Field properties of multiple coherently combined lasers

Field properties of multiple coherently combined lasers

Hal E. Hagemeier and Stanley R. Robinson

The coherent combination of several single-mode lasers can produce a field similar to that of a mode-lockedlaser but with more flexibility. The field for a quasi-monochromatic wave is considered a complex coher-ence-separable random process. The ensemble mean and covariance are determined for the case of a tempo-rally stabilized amplitude and a temporal phase that, with appropriate assumptions, is a stationary, Gauss-ian random process. Mean fields are used throughout as the signals of interest. The Huygens-Fresnel prin-ciple is used to investigate the field properties in the Fraunhofer region for two cases.. The first case is Nbeams superimposed with optical axes coincident. Range measurement precision is found to be proportion-al to 1/(N 3Af)'/ 2 , where Afis the frequency difference between adjacent lasers. Velocity measurement pre-cision is found to be proportional to (NAf3)"/ 2 . The second case is N beams in a linear array. The far-fieldresult is a scanning beam that in certain cases can be steered. An array of about 1000 lasers is needed forreasonably low sidelobes.

1. Introduction

An inhomogeneously broadened laser typically os-cillates at several longitudinal modes, which are atfrequencies separated by an amount that is inverselyproportional to the length of the cavity. Generally,these modes have random phases with respect to eachother, and the resultant total laser output power isnearly constant and proportional to the number ofmodes. If, however, the individual modes maintain aconstant relative phase, the laser output power is peri-odic with a period proportional to the cavity length, andthe output peak power is now proportional to the squareof the number of modes. The process of aligning therelative phase of the individual modes is known asmode-locking and is often done in applications thatrequire very short pulses and high peak powers.Mode-locking can also be accomplished for transversemodes; this produces a spatial periodicity, i.e., a scan-ning beam.1

Unfortunately, since the pulse properties of themode-locked laser are determined essentially by thecavity length, variations in pulse format are not easilyaccomplished. To obtain more flexibility, it is rea-sonable to consider combining the outputs of severalsingle-mode lasers. In this case, the phase, frequency,

Both authors are at Wright-Patterson Air Force Base, Ohio 45433.H. E. Hagemeier is with Aeronautical Systems Division (AFSC), andS. R. Robinson is with Air Force Institute of Technology (ATC), De-partment of Electrical Engineering.

Received 29 March 1978.0003-6935/79/030270-11$00.50/0.© 1979 Optical Society of America.

and amplitude of each laser can be chosen to synthesizedesirable combined beam properties. If the frequencydifference between any two lasers is an integer multipleof a constant frequency offset, and if the fields arecombined coherently, the result duplicates the outputof a mode-locked laser. Now, however, the frequencyoffset can be varied over a much larger range than pos-sible with a mode-locked laser so that the pulse prop-erties of the combined beam can be chosen to meet morediverse requirements. The number of beams that arecombined can also be varied to change the output pulsewidth and amplitude. In addition, the lasers could bespatially separated, i.e., placed in an array configuration,so that spatial properties analogous to the temporalproperties can be duplicated.

Analysis of the coherent combination problem hasbeen largely experimental and based only on deter-ministic models of the individual laser outputs.2-5

There has also been some speculation on the applica-bility of a phased array of lasers.6 Although these arenecessary first steps, a more complete analytical ap-proach, which considers the instabilities of the variouslasers and examines the resulting electromagnetic fielddistribution in the far field, is needed. This paper isbased on a recent study 7 which addressed the problem;the results are presented in three major sections.

Section II presents a statistical model of the indi-vidual laser modes. For simplicity, scalar fields areused, and the field descriptions are classical.8-10 Themode amplitudes are considered deterministic; the laserinstabilities are due to random phase fluctuations ofeach beam. The desired result is a first and secondmoment description of the field as a random process.The ensemble mean and covariance of the field arepresented as functions of time and space. The mean

270 APPLIED OPTICS / Vol. 8, No. 3 / I February 1979

of the field is used as the signal of interest. The co-variance is interpreted as a measure of how closely theactual field sample functions will approach the meanand is similar to the mutual coherence functions de-scribed by other authors.11-'5 Far-field results aredetermined using the Huygens-Fresnel integral, and theduality of time and space variations is shown as a gen-eral case when several beams are combined.

Section III considers two special cases of the gener-alized field results: the coaxial case, and the side-by-side array case. The means and covariances are de-termined for both cases, and some characteristics of thesignal field are examined. Although most of the resultsare generally applicable to any single-mode laser, we useproperties of the waveguide carbon dioxide (CO)2 laserfor illustration.16 For the coaxial case, coherent (het-erodyne) detection and a matched filter receiver areassumed, and the ambiguity function of the combinedfield mean is determined. This shows explicitly thedependence of the potential measurement precision ofthe signal as a function of the number of lasers, thefrequency differences between lasers, and the relativephases of the lasers. For the array case, the scanningcharacteristics of a periodic array are considered.Aperiodic arrays are studied in an attempt to reducehigh sidelobes.

Section IV contains a discussion of alignment errorsand their effect on the far field.

11. Field Models

We use the classical representation for the opticalfields throughout this paper. Typically, a monochro-matic linearly polarized electric or magnetic field maybe written in complex form as

u(x,y,z,t) = Re[U(x,y,z,t) exp(-j27rfot)],

where Re means the real part of, and U(x,y,z,t) is thecomplex envelope of u(x,y,z,t).

U(x,y,z,t) = A(x,y,z,t) expLk(x,y,z,t)],

where A is the amplitude of the wave, 0 is the phase, andx,y,z, and t are the space and time coordinates. We willassume the following: (a) U is a complex random pro-cess; (b) z is the direction of propagation, and the fieldUO will be considered at a particular plane z = zo; (c) theamplitude A is spatially coherent and separable into adeterministic (and assumed constant'7-2l) spatial partand a random temporal part22 23; (d) the phase 0 is alsoseparable; and (e) the spatial part of the field U is theTEMoo mode, i.e., the standard Gaussian sphericalwave. Therefore,

Uo(x,y,t) = AUo,(x,y) expko(t)]

= A /-exp [-i A(X2 + y2)J} explj[0 + ¢(t)]}, (1)

where U0 is the field at some initial plane in front of thelaser, U (xy) is the spatial part of the Gaussianspherical wave (normalized for unit power flow), 0 hasbeen added as an arbitrary phase angle that is control-lable (as is A), w is the spot size of the Gaussian ampli-tude distribution, and q is the complex radius of cur-vature defined by

COMBINER

FIELD TO BEOBSERVED

Fig. 1. Coherent (heterodyne) detection of a fie]

1 1 .

