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Two-frequency Fluctuations of LightIntensity in a Turbulent MediumA.S. Gurvich a , V. Kan a & Vl.V. Pokasov aa Institute of Atmospheric Physics, Moscow, U.S.S.R.Published online: 16 Nov 2010.

To cite this article: A.S. Gurvich , V. Kan & Vl.V. Pokasov (1979) Two-frequency Fluctuations of LightIntensity in a Turbulent Medium, Optica Acta: International Journal of Optics, 26:5, 555-562, DOI:10.1080/713820035

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OPTICA ACTA, 1979, VOL. 26, NO . 5, 555-562

Two-frequency fluctuations of light intensity in a turbulentmedium

A. S . GURVICH, V. KAN and VL . V . POKASOV

Institute of Atmospheric Physics, Moscow, U .S .S.R .

(Received 30 January 1979)

Abstract. The results of measuring the correlation of intensity fluctuationsfrom two independent monochromatic light sources are given . The experiment iscarried out in the region of strong fluctuations . It is shown that in this case theassumption of a gaussian field distribution leads to results essentially differentfrom experiment .

1 . IntroductionUsing the method of smooth perturbations (MSP), Tatarskii and Zhukova [1],

Rytov [2], Fuks [3] and others studied the correlations of the amplitudes of wavespropagating on different frequencies in a randomly inhomogeneous medium, andalso frequency correlations . The results of these works are applicable to the region ofsmall fluctuations of light intensity . In calculations on the chromatic scintillation ofstars [1], the decorrelation of light intensity fluctuations on different wavelengths isdue mainly to the difference of the trajectories of beams in the atmosphere caused bythe dependence of atmospheric refraction on wavelength, i .e. by the dispersion ofrefraction . Results of observations [1] obtained in conditions in which the MSPcould be applied are in good agreement with these calculations . In the region ofsaturated fluctuations of light intensity, frequency correlations were analysed byAlimov and Eruchimov [4] for the geometrically thin layer (phase screen) and byEruchimov and Uriadov [5], Shishov [6] and Fante [7] for the random layer of therandomly inhomogeneous medium . These papers considered the case when thetrajectory of the beams of different wavelengths coincided . In calculations of thefluctuations of light intensity-the fourth moment of the field-it was assumed thatin the region of saturated fluctuations the gaussian distribution for fields is valid . Itfollows from this assumption (see, for instance, [7]) that with the broadening of thetransmitting emission spectrum a considerable decrease in intensity fluctuations is tobe expected with the propagation of this radiation in particularly turbulent mediumat a sufficiently large distance . However, in experiments conducted by Gracheva andGurvich [8] and Gracheva [9] with a non-coherent light source, no such decrease inintensity fluctuations was recorded .

This paper presents the results of measuring the correlation of intensityfluctuations from two monochromatic sources of light with wavelength A1 =0 .63,umand '12=0.44µm . Measurements were made in the regions of weak and strongfluctuations of intensity with wide beams of light . The optical axis of the beams wasthe same . The experiment consisted of two parts . In the first, measurements weremade over long atmospheric paths of 650 and 1750m, close to the ground-theheight H above the Earth's surface ranging from 0 . 5 to 3 m . In the second part, shortpaths 0 .35 and 1 .05 m long were used in, as a randomly inhomogeneous medium, alayer of convectionally turbulent water . In liquid it is possible to create conditions

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for strong intensity fluctuations of propagating emission at distances of several tensof centimetres, a fact which was used by Gurvich et al . [10] and Bissonnette [11] . Inthe course of making the measurements, the characteristics of the turbulent mediumwere measured independently .

