target-plane intensity approximation for apertured gaussian beams applied to heterodyne backscatter...

Target-plane intensity approximationfor apertured Gaussian beams appliedto heterodyne backscatter lidar systems

Waltraud Pichler and Walter R. Leeb

With an application in lidar systems in mind, we investigate the effects of transmit-aperture truncation ofGaussian beams by employing the extended Huygens-Fresnel principle. We derive an approximation tothe top-hat aperture-transmission function by defining an abstract Gaussian aperture-transmissionfunction. The two fitting parameters of the latter are found when the beam radius and the on-axisintensity for both aperture cases are equated in the observation plane. Bounds for the applicability ofthe approximation are established, and its accuracy and usefulness is demonstrated through applicationto the calculation of the return signal of a heterodyne lidar system.

Key words: Diffraction, laser radar, lidar, Gaussian lidar.

1. Introduction

The effects of beam truncation caused by finiteaperture diameters are important in the design of theoptical subunit of a lidar system, and they must beincluded in the mathematical system description.Depending on the type of aperture-transmission func-tion selected, numerical or analytical methods mustbe applied for the evaluation of typical system param-eters, such as the received signal, the signal-to-noiseratio, or the heterodyne efficiency.

The lidar source beam is usually modeled quite wellby a fundamental transverse mode (TEMOO), which ischaracterized by a Gaussian field distribution. Wemay define an effective beam radius a as that trans-verse distance where the intensity has dropped toexp(-1) of its on-axis value. It was found thattruncation effects, which are typically caused by thetransmitter optics, may safely be neglected for aphysical aperture radius that is > 2.8 times theeffective beam radius.' In this case, the field distri-bution remains Gaussian throughout the lidar sys-tem, and the mathematics involved allow an analyti-cal treatment. Descriptions of coherent lidar systemsthat are based on untruncated Gaussian beams have

The authors are with the Institut ffir Nachrichtentechnik undHochfrequenztechnik, Technische Universitit Wien, Guss-hausstrasse 25/389, A-1040 Wien, Austria.

Received 7 July 1993; revised manuscript received 19 November1993.

0003-6935/94/214761-10$06.00/0.C 1994 Optical Society of America.

been obtained long ago2 4 and have been termedGaussian lidar.

In practice, however, an aperture modeled by acircular top-hat function will considerably truncatethe Gaussian beam. Then the mathematical descrip-tion requires numerical treatment, which leads toproblems resulting from the infinite integration lim-its and causing rounding errors. In recent yearssuch numerical calculations concerning rather simplelidar configurations have been published.5-' 0

In this paper we approximate the effects of top-hataperture truncation by introducing an aperture witha properly specified Gaussian aperture-transmissionfunction. The fields then remain Gaussian just as inthe case of negligible truncation, and analytical treat-ment is possible. First in Section 2 we formulate thediffraction of a Gaussian beam by a finite aperturebased on the Huygens-Fresnel principle.11"2 Allapertures are assumed to be circular. We also estab-lish equations covering the top hat and the Gaussianaperture.

In Section 3 we develop the approximation concept.We specify a Gaussian-shaped aperture-transmissionfunction by means of two fitting parameters. Theyare determined when the on-axis value and theeffective beam radius of the diffracted field are equatedin the observation (target) plane for the case oftop-hat truncation and of the Gaussian aperture.Typical errors in the intensity distribution arisingfrom the approximation are given. Conditions forthe applicability of the Gaussian approximation areestablished in Appendix A.

20 July 1994 / Vol. 33, No. 21 / APPLIED OPTICS 4761

As an application of our Gaussian approximation, aheterodyne lidar system with a finite aperture isconsidered in Section 4. We calculate the returnsignals from a diffuse target for both aperture-transmission functions and show the accuracy of theapproximation. Several suggestions for further ap-plications of our approximation follow in Section 5.

2. Effect of Diffracting Apertures

When treating the diffraction of laser fields within thelidar system, we use the scalar notation that follows.The optical field propagating in the z direction isdescribed by

l(r, z, t) = U(r, z, t)exp(jkz -jwt), (1)

where r and t are the transverse coordinate and time,respectively, k = 2r/X is the wave number, is thewavelength, v and X = 2rv are the light frequencyand angular frequency, respectively, and U(r, z, t)[dimension (Wm-2)/ 2] is the reduced field. The sca-lar fields are normalized such that, in the absence ofabsorption,

fI (r, z, t) 12

d2 r = f U(r, z, t) 12d2r = PT(t -zc)

(2)

where the vertical bars denote the absolute value, PTis the transmitted laser power, c is the speed of light,and d2r denotes the two-dimensional integration overthe plane defined by z = constant. The integration isperformed over the entire transverse plane. Forease of notation, infinite integration limits are alwaysomitted. As the absolute values of the optical fieldsare not of interest in our paper, we use a normalizedform u of the reduced optical fields defined by

u(r, z, t) =(P t))"2 (3)

where PL(t)) is the average laser power. Note thatthis normalization refers the lidar performance to thelaser source power (PL(t)) and not to the laser poweractually transmitted (PT(t)) (see Fig. 1). Below weassume temporal stationarity and omit the z coordi-nate, as the symbol used for the two-dimensionalvector will unmistakably denote the z coordinate.