4 R rw2

SIGNAL

[d.

(2)

for R the radius of curvature of the spherical wave phasefront and X the wavelength.24

There is an issue here regarding the stationarity of0(t). For a free-running laser, the random phase is aWiener-Levy process, i.e., it is zero mean, Gaussian, andnonstationary. 2 5 26 However, if we require that the fieldbe observed with an optical heterodyne detector,27 thereceiver is only observing the difference between thefield's phase 0(t) and the local oscillator phase kr(t) (seeFig. 1). For convenience, the local oscillator is assumedto have unit amplitude. Now, 0r(t) can be made totrack 0(t), so that the difference phase can be consid-ered a stationary Gaussian random process as long asthere is a high enough SNR in the phase-locked trackingloop. 28

Now, using the well known Huygens-Fresnel inte-gral,8 29 we can compute the field U1 a distance z 1 downthe optical axis from the output field U0 of the laser.For simplicity, we impose the Fraunhofer condition andobtain

U,(x,,yit) = P(x1,y,)51XY[Uo(xyt)], (3)

where P(x1 ,y,) is the set of phase factors from the Hu-ygens-Fresnel integral given by

P(xlyl) = exp(kz1 ) exp j2 (X2 + y2) (4)P jy, Xz 1 1i 2z, 1

and Wxy [Uo(x,y,t)] is the 2-D spatial Fourier transformof UO on x and y evaluated at the spatial frequencies &= xi/(Xzi) andfy = yl/(XZ1 ).8 Aperture functions arenot considered since the diffraction effects of a con-servative aperture design are completely negligible.30

The far field of a single beam becomes

U,(xj,yj,t) = P(xj,y,)Uj,(x,,y)A exp ti[O + 0(t)]},

where

Ui,(xiyl) = 5IXY[Uos(xy)]

( xq1 m j q (2 2)ep XZ2 +,

(5)

(6)

For multiple beams with identical spatial distributionscombined coaxially (see Fig. 2), we add an index on A,0, and 4, and a frequency offset term fi to the time de-pendent part of each beam to obtain

1 February 1979 / Vol. 18, No. 3 / APPLIED OPTICS 271

I I BEAM COMBINEDl COMBINING - OUTPUT BEAM

a i l | ., OPTICS UIx.y.I)

INDIVIDUAL

OUTPUT BEAMS

Fig. 2. Multiple beams combined coaxially.

+ 2kXL + dikyLfikAZ fik XZ

Using known results for the characteristic functionof a Gaussian random variables we have, for the singlebeam of Eq (5),

E[U1(x1,y1,t)J = PU,,AE[expj[0 + 0(t)]}]

= PUIA exp(jO)MO,(v)I1=I

= APUI, exp (I 2-)

y

n.IEII

I1 _LD bE 132 X

d1D d12

L Do - D0D-2

0 1 2

w1

w -

Fig. 3. Front view of laser array.

N-iU,(x,,y1,t) = PUs Ai expU[2rfit + O + 4Pi(t)JJ

i=o

where E[-] means to compute the ensemble average ofthe quantity in brackets, M,,(v) is the characteristicfunction of 0(t), defined to be E~exp[jvq0(t)]1, and U.2 isthe variance of the random process 0(t). Since 0(t) isconsidered stationary, U

2 is a constant.For the multiple coaxial beams, Eq. (7), we have

E[UI(x,,y,,t)] = PUI, A; Ai exp [(2irfit + Oi) - I , (11)

where a2 is the variance of the phase i (t) of the ithbeam, and the i (t) are assumed independent.

The covariance of the field in Eq. (7) can be writ-ten

Cov(U,Ul) = E[U1(xy,,t)Ul(xyt')]- E[Ul(xlylt)]E[Uj(x,',ylt')]

= PP*UlJUls E Ai

X exp(j2irfr - a)1exp[Rbi(r)] - 1 (12)

where P* and U*, are the complex conjugates of P andU1, and are functions of xl and y', and R5i(T) is thecorrelation function of qi (t) and depends only on r =t - t'.

(7)

The number of lasers N that are used is only limited bythe total bandwidth,

N-1E fi,

i=O

since the total bandwidth must be less than or equal tothe gain linewidth of the lasers being used.

It may not be possible or desirable for physical oreconomic reasons to combine the individual laser beamscoaxially. The lasers may be aligned side by side in a1-D or 2-D array similar to the antennas in a phasedarray radar.6 Figure 3 illustrates a rectangular arraywhere Dik is the (center-to-center) spacing in the x di-rection, dik is the spacing in the y direction, and Dik anddik are longer than the spot size w for all i and k. Thefield can now be written

n-I .- IU,(x,,y,,t) = PUl, E E Aiktexpj[27rfikt, + ik + Iik(t)It, (8)

i=O k=O

where mn = N and t,} are geometry dependent timedelays. The lasers could be arranged in a circle, spiral,or any other geometrical shape, but for simplicity thearray is considered rectangular. Equation (8) is thenessentially identical to Eq. (7) except for t, which isdependent on time and space,

isThe mean of the far field for the array case, Eq. (8),

[ ~~~~~2 1

E[Ui(x,,y,,t)J = PUI, E A: Aik exp (2

7fikt + Oik)- 2 ] (13)k 1 2 where t is defined in Eq. (9). The covariance for thiscase becomes

Cov(U1 ,U'1) = PP*UsU;s E E A 2 expU2rfik(t, - t) - ao]ikX exp[R,,,(r)J - 1, (14)

where t' is a function oft', xl, and yl. For the general2-D arrays, Eqs. (13) and (14) cannot be evaluated an-alytically without numerous assumptions. We seek todo that in a later section, however, before leavingmodeling, we can make one further generalization ofEqs. (8) and (13).

