Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1287

Electromagnetic theory of range-Doppler imaging inlaser radar. I: Scattering theory

Arden Steinbach

Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02173-9108

Received August 11, 1990; revised manuscript received February 27, 1991; accepted March 8, 1991

An electromagnetic theory of range-Doppler image formation is presented. The new formalism replaces scat-tering from the diffusely reflecting surface of a moving target by scattering from N noninteracting, classicalatoms uniformly distributed over the visible surface of the original target. The total field scattered by the col-lection of moving atoms is then calculated with the use of the Lignard-Wiechert field formalism. It is demon-strated that the new theory, by comparison with currently available scalar theories, gives a much more accuratedescription of the important features of the image-formation process. In particular, the new theory takes intoaccount diffraction of the transmitter beam and movement of the target during illumination and is capable ofdescribing wave-front aberration and the bistatic Doppler shift. It also permits the simulation of the time de-pendence of (Gaussian) speckle in the receiver aperture plane. Because of the simplicity of the scatteringmodel, the theory cannot account for depolarization of the radiation scattered by real surfaces.

1. INTRODUCTION

With a laser radar imaging technique known as range-Doppler imaging, one seeks to form a plot or image thataccurately represents the distribution in both range andradial velocity of the points of the diffusely reflecting sur-face of some distant, moving target. The targets of inter-est are typically either so small (maximum dimension lessthan a few tens of meters) or so distant (range up to thou-sands of kilometers) that resolution in the direction per-pendicular to the radar line of sight is not obtainable. Insuch cases the additional radial-velocity information pro-vided by range-Doppler imaging frequently makes it pos-sible for one to distinguish among various objects, such asballistic missiles, aircraft, and satellites, on the basis ofthe motion of the reflecting surface of the object with re-spect to the object center of mass. One obtains therange-Doppler image by (a) flood illuminating the distanttarget with a pulse, or series of pulses, of frequency-modu-lated laser radiation, (b) heterodyne detecting the scat-tered radiation collected by the receiver aperture, andfinally (c) subjecting the demodulated signal to cross-correlation processing, in which each one-dimensionalcross correlation generates a horizontal line (correspond-ing to constant Doppler shift or radial velocity) in therange-Doppler image.

To describe the formation of a range-Doppler imagemathematically, one currently uses either the elementarytheory' of Woodward or the later, much more sophisti-cated, theory of Kelly and Wishner.2 As the transmitterwavelength shrinks to optical wavelengths, both formula-tions exhibit severe shortcomings, including the follow-ing: First, both theories are inherently scalar in natureand are therefore incapable of taking into account polar-ization, even at an elementary level. Second, both theo-ries ignore diffraction of the transmitter beam andtherefore fail to describe those speckle effects that dependon the radius of curvature of the wave front incident uponthe target.3 Third, neither of the theories available at

present takes into account wave-front aberration, i.e., theapparent change in direction of the radiation receivedfrom a target that has a nonzero component of velocityperpendicular to the nominal lines of sight of either thetransmitter or receiver. At microwave wavelengths,wave-front aberration effects are so small as to be unob-servable. At much smaller wavelengths, however, aberra-tion becomes a severe problem, as is demonstrated inSection 3. Fourth, the scalar theories do not give an ex-pression for the bistatic Doppler shift, a quantity that willassume great importance once laser radar experimentsare performed at large bistatic angles. And finally, cur-rently available theories are unable to simulate dynamicspeckle effects in the receiver aperture plane.

The remainder of this paper is organized as follows: InSection 2 the basic elements of an electromagnetic theoryof range-Doppler imaging as applied to the general bistaticimaging configuration are summarized. The new for-malism replaces scattering from the diffusely reflectingsurface of a moving target by scattering from N noninter-acting, classical atoms uniformly distributed over thevisible surface of the original target. The total field scat-tered by the collection of moving atoms is then calculatedwith the use of the Li6nard-Wiechert field formalism.The emphasis here is on concepts rather than on detailedcalculation. It is shown in Sections 2 and 3 that thesimple electromagnetic theory introduced here is alreadysufficiently powerful to overcome the five shortcomings ofthe scalar theories described above. In all fairness, how-ever, it must be pointed out that the new theory-becauseof the simplicity of the scattering model employed-hassignificant limitations of its own. In particular, the the-ory (a) cannot give correctly the absolute magnitudes ofthe scattered fields measured at the receiver aperture be-cause N cannot be made large enough and (b) cannot prop-erly describe cross-polarization effects in the scatteredradiation because the scattering model employed ignoresmultiple scattering. In Section 3 the power of the newtheory is demonstrated in a calculation of the fields scat-

0740-3232/91/081287-09$05.00 i 1991 Optical Society of America

Arden Steinbach

1288 J. Opt. Soc. Am. A/Vol. 8, No. 8/August 1991

tered by a single-atom target, which is imaged in a simpli-fied bistatic configuration. This example is then followedby a few conclusions in Section 4.