2 . Description of the experiment2.1 . Measurements made overlong atmospheric paths

Figure 1 shows the block diagram of the experiment . The radiation of the He-Nelaser L1 with ) l =0 .63ym and the radiation of the He-Cd laser L2 with ~2 =0 .44µm

A

M5 N M4 F2I(r,)

P2VIM"

lI

1~©

11

ED

M5

F1

PR

Figure 1 . Block diagram of the experiment . L1, L2, lasers; Ml, M3, mirrors ; M2, M4, beamsplitters ; A, randomly inhomogeneous medium ; C, collimator ; M5, glass plane plate; N,mask; F1, F2, F3, interference light filters; P1, P2, P3, photomultipliers ; 0, objective :S, slit; M, motor; ED, electronic device; RC, integrator; PR, pen recorder .

were combined by the mirror M 1 and the beam splitter M2 . Beams broadened by thecollimator C with an effective diameter d passed through the turbulent medium A,arriving at the mask N in which there was a pinhole of radius a . (The values of d, aand the Fresnel number of the transmitting aperture a1, 2 = d2k1, 2/4x are given in thetable. Here kl, 2=e112 27r/Al,

2, where a is the average permittivity of the medium and -x the length of the path .) The emission, having passed through the pinhole, wasdivided by the beam-splitter M4. The two beams (one deflected by the mirror M3)passed through the interference light filters Fl and F2 to arrive at the photo-multipliers P1 and P2 . The signals from the photomultipliers, proportional to theinstantaneous intensities of the laser beams I(2 l ) and I(2 2 ), were processed by theelectronic device ED. As a result of processing, the values

RZ(~1)=<(I(2l)-<I(21)>)2>

<I(Al)>2

and

112(2z)=<(I(22)-<I(22)>)2><I(22)> 2

were obtained for the normalized variances of the intensity fluctuations at thewavelengths A l and A2, and also the value

11 2 (21, 22) _ <(I(2 1) -<I(2l)))(I(22) -<I(12)>)>

(I(2l)><I(22)>

for the correlation of the intensity fluctuations at the wavelengths (the two-frequencycorrelation of intensity fluctuations) .

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Fluctuations of light intensity in a turbulent medium

557

A portion of the laser emission which passed through the turbulent medium wasdirected by the glass plate M5 to the equipment used to simultaneously measure thespatial spectrum of the two-point function of the coherency of the emission on thewavelength by the technique described by Artem'ev and Gurvich [12] . This wasachieved by moving the narrow vertical slit S in a horizontal direction in the focalplane of the objective 0 by the motor M . The emission which passed through the slitarrived at the photomultiplier P3 after passing through the interference light filterF3 (2=0 .63 pm) . The signal was then averaged by the integrator RC, and the penrecorder PR registered the averaged intensity profiles on the focal spot .

In measurements on the atmospheric path x=650m for processing the signalsfrom photomultipliers P1 and P2, the analogue thermal correlator described byGracheva et al . [13] was used . For the atmospheric path x =1750 m, signals fromphotomultipliers P1 and P2 were recorded on magnetic tape . The autospectra andmutual spectra of the intensities for two wavelengths and histograms for eachwavelength were obtained by computer-processing these recordings . From thesedata the normalized variances of intensity fluctuations for each wavelength, and alsofrequency correlations for the two wavelengths, were determined . It should bepointed out that on atmospheric paths a certain influence on the frequencycorrelations of intensity fluctuations can be exerted by dispersion of refraction by thevertical temperature gradient in the air . The distance between the trajectories ofbeams A 1 and A2 on the path x=1750m is estimated to be 0 .5 cm . For the sake ofcomparison, let us note that the Fresnel zone pF = .J(A 1x) on this path reached3 .32 cm .

During measurements on short paths, laser emission was propagated in waterbetween two horizontal heat exchangers in a thermally insulated vessel 35 cm long .The inhomogeneities of permittivity were caused by the turbulent convection due tothe temperature difference AT between the heater (the lower heat exchanger) and thecooler (the upper heat exchanger) . Convective turbulence appears with sufficientlylarge Rayleigh numbers Ra (see, for instance, [14] or [15]) and in the experiment

Ra,> 10' . The turbulence was regulated by the changes 0 T and Ah (Ah is the distancebetween the heat exchangers) . The relatively large values of OE/0T in water, ascompared with air, make it possible to create conditions of strong fluctuations of theintensity of the emission propagating along the short path . The temperaturedifference AT varied from 30 to 65 ° C and Oh from 5 to 10 cm . The electronicanalogue device used in measurements on short paths is described by Gurvich et al .[16] .-