Suppose that, due to misalignment, the array posi-tions Dik and dk are random variables in addition to theinstabilities kik (t). If Dik and dik are statistically in-dependent of each other and of 0ik(t), Eq. (13) be-comes

E[U1(x1,y1,t) = PUI,, E E Aik exp(j24ikt + Oik)i k

X Mdih(2r Mdi, 27r t ) Moik(l). (15)

It is reasonable to assume that each laser is located in

272 APPLIED OPTICS / Vol. 18, No. 3 / 1 February 1979

NSINGLEMODE

LASERS

(9)

(10)

.[] F1

the vicinity of a desired point described by Dik and dik-Thus, we can interpret the D's and d's as means of therandom positions. If we further suppose that the pos-itional errors in x and y are each Gaussian and identi-cally distributed with variances a 2 and T2, Eq. (15)becomes

E[Ui(x,,y,,t)] = PU1, exp { /2 [At)_a2 + -)2a +

22r~ 2lr~ikX 2irdikX E Aik exp 2arikt + X + + OikI

i k 1, Xz, Xz, I

(16)

This is the desired general result. If i, k, aD, d, Dik,dik, and fik are identically zero, Eq. (16) represents thesingle-beam mean of Eq. (10). If aD, aod, Dik, and dik arezero, the result is the coaxial case similar to Eq. (11) withthe double sum rewritten as a single sum. If aD and adare zero, the result is the 2-D array case as in Eq. (13).Finally, if i = 0, the result is a 1-D linear array. Equa-tion (16) will be the starting point for each applicationconsidered in the following sections.

Ill. Systems ApplicationsWe now examine two special cases of Eq. (16) in more

detail: the coaxial case and the array case. Equation(16) will also be useful in examining the effects ofalignment errors on the performance of the two typesof systems.

A. Coaxial Case: Signal Detection and EstimationResults

As noted earlier, Eq. (11) is a special case of Eq. (16).Since the Ai are controllable, they are assumed to be thesame for convenience. If the frequencies between thelasers are the same (as in a mode-locked laser), fi = if,where Af is the constant frequency spacing. A similarassumption can be made for Oi, but the results are notchanged significantly, so the 0i are considered zero.Now we have

E[U1(x1,y1,t)] = PU1,A exp - - + j - 2i~tr2 2 J

sinNirAftX , (17)sisAft

where 2= a The (sinNx)/(sinx) term (the typical

result of a mode-locked laser3 2 ) is a periodic pulse trainin time, as shown in Fig. 4. The pulses have a null-to-null pulse width of

tp = 2(NAf) (18)

and a period of

Tp= 1/(Af). (19)

For the waveguide CO2 laser (whose gain line width isabout 700 MHz) NAf must be less than or equal to 700MHz. Therefore, the minimum achievable pulse width,tp, is about 3 nsec5 . Tp depends on Af which is limitedby the total number of lasers used (for fixed NAf). Thepeak amplitude of the (sin Nx )/(sin x) function is justN as expected for coherent superposition of N sources.The apparent frequency of the field is shifted from fo

N= 8

27rN

sin (wN/2)sin (/2) (N=8)

27rN

27r (I

Fig. 4. Plot of sin[N(w/2)]/sin (/2) for N = 8, w = 27rAft.

by ((N-1)/2)Af. The intensity of the field in Eq. (11)is just

E[Ii] = E[U1(x,,y,,t)Ul(x,,y,,t)]

2(a2 +b2 ) 2rb 2

((1 - exp(-a 2))N + exp(-U2) s . Aft

where from Eq. (2) q has the form a + jb with

zl(7rw2 )2

(7rW2)2 + (Az,)2

(20)

b = irX(WZI)2 (rW

2)

2+ (AZ 1)

2

and, in the far field case, R -'z 1. Note that in Eq. (20)as a2

- co, exp(-a 2) - 0 and E[I1] becomes a constantwith respect to time. This is equivalent to the result ofa multimode laser with no mode locking. If the phasecontrol is very good and 2 - 0, then exp(-a 2 ) - 1 andE [I] is a pulse train, i.e., E [I,] - IE [Ui] 12 for a-2 - 0.This is equivalent to a perfectly mode-locked laseroutput.

The random phase fluctuations oi(t) simply causethe amplitude of the field in Eq. (17) to be reduced byexp[- (a2/2)]. From this relation, the degree of controlto the lasers can be calculated. If the desired maximumattenuation is i7,

a = (-2 1n?7)1/2. (21)

For example, if w7 is desired to be no larger than 1 dB(about 0.80), a must be about 0.67 rad. Similarly, forphase control on the order of 1 rad, w7 is about 50%. Themean power flow through the z = z plane (with com-plex envelopes suitably normalized) for t = 0 (pulsepeak) and A = 1 becomes

E[Power] = N[ - exp(-a2 )] + N2 exp(- U2). (22)

Thus as 2- , E [Power] approaches N (incoherent

superposition), while as 2 - 0, E [power] approachesN 2 (coherent superposition).

It is also instructive to examine the behavior of thecovariance of U1 . For Eli = a,

Cov(U,,Ul) exp(-a2)exp[R,&r)] - 1. (23)

Since R, (0) A 2 , 33,34 then at T = 0, the covariance of

U, becomes the variance a 2 of U1 where

a 1 - exp(- 2).

By the Chebyshev inequality,3 5 we obtain

(24)


Pr [I Il,-E[LJI] >e]< -a 1- exp(-a 2 )

Now as 2- 0, Pr[ U, - E[U1] I Ž ] approaches zero,

and U, is equal to its mean with probability one. As Tapproaches infinity, the two random variables 0(t) and0(t + -) become more and more uncorrelated. Typi-cally Ro(r) = 0 for all r > T, where we define T, as thecoherence time of the process 0(t). From Eq. (23), T >T, gives

Cov(U,,U) = 0,

i.e., the coherence time of the field U, is roughly thecoherence time of the phase s0(t).

One further comment on Eq. (11) concerns the beamspot size. Since the individual beams are added coax-ially, the spot size of the total beam propagates as thespot size of any single beam. Therefore, the beam spotsize w at a distance z is

1VI = (Az)/(ru)), (25)

where w is the spot size at z = z. The equality holdswhenever the Fraunhofer condition is met.

The temporal pulse train of Eq. (17) suggests rangingapplications, and, for reasonable assumptions, the re-sults are very much like the microwave radar case. Ifwe consider heterodyne detection with a strong localoscillator, use matched filter postdetection process-ing3 6-3 8 and ignore certain physical phenomena such aspath losses and field alignment that affect any signal inthe same manner, then we can determine the perfor-mance of our signal of interest. Since the detector doesthe spatial integration of the field, we will consider anapproximation over a single period of Eq. (17) to be thesignal of interest,

s(t) A exp (_ al sinN~rAft2 irAft

(26)

The potential performance can be determined from themean log likelihood function (or ambiguity function),x(r,v), which, for our case, is the cross correlation of s (t)with a time and frequency v shifted version of it-self. 3 9-41

IX(rv)I Ix(o,o)I=A' N

S ~~~~~~~~i '\r.F

- io

Fig. 5. Ambiguity function of sin(NirAft)brAft.

x(rv) = f- s(t)s*(t - ) exp(j27rt)dt.

