probability density of irradiance fluctuations observed over terrestrial ranges

Probability density of irradiance fluctuations observedover terrestrial ranges

Rita Mahon,1,* Christopher I. Moore,2 Harris R. Burris,2 Mike Ferraro,1

William S. Rabinovich,1 Michel Suite,2 and Linda M. Thomas2

1Code 5654, Naval Research Laboratory, Overlook Ave, SW Washington, DC 203752Code 8123, Naval Research Laboratory, Overlook Ave, SW Washington, DC 20375

*Corresponding author: [email protected]

Received 15 June 2011; revised 9 September 2011; accepted 12 September 2011;posted 13 September 2011 (Doc. ID 149281); published 2 December 2011

The irradiance fluctuations imposed on a laser beam that has propagated over horizontal terrestrialpaths in the range of 2 to 24km are compared to lognormal (LN) and gamma-gamma (GG) distributions.For the direct links reported here the irradiance fluctuations follow a LN distribution except in cases ofweak turbulence, characterized by a scintillation index of less than 1 and a Fried parameter larger thanthe receiver aperture, in which case the GG distribution gives an improved fit. In very weak turbulencethe difference between the two distributions is insignificant. © 2011 Optical Society of AmericaOCIS codes: 010.1330, 010.1300, 030.7060, 060.2605.

1. Introduction

A knowledge of the parameters that describe the ef-fects of atmospheric turbulence on a free-space opti-cal link are important in optimizing protocols andtheir implementation. Data loss due to turbulencepresents itself in the form of burst errors and so faderates and durations at a given signal level are ofgreat significance. The form of the probability den-sity function (PDF) of the irradiance fluctuations, im-posed on an optical beam as it propagates throughthe atmosphere, is one of the fundamental param-eters involved in predicting the performance of an op-tical communications link. The size of the receiveraperture plays a role in the degree of averaging thatoccurs over the spatial fluctuations, which in turn de-pend upon the scale sizes of the eddies encounteredalong the propagation path. For a range of length, L,and wavenumber, k, the so-called inner scale of tur-bulence, l0, that is smaller than either the Fresnelzone ðL=kÞ1=2 or the spatial coherence radius ρ0,causes diffractive scattering, while turbulence cellscloser to the outer scale of turbulence, L0, which is

larger than the Fresnel zone or the scattering diskL=kρ0, result in refractive effects on the propagatingwave. For a given sized receiver and range, a varietyof turbulence conditions will require the communica-tions system to adapt to providing the optimum pack-et length and transmission protocols as changesoccur in prevailing meteorological conditions.

At present much reliance is placed on analyticalmodels [1–4] that have been extensively developed,but with little experimental data available, verifica-tion has been limited. At the Naval Research Labora-tory a 6-month-long investigation [5] of a 16km linkacross the Chesapeake Bay proved to be invaluable,giving results that could not have been produced inthe absence of continuous 24h monitoring. It wasshown that the atmospheric turbulence does notshow a diurnal variation in the maritime environ-ment, in contrast to the terrestrial case. Also, a strik-ing dependence of the turbulence parameters wasseen to depend upon the temperature gradient exist-ing above the water surface, being aminimum at zerotemperature difference between the air and water.The sign of the temperature gradient indicates eithera well-mixed convective atmosphere when the air iscolder than the water or the presence of stable layerswhen the air is warmer than the water surface,

0003-6935/11/356476-08$15.00/0© 2011 Optical Society of America

6476 APPLIED OPTICS / Vol. 50, No. 35 / 10 December 2011

leading to a more chaotic link in the latter case. In aneffort to extend the studies to terrestrial links, a por-table atmospheric receiver system has been devel-oped. The Transportable Atmospheric Testing Suite(TATS) consists of sensors to monitor atmosphericturbulence and meteorological parameters over bothdirect and retroreflected free-space optical links.In this paper, irradiance fluctuation data acquiredat China Lake during the period 6–10 December2010 are analyzed with regard to the PDF and thedegree to which the mathematical models agree withthe recorded data. The location provides a uniform,flat, low humidity, desert environment with justabove ground line-of-sight to 24km as well as a19km link from a mountain range. For moderate ir-radiance fluctuations the results strongly supportthe conclusions of Vetelino et al. [2] that the size ofthe receiving aperture, in relation to the spatial co-herence width, determines whether the PDF is bet-ter fit to a single or double stochastic model. In thesaturation regime this transition is not apparent.