Estimates have shown that for short paths the maximum distance between thebeams A 1 and '2' due to the dispersion of refraction in water, did not exceed 10pm,which enabled us to exclude from our consideration the effects caused by thedifference of trajectories .2 .2 . Measurements made in a randomly inhomogeneous medium

The randomly inhomogeneous medium is determed by the spectrum of theturbulent fluctuations of permittivity 0,.(K) . Following Tatarskii [17], we describedthe spectrum in the form

KZ(p~(K)=0 .033CEK -11 /3 exp -

-K,

(1)m

where K is the spatial wavenumber, KM =5.92/l0 (l0 is the inner scale) and CE is the

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A . S . Gurvich et al .

structure coefficient of the fluctuations of permittivity . In a turbulent liquid

Pr = vm »1,xm

where Pr is the Prandtl number, vm the kinematic viscosity and Xm the thermaldiffusivity of the liquid . The spectrum of a, as has been shown by Batchelor [18],differs from the Kolmogorov spectrum in the region of high frequencies . However,in the conditions of our experiment the Prandtl number in water was not high(Pr<6) and we used equation (1) as the expression for the spectrum .

During the propagation of laser beams at a single wavelength, the values /30=4< 2> were used as a longitudinal parameter . They were calculated to a firstapproximation by the MSP (this is described in more detail by Gurvich et al . [10]and Gurvich and Tatarskii [19]) . By analogy with this as a longitudinal parameter inthe given problem, the values /3001,)2)=4<x(.1)x('.z)>, calculated to a firstapproximation by the MSP, were used . On the basis of the results of [3], for thespectrum equation (1) we have that

2zt 2k2x ('°° [ sin (S2x2x/k)2P (A1, A2)sin (K2x/k)

z x/k

0,(K)x dxK1 -c2 J 0 L QK 2x/k -

0 .31 CEk2xxm 1/3 _ 11 I

1- S22 r'

6

(1 +Q2D2)11/12

11-sin

\ 6arc tan SID

nD

2

D

)11/12(1 +Dsin (6 arc tan D

(2)

Here k=nw/c, where n is the refractive index of the medium, c is the speed of light,w=2w1w2/(w 1 +(0 2 ), where w 1 and w2 are the frequencies of the propagating laseremissions, SZ=(w2-w1)/(w2+ (01) is the frequency detuning (in our case Q=0 .178),D=K22,x/k is a wave parameter and F(- 6) is a gamma function . Assuming that f1=0in equation (2), the expression for f1 (), 1 ) or #0(' .2) at a single wavelength can befound .

According to [17], the values of the inner scale 10 necessary for the calculation of

Km are determined by the ratio

1 0 =(3CB) 3141jKPr-314 ,

( 3)

where C2 is the universal constant and rJK=Vm/4EK 114 is the Kolmogorov scale ofspeed fluctuations, FK being the dissipation speed of kinetic energy of turbulence .Following Monin and Yaglom [20], the value of Ce can be assumed to be 2 .8 .Measurements by Zubkovsky and Koprov [21] have shown that in the atmosphereclose to the ground, with average conditions, rK x 100 cm2/s 3 . Using this value andtaking into account that for air, vm =0.15 cm 2/s and Pr =0 .7, it can be found fromequation (3) that 10 0 .4 cm. With such a value of 1 0 for our atmospheric paths, theconditions D>> 1 and .)D>> I are fulfilled . Using these conditions, equation (2) can besimplified as

I - 516#0(Ai,'2)= 1-522 0 .31 CC k'16x11/6 •

(4)

Assuming in this expression that S?=0, it is possible to obtain the well-knownformula for the parameter #o at a single wavelength .

