Optical propagation in laboratory-generated turbulence

Optical propagation in laboratory-generated turbulence

Richard A. Elliott, J. Richard Kerr, and Philip A. Pincus

A versatile and useful facility for simulating the effects of atmospheric turbulence on optical propagation is

described, and the relevant system parameters are characterized. The scattering medium is a turbulent liq-

uid (ethanol) with the turbulence created by unstable convection generated by a strong vertical thermal gra-

dient. Measurements of the structure function and the spatial spectrum of the resulting refractive index

fluctuations are presented and compared with the theoretically predicted forms. The effects of this scatter-

ing medium on laser beam propagation are determined and compared to the first-order Rytov theory. In

particular, probability density functions, moments, and spatial covariance functions of the irradiance re-

sulting from propagation through the system with a variety of turbulence levels and path lengths are pre-

sented.

1. Introduction

The deleterious effects that propagation througha randomly inhomogeneous medium has on an initially

coherent optical signal can only be described in statis-

tical terms. The probability distribution of the irra-diance fluctuations, its moments, and the spatial andtemporal covariance functions are examples of quan-

tities which can be measured and which are important

for designing optical systems operating under suchconditions. A considerable effort has been expendedin recent years in attempts to understand the relation-ship between the statistical properties of the irradianceand measurable characteristics of the random medium.These characteristics are again statistical propertiessuch as the variance of the index of refraction inhomo-geneities and their spatial spectrum.

The results of this effort have been mixed. Themethod of smooth perturbations or Rytov theory' hasbeen highly successful for weak scattering and relatively

short propagation paths, where the normalized varianceof the irradiance ' << 1, but it is entirely erroneous forlonger paths where the experimental results show Aosaturating to a value of order unity rather than in-creasing monotonically. 2-4 More modern works based

on the extended Huygen-Fresnel principle 5 -7 or ap-

proximate solutions for the fourth-order coherence

When this work was done all authors were with Oregon Graduate

Center, Beaverton, Oregon 97005; Philip Pincus is now with Floating

Point Systems, P.O. Box 23489, Portland, Oregon 97225.

Received 20 April 1979.

0003-6935/79/193315-09$00.50/0.©0 1979 Optical Society of America.

function8 9 have certainly elucidated the problem, butthe analytical expressions are complex and it is difficultto obtain precise results without extensive computa-tion.

There are also problems involved in obtaining good

experimental data. Most of the experimental work hasbeen done in the atmosphere because of the potentialimpact on optical communication and target illumina-tion systems. However, it is difficult to realize a uni-form, stable, well-characterized propagation path ofsufficient length to generate strong scattering. For thisreason a model system that allows experiments in thelaboratory under precise control is desirable.

There has recently been significant activity in thisarea. Experiments on laboratory-generated turbulencehave been performed by Bissonette' 0 and Gurvich etal. 1 Both groups heated a liquid from below andstudied the effect on a laser beam propagating throughthe convective turbulence. The use of a liquid ratherthan a gas is advantageous because the change in indexof refraction with temperature (dn/dT) is greater byseveral orders of magnitude, leading to much strongerscattering. A study on a turbulent heated air flow hasalso been reported recently,12 but no optical measure-ments were made on that system. Bissonette measuredthe temporal spectrum of the temperature fluctuationsand the structure function of temperature, DT(r) =

I T(ro + r) - T(ro) 12, and reported no disagreementwith the Kolmogorov spectrum. However, since therewas no net flow of the water in his scattering tank, theinferred spatial spectrum is questionable becauseTaylor's hypothesis is not valid.13 Also, the probes with

which the temperature fluctuations were recorded couldnot resolve scale sizes less than 2 mm, while scale sizes

as small as 1 mm may well be present. Unfortunatelythe Russian group made no attempt to measure the

1 October 1979 / Vol. 18, No. 19 / APPLIED OPTICS 3315

spatial spectrum of temperature fluctuations directlybut assumed a form for the spectrum and determinedthe parameters involved by measuring the thermalgradient and heat flux through the cell.

