Aperture averaging of the two-wavelength intensity covariance

Aperture averaging of the two-wavelength intensity
covariance 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 experimentally determined for visible (0.63 Arm)and infrared (1.06 Am) collinear, approximately spherical beams which
propagated through the earth's turbulent atmosphere. Range varied from 1300 to 3250 m, and due to the
prevailing atmospheric conditions, most measurements were made in the strong turbulence regimes. Results 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 experimental 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, experiences intensity fluctuations which significantly limit
the performance of line-of-sight communication systems and certain lidar configurations. The normalized
variance of the fluctuations of the intensity I(Xi), also
called 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-index
variations in the atmosphere, up to a saturation value
of the order of unity.4 In practice, the near-unity saturation value of a amounts to many decibels (dB) of
fluctuations, and systems designers must do their best
to reduce the adverse effects of intensity fluctuations
on the dynamic range and sensitivity of their systems.
One way to average over these fluctuations, thereby
reducing their corresponding U2, is to use large optical
apertures at the receiver end of the system. In weak
turbulence conditions, Homstad et al.5 have succeeded
in correlating theoretical predictions, based on the
M. Tur is with Tel AvivUniversity, Schoolof Engineering, Ramat
Aviv, P.O. Box 39040, Tel Aviv 69978, Israel; the other authors are
with 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 to
show 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) was
found to be a decreasing function of the ratio R
/XE,
where X is the optical wavelength, L is the propagation
length, and \/XE is the spatial correlation distance of
the 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 turbulence conditions, large receiving apertures can indeed
improve the signal-to-noise ratio of the receiver.
However, this technique cannot be extended to propagation ranges falling in the strong turbulence regime.
It has been determined, both theoretically6 and experimentally,7 8 that the covariance function of the intensity fluctuations for large integrated-path turbulence
is characterized by two transverse scale sizes: a short
one that governs the initial fast drop of the covariance
function and a fairly long scale that characterizes the
long tail of that function. This long tail defeats effective aperture averaging of strong turbulence signals and
renders it economically impractical.
A more suitable (but complicated) way to avoid the
deteriorating effects of the random index-of-refraction
variations of the turbulent atmosphere is to synchronize
the transmission of high-density packets of information
to those instants when the atmospheric random
index-of-refraction field attains a fairly uniform realization which exhibits very good transmission. Thus,
if a continuously monitored probe beam was found to
pass the relevant propagation path with an instantaneous high signal-to-noise ratio, a burst of the useful
information should be immediately sent to take advantage 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 of
Hufnagel and Fried 9 10 : A series of short exposures of
an atmospherically distorted image contains, with a
small but finite probability, some images which are essentially diffraction-limited. Successful and practical
exploitation of the synchronization method heavily
depends on the ability of the transmission system to
respond to the results of the probe beam within a time
interval much shorter than the period, tc, over which the
He - Ne
and signal beams would come from different lasers with
possibly different wavelengths. The probe beam source.
could be a cheap low-power He-Ne laser while a pulsed
Nd3+:YAGsystem would be used for the transmission
of the information. Two lasers with different wavelengths can be used only if there is a substantial correlation between the intensity fluctuations of the two
beams. This work is concerned with the experimental
investigation of the effect of aperture averaging on the
degree of correlation between the intensity fluctuations
of waves with different wavelengths, propagating
.3
Nd A:YA
spatial two-wavelength correlations.
Fuks' 2 calculated
the bichromatic covariance function as detected by a
single point detector. Baykal and Plonus13 derived the
two-wavelengthstructure functions of the log amplitude
and phase for two spatially separated point sources and
two different observation points. They showed that the
two-wavelength structure function could be obtained
in terms of known single-wavelength structure functions. Recently, Tamir et al.14 derived closed form
expressions that represent the aperture averaged
spectral correlation coefficient in the weak fluctuation
regime. Experimentally, Gurvich et al. 15 measured the
normalized spectral cross covariance as given by
cov(X1 ,X2)=
32
over atmospheric
= ([I(X1) - (I(X 1))][I(X 2 ) - ((X 2))])
(I(XlO)
((X2))
(2)
paths of 650 and 1750 m as well as
through 0.35- and 1.05-m layers of convectionally turbulent water. For Xi = 0.63 Am and X2
=
0.44 Am, they
found that the dependence of :3on the strength of turbulence is very similar to that of o,
c- namely, initially:
increases with the strength of turbulence, in quantitative agreement with the Rytov based theoretical predictions, only to saturate for large integrated-path
turbulence. The saturation regime could not be accounted for by the available theoretical treatments, all
of which erroneously assumed the fluctuating electro2402
APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985
;zz-
D
) X
Fig. 1. Experimental setup: L1,L2, lasers; SL, beam forming lens;
M, plane mirror; S1,S2, dichroic beam splitters; D, variable circular
aperture; F1,F2, narrowband interference filters; T1,T2, focusing
lenses; P1,P2, photodiodes; A1,A2, amplifiers; ADC 1,ADC2 , analogto-digital converters; Rockwell AIM 65, microcomputer.
