Effects of C2 on a vertically pointing diffraction-limitedlidar
Ronald L. Schwiesow
Examples of different C2 profiles lead to substantially different profiles of lidar image radius in a study of thecalculated performance of a diffraction-limited lidar system. The differences in image radii indicate theusefulness of a ground-based lidar for measurement of C2 profiles used to predict optical propagationphenomena. We conclude that the overall strength of the C2 profile and its general altitude dependence canbe determined from inspection of the image radius profile. Approximate calculations of available andrequired SNRs show that a lidar with a telescope aperture of 0.5 m and a few pulses of 1-J total transmittedenergy will provide useful image radius data to an altitude of 20 km under daytime conditions. The weightingfunction for sensitivity of the fractional increase in image radius to changes of C2 on a logarithmic altitudescale is approximately constant with height.
1. Objective and Summary
Inhomogeneities in the index of refraction of theatmosphere affect the propagation of light involved insuch activities as imaging and free-atmosphere opticalcommunication links. Measurements of optical tur-bulence, usually characterized by a structure parame-ter C2 can be used to predict the performance of opti-cal systems operating through the atmosphere. Inparticular, complete site surveys require measurementhardware that is easy to transport and has daytimemeasurement capability.
The purpose of the study discussed here is to calcu-late the effect of various assumed C2 profiles on avertically pointing diffraction-limited lidar to deter-mine if atmospheric optical inhomogeneities have alarge enough effect on the lidar to be easily measur-able. As a basis for the study and interpretation of theresults, we give details of an appropriate lidar configu-ration for obtaining some (not necessarily complete)information oln the C2 profile.
We conclude that a diffraction-limited lidar is usefulfor C2 measurements because the height profile of theimage radius from a given scattering volume is clearlydependent on the C2 profile as an input parameter. Agraph of the logarithm of altitude vs image radius isapproximately linear for typical C2 profiles. The slope
The author is with National Center for Atmospheric Research,P.O. Box 3000, Boulder, Colorado 80307-3000.
of the graph depends on the value of C2 aloft (above 1km), and the intercept depends mostly on Cn at lowerlevels. The sensitivity of a representative lidar at 500-nm wavelength is indicated by a change in slope by afactor of 4 for a change of C2 by a factor of 25 aloft andby a change in intercept by a factor of 102 for a changeof Cn by a factor of 25 at lower levels. Using a reason-able estimate for the required SNR, we can get anadequate SNR to 20 km in the daytime with -1 J oftransmitted pulse energy and a 0.5-m transceiver aper-ture.
In this paper we address the problem of obtaining animage radius profile, given a Cn profile and the depen-dence of image radius on C2. The inverse problem ofdetermining Cnprofiles in detail, given an image radiusprofile, is not treated here; however, we do demon-strate how the general shape of the image radius profileallows one to choose between representative Cn pro-files.
11. Technique
A. Background
One way to obtain information about C2 is to exam-ine the image of a remote source, such as a star, eitherby photography' or with a reticle photometer2 to deter-mine the modulation transfer function for the atmo-sphere. Although stellar photography can be used indaylight, it provides only a single value to characterizethe C2 profile. More sophisticated spatial filtering 3
gives more data about Cn but requires observatory-quality telescope pointing and tracking to operate inthe daytime. When both ends of a path are accessible,
a separate transmitter and receiver can be used as ascintillometer to measure C' along the path.4
Using the scattering volume of a lidar system as avirtual monochromatic source that can be located atany altitude by time-gating the receiver (using a pulsedtransmitter), we can combine aspects of both stellarimagery and scintillometer geometry. Belenkii andMironov5 and Belenkii et al. 6 have proposed such alidar backscatter system using scattering by aerosols.Their proposal is based on a cw laser and bistaticgeometry, but the possibility of a pulsed lidar is men-tioned.
In contrast to previous work, our results are based ona pulsed lidar with a common telescope for both trans-mit and receive functions. We also present the imageradius profile to 20 km for various types of C2 profile,and we calculate an approximate daytime SNR forrepresentative lidar system parameters that dependon the conservative assumption of molecular scatter-ing.
