Influence of turbulence strength on temporalcorrelation of scintillation

Jaime A. Anguita* and Jaime E. CisternasCollege of Engineering and Applied Sciences, Universidad de los Andes,

Chile San Carlos de Apoquindo 2200, Santiago 7620001, Chile*Corresponding author: janguita@miuandes.cl

Received April 1, 2011; accepted April 6, 2011;posted April 11, 2011 (Doc. ID 145034); published April 29, 2011

Through extensive laboratory experimentation we demonstrate that the temporal frequency content of turbulence-induced scintillation strongly depends on the temperature gradient exerted at the propagation path of a collimatedlaser beam. We find a power law relating the turbulence strength induced by convection with the vertical tempera-ture gradient and we show that the cutoff frequency of scintillation shows an approximately linear growth withturbulence strength, measured by angle-of-arrival fluctuations. The impact of these findings are discussed inthe context of free-space optical communications. © 2011 Optical Society of AmericaOCIS codes: 010.1300, 010.1330, 060.2605, 010.3310.

The performance of the detection process in a free-spaceoptical (FSO) communication link depends not only onthe magnitude of signal fluctuations (i.e., the scintillationindex), but also on the temporal evolution of thosefluctuations. The time span of deep signal fades affectsthe system’s ability to maintain the link, particularly athigh data rates. Most of the published work on temporalcorrelation of the FSO channel assumes the validity ofthe Taylor’s frozen turbulence principle [1,2]. However,previously published reports based on terrestrial experi-ments show that it is difficult to find a functional relationbetween lateral wind velocity and temporal correlation[3,4]. There have been some relevant research studiesthat couple the frequency content of signal fluctuationswith the strength of the turbulence [5,6] through Fried’sparameter r0 [2].In this Letter, we report our findings in a laboratory

experiment consisting of the temporal analysis of theintensity fluctuations resulting from the propagation ofa continuous-wave, expanded laser beam over a hotsurface. We show that the temporal frequency contentof the signal fluctuations exerted by turbulence showsa marked dependence on the temperature gradientimposed at the beam propagation path, and we show thatthe strength of the turbulence is related to the tempera-ture gradient through a power law with a coefficient ofapproximately 1.8.The purpose of the experiment is to reproduce the

horizontal propagation of a collimated laser beamthrough the atmosphere for a wide range of turbulenceconditions. The setup is now described. The output ofa continuous-wave pigtailed single-mode laser sourcewith wavelength λ ¼ 650 nm, is collimated using anaspherical lens. The collimated beam is afterwards ex-panded to achieve a beam diameter of 3 cm. The ex-panded beam is propagated along the optical table andis reflected off two 5 cm flat mirrors to achieve a totalpropagation distance of 7:3m, as depicted in Fig. 1.The receiver includes a 5 in: telescope with focal length1:25m, a collimating lens, and a fiber coupler. Thiscoupler focuses the light into a multimode graded-indexfiber, with 65 μm core diameter, that is connected toa large-bandwith pigtailed PIN detector. A digital

oscilloscope is used to sample the detector’s outputand store the signal sequences.

A 74 cm × 40 cm electric hot plate is placed under thepropagation path to generate turbulence by convection.The plate generates distributed heat beneath one-third ofthe total propagation length. To avoid generating air vor-tices other than those of the naturally occurring convec-tion, all fan-equipped devices were placed away from theexperiment area. Two thermistors were used to measurethe temperature gradient: one was located about 5mmabove the hot plate and the second was placed 25 cmabove the other, next to the optical beam. Given thatthe room’s temperature is lower than that of the plate,convection would induce a vertical turbulent flow overthe plate, similar to that of a laser communication linknear the ground.

We denote the received optical power sequences astime functions. At the beginning of every experimentalsession, the plate was heated up to near its maximumtemperature, and turned off afterward. Time functionsspanning 20 s were recorded with a digital oscilloscopeevery 90 s to sample the fluctuating optical signal astemperature decayed. The sampling rate of the oscillo-scope was kept at 5 kHz. Temperature readings weresimultaneously logged every few seconds during signalsampling.

An instructive way of representing the correlation timeand its dependence on temperature is through thetemporal spectra of irradiance fluctuations. The first con-tribution of this Letter is the study of the effects of tem-perature gradient on the shape of the spectrum. It isconvenient to plot the spectrum’s amplitude in semi-log scale, as it features a linear profile in the pass band.We have picked three sample cases, (i) ΔT ¼ 55 °C,(ii) ΔT ¼ 25 °C, and (iii) ΔT ¼ 5 °C, that make the

Fig. 1. (Color online) Diagram of the laboratory experimentalsetup.

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distance between curves more apparent and emphasizethe large range of temperature gradients studied. Thespectra are illustrated in Fig. 2. This plot also servesto prove that the irradiance fluctuations obtained inour experiment reproduce those seen in atmosphericpropagation experiments [6]. The features of the spectraare qualitatively the same for different values of ΔT , butthe slope of the falloff decreases in magnitude withincreasing temperature gradient, and the cutoff fre-quency grows accordingly. We have fitted an exponentialcurve of the form