2. ELECTROMAGNETIC THEORY OFRANGE-DOPPLER IMAGINGAs indicated in Section 1, a full theoretical treatment ofrange-Doppler imaging requires a mathematical descrip-tion of (a) the scattering from the moving target, (b) theheterodyne detection of the return signal, and, finally, (c)the cross-correlation processing of the demodulated sig-nal. Since the first of these three steps is by far the mostimportant and also the most complicated, attention in thispaper will be confined to it.

A. Scattering from a Moving Target: GeneralObtaining a mathematical description of the scatteringfrom a diffusely reflecting target of the type considered inrange-Doppler imaging is difficult for a number of rea-sons. In the first place, describing the scattering fromthe rough, but essentially planar, surface of an object thatis stationary with respect to the source of the incident ra-diation is in itself an extremely complicated problem. Inthe case of radar imaging, this stationary scattering prob-lem is complicated further by the fact that many targetshave curved surfaces. In addition, such targets are oftenmoving, sometimes in a complicated fashion. If thetarget motion consists exclusively of uniform translation,with all points of the scattering surface moving with thesame constant speed along parallel straight-line paths,then a simple Lorentz transformation to the rest frame ofthe target reduces the problem to the preceding one ofscattering from a stationary object. Typically, however,the motion of targets such as ballistic missiles consists ofa superposition of translation (<7 km/s), rotation (1-3 Hz), and low-frequency vibration. In this case the vari-ous scattering points on the target surface move withdifferent velocities, and the preceding simplifying methodcannot be used. However, if, for the purpose of approxi-mate calculation, the diffusely reflecting surface of themoving target is replaced by a set of N noninteracting,classical atoms uniformly distributed over the visible sur-face of the original target, then the preceding Lorentztransformation strategy can be applied to each atom indi-vidually. In that case, the total scattered field can bewritten simply as the sum of the fields scattered by the Natoms.

If we now adopt the above model, the scattering problemreduces to that of determining the field scattered by asingle, moving atom. Consider, therefore, the generalbistatic radar configuration illustrated in Fig. 1. Assumethat the exit aperture of the laser transmitter is circularand that it lies centered on the ZT axis in the XTYT plane ofthe transmitter reference frame T as shown. Since thereceiver aperture in the bistatic configuration can be lo-cated anywhere in space, it has been omitted for clarity.At time t = 0, as measured in frame T the leading edge ofa highly collimated pulse of laser radiation of total dura-tion To leaves the exit aperture of the transmitter.Shortly thereafter, the pulse passes over the atom as itmoves along a curved path in space through the transmit-ter diffraction tube. According to the Lorentz force

equation, the oscillating electric field of the incident pulseforces the electrons and protons within the atom to exe-cute small oscillations about the unperturbed path of theatom. However, the oscillating magnetic field of thepulse has essentially no effect because of the extremelysmall peak power densities with which targets are typi-cally illuminated by a laser radar. As the electrons andprotons oscillate, they emit radiation. Because of thelarge mass difference between electron and proton, thecontribution of the protons to the total emitted radiationis negligible.

As pointed out above, the problem of computing the fieldscattered by a moving atom can be greatly simplified byLorentz transformation to the rest frame of the atom,whereupon the problem reduces to the much simpler oneof calculating the field scattered by a stationary atom.How to apply this idea to the present case is not immedi-ately obvious inasmuch as the target atom moves, in gen-eral, with a nonuniform speed along a curved path inspace.

Within the framework of special relativity theory, onetreats such a problem by first approximating the curvedpath between its end points by a series of straight-linesegments, along each of which the atom is assumed tomove with a constant speed equal to the mean speed of theatom along the associated curved segment. Later, aftersolving the scattering problem for each straight-line seg-ment separately and summing the contributions from allsegments, one allows the number of segments to go to in-finity at the same time that the length of each goes tozero. The justification for using the above approximationprocedure is the so-called clock hypothesis5 of special rela-tivity, namely, the assumption that an ideal clock movingnonuniformly through an inertial frame runs at an in-stantaneous rate that depends only on its instantaneousspeed along the path. In particular, the acceleration ofthe clock should have no effect on its rate. The abovehypothesis was beautifully confirmed in muon-lifetimemeasurements in a storage ring at the Centre Europeende Recherches Nuclgares in 1968.6

