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-Madani
Director 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.
b
Probability(a  h  b)   W (h)dh
W(h)
a

0
a b
h
h(l )   W (h)hl dh
0
• 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)
EXPERIMENT
Experimental set up at the source
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.
Beam expander x20
used to reach 1 cm
beam diameter
adequate for long
distance propagation.
Red He-Ne 2
mW laser with
0.8 mm beam
diameter.
The rest of the
beam (50%) is sent
to the ccd sensor.
This camera records
the statistics of the
beam at the source.
SLM – PHASE SCREENS
Laser
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
(Gamma_phi )2
~Speckle Size
(mm)
1
0.015
10
0.047
100
0.15
150
0.18
300
0.26
Experimental set at the receiver
The laser beam
recorded using
camera
capable to record
4096 different
levels of light
intensities at the
rate of 1000
frames per second
.
Weather station
records the
atmospheric
conditions.
.
THEORY
Post-processing procedure
1. Calculation of statistical moments of fluctuating intensity from data
k max
hk ( x, y) l
k 1
k max
h ( x, y )  
(l )
Fluctuating intensity
h
k
Index of realization
k max Total number of realizations
( x, y ) Coordinates of the pixel
2. Fitting the moments into the Probability Density Function
Note:
h (l )   W (h)h l dh
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 Model
Barakat
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
SI 0.0034
SLM 300
NO SLM
4000
SLM 0.001
3000
3500
3500
3000
2500
3000
2500
2000
2500
2000
2000
1500
1500
1500
1000
1000
1000
500
500
500
0
400
500
600
700
800
900
1000
0
500
600
700
800
PDF vs Histogram
900
1000
1100
1200
400
450
PDF vs Histogram
6
SI 0.0120
600
SI 0.0034
7
3
4
550
8
3.5
SI 0.0066
500
PDF vs Histogram
4
5
0
350
6
2.5
5
2
3
4
1.5
3
2
1
2
0.5
1
1
0
0
-0.5
0
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
SLM
0.001
0.01
0.1
1
10
15
100
150
300
NO SLM
MIN
380
380
400
375
310
400
360
425
575
550
MAX
540
540
700
650
500
700
700
725
1000
850
Peak PDF Scintilation index
7.5
0.0034
7.3
0.0042
4.2
0.0105
4.2
0.0101
5
0.0077
4.2
0.0105
3.7
0.0128
4.1
0.0098
3.7
0.012
5.3
0.0066
Peak PDF
8
7
6
5
4
3
2
1
0
0.0001
Range of Intensities
0.01
NO10000
SLM
0.014
1000
0.012
800
0.01
0.008
600
0.006
400
0.004
200
0.002
NO SLM
0
0.01
100
Scinitilation index
1200
0.0001
1
1
100
10000
0
0.0001
0.01
1
100
10000
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
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.