Proposed Method of Measuring the Current Distributionin a Tokamak Plasma

Norton Bretz

It is shown that the current distribution in a typical Tokamak plasma can be measured by a light scat-tering technique. The direction of the total magnetic field is measured accurately enough that the mag-nitude of the small poloidal component can be found. The field direction is measured by observing the

scattered frequency spectrum of CO2 laser light. The usual Gaussian spectrum becomes modulated atthe electron cyclotron frequency when the difference between the incident and scattered wave vectors is

nearly perpendicular to the magnetic field. The harmonics can be superimposed with a Fabry-Perot in-terferometer and their collective width resolved as the scattering direction is changed. The SNR is highonly when the detector is shielded against background radiation.

I. Introduction

A knowledge of the current distribution in a Toka-mak plasma is crucial to the understanding of thesedevices. At present no accurate measurements ofthis parameter have been reported; however, severalexperiments have been proposed,1 -5 and most ofthese are presently being pursued. All these meth-ods measure the current distribution indirectly bymeasuring the direction of the total magnetic fieldB. In a Tokamak this field is made up primarily ofan externally imposed toroidal component Bt and amuch smaller poloidal component Bp created by theOhmic heating current. In the ST Tokamak,6 forexample, when I = 50 kA and Bt = 30 kG, one hasqp Bp/Bt < 2.6 x 10-2 (op < 1.5°). So in orderto measure the current distribution accurately thefield direction must be measured to an accuracy of0.1-0.20. This is especially true near the center ofthe current channel where accurate measurements ofthe plasma resistivity and quality factor q(r) = (r1R)[Bt/Bp(r)] are important (r is the minor radial coor-dinate, and R is the major radius of the torus).

A scheme similar to that described by Perkins7

will be examined here. The measurement of thefield direction is accomplished by observing the fre-quency spectrum of 10.6-pum CO2 laser radiationscattered by the plasma. The usual Gaussian spec-trum becomes modulated at the electron cyclotronfrequency when the difference (K) between the inci-dent and scattered wave vectors is nearly perpendic-ular to the magnetic field. When the scattering

The author is with the Plasma Physics Laboratory, PrincetonUniversity, Princeton, New Jersey 08540.

Received 26 September 1973.

angle is 900 the modulation appears when K is with-in about 10 of the perpendicular plane. As K-B be-comes smaller the harmonics become narrower. AFabry-Perot interferometer can superimpose the har-monics and measure the narrowing as the scatteringdirection is changed. By this method angular devia-tions from a purely toroidal field as small as 0.1° canbe resolved.

A discussion of the detection problem will be givenshowing that the SNR can be large only if the detec-tor is properly shielded against unwanted back-ground radiation. Once this is done, however, theSNR ranges from 40 to 10 for a practical laser-detec-tor system and for temperatures and densities typi-cal of Tokamak plasmas.

II. Scattered Spectrum

The scattered spectrum exhibits a change when K= Ks - Ki = [47r sin(Os/(2)]Xi (K/K) is near to theplane perpendicular to B. It happens that for Xi =10.6 Aim, Os - 90°, and temperatures typical of Toka-mak discharges; the field induced change only occursfor a narrow range in the direction of K.

The Thomson scattering spectrum of a plasma in amagnetic field8 shows that the usual Gaussian spec-trum becomes modulated at a frequency correspond-ing to the electron cyclotron frequency when the con-dition ac 3 wc/kve > 1 is met. We have taken wo,= eB/mc, k, = K sink, sink = K.B/KB, and Ve =(2kTe/me)"/ 2 .

The scattering form factor S(K,w) for 1/KXD < 1 is

S(k, w) = 7.1/2,,, exp(-z) E In(z) exp(-a, 2 )[(c/) -n]2

where z = (k I 2Ne2)/(2Wc2), hi = K cos4, o = US -

Wi, XD = [kTe/(47rne2)]/2, and In(z) is the modified

1134 APPLIED OPTICS / Vol. 13, No. 5 / May 1974

These modulations have been observed by Evansand Carolan 9 and by Kellerer.10

I11. Field Direction Measurement

We now discuss how the field direction measure-ment might be made on a Tokamak plasma. Figure3 shows the 900 scattering geometry and its relationto the magnetic field. Experimentally the modula-tion is observed by scanning the angle s, where s

- -[(Ks-Bp)/KsBp]. From the geometry = (s- )/(2).1 /2 Figure 4 shows how the scattering ex-

periment would be done on a Tokamak. The designthat has been considered here envisions scanning theplasma along a line of constant Bt (vertically) sothat the spacing between the scattered harmonics re-mains the same.