Using Eq. (26) we obtain 4l

A 2 ( lvIIAf2 NA/I

sinN~rAf r - II)IX]NT~T( -N-) I

'r(1 - I) I

= 0, elsewhere,

IvI NAf (27)

which is sketched in Fig. 5. If multiple pulses are con-sidered, the resulting ambiguity function is just a sumof terms with the form of Eq. (27) shifted on the delayaxis by intervals of 1/(Af) and multiplied by a weightingterm. However, as long as the round trip time (delay)of a pulse is less than the pulse rate, there is no ambi-guity as to which peak on the axis is involved. Thefine structure in Doppler affects the resolution of targetswith similar Doppler velocities, but it does not affect themeasurement precision.40

It is now possible to obtain Cramer-Rao lower boundsfor the variance of the estimates ofT and v. The actualvariances approach these bounds with near equalitywhen there is a sufficient SNR.4 0 4

1 If N0/2 is the powerspectral density of the white Gaussian measurementnoise, and the temporal signal energy E is determinedfrom Eq. (17) to be

E = A 2 exp(- U2

) [N/(Af)], (28)

the range precision aR and range rate precision aR fora finite approximation to Eq. (26) can be shown to be

c(3No)1/ 2

UfR >I[8 r 2A2 exp(-a2)N 3 Af]1/2

A(NoNAf3)1/2

[16A2 exp(-U2)]1/ 2

(29)

(30)

where c is the speed of light. If the total bandwidthNA/ fills the available 700 MHz (i.e., is constant), oR 1/N and a Af. Thus as N increases, aR decreases.Since NAf is constant, Af decreases as N increases,which implies that aq decreases as well. Therefore, byusing more lasers, the range and range rate precision cansimultaneously be made as accurate as desired. It isalso possible to improve oR and a at the same time byincreasing the number of pulses that are collected. Thisapproach increases the effective signal and is only lim-ited by the number of pulses that can be received andby the assumption that the target range and Doppler donot change appreciably during the observation period.4 0

Note from Eq. (28) that E increases with N, hence, foran NAf that does not fill the entire bandwidth, thesystem can stay above the threshold necessary for theCramer-Rao bounds to apply despite the increasedbandwidth that may result as N increases. Conversely,for constant NAf values, small values of N may lead tobelow threshold operation.


B. Array Case: Space-Time Field Properties

Most of the results we consider can be extended totwo dimensions, if desired. Nevertheless, the importantresults can be demonstrated with a linear array. Webegin by assuming the positions of the lasers are pre-cisely known, i.e., arD = 0, fi = iAf, the Oi are zero, theapertures are uniformly excited, i.e., Ai = A, and thearray itself is uniform, i.e., Di = iD. Therefore, Eq. (16)becomes

E[Uj(xj,yj,t)] = PU8A exp - 2 + j (N I2irAft + 2irDx2 /_ I

sinNirlAft + [(Dxj)/(Xzj)Ii (31)

sinirlAft + [(Dxj)/(z()J)As before, the exponential factor indicates a shift of thecenter optical frequency, but it also indicates that thereis now a spatial frequency dependence. In fact, the(sinNx)/(sinx) term has the form of a classical travelingwave with temporal pulse width and period as given inEqs. (18) and (19), phase velocity of (1/D) m/sec, spatialperiod of [(Az1)/D] m, and spatial pulse width of[(2Xzl)/ND] m.

The spatial beam profile at t = 0 is identical (withina scaling constant) to the temporal profile at x 1 = 0, asshown in Fig. 4. Therefore, in addition to the temporalpulse train, there is now a spatial pulse train scanninglinearly in xl at a rate of Af Hz beneath the Gaussianenvelope. From Eq. (25), for A 10.6 gm, z1 = 1 km,and w = 1 mm, the spot size at z1 is about 3.5 m. Thebeamwidth (spatial pulse width) for an array dimension

GAUSSIANENVELOPE

- 4 - U Z q METERS

N =100, D-l m, Z, I km

Fig. 6. Example of far-field mean beam pattern for N = 100, D =

1 cm, and = 1 km.

PEAK

FIRSTSIDELOBE 4

Fig. 7. Peak-to-first-sidelobe ratio vs N for sin(Nx)/sin(x).

ND = m is about 2 cm. For D = 1 cm (this would befor an array of 100 lasers spaced 1 cm apart), the spatialperiod is about 1 m. The result is illustrated in Fig. 6.The small beams move right or left (depending onwhether the frequency offsets step up the array or downthe array) at a velocity of [(AfAzl)/D] m/sec, and thepattern appears to repeat every 1/(Af) sec.

This is a useful result for a system that does not re-quire resolution of close targets to less than about w1and does require multiple target returns (for example,to reduce glint or other transient phenomena). It alsocomplicates the ambiguity problem as there is no wayto distinguish which spatial pulse (commonly calledgrating lobes) is providing the return.

Before dealing with the grating lobes, it is interestingto consider the results of nonuniform laser amplitudesAi. For Eq. (31) the first sidelobe is the highest; Fig.7 shows the peak-to-first-sidelobe ratio as a function ofN. As N increases past about 20, the first sidelobegrows linearly with the peak. For nonuniform Aijf =iAf, Di = iD, and i = iO, Eq. (16) in the 1-D case be-comes a function of

YI Aipi,

where

P = exp L(27rAft + l + 0 ]

= exp [j()(-D -sinoo)J (32)

and -sin0 0 = [(XAft)/D] + [(X0)/(2wD)]j (see Fig. 8).By manipulating the Ai, it is possible to choose the rootspi of the polynomial to yield desirable beamwidth orsidelobe properties.42-45 Although the array can bedesigned so that the sidelobes are always below a desiredvalue, there are still grating lobes.

We now have an interesting use for 0. If in Eq. (32)Af = 0, the beam position is only a function of 0 and notof time. This represents the case of a monochromatic(cw) resultant, where all N lasers are oscillating atprecisely the same frequency. For small 0 -00 -(X0)/(2-rD), so that the main beam may be pointedanywhere in the Gaussian envelope by proper choice of0. (Of course, because of the grating lobes, one mainbeam is indistinguishable from its neighbor, and in factall the lobes move simultaneously with 0. Nevertheless,a main beam will be referred to for convenience.) Themain beam may be made to scan sinusoidally for

0 = [(2-rDr)/XJ sin2irfst,

where r is the scale factor that determines the limits ofthe scan displacement, and /, is the scan rate.