magnetic field to obey a Gaussian probability distribution. Gurvich et al. 15 also used their experimental
data to calculate the correlation coefficient (our p is
their K2 )
P ¢IGA1)al(X2)
P(X\1,X2) = cov(X1,X2)
through atmospheric paths of 1300 and 3250 m in strong
turbulence conditions.
Several theoretical relevant results, applicable to
unsaturated paths, are known. Ishimaru"l formulated
general expressions for the temporal frequency spectra
of plane, spherical, and beam waves operating at two
different wavelengths and showed how these spectra
could be used to infer the wind velocity and the indexof-refraction structure constant C2. Using the Taylor
forzen-in hypothesis, his results are also pertinent to
.-_
TURBULENT
ATMOSPHERE
atmosphere stays in its supertransmission status. Since
tc is of the order of a millisecond, the bursts must be
very short and, for high data rate transmission, also very
powerful. Thus, it could be very convenient if the probe
SL
(3)
and found it to decrease from near unity at weak turbulence to -0.5 in strong turbulence conditions. These
relatively high values of the correlation coefficient
justify further study of the two-wavelength spectral
covariance function.
This paper presents the results of an experimental
investigation of the spectral degree of correlation between the intensity fluctuations of two beams emanating from two independent monochromatic sources:
One emitting at Xi = 1.064gim and the other at X2 =
0.6328 jam. Measurements were made in the regime of
strong fluctuations and the effect of aperture averaging
on both cov(X1.
06,X0.63), and various conditional twowavelength probability density functions were monitored. Section II describes the experimental arrangement, and the results are presented and discussed in
Sec. III.
II.
Experimental Arrangement
A.
Experimental Setup
The experimental setup is shown in Fig. 1. The
transmitter incorporates two cwlasers, both operating
at their TEMOO modes.
L
is a 120-mW, 1.06-um
Nd3 +:YAG laser emitting with a divergence angle of 3
mrad. L2 is a 16-mW, 0.6328-gm He-Ne laser having
an initial divergence of 1 mrad which is then transformed by lens SL to a 3-mrad beam. The emission
from L2 is combined with the emission from LI by plane
mirror Ml and beam splitter SI, and the two beams are
then transmitted to the atmosphere with a diameter of
-1.2 mm. For the relevant path length of 1-3 km, both
beams can be considered spherical.
The receiver aperture is circular with a variable diameter, ranging from 5 to 30 mm. The two beams are
separated in the receiver by beam splitter S2 and each
passes through the appropriate narrowband interference filter Fl (1.064 gm) or F2(0.6328,um).
x
The beams
are focused by two telescopes Ti and T2 (f = 135 mm)
on PIN photodiodes with integrated preamplifiers P1
and P2, whose outputs are proportional to the instantaneous intensities of the laser beams I(X106 ) and
I(X0 .63 ). In the focal planes of Ti and T2 we placed
0.3-mm diam circular apertures to reduce the background radiation. P1 and P2 are followed by linear
amplifiers Al and A2, which feed two 8-bit analog-
- 01'J2 (0.
each signal record was 5 sec, and each measurement of
the received signals was followed by background noise,
system noise, and offset measurements.
Data Handling and Processing
B.
D5= mm
6 3 )-D
5mm
I ( 1.06)-D =25mm
4
Oi2(
X2
0.63 )
D =25mm
0
r-2
x
0
X
_
2
0
oX
X
0
to-digital converters, ADC, and ADC2 , having a conversion time of 15 gsec. The sampling rate was 2000 Hz
for each ADC, and conversion started simultaneously
on both channels under the control of a microcomputer,
which also stored the data on a diskette. The length of
2 ( X1.06)
*
l
0
I
l
2
I
4
l
6
Fig. 2. Measured normalized variance of intensity is plotted as a
function of the parameter 13o = [0.5C.k 7 /6L11 /6 11/2 for two aperture
sizes.
