optimal truncation and optical efficiency of an apertured coherent lidar focused on an incoherent...

Optimal truncation and optical efficiency of anapertured coherent lidar focused on an incoherentbackscatter target

Barry J. Rye and Rod G. Frehlich

Two earlier computations of the optimal truncation of Gaussian beams for a simple, focused, coherentlidar that used an incoherent backscatter target with identical circular transmitter and receiver aperturesdiffer because they refer to different receiver geometries. The definitions of heterodyne and system-antenna efficiencies are reviewed in light of the discrepancy and are used to compare the opticalperformance of systems with apertures illuminated by beam profiles that are not Gaussian. Theheterodyne efficiency is less than 0.5 for all cases considered here.

1. Introduction

Analytic expressions were obtained long agol2 for thereturn signal from a coherent lidar that had thefollowing properties: Gaussian beam profiles, mono-static geometry (common transmitter-receiver axis),an incoherent-backscatter target of negligible depthalong the lidar axis and an infinite extent perpendicu-lar to the lidar axis, infinite optical apertures andaberration-free optics, an infinite area of photodetec-tor with spatially uniform response, and no refractiveturbulence. In two papers by Rye3 and Wang4 thatare compared here, the assumption of infinite opticalaperture was replaced by the more realistic assump-tion of hard circular apertures; truncation at theseapertures, but no other possible loss in the transmit-ter and receiver optics, was taken into account. Theoptimal truncations of the transmitter and the localoscillator beam profiles are computed, assuming eachis initially Gaussian. Even with the further simplify-ing assumptions of identical apertures, i.e., that thesizes of the transmitter and receiver apertures are thesame and that both transmitter and receiver areproperly aligned and focused on the target (theformer making redundant the assumption that thesystem is monostatic), the results of the two calcula-tions differ and the reasons for the discrepancy have

Barry J. Rye and Rod G. Frehlich are with the CooperativeInstitute for Research in Environmental Sciences, University ofColorado-National Oceanic and Atmospheric Adminstration, Boul-der, Colorado 80309-0449.

Received 11 June 1991.0003-6935/92/152891-09$05.00/0.© 1992 Optical Society of America.

not been resolved. Although the discrepancy is notlarge, the questions involved are fundamental tocoherent lidar and are relevant to future computa-tions for more complex geometries.

This paper should review and clarify the situation.The discussion is simplified by retaining all the as-sumptions mentioned above except, where stated, ofequal transmitter and receiver radius and of Gaussianbeams. After determining the cause of the differ-ence in the values obtained for the optimal truncationin Section II, we review the definition of opticalefficiency for heterodyne lidars using incoherent back-scatter, which is a desirable precursor to decidingwhat is meant by "optimal." We contrast the com-puted results of Refs. 3 and 4 and comment briefly onother computations of a similar nature 5-8 in SectionIII. A condensed discussion of these topics haspreviously appeared in conference proceedings.9 InSection IV, we investigate some systems that haveeven better efficiency than that obtained with Gauss-ian beams, in part to set in context the question ofwhat the upper limit for the heterodyne efficiencymight be.

II. Difference between the Two Treatments

First we summarize the receiver output, using thenotation developed previously.3 An expression forthe mean-square heterodyne photodetector outputcurrent can be written:

(i2 ) = 2R 2 TLPL(PR), (1)

where Ri = qel(hv) is the current responsivity ( isthe quantum efficiency, e is the electronic charge, h is

20 May 1992 / Vol. 31, No. 15 / APPLIED OPTICS 2891

Planck's constant, and v is the optical frequency).The optical local oscillator power incident on thephotodetector is PL. When we evaluate differentbeam geometries to determine the optimum, PL willbe regarded as a constant in the comparison. Itdetermines inter alia the optical bias of the photode-tector and an important source of noise, the localoscillator shot noise. As is well known, in a coherentreceiver it is necessary for the local oscillator andoptical signal beams to be aligned and to have aconstant phase relation if they are to mix optimally.Because this optimal mix cannot be achieved in a lidarunless the return is spatially coherent over thereceiver area, and then only for the return from asingle range, (PR) is defined as the effective averageoptical power that is received by the system. It maybe regarded as that part of the total received powerproducing a heterodyne output current at the photo-detector. The angle brackets used here emphasizethat (PR) is an ensemble average over the random(speckle) fluctuations that are expected when thebackscatter field is incoherent at the target. (PR ) canbe written in terms of an overlap integral of opticalfields in any plane in the system.10 Within thisintegral the local oscillator contribution is obtainedby imagining a reciprocal receiver, which is analogousto the transmitter and which propagates a beaminitiated by the phase conjugate of the local oscillatorat the photodetector backward through the receiveroptics. This equally imaginary beam is often termedthe backpropagated local oscillator (BPLO). TL rep-resents the power loss that the BPLO would have if itwere truncated at the receiver (or reciprocal receiver)aperture. The BPLO beam is depicted in the simplereceiver geometry of Fig. (a). The receiver aper-ture can be regarded as a spatial filter; the part of theBPLO striking it represents spatial frequencies (or,more physically, angles of arrival on the photodetec-tor) within the local oscillator that are not present inthe received optical signal, so this portion of the localoscillator is wasted.