Fig. 1. Scattered spectrum for ac >> 190°.

I-

05

when kTe = 1.0 keV, O =

I

ksIIl

OS

nwc(0

Fig. 2. Shape of a single harmonic for kTe = 1.0 keV, B = 30kG, and s = 90°. The scale has been normalized to unity when

= 7r/2.

Bessel function of the first kind. The normalization

f S(k,w)dw= 1.

Figure 1 shows the qualitative shape of the spectrumwhen a >> 1. The spectrum consists of a largenumber of delta functions under a Gaussian enve-lope. Figure 2 shows the shape of a single harmonicfor several values of ac when B = 30 kG, kTe = 1.0keV, and 0 = 900. In general, as becomes smaller(ac larger), the individual harmonics become nar-rower.

i

P

/

Fig. 3. Scattering geometry for O = 900.

SIDE VIEW AO

L=

LASER BEAM

TOPVIEW

B

Fig. 4Sctei LASER BEAMIo OUT OF PA PER

I\\I S Sf

Fig. 4. Scattering system on a Tokamak.

May 1974 / Vol. 13, No. 5 / APPLIED OPTICS 1135

S ( k,W)

0 0.5IL~i

1.5t Ia)e

1.0

W(x 1013sec I

i i g B l l l l W

L

t

The resolution of a single harmonic would be diffi-cult because there are relatively few scattered pho-tons per peak. For example when the electron tem-perature is 1.0 keV there are about sixty peaks be-tween the 1/e points of the Gaussian profile. It ispossible, however, to superimpose the harmonics andmeasure their collective width using a Fabry-Perotinterferometer.

The interferometer has a transmission given by

TF-P = 1/(1 + [(4p)/(1 - p)2

] sin21[w(t/c) cosG]).

where p is the reflectivity of the plates, t is their sep-aration, and 0 i is- the incidence angle. This trans-mission function has maxima separated by F-p =

(irc)/(t cos6s) whose width is determined by thevalue of p. The ratio of the separation of the maxi-ma to their width at the half-intensity points iscalled the finesse F and is related to the reflectivityby F = (rpl/ 2 )/(l - p). When the light incident onthe interferometer is unmodulated the transmissionis given by Tavg = (1 - p)/(1 + p).

If the incident light has narrow peaks separated byc and if one chooses (F-p = w°, the transmission

approaches unity. Thus if light that is sufficientlyparallel is sent through the interferometer, the trans-mitted intensity as a function of Os will exhibit a res-onance near Os = Op. Increasing the finesse narrowsthe resonance (and the solid angle available for theobservation). However, if the SNR is sufficientlyhigh, the direction of B can be measured to an arbi-trarily high degree of accuracy.

We now turn to a discussion of effects that limitthe maximum finesse and therefore also limit the ac-curacy of the field direction measurement. Theleast important limitation is the laser line width. Aslong as the CO2 laser is constrained to lase on a sin-gle rotational line, the line width is determined bypressure broadening for atmospheric pressure sys-tems. This is Awil/w < 5 x 10-5 and will be negli-gible for this discussion. More importantly, therewill be small deviations in the magnitude of themagnetic field along a vertical scan due both to thepoloidal field and diamagnetic fields and to shot-to-shot variations in the toroidal field. The result ofthese deviations in the constancy of the total fieldwill be a mismatch between the electron cyclotronfrequency c and the free spectral range of the inter-ferometer WF-p. The contribution of the poloidalcomponent to the magnitude of the total field is seenfrom

B B,[1 + 1/2(BP2 /B 2)].

For most Tokamaks Bp/Bt < 0.1, so that the fieldchange AB/B is less than 0.5%. The field deviationdue to diamagnetic currents is the order of A =(87 rnkTe)/B 2 which is limited by the restriction onthe poloidal beta fUp • 1 to about 0.2%. Shot-to-shot variations in the toroidal field due to circuitfluctuations generally have AB/B < 0.5%. Finallythere will be a broadening of the scattered reso-nances due to a finite transit time effect. The finite

time required by an electron whose velocity is Ve tocross a beam whose diameter is D results in a fre-quency spread in the scattered radiation given byAwt 2(ve/D). The finesse is limited by the con-dition F < wC/Awt. When D 0.1 cm, this restric-tion on F is about the same as that due to fieldchanges. The above considerations require F to beno greater than about 6 for kTe 1.0 keV and B30 kG.

