Interference of backscatter from two droplets in afocused continuous-wave CO2 Doppler lidar beam

Maurice A. Jarzembski and Vandana Srivastava

With a focused continuous-wave CO2 Doppler lidar at 9.1-mm wavelength, the superposition of back-scatter from two ;14.12-mm-diameter silicone oil droplets in the lidar beam produced interference thatresulted in a single backscatter pulse from the two droplets with a distinct periodic structure. Thisinterference is caused by the phase difference in backscatter from the two droplets while they aretraversing the lidar beam at different speeds, and thus the droplet separation is not constant. Thecomplete cycle of interference, with periodicity 2p, gives excellent agreement between measurements andlidar theory. © 1999 Optical Society of America

OCIS codes: 010.3640, 280.3640, 290.1350, 260.3160, 350.5030, 030.0030.

latpp

1. Introduction

Interference can be observed in many physical situ-ations, including most notably in optics, electromag-netic fields, and acoustics, and it arises because of thecombination of waves. For the specific case of elec-tromagnetic scattering, a theory for scattering fromtwo spheres has been formulated,1,2 and measure-ments2 of the effect of interference for the combinedelectromagnetic scattering from two particles havebeen made in the microwave regime. This interfer-ence2 is a function of the wavelength of the radiation,the distance by which the particles are separated, theorientation of the particles with respect to the direc-tion of the incident radiation, and the angular direc-tion at which the scattering event is viewed.Interference that is due to electromagnetic scatteringfrom two particles for short wavelengths, however,can be difficult to observe because of the limitationsplaced on both the particle size and the particle sep-aration with respect to the wavelength of the incidentradiation. For radio waves and microwaves, it couldbe relatively easy to construct an experiment to ob-serve the interference that is due to the combined

The authors are located at the Global Hydrology and ClimateCenter, 977 Explorer Boulevard, Huntsville, Alabama 35806.M. A. Jarzembski is with the NASA Marshall Space Flight Center;his e-mail address is maurice.jarzembski@msfc.nasa.gov. V.Srivastava is with the Universities Space Research Association;her e-mail address is vandana.srivastava@msfc.nasa.gov.

Received 15 October 1998; revised manuscript received 10 Feb-ruary 1999.

0003-6935y99y153387-07$15.00y0© 1999 Optical Society of America

scattering from two particles; however, for shorterwavelengths one has to rely primarily on chance tocapture the event because it is so difficult to controltwo particles simultaneously. However, by the useof a continuous-wave ~cw! Doppler lidar and a parti-cle generator one can readily observe this effect, evenby chance, without too much effort for micrometer-sized particles in the shorter-wavelength regimes ofthe electromagnetic spectrum. In this paper, inter-ference that is due to the superposition of backscatterfrom two micrometer-sized droplets obtained with aNASA Marshall Space Flight Center cw CO2 Doppleridar at 9.1-mm wavelength is reported. The result-nt single backscatter signal from both droplets con-ained an interference structure with a well-definederiodicity that was accurately measured and com-ared with cw lidar theory.3 The agreement be-

tween measurements and theory is excellent,indicating that the interference arises because thedroplets are moving at different speeds and, there-fore, the relative droplet separation is not constant.These results show a well-defined periodic structureof the superimposed backscatter signal from bothdroplets in the lidar beam moving in and out of con-structive and destructive interference.

2. Measurements

Full details of this laboratory experimental setup,particle generation method, measurement technique,and the cw lidar can be found elsewhere4,5; thereforewe shall refrain from an effort to digress into detailshere. Figure 1 shows measurements of time-resolved signal pulses with a high-speed digitizingLeCroy oscilloscope from single silicone oil droplets