One is left with the problem of calculating the scatteredelectric and magnetic fields at an arbitrary point inspace and time in the transmitter frame T that are dueto the interaction of the incident laser pulse with thetarget atom while it moves along the straight-line seg-ments (P1 - P2), (P2 P3), etc. (see Fig. 1). The pointsPI, P2, ... , PM are all points along the original curved pathand represent the position of the target atom at timest = t1, t2 ,..., tM, respectively. (Note in this connectionthat the positions of all N atoms used to approximate thescattering surface of the original target are assumedknown as a function of time t.) Consider now the calcula-tion for the line segment (P1 - P2). Imagine a movingreference frame, denoted K12%, with its origin coincidentwith the target atom as it moves from P to P2. Assumealso that frames T and K12' have the standard orientationwith respect to each other; that is, x' is always parallel toXT, Y' to YT, and z' to ZT. To reduce the scattering problemat hand to the simpler one of scattering from a stationaryatom, one ultimately needs to transform the space andtime coordinates and also the incident electric and mag-netic fields of the laser pulse in frame T into correspond-ing quantities in frame K12'. This cannot be done

Arden Steinbach

Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1289

TRANSMITTERDIFFRACTIONTUBE

MOVING FRAME, K'12

(X', Y, Z1 , t ')

I- -o PATH OF ATOM

L = (Pl, 012)

\ Z

/°1012 Y

INTERMEDIATE FRAME, K1 2

(X, Y, Z t)

XT

TRANSMITTERFRAME, T

X

(XTYT ,ZTt)Fig. 1. Scattering of a laser pulse by a single atom moving along a curved path.

directly since, if K12' continues moving along the linejoining P1 with P2, it will never coincide with T. Unfor-tunately, the Lorentz transformation and the field trans-formation equations are derived under the assumptionthat the two relatively moving reference frames coincideexactly at time t = t' = 0.

As a first step toward circumventing this problem, passa line through points P1 and P2 and note the point of inter-section of this line with the XTYT plane of frame T. Callthis point 012 and erect axes in standard orientation; callthe new (intermediate) reference frame K12. Note thatspatial coordinates in the frames T and K12 are related bya simple spatial translation. Now K12 and K12' are refer-ence frames having the property that the two frames docoincide at some time. Unfortunately, it is the wrongone: If one lets L1 be the separation between P1 and 012and V12 be the constant speed of the target atom along theline segment (P1 -> P2), then frames K12 and K12' coincideat t = t + Ll/V12 - At12 > 0-

Fortunately, the equations of electrodynamics (and theequations of physics, in general) are invariant under timetranslation. In the present case, this means that one canshift the T-frame time scale forward by At12 [in which casethe laser transmitter radiates during the time period(-At 1 2, -At12 + To) rather than (0, To)], do the scatteringcalculation, and then at the end shift the time scale in theopposite direction, i.e., backward. As Rindler7 has noted,the above spatial translation and time translation opera-tions are the key elements in the inhomogeneous Lorentztransformation (or Poincare transformation), which en-ables one to connect space and time coordinates and alsoelectric and magnetic fields in two inertial frames thatmove with respect to each other in other than the simpletextbook fashion. The above ideas now enable one to

sketch out a detailed plan for solving the moving scat-terer problem.

B. Calculation of the Scattering from a MovingScatterer: An OutlineTo calculate the scattered electric and magnetic fields atan arbitrary point XT in frame T as a function of time tthat are due to the interaction of the incident laser pulsewith the target atom as it moves along the straight-linepath between P1 and P2 (see Fig. 1), one should proceed asdescribed in Subsections 2.B.1-2.B.8.

1. Vector Diffraction Theory: Incident Fields[E(xT, t), B(xT, t)] in Transmitter Frame TDetermine the incident electric and magnetic fields in thetransmitter frame T by applying the vector diffractionformula8 9 of Franz to the frequency-modulated pulseemitted by the laser radar transmitter during the timeinterval t E (0, To). This calculation cannot be donedirectly since the Franz formula [Eq. (9.156) of Ref. 8],like all diffraction formulas, applies to only a single-frequency component of the time-varying fields. Rather,one must first decompose the electric and magnetic fieldsat the exit aperture of the transmitter-i.e., [E(XTt),

B(XT, t)]z= 0 -into their frequency components with aFourier transform, then use the Franz formula to propa-gate each frequency component from the plane ZT = 0 tosome arbitrary later plane ZT > 0, and, finally, obtain thecomplete expression for the incident fields [E(XT, t),B(XT, t)] with an inverse Fourier transform.'0 In prac-tice, the final results turn out to be simpler than the aboveprocedure might suggest. Diagrammatically, one has

Arden Steinbach

1290 J. Opt. Soc. Am. A/Vol. 8, No. 8/August 1991

[E(XT' F1T=

f FE Franz

[e(XT, w)T=0

where

[E(X, t)JT>o

[e(XT, (OIT>O

e(xT, co) = - f E(XT, t)exp(iwt)dt,

E(XT, t) = f e(xT, co)exp(- iwt)dw,

and an analogous result holds for B(XT, t).Alternatively, if one wishes to ignore diffraction of the

incident pulse entirely, one can obtain the associated inci-dent fields in the transmitter frame simply by making thesubstitution t t - zT/c in the original expressions forthe electric and magnetic fields in the exit aperture (i.e.,at ZT = 0) of the transmitter.

