Aperture averaging of the two-wavelength intensitycovariance function in atmospheric turbulence

Z. Azar, H. M. Loebenstein, G. Appelbaum, E. Azoulay, U. Halavee, M. Tamir, and M. Tur

The influence of aperture averaging on the two-wavelength intensity covariance function was experimental-ly determined for visible (0.63 Arm) and infrared (1.06 Am) collinear, approximately spherical beams whichpropagated through the earth's turbulent atmosphere. Range varied from 1300 to 3250 m, and due to theprevailing atmospheric conditions, most measurements were made in the strong turbulence regimes. Re-sults show that (1) the covariance function monotonically decreases as the receiver aperture size increases;(2) the correlation coefficient attains high values (-0.7) even for a relatively small aperture size of 5 mm; (3)while the single wavelength probability distribution of the intensity is approxiamtely lognormal, the experi-mental two-wavelength conditional probabilities are higher than those predicted by the lognormal model.

1. Introduction

It is well knownl-3 that an electromagnetic wave,propagating through the turbulent atmosphere, expe-riences intensity fluctuations which significantly limitthe performance of line-of-sight communication sys-tems and certain lidar configurations. The normalizedvariance of the fluctuations of the intensity I(Xi), alsocalled the scintillation index,

2 (12)- (I)2

grows with (a) the range; (b) the wave optical frequency;and (c) the strength of the random refractive-indexvariations in the atmosphere, up to a saturation valueof the order of unity.4 In practice, the near-unity sat-uration value of a amounts to many decibels (dB) offluctuations, and systems designers must do their bestto reduce the adverse effects of intensity fluctuationson the dynamic range and sensitivity of their sys-tems.

One way to average over these fluctuations, therebyreducing their corresponding U2, is to use large opticalapertures at the receiver end of the system. In weakturbulence conditions, Homstad et al.5 have succeededin correlating theoretical predictions, based on the

M. Tur is with Tel Aviv University, School of Engineering, RamatAviv, P.O. Box 39040, Tel Aviv 69978, Israel; the other authors arewith Soreq Nuclear Research Center, Yavne 70600, Israel.

Received 21 December 1984.0003-6935/85/152401-07$02.00/0.© 1985 Optical Society of America.

Rytov approximation, with their experimental data toshow that aperture averaging significantly reduces a.More specifically, the aperture averaging factor Q(R)= c(R)/U2 (R = 0) (R is the radius of the aperture) wasfound to be a decreasing function of the ratio R /XE,where X is the optical wavelength, L is the propagationlength, and \/XE is the spatial correlation distance ofthe intensity fluctuations.' For L = 430-860 m,Homstad et al.5 found that typically, Q[R/KE = 1]- 0.1 and Q[R/KIJ = 4] 0.01. Thus, in weak tur-bulence conditions, large receiving apertures can indeedimprove the signal-to-noise ratio of the receiver.However, this technique cannot be extended to propa-gation ranges falling in the strong turbulence regime.It has been determined, both theoretically6 and exper-imentally,7 8 that the covariance function of the inten-sity fluctuations for large integrated-path turbulenceis characterized by two transverse scale sizes: a shortone that governs the initial fast drop of the covariancefunction and a fairly long scale that characterizes thelong tail of that function. This long tail defeats effec-tive aperture averaging of strong turbulence signals andrenders it economically impractical.

A more suitable (but complicated) way to avoid thedeteriorating effects of the random index-of-refractionvariations of the turbulent atmosphere is to synchronizethe transmission of high-density packets of informationto those instants when the atmospheric randomindex-of-refraction field attains a fairly uniform reali-zation which exhibits very good transmission. Thus,if a continuously monitored probe beam was found topass the relevant propagation path with an instanta-neous high signal-to-noise ratio, a burst of the usefulinformation should be immediately sent to take ad-vantage of this short-lived atmospheric condition. This

