1. Introduction

The performance of a laser radar or laser communication system can be significantly degraded by turbulence-induced scintillation resulting from beam propagating through the atmosphere [1–4]. Specifically, irradiance scintillation can lead to power losses at the receiver and eventually to fading of the received signal below a prescribed threshold. Over the past decades, several atmospheric turbulence spectral models have been developed and applied in the research of the irradiance scintillation index associated with Gaussian-beam wave propagating through Kolmogorov atmospheric turbulence [5–10]. Only the modified atmospheric spectral model [11] can feature the high frequency enhancement property (also called “bump” property, which is caused by the turbulence inner scale) in the irradiance scintillation index [12].

For non-Kolmogorov turbulence existing in certain portions of the atmosphere [13–18], the general non-Kolmogorov spectral model has been proposed and used to investigate the irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence [19,20]. However, this spectral model does not consider the influence of finite turbulence inner and outer scales. To study non-Kolmogorov atmospheric turbulence, some theoretical spectral models were also developed, such as the generalized von Karman spectrum [21] and the generalized Exponential spectrum [22], which consider finite turbulence inner and outer scales and have general spectral power law values instead of the standard power law value of 11/3. The generalized modified atmospheric spectral model [23] for the non-Kolmogorov turbulence can characterize the high frequency enhancement property, but some coefficients in this spectral model should be reevaluated and justified by future experimental data.

In this study, the generalized von Karman spectral model is used to investigate the irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence. And then, the impacts of turbulence inner scales, outer scales and spectral power law values on the irradiance scintillation index have been analyzed.

2. Generalized von Karman spectrum

The generalized von Karman spectral model [21] has the following form

3. Irradiance scintillation index for Gaussian-beam wave

The irradiance scintillation index for Gaussian-beam wave propagating through weak atmospheric turbulence is given by [24]

For interpretation purpose, ${\sigma }_I^2\left(\rho \right)$ is expressed as a sum of radial and longitudinal components

Physically, ${\sigma }_{I,r}^2\left(\rho \right)$ and ${\sigma }_{I,l}^2$ describes the off-axis and on-axis contribution to the irradiance scintillation index, respectively. ${\sigma }_{I,r}^2\left(\rho \right)$ varies with ρ and becomes zero at the beam centerline ($\rho =0$). ${\sigma }_{I,l}^2$ is a constant in the transverse plane at . When $\Lambda =0$and$\Theta =1$, ${\sigma }_{I,r}^2\left(\rho \right)$becomes zero ($I_0\left(0\right)-1=0$) and ${\sigma }_{I,l}^2$ reduces to the expression of irradiance scintillation index for plane wave. When $\Lambda =\Theta =0$, ${\sigma }_{I,r}^2\left(\rho \right)$becomes zero and ${\sigma }_{I,l}^2$ reduces to the expression of irradiance scintillation index for spherical wave.

To consider the influences of finite turbulence inner scale, outer scale and spectral power law values, in the next section, ${\sigma }_I^2\left(\rho \right)$ is replaced with${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$, and its theoretical expression for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path will be derived.

3.1 The radial component of the irradiance scintillation index

For non-Kolmogorov turbulence, substituting Eq. (1) into Eq. (7), ${\sigma }_{I,r}^2\left(\rho \right)$ becomes

Expanding $I_0$ within a Maclaurin series [25]

Equation (9) can be expressed as

Integrating with respect to ξ, Eq. (11) becomes

Using the confluent hypergeometric function of the second kind $U\left(a;c;z\right)$ and its property [25]

Equation (12) can be expressed as

For analysis purpose, Eq. (15) is divided into two parts

For.., Eq. (17) can be approximated by the simpler expression

For mathematical analysis convenience, defining outer scale parameter $Q_0=\frac{L{\kappa }_0^2}{k}$, Eq. (19) becomes

Defining inner scale parameter$Q_l=\frac{L{\kappa }_l^2}{k}$, using the gauss hypergeometric function ${}_{2}F{}_1\left(A,B;C;Z\right)$ [25]

Substituting Eqs. (20) and (23) into Eq. (16), ${\sigma }_{I,r}^2\left(\rho ,\alpha ,l_0,L_0\right)$ can be expressed as

3.2 The longitudinal component of the irradiance scintillation index

For non-Kolmogorov turbulence, substituting Eq. (1) into Eq. (8), ${\sigma }_{I,l}^2$ becomes