First, the dc background levels were subtracted from
the laser signals and the corrected records were used to
compute the variance of each wavelength:
(X) = (I)()) -()2
i = 1,2
URM
= ~(I(Xi))2
as well as the spectral covariance function
cov(X1.06 ,X0.63) [Eq. (2)] and the spectral correlation
coefficient p [Eq. (3)].
We tested the experimental setup by replacing the
1.064-gm narrowband interference filter Fl with a
narrowband 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 path
length of 1300 m with similar results for a 5-m path.
Thus, any decrease in the measured value of the spectral
correlation coefficient is completely due to the atmospheric turbulence.
Simultaneously with the spectral correlation measurements we also recorded the index-of-refraction
structure constant C2 by monitoring the normalized
variance U2of a He-Ne laser beam propagating over a
short path length. C2was then derived from U2using
the Rytov expressions
C2-
n
ln(1 + UY)
(5)
(0.5k7 6LI1 /6)
where L = 120 m and k = 2-7r/X,X = 0.6328 gm. Note
that Eq. (5) is accurate only for asymptotically large
Fresnel zone size /L/»
>> 1. However, since in our
case 3 _ -\/uL/b _ 9, the accuracyof the above estimate
for C2 is .25%16
111. Results and Discussion
A.
Spectal Covariance Function
day was clear and wind velocity was 2-5 m/sec. We
used the same 1300-mpath and C2 10-1210-11 in 2 /3 .
The third day was also clear and wind velocity was 3-5
m/sec. The experiment was carried out over a 2-30-m
high, 3250-m long propagation path with C2 1012
m-2/3. During the experiments, the Rytov parameter
Oo(Xm) = [0.5C2k 6L11/ 6 ]1/2 assumed values between
1 and 12 [Xm = 2/km = (X1 .06 + XO.63)/2]. In these
experimental conditions, the beam diameters at the
receiver, in the absence of turbulence, were 4-10 m depending on range, and the wave coherence length, Pp,2
was of the order of 1 mm, which is also a repesentative
value for the turbulence inner scale lo.
Figure 2 depicts the measured uI(X 1.o6), oX(o.63)as
a function of the appropriate
/0 for two aperture sizes.
It is seen that most of the measurements were made in
strong turbulence conditions, 00 _ 1, where saturation
limits the magnitude of the oJto a value between 1 and
5. Such high values for the normalized variance (higher
than those of Gurvich et al. 15) were also recently reported by other authors.'17 8 While the similarity
theory' 9 predicts that U2 is a single-valued function of
1%,the large scatter in Fig. 2 is consistent with previous
experimental studies. 1 7' 1 8
Since our motivation is basically practical, in Fig. 3
we plot the dependence of aI(X.0 6 ), oi(XO.63 ), and
cov(X1 .06 ,X0.63) on the receiver aperture, averaged over
all the pertinent data sets. Note that the vertical scales
in Fig. 3 are logarithmic and the plotted data approximately fit straight lines. The dependence of a2 (X)on
D is determined by the structure of the transverse
spatial covariance function ([I(rl) - (I)] [I(r2) - (I)]),'
where I(rn) and I(r2 ) are single-wavelength intensities
in the receiver plane. In strong turbulence, this co-
The measurements of the intensity spectral correlations wereperformed over three days. The first day was
cloudy and wind velocity was 5-8 m/sec. The experi-
variance function has two scales2 0 which can be ex-
ment was made over 1300 m of moderately uniform path
pl, which is of the order of min(l 0 ,pp), and a long scale,
length. The height above the earth's surface ranged
P2, which is proportional
from 2 to 8 m, and C2 _ 10-13-10-12 m-2/3. The second
pressed in terms of the atmospheric turbulence inner
scale lo and the wave coherence length pp: a small scale,
to L/kpp. As the receiver diameter D increases from its zero value, there will be a
1 August 1985 / Vol. 24, No. 15
APPLIED OPTICS
2403
10
I.O-
L = 1300m
0.81
9
*
.1
'A
0
0X
°
X
0
X
0*
0.6
0
B
B-
I
0.41
X
Nb
0
A7 ( X 1.06)
1712
( 0 .6 3 )
-L
1300m
0.2
* Cov.(X1 06A0. 63 )
0
.
.
I
.
.
10
I
.
.
20
.
.
I
(a)
I
30
D (mm)
l
0.1
5o
I
I
(a)
I
10
20
30
D (mm)
101
L=3250m
O.8_
0
0
B
B
*
U
ID0.6
b
0~~~~
0
B
0
B.
