Comparison of probability density functions for aperture-averaged irradiance fluctuations of a Gaussian beam with beam wander

Melissa Beason; Melissa Beason; Szymon Gladysz; Larry Andrews

doi:10.1364/AO.391938

Applied Optics
Vol. 59,
Issue 20,
pp. 6102-6112
(2020)
•https://doi.org/10.1364/AO.391938

Comparison of probability density functions for aperture-averaged irradiance fluctuations of a Gaussian beam with beam wander

Melissa Beason, Szymon Gladysz, and Larry Andrews

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Melissa Beason, Szymon Gladysz, and Larry Andrews, "Comparison of probability density functions for aperture-averaged irradiance fluctuations of a Gaussian beam with beam wander," Appl. Opt. 59, 6102-6112 (2020)

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About this Article
History
- Original Manuscript: March 3, 2020
- Revised Manuscript: June 2, 2020
- Manuscript Accepted: June 3, 2020
- Published: July 9, 2020

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Abstract

Over the years, there has been much interest in the use of optical wavelengths for communication because of the potential for high data rates. However, the performance of these systems can become significantly degraded due to turbulence-induced signal fluctuations. These fluctuations can be minimized by enlarging the receiving aperture, thereby averaging the fluctuations. There is extensive interest in developing probability density functions (PDFs) describing these intensity fluctuations so as to accurately predict system performance. This work examines several PDF models that have been suggested to represent fluctuations by comparing them to simulations of realistic propagation scenarios of a collimated Gaussian beam with centroid wander. Unlike with an infinite plane or spherical wave, the empirical PDF shape in these simulations changed significantly with increased aperture, going from a positively skewed to a negatively skewed distribution; therefore, the PDF model that describes it also must change. This change in skew can have serious consequences on the metrics of an optical communication system, e.g., number of fades and bit-error rate. In this work, we examine the evolution of the empirical PDF with aperture size and the fit of potential PDF models under various strengths of turbulence. We show that the change in skew with increasing aperture size occurred for all strengths of turbulence and that the currently used PDF models do not adequately characterize this for realistically sized apertures where beam wander is present.

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Simulated $C_{n}^{2} (m^{- 2 / 3})$	# of Screens	Grid Sample (mm)	Beam Radius (mm)	$σ_{R}^{2}$	Strehl Ratio	$σ_{I, l}^{2}$
$1.4 \times 10^{- 15}$	5		49	0.1	0.95	0.046
$7 \times 10^{- 15}$	5	0.52	54	0.5	0.76	0.26
$7 \times 10^{- 14}$	10	1.12	104	5	0.20	2.50
$2 \times 10^{- 13}$	16	1.97	171	14.2	0.07	3.55

D (mm)	CDF	Data	LN	GG	IGG	N	EW	W3
0.53	$P (I < 0.48)$	$1.1 \times 10^{- 3}$	$3.7 \times 10^{- 4}$	$1.0 \times 10^{- 3}$	$1.0 \times 10^{- 3}$	$7.4 \times 10^{- 3}$	$1.4 \times 10^{- 3}$	$3.3 \times 10^{- 3}$
47.4	$P (I < 0.66)$	$1.3 \times 10^{- 3}$	$4.2 \times 10^{- 4}$	F	$5.9 \times 10^{- 4}$	$2.8 \times 10^{- 3}$	$2.1 \times 10^{- 3}$	$3.1 \times 10^{- 3}$
56.8	$P (I < 0.70)$	$1.1 \times 10^{- 3}$	$2.6 \times 10^{- 4}$	F	F	$1.6 \times 10^{- 3}$	$1.4 \times 10^{- 3}$	$3.3 \times 10^{- 3}$
141.2	$P (I < 0.98)$	$1.1 \times 10^{- 3}$	$4.7 \times 10^{- 6}$	F	F	$5.8 \times 10^{- 6}$	$4.0 \times 10^{- 3}$	$9.7 \times 10^{- 4}$

