Comment on “Heterodyne lidar returns in theturbulent atmosphere: performance evaluationof simulated systems”

Rod Frehlich and Michael J. Kavaya

The explanation proposed by Belmonte and Rye �Appl. Opt. 39, 2401 �2000�� for the difference betweensimulation and the zero-order theory for heterodyne lidar returns in a turbulent atmosphere is incorrect.The theoretical expansion the authors considered is not developed under a square-law structure-functionapproximation �random-wedge atmosphere�. Agreement between the simulations and the zero-orderterm of the theoretical expansion is produced for the limit of statistically independent paths �bistaticoperation with large transmitter–receiver separation� when the simulations correctly include the large-scale gradients of the turbulent atmosphere. © 2002 Optical Society of America

OCIS codes: 010.1330, 010.3640, 040.2840, 120.0280, 280.3640.

1. Introduction

The effects of refractive turbulence on heterodyne orcoherent Doppler lidar have been investigated by the-oretical methods1–5 and numerical simulations.6–10

The theoretical results are important for verificationof the simulation algorithms and for calculations inparameter regimes where simulations are not feasi-ble such as conditions of large path-integrated refrac-tive turbulence �strong scattering�. The twomethods are complementary, and the valid parame-ter space for each method is a critical issue for eval-uation of Doppler lidar performance.

The analysis of coherent Doppler lidar performancesimplifies for diffuse or aerosol targets with a con-stant backscatter coefficient at a range R. An im-portant statistical quantity is the signal-to-noiseratio �SNR� that can be written as4,11

SNR�R� � C�R� �AR�S�R�

R2 (1)

R. Frehlich �rgf@cires.colorado.edu� is with the Cooperative In-stitute for Research in Environmental Sciences, Campus Box 216,University of Colorado, Boulder, Colorado 80309. M. J. Kavaya�m.j.kavaya@larc.nasa.gov� is with the National Aeronautics andSpace Administration Langley Space Flight Center, Mail Stop 472,Hampton, Virginia 23681.

Received 5 December 2000; revised manuscript received 20 Au-gust 2001.

0003-6935�02�091595-06$15.00�0© 2002 Optical Society of America

for identical transmit and receiver apertures whereC�R� is the coherent responsivity, AR is the aperturearea, and �S�R� is the system efficiency that is givenby

�S�R� �R2�2

AR �

� jT�p, R� jBPLO�p, R��dp, (2)

where � is the laser wavelength; jT�p, R� and jBPLO�p,R� are the random normalized intensities of thetransmit and backpropagated local oscillator �BPLO�beam, respectively; � � denotes ensemble average overthe random refractive turbulence and the randomphases of the backscattered fields at the target; pdenotes the two-dimensional transverse coordinateat the target; and dp denotes two-dimensional inte-gration. Here, jT�p, R� �eT�p, R��2 and jBPLO�p, R� �eBPLO�p, R��2, where

eT�p, R� � �

eL�u, 0�WT�u�G�p; u, R�du, (3)

eBPLO�p, R� � �

eLO*�u, 0�WR�u�G�p; u, R�du,

(4)

and eL�u, 0� is the normalized laser transmit field atthe transmit telescope aperture WT�u�, eLO�u, 0� isthe normalized local oscillator field at the receivertelescope aperture W �u�, and G�p; u, R� is the

R

20 March 2002 � Vol. 41, No. 9 � APPLIED OPTICS 1595

Green’s function for propagating the fields through arandom atmosphere under the Fresnel approxima-tion that can be written as a Feynman pathintegral.4,12–14 Combining these equations produc-es4

�S�R� �R2�2

AR �

�

�

�

�

eT�u1, 0�eT*�u2, 0�

� eBPLO�u3, 0�eBPLO*�u4, 0�

� �4�p, u1, u2, u3, u4, R�du1du2du3du4dp,

(5)

where

�4�p, u1, u2, u3, u4, R� � �G�p; u1, R�G*�p; u2, R�

� G�p; u3, R�G*�p; u4, R��

(6)

is the fourth-moment Green’s function for wave prop-agation through the random atmosphere. For typi-cal atmospheric conditions, there is no exact solutionfor �4, and various approximations and series expan-sions have been produced. One of the most basicseries expansions is a Taylor-series expansion.

