Molecular Backscatter Heterodyne Lidar: A Computational Evaluation
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observed spectra to theory.1 Atmospheric lidar sys-tems that exploit molecular scattering are more ro-bust because the backscatter coefficient is morestable than that of aerosols, especially in the free
sn 5 N, (1)
where L 5 ly@2 sin~uy2!# is both the scattering scalelength that is conventionally used in Rayleigh-troposphere.All approaches that have been proposed or at-
tempted to date have used a short optical wavelength~l! to take advantage of the well-known dependencef Rayleigh-backscattered power PR } 1yl
4 or thedetector photocount N } PRy~hn! } 1yl
3 ~where h islancks constant and n is the optical frequency!.he need for eye safety in an atmospheric lidar en-ails operation in the ultraviolet ~use of e.g., the third
harmonic of Nd:YAG at l 5 355 nm!. In the short-
scattering theory3,4 and the proportionality constantin Eq. ~1! for the velocity component observed in alight-scattering experiment, n 5 LF1. Here I con-sider only backscatter lidar, so that the scatteringangle u 5 180 and L 5 ly2. For thermal scatterLF2 ' 300 mys, and a lidar requiring sn , 1 mys callsfor N . ~300!2 ' 105 photocounts or the equivalentanalog signal. The CRLB for temperature measure-ment is sT
2 5 2T2yN,4 so that a slightly greatersignal energy is needed to obtain sT , 1 K. In bothcases, lidars of significant pulse energy and antennasize with considerable pulse and range gate averag-ing are likely to be needed, and spectral modulationarising from atmospheric fluctuations and back-ground light can contribute to error. Moreover,the assumption of a Gaussian spectral profile hasbeen criticized5 because the effect on the spectrum ofcollisions might bias atmospheric temperature esti-mates. This is also likely to apply to wind measure-ments, in which optimal estimators of frequency shiftmake use of fitting the observed spectrum to an as-
The author is with the Cooperative Institute for Research inEnvironmental Sciences ~University of Colorado and National Oce-anic and Atmospheric Administration!, Environmental Technol-ogy Laboratory, RyEyET2, 325 Broadway, Boulder, Colorado80303
Received 20 October 1997; revised manuscript received 5 June1998.
0003-6935y98y276321-08$15.00y0 1998 Optical Society of America
20 September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6321Molecular backscatter hetera computational evaluation
Barry J. Rye
The application of heterodyne lidRayleigh cross section, infraredthose of current and proposed dirin the visible or ultraviolet. Rayregimes encountered in the infrara triplet of relatively narrow linesments. 1998 Optical Society o
OCIS codes: 280.3640, 280.13
Use of molecular scattering as a target for environ-mental lidar systems has been of interest for manyyears. It is capable of supplementing aerosol back-scatter that is conventionally used in Doppler windmeasurements and makes possible temperature pro-filing. Molecular density and hence atmosphericpressure can also be deduced in principle either bymeasuring the backscatter coefficient or by fittingyne lidar:
observe molecular scattering is considered. Despite the reducedms are predicted to require mean power levels comparable withetection lidars that operate with the thermally broadened spectraBrillouin scattering in the kinetic and hydrodynamic ~collisional!of particular interest because the observed spectrum approachesare more suitable for wind, temperature, and pressure measure-
erica80.3420, 290.5870, 290.5830.
wavelength limit, the scattered light has a thermallybroadened spectrum that is described approximatelyby the Gaussian function exp@2~F 2 F1!
where F1 is the Doppler frequency shift and F2 is theignal bandwidth. The CramerRao lower boundCRLB! on the variance of an optical estimate of windpeed limited only by signal shot noise is then2
6sumed theoretical model. It is therefore worth con-sidering the alternative of using the more highlystructured spectra that are available when collisionaleffects are dominant at longer wavelengths. Atthese wavelengths, use of optical heterodyne receiv-ers is advantageous.
2. Molecular-Scattering Spectrum
The spectrum of a Rayleigh return is characterized3,4by the parameter
y 5 ~LyLmfp!, (2)
which is referred to as the normalized collision fre-quency. The wavelengths for which y 5 1 are shownn Fig. 1 as a function of altitude, using values for theollision mean free path Lmfp calculated from theumerical formula Lmfp 5 2.17T~K!