20 May 1999 y Vol. 38, No. 15 y APPLIED OPTICS 3387

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responding to the Gaussian lidar beam intensity atthe lidar beam focus. Also, for the special case ofsimultaneous backscatter from two droplets in thelidar beam, a periodic interference structure can beobserved as the combined pulse profile. The drop-lets were ;14.12 mm in diameter. The stream ofsilicone oil droplets resided at a Doppler-shift centerfrequency of fD ; ~3.4 6 0.2! MHz. The dropletspeed at the center frequency was calculated to be;21.9 ms21, with the droplet speeds ranging from;20.6 to ;23.2 ms21. The strength of backscatteraries according to the placement of the droplets inhe lidar beam; that is, droplets that reside near thexis of propagation give stronger backscatter pulseshan those away from the axis. Also shown on aeparate channel of the oscilloscope is the corre-ponding signature when an amplitude demodulatorircuit designed to detect the amplitude envelope ofhe Doppler-shift frequency within each pulse profiles used. The use of the filter helps in observation ofubtle details of the interference that otherwiseight not be easily decipherable.Backscatter from simultaneous droplet events

hows a complete cyclic interference structure ofaximum and minimum. As one can see by using

he amplitude demodulator circuit, the structure re-embles a ripplelike structure that resides on top ofhe pulse profile. This interference is due to theresence of two or more droplets in the lidar beam athe same time with slightly different speeds such that

Fig. 1. Time-resolved signal pulse measurements with a high-speas from two closely spaced droplets that show a structure of construan amplitude demodulator circuit is also shown for each figure. Fscale is 1 Vydivision.

388 APPLIED OPTICS y Vol. 38, No. 15 y 20 May 1999

backscatter from the two droplets can interfere. Asa result, the periodic process of constructive and de-structive interference is established, dependingwhether the backscatter from each droplet is in phaseor out of phase, respectively. Figures 1~a!–1~c! showhree distinct backscatter interference events forhich the average period T of the complete cycle of

interference is 13.02 6 0.39, 24.39 6 0.76, and31.29 6 1.66 ms, respectively. As will be shown be-low for these cases, the relative speed difference Dnbetween the two droplets decreases with increasingT. The droplets for these three cases are probablyquite near the axis of the beam of propagation be-cause the signal is large as compared with those ofsingle-droplet events also documented. Figure 1~a!shows that the interference disappears toward theright-hand edge of the profile, which occurs becauseone of the droplets leaves the lidar beam while theother one remains in the beam, thus giving backscat-ter for a single droplet. Figure 1~d! shows an inter-ference structure where T appears not to be aconstant. In this case, at the beginning of the inter-ference T is ;18.5 ms, whereas at the end of theinterference T is ;12.7 ms. This may show a situa-tion in which the relative speed between the dropletsis not constant because acceleration or deceleration ofthe droplets or both could be occurring or perhapsbecause the trajectories of the droplets in the lidarbeam may be skewed with respect to each other.Figure 1~e! shows that a single pulse for two droplets

nsient LeCroy oscilloscope from single silicone oil droplets as welland destructive interference. A time scan for the signal fed into

ll figures, the time scale is 50 msydivision, whereas the amplitude

ed tractiveor a

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traversing toward the edge of the lidar beam haslower signal strength, which means that the interfer-ence structure may still be detected here. Finally,Fig. 1~f ! shows a slightly more complicated interfer-ence structure in which it appears that the interfer-ence consists of at least three droplet events in thelidar beam, each having a separate and distinctspeed. In this case, a smaller peak can be observedsuperimposed within the interference structure; con-sequently, T is not constant. Of the six cases thatshow interference here, Figs. 1~a!–1~c! offer good com-parison with fundamental cw lidar theory for thiscoherent detection.

