bistatic coherent laser radar signal-to-noise ratio

Bistatic coherent laser radar signal-to-noise ratio

Eric P. Magee and Timothy J. Kane

We investigate the signal-to-noise ratio �SNR� for a bistatic coherent laser radar �CLR� system. With abistatic configuration, the spatial resolution is determined by the overlap of the transmit beam and thevirtual backpropagated local oscillator beam. This eliminates the trade-off between range resolutionand the bandwidth of the transmitted pulse inherent in monostatic systems. The presented analysis iscompletely general in that the expressions can be applied to both monostatic and bistatic CLR systems.The heterodyne SNR is computed under the assumption of untruncated Gaussian optics and untruncatedGaussian beam profiles. The analysis also includes the effects of refractive turbulence. The resultsshow that, for maximum SNR, small transmit and local oscillator beam profiles �e�1 intensity radius� aredesired. © 2002 Optical Society of America

OCIS codes: 010.3640, 010.1300, 280.3640, 290.4020.

1. Introduction

Wind velocity is estimated with a lidar system by mea-surement of the Doppler frequency shift, relative to thecarrier frequency, undergone by laser radiation scat-tered by particles suspended in the flow. This fre-quency shift can be measured with either a coherent�heterodyne� or an incoherent �direct� detectionscheme. For a coaxial system, the return signal isDoppler shifted by an amount proportional to the ve-locity of the aerosols along the line of sight of the trans-mitter. For sensing three-dimensional vector winds,measurements from at least three different directionsare required. Typically for the monostatic case, atechnique called velocity azimuth display lidar is usedto obtain the vector wind.1,2 This technique, as wellas most Doppler techniques, must sacrifice spatial andtemporal resolution to measure the horizontal velocity.In effect, the horizontal velocity is averaged over across-sectional area of the order of 0.1 km2 for a 1-kmaltitude and a 20-deg cone angle. To obtain rangeresolution, most monostatic lidar systems use a pulsedlaser. This imposes a fundamental limit for therange–velocity resolution product.2,3

When this research was performed, E. P. Magee �[email protected]� and T. J. Kane were with the Department of ElectricalEngineering, Pennsylvania State University, University Park,Pennsylvania 16802. E. P. Magee is now with the Department ofElectrical and Computer Engineering, Air Force Institute of Tech-nology, Wright-Patterson Air Force Base, Ohio 45433.

Received 25 April 2001; revised manuscript received 27 Septem-ber 2001.

0003-6935�02�091768-12$15.00�0© 2002 Optical Society of America

1768 APPLIED OPTICS � Vol. 41, No. 9 � 20 March 2002

The spatial resolution, range resolution, and veloc-ity estimation error limitations can be addressedwhen a bistatic configuration is implemented with arelatively long pulse or continuous-wave �cw� lasertransmitter.4 In this configuration, the spectralwidth of the laser is narrow compared with a short-pulse system, which results in improved velocity ac-curacy. However, the range resolution is notaffected because the scattering volume is defined bythe overlap of the transmit and virtual backpropa-gated local oscillator �BPLO� beams.

In a bistatic configuration the measured Dopplershift is proportional to the component of velocityalong the bisector of the bistatic angle. The three-dimensional velocity vector can be determined by useof a multistatic �multiple receivers� configurationwith each receiver measuring a different componentof the velocity within the same scattering volume.5,6

Knowledge of the three-dimensional winds within asmall volume can be useful in a number of researchareas, such as aviation, weather prediction, meteoro-logical modeling, and severe weather detection. Themain thrust of this research is to provide a theoreticalbasis for the design and implementation of a multi-static Doppler lidar system for measurement of three-dimensional vector winds within the atmosphericboundary layer.

The primary metric used in the analysis of a coher-ent laser radar �CLR� is the power signal-to-noiseratio �SNR�, defined as

SNR�t� ��is

2�t��i 2�

, (1)
N

where � � � denotes the ensemble average, �is2�t��

�amp2� is the intermediate-frequency signal power attime t �s�, and �iN

2� is the average noise power. Themajority of CLR systems operate in a monostatic con-figuration; thus the analysis available in the litera-ture focuses primarily on monostatic operation.1,7–17

The general results derived by Frehlich and Kavayaare valid for bistatic systems when the aperture sep-aration is much less than the target range.14 Thepurpose of this research is to extend the analysis ofFrehlich and Kavaya to a bistatic system withoutconstraints on the baseline separation.

The general expressions used in the analysis of abistatic Doppler lidar system are derived in Section 2.This includes the specific geometry and coordinatesystems used throughout the paper, which are im-portant because the added complexity in a bistaticsystem is due primarily to the geometry. The gen-eral expressions for the SNR �including the effects ofrefractive turbulence� are given for a distributedaerosol target.

2. Theory

A. Geometry

The geometry for a bistatic CLR system is shown inFig. 1. The primary coordinate system used to lo-cate the receiver, transmitter, and target is a fixedreference system �x, y, z� centered at the transmitter.The z axis of this coordinate system is in the verticaldirection. The fixed coordinate system is also usedto define pointing directions of the transmitter andreceiver optics. The north referenced coordinatesystem is centered at the transmitter with the xn–znplane in the scattering, or bistatic, plane.18,19

Transmitter, receiver, and target referenced coordi-nate systems are also used.

The receiver and transmitter are separated by thebaseline, B � �bx

2 � by2 � bz

2�1�2 �m�, where bx, by,and bz are the coordinates of the receiver in the fixedcoordinate system. The center of the target is lo-cated at a point �px, py, pz� in the fixed coordinatesystem. The angles �nt and �nr are, respectively, thetransmitter and receiver look angles with respect tothe north referenced coordinate system, which arepositive when measured clockwise from zn. The an-gles r, t, �r, and �t are the standard spherical co-ordinate azimuth and elevation angles �t for thetransmitter and r for the receiver�. RT �m� is theEuclidean distance to the target measured from thetransmitter, and RR �m� is the Euclidean distancefrom the receiver to the target. Using Fig. 1, we findthe relationship between the receiver and the targetcoordinate systems to be20

�xr

yr

zr

� � � cos �S 0 sin �S

0 1 0�sin �S 0 cos �S

��xp

yp

zp

� � � 00

RR

� , (2)

where �S � �nt � �nr is the bistatic angle. Likewise,the relationship between the target and the trans-mitter coordinate systems is given by

