Probability density function of partially coherent beams

propagating in the atmospheric turbulence

Olga Korotkova, Physics Department, University of Miami, FL

Charles Nelson Electrical and Computer Engineering Department, USNA

Svetlana Avramov-Zamurovic, Weapons and Systems Department, USNA

Reza Malek-MadaniDirector of Research, Mathematics Department, USNA

Sponsored by ONR

Goal:Reconstruction of the Probability Density Function of a partially coherent beam propagating in turbulent atmosphere

• Experiment:

• On-field experiments are set up at the United States Naval Academy• A partially coherent beam with controllable phase correlation is produced

with the help of the reflecting SLM• Measurement of the intensity statistics of the beam in its transverse

cross-section is made using a ccd sensor

• Theory:

• Statistical moments of fluctuating intensity from the data are found and intensity histograms are constructed

• PDF reconstruction model is applied• Comparison among the models and data sets is made

Probability Density Function

• PDF of fluctuating intensity W(h) shows with which chance the beam’s intensity attains a certain level.

• Determination of the PDF from moments is an academically noble problem: (famous Hausdorff moment problem)

• Knowledge of the PDF of the intensity is crucial for solving inverse problems of finding the statistics of a medium

• Knowledge of the PDF is necessary for calculation of fade statistics of a signal encoded in a beam (BER errors in a communication channel)

h

W(h) Probability( ) ( )b

a

a h b W h dh

0 a b

( )

0

( )l lh W h h dh

EXPERIMENT

Experimental set up at the source

Red He-Ne 2 mW laser with 0.8 mm beam diameter.

The laser beam is reflected from the SLM to cerate partially coherent beam and sent to beam splitter.

Beam splitter distributes part of the beam to be sent through the atmospheric channel across the water.

The rest of the beam (50%) is sent to the ccd sensor. This camera records the statistics of the beam at the source.

Beam expander x20 used to reach 1 cm beam diameter adequate for long distance propagation.

SLM – PHASE SCREENS

Laser

(Gamma_phi )2

~Speckle Size (mm)

1 0.015

10 0.047

100 0.15

150 0.18

300 0.26

T. Shirai, O. Korotkova, E.Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 7 (2005) 232-237

Experimental set at the receiver

The laser beam recorded using cameracapable to record 4096 different levels of light intensities at the rate of 1000 frames per second.

Weather station records the atmospheric conditions..

THEORY

1. Calculation of statistical moments of fluctuating intensity from data

2. Fitting the moments into the Probability Density Function

Note: dhhhWh ll )()(

max

1 max

)( ),(),(

k

k

lkl

k

yxhyxh

Index of realization

Total number of realizations Coordinates of the pixel

Fluctuating intensity

k

maxk

),( yx

h

Post-processing procedure

Probability Distribution Function Reconstruction Method

• Barakat: Gamma-Laguerre distribution ▫Medium and source independent ▫Uses first n moments of detected intensity▫Valid in the presence of scatterers▫Valid anywhere in the beam

Gamma-Laguerre ModelBarakat

R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments, J. Opt. Soc. Am. 16, 2269(1999)

RESULTS

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0 0.5 1 1.5 2 2.5 3-0.5

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4PDF vs Histogram

NO SLM SLM 300 SLM 0.001

0 0.5 1 1.5 2 2.5 30

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6PDF vs Histogram

0 0.5 1 1.5 2 2.5 30

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8PDF vs Histogram

SI 0.0034

SI 0.0034SI 0.0066

SI 0.0120

SLM MIN MAX Peak PDF Scintilation index0.001 380 540 7.5 0.00340.01 380 540 7.3 0.00420.1 400 700 4.2 0.01051 375 650 4.2 0.0101

10 310 500 5 0.007715 400 700 4.2 0.0105

100 360 700 3.7 0.0128150 425 725 4.1 0.0098300 575 1000 3.7 0.012

NO SLM 550 850 5.3 0.0066

NO SLM

NO SLM

NO SLM

Reflections on data analysis

• The normalized intensity PDF of a partially coherent beam changes its shape with the change in the initial phase coherence length.

• For weakly randomized beams (SLM 0.1 - SLM 300) the intensity fluctuations are enhanced leading to larger scintillation index. As the laser beam gets strongly randomized (SLM 0.001 - SLM 0.01) the intensity fluctuations drop fast, leading to a much smaller scintillation index

• The shape of the PDF remains Gamma-like for laser beam (no SLM) and for weak and moderate SLMs (SLM 300 – SLM 0.01). Only in the case of strong SLM, (SLM 0.001) which corresponds to completely incoherent beam the PDF takes the Gaussian form, i.e. it can resist to atmospheric fluctuations.

SUMMARY

• Based on Gamma-Laguerre model by Barakat we reconstructed from the collected data the single-point Probability Density Function (PDF) of the fluctuating intensity of a partially coherent beam propagating through the atmospheric turbulence

• The dependence of the PDF on the initial phase correlation has been examined. We found that the structure of the PDF is Gamma-like for weak SLMs and becomes more Gaussian-like for strong SLMs. Also we found that compared to laser beam (no SLM) the scintillation index of partially coherent beams is somewhat larger for weak SLM beams but much lower for strong SLM beams.

• Our results are fundamental for understanding of interaction mechanism and optimization of semi-random radiation energy transfer in natural environments. This research may also find uses for solving inverse problems (sensing) and for communications through turbulent structures.

Summary