1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS 2401

idea is somewhat similar to the "lucky shot" concept ofHufnagel and Fried910: A series of short exposures ofan atmospherically distorted image contains, with asmall but finite probability, some images which are es-sentially diffraction-limited. Successful and practicalexploitation of the synchronization method heavilydepends on the ability of the transmission system torespond to the results of the probe beam within a timeinterval much shorter than the period, tc, over which theatmosphere stays in its supertransmission status. Sincetc is of the order of a millisecond, the bursts must bevery short and, for high data rate transmission, also verypowerful. Thus, it could be very convenient if the probeand signal beams would come from different lasers withpossibly different wavelengths. The probe beam source.could be a cheap low-power He-Ne laser while a pulsedNd3+:YAG system would be used for the transmissionof the information. Two lasers with different wave-lengths can be used only if there is a substantial corre-lation between the intensity fluctuations of the twobeams. This work is concerned with the experimentalinvestigation of the effect of aperture averaging on thedegree of correlation between the intensity fluctuationsof waves with different wavelengths, propagatingthrough atmospheric paths of 1300 and 3250 m in strongturbulence conditions.

Several theoretical relevant results, applicable tounsaturated paths, are known. Ishimaru"l formulatedgeneral expressions for the temporal frequency spectraof plane, spherical, and beam waves operating at twodifferent wavelengths and showed how these spectracould be used to infer the wind velocity and the index-of-refraction structure constant C2. Using the Taylorforzen-in hypothesis, his results are also pertinent tospatial two-wavelength correlations. Fuks' 2 calculatedthe bichromatic covariance function as detected by asingle point detector. Baykal and Plonus13 derived thetwo-wavelength structure functions of the log amplitudeand phase for two spatially separated point sources andtwo different observation points. They showed that thetwo-wavelength structure function could be obtainedin terms of known single-wavelength structure func-tions. Recently, Tamir et al.14 derived closed formexpressions that represent the aperture averagedspectral correlation coefficient in the weak fluctuationregime. Experimentally, Gurvich et al. 15 measured thenormalized spectral cross covariance as given by

cov(X1,X2) = 32 = ([I(X1) - (I(X 1))][I(X 2 ) - ((X 2))]) (2)(I(XlO) ((X2))

over atmospheric paths of 650 and 1750 m as well asthrough 0.35- and 1.05-m layers of convectionally tur-bulent water. For Xi = 0.63 Am and X2 = 0.44 Am, theyfound that the dependence of :3 on the strength of tur-bulence is very similar to that of c- o, namely, initially:increases with the strength of turbulence, in quantita-tive agreement with the Rytov based theoretical pre-dictions, only to saturate for large integrated-pathturbulence. The saturation regime could not be ac-counted for by the available theoretical treatments, allof which erroneously assumed the fluctuating electro-

He - Ne SL

.3Nd A:YA

TURBULENTATMOSPHERE

;zz- D.-_ ) X

Fig. 1. Experimental setup: L1,L2, lasers; SL, beam forming lens;M, plane mirror; S1,S2, dichroic beam splitters; D, variable circularaperture; F1,F2, narrowband interference filters; T1,T2, focusinglenses; P1,P2, photodiodes; A1,A2, amplifiers; ADC 1,ADC2 , analog-

to-digital converters; Rockwell AIM 65, microcomputer.

magnetic field to obey a Gaussian probability distri-bution. Gurvich et al. 15 also used their experimentaldata to calculate the correlation coefficient (our p istheir K2)

P(X\1,X2) = cov(X1,X2)P ¢IGA1)al(X2)(3)

and found it to decrease from near unity at weak tur-bulence to -0.5 in strong turbulence conditions. Theserelatively high values of the correlation coefficientjustify further study of the two-wavelength spectralcovariance function.

This paper presents the results of an experimentalinvestigation of the spectral degree of correlation be-tween the intensity fluctuations of two beams ema-nating from two independent monochromatic sources:One emitting at Xi = 1.064 gim and the other at X2 =

0.6328 jam. Measurements were made in the regime ofstrong fluctuations and the effect of aperture averagingon both cov(X1.06,X0.63), and various conditional two-wavelength probability density functions were moni-tored. Section II describes the experimental arrange-ment, and the results are presented and discussed inSec. III.