�xt

yt

zt

� � � xp

yp

zp � RT

� . (3)

B. Signal-to-Noise Ratio

In this subsection, the generalized coherent lidarSNR expressions are given. The derivations �basedon the coordinate systems shown in Fig. 1� closelyfollow those of Frehlich and Kavaya14 with the gen-eralization for a bistatic geometry with baseline sep-arations much greater than the aperture sizes. Aschematic for a bistatic coherent lidar system isshown in Fig. 2. The transmitter lens is located inthe plane defined by zt � 0 and by position vector u�m�, and the receiver lens is located in the planedefined as zr � 0 and by position vector v �m�. Theposition vector p �m� defines the transverse plane�zt � RT� at the target location. The detector is lo-cated at transverse coordinate w �m� and distance L�m� from the receiver plane �a positive L implies anegative zr�. The receiver and transmitter lensesare described by dimensionless response functionsWR�v� and WT�u�, respectively.

The general expression for the detector plane SNRis given by

SNR�t� �1

hBw�PLOD� �D�

D

�Q�w1��Q�w2�

� MS�w1, w2, L, t�MLO*�w1, w2, L�d2w1d2w2, (4)

Fig. 1. Geometry for a bistatic coherent detection laser radarsystem.

20 March 2002 � Vol. 41, No. 9 � APPLIED OPTICS 1769

where �Hz� is the optical frequency, h � 6.626 10�19 �J�s� is Planck’s constant, Bw �Hz� is the elec-trical bandwidth, �D denotes integration over the de-tector surface, �Q�w� �electrons�photon� is thedetector quantum efficiency as a function of locationon the detector,

MS�v1, v2, L, t� � �ES�v1, L, t�ES*�v2, L, t�� (5)

is the mutual coherence function �W m�2� of the scat-tered field ES�v, z � L, t� �W1�2 m�1�, MLO�v1, v2, L�is the mutual coherence function of the local oscillator�LO� field, and PLOD�w� is the average effective LOpower measured by the detector.

It is often convenient to perform the above calcu-lations in a plane other than the detector plane.These calculations can be constructed with a tech-nique first described by Siegman.21 This techniqueinvolves the backpropagation of the LO field to theplane where the calculations are to be performed.The backpropagated local oscillator �BPLO� calcula-tions were formalized by Rye.7,22 The target planerepresentation of the SNR �when we assume an idealshot-noise-limited uniform photovoltaic detector� foran infinite uniform aerosol target is given by

SNR�t� ��Q

hBw ��RT

�

K2� zp��PL�t � zt�c��

� �� zp, �S�C� zp, t�dzp, (6)

where

C� zp, t� � �2 ��

�

�jT�p, zt, t � zt�c� jBPLO�p, zr��d2p

(7)

is the target plane representation of the dimension-less coherent responsivity. Note that the coherentresponsivity is proportional to the overlap area of thetransmitted and BPLO random irradiance profilesjT�P, z, t� �m�2� and jBPLO�p, t� at the target. K�zp�denotes the one-way atmospheric extinction andK2�zp� represents the two-way atmospheric extinc-tion �which is considered to be constant over the scat-tering volume�; zp is the range along the z axis of thetarget coordinate system �zp � 0 implies zt � RT�; ztand zr are functions of zp, as defined in Eqs. �2� and�3�; and ��zp, �S� is the total volume scattering coef-ficient �m�1 sr�1� at wavelength � as a function ofscattering angle �S. The laser fields, both transmit-ter and LO, are normalized by

EL�u, z, t� � ��PL�t�� eL�u, z, t�,

ELO�v, z� � ��PLO� eLO�v, z�, (8)

where PLO �W� is the LO total power and eL�u, z, t��m�1� and eLO�v, z� �m�1� are the normalized fields forthe transmitter and the LO, respectively. This nor-malization is used because the coherent lidar perfor-mance can now be referenced to the average laserpower �PL�t�� instead of the average transmittedpower �PT�t��. It should be noted that the time de-pendence indicated in eL�u, z, t� is the spatial varia-tion of the laser field as a function of time. Such avariation could be manifested by the interference ofspatial modes. It is generally assumed that the spa-tial characteristics of the laser field do not vary withtime, or change slowly with respect to the observationtime. Then the random irradiance profiles of thenormalized transmitter and BPLO fields at the targetare given by

jT�p, zt, t� � �eT�p, zt, t��2,

jBPLO�p, zr� � �eBPLO�p, zr��2. (9)

The BPLO field is given by

EBPLO�v, 0� � �Q ELO*�v, 0�WR�v�, (10)

where we assumed an infinite uniform detector��Q�w� � �Q�.

C. Refractive Turbulence Effects

The primary effect of refractive turbulence is a deg-radation of the spatial correlation of the transmittedand scattered waves. This spatial decorrelation ofthe waves results in an overall reduction in the co-herent signal power, which has been experimentallydemonstrated by Chan et al.23

The refractive turbulence effects are accounted forby the fourth moment of the Green’s function forpropagation in inhomogeneous media:

�4�p; u1, u2, v1, v2, zt, zr�

� �G�p; u1, zt�G*�p; u2, zt�

� G�p; v1, zr�G*�p; v2, zr��. (11)

Fig. 2. Schematic for a bistatic coherent detection laser radarsystem. The dotted lines represent the BPLO field.