2y@~T~K! 1111!p~atm!# nm,6,7 with pressure p and temperaturetaken from the U.S. Standard Atmosphere model.The curve marks the transition between weak andstrong collisional contributions and lies in what isknown in Rayleigh-scattering theory as the kineticregime. Evidently, a lidar using tropospheric back-scatter at wavelengths 1 mm , l , 3 mm operates inhis regime. Calculation of the scattered spectrumsing Tentis S6 theory for molecular scattering8 at
two altitudes and three wavelengths of interest re-sults in the spectra given in Fig. 2. The sharpeningof the spectrum and its separation into three compo-nents at longer wavelengths when collisions domi-nate, in what is known as the hydrodynamic regime,is most marked at lower altitudes. To my knowl-edge, molecular-scattering spectra in the kinetic andhydrodynamic regime have not been reported with aninfrared laser source even in laboratory experiments,but they have been observed with visible lasers usingslightly forward scattering from gases including N2at high pressure.9,10 CO2 laser systems ~l ; 10 mm!
ith heterodyne receivers have been used to observe
Fig. 1. Altitude at which the Rayleigh-scattering parameter y 51 for backscatter lidar @Eq. ~2!#, or ~equivalently! at which l 52Lmfp, where the collisional mean free path Lmfp is calculatedusing temperature and pressure values obtained from the U.S.Standard Atmosphere model. To the right of the curve, scatteringis in the collisional or hydrodynamic regime whereas to the left ofthe curve it is relatively unaffected by collisions.
322 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998trongly forward scattering from plasmas in theomewhat analogous collective or coherent regime.he signal spectrum then contains features arising
rom organized density fluctuations that includecoustic modes.11,12
Fig. 2. Spectra predicted using Tentis S6 theory for the back-scatter signal at two altitudes with pressure and temperatureobtained from the U.S. Standard Atmosphere model. The alti-tudes are 3 km ~solid curve! and 10 km ~dashed curve!. The lidar
avelengths are ~a! 0.355, ~b! 2.1, and ~c! 10.6 mm. The atmo-phere is assumed to consist entirely of N2. At shorter wave-
lengths and higher altitudes, where y , 1 @Eq. ~2!# and collisionshave little influence, the spectral profile tends toward a thermallybroadened Gaussian. At longer wavelengths and lower altitudes,collisions cause the spectrum to become, in the limit, a triplet ofthree Lorentzians, of which the outer ~Brillouin! curves representscattering from thermally generated sound waves.
photon-limited sensitivity resulting from amplifica-tion of the signal by optical mixing, low transmissionloss, and high rejection of thermal background. Per-
fpssvaThe advantage of using narrow spectral featuresfor wind measurements of frequency shift is quanti-fied by Eq. ~1!. When the spectrum has become atriplet, the temperature can be found from the sepa-ration of the two Brillouin sidebands ~Fig. 3!, whichorresponds to twice the speed of sound ~approxi-ately 600 mys! and is proportional to =T. If the
Doppler shift of each sideband can be found indepen-dently to 6~1y=2! mys, the separation would be
nown to approximately 61 mys, or 1 part in 600, andthe temperature to approximately 1 part in 300, or 1
Fig. 3. Frequency separation of the peaks of the Brillouin side-bands ~expressed as a velocity! calculated using Tentis S6 theoryor a 10-mm backscatter lidar as a function of temperature. Theoints correspond to temperatures in the U.S. Standard Atmo-phere at different altitudes ~shown in kilometers!. The pres-ures assumed correspond to the U.S. Standard Atmospherealues at the same altitudes ~continuous line! and pressures 10%bove an below these values ~dashed lines!. The lines show that
sideband separation is essentially a function of temperature and isalmost independent of pressure.K; if the width of a sideband is one tenth of the widthof the thermally broadened singlet ~see Section 1!,
20formance with Rayleigh backscattering is evaluatedhere by calculating the parameters of various sys-tems that can achieve a particular standard devia-tion, shet for the Doppler-shift estimate.
There are various formulas that have been derived,based on various assumptions and mainly in theDoppler radar literature, for the CRLB on the vari-ance of a heterodyne estimator of frequency shift andreturn power. The results have in common that thevariance is not inversely proportional to N as in Eq.~1! but has a more complex dependence that is attrib-utable to signal fading and small signal suppression;these are discussed in general terms in Appendix A.2.In the derivation of the formulas, it is assumed thateach single-pulse lidar return is digitized at a fre-quency FS corresponding ~in accordance with
yquists theorem! to the receiver bandwidth. Aample of duration t containing M discrete dataoints that is drawn from this digitized time seriesepresents the return from a single range gate. Aimple formula for CRLB is available under the as-umptions of stationary conditions within the rangeate, white receiver noise, and Gaussian statistics foroth signal and noise.13 The autocorrelation func-
tion of the return signal at the kth lag is assumed tohave the form C~k, F1! 5 a~k!exp@ j2p~F1yFS!k#,where F1 appears only in the frequency-shift term.The covariance matrix G of the signal is defined withcomponents gi,k 5 a~k 2 i!. If the matrices Q 5 Gd 1I and G 5 Q21 have, respectively, components q and g,where d is the wideband signal-to-receiver noise ratioand I is an identity matrix, the CRLB obtained is
~i 2 k!2qikgki
For most calculations in both radar and aerosolbackscatter lidar, a~k! has been assumed to be Gauss-ian. Here, a~k! was found numerically by Fouriertransforming theoretical Rayleigh spectra calculated,like those in Fig. 2, with the Tenti algorithm. Thespectra were computed over a receiver bandwidth FScorresponding to the band 21.5 , x , 1.5. x is anormalized frequency used in the Rayleigh-scattering literature3,4 that relates the measured ve-locity component LF to the thermal speed using
~2kB Tymmol!1y2F, (4)
where kB is Boltzmanns constant and mmol is themolecular mass.