3. Theory

It has been shown that the heterodyne backscattersignal current is produced by a single scatterer nearfocus behaves as3

is } expF22SpRrlL D2Gcos~f!, (1)

where f is given by

f 5 2kL 1 fD t 2 wR 1 2w 1 2 tan21FpR2

lL S1 2LFDG ,

(2)

here R is the ~1ye!2 intensity radius at the telescopeprimary mirror, L is the range measured from thetelescope primary mirror to the particle, F is therange to center of the focal volume, l is the photonradiation ~k 5 2pyl!, fD is the Doppler-shift fre-uency caused by the moving scatterers ~heterodyne

beat frequency between the backscatter radiation fre-quency and the reference local-oscillator frequency!, rs the position of the scatterer at distance L at a given

time t, wR is the phase of the reference local-oscillatorsignal relative to the transmitted signal, and w is afunction3 that depends on lidar parameters L, F, R, l,and r. In relation ~1! the exponential term gives theoutline of a particle’s overall pulse profile in the lidarbeam, as one can observe from Fig. 1. Because back-scatter was measured at the focus, L 5 F, for all thedroplets shown in Fig. 1, a range-dependent term3 inthe argument of the exponential function as well asan additional term3 multiplied by the exponentialfunction, which gives the range response of backscat-ter, have both been neglected.

The cosine term, on the other hand, gives the phaserelationship information, and it is this term that isresponsible for fD observed within each droplet pulseprofile ~Fig. 1! as well as for the interference that isdue to the superposition of more than one backscatterpulse. Consequently, for pedagogic purposes, therange-dependent terms, although they are small, arenot neglected in the argument of the cosine term,even though measurements were made at focus.When the signal power is evaluated3 for an ensembleof droplets and for time periods that are long withrespect to 1yfD, the square of this term is 1y2. How-

ever, for the case of two droplets, this term needs to bepresent. This is so because the argument of the co-sine term is slightly different for two droplets at dif-ferent positions L that are moving at different speedsin the lidar beam and have slightly different valuesfor fD. Therefore each droplet in the beam has adifferent value for f given in Eq. ~2! that is a compli-cated function continuously changing in time. So,for complete analysis of the two-droplet interferenceproblem, f1 and f2 corresponding to droplets 1 and 2would have to be solved simultaneously. However,to investigate this interference effect more simply,rather than determining each term in f1 and f2, it isbest to view the first droplet as having a value f whilehe second droplet moving with respect to the firstroplet has a value f 1 d. Here d can be expressed

as a small fractional value of f such that d 5 pf,where p is a dimensionless constant that needs to bedetermined. In this case d is not a constant, but, justas f changes in time, d also changes in time. There-fore, for two droplets moving with slightly differentspeeds in the lidar beam, the combined backscatteredradiation from both of the droplets undergoes changein phase with time, resulting in periods of construc-tive and destructive interference. This change isgoverned by the added phase d with respect to f fordroplet 2 with respect to the phase f for droplet 1.Therefore the backscatter signal current can be ex-pressed as the sum of the individual backscatter cur-rents is1 and is2 for droplets 1 and 2, respectively,which is given by the addition of two harmonic wavesas

is } cos~f! 1 cos~f 1 d!. (3)

n relation ~3! it is assumed that the exponential termor both droplets is the same, as we are not too con-erned here with the amplitude of the combined pro-le because it does not contain phase properties ofackscatter. Now is pulsates with a beat frequency

that depends on the value of d with respect to f. Fora given d 5 pf, it can be seen that within one periodor one beat of is the number of wavelengths is simplyp21.

Now a relationship for both droplets involving bothd and f, the average Doppler frequency shift fD, and

eriod T can be determined. Each droplet has apecific Doppler frequency shift, fD1 and fD2; the av-

erage is simply fD, which for this case corresponds tothe average fD within the single backscatter pulsecontaining the interference pattern for both dropletsshown in Fig. 1. The difference of these two frequen-cies would, therefore, correspond to the beat fre-quency fbeat, which is the inverse of T. Thus thenumber of wavelengths corresponding to fD that areccurring within T is given by fDT. In the same

manner, the ratio dyf governs the number of wave-lengths, p21, that occur within one period. Inas-much as the superposition of backscatter from thetwo droplets is governed by relation ~3! and becauseyf controls the pulsation period T that was found in

20 May 1999 y Vol. 38, No. 15 y APPLIED OPTICS 3389

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our measurements, the interference structure is gov-erned by the equation

dyf 5 1yfD T. (4)

In Eq. ~4!, the right-hand side contains the two mea-surable experimental input parameters fD and T that

efine the phase relationship between d and f on theleft-hand side, which needs to be substituted into Eq.~3!. Equation ~4! is therefore a simple relation in-volving the phase relationship of d with respect to fin relation ~3!. This phase relationship governs thecomplicated interference structure from the two drop-lets. Thus, if the speed difference between two drop-lets is small, this will imply a small d, which from Eq.~4! shows that as d decreases, the period of pulsationT increases or fbeat decreases.