Another attractive option is to numerically simu-late many realizations of the random intensities jT�p,R� and jBPLO�p, R� and determine �S�R� from Eq. �2�.This method has been successfully employed formany problems of wave propagation in randommedia.15–19 For coherent Doppler lidar applications,the results generally disagree with the first-orderterm of the theoretical expansions6–8 because higher-order terms of the expansion are important �thesehigher-order terms are numerically intensive and dif-ficult to calculate5�. An explanation proposed byBelmonte and Rye7 for the disagreement of the sim-ulations and approximations to the zero-order theoryis that the series expansion is valid only for a randomatmosphere described by a square-law structurefunction that is equivalent to an atmosphere com-posed of random wedges20 �see Belmonte and Rye,7pg. 2408: “Also shown are the simulation output�dashed curves�, for which the same simplifying as-sumptions that are used to derive the analytical re-sults were employed. We took the random-wedgeatmosphere into consideration by using an exponentof 4 instead of the classical 11�3 in the definitionof the von Karman spectrum that one needs to con-struct the phase screens.”� With this interpretationand modified simulation algorithm based on an ap-proximation for a random-wedge atmosphere, Bel-monte and Rye7 claimed agreement between thesimulations and the approximations to the zero-ordertheoretical expansion for the case of statistically in-dependent paths �e.g., bistatic operation with a largeseparation between the transmitter and the receiversuch that the transmit and receive paths experiencestatistically independent turbulence�. This is an in-

correct interpretation because such an atmospherewould produce only beam tilts and no beam distor-tion.20 Therefore the transmit intensity jT�p, R� andthe BPLO intensity jBPLO�p, R� would both containthe same random shift, and the system efficiency�S�R� �Eq. �2�� would be equal to the free-space cal-culation �see Appendix A�. This interpretation wasfirst stated by Wandzura20: “The meaning of thequadratic structure function �QSF� is that the phase-front aberration it describes is flat and randomly ori-ented with a prescribed slope variance. It cannotmodel situations where higher-order aberrations arecrucial. These include �ii� processes involving reflec-tion of transmitted light back to a collocated receiver�e.g., lidar systems�, allowing the effects of tilt tocancel out in the received signal.” We show that, fortypical atmospheric conditions, agreement is pro-duced for the bistatic limit of the zero-order term ofthe series expansion, and we clarify the procedureused to generate the series expansion.

2. Theory

The key quantity �4 can be written under the Markovapproximation and narrow angular scattering as14

�4�p, u1, u2, u3, u4, R�

� G4f�p, ui, R� � Dr1 � Dr2 � Dr3 � Dr4

� exp�12 �

0

R

�d�r1 � r2, z� � d�r3 � r4, z�

� d�r1 � r4, z� � d�r2 � r3, z� � d�r1 � r3, z�

� d�r2 � r4, z��dz� , (7)

where G4f �p, ui, R� is the fourth-moment free-space

Green’s function, and

d�s, R� � 4�k2 �

�1 � cos�s � q��

� �n�qx, qy, qz � 0, R�dq, (8)

where k 2��� is the laser wave number, q �qx,qy�, and �n�qx, qy, qz, R� is the three-dimensionalspectrum of refractive-index fluctuations at range Rand Dri denotes the infinite number of paths ri�z�from the transmitter–receiver plane z 0 with ri�0� ui to the target plane z R with ri�R� p. TheMarkov approximation is equivalent to the randomatmosphere being replaced by a series of statisticallyindependent random refractive-index screens trans-verse to the propagation direction and is an excellentapproximation for intensity statistics such as �4.21

For weak scattering conditions that emphasize thelow spatial frequencies �lf � of the scintillation pro-cess, the standard series expansion is a Taylor seriesof the quantity exp�Q�2� �Refs. 4 and 14� where Q

1596 APPLIED OPTICS � Vol. 41, No. 9 � 20 March 2002

�0R�d�r1 r4, z� � d�r2 r3, z� d�r1 r3, z� d�r2

r4, z��dz. The zero-order term is given by

�S0lf�R� �

1AR �

OT�s, R�BPLO*�s, R�

� exp�DS�s, R��ds, (9)

where

DS�s, R� � �0

R

d�s�1 � z�R�, z�dz (10)

is the spherical-wave structure function,

OT�s, R� � �

eT�r � s�2, 0�eT*�r � s�2, 0�

� exp�ikr � s�R�dr, (11)

OBPLO�s, R� � �

eBPLO�r � s�2, 0�eBPLO*�r � s�2, 0�

� exp�ikr � s�R�dr. (12)

This expression is the receiver-plane version. Thetarget-plane version simplifies to4

�S0lf�R� �

�2R2

AR �

� jT�p, R��� jBPLO�p, R��dp, (13)

which contains the product of the average normalizedintensity of the transmit and BPLO fields at the tar-get �the statistically independent path result that isone of the few numerically tractable theoretical re-sults�. This series expansion was presented in Ref. 4with the following caution: “Note that our theorywas not developed under a square law structure func-tion approximation. That pathological case corre-sponds to an atmosphere composed of randomwedges,20 which implies there is only beam wanderand no scintillation, and wave-front tilts will be self-correcting for monostatic lidar.” The mathematicalderivation of this result is produced in Appendix A.

