receiving efficiency of monostatic pulsed coherent lidars 1: theory

Receiving efficiency of monostatic pulsed coherent

lidars. 1: Theory

Yanzeng Zhao, Madison J. Post, and R. Michael Hardesty

The receiving efficiency 1 as a function of range z is investigated for pulsed coherent lidars using a theory that

relates ,(z) to the transmitted laser intensity and the point-source receiving efficiency fl,(r,z). The latter can

be calculated either by a forward method, or by a backward method that employs the back-propagated local

oscillator (BPLO) approach. The BPLO method is efficient and accurate provided that cascaded diffraction

effects inside the lidar system are properly taken into account. The theory is applied to the ideal case to

examine the optimization of the system when both transmitted and BPLO fields at the antenna are Gaussian,including optimum telescope aperture.

1. Introduction

For an atmospheric sounding monostatic coherentlidar system with a short transmitted laser pulse themean power of the ac detector signal Pac due to theatmosphere-backscattered signal (ABS) from range zis

P.= (li (Z)12) TSTO(z)e2,dMR,ac = I~z)R = (nV)2

X [J VS(zP)V(P)dPj[J V(zP)VL(P)dP]) (1)

where Pac is in watts, is is the detector current (A); R isthe output resistance of the detector (Q); T, is the two-way transmittance of the optical system; Ta(z) is thetwo-way transmittance of the atmosphere; e is theelectron charge (C); fld is the detector quantum effi-ciency which is assumed to be uniform over the detec-tor area; M is the gain of the detector; h is Planck'sconstant (J s); v is the optical frequency of the trans-mitted laser (Hz); VL(P) and V5(z,P) are respectivelythe complex amplitudes of electric field (V/m) at apoint P on the detector, due to the local oscillator (LO)and the atmosphere-backscattered signal (ABS)(without atmospheric attenuation) from range z; andAD is the detector area (m2). The angle brackets standfor the statistical average of the variable enclosed.Complex conjugation is indicated by *.

Yanzeng Zhao is with Cooperative Institute for Research in Envi-

ronmental Sciences, University of Colorado/NOAA, Boulder, Colo-

rado 80309-0449; the other authors are with NOAA/ERL/WavePropagation Laboratory, Boulder, Colorado 80303.

Received 12 May 1989.

Equation (1) can be rewritten asT~e2XdMRcEGPL ( ()Q)()

Pac(Z) = T 7McPLT (z)/3z)W2Z)~),2(hv)2 a

(2)

where c is the speed of light (m/s); E0 is the pulseenergy of the laser; PL is the LO power incident on thedetector (W); O3(z) is the volume scattering coefficientof the atmosphere (m- 1 sr-1 ); Q(z) = AR/Z 2 is the solidangle of ABS subtended by the antenna (telescope) ata point in the scattering plane (sr); (AR is the antennaarea); and q(z) is the dimensionless system receivingefficiency. Comparing Eq. (2) with Eq. (1),

(([JAn L(ZP) [JAD V (zP)VL(P)dP] (

wherecE0

PS = 2 O'Q(2

is the power (W) of ABS excluding atmospheric atten-uation. Equation (3) can also be written as

7(Z) = ([fAD Vsn(zP)VL(P)dP1][j VS((zPP)VLn(P)dP]>)X (5)

where Vsn and VLn are the power-normalized signaland local oscillator fields (m- 1 ), respectively. It can beseen from Eq. (3) that 1(z) is the ratio of the meanpower of the real ac signal, to the mean power of the acsignal in an ideal case when the ABS power received bythe antenna is completely received by the detector, andthe LO and ABS diffraction pattern at the detector areperfectly matched with each other (i.e., the power nor-malized amplitude distributions and the phase distri-butions of the LO and the ABS are exactly the same).i1(z) can be close to 1 when the signal is backscatteredfrom a signal far-range axial point (or a very small area

(4)

1 October 1990 / Vol. 29, No. 28 / APPLIED OPTICS 4111

around the axis) and the LO size is matched well withthe Airy pattern of the signal. For realistic atmo-spheric sounding monostatic coherent lidars, however,this ideal cannot be achieved. Atmosphere-backscat-tered signals are from particles distributed throughoutthe volume of a finite-sized transmitted beam; the sizeof the transmitted beam is about the same as that ofthe receiver. Only for those on-axis particles can n beclose to 1. For the off-axis particles, the diffractionpattern of each scattering point at the detector is alsooff-axis and does not match that of the LO, thus thereceiving efficiency is less than unity. The more off-axis the scattering particles are, the lower their receiv-ing efficiency. As a result, the overall receiving effi-ciency 1(z) is much <1. Even for a coaxial lidar systemwith ideal Gaussian beams and a common telescope forboth transmitting and receiving, can never exceed0.4, as we shall discuss in Sec. III. From anotherperspective, (z) may be defined as the ratio of theeffective coherent power to the total (incoherent) pow-er of the ABS. Since the area of coherence of the signalfrom any scattering volume is always less than the areaof the receiver/transmitter, 77(z) must always <1.