For an incoherent backscatter heterodyne lidar,
September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6323determination of its shift to 6~1y=2! mys should re-quire N ' 2 3 103, not 105. Moreover, the measure-ment of a frequency difference would be lesssusceptible to laser frequency or wind speed fluctua-tions, compared with either wind speed or singletspectral width measurements, because both side-bands are displaced andyor broadened equally.
Brillouin scattering is interpreted as arising fromthermally excited sound waves.3 Atmospheric tem-
erature measurement with a precision of approxi-ately 61 K by scattering from sound waves is
lready well established by use of the radio-acousticounding system. This is, in principle, an all-eather system and has an important synergy withoppler radar profilers that are widely deployed, but
t requires active generation of a high-power audibleulse. In clear weather, the lidar technique is po-entially capable of longer range.
3. Heterodyne Lidar Performance
In comparison with direct detection, heterodyne lidarhas advantages of simplified spectral resolution,
of the art. I assumed that the losses in each systemare the same, and I adjusted ATR so that each system
three spectral features separately, discarding the in-termediate regions of the spectrum where the signalis low @Fig. 2~c!#.
Table 1. Parameters of the Three Heterodyne Lidarsa
Wavelength, l ~mm! 0.355 2.1 10.6Laser-pulse energy, E ~J! 0.1 0.5 1
6gives approximately the same values in Fig. 4 for thepulse count needed to achieve shet 5 1 mys. It ap-pears that measurements to a precision of 61 myscould be obtained using any of the systems with ap-proximately 1000 pulses, or within 100 s at 10 Hz, ataltitudes as high as approximately 7 km with a 1-kmgate or as high as 5 km with a 100-m gate. No
Fig. 4. Number of lidar pulses n needed under stationary condi-ions to secure a standard deviation in the Doppler-shift estimatef 1 mys by use of a ground-based heterodyne lidar. For the solidurve plots, the lidar parameters are those listed in Table 1. Theavelengths are 0.355 mm ~xs!, 2 mm ~filled squares!, and 10.6 mm
ith an output energy of 1 J ~open squares!. The dashed curve
corresponds to the same 10.6-mm system with an output energyexplicit account is taken of the effects of wind ortemperature shear within the range gate or of refrac-tive turbulence.
The signal energy needed to make these measure-ments over n pulses, expressed as an effective accu-mulated photocount, N 5 nFStd 5 nMd, is shown inFig. 5. This plot confirms the lower photocountneeded at 10 mm, and to a lesser extent at 2 mm, as aresult of the more highly structured return spectrum.Comparing the two extreme wavelengths, approxi-mately 4050 times as many photocounts are neededat 0.355 mm as at 10 mm. The results for 10 mm maybe conservative because the width of each of the threespectral peaks is a factor of over 10 less than the~Gaussian! spectrum at 0.355 mm, which should re-sult in a reduction of over 100 in the photocount @Eq.~1!#. In practice, greater advantage could be takenof spectral narrowing by filtering and processing the
Fig. 5. Effective photocount needed to obtain the results of Fig. 4,showing that ~i! the photocount needed to make a measurement isreduced by using lidar wavelengths in the collisional regime ~y .! where the spectra are more highly structured; and ~ii! for the
infrared systems, the minima that indicate that the proposed lidarparameters encompass the optimal operating point for a hetero-dyne system ~see Appendix A!. The symbols represent the samelidar parameters as in Fig. 4.where PS is the detected signal power calculated by alidar equation in which ATR is the area of a ~common!ransmitreceive antenna, b is the backscatter coef-cient, E the laser-pulse energy, c is the velocity of
ight, and r is the range. The lidar is assumed to beocused at each range to maximize the heterodyneeturn so that the results are not always those ex-ected of a lidar operating with a collimated outputeam. The overall lidar efficiency hL ~assumed here
to be 4%! is intended to take account of all systemlosses including photodetector quantum efficiency,antenna effi...