receiving efficiency of monostatic pulsed coherent lidars 2: applications

Receiving efficiency of monostatic pulsed coherentlidars. 2: Applications

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

Using the theory developed in Part 1, the receiving efficiency as a function of range, q(z), is calculated underdifferent conditions for the NOAA/ERL/Wave Propagation Laboratory CO 2 Doppler lidar. Theoreticalanalyses, numerical calculations, and experimental measurements are carried out to quantify the sensitivityof 1(z) to transmitted laser beam quality, telescope focal setting, telescope power, scanner astigmatism, LObeam divergence, and system misalignment. These results bring insight to the design of practical coherentlidar systems.

1. IntroductionIn Part 1 of this work,' the authors describe the

theory of calculating the receiving efficiency for mon-ostatic coherent lidars. Here we apply the theory tocalculating receiving efficiency for the NOAA/ERL/Wave Propagation Laboratory (WPL) coherent CO 2lidar. The WPL CO2 Doppler lidar is a coaxial system.An off-axis parabola-parabola (Mersennes type) tele-scope is used for both transmitting the laser pulse andreceiving the aerosol backscattered signal; the focallength of the primary mirror is 202 cm. The detaileddescription of the system can be found in papers byPost, 2 Lawrence et al.,3 and Hall et al.4 In the secondgeneration of this lidar system the transmitter hasbeen upgraded. 5 A more powerful injection-lockedTEA laser has replaced the old hybrid laser withoutsignificant redesign of the existing optics. Since thenew laser has an unstable resonator and the beam sizeis larger than the old laser, a beam expander is used inreverse to reduce the beam size so that other opticalparts in the transceiver need not be changed. Theoptical layout of the transmitting branch of the lidar isschematically shown in Fig. 1.

The system's receiving efficiency (z) is calculatedaccording to Eq. (18) in Ref. 1:

(z)= j I,(Qz)n,(Qz)dQ, (1)

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.

where (z) is the receiving efficiency at range z, Itn(Q,z)is the power-normalized intensity of the transmittedbeam at a point Q in the scattering plane, and n,(Q,z) isthe single-point receiving efficiency at Q, which isproportional to the normalized intensity of the BPLOfield ILN(Q,Z).

The descriptions for calculating Itn(Q,z), 8(Q,z) and7(Z), with discussions about the intermediate results ateach step, are given in Sec. II. The effects of transmit-ted beam quality, LO quality, telescope power, tele-scope focal setting, system misalignment, and scan-ner-mirror introduced astigmatism are analyzed inSec. III. From the analyses we gain insights to im-prove and optimize the system, as discussed in Sec. IV.A more detailed description of the numerical calcula-tions and a package of programs will be publishedshortly in a NOAA Technical Memorandum.

II. Numerical Calculation of n, ls and 7t for the WPLCoherent Lidar

A. The Power-Normalized Intensity Distribution n of theTransmitted Beam

1. Laser Output PatternAt present the injection-locked CO2 laser has a con-

focal unstable resonator. The concave, totally reflect-ing back mirror has a focal length of 15.0 m; and theconvex, partially reflecting output mirror has a focallength of-11.9 m. The reflectivity of the 3.0-cm diamoutput mirror is parabolically tapered from 70% at thecenter to zero at the edge. The magnification of the3.1-m long cavity is 1.26, and the equivalent Fresnelnumber for 1P(20) line is 0.893. The cavity is foldedby two flat mirrors, one of which is used to introducethe injecting laser into the main laser cavity via apinhole or a partially transmitting reflective coating.

Unstable resonator modeling has been done for this

4120 APPLIED OPTICS / Vol. 29, No. 28 / 1 October 1990

Fig. 1. TEA laser output patterns (with injection pinhole): (a)intensity; (b) phase. Horizontal size: 1.89 X 1.89 cm2. Arbitraryscale for the intensity. Phase values (in radians): 0.0 at the center;

-0.751 at the maximum; 0.32 at the edge.

cavity without gain, using the FFT method developedby Sziklas and Siegman.6 The number of samplingpoints is 256 X 256. The output of the laser is verydifferent from that of the old hybrid CO 2 laser, whichhad a stable resonator and Gaussian output. Modelcalculations show that diffraction from the pinholeintroduces further complexity into the laser mode,causing structures with high spatial frequency compo-nents. The intensity and phase patterns of the laseroutput with a pinhole for injection are shown in Fig. 1.Output patterns of the same cavity without the pinholeare shown in Fig. 2 for comparison. These computa-tional results agree very well with the experimentalburn patterns of the laser. More about resonatormodeling is supplied in a separate paper 7

Fig. 2. TEA laser output patterns (without pinhole): (a) intensity;(b) phase. Horizontal size: 1.89 X 1.89 cm2. Arbitrary scale for theintensity. Phase values (in radians): 0.0 at the center; -0.114 at

the maximum; 0.581 at the edge.

(2) Laser Propagation Through the TransmittingSystem

Calculations have been made for laser propagationfrom the resonator output through the transceiver sys-tem, using circular symmetric Fresnel diffraction for-mulas. The FFT method is not used because the laserhas been sharply truncated, and the spatial spectrumthus contains very high frequency components. Usingthe FFT method would require an extremely largenumber of sampling points for a 2-D calculation, andwould take even more computer time than is requiredfor a direct 1-D diffraction calculation.