Equation (3) can also be written as

Pd(Z) ([IAD VS(z)P)V P d[J V(zP)V (P)dPIn A) P. L ., a = 71pfmix, (6)

wave at the telescope is a uniform plane wave and thusis completely coherent), which is not the case for atmo-spheric soundings in general. Sonnenschein and Hor-rigan2 calculated the mixing at the antenna plane.They were also able to use the analytical expressionsfor the fields in the calculation by assuming untruncat-ed Gaussian distributions for both the LO and trans-mitted beams. (Actually they ignored the truncationeffect of the beam at the antenna.) However, in mostcases beam propagation within lidar systems is verycomplicated, and hence the rigorous calculation of x1(z)from Eqs. (3) or (5) requires extensive numerical com-putations. The amount of computation can be short-ened considerably by using a back-propagated localoscillator (BPLO) field based on Siegman's theorem.3Rye 6 and Wang7 mathematically propagated the LOfield backward from the detector and mixed it with thesignal in the scattering plane instead of on the detec-tor, thus significantly simplifying the calculation.The method has been widely used (e.g., Post,8 Sha-piro,9 and Tratt and Menzies10) but with a generaltendency that ignores the truncation (and hence thediffraction) of the BPLO field within the receiving

where p is the ratio of the ABS power Pd(z) fallingwithin the detector area to that received by the anten-na, P = (cEoi2)#(z)Q(z); and i7mix is the mixing effi-ciency.

From Eq. (6) one can see that the receiving efficiency17(Z) is determined by two facts:

(1) Power loss-represented by 7p in Eq. (6), whichshows the effect of spillover of the ABS power beyondthe aperture of the detector and/or other optical ele-ments between the detector and the receiving tele-scope;

(2) Mixing-represented by 77mix, which is the ratioof the averaged effective coherent power to the totalincoherent power received by the detector, indicatingthe averaged effect of the matching of the amplitudeand phase patterns of the LO with those of the ABS atthe detector.

Another range-dependent system characteristicfunction, the range response function R(z) ( 2 ) of acoherent lidar, is often used. It is related to 77(z) by

R5(z) = 2(z) (7)z2

The two range-dependent functions (z) and R(z) arevery important to the design, calibration, and applica-tion of coherent lidars, but currectly calculating thesefunctions is difficult. In early work, simplified modelswere used so that analytical expressions would apply.Degnan and Klein' used an analytical form for thediffraction pattern of the ABS on the detector andsubsequently calculated the mean power of the hetero-dyne signal. However, their formulas are valid onlyfor very distant axial point objects (when the incident

system, except for the diffracting effect of the antennaaperture; only Shapiro9 considered the effect of thefinite size of the detector. It is usually assumed thatthe passage of light between the entrance and exitplanes in an imaging system can be adequately de-scribed by geometrical optics, and therefore that dif-fraction effects play a significant role only during pas-sage of light from the object to the entrance pupil andfrom the exit pupil to the image plane (see Good-man'1 ). This is often not the case for realistic lidartransceivers where the signal beams may traverse alengthy path through a cascaded optical system withmany optical elements (such as lenses, polarizers, andsteering mirrors). Thus the beam is often subject tosignificant truncation and diffraction. (Optical beampropagation through complex optical systems was dis-cussed by Yura and Hanson.12) We have found thatneglecting such diffraction effects within the systemcan cause major errors in calculating 11(z).

To clarify how we calculate the receiving efficiency,we give a detailed derivation for (z) in Sec. II withemphasis on the diffraction effects in the receivingsystem. In Sec. III we apply the theory to the idealcase where both the transmitted and BPLO fields atthe antenna are Gaussian, and discuss the optimiza-tion of receiving efficiency, SNR, and the antennaradius. In a separate paper (i.e., the second part of thiswork) we apply the theory to the numerical calcula-tions of 77(z) for the NOAA/ERL/Wave PropagationLaboratory coherent CO2 lidar. In these two papers,the effects of atmospheric refractive turbulence is notincluded.