0
0.4k
0
-
x a- 2 (
-L
b
-0 a(X
0.6 3 )
* cov.0106
L3250m
0.2 _
o.63
(b
0
10
D
(b)
0.1'
.
(o
I
.
I
.
10
I
I
.
.
20
I
.
I
.
30
D (mm)
20
(mm)
30
Fig. 4. Experimental correlation coefficient for (a) L = 1300 m and
(b) L = 3250 m as a function of the aperture size. The data points
represent averages over all sets of measurements, and the resulting
standard deviations are indicated by the bars.
Fig. 3. Measured normalized variance and the two-wavelength covariance for (a) L = 1300 m and (b) L = 3250 m as a function of the
aperture size. The data points shown represent averages over all sets
of measurements, with a normalized standard deviation of -15%.
than
significant reduction in the intensity fluctuations
whenever D passes through pl. In our experiments,
with the range for a given R. In our measurements, Q(R
= 30) 2.5, L = 1300 m and Q(R = 30) 4.5, L = 3250
m. The covariance function cov(X.o6 ,X0 .63) monoton-
though, since p,
1 mm, while D
5 mm, this initial
aperture averaging, which is due to the scale pl, cannot
be observed. The effects of aperture averaging shown
in Fig. 3 are due to the longer scale, P2. In the conditions of our experiments, P2 is of the order of 100 mm or
so and approximately independent of wavelength (pp
c k - 7 /6 ). Thus forD
30 mm, a2 vs D, Fig. 3 shows a
moderate decrease, which is almost equally steep for
both wavelengths. It was also observed that for small
values of D, (X,.0 6) decreases somewhat more rapidly
2404
APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985
(X0.63),
while the converse is true for larger values
of D.7'5 The aperture averaging ratio Q(R) decreases
ically decreases with D. Our results for cov(X.06,X0.63)
are higher than those observed by Gurvich et al. ,15 but
this 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 monotonically
increasing function, which attains fairly high values
even for relatively small aperture
sizes and shows a
characteristic scale of the order of 1-1.5 cm. Another
illustration of this high degree of correlation is given in
Fig. 5, where a sample time record of the two signals is
only in the weak turbulent regime, its deviations from
strong turbulence results are quite small, especially for
receivers with finite apertures. 8 Indeed, it is evident
from Fig. 6 that for practical applications the lognormal
model adequately describes the single-wavelength experimental data.
The bivariate lognormal density function is given
by
-4
8
0
.
P (I1.06,io.63,P)
= [27rUln sl6lnIo.,sIl.0610.63]
-1
L
1 2
X (1- p2 )- ' exp
F
I-2(1
Z-B
2
1
(9)
.
(10)
- p2) J
where
4-
F2 =
n1s1.06- (
*1.06)
2 + [In0.63- (nIO.63) 2
0.05
0
0.15
0.10
TIME
0.20
0.25
-2 p
0.30
[lnIi.06- (nIi.O6)0
L
(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.
B.
In the previous subsection, we experimentally established that even in strong turbulent conditions p is
close to unity. However, from a system design point of
view, it is extremely important to know the conditional
probability distribution
_ I1.06; given that IO.63 _
YO.631.
(6)
In other words, once the probe intensity IO.63exceeds
threshold 10.63,P(1.
a predetermined
0 6 17o.6 3 ,p)
gives
the probability that the signal beam intensity will exceed another threshold, 11.06 for a given spectral correlation coefficient p (assuming no time delay between the
propagation of both beams).
Figures 6(a)-(c) show the experimental results for
P(I,.0 6 1Io.6 3 ,p) as well as for the single-wavelength
probabilities P[I(Xi);I] = prob[I(Xi) _7]. The single-
wavelength probabilities are presented in terms of the
parameter a defined by Eq. (7):
I(X) - (1(ki))
a I~xs)
(7)
Superimposed on these single-wavelength data are
theoretical curves of the lognormal probability, the
density of which is given by
p(I)
where ai
=
t
(8)
r2nrI.'-exp[-(lnI - (lnI))2/(2oU2j)],
ln(1 + a ) and (lnl) = ln(I) - /2ei?2iand
= [v
-
(nIo.63)1
O(Io.63
J
f
dI1.0 6
f
dlo. 63 p(U1.06 ,Io.6 3,P)
(11)
I.63
and the conditional probability of I1.06to 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 probability of 11.06 to exceed (Il.or,) [1 + aoa1(X1.oo)],given that
IO.63exceeds (0.63) [1 + aou 1 (X0.6 3 )] (with the same a)
Conditional Probability Distribution
P(I.06Io. 63 ,p) = prob[I1.06
I I*
JL
In1.06
fIL.6
with the results of Gurvich et al. 15
llo.63
The joint probability distribution that both 11.06_ I1.06
and IO.63Ž 10.63 is given by
P(I.06,Io.63,p) =
shown for L = 1300 m. The results of the experiment
over the 3250-m path [Fig. 4(a)] follow the same pattern,
and PI3250mis smaller than PI 1300monly by 10%, indicating a weak range dependence of p in agreement
nlI.63
I
Ulnhm.6
the measured variance was used for the calculation of
2
O,1nI-
While it has been established 2 7 '21that the lognormal
distribution is a good approximation to measured data
is shown in Figs. 6(a)-(c) for three different aperture
sizes in some typical runs, in comparison with the lognormal theory. It is clear that the experimental data
show higher conditional probability than that predicted
by the lognormal model. As expected, the conditional
probability increases with the correlation coefficient.