II. Experimental Arrangement

A. Experimental Setup

The experimental setup is shown in Fig. 1. Thetransmitter incorporates two cw lasers, both operatingat their TEMOO modes. L is a 120-mW, 1.06-umNd3+:YAG laser emitting with a divergence angle of 3mrad. L2 is a 16-mW, 0.6328-gm He-Ne laser havingan initial divergence of 1 mrad which is then trans-formed by lens SL to a 3-mrad beam. The emissionfrom L2 is combined with the emission from LI by planemirror Ml and beam splitter SI, and the two beams arethen transmitted to the atmosphere with a diameter of-1.2 mm. For the relevant path length of 1-3 km, bothbeams can be considered spherical.

The receiver aperture is circular with a variable di-ameter, ranging from 5 to 30 mm. The two beams areseparated in the receiver by beam splitter S2 and each

2402 APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985

passes through the appropriate narrowband interfer-ence filter Fl (1.064 gm) or F2(0.6328,um). The beamsare focused by two telescopes Ti and T2 (f = 135 mm)on PIN photodiodes with integrated preamplifiers P1and P2, whose outputs are proportional to the instan-taneous intensities of the laser beams I(X106) andI(X0.63). In the focal planes of Ti and T2 we placed0.3-mm diam circular apertures to reduce the back-ground radiation. P1 and P2 are followed by linearamplifiers Al and A2, which feed two 8-bit analog-to-digital converters, ADC, and ADC2 , having a con-version time of 15 gsec. The sampling rate was 2000 Hzfor each ADC, and conversion started simultaneouslyon both channels under the control of a microcomputer,which also stored the data on a diskette. The length ofeach signal record was 5 sec, and each measurement ofthe received signals was followed by background noise,system noise, and offset measurements.

B. Data Handling and Processing

First, the dc background levels were subtracted fromthe laser signals and the corrected records were used tocompute the variance of each wavelength:

(X) = (I)()) -()2URM = ~(I(Xi))2 i = 1,2

as well as the spectral covariance functioncov(X1.06,X0.63) [Eq. (2)] and the spectral correlationcoefficient p [Eq. (3)].

We tested the experimental setup by replacing the1.064-gm narrowband interference filter Fl with anarrowband interference filter centered on 0.6328 gm.The measured spectral correlation coefficients were p(D= 10 mm) = 0.999 and p(D = 5 mm) = 0.996 for a pathlength of 1300 m with similar results for a 5-m path.Thus, any decrease in the measured value of the spectralcorrelation coefficient is completely due to the atmo-spheric turbulence.

Simultaneously with the spectral correlation mea-surements we also recorded the index-of-refractionstructure constant C2 by monitoring the normalizedvariance U2 of a He-Ne laser beam propagating over ashort path length. C2 was then derived from U2 usingthe Rytov expressions

C2- ln(1 + UY) (5)

n (0.5k7 6LI1 /6)

where L = 120 m and k = 2-7r/X, X = 0.6328 gm. Notethat Eq. (5) is accurate only for asymptotically largeFresnel zone size /L/» >> 1. However, since in ourcase 3 _ -\/uL/b _ 9, the accuracy of the above estimatefor C2 is .25%16

111. Results and Discussion

A. Spectal Covariance Function

The measurements of the intensity spectral correla-tions were performed over three days. The first day wascloudy and wind velocity was 5-8 m/sec. The experi-ment was made over 1300 m of moderately uniform pathlength. The height above the earth's surface rangedfrom 2 to 8 m, and C2 _ 10-13-10-12 m-2/3. The second

4r-2

2

0 2 4 6

Fig. 2. Measured normalized variance of intensity is plotted as afunction of the parameter 13o = [0.5C.k 7 /6L11 /6 11/2 for two aperture

sizes.

day was clear and wind velocity was 2-5 m/sec. Weused the same 1300-m path and C2 10-1210-11 in 2 /3 .The third day was also clear and wind velocity was 3-5m/sec. The experiment was carried out over a 2-30-mhigh, 3250-m long propagation path with C2 1012m-2/3. During the experiments, the Rytov parameterOo(Xm) = [0.5C2k 6L11/ 6]1/2 assumed values between1 and 12 [Xm = 2/km = (X1.06 + XO.63)/2]. In theseexperimental conditions, the beam diameters at thereceiver, in the absence of turbulence, were 4-10 m de-pending on range, and the wave coherence length, Pp ,2was of the order of 1 mm, which is also a repesentativevalue for the turbulence inner scale lo.