Consider a system layout as given in Fig. 1, whereuo(r) represents the source field just before the aper-ture, u(r) represents the transmitted field just afterthe aperture, and ui (p) represents the field within theobservation plane. We assume that an untruncatedmonochromatic Gaussian beam with a field pattern of

apertureW(r)

as er * -- - - - - --,~~~~~~~~~~~~~~~~~~~~LoI PTd

. _ r . . . . . . . . . . .

Ue() U()

I

target distance z

transmitter plane(r plane)

observation plane

(p plane)

Fig. 1. Optical layout: A laser beam of field u0 and power PL

impinges onto an aperture positioned in the r plane. The result-ing optical field ui of power PT is investigated in the observation pplane at a distance z.

If the effective beam radius a approaches infinity, theGaussian amplitude distribution becomes uniform.The phase curvature of the beam is characterized byf,which is positive for convergent beams and negativefor divergent beams. For collimated beams f isinfinite.

The optical field just behind the aperture is relatedto the incident field by

ue(r) = uo(r)W(r), (5)

where W(r) is a dimensionless aperture-transmissionfunction. In this paper we cover two often-employedaperture functions-the top-hat aperture and therather abstract Gaussian aperture.

Top-hat aperture and hard aperture are synonymsfor an aperture-transmission function War) describedby

Irl < d/2

Irl > d/2'

where d is the aperture diameter. To characterizethe degree of truncation caused by this spatial filterwe define a beam-truncation ratio m by

m = d/2a. (7)

According to Dickson,' truncation effects may beneglected form > 2.8.

Although the top-hat aperture is reasonably realis-tic for an actual lidar system, a fictive Gaussianaperture-transmission function will prove to be ofpractical importance because it allows analytic solu-tions of the diffraction effects. The Gaussian aper-ture-transmission function is defined as

1 [ r2 jkr2 \uo(r) = (Tr rx2)1/2 expk 2 2 2f

W g ) exp(~ 2r )2Wg(r) = exp dg2), (8)

(4)

impinges onto an aperture located in the r plane.Here dg, the effective Gaussian aperture diameter, isdefined by the exp(- 1) reduction in transmittance.

4762 APPLIED OPTICS / Vol. 33, No. 21 / 20 July 1994

(6)1

W,(r = 0

The influence of a lens placed in the aperture planecan be easily included by modification of Eq. (5) to

ue(r) = uo(r)F(r), (9)

where F(r) is the lens-transmission function given by

F(r) = W(r)exp[-j4)(r)]. (10)

The factor 4(r) characterizes the phase shifts causedby the lens. Because truncation and focusing effectsare separated in Eq. (10), either of the transmissionfunctions may be inserted. An ideal lens of focallength fL adds a phase shift, which is given by

kr2

+)(r) 2L (11)

For a converging lens fL is positive. Without anyrestriction to generality, we omit the phase factor ofthe lens by setting fL = oo. In the case of fL • ao, thesymbol f must be replaced by ( f -' + fL-')-.

From the Fresnel-Kirchhoff diffraction-integral so-lution to the scalar Helmholtz-wave equation,' 2 inthe paraxial approximation the field u(p) in theobservation plane and at distance z is given by

- k exp(jkz) fu()-2 mjz d2rue(r)expiz IP

where r and p are vectors in the respective planes.Inserting Eqs. (4)-(6) into Eq. (12) yields the optical

field in the observation plane for a top-hat aperturefunction in the form

_ k exp(jkz) (jkp2 \Uti(P) = jz(ira 2)1/2 exp

x /2 |r 2 jkr 2 (1 1 )x J rP[-.262 2 kf zoZI (13)

where Jo(x) is the Bessel function of zeroth order.The integration of Eq. (13) must be performed numeri-cally. The observation-plane field for a Gaussianaperture function can be calculated with Eqs. (4), (5),(8), and (12), yielding

transmitter plane

I 1 U012 i Wt(r)

(a) *..' - - -

I0 tlU.12 ._ W(r)

2t Kd9

(b) J- . -.-

(A

observation plane

- lutil2

Fig. 2. Gaussian approximation concept: Source intensity distri-bution, aperture-transmission function, and observation-plane in-tensity are sketched for (a) a top-hat aperture and (b) a Gaussianaperture.