Note that even when the departure from the average
value is extremely high, a 5, the conditional probability is still high. This conclusion is also supported by
the sample records of Fig. 5: the high peaks are nicely
correlated. Therefore, as long as only the conditional
probability is considered, there is almost no penalty in
working with high a, i.e., high signal-to-noise conditions.
However, since the probability of IO.63 to exceed (10.63)[1
+ aaI(X0.6 3)] 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 an
experimental investigation of the spectral degree of
correlation between the turbulence-induced intensity
fluctuations of two collinear, approximately spherical
1.064-gm and 0.6328-gumbeams propagating through
strongly scattering atmospheric paths. It was established that in spite of the sizable wavelengthseparation
and the long propagation paths, the correlation coefficients between the wavelengths attained fairly high
values, even for small apertures. Moreover, the measured two-wavelength conditional probabilities were
found high enough as to justify further pursuit of the
synchronization transmission concept. The experi1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS
2405
I1
(b)
RUN 365
20mm
OD
or2 ( 110 6 )
0.8[
0.6f
Prob.
I
'.81780
1643
(X.63)
p 0 .8 9
\\`a
l \*
- I\\s eI
ON
X~
0.4
log - normal models
0.2[
I
0
2
a
a
4
6
8
0.61
Prob.
Fig. 6. Probability, prob(Io.63 (10.63) [1 + aoI(X.6 3 )]}, (X), and the
conditional probability of I1.06 to exceed (I.06) [1 + aI(X. 0 6)] given
that I0.63 exceeds (0.63) [ + a 1(Xo.63 )] (with the same a) (solid circles) as a function of a for L = 1300 m. The solid and dashed lines
represent the lognormal models of Eqs. (8) and (12), respectively.
Each record length was 10,000points. Also note that the received
intensity is always positive.
a
mental results only apply to wide enough beams, where
beam wander effects can be neglected. Experimental
temporal correlations, also very important to the validity of the above-mentioned concept, will be the
subject of a future publication.
References
1. V. I. Tatarski, "The Effect of the Turbulent Atmosphere on Wave
Propagation," Israel Program for Scientific Translations, Jerusalem (1971).
2. R. Fante, "Electromagnetic Beam Propagation in Turbulent
Media," Proc IEEE 63, 1669 (1975).
3. R. Fante, "Electromagnetic Beam Propagation in Turbulent
Media:
An Update," Proc IEEE 68, 1424 (1980).
4. A. Ishimaru, Wave Propagation and Scattering in Random
Media (Academic, New York, 1978), Vol. 2.
2406
APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985
(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 a
Random Medium," J. Opt. Soc. Am. 62, 966 (1972).
7. J. R. Dunphy and J. R. Kerr, "Scintillation Measurments for
Large Integrated-Path Turbulence," J. Opt. Soc. Am. 63, 981
(1973).
8. D. L. Fried, G. E. Mevers, and M. P. Keister, Jr., "Measurements
of Laser-Beam Scintillation in the Atmosphere," J. Opt. Soc. Am.
57, 787 (1967).
9. R. E. Hufnagel, "Restoration of Atmospherically Degraded Images" (National Academy of Sciences, Washington, D.C., 1966),
Vol. 3, Appendix 2, p. 11.
10. D. L. Fried, "Probability of Getting a Lucky Short-Exposure
Image Through Turbulence,"
J. Opt. Soc. Am. 68, 1651 (1978).
11. A. Ishimaru, "Temporal Frequency Spectra of Multifrequency
Waves in Turbulent Atmosphere," IEEE Trans. Antennas Propag. AP-20, 10 (1972).