If we define Uo(ro) as the input field and U(r) as theoutput field in the diffraction calculation, we have

1 October 1990 / Vol. 29, No. 28 / APPLIED OPTICS 4121

Fig. 3. Transmitting laser patterns at telescope secondary: (a)intensity; (b) phase. Horizontal size: 3.06 X 3.06 cm2. Arbitraryscale for the intensity. Phase values (in radians): 0.0 at the center;


2p(r a /jrr 2\ /orJ ( 2 rrr, U(r) = 27rp~)XZ J Uo(r0) exp ( Z p0 (r0 )J0 ( J r0dr0, (2)

where p(r) = exp[-jirr2/(Xf)] with a lens of focallength f on the output side, and p(r) = 1 when there isno lens; po(ro) = exp[-jrr2/(Xf)] with a lens at inputside, and po(r) = 1 without a lens; Jo is the zero orderBessel function of the first kind; a is the radius of thediffracting aperture. These equations are usedthroughout the propagation process from the laser out-put mirror to the telescope primary and beyond intothe atmosphere. (Telescope mirrors are treated asideal, aberration-free lenses, because they are parabol-ic.) The value of Jo is calculated from the series ex-pansion for Bessel function. When the argument is>25, Jo is calculated from asymptotic series. All theformulas are given in Ref. 9. The computational accu-racy for J is better than 1 X 10-l1. To maintain highcomputational accuracy for U(r), the increment in rand r is automatically controlled so that the incre-

Fig. 4. Transmitting laser patterns at telescope primary (OX):(a) intensity; (b) phase. Horizontal size: 15 X 15 cm2. Arbitraryscale for the intensity. Phase values (in radians): 0.0 at the center;


ment in the argument of the Bessel function Jo issufficiently small and the number of samples in eachperiod of the function is sufficiently large. When z issmall and the Fresnel number is high (e.g., LO beampropagation from the laser to the detector), accuracy isdouble-checked by varying the increment in r and ro.

After propagating through the beam expander, therelative intensity and phase distributions of the laserunchanged, but the size of the beam is halved. Howev-er, it changes remarkably during the propagation fromthe beam expander to the secondary. Since the pathlength is 4.47 m and the beam radius is only -0.7 cm,the Fresnel number is -1. At the secondary, the in-tensity pattern becomes smooth and peaks at the cen-ter, appearing to be nearly Gaussian [Fig. 3(a)]. Thehigh spatial frequency components of the original pat-tern are displaced far into the wings of the beam as itpropagates, and they are truncated by the telescopeprimary.

If the secondary mirror has a focal length of -20 cm


Fig.5. Transmittinglaserpatternsattelescopeprimary (20X): (a)intensity; (b) phase. Horizontal size: 15 X 15cm2. Arbitrary scalefor the intensity. Phase values (in radian): 0.Oatthe center;-0.271

at the maximum; -0.115 at the edge.

(thus the power of the afocal telescope is 10X), thelaser energy transmitted from the telescope will be 95%of the laser output, assuming there are no additionallosses associated with intermediate optical compo-nents. The corresponding patterns of the laser at theprimary are shown in Fig. 4. When the focal length ofthe secondary is -10.3 cm (the power of the telescope is20X) as in the old system, the transmitted energy willbe only 79% of the original laser output. These pat-terns are shown in Fig. 5. Comparing the patternswith and without the intracavity pinhole, we can seethat, although the differences of the beam profiles atthe laser output mirror are remarkable, the patterns atthe secondary are almost the same, and the percentageof the transmitted energy through the telescope is alsothe same. This implies that the pinhole in the lasercavity does not significantly affect the lower spatialfrequency components of the laser output energy (themain contributors to the energy), and its effects ontransmitted energy and system receiving efficiency arenegligible. Unfortunately, it does significantly affect

the transverse mode content, and hence the frequencystability of the laser.

The phase patterns show that the transmitted beamleaving the telescope primary is slightly convergent fora 20X telescope-the average radius of curvature beingabout -12 km. For a 1OX telescope, however, theoverall wave front is convex; it is concave only in thecentral area with a radius of curvature of about -3 km.These phase patterns have a significant effect on beamdivergence, and hence on system receiving efficiency.

3. Laser Intensity Pattern at Different RangesThe laser field obtained at the primary mirror

(above) is input to the next diffraction calculation forlaser intensity pattern Itn(Q,z) at different ranges inthe atmosphere. For a circular symmetric transmit-ter, Itn(Q,Z) can be written as Itn(r,z), where r is the off-axis distance. Itn is then used to calculate 1(z), usingEq. (1). When the transmitting and receiving axes areperfectly aligned, Eq. (1) becomes

adz) = 2r I,. (r,z)n,,(r,z)rdr. (3)

Since the beam size changes with range z, the sam-pling interval of r (i.e. Ar) in the numerical calculationof Itn(r,z) needs to be changed accordingly. To com-pare the beam divergence of the new laser with that ofthe old hybrid CO2 laser, we chose the spot sizes of theuntruncated Gaussian beam of the old laser, W(z) (seeTable I), as the reference scales. At each range, thesampling interval Ar = 0.01 W(z), and the total numberof sampling points is 200. The same scale is used in ascalculations, so the two functions in the integrand ofEq. (3) have the same sampling intervals.