For this monochromatic case we can also obtain ananalytical result for the planar array. If CD = d = fik

= 0, Aik = A, Dik = iD, dik = id, and ik = ia., + kay, Eq.(16) becomes a function of

P x

where


nI

x

D

x l

Zi

, 0

zo0 = LASERS

Fig. 8. Geometry of linear array and far field.

x 0

D

Xl

.

DIRECTION OF MAIN BEAM

Zi

0 = LASERS

Y.

Fig. 9. Geometry of planar array and far field.

px expi ~')D(-X'+3) ,I ZI A 33 2D

ex I (Y1 + Xad)]

In spherical coordinates, x1 = z 1 sin/ cost and Y1 = sink sing (see Fig. 9).46 A conical scan, useful fortracking purposes, can be synthesized if D d (or if a,,and ay are proportional to D and d, respectively) for

ax = r cos2irftt, ay = r sin2irft. (34)

Because of the grating lobes, the resulting scan in thex1, Yi plane will look like a series of concentric circleswith the amplitudes determined by the over-allGaussian envelope.

Thus far, we have discussed equally spaced (periodic)arrays, which can be designed to operate in certain de-sirable ways. The grating lobes may not be objection-able in certain applications. However, to reduce am-biguities in target location and to obtain the resolutioninherent in the spatial pulse width, the grating lobesmust be removed. Since these lobes exist for any pe-riodic array with D > (A/2),42 and since it is not physi-cally possible to choose w or D such that the center lobemay be selected, the array must be made aperiodic.

Unfortunately there is no general theory governingthe design of aperiodic arrays. For a linear array, thelowest sidelobe levels are generally from space taperedarrays.4 7 However, the side lobes tend to get higher forangles further from the main beam.48 It appears that

at optical wavelengths it is not possible to reduce thesesidelobes significantly. To complete our discussion oflaser positioning as a means of reducing sidelobes, weconsider the entire class of laser arrays using randomarray theory.

It is possible to make certain probabilistic statementsthat are valid for a class of arrays whose elements maybe positioned in some random manner. Such state-ments are merely existence statements and are no helpat all in realizing a specific array. As shown in theAppendix, the average sidelobe level with respect to thepeak for the 1-D array where hi (t) and Di are consideredGaussian random variables is 1/N.48 It is still possiblethat there will be some peak sidelobe much higher thanthe average. The ratio of the peak sidelobe to the mainbeam is given in the Appendix as

(35)

where ,3 is the probability (confidence level) that no si-delobe will be above a desired value, and n is an arrayparameter given by n = L/A. The primary restrictionon Eq. (35) results from an assumption that the field ofinterest had real and imaginary parts that wereasymptotically jointly Gaussian. This approximationis valid for [N/(7rh3 )]1/2 << 1.49 A plot of N vs h is givenin Fig. 10 for q = 1,5, where L = lOqX and f = 90%. Thefigure indicates that as the length of the array L in-creases, the peak sidelobe is also likely to increase butnot a great deal.

Figure 10 also indicates a very striking result. Fora reasonable peak sidelobe level of -20 dB, a 1-m array(q 5) must consist of about 1400 lasers. As notedearlier, these probabilistic statements are existencestatements for an infinitely large class of arrays whoseelements are positioned according to some underlyingprobability density. Therefore, there are at least somearrays which have about 1400 lasers whose sidelobes areless than -20 dB with a confidence level of 90%. Thisis not true for most of the arrays in the class. Fur-thermore, this is not a design procedure for a specificarray. Thus we conclude that practically there is noeasily realizable spatial pattern which can be used toeliminate effectively the grating lobes.

N /

10'

10

! A~~~~~~~/ H10' / /

10 h db

Fig. 10. Peak sidelobe level vs N.


l rI'

00I

.h = -(11N) In(I - #I/n),.

.

IV. Alignment Considerations

There are three sources of error that are a result of thephysical positioning of the laser elements. These areerrors in the element location in x for a linear array (xand y for a planar array), errors in the alignment of theindividual optical axes with each other, and errors in theelement locations in z. The first problem can be con-sidered in a manner completely analogous to the resultsfor the phase variance discussed in connection with Eq.(21). For the linear array, each laser may have an errorrepresented by the difference between its actual locationand Di. As we assumed before, it is reasonable that thiserror may be distributed in a Gaussian manner so thatfrom Eq. (16) the variance of that error is aD. Now aDcan be determined as a function of x1 and z1 so that theresult has a specified effect on E[Ul]. This case is theopposite of the random array results where we wanteda large aD (see the Appendix). Here we want a smallaD to reduce the attenuation of the far-field pattern.For example, if x1 = 1 m, A = 10.6 ,im, z 1 = 1 km, and aD= 1 mm, then expj- 1/2[(2irx1)2/(Az1)2 aD = exp(-0.18)= 0.84. For a specified amplitude reduction s (for ex-ample, 80% or about 1 dB) and a given range zj, theaccuracy aD with which the laser positions must bemade can be found. The result is

aD = [X/(2irxj)]zj(-2 n,7)'/2. (36)

As z1 increases, the necessary accuracy of the locationof the elements decreases. A similar result could bederived for ad or a planar array with aD and ad.

The second physical positioning problem occurs whenthe optical axes of the lasers are not parallel. In thiscase, for a linear array with oi the angle of the ith opticaxis with respect to the normal to the array and for smallangles (sinoi oi), the far field becomes

U1(x1,y1,t) - UI(x - 27rZi,Y1t), (37)

where the frequency shifting property of Fouriertransforms has been used. For the coaxial case, theGaussian envelope behaves as in Eq. (25). Therefore,using, for example, the Rayleigh criterion50 the maxi-mum oi should be on the order of w, or about 3 mradand no larger. Similar results are obtained for y in the2-D case. For the linear array case, oi must be on theorder of 10 grad, since the spatial pulse width is sosmall.

One final, rather serious, problem must be addressed.Thus far our far-field analysis has been derived on thebasis that all N beams that were superimposed propa-gated a distance z 1 into the far field. This assumes thatz, is measured from precisely the same point in eachoutput beam (for example, the beamwaist). If this isnot true, i.e., if the waists are not perfectly aligned, thez 's are not the same. This may be represented as z 1 +Azi, where Azi is the offset of the ith beam. Substi-tuting this new distance into Eq. (16) indicates that Azimakes a negligible contribution everywhere except theexp(jkAzi) term in P(x1 ,y1 ) [see Eq. (4)]. If kAzi isfolded into the interval 0 to 27r and is considered aGaussian random variable independent of ai(t), then

E[exp(jkAzi)] = exp[- (az)2 /2], where a2 is the varianceof k Azi. Even if the beamwaists could be perfectlyaligned so that Azi was zero, vibrations, thermal ex-pansion, and other effects would combine to create aAzi. Presumably this additional phase variation couldbe combined with 4i (t) and compensated appropriately.However, even this requires that the optical pathlengthsfrom the point at which the beams are sampled to thedetectors that form the first part of the phase lockedloop must be equal. The bottom line is that for coher-ent combination, the phase of the lasers must be con-trolled precisely, and at optical wavelengths that is avery stringent requirement.

