1. Introduction

Propagation of optical pulses through atmospheric turbulence has attracted much attention over many years by reason of its importance in practical applications, e.g., high-speed free-space optical (FSO) communications, lidar, ranging, etc. Atmospheric turbulence causes such pulse distortions as temporal broadening and time-of-arrival fluctuations; a deep understanding of them is useful in analyzing and designing engineering systems involving optical pulse propagation in the earth’s atmosphere. Over the past decades, numerous researchers have studied various aspects in the problem of pulse propagation through atmospheric turbulence [1–10]. However, the existing works with respect to these topics generally build themselves on an assumption that refractive-index fluctuations in the atmosphere are statistically homogeneous and isotropic.

Anisotropy of atmospheric turbulence has been observed by many experimental measurements [11–14]. This behavior is often attributed to the phenomenon that turbulent irregularities in some portions of the atmosphere are elongated along the earth’s surface [15–17]. For instance, the scale of large turbulent irregularities along the horizontal direction in the upper troposphere and stratosphere may be much greater than that along the vertical one [16,17]. Consequently, the widely made assumption of isotropic turbulence in the treatments of optical-wave propagation through the atmosphere does not always hold true in practical situations. Up to now, considerable efforts have been made to elucidate the effects of anisotropic atmospheric turbulence on optical wave propagation. Consortini et al. [18] studied the impact that anisotropic atmospheric turbulence has on the mutual dancing of two parallel propagated narrow laser beams. Manning [15] theoretically examined the mutual coherence function (MCF) for a monochromatic plane wave propagating in anisotropic atmospheric turbulence, and found that the horizontal MCF differs from the vertical one. Gurvich and BelenKii [19] analyzed the degradation of star image caused by stratospheric turbulence, which is usually anisotropic. Kon [20] developed a qualitative theory to analyze the amplitude and phase fluctuations of waves propagating in anisotropic atmospheric turbulence, and later these issues have been quantitatively dealt with by others [21,22]. The spreading and scintillation index of a beam wave travelling in anisotropic atmospheric turbulence have also been formulated recently [23,24]. These works reveal that anisotropy may play an important role in analysis of optical wave propagation through atmospheric turbulence. However, there have, to our knowledge, been no reports concerning the problem that how the aforementioned temporal pulse broadening behaves when the assumption of isotropic turbulence is no longer valid. Thus, a detailed investigation into it is imperative.

The purpose of this paper is to understand the effects of anisotropy on the temporal broadening of optical pulses propagating along a horizontal path in the earth’s turbulent atmosphere. First, by employing the extended Huygens-Fresnel principle, we formulate the two-frequency MCF for beam waves travelling horizontally in anisotropic atmospheric turbulence. Then, with the help of this formulation, the expression for the mean square temporal width of Gaussian-beam-wave pulses is developed based on the temporal moment method. Finally, numerical calculations are carried out, and according to the results obtained, we address the question as to how anisotropy of turbulence affects the temporal broadening behavior of optical pulses.

2. Theoretical formulations

As in [16,17,20,24], we assume that anisotropic turbulent irregularities, in the mean, can be visualized as an oblate spheroid generated by ellipse rotation around the minor axis. In the earth’s atmosphere, the short axis of anisotropic turbulent irregularities is indeed perpendicular to the horizontal plane, which corresponds to the situation where anisotropy of turbulence is only present over planes perpendicular to the horizontal plane. Here, let us consider that optical pulses propagate along a horizontal path in the earth’s atmosphere. Moreover, we specify that the z-axis and propagation path coincide, and the source and observation planes are positioned at z = 0 and z = L, respectively. With this propagation geometry in mind, the circular symmetry in planes perpendicular to the propagation path reported in [23,24] is no longer maintained. Thus, in what follows, we begin by developing new fundamental expressions for optical pulse propagation in anisotropic atmospheric turbulence instead of building our derivations on existing formulations [1-10,25,26], which have been obtained by making use of the said circular symmetry.