The normalized transmitted beam intensity distri-bution as a function of range for the 20X telescope isshown in Fig. 6. The curve labeled GA is the untrun-cated Gaussian distribution of the old laser beam. Wecan see that It(r,z) is much narrower than Itn(r,z) forthe old laser at near ranges, while it is very close to thatof the old laser in the far range.

B. The Single-Point Receiving Efficiency 7As mentioned in Sec. II, there are two ways of calcu-

lating 77. The forward method is to calculate thenormalized single-point diffraction pattern Kn(QPz)at the detector, using Eq. (30) in Ref. 1, and to obtain

Table I. The Spot Sze of the Untruncated Old Hybrid CO2 LaserTransmitted Through a 19.6X Telescope

z(km) W(m) z(km) W(m) z(km) W(m)

1.0 0.1668 11.0 0.4736 21.0 0.79962.0 0.1940 12.0 0.5060 22.0 0.83243.0 0.2228 13.0 0.5384 23.0 0.86524.0 0.2526 14.0 0.5709 24.0 0.89815.0 0.2832 15.0 0.6034 25.0 0.93096.0 0.3143 16.0 0.6360 26.0 0.96387.0 0.3457 17.0 0.6687 27.0 0.99678.0 0.3774 18.0 0.7014 28.0 1.02969.0 0.4093 19.0 0.7341 29.0 1.0625

10.0 0.4414 20.0 0.7668 30.0 1.0954


.z7 6,

0 .2 .4 .6 .8 1.01.21.41.61.82.02.22.4r/WO

Fig. 6. Normalized intensity of the transmitted beam, It.(r,z), as afunction of range. Telescope power is 2X.

Receiving Optics DiagramNOAA Doppler Lidar

HeterodyneLO Laser Expander Frsnel 4 Iris Signalo

Ll L2 PWG; TR I2 WG2 L3 Dl

(not to scale)/

/ ~~SecondaryNi (Parabola) Fou

__ _ (To Focus)- Primary

(Parabola) | (To Focus) (From Scanner)

Fig. 7. Optical layout of the receiving branch in WPL lidar.

the function G(Q,z) from Eq. (15) in Ref. 1 (the inte-gration starts in the antenna plane and ends in thedetector plane). The backward method is to calculatethe normalized BPLO field in the scattering plane,using Eq. (32) in Ref. 1 (the integration starts in thedetector plane and ends in the antenna plane), and toobtain as from the intensity of the BPLO, using Eq.(33) in Ref. 1. For this work, we calculate U using theBPLO method as the standard procedure, occasionallycross-checking special cases by using both methods.Although very time-consuming, the forward approachis at times the method of choice because it gives thetransfer function K(Q,P,z) from point source Q topoint P in the detector plane where the actual mixingprocess takes place. Only with the forward methodcan we see directly the spillover of the signal power atthe detector and the match of the intensity and phasepatterns of the atmosphere backscattered signal(ABS) from each point Q with those of the LO. Suchcomparisons are generally more intuitive than insightsattempted using the fictitious BPLO patterns. Also,the forward method is more useful for estimating theoptimized LO size in the detector plane.

The receiving system optical layout is shown sche-matically in Fig. 7. Diffraction calculations for propa-gation of the LO field up to the detector lens, and forthe BPLO fields inside the receiver and in the atmo-

sphere are carried out using Eq. (2). The number ofsampling points at each step is determined by theaccuracy needed. The forward method uses 2-D dif-fraction calculations, and the number of samplingpoints is a compromise between computational accura-cy and computer time.

1. LO Beam PropagationThe original LO beam is a Gaussian beam with a

0.18-cm spot radius at its waist. It grows to 0.42-cmradius at the input of a 4X beam expander that colli-mates it. The output aperture of the beam expander is3.50 cm; thus the LO is truncated approximately at itse-2 point. After propagation through 26 cm, the LObeam passes through the half-wave plate POLI, andarrives at the first wire grid polarizer WG1 (Fig. 7).The intensity pattern (Fig. 8) shows high spatial fre-quency fluctuations, both theoretically and experi-mentally. The aperture of WG1 is only 2.0 cm, so thebeam is again seriously truncated. After propagationfor 65 cm, the beam pattern at the detector lens is evenmore deeply modulated, quite differently from aGaussian beam (Fig. 9). (The patterns in Figs. 8 and 9are very similar to the near field patterns of a truncat-ed Gaussian beam shown in Ref. 8, Chapter 18, Sec. 4.)The aperture of the 10-cm focal length detector lens L3is 2.53 cm, allowing the incident LO to pass through itwithout serious truncation. Finally, the beam is fo-cused at the detector, where its diffracted intensitypattern is much wider than it would have been (19.3,Mm) had the LO beam been untruncated. There is aphase change of r at the first zero of the intensitypattern (Fig. 10). To emphasize differences betweentruncated and untruncated LO beams, the intensity inFigs. 8-10 are all referenced to the maximum intensity

2.0 . . . . . . .

