aerosol speckle effects on atmospheric pulsed lidar backscattered signals

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Aerosol speckle effects on atmospheric pulsed lidar backscattered signals S. R. Murty The effects of refractive turbulence along the path on the aerosol speckle field propagation and on the decorrelation time are studied for coherent pulsed lidar systems. 1. Introduction Lidar systems using atmospheric aerosols as target exhibit return signal amplitude and power fluctua- tions which indicate speckle effects. 1 ' 2 The speckle effects are manifested statistically as Rayleigh-distrib- uted amplitude and exponentially distributed power of the backscattered signals. The signals from the various scatterers which make up the target combine to form a speckle pattern at any point along the back- scattered beam. The speckle size is -Xz/b, where z is the range from the target, Xis the wavelength, and b is the radius of the illuminating beam on the diffuse target. Therefore, as the beam size on the diffuse target increases, the speckle size at the receiver de- creases. Since the field across a speckle pattern is produced by the addition of a number of signals of random phase, the speckle pattern changes as each particle moves, thereby generating spatial and tempo- ral changes in intensity. The particles move primarily due to atmospheric turbulence, and the changes in the speckly pattern are caused by the velocity turbulence in the illuminated region of the atmosphere which contributes to the backscattered signal at the receiver. For a pulsed laser, the backscatter region of the atmo- sphere is ctp/2 around the range point where tp is the pulse duration and c is the speed of light. In addition to producing signal intensity fluctua- tions, aerosol-generated speckle causes a decorrelation of the backscattered signals, as the aerosols are free to move about relative to each other along the beam axis and transverse to it. The resulting decorrelation time determines the temporal window of the return signal The author is with Alabama A&M University, Schoolof Engineer- ing & Technology, Normal, Alabama 35762. Received 30 November 1987. 0003-6935/89/050875-04$02.00/0. ©1989 Optical Society of America. from which the power spectrum can be calculated to obtain wind speed. A laser beam experiences fluctuations in intensity and a reduction in coherence due to propagation in the refractive turbulence of the atmosphere between the telescope and range point as shown in Fig. 1. Conse- quently the received signal characteristics are affected by a combination of refractive turbulence and speckle effects. The objective of this paper is to examine the effect of refractive turbulence along the path on the aerosol speckle field propagation and on the decorrela- tion time. Churnside and Yura 3 have developed the basic theo- retical formulation of the spatiotemporal correlation function for the aerosol return-signal field. Ancellet and Menzies 4 presented results of measurement using a pulsed TEA CO 2 laser and found a decorrelation time of 2-2.5 Ars over a 1.3-km range. Laser pulses shorter than -150 ps are broadened due to atmospheric turbu- lence, but the pulse spreading and wander in turbu- lence decrease as the pulse becomes longer. 5 ' 6 In this work, the pulse is represented as a Fourier integral, and we make use of the two-frequency mutual coher- ence function to study propagation of the aerosol speckle field. II. Analysis The transmitter and receiver are assumed to be colo- cated having a common telescope aperture diameter d. The laser transmitter is assumed to be pulsed with a pulse duration tp and a transmitted power PT(t) at time t. The complex field of the pulse propagating in the turbulent medium at any range point z is given by U 1 (z,p,t) = J A(w)uI(p,k,t) exp(icot)dw, (1) where A(w) is the Fourier spectral amplitude, u is the monochromatic field amplitude at a distance z from the transmitter, p is a 2-D transverse vector in the aerosol target plane, and k is the optical wave number. Using the paraxial approximation and the extended Huygens-Fresnel principle, 7 u can be expressed as 1 March 1989 / Vol. 28, No. 5 / APPLIED OPTICS 875

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Aerosol speckle effects on atmospheric pulsedlidar backscattered signals

S. R. Murty

The effects of refractive turbulence along the path on the aerosol speckle field propagation and on thedecorrelation time are studied for coherent pulsed lidar systems.

1. Introduction

Lidar systems using atmospheric aerosols as targetexhibit return signal amplitude and power fluctua-tions which indicate speckle effects.1' 2 The speckleeffects are manifested statistically as Rayleigh-distrib-uted amplitude and exponentially distributed powerof the backscattered signals. The signals from thevarious scatterers which make up the target combineto form a speckle pattern at any point along the back-scattered beam. The speckle size is -Xz/b, where z isthe range from the target, X is the wavelength, and b isthe radius of the illuminating beam on the diffusetarget. Therefore, as the beam size on the diffusetarget increases, the speckle size at the receiver de-creases. Since the field across a speckle pattern isproduced by the addition of a number of signals ofrandom phase, the speckle pattern changes as eachparticle moves, thereby generating spatial and tempo-ral changes in intensity. The particles move primarilydue to atmospheric turbulence, and the changes in thespeckly pattern are caused by the velocity turbulencein the illuminated region of the atmosphere whichcontributes to the backscattered signal at the receiver.For a pulsed laser, the backscatter region of the atmo-sphere is ctp/2 around the range point where tp is thepulse duration and c is the speed of light.