Using$\cos \left(x\right)=\mathrm{Re}\left[e^{-ix}\right]$, Eq. (26) becomes

Using Eqs. (13) and (21), Eq. (27) becomes

Following the same procedure as [7], the approximate expression of Eq. (28) can be expressed as

Considering$\tilde{\Theta }=1-\Theta$, Eq. (29) becomes

Here, the part of $\mathrm{Re}(\cdot )$in Eq. (30) can be expressed as

Substituting Eq. (31) into Eq. (30), ${\sigma }_{I,l}^2\left(\alpha ,l_0,L_0\right)$ becomes

3.3 Irradiance scintillation index for Gaussian-beam wave

For non-Kolmogorov turbulence, substituting Eqs. (24) and (33) into Eq. (6), the expression of ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$ is obtained

Since plane and spherical waves are both characterized by the limiting condition$\Lambda =0$, in this case ${\sigma }_{I,r}^2\left(\rho ,\alpha ,l_0,L_0\right)$vanishes, ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$ is given by the longitudinal component

4. Numerical results

In this section, simulations are conducted to analyze the influences of $l_0$,$L_0$ and α on the irradiance scintillation index ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$. To avoid the mutual interferences between parameters, in the following simulations, two of the three parameters are fixed and only one parameter’s influence on ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$ is analyzed.

To remove dependence on the structure constant${\widehat{C}}_n^2$, the scaled irradiance scintillation index ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)/{\sigma }_{I\_pl}^2\left(\alpha \right)$ as a function of ${\Lambda }_0$will be plotted just as the Kolmogorov case [7]. Since the path length L and optical wavelength λ are fixed ($L=250m$and$\lambda =1.06\mu m$, then $\sqrt{L/k}=0.0065$. In fact, other values can also be chosen), all changes at the transmitter ${\Lambda }_0=2L/k{W}_0^2$ correspond to variations in the transmitter beam radius$W_0$.

4.1 Effect of inner scale’s variation on the irradiance scintillation index

To analyze turbulence inner scale’s influence on the irradiance scintillation index, α and $L_0$ are assigned to constant values (any α satisfies $3<\alpha <4$ can be chosen, in this simulation, $\alpha =10/3$. For the atmospheric turbulence, usually $L_0$ is in the order of magnitude of meter. $L_0=1.7m$ is chosen like [7]). For analysis purpose, different $Q_l$ values (introduced immediately following below Eq. (20), it is related to $l_0$ and represents the square ratio of the Fresnel zone $\sqrt{L/k}$with $l_0$) are chosen instead of various inner scale sizes. Here, $Q_l=10$($l_0$ is comparable with the Fresnel zone), $Q_l=100$($l_0$is smaller than the Fresnel zone) and $Q_l=1000$($l_0$is much smaller than the Fresnel zone and near to zero) are adopted. Other $Q_l$ values can also be set in the numerical analysis.

Figure 1 shows ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)/{\sigma }_{I\_pl}^2\left(\alpha \right)$ as a function of ${\Lambda }_0$. The lower set of curves represents the on-axis ($\rho =0$) irradiance scintillation index, while the upper set of curves denotes irradiance scintillation index levels at the diffractive beam edge ($\rho /W=1$). Also, different types of Gaussian-beam wave are chosen (here collimated beam ${\Theta }_0=1$and convergent beam ${\Theta }_0<1$are chosen). According to Eq. (4), when ${\Lambda }_0=0$and ${\Theta }_0=1$, then $\Lambda =0$and $\Theta =1$, it corresponds to the plane wave. When${\Lambda }_0\to \infty$, then $\Lambda =0$and $\Theta =0$, it denotes the spherical wave no matter what kind of Gaussian-beam wave is chosen.

 

Fig. 1 Scaled irradiance scintillation index as a function of ${\Lambda }_0$with different $Q_l$values. (a): collimated beam (${\Theta }_0=1$); (b): convergent beam (${\Theta }_0=0.7<1$); (c): convergent beam (${\Theta }_0=0.1<1$).

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Figure 1 shows that inner scales effects on the irradiance scintillation index are similar to the Kolmogorov turbulence case ($\alpha =11/3$) [7] for ${\Lambda }_0$ in the range 0 to 100. As $l_0$ increases ($Q_l$decreases), ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$ decreases. This can be explained from the definition of the generalized von Karman spectral model${\Phi }_n\left(\kappa ,\alpha ,l_0,L_0\right)$. When$l_0$increases, ${\Phi }_n\left(\kappa ,\alpha ,l_0,L_0\right)$decreases, and that makes ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)$ decreases.