B. Atmospheric Effects
The size of the beam waist of a focused upward-propagating laser pulse depends on diffraction at thetransmitter aperture and on refraction by atmosphericrefractive-index inhomogeneities. Forward scatter-ing by aerosols also affects the beam size but on a scalegenerally larger than that resulting from refractiveeffects. A target at the beam focus, whether it be adiffuse scattering plate or a gaseous molecular medi-um, produces a source with spatial extent and radiantflux density distribution determined by both diffrac-tion and atmospheric refraction. The image size atthe receiver focal plane of the scattering source, whichis located at the focus altitude, depends on diffractionat the receiver aperture and on atmospheric refractiveinhomogeneities along the return path as well as on thespatial extent of the virtual source.
To estimate the image broadening that results fromforward scattering by aerosols, we can consider mea-surements of the solar aureole. The radiance of theaureole is _10-4 times that of the sun, and the half-angle of the aureole peak at the le radiance point is-60 mrad.7 Compared to a diffraction-limited imageof <2 ,urad for a 50-cm aperture at 500-nm wavelength,the light from aerosol forward scattering of the back-scattered signal can be considered to be a uniformlydistributed low-level background that can be includedwith the general skylight background. Studies of for-ward scattering on a horizontal path show that higherspatial frequencies in an image are determined almostentirely by turbulence (refractive-index inhomogenei-ties), whereas aerosols affect only the lowest spatialfrequencies.8 This observation supports the conclu-sion that aerosol forward scattering can be neglected inan analysis of image size and should be included in thebackground light level, which is independent of loca-tion in the image plane.
It is obvious then that the image radius at the lidarreceiver focal plane depends on inhomogeneities inatmospheric refractive index, characterized by C2
both on the upward and downward optical paths. Wequantify this dependence in the performance sectionof the paper.
The image radius for a coaxial-beam lidar systemdepends on what is commonly called beam spreadingbut not on beam wander. Large-scale inhomogenei-ties that cause beam wander (refractive wedges) affectcoaxial transmit and receive paths identically and sodo not cause image motion. In other words, the scat-tering source moves to compensate for beam wander sothat the image position is fixed because the time be-tween the transmit pulse and received signal is shortcompared with the time scale of changes in the atmo-spheric refractive-index distribution. On the otherhand, small-scale inhomogeneities that cause beamspread (refractive corrugations) affect both the size ofthe scattering source and additional image blurring onreturn. Thus only part of the entire spectrum of tur-bulent optical fluctuations affects a coaxial lidar sys-tem. 9
C. Lidar Elements
A lidar that is useful for studying the effects of Cnmust be diffraction-limited if atmospheric refractive-index inhomogeneities are to have a larger influence onimage radius than imperfections in the optics have.The best way of maintaining the alignment of diffrac-tion-limited transmit and receive optics and of match-ing the scattering spot size to the diffraction-limitedreceiver field of view is to use a common telescope fortransmitting and receiving. Dividing the outgoingand incoming beams can be done by polarization cod-ing (as is common for cw infrared Doppler lidars), bytime gating, or by a combination of both methods.
The laser transmitter must also be diffraction-limit-ed (single transverse mode) and have, for example, apulse length of <300 ns for a 50-m range resolution. Agas laser is most likely to have adequate beam qualityfor a master oscillator to drive an amplifier or injec-tion-locked oscillator, but we have not specified a par-ticular type of laser for the performance calculations.
The receiver images the lidar scattering volume atthe focal plane of the transceiver telescope. For bestsensitivity, the telescope must be focused at the heightof interest, which also corresponds to the time delaybetween transmitting the pulse and turning on thereceiver. At a sufficiently large range, the diameter ofthe beam waist is not reduced by focusing, and thetelescope may be set for a collimated beam. Imageradius data are then available from many altitudeswith a single pulse. At closer ranges, at least one laserpulse is required for each altitude. Because there isless need for height resolution at higher altitudes, wefound it useful to set the range gate to the larger valueof either 50 m or 10% of the altitude for which data arebeing obtained and to graph the image radius as afunction of the logarithm of the altitude.