4112 APPLIED OPTICS / Vol. 29, No. 28 / 1 October 1990

-L d-, s

11. Single-Point Receiving Efficiency q,, and Overall

Receiving Efficiency q

A. The Transmission Function K,(Q,P,z) and the Single-Point Receiving Efficiency 77(Qz)

The instant random ABS field (excluding atmo-spheric attenuation) V(P,z) in the detector planefrom a scattering layer in the atmosphere is the vectorsum of the backscattered radation field from all theparticles within that layer. If the c'oncentration of thescattering particles is high enough (it is usually thecase in the atmosphere), the sum can be expressed asan integral:

V5(PZ) = f j U,(Qz)B(Qz')K(QPz')dz'dQ, (8)

where Q is a typical point in the scattering plane S at

and substitute Eqs. (11) and (12) into Eq. (3), theexpression for 7(z) becomes

(13)(Z) = (A(z)A*(z))cEO(Z)/2 '

where

A(z) = !A|j [| w(z')dz' I| Utn(Q~z')B(Qsz)Kn(QPsz')dz'dQ]

X VL(P)dP. (14)

Changing the order of the integral and introducing afunction

G(Q,z) = J|r) Kn(QPz) VLn(P)dP.

and letting F(Qz) = Utn(Qz)G(Qz), we have

(15)

w(z)dz B(Qz')F(Qz)dQ][j w(z')dz' B*(Qz')F*(Qz)dQ)

77 = cEf3(z)/2

Izl Jz2 I w(z')w(z') (B(Q,z')B* (Q',z'') )F(Q,z)F* (Q',z)dQdQ'dz'dz'`

. (16)cE0 f3(z)12

range z', and P is a typical point in the detector plane D(see Fig. 1), Ut is the field of the transmitted beam,B(Q,z') is the amplitude backscattering coefficient ofthe atmosphere (sr-' m), a random function, andK(Q,P,z') is the complex transmission function1 3 ofthe medium (excluding atmospheric attenuation).K(Q,P,z') thus represents the complex amplitude ofthe signal at a point P in the detector plane, due to amonochromatic point source of unit strength and ofzero phase, situated at (Q,z').

If the laser pulse length r is short, then the scatteringlayer from which the backscattered signals simulta-neously arrive at the antenna is thin. If the thicknessof the layer, cri2, is also small compared with z, thenthe normalized transmitted field in the layer does notchange appreciably with z'; i.e.,

U,(Qz')=wUtz'Z) (9)

where w(z') = [po(z)] 1/2 is the square root of the power

P0 emitted by the laser at time t' = 2(z - z')ic. Forsimilar reason the transmission function K(QPz') =K(QPz) if the longitudinal phase factor is ignored. Ifwe normalize K(Q,P,z) by Q1/2,

Since B(Qz') at different points in the scatteringvolume is formed by randomly located aerosols, it isa zero mean complex Gaussian random variable.'4

The values of B(Qz) at different points are statis-tically independent. Thus (B(Qz')B*(Q',z') =B(Q,')) (B*(Q',z'') = 0 if Q F4 Q' and z' F# z". We

now have

(B(Qz')B*(Q',z')) = f3(Qz')6(z',z')(QQ'), (17)

where ,B is the power backscattering coefficient, and 6 isthe Dirac delta function. Thus if is considered uni-form across the laser beam, i.e., /(Qz') = /3(z), Eq. (16)becomes

=cEo(z)2 fz PO(z')dz' L In(Qz)IG(Qz) 2dQ

= I(Qz)ns(Qz)dQ,

where

J Po(z')dz' = |z Iw(z')I2dz' = f Po(t')cdt'/2 = cEO/2,

Itn(Q,z) = Utn(QZ)Ltn(Q1Z)

i75(Qz) = G(Qz)G*(Qz) = fD K2(QPz)VLn(P)dp

(18)

(19)

(20)

(21)

K,(QPz) = K(Q,P,z) K(QPz)

z

then we have

V,(Pz) = CO f w(z')dz' I

where z1 = z - 2cr. Using the normalized LO field

VLn(P) = VLP )

(10)

is defined as the single-point receiving efficiency.The above derivation for 7(z) in an atmosphere with-

out refractive turbulence reveals several importantfacts:

[1) (1) The signal source field in the scattering plane,U8(Qz) = Utn(Qz)B(Qz), is spatially incoherent,the mutual coherence (Us(Q',z') Us(Q',z')) = (Utn-

(Q',z')B(Q',z') Utn(Q",z")B* (Q',z') ) = Itn(Qz)6(Q' -12) Q",z' - z') is a delta function because of the indepen-

dence of the scattering particles. Thus the mutual


-

U,(Qz')B(Qz')K,,(QPz)dQ,

coherence of the transmitted laser beam does not af-fect the mutual coherence of the atmosphere-back-scattered radiation in the detector plane.

We now rewrite Eq. (18) as

q(z) = | Itn(Qz)dQ fAD Kn(QPz)'VL(P)dP 2

IA) |l [| Itn(QZ)dQK(QP'z)K(QP"z)j

x VL*(P')VLn(P )dP'dP1. (22)

Comparing Eq. (22) with Eq. (5) we can see that themutual coherence of the signal in the detector plane

(WV5 (P',z) V*n(P",z)) = ItD(Qz)Kf(QP',z)K*(QP",z)dQ (23)

is basically the mutual coherence of the transfer func-tion Kn(QPz) weighted by the transmitted laser in-tensity.