There has also been progress on the analytical side.Hill had made a more complete study of the spatialspectrum of index of refraction fluctuations in turbulentfluids. He has considered not only fluctuations due totemperature variations in air and liquids, but also thosedue to humidity fluctuations in air and salinity fluctu-ations in water.14,15

From out point of view the most significant of hisfindings is that the ubiquitous Kolmogorov K-513 spec-trum does not adequately describe the index of refrac-tion spatial spectrum but must be modified at high wavenumbers. That is, the -5/3 inertial-convective sub-range of the spectrum extends up to K (10,q)-, where11 = ( 3 /E)1/4 is the Kolmogorov microscale, v being thekinematic viscosity and e the rate of dissipation of me-chanical energy. Beyond this point and up to K (Pr)1 /2/5.51 there is a viscous-convective subrange K- 1. For higher wave numbers, in the viscous-dissipa-tive subrange, the spectrum falls off approximatelyexponentially. The quantity Pr is the Prandtl number,Pr = vD, where D is the diffusivity of the quantitycausing the index change. Hill and Clifford have alsocalculated the effect this spectral change induces in thestatistical properties of the irradiance and show that itcan often be significant in cases of practical in-terest. 16

11. Experimental SystemThe scattering apparatus is illustrated schematically

in Fig. 1. The light source is a 50-mW He-Ne laser(Spectra Physics model 125) followed by a beam-ex-panding telescope. A collimated beam 1.2 cm in diampropagates through the scattering cell, a 25-cm X 25-cmX 50-cm tank filled with ethanol, at a height of 14 cm.For multiple-pass experiments mirrors can be insertedto provide up to five passes of the beam through inde-pendent regions of the turbulent liquid. Path lengthsof 0.5 m, 1.5 m, and 2.5 m are thus available. The lightleaving the cell is projected onto the detection systemof sixteen photodiodes in two linear arrays of eight each.The effective interdiode spacing and aperture size iscontrolled by the projection optics. In the experimentsdescribed below the aperture size ranged from 50 gm to500 gm. The minimum distance between diodes wasvaried from 92 gm to 875 gm, with the largest effectiveseparation being 6.13 mm. The signals from the diodesare recorded by a microprocessor-controlled sixteen-channel, 14-bit ADC and digital tape recorder. Thedata collection system has a dynamic range of 80 dB,and the sampling rate can be as great as 200 Hz.

Turbulence was created in the scattering cell bymaintaining an unstable temperature gradient in theethanol. The tank is heated from below by an alumi-num plate resting on electrical heaters rated at 1 kW.The temperature gradient was maintained, and themean temperature controlled, by circulating thermo-statically controlled cold water through a copper tubing

Fig. 1. Schematic illustration of scattering apparatus.

heat exchanger resting on a flat aluminum plate 1 cmbelow the liquid surface. In order to prevent largeconvection cells from forming, vertical Plexiglas dividers3 cm high and 10 cm apart run transversely and long-itudinally across the bottom of the tank.

The properties of the turbulent liquid which are im-portant to optical propagation are the spatial spectrumand the structure function of the refractive index fluc-tuations. Since these fluctuations are solely due totemperature fluctuations and the rate of change of therefractive index with temperature dn/dT is known,measurement of the spatial spectrum 47X(K) and thestructure function DT(r) = T(r + r) - T(ro) 12 of thetemperature is sufficient.

Structure functions were determined by means of two2.5-gm diam, 1.5-mm long platinum wire probes ofnominal 40-Q resistance, a Contel MT-2 microthermalanalyzer, and the digital recording apparatus describedabove. The probe spacing was varied from 2 mm to 64mm. The spatial spectra were recorded with a singleprobe (TSI model 1276) consisting of a 25-,um diamglass fiber plated with platinum. The length of theactive region of the probe is 250 gm, and it is presumedthat it can resolve scale sizes of this order. The timeresponse of the probe was measured to be <10 msec. Areversible-drive lead screw assembly controlled by themicroprocessor was used to translate the probe throughthe liquid, and the temperature was recorded digitally.The bandwidth of this recording system (100 Hz)exceeds by about an order of magnitude that of thetemperature fluctuations observed in our experi-ments.