An expression for (PR) can be written in a formatthat is similar to the ordinary lidar equation byincluding the overlap integral within an effectivereceiver area (A), i.e.,

(PR) (A)p TTPT

where

f2 IT(s)IL(s)d2S

(A) = (r) 2 TLPTLPL (3)

In these equations, PT is the power of the transmitterlaser before truncation; the transmission factor TT isthe fraction of PT that is transmitted through thetransmitter aperture, in exact analogy with the trans-mission factor TL of the BPLO described above; p isthe target reflectance and r is its range; A is the

(a)

RECEIVERAPERTURE

PHOTODETECTOR

(b)

LOCAL OSCILLATOR

l

<\ I T

(2) < 1 -1

BEAMSPLITTER \\

RECEIVERAPERTURE

E~_

POTODETECTOR

(c)Fig. 1. Receiver geometries in schematic form. (a) GeometryI: the untruncated local oscillator is combined with the receiversignal. (b) Geometry II: the local oscillator beam is truncated insecondary optics designed to optimize the BPLO field by avoidingits truncation at the receiver aperture. (c) Alternate geometryII: the theoretically optimum geometry. The local oscillator istruncated directly by the receiver aperture, but the BPLO is againuntruncated.

2892 APPLIED OPTICS / Vol. 31, No. 15 / 20 May 1992

[SIGNAL]

wavelength; and s is the displacement in the targetplane. The effect of the terms in the denominator ofEq. (3) is that the irradiances IT(S) and IL(S) areratioed to the powers of the beams passing throughthe transmitter and reciprocal receiver apertures.(A) is written in terms of target-plane irradiances andthe round-trip atmospheric attenuation is neglectedin the lidar equation, so these expressions can becompared directly with those used by Wang.4 If Eqs.(1) and (2) are combined, the result is

(i) = 2R2 (A) P TLPLTTPT (4)

Wang4 uses an expression, his Eq. (4), for the meanreceived signal (Sd), which in the present notation is

(Sd) = vPT _ I o (S) 121 UL (S) 12d2s, (5)

where the target-plane fields Uo'(s) and UL'(s) aregenerated by expressions for truncated Gaussianfields in the transmitter and reciprocal receiver exitpupil planes using the Huygens-Fresnel equationsand are normalized to the transmitter and local oscil-latorpowers. Writing|Uo'(s) 2 =IT(S)IPT, IUL'(S)12

=

IL(S)IPL and using Eq. (3), we find

(S) (A) P TLTTPT (6)

which is the same as Eq. (4) apart from the factor2Ri 2PL, regarded here as constant. Therefore, thecause of the difference between the results obtainedin Refs. 3 and 4 does not lie in the equations that havebeen used, but in their interpretation.

For the receiver geometry of Fig. 1(a), which we callgeometry I, the receiver is symmetrical with thetransmitter and, as we have seen, BPLO power is lostthrough truncation just as transmitter power is lost.For the two alternative geometries of Figs. (b) and1(c), which we collectively call geometry II, thisunnecessary loss is removed by shaping the localoscillator beam with either secondary optics [Fig.1(b)] or the receiver aperture [Fig. 1(c)] in such a waythat the BPLO is not truncated and TL = 1. Figure1(c) is the same as Wang's Fig. 1(a)4 and describes thegeometry that he intended to consider. The BPLOprofile is the same in geometry II as in geometry Ibetween the receiver aperture and the target. Be-tween the receiver aperture and the photodetector,however, there is no longer any symmetry with thetransmitter optics. Throughout the region betweenthe beamsplitter, Fig. 1(b), or the receiver apertureFig. 1(c), and the photodetector, the BPLO is thephase conjugate of the local oscillator. Neither figurehas a Gaussian profile within this region, but becausethe BPLO is not truncated, the local oscillator con-tains no spatial frequency components that are notuseful for photomixing with the optical signal.