This phenomenon can also be explained from the physical point of view: the irradiance fluctuations are contributed mostly by small-scale ($<\sqrt{L/k}$) turbulence cells [10,12]. When $l_0$ increases ($Q_l$decreases), the optical wave meets less small-scale ($<\sqrt{L/k}$) turbulence cells along its propagation path, which makes the irradiance scintillation index decreases.

4.2 Effect of outer scale’s variation on the irradiance scintillation index

In this section, $\alpha =10/3$, $Q_l=1000$(corresponding to very small $l_0$ value near to zero, and other $Q_l$can also be chosen). Figure 2 shows ${\sigma }_I^2\left(\rho ,\alpha ,l_0,L_0\right)/{\sigma }_{I\_pl}^2\left(\alpha \right)$ as a function of ${\Lambda }_0$for outer scale $L_0$values of $1m,2m$and$3m$($L_0\gg \sqrt{L/k}$is satisfied).

 

Fig. 2 Scaled irradiance scintillation index as a function of ${\Lambda }_0$with different outer scale values. (a): collimated beam (${\Theta }_0=1$); (b): convergent beam (${\Theta }_0=0.7<1$); (c): convergent beam (${\Theta }_0=0.1<1$).

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For weak atmospheric turbulence, the irradiance scintillation index for plane wave and spherical wave is not greatly influenced by the outer scale [7,24]. Figure 2 shows that the outer scale has no noticeable effect on the irradiance scintillation near the center of the beam ($\rho /W\approx 0$), similar to the limiting case of plane wave and spherical wave. This can be explained from the physical definition of the irradiance scintillation index for Gaussian beam. When $\rho /W\approx 0$, the radial component of the irradiance scintillation index disappears, and the expression of longitudinal component of the irradiance scintillation (see Eq. (33)) is independent of outer scale $L_0$. However, a finite outer scale of the order of 1-2 m can lead to variation in the irradiance scintillation away from the center of the beam near the diffractive beam edge ($\rho /W\approx 1$), particularly for beams with transmitter diameter ranging from $\sqrt{0.1}$ to $\sqrt{10}$ times the size of the Fresnel zone, that is $0.1<{\Lambda }_0=\frac{2L}{k{W}_0^2}<10$. This phenomenon is consistent with the Kolmogorov turbulence case ($\alpha =11/3$) [7]. It can be explained similar to the physical explanation in section 4.1, here the outer scale’s influence is analyzed instead. When $L_0$ increases, the Gaussian-beam wave meets more small-scale ($<\sqrt{L/k}$) turbulence cells along its propagation path, which makes the irradiance scintillation index increases, especially when the Gaussian-beam width $W_0$is thought to be comparable with the Fresnel zone $\sqrt{L/k}$ (in this case, it will produce larger irradiance scintillation index with respect to the case of other Gaussian-beam width [10]).

4.3 Effect of α’s variation on the irradiance scintillation index

In this simulation test, $L_0$ = $1.7m$, $Q_l$ = 1000. Figure 3 shows that the scaled irradiance scintillation index firstly increases with the increase of α, and then decreases with increasing ${\Lambda }_0$. Different values of α lead to more obvious variation in the irradiance scintillation index away from the center of the beam near the diffractive beam edge ($\rho /W\approx 1$) than the case near the center of the beam ($\rho /W\approx 0$).

 

Fig. 3 Scaled irradiance scintillation index as a function of ${\Lambda }_0$with different α. (a): collimated beam (${\Theta }_0=1$); (b): convergent beam (${\Theta }_0=0.7<1$); (c): convergent beam (${\Theta }_0=0.1<1$).

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5. Conclusions

In this study, theoretical expression of the irradiance scintillation index with finite inner scale, finite outer scale and general spectral power law is derived under the assumption of weak-fluctuation theory for Gaussian-beam wave propagating through non-Kolmogorov atmospheric turbulence with horizontal path.

Simulation results show that variable turbulence outer scale produces obvious effects on the irradiance scintillation index away from the center of the beam near the diffractive beam edge, and produces ignorable effects on the case near the center of the beam, and this conclusion is different from the cases for plane wave and spherical wave [7,24]. Variable turbulence inner scale brings obvious impacts on the final model, as the turbulence inner scale size increases, the irradiance scintillation index decreases. Different power law α produces obvious effects on the irradiance scintillation index especially for the case near the diffractive beam edge. The results in this study will help to better investigate the effects of turbulence on the Gaussian-beam wave propagating through weak non-Kolmogorov atmospheric turbulence with horizontal path.