International Journal of Optics

Volume 2016 (2016), Article ID 8730609, 8 pages

http://dx.doi.org/10.1155/2016/8730609

## Irradiance Scintillation Index for a Gaussian Beam Based on the Generalized Modified Atmospheric Spectrum with Aperture Averaged

^{1}School of Astronautics and Aeronautic, University of Electronic Science and Technology of China, 2006 Xiyuan Avenue, Chengdu 611731, China^{2}School of Accounting, Southwestern University of Finance and Economics, 555 Liutai Avenue, Chengdu 611130, China

Received 18 August 2015; Revised 5 January 2016; Accepted 21 January 2016

Academic Editor: Fortunato Tito Arecchi

Copyright © 2016 Chao Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the aperture-averaged irradiance scintillation index of a Gaussian beam propagating through a horizontal path in weak non-Kolmogorov turbulence. Mathematical expressions are obtained based on the generalized modified atmospheric spectrum, which includes the spectral power law value of non-Kolmogorov turbulence, the finite inner and outer scales of turbulence, and other optical parameters of the Gaussian beam. The numerical results are conducted to analyze the influences of optical parameters on the aperture-averaged irradiance scintillation index for different Gaussian beams. This paper also examines the effects of the irradiance scintillation on the performance of the point-to-point optical wireless communication system with intensity modulation/direct detection scheme.

#### 1. Introduction

Optical wireless communication technology has drawn much attention for its significant technological challenges and prospective applications. It uses beams of laser propagating through the atmosphere to transmit data wirelessly at high speed [1, 2]. However, the atmosphere is full of numerous turbulence eddies, which has great degrading impacts on the performance of the communication system. Irradiance scintillation, one typical effect of atmospheric turbulence, results from stochastic redistribution of the optical energy within a cross section of the laser beam [3, 4]. The degree of irradiance scintillation for optical propagation on the receiving antenna can be characterized statistically by the irradiance scintillation index.

In the past few decades, various power spectrum models of refractive index have been proposed to analyze the irradiance scintillation index for different situations [5–15]. Generally speaking, these models can be classified into two categories: Kolmogorov models and non-Kolmogorov models. The former have a fixed power law value of 11/3 whereas the latter allow the power law value to vary in the range from 3 to 4. Practically, most non-Kolmogorov models can be generalized from corresponding Kolmogorov models [16]. Among the non-Kolmogorov models, the generalized modified atmospheric spectrum not only considers the finite inner and outer scales of turbulence, but also features the small rise at a high wave number, which is clearly seen in temperature data recorded by sensors [17, 18]. These properties make the generalized modified atmospheric spectrum useful in the investigation of the irradiance scintillation index along the radial and longitudinal propagation [10, 11, 13].

In this study, the generalized modified atmospheric spectrum is used to investigate the aperture-averaged irradiance scintillation index of a Gaussian beam in weak non-Kolmogorov turbulence along a horizontal path. The Gaussian beam, whose transverse electric field and intensity are normally distributed, is a typical kind of electromagnetic wave. The rest of the paper is organized as follows. Section 2 introduces the generalized modified atmospheric spectrum and the irradiance scintillation index of a Gaussian beam in weak turbulence. Section 3 presents a detailed expression reduction. The influences of the inner and outer scales of turbulence on the irradiance scintillation index of a Gaussian beam are analyzed in Section 4, followed by conclusions in Section 5.

#### 2. Theoretical Models

##### 2.1. Generalized Modified Atmospheric Spectrum

The generalized modified atmospheric spectrum takes the form [17]where is the angular wavenumber of the turbulence scale, is the spectral power law value, is the generalized atmospheric structure parameter, and and are the inner and outer scales of turbulence. is a function related to :where is the gamma function.

Letwhereand the constant coefficients , , and in (4) are usually set as , , and . The term in (1) is given by For the convenience of mathematical analysis, (5) is rewritten aswhere the coefficients are , , , , , , , and .

It must be pointed that the values of these coefficients , , and are based on the experimental data for the classic Kolmogorov turbulence. Besides, the generalized modified atmospheric spectrum will turn into the generalized exponential spectrum if or the generalized Kolmogorov turbulence if and .