1.8

1.6

1.4

>1.2

E 1.0

~ 8.6

.4

.2

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0r (cm)

Fig.8. Relative intensity of LO at the first wire grid polarizer WG1,where the aperture radius is 1.0 cm. Before WG1, the LO beam isfirst truncated at the output of the beam expander, where theaperture radius is 1.75 cm, and the e-2 size of the untruncated LObeam is 1.744 cm. At the center of WG1, the ratio of the LOintensity to the maximum intensity of the untruncated LO, indicat-

ed by Imax, is 1.663.


< 1.2

1.0

.8

.6

.4

.2

00 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

r(cm)

Fig.9. Relative intensity of LO at detector lens, where the apertureradius is 1.265 cm. At the center of the lens, the ratio of the LOintensity to the maximum intensity of the untruncated LO, Imax, is

1.473.

E

40 50 r (I m)

cc

'a

654

3

21

-1 -

-2--3 --4-

-5-6-

0 10 20 30 40 50 60r (ML m)

70 80 90 100

Fig. 10. LO patterns in detector plane: (a) relative intensity I(r)/Imax, where Imax is the maximum intensity of the untruncated LO; (b)phase (in radian). The e 2 size of the untruncated LO at thedetector is 19.3 ,um, which is much smaller than the size of the Airy

pattern of the truncated LO, -46 gm.

of the untruncated LO, Imax, at each location. Thepeak intensity of the truncated LO at the detector isonly 7.8%, and the total power received by the detectoris only 45% of an untruncated Gaussian pattern. Afternormalizing to the total LO power illuminating thedetector chip, the peak intensity of the LO, ILN(O), is

Fig. 11. BPLO patterns at detector lens: (a) intensity; (b) phase.Horizontal size: 2.53 X 2.53 cm2. Arbitrary scale for the intensity.

Phase values (in radian): 0 at the center; -0.768 at the edge.

still only 17.3% of the untruncated Gaussian. Thecentral part of the power-normalized LO patternILN(P) is nearly Gaussian with a spot radius of 46 Am,which is much smaller than the half-size of the detectorchip (100 Am). The high spatial frequency compo-nents of the LO beam incident on the lens fall well offthe detector so they will not reappear in the BPLOfield.

2. BPLO Field Propagation Inside the ReceiverThe BPLO field is first calculated at the detector

lens, using the conjugated LO field incident on thedetector as input. Although the detector chip issquare shaped, radial symmetric formula is used tocalculate the BPLO. As mentioned above, the LOspot size is much smaller than the detector chip, socontributions from the corners of the chip are negligi-ble.

The BPLO intensity pattern at the lens [Fig. 11(a)]is dramatically different from the pattern of the for-ward-propagating LO at this point (Fig. 9). Removal


................ -

I01

Fig. 12. BPLO patterns at telescope secondary: (a) intensity; (b)phase. Horizontal size: 3.06 X 3.06 cm 2. Arbitrary scale for theintensity. Phase values (in radian): 0.0 at the center; -0.342 at the

maximum; 1.96 at the edge.

of high spatial frequency components by detectortruncation, and hence smoothing of the pattern, isobvious. The wave front is flat in the central zone andconcave at the edge, and the intensity profile is notGaussian at all.

Although the BPLO beam passes through many op-tical elements within the receiver, their apertures aresufficiently large to allow diffraction effects to be ne-glected. Therefore, the diffraction integral in Eq. (2)can be calculated only once from the detector lens tothe secondary of the telescope. During propagationthe BPLO intensity pattern develops more structurein the center, and the wave front changes from concaveto generally convex, with slight concavity in the centralarea (Fig. 12).

Similar to the transmitted laser pattern, the BPLOpattern at the antenna is also strongly affected bytelescope power. For the 1OX telescope, the intensitypattern is truncated by only 2% [Fig. 13(a)]. Theoutput wave front is convex, with a concave centralarea, the radius of curvature of which is about -2.5 km

Fig. 13. BPLO patterns at telescope primary (10X): (a) intensity;(b) phase. Horizontal size: 15 X 15 cm 2. Arbitrary scale for theintensity. Phase values (in radian): 0 at the center; -0.358 at the

maximum; and 1.52 at the edge.

[see Fig. 13(b)]. For the 20X telescope, however, theintensity pattern is seriously truncated [Fig. 14(a)]; thetotal BPLO power passing through the primary is only74% of the original. The phase front is totally concavewith an average radius of curvature of about -10 km[Fig. 14(b)].

3. BPLO Propagation in the Atmsophere and theSingle-Point Receiving Efficiency qt

The procedure for calculating BPLO propagation inthe atmosphere parallels calculations for the transmit-ted laser beam. -s is calculated using Eqs. (32) and(33) in Ref. 1. A typical qs distribution for a collimatedLO and a 20X telescope is shown in Fig. 15.