For the calculations that follow we choose F = 6 (p_ 0.6). Figure 5 shows the transmission functionfO)

f(0) = If' T,(w)S(k, w)dwI,,

in an ideal case where there are no mismatches be-tween WF-p and wc, the alignment of the interferom-eter is perfect, and O = 0. Resonance curves areshown for several temperatures, but in each case theintegration is taken to the 1/e points of a 1.0-keVGaussian (ile). The sensitivity of the resonance isweak since ac 1/(kTe)1 2, but generally if the res-onance is measured with channels whose width is0.1-0.2°, the location of the maximum can be re-solved to about 0.10. Finally Fig. 6 shows the effectof a small field change on the resonance shape of thefirst harmonic (n = 1) for kTe = 1.0 keV, Bt = 30kG, and F = 6.

For larger toroidal fields the resonance shapes canbe very similar to the ones shown in Fig. 5. For ex-ample, if B is doubled (B = 60 kG), the number ofpeaks under a 1.0-keV Gaussian is halved, and onemay use a larger finesse (F - 12) to resolve the har-monics.

1.0

kTeO 025

f(4,I 050 \

f (0

.0.7

0.5 1.0

1.5205

0 0.01 H 0.02

Fig. 5. The output of the Fabry-Perot interferometer

f (O) = J TF-pS(k, w)dw vs X,

taking the integration in all cases to the l/e points of a 1.0-keV

Gaussian. The fixed parameters are 0 = 90°, B = 30 kG, and F

= 6.

1136 APPLIED OPTICS / Vol. 13, No. 5 / May 1974

0.8AOi =.000

:0150.6 020

f(4,

0.4

.025

0.2

0 , , , I

Fig. 6. The output of thefirst scattered harmonic

4,

Fabry-Perot interferometer for the

NO) = L cTF-PS(k, w)dw vs X0 5wC

for the conditions kTe = 1.0 keV, B = 30 kG, and F = 6. The ef-fect of a small change in the field AB/B is shown. f(o) has beennormalized to 0.84 for convenient comparison to the other figures.

IV. Signal-to-Noise Ratio

In order to assess the practicality of this experi-ment, it is necessary to calculate the SNR for a reallaser-detector system.

The number of photons ANs scattered at Os = 900into a solid angle AQ by a plasma whose density is neis

AN = Nnero2LAQ,

where Ni is the number of incident photons, r is theclassical electron radius (r 2 = 7.9 X 10-26 cm 2 ), Lis the length of the scattering volume, and it hasbeen assumed that the incident beam is polarized.

The element of solid angle A = AOAO, is partlydetermined by our requirement on the accuracy ofthe field direction measurement. We choose s =0.004 (0.23°). The orthagonal element As may bemuch larger. Since ac 1/sin(Os/2) a choice of As= 0.20 (11.50) does not alter the shape of f(O) appre-ciably. Then A Q = 8.0 x 10-4 sr.

Commercial lasers are available with the followingcharacteristics:

pulse energy 10 J;beam divergence 2 mrad;beam diameter 3 cm;pulse width 0.5 sec.

So we will take 10 J of input energy, ne 2 1013cm- 3 , and L = 1 cm to obtain ANs = 6.6 105 pho-tons.

The number of photons that actually reaches thedetector is determined by various transmission loss-es, lumped into T, and the function f(O) that charac-terizes the Fabry-Perot interferometer transmission.

We will consider the Ge:Hg photoconductor to de-tect the scattered signal. The following characteris-tics will be taken as representative of this detector' 2:

operating temperature 4 K;detector resistance RD 0.3 MQ;detector area AD 0.8 X 10-2 cm 2 (0.1 = cm

diam);field of view OD 60°;stray capacity Cs 40 pf;responsivity at 10 gm IR 105 V/W;detectivity at 10 Am D*Lo em = 2 x 1010 cm

Hz'/ 2/W.Figure 7 shows a drawing of the detector and itsequivalent circuit. The change in voltage AVL de-veloped across the load resistor RL in response to achange A exp(iw't) (photons/sec) incident on thedetector is

AVL

VORL (l ARD\A(RL + RD)[(RL + RD)2

+ W' 2CS

2RL

2RD

2]

1 2 AX

when RL < RD,

RL V0 (AR,'AVL = RD2 (1+ /2C 2RL2)1/2 ( A O A.