This discussion can be further clarified by use ofthe schematic shown in Fig. 2, which represents ageometrical interpretation, or a classic interferomet-ric explanation, of the situation. ~Note, however,that the drawing is not to scale, as the distance cov-ered by the droplets relative to droplet size and l ismuch larger than that indicated here.! Two drop-lets, labeled 1 and 2, are shown traversing a portionof the lidar beam at angle u to the direction of beam

ropagation. Droplet 1 is moving faster than drop-et 2. At one point at time t 5 t0 within the traversal

of the lidar beam, backscatter from both droplets in-terferes constructively; thus, enhanced backscatter isobserved at the detector. This situation is indicatedby a wavelength difference of 0l along the lidar beam.After a time t 5 t0 1 T, droplet 1 moves a distance d1with speed n1 while droplet 2 moves a distance d2

Fig. 2. Schematic showing the condition for interference of back-scatter from two droplets moving at different speeds traversing thelidar beam.

Table 1. Calculated Interfere

Parameter Fig. 1~a!

T ~ms! 13.02 6 0.39fDT 44.27 6 2.92dyf ~%! 2.26 6 0.15f for 1T 278 ~88.49p!d for 1T 2.000p 6 6.6%fyT ~radyms! 21.35 6 3.0%Dn ~ms21! 0.49 6 3.0%K 1248 ~397.3p!

390 APPLIED OPTICS y Vol. 38, No. 15 y 20 May 1999

with speed n2 such that after the droplets arrive attheir new positions L, indicated by 19 and 29, theirbackscatter interferes constructively again, whichcorresponds to a wavelength difference of ly2 alongthe lidar beam. This process will continue for wave-length differences of l, 3ly2, 2l, . . . , etc. until drop-let 1 exits the lidar beam, leaving only droplet 2 andthus giving only backscatter for droplet 2. Becausethe complete cycle of each consecutive interference isgoverned by the distance along the lidar beam thatseparates the two droplets to be half-integral multi-ples of l, the total optical path difference between thecombined incident radiation and backscatter fromboth droplets is twice this amount, or integral mul-tiples of l, that is, 0l, 1l, 2l, 3l, . . . , etc. For thiscondition the detector records integral numbers ofcomplete cycles of interference. Shown in Fig. 2 isthe interference condition derived from the simpleequations that govern this motion. This interfer-ence condition shows the relationship of the speeddifference Dn between droplet 1 and 2 in terms of l, T,and u and the inverse relationship of Dn with respectto T. The same equation can also be derived fromthe Doppler interpretation, where fD from a singledroplet along the lidar beam axis is given by6 fD 5 2ncos uyl. Here the time interval T is given by 1yT 5fbeat 5 fD1 2 fD2 5 2Dn cos uyl. It is noteworthy thatthe application of this equation to the interference ofbackscatter from two droplets shows similar charac-teristics to the simple interference conditions foundoften in optics, for example, in thin films. Calcula-tions of Dn for the cases shown in Figs. 1~a!–~c! can,herefore, easily be performed.

4. Discussion of Measurements

Table 1 is a compilation of the calculations for thethree cases shown in Figs. 1~a!–1~c!. However, onlyhe case of Fig. 1~a! is discussed in detail here. Fig-re 3~a! shows calculation of relation ~3! as a function

of f with d 5 0.0226f for the case corresponding tohat in Fig. 1~a!. If d were zero, the function wouldimply be a cosine curve; however, with the additionf d, the function pulsates with a period that is indic-tive of T or fbeat. Figure 3~b! shows the same func-

tion for only one period, showing a higher frequencycontained within the envelope. Determination of dfor one T can now be made. For fD 5 ~3.4 6 0.2!