Ill. Turbulence Properties

The turbulent temperature fluctuations observedwith the system described above exhibit virtually thesame properties observed in atmospheric thermal tur-bulence. These include the spikey randomness char-acteristic of small-scale turbulence variables and bothspatial and temporal intermittency often seen in theatmospheric variables. The only significant qualitativedifference we have noted is the time scale of the fluc-tuations. In the atmosphere the bandwidth of thefluctuations is determined by small eddies of the orderof millimeters being transported by the wind at veloci-ties of the order of meters per second and ranges up toseveral kilohertz. In the tank, the smallest-scale sizesare also in the millimeter range, but the characteristicvelocities are centimeters per second or less, thus

3316 APPLIED OPTICS / Vol. 18, No. 19 / 1 October 1979

0.1

0.05-

.C

0.0

-0.05

I~~~~~~~~I Icm .1

Fig. 2. Temperature as a function of position recorded with moving

250-Atm X 25-rem diam cylindrical thin-film probe. Probe speed: 1cm/sec.

I0,

#T(K) 102

10I.14

I..

1.0 3.0 10.0 30.0 100.0

K (cm'I)

Fig. 3. Spatial spectrum of temperature fluctuations obtained byFourier-transforming temperature recording.

51 0

I-

--

W)

102

-PI,k...~~~~~~~~~. .. *2 .,_

. -. .

.....

.0 3.0 10.0

K (cm 1)

30.0 100.0

Fig. 4. Experimentally determined spatial spectrum multiplied by

yielding signal bandwidths of the order of 1-10 Hz.Figure 2 displays a single short trace of the thermalfluctuations observed with the platinum thin-filmprobe. In this example, the probe was moving throughthe liquid at 1 cm/sec. The fluctuations observed areon the order of 0.1° and indicate scale sizes of significantenergy in the range 1 mm-1 cm.

Power spectra of the thermal fluctuations were ob-tained by Fourier-transforming recordings of the tem-perature like the one displayed in Fig. 2. Spatialspectra were inferred from these by dividing the fre-quency by the probe velocity (1 cm/sec) and multiplyingby 27r. An example of such a spatial spectrum whichresulted from data obtained from sixty traverses of theprobe is shown in Fig. 3. We note that the energy isdown by four orders of magnitude for K > 100 cm-', i.e.,for scale sizes less than 0.6 mm. Data for higher wavenumbers which are masked by noise have been deleted.The noise power is -0.5 on the scale shown.17

Once a spectrum has been obtained by this methodand stored in the computer, it is a trivial matter to fitit to curves derived from the various theoretical models.If the measured spectral density is divided by the the-oretical value and replotted, the resulting values shouldlie on a horizontal straight line, and the parameters ofthe theoretical spectrum can be adjusted to give the bestfit. As mentioned above, Hill1 4 and Hill and Clifford16

have determined that 1-D spectra for high Reynoldsnumber flows in fluids with Pr > 1 should have an in-ertial-convective range where it behaves as K-/, a vis-cous-convective range varying as K-1 , and a viscous-dissipative range which decays exponentially. Multi-plying the experimentally determined spectra by K5/3

or K and replotting should therefore reveal flat portionscorresponding to the inertial-convective or viscous-convective ranges, respectively. Figure 4 displays thespectrum of Fig. 3 multiplied by K5/3. It is evident thatthe small wave-number portion of the data is not flat,indicating that there is not a well-defined inertial-convective range.

The form of the viscous-convective and viscous-dissipative ranges of the 1-D spectrum in a system witha large Prandtl number iS16

XT(K) = AK- 1 exp(-KlI),where the scale 11 is related to the Kolmogorov micro-scale 7j by -q = lPr 1 /2 /5.5. The best least-squares fit ofthis curve to the data used in Figs. 3 and 4 is shown inFig. 5; Fig. 6 displays a fit of the same data to the strictlyempirical spectrum T(K) = B(K + b 2K 3)-'; and Fig. 7shows the best fit provided by qT(K) = C[K-1 exp(-ac-172K3) - Vr erfc(V/a-qK)], the Batchelor spectrum18

which was used by Gurvich et al."1 In each case thedata have been divided by the theoretical value andshould plot as a straight line. The empirical spectrumB(K + b2K 3 )-l clearly agrees best with the experimentaldata; however, since it has no theoretical basis and in-deed cannot be correct for asymptotically large K, theHill spectrum which also has the advantage of leadingto simple analytical results will be used below.