(2) Equations (18) and (21) show that only the sin-gle-point diffraction pattern Kn(QPz) is involved inthe mixing process with LO in the detector plane. Thesingle-point receiving efficiency is a characteristicfunction of the receiver only, being independent of thetransmitted beam and the axial alignment of the trans-mitter and the receiver. The overall receiving effi-ciency is the sum of the single-point receiving efficien-cies, weighted by the intensity of the transmittedbeam. Thus the mixing process is carried out on apoint-by-point basis without phase relationships be-tween the points. In fact, Eq. (22) can be rewritten as

7(z) = L It,(Qz)dQ J Kn(QPz)VLV(P)dP

= dQ It.(Qz)K(QP',z) VL(P)dP

=I dQ V (QPz)VL(P)dP (24)

where V(Q,P,z) is the average signal field strengthpattern at the detector of a point source at (Qz) withamplitude [Itn(Q,z)]l"2 and zero phase. This againshows the nature of the source as a group of indepen-dent points.

B. The Single-Point Receiving Efficiency 77,(Q,z) and theBPLO field

The next step is to develop the relation between thesingle-point receiving efficiency i7(Q,z) and the BPLOfield. The central problem here is to derive the ex-pression for K(QPz), the normalized point sourcediffraction pattern (or the normalized transmissionfunction) in the detector plane.

The receiving system of a coherent lidar is schemati-cally illustrated in Fig. 2. It consists of an antenna, ortelescope (a primary mirror in plane A and a secondarymirror in plane B with common foci in plane C), adetector lens (in plane L), a detector (in plane D), and aseries of optical components, such as steering mirrorsand polarizers, located between B and L at planes I,,I2 . . , and I,

A

S

I' Z

Fig. 1. Scattering plane (S), antenna plane (A), and detector plane(D).

Fig. 2. Conceptual optical layout of a receiving system in a coherentlidar.

Under Fresnel approximation the electric field atthe antenna plane due to a point source at Q with unitstrength is

U.(QQA) =exp(jz) exP[j (r - rA )2

where QA is a typical point in the plane of the primary;r and rA are the radial vectors of Q and QA, respective-ly. If we ignore the phase factor exp(jkz) and normal-ize the field strength by [(z)]1/ 2 , we have the normal-ized field

USf(QQA) = expkz - (r -rA) ._AR LZ Jz I

(26)

We then define H(F,G) as the point response func-tion in the Huygens-Fresnel integral

U(G) = r UO(F)H(F,G)dF, (27)

where UO(F) is the input field, U(G) is the output field,and H(F,G) represents the disturbance at point G dueto a point source F of unit strength and zero phase.Under Fresnel approximation H(F,G) can be writtenas 12 ,1 3

H(F,G)- = AR [ (rG rF)2H(,G =XRexp[ XR (28)

where rF and rG are the radial vectors at the planescontaining G and F, and R is the distance between thetwo planes. If a lens is involved, an additional phasefactor,

r (229exp- f , (29)


(25)

will multiply U0 or U, where r will be rF or rG depend-ing on where the lens is. This factor is also included inH(F,G) in our derivation below.

Using the normalized field Usn(Q,QA) in Eq. (26) asthe input field, and using the point response functionH(F,G) defined above, we have the normalized trans-mission function from a point Q in the scattering planeS to a point P in the detector plane

Kn(QPz) = L HL(PL,P)dPL

X J HN(PNPL)dPN... JB HB(QBPI)dQBIN

X J HC(QCQB)dQC A HA(QAQC)USn(QQA)dQA. (30)

From Eqs. (15) and (30), we have

G(Q,z) = J V(P) [J HL(PL,P)dPL

X j HN(PN,PL)dPN-... JB HB(QBPI)dQBIN

x I (QQB)dQJ HA(QA.QC)Usn(Q.QA)dQA]dP. (31)C ~ ~~~ I

Changing the order of the integrals and invoking thereciprocity theorem, i.e., H(F,G) = H(G,F), and notingthat Usn(QQA) = (Xz/iAl 1 2)H(QQA), we can rewriteEq. (31) as

G(Q,z) = Xz H(QAQ)dQA HA(QCQA)dQ, J H(QBQ,)dQBjFAR A I B"

X J HB(P1,QB)dP1... JL HN(PL,PN)dPL JA HL(P,PL)Vifn(P)dP.