4. Forward Method of Calculating asThe forward method first calculates the single-point

signal diffraction pattern at the detector, or the nor-malized transmission function Kn(Q,P,z), defined inEq. (10) and (30) in Ref. 1. First, we calculate the


Fig. 14. BPLO patterns at telescope primary (20X): (a) intensity;(b) phase. Horizontal size: 15 X 15 cm2. Arbitrary scale for theintensity. Phase values (in radian): 0 at the center; -0.356 at the

maximum; -0.348 at the edge.

signal field at the secondary for a single-point sourceQ', which is the image of a point source Q in thescattering plane S through both elements of the tele-scope. The position of Q' is calculated using geometricoptics. The point source signal beam from Q' is trun-cated at the secondary by the exit pupil, which is theprojection of the primary on the secondary. As wementioned in Ref. 1, the signal field for off-axis Q is notcircularly symmetric and 2-D calculations are inevita-ble. Therefore, a 2-D FFT routine using 256 X 256sampling points has to be employed to calculate dif-fraction for the signal beam propagating from the sec-ondary to the detector lens. Propagation from thelens to detector, however, is calculated using a 2-DFresnel diffraction integral. Thus the number of sam-pling points at the lens can be reduced to 32 X 32, andthe number of sampling points at the detector to 10 X10. s is then calculated using Eqs. (15) and (21) inRef. 1. For each point Q in the scattering plane wehave to go through this entire process. The number of

.6

.4

.3

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6

r/WO

Fig. 15. Single-point receiving efficiency Os(rz) as a function ofrange. LO is collimated. Telescope power is 20X.

radial samples in the scattering plane is seventeen, andthe radial increment Ar = 0.1 W(z). Us is then interpo-lated to increase the number of points to thirty-threeto improve the accuracy in calculating 7(z) from Eq.(3). The numbers of sampling points in the 2-D calcu-lations are limited by the computer.

As an example, diffraction patterns at the detectordue to single-point sources from a scattering plane at a30-km range are shown in Fig. 16. When the sourcepoint Q moves off axis, the whole diffraction patternK,(Q,P,z) moves off the center of the detector; hencepower spills outside the detector and the pattern ismismatched with the LO pattern, resulting in a de-creased 17(Qz). However, this effect is not as signifi-cant in the near range. In this case the image plane ofthe signal is far behind the focal plane of the detectorand the size of any point source signal diffraction pat-tern at the detector is greater than the size of thedetector chip itself. Then the profile of 77,(Qz) is lowand flat, and as Q moves off axis in the scatteringplane, 77,(Q,z) does not change significantly.

The forward method follows the physical process ofsignal propagation (from the scattering plane to thedetector) and uses the real diffraction patterns of thesignal in the detector plane from single-point sources,thus enabling us to better understand the factors af-fecting heterodyne detection. The only disadvantageof this method is the vast amount of computation thatis required, which in turn necessitates a trade-off be-tween computing time and accuracy. Nevertheless,the forward method provides reasonable accuracy forfs and 7(z) with modest requirements on computingtime. We have found that for far field calculations,the central part of the as distribution is calculatednearly identically using the two different methods,whereas at the far edge of the distribution the forwardmethod gives lower values than the BPLO method.This is because the number of sampling points in theforward method is not large enough to deal properlywith high spatial frequency components. Since errorsin US are mainly in the wings of the distribution wherenormalized intensities are low, the overall error in cal-culating receiving efficiency beyond 5 km with theforward method is only 1-3%.


t35

>.30 2

25 .2-5;.20.

.15

.10

.05

0 2 4 6 8 1012141618202224262830R(km)

Fig. 17. Effect of injection pinhole on system receiving efficiency:(1) laser with injection pinhole; (2) laser without pinhole. LO is

collimated and detector is at LO waist.

range is beyond 1 km. The effects on system perfor-mance on varying parameters of the transmitting laser,of the LO, and of the telescope are quantified in theplots of receiving efficiency vs range.

11. Effects of Various Parameters on the SystemReceiving Efficiency

A. The CO2 TEA Laser

Fig. 16. Signal diffraction patterns at the detector due to a single-point source at 30 km: (a) r = 0; (b) r = 0.5W; (c) r = 1.0W; (d) r =1.5W. Horizontal size: 200 X 200 gM2. Arbitrary scale for the

intensity.

C. The Overall Receiving Efficiency i7(z)

The receiving efficiency as a function of range isobtained using Eq. (3) after as and Itn are calculated.Generally the computational accuracy for -q is betterthan 0.1% if the backward method is used and the

1. Injection PinholeAs we mentioned earlier, the pinhole affects only

near field, not far field patterns of Itn(r,z). The trans-mitted beam pattern for the case without a pinhole isslightly narrower in the far field, resulting in a smallincrease in receiving efficiency (- 6%). (See Fig. 17.)

2. WavelengthMost of our calculations are carried out for a laser

wavelength of 10.59 Am. For a wavelength of 9.25 Am,the near field pattern changes somewhat, but receivingefficiency increases only 1-2% at far ranges, a negligi-ble change.