The value of 7j determined by the value of 11 requiredto fit the Hill spectrum to the experimental spectrummay be compared with that calculated on the basis of


I I

104

103

I

10.0

IC

W 1.0 ....... .

0.1

1.0 3.0 10.0 30.0 100.0

K (m 1)

Fig. 5. Ratio of experimental spatial spectrum to Hill spectrum, XT(K)

= A-1 exp(-Kll); 11 = 0.055 cm.

U

N

k

CL

N

0.

C1I-

-6-IC

10.0 F

I.o

. . i.. . .

.- 4

1.0 3.0 10.0 30 0 100.0

K (cm 1)

Fig. 7. Ratio of experimental spectrum to Batchelor spectrum, kT(K)= C[K-1 ep(-a17 2K2) - /7r erfc(VaqlK)]; -qV\a = 0.023 cm.

Table I. Properties of Ethanol

Density - p 0.79 g/cm 3

Index of refraction n 1.361dn/dT -4 X 10-4/C

Viscosity v 1.5 X 10-2 cm2 /secCoefficient of thermal diffusivity D -9.4 X 10-4 sec/cm2

Coefficient of thermal expansion BT 0.0011/0 CSpecific heat Cp 0.574 ca1/g1CPrandtl no. Pr 16

K (cm 1)

Fig. 6. Ratio of experimental spectrumb = 0.114 cm.

.0

the physical parameters of the fluid and the amount ofpower put into the tank. The Kolmogorov microscaleis given by

(V3i 1/4

where v is the viscosity and the energy dissipation rate.The dissipation ratel is

e( (BTqT)/(CpP),

where g = the gravitational constant,BT = coefficient of thermal expansion,qT= turbulent heat flow related to the heating

power applied to the tank,Cp = specific heat, and

p = density.

For an input power of 900 W, tank area 1250 cm2, andthe parameter values appropriate to ethanol given inTable I, we have = 0.09 cm2 /sec3 . This in turn yieldsa length scale of q = 0.054 cm. This agrees quite wellwith the experimentally determined value of 7 =Il(Pr/30)'/2 = 0.04 cm. It is noteworthy that the lengthscale characteristic of the spectrum depends on -1 /4 .Thus, all other parameters and conditions being equal,the position of the spectrum of the thermal fluctuationsis very insensitive to the power in the turbulence; forexample, a tenfold increase in heat flow qT yields atenfold increase in the dissipation rate , but only anincrease of the order of 1.75 in the value of 1. Thusalthough the characteristic length scales 77 and 1 varywith the power input, the variation is weak enough thata single value of 1 is applicable to a substantial rangeof heater power levels.

At this point it may be noted that Bissonette' 0 re-ported observations indicating that a -5/3 slope iner-tial-convective range spectrum did obtain in his ex-periments in turbulent water. However, his spectrawere determined with a stationary probe. We show inFig. 8 an example of a power spectrum derived fromstationary probe recordings of the thermal fluctuationsin our ethanol tank. These data have been multipliedby f5/3 and clearly exhibit a flat portion at the low-fre-quency end. We infer from this that the -5/3 slopemay be an artifact of measuring the spectra with a sta-tionary probe.


o o -

-6-

W.0

0.l

-.. .. r.. s. .. .. VY -- A.. _

* . . A

I I I I1.0 3.0 10.0 30.0 100

to OT(K) = B(K + b 2 K3)-l;

The spatial spectrum of index of refraction fluctua-tions is just the spectrum of temperature fluctuationsmultiplied by n-2(dn/dT)2 , in this case 8.7 X 10-80 C-2.Although all the information about the turbulent fieldthat is relevant to optical propagation is contained inthis function, propagation theories are often couchedin terms of the structure function, which may be cal-culated from the 3-D spectrum

1 dn(K)An (K) =-

471rK dK

by means of the relation

Dn(r) = Air f (i -sinKr) n(K)K2d (1)

0.1 1.0 10.0

f (hz)

Fig. 8. Power spectrum of temperature obtained with stationaryprobe multiplied by f 5

/3

.