Hz and T 5 ~13.02 6 0.39! ms, fDT is calculated tobe ~44.27 6 2.92!. This is actually the number of

arameters for Figs. 1~a!–1~c!

Fig. 1~b! Fig. 1~c!

24.39 6 0.76 31.29 6 1.6682.93 6 5.52 106.39 6 8.431.21 6 0.08 0.94 6 0.07

520 ~165.52p! 668 ~212.63p!2.003p 6 6.7% 1.999p 6 7.9%

21.32 6 3.1% 21.38 6 5.3%0.26 6 3.1% 0.21 6 5.3%

1246 ~396.6p! 1250 ~397.9p!

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wavelengths within one T, as shown in Fig. 3~b!.Therefore, from relation ~4!, d has changed relative tof by a factor of 0.0226. In Fig. 3~a! or 3~b!, one

eriod T was completed for f ; 278 ~;88.49p! rad,hich leads to fyT ; 21.35 ~;6.80p! radyms. For

ne complete period, d ; 0.0226 3 278 rad, or ~2.000p6.6%! rad, which shows that Eq. ~4! gives excellent

greement with the expected value of 2p rad. Thisdvancement of d by 2p for each complete cycle ofnterference is indicative of a difference of one wave-ength for the difference in the total optical pathength ~incident radiation plus backscatter radiation!etween the two droplets, as shown in Fig. 2.

Fig. 3. Calculation of the function @cos~f! 1 cos~f 1 d!# for d 5.0226f as a function of f for ~a! 11 periods and ~b! 1 period with

higher resolution, where the number of wavelengths containedwithin one period can be distinguished. Calculation of the samefunction modulated by exp@22~fyK!2# for ~c! d 5 0.0226f and K 51248 to compare with Fig. 1~a! for 1 ms ' 21.35 rad, ~d! d 5 0.0121fand K 5 1246 to compare with Fig. 1~b! for 1 ms ' 21.32 rad, ~e! d5 0.0094f and K 5 1250 to compare with Fig. 1~c! for 1 ms ' 21.38rad.

ence, within one complete cycle of interference con-ained in each T increment at an average fD for both

droplets, f has changed by 278 rad, whereas at thesame time d has changed in increments of 2p gov-erned by Eq. ~4!. Determination of f and d from Eq.~2! and relation ~3! is quite complex; nevertheless, Eq.4! facilitates the estimation of phase relationship ofhe two droplets by measurement of fD and T. Fur-hermore, from the condition for interference given inig. 2, the relative speed difference Dn between the

wo droplets is calculated to be ~0.49 6 3.0%! ms21,well within the difference of ;2.6 ms21 for expectedspeeds, which may range from 20.6 to 23.2 ms21.

In addition, a calculation of relation ~3! modulatedy the exponential term given in relation ~1! can be

performed that will give a representation of the pulseshape for two droplets as displayed on the oscillo-scope. This calculation can be done in the time do-main or the phase domain. However, to study theinterference of backscatter pulses in the phase do-main one must rewrite the exponential term in termsof f. This is a two-step process: First, in relation~1!, this exponential term is of the form exp~22r2yC2!, with C 5 lFypR written in the spatial domain r.It can be rewritten in the time domain4 because thedroplets are moving at constant speed n to yieldexp~22t2yD2!, with D 5 lFypRn sin u, where for thisanalysis it is assumed that the droplets traverse thelidar beam through the axis of propagation. Next,as t progresses, f also progresses at a constant ratesuch that at a given position of the droplets in thelidar beam there is a continuous well-defined f at anygiven t. For example, for the case shown in Fig. 1~a!,for every 1 ms in t, f advances by 21.35 rad. There-ore the exponential term can be expressed in termsf f as exp~22f2yK2!, where K is a constant thatefines the standard deviation of the exponentialerm in the phase domain. One can determine theonstant K by making both exponential expressionsquivalent; that is, K 5 lFfypRnt sin u. Substitut-