At this stage of our research it is not possible tospecify the best method for measurement of the imageradius or its derivative with respect to the logarithm ofheight. Possible measurement methods are streak or
gated photography, an imaging photomultiplier tubewith digital postdetection processing, or a cylindricallens and linear detector array. A very simple ap-proach is to divide the received energy into central andperipheral portions by a reflective field stop with azi-muthly averaged reflectivity changing in the radialdirection in the image plane. With proper reflectivityweighting, the ratio of transmitted-to-reflected signalis a measure of the second moment of the energy distri-bution in the image. Harris10 discussed an opticalprocessor to compute the second radial moment of alaser intensity distribution in two dimensions using aparabolic transmission mask.
D. Computational Method
For typical C2 profiles, we adopted a model of theform
C2(z) = aC2(z)zb,
where C20(z) is Hufnagel's model1 without the ran-dom factor, z is altitude, a is a constant in the 0.2-5range, and b is a constant in the range 0.4. The othertype of C2 profile used was a constant value plus a spikeextending over four range gates centered at variousaltitudes. This type of profile was used for weightingfunction determination.
The basic way C2 affects image radius r is throughthe lateral coherence length po(z) for which Yura'2gives a useful formula:
Po(z) [1.45k C I(S)(s/z)5'/ds]
where k is the optical wave number. Length po(z)depends on an integral over C2(s), where s is an integra-tion variable representing path position between thescattering volume and the surface (lidar location).Note that s = 0 is the location of the scattering volume,and s = z is the surface. The radius at the le point onthe scattering-source Gaussian distribution is also giv-en by Yura12 in terms of the Gaussian radius of thetransmit beam g and p0 as
r2(z) = (z/kg)2 + (2z/kpo)2
for a focused beam. To approximate the overall imageradius, we use a root of the sum of the squares of (1) thepartial image radius corresponding to a point at thealtitude of interest and (2) the geometric image of thelie scattering source radius in the form
r2(z) (1.22 Xf/A) 2+ (rtf/z)2,
where f is the effective focal length of the transceivertelescope, and A is the smaller value of either thetelescope aperture diameter or p0 at z. Because thereceiver aperture is uniformly illuminated, we calcu-late the image radius from an ideal point source on thebasis of an Airy disk. This form of the calculationassumes statistically independent propagation on theupward and downward paths. The validity of theassumption is mentioned later under sources of error.
The important p0 for a lidar application is the short-term value pCt, rather than the long-term pCt, obvi-
Table 1. Lidar Parameters Used to Calculate Performance
Fig. 1. Sample C2 profiles with different constant multipliers butthe same altitude dependence. Profile 0 is Hufnagel's average
model.
ously because a monostatic lidar is insensitive to wavefront tilt.9 The short-term turbulence-induced beamspread pst is always less than or equal to the long-termbeam spread pIt because pot > pit. Theoretical calcula-tions show that in the near field the ratio pst/pIt is afunction of po/D, where D is the telescope aperturediameter. 1 3 Furthermore, for all plo/D ' 1.0, pst/pI is>0.63, which is not a major difference. In the far field,pst/pIt is -1.0. Direct experimental observations14
show a pst/pIt 0.8 for pit/D 0.5; another type ofexperiment15 implies that p/p' < 4 for p/D 1.Thus these results support the theoretical estimates.Because pst and pit are not substantially different, evenin the near field, we have followed Yural 2 and per-formed the calculations with pit. Use of the long-termvalue removes the complication of a po /D dependenceand results in overestimating image size by 37% atmost, where the error decreases with increasing range.
For illustrative purposes we have chosen lidar pa-rameters given in Table I. Values for atmosphericmolecular backscatter coefficient as a function ofheight and daytime sky brightness are taken from anearlier survey of the literature.16
Ill. Performance
A. Representative Image Radius Profiles
Three of the C profiles used to calculate imageradius profiles at the lidar transceiver focal plane areshown in Fig. 1. These profiles differ only by a con-stant multiplier. The resulting image radius profilesare shown in Fig. 2. Note that the image radius pro-files display significant differences from each other inslope and in the altitude at which the curve breaksfrom essentially constant radius (dominated by aper-ture diffraction) to an approximately linear increase in
Fig. 2. Image radius profiles for each of the C' profiles in Fig. 1.Note the differences in slope and the differences in the altitudewhere refractive effects begin to dominate diffraction (e.g., at -700
m for curve 1).