S ¼ 10ðaþbf Þ ð1Þ

to each spectrum within the decaying range. For the ex-ample above the slopes are b ¼ −0:0203 s forΔT ¼ 55 °C,b ¼ −0:048 s for ΔT ¼ 25 °C, and b ¼ −0:0905 s forΔT ¼ 5 °C.We define cutoff frequency as

f 0 ¼ 2=jbj: ð2ÞAnalogously, we define correlation time [6] as

τ0 ¼ jbj=2: ð3ÞFor the examples shown in Fig. 2, f 0 ¼

f98:5; 42; 22:1gHz, respectively, and the correlation timefor each case is τ0 ¼ f10:2; 23:8; 45:2gms, respectively.We have observed on every experimental session thattemporal correlation decreases monotonically with in-creasing temperature gradient. This observation matchesthose we have obtained from atmospheric propagation incases where lateral wind is negligible. We conjecture thatlateral wind velocity increases the cutoff frequency of thefluctuations as a result of the vector addition of convec-tion and wind velocities.The second contribution of this Letter is illustrated by

the functional relation between cutoff frequency, f 0, andtemperature gradient,ΔT . In Fig. 3 we have plotted f 0 forthree independent datasets (i.e., measured on differentdays). The linear correspondence between f 0 and thetemperature gradient is apparent in all cases. A straightline has been fitted to each dataset and it can be observed

that the slopes differ by a very small fraction. The aver-age slope is 1:45Hz=°C (and the standard deviation of theslope estimation is 0.06). The correlation coefficientbetween observations is found to be ρ ≥ 0:96 for all data-sets. The graph in Fig. 3 provides strong evidence that thefrequency content of the intensity fluctuations is essen-tially coupled to the strength of the turbulence, inducedby the temperature gradient.

The third contribution of this Letter consists of relatingthe cutoff frequency of scintillation with the turbulencestrength. We have determined a relation between the re-fractive index structure parameter, C2

n, and the tempera-ture gradient by means of the angle-of-arrival fluctuations[3,7]. The latter are derived from the geometry of the pro-pagation system and provide a relation between the var-iance σ2β of the random angle β (extended by the centroidof the point-spread function at the focal plane, the centerof the aperture, and the optical axis) and C2

n. Assumingthat the beam is modeled as a spherical wave at the re-ceiver and that the turbulence has a Kolmogorov spec-trum, the structure parameter of turbulence is given by

C2n ¼ 0:913σ2βD1=3=z; ð4Þ

where D is the receiver aperture diameter and z ispropagation length [7]. In our experiment, D ¼ 3 cmand z ¼ 7:3m. To determine the angle-of-arrival fluctu-ations, we recorded 300 images (600 for weaker turbu-lence conditions) of the received point-spread functionevery 30 s using a CCD camera while we logged the tem-perature gradient. We performed these measurements intwo independent experiments to ensure the consistencyof the results.

Our measurements delivered values of turbulencestrength that varied from C2

n ¼ 1:0 × 10−12 m−2=3 to C2n ¼

9:5 × 10−11 m−2=3. These values are plotted against ΔT inthe log-scale scatter plot of Fig. 4. There is a good con-sistency among the two datasets and the scattering ishighly correlated, with a clear power relation betweenΔT and C2

n. The first set delivers a power coefficientof 1.95, while the second coefficient is 1.63. The aggre-gated data is well modeled by the expression

Fig. 2. (Color online) Power spectra of time functions withΔT ¼ 55 °C, ΔT ¼ 25 °C, and ΔT ¼ 5 °C. An exponential curveis fitted to each spectrum in the range within which theresponse is decaying.

Fig. 3. (Color online) Cutoff frequency of scintillation as afunction of ΔT in degrees Celsius.

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C2n ¼ 3:3 × 10−14ðΔTÞ1:82; ð5Þ

which is also plotted in Fig. 4.With these results one can quantitatively compare the

behavior of the temporal correlation with that of theturbulence strength. The latter may be conveniently ex-pressed using the nondimensional Rytov variance [2],σ2R ¼ 1:23C2

nk7=6z11=6, in which k is the wave number.The relation between f 0 and σR is depicted in Fig. 5, in

which we have included the values for the three datasetsconsidered in Fig. 3. Another metric that is relevant toour experiment is the unitless ratio D=r0, where r0 ¼ð0:16C2

nk2zÞ−3=5 is the Fried’s parameter [2]. BecauseD=r0 is not a linear function of σR, the scale of the formeris not linear in Fig. 5. The graph of Fig. 5 may be useful forestimating the temporal correlation at other link rangesor turbulence strength conditions. Although we havefitted straight lines to the datasets, f 0 and σR would relatethrough an estimated 0.91 power coefficient, as obtainedby combining σ2R with (5).

In summary, we provided an experimental proof thatthe temporal content of the intensity fluctuations of alaser propagation link through atmospheric turbulencedirectly depends on the turbulence strength, measuredin this case by angle-of-arrival fluctuations. We found thatthe cutoff frequency of the temporal spectrum of inten-sity fluctuations is an increasing linear function of thevertical temperature gradient (within the temperaturerange studied) and an approximately linear function ofthe unitless metrics σR and D=r0. Our measurementswere obtained by means of exerting vertical heat convec-tion with temperature gradients up to 80 °C, and whereno lateral air flow existed. We conjecture that in caseswhere a (moderate) lateral wind is present, the correla-tion time of the intensity fluctuations will depend on avector superposition of the flows produced by tempera-ture gradient and lateral wind.

This work has been supported by the Chilean AgencyConicyt under grant Fondecyt-1090709 and by Universi-dad de los Andes under grant FAI-ICIV001-09.

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Fig. 5. (Color online) Cutoff frequency of scintillation(in hertz) as a function of σR (bottom abscissa) and D=r0 (topabscissa).

Fig. 4. (Color online) Experimentally estimated C2n as a func-

tion of ΔT (in degrees Celsius) using angle-of-arrival fluctua-tions, for two independent datasets.

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