We use the Hill spectrum On(K) = n- 2 (dn/dT) 2AK-1

exp(-K1l) with 11 = 0.55 mm as determined above, sinceit leads to a simple analytic result and since it has sometheoretical justification. The 3-D spectrum is in thiscase

02 ¢n(K) =-K K 3(1 + K) exp(-Kl1),

47r

D (r) = C2 In [1 + (r/11)21,

K

40

3.0

Dn Cr)

X 010

2.0

I.0

. 20 4.0 8.0 16.0 32.0 64.0

r (mm)

Fig. 9. Structure function of refractive index fluctuations.

10.0F

5.01-

2.0-2cnx 1010

a.

0.5

0.2

o4l

(2)

where C = n-2(dn/dT) 2 A is a dimensionless parameterdescribing the strength of the index of refraction fluc-tuations and may be compared to the usual structureparameter C. Figure 9 shows a plot of the derivedstructure function given in Eq. (2), together with valuesof (In (ro + r) - n(ro) 12) obtained from half-hour av-erages of the square of the temperature difference.This exhibits nice agreement over the central portionof the curve. The deviation at small separations can beexplained by the inability of the 1.5-mm-long probes toadequately resolve scale sizes of this magnitude.

The strength of the turbulence can be characterizedby the constant C2, and this is readily measured bydetermining Dn (r) for some convenient value of r.Figure 10 is a plot of c2 determined in this way, with r= 1 cm, as a function of heater power. The values of 2range over nearly two orders of magnitude from -2.6 X

10-"1 to -10-9.

IV. Optical Propagation Results

A. First-Order Statistics

Records of the irradiance recorded at a single detectorfor the weakest and strongest path-integrated turbu-lence are shown in Fig. 11. The values of a' shown onthe figure provide a measure of the path-integratedturbulence and are the log-irradiance variances calcu-lated in the Rytov approximation,2 i.e.,

R 2= 87r2k 2L -(1- -- sin k &n (K)KdK,J o k K 2L k J0 500 600 700 800 900 1000

HEATER POWER (Watts)

Fig. 10. Strength of turbulence Cn vs heater power.

(3)

with k = 2irn/Xo,Xo = the free-space optical wavelength,

K = the index of refraction wave number,L = path length, and

tn(K) = Cn(1 + K11) exp(-Kl)/47rK 3

with 11 = 0.55 mm.These traces are qualitatively similar to the irradiance


104

103-

M

8,

(/)

1021

1'0 -

o

I I I I I I

.. I:."r115-P

"'k

I

.0

2aR210

I I sec I

Fig. 11. Point detector recording to irradiance.

strong (bottom) path-integrated turbulence. log-irradiance variance in the Rytov appi

2I.

me

0.5

02

I 02 05 1 0 2.0 50 2

JR20

Fig. 12. Variance of log irradiance plotted agai

value calculated in the Rytov approximation (str

agation path lengths: A, 0.5 m; 0, 1.5 n

50,

5 0

a-'

2.0

21.0

0.5-

0.2 A

0.1 I li l l l l l

0.1 0.2 0.5 1.0 2.0 5 0 10 20R

Fig. 13. Normalized variance of the irradiance vs

of the log irradiance. Propagation path lengths:., 2.5 m.

fluctuations observed in the atmosphere, although inthese laboratory results the time scale of the fluctua-tions is significantly greater, i.e., 10 msec.

In Fig. 12 the measured variance of the log irradianceis compared to the calculated Rytov value for a varietyof turbulence strengths and for the three path lengthsused in our experiments. Satisfactory agreement isobserved for CR < 1, and we note the saturation of ol nto values <2 and its asymptotic trend to a value -0.7 forvery large CR in the manner observed in atmosphericpropagation experiments.

Figure 13 is a plot of the measured normalized vari-ance of the irradiance, U = (I - 7)2/72, against -2, theRytov log-irradiance variance for the same cases dis-played in Fig. 12. Here we note the rise of rr with in-creasing turbulence strength, saturation at a value ofnearly 4, and an asymptotic trend to 1. The value unityis that which would be observed if the probability den-sity function of the irradiance fluctuations were expo-nential. The peak value of 3.9 is in agreement with

or weak (top and measurements made by Gurvich et al. " also usingis the calculated ethanol but is somewhat larger than the normalizedR i thcca variance observed in the atmosphere.

