## Exponentiated Weibull distribution family under aperture averaging Gaussian beam waves: comment |

Optics Express, Vol. 20, Issue 18, pp. 20680-20683 (2012)

http://dx.doi.org/10.1364/OE.20.020680

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### Abstract

Recently, an exponentiated Weibull distribution model was presented for describing the effects of aperture averaging on scintillation of Gaussian beams propagating through atmospheric turbulence. The model uses three parameters that are derived from physical quantities so that in principle the model could be used to predict optical link performance. After reviewing this model, however, we find several inconsistencies that render it unusable for this purpose under any scintillation conditions.

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1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express **20**(12), 13055–13064 (2012). [CrossRef] [PubMed]

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express **20**(12), 13055–13064 (2012). [CrossRef] [PubMed]

*α*,

*β*, and

*η*:where

*I*denotes the irradiance normalized to its mean. The authors relate these parameters directly to observable quantities (i.e., the receiver aperture diameter,

*D*, the aperture averaged scintillation index,

_{σI2}, and the atmospheric coherence radius,

*ρ*

_{0}) according to Eqs. (10)–(12) of their publication as shown below: where

_{g(α,β)=∑i=0∞(−1)i(i+1)−(1+β)/βΓ(α)i!Γ(α−i), }and

_{Γ(⋅)}is the gamma function. Note, in particular, when

*α*= 1 and

*β*= 1, Eq. (12) yields

*η*= 1 and the EW distribution in Eq. (7) becomes a negative exponential distribution, exp[–

*I*] (in this regard see below).

*D*/

*ρ*

_{0}= 0.32 for both simulation and experiment, whereas the values of

_{σI2}differ by less than 2% (0.486 vs. 0.477). We would therefore expect the two sets of fit parameters,

*α*,

*β*, and

*η*, to be nearly identical. This, however, is not the case as

*α*,

*β*, and

*η*differs by about 69% (2.4 vs. 4.05), 12% (0.95 vs. 0.84), and 53% (0.55 vs. 0.84), respectively.

_{σR2}. Now Eqs. (10)–(12) are functions of

*D*/

*ρ*

_{0}, and the aperture averaged scintillation index,

_{σI2}

_{.}Although

*ρ*

_{0}can be expressed as a function of

_{σR2}

_{,}we are not aware of any direct relationship between

_{σI2}and

_{σR2}and are at a loss as how the estimated values of

*α*,

*β*, and

*η*shown in Fig. 3 were obtained. Further, as stated in the text, we find that the parameter values obtained from the fits and those obtained from Eqs. (10)–(12) are not consistent.

1. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express **20**(12), 13055–13064 (2012). [CrossRef] [PubMed]

*exp*[-

*I*] if and only if both

*α*= 1 and

*β*= 1. Further it is well known that under these conditions not only does the PDF of irradiance become a negative exponential but also

_{σI2}for a point receiving aperture becomes unity, consistent with Eq. (11), for

*α, β*= 1. From Eq. (10), however, the condition

*α*= 1 implies that

*D*/

*ρ*

_{0}= 14.0, which by definition indicates significant aperture averaging and which must necessarily yield a non-unity value of the aperture averaged scintillation index,

_{σI2}. Thus, the EW theory leads one to a physically inconsistent conclusion and is likewise inappropriate for this turbulence regime as well.

## Acknowledgment

## References and links

1. | R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: June 26, 2012

Revised Manuscript: July 20, 2012

Manuscript Accepted: July 23, 2012

Published: August 23, 2012

**Citation**

H. T. Yura and T. S. Rose, "Exponentiated Weibull distribution family under aperture averaging Gaussian beam waves: comment," Opt. Express **20**, 20680-20683 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20680

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