When the angular deviation of propagating wavesthat is due to refractive turbulence is small, theGreen’s function is given by a Feynman path inte-gral.14,24,25 If the temporal variations of the fieldswith propagation distance are slow, the Markov ap-proximation is valid.26 Under the Markov approxi-mation, refractive turbulence behaves as anuncorrelated process in the propagation direction forthe moment of interest. Under the above conditions,the fourth-moment Green’s function can be expressedas a series.24 In the bistatic case, the fourth-moment Green’s function reduces to two second-moment Green’s functions because the randomturbulence effects along each path are statisticallyindependent. With the above conditions, the fourthmoment of the Green’s function is given by

�4 � �G�p; u1, zt�G*�p; u2, zt��

� �G�p; v1, zr�G*�p; v2, zr��, (12)

with

�G�p1; u1, R�G*�p2; u2, R�� k2

�2�R�2

� exp� ik2 R

��p1 � u1�2 � �p2 � u2�

2�� exp�� 1

2 �0

R

D��u1 � u2��1 �zR

� �p1 � p2�zR

, z�dz� , (13)

where

D��x, z� � 4�k2 ��

�

�1 � cos�� x��

� �n��, �z � 0, z�d2� (14)

is the structure-function density �m�1�, �n��, �z, z��m3� is the local three-dimensional spectrum of therefractive-index fluctuations at range z, and � �radm�1� is the spatial wave vector. When we use theKolmogorov spectrum for refractive-index fluctua-tions,27,28

�0

R

D��u1 � u2��1 � z�R�, z�dz � ��u1 � u2��0�R�

�5�3

,

(15)

where

�0�R� � �2.91438k2 �0

R

Cn2� z��1 � z�R�5�3dz��3�5

(16)

is the transverse-field coherence length �m� of a pointsource located at range R and Cn

2�z� �m�2�3� is therefractive-index structure constant at range z. Re-placing the 5�3 power with 2 in Eq. �16� is a useful

approximation that produces little error.29 It shouldbe noted that, when we used the Kolmogorov spectralmodel in Eq. �14�, we assumed that the inner scale iszero. We could easily account for the effects of innerscale by choosing a spectral model that includes theinner scale parameter. Under weak turbulence con-ditions the effects of inner scale on the transversecoherence length �0 are generally quite small.30,31

3. Results

The SNR results presented in Section 2 are generalfor any multistatic or bistatic coherent lidar system.In this section we present the results of applyingspecific system parameters to the performance equa-tions given in Section 2. To obtain closed-form ex-pressions, a lidar system with untruncated Gaussianfields and apertures is assumed. With the generalexpressions derived for a Gaussian lidar system, thescattering coefficient is calculated by use of Mie the-ory.32 Next, the refractive turbulence effects arepresented. The general Gaussian lidar expressionsare then applied to a specific geometry.

A. Gaussian Lidar System

To determine the performance of a coherent lidar sys-tem, analytic expressions for the SNR, as well as otherparameters, are desired. This is best accomplishedwhen we describe the main components of the coherentlidar as untruncated complex Gaussian functions.This representation allows analytic solutions and con-tains all the physics of the system. In the followingcalculations, we also assumed that the transmitter andLO fields are deterministic, the detector response isuniform, the detector collects the entire incident LOand scattered power �infinite detector�, and the trans-mitted and scattered fields propagate along indepen-dent paths. It has been shown that the independentpath assumption is valid when the baseline separationis larger than the aperture size,15 which is the case fora practical multistatic system.

The laser field at a plane defined by zt � 0 �justprior to the lens� is described as an untruncatedGaussian,

eL�u, 0, t� �1

�L�� exp�� u2

2�L2 �

iku2

2FL , (17)

where u2 is u � u, �L �m� is the 1�e intensity radius ofthe laser beam, and FL �m� is the phase curvature ofthe laser beam �FL � 0 if focused at positive distancez�. If the aperture size is equal to �L, 63% of thebeam power is transmitted; if the aperture size isequal to 1.5�2 �L, 99% of the laser beam power istransmitted. The truncation effects by an aperturecan be neglected if the physical aperture size isgreater than 4�2 �L.33

The transmitter lens response for a scalar field isalso described with an untruncated Gaussian func-tion. This lens function is given by

WT�u� � exp�� u2

2�T2 �

iku2

2FT , (18)


where �T �m� is the 1�e intensity radius of the trans-mitter lens and FT �m� is the phase curvature of thetransmit lens �FT � 0 for the focusing lens�. Al-though this representation is not realistic for a cir-cular aperture, it allows analytic solutions and at thesame time preserves a size parameter for the lens.Using Eqs. �17� and �18�, we can write the normalizedfield at the exit of the lens as

eT�u, 0� �1

�L�� exp�� u2

2�TE2 �

iku2

2FTE , (19)

with

1�TE

2 �1�L

2 �1�T

2 , (20)

where �TE �m� is the 1�e intensity radius of the trans-mitted field, and with

1FTE

�1FL

�1FT

, (21)

where FTE �m� is the phase curvature of the transmitbeam. When we insert Eq. �19� into Eqs. �9�, theaverage irradiance of the normalized transmitterfield �with respect to the transmitter referenced co-ordinate system� at the target plane becomes6

�jT�xt, yt, zt�� TE

2

��L2�BT

2� zt�exp�xt

2 � yt2

�BT2� zt�

� , (22)

where

�BT2� zt� � �TE

2�1 �zt

FTE2

�zt

2

k2�TE2 �

2zt2

k2�02� zt�

. (23)

In other words, the average normalized irradiance ofthe transmitted field in the target plane has a Gauss-ian profile with a 1�e radius of �BT �m�. In derivingEq. �22�, we used a square-law structure function,replacing 5�3 with 2 in Eq. �16�.

The receiver lens and LO fields are described in amanner similar to that of the transmitter lens and

transmitted field. The BPLO irradiance profile isgiven by

�jBPLO�xr, yr, zr�� RE

2

��LO2�BR

2� zr�exp�xr

2 � yr2

�BR2� zr�

� ,

(24)

where

�BR2� zr� � �RE

2�1 �zr

FRE2

�zr

2

k2�RE2 �

2zr2

k2�02� zr�

(25)

is the 1�e intensity radius of the virtual BPLO in thetarget plane. Here, �RE �m� and FRE �m� are definedin a manner identical to Eqs. �20� and �21� with theappropriate change of subscripts for the receivercharacteristics.