E
1-J
10
10
100
-10F
10-17 10-'6 10 5
C2 (m--)
Fig. 4. Sample C2 profiles with different altitude dependences butthe same constant multiplier. Profile 0 is Hufnagel's average mod-
el, as in Fig. 1.
E
I--J
1.0 2.0
CHANGE IN IMAGE RADIUSQpm)Fig. 3. Profile of the change in image radius between range gates
proportional to dr/d(logz) for each sample profile in Fig. 1.
radius with the logarithm of altitude (dominated byatmospheric refractive inhomogeneities). Anotherway of looking at the image radius profile is to considerthe change in image radius between range gates, asgraphed in Fig. 3. Because the range gate spacingincreases with altitude, the radius change profile isactually proportional to dr/d(logz). The radiuschange profiles also show the significantly differenteffects that various C' profiles have on a lidar. Thebreak in the radius curve corresponds to a suddenincrease in the radius change curve, and the slope ofthe radius curve correlates with the magnitude of theradius change curve.
For a given C2 profile, the altitude of the breakbetween a diffraction-dominated and a refraction-dominated image depends on the transceiver aperturediameter. Figures 2 and 3 demonstrate that a 0.5-maperture results in a break just above 100 m for theaverage C2 profile labeled 0. A smaller aperturewould allow a more compact mobile lidar but at thecost of a higher break altitude. Below the break, alldetail on the C2 profile is lost; however, the height ofthe break gives a measure of C2 below the break in someaverage sense. Use of a shorter wavelength lowers thebreak for a given C2 profile and transceiver aperture.
Profiles of C2 in Fig. 4 differ from those in Fig. 1,because in Fig. 4 the profiles have the same constantmultiplier but different altitude dependence. Figures5 and 6 show the resulting profiles of image radius andchange in image radius. As is the case for C2 profileswith different multipliers, profiles with different
0 20 40IMAGE RADIUS (um)
60
Fig. 5. Image radius profiles for each of the C2 profiles in Fig. 4.Note the differences in slope but the similarities in the altitudewhere refractive effects begin to dominate (i.e., altitude intercept).
104
w
0 1.0 2.0CHANGE IN IMAGE RADIUS (pm)
Fig. 6. Profile of the change in image radius between range gatesproportional to dr/d(logz) for each sample profile in Fig. 4. Note thesimilarity to Fig. 3 except for differences in the knees of the curves.
altitude dependences result in significantly differentimage radius profiles. Image radius profiles for thecase of variable altitude dependence show slopes simi-lar to those for the case of variable multiplier, but thealtitude at profile break is approximately the same forall image radius profiles that correspond to C2 profileswith different altitude dependence but the same con-stant multiplier.
Comparison of the results in Figs. 2, 3, 5, and 6 leadsus to two general conclusions.
(1) Image radius profiles for profiles differing inconstant multiplier by a factor of 5 or in height-depen-dent power law by a power of 0.4 are sufficiently un-alike to allow clear and unambiguous identification ofthe parent C2 profile.
(2) The slope of the image radius profile is primarilydependent on C2 values at higher levels (above thelarger value of either a few hundred meters or thealtitude at the curve break), and the altitude at thecurve break is primarily dependent on C2 values atlower levels (below the break).
B. Signal Levels
The lidar SNR required to measure the image radiusprofile with sufficient accuracy to infer useful informa-tion about C2(z) depends on details of the detector anddata processing as well as on how much informationabout C2 is desired and the effectiveness of a possiblemathematical inversion scheme. For example, if aradially weighted reflective aperture is used in theimage plane, the signal of interest is the ratio of trans-mitted-to-reflected energy integrated over some rangeinterval. However, a generalized criterion for re-quired SNR is that the system be able to sense thechange in image radius between adjacent range gates.If this is the criterion, the required SNR is of the orderof the reciprocal of the fractional change in imageradius between range gates. This is a reasonable ap-proach because a radius change of 2 m is more diffi-cult to detect on a 100-Am image than on a 10-,umimage. The approximate required SNR for each of thefive different C' and image radius profiles is graphed inFig. 7. Above the break, the required ratios are be-tween 10 and 100.