##### 2.2. Aperture-Averaged Irradiance Scintillation Index of a Gaussian Beam

The mathematical model of a Gaussian beam depends on the radius and the phase front radius at the transmitter. Let be the propagation path length, and let be the angular wavenumber of the Gaussian wavewhere is the wavelength of the Gaussian wave. Thus, the curvature parameter of the Gaussian wave at the transmitter and the Fresnel ratio of the Gaussian wave at the transmitter are defined as [19], , and are nondimensional parameters of the Gaussian wave at the receiver:

The aperture-averaged irradiance scintillation index of a Gaussian beam propagating through weak atmospheric turbulence in the absence of beam wander takes the form [8]where is the normalized path coordinate is the coordinate along the propagation direction, is a nondimensional parameter characterizing the relative radius of the collecting lensand is the collecting lens diameter.

##### 2.3. Optical Wireless Communication Link Performance

The aperture-averaged irradiance scintillation index is relevant to the point-to-point optical wireless communication system with intensity modulation/direct detection scheme. The probability of fade, the mean signal-to-noise ratio, and the mean bit-error rate are often used to judge the performance of the link [15].

The probability of fade describes the percentage of time; the irradiance of the beam at the receiver is below certain threshold value . For weak turbulence, the probability of fade is defined bywhere is the error function and is the fade threshold parameter:The fade parameter stands for the mean irradiance in decibels below the threshold value .

The mean signal-to-noise ratio is a measure which compares the level of the received beam to the level of background radiance. For weak turbulence, the mean signal-to-noise ratio is defined bywhere is the signal-to-noise ratio in the absence of turbulence.

The mean bit-error rate is the number of bit errors in unit time. For weak turbulence, the mean bit-error rate is defined bywhere is the probability density function of lognormal distributionand is the complementary error function.

#### 3. Expression Reduction

This section mainly discusses the reduction of (10).

Without loss of generality, we defineBased on Euler’s formula, we can get [20, 21]Thus, (10) can be rewritten asSubstituting (1) into (20), it follows thatTo reduce the iterated integral in (21), it is necessary to expand the integrand by (6):Based on the equation for , , and [21, 22]the improper integral of can be rewritten asBased on Euler’s formula and de Moivre’s formula, we can get [20, 21]Thus, the iterated integral in (21) has been reduced to the definite integral of the real variable in the bounded interval , which can be easily computed by methods of numerical integration with arbitrary precision.

#### 4. Numerical Simulations

This section conducts numerical simulations to analyze the influences of , , , , and on the aperture-averaged irradiance scintillation index of different Gaussian beams. The simulations performed in this paper, however, should be only considered as arbitrary examples to indicate certain trend of results. Unless specified otherwise, all the numerical simulations are conducted with the default settings: m, rad/m, m, , m, and . Of course, other values which satisfy (12) can also be chosen.

Figure 1 depicts the aperture-averaged irradiance scintillation index for different Gaussian beams as a function of the spectral power law for several pairs of the inner and outer scales and . As shown for the convergent beam () in Figure 1(c), the aperture-averaged irradiance scintillation index first increases and then decreases with an increase in the spectral power law when other parameters are fixed, which acts in accordance with that under the generalized non-Kolmogorov spectrum and the generalized scale-dependent anisotropic spectrum, as presented in [8, 15]. Similar phenomena can be also found for the focused beam () in Figure 1(a), the convergent beam () in Figure 1(b), and the divergent beam () in Figure 1(d), respectively. According to Figure 1, it also shows that the aperture-averaged irradiance scintillation index is more sensitive to the outer scale than to the inner scale , and an increase in the outer scale will lead to an increase in the aperture-averaged irradiance scintillation index when other optical parameters are fixed. These phenomena can be physically explained by the fact that the irradiance scintillation can be decomposed into radial and longitudinal components. The former is sensitive to the outer scale , while the latter is sensitive to the inner scale . The longitudinal component of the irradiance scintillation only exists at the beam center, and thus the radial component occupies the dominant position in the influences on the aperture-averaged irradiance scintillation index . As the outer scale of turbulence increases, the Gaussian beam will meet more turbulence eddies along its propagation.