2. Spatial Translation: Incident Fields E1(x, t), Bi(x, t)] inthe Intermediate Frame K12Since coordinates in the two frames T and K12 are simplyrelated by

XT = X + a]YT = y + b XT = x+d (1)ZT = Z J

where d = (a, b, 0) represents the displacement of 012 withrespect to the origin of frame T write the incident fieldsobserved in the intermediate frame K12 immediately as

[Ei(x, t), Bi(x, t)] = [E(x + d, t), B(x + d, t)]. (2)

3. Time Translation: Time-Translated, Incident FieldsIEiA(x, t), B,,I,(x, t)] in Frame K12

Shift the time scale in both frames T and K12 forwardby an interval At,2 = t + Ll/v 2 , as determined previ-ously. Then the transmitter radiates during the periodt E (-At12 ,-At 2 + To) instead of t E (0,To). The time-translated, incident fields in K12 are written as

[EiA(x, t), BjA (x, t)] = [Ei(x, t + At2 ), Bi(x, t + At,2 )]

= [E(x + d,t + At,2),B(x + d,t + A 2)]. (3)

4. Lorentz Transformation and Field TransformationEquations: Time Translated, Incident Fields [EA(x', t'),B, A(x', t')] in Moving Frame K12'Because (a) K12' translates with uniform velocity v2 withrespect to K12 and (b) K12' and K12 coincide at t = t' = 0(as a consequence of the preceding time-translation opera-tion), space and time coordinates and also electric andmagnetic fields in the two frames are simply related toone another. Write the time-translated, incident fields inthe moving frame K12' by invoking the field transforma-tion equations"

Et!,X = yEj,A + C X B + ( -) , 2 V12

Bi = (Bi, - x EA + (1 t) B 2 V12, (4)= Y\BiA c 'IV1 2

2V

wherey= 1/[1 - ()12

Then write the independent variables x and t in the ex-pressions on the right-hand side of Eqs. (4) in terms of x'and t', using the homogeneous Lorentz transformation 2

X=' V12XI + (Y 1 2~V 'V2'

t = (t + x' v12 ) (5)

Note that both Eqs. (4) and (5) can be inverted simply byinterchanging primed and unprimed quantities and let-ting v12 - - V12.

5. Li6nard-Wiechert Field Formalism: Time-Translated,Scattered Fields El,, (x', t), B. (x', t')] in Frame K12 'The incident, time-translated electric field in the movingframe E ',A (x', t') drives oscillation of the electrons (pro-tons ignored for reasons given previously) within thetarget atom fixed to the origin of K12'. For incidentpower densities typical of lasar radar, the maximum am-plitude of oscillation of the electrons within the targetatom is much smaller than a nuclear diameter. Thereforesolve for the motion of the electrons by assuming that theelectrons satisfy the equation of a damped, driven har-monic oscillator with Ea (0, t') as the driving term.{Note that the driving force, since it is pulsed, is nonzeroonly during some time interval (t,,w', tm'). Since this in-terval will generally not coincide exactly with the interval(t1',t2 '] = [-L,/yvo,(-L,/yvo) + (t2 - t)/y]duringwhichthe target atom moves along the straight-line path be-tween P, and P2, radiation will be emitted only during theperiod common to both intervals.} After solving for theposition of the electrons as a function of time t', differen-tiate twice to obtain both the velocity and acceleration.Insert these three quantities into the Lignard-Wiechertfield equations 3 to obtain the time-translated, scatteredfields Es,,'(x', t'), B8,A'(x', t')] as measured by an observerin the moving frame K12'.

6. Inverse Lorentz Transformation and Inverse FieldTransformation Equations: Time-Translated, ScatteredFields [E, a(x, t), Bs, (x, t)] in Frame K12

Apply the inverse of each transformation given in Eqs. (4)and (5) above.

7. Inverse Time Translation: Scattered Fields[E,(x, t), B,(x, t)] in Frame K12

The time-translated, scattered fields [Es,j(x, t), B., a(x, t)]in K12 are those obtained under the assumption that thetransmitter emitted during the period t E (-At 2 , -At, 2+ T). To obtain the scattered fields assuming that thetransmitter emitted during the period t E (0,To), shiftthe time scale backward by At12:

[E,(xt),B(xt)] = [E1,,(xt - At 2),B3,,A(xt - At,2 )].

(6)

8. Inverse Spatial Translation: Scattered Fields[ET(XT, t), B T(xT, t)] in Frame TFrom Subsection 2.B.2 above, XT = x + d. Therefore

Arden Steinbach

Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1291

incident laser pulse from the target atom as it moves alongthe path between P, and P2 are then obtained by followingthe eight steps in Subsection 2.B. For simplicity, let thetransmitted laser pulse have the form of an uncoded sinu-soid of length To; i.e., take the electric and magnetic fieldsin the plane of the exit aperture of the laser transmitterto be

XT X

[E(XT, t)]ZT=0 = E0 sin 6ut

x circ[(XT +YT 2) lrect(t_- T/2 ex,D/2 J = T0 /

[B(XT, t)]-T=O = jZ X [E(XT, t)]zToI

Fig. 2. Scattering of a laser pulse by an atom moving along thestraight-line path between Pi and P2.

write immediately

[ET,S(xTt), BTS(XT, t)] = [ES(XT - d, t), BS(XT - d, t)], (7)

which gives the scattered fields ultimately observed inframe T.