The form of the probability density function (PDF)

of the irradiance is still open to question. It is generallyaccepted that it is log normal in weak turbulence, andthis is borne out by experiments. The situation is lessclear in stronger turbulence conditions, but there is goodreason to believe that the PDF should be asymptoticallyexponential,'9 although this has never been confirmedby experiment.

Figures 14, 15, and 16 show measured PDFs of theirradiance in progressively stronger path-integratedturbulence conditions. The curves drawn in each figureare the maximum likelihood fitted log-normal and ex-ponential distributions, P (I) = [aI(2r)1/ 2]-1 exp[ln(I

L ,L , , - b)2 /2a2] and Pe (I) = - exp(-I/), with a2 = In, and50 200 500 b = nI being the measured variance and mean of the

log irradiance and I the measured mean irradiance. Innst the theoretical each case the vertical scale is logarithmic, and the ex-aight line). Prop- ponential function plots as a straight line. The weak; *, 2.5 m. turbulence short path case, r2 = 0.22, in Fig. 14 shows

reasonable but imperfect agreement with the log-nor-mal distribution. Figure 15 is a stronger turbulence

case R = 4.2. Here the scintillations are near theirpeak, o = 1.66, and the PDF is still qualitatively lognormal. Finally Fig. 16 shows the distribution for thecase of the greatest path-integrated turbulence ob-served, oR = 210. In this case the data points fall be-tween the exponential and log-normal curves, but ap-parently we have not yet achieved great enough path-integrated turbulence to observe unambiguously ex-ponentially distributed irradiance fluctuations. Acomparison of the measured and calculated momentsfor the strongest turbulence case is provided in Table

II and bears out the above observation. The quantitiesiO 100 200 500 listed are the normalized variance, ol = (I/I-1)2; the

skewness, yl; and kurtosis, 2. The latter are definedthe Rytov variance in terms of the third and fourth central moments, 3 =

A, 0.5 m; , 1.5 m; (I/I-1)3 and 4 = (I/I-1)4 by the relations y = 3/

and y2 = g4/0-J - 3.


us l- X ,. . .

P(I/I)~ |. ' I

0 2 3 4 5

I/I

Fig. 14. Probability density function of irradiance. Weak turbu-lence, aR = 0.22; al = 0.72.

PaI/I)

0.01

B. Second-Order Statistics

Spatial covariance functions were constructed fromsimultaneous recordings of the irradiance at the sixteendetectors and calculation of the correlation coefficientsbetween the various relevant pairs of detectors. Dif-ferent spacings and aperture-averaging effects have alsobeen investigated by varying the magnification of theoutput beam before the detector array and thus varyingthe effective size of the individual detectors and theirspacing.

A typical set of covariance curves observed at oneturbulence level for three different path lengths isshown in Fig. 17. c2 is approximately 4.5 X 10-11, andthe observed normalized irradiance variances are L =0.5 m, a2 = 0.72; L = 1.5 m, cr2 = 2.5; L = 2.5 m, o-2 = 1.20.Thus this example encompasses an unsaturated con-dition, a nearly peak scintillation level condition, anda strong saturation condition. To compare these to afirst-order theoretical covariance function, the sepa-ration coordinate is rescaled, and the data of Fig. 17 arereplotted in Fig. 18. The curves are those obtained fora plane wave in the Rytov approximation,2

I.0o

0.8

1/I

Fig. 15. Probability density function of irradiance. Moderateturbulence, aR = 4.2; a = 1.66.

CI(d)

0.

I.0

P(I/I)

* L 0.5me L = 1.5mA L = 2.5m

0 0.15 (.05d (mm)

Fig. 17. Spatial covariance of irradiance for propagation path lengthsof 0.5 m, 1.5 m, and 2.5 m. 2= 4.5 X 10-11.

0.01

2 3

I/I

Fig. 16. Probability density function of irradiance. Strong turbu-lence, aR = 210; al = 1.07.