B. LO DivergenceIf the LO beam is not collimated, its waist after

passing through the detector lens is behind or in frontof the focal plane of that lens, depending on whetherthe LO beam is divergent or convergent. This signifi-cantly affects receiving efficiency (see Fig. 18). Usual-ly the detector is located at the LO waist by experimen-tal procedures. When the LO waist is very slightlybehind the focal plane of the lens (as is the detector),Receiving efficiency increases slightly at finite ranges(e.g., curve 2 in Fig. 18). This is because the imageplanes of all finite range scattering planes are behindthe focal plane of the lens for infinity and becausewave-front changes in signal beam during propagationfrom the secondary to the detector lens extend theimage planes even farther behind the focal plane.Thus if the LO waist is slightly behind the lens's focalplane, the signal pattern matches the LO pattern bet-ter. If the LO is more divergent yet, pushing the waiststill farther behind the focal plane (e.g., curves 3-5 in


z .35

' .30 2

20

1 5 . - 6…

.10

.05: /

C0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30R(km)

Fig. 18. Effect of LO beam divergence on system receiving efficien-cy (20X telescope): (1) collimated LO; (2)-(5) uncollimated LO.Distance from focal plane to the detector: (2) 0.5 mm; (3) 1.0 mm;

(4) 1.5 mm; (5) 2.0 mm; (6) -1.0 mm. Detector is at LO waist.

Fig. 18), the near range receiving efficiency increaseswhile the far range efficiency decreases. When the LOwaist is behind all realistic image planes, one can seethe decrease of X7 at all ranges (e.g., curve 5). If LObeam is convergent (and thus the waist is in front of thedetector, e.g., curve 6 in Fig. 18), x at all ranges is lowerthan that in the collimated LO case. Since the imageplane of the signals approaches the LO waist as rangeincreases, iq monotonally increases with range.

These calculations show that the collimation of theLO beam is crucial to the system performance. In ourcase, the detector lens has a focal length of 10 cm; thusa 0.5-mm misfocus is caused by an incident LO beamwith a radius of curvature of 20 m. Even with interfer-ometric testing it is difficult to set the LO collimationthis well. At a 20-m radius of curvature, the effect onfar range receiving efficiency is of the order of 10%,whereas for the near range (e.g., 2.4 km), the effect is ofthe order of 50% or worse.

C. Effect of Telescope Power and Telescope FocalSetting

The effect of telescope power is twofold. For thetransmitting branch, decreasing the telescope powerincreases the transverse phase difference across thelaser beam at the primary; hence it increases the diver-gence of the transmitted beam and decreases the re-ceiving efficiency. For the receiving branch, decreas-ing the telescope power increases the signal beam sizeafter the secondary, thus increasing the spilled-overpower at the detector lens and decreasing the receivingefficiency. On the other hand, decreasing the tele-scope power increases the distance from the imagepoint source Q' to the detector lens, which decreasesthe distance between the detector and the image planeof the signal and improves the matching of the LO tothe signal. The latter effect opposes the former. Thenet result is a decrease of receiving efficiency at mostranges but an increase at the very near ranges. Thiscan be clearly seen in Fig. 19, in which the telescopepower is 1OX.

For similar reasons, the effect of LO beam diver-gence on X is different for 1OX and 20X telescopes.

.50

.45

.40

Z .35U-i

, .30L .25

> .20

" .15.10

.05

'0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30R(km)

Fig.19. Effect of LO beam divergence on system receiving efficien-cy (10X telescope): (1) collimated LO; (2), (3) uncollimated LO.Distance from focal plane to the detector: (2) 0.5 mm; (3) 1.0 mm.

With a lower telescope power, the image point sourceQ' is farther from the detector lens, and it is focusedcloser to the focal plane of the detector lens. There-fore, when the LO waist (hence the detector) is situat-ed behind the focal plane, decreases more at farranges, but it increases little at near ranges.

The dependence of receiving efficiency on telescopefocal setting concerns both changes in the transmittingbranch and changes in the receiving branch. In thetransmitting branch, focusing the system at a finiterange means adding a negative phase factor [p2 (ro) inEq. (2)] to the laser field at the primary, thus changingthe laser beam wave front at the primary. This ofcourse changes the intensity distribution of the beamat all ranges. In the receiving branch, the same thinghappens to the BPLO field at the primary, thus chang-ing the value and the distribution of the single-pointreceiving efficiency as. We may also explain this effectin terms of the forward method: altering the focalsetting changes the distance between the image planeof the signal and the LO waist (at the detector), thuschanging fS. However, it is easier to explain the focalsetting effect in terms of BPLO method as follows:

Since the wave front of the transmitting laser or theBPLO changes with telescope power, the dependenceof X on the system focal setting is closely related totelescope power. For the 20X case, the transversephase difference from the center is negative through-out the primary area for both the transmitted beamand the BPLO. Focusing the system to a finite rangeadds more negative phase difference, and thus bettercompensates for the exponential term expLbrr2/(Xz)]in Eq. (2) for the near range, but over-compensates forit in the far range, causing an increase of 77(z) in thenear range and a decrease in the far range [Fig. 20(a)].For the 1OX telescope, however, the phases of thetransmitted laser field and the BPLO field at the pri-mary are negative only in the central three-fifths ofprimary radius; they are positive beyond three-fifthsof radius. Focusing the system at a finite range be-yond 3 km compensates for these phase differences.Thus setting the telescope focus to a finite range in-creases the receiving efficiency at all ranges of interest


3

11 -

z .35

0a .30

.25CD5 .20

t) .15

.10

.05

.50

.45

40)

z .35

Eu .30L .25CD

5 .20

Uj .15

.10

.05

t3

z-LUC-.u-

R(km)

0 4 6 8 10 12 14 16 18 20 22 24 26 28 30R(km)

Fig. 20. Focal setting effect: (a) 20X telescope: (1) focused atinfinity; (2) focused at 2.4 km; (3) focused at 5 km; (4) focused at 10km. (b) telescope lOX: (1) focused at 2.4 km; (2) focused at 3.5 km;

(3) focused at 5 km; (4) focused at 10 km; (5) focused at infinity.