12. I. M. Fuks, "Correlation of the Fluctuations of Frequency Spaced
Signals in a Randomly Inhomogeneous Medium," Izv. Vyssh.
Uchebn. Zaved. Radiofiz. 17, 1665 (1974).
13. Y. Baykal and M. A. Plonus, "Two-Source, Two-Frequency
Spherical Wave Structure Functions in Atmospheric Turbulence," J. Opt. Soc. Am. 70, 1278 (1980).
14. M. Tamir, E. Azoulay, S. Tsur, and U. Halavee, "Aperture-Averaged Spectral Correlations of Beams in a Turbulent Atmosphere," Appl. Opt. 23, 2359 (1984).
15. A. S. Gurvich, V. Kan, and V. Pokasov, "Two-Frequency Fluc-
Temperature Fluctuations and Its Application to Optical Propagation," J. Opt. Soc. Am. 68, 892 (1978).
17. R. L. Phillips and L. C. Andrews, "Measured Statistics of
Laser-Light Scattering in Atmospheric Turbulence," J. Opt. Soc.
Am. 71, 1440 (1981).
18. G. Parry, "Measurement of Atmospheric Turbulence Induced
Intensity 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 Intensity of Light Propagating in a Turbulent Medium," Sov. Phys.
JETP 40, 1011 (1975).
20. R. J. Hill, "Theory of Saturation of Optical Scintillation by Strong
Turbulence: Plane-Wave Variance and Covarianceand Spher-
tuations of Light Intensity in a Turbulent Medium," Opt. Acta
26, 555 (1979).
16. R. J. Hill and S. F. Clifford, "Modified Spectrum of Atmospheric
ical-Wave Covariance," J. Opt. Soc. Am. 72, 212 (1982).
21. G. Parry and P. N. Pusey, "K Distributions in Atmospheric
Propagation of Laser Light," J. Opt. Soc. Am. 69, 796 (1979).
Pattercontinuedfrompage2368
B
Complementary-logicfault detector
A circuit for checkingtwo-line complementary-logicbits for single faults is
used as a building block for a self-checkingmemory interface for Hammingcoded data. It is intended for such applications as fault-tolerant computing,
data handling, and data transmission. The circuit performs an exclusive-OR
function.
Two-line complementarylogic uses redundancy to provide an error check.
Suppose that two signals are denoted A and B, with the four conductors and
their logicstates denoted ao, 0l, bo,and b1, respectively. Line A is said to be
in the logic1 state when the voltage on its conductors corresponds to ao = 0,
a1 = 1. The logic0 state of line A is a0 = 1,a= 0. Similarly,the logic1 and
Ostates of line B are bo = 0, bl = 1 and bo = 1, b1 = 0, respectively. All other
states indicate a fault condition.
When the circuits are operating correctly, ao and a1 or boand b, are in opposite logicstates at all times. Erroneous conditions include both lines in the
0 state, both lines in the 1state, or both conductors in the same line at the 0 or
1 voltagelevel. The onlycorrect states are the two shownat the top of Fig. 19.
Any other combination of logic levels on the four conductors represents a
fault.
The circuit shownin Fig. 20accepts inputs from lines A and B and givesan
output on line C depending on the condition of the inputs. If lines A and B are
in either of the twocorrect states, the output is A exclusive-OR B. An erroneous
input combinationgivesrise to one of the output error indications shownin the
table.
The circuit can alsobe used to complement(invert)a bit signal. This feature
can be used to correct an error if the error can be attributed to the proper bit
line. With the proper Hamming code, a single error can be corrected by converting the data to the configuration that requires the least change from the
erroneous configuration. Many such circuits can be combined to produce a
complete memory interface with both detection and correction abilities.
NEW
EXCLUSIVEOR
bo
b1
CONVENTIONAL
EXCLUSiVEOR
Ai
h1
9
-cg
a00-
,
ntransistor conducts when gate Is high.
p tranalstor conducts when gate Is low.
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).
Fig. 20.
Fig. 19. Only two logicstates are correct for the four conductors in
lines A and B. These are shown at the top of the table. Any other
This work was done by John C. Wawrzynekof Caltech for NASA'sJet Propulsion Laboratory. Refer to NPO-15410.
combination of logic levels signifies an error in data processing or
transmission. The circuit of the figure produces an output indicative
of the correctness or of the type of error in the logic levels on the
four conductors.
continuedonpage2422
1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS
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