Dat. Theo-fi1c

L S 0 5m - L OSm* L 1.5 m --- L * 1 5mA L . 2.5m . -. L . 2 5m

0.2 0.4 0.6

d/4TL

Fig. 18. Spatial covariance of irradiance. Same data as Fig. 17. See

text for explanation of theoretical curves.


Table II. Comparison of Experimental and Theoretical Moments

Measured Log normal Exponential

Variance, al 1.07 1.21 1Skewness, ,y 3.51 4.64 2Kurtosis, Y2 23.5 54.4 6

0

0.8

06

Cj(d)

L = 0.5m

Detector Size* - 0.053mm* - 0.1mm

- 0 17mm

0.4 -

0.2 I I I

0 0.2 0.4 0.6 0.8 1.0d (mm)

Fig. 19. Spatial covariance of irradiance illustrating the effect of detector size for short-path, unsaturated scintillation conditions.

2 r k .2 KLICl(d) = [aR1-87r 2K2L J -- sin-I

K2L k~X 1P,,(K)J(~lcdK (4)

Here again, 4',(K) = (Cn/47K3)(1 + K11) exp(-Kll), theHill spectrum, has been used. The data for all threepath lengths deviate significantly from the theoreticalcurves with the longer path data differing by the largestamount. The discrepancy is unexpected in the shortpath case but may indicate that the Rytov theory breaksdown at smaller path-integrated turbulence values thanheretofore expected. The covariance should of coursebe more sensitive to any inadequacy in the theory thanwould the variance. It may be noted that the experi-mental data in all cases falls off more quickly at smallseparations than the theoretical curves and they decaysmore slowly. This is the qualitative behavior expectedfor strong path-integrated turbulence conditions.

We next consider the effect of receiver aperature sizeon the covariance function measurements. Figures 19and 20 show covariance functions obtained under thesame turbulence conditions with path lengths of 0.5 mand 2.5 m, respectively. The three curves on each figureare taken with different effective receiver aperture sizesranging from 0.053-mm to 0.17-mm diam.

For Fig. 19, the turbulence conditions and pathlength, 0.5 m, yield unsaturated scintillation conditions.The scintillation patch size is large, and the aperturesizes used in the experiment were all small compared tothis patch size. Thus aperture averaging has negligibleeffect on the observed covariance functions. In the caseof Fig. 20, L = 2.5 m, and the scintillation conditions aresaturated. The smallest aperture covariance functionshows a rapid decrease, indicating a small covariancepatch size. With increasing receiver aperture size thecovariance function increases significantly, indicatingaveraging over several scintillation patches.

IV. Conclusion

A versatile and useful facility for simulating the ef-fects of atmospheric turbulence on optical propagationhas been constructed, and its parameters have beencharacterized. The strength of the turbulence gener-ated by unstable convection has been determined as afunction of power input, and the structure function andspatial spectrum of the resulting index of refractionfluctuations have been recorded and compared with thetheoretically predicted form.

The effects of this scattering medium on opticalpropagation have been studied in some detail. ThePDF of the irradiance has been determined for a varietyof turbulence levels and path lengths with the resultsconforming in a qualitative way to the theory. ThePDF does not appear to be exactly log normal even atweak turbulence levels and short path lengths. It doesappear to become progressively closer to an exponentialdistribution with increasing path-integrated turbulenceas predicted, but in no case is the functional form un-ambiguous. Measurement of the normalized variance,skewness, and kurtosis of the irradiance confirms thisbehavior. Further work in this area should includeexperiments at longer path lengths, and also the mea-sured PDF should be compared with the exponential-Bessel distribution recently proposed by Bissonnetteand Wizinowich.20

The measured spatial covariance functions of theirradiance show considerable deviation from the first-order (Rytov) theory even at weak turbulence levels,which together with the deviation of the PDF from lognormal may indicate that the first-order approximationis invalid sooner than expected. This should certainlybe investigated in more depth and the data comparedwith predictions derived from the accurate multiplescattering theory.56 Apart from this possible expla-


1.0

0.8

Cl(d)

0.2

L= 2.5 m

Detectar Size

* - 0.053 mm* - 0.mm

- 017mm

.2 1.4 1.6

d (mm)

Fig. 20. Effect of detector size on spatial covariance of irradiance. Long-path, saturated scintillation.

nation there appears to be no reason for the discrepancy.Aperture averaging effects can be ruled out, as theyaffect the covariance in the opposite manner, namely,the measured values would be greater than the truevalues. We have also determined the effects of thedetector aperture size directly (see Figs. 19 and 20) andfound that the scintillation patch size was always greaterthan the smallest effective aperture used (53 gm).