Because there are no differences between geome-

tries I and II that affect the BPLO beam between thereceiver aperture and the target, (A) and (PR) are thesame for both. This is not immediately obvious fromEq. (3), in which we must set TL = 1 for geometry II,but the normalized factor IL(s)/(TLPL) remains thesame as for geometry I. In Eq. (1) for (i2), however,TL is calculated with the truncation of the GaussianBPLO beam for geometry I, whereas TL = 1 forgeometry II. The geometries can therefore be con-trasted in two respects. First, in geometry I but notin geometry II, the optics and the equations aresymmetrical with respect to the transmitter and thereceiver. Second, for given BPLO truncation, theoutput heterodyne current can be expected to besmaller, by the factor TL, for geometry I than forgeometry II.

Wang's Eq. (4), reproduced here in different nota-tion as Eqs. (5) and (6), is symmetrical between trans-mitter and receiver and contains a computed BPLOtruncation factor TL; therefore it must apply to geom-etry I. A matched geometry is considered, in whichthe transmitter and BPLO beams within the aper-tures have the same profiles and phase curvatures.For identical apertures this symmetry implies TL = TT.

Rye3 computed (A) using the following aperture-plane expression, which can be shown to be equiva-lent to Eq. (3), most easily by use of the powertheorem of Fourier transforms, within the Fouriertransform formulation of the Huygens-Fresnel prop-agation equations:

(A) = fr ILT(b)IL*(b)d'b. (7)

Here p,(b) are autocorrelation functions of the (trun-cated) transmitted and BPLO beams, b is the displace-ment from the origin in the aperture plane. (A) isjust the integral scale of the dimensionless product

J.T(b)pLL*(b). It has a direct physical interpretationas the average area in the receiver aperture planeover which the return (random) optical signal mixeswith the local oscillator, which is why it appears in thelidar equation, Eq. (2), in place of the receiver aper-ture area. The properties of the autocorrelationfunctions and (A) are recapitulated in Appendix A,because evidently they are central to any discussion ofthe optics of coherent lidar systems, including thedefinitions of efficiency discussed in Section III. Forour present purpose, we note that it is more immedi-ately obvious from Eq. (7) than from Eq. (3) that (A)has the same value for both geometries I and II.Substitution of Eq. (7) into Eqs. (1) and (2) leads to

(i2 ) = 2Ri2 P TTPT2TLPL r T(b)pLL*(b)d b.'T r

(8)

This equation is consistent with the argument givenin the paragraph following Eq. (6), which is that theonly difference between the two geometries is in thevalue of the factor TL. Geometry II is chosen bysetting TL = 1.


Therefore, the difference between the two treat-ments is that Wang4 determined the optimal trunca-tion for geometry I, with TL < 1, whereas Rye3determined the optimal truncation for geometry II,with TL = 1. The heterodyne output power (i2) issmaller for geometry I by the factor TL at a giventruncation.

Ill. EfficienciesThe beam geometry-dependent factor is TTTL(A),which appears in both Eqs. (4) and (6). By the term"optimal truncation" used here, we mean the trunca-tion at which this factor is maximized. (A) describesthe mixing within the area of overlap between thetransmitted and BPLO beams on the target side ofthe receiver aperture. At the photodetector, thelocal oscillator power may be greater than that of theBPLO propagated beyond the receiver aperture aftertruncation; the loss in mixing efficiency caused bythis truncation is contained in the term TL, which canbe adjusted by proper receiver design to have itsmaximum value of unity, at least in principle. Thereason there is a maximum in the remaining productTT(A) is that as the 1/e2 radius of the transmittedbeam is increased, there is a tradeoff between (1)reducing the illuminated area on the target, whichincreases the spatial coherence of the return signaland increases (A), and (2) increasing the power losscaused by truncation, which reduces TT.

We made TL unity in geometry II, but the questionarises whether the loss caused by local oscillatortruncation at the aperture in either Fig. 1(b) or 1(c)should also be included in the optimization. Inpractice, this truncation, say TL', entails that to keepthe local oscillator power constant at the photodetec-tor, the power of the local oscillator laser must beincreased by a factor of 1/TL'. If the power increasewere significant, then TL' would have to be includedin the equations for geometry II. Comparison of Fig.1(a) with 1(c) shows that TL' would be the same as TLfor given truncation, which would remove the differ-ence between the calculations for the two geometries.The view taken here and in a previous study3 is thatthe increase in local oscillator power caused by prop-erly shaping its beam is a negligible engineering cost,that even the total local oscillator power (much lessits truncation) is not usually considered a significantconstraint when designing a lidar system, and thatthe difference between geometries I and II shouldtherefore be taken into account.