2. Background

A. Scintillation

The fluctuation statistics of a laser communicationssignal is measured by the scintillation index, σ2I , orthe normalized irradiance variance. The temporalbehavior of the received intensity is seen to have fluc-tuations in strong turbulence showing a knee in thepower spectra at a few hundred hertz and residuallevels out to 1kHz. [6,7] Hence, the bandwidth of apoint receiver system needs to be at least 2kHz toencompass most atmospheric situations. The dy-namic range and linearity of the recorded intensitiesare also dependent on the level of turbulence and willlimit the accuracy of the scintillation index measure-ment. Another way of displaying the fluctuation sta-tistics is in the form of a histogram, which is a sampleestimate of the PDF. The temporal fluctuations re-corded over a period of time, typically of the orderof tens of seconds, are normalized to the mean of thatset of data and binned. Saturation of the detector willresult in an overpopulation in the highest intensitybin. Excessive noise in the detector will result in thelowest intensity bins being overpopulated. The formof the PDF in these lowest intensity regions is centralto finding an analytical model, applicable in all situa-tions, that can be used for predicting fade statistics.In low turbulence the PDF will be approximatelyGaussian since the fluctuations are primarily causedby turbulence cells close to the size of the Fresnelzone. In regimes of stronger irradiance fluctuationsthe scale size of dominant eddies is no longer thatof the Fresnel zone but has two contributions fromboth a small-scale, of order the coherence radius,and a large-scale component of order the scatteringdisk. It has long been accepted that LN statistics areadequate to describe the irradiance fluctuations inweak turbulence. A large number of distributions [1]have been considered for use in the moderate-to-

strong fluctuation regimes, but the consensus PDFis the GG distribution.

Scintillation is due to both temporal variations inthe received irradiance as well as spatial variationsover a receiver’s aperture. Current discussions as tothe applicability of LN or GG fits to the probabilitydensity of irradiance fluctuations center on the de-gree to which the spatial variations are averagedover the receiving aperture [2]. Since the GG distri-bution describes a double stochastic situation wherethe mean of the small-scale irradiance is modulatedby the distribution of the large-scale fluctuations,then it is going to apply in situations where thereceiving aperture is smaller than both the Fresnelsize and the coherence radius. When the aperture islarger than the coherence radius, the small-scalescintillations are averaged out and the distributionbecomes that of the large-scale fluctuations onlyand reverts to being LN. Since the term scintillationindex is defined as the normalized variance of a pointreceiver, atmospheric turbulence measurements areusually carried out with the smallest aperture thatgives a signal that can be matched to the dynamicrange of the detector. As such, the aperture will beof a minimal size and is more likely to reflect theneed for GG statistics. However, in practical recei-vers used in optical communications systems singlereceiver systems are likely to have apertures of theorder of 75–100mm in diameter and so the small-scale scintillations will be spatially averaged andLN distributions are then expected. Alternatively,for multiple receiver systems that take advantageof spatial diversity and fiber amplifier-based detec-tors, the smaller apertures may lead to situationswhere the irradiance fluctuations are better de-scribed by GG statistics.