Figure 2 depicts the measured uI(X1.o6), oX(o.63) asa function of the appropriate /0 for two aperture sizes.It is seen that most of the measurements were made instrong turbulence conditions, 00 _ 1, where saturationlimits the magnitude of the oJ to a value between 1 and5. Such high values for the normalized variance (higherthan those of Gurvich et al. 15) were also recently re-ported by other authors.'178 While the similaritytheory' 9 predicts that U2 is a single-valued function of1%, the large scatter in Fig. 2 is consistent with previousexperimental studies. 1 7' 1 8

Since our motivation is basically practical, in Fig. 3we plot the dependence of aI(X.0 6), oi(XO.63), andcov(X1.06,X0.63) on the receiver aperture, averaged overall the pertinent data sets. Note that the vertical scalesin Fig. 3 are logarithmic and the plotted data approxi-mately fit straight lines. The dependence of a2 (X) onD is determined by the structure of the transversespatial covariance function ([I(rl) - (I)] [I(r2) - (I)]),'where I(rn) and I(r2 ) are single-wavelength intensitiesin the receiver plane. In strong turbulence, this co-variance function has two scales2 0 which can be ex-pressed in terms of the atmospheric turbulence innerscale lo and the wave coherence length pp: a small scale,pl, which is of the order of min(l 0 ,pp), and a long scale,P2, which is proportional to L/kpp. As the receiver di-ameter D increases from its zero value, there will be a

1 August 1985 / Vol. 24, No. 15 APPLIED OPTICS 2403

x 2 ( X1.06) D = 5 mm

- 01'J2 (0. 63)-D 5mm

I ( 1.06) -D =25mm2( X2 ) D = 25mmOi 0.63

0x 0

X oX0

0 _ X

*

l I l I l

10L = 1300m

I

Nb

0.1

101

0

b

5o

0.1'

10

10

20 30D (mm)

20D (mm)

30

Fig. 3. Measured normalized variance and the two-wavelength co-variance for (a) L = 1300 m and (b) L = 3250 m as a function of theaperture size. The data points shown represent averages over all sets

of measurements, with a normalized standard deviation of -15%.

significant reduction in the intensity fluctuationswhenever D passes through pl. In our experiments,though, since p, 1 mm, while D 5 mm, this initialaperture averaging, which is due to the scale pl, cannotbe observed. The effects of aperture averaging shownin Fig. 3 are due to the longer scale, P2. In the condi-tions of our experiments, P2 is of the order of 100 mm orso and approximately independent of wavelength (ppc k -7 /6 ). Thus forD 30 mm, a2 vs D, Fig. 3 shows amoderate decrease, which is almost equally steep forboth wavelengths. It was also observed that for smallvalues of D, (X,.06) decreases somewhat more rapidly

0.81

.1'A 0.60

0

B-

0.41

0.2

0

ID 0.60

0

B.

10 20

D (mm)

30

0.4k

-L L3250m

(b0 10 20 30

D (mm)

Fig. 4. Experimental correlation coefficient for (a) L = 1300 m and(b) L = 3250 m as a function of the aperture size. The data pointsrepresent averages over all sets of measurements, and the resulting

standard deviations are indicated by the bars.

than (X0.63), while the converse is true for larger valuesof D.7'5 The aperture averaging ratio Q(R) decreaseswith the range for a given R. In our measurements, Q(R= 30) 2.5, L = 1300 m and Q(R = 30) 4.5, L = 3250m. The covariance function cov(X.o6 ,X0 .63) monoton-ically decreases with D. Our results for cov(X.06,X0.63)are higher than those observed by Gurvich et al. ,15 butthis is also true for the normalized variances.

Figure 4 shows p (X1.0 6 ,X0 .6 3 ) as a function of the re-ceiver aperture D. As expected, it is a monotonicallyincreasing function, which attains fairly high valueseven for relatively small aperture sizes and shows acharacteristic scale of the order of 1-1.5 cm. Anotherillustration of this high degree of correlation is given inFig. 5, where a sample time record of the two signals is

2404 APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985

0.2 _

90 X

* ° X

0 X* 0 B

X A7 ( X 1.06)

0 1712 ( 0 .6 3 )

* Cov.(X1 06A0. 63 )

(a)l I I I

-L 1300m

(a)I . I . I

L=3250m

0

0

B B

0~~~~* U b B

0

- x a- 2 (

-0 a(X 0.6 3 )

* cov.0106 o.63

(b)I I I I I I

I.O-

. . . . . . I

. . . . . . .