2.1. Two-frequency MCF for beam waves

The two-frequency MCF is a fundamental quantity for determining the basic properties related to pulse propagation through atmospheric turbulence. Although the two-frequency MCF for beam waves travelling in isotropic turbulence has been investigated by many researchers [1,2,4-10,26], it is rather unexplored for the case of anisotropic turbulence. By employing the extended Huygens-Fresnel principle, the two-position, two-frequency MCF for beam waves in atmospheric turbulence can be written by [26]

Over the years, various forms for the spatial power spectrum of refractive-index fluctuations in anisotropic atmospheric turbulence have been suggested by different authors [13,15-17,20-22]. For simplicity, most of the theoretical studies in the literature involving optical wave propagation through anisotropic atmospheric turbulence have supposed that the degree of anisotropy is constant for all scales of turbulent irregularities, and used a spatial power spectrum of refractive-index fluctuations with the same anisotropic factor for various scales of turbulent irregularities to develop the corresponding mathematical models. Nevertheless, as pointed out by many authors [16,17,23,24], the degree of anisotropy actually varies with the specific scales of turbulent irregularities; viz., anisotropy prevails mainly at large scales and the turbulent irregularities near the size of the inner scale become isotropic. Thus, it is desirable to characterize optical wave propagation in anisotropic atmospheric turbulence based on a spatial power spectrum of refractive-index fluctuations which takes into account the variation in the degree of anisotropy at various scales of turbulent irregularities. Recently, Toselli [29] proposed a spatial power spectrum of refractive-index fluctuations, which introduced a rescaling of turbulent irregularities owing to anisotropy for various scales; it has been pointed out that this spectrum is developed under the assumption that anisotropy of turbulence exists only along the propagation path, i.e., the z-axis. As opposed to the case dealt with by Toselli [29], it can be deduced from the previous descriptions that, for the propagation geometry considered in this paper, anisotropy of turbulence is indeed present in planes perpendicular to the propagation path. Hence, we adopt an expression for the spatial power spectrum of refractive-index fluctuations, which is slightly different from the original one given by Toselli [29], in the form

To develop a theoretical model for the mean square temporal pulse width, only the single-position, two-frequency MCF is needed; therefore, we will specify r_d ≡ 0 in the derivations below. It is noted that analytically evaluating the integral in Eq. (7) is formidable under general conditions. For mathematical tractability, here we confine our attention to the case of strong atmospheric turbulence, i.e., ${\tilde{C}}_n^2{\overline{k}}^{2}L{l}_0^{\alpha -2}\gg 1$ [30], where $\overline{k}$ denotes the optical wave number corresponding to the carrier frequency of the narrowband pulses. In this case, we can apply the formula ₁F₁(b;c;−x) ~ 1 − bx/c with |x| ≪ 1 to simplify Eq. (7) and obtain

To proceed further, we take into account a completely coherent pulsed beam-wave field, radiated from a narrowband source. In this case, the two-position, two-frequency MCF at the source plane can be given by ${\mathrm{\Gamma }}_2^{(0)}({\mathbf{s}}_1,{\mathbf{s}}_2;k_1,k_2)=u_0({\mathbf{s}}_1,k_1)u_0^{*}({\mathbf{s}}_2,k_2)$ with $u_0({\mathbf{s}}_m,k_m)=a_0(k_m)\exp (-s_m^2/w_0^2)$ denoting the wave field of a collimated Gaussian beam at the source plane, where the asterisk represents the complex conjugate, w₀ is the initial beam radius, a₀(k_m) is an amplitude function depending on the optical wave number k_m (m = 1, 2), and s_m = |s_m| (m = 1, 2). For tractability reasons, here we assume that the temporal shape of the optical pulses is Gaussian with a half-width T₀ at the source plane, which is defined by the e⁻¹ point of the temporal pulse-shape function ([28], p. 742, Eq. (23)). Without loss of generality, we specify that $a_0(k_m)={\pi }^{1/2}{T}_0\exp [-{(k_m-\overline{k})}^2{c}^2{T}_0^2/4]$ with c being the speed of light. In the case of r₁ = r₂ = 0, by substituting Eq. (6) with Q₂ given by Eq. (9) into Eq. (1), and evaluating the resultant four-dimensional integral, one can find the on-axis two-frequency MCF given by