To perform the target plane integration, the trans-mitter and receiver referenced coordinates must betransformed to the target plane coordinates. Apply-ing the required coordinate transformations �see Eqs.�2� and �3�� and substituting the transmitter and re-ciprocal receiver truncation ratios,

TT ��TE

2

�L2 , TR �

�RE2

�LO2 , (26)

into Eqs. �22� and �24�, we obtain the irradiance pro-files �with respect to the target referenced coordinatesystem�

�jT�xp, yp, zp, RT�� TT

��BT2� zp, RT�

� exp�� xp2 � yp

2

�BT2� zp, RT�

� , (27)

�jBPLO�xp, yp, zp, RR�� TR

��BR2�xp, zp, RR�

� exp�� xp cos �S � zp sin �S�2 � yp

2

�BR2�xp, zp, RR�

� , (28)

where

�BT2� zp, RT� � �TE

2�1 �zp � RT

FTE2

�� zp � RT�

2

k2�TE2 �

2� zp � RT�2

k2�0T2

�TE2�1 �

RT

FTE2

�RT

2

k2�TE2 �

2 RT2

k2�02�RT�

,(29)

�BR2�xp, zp, RR� � �RE

2�1 ��xp sin �S � zp cos �S � RR

FRE2

��xp sin �S � zp cos �S � RR�

2

k2�RE2 �

2��xp sin �S � zp cos �S � RR�2

k2�0R2

�RE2�1 �

RR

FRE2

�RR

2

k2�RE2 �

2 RR2

k2�02�RR�

, (30)


and �0R and �0T are the transverse-field coherencelengths with respect to the receiver and transmitter.The approximations in Eqs. �29� and �30� are thezero-order terms of the Taylor-series expansion aboutxp � yp � zp � 0, which is equivalent to one assumingthat the 1�e radii do not change significantly withinthe overlap of the transmit beam and the imaginedBPLO beam. It can be shown that this approxima-tion is valid for the small aperture sizes consideredhere.

When we insert Eqs. �26�–�28� into Eq. �6� and usethe first-order approximations in Eqs. �29� and �30�,the coherent responsivity becomes

where �BT0 and �BR0 indicate the zero-order term inthe Taylor-series expansions �see Eqs. �29� and �30��.For the monostatic case, �RT � RR � R and �S � 0�,Eq. �31� reduces to

C�R� ��2TT TR

��BT2�R� � �BR

2�R��, (32)

which is identical to the independent path calculationobtained by Frehlich and Kavaya.14 At this point wemodel the temporal profile of the transmitted laserpulse as an untruncated Gaussian,

�PL�t�� UL

�p��exp�� t2

�p2 , (33)

where �p �s� is the 1�e pulse width and UL �Js� is thelaser-pulse energy. When we insert Eqs. �31� and�33� into Eq. �6� with the assumption that � and K donot change significantly over the scattering volume,i.e.,

K2� zp� � K�RT�K�RR�, �� zp, �S� � ��RT, �S�,(34)

the target plane SNR becomes

SNR�RT, RR, t� �UL�Q K2�RT, RR��RT, �s��

3TT TR

�hBw�eff2�RT, RR�

� exp�� t � RT�c�2

�p2

� �1 � ��RT, RR�� , (35)

where

�eff2�RT, RR� � ��BR

2�RR� � �BT2�RT��

� ��p2c2 sin2 �s�BR

2�RR�

� �BT2�RT�cos2 �s��

1�2, (36)

ε�RT, RR� �

�BR2�RR� � �BT

2�RT�cos2 �s

�p2c2 sin2 �s � �BR

2�RR� � �BT2�RT�cos2 �s

. (37)

For the case of a cw laser transmitter ��p 3 ��, the

SNR simplifies to

SNR�RT, RR� �

�PL��Q K2�RT, RR��RT, �s��2TT TR

hBw sin �S��BR2�RR� � �BT

2�RT��1�2 , (38)

where �PL� is the average output power of the cw lasertransmitter.

Now that the general expressions for the SNR havebeen applied to a Gaussian system, the scatteringproperties of the target �aerosol particles in this case�need to be determined. We address this in Subsec-tion 3.B using Mie scattering theory.

B. Volume Scattering Coefficient

Before we can calculate the SNR, the functionality ofthe volume scattering coefficient ��zp, �S� must bedetermined. We accomplished this by assumingspherical scattering particles and using the well-known Mie scattering formulas.32,34 Although thespherical particle assumption may not be wellfounded, it serves as a good first-order approximationfor the scattering from aerosol particles. Koepkeand Hess have studied the effects of nonsphericalparticles on the scattering function and they con-cluded that “the scattering functions of the aerosoltypes continental and urban can be sufficiently cal-culated with Mie theory.” 35

If the number of dielectric spheres with refractiveindex m and radius in the interval �r, r � dr� is givenas N�r� dr, then the volume scattering coefficientsare36

��,��S� �1k2 �

r1

r2

i�,��S�N�r�dr, (39)

where i�,� corresponds to the scattered intensity �perunit irradiance� in either the s-polarized or the

C� zp, RT, RR� ��2TT TR

��BR02�RT� � �BT0

2�RR��BR02�RR� � �BT0

2�RT�cos2 �S��1�2

� exp��zp2� sin2 �S

�BR02�RR� � �BT0

2�RT�cos2 �S�� , (31)


p-polarized state. It is important to note that thescattered intensities have an implicit dependence onwavelength, size parameter, and index of refraction.

Although this model for � is not entirely accurate,it does provide a good approximation to the angulardependence of aerosol scattering. By using Eq. �39�,we assumed that the refractive index of all the con-tributing aerosols is an average effective index of allthe scattering particles. It is important to note that� is the critical engineering parameter for determi-nation of the SNR. In an actual fielded system, aneffective � can be estimated from the return signalpower. The operation of this system does not re-quire knowledge of the actual aerosol density, scat-tering cross section, or size distribution.

For the calculations in this study, we use the vec-torized FORTRAN code developed by Wiscombe37,38 witha rural aerosol model described by Shettle andFenn.39 The variation of aerosol density with alti-tude, in reality, will exhibit spatial and temporalvariations, the modeling of which is not includedhere. An average profile, represented by an expo-nential, is used in this analysis. This profile is givenby

Nz� z� � N0 exp�� zH� , (40)

where Nz �cm�3� is the particle concentration, N0�cm�3� is the concentration at ground level, z �m� isaltitude, and H �m� is the characteristic scaleheight.40 Typical values of H range from 1 to 1.4km.41

Our next step in applying the SNR expressions tospecific configurations is to determine the refractiveturbulence effects. A common approach is to com-pare the calculated SNR without refractive turbu-lence with the SNR with refractive turbulence. InSubsection 3.C, the analysis of �0 is given for a spe-cific Cn

2 model. Because �0 depends on the actualpropagation path, there will, in general, be differentvalues of �0 for each receiver and transmitter. Once�0 is calculated, the reduction in SNR that is due torefractive turbulence is addressed.