A realistic model for the atmospheric spectrum isthe Hill spectrum,22,23

�n�q, z� � ACn2� z�q11�3f �ql0� z��, (14)

where A 0.0330054, q �qx2 � qy

2 � qz2�1�2 is the

magnitude of the three-dimensional wave vector, Cn2

is the refractive-index structure constant, l0 is theinner scale, and

f � x� � �1.0 � 0.70937x � 2.8235x2 � 0.28086x3

� 0.08277x4�exp�1.109x�. (15)

The form of the spectrum at the high-wave-numberregion is critical for laser scintillation experiments.The Gaussian model for f �x� with the same definitionof inner scale is given by24

f � x� � exp�x2�34.91815� (16)

and produces errors for the scintillation intensityvariance25 that is related to C�R� and �S�R�.4,23 Forconstant Cn

2 and l0 �Ref. 26�,

DS�s� � 8�2ACn2k2l0

5�3RH�s�l0�, (17)

where

H� z� � �0

1

g� xz�dx, (18)

g� x� � �0

q8�3f �q��1 � J0�qx��dq. (19)

The field coherence length �0 is defined by

DS��0� � 1, (20)

and a useful approximation that we call the square-law structure-function approximation for the seriesexpansion �not to be confused with the square-lawstructure-function approximation for the random at-mosphere or, equivalently, the random-wedge atmo-sphere� is

DS�s� � s2��02, (21)

which has been shown to have small error for calcu-lations of the average beam intensity.1 If �0 �� l0,this is a good approximation. The main motivationfor this approximation is analytic expressions for im-portant quantities such as C�R� and �S�R� with aGaussian lidar model.1,2,4,5

For strong scattering, the scintillation process con-sists of two components: a low-spatial-frequency�large-scale� component �lf � and a high-spatial-frequency component �hf �. Therefore another seriesexpansion is required based on the high-spatial-frequency component of the scintillation.4,14 Thisexpansion is a Taylor series in the quantityexp�P�2� where P �0

R �d�r1 r2, z� � d�r3 r4, z� d�r1 r3, z� d�r2 r4, z��dz. For monostaticlidar with matched transmit and BPLO beams, thefirst term �S0

hf �S0lf �Ref. 4� and the total strong-

scattering expression is

�SS�R� � �S0lf�R� � �S0

hf�R� � 2�S0lf�R�, (22)

which is equivalent to the Gaussian fields assump-tion in strong scattering14 and provides the lowerbound to Doppler lidar performance.

3. Numerical Simulation

We determine the effects of refractive turbulence fortypical ground-based atmospheric conditions usingnumerical simulations of the target-plane intensitypatterns jT�p, R� and jBPLO�p, R� �see Eq. �2��. Thestandard algorithm7,15–17 propagates the fieldthrough a series of statistically independent random-phase screens �the Markov approximation� that areproduced by the fast Fourier-transform �FFT� algo-rithm. The phase screens replace the continuousrandom media over a propagation distance �z. The

20 March 2002 � Vol. 41, No. 9 � APPLIED OPTICS 1597

structure function of the phase screens at range R isgiven by �z d�s, R� �see Eq. �8��. Because the phasescreens produced by the FFT algorithm are periodic,the spatial correlation is periodic and the simulationof the fields have errors for those statistics that aresensitive to large-scale turbulent fluctuations �seeFig. 1 of Ref. 6�. Improved algorithms to generatephase screens with the correct large-scale statisticshave been produced6 with random subharmonics�SH�.18 This algorithm �FFT–SH� has been appliedto the simulation of wave propagation in random me-dia and heterodyne lidar performance.6,8 The large-scale phase perturbations have been shown toproduce a small effect for optimally designed mono-static coherent Doppler lidar.6 The statistically in-dependent path �bistatic operation� limit can beproduced through generation of uncorrelated phasescreens for the transmit and BPLO fields.