So the frequency response of the circuit is about w' =(RLCs)-'. The effective response of the detector cir-cuit (volts out per watt in) is called the responsivityIRL, where

IRL = (1/Ihi)(AVL/1AO)

The responsivity given by the manufacturer refers tothe matched load (RL = RD) circuit and a choppedsignal for which 'RDCS < 1. Holding the voltageacross the detector constant, the responsivity R isrelated to the responsivity IRL for RL < RD andw'RLCS 1 by

IRL = 2(RL/RD)1IR/[(1 + W12C 2RL2)1/2 ]1

e Hg DETECTOR

Fig. 7. Detector and equivalent circuit.

May 1974 / Vol. 13, No. 5 / APPLIED OPTICS 1137

X (cm)

Fig. 8. Room temperature background radiation spectrum PBand detector responsivity IR.

So in order to resolve a signal whose rise time is Tone must choose CsRL T w C/. For the laserpulse in question T 0.5 lisec, so that RL 10 kU.The actual responsivity of the circuit is

IRL = f2(RL/RD) R.

The power scattered to the detector is given by

AP, = ANTf(0)(-lhojT)

giving a signal voltage of

AV, = IRLAPS = V21[ANsTf(0)]1/Ttji|iRL-

Taking T 0.25 for our typical detector-laser sys-tem we have AV, 8 3.6 X 10-5 f(o) V.

The foregoing discussion relates to the SNR onywhen external noise sources limit the smallest signalsthat can be observed. A measure of the noise inher-ent to detector is called the detectivity D*lo ;m Theincident power required to make a signal equal tothe root mean square noise signal is called the noisepower APN and is related to the detectivity by

APN = [(AD /2)/(D j0,.)](Aw/2r)1/2

where Aw is the bandwidth of the observation.Choosing Aw = W' = T-1 we find the signal-to-noiseratio (S/N)B to be

' APN = ANTf (()hD*O 0,, (A2T)Thus we have (S/N)B = 2.5f(k) for our model sys-tem.

Fortunately the SNR can be much higher becausethe detector noise is mainly caused by fluctuations inthe room temperature blackbody background whichimpinges on the detector through its aperture to theoutside world. The background spectrum is highestin the region where the detector is most sensitive.The parameters RD, IR, and D*lOum are implicitfunctions of the background power PB (Ref. 13); RD

- 1/PB, R - PB, and D*lom - 1/PB'/ 2 . As a re-sult IRL is independent of PB, but D*1iom can be in-creased in three different ways: (1) a cooled band-pass filter can cut out all background that is not inthe region near W1 where scattering will be observed;(2) a cooled polarization filter can eliminate half ofthe unpolarized background; and most importantly(3) the scattered etendue LDAQ where D is the beamdiameter can be matched to the etendue of the de-tector ADAUD = AD sin 2 (OD/2 ). The increase inD*o0 m from (1), (2), and (3) can be estimated easi-ly. The room temperature background spectrumand the detector responsivity are shown in Fig. 8.The transmission of a cooled bandpass filter isshown. The transmission limits are taken to the 1/epoints of a 1.0 = keV Gaussian. A numerical esti-mate shows that the background power is reduced bya factor of about 3.5. For (3) a set of cooled aper-tures is required. The apertures define the etendueof the detector excluding the unwanted background.It is evident that the beam diameter should be cho-sen as small as practical; the reason for this beingthat one wants the brightness of the source of scat-tered radiation to be greater than the brightness ofthe background. If we take D = 0.1 cm, the ratio ofthe etendue of the detector to that of the scatteredradiation is about 79. The improvement in D*,o ,mfrom (1), (2), and (3) is a factor of (3.5 X 2 x 79)1/2

= 23.5, making the background signal-to-noise forthe apertured, filtered detector-laser system to be(S/N)B = 59f(O). And when F = 6 we have approxi-mately that 1 > f(Ot) >~ Y4.