(32)

Note that the multifold integral in the right-handside of Eq. (32) is the process that calculates the nor-malized diffraction field in the scattering plane S, fromthe source field VLn (P) in the detector plane. Mathe-matically, it is equivalent to a propagation process:the local oscillator beam starts from the detector planeand propagates back to the scattering plane. Theresultant field from the integral is just the normalizedback-propagating LO (BPLO) field LLn in the scatter-ing plane S. We then have G(Q,z) = aU 0*Ln(Q,z), and a= Xz/(jAj'2). Consequently, according to Eq. (21) thesingle-point receiving efficiency qs(Q,z) =G(Q,z)G*(Q,z) is proportional to the intensity of thenormalized BPLO field

n,5(Qz) = IaULn(Qz)I = A ILn(QZ). (33)AR

There are several remarks we need to make here:(1) The only difference between Eqs. (31)and (32) is

the order of the integrals. To calculate G(Q,z) fromeither Eq. (31) or Eq. (32) one must go through all theintegrals step by step; no step can be ignored. Thus atfirst glance there seems to be no advantage in using the

BPLO treatment. However, when using Eq. (31), wewill find that in every integral the integrand is a func-tion of Q (the point in the scattering plane), thus tocalculate q, we have to repeat the integration from theright-most integral in the equation for each Q. Wealso note that the integrand for off-axis Q is not circu-larly symmetric. Thus the calculation is always 2-Dand time consuming. If the number of samples in thescattering plane is M, the number of samples in eachintegral is N X N, and the number of integrals is L,then the total time of computation is proportional to(L - 1)N4 + MN2. On the contrary, the integrand ineach step of Eq. (32), except for the last one, is not afunction of Q; thus the result of the first several stepsof integration is common to all points in the scatteringplane and needs only to be calculated once. In otherwords, Eq. (32) can be rewritten as

G(Q,z) = JAH(QAQ)W(QA)dQA, (34)

where

W(QA) = J HA(QCQA)dQC JB HC(QBQC)dQB HB(P1,QB)dP1 ...

J HN(PL,PN)dPL fA HL(PPL) VLn(P)dP (35)L AD

is a function of QA in the antenna plane only, and thusit needs to be calculated only once. In addition, if thesystem is well aligned and free from astigmatism, cir-cular symmetric properties of the integrand furthersimplify the integration to a 1-D calculation. Thus ifthe number of samples is Ml in the scattering planeand the number of samples is N, in the L - 1 integralsin Eq. (32), the total amount of computation is propor-tional to (L - 1)N21 + MNl. Even if Ml and N, aremuch greater than M and N, respectively, the time forcomputation is still much less than that for using Eq.(31). Thus the BPLO treatment greatly reduces theamount of computation.

(2) The integration in each step of Eqs. (31) or (32)is over a limited area. That is, there is a pupil functionin each integral that affects diffraction and cannot beignored. In particular, the calculation of the BPLOfield should start at the detector lens and includetruncation by the detector chip. When truncationoccurs, the BPLO at the detector lens that propagatesbackward toward the telescope is not the same as thereal LO beam at the lens propagating toward the detec-tor. Also, because of vignetting by the detector, thesize of the BPLO at the lens likely is larger than theforward-propagating LO and comparable to the lensaperture, causing truncation at the lens and furtherdiffraction. Similar effects are induced by other opti-cal elements between the lens and the primary. Thus,the final BPLO field at the antenna may be very differ-ent from the Gaussian distribution one would haveexpected for a system without truncation.

The LO field at the detector should also be calculat-ed very carefully. Usually the output beam profile ofthe LO laser is a Gaussian. After passing through


several optical elements and the detector lens it formsa waist where the detector should reside. The LObeam might be truncated by the detector lens and/orother optical elements between the LO laser and thelens; thus the field at the waist is usually different froman ideal Gaussian distribution.

It is worth pointing out that the criterion for ignor-ing the diffraction effect of the sharp-edged aperturesfor a Gaussian beam is quite stringent. As Siegmanl 5

indicated, even though an aperture may cut off only avery small fraction of the total power in an opticalbeam, it may produce aperture diffraction effectswhich significantly distort the transmitted beam inboth the near field and far field regions. For example,truncating an ideal Gaussian beam with a sharp-edgedcircular aperture of diameter D = 7rw (w is the e-2radius of the beam) will cut off only 1% of the totalpower, but it will cause near field diffraction rippleswith an intensity variation AII 17%, along with apeak intensity reduction of -17% on axis in the farfield. The diameter of the sharp-edged aperture re-quired for -1% diffraction ripple effects is D - 4.6w.

(3) As a logical consequence, in designing coherentlidars we should try to avoid truncation as much aspossible. If truncation is unavoidable, we should firstmodel and understand the effects, then compromisebetween various parameters to optimize the system.

Ill. 7(z) for a System with Ideal Gaussian Beams-theOptimization of 7, SNR, and Antenna Radius

In this section, our theory is applied to the ideal casewhere both the transmitted and BPLO fields at theantenna of a coaxial monostatic system have Gaussianamplitude beam profiles with flat wave-fronts. It im-plies that truncations inside the system for both trans-mitted and LO beams are negligible. Under theseassumptions it is possible to determine the receivingefficiency as a function of the Fresnel parameter F =rRi/(Xz) (where R is the antenna radius) and other

parameters, and find the maximum receiving efficien-cy for each F. Then the system's SNR can also beoptimized at the maximum detection range.