In addition to producing signal intensity fluctua-tions, aerosol-generated speckle causes a decorrelationof the backscattered signals, as the aerosols are free tomove about relative to each other along the beam axisand transverse to it. The resulting decorrelation timedetermines the temporal window of the return signal

The author is with Alabama A&M University, School of Engineer-ing & Technology, Normal, Alabama 35762.

Received 30 November 1987.0003-6935/89/050875-04$02.00/0.© 1989 Optical Society of America.

from which the power spectrum can be calculated toobtain wind speed.

A laser beam experiences fluctuations in intensityand a reduction in coherence due to propagation in therefractive turbulence of the atmosphere between thetelescope and range point as shown in Fig. 1. Conse-quently the received signal characteristics are affectedby a combination of refractive turbulence and speckleeffects. The objective of this paper is to examine theeffect of refractive turbulence along the path on theaerosol speckle field propagation and on the decorrela-tion time.

Churnside and Yura3 have developed the basic theo-retical formulation of the spatiotemporal correlationfunction for the aerosol return-signal field. Ancelletand Menzies4 presented results of measurement usinga pulsed TEA CO2 laser and found a decorrelation timeof 2-2.5 Ars over a 1.3-km range. Laser pulses shorterthan -150 ps are broadened due to atmospheric turbu-lence, but the pulse spreading and wander in turbu-lence decrease as the pulse becomes longer.5 '6 In thiswork, the pulse is represented as a Fourier integral,and we make use of the two-frequency mutual coher-ence function to study propagation of the aerosolspeckle field.

II. Analysis

The transmitter and receiver are assumed to be colo-cated having a common telescope aperture diameter d.The laser transmitter is assumed to be pulsed with apulse duration tp and a transmitted power PT(t) attime t. The complex field of the pulse propagating inthe turbulent medium at any range point z is given by

U1 (z,p,t) = J A(w)uI(p,k,t) exp(icot)dw, (1)

where A(w) is the Fourier spectral amplitude, u is themonochromatic field amplitude at a distance z fromthe transmitter, p is a 2-D transverse vector in theaerosol target plane, and k is the optical wave number.Using the paraxial approximation and the extendedHuygens-Fresnel principle, 7 u can be expressed as

1 March 1989 / Vol. 28, No. 5 / APPLIED OPTICS 875

P. = p(t + r) = p(t) + VTT.

The backscattered field at time, t + ar, is given by

Us(qt + r) = exp[iw0 (t + r - z,)] J__ A(w + wo)

X u(q,w + o0,t + T)ATMOSPHERIC MOVING AEROSOL TARGET

LASER TRANSMITTER TURBULENCE

Fig. 1. Sketch of pulsed lidar propagating through atmosphericturbulence using moving aerosols as a target.

u1 (p,k,t) = k exp(ikz) J d2ru((rt -Z

X exp[.2 (p - r)2 + (rpkt)]* (2)

In Eq. (2), r is a transverse vector in the transmitterplane, (r,p,k,t) is the sum of the logamplitude andphase perturbation experienced by a spherical wavepropagating from r to p through the atmospheric tur-bulence, and UT represents the field amplitude at thetransmitter given by

[2PT(t) 1/2 1 ik ]UT(r,t) = F P~)1R2exp -!+Žr2

L ira2 J -Lka2 2f/J

where a is the beam radius at exp(-2) intensity, andthe geometric focus of the telescope.

The complex field Uj(z,p,t) is assumed to be baiscattered at range z by an atmospheric aerosol pres(in the illuminated volume. The scattered pulse pr(agates back to the receiver through atmospheric turllence, and its field is given by

US(q,t) = | A(w)u,(q,k,t) exp(iwt)dw,

(3)

X exp[iw(t + T -zT/)]dw, (7)

and the monochromatic wavefield at time, t + , isgiven by

U,(qwt + r) =kS exp(2ikz,) dr2u r t +r -)

X exi (p,4 - r)2 + (q -P,)2]

+ T(r,p,,k,t + T) + 'T'(p,,qkt + r). (8)

The cross-product of the received fields at time t and t+ is given by

U3 (q1,t) V(q 2,t + r) =f dw, J dW2Ae(wl)