It is important to note that the image radius mea-surement using any of the techniques mentioned pre-viously is essentially a ratio measurement where someindication of the second moment of the energy distri-bution in the image is compared to the integratedenergy in the image. Thus the amount of energy in theimage is not important in an absolute sense except thatthe energy must be sufficient to measure the change inimage radius between range gates. The two principalsources of noise to consider in evaluating the availableSNR are shot noise (photon arrival rate statistics) onthe mean signal and speckle noise17 18 (Rayleigh phasormodel) from the random position of the molecularscatterers serving as a diffuse target.
To evaluate the mean signal-to-mean-noise ratio(often called carrier-to-noise ratiol8), it is appropriateto apply Poisson statistics to estimate the shot noise onthe photon arrival rate. Using the lidar parameters inTable I and appropriate atmospheric values, 6 we cancalculate both the signal from molecular scattering andthe noise as the square root of signal plus daytime skybackground. The available carrier-to-noise ratio for asingle 1-J pulse is shown as the dashed line in Fig. 7.Near 20-km altitude, the carrier-to-noise ratio falls offrapidly because of reduced molecular density, in-creased range-gate length leading to more sky back-ground photons per range gate, and the z 2 decrease incollected solid angle.
The presence of speckle noise limits the SNR to 1 fora single measurements even if the carrier-to-noiseratio is large. Because the image radius measurementdepends on an energy ratio rather than an energy
I-
I-1
10 100REQUIRED SIGNAL-TO-NOISE
1000
Fig. 7. Required signal-to-noise ratio for each of the profiles inFigs. 3 and 6. The dashed line is the available daytime signal-to-noise ratio for a lidar with 0.5-m transceiver diameter, -J pulseenergy, and range gate of 10% of the range, which uses only molecular
scattering for a target.
magnitude, the speckle noise is not a serious problem.A sufficient number of measurements must be aver-aged to insure that the total number of detected photo-electrons is approximately the mean value expectedfrom the carrier-to-noise ratio so that the ratio of ener-gy in central, and peripheral parts of the image can bedetermined with the necessary accuracy. When thedesired range cell is longer than a single 50-i rangegate, several independent realizations of the signal canbe obtained from a single pulse.
The assumption of statistically independent propa-gation on upward and downward paths, while usefulfor simplified calculations, needs study. Working in adifferent wavelength region, Menyuk and Killinger19
have shown a time to independence of -1 ms for cer-tain conditions and lidar system parameters. Time toindependence will depend on the number of phase cellsinvolved, wind, aerosol scattering, and other variables.Independent realizations are needed, as for specklenoise, to provide adequate carrier-to-noise ratios andan accurate measure of the average image radius.
The results summarized in Fig. 7 lead to an addition-al conclusion.
(3) A lidar system with the parameters chosen forthis study can measure image radius data with anadequate carrier-to-noise ratio to an altitude of -20km with -1 J of transmitted energy. The existence ofspeckle noise and correlated structure in the atmo-sphere requires measurement averaging to insure ade-quate detected energy.
C. Weighting Function
Here weighting function means the relative effect(as a function of altitude) of a perturbation of the C2
value on the image radius profile. Appropriate profiles to study the weighting function are shown inFig. 8. The magnitude of the constant-value C2 refer-ence profile (shown dashed) was arbitrarily chosen toplace the radius profile break just below 100-m alti-tude for the example lidar system, but the reference Cnvalue is close to the average atmospheric C2 profile 0 inFigs. 1 and 4. The increases in by a factor of 10extend over four range gates, which means that theactual thickness of the enhanced C2 layer is larger at
Fig. 8. Sample C2 profiles used to explore the weighting functionfor effects of refractive turbulence on a diffraction-limited lidar.The value of the constant profile (shown dashed) is chosen to place
the break in the image radius profile just below 100 m.
E
I-.
0 50 100 150IMAGE RADIUS(pm)
200
Fig. 9. Image radius profiles for each of the C2 profiles in Fig. 8.After an initial increase in radius over the constant C2 profile fromthe C2 spike, the image radius approaches the constant C2 profile
(shown dashed).
higher altitudes but that the fractional thickness of thelayer is constant.