Hence, for geometry I we maximize the productTTTL(A) for given system parameters; for geometry IIwe maximize TT(A) as TL = 1. To avoid presentingresults in arbitrary units, we find it useful to combinethe products into dimensionless efficiency parame-ters. When considering geometry II, Rye3 defined aquantity termed the "antenna efficiency" by normal-izing TT(A) to the (geometrical) transmitter area AT,because it is the transmitter that determines thestatistical properties of the incoherent backscatter-return signal in the absence of propagation fluctua-

tions. We now generalize this definition so thatgeometry I can be included and write the system-antenna efficiency as

TT(A)= TL AT(9)

AT

This expression is similar to the system-receivingefficiency defined by Zhao et al.8 and given in their Eq.(45), in which AR appears rather than AT; for identicalapertures the two definitions are the same. Frehlichand Kavaya12 have generalized the system-receivingefficiency expression for arbitrary target statistics.

Before checking the results obtained for geometriesI and II, we find it useful to contrast these definitionsof system efficiency, which are peculiar to coherentlidar, with the heterodyne efficiency, which has thepurpose of comparing the heterodyne receiver outputwith that of a direct-detection receiver, for the samereceiver optics and input signal. The concept ofheterodyne efficiency was introduced in the context ofa class of communication systems,13-16 for whichheterodyne receivers were to be used to observemodulation that was superimposed on transmittedlaser beams. These communication systems differfrom those considered here in a number of importantrespects.

1. The optical signal is often assumed to be deter-ministic (i.e., propagation fluctuations are neglected),so the mixing signal can be calculated directly fromthe optical field amplitudes, and there is no need tocompute ensemble-averaged covariance functions.

2. The input beam occupies a limited region ofspace, whereas light is returned in all directions in anincoherent backscatter lidar. Therefore, for a direct-detection lidar with an infinitely large photodetector,the detected signal increases without limit as thereceiver aperture area is increased.

3. The signal is regarded as predefined, so thequestion of optimizing it to suit a particular receivergeometry does not arise, and there is no term analo-gous to TT in the heterodyne efficiency.

Nevertheless, the notion of comparing the hetero-dyne with the direct-detection output is useful intu-itively and may be of practical value in calibrating andevaluating coherent receiver systems. Frehlich andKavaya12 adapted Fink's14 definition of heterodyneefficiency to make it suitable for incoherent backscat-ter heterodyne lidar. Their definition is based onthe ratio of the mean-square current [Eq. (1)] to theproduct of the signal currents obtained if the backscat-ter return and the local oscillator were separatelydetected by using direct detection. Referring to Eqs.(1) and (2), we would find these to be, respectively,

p TTPT(iR) = RAR r' iL =RiPL, (10)

where AR is the (geometric) receiver aperture area.


(9)

The heterodyne efficiency is then

nh =2(i0) T L(A) (11)=2(iR)iL AR

Clearly 'qa = TT(ARIAT)qh, and for the identicalaperture lidar systems with AR = AT that are consid-ered here, the system-antenna efficiency is smallerthan the heterodyne efficiency by the factor TT.

These formulas conveniently summarize some ofthe main points made here. The heterodyne effi-ciency depends on two truncation-dependent terms,TL and (A); of these, TL can be made unity by properreceiver design (geometry II). It is not apparent inthe calculations of heterodyne efficiency for communi-cations systems, in which the computations wereoften done by using fields in the photodetector plane,that these two terms were so distinguished. Thesystem-antenna efficiency depends on the heterodyneefficiency and on the further truncation-dependentterm TT. In principle the heterodyne efficiency canbe measured directly from the average heterodyneand direct-detection signals,12 and the system-an-tenna efficiency can be inferred from it, given TT(ARIAT)-

We recomputed the results of Rye3 and Wang4 forTable I. We computed (A) using the target-planeexpression, Eq. (3). The results for both heterodyneand system-antenna efficiency were computed. Forgeometry I, the optimal truncation for maximizingsystem-antenna efficiency was essentially that speci-fied by Wang, whereas for geometry II it was the sameas that obtained by Rye. This finding confirms boththe accuracy of the computations in each case and theinterpretation of the difference between them pre-sented here.

As we would expect following our earlier discus-sion, the data of Table I show that even for matchedsystems, in which the profiles and radii of the trans-

mitter and BPLO beams are the same, geometry IIprovides greater system-antenna efficiency at higheroptimal truncations than geometry I. Because geom-etry II is not symmetrical between transmitter andreceiver, the system-antenna efficiency is maximizedwhen the system is not matched3; this case forGaussian beams has also been recomputed, and theearlier results have been confirmed in Table I. Themaximum system-antenna efficiency is obtained withunmatched optics and gives an average return signalof 10% (0.4 dB) greater power compared with thematched optimal arrangement with a Gaussian localoscillator at the photodetector considered by Wang.

References are also given in Table I to authorsknown to us who have performed similar computa-tions and whose values, without exception but some-times after some interpretation, appear to be inagreement. Some of these authors used receivergeometry II (TL = 1) and some of them used receivergeometry I (TL < 1). All included transmitter trun-cation TT in the optimization, so all maximized thesystem-antenna efficiency (or an extended form of it,see the fourth paragraph below) rather than theheterodyne efficiency.