There are, of course, more experiments which can be.done to extend our understanding of the basic proper-ties of the scattering medium and the effects on opticalpropagation, particularly in the multiple scatter regime,and these should be pursued. However, the major usefor a facility such as the one described here will probablybe in the investigation of such systems and phenomenaas speckle propagation through turbulence; beamwander-tracking and cancellation systems; turbulenceeffects on radiation reflected from glints and diffusetargets; propagation of partially coherent light; andpulse stretching. Investigations of this type will bemuch easier to perform, and the results will be lessambiguous because of the completely characterizedscattering medium.

Research sponsored in part by the Air Force Officeof Scientific Research, Air Force Systems Command,USAF, under Grant AFOSR-77-3401. The UnitedStates Government is authorized to reproduce anddistribute reprints for Governmental purposes not-withstanding any copyright notation hereon.

References1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium

(McGraw-Hill, New York, 1961).2. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523

(1970).3. J. R. Kerr, J. Opt. Soc. Am. 62, 1040 (1972).

4. J. R. Dunphy and J. R. Kerr, J. Opt. Soc. Am. 63, 981 (1973).5. H. T. Yura, J. Opt. Soc. Am. 64, 59 (1974).

6. S. F. Clifford, G. R. Ochs, and R. S. Lawrence, J. Opt. Soc. Am.64, 148 (1974).

7. M. H. Lee, R. A. Elliott, J. F. Holmes, and J. R. Kerr, J. Opt. Soc.

Am. 66, 1389 (1976).8. A. M. Prokhorov, F. V. Bunkin, K. S. Goshelashvily, and V. I.

Shishov, Proc. IEEE 63, 790 (1975).9. R. L. Fante, Proc. IEEE 63, 1669 (1975).

10. L. R. Bissonnette, Appl. Opt. 16, 2242 (1977); "Average Irradianceand Irradiance Variance of Laser Beams in Turbulent Media,"Canadian Defense Research Establishment Valcartier, ReportDREV 4104/78 (1978).

11. A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel', Izv. Vyssh.Uchelon. Zaved. Radiofiz. 20, 1020 (1977).

12. H. Gamo and A. K. Majumdar, Appl. Opt. 17, 3755 (1978).13. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric

Turbulence (Wiley, New York, 1964), p. 56.14. R. J. Hill, Radio Sci. 13, 953 (1978).15. R. J. Hill, J. Opt. Soc. Am. 68, 1067 (1978).16. R. J. Hill and S. F. Clifford, J. Opt. Soc. Am. 68, 892 (1978).17. An anonymous reviewer has pointed out that corrections to the

recorded spectra to remove the effects of the probe length andthe inaccuracy of Taylor's hypothesis due to the finite probe ve-locity can be calculated. [See, e.g., J. C. Wyngaard, Phys. Fluids14, 2052 (1971), and J. C. Wyngaard and S. F. Clifford, J. Atmos.Sci. 34, 922 (1977).] Since the two effects tend to cancel eachother at high wave numbers and are less significant at low wavenumbers, and since the rms velocity fluctuation which is neededto make the corrections is unknown for our system, no attemptto calculate these corrections has been made.

18. H. L. Grant, B. A. Hughes, W. M. Vogel, and A. Moilliet, J. Fluid

Mech. 34, 423 (1968).19. J. W. Strohbehn, T. Wang and J. P. Speck, Radio Sci. 10, 59

(1975).20. L. R. Bissonnette and P. L. Wizinowich, "Irradiance Probability

Distribution in Saturation Regime of Optical Waves Propagatingin Turbulence," Canadian Defense Research EstablishmentValcartier, Report DREV 4114/78 (1978).


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Optical propagation in laboratory-generated turbulence