3. Intensity Distribution Approximation

A. Approximation Procedure

In Fig. 2 we show the intensity distribution of thesource I uo(r) 12, the applied aperture-transmissionfunction W(r), and the corresponding observation-plane intensity distribution I u,(p) 12 for the casesintroduced in Section 2. On the left side of Fig. 2(a)the Gaussian beam and the top-hat aperture-transmis-sion function constitute the source. The intensitypattern in the observation plane is sketched on theright. The analogous plots for the Gaussian-aper-tured system are given in Fg. 2(b). Although thepatterns differ, their main properties may be charac-terized by two fundamental parameters: the on-axispeak intensity u(0) 12 and the exp(- 1) intensityradius q. The idea of our approximation is to find aGaussian aperture that yields values of q and u(0) 12identical to those resulting from the top-hat case.For this two-point matching procedure two degrees offreedom in the Gaussian aperture-transmission func-tion are needed. Thus we redefine the Gaussianaperture-transmission function Wg(r) given in Eq. (8)by introducing an on-axis transmission factor A in theform of

Ugi(p) =

k exp(jkz)exp( 2z)

j2z(rra 2 )1/2 2 + 1 - IIdo2 g

2 2 f z

X 1 2 ~~k i!\] (14)

I [2 g2 2 f z)J

If we put dg = 0 for the Gaussian aperture, Eq. (14)represents the untruncated beam in the observationplane.

2r2Wg(r) = A exp dg2 (15)

It is expedient to relate dg to the top-hat diameter dby the introduction of

8 = dg/d, (16)

where is the equivalent-diameter ratio. In theremainder of this section we will show how the tworeal-valued fitting parameters A and 8 can be found.


- r 12 , (12)

Note that for this purely mathematical equivalencythe parameter A may also assume values > 1.(Below we call systems in which the Gaussian approxi-mation is applied the matched Gaussian case.)Because of the introduction of A, Eq. (14) must bemodified by a factor A and now reads as

Ui(P) =kA exp(jkz)

j2z(Trra 2)'/ 2

X exp-

4z2

jkp2Vx 2-~z

1 2 jk 12o 2 dg2 2 f z

k 2 p 2

1 2 jk 11 2u.2 dg2 2 f Z 1*

(17)

For the further procedure we use a normalized,dimensionless representation of the optical fields bydefining

j(,Tot2)1/2

lr =*P2 'u, i (18)exp(Jkz + jp2)

Applying Eq. (18) to Eq. (17), we find that thedimensionless optical field in the observation plane is

Urgi(Pd) =

2[282m2 + 2 + (e - n)

truncation ratio m represent severe truncation. Form >> 1 truncation effects will vanish. For largervalues of m the untruncated-beam concept, where theaperture-transmission function equals unity, may beapplied. The target distance is included in theparam-eter n. The value n = 0 (or n << 1) corresponds tofar-field targets, whereas n >> 1 applies to thenear-field case at least as long as the paraxialapproximation of the diffraction formula Eq. (12) isapplicable. The quantity is a measure for thephase curvature of the transmitted beam in thetransmitter plane. For a collimated beam, 6 equalszero. The normalized observation-plane field distri-bution of the top-hat-apertured system is found bycombining Eqs. (13), (18), and (21), which yields

Urti(pd) = f rJo(rpdn)exp(- r2{2m2 + (t - n)]}dr.

(22)

First we find the normalized beam radius q in theobservation plane for the case of the top-hat apertureby solving

I uti(0) II exp(-1) = rti(q) 12 . (23)

Using Eq. (22) we obtain values of q numerically forgiven m, , and n. The Pegasus algorithm, amodification of the regula falsi, is one convenient andfast converging method to use to solve for q.However, the knowledge of a solution interval isrequired a priori.

Next we turn to the matched Gaussian case and usethe value of radius q to find the equivalent apertureparameter 8 by setting

P 2

4 282m2 + i_ (a - n) + 2

(19)

jurgi(0) 12 exp(-1) = urgi(q) 2, (24)

and employing Eq. (19). Solving the biquadraticequation and excluding negative solutions, one findsfor the diameter ratio

8~q2 -'m [+ (n2q2- 8M2)2 - 16[4m4 + (e 2 _)- (mnq)2\ 1/22 4m4 + _ 2 ) - (mnq)2 -

(25)

where, in addition to the normalized transverse coor-dinate

pVd - d ' (20)

three more dimensionless parameters,

d kd2m = -X n = -,

2a z

kd 2

A= f 'were used. Small values of the already-defined beam-

One may distinguish three classes of solutions of 8.The first class is characterized by the two real-valuedsolutions for 8. Either solution is applicable for theapproximation purpose. The second class comprisesone real and one complex-valued solution. Here thereal one represents the value of the fitting parameter.Finally, the third class is characterized by two com-plex solutions for 8. As real values have been pre-sumed, this class provides no valid solution for thefitting parameter, and it is apparent that some limita-tions concerning the applicability of the approxima-


X exPt-

.

tion do occur. A detailed discussion of the validitybounds is given in Appendix A.