First, the earliest computation to include trunca-tion appears to have been by diMarzio as reported byHuffaker et al., 5 but this computation was neverpublished in the open literature. (PR) is calculatedfor matched and focused Gaussian beams of differentradii within an aperture of given area AT = AR and isnormalized to the power received by a referencesystem having no truncation, but having beams witha /e2 irradiance radius bo equal to the given apertureradius, so rb0

2 = AT = AR. For matched beamswithout truncation, Eq. (A.2) of Appendix A showsthat (A) = 0.5 AT' = 0.5 AR'; also, if these beams arefocused, AT' = AR' = 2rbo2 (Ref. 3). Combiningthese equations, we find that the effective receiverarea for diMarzio's untruncated reference system

Table 1. Recomputed Parameters for Maximum System Efficiency and the Corresponding Heterodyne Efficiency in a Truncated Gaussian Beam Lidara

Heterodyne Efficiency System-Antenna EfficiencyTransmitter Receiver '9h= (TLA)VAR 'ia= TL(TTA)IAT

Receiver Geometry 'YT rtrunc YR rtruc dB % dB %

I, matched 0 .8 0 2 b 1.763c 0 .8 0 2 b 1.763c -3.77 42.0 -3.97 40.1dbII, matched 0.870ef 1.626 0.870ef 1.626 -3.40 45.7 -3.72e 42.4II, maximum fla 0.815eg 1.736 1.186eg 1.192 -3.36 46.1 -3.58eg 43.8

aWe used identical transmitter and receiver apertures and focused on an incoherent backscatter target. The parameters used in thestudies by Rye3 and Wang" can be compared as follows. Rye described the truncation using the ratio YT = bo/bT for the transmitter, wherebo is the 1/e2 irradiance radius of the Gaussian beam and bT is the radius of the transmitter aperture (with a corresponding expression forthe reciprocal receiver). Wang4 used a for the l/el irradiance radius and described the transmitter truncation using the truncation ratiortn,= dl/2ci, where dl is the diameter of the transmitter aperture. The relation between the two truncation parameters is therefore

= /2/rtrnc. Tratt and Menzies7 (in their Figs. 2 and 3) described the truncation using the factor a/W = 1 /,y which is not shown here.Zhao et al.8 (in their subsec. IIIB) used PM and "lm for Yh and la, respectively. DiMarzio's calculation (Ref. 5, Figs. 4-10) is discussed at theend of Section III.

bRef. 18.CRef 4.dRef. 5.eRef. 3.fRef. 6.gRef. 7.


(the area to which he has normalized his computedreturn power) is therefore (A) = 0.5 2 = AR.

Because his equations are symmetrical between trans-mitter and receiver, we conclude that he has in effectcalculated TTTL(A )/AR wvith TL = TT, i.e. [Eq. (9)], thesystem-receiver efficiency for geometry I, which isconsistent with his optimum result quoted in Table I.

Second, in a paper whose primary purpose was todemonstrate the value of the numerical computationof heterodyne signals in real (nonideal) systems con-taining distributed truncations, Zhao et al. 8 alsoobtained the optimal truncation and maximum sys-tem-receiver efficiency for identical apertures in geom-etry I.

Third, to the best of our knowledge, the optimaltruncation for geometry II was originally determinedby Priestley6 for the case of matched-beam profiles.3However, this information also did not appear in theopen literature.

Fourth, Tratt and Menzies7 also used geometry IIfor the receiver and pointed out that the notion ofsystem efficiency could be usefully extended to ac-count for losses in the laser that provides the beamprofile, as well as for losses caused by the subsequenttruncation in the optics. They introduced a figure ofmerit for lidar design that includes not only theheterodyne efficiency and the transmitter beam trun-cation, as in Ta, but also the laser transmitter extrac-tion efficiency for a given transmitter beam profile.This profile is not necessarily Gaussian, e.g., forlasers constructed around unstable resonator cavi-ties. In the course of their work, Tratt and Menziesinvestigated, as a limiting case, the optimal trunca-tions for unmatched systems and included the compu-tation oflai with the Gaussian beam profiles consid-ered here.

IV. Maximizing the Efficiencies

It follows from our discussion so far that the hetero-dyne signal and the system-antenna efficiency aremaximized if TL = 1 and if TT and (A) are maximized.In seeking this maximum it is not necessary torestrict ourselves to Gaussian local oscillator beams,because losses that arise from shaping or profiling thelocal oscillator beam can be neglected, as argued whenjustifying geometry II. The restriction on the trans-mitter beam is that its profile should be realizable,i.e., it should be obtainable from the laser source afterpropagation to the transmitter aperture plane so thatTT can be calculated.