Equation (10) shows our first theoretical contribution, which lays down the necessary groundwork for determining the on-axis temporal broadening of Gaussian-beam-wave pulses due to strong anisotropic atmospheric turbulence. Notice that, in arriving at Eq. (10), we have used the narrowband approximation ${\tilde{k}}_{12}^2-k_{12}^2/4\approx {\tilde{k}}_{12}^2$. In addition, for later mathematical convenience, the on-axis two-frequency MCF is written in the form of Eq. (10). In fact, Eq. (10) is consistent with Eq. (9) of [7] when ζ_e = 1 and α = 11/3 provided that we exclude the effects due to the introduction of the amplitude function a₀(k_m). By the way, the author of [7] has noted that Eq. (9) therein agrees with Eq. (50) of [26] except for a different definition for ρ₀; this distinction in ρ₀ actually arises from the different conditions under which the expression indicated by Eq. (7) is simplified.

2.2. Mean square temporal pulse width

The mean temporal pulse width can be readily determined if an analytical expression for the time-dependent mean intensity of the pulses has been developed. Theoretically, we can calculate the time-dependent on-axis mean intensity of the pulses by taking the Fourier transform of the on-axis two-frequency MCF given by Eq. (10). However, Liu and Yeh [1,3] have suggested the temporal moment method for computing the mean square temporal pulse width, which does not need the said Fourier-transform operation, and has been further used by their followers [4,7-9]. In accordance with [1,3,4,7-9], the n^th temporal moment of the pulses can be written by

Based on Eqs. (10), (17) and (18), one can find (see Appendix B)

Equation (19) as our second theoretical contribution indicates that the mean square temporal width of a Gaussian-beam-wave pulse does depend on the anisotropy of turbulence. The expression for the mean square temporal pulse width given by [7] has been developed by making a baseband assumption; nevertheless we have not made such baseband assumption to develop Eq. (19). Indeed, our derivations have been based on the premise that the propagated pulse is a modulated signal with carrier frequency ${\omega }_0=\overline{k}c$. Furthermore, when ζ_e = 1 and α = 11/3, it is interesting to find that the sum of the first two terms in Eq. (19), i.e., $T_0^2/4+2\delta /c^2$, is exactly equivalent to the expression for the mean square temporal pulse width developed by Young et al. ([4], Eq. (20)) based on the Rytov approximation (Notice that κ₀ = 1/L₀ in [4]). It should be stressed that the mean square temporal pulse width shown by Eq. (18) builds on the concept of the pulse arrival time, which is defined in a manner that the “time centroid” of a pulse is regarded as the instant of time when it arrives at the observer. It has been shown that the mean square temporal pulse width derived from Eq. (18) is equal to 1/4 times the mean square temporal pulse half-width defined according to the e⁻² point of the time-dependent mean intensity of optical pulses [4].

3. Numeral calculations and analysis

In this section, we examine the effects that anisotropy of turbulence has on the temporal broadening behavior of Gaussian-beam-wave pulses propagating along a horizontal path in the atmosphere. Notice that, the parameter ${\tilde{C}}_n^2$ appearing in Eq. (5) is in units of m^3−α. To avoid the unit choice inconsistency in representation of calculation results pointed out by Charnotskii [31], here we use the nondimensional parameter $q={\tilde{C}}_n^2{\overline{k}}^{2}L{l}_0^{\alpha -2}$ as a measure of turbulence strength. In addition, the Fresnel ratio ${\Lambda }_0=2L/(\overline{k}{w}_0^2)$ defined in [28] will be employed to describe a collimated Gaussian beam; a smaller Λ₀ means that the beam acts more like a plane wave, whereas a larger Λ₀ implies that it acts more like a spherical wave. For purposes of later description convenience, we define the turbulence-induced increment of mean square temporal pulse width as ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)={\sigma }_{t,1}^2(\alpha ,{\zeta }_e)+{\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ with ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)=2\delta /c^2$ and ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)=\mathrm{\Delta }/c^2$, where the dependence of ${\sigma }_{\text{turb}}^2(\cdot )$, ${\sigma }_{t,1}^2(\cdot )$ and ${\sigma }_{t,2}^2(\cdot )$ on the parameters α, ζ_e and Λ₀ is shown explicitly; note that δ and Δ are related to the parameters α, ζ_e and Λ₀ by Eqs. (15), (16) and (20). It is apparent that ${\sigma }_{\text{turb}}^2(\cdot )$ actually characterizes the amount of temporal pulse broadening caused by atmospheric turbulence. To begin, the basic parameters used in the calculations below are given as follows: L = 20 km, q = 15, L₀ = 10 m, l₀ = 2 mm and $\overline{k}=2\pi /{\lambda }_0$ with λ₀ = 800 nm.