Assume that the normalized transmitted and BPLOfields at the common antenna are

2 ex(r~,2U,.(r.) = -/ expan - )

ULa(r.) = 2 exp - )TL 2 1 L

(36)

(37)

where RT and RL are the e 2 radii of the Gaussianintensity distribution of the transmitted beam and theBPLO, respectively. We then have the transmittedfield and the BPLO field in the scattering plane atrange z when the system is focused at zf:

Urz)=27r fR 0,U,( /j rr; \ 2rrr drUt5(r2Z) = A UJ , U(ra) exp A XA )radras

27r = jrr2a\Iiirr\UfL,,(r,z) =Z---J U,,(ra) exp NZ)JAz 0 rdr,

(38)

where z-jl = z 1 - zf' is the equivalent range. Jo is thezero-order Bessel function of the first kind. The finiteupper limit Ra truncates the beams at the edge of theantenna. Substituting Eq. (36) into Eq. (38),

Ut(rrz) = 87R1 J- 21 + j uZr0\UtzRT 10 =expi - + -~ )r IJ ZI radra.LzR k, R Xr5J " \ A z/(40)

Let x = raiRa, y = rRa, PT = RT/Ra, PL = RLiRa, F =rRa (z), and Fe = rRai( ze). We then have

Ut5 (Yz) = JO exp[(- - + jFe)X2]J(2Fxy)xdx. (41)Tzn T f PT / J

The intensity of the transmitted beam isI") 8F ff 11 I~ 2lJ2F 2~

tl.(YZ = 2zp If exP[(- 2 + jF, X JO(2Fxy)xdx

Similarly, the intensity of the BPLO is

ZPL Jo p[( PL + jF 2 X2]Jo(2Fxy)xdx}-

Let

A(,F,Fe,y) = {J exp[(-t + jFe)x2IJO(2Fxy)xdx}.

Then

AR L( JO Itn(r.Z)IL(rz)rdrj

128F2 [ A(PT2 FFeY)A(PE 2 ,FFeY)YdY

PTPL

= 128F2ab 7 A(a,FFe,y)A(b,FFe,y)ydy,

where a = P-2, b = p 2. Thus

n(z) = (F,Fe,a,b).

To maximize for a given F and Fe,equations should be satisfied:

(42)

(43)

(44)

(45)

(46)

the following

d = 128F2b A(b,FFe,y)

X [A(aFF y) + a (aFFey) lvdy = 0,

(47)

d = 128F2a A(a,FF,y)Ob Jo

X A(bFFey) + b Ab eF,)dy = o.I I ~ ~ ~~ + AbFe) 0.

Since the two equations in Eq. (47) should be ap-plied simultaneously, we must have

[p A(brFFesY) A(amFFesy) + am dA(aFF5 ,y) | ydy = 0,

J A(amFFey)A(bmFXFesy + bm b bbmlydy = .

(48)

The two equations are identical in form, hence am = bm(39) = c = 1/PM, and PM is a function of F and Fe. Conse-

quently, the maximum of is


n = 128F2 [c(F,F) 2 J0[A(c,FFe,y)] 2ydy = nM(F,Fe), (49)

or by introducing a factor g = FeiF,

n = qM(Fg). (50)

We can now investigate the maximum receiving effi-ciency under different conditions:

A. The System is Focused at Infinity

In this particular case, Ze = Z, Fe = F, and g = 1.Then according to Eq. (50), 7M and PM only depend onF. These analytical inferences have been verified bynumerical calculations in three cases: (1) z = 30 km, X= 10.59,vm; (2) z = 1080 km, X = 10.59 Am; (3) z = 30km, X = 1.06 Aim. The numerical results have con-firmed that:

1. At fixed X and z, for each antenna radius Ra, 1 is afunction of PT and PL- When PT and PL - 0 (the idealuntruncated Gaussian case), 7 -> 0. Thus untruncat-ed Gaussian cases that are achieved by reducing beamsize or increasing aperture size indefinitely are veryinefficient.

2. For each Ra, there is a maximum of n,i7M. 7M islocated at PT = PL = PM-

3. 'iM and PM are functions of F only for a collimatedsystem (focused at infinity). They are plotted againstF in Fig. 3. For a coaxial system, the maximum valueof 7M is 0.4 when F < 0.02. For larger values of F, NMdecreases because of the destructive effect of both thephasor exp(jFx2) and Jo(jFxy) in Eq. (44). However,the decrease in 77M is slow at large F values, because PMalso decreases, ultimately permitting the Gaussianfunction to become dominant in the integral of Eq.(44).

These results lead us to an interesting criterion forthe maximum size of the antenna in a collimated sys-tem. We define the optical sensitivity S of a system as

rrR2S(Ra) = 2rR 7(z) = ffiq(z), (51)

which includes all the important geometric parametersof the lidar measurement that influence the SNR. It is

also the effective solid angle Qe = f727(z) within whichthe backscattered radiation contributes to the signal.