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Fluctuations of light intensity in a turbulent medium

559

The structure coefficient CE is related to the width of the blurred focal spot, y 0 . 5 ,on the focal plane of the objective 0 at the half-intensity level for paths x =1750mand 650m [12] by the expression

CE =Y0.s/2.39F513k li3x

where F is the focal distance of the objective 0 (see figure 1) .In the experiment on short paths the inner scale 1 0 was determined from equation

(3) . The speed of dissipation was calculated from measurements of the heat fluxgenerated by the heater [20] . The value of the structure coefficient CE were foundfrom measurements of the width of the blurred focal spot y o . 5 . For the experimentwith short paths, the conditionD 1 (la ) >> 1 was fulfilled (D 1 is the structure function ofthe complex phase [17]) . In this case, CE depends on yo . 5 according to the relation

Ce =YO.5/2-51EF2xKm3

The values ofD and CE are given in the table .

!,0

0 5 !0 PFigure 2. The dependence of /J on the longitudinal parameter fi o . Measurements along

atmospheric paths : 1, fi(2 1 ,1 2) ; 2,

3, X.12). Measurements along short paths :4, fi(A 1 , 2 2 ) ; 5, #( .1 1 ) ; 6, fi(A 2 ) . Calculation by the method of smooth perturbations :7, /3=[exp(fo)-1]1/2 .

3. Results and discussionFigure 2 shows values of #(A 1 ), 13(12) and P(A1, .1 2 ) obtained in the experiment as a

function of the corresponding parameters / 30(A1), l0(12) and NAI, .1 2 ) . The values of/30 for the atmospheric experiment were calculated from equation (4) . Data for which/30 > 3 were obtained along the 1750 m path. For the experiment with the short pathvalues of #0 were calculated from equation (2) . In this case data for which f0 > 4 areobtained on the path x=1 .05m .

It is evident from figure 2 that for small values of#0, the experimental data agreewith the calculated ones whitin the MSP approximation. In case of intermediatevalues # 0 : 1, for dependences of #(A 1 ) and 13(12) on #0, a maximum is observed-this

x(m) d(cm) a(mm) a l a2 D CE (cm -2 /3 )

650 10 0. 3 0 . 38 x 10 2 0 . 55 x 102 130 (0 . 8-3 . 6) x 10 -141750 35 0. 3 1 . 75 x 10 2 2 . 5 x 10 2 360 (0.9-2 . 8) x 10 -14

0.35 3 . 5 0 . 02 1 . 2 x 10 4 1 . 6 x 10 4 1-4 (0 . 06-5. 3) x 10 -81 .05 3 . 5 0. 01 0. 4 x 10 4 0. 6 x 10 4 5-11 (1 .2-7. 6) x 10 -8

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is a region of random focusing of emission . This is in agreement with the knownmeasurements by Gracheva and Gurvich and with modern theoretical concepts (see,for instance, the review by Prokhorov et al . [22] and papers by Yakushkin [23], Fante[24] and Zavorotnyi et al . [25] .

For the values of the two-frequency correlation of intensity fluctuations f(A1A2)in the region of random focusing, a peak is also observed which is more sharplypronounced for measurements made along the short path. The values of /3(21, A2)slowly decrease with increasing fl o . It should be noted that the greater values of /3(x, 1 )and 13(A 2 ) obtained on the short path, as compared to the atmospheric values in figure1 are partly attributable to the considerable difference in wave parameters and therelationships between t o and the radius of coherence p. [23] (p, is the distance atwhich the function of mutual coherence decreases by a factor of e) . Similarly, thevalues of /3(A 1 , A 2 ) obtained on the short path are greater than the atmospheric valuesfor, most probably, the same reasons . It is also probable that decorrelation occursdue to the dispersion of refraction in the atmosphere .

In papers dealing with the correlation of intensity fluctuations at two wave-lengths in the region of strong scintillations, the correlation of complex fields thetwo-frequency function of the mutual coherence of fields, F2(w 1i awe)-is usuallycalculated . In treating the correlation of intensity fluctuations-the fourth momentof the complex field-it is assumed that in the region of strong scintillations there is agaussian distribution of fluctuating fields, similar to that occurring with saturatedscintillations at a single wavelength [22] . If we assume a gaussian distribution offields and the results obtained by Fante [7], /3(x,1, A 2 ) may be represented as