First, we examine the change in the turbulence-induced increment of mean square temporal pulse width with varying values of the effective anisotropic factor ζ_e. Figure 1 illustrates the scaled turbulence-induced increment of mean square temporal pulse width as a function of the effective anisotropic factor ζ_e with various combinations of Λ₀ and α. It is observed from Fig. 1 that severe temporal broadening may occur for optical pulses propagating in strong anisotropic atmospheric turbulence. It is seen from Fig. 1 that, for all combinations of Λ₀ and α specified there, basically the scaled turbulence-induced increment of mean square temporal pulse width grows linearly with increasing ζ_e. By recalling the definition of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, it is easy to find that its first term, i.e., ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, is proportional to ζ_e; on the other hand, we note that its second term, viz., ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, does not vary linearly with ζ_e. Hence, it can be inferred from the above observations that ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is dominated by its first term for the nominal parameter values used in creating Fig. 1. It is also found that although the slopes of the curves shown by Fig. 1 depend heavily on α, they have little dependence on Λ₀. This phenomenon is actually due to the reason that ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, which does not depend on Λ₀, dominates ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$. Moreover, it is noted from Fig. 1 that with a fixed value of ζ_e, a greater α leads to a larger turbulence-induced increment of mean square temporal pulse width under the condition that the nondimensional turbulence-strength parameter q remains constant. Therefore, the temporal broadening behavior of optical pulses also depends heavily on the spectral index α besides the effective anisotropic factor ζ_e.

 

Fig. 1 Scaled turbulence-induced increment of mean square temporal pulse width as a function of the effective anisotropic factor ζ_e with T₀ = 100 fs.

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The scaled turbulence-induced increment of mean square temporal pulse width in terms of the Fresnel ratio Λ₀ with various combinations of ζ_e and α is shown in Fig. 2. As before, we find from Fig. 2 that when other parameter values are fixed, enlarging the value of either α or ζ_e results in an increase in ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$. However, it is also seen from Fig. 2 that although the variation in ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ with varying Λ₀ is nearly unobservable when Λ₀ > 10⁻², it does become appreciable as Λ₀ decreases below 10⁻² for the curves associated with relatively small values of both α and ζ_e; i.e., steadily reducing Λ₀, meaning that the beam behaves more and more like a propagating plane wave, together with keeping other parameter values fixed may cause a significant variation in ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ if both α and ζ_e are relatively small. This fact manifests that when the Fresnel ratio Λ0 becomes tinier and tinier, it can play an important role in determining the temporal pulse broadening provided that both α and ζe are relatively small. A physical interpretation of this finding is as follows. To begin, we let

 

Fig. 2 Scaled turbulence-induced increment of mean square temporal pulse width in terms of the Fresnel ratio Λ₀ with T₀ = 100 fs.

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Equation (21) is indeed the on-axis two-frequency MCF shown by Eq. (10) with the term $a_0({\tilde{k}}_{12}+k_{12}/2)a_0^{*}({\tilde{k}}_{12}-k_{12}/2)$ removed; based on it, we can calculate the wave-number coherence bandwidth of turbulent atmosphere. In accordance with Fante ([26], p. 1449), the wave-number coherence bandwidth can be defined according to the value of k₁₂ at which |Γ_turb(k₁₂)| is equal to e⁻¹ of its value at k₁₂ = 0. Figure 3 shows |Γ_turb(k₁₂)|/Γ_turb (0) in terms of k₁₂ with various combinations of ζ_e, Λ₀ and α. Examination of Fig. 3(a) reveals that with a fixed ζ_e, the wave-number coherence bandwidth decreases as Λ₀ changes from 1 to 10⁻³, and a smaller ζ_e corresponds to a greater magnitude of this decrease. Further, it is observed from Figs. 3(b) and 3(c) that the magnitude of the said decrease becomes slighter as α grows larger. Thus, a statement can be made that when both α and ζ_e are relatively small, the wave-number coherence bandwidth of turbulent atmosphere narrows appreciably as Λ₀ varies from a large to a tiny value. As is well known, a narrower coherence bandwidth of turbulent atmosphere leads to greater temporal broadening of optical pulses passing through it. This is just the reason behind the aforementioned finding. Incidentally, we specify T₀ = 100 fs in creating the curves shown in Figs. 1 and 2; however the use of a fixed T₀ to create these curves does not cause any difficulty in employing these results to analyze the turbulence-induced increment of mean square temporal pulse width in other situations, where T₀ may be on the order of several picoseconds or even longer, because T₀ is used only as a scaling factor to make the quantity associated with the vertical axis in these plots a nondimensional quantity. In fact, based on the curves shown by Figs. 1 and 2, one can easily obtain the values of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, which is independent of T₀.