C. Refractive Turbulence Effects

The general expressions for refractive turbulence ef-fects are detailed in Section 2 and are accounted for inthe coherent responsivity and SNR with thetransverse-field coherence length �0 given in Eq. �16�.Because Eq. �16� is a path-dependent integral, thevalue of �0 will, in general, be different for the trans-

mitter and the receiver paths. If we model therefractive-index variations with altitude as

Cn2� z� � Cn

2� z � 1 m� z�4�3, (41)

then the transverse-field coherence length, as viewedfrom the transmitter, is given by

�0�RT� � �2.91438k2Cn2� z � 1 m� �

0

RT

� zt cos �nt

� 5��4�3�1 � zt�Rt�5�3dzt��3�5

, (42)

where the transmitter is arbitrarily placed 5 m abovethe ground to evaluate the integral in Eq. �16�. Sim-ilarly, for the receiver,

�0�RR� � �2.91438k2Cn2� z � 1 m�

� �0

RR

� zr cos �nr � 5��4�3�1 � zr�Rr�5�3dzr��3�5

, (43)

and again, the receiver is located 5 m above theground. This integral is evaluated numerically andthe results are used in Eqs. �29� and �30�.

The overall effect of refractive turbulence is a re-duction in the SNR. This effect is quantified whenwe take the ratio of the SNR with refractive turbu-lence effects included, designated SNR�0

, to the SNRwithout refractive turbulence effects, designatedSNR�0

3 �. The peak SNR, including refractive tur-bulence effects, can be written as

SNR�0

1�eff

2�RT, RR�, (44)

where �eff2�RT, RR� is defined in Eq. �36� �with the zero-

order approximations of �BR2 and �BT

2� and includes therefractive turbulence effects. We define �BR

2�RR� and�BT

2�RT� as the 1�e intensity radii of the transmitted andvirtual BPLO beams in the target plane neglecting theeffects of refractive turbulence, i.e.,

�BR2�RR� � �RE

2�1 �RR

FRE2

�RR

2

k2�RE2 ,

�BT2�RT� � �TE

2�1 �RT

FTE2

�RT

2

k2�TE2 . (45)

Then the SNR reduction factor F is given by

FP �SNR�0

SNR�03�

� ��BR2 � �BT

2��p2c2 sin2 �s � �BR

2 � �BT2 cos2 �s�

��BR2 � �BT

2��p2c2 sin2 �s � �BR

2 � �BT2 cos2 �s�

�1�2

(46)


for the pulsed case. For the cw case ��p 3 ��, theSNR reduction can be expressed as

Fcw � ��BR2 � �BT

2

�BR2 � �BT

21�2

. (47)

If the refractive turbulence effects are small, FP !Fcw ! 1, and the refractive turbulence will have littleeffect on the SNR.

Now that all the specific parameters �other thangeometry� have been addressed, the SNR expressioncan be applied to specific cases. In Subsection 3.D,the results obtained above are applied to a specificgeometry as well as to the monostatic geometry forcomparison with previous results.

D. Specific Geometry Considerations

The results derived above can be applied to any bi-static �independent path�Gaussian lidar system. Inthis subsection these results are applied to a specificgeometry and system parameters. The specificcases considered here are for the transmit laser di-rected vertically and the receiver scanned along thebeam �see Fig. 1� and for the monostatic with thetransmitter directed vertically, for comparison withthe above results. Other alignments have been con-sidered in a previous study.6

The system parameters used in the calculationsare given in Table 1. For these calculations, it isassumed that the transmit lens and receiver lens areidentical, �R � �T. For the 1�e laser and LO inten-sity radii, the optimum monostatic truncation valuesof 0.707 �T and 0.707 �R are chosen for simplic-ity.10,14

Plots of the pulsed SNR for the single transmitterand receiver pair, single polarization, and differentaperture sizes are shown in Fig. 3. It is worth notingthat there is an optimum aperture size for eachrange. We can determine this aperture size by dif-ferentiating the expression for SNR with respect to�R � �T. For the pulsed case, we rewrite Eq. �36� as

�eff2 � �p c sin �s��BR

2 � �BT2��1 � "��1�2, (48)

where

" ��BR

2�RR� � �BT2�RT�cos2 �s

�p2c2 sin2 �s

. (49)

The parameter " is shown to be essentially zero,6especially in the near field; therefore the optimiza-tions for the pulsed and cw cases are nearly identical.Performing the required differentiation yields an op-timum aperture size given by

�Ropt � �9�2 RT

2 � B2�

2k2 �1�4

� 1.4565�2 RT2 � B2

k2 1�4

. (50)

In the near-field limit �RT 3 0�, the optimum aper-ture size �R

opt approaches 5.8 mm for B � 100 m and� � 1 #m.

A plot of the coherent responsivity shows the sametrend in terms of optimum aperture sizes �see Fig. 4�.As can be seen from the plot, the coherent responsiv-ity is optimized in the near field for an aperture sizeof approximately 5 mm. This optimization indicatesthat the transmitter and LO beam sizes can also besmall. Recall that �LO � 0.707 �R and �L �0.707 �T. Because small beam sizes can be used,the analysis can be simplified somewhat when weassume that the transmitter and receiver lenses arelarge compared to the beam sizes. At these beamsizes �$4 mm� the lenses need to be approximatelyonly 25 mm before truncation effects can be ignored.In this case, the lens aperture sizes can be set toinfinity, and Eqs. �29� and �30� can be expressed as

�BT2� zp, RT� � �L

2 �RT

2

k2�L2 �

2 RT2

k2�02�RT�

,

�BR2�xp, zp, RR� � �LO

2 �RR

2

k2�LO2 �

2 RR2

k2�02�RR�

. (51)

For infinite apertures, the transmitter and reciprocalreceiver truncation ratios become unity. Assumingmatched transmit and LO beams ��L � �LO�, the

Fig. 3. Pulse SNR �s-polarization� plots for various aperture sizes.Refractive turbulence effects are not included. �R � 100 mm�solid curve�; �R � 50 mm �dotted curve�; �R � 10 mm �dashedcurve�; �R � 5 mm �dashed–dotted curve�; �R � 1 mm �dashed–double-dotted curve�.

