Horrigan 9 haveobtained analytic expressions for the return signalarising from atmospheric backscattering in a heterodynelidar system as well as for that from a nonlidar systemwhere a continuous wave source is used with no rangeresolution. However, their results are restricted to thecase where the transmitted beam profile is an untrun-cated Gaussian, and the receiver aperture is a disk.Where high power laser pulses are transmitted, thetransmitter and receiver paths may have to be spatiallyseparated to isolate the detector from the transmittedbeam, in which case antenna obscuration would haveto be taken into account in a coaxial configuration.This has been done by Degnan and Klein10-' 2 in a seriesof papers relating to coherent backscatter lidar, i.e., itis assumed that backscattering is obtained from a singleunresolvable point target rather than from a randomarray of such targets. A complete parametric descrip-tion of a heterodyne receiver would contain the conceptsof conversion gain for a coherent input signal and thequantum noise figure introduced by Arams et al., 13 butthese are omitted here as it is intended to consider onlygeometrical effects.

The layout of the paper is that the standard formulasfor the output current of a photodetector illuminatedby incoherent light are given in Sec. II; it is shown thatthe nontrivial geometrical parameters relating to thisoutput are integrals of the mutual intensity across thedetector surface so that this quantity is calculated andrelated to the complex degree of coherence of the fieldat the receiver aperture which is determined in Sec. III;the parameters that can then be defined are listed inSec. IV and the results discussed there and in Sec. V.

11. Mixing Signal

A. Receiver OutputIn a heterodyne receiver the photodetector is illu-

minated by a local oscillator (l.o.) beam (assumed co-herent) as well as by the signal and acts as a mixer, theac coupled output of which is detected using techniquesappropriate to the intermediate frequency (i.f.).14 Ifthe i.f. detector is a square-law rectifier, its currentoutput is proportional to the mean square ac output ofthe photodetector the spectral density of which can beshown to be (see Appendix A)

i(V)2)- e2F, {7('II)6(T) + 2 Re (-)IgS(T)l exp(jcOT)

x JAD SJD JR(al:a 2)JL(al:a 2)dajda2, (1)

where the Fourier transform operator is defined by theintegral'5 F^{f (-)) = Sf(T) exp(j27rrr)dT, the functiongS(T) is the normalized temporal correlation functionof the scattered field, and JR (al: a2 ) and JL (al:a 2 ) de-scribe the mutual intensities of, respectively, the re-ceived signal and the l.o. beam on the detector surface,of which the area is AD and where position is defined bythe 2-D vector a. It is assumed in deriving Eq. (1) thatthe l.o. photon flux on the detector surface (L) isgreater than the flux due to the received field and thatthe photodetector gain is unity, while recombinationnoise is neglected. Writing the subscripts 0, L, and Sto refer, respectively, to the laser output, the local os-cillator and the scattered beams, the angular frequencyW = WL - US

ANTENNAPLANE

DETECTORPLANE

Fig. 1. Geometry of the local oscillator beam.

detector is simply placed a distance v beyond the > AT sothatAR/ni f i's(0)d = AC =AT

For any other geometry, 7 a is less than one. It isquite generally maximized of course when AR/n, 1min(Ac,AR) is maximized; Eq. (31) shows that thiscondition is satisfied when the scattering is producedin the transmitted far field, and AC is approximately thearea occupied by the transmitted beam [see also Eq.(22)]. For many beam profiles (including the Gaussian)filling the transmitter aperture to maximize AC wouldentail unacceptable transmission losses. Likewise di-viding a given over-all antenna area to make AR >> ATas in (3) is wasteful since it also reduces the far-fieldvalue of AC. There will be an optimum far-fieldtrade-off between TT, AT, and AR/ni for a given beamprofile.

In atmospheric backscatter lidar, returns from thenear field of the transmitter have also to be considered.In general the usual interpretation ' 18 of the van Cit-tert-Zernike theorem by which the speckle element areaAC is approximately that occupied by a far-field dif-fraction pattern formed by a converging beam with theamplitude profile of the source intensity indicatesthat

ACQS - X2, (35)where Qs is the solid angle subtended by the scatteringsource at the antenna plane. In a lidar system designedfor general surveillance over a varying range it wouldseem sensible to use a transmitted beam which is (apartfrom diffraction) nondivergent so that the return fromthe farthest distances (in the far field of the transmitter)can be maximized. Then n - AR/Ac - (AR/X2 )QSvaries as 1/r2 , and the receiver output given by Eqs. (18)and (26) depends on range only through the termsexp(-2ar), s, and 77h. Such an arrangement, whichminimizes the signal variation with range, is convenientfor signal processing. A similar transmitter configu-ration has recently been proposed22 for use with inco-herent detection systems, the number of modes ob-served being physically restricted to a constant valuein this case by spatial filtering in the focal plane of thereceiving antenna; the coherence area in this plane is ofcourse approximately that occupied by the diffractionpattern of the receiver aperture.2 It might perhaps becommented that where direct detection is used re-striction of the number of speckle elements observed hasthe potential drawback of enhancing the intensityfluctuation noise (see Sec. I and Ref. 5).