D (mm)	CDF	Data	LN	GG	IGG	N	EW	W3
0.59	$P (I < 0.097)$	$1.03 \times 10^{- 3}$	$31.9 \times 10^{- 6}$	$4.6 \times 10^{- 4}$	$4.7 \times 10^{- 4}$	$3.8 \times 10^{- 2}$	$4.8 \times 10^{- 4}$	$5.7 \times 10^{- 3}$
36.3	$P (I < 0.22)$	$1.08 \times 10^{- 3}$	$1.3 \times 10^{- 5}$	F	$5.0 \times 10^{- 4}$	$1.4 \times 10^{- 2}$	$1.0 \times 10^{- 3}$	$5.1 \times 10^{- 3}$
67.9	$P (I < 0.4)$	$1.07 \times 10^{- 3}$	$2.6 \times 10^{- 6}$	F	F	$1.3 \times 10^{- 3}$	$1.5 \times 10^{- 3}$	$1.6 \times 10^{- 3}$
92.1	$P (I < 0.56)$	$9.9 \times 10^{- 4}$	$1.3 \times 10^{- 7}$	F	F	$4.2 \times 10^{- 5}$	$1.5 \times 10^{- 3}$	$1.4 \times 10^{- 3}$

D (mm)	CDF	Data	FE	GG	IGG	N	EW	W3
1.3	$P (I < 4 \times 10^{- 4})$	$1.02 \times 10^{- 3}$	$1.3 \times 10^{- 3}$	$1.2 \times 10^{- 3}$	$2.1 \times 10^{- 3}$	0.26	^a $1.0 \times 10^{- 2}$	$4.5 \times 10^{- 3}$
28.0	$P (I < 0.015)$	$9.6 \times 10^{- 4}$	$2.1 \times 10^{- 2}$	$6.1 \times 10^{- 3}$	$1.1 \times 10^{- 2}$	0.20	$2.4 \times 10^{- 3}$	$1.5 \times 10^{- 2}$
99.9	$P (I < 0.05)$	$1.1 \times 10^{- 3}$	$6.5 \times 10^{- 2}$	$4.3 \times 10^{- 5}$	$5.0 \times 10^{- 5}$	$2.8 \times 10^{- 2}$	^a $9.2 \times 10^{- 3}$	$6.4 \times 10^{- 3}$
169.6	$P (I < 0.14)$	$9.3 \times 10^{- 4}$	–	F	F	$4.9 \times 10^{- 5}$	^a $1.2 \times 10^{- 6}$	$1.9 \times 10^{- 3}$

D (mm)	CDF	Data	FE	GG	IGG	N	EW	W3
2.2	$P (I < 4 \times 10^{- 4})$	$1.1 \times 10^{- 3}$	$2.2 \times 10^{- 3}$	$6.1 \times 10^{- 3}$	$1.7 \times 10^{- 3}$	0.30	$8.6 \times 10^{- 4}$	$5.5 \times 10^{- 3}$
49.4	$P (I < 0.028)$	$1.1 \times 10^{- 3}$	$2.9 \times 10^{- 2}$	$5.9 \times 10^{- 3}$	$7.5 \times 10^{- 3}$	0.17	$2.6 \times 10^{- 3}$	$1.5 \times 10^{- 2}$
160.6	$P (I < 0.06)$	$1.0 \times 10^{- 3}$	–	$4.5 \times 10^{- 5}$	$5.2 \times 10^{- 5}$	$2.4 \times 10^{- 2}$	^a $9.5 \times 10^{- 3}$	$5.2 \times 10^{- 3}$
398.1	$P (I < 0.4)$	$1.0 \times 10^{- 3}$	–	F	F	$1.7 \times 10^{- 12}$	$4.2 \times 10^{- 38}$	$2.2 \times 10^{- 6}$

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Applied Optics