XA1,A2) = jF2( (0 l,w2)I = expC- (w28c2 )2n2X A(0)1 lexp(/')l,

where the first exponent is a `refractive' factor and the second is a 'diffractive' factor .Let us evaluate the `refractive' factor, taking into account that the value of the'diffractive' factor does not exceed unity [7] . A(0) is determined by the outer scale ofturbulence Lo [17] and is given by A(0) z C2L05/3 . If we assume that Lo equals Ah/2= 2. 5 cm for the experiment on the short path, and H/2 x 0 .5 m for the atmosphericexperiment, the argument of the exponent of the `refractive' factor in both cases canbe approximated by

(w2-(ol)2n2xA(0) : 1038c2

(5)

for the largest values of f 0(21 , A2) in our measurements . It is worth noting that, forthe same values of /3 0i the assumption of a gaussian distribution of the field gives adifference with experiment of not more than 20 per cent for #(A 1 ) and 13(,2) . At thesame time, for /3(A 1 , A 2 ) the experimental values and the estimate of only the`refractive' factor according to equation (5) are incomparable, even by order ofmagnitude . This seems to show that, to calculate the frequency correlations ofintensity fluctuations, the conditions under which a gaussian distribution of fieldsmay be assumed should be specified .

As has been noted above, in the region of strong fluctuations for fixed fo thevalues of normalized variance, 1 2(A 1 ) and /32012), and correlations, fl 2(Ai, A2),obtained on the short path surpass similar atmospheric values . In this connection

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it is interesting to investigate the dependence of the

K2=fJ 2 () 1, 22)/f(2 1)f(22)on the parameter characterizing the `intensity' of fluctuations. The parameter#0(A1, 22) can be chosen .

To a first approximation by the MSP, the value

KMSP - l'0(t1,'2)/P0('1)P0( 112)

is determined by the detuning 52 and the wave parameter D, and does not dependexplicitly on #0 (see, for instance, [3]) . With the change ofD from D-+O (geometricaloptics) to D-+oc, the value KMSP varies within corresponding limits :

1 iKMSPi(I -Q 5 6)(1 -f12)-5/12

In the conditions of our experiment, (1 -525/6) (1 _j22 ) -5/12_ 0.77

KI'D

0,5

Fluctuations of light intensity in a turbulent medium

561

,

% xXx x

• •X x xxx

X XX S

correlation coefficient

5

Po (T, A2)

Figure 3 . The dependence of K=#(A1,22)/[fi(A1)fi(22)]1/2 on the parameter $o . 1, atmos-pheric paths ; 2, short paths . The dashed lines represent the range of K„, P variations .

Figure 3 represents the dependence K on 130(x . 1 , 22 ) . The dashed lines indicate thelimiting values of KMSP for our experiments . It is evident from figure 3 that the valuesof K obtained along short and long paths are quite close, despite the fact that thelengths of paths differed by several orders of magnitude . With increasing fl o , Kdecreases to values much lower than those predicted to a first approximation by theMSP.

In conclusion, it should be noted that the measurements taken estimate the valuesof frequency correlation in the region of strong intensity fluctuations . The resultsobtained show that the MSP produces an overestimated result, while the assumptionof a gaussian distribution of the field leads to a considerable underestimation of thevalues .

AcknowledgmentThe authors express their gratitude to V. I . Tatarskii, V . I . Shishov and V . U .

Zavorotnyi for useful discussions .

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Fluctuations of light intensity in a turbulent medium

On presente les resultats de la mesure de la correlation des fluctuations d'intensite de deuxsources lumineuses monochromatiques independantes . L'experience est effectuee dans laregion de fortes fluctuations . On montre que dans cc cas I'hypothese d'une distributiongaussienne du champ conduit a des resultats essentiellement differents de ceux obtenusexperimentalement .

Es werden Mef3ergebnisse der Korrelation von Intensitatsfluktuationen zweier unabhang-iger monochromatischer Lichtquellen mitgeteilt . Das Experiment wurde im Bereich starkerFluktuationen durchgefuhrt . In diesem Falle fuhrt die Annahme einer gauf3schen Feldverteil-ung theoretisch zu ganzlich anderen Ergebnissen .

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