Since SNR is proportional to S, optimizing S isequivalent to optimizing SNR with respect to the geo-metric parameters of the system. For a coherent lidarsystem designed for a maximum detection range zMbut focused at infinity, the optimum optical sensitivityof the system S at z is

ir2

S(Ra) = - nM() = - FM(F) = -Y(F)ZM ZM ZM

where

Y(F) = FnM(F).

(52)

(53)

Y(F) is plotted against F in Fig. 4. When F is small,Y(F) increases linearly with F (therefore, with antennaarea). Then the rate of increase slows due to thereduction of 77M(F) with F, until a maximum value of0.524 is reached at F = FM = 2.47. After the maxi-mum, Y(F) slowly decreases to -0.5 between F = 4 andF = 10. The maximum antenna radius RM corre-sponding to the maximum Y(F) satisfies

irR2M = 2.47, (54)

XZM

thus

RM = 0.89jXM-. (55)

Since the system's SNR is proportional to S(Ra),increasing Ra beyond RM no longer increases SNR, butslightly decreases it. Thus we may consider FM as theFresnel number where the system SNR is saturated.

For cost considerations, the maximum size of theantenna should even be smaller than RM. We canchoose F = 1 to approximate where Y(F) first signifi-cantly deviates from a straight line. (When F > 1,Y(F), hence S(RG), does not increase with the antennaarea but at a much slower rate.) In this case, thecorresponding antenna radius can be referred to as theoptimum radius Ropt, and

Ropt = 0.546JXM-. (56)

1.0.9X.8

.7\

X .6\

.5M- .4

- .3

.2

.1 1

0 1 2 3 4 5 6 7 8 9 10F

Fig. 3. 7M and PM as functions of the Fresnel number F = 7rR/(Xz)

for an afocal system. 1, M; 2, PM.

.- 5 C

/w i.4 I

/3 -

.2 1 I

C Io1 C

0 1 2 3 4 5 6Fopt FM F

Fig. 4. Y(F) = FnM(F) for afocal system.tangent line at F = 0.


7 8 9 10

The dashed line is the

The corresponding value of Y and S areYopt = 0.355,

- 0.355XzM (57)

As examples, we list RM and Ropt for some specialcases in Table I.

If the atmospheric backscattering coefficient is pro-portional to Xn (1 < n < 4), and if shot-noise of thedetector is dominant, then the optimum SNR will beproportional to X-nSpt, thus

SNROpt K 0.355XC1ZM

B. The System is Focused at ZM

For this case, zf = zM, thus Fe = 0, and g = 0. Thephase difference across the antenna p(r) = exp(jFex2)in Eqs. (41)-(44) will be unity, resulting in a constant77M = 0.4 and a constant PM = 0.8 for F = 0 to '10.Thus, there will be no theoretical limitation on theantenna radius to increase the SNR. In this case, thesystem optical sensitivity,

0.4XFS = 0-4S = 0.4Q, (59)

ZM

is proportional to F, or proportional to the solid angleQ.

C. General Cases

For general cases, Fe Fd F, g Fd 0. We calculatenM(F,g) (Fig. 5) and Y(Fg) = FM (Fig. 6) for various gand F. From these figures we can see:

1. M decreases with F, thus there is still a maximumvalue Y (and a corresponding FM) for Y. The smallerthe value of g, the higher the YM and FM (hence, themaximum antenna radius increases). Similar to Eqs.(56) and (58), the maximum radius of the antenna =(FMXzM/r)1/2, and the maximum SNR is proportionalto M/(Xn-ZM).

2. NqM and Y decrease with g. The rate of decrease ismost rapid at g = 0, and it lessens as g increases.

3. For small F values, M and Y at different g valuesare not very different. At large F values, however,Y(F,O) is much higher than Y(F,1). This means that atlarge F values. (with short zM or X, or large Rj) focusingthe system at zM will get much higher SNR than focus-ing it at infinity.

Table 1. The Maximum Antenna Radius R and Optimum Antenna RadiusR0pl as Functions of Wavelength and Maximum Detection Range ZM

Wavelength (,um) Zm (km) Rm (cm) Ropt (cm)

10.59 1 9 610.59 30 50 3210.59 200 130 8110.59 1000 290 183

1.06 1 3 21.06 30 15 101.06 200 41 261.06 1000 91 58

(58)

.5 0 F I I I , . I . I . . . . .

.45

.40 ----- - I g=

.35 2

.30

.4.25 \\\\ .X .5 .20 .8..6

.15 1.0

.10 2.01.5125.0

.05

C_0 1 2 3 4 5 6 7 8 9 10F

Fig. 5. nM as a function of F and g in general cases.

2

1

1

1

LL 1

1 2 3 4 5 6 7 8 9 1 OF

Fig. 6. Y = FM as a function of F and g in general cases.

