1. 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

2. 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

3. 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) 0 ab

4. EXPERIMENT

5. 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.

6. SLM – P HASE S CREENS Laser (Gamma_phi ) 2 ~Speckle Size (mm) 10.015 100.047 100 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

7. 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..

8. THEORY

9. 1. Calculation of statistical moments of fluctuating intensity from data 2. Fitting the moments into the Probability Density Function Note: Index of realization Total number of realizations Coordinates of the pixel Fluctuating intensity Post-processing procedure

10. 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

11. 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)

13. RESULTS

14. NO SLM SLM 300SLM 0.001 SI 0.0034 SI 0.0066 SI 0.0120

15. SLMMINMAXPeak PDFScintilation index 0.0013805407.50.0034 0.013805407.30.0042 0.14007004.20.0105 13756504.20.0101 1031050050.0077 154007004.20.0105 1003607003.70.0128 1504257254.10.0098 30057510003.70.012 NO SLM5508505.30.0066 NO SLM

16. 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.

17. SUMMARY

18. 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