4. Results

The results of simulations and theory for the Hillspectrum �Eq. �15�� and the statistically independentpath case �bistatic� are shown in Fig. 1 for the samelidar parameters of Fig. 16 in Ref. 7 �circular tele-scope with diameter D 0.14 m, matched collimatedtransmit and BPLO Gaussian beams with optimal27

1�e intensity radius �L �LO 0.2836D 0.39704m and perfectly aligned at the target, Cn

2 1012

m2�3, l0 0.01 m, � 2.0 �m�. �Note: Belmonteand Rye7 normalized the system efficiency by theno-turbulence case that shifts only the location of thecurves and not the relative spacing because of the logaxis.� The results from the most robust simulation

algorithm �FFT–SH� agree well with the exact calcu-lation of the zero-order term �Eq. �9��. The resultsfrom the traditional simulation algorithm �FFT� donot agree for large ranges because the large-scale tiltsare not correctly represented. This is shown in Fig.7 of Ref. 10 where the average beam width from thetraditional simulation is smaller than the theoreticalcalculation for ranges greater than 1000 m. The er-ror is larger for the focused beam case.6 The square-law structure-function approximation for theexponential expression of the zero-order term alsohas little error �see Eqs. �9�, �10�, �21�� �note that thedotted curve is close to the solid curve�. This is to beexpected because the coherence length �0 is less thanthe inner scale l0 for all ranges greater than R 200m. The most likely explanations for the disagree-ment in Figs. 16–21 of Ref. 7 are the assumption of aGaussian transmittance profile for the telescope ap-erture and the approximation of the coherence length�0 �Eq. �20�� by the zero inner-scale limit of the Kol-mogorov spectrum �see Eq. �165� of Ref. 4�.

Belmonte and Rye7 produced agreement with a cal-culation of the bistatic limit by introducing a turbu-lence spectrum with a power-law slope of 4 toapproximate a random-wedge atmosphere. Thisspectrum has a singularity for the phase structurefunction that can be demonstrated with the turbu-lence spectrum

�n�q, z� �B

�L02 � q2�2 , (23)

where B is a constant and L0 is the outer scale ofturbulence. For q �� L0

1, �n�q, z� Bq4. Thestructure function, Eq. �8�, then becomes

d�s� � L02�1 �

sL0

K1�s�L0�� , (24)

where K1�z� is the modified Bessel function of order 1.For s �� L0,

d�s� � �0.307966 � 0.5 ln s � 0.5 ln L0��s2, (25)

which is approximately a square-law structure func-tion but depends on the outer scale L0. For fixed sand large L0, d�s� � 0.5s2 ln L0, which has a square-law dependence. However, d�s� becomes infinite inthe limit of infinite L0. The Kolmogorov spectrumwith a slope of 11�3 does not have a contributionfrom an outer scale for s �� L0, and therefore thestructure function is well defined for an infinite L0.In addition, the intensity statistics of the lidar beamsare independent of the outer scale when L0 is suffi-ciently large and the limit of infinite L0 is valid forboth theory and simulations. A numerical simula-tion of phase screens based on the FFT algorithmimposes an outer scale of approximately the dimen-sions of the simulation. The simulation results ofBelmonte and Rye7 who used the spectrum with apower-law slope of 4 may have produced agreementwith the bistatic theory by a fortuitous choice of the

Fig. 1. System efficiency �S�R� versus range R for a coherentDoppler lidar with the Hill spectrum �Eq. �15�� and with statisti-cally independent paths for the transmit and BPLO beams �bi-static limit�. The exact theoretical prediction �solid curve� �Eq.�9�� and the square-law structure-function approximation �dashedcurve� �see Eqs. �9�, �10�, �21�� are compared with the results fromthe robust simulation algorithm �FFT–SH� and the traditionalsimulation algorithm �FFT�. The case of no turbulence is shownas a dashed curve.

1598 APPLIED OPTICS � Vol. 41, No. 9 � 20 March 2002

simulation dimensions and therefore the effectiveouter scale L0.

The results for monostatic operation are shown inFig. 2. The results from both simulation algorithmsagree �see also Refs. 6 and 8� and are considerablylarger than the zero-order strong-scattering theory.This is typical of intensity scintillation,15,16,19 espe-cially with an inner scale of turbulence. It is diffi-cult to approach the theoretical strong-scatteringlimit, and therefore higher terms of the theoreticalexpansions are required. However, for the focusedbeam geometry and strong scattering, the simulationresults converge more quickly to the theoretical pre-dictions �see Fig. 10 of Ref. 8�. This is a parameterregime where improved theoretical expansion couldbe valuable. With the development of stable coher-ent Doppler lidar, the measurement of SNR for hardtargets has become accurate because the relative ac-curacy from the speckle process is 1��N where N isthe number of lidar shots processed. It is importantto have accurate predictions from theory and simu-lations to understand the effects of refractive turbu-lence. The results for the exponential spectrummodel �Eq. �16�� do not agree with the Hill model.This has been observed for other scintillation statis-tics.25