To obtain the scattered electric and magnetic fields ob-served in frame T, as a consequence of the interaction ofthe laser pulse with the target atom as it moves along thestraight-line path between points P2 and P3, one need onlyrepeat the procedure described in Subsections 2.B.1-2.B.8above after making the following replacements:

012 023,

Li L2 = (P2,0 23),

K 12, K12 ' K2 3, K2 3 ',

At12 At2 3 = t2 + L 2 /V23 .

The extension of the calculation to an arbitrary straight-line segment is then straightforward. After summing thescattered fields due to all straight-line segments, one re-peats the calculation for all remaining N - 1 target atomsand adds the individual contributions. At the conclusionof the calculation, one then has the fields scattered by Nmoving target atoms, in which the trajectory of each atomis approximated by a set of straight-line segments.

3. SAMPLE CALCULATIONThe theory sketched in Subsection 2.B will now be appliedto the problem of calculating the electric and magneticfields scattered by a single-atom target, which is imagedin the simplified bistatic configuration illustrated inFig. 2. Assume that the target atom, which sits at theorigin of reference frame K12', moves with constant veloc-ity V12 = -VOZ(vo > 0) along the straight-line path be-tween points Pi and P2. Since both points lie on the ZTaxis of the transmitter frame T, reference frames T andK1 2 coincide, which gives x = XTY = YT, and z = ZT.

Let the emission of the laser pulse of length To begin att = 0. At time t = t, the target atom is at P, at a dis-tance L, from the origin of T, K12. For simplicity, chooset = L1/c so that the front edge of the laser pulse meetsthe atom at point PF.

The electric and magnetic fields at an arbitrary point inspace and time (XT, t) that are due to the scattering of the

(8)

where rect(x) = 1 for -1/2 < x < 1/2 and 0 otherwise,circ(p) = 1 for 0 s p s 1 and 0 otherwise, and D is thediameter of the laser transmitter exit aperture. For sim-plicity, ignore diffraction at the exit aperture entirely.Then, the incident fields in the transmitter frame aregiven by

E(XT, t) = Eo sin[w(t - ZC)]circL D/2 ]

x rect( - ZT/C - T/2 ,X r e c \ T

(9)

Since frames T and K12 coincide, the incident fields in K12are given by

Ei(x, t) = E0 sin[o(t - z/c)]circX D/21]

X ett- zc - To/2 \X Ix rect( /T°.

Bi(x, t) = &z x Ei(x, t) . (10)

Before Lorentz transforming to the moving frame K12'one must shift the time scale in K12 forward by

Li L, Li vAt12 = t + Ll/v2 = + 1 + C)

C uO uO c

(11)

For notational simplicity, set At At,2 . Then, the time-translated, incident fields in K12 are given by (see Subsec-tion 2.B.3)

EiA(x, t) = E0 sin[w(t - zc + At)]circ[ D/2 ]

x rect(t - ZC - To/2 )

Bi,,(x,t) = e_ x Ei,A(x,t).

The homogeneous Lorentz transformation and the fieldtransformation equations [Eqs. (4) and (5)] give the time-

Z. K'12

LASER PULSE

Arden Steinbach

B(XT, t = �z X E(XT, t) -

(12)

1292 J. Opt. Soc. Am. A/Vol. 8, No. 8/August 1991

translated, incident fields in the moving frame K12' as where

(1 + V)E0 sin [y(1 + VO) (t' - + CoAt

X circ [X (x 2 + y'2

)1"2 1

D/2

rect{[Y(1 + '- Z) + - TO/21

Bt! A(x', t') = z E (X, t') (13)

where

V = 1 1 - (o)2]

f(xt') = sin[y(1 + ) (t - ) +wAt

rej[Y(1 + -_ •1) + At - To/2]1

and A1 is a collection of constants giving the amplitude ofthe scattered radiation. Equation (4) is easily inverted togive the time-translated, scattered fields observed in theintermediate frame K12:

( ~~~~~E ' A'V 12E.,A(xt) = ;yEsA -- x B'sA + (1 - ) ,j 2 V12

Y[E, (Xvt2) - - x Bs,(Xc '