B. Summary of Recent Relevant PDF Studies

Vetelino et al. [8] studied the irradiance fluctuationsover a 1:5km direct link using receivers of threedifferent aperture sizes. The simultaneous measure-ments were used to infer the atmospheric structureconstant,C2

n, the inner scale of turbulence, l0, and theouter scale of turbulence,L0. In a later paper [2], theycompared the LN and GG PDFs for both the experi-mental data as well as random phase screen-basedsimulation data generated using the same C2

n asthe experimental datasets. Considering the six datapoints from the experimental data, their results in-dicate that for receiver apertures smaller than ρ0 theirradiance fluctuations are GG distributed, but forapertures larger than, or equal to, ρ0 the LN distri-bution holds. The advantage in having the simula-tion dataset is that it extends the dynamic rangebeyond what is available in the experimental data-set. The comparison of the simulation data withthe LN and GG distributions supports the conclusionthat for aperture sizes greater than ρ0, the small-scale contributions are averaged out and the LN dis-tribution reflects the distribution of the large-scalefluctuations.

10 December 2011 / Vol. 50, No. 35 / APPLIED OPTICS 6477

Lyke et al. [3] also carried out numerical wave-optics simulations of a laser propagation link andanalyzed the PDF of the aperture-averaged irradi-ance fluctuations at the receiver. The link rangesused in [3] are much longer than those in [2], being50, 75, and 100km, but theC2

n values are correspond-ingly smaller—1 × 10−18 to 1 × 10−15—so that the Ry-tov variance σ2R ¼ 1:23C2

nk7=6L11=6 covers the rangetraditionally considered to describe weak scintilla-tion σ2R < 1 through strong scintillation σ2R > 1. Therange of σ2R covered in the simulations of [3] is from0.026 to 92.4. The simulations of [3] are in closeagreement with the simulations of [2] at the two spe-cific Rytov variances of σ2R ¼ 2:7 and 19.2 pertinent tothe conditions covered in [8]. The simulations in[3] confirm that in the weak scintillation regime(Fresnel zone is smaller than coherence radius),the irradiance fluctuations are lognormally distribu-ted for receiver apertures larger than the Fresnelzone size. They also state that the GG distributionis an equally good fit in the weak fluctuation regimefor all aperture sizes. In strong scintillation con-ditions the simulations are in agreement with theresults in [2] in that irradiance fluctuations for aper-tures smaller than the coherence radius are GG dis-tributed while in those for which the aperture islarger or equal in size to ρ0 the LN distribution holds.However, in moderate scintillation conditions thesimulations of [3] show the irradiance fluctuationsto be GG distributed for all sized receivers from pointlike to 3ρ0=2.

3. Experiment

The TATS system, shown schematically in Fig. 1, iscomprised of a 125mm diameter telescope that re-cords both spatial entrance pupil images at ratesof 140 fps or 328 fps and simultaneously measuresthe angle-of-arrival (AOA) of the transmitted beam.The images give spatial correlation information,while the AOA gives a measure of C2

n. There is apickoff mirror in front of the corrector plate of thetelescope that directs a 30mm diameter centralportion of the incoming beam to a photodiode thatmeasures the intensity variations and gives a mea-sure of the scintillation index. Neutral density andbandpass filters reduce this aperture to 22mm.A retroreflector can either be placed behind the par-tially reflecting pickoff mirror, or just above the tele-scope in order to measure the effective scintillationindex back at the interrogating beam source. The in-terrogator used in these experiments has been adual-mode optical interrogator (DMOI) developedby NovaSol. The DMOI is a bistatic system with100mm diameter transmit and receiver apertureswith variable divergence on the 1:55 μm transmitbeam. Beam pointing to within �1° is achieved witha gimbal and then fine pointing uses fast steeringmirrors with the offsets optimized by cone trackingthat is turned off for the turbulence measurements.Weather stations are set up at the receiving TATSstation and either at the interrogator site or midway

along the link path. In addition to the normal meteor-ological parameters (temperature, humidity, precipi-tation rate, solar fluence, wind speed and direction),the weather stations also record temperature gradi-ents above the ground. A visibility monitor is locatedon one weather station. Data were recorded synchro-nously and all raw data was saved, with backgroundlevels being taken at regular intervals. Intensitydata were recorded at a 5kHz rate while AOA wasat the camera framing-rate of 60Hz. The imageswere taken with a windowing InGaAs camera andusually at the 328 fps for a 128 × 128 pixel image,unless the turbulence level was low and the full im-age could then be recorded at 140 fps in a 200 × 200pixel image.