V. Conclusions

We have considered the problem of coherently com-bining the fields of N single-mode lasers. Our generalresult, Eq. (16), included the effects of laser instabilityand positional (alignment) errors; the space-time du-ality of the effects was noted.

A few simplifying assumptions produced a far-fieldresult almost identical to that of a mode-locked laser.However, there was additional flexibility in that thepulse widths and pulse rates could be easily varied bychanging the frequencies of the individual lasers. Theminimum pulse width was determined to be about 3nsec using CO2 waveguide lasers. For the reasonableassumptions of coherent detection and a matched filterreceiver, the performance of the mean of the far fieldwas seen to be described in terms of the ambiguityfunction. For a constant NAf, the measurement pre-cision of range and rate could be made arbitrarily goodby increasing the number of lasers or using more thanone pulse in the measurement. This coaxial case hasapplication to target detection and tracking problemsas well as terrain mapping. The system is very flexiblebecause of the capability to change pulse characteristics,thus changing measurement precision when neces-sary.

The case of a linear array of lasers was analyzed andfound to have a spatial pulse that scanned linearly be-neath the over-all Gaussian envelope. For the mono-chromatic case (Af = 0), the spatial beam could be madeto scan sinusoidally, and a planar array was seen topermit conical scanning. These results make the sys-tem applicable to target tracking and even track-while-scan since the beams's relative phases can bechanged electronically. For some applications, how-ever, the secondary spatial beams (grating lobes) areundesirable, so the cases of aperiodic and random arrayswere considered briefly. Of the aperiodic arrays, thespatial taper was found to provide some of the lowestsidelobes, but for optical wavelengths the beam patternaway from the main peak became less controllable. Theclass of arrays with element locations determined in arandom manner was found to have a ratio of averagesidelobe intensity to peak intensity inversely propor-tional to the number of lasers. Furthermore, to obtaina reasonable high probability of low sidelobes (i.e., nograting lobes), the array was found to require a largenumber of lasers (>1000). Thus spatial positioning was


shown to be impractical as a means of reducing gratinglobes for a linear array.

Three cases of misalignment were considered.Variations in laser spacing produced results similar tothe effect of the phase variance. In each case the re-quired standard deviation for a desired effect on the farfield was determined. Misalignment of the laser axeswas also considered and found to require alignmentaccuracy to within milliradians for the coaxial case andto with 10 grad for the array case. One other probleminvestigated was that of the misalignment of the fieldsin the direction of propagation. This was seen to pro-duce a nonnegligible phase variation in the far field thatrequired physical alignment of the lasers to within anoptical wavelength to eliminate the variation. Thestatistics resulted in another amplitude reduction of thefar field proportional to the variance of the misalign-ment.

The statistical models used indicate that given a goodphase-locked-loop design for heterodyne detection, thefields may be combined and detected coherently.Misalignment problems in the plane of the array and inoptic axis angle, though difficult, also seem to be withinthe realm of engineering solutions. The waist align-ment problem (in the direction of propagation) is muchmore serious and may represent the most seriousstumbling block to implementing coherent combinationof two or more lasers. The flexibility of such a systemis desirable for ranging applications and possibly forvelocity measurement as well, since the pulse formatmay be changed easily. The coaxial case requires somesort of combination scheme, however, which may belossy as well as expensive. A linear array may find ap-plication in cases where beam scanning or pointing isnecessary. However, unless it is possible to build arraysof more than several hundred lasers within a length ofa few meters or less, the multiple lobe problem cannotbe eliminated. This will certainly restrict the appli-cation of such a system even though the high resolutionof the coherent result (- cm) may be desirable.Appendix

From Eq. (16) we consider the 1-D array with t = = 0 and obtain

E[U1(x1,y1,t)I = PU1, exp j-l/2 F,2

+ (27rA-Z 2 r2]} e p ( Il rx)

There is still, however, a finite probability that somesidelobe will be much higher than the average sidelobelevel of 1/N. This is true since the far-field pattern ofany given array is simply one possible sample functionof all those under consideration, and any individualsample function may have very high or very low side-lobes. Equation (Al) indicates that the mean of U1 canbe made to attenuate the grating lobes for xl id 0 asmuch as desired. The field, however, may deviate fromthis mean by a considerable amount. To investigatethis deviation a new definition is used so that the fieldunder consideration is zero mean and unit variance,

U'(xi y t) = UI(xi,yi,t) - E[U1(x1 ,y1,t)JU1(X1' 1'0 - Var(Ui,Ui)

(A2)

In general, Uj is a complex random process with real andimaginary parts. For a large number of elements (la-sers), these parts may be considered asymptoticallyjointly Gaussian random processes by the central limittheorem.48 49 With this assumption, the joint proba-bility density of the real and imaginary parts of the fieldU'l is known, and the probability that any sidelobe (agiven amplitude of the field) will not exceed somespecified threshold can be determined. (This is not thesame as the assumption that was made earlier that onlythe statistics of the phase were known and that theamplitude was deterministic. In this case, both theamplitude and phase of the far field UI are consideredrandom for this class of arrays since the elements mayhave locations determined by some unspecified prob-ability density.) Since U1 is zero mean, its real andimaginary parts are identically distributed as well asindependent. This is because the element locations areconsidered to have a common underlying probabilitydensity, so that the element positions are identicallydistributed about their means Di. Now Var[U1Ul] =N [as noted following Eq. (Al)] since E[I] = E[U U]= Var[U1 U*], and a' is large enough. The resulti48,49

Pr (UII > iN) = 1 -Pr (lUld AoI))

P'~~,U,'1> A.~=,_P'~,U, 2) oJ

= exp (- N°)'

Pr(Ui2> ) =exp_ )