The last step is to equal the on-axis intensities result-ing from top-hat and Gaussian aperture functions.The peak transmission factor A can be found from

Iurgi(O)1 2 = IUrt(0)12, (26)

by insertion of Eqs. (19) and (22). After some ma-nipulation, A follows as

in Fig. 4. Again the traces nearly coincide. Some exactvalues for 8 and A are given in Tables 1 and 2. Fornegligible beam truncation (m > ) the on-axis transmis-sion factor A converges to unity. Therefore one can setthe fitting parameters to 8 = - and A = 1 for large valuesof m, as anticipated. We have also calculated the fittingparameters 8 and A for the case of focused beams (n =t) for different observation-plane ranges n (see Table3). They turned out to be independent of n.

[4(82M2 + 1)2 + 84(2 1)i + exp(-m2 ) - 2 exp( 2 )( )]/I = 214

|~~~~~~8 4M ( 2)]

where the negative solution of the square root is to bediscarded. For real 8, Eq. (27) yields a valid solution.

The fitting parameters 8 and A represent thediameter ratio and the peak transmission, respec-tively, of the equivalent Gaussian aperture. Theydepend on the parameter triple m, n, and i, and on q,which in turn depends on m, n, and e as well.

B. Examples for Fitting Parameters

We present calculated fitting parameters 8 and A forsome representative parameter values including thenear-field (n = 5) and far-field (n = 0.01; 0.1) cases inFigs. 3 and 4. The beam-truncation ratio m wasrestricted to values less than 6. In Fig. 3 we noticean extremely small dependence of the diameter ratio 8on the normalized range n. Further one finds that 8becomes infinite for large values of the beam-truncation ratio m. This is to be expected, because8 = - corresponds to the untruncated-beam case.

We plot the peak transmission factor A versus thebeam-truncation ratio m for collimated beams ( = 0)

4. Exemplary Application:Heterodyne Lidar

0

Incoherent Backscatter

To demonstrate the effectiveness of the Gaussianaperture-transmission approximation we will apply itto a heterodyne lidar system that detects the meanreceived signal from an incoherent backscatter tar-get.

The optical layout of the system under investiga-tion is sketched in Fig. 5. We assume a symmetricaltransmitter-receiver configuration, where the trun-cated local oscillator (LO) is combined with the re-ceived signal. Further we have assumed a mono-static configuration and a detector with an area largeenough to collect the entire optical fields with uni-form quantum efficiency. This system has beenanalyzed by Wang5 for optimum beam truncation forthe case of a Gaussian source and LO beams. Adetailed discussion of the receiver configuration canbe found in Ref. 7, and a complete derivation of themean received intermediate-frequency signal can befound in Refs. 5, 10, and 14. Here we first recall the

1 0000 T

0

a)

EI0

a

0~

a)

1000 +

100 +

10 +

0 2 4 6

1

0.1

beam truncation ratio mFig. 3. Equivalent-diameter ratio versus beam-truncation ratiom for collimated beams ( = 0) and different values of the rangeparameter (n = 0.01, 0.1, 1,5). For the values of n given, b showsa negligible dependence on n. Within the solid line that is plotted,the individual curves cannot be resolved.

2

< 1.8

t 1.6( ,c 1.40cii 1.2

U)

c 0.80.6

a) 0.40.0.202

- n=0.01; 0.1; 1

....... n=5

2 2.8 4

beam truncation ratio mFig. 4. Peak transmission factor A of the Gaussian aperturefunction versus the beam-truncation ratio m for a collimated beam( = ) and several values of the target-distance parameter n. Form > 2.8 truncation effects cease, as indicated by Dickson.'


(27)

6

II

III

IIII

II- l

Table 1. Equivalent-Diameter Ratio 8 for Collimated Beams ( = 0),Different Observation Plane Distances (n = 0.01, 0.1, 1, 5), and Several

Values of the Beam-Truncation Ratio m

m n = 0.01 n = 0.1 n = 1 n = 5

0.1 0.5227 0.5227 0.5239 0.56060.5 0.5344 0.5344 0.5356 0.57351.0 0.5745 0.5745 0.5759 0.61781.5 0.6560 0.6560 0.6577 0.70812.0 0.8112 0.8112 0.8136 0.88082.5 1.1190 1.1190 1.1226 1.22393.0 1.7820 1.7821 1.7883 1.9626

most essential equations describing the system.Then we calculate the intermediate signal in thereceiver for (a) the top-hat model by numerical integra-tion and (b) the Gaussian approximation. Lastly wecompare the results.

