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 rangea