Since data are recorded for varying ranges andperspectives using the DMOI and TATS receivers, acalibration interrogator was set up to constantlymonitor a 1:1km link to a 25mm diameter retrore-flector. This system was always at the same locationas the TATS receiver and operated essentially alongan orthogonal path so there was never any opticalinterference. This system used amonostatic arrange-ment with the outgoing 0:85mrad divergence beamexiting from a 5mm diameter hole in the center ofa 30mm minor diameter elliptical receive mirror.

The photodiode receivers in all three cases wereidentical 350 μm diameter InGaAs detectors opti-mized for large dynamic range (40dB) and low noisewith a bandwidth of 1MHz. The electronic noise levelcorresponded to less than 1nW incident power. Evenwith wavelength bandpass filtering and baffles, itwas still necessary to record the background photo-diode signals at regular intervals during the day-light. Lasers were turned off every 30 min whilebackground data were recorded.

Intensities were digitized using a 16 bit depthanalog-to-digital conversion, at a 5kHz rate. Scintil-lation index and AOA derived C2

n were displayed inreal time, while the raw data was saved for subse-quent use in laser communication simulation work.The intensity data were monitored to ensure thelevels were utilizing the available dynamic rangeof the receiver. All data laptops were synchronizedfor accuracy in correlating the many data inputs.

4. Irradiance Data

A. Overview of σ2I Measurements

The short-range retroreflector irradiance datashowed reproducibility from day to day. A well-defined minimum in σ2I and AOA variance occurredin the hour before sunset. This occurred at a tem-perature difference of 0:2–0:4C°, as measured be-tween ground and 15 ft elevation and persisted forabout 20 min. The scintillation index dropped from1 to 0.01 and back to 1 over a period of 1h. The tem-perature difference at sunrise was not so stable, fluc-tuating in the range of �1C°, and the turbulenceminimum was likewise not so well-defined. The scin-tillation index measured on this 1:1km retroreflector


link was in the range of 8–12 for the 3h aroundmidday.

The direct links carried out down the road to 2 and8:3km also showed a diurnal minimum in σ2I , as didthe corresponding retroreflected links back to theDMOI. However, the long-range, just above grounddirect link, of 24km did not show the minimum atsunrise or before sunset. The 19km Slate Range tovalley direct link did show a reduction in turbulencenear sunrise, with σ2I rising from 0.3 to 2 in the hourafter sunrise. The σ2I levels on this direct link per-sisted in the range of 2–3 for the next 6h even whilethe local 1:1km retroreflector link was recordingvalues of σ2I close to 0.1 in the quiescent period beforesunset.

B. Supporting Data

In order to investigate the datasets analyzed inthis paper with regard to the relative size of spatialfluctuations and receiver aperture, various spatialparameters, such as the Fresnel zone size, the scat-tering disk radius, and the coherence radius ρ0, areall considered. The Fresnel zone size, ðL=kÞ1=2, is lar-ger than the 22mm receive aperture for all exceptthe 2km range data, for which case they are of com-parable size. The spatial coherence radius, ρ0, or theclosely related Fried parameter [4], r0 ¼ 2:1 ρ0 ¼ð0:423C2

nk2LÞ−3=5 is determined from the measuredvalue of C2

n deduced from the AOA data recordedat the same time as the irradiance measurements.