A Gaussian beam uo(r) defined by Eq. (4) passes thetransmitter aperture of diameter di. After trunca-tion the field propagates to a diffuse target located inthe p plane at distance z. The incident beam uj(p) isscattered by the diffuse target. Backscattered andincident fields are related by

u5(p) = uj(p)R(p), (28)

where the complex reflection coefficient R(p) charac-terizes the target shape and surface roughness and is,in general, stochastic. A perfectly diffuse target isoften regarded as a collection of numerous indepen-dent scatterers within the illumination spot. Thediffusely reflected fields are spatially uncorrelated sothat R(p) possesses the following property'5 :

(R(p,)R*(p2 )) = R X (P - PA7 (i P)

Table 3. Parameter Values of the Equivalent-Diameter Ratio a and thePeak Transmission Factor A for Focused Beams (n = g) and Several

Values of the Beam-Truncation Ratio m

m 8 A

0.1 0.5227 1.83070.5 0.5344 1.76351.0 0.5745 1.58561.5 0.6560 1.37302.0 0.8112 1.19322.5 1.1190 1.07823.0 1.7820 1.02353.5 3.3669 1.00504.0 7.6318 1.00074.5 20.7428 1.00015.0 67.2793 1.0000

After truncation it is combined with the LO beam,defined by a field distribution ULO and an opticalangular frequency WLO, and focused onto a photodetec-tor located in the w plane.

We denote by ( 2(t)) the ensemble average of thesquare of the intermediate frequency (IF) signalcurrent at frequency Ax = WLO - W << W. It is givenby 4

(is 2 (t)) = 2(-) 2 f d2 wid 2 w 2

X (Ur(Wl)Ur*(W 2 )ULO(Wl)ULO*(W 2)) (30)

where -q is the detector quantum efficiency, e is theelectronic charge, and h is Planck's constant. Theintegrations are performed over the detector area AD,and the brackets indicate the ensemble average result-ing from the random nature of the beam-target

(29)

where Rd (dimension -2 ) is the target-reflectivitycoefficient (mean directional reflectivity), pi and P2 aretwo vectors in the target plane, and 8( . ) symbolizesthe Dirac delta function. The brackets ( ... ) denotean ensemble average over the target statistics. Thescattered beam us(p) travels back to the lidar systemand impinges on the receiver aperture of diameter d2.

Table 2. Peak-Transmission Factor A for Collimated Beams (g = 0),Different Observation Plane Distances (n = 0.01, 0.1, 1, 5), and Several

Values of the Beam-Truncation Ratio n

A

m n = 0.01 n = 0.1 n = 1 n = 5

0.1 1.8307 1.8306 1.8254 1.68210.5 1.7635 1.7635 1.7587 1.62771.0 1.5856 1.5855 1.5818 1.48271.5 1.3729 1.3729 1.3705 1.30782.0 1.1932 1.1932 1.1918 1.15902.5 1.0782 1.0782 1.0777 1.06403.0 1.0235 1.0235 1.0233 1.0191

transmitter plane(r plane)

(a) transmitted field -*

target distance: z

*- scattered field(b)

BPLO -4

beam (w plane)splitter I

receiver plane(p plane)

target plane(p plane)

Fig. 5. Optical schematics of a heterodyne lidar: (a) transmitterand (b) coherent receiver.


interaction. Please note that in Eq. (30) the normal-ized optical fields are replaced by the absolute ones, asdefined in Eq. (3). We call (j 8 2(t)) the mean receivedsignal of the heterodyne system.

By invoking Siegman's theorem 6" 7 of back-propa-gated local oscillator (BPLO), which states that themixing of the signal wave front and LO beam, andthus ( 2(t)), can be found in any plane along theoptical signal path, we can formulate the mixingprocess in the plane of the receiver aperture as

(i82(t)) = 2(-) PLPLO Jf d2pd 2P2

X (r(pl)Ur*(p2)ULO(p1)ULo*(P2)), (31)where PL and PLO0 are the laser power and the BPLOpower before beam truncation at the transmitter andreceiver aperture, respectively, and r and ULO arenormalized fields. Again invoking Siegman's BPLOapproach and using Eq. (28), we find the meanreceived signal in terms of target-plane coordinates as

(i 2(t)) = 2qe 2PLPLO ff d2pid2p2

x (R(pI)R*(P 2 )Ui(Pl)Ui*(P2)ULO(Pl)ULO*(P2))-

(32)

Upon insertion of Eq. (29) into Eq. (32) the meanreceived signal becomes

(is2(t)) = 2(y- ) P L P LO RdX 2 d 2p Ui(P) I ULOW 12.