Pr ( U 2 > B) = exp (-B),(Al)

where the Di are the means of the element locations, anda 2is their variance (since they are considered to haveidentical variances). For Di = iD, the series in Eq. (Al)has the spatial grating lobes shown in Fig. 6. It can beseen immediately that for x 0, a can be chosenappropriately large to attenuate the lobes. In a manneranalogous to the temporal results of Eq. (20), the meanfar-field intensity is proportional to N for x1Is 0. Atx = 0, however, the main beam peak is proportional toN2 . Therefore, the power ratio of the average sidelobelevel to the main beam peak is N/N 2

= 1/N, so that theaverage sidelobe level is inversely proportional to thenumber of lasers in the array.4 8

(A3)

where B = (A2)/N is the power ratio of the desiredthreshold to the average sidelobe level. The exponen-tial dependence is obtained from the exponential dis-tribution of the amplitude of I UI 1. The probabilitythat the peak sidelobe is less than B is just 1 - exp(-B).Equation (A3), however, is true for only a single samplevalue of U; at some x 1. To find the probability that nosidelobe anywhere in the pattern exceeds B, it is nec-essary to take many spatial samples of Ul. The issuenow is how many samples should there be and whereshould they be. The latter problem is reduced bysimply choosing the distance between samples to be thesame. To make the computations simpler, the number


of samples may be constrained by the requirement thatthe samples be statistically independent. Crudely,then, the number of samples needed to describe (spa-tially) the random process U1 is a set of independentrandom variables. For such a case, the samples areseparated by no less than the coherence distance of thepattern, and the pattern is considered piecewise con-stant over a coherence length. The total probabilitythat U (xi) is less than B is just the product of theprobabilities that each individual sample is less thanB,

/ = Pr[lI U(xi)I12 < B]

=Pr[IU'(x)12 <Band Ul(x2)12 < Band ... Ul(Xj12 < B]

= [1 - exp(-B)]n, (A4)

where n is the number of samples and is typically calledthe array parameter. As the coherence distance of thepattern gets small, n gets large. Equation (A4) can berewritten

B = -ln(1 - (3's"). (A5)

Now if the probability (confidence level) : is chosen andn is known, the peak sidelobe level (with respect to theaverage sidelobe level 1/N) can be determined. Forexample, for a 90% probability that no sidelobe exceedsB and for n = 10, then B = 4.56, i.e., with 90% confi-dence, the peak sidelobe of U 12 is 4.56 times as highas the average sidelobe level 1/N. It should be noted,too, that the underlying distribution of the array ele-ment locations is not a factor in this analysis; this is onereason why the analysis is general for a large class ofarray configurations.

The determination of the array parameter n can bemade in several ways, and the result can be givenby4 8

n = L/A, (A6)

where L is the length of the array. This is intuitivelycorrect since A/L is roughly the angular lobe width of thepattern. Therefore, the necessary sample rate shouldbe proportional to A/L. For optical wavelengths (par-ticularly for CO2 lasers), n is on the order of 105. Theeffects on n of a main beam at some location other thanx1 = 0, unequal element excitations, nonisotropic ele-ments, and signal bandwidth are all negligible (to withinan order of magnitude) for this case.51 Since B is theratio of the peak sidelobe to the average sidelobe, theratio of the peak sidelobe to the main beam is givenby

h = B/N = -(1/N) In(1 - /1/M). (A7)

References1. P. W. Smith, Proc. IEEE 58, 1342 (1970).2. L. J. Aplet et al., Laser Array Techniques, Quarterly Progress

Letter 3A (Hughes Aircraft Co., Culver City, Calif., 1960), AD821235.

3. L. G. Komai et al., Multi-Laser Element Techniques, AFAL TR66-134 (Air Force Avionics Laboratory, Wright-Patterson AirForce Base, Ohio, 1966).

4. L. G. Komai et al., Laser Array Techniques, AFAL TR 66-257(Air Force Avionics Laboratory, Wright-Patterson Air Force Base,Ohio, 1966).

5. C. L. Hayes and L. M. Laughman, Appl. Opt. 16, 263 (1977).6. V. J. Corcoran, IEEE Trans. Microwave Theory Tech. MTT-22,

1103 (1974).7. H. E. Hagemeier, "Field Properties of Multiple, Coherently

Combined Lasers," Masters Thesis, Air Force Institute ofTechnology, Wright-Patterson Air Force Base, Ohio, GEO/EE/77D-3 (December 1977), AD-A053349.

8. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1975), pp. 10-36.

9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968), pp. 36-61.

10. R. F. Harrington, Time-Harmonic Electromagnetic Fields(McGraw-Hill, New York, 1961), pp. 11-26.

11. A. G. Arutyunyan et al., Sov. Phys. JETP 37, 764 (1973), p.765.

12. Ref. 8, pp. 499-518.13. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965), pp.

235-242.14. A. Papoulis, Systems and Transforms with Applications in

Optics (McGraw-Hill, New York, 1968), pp. 359-399.15. B. Pincinbono and E. Boileau, J. Opt. Soc. Am. 58, 784 (1968).16. J. J. Degnan, Appl. Phys. 1, 1 (November 1976).17. F. T. Arecchi and A. Berne, Phys. Rev. Lett. 16, 32 (1966).18. W. E. Lamb, Jr., "Theory of Optical Masers," in Quantum Optics

and Electronics, C. DeWitt et al., Eds., (Gordon and Breach, NewYork, 1965), p. 378.

19. B. Pincinbono, Phys. Rev. A 4, 2398 (1971), p. 2399.20. Ref. 15, p. 784.21. A. E. Siegman, Introduction to Lasers and Masers (McGraw-Hill,

New York, 1971), p. 63.22. Ref. 11, pp. 768-770.23. Ref. 13, p. 236.24. Ref. 21, p. 307.25. Ref. 18, pp. 370-371.26. A. E. Siegman et al., IEEE J. Quantum Electron. QE-3, 180

(1967).27. R. M. Gagliardi and S. Karp, Optical Communications (Wiley,

New York, 1976).28. A. Viterbi, Principles of Coherent Communications (McGraw-

Hill, New York, 1966), pp. 86-92.29. Ref. 8, pp. 370-386.30. Ref. 21, pp. 312-313.31. A. Papoulis, Probability, Random Variables, and Stochastic

Processes (McGraw-Hill, New York, 1965), pp. 254, 474-510.32. A. Yariv, Quantum Electronics (Wiley, New York, 1975), p.