C. Data Analysis

The irradiance data analyzed in this paper are fromthe direct links only. More than 10,000 samples ofraw data, each of 10 s duration, were recorded overthe course of the 5-day experiment. Randomly se-lected samples of data, chosen from the differentrange optical links, have been evaluated with regardto the PDF of the received irradiance. Intensity var-iations over 10 s time intervals (5 × 104 points) wereinitially corrected for the background signal due tosunlight that varied depending on the time of day.The data were normalized for mean intensity hIi ¼ 1,and the LN distribution of the normalized irradiancewas also normalized so that the integrated probabil-ity was 1. The calculated log-irradiance variance σ2ln Iwas evaluated for the dataset. The latter was thenused as the only variable in the LN distribution ofthe normalized irradiance [2–4]:

pðIÞ ¼ 1

Iffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πσ2ln I

q exp�−

�lnðIÞ þ 1

2 σ2ln I

�2σ2ln I

2�; ð1Þ

where σ2ln I is the natural log-irradiance variance.To compare the PDF of the data with Eq (1), a trans-formation of variables was used: z ¼ lnðIÞ so thatpzðzÞ ¼ ezpIðezÞ.

The corresponding PDF for a GG distribution [2]has the form

pðIÞ ¼ 2ΓðαÞΓðβÞI ðαβIÞ

αþβ2 Kα−β

�2

ffiffiffiffiffiffiffiffiαβI

p �; ð2Þ

where I is again the normalized irradiance, ΓðxÞ isthe gamma function, andKμðxÞ is themodified Besselfunction of the second kind. The parameters α and βare the inverse of the large-scale and small-scalescintillation values σ2x and σ2y , respectively, and, byextension, functions of the large-scale and small-scale log-irradiance variances:

α ¼ 1

σ2x¼ 1

expðσ2ln xÞ − 1; ð3Þ

β ¼ 1

σ2y¼ 1

expðσ2ln yÞ − 1: ð4Þ

The scintillation index is likewise related to thelarge-scale and small-scale log-irradiance variancesresulting in a constraint on α and β:

σ2I ¼ expðσ2ln x þ σ2ln yÞ − 1 ¼�1þ 1

α

��1þ 1

β

�− 1: ð5Þ

Hence, knowing the scintillation index of the ac-tual dataset, the best fit to a GG profile is determinedby varying a single parameter.

5. Sample Probability Density Distributions

Figure 2 shows resulting PDFs of irradiance data ac-quired in experiments that were carried out beside adirt road with the interrogator at a height of 2m andthe TATS receiver at a height of 5 ft above ground.Experiments were carried out at a number of rangesbetween 2 and 11km. Data were acquired for periodsof 1–2h at each range. Each plot of irradiance data isfrom a 10 s randomly chosen set of data. The date andtime (UTC) are given on each plot.

Likewise, Fig. 3 presents sample PDFs from thelong-range links. At a range of 24km experimentswere carried out using a man lift to vary the heightof the interrogator from 16.5 to 67 ft, while the TATSreceiver stayed at 4 ft. Further experiments were car-ried with the interrogator at an altitude of 2564 ftwith respect to the TATS receiver and at a rangeof 19km. Again, each plot of irradiance data is froma 10 s randomly chosen set of data.

Fig. 1. (Color online) Schematic diagram of the TransportableAtmospheric Testing Suite receiver system.


Figures 2(b) and 2(e) show data that have an im-proved fit with a GG distribution. These moderateirradiance fluctuation data support the conclusionsof Vetelino et al. [2] that the size of the receiving

aperture, in relation to the coherence width,determines whether the PDF is better fit to asingle or double stochastic model. However, strongerirradiance fluctuation data, such as shown in

a

4 3 2 1 0 1 2

0.001

0.01

0.1

ln I

PDF

Range 2.4km ;101206_2208452978;

22mm dia receiver; Fresnel zone 24mm; r0 9.5mm;σΙ2 1.76; σ LnI2 1.17; σ R2 14

b

4 3 2 1 0 1

0.001

0.005

0.010

0.050

0.100

0.500

ln I

PDF

Range 8.3km ;101207_0015048180;

22mm dia receiver; Fresnel zone 45mm; r0 27mm;σΙ2 0.849; σ LnI2 0.96; σ R2 6.7; 4.3; 2

c

4 2 0 2

0.001

0.002

0.005

0.010

0.020

0.050

0.100

0.200

ln I

ln I

PDF

Range 11km;101210_1650022343;