[Fig. 20(b)]. Contrary to expectations, the far range?i(z) can increase up to 20% when the system is focusedto 3.5 km (the optimal range). Further decreasing thefocal range will increase the near range 17(z) but de-crease the far range 7(z).

D. The Misalignment Effect

When there is a misalignment a between the trans-mitting and receiving axes outside the primary of thetelescope, the single-point receiving efficiency distri-bution and the intensity distribution of the transmit-ted beam no longer have the same center. Receivingefficiency is calculated using the following equation:

2,(Za) = ',(r,z)rdr It[r(zar,q)1dP. (4)

For a fixed misaligned angle a the effect is moresevere for the 20X telescope than for the 1OX telescope(Fig. 21). The relative decrease for a 50-Arad mis-alignment in the 20X case is 11.0 dB at far ranges, whilein the 1OX case it is only 4.8 dB. However, since oursystem uses the same telescope, misalignment outsidethe telescope is solely determined by the angle ai be-tween the axes of the transmitted beam and the receiv-er on the detector side of the telescope secondary; a =as/M, where Mis the telescope power. Thus for a fixedai, the resultant a is twice as large in the 1OX case thanin the 20X case, resulting in a greater relative decrease

.50

.45

.40

z.35:

, .30

O .25

20C-' 1 5 2u-i 3

.10 4

n~~~~~~~~~~~.055

CQ 2 4 6 8 10 12 14 16 18 2022 24 26 28 30R(km)

Fig.21. Misalignment effect: (a) telescope 20X; (b) telescope lOX.LOis collimated and detector is at LO waist. Solid lines: (z) withmisalignment. Angle of misalignment between transmitter andreceiver: (1) 10,urad; (2) 20 grad; (3) 30,urad; (4)40 grad; (5) 50 grad.

Dashed lines: x(z) with perfect alignment.

in n. For example, if ai is 200 grad, the relative de-crease at far ranges in the 1OX case is 0.75 dB, while inthe 20X case it is 0.43 dB. If ai is 500 grad, the relativedecreases are 3 dB and 1.7 dB, respectively. It is clearthat the 20X telescope is less sensitive to system mis-alignment in terms of ai.

E. The Scanner Astigmatism EffectExperiments show that the two scanning mirrors in

our system are not perfectly flat. The beam scanningsystem has an overall focal length of -1.2 km, whichcan be compensated by a 0.35-mm shift in the primaryto secondary separation. However, even after com-pensation, it has a residual focal length of -4.9 km inthe horizontal direction and 5.5 km in the vertical. Inthis case, the transmitted beam and the BPLO in theatmosphere are no longer circular-symmetric. Equa-tion (2) is thus modified:

U(r,o) = p(r) f Uo(ro) exp(jpi-Ž) po(ro)rodro

( 22r ex jr(ro cosO)1 Fx jr(ro sinO)1X j exp[ - °;f exp[ oi

X exp[- 2rrr cos(O - O) dO. (5)


. � I � � I . . . . . .

P

=_05.I

-z

u-

US

wCD

LIlL

R(km)

30 40 50 60 70LO SPOT SIZE (m)

zci5

zU-i

I

.4

.2

.2

.2

.1

.0

_0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30R(km)

Fig. 22. Scanner astigmatism effect: (a) 20X telescope; (b) 1OXtelescope. (1) (z) with perfect scanning mirrors; (2) (z) with

astigmatic scanners.

The astigmatism of this scanner has a strong effecton the receiving efficiency with the 20X telescope [Fig.22(a)]. The far range receiving efficiency is decreasedby 27%. For the lOX telescope, x(z) is not affected somuch (only 8% decrease results) because both thetransmitting beam and BPLO at the primary have amore curved wave front than they have with the 20Xtelescope; hence they are relatively less sensitive to theadditional distortion. Since astigmatism distorts thewave front of both the transmitted laser and theBPLO, the advantage of using higher telescope powerto reduce the transverse phase difference at the prima-ry is almost negated, except at the near ranges (<2km). The effects of telescope focal settings are alsoweakened for similar reasons. In agreement with thetheoretical calculations, experiments show but littledifference in the far field signals with change of sec-ondary focal length or telescope focal setting.

IV. Optimization Design Considerations of the SystemOn the bases of the above analyses, we can enhance

the receiving efficiency, and hence the overall systemperformance, in the following ways:

(1) Improve the optical quality of the scanningmirrors and redesign the mirror mount to eliminateastigmatism. This will give us -30% increase in 7,

using the 20X telescope.(2) Optimize the LO size and quality. We have

used the normalized signal pattern Kn(QP) at the

Fig. 23. Receiving efficiency at 30 km as a function of LO spot sizeat the detector. LO is Gaussian and collimated.