There are several other sources of noise that com-pete with the noise due to the room temperaturebackground. First we recognize that limiting thefield of view will decrease the noise in the detectoronly up to the point where fluctuations in the 4 Kenvironment begin to compete with fluctuations inthe room temperature radiation arriving through therestricted aperture. This limit has not been reachedin the system that has been described; in fact thetemperature surrounding the detector would have tobe raised to about 100 K before the two noise sourceswould be equal. A second fundamental restrictionon the SNR is due to shot noise. This is the noisesource important in photomultiplier tubes. It iscaused by the statistics involved when a finite num-ber of electrons carry the signal information. Theshot SNR is (S/N), = ,nTANf(0), where n is thequantum efficiency of the detector. Typically ,- >0.5 SO that S/N), > S/N)B. A more importantsource of noise is the Johnson noise in the load resis-tor. The magnitude of this noise at the input to thepreamp is

AVj = [(2/r)KTRLAo]"1281/[(1 + w'2RL2CS2)1/2]I,

taking W'RLCs = 1 and w' = Aws as before,

AV.j = (kT/rC 1/2

so that (S/N)j = (AVs)/(AVj) = 53f(q5) when T = 4K. Obviously it is quite important to cool the load

1138 APPLIED OPTICS / Vol. 13, No. 5 / May 1974

Fig. 9. Side view of optical scheme for a single channel system.

0.8

0.6

0.4

0.2

00

Fig. 10. Output of Fabry-Perot interferometer

rU,f() = f TF.PS(k, W)dW vs X,

where kT = 1.0 keV, B = 30 kG, and F = 6. The effect of raysthat differ from the normal by A01 is shown.

resistor. Finally any experiment must contend withreal amplifiers and their own inherent noise. Thiscan be a particular problem with a cooled detectorwhose noise has been systematically depressed. Thesignal that is already small remains the same.Room temperature amplifiers are commerciallyavailable, however, that have root mean squarenoise signals of no more than about 5 10-7V and are otherwise compatible with the system thathas been described. The amplifier signal-to-noise isexpected to be greater than S/N)j.

The essential features of this noise discussion havebeen well established experimentally. In fact theimprovement in D*lom by filtering, aperturing, andcooling the background has been used by Chen14 tocarry out a Thomson scattering experiment using aGe:Hg detector.

The signal-to-noise established in this section hasbeen deliberately constrained by considering only fil-tering and aperturing commercial detectors. Theoptimal detector could have a much larger value ofIR. This may be seen from the equation,

IR= (RD/li cA)ieGi1/[(1 + W12

TR2)1/

2]1,

where G = (ED11TR/b), -TR is the electron-hole recom-bination time, gi is the mobility, ED is the fieldacross the detector, and b is the separation betweenthe biasing electrodes (see Fig. 7). In the presentcase TR - 10-8 sec so that 'TR < 1. The value ofTR can be increased by reducing the residual impuri-ty concentration. Present purification techniqueslimit the maximum value of TR to about 3 x 10-7sec, which is still smaller than the time response re-quired here. Some advantage may also be gained byreducing b. The image of the laser beam segmentthat will be focused on the detector is 0.2 cm 0.02cm, a factor of 5 demagnification. The small dimen-sion should be chosen to be b (b 0.1 cm had beenassumed previously). The responsivity could be im-proved so that the SNR would be limited only bybackground noise.

V. Optical System

A sketch of the lenses, filters, and apertures neces-sary to carry out the measurement is shown in Fig. 9.A single channel system is shown that views a solidangle As x A0s = 0.004 x 0.2 (i.e., 0.23 X 11.5°).

Radiation is collected by the first lens which has afocal length f and an aperture with dimensions D:x L. Clearly one must have D = fA50 and L =fAOs. This lens introduces a set of nearly parallelrays to the Fabry-Perot interferometer. The effectof rays that differ from normal incidence by A6i isshown in Fig. 10 for the case where F = 6, kTe = 1.0keV, and B = 30 kG. Apparently it is necessarythat A < 0.015.

Fig. 11. Top view of optical scheme for a multiple detector sys-tem.