5. Summary and Discussion

The theoretical series expansion for SNR ��S�R�,C�R�� is produced as a Taylor-series expansion anddoes not assume a square-law structure-function ap-proximation for the random atmosphere �random-wedge atmosphere�. The exact calculation of the

zero-order term �Eq. �9�� of the theoretical expansionof heterodyne system efficiency �S�R� for the case ofstatistically independent paths for the transmit andBPLO beams �perfectly aligned bistatic lidar withlarge separation between the transmitter and thereceiver� agrees with the robust simulation method�FFT–SH� �see Fig. 1� and also agrees well with thesquare-law structure-function approximation to theexponential term of the expansion �see Eqs. �9�, �10�,�21��. The traditional FFT simulation algorithm haserrors because the large-scale phase tilts are not cal-culated correctly, and therefore this algorithm doesnot accurately predict performance for bistatic oper-ation and the improved FFT–SH algorithm is re-quired. The FFT bistatic simulation results ofBelmonte and Rye7 should have been within 40% ofthe exact calculation �Eq. �9��. The most likelysource of the larger disagreement is use of the ap-proximate Gaussian lidar expressions for the theo-retical calculations. The agreement they producedwith the simulation results using an approximationto a random-wedge atmosphere is most likely a for-tuitous choice of the simulation dimensions that im-poses an effective outer scale L0 that determines themagnitude of the structure function �see Eqs. �24�and �25��.

For monostatic lidar �see Fig. 2�, the results of thesimulations are larger than the predictions of thezero-order theory for weak scattering �Eq. �9�� andalso strong scattering �large range R� �Eq. �22��.This parameter regime requires more terms of thetheoretical expansion or a better expansion. Nu-merical simulation of lidar performance is an attrac-tive method for those parameter regimes where thesimulations are valid. For other parameter regimes,better theoretical expansions may be required for ac-curate predictions. The exponential model �Eq. �16��for the inner-scale region of the turbulence spectrumdoes not agree with the Hill model.

Appendix A

The mathematical derivation of the results for thesquare-law structure function requires previouspath-integral expressions. A different representa-tion of Eq. �7� is given by Eq. �70� in Ref. 14. Sub-stituting the square-law structure function d�s, z� T�z�s2 into Eq. �70� produces

�4�p, u1, u2, u3, u4, R� �k2

�2�R�2

� exp�ikR

��� � �� � �� � �� � 2p � ����� exp�2��2 �

0

R

�1 � z�R�2T� z�dz�� � D�1 � D�1 exp�ik �

0

R

�1 � �1dz� , (A1)

Fig. 2. System efficiency �S�R� versus range R for a monostaticcoherent Doppler lidar with the Hill spectrum �Eq. �15��. Thezero-order term of the theoretical expansion for weak scattering�solid curve� �Eq. �9�� and strong scattering �dashed curve� �Eq.�22�� are compared with the results from the robust simulationalgorithm �FFT–SH� and the traditional simulation algorithm�FFT�. The results for the exponential spectrum �Eq. �16�� andthe traditional simulation algorithm �FFT–exp� are also shown forcomparison.

20 March 2002 � Vol. 41, No. 9 � APPLIED OPTICS 1599

where �1�z� and �1�z� are transformed path-integralvariables, �� �u1 u2 � u3 u4��2, �� �u1 � u2 �u3 � u4��2, �� �u1 � u2 u3 u4��2, �� �u1 u2 u3 � u4��2 �see Eqs. �65� and �24� of Ref. 14�.The first exponential term of Eq. �A1� is the free-space Green’s function, and the second exponentialterm is the contribution from the random-wedge at-mosphere. The system efficiency, Eq. �5�, containsan integration over the target-plane coordinate pthat produces the two-dimensional delta function�����. The integration over �� removes the exponen-tial term containing T�z�, and the remaining termsare the free-space calculation, i.e., there is no effectfrom the square-law structure function. Note thatthe integration over p assumes that the target hasuniform backscatter over the laser-illuminated re-gion of the target plane.

This research was supported by the National Aero-nautics and Space Administration, Michael J.Kavaya, Technical Officer. The authors acknowl-edge the suggestions of the anonymous reviewers.

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1600 APPLIED OPTICS � Vol. 41, No. 9 � 20 March 2002