 

Fig. 3 Variation of |Γ_turb(k₁₂)|/Γ_turb(0) with k₁₂, where ${\tilde{k}}_{12}=\overline{k}$. (a) α = 3.3; (b) α = 11/3; (c) α = 3.8. The horizontal black dashed line denotes |Γ_turb(k₁₂)|/Γ_turb(0) ≡ e⁻¹.

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Figure 4 illustrates the ratio of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ to ${\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ in terms of the spectral index α with various Λ₀. Note that, ${\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ denotes the increment of mean square temporal pulse width caused by isotropic atmospheric turbulence; consequently, for the specified values of α, ζ_e and Λ₀, the ratio of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ to ${\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ actually quantifies the effects of anisotropy on the temporal pulse broadening; in other words, a value of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)/{\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ closer to 1 means a weaker impact that anisotropy of turbulence has on the temporal pulse broadening. It is seen from Fig. 4 that, for large values of Λ₀ (e.g., Λ₀ = 1, 10, 10²), ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is basically always equal to ζ_e times ${\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ as the spectral index α varies from 3 to 4, implying that the increment of mean square temporal pulse width caused by anisotropic atmospheric turbulence is approximately ζ_e times its isotropic counterpart. The result in this case resembles that obtained by Kon ([20], Eq. (33)) concerning the effects that anisotropy of turbulence has on the intensity-fluctuation variance. However, if Λ₀ becomes relatively tiny (e.g., Λ₀ = 10⁻²), the effects of anisotropy on the temporal pulse broadening may weaken significantly when the spectral index α approaches 3. By examination of the expression for ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, we recognize that ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)={\zeta }_e{\sigma }_{t,1}^2(\alpha ,1)$ and the Fresnel ratio Λ₀ affects the value of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ only through the term ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$. The above discussions imply once again that ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is dominated by ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ when either Λ₀ or α is relatively large, whereas the picture changes when both Λ₀ and α become relatively small.

 

Fig. 4 Ratio of ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ to ${\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$ in terms of α. (a) ζ_e = 5; (b) ζ_e = 10.