2. Far-Field Contribution to the HeterodyneEfficiency

Suppose first that the .o. wavefront curvature isadjusted to make d = 0. In the antenna plane thebackpropagated .o. profile is then the far-field dif-fraction pattern formed by a beam initially convergingtoward the center of the antenna and having the profileof the actual l.o. beam within the aperture of the pho-todetector surface. Formally we show this by substi-tuting in Eq. (9)

Evu2(a) = IuL(a)I,

so UL(b) becomesuL(b) = Ec(b)lu(b)l,

IuL(b)I = - H,,(b/v)fD(a)IUL(a) I.

(36)

(37)

Further we suppose that the scattering surface isimaged on the photodetector, i.e., that r = u where

(1/v) + (/u) = 1/f.Then in Eq. (25), fR(b)er(b)UL(b)IfR(b)I I uL(b)I, and L(O) becomes

/Lb(U) = fRu(b)I*IfRuL(b),while the heterodyne efficiency is

t7b = [S(lOsU3) IULb()d]/[Ips(U)IgR()d#].

becomes

(38)

(39)This will have the value unity provided i'Lb (/3) = iR (/)over the spatial extent of l's (d) l, i.e., if u ~(b) = AR 1/2over values of b satisfying 0 < u-b2 < AR/n1.

The computations of heterodyne efficiency presentedby Degnan and Klein1 0 and by Fink23 are essentially cal-culations of 'lb in the case where the return is coherent,i.e., where Ii's (/3 I = 1; then ab = 1 if i'Lb (/) = i'R (3) overAR, i.e., if the aperture-limited l.o. profile described inSec. (IV.A) is employed. In Refs. 10 and 23 this resultis derived by maximizing the signal to l.o. noise ratio,whereas here the l.o. flux (L) is held constant so theresult appears as the means of maximizing the signal.For partially coherent illumination of the antenna ap-erture Eq. (39) shows that while use of the aperture-limited profile is sufficient to ensure "lb = 1 it is notnecessary; moreover mismatch of the .o. profile de-scribed by this contribution to the efficiency might beexpected to be even less critical for incoherent back-scatter than for a coherent return because of the effectof the wings of I s in the integral in Eq. (39).

3. Contribution from the Depth of Field of theReceiver

If we now consider the return from other ranges butretain the assumption d = 0, in Eq. (25)

fR(b)Er(b)UL(b) = fR(b)Er(b)UL(b).Writing

ALc(O) = fRErUL(b)fRGrUL (b), (40)

1 May 1979 / Vol. 18, No. 9 / APPLIED OPTICS 1395

the wavefront curvature component of the heterodyneefficiency can be defined as

?CllsS~c$, (41)CS S(al)|ALb(3)dfa

Comparing Eqs. (37), (38), and (40) it can be seen thataqc - 1 if the argument of fR(b)Er(b)UL(b),

kb2 1 AT/X,which is also satisfied by definition. If follows that forthis configuration any minimum in 77c must occur at rt AT/X. Criteria such as (42) are of course approxi-mate, and no account can be taken here of nonoverlapbetween the transmitted beam and the acceptancepattern of the receiver at short range.4. Local Oscillator Depth of Focus

Finally, to allow for incorrect adjustment of the l.o.curvature (d Ed 0), a third component of the heterodyneefficiency can be defined as

?d = [S|S(f)IL(fl)dit/[fgs(fl)ALc(0)dfl]. (43)This contribution is of order unity if over AR/ni,

i'Lc() i'L(), i.e., comparing Eqs. (25) and (40), ifUL(b) c u(b). A more strict condition, namely, thatUL(b) u (b) foi'all b can be specified by comparingEqs. (10) and (37) using Eq. (36), which yields

ka 2 ! 1 I r2 v v+d 2

over AD, i.e., if

I1.' 1 < " X I.(44)v v+d ~ADI

As might be expected this inequality is not especiallyrestrictive where the small detector areas associatedwith aperture-limited l.o. beams (Sec. IV.A) are used.For example an l.o. beam having the aperture-limitedprofile may be produced by placing the photodetectorin the focal plane of the receiving antenna and bringingan l.o. beam initially having a uniform profile to a focuson the photodetector using a small ancillary optical

system having the same F-number Nf as the receivingantenna; the pupil function of this ancillary systemwould be a scaled-down form of that of the antenna.For such an arrangement v + d equals the focal lengthof the focusing lens in the ancillary optics so 0 < v + d< v = f, and (44) becomes v + d >> AD/X Since theeffective photodetector area in this case (i.e., the areailluminated by the l.o.) is AD - N'X 2, (44) becomesapproximately v + d >> NFX, which should be simple tosatisfy. Vignetting of the l.o. profile by the edge of thephotodetector merely relaxes this condition further.

If larger detector areas are used to take in more of thearea occupied by the received field, according to th...

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