According to Subsection 2.A.5, the quantity Ez'(O, t')drives oscillation of the electrons within the target atomfixed to the origin of frame K12'. If one inserts Eq. (11)into the argument of the rect function in Eq. (13), onefinds readily that E,A(O, t') is nonzero only for t E (t'ta,t'fin) = {-L/yvo,(-Li/yvo) + To/[y( + v/c)]}. On theother hand, the target atom moves along the straight-linepath between points P and P2 during the time interval(tl',t2 ') = {-Lil/yvo,(-Ll/yvo) + (t2 - t)/y}. To keep theremainder of this calculation simple, assume that t2' >tfin'. Then radiation will be emitted during the entire in-terval (trta, ti'). The Li6nard-Wiechert field formalismthen gives the time-translated, scattered fields in K1 2' as

E ,A(x' t') = Alf(x' t')

[(y,)2 + (Z)2]e - xy - x'z'ex,

[(X)2 + (y')2 + () 2]3 /2

BS A(x', t') = Alf(x t) (X)2 + (y')2 + (Z)2 ] (14)

B8,,(xt) = Y(Bs A + x Es') + (1B' V12

- '}') 2 V 1 2V12

= y[Bs',A(xt') + - x Es,A(Xt),c

(15)

where the terms involving (1 - y) have been dropped be-cause (1 - ) is of order ( 12/c)2. To complete the calcula-tion, one must write the variables x, t' on the right-hand side of expressions (15) in terms of x, t. WithV12 = -voj,, the expressions in Eq. (5) are easily invertedto give

x = ,

Y = Y,

Z' = 'Y( + ot),

t= Y(t + voz) (16)

A simple calculation then gives, correct to order (voic),

[X2+y2 + 2(Z + Vt) 2]3 2 Y+ y2(z + t)2

X e - Xyey - xy(z + Vot) z)I

BsA(x, t) = yAg(x, t) t)2]3 2 ( - XY + (Z +

X - y[X 2 + y2 + 21t2] + /2'X -_y[X2 + y + Y2(Z +Vot)2]1e2j, I

Voy(z + ot)[x 2 + y 2 + 2 (Z + Vot)2 ] 12e

Vot)[X2 + y2 + 2(Z + ot)2]112 _ °[y2 + 2 (Z + Vt)2]lcJ

(17)

where

g(x t) = f[(xt') - (,t)] = sin(Y(1 + ){ Y (t+ ) - [X2 + 2 + y2(Z + Vot)21/2} + CoAt

( c+ VoZ -- 1 [X2 + y2 + 2(z + Vt)2]1I2} + At-T 0/2)

X rect

(18)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Arden Steinbach

x rect (18)

Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1293

0 - Z 0 Pl AP

0 .... Li 1 )oa

YT Li

OB Observer

Fig. 3. Effects of wave-front aberration.

The time-translated, scattered fields [E,,A(x, t),B,,A(x, t)] in K12 are those obtained under the assumption

tion in Eq. (21) reduces to

( _ CO( + At)

= (1 + )({(t - At) c

x [L12 + y2V,2(t - At)2]1/2 + )At. (22)

Since the above phase is an exact linear function of time tonly for observation points lying on the ZT axis, it is usefulto compute the instantaneous frequency

Finst at 2(1 + - () - At) + Y2( )(t-

that the transmitter emitted during the period t E(-At, -At + To). To obtain the scattered fields, assumingthat the transmitter emitted during the period t E (0, To),one need only shift the time scale backward by At, yielding

[E5(x, t), B 8(x, t)] = [EsA(x, t - At), B,,A(x, t - At)]. (19)

Since the two coordinate systems K12 and T are coinci-dent, the inverse spatial translation required at this point(see Subsection 2.B.8) is trivial. The scattered fields ulti-mately observed in frame T are therefore given by

[ET,S(XT, t), BTs(XT, t)] = [ESA(XT, t - At), BSI(XT, t - At)].(20)

These fields are those at an arbitrary point XT in frame Tin response to the collision of the incident, pulsed field ofEqs. (8) with a classical atom moving along the ZT axis be-tween points PI and P2. Two important characteristics ofthe scattered fields should be noted at this point. First,the scattered fields have an amplitude modulation that issmall for points of observation XT, which are sufficientlyfar away from the exit aperture of the transmitter. Andsecond, the rapidly varying part of each field [see argu-ment of sine function in Eq. (18)] is exactly harmonic onlyfor observation points lying on the ZT axis.

A. Bistatic Doppler ShiftBoth ET,S(XT, t) and BTs(XT, t) contain the rapidly varyingfunction

g(XT,t - At) = sin[w(ir + At)]rect + T°/ ), (21)

where

T = (1 + )((t - At + c )

{2 + YT2 + Y2[ZT + Vo(t - At)]2}1I2)

It follows that the frequency of the scattered wave re-ceived by an observer in frame T is shifted by an amountthat depends on both the velocity of the scatterer and theobserver position. For simplicity, let the observationpoint lie on the YT axis (see Fig. 3) and have coordinatesXT = (0, YT = Li, 0). Then the argument of the sine func-