X u,(qjwjt)Ae(w 2)US(q2W2,t + r)

X exp[i(Wl - 2)t11 (9)

where Ae(w) = A(w + coo).Equation (9) will be ensemble-averaged over the

propagation path for refractive turbulence and overthe illuminated volume for velocity turbulence to ob-tain the pulse spatiotemporal correlation function.This process is represented by double angular brack-ets, and the correlation function of the received opticalfields is given by

(4) <<US(q,t)7*1(q 2,t + -)>> =f d-do, X dW2A,(wj)A;(W2)

where the monochromatic wave field u(qkt) is givenby 9

u,(q,k,t) = ( ) u -pkt-

X exPikz + ik (q - p)2

+ 1(pqkst)]* (5)

In Eq. (5), q is a transverse vector in the receiver planeand S(k) is the monochromatic backscatter coefficientof the aerosol.

The pulse is assumed to have a center carrier angularfrequency coo and optical wavenumber ko and is nar-rowband. It is convenient to write Eq. (4) as

Us(qt) = exp[i(w0t - kz)] J dA( + o)

X u(q,w + wot) exp(icot). (6)

During a small time interval of <1 ms, the aerosolposition is assumed to change due to a radial velocityvz, a transverse velocity VT to a new position given bythe following relations:

Z= z(t + r = z(t) + VT;

x r(qjq2@01X2)

X exp[i(Wl - 2)t] (10)

where r(q,q 2,w1 ,02) = <<us(q,1 ,t)us(q2 ,w2,t + -r)>> isa two-frequency mutual coherence function. The de-termination of the effects of aerosol speckle, refractiveturbulence, and pulse shape are now dependent on theevaluation of the function r.I11. Ensemble Averaging

Following Churnside and Yura3 for the averagingover velocity turbulence, the aerosol velocities are as-sumed Gaussian-distributed within the inertial sub-range, and we obtain the expected value of the mutualcoherence function as

r(q,q2, 1W12 ) = L dv J d2VT J dzd 2p

X (u,(qj,wj,t)u*(q 2,W2t + r))

X P(V7 )P(VT)/V. (11)

In Eq. (11), V is the illuminated volume, and theprobability density functions of the velocity compo-nents are given by

876 APPLIED OPTICS / Vol. 28, No. 5 / 1 March 1989

P(VZ) = 1 [- (v -

P(VT) = 2 exp[-(VT- T],27rafT

where v,, VT and aZ, U2 are the means and variances ofthe wind components along the z axis and transverse toit.

The ensemble average over path turbulence is givenby

(u8(q1,w1,t)u:(q2w2,t + r)) =jk2S(-1)2( exp[2i(k1 - k2)Z(2irZ2)2

- 2ik2 Vir] Jf d2rld2r2U (rlt - 2z

Cij = 0.132ir2kikjzC2 J dt f U-5 8/3J0

x [ul(r - rj)( - CA] SirU t(2 1 - t)zI

[ut(r i - 2k(16)

The integrations of Eq. (12) are lengthy and cannotbe performed in closed form. We present the approxi-mate results in this work for the simplified case of acollimated beam and negligible transverse velocitycomponent and consider points along the beam axis.It is also assumed that A1 2 << 1, and we make use of aquadratic approximation for the structure function.With these simplications, the integrations of Eq. (12)can be carried out, and the result is

X r4(r2,t + T- C ) exp[ (p -r) (U,(O11Ut)u(oW2,t + r)) =2S2 F2

a2z 2 (1 + F2 + 2a2 /p )ir

+ i (q P)2 - 2 (P + VTT-r2)

-2 q2 - P- VTT)I2z I

X H(q1 ,q2 ,r,r 2 ,k1 ,k2 ), (12)

where H is the fourth-order mutual coherence functiongiven by

H = (exp[I(p,r1,kj,t) + '(p,q1 ,kj,t) + **(p + VTTr2,k2,t + r)

+ I*(p + VTr,q2 ,k2,t + T)]). (13)

The change in the aerosol position has very littleaffect on the refractive turbulence encountered alongthe path for small r and will be neglected in evaluatingthe mutual coherence function, which can be writtenas8

H = exp[-(1/2)(D1 2 - D13 + D14 + D23 + D34 - D24 + D34)

+ 2CxU + 2Cx.I. (14)

In Eq. (14) Dij = D(Iri - rjI) is the two-frequency wavestructure function of a point source given byl0