259.33. D.. Middleton, Introduction to Statistical Communication

Theory (MGraw-Hill, New York, 1960), p. 126.34. Ref. 31, pp. 336-338.35. Ref. 31, p. 150.36. C. E. Cook and M. Bernfeld, Radar Signals (Academic, New York,

1967), pp. 18-32.37. Ref. 33, pp. 335-395.38. R. E. Ziemer and W. H. Trantor, Principles of Communications

(Houghton Mifflin, Geneva, Ill., 1976), pp. 311-317.39. Ref. 36, pp. 59-108.40. A. W. Rihaczek, Principles of High Resolution Radar

(McGraw-Hill, New York, 1969), pp. 70-75, 118-158, 191.41. H. L. Van Trees, Detection, Estimation, and Modulation Theory,

Part 3 (Wiley, New York, 1971), pp. 275-313.42. B. D. Steinberg, Principles of Aperture and Array System Design

(Wiley, New York, 1976), pp. 89-90.43. W. F. Richards and Y. T. Lo, IEEE Trans. Antennas Propag.

Ap-23, 165 (1975).


44. W. L. Stutzman and E. L. Coffey, IEEE Trans. Antennas Propag.AP-23, 764 (1975).

45. Ref. 42, pp. 95-110.46. R. S. Elliot, "Theory of Antenna Arrays," in Microwave Scanning

Antennas, Vol 2, R. C. Hansen, Ed. (Academic, New York, 1966),p. 36.

47. Y. T. Lo and S. W. Lee, IEEE Trans. Antennas Propag. AP-14,22 (1966).

48. Ref. 42, pp. 130-156.49. Y. T. Lo, IEEE Trans. Antennas Propag. AP-12, 257 (1964).50. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass.,

1974), pp. 354, 360.51. Ref. 42, pp. 171-184.

March

5-7 Preservation and Restoration of Photographic Imagesseminar, Graphic Arts Res. Ctr., RIT, RochesterGraphic Arts Res. Ctr., RIT, 1 Lomb Memorial Dr.,Rochester, N.Y. 14623

5-9 1979 Pittsburgh Conf. on Analytical Chemistry and Ap-plied Spectroscopy, Cleveland Convention Ctr.Pittsburgh Conf. on Anal. Chem. and Appl. Spectrosc.,Suite 215, Whitehall Ctr., Pittsburgh, Pa. 15227

5-9 High Energy and High Power Lasers course, AnaheimLaser Inst. of Am., P.O. Box 9000, Waco, Tex. 76710

5-9 2nd Internat. Conf. on Infrared Physics, Zurich F.Kneubeihl, Solid State Phys. Lab., ETHHoenggerberg,CH-8093, Zurich, Switzerland

Meetings Calendar

1979

February

5-9 Fiber and Integrated Optics course, Orlando, Fla. Con-tinuing Eng. Educ., Geo. Washington U., Washington,D.C. 20052

5-9 Digital Image Processing of Earth Observation SensorData course, Geo. Washington U. Continuing Eng.Educ., Geo. Washington U., Washington, D.C.20052

12-16 Remote Sensing and Digital Information Extractioncourse, Geo. Washington U. Continuing Eng. Educ.,Geo. Washington U., Washington, D.C. 20052

12-16 Quality Control for Photographic Processing course, Coll.of Graphic Arts and Photography, RIT, Rochester W.D. Siegfried, Graphic Arts Res. Ctr., 1 Lomb MemorialDr., Rochester, N. Y. 14623

12-16 Modern Optics for Engineers course, Albuquerque LaserInst. of Am., P.O. Box 9000, Waco, Tex. 76710

12-16 Xerography and Electrographics workshop, RIT, Roch-ester College of Continuing Educ., 1 Lomb MemorialDr., Rochester, N.Y. 14623

18-22 Theory of Alloy Formation, APS topical mtg., New Or-leans W. W. Havens, Jr., 335 E. 45th St., New York,N.Y. 10017

19-23 Applied Oceanography and Meteorology for Engineersand Program Managers course, Geo. Washington U.Continuing Eng. Educ., Geo. Washington U., Wash-ington, D.C. 20052

20 OSA-SAS Detroit Section mtg. R. E. Michel, Law-rence Inst. of Technol., 21000 W. Ten Mile Rd.,Southfield, Mich. 48075

6-8 Optical Fiber Communication, OSA-IEEE TopicalMeeting, Shoreham Americana Hotel, WashingtonD.C. OSA, 2000 L St. N. W., Suite 620, Washington,D.C. 20036

12-13 Quality Assurance in Air Pollution Measurement, APCAspecialty mtg., Grand Hotel, New Orleans G. VonBodungen, Louisiana Air Control Comm., P.O. Box60630, New Orleans, La. 70160

12-14 1979 Particle Accelerator Conf., APS topical mtg., SanFrancisco R. B. Neal, SLAC, P.O. Box 4349, Stanford,Calif. 94305

12-15 6th Iranian Conference on Electrical Engineering, ShirazM. Rafian, 6th ICEE, P.O. Box 737, Shiraz, Iran

18-23 ASP-ACSM Ann. Spring Mtg., Washington Hilton Hotel,Washington, D.C. T. J. Lauterborn, U.S. GeologicalSurvey, 507 National Ctr., Reston, Va. 22092

19-22 APS Div. of High Polymer Physics, Chicago W. W.Havens, Jr., 335 E. 45th St., New York,N.Y. 10017

19-23 APS mtg., Chicago W. W. Havens, Jr., 335 E. 45th St.,New York, N.Y. 10017

19-23 Lasers for Engineers course, Dallas Laser Inst. of Am.,P.O. Box 9000, Waco, Tex. 76710

25-27 Applications of Optical Instrumentation in Medicine VII,Hotel Toronto, Toronto SPIE, P.O. Box 10, Belling-ham, Wash. 98225

26-30 Photographic Science Course, Rochester Inst. ofTechnology, 1 Lomb Memorial Dr., Rochester, N.Y.14623

April

1-6 Polymers for Optical Fiber Systems, ACS symp., Hono-lulu M. J. Bowden, Bell Labs., 600 Mountain Ave.,Murray Hill, N.J. 07974

2-5 SPIE Tech. Symp. East '79, Hyatt Regency Hotel,Washington, D.C. C. W. Haney, U.S. Naval Air Dev.Ctr., Warminster, Pa. 18974

2-6 6th Internat. Vacuum Metallurgy Conf. on SpecialMelting and Metallurgical Coatings, San Diego R. W.Buckman, Jr., P.O. Box 18006, Pittsburgh, Pa. 15236

26-2 Image Evaluation seminar, RIT, Rochester College ofMarch Continuing Educ., 1 Lomb Dr., Rochester, N.Y.

14623


continued on page 289

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Field properties of multiple coherently combined lasers