Table 1. System Parameters

Parameter Symbol Value

Baseline separation B 100 mSystem bandwidth Bw 100 MHzWavelength � 1 #mQuantum efficiency �Q 0.51�e pulse width �p 250 nsPulse laser energy UL 1 J1�e laser intensity radius �L 0.707 �T

1�e LO intensity radius �LO 0.707 �R

Aerosol density at ground N0 5000 cm�3

Index structure constant Cn2�z � 1 m� 10�12 m�2�3

Transmitter focus FTE �Receiver focus FRE �


optimum beam size �when infinite apertures are as-sumed� is given by

�Lopt � �2 RT

2 � B2

2k2 1�4

� 0.841�2 RT2 � B2

k2 1�4

. (52)

In the near-field limit �RT 3 0�, the optimum beamsize approaches 3.3 mm for B � 100 m and � � 1 #m,which in turn implies that the lens sizes need beapproximately only 20 mm before the truncation ef-fects can be neglected.

The pulsed SNR with infinite apertures is shown inFig. 5. As can be seen in this plot, the SNR is im-proved. This is because there are no power trunca-

tion effects owing to the lens apertures �see Eqs. �26��.The remainder of the calculations are performed withthe assumption that the aperture sizes are large com-pared to the transmitter and LO beams.

From Eq. �52� it is evident that the optimum beamsize increases with range RT. This trend can also beseen in the plots of SNR and especially coherent re-sponsivity. Because the geometry in this type of sys-tem limits the accuracy of sensing horizontal windsat high altitudes, optimization of the near-field SNRis desired.

An advantage of the small beam sizes is the reducedeffect of refractive turbulence. The reduction in SNRbecause of refractive turbulence is quantified with theSNR reduction factor F �see Eqs. �46� and �47��. Be-cause the value of " �see Eq. �49�� is approximatelyzero, we can argue that, in the near field, the SNRreduction will be the same for the cw and pulsed sys-tems. Figures 6 and 7 show the SNR reduction factorfor this geometry and various aperture sizes and for�L � 5 mm with various index structure constants atground level. From Fig. 7 it is evident that the re-fractive turbulence has a minimal effect on the SNR fortypical turbulence strengths. Adverse effects are notpresent until the turbulence strength at the groundapproaches 10�10 m2�3, which is considered to bestrong refractive turbulence. Typical values for Cn

2

near the ground vary from 10�13 to 10�17 m2�3, with10�15 m2�3 often quoted as a typical average value.42

E. Monostatic ConfigurationThis configuration is added as a test case to verify theaccuracy of the equations derived above. The SNR

Fig. 4. Coherent responsivity for various aperture sizes. �R �100 mm �solid curve�; �R � 50 mm �dotted curve�; �R � 10 mm�dashed curve�; �R � 5 mm �dashed–dotted curve�; �R � 1 mm�dashed–double-dotted curve�. As can be seen, the optimum ap-erture size is approximately 5 mm.

Fig. 5. Pulsed SNR �s-polarization� plots for various beam sizesand infinite apertures. Refractive turbulence effects are not in-cluded. �L � 100 mm �solid curve�; �L � 50 mm �dotted curve�;�L � 10 mm �dashed curve�; �L � 5 mm �dashed–dotted curve�;�L � 1 mm �dashed–double-dotted curve�. Again, the optimumaperture size is approximately 5 mm.

Fig. 6. SNR reduction factor for various beam sizes. �L � 100mm �solid curve�; �L � 50 mm �dotted curve�; �L � 10 mm �dashedcurve�; �L � 5 mm �dashed–dotted curve�; �L � 1 mm �dashed–double-dotted curve�.

Fig. 7. SNR reduction factor for a fixed beam size ��L � 5 mm�and various turbulence strengths.


for this case is shown in Fig. 8. Because the volumebackscatter coefficient � is different for our applica-tion �Frehlich and Kavaya assume a constant � as afunction of altitude14�, the SNR does not comparedirectly. However, because the coherent responsiv-ity does not depend on such system parameters, adirect comparison can be made. The coherent re-sponsivity for the monostatic case is shown in Fig. 9.These results agree with the Frehlich and Kavayaresults for the 10-cm aperture, collimated case.14

F. Spatial Resolution

An advantage of the small beam sizes, other than theobvious design advantages, is the spatial resolutionthat is achievable, at least in the near field. Recallthat, for this system, the spatial resolution, both inrange and in the transverse direction, is determinedby the overlap of the transmitted laser and the imag-ined BPLO. The approximate scattering volume isthe transverse scattering area �in the xp–yp plane�

multiplied by the extent in the zp direction. Thetransverse area is set by the transmit beam, and thezp extent is set by the BPLO beam at xp � yp � 0.The approximate volume is then given by

V ��

�

��

�

��

�

exp�� xp2 � yp

2

�BT2

� exp�� zp2 sin2 �S

�BR2 �dxpdypdzp

��3 �BR�BT

2

sin �S. (53)

A plot of the scattering volumes for various beamsizes is shown in Fig. 10. From this plot it can beseen that, for altitudes less than 1 km, the scatteringvolume is less than 100 cm3 for the 5-mm beam andB � 100 m.

It is interesting to compare the scattering volumefor the monostatic configuration with that of the bi-static configuration. In the monostatic case, the zpextent is determined by the pulse length. In thiscase, the scattering volume is given as

V ��BT

2c"t2

� ��BT2c�p�ln 2 , (54)

where "t � 2�ln 2�1�2�p is the effective pulse lengthfor the pulse profile given in Eq. �33�. A plot of thescattering volume is shown in Fig. 11. As can beseen, the monostatic scattering volumes for this pulselength are much larger than the scattering volumesin the multistatic case.

Fig. 8. Monostatic SNR for a pulsed coherent lidar system forvarious aperture sizes. �R � 100 mm �solid curve�; �R � 50 mm�dotted curve�; �R � 10 mm �dashed curve�; �R � 5 mm �dashed–dotted curve�; �R � 1 mm �dashed–double-dotted curve�.

Fig. 9. Coherent responsivity for the monostatic case and variousaperture sizes. �R � 100 mm �solid curve�; �R � 50 mm �dottedcurve�; �R � 10 mm �dashed curve�; �R � 5 mm �dashed–dottedcurve�; �R � 1 mm �dashed–double-dotted curve�.