ing for the various known parameters R 5 0.0305 m,F 5 9.53 m, l 5 9.1 3 1026 m, u 5 45°, and n ; 21.9ms21, with fyt for this interference case being 21.35radyms, yields a value of K of ;1248 ~;397.3p! rad.Figure 3~c! shows calculation for relation ~3! modu-lated by the exponential term for K 5 1248 rad; thatis, exp@22~fy1248!2# 3 @cos~f! 1 cos~1.0226f!#. Thegreement of Fig. 3~c! with the measurement @Fig.~a!# of two simultaneous droplets in a lidar beam isxcellent. Also, the relative placement of the peakswing to the interference expression that containshe summation of the two cosine terms is excellent.evertheless, the measured width of the whole back-

catter pulse is slightly wider than the calculation.his could be attributed to the standard deviation of

D, which gives a velocity distribution of droplets of~21.9 6 1.3! ms21. Thus, if n is smaller than 21.9ms21, K will increase, which will widen the exponen-tial term. Also, this slight increase in the width ofthe pulse was observed for single-droplet pulses,4which may be caused by the optical quality of thelidar.

20 May 1999 y Vol. 38, No. 15 y APPLIED OPTICS 3391

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Similarly, for the cases in Figs. 1~b! and 1~c! thealues of all parameters were determined ~Table 1!,nd again for one period, Eq. ~4! gives values2.003p 6 6.7%! and ~1.999p 6 7.9%! rad, respec-ively, which agree well with the expected value ofp rad. The corresponding measured values for Dnre ~0.26 6 3.1%! and ~0.21 6 5.3%! ms21, respec-ively. The calculated profiles for these two casesre shown in Figs. 3~d! and 3~e!. Although fullrofiles were not captured in their entirety by thescilloscope traces shown in Figs. 1~b! and 1~c!, thegreement of calculations with measurements isevertheless quite good. Moreover, there is excel-

ent agreement for the values for fyT and K for thehree cases listed in Table 1. However, similarlyo the case shown in Fig. 1~a!, the measured widthsf the profiles shown in Figs. 1~b! and 1~c!, espe-ially that shown in Fig. 1~c!, appear to be greaterhan those in the calculations. This differenceay be attributable to the possibility that the drop-

ets may be moving more slowly than the meanpeed of 21.9 ms21, or perhaps the optical quality of

the lidar may cause the observed discrepancy.However, for the case shown in Fig. 3~c! there may

ctually be a third droplet with a smaller amplitudet the beginning of the profile. The addition of ahird droplet here may also explain the higher un-ertainty in T ~as well as in the other parameters inable 1!, the jagged second interference maximums shown from the time scan with an amplitude-emodulator circuit, and the seemingly very longime duration of the whole pulse @Fig. 1~c!#.

Discussion of these measurements and calcula-ions can be summed up with the schematic shown inig. 4. This figure shows a vector representation ofhe addition of two harmonic waves pertaining to theackscatter situation of Fig. 1~a!. Such a figure canepict the stages of interference of combined back-catter from two droplets, including total construc-ive and total destructive interference, which caneadily be observed by the resultant vector. Dropletcurrent is labeled is1 and droplet 2 current is labeled

is2; the total current is is represented by the resultantvector. A phase of f 5 f0 was arbitrarily estab-

Fig. 4. Vector addition of the harmonic wave functions is1 and is2

for each droplet and the resultant harmonic wave function is @re-lation ~3!# for the case shown in Fig. 1~a! for an arbitrary initialcondition of f 5 f0 with ~a! d 5 0.0p ~totally constructive inter-erence!, ~b! d 5 0.226p ~nearly constructive interference!, ~c! d 5

0.904p ~nearly destructive interference!.