May 1974 / Vol. 13, No. 5 / APPLIED OPTICS 1139

The second lens is identical to the first. Its pur-pose is to send the beam into a cooled enclosurewhere the etendue can be defined by two cooled ap-ertures. It is only necessary that the apertures becooled to below about 100 K and may themselves bepart of a liquid nitrogen cooled (77 K) heat shield forthe liquid helium container. The dimensions of thecooled apertures A2 and A3 are D, x L,. and D x L,respectively. The cooled bandpass and polarizationfilters can be placed in this region also; these canalso be cooled to 77 K.

Once the etendue has been defined the image of A3

must be reduced by the lens f onto the detector.Since detectors are small a factor of about 5 demag-nification is necessary.

A set of dimensions that is consistent with the re-quirements outlined is shown below:

lens 1: f = 50 cm, D, = 0.2 cm, L, = 10 cm;Fabry-Perot: diam = 10 cm, AO < 0.01;lens 2 (uncooled): f1 = f2;

aperture A2 (cooled): D2 = 0.2 cm, L2 = 10 cm;apertureA3 (cooled): D3 = 0.1 cm, L3 = 1.0 cm;lens 3 (cooled): x = 12.5 cm, Y = 2.5 cm, f3 = 2.1

cm;image of A3 on detector: 0.02 cm X 0.2 cm.This single detector system can scan the angle lbs

by moving the apertures Al and A2 . However, amultiple aperture, multiple detector system could beset up in much the same way. The system describedhere may be regarded as one of a number of parallelchannels all of which use adjacent slots in the inter-ferometer and associated lenses. Figure 11 showsthe top view of the optical scheme for a multiple de-tector system. The lens aperture on f, has been re-moved. The cooled apertures A2 and A3 consist ofarrays of slots each with the same dimensions con-sidered previously. Each channel is focused on aseparate Ge:Hg detector. The five-channel systemshown here covers a range in Os of about 1.10.

VI. Conclusion

It is possible to measure the current distribution ina Tokamak plasma with considerable accuracy.Furthermore this technique is not constrained by a

narrow range of operating parameters. It can beused in the parameter space found in most presentand proposed Tokamak facilities. The technique ismost accurate near the center of the discharge wherethe temperature is highest and where the accuracy ismost important. It may also be noted that someTokamaks are run at near liquid nitrogen tempera-tures because of constraints on the available electri-cal power. Were this experiment done on one ofthese devices the detector noise problem would beautomatically eliminated, and much of the cooledaperture system could be removed.

I thank D. Dimock, C. Daughney, and F. Perkinsfor helpful advice and encouragement.

This work was supported by U.S. Atomic EnergyCommission Contract AT(11-1)-3073, and use wasmade of computer facilities supported in part by Na-tional Science Foundation Grant NSF-GP579.

References1. F. W. Perkins, Bull. Am. Phys. Soc. 11, 1418 (1970); also

Princeton Plasma Physics Laboratory MATT-818.2. R. Cano, I. Fidone, and M. J. Schwartz, Phys. Rev. Lett. 27,

783 (1971).3. R. Cano, C. Etievant, and J. Hosea, Phys. Rev. Lett. 29, 1302

(1972).4. F. C. Jobes and R. L. Hickok, Nucl. Fusion 10, 195 (1970).5. M. Murakami, J. F. Clarke, G. G. Kelley, and M. Lubin,

Bull. Am. Phys. Soc. 11, 1411 (1970); also Oak Ridge Nation-

al Laboratory ORNL-TM-3093.6. J. Sheffield, Plasma Phys. 14, 385 (1972).7. D. Dimock et al., Plasma Physics and Controlled Nuclear Fu-

sion Research (International Atomic Energy Agency, Vienna,1971), Vol. 1, p. 451.

8. T. Laaspeere, J. Geophys. Res. 65, 3955 (1960).9. D. E. Evans and P. Carolan, Phys. Rev. Lett. 25, 1605 (1970).

10. L. Kellerer, in 4th European Conference on Controlled Fusionand Plasma Physics (Comitata Nazionale per l'Energia Nu-cleare, Rome, 1970), p. 125.

11. Lumonics Research Ltd., Ottawa, Ont.12. Santa Barbara Research Center, Goleta, California.13. D. W. Kruse, L. D. Mc Glauchlin, and R. B. Mc Quistan, El-

ements of Infrared Technology (Wiley, New York, 1962), p.

365.14. K. Chen, Ph.D. Thesis, Massachusetts Institute of Technolo-

gy (1972).

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1140 APPLIED OPTICS / Vol. 13, No. 5 / May 1974