Fig. 10. Scattering volumes for various beam sizes. �L � 100mm �solid curve�; �L � 50 mm �dotted curve�; �L � 10 mm �dashedcurve�; �L � 5 mm �dashed–dotted curve�; �L � 1 mm �dashed–double-dotted curve�.


4. Conclusions

The feasibility of a multistatic, pulsed coherent Dopp-ler lidar system to measure winds within the bound-ary layer has been presented. This system will becapable of detecting high-spatial-resolution wind pro-files for all three components of the wind velocitywithin the boundary layer. The multistatic config-uration also decouples the vertical resolution and fre-quency estimation accuracy. It has also been shownthat the transmitter and receiver optics are small foruseful operation. In fact, the performance is im-proved with use of smaller apertures. This enablesthe design of a compact system.

Previous research showed that, for an averagepower of 5 W, the collimated cw system does notappear feasible for the observation times desired.6The pulsed case, on the other hand, has adequateSNR for frequency estimation in a single shot. Infact, the SNR is high enough that the question shouldbe raised as to whether it is better to use a systemthat has high peak power and low pulse repetitionfrequency or lower peak power and higher pulse rep-etition frequency. This question was posed by Ryeand Hardesty43 and addressed by Frehlich and Yad-lowsky.44 They concluded that in the high SNR re-gime, it is better to transmit many low-energy pulsesinstead of one pulse with the same energy. This factshould be taken into account in the design of thesystem.

The results also show that there is an optimumaperture size as a function of target range. Thenear-field optimum aperture size for the cases eval-uated is of the order of 5 mm, much smaller than

might be expected. This is the case only within thenear field of the transmitter; the transmit beam ra-dius does not grow rapidly, giving rise to small scat-tering volumes. This small scattering volume canbe thought of as a small incoherent source. The spa-tial coherence of the field at the receiver �in the ab-sence of turbulence� increases as the size of theincoherent source decreases �Van Cittert–Zerniketheorem�.42 This increased spatial coherence in-creases the coherent responsivity, and consequentlythe SNR.

Because the apertures sizes, and thus the trans-mitter and LO beams, are small for this system, per-formance can be improved when the lenses are mademuch larger than the beam sizes. If the lenses aredesigned to be five to six times larger than the beamsizes, the effects of truncation can be ignored and thelens sizes can be assumed infinite in extent. Thetransmitter and LO beam sizes then set the effectiveaperture sizes. The optimum beam size is of theorder of 3 mm; therefore if the lenses are larger than15 mm, the truncation effects are negligible. Theadvantages of a smaller effective aperture are com-pact design, small scattering volume �in the nearfield�, and minimal refractive turbulence effects.

In the cases examined, the effects of refractive tur-bulence on the SNR have been shown to be negligiblefor typical turbulence strengths. Only in strong tur-bulence is the SNR degraded. The small effectiveapertures minimize the effects of refractive turbu-lence. This is because the transverse-field coher-ence diameter of the scattered radiation that is due torefractive turbulence will typically be larger than theeffective aperture diameter. This implies that thespatial coherence of the scattered fields will be main-tained over the small effective aperture of the re-ceiver.

A big advantage of the multistatic configuration isthe high spatial resolution. Typical Doppler lidarsystems are scanned to obtain information about thewind direction. This is done at the sacrifice of spa-tial resolution. Even when we are detecting only thevertical wind profile with a monostatic system, therange resolution can be improved only by use ofshorter pulses. This is at the expense of estimationaccuracy. In the multistatic configuration, the spa-tial resolution is determined by an overlap volumeand not tied to the pulse length of the transmit laser.For the optimum apertures, the scattering volume isless than 100 cm3 out to an altitude of 1 km. Com-pare this to scattering volumes of the order of 105 cm3

for the pulse length used in this analysis �250 ns�.The small effective aperture sizes that can be usedalso contribute to a reduction of the scattering vol-ume in the near field.

In this paper we did not address the detailed designissues involved with building the system. Some is-sues that need to be addressed in this area are gettingthe LO beam to the receivers, optical design of thetransmitter and receiver, alignment, and numericaleffort for real-time estimates. The LO beam couldbe routed to the receiver by single-mode optical fiber

Fig. 11. Effective scattering volume for the monostatic configu-ration and various aperture sizes. The 1�e pulse width is 250 ns.�R � 100 mm �solid curve�; �R � 50 mm �dotted curve�; �R � 10mm �dashed curve�; �R � 5 mm �dashed–dotted curve�; �R � 1 mm�dashed–double-dotted curve�.


links, but frequency stability may be an issue forlarge baselines. The most difficult problem in thedesign stage may be the alignment and scanning,especially for coherent detection. The optimum lo-cation of the transmitter and receivers also needs tobe addressed. Because this is an application-specificoptimization, it is not addressed in this research.

We thank Rod Frelich for the many technical dis-cussions between us during the completion of thisresearch. We also thank the reviewers for their in-sightful comments. The views expressed in this ar-ticle are those of the authors and do not reflect theofficial policy or position of the U.S. Air Force, theU.S. Department of Defense, or the U.S. government.

References1. J. G. Hawley, R. Targ, S. W. Henderson, C. P. Hale, M. J.

Kavaya, and D. Moerder, “Coherent launch-site atmosphericwind sounder: theory and experiment,” Appl. Opt. 32, 4557–4568 �1993�.

2. S. F. Clifford, J. C. Kaimal, R. J. Lataitis, and R. G. Strauch,“Ground-based remote profiling in atmospheric studies: anoverview,” Proc. IEEE 82, 313–355 �1994�.

3. A. V. Jelalian, Laser Radar Systems �Artech House, Norwood,Mass., 1992�.

4. M. Harris, G. Constant, and C. Ward, “Continuous-wave bistaticlaser Doppler wind sensor,” Appl. Opt. 40, 1501–1506 �2001�.

5. R. M. Huffaker, “Laser Doppler detection systems for gas ve-locity measurement,” Appl. Opt. 9, 1026–1039 �1970�.

6. E. P. Magee, “Performance analysis of a multistatic coherentDoppler lidar,” Ph.D. dissertation �Pennsylvania State Univer-sity, University Park, Pa., 1998�.