392 APPLIED OPTICS y Vol. 38, No. 15 y 20 May 1999

lished as an initial condition. Both f and d varycontinuously with time. The vectors that representthe two harmonic wave functions is1 and is2 and theresultant vector that represents the resultant har-monic wave function is rotate; however, the relativepositions will also change because all the vectors ro-tate with different angular velocities. Figure 4~a!hows current from both droplets in phase with eachther, giving the vector sum of both currents as theighest possible and, thus, total constructive inter-erence. As time advances, both f and f 1 d ad-ance at different rates. After a certain time, suchhat f has advanced by 10p rad for droplet 1, f 1 das advanced by 10.226p rad for droplet 2 @Fig. 4~b!#.he vectoral sum is now nearly total constructive

nterference. Finally, the case is shown for f ad-ancing by 40p rad for droplet 1, while f 1 d hasdvanced by 40.904p rad for droplet 2 @Fig. 4~c!#.ere the vectoral sum is now nearly total destructive

nterference, depicted by the short length of is. Totalestructive interference will occur for d 5 p rad orhen f has advanced by ;44.27p rad. This corre-

sponds to one half-period, which is exactly half of f 5278 rad, as determined.

5. Conclusion

The interference of the superposition of backscatterfrom two droplets was measured with a NASA Mar-shall Space Flight Center focused cw CO2 Dopplerlidar at 9.1-mm wavelength for the first time to ourknowledge. These measurements show not onlythat this cw lidar is a sensitive instrument to detectmicrometer-sized droplets but that detection of theinterference structure of backscatter from two drop-lets on time scales of tens of microseconds can easilybe made with high resolution. The periodicity ofthis structure on the whole pulse profile agrees ex-tremely well with fundamental cw lidar theory ofcoherent detection of backscatter from two droplets inthe lidar beam at the same time, indicating that thecombined backscatter from two droplets can inter-fere. There is a complete cycle of constructive anddestructive interference separated by 2p rad for eachonsecutive maximum or minimum, respectively, onhe combined backscatter pulse profile. Because in-erferometric investigations provide valuable as wells high-resolution information about physical pro-esses, it is possible that this investigation may pro-ide useful information on the dynamics of complexow fields with particles used as tracers and a cwoppler lidar as a diagnostic tool. This technique,

herefore, not only can make absolute velocity mea-urements of the tracers but can also determine theifferential velocity between the tracers.

This research was supported by the NASA Mar-hall Space Flight Center under the Center Director’siscretionary Fund. The authors are grateful toike Stewart, who designed the amplitude demodu-

ator circuit, and acknowledge the help of Amy Jai-ien Lai, who was sponsored by NASA’s Summer

tionships for coaxial systems that heterodyne backscatter from

High School Apprenticeship Research Program andparticipated in work in the laboratory.

References1. G. W. Kattawar and C. E. Dean, “Electromagnetic scattering

from two dielectric spheres: comparison between theory andexperiment,” Opt. Lett. 8, 48–50 ~1983!.

2. K. A. Fuller, G. W. Kattawar, and R. T. Wang, “Electromagneticscattering from two dielectric spheres: further comparisonsbetween theory and experiment,” Appl. Opt. 25, 2521–2529~1986!.

3. C. M. Sonnenschein and F. A. Horrigan, “Signal-to-noise rela-

the atmosphere,” Appl. Opt. 10, 1600–1604 ~1971!.4. M. A. Jarzembski, V. Srivastava, and D. M. Chambers, “Lidar

calibration technique using laboratory-generated aerosols,”Appl. Opt. 35, 2096–2108 ~1996!. @Note that l should be addedto the numerator of Eq. ~8!#.

5. J. Rothermel, D. M. Chambers, M. A. Jarzembski, V. Srivastava,D. A. Bowdle, and W. D. Jones, “Signal processing and calibrationof continuous-wave focused CO2 Doppler lidars for atmosphericbackscatter measurement,” Appl. Opt. 35, 2083–2095 ~1996!.

6. J. D. Briers, “Laser Doppler and time-varying speckle: a rec-onciliation,” J. Opt. Soc. Am. A 13, 345–350 ~1996!.

20 May 1999 y Vol. 38, No. 15 y APPLIED OPTICS 3393