D(p) = 2(p/po)51 3 - 12/0

po = (0.545koC2z)-3/5,

-_ 0.39kCnL'/3z, A12 = Ik1- k2/k 0, (15)

where L0 is the outer scale of turbulence, c is the speedof light, and C2 is the turbulence structure constantassumed uniform along the path. C = C.(dri - rjl) isthe two-frequency covariance of the logamplitude of apoint source given by1l

x[Pt -- ) P4t + 2z)]1/2

X exp{iW - W2)Td - (1- -2)2/2

- (k1 - k2)p2/2z - 2( 2z ) P2/

[1 + ( _ )]2 2iklz - 2ik2ZT} (17)

where F = koa2 /2z is the Fresnel number of the source,Td is the time delay given by

F /1+a Td I'ck, 1 + F2 + 2a2 /p 0/

the coherence bandwidth is given by-2 39CL5/3Z/C2,

(18)

(19)

and Lo is the outer scale size of the turbulence, and c isthe speed of light. It may be noted that the pulsebroadening is inversely proportional to the coherencebandwidth. We will now carry out integrations indi-cated in Eq. (11) over the velocity spectrum and sens-ing volume V and replace S2N/V by the aerosol back-scatter coefficient fl, where N is the total number ofaerosols within the volume V. The result is

r(oow 1 ,w2 ) = J t [ T C) C )]

X expi(Wl - w2)Td - (1 2)2/U2

+ 2i(k1 - k2)Z- 2ik2 vZr - 2ozk2T 2 ]dz. (20)

The integrations over w, and W2 are carried out for aGaussian pulse with frequency spectrum given by

A(w) = (27mo-1/2 exp[(w - o)2/2a,2, (21)

where , is the bandwidth of the pulse. The variablesare changed using 2w'0 = W1 + W2, 5 = W1 - W2-

1 March 1989 / Vol. 28, No. 5 / APPLIED OPTICS 877

The spatiotemporal correlation function is calculat-ed for a Gaussian transmitted power distribution givenby

PT(t) = - exp(-t2/t2),vWtp

where E is the total energy in the pulse, and the pulsewidth is related to bandwidth as tp = 2cr,,j. Theintegration over z is performed designating arbitrarilythe time of peak power by t = 0, and the result is

<< U (o t)U~ t T>= Te

«U8 Uo+) 2zo (Q2 + 4 Y)1

/2

(T2 + t2)1/2

X1 + 2a2/p2 exp[-2 k2r2

2 ~~~~~~~2- (r + 2UZazT/c) 2Te + t2

X 2/4 + (T d+ L4 tp Te

- i(oT + 2kovzT)] (22)

where = Uf-2 + 4Q-2 and z is the range. Equation

(22) is the main result of this paper, which exhibits theeffects of refractive path turbulence on pulse spread-ing and aerosol speckle. The first term in the expo-nential gives the decorrelation effect due to velocityturbulence along the beam axis; the second and thirdterms give the effect of refractive turbulence and ve-locity turbulence on pulse broadening and its contri-bution to the decorrelation of aerosol scattered signals.

IV. Discussion

The spatiotemporal correlation function obtainedby Churnside and Yura3 has been extended to includethe spectral effects of pulsed signals and refractiveturbulence along the propagation path. An approxi-mate result for a collimated beam along the beam axisis presented in Eq. (22) for the combined effects as-suming a quadratic approximation for the structurefunctions. These approximations permit us to bringout explicitly the effect of time delay Td and coherencebandwidth Q on the correlation function.

We consider a long path wind-measuring pulsed li-dar having a pulse duration of 100 ns operating at 9.1-,m wavelength. The path length is assumed as 100 kmthrough a turbulent atmosphere with C2 = 6.4 X 10-15m-2/3 corresponding to a clear day in the lower atmo-

sphere with an outer scale length Lo = 100 m and a, = 1m/s. The beam radius at exp(-2) intensity is taken as1 m, which gives the Fresnel number of the source as F= 3.45. The transverse coherence length for this pathis po = 0.0465 m.

The coherence bandwidth Q and the time delay Tdcorresponding to these parameters are Q = 4.1 X 1011rad/s and Td = 1.65 X 10-14 s. Using these values, weobtain a, = 1.4 X 1010 rad/s and - = 70.9 ns. Thesetime scales are shorter than the correlation time of theorder of 1 us corresponding to moderate turbulence-induced aerosol dephasing, and the effect of pulsebroadening is negligible for this case. When the corre-lation time due to aerosol dephasing approaches thenanosecond range, pulse broadening effects need to beconsidered.

This work is supported by NASA grant NSG 8037.It is a pleasure to acknowledge the discussions withJames W. Bilbro of NASA Marshall Space Flight Cen-ter.

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878 APPLIED OPTICS / Vol. 28, No. 5 / 1 March 1989