7. B. J. Rye, “Antenna parameters for incoherent backscatterheterodyne lidar,” Appl. Opt. 18, 1390–1398 �1979�.

8. T. J. Kane, W. J. Kozlovsky, R. L. Byer, and C. E. Byvik,“Coherent laser radar at 1.06 #m using Nd:YAG lasers,” Opt.Lett. 12, 239–241 �1987�.

9. R. T. Menzies and R. M. Hardesty, “Coherent Doppler lidar formeasurements of wind fields,” Proc. IEEE 77, 449–462 �1989�.

10. R. G. Frehlich, “Conditions for optimal performance of mono-static coherent laser radar,” Opt. Lett. 15, 643–645 �1990�.

11. S. W. Henderson, C. P. Hale, J. R. Magee, M. J. Kavaya, andA. V. Huffaker, “Eye-safe coherent laser radar system at 2.1#m using Tm,Ho:YAG lasers,” Opt. Lett. 16, 773–775 �1991�.

12. R. Targ, M. J. Kavaya, R. M. Huffaker, and R. L. Bowles,“Coherent lidar airborne windshear sensor: performanceevaluation,” Appl. Opt. 30, 2013–2026 �1991�.

13. M. J. Kavaya and P. J. M. Suni, “Continuous wave coherentlaser radar: calculation of measurement location and vol-ume,” Appl. Opt. 30, 2634–2642 �1991�.

14. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar per-formance for general atmospheric refractive turbulence,” Appl.Opt. 30, 5325–5352 �1991�.

15. R. G. Frehlich, “Effects of refractive turbulence on coherentlaser radar,” Appl. Opt. 32, 2122–2139 �1993�.

16. R. G. Frehlich, “Optimal local oscillator field for a monostaticcoherent laser radar with a circular aperture,” Appl. Opt. 32,4569–4577 �1993�.

17. R. G. Frehlich, “Heterodyne efficiency for a coherent laserradar with diffuse or aerosol targets,” J. Mod. Opt. 41, 2115–2129 �1994�.

18. M. C. Jackson, “The geometry of bistatic radar systems,” IEEProc. F 133, 604–612 �1986�.

19. N. J. Willis, Bistatic Radar �Artech House, Boston, Mass.,1991�.

20. M. Fogiel, ed., Handbook of Mathematical, Scientific, and En-

gineering Formulas, Tables, Functions, Graphs, Transforms�Research & Education Association, Piscataway, N.J., 1994�.

21. A. E. Siegman, “The antenna properties of optical heterodynereceivers,” Appl. Opt. 5, 1588–1594 �1966�.

22. B. J. Rye, “Refractive-turbulence contribution to incoherentbackscatter heterodyne lidar returns,” J. Opt. Soc. Am. 71,687–691 �1981�.

23. K. P. Chan, D. K. Killinger, and N. Sugimoto, “HeterodyneDoppler 1-#m lidar measurement of reduced effective tele-scope aperture due to atmospheric turbulence,” Appl. Opt. 30,2617–2627 �1991�.

24. J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, andF. S. Henyey, “Solution for the fourth moment of waves prop-agating in a random media,” Radio Sci. 21, 929–948 �1986�.

25. R. G. Frehlich, “Space-time fourth moments of waves propa-gating in random media,” Radio Sci. 22, 481–490 �1987�.

26. V. I. Tatarski, The Effects of the Turbulent Atmosphere onWave Propagation �Keter, Jerusalem, 1971�.

27. V. A. Banakh and V. L. Mironov, Lidar in a Turbulent Atmo-sphere �Artech House, Dedham, Mass., 1987�.

28. R. G. Frehlich, “Intensity covariance of a point source in arandom medium with a Kolmogorov spectrum and an innerscale of turbulence,” J. Opt. Soc. Am. A 4, 360–366 �1987�.

29. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systemsin the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 �1979�.

30. C. Y. Young and L. C. Andrews, “Effects of a modified spectralmodel on the spatial coherence of a laser beam,” Waves Ran-dom Media 4, 385–397 �1994�.

31. L. C. Andrews, Laser Beam Propagation through Random Me-dia �SPIE, Bellingham, Wash., 1998�.

32. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles �Wiley, New York, 1983�.

33. L. D. Dickson, “Characteristics of a propagating Gaussianbeam,” Appl. Opt. 9, 1854–1861 �1970�.

34. H. C. van de Hulst, Light Scattering by Small Particles �Wiley,New York, 1957�.

35. P. Koepke and M. Hess, “Scattering functions of troposphericaerosols: the effects of nonspherical particles,” Appl. Opt. 27,2422–2430 �1988�.

36. R. M. Measures, Laser Remote Sensing: Fundamentals andApplications �Wiley, New York, 1984�.

37. W. J. Wiscombe, “Mie scattering calculations: advances intechnique and fast, vector-speed computer codes,” Tech. NoteNCAR�TN-140�STR �National Center for Atmospheric Re-search, Boulder, Colo., 1979�; updated version available atftp:��climate.gsfc.nasa.gov�pub�wiscombe�Single_Scatt�Ho-mogen_Sphere�Exact_Mie��.

38. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl.Opt. 19, 1505–1509 �1980�.

39. E. P. Shettle and R. W. Fenn, “Models for the aerosol of thelower atmosphere and the effects of humidity variations ontheir optical properties,” Tech. Rep. AFGL-TR-79-0214 �U.S.Air Force Geophysics Laboratory, Hanscom Air Force Base,Mass., 1979�.

40. E. J. McCartney, Optics of the Atmosphere, Scattering by Mol-ecules and Particles �Wiley, New York, 1976�.

41. K. Parameswaran, K. D. Rose, and B. V. K. Murthy, “Aerosolcharacteristics from bistatic lidar observations,” J. Geophys.Res. 89, 2541–2552 �1984�.

42. J. W. Goodman, Statistical Optics �Wiley, New York, 1985�.43. B. J. Rye and R. M. Hardesty, “Discrete spectral peak estima-

tion in incoherent backscatter heterodyne lidar. II: Corre-logram accumulation,” IEEE Trans. Geosci. Remote Sens. 31,28–35 �1993�.

44. R. G. Frehlich and M. J. Yadlowsky, “Performance of mean-frequency estimators for Doppler radar and lidar,” J. Atmos.Oceanic Technol. 11, 1217–1230 �1994�.


bistatic coherent laser radar signal-to-noise ratio

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