If TT is neglected, the beam profile that optimizesthe heterodyne efficiency can be found. In two cases,Frehlich.'7 gave integral equations to be satisfied bythe optimal profiles within identical transmit-receiveapertures: (1) for arbitrary transmitter and BPLOprofiles, when it can be shown that the profiles mustbe matched, and (2) for a given transmitter profile,when the BPLO profile is to be determined. How-ever, solutions to the equations in al but the simplestcase (untruncated Gaussian beams or Gaussian beamsand apertures) remain to be found. Here we give a

qualitative discussion of the maximization problem,illustrated by further numerical computations ofefficiencies for various geometries.

In Appendix A the properties of A) and otherrelated coherence areas are reviewed with specialreference to their definitions in the aperture plane.It is convenient to discuss the maximization of (A)with reference to the known properties of the spatialautocorrelation functions pL(b) that appear in thesedefinitions, i.e., that they are real, that 0 pLb) 1,that p,(O) = 1, and that for b greater than twice theaperture radius, they are zero.

The effective areas of the transmitter AT' andreciprocal receiver AR' are the integral scale of pLT(b)and [LL(b), respectively. Because the integral of afunction is proportional to the value of its Fouriertransform at the origin, maximizing these areas is thesame as maximizing the on-axis irradiance of thelidar on the target, or, in radar terminology, maximiz-ing the antenna gain.3" These maxima are ob-tained by illuminating the apertures uniformly.'9In this case the pL(b) are identical to the opticaltransfer function of a simple open aperture, andAT' = AT and AR' = AR. The function IIm(b) ob-tained for uniform illumination can therefore beregarded as the optimal realizable autocorrelationfunction of the optical field within a given aperture, inthe sense that it is the autocorrelation of a field thatcan be produced in practice, and it maximizes theeffective area (its own integral), which for uniformillumination in fact becomes equal to the geometricalarea of the aperture. It is plotted for the circularapertures considered here in Fig. 2, where it iscompared with the functions l.LT(b) and ~..L(b) ob-tained for the unmatched Gaussian profles of Table Ithat produce maximum system-antenna efficiency.

The theorems underlying the optimal value of(A)= f pITWb)1LL(b)d 2b are less obvious. The product

1 -... . .. . ................... .. .. ..... ..................... ................ .................... ..

0 .9 ..... N.------- .. ...... .. ... ............................ ............................. .............

0 .8. . -.........I.................................... ................. . ..........

~~ 0.7 . -.. ~~~~~~0 .6 .......- ............ ............. .......................................... .............

0 . ... . .. .. ...... .. ........... ............ ...................... ..........................................

0

0 0.2 0.4 0.6 0.8 1 12 1.4 1.6 1.8 2b (normalized to aperture radius)

Fig. 2. Autocorrelation functions of optical fields in the referencesphere of the aperture planes (see Appendix A or Ref. 3) for focusedbeams: a, pLm(b) for uniform illumination (also the optical trans-fer function for simple aperture); b, PTr(b) for Gaussian transmit-ter with YT = 0.815 (see Table I); c, p1L(b) for Gaussian receiverwithYL = 1.186.

2896 APPLIED OPTICS Vol. 31, No. 15 20 May 1992

-LT(b)L(b) also has the normalization properties ofan autocorrelation function, but it may not be realiz-able, i.e., it may not be the autocorrelation function ofa single optical field that could be produced in prac-tice. If it is realizable, then (A) is maximized if11T(b)PL(b) = AM(b). This condition can be achievedapproximately for unequal apertures, by either (1)making fLT(b) = Im(b) and L(b) 1; i.e., the trans-mitter is uniformly illuminated and the receiver aper-ture is much larger than the transmitter, so (A) =AT < AR; or (2) making PL(b) = pM(b) and pt (b) 1;i.e., the BPLO is uniform at the reciprocal receiveraperture, and the transmitter aperture is much largerthan the receiver, so (A) = AR << AT. According toEq. (11), point (2) corresponds to making the hetero-dyne efficiency unity for geometry II, whereas fromEq. (9), point (1) would make the system-antennaefficiency unity if in addition TT = 1.

For the problem considered in this paper in whichthe transmitter and receiver apertures are con-strained to be identical, these maxima are unachiev-able. It is possible, e.g., to make T(b) greater thanlm(b) over a limited range of b (see Fig. 2) and toadjust L(b) so that *T(b)PL(b) = pM(b). At somepoint, however, AT(b) must become less than pM(b),so [LL(b) > 1, which is not permitted because autocor-relation functions have their maximum value at theorigin. The requirement that the effective areascannot be larger than their geometrical areas is aneven more stringent condition on the [L(b) and on themaximum value of (A). The ii(b) products can beillustrated by plotting the integrand in the aperture-plane expression for A) as a function of b. Becausethe (b) considered here are circularly symmetric,this integrand can be written 2bp-T(b)4L(b). It iscompared with 2rbviM(b) in Fig. 3 for uniform illumi-nation and for the Gaussian beams giving optimaltruncation. (A) becomes much smaller than the

3

2 . ... ......... .... . 'S...................... .. .....................................................................