O.8_

o(

-4

8 0 .

Z-B

4-

0 0.05 0.10 0.15 0.20 0.25 0.30

TIME (secI

Fig. 5. Sample time records of the two signals: (a) X = 1.064 Am,(b) X = 0.6328 /Am. The range was 1300 m and the aperture size was

20 mm.

shown for L = 1300 m. The results of the experimentover the 3250-m path [Fig. 4(a)] follow the same pattern,and PI 3250m is smaller than PI 1300m only by 10%, in-dicating a weak range dependence of p in agreementwith the results of Gurvich et al. 15

B. Conditional Probability Distribution

In the previous subsection, we experimentally es-tablished that even in strong turbulent conditions p isclose to unity. However, from a system design point ofview, it is extremely important to know the conditionalprobability distribution

P(I.06Io. 63,p) = prob[I1.06 _ I1.06; given that IO.63 _ YO.631. (6)

In other words, once the probe intensity IO.63 exceedsa predetermined threshold 10.63, P(1. 0 617o.6 3 ,p) givesthe probability that the signal beam intensity will ex-ceed another threshold, 11.06 for a given spectral corre-lation coefficient p (assuming no time delay between thepropagation of both beams).

Figures 6(a)-(c) show the experimental results forP(I,.0 6 1Io.6 3 ,p) as well as for the single-wavelengthprobabilities P[I(Xi);I] = prob[I(Xi) _7]. The single-wavelength probabilities are presented in terms of theparameter a defined by Eq. (7):

I(X) - (1(ki))a I~xs) (7)

Superimposed on these single-wavelength data aretheoretical curves of the lognormal probability, thedensity of which is given by

p(I) = [v t r2nrI.'- exp[-(lnI - (lnI))2/(2oU2j)], (8)

where ai = ln(1 + a ) and (lnl) = ln(I) - /2ei?2i andthe measured variance was used for the calculation of

2O,1nI-

While it has been established 2 7 '21 that the lognormaldistribution is a good approximation to measured data

only in the weak turbulent regime, its deviations fromstrong turbulence results are quite small, especially forreceivers with finite apertures.8 Indeed, it is evidentfrom Fig. 6 that for practical applications the lognormalmodel adequately describes the single-wavelength ex-perimental data.

The bivariate lognormal density function is givenby

P (I1.06,io.63,P) = [27r Uln sl 6lnIo.,sIl.0610.63] -1

X (1- p2 )-1 '2 exp L F2 1I-2(1 - p2) J (9)

where

F2 = n1s1.06 - ( *1.06) 2 + [In0.63 - (nIO.63) 2Ulnhm.6 I nlI.63

[lnIi.06- (nIi.O6)0 llo.63 - (nIo.63)1-2 p I I* . (10)L In1.06 J L O(Io.63 J

The joint probability distribution that both 11.06 _ I1.06and IO.63 Ž 10.63 is given by

P(I.06,Io.63,p) = f dI1.0 6 f dlo.63 p(U1.06 ,Io.6 3,P) (11)fIL.6 I.63

and the conditional probability of I1.06 to exceed 11.06,given that IO.63 _ 10.63, can be expressed by

P(I.06170.63,P) = P(7.0670.63,P)/P(7.63)- (12)

The experimentally determined conditional proba-bility of 11.06 to exceed (Il.or,) [1 + aoa1(X1.oo)], given thatIO.63 exceeds (0.63) [1 + aou1 (X0.6 3 )] (with the same a)is shown in Figs. 6(a)-(c) for three different aperturesizes in some typical runs, in comparison with the log-normal theory. It is clear that the experimental datashow higher conditional probability than that predictedby the lognormal model. As expected, the conditionalprobability increases with the correlation coefficient.Note that even when the departure from the averagevalue is extremely high, a 5, the conditional proba-bility is still high. This conclusion is also supported bythe sample records of Fig. 5: the high peaks are nicelycorrelated. Therefore, as long as only the conditionalprobability is considered, there is almost no penalty inworking with high a, i.e., high signal-to-noise conditions.However, since the probability of IO.63 to exceed (10.63) [1+ aaI(X0.63)] decreases very rapidly with a, the fre-quency of events with large a will be very small.