Thus the instantaneous frequency of the scattered wavereceived at an observation point lying on the YT axis isnearly a linear function of time; i.e., the received signalcontains a linear frequency modulation. Inasmuch asthis unwanted frequency modulation poses a potentialproblem in high-resolution laser radar, it is important todetermine its approximate magnitude. The quantity ofinterest is

Aw inst - 'Winst(tend) - inst(tbeg), (24)

where tbeg and tend are the times at which reception of thescattered pulse at point XT = (0, Li, 0) in frame T beginsand ends, respectively. A straightforward calculationfrom the argument of the rect function in Eq. (21) gives

tbeg = (2 + 1)-,

tend =To

C 1 + (Vo/C)

(25)

Taking L, = 3 x 106 m, To = 10-5 s, v0 = 7 x 103 m/s,and 'c = 27r x 3 x 10'4 rad/s and then inserting the re-sults of Eq. (25) into Eqs. (23) and (24), one obtains

AVinst = 2w -115 Hz,

which is a negligible frequency shift for a laser radar.Hence one makes an error that is no larger in order of

magnitude than (V0/C)2 by assuming the received pulse tohave a constant angular frequency Wrec given approxi-mately by

Wrec ̀ inst(tbeg)

= W 11 + " Cos( 4]/( c) (26)

The above result should be compared with the Landau-Lifshitz result' for the frequency a' of the light scatteredby a moving charge:

1 - (v/c)cos 11 - (v/c)cos'0J

where 0 and 0' are the angles made by the incident and

(23)

Arden Steinbach

(L,)2 + Li VO To 2 12

C C C 1 + (Vo/c)

1294 J. Opt. Soc. Am. A/Vol. 8, No. 8/August 1991

scattered waves, respectively, with the direction of mo-tion (v is the velocity of the charge). In the notation ofthis paper, the Landau-Lifshitz result reduces to

(27)

which agrees with expression (26) to order (vo/c). How-ever, Landau and Lifshitz are incorrect in implying thattheir result is exact. In fact, the exact results are con-tained in Eqs. (20)-(23) above. Equation (23), in particu-lar, enables one to calculate the received instantaneousfrequency arbitrarily accurately for the imaging configu-ration illustrated in Fig. 3.

B. Wave-Front AberrationIt will now be demonstrated, for the bistatic imaging con-figuration illustrated in Fig. 3, that the scattered wavefront received by an observer at point OB is aberrated inthe sense that it appears to come from the direction ofpoint AP rather than from points lying along the scat-terer's path between points PI and P2. Wave-front aberra-tion is an old phenomenon, having been discovered byJames Bradley in 17255 as he sought an explanation forthe small seasonal change in the positions of the stars.After lengthy consideration of his experimental data,Bradley concluded that the seasonal change is only appar-ent and that the angle 0 between a given star's actual andapparent positions is given by

0. = V', (28)C

where v is the component of the Earth's velocity perpen-dicular to the line of sight to the star and c is the speed oflight. Clearly, if c were infinite there would be no aberra-tion. In bistatic radar the aberration problem is morecomplicated than in the foregoing case of stellar aberra-tion because the path from transmitter to receiver in-volves propagation along two different directions withrespect to the velocity vector of the target.

Consider now the bistatic imaging configuration ofFig. 3. At time t = 0, the leading edge of a pulse of laserradiation of total duration To [Eqs. (8) and (9)] leavesthe transmitter aperture and moves outward along the+ZT axis. At time t = L/c, the leading edge ofthe pulse collides at point P with an atom moving in the- ZT direction with uniform speed v. The atom there-upon begins to radiate and continues to do so until timet = L/c + To/(l + v0/c) [deducible from Eq. (25)].During this time interval, the atom moves a distanced = voTo/(1 + vo/c). For v = 7 x 103 m/s and To =10- s,d 7 x 10-2 m. The direction of arrival of thescattered wave front at point OB is obtained by calculat-ing the direction of the Poynting vector

cS = ETs(XT t) x BT,S(XT, t) (29)

during the period of arrival of the pulse at OB. SettingXT = (,L ,0) and t = tbeg = (V2 + 1)Li/c [Eq. (25)]and inserting Eq. (20) into Eq. (29), one finds that thePoynting vector points along the line AP -* OB, as shown

in Fig. 3. The aberration angle shown satisfies

tan(4+ 0a 1 + V 2, (30)

which is easily solved to yield

2 (c)(31)

For v = 7 x 103 m/s, one finds that Oa 17 X 10-6 rad.If, in addition, one sets LI = 3 x 106 m in Fig. 3, one read-ily finds that points AP and PI are separated by 102 m.Thus the scattered radiation received at point OB doesnot appear to come from those points along the atom'spath at which radiation was actually emitted, i.e., frompoints between PI and a point 7 x 10-2 m to the left of P.Rather, the scattered pulse appears to come from a pointAP, which lies a full 102 m behind point P, in a "projectedprecursor position"'6 of the radiating atom.