U 1~~~~~~~~~~~~~~~~~~.5-.. .... . . ...................... ................C . _ /Z . _ X\\ ...... , '

Ca~~~~

0 ~~~~~~~~a

C, ~~~~~~~~~~~~~~~ .-~~~~~.............

b

0 0.2 0.4 0.6 0.8 i 1.2 1.4 1.6 1.8 2b (normalized to aperture radius)

Fig. 3. Integrand of the effective area functions given as follows:a, AT' or AR' [Eq. (Al)] with uniform illumination; the area underthis curve is the aperture area, which is r because the apertureradius is normalized to unity; b, (A) [Eq. (7)] for uniformillumination; c, (A) for optimal unmatched Gaussian beam geom-etry, using autocorrelation functions shown in b and c of Fig. 2.

integral of 27rb IIM(b), and it is the contribution withinthe integrand from higher spatial frequencies that issuppressed.

The results from a number of further computa-tions for different geometries are summarized inTable II. First, for completeness, the truncation formaximum heterodyne efficiency using Gaussian beamswas computed. For geometry II, to maximize (A) fora given AR is to maximize the heterodyne efficiency.This maximization is not achieved at the same trunca-tion as that for the system-antenna efficiency. Be-cause of the symmetry between transmitter andlocal-oscillator beam profiles in the equations for (A),the maximum heterodyne efficiency occurs for amatched system.'7

We also investigated whether improved efficiencies(of either type) can be obtained by using beam profilesthat are not Gaussian. With uniform illuminationof both the transmitter and the reciprocal receiver,we see from Table II that the heterodyne efficiency isless than the best result for the matched Gaussianprofile (and the system-antenna efficiency is undefin-able because the transmitter truncation loss is notspecified). Tratt and Menzies7 have pointed out thatsystem-antenna efficiencies superior to the optimumGaussian profile can be obtained by using the trans-mitter profile of an unstable resonator laser with agraded-reflectivity output coupler, together with aGaussian local oscillator. The irradiance for thetransmitter geometry in this case is the difference oftwo Gaussian functions, the narrower of which isreduced in irradiance by a reflectivity factor. Theirresult was confirmed and is reproduced in Table II.We found that further improvement in the system-antenna efficiency can be obtained if the transmitterfield has a super-Gaussian profile that is proportionalto exp(-bn/Bon), where bo is the 1/e2 irradianceradius. We investigated value of n up to 20, withsuper-Gaussian local oscillators having n up to 8, andthe optimum we found had n for the transmitterequal to 16 and the local oscillator Gaussian; thisfinding is also given in Table II. This geometry givesthe largest system-antenna efficiency, Tia = 0.472,that we have discovered. The further question, firstdiscussed by Tratt and Menzies7 for the unstableresonator cavity, concerning the overall efficiency of alaser transmitter-antenna system (including laserenergy extraction efficiency) for an output profile ofthis type, is beyond the scope of this paper. Thelargest heterodyne efficiency we have found is thatgiven in Table II for the matched Gaussian profile,with Th = 0.485, but as we see, this geometry gives asystem-antenna efficiency that is not useful.

A feature of interest in Table II is that we did notfind a geometry that gave a heterodyne efficiencygreater than 0.5 (and system-antenna efficiencies areof course somewhat less than the heterodyne effi-ciency). This suggests that the behavior of aper-tured systems parallels that of untruncated Gaussiansystems, as described by Eq. (A2), in the followingcurious manner. If the transmitter is much larger


Table II. Heterodyne and System-Antenna Efficiencies for Different Transmitter and Receiver Beam Profilesa

Heterodyne Efficiency System-Antenna EfficiencyTransmitter Receiver l9h = (TLA)/AR 'qa = TL(TTA)IAT

Receiver Geometryand Beam Profile YT rtrnc 'YR rtrune dB % dB %

II, Gaussian, matched, 1.275 1.109 1.275 1.109 -3.14 48.5 -4.65 34.3maximum lsh

Uniform - - - - -3.38 46.0 - -II, difference of Gaussian 0.69, 0.60 2.05, 2.36 1.186 1.192 -3.27 47.1 -3. 4 4b 45.3

profiles, maximum la (0.9) (0.9)II, super-Gaussian trans- 0.978 1.114 1.263 1.120 -3.36 47.7 -3.26 47.2

mitter (n = 16), Gaus-sian receiver

aThe difference of Gaussian profile, which is applicable to some unstable resonator laser outputs, 7 is specified by the two values of y andrtrun, and by the reflectivity factor (R in Ref. 7). For a super-Gaussian beam of order n (see text), the product of y (which is based on the l/e 2

irradiance radius) and rtr,,, (based on the l/e irradiance radius) is the nth root of 2.bRef. 7.