Knowing the behavior of M for general cases helpsin understanding the effects of any focal-setting errors.Uncertainty in a focal-setting changes the effectiveFresnel number Fe (thus it changes g), resulting in anerror in both 77M and Y.

Assume that the focal length of the primary mirror isf, the distance from the focal plane of the secondary tothe primary is s, and the uncertainty in setting s is e. Ifthe system is focused at zf, then A (/zf) = -A(1/s) = /S2 e/f2, and A(1/ze) = -A(1/zf) = -e/f2. It followsthat

Ag = F = z )

F \Z'. f2 (60)

where L = zM/f is the relative range, and 6 = elf is therelative focal setting error. From Eq. (60) we see thatfor the same relative accuracy in focal setting, thelonger the maximum range, the greater the error in g,and thus the greater is the uncertainty in M and Y.For example, if = 1 X 10-4 , L = 1 X 103, Ag = 0.1; if L =1 X 104, Ag = 1. If the system is nominally focused atzM(thusg = ),whenF= 5, flMwill be 0.39-0.4 forL = 1X 103 and 0.1-0.4 for L = X 104, and Ywill be 1.94-2.0for the former and 0.5-2.0 for the latter. For theformer case, since the uncertainty is very small, F = 5 isstill not an upper limit for Y. For the latter case,


.0~~~~~~~~~~~.

/l4 5.0

-9,

however, an uncertainty of a factor of 4 in systemperformance seems unacceptable. Thus reducing F to1 - 2 is more reasonable. This is almost the sameupper limit as for a system focused at infinity.

The above discussion shows that if the maximumrange zM is long, even for a system with a nominalg = 0,uncertainties of the focal setting can still limit theantenna radius, and the requirement for the accuracyof focal setting is quite stringent. For example, if zM is500 km, f = 2 m, L = 2.5 X 105, and if Ag = 1 (hence FM= 1 2) is desired, e must less than 8 im. Carefulthermal and vibrational design may be necessary.However, for a more modest zM = 30 km, L = 1.5 X 104,if Ag = 1 is required, e needs only to be <133 im. Inthe latter case, improvement in SNR with aperturesgreater than that in Table I is indeed practical. Thusif zM is great and the focal-setting accuracy is not highenough, focusing the system at the maximum rangedoes not have much advantage.

IV. Conclusions

Pulsed coherent lidars are usually complex, cascad-ed optical systems. The system's receiving efficiencymay be affected significantly by the diffraction withinthe system, especially when either the transmitted orthe BPLO beam is truncated by the optical elements inthe system. Even for an untruncated Gaussian beam,diffraction effects still exist. The size and wavefrontof the beam at the secondary of the telescope dependon the distance the beam propagates in the system;hence, the telescope should be adjusted to force theprimary mirror to be locted at the waist of the outgoingGaussian beam (i.e., the output wavefront at the pri-mary should be flat).

Due to the finite extent of the scattering volume, thecoherent area of the atmospheric backscattering signalis always smaller than the antenna area. Consequent-ly, the overall receiving efficiency of a coherent lidarsystem can never approach 1. In fact, for ideal coaxialsystems with Gaussian beams and a common telescopefor transmitting and receiving, receiving efficiency cannever exceed 0.4. In general, the system receivingefficiency -q is a function of the Fresnel number F, thebeam-to-antenna ratio p, and a factor g that indicatesthe ratio of the maximum range zM to the equivalentrange ze. After optimizing p, the maximum receivingefficiency 77M and the corresponding function Y arefunctions of F and g. We have discussed the optimiza-tion of such systems for three different conditions:

1. For an afocal system (g = 1), the receiving effi-ciency is solely a function of the Fresnel number F; butit decreases with F for large F. For a fixed maximumdetection range zM and a given wavelength, receivedpower is also proportional to antenna area (hence, F).These competing trends result in a maximum in SNRfor an afocal system at F = 2.47, and an optimumantenna radius that is proportional to the Fresnellength (Xz) 112 .

2. For a system focused at the maximum detectionrange zM (g = 0), the maximum receiving efficiency willbe at its highest value 0.4 as long as F < 10. There is notheoretical limitation on antenna radius to increasethe SNR.

3. In general cases, g szd 0. The maximum obtain-able SNR and the corresponding maximum Fresnelnumber FM (hence, the maximum antenna radius) in-crease with decreasing g.

In practice, however, the system's performance isaffected by the focal-setting error e by causing an un-certainty in g. The uncertainty in g is proportional tozM and increases with F, which can limit the usableantenna radius even when the system is focused at themaximum range zM (g = 0).

This work is partially supported by NASA undercontract H-98023B. The authors are indebted to T. R.Lawrence and B. J. Rye for very helpful discussions.Special thanks are due to R. Frehlich, R. J. Lataitis,and again to B. J. Rye, for their critical review of thispaper and valuable suggestions.

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receiving efficiency of monostatic pulsed coherent lidars 1: theory

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