The departures of the perturbed image radius pro-files from the reference profile are shown in Fig. 9. Asexpected, the influence of the perturbation decreaseswith increasing height above the perturbation becausethe perturbation in C2 becomes less significant com-pared with the integrated effect of the constant part ofthe C2 profile. The effect of the spike in c2 is seenmore clearly in the change in image radius profile ofFig. 10, but the most meaningful measure of the effectof the C2 perturbation is the fractional increase inimage radius graphed in Fig. 11. Based on the testprofiles and range gate values used in the study, weobserve an almost uniform weighting function for thesensitivity of the lidar to C2 perturbations over thealtitude range from 100 m to 20 km.
From the results in Fig. 11, we draw another conclu-sion.
(4) The weighting function for the effect of a pertur-bation in C2 on the lidar is approximately uniform withheight when the fractional change in image radius isconsidered on a logarithmic altitude scale.
IV. Comments
Because the results of this study show that differentCn profiles have substantially different effects on avertically pointing diffraction-limited lidar, it isworthwhile to consider the problem of inferring more
10
10
100 2 4 6 8 10 12
CHANGE IN IMAGE RADIUS(prm)
Fig. 10. Slope of the image radius profile, dr/d(logz), for each of thespike C2 profiles. The reference profile (constant C) is again shown
dashed.
E0
I-
0.02 0.04 0.06 0.08 0.10 0.12FRACTIONALCHANGE IN IMAGE RADIUS
Fig. 11. Fractional change in image radius profile, (l/r) dr(logz), foreach C2 spike. The reference profile is dashed.
detailed information on the C2 profile from a givenimage-radius profile or its derivatives. At this stage, itis possible to select one of five typical types of cnprofile as being most consistent with a given image-radius profile; however, a mathematical inversion pro-cess should result in more C2 information.
A comparison of available and required SNR showsthat a lidar to measure Cn is practical given 1-J laserpulses at 500 nm or lesser power at shorter wave-lengths. Although a 0.5-m transceiver primary mirroris large and heavy, the telescope requires no scanningmount and no pointing accuracy. Such a telescopeshould be easily mobile because the heaviest elementcan lie flat on a fixed horizontal bed.
Although lidar image size is related to C2, we note inpassing that other information is available from thebackscatter return. The backscatter intensity profileis related to aerosol loading, and the aerosol profilebreak can indicate the height of the mixed layer.Modifications to the lidar may give temperature16' 20
and wind2l'22'23profiles, although these advanced tech-niques require further experimental verification andtesting.
In addition to inversion studies on the data, futureresearch should cover at least two topics. First, detailsof the image radius measurement scheme must bedeveloped and analyzed for effectiveness. Second, alimited field trial of the lidar (backscatter) techniquecould be done, using a low-power laser and diffuse(hard) scattering targets on a horizontal range.
We thank S. F. Clifford, Ting-i Wang, R. Frehlich,and R. E. Good for helpful comments during the re-search. The National Center for Atmospheric Re-
search is sponsored by the National Science Founda-tion.
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HIGH-RESOLUTION MEASUREMENTS OF THIN-FILM INTERFACESThe "microstructure" of the interface between a thin filmand a bulk material is of extreme importance to thesemiconductor industry because it often controls theelectrical behavior of a semiconductor device. It isdifficult to measure, however: surface science techniquescan only probe a solid interface destructively, by diggingdown to it. With more penetrating probes, the informationfrom the interface is lost in the mass of responses fromthe bulk material. Now researchers from NBS, theUniversity of Tennessee, Oak Ridge National Laboratory, andLawrence Berkeley Laboratory have tested a technique toobtain greatly enhanced responses--up to 50 times, so far--from thin-film interfaces. The trick is tomake multilayer"sandwiches" of alternating substrate and film, and usesoft x-ray fluorescence, exciting the core electrons of theatoms to measure position and chemical species.Feasibility studies on an ultra-high sensitivityspectrometer in the NBS facility at the NationalSynchrotron Light Source (Brookhaven) used silicon andcarbon and could discern the interface characteristics offilms as thin as 3 angstroms.