22mm dia receiver; Fresnel zone 52mm; r0 23mm;σΙ2 2.26; σ LnI2 1.94; σ R2 11.3

d

4 3 2 1 0 1 2

0.001

0.005

0.010

0.050

0.100

ln I

PDF

Range 2km ;101210_2220014062;


e

3 2 1 0 1

0.001

0.005

0.010

0.050

0.100

0.500

1.000

PDF

Range 2km ;101210_2330058281;

22mm dia receiver; Fresnel zone 22mm; r0 22mm;σΙ2 0.28; σ LnI2 0.28; σ R2 2.92; 10.2; 6

f

1.0 0.5 0.0 0.5 1.0

0.005

0.010

0.050

0.100

0.500

1.000

ln I

PDF

Range 2km;101211_0000079687;

22mm dia receiver; Fresnel zone 22mm; r0 52.4mm;σΙ2 0.087; σ LnI2 0.085; σ R2 0.69; 22; 25

Fig. 2. (Color online) Irradiance data, recorded at various ranges between 2 and 11km, are compared with a lognormal probability den-sity function (dashed curve) and, where applicable, with a gamma-gamma probability density function (continuous curve). All relevantparameters are listed with each plot.


Figs. 2(c), 3(c), and 3(f), are well fit by their self-consistent LN distributions even when their coher-ence widths exceed the aperture size.

Forty-four PDFs of irradiance data recorded dur-ing the field test have been evaluated. Most of thedata were acquired over long ranges or during highscintillation periods typical of daytime terrestrialatmospheres and follow a lognormal distribution

function. Since lower scintillation was only foundnear sunrise and sunset, data from these periods aremore heavily represented in the cumulative depen-dencies shown in Figs. 4 and 5. It is seen from Fig. 4,where the scintillation index is plotted against theRytov variance, that for weak fluctuations, σ2R < 1the LN and GG functions are equivalent. For 1 <σ2R < 10 and for σ2I < 1, the PDFs are better described

a

3 2 1 0 1 2

0.001

0.005

0.010

0.050

0.100

0.500

1.000

ln I

PDF

Range 24km;101207_2120049668;

22mm dia receiver; Fresnel zone 77mm; r0 16mm;σΙ2 0.707; σLnI2 0.53; σR2 39.7

b

3 2 1 0 1 2

0.001

0.005

0.010

0.050

0.100

0.500

ln I

PDF

Range 24km;101207_2155058262

22mm dia receiver; Fresnel zone 77mm; r0 18.8mm;σΙ2 1.26; σ LnI2 0.9; σ R2 30.4

c

6 4 2 0 2

0.001

0.01

0.1

ln I

PDF

Range 24km ;101208_0005047168;


d

4 3 2 1 0 1 2

0.001

0.01

0.1

ln I

PDF

Range 24km;101208_1613040956;

22mm dia receiver; Fresnel zone 77mm; r0 13.3mm;σΙ2 1.62 σ LnI2 0.98; σ R2 54

e

6 4 2 0 2

0.001

0.01

0.1

ln I

PDF

Range 19km;101209_2250008906;

22mm dia receiver; Fresnel zone 68mm; r0 17.6mm;σΙ2 2.235; σLnI2 1.71; σR2 27.7

f

8 6 4 2 0 2

5 10 4

0.001

0.005

0.010

0.050

0.100

ln I

PDF

Range 19km;101209_2355599687;


Fig. 3. (Color online) Irradiance data, recorded for the long-range links, are compared with a lognormal probability density function. Allrelevant parameters are listed with each plot.


by a GG model in which the small-scale diffractivefluctuations are themselves modulated by thestatistically independent large-scale refractive fluc-tuations. Moving into the saturation regime, charac-terized by σ2R → ∞ or for σ2I > 1, the data show thatthe LN dependence of the irradiance fluctua-tions holds.