IV. Summary

This paper presented and discussed the results of anexperimental investigation of the spectral degree ofcorrelation between the turbulence-induced intensityfluctuations of two collinear, approximately spherical1.064-gm and 0.6328-gum beams propagating throughstrongly scattering atmospheric paths. It was estab-lished that in spite of the sizable wavelength separationand the long propagation paths, the correlation coeffi-cients between the wavelengths attained fairly highvalues, even for small apertures. Moreover, the mea-sured two-wavelength conditional probabilities werefound high enough as to justify further pursuit of thesynchronization transmission concept. The experi-

1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS 2405

0 2a

(b) I

4 6 8

0.61

Prob.

a

mental results only apply to wide enough beams, wherebeam wander effects can be neglected. Experimentaltemporal correlations, also very important to the va-lidity of the above-mentioned concept, will be thesubject of a future publication.

References1. V. I. Tatarski, "The Effect of the Turbulent Atmosphere on Wave

Propagation," Israel Program for Scientific Translations, Jeru-salem (1971).

2. R. Fante, "Electromagnetic Beam Propagation in TurbulentMedia," Proc IEEE 63, 1669 (1975).

3. R. Fante, "Electromagnetic Beam Propagation in TurbulentMedia: An Update," Proc IEEE 68, 1424 (1980).

4. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic, New York, 1978), Vol. 2.

Fig. 6. Probability, prob(Io.63 (10.63) [1 + aoI(X.6 3 )]}, (X), and theconditional probability of I1.06 to exceed (I.06) [1 + aI(X. 0 6)] giventhat I0.63 exceeds (0.63) [ + a 1(Xo.63 )] (with the same a) (solid cir-cles) as a function of a for L = 1300 m. The solid and dashed linesrepresent the lognormal models of Eqs. (8) and (12), respectively.Each record length was 10,000 points. Also note that the receivedintensity is always positive. (a) D = 10 mm; (b) D = 20 mm; (c) D

30 mm.

5. G. E. Homstad, J. W. Strohbehn, R. H. Berger, and J. M. Hene-ghan, "Aperture-Averaging Effects for Weak Scintillations," J.Opt. Soc. Am. 64, 162 (1974).

6. W. P. Brown, Jr., "Fourth Moment of a Wave Propagating in aRandom Medium," J. Opt. Soc. Am. 62, 966 (1972).

7. J. R. Dunphy and J. R. Kerr, "Scintillation Measurments forLarge Integrated-Path Turbulence," J. Opt. Soc. Am. 63, 981(1973).

8. D. L. Fried, G. E. Mevers, and M. P. Keister, Jr., "Measurementsof Laser-Beam Scintillation in the Atmosphere," J. Opt. Soc. Am.57, 787 (1967).

9. R. E. Hufnagel, "Restoration of Atmospherically Degraded Im-ages" (National Academy of Sciences, Washington, D.C., 1966),Vol. 3, Appendix 2, p. 11.

10. D. L. Fried, "Probability of Getting a Lucky Short-ExposureImage Through Turbulence," J. Opt. Soc. Am. 68, 1651 (1978).

2406 APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985

0.8[

0.6f

RUN 365OD 20mm

or2 ( 110 6 ) '.8

\ \`a (X.63)l \* p 0.8 9

- I \\s eI

ON X~log - normal models

I

Prob.

0.4

0.2[

a

I 1

17801643

11. A. Ishimaru, "Temporal Frequency Spectra of MultifrequencyWaves in Turbulent Atmosphere," IEEE Trans. Antennas Pro-pag. AP-20, 10 (1972).

12. I. M. Fuks, "Correlation of the Fluctuations of Frequency SpacedSignals in a Randomly Inhomogeneous Medium," Izv. Vyssh.Uchebn. Zaved. Radiofiz. 17, 1665 (1974).

13. Y. Baykal and M. A. Plonus, "Two-Source, Two-FrequencySpherical Wave Structure Functions in Atmospheric Turbu-lence," J. Opt. Soc. Am. 70, 1278 (1980).