than the receiver, then for both systems the effectivearea (A) approaches the effective receiver area AR'; forthe apertured system with optimal (uniform) beamprofiling, (A) equals the actual receiver area AR, andthe heterodyne efficiency for geometry II approachesunity. In contrast, if the transmitter and receiverareas are the same, then for the unapertured system,(A)is reduced to AR'/2, or half its maximum value.However, for the apertured system, the heterodyneefficiency (A )/AR would appear to be slightly less than0.5 at most. We have yet to find a proof to supportthe conjecture that the maximum value of heterodyneefficiency for any matched, focused, and truncatedsystem with identical transmitter and receiver aper-tures may be 0.5.

Appendix A. Antenna Plane Formulation for (A)

In radar, we can usually assume that the stops in thetransmitter and receiver optics are positioned at theantenna apertures, which are the largest componentsin the optical system. If this is not the case, then Eq.(7) applies in the pupil plane rather than in theaperture plane. This equation has been simplifiedby supposing that the p(b) are calculated by usingoptical fields on reference spheres centered in theantenna pupils, rather than on the planes them-selves, to remove curvature terms, which are of thegeneral form exp[-jb2 /(Xr)].3

The interpretation of (A) as an area leads toproblems if the stop in the optical system lies in thefocusing plane, in which case the pupil is at infinitedistance and (A) is infinitely large. In this case (A) isusefully replaced by the solid angle fl = (A)/r2 . Sucha system was analyzed in the context of a laboratoryexperiment in which a focal-plane stop was used as aspatial filter for both heterodyne and direct-detectionreceivers.2 0

The normalization of the autocorrelation functionspL(b) is such that they are dimensionless, and theirreal part is equal to unity at b = 0. Because theirFourier transforms are real (but normalized) irradi-ances, the pL(b) are in general Hermitian, with pL*(-b) =IUL(b) and imaginary parts equal to zero at b = 0.

(A) is just the integral scale of the product of thesefunctions, which are also Hermitian, so (A) is alwaysreal, as is obvious also from the alternative andequivalent target-plane expression in Eq. (3). Forthe present problem, with both transmitter andreceiver focused on the target, the p(b) are them-selves real functions, so in Eq. (7) the complexconjugate AL*(b) = PL(b).

Apart from being more compact, the expression inEq. (7) can be related to other parameters relevant inboth conventional and laser radar. Equation (7) hasobvious similarities with the expressions for theeffective areas of the transmitter, AT', and the recipro-cal receiver, AR', given by

AT' = EI T F(b)d2b, AR' = I' 2L*(b)db,(Al)

that appear in the standard radar definitions ofantenna gain.3 18 In the limiting case of AT' >> AR'

then (A) AR', and vice versa. For the problem ofcalculating the return with untruncated Gaussianbeam profiles,"2 we can most simply express theresult by using the relation3

1 1 1

(A) AT + '(A2)

The largest value that (A) can have is therefore AT' orAR', whichever is smaller, if there is a large disparitybetween the magnitudes of these areas. For AT' =AR' = ATR', then (A) is reduced to 0.5 ATR'.

For incoherent backscatter and in the absence ofrefractive-turbulence effects on the propagation, whichhas been assumed here throughout, AT' can be identi-fied with the (ensemble average) coherence area of asingle speckle in the partially coherent return. Theheterodyne signal is determined by the mutual coher-ence function that determines this coherence areaand not simply by the return irradiance, which isspread uniformly on average over the whole receiveraperture.

From the point of view of Fourier optics, thedeterministic functions [(b) can be identified with


the optical transfer function of an optical imagingsystem, in which the arbitrary, apertured beam pro-files considered here are replaced by an aperturedmask having the arbitrary transmission profile andilluminated uniformly. The optical transfer func-tion finds extensive use in the analysis of opticalsystems, partially because of its simple normalizationproperties, which are the same as those for [L(b), andalso because of its finite spatial extent, as it is zero forb greater than twice the aperture radius. Like p(b),for focused systems it is a real, nonnegative functionof b that decreases to zero, but not necessarilymonotonically. 21

R. Frehlich acknowledges the support of the Na-tional Science Foundation under grant ECS-8819375.

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optimal truncation and optical efficiency of an apertured coherent lidar focused on an incoherent...

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