1.0

.9/

.8 '

.7

.5-

.4

.3

.2

.1

0 10 20 30 40 50 60 70 80 90 100r(m)

Fig. 24. Signal diffraction pattern (from 30 km) at the detector:(1) total intensity of the incoherent signal; (2) intensity pattern dueto an axial point source; (3) phase pattern due to an axial pointsource.

detector from a 30-km scattering plane to calculate thereceiving efficiency as a function of LO spot size at thedetector. The LO field is assumed to be Gaussian.The result (Fig. 23) shows that the optimized LO radi-us at e-2 point is 53 ,m, which is about the same size asthe Airy pattern of a single-point source (Fig. 24), notthe size of the total signal intensity. (Note that thereceiving efficiency in our system is 0.268, which is 6%lower than that in Fig. 23 at Wo = 46 ,gm, owing to thedeparture of our LO from a pure Gaussian distribu-tion.)

The optimized LO diameter at the detector lens inour system is 1.2 cm, obtainable with a 1.4X powerbeam expander provided the beam is not truncated ateither the polarizer or the detector lens. Then the LOpattern at the detector will be Gaussian with a sizematching the far range signal patterns. This kind ofoptimization will give us a 16% increase in i7.

(3) Remove the pinhole in the laser cavity, whichwill produce an additional 6% increase in far range t,

and may improve the frequency stability of the lasergreatly. We have recently implemented this improve-ment by injecting through a 99% reflecting, 1% trans-mitting cavity folding mirror.

(4) Make the laser output and the detector lens


.25

- 20LU

z .15

W .v

.05

CO2C 80 90 100

IV S . ., . . . .

45

40

'5

0 2 1 ______________

5

0

15

. . 4 . . . .A A A A A .A r AA .

-. I . . .

n

.30

'O

much closer to the secondary to eliminate propagationdiffraction loss, if possible. An early calculationshows that if the diffraction effect from the secondaryto the detector lens were ignored, the far range receiv-ing efficiency would increase 27% (the near-rangewould increase even more).

If all the above improvements can be made, in-creases of 2-3 dB in n are obtainable.

V. Summary

Theoretical analyses and numerical calculationshave quantified the sensitivity of (z) to our transmit-ted laser beam pattern at the primary, and to thetelescope focal setting, telescope power, scanner astig-matism, LO beam divergence, and system misalign-ment. We have obtained insight into the practicaldesign of monostatic coherent lidar systems. Furtheroptimization could result in an increase of 2-3 dB in 7at far ranges. We have observed that the optimizedsize of LO at the detector that gives the maximumreceiving efficiency at far ranges is not the one thatmatches the size of the total ABS at the detector, butthe one that matches the Airy pattern of a remote axialpoint source, which supports our discussions in Ref. 1.Also, a slightly divergent LO yields higher X values, butit is hard to set this divergence properly in practice.

This work is partially supported by NASA undercontract No. H-98023B. The authors wish to thank R.E. Cupp for providing experimental information, K. R.Healy and R. A. Richter for help in computer graphicsprogramming, and T. S. Balzer for help in computer

usage. Special thanks are due to R. Frehlich, R. J.Lataitis, and B. J. Rye, for their careful review of thepaper.

References1. Y. Zhao, M. J. Post, and R. M. Hardesty, "Receiving Efficiency of

Monostatic Pulsed Coherent Lidars. 1: Theory," Appl. Opt. 00,000-000 (1990).

2. M. J. Post, "Atmospheric Infrared Backscattering Profiles: In-terpretation of Statistical and Temporal Properties," NOAATechnical Memorandum ERL WPL-122 (1985).

3. T. R. Lawrence, R. M. Hardesty, M. J. Post, R. A. Richter, R. M.Huffaker, and F. F. Hall Jr., "Performance characteristics of theNOAA Pulsed Doppler Lidar and Its Application to AtmosphericMeasurements," in Proceedings, Fifth Symposium on Meteoro-logical Observations and Instrumentation, (Toronto, Canada,(April 11-15, 1983) p. 481.

4. F. F. Hall, Jr., R. E. Cupp, R. M. Hardesty, T. R. Lawrence, M. J.Post, R. A. Richter, and B. F. Weber, "Six Years of Pulsed-Doppler Lidar Field Experiments at NOAA/WPL," in Proceed-ings, Sixth Symposium on Meteorological Observations and In-strumentation, New Orleans, (January 12-16, 1987) p. 11.

5. M. J. Post and Richard Cupp, "Optimizing a Pulsed DopplerLidar," Submitted to Appl. Opt. 28, 4145-4158 (1990).

6. E. A. Sziklas and A. E. Siegman, "Mode Calculations in UnstableResonators with Flowing Saturable Gain. 2: Fast FourierTransform Method," Appl. Opt. 14, 1874-1889 (1975).

7. Y. Zhao, M. J. Post, and T. R. Lawrence, "The Effects of InjectionPinhole, Mirror. Tilt, and Reflectivity Function of the TaperedOutput Mirror on the Performance of a CO2 TEA Laser withUnstable Resonator," to be published.

8. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA1986).

9. British Association for the Advancement of Science: Mathe-matical Tables, Vol. 6: Bessel Functions, Part I, Functions ofOrder Zero and Unity (University Press, Cambridge, 1950).


receiving efficiency of monostatic pulsed coherent lidars 2: applications

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