Because of the diffuse target we have assumed, theexpression for (i"2(t)) conveniently factors into thecontributions that are due to target, signal, and LO.The signal and the BPLO target-plane intensitiesenter Eq. (33) independently. This feature makes itpossible to apply the intensity-distribution approxima-tion described in Section 3.

In the following we use Eq. (33) to express thereceived signal for the case of a top-hat aperture and

1

For the LO field in the receiver plane we assume aGaussian distribution with spot size and phase curva-ture identical to that of the signal beam in thetransmitter plane.

First we evaluate the mean received signal for thetop-hat aperture using Eqs. (34) and (33). Signaland BPLO fields may be expressed in target-planecoordinates according to Eq. (13). After some ma-nipulations the mean received signal for the top-hataperture reads as

(SI) = J dpd2pd2Is2lILol2, (35)

with

1s=J'drrJ(nlvpd2r)expL2 i2r2 + (n,-(,)r 2],

(36)

ILO= drrJ(n2Pd2r)exp222 + (n2 -,)r

(37)

where we have used

d2 PVT di' Pd2 d'

dime, = 2a'

kdi2

ni =Z ,

kd,2

7i =-, i = 1, 2. (38)

The parameter VT defines the receiver-to-transmitterdiameter ratio. The index 1 denotes the parametervalues of the transmitter configuration, and the index2 indicates the BPLO characteristics. Evidently themultiple integrations and the infinite upper-integra-tion limit in Eqs. (35)-(37) make a numerical evalua-tion not an easy task.

To obtain the received signal (Sg) for the Gaussianaperture according to Eq. (33), we apply Eq. (19) forboth the signal and the BPLO field. After integra-tion one finds

A 2& 2 m.2m. 2 1y,$2

ml2 + 8)[(m22 + 2 + (fl2 2)]22 2+ M)2 + 7 (f - )2]' (39)

for a Gaussian aperture. For this purpose we use adimensionless representation of the mean receivedsignal by defining

(S) = (j 2 ( Rd\( 2 (34)2PLPLOI ii i1z )V ir i

where 81, Al, 82, and A2 are the fitting parameters ofthe signal path and the BPLO path, respectively.They can be calculated with the help of the approxima-tion procedure described in Section 3.

For the comparison that follows we assume afar-field target (n = 0.1) and collimated beams(E = 2 = 0). The mean received signals versus thebeam-truncation ratio of the transmitted beam ml

20 July 1994 / Vol. 33, No. 21 / APPUED OPTICS 4767

are evaluated for several values of the receiver-to-transmitter diameter ratio (VT = 0.5, 1, 2, 3) with thehelp of Eqs. (35)-(37) and (39). In the matchedGaussian case, the fitting parameters 61 and Al forthe transmitted beam have to be computed only once,whereas the numerical integrations required for thetop-hat aperture have to be evaluated separately forany parameter set (ml, ni, Al, and VT) of interest.

The mean received signal was evaluated for bothaperture concepts and is sketched in Fig. 6. Becauseof the very small differences in the signals of top-hatand Gaussian apertures, the corresponding tracescannot be resolved in Fig. 6. The trace labeled VT = 1corresponds to a common-aperture system. Themaximum return signal approaches an asymptoticvalue when VT increases beyond two.

To demonstrate the accuracy of our approximation,we define the percent approximation error E as

-(Se)) 100,

where (S) is the mean received signal of the hetero-dyne system, the index g indicates the Gaussianaperture and the index t indicates the top-hat aper-ture.

Figure 7 shows the approximation error for differ-ent values of the receiver-to-transmitter diameterratio VT. Of course the error may be made up notonly of errors resulting from the Gaussian approxima-tion but also of errors in the numerical calculation forthe top-hat case. When evaluating the latter, theinfinite upper-integration limit has to be replaced by areasonable finite value, which in connection with thenumerical integration algorithm severely influencesthe result. The integration efforts escalate with anincreasing upper limit and a decreasing tolerance inthe numerical integration algorithm. To ensure thatthe numerical errors are at least two orders ofmagnitude smaller than the approximation erros, wehave taken much care in the computations.