The data are also represented in Fig. 5, where thescintillation index is plotted against the atmosphericcoherence width, or Fried parameter, ro ¼ 2:1ρ0.Since the current data were all acquired with a recei-ver of diameter 22mm, it is seen that the irradiancedata that follow a GG distribution are all in the mod-erate fluctuation regime and have values of ro great-er than the aperture diameter. According to Vetelinoet al. [2], the transition should occur for aperturesizes greater than ρ0. The difference of a factorof ∼2 with [2] might be explained if that workused the spherical wave case r0 ¼ ð0:16C2

nk2LÞ−3=5,whereas the present work uses the plane wave re-sult, as in [4]. The simulations of Lyke et al. [3] showthat the changeover occurs in high scintillation con-ditions at ρ0 and in moderate scintillation conditionsat 3ρ0=2. The present results show that for high scin-tillation conditions the irradiance fluctuations canusually be represented by LN statistics.

6. Discussion

The preponderance of the irradiance data acquiredduring the 5-day test at China Lake fits well to aLN distribution with the log-intensity variance ofthe data itself as input to the PDF. The situationswhere the GG function provides a better fit occurfor moderate fluctuation levels with σ2I < 1 and a co-herence width r0 larger than the diameter of the re-ceiver aperture. The even lower turbulence data fitsan LN or GG distribution equally well. These situa-tions occurred in the quiescent atmosphere just priorto sunset and for medium ranges. In these cases thechoice between LN and GG fits is in agreement withrecent work [2,3] that shows a sufficiently large re-ceiving aperture as averaging the small-scale fluc-tuations, leaving the large-scale fluctuations aloneto determine the form of the PDF. There are almostno instances in which the long-range data showedimprovement with a GG fit.

In summary, in very weak turbulence the LNand GG distributions give equally good fits to themeasured irradiance distributions. In less weakturbulence, where the coherence width exceeds theaperture size, the GG distribution provides an im-proved fit over the LN. During daylight hours thelong-range links covered in this paper were typicalof strongly saturated irradiance fluctuations and,as such, even the 22mm diameter receiver was suffi-cient to ensure averaging over the small-scale scin-tillation, leaving the PDF to represent the large-scale fluctuations that are lognormally distributed.

This work was supported in part by the U.S. Officeof Naval Research (ONR).

References1. L. C. Andrews and R. L. Philips, Laser Beam Propagation

through Random Media (SPIE Press, 2005).2. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aper-

ture averaging effects on the probability density of irradiancefluctuations in moderate-to-strong turbulence,” Appl. Opt. 46,2099–2108 (2007).

Fig. 4. (Color online) A plot of scintillation index versus Rytovvariance for irradiance measurements acquired over ranges from2 to 24km is shown. The legend designates those data for whichthe probability density profiles are better fit to a lognormal distri-bution or a gamma-gamma distribution.

Fig. 5. (Color online) A plot of scintillation index versus Friedparameter r0 for irradiance measurements acquired over rangesfrom 2 to 24km is shown. The diameter of the receiver apertureis also indicated. The legend designates those data for whichthe probability density profiles are better fit to a lognormal distri-bution or a GG distribution.


3. S. D. Lyke, D. G. Voelz, and M. C. Roggemann, “Probabilitydensity of aperture-averaged irradiance fluctuations for longrange free space optical communication links,” Appl. Opt. 48,6511–6527 (2009).

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5. R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell,M. R. Suite, and L. M. Thomas, “An analysis of long-termmea-surements of laser propagation over the Chesapeake Bay,”Appl. Opt. 48, 2388–2400 (2009).

6. R. Rao, S. Wang, X. Liu, and Z. Gong, “Turbulence spectrumeffect on wave temporal-frequency spectra for light propagat-ing through the atmosphere,” J. Opt. Soc. Am. A 16, 2755–2762 (1999).

7. R. Mahon, C. I. Moore, H. R. Burris, W. S. Rabinovich, M. Stell,M. R. Suite, and L. M. Thomas, “Power spectra of a free spaceoptical link in a maritime environment,” Proc. SPIE 7464,746407 2009).

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probability density of irradiance fluctuations observed over terrestrial ranges

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