14. M. Tamir, E. Azoulay, S. Tsur, and U. Halavee, "Aperture-Av-eraged Spectral Correlations of Beams in a Turbulent Atmo-sphere," Appl. Opt. 23, 2359 (1984).

15. A. S. Gurvich, V. Kan, and V. Pokasov, "Two-Frequency Fluc-tuations of Light Intensity in a Turbulent Medium," Opt. Acta26, 555 (1979).

16. R. J. Hill and S. F. Clifford, "Modified Spectrum of Atmospheric

Temperature Fluctuations and Its Application to Optical Prop-agation," J. Opt. Soc. Am. 68, 892 (1978).

17. R. L. Phillips and L. C. Andrews, "Measured Statistics ofLaser-Light Scattering in Atmospheric Turbulence," J. Opt. Soc.Am. 71, 1440 (1981).

18. G. Parry, "Measurement of Atmospheric Turbulence InducedIntensity Fluctuations in a Laser Beam," Opt. Acta 28, 715(1981).

19. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, and V. V. Pok-asov, "Similarity Relations for Strong Fluctuations of the In-tensity of Light Propagating in a Turbulent Medium," Sov. Phys.JETP 40, 1011 (1975).

20. R. J. Hill, "Theory of Saturation of Optical Scintillation by StrongTurbulence: Plane-Wave Variance and Covariance and Spher-ical-Wave Covariance," J. Opt. Soc. Am. 72, 212 (1982).

21. G. Parry and P. N. Pusey, "K Distributions in AtmosphericPropagation of Laser Light," J. Opt. Soc. Am. 69, 796 (1979).

Patter continued from page 2368

Complementary-logic fault detectorA circuit for checking two-line complementary-logic bits for single faults is

used as a building block for a self-checking memory interface for Hamming-coded data. It is intended for such applications as fault-tolerant computing,data handling, and data transmission. The circuit performs an exclusive-ORfunction.

Two-line complementary logic uses redundancy to provide an error check.Suppose that two signals are denoted A and B, with the four conductors andtheir logic states denoted ao, 0l, bo, and b1, respectively. Line A is said to bein the logic 1 state when the voltage on its conductors corresponds to ao = 0,a1 = 1. The logic 0 state of line A is a0 = 1, a= 0. Similarly, the logic 1 andO states of line B are bo = 0, bl = 1 and bo = 1, b1 = 0, respectively. All otherstates indicate a fault condition.

When the circuits are operating correctly, ao and a1 or bo and b, are in op-posite logic states at all times. Erroneous conditions include both lines in the0 state, both lines in the 1 state, or both conductors in the same line at the 0 or1 voltage level. The only correct states are the two shown at the top of Fig. 19.Any other combination of logic levels on the four conductors represents afault.

The circuit shown in Fig. 20 accepts inputs from lines A and B and gives anoutput on line C depending on the condition of the inputs. If lines A and B arein either of the two correct states, the output is A exclusive-OR B. An erroneousinput combination gives rise to one of the output error indications shown in thetable.

The circuit can also be used to complement (invert) a bit signal. This featurecan be used to correct an error if the error can be attributed to the proper bitline. With the proper Hamming code, a single error can be corrected by con-verting the data to the configuration that requires the least change from theerroneous configuration. Many such circuits can be combined to produce acomplete memory interface with both detection and correction abilities.

Fig. 19. Only two logic states are correct for the four conductors inlines A and B. These are shown at the top of the table. Any othercombination of logic levels signifies an error in data processing ortransmission. The circuit of the figure produces an output indicativeof the correctness or of the type of error in the logic levels on the

four conductors.

Fig. 20. Signals A(ao, al) and B(bo, bl) are fed to the new circuit.An error indication is obtained for any input combination other than

(ao, a,; bo, bi) = (0, 1; 1, 0) or (1, 0; 0, 1).

This work was done by John C. Wawrzynek of Caltech for NASA's Jet Pro-pulsion Laboratory. Refer to NPO-15410.

continued on page 2422

1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS 2407

CONVENTIONALEXCLUSiVE OR

B

NEWEXCLUSIVE OR

bo b1

a00-

h19

-cg

, n transistor conducts when gate Is high.p tranalstor conducts when gate Is low.

Ai