The approximation errors plotted in Fig. 7 arenearly zero for beam-truncation ratios exceeding

A

cV

,

.)a0

'aC-,

C(C,

E

0 1 2 3 4 5 6normalized transmitter radius m,

Fig. 6. Mean received signal (S) versus transmitter beam-truncation ratio ml for collimated beams ( = 0 and t2 = 0), adiffuse far-field target (n1 = 0.1), and several parameter values ofthe receiver-to-transmitter diameter ratio VT.

we-

0 1 2 3 4 5 6

normalized transmitter radius ml

Fig. 7. Gaussian aperture-approximation error in mean receivedsignal (S) versus transmitter beam-truncation ratio ml for severalvalues of the receiver-to-transmitter diameter ratio VT.

three. This is to be expected, as only minor beamtruncation occurs. Please note that the error shownin Fig. 7 is plotted versus the transmitter beam-truncation ratio ml. In the case of VT < 1 the valueof the receiver beam-truncation ratio m2 is smallerthan ml. Thus for T < 1, m2 < 3 is the relevantcondition. Keeping this is mind, we note that forsmall beam-truncation ratios the approximation er-ror is less than 3%.

5. Conclusions

Our Gaussian approximation method to top-hat aper-tures is a valuable tool for treating systems exhibitingpronounced aperture-truncation effects because theanalytical Gaussian lidar concept may be applieddirectly after the two fitting parameters have beencalculated. For many configurations of practical im-portance, an equivalent Gaussian aperture functioncan indeed be found.

Errors caused by numerical effects, like roundingerrors and problems with inifite integration limits,are eliminated. Consequently, multidimensional in-tegrations can be performed analytically. Ourmatched-system approach offers an ample field ofapplication. The signal-to-noise ratio, heterodyneefficiency, Wang's 5 optimum beam truncation, therange-weight function described by Lading et al., 18and the speckle treatment described by Churnsideand Yura19 are only some of the systems characteris-tics to be revaluated.

Appendix A: Validity Bounds

Because of the definition of the Gaussian aperture,the fitting parameters and A are positive reals.Complex values of S and A indicate that no Gaussianapproximation is possible for the specific set of systemparameters m, n, and I.

The limits of the approximation can be found bycomparison of the beam radius of the top-hat aper-ture in the observation plane with the realizablebeam radius of the Gaussian aperture. For theGaussian case, we calculate the effective beam radiusin the observation plane by first inserting Eq. (19)into Eq. (24). After some manipulation we may


4

2-VT=1

0 ' - - .-../ I

VT=22 _. / -'VT=O 5

.4- . i i | .

-

E'7,, = 1

140T

w'a

Cu

120t

100.E

.0

V 60

Ug 40

20

0

Cu

l 2

.0to

ZIDa,

.'

q

qmin- -_'

2 3 4

beam truncation ratio m6

Fig. 8. Required (q) and minimum achievable (qin,,) beam radiiversus the beam-truncation ratio m for n = 0.1 and e = 0.

express the functional dependence of q on 8 as

4(82in 2 + 1)2 + 84( 2

n 282(62M2 + 1)

Analyzing Eq. (Al) we find that the obsernbeam radius q has no upper limit but lower limit. The minimum value of q obthe Gaussian aperture is found when the Eq. (Al) with respect to S is set equal to z

[M2 + 4m 2 n 2 1 if 4M 2- I

qmin [4(thM2 + 1)2 + th4( 2 )] /2

th 2n2At2M2 + 1)

where th is given by

t2=_ 4h 4m 2 I - n I

In Eq. (A2) qmin represents the smallest eflradius that can be approximated withaperuure-Lransmssion uncuion or a specinc m, a,and parameter triple. If the beam radius of thetop-hat aperture system q is smaller than qin, theGaussian aperture approximation may not be applied.The existence of a limit of the Gaussian concept isplausible, as the radius of the transmitted Gaussianbeam cannot be narrowed beyond its diffraction limit.

By plotting the dependence of the required radii qand qmin on m for given values of t and n we may findthe validity regions for the Gaussian aperture approxi-mation. The first example, given in Fig. 8, shows thevalidity bounds for a collimated beam ( = 0) in thefar-field case (n = 0.1). We notice that the conditionqmin < q holds within the entire interval of minvestigated (0.1 < m < 6.0). A slightly differentcase is treated in Fig. 9. Here a collimated beam(e= 0) and the near-field case (n = 10) are investi-gated. We may distinguish two regions, namely F

(Al

0 mmf 2 4

' beam truncation ratio

F @ A

6

Dm

Fig. 9. Validity regions of the Gaussian aperture approximationfor a near-field target (n = 10 and = 0). The values of therequired q and the realizable qmj, beam radii in the observationplane are plotted versus the beam-truncation ratio. The allowedregion is denoted byA, and the forbidden one by F.