The above results have serious consequences for short-wavelength laser radars for the following reason. By theSiegman antenna theorem,'7 a laser radar receiver oper-ated in the heterodyne detection mode will efficiently de-tect only radiation whose plane-wave components havepropagation vectors lying within the receiver receptioncone; that is, within a cone of half-angle r A/Dr aboutthe receiver line of sight, where Dr is the diameter of thereceiver aperture. The angle 0 r, which is normally re-ferred to as the receiver beamwidth, therefore representsthe angular resolution of the receiver. If a CO2 laserradar receiver with beamwidth Or 10' rad (A = 11 gtm,Dr = 1 m) is aimed at point PI in Fig. 3, the return pulsewill be received, but at relatively low efficiency, sincethe return signal is aberrated by 1.7 x 10-5 rad. If, how-ever, a solid-state laser radar receiver with beamwidthOr 10-6 rad (A = 1 m, Dr = 1 m) is aimed at the samepoint in receive mode, no return pulse will be seen at allsince the aberration angle exceeds the beamwidth of thereceiver by a factor of 17. Note that the aberration prob-lem afflicts laser radars operated in both the monostaticand bistatic modes, although the problem is much morecomplicated in the latter case.

4. CONCLUSIONSThis paper has (a) described in detail a method for calcu-lating the scattering of a pulse of electromagnetic radia-tion from a classical atom moving along a curved path inspace and also (b) illustrated the use of the theory by itsapplication to a specific simple case. These theoreticalresults form the most important part of a new theory ofrange-Doppler imaging, which is based on rigorous elec-tromagnetic theory. As demonstrated in Sections 2and 3, the new theory takes into account diffraction of thetransmitter beam, movement of the target during illumi-nation, the bistatic Doppler shift, and wave-front aberra-tion. Although it has not been explicitly demonstrated inthis paper, the formalism described here should also en-able one to simulate the time dependence of (Gaussian)speckle in the receiver aperture plane because of the highaccuracy with which the time dependence of the receivedfields can be computed. The number N of scatterers re-

Arden Steinbach

Wrec = ( 1 VO) 1 - V0 Cos 7rc C (4) 1

Vol. 8, No. 8/August 1991/J. Opt. Soc. Am. A 1295

quired for an adequate simulation of the range-Dopplerimaging process is unknown at present.

ACKNOWLEDGMENTSI thank Kent Edwards and Charles Meins for their stimu-lating comments and suggestions and especially for theirunfailing support during the course of this research. Ialso thank Susan Clark for her careful and precise prepa-ration of the manuscript and Richard Bourbeau for hisfine work in preparing the figures. This research wassponsored by the U.S. Department of the Army.

REFERENCES1. G. W Deley, "Waveform design," in Radar Handbook, M. I.

Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 3.2. E. J. Kelly and R. P. Wishner, "Matched-filter theory for

high-velocity, accelerating targets," IEEE Trans. Mil. Elec-tron. MIL-9, 56-69 (1965).

3. T. Asakura and N. Takai, "Dynamic laser speckles and theirapplication to velocity measurements of the diffuse object,"Appl. Phys. 25, 179-194 (1981).

4. M. Nieto-Vesperinas, "Depolarization of electromagneticwaves scattered from slightly rough random surfaces: a

study by means of the extinction theorem," J. Opt. Soc. Am.72, 539-547 (1982); J. A. Ogilvy, "Wave scattering fromrough surfaces," Rep. Prog. Phys. 50, 1553-1608 (1987).

5. W Rindler, Essential Relativity: Special, General, and Cos-mological, 2nd ed. (Springer-Verlag, New York, 1977),Sec. 2.13.

6. D. H. Perkins, Introduction to High Energy Physics (Ad-dison-Wesley, New York, 1972), p. 192.

7. Ref. 5, Sec. 2.6.8. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley,

New York, 1975), Sec. 9.12.9. C. -T. Tai, "Kirchhoff theory: scalar, vector, or dyadic?"

IEEE Trans. Antennas Propag. AP-20, 114-115 (1972).10. M. Born and E. Wolf, Principles of Optics, 4th ed. (Perga-

mon, New York, 1970), Sec. 8.3.1.11. Ref. 8, Sec. 11.10.12. Ref. 8, Sec. 11.3.13. Ref. 8, Eqs. (14.13) and (14.14).14. L. D. Landau and E. M. Lifshitz, The Classical Theory of

Fields, 4th ed. (Pergamon, New York, 1975), p. 220.15. C. Kittel, W D. Knight, and M. A. Ruderman, Mechanics-

Berkley Physics Course (McGraw-Hill, New York, 1965),Vol. 1, Chap. 10.

16. J. Van Bladel, Relativity and Engineering (Springer-Verlag,New York, 1984), Sec. 3.11.

17. A. E. Siegman, "The antenna properties of optical hetero-dyne receivers," Proc. IEEE 54, 1350-1356 (1966).

Arden Steinbach