' (forbidden) andA (allowed), concerning the validity inFig. 9. Within F one finds qnin > q, and the approxi-

ration-plane mation concept fails. In both examples the observa-loes have a tion-plane beam redius q approaches qmin for increas-

Stainable for ing values of m. For large m, a failure of thederivative of approximation concept may be pretended sometimes.;ro, yielding However, the reason for failure is inaccurate numeri-

cal calculation of q. This does not constitute a realn < problem because for large values of m the influence of

the aperture may be neglected, and the fitting param-eters are to be set 8 = o and A = 1.

The value m = min separating the allowed and theotherwise forbidden regions of the Gaussian approximation can

be read from Figs. 8 and 9 for the cases n = 0.1 and= O and for n = 10 and = 0. It amounts to Mmin =

(A2) 0 and min = 1.75, respectively. A more generalstatement concerning min and the applicability ofthe Gaussian approximation was found by numericalanalysis. It turned out that n and e enter imin only

(A3) as the absolute value of the difference of the target-range and focal-distance parameters, i.e., as In - t1.The result is shown in Fig. 10. For In - < 7.38

.ective beam one finds m.,, = 0. It follows that the Gaussiana Gaussian aperture approximation is always applicable for fo-

2

C'SE

I1

00 2 4 6 8 10 12

In-tI

Fig. 10. Applicability of the Gaussian aperture approximation:minimum beam-truncation ratio mmin versus I n - .1


61 1 1 1 1 I

q2(8)

10

cused beams (n = ) and for collimated beams withthe observation plane in the far field (n << 1 and

= ).

References1. L. D. Dickson, "Characteristics of a propagating Gaussian

beam," Appl. Opt. 9, 1854-1861 (1970).2. C. M. Sonnenschein and S. A. Horrigan, "Signal-to-noise

relationships for coaxial systems that heterodyne backscatterfrom the atmosphere," Appl. Opt. 10, 1600-1604 (1971).

3. T. R. Lawrence, D. J. Wilson, C. E. Craven, I. P. Jones, R. M.Huffaker, and J. A. L. Thomson, "A laser velocimeter forremote wind sensing," Rev. Sci. Instrum. 43, 512-518 (1972).

4. R. G. Frehlich and M. J. Kavaya, "Coherent laser radarperformance for general atmospheric refractive turbulence,"Appl. Opt. 30, 5325-5352 (1991).

5. J. Y. Wang, "Optimum truncation of a lidar transmittedbeam," Appl. Opt. 27, 4470-4474 (1988).

6. J. Y. Wang, "Detection efficiency of coherent optical radar,"Appl. Opt. 23, 3421-3427 (1984).

7. B. J. Rye and R. G. Frehlich, "Optimal truncation and opticalefficiency of an apertured coherent lidar focused on an incoher-ent backscatter target," Appl. Opt. 31, 2891-2899 (1992).

8. B. J. Rye, "Antenna parameters for incoherent backscatterheterodyne lidar," Appl. Opt. 18, 1390-1398 (1979).

9. Y. Zhao, M. J. Post, and R. M. Hardesty, "Receiving efficiencyof monostatic pulsed coherent lidars. 1: Theory," Appl.Opt. 29, 4111-4119 (1990).

10. J. Y. Wang, "Heterodyne laser radar SNR from a diffuse targetcontaining multiple glints," Appl. Opt. 21, 464-476 (1982).

11. R. F. Lutomirski and H. T. Yura, "Propagation of a finiteoptical beam in an inhomogeneous medium," Appl. Opt. 10,1652-1658 (1971).

12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1958), Chaps. 3 and 4.

13. G. Engeln-MUIlges and F. Reutter, Formelsammlung zurnumerischen Mathematik (Wissenschaftsverlag, Mannheim,Germany, 1987), p. 25.

14. B. J. Rye, "Refractive-turbulence contribution to incoherentbackscatter heterodyne lidar returns," J. Opt. Soc. Am. 71,687-691 (1981).

15. M. H. Lee, J. F. Holmes, and J. R. Kerr, "Statistics of specklepropagation through the turbulent atmosphere," J. Opt. Soc.Am. 66, 1164-1172 (1976).

16. A. E. Siegman, "The antenna properties of optical heterodynereceivers," Proc. IEEE 54, 1350-1356 (1966).

17. R. H. Kingston, Detection of Optical and Infrared Radiation,Vol. 10 of Springer Series in Optical Sciences (Springer-Verlag,Berlin, 1978), Chap. 3.

18. L. Lading, S. Hanson, and A. S. Jensen, "Diffraction-limitedlidars: the impact of refractive turbulence," Appl. Opt. 23,2492-2497 (1984).

19. J. H. Churnside and H. T. Yura, "Speckle statistics of atmo-spherically backscattered laser light," Appl. Opt. 22, 2559-2565 (1983).


target-plane intensity approximation for apertured gaussian beams applied to heterodyne backscatter...

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