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At this point, it is interesting to investigate which one of the two terms, that is, ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ and ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, dominates the turbulence-induced increment of mean square temporal pulse width under various conditions. To this end, we focus our attention on the ratio of ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ to ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$; it is noted that if this ratio is much smaller than 1, ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is dominated by ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ and hence ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ can be ignored without significant error; conversely, ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ becomes negligible when this ratio is much larger than 1. Figure 5 shows the contours of ${\log }_{10}[{\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)/{\sigma }_{t,1}^0(\alpha ,{\zeta }_e)]$ as functions of α and Λ₀, where the regions shaded using white roughly represent ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\ll {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, those using cyan approximately denote ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\gg {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, and those using yellow and green indicate the transition regions. It is found from Figs. 5(a) – 5(d) that ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\ll {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ holds true in most situations of interest, whereas the case of ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\gg {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ occurs only in a narrow region. As stated previously, when ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\ll {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, one can ignore the contribution of ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ to the turbulence-induced increment of mean square temporal pulse width without significant error; under this condition, it is easy to find that ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\approx {\zeta }_e{\sigma }_{\text{turb}}^2(\alpha ,1,{\Lambda }_0)$, which has actually been confirmed in the analysis of Fig. 4. Similar reasoning can lead to another statement that if ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)\gg {\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is related to ζ_e primarily by Eq. (20). Furthermore, by comparing Figs. 5(a) – 5(d), it is seen that with other parameter values fixed, enlarging the effective anisotropic factor ζ_e results in wider regions where ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ is dominated by ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$. The reason for this fact is the following: if ζ_e grows so large that ${\overline{k}}^2/({\beta }^{2}L)\ll 1/(w_0^{2}L)$ and $8{\overline{k}}^2/(w_0^2{\beta }^2)\ll {\overline{k}}^2/L^2+4/w_0^4$, the quantity Δ shown by Eq. (20) nearly does not change with a further increase in ζ_e; in this case, keeping δ ∝ ζ_e in mind, it can be readily concluded that ${\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$ will dominate the turbulence-induced increment of mean square temporal pulse width as ζ_e becomes large enough. Finally, we examine ${\sigma }_{\text{turb}}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$ in two limiting cases. In the limit of Λ₀ = 0, i.e., w₀ → ∞ with $\overline{k}$ and L fixed, $f_m(k_{12})$ indicated by Eqs. (B6) – (B8) can be significantly simplified (m = 1, 2, 3); under this condition, we can analytically evaluate the integrals in Eqs. (B3) – (B5), and Eq. (B10) reduces to ${\sigma }_{t_a}^2\approx T_0^2/4+{\sigma }_{t,1}^2(\alpha ,{\zeta }_e)+{\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, where ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)=2L^2(1+{\zeta }_e^{-4})/(c^2{\beta }^4)$ in this case. This result can also be obtained according to Eq. (19) by letting w₀ → ∞. On the other hand, when w₀ → 0, one finds that $f_1({\tilde{k}}_{12})\sim {\tilde{k}}_{12}^2{w}_0^4/4$, $f_2({\tilde{k}}_{12})\sim {\tilde{k}}_{12}^2{w}_0^6/(8L)$ and $f_3({\tilde{k}}_{12})\sim {\tilde{k}}_{12}^2{w}_0^8/(8L^2)$, which in turn lead to ${\sigma }_{t_a}^2\sim T_0^2/4+{\sigma }_{t,2}^2(\alpha ,{\zeta }_e)+{\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)$, where ${\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)=w_0^4/(4L^2{c}^2)$ in this case. Hence, in the limit of Λ₀ → ∞, i.e., w₀ = 0 with $\overline{k}$ and L fixed, it follows that ${\sigma }_{t_a}^2=T_0^2/4+{\sigma }_{t,1}^2(\alpha ,{\zeta }_e)$, meaning that the last term in Eq. (19) vanishes completely. This result is indeed consistent with that presented in [10] about the temporal broadening of spherical-wave pulses if we let ζ_e = 1 and note that κ₀/κ_m ≪ 1.

 

Fig. 5 Contours of ${\log }_{10}[{\sigma }_{t,2}^2(\alpha ,{\zeta }_e,{\Lambda }_0)/{\sigma }_{t,1}^2(\alpha ,{\zeta }_e)]$ as functions of α and Λ₀. (a) ζ_e = 1; (b) ζ_e = 5; (c) ζ_e = 10; (d) ζ_e = 15.

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As a final comment, because the exact functional form of the anisotropy distribution function for various scales has not yet been known, we used nominal values of ζ_e in the previous calculations. Determining practical values of ζ_e under various conditions is beyond the scope of this paper. However, our formulation is applicable to analysis of the dependence of temporal pulse broadening on anisotropy of turbulence provided that ζ_e is determined. Furthermore, it should be noted that the spatial power spectrum of refractive-index fluctuations for anisotropic atmospheric turbulence proposed in [29], which has been adopted to develop the theoretical models in Sec. 2, still needs to be tested further by experimental measurements under different conditions. As a result, the work reported here should be regarded as an initial effort to understand the impact of anisotropy on the temporal pulse broadening.

4. Conclusions

In this paper, a novel expression for the on-axis two-frequency MCF of Gaussian beam waves propagating along a horizontal path in strong anisotropic atmospheric turbulence has been derived by making use of the extended Huygens-Fresnel principle. According to the expression obtained, by employing the temporal moment method, a closed-form formula for the mean square temporal pulse width due to strong anisotropic atmospheric turbulence has been further developed. Based on the achieved theoretical results, the temporal broadening of optical pulses passing horizontally through strong anisotropic atmospheric turbulence can be easily determined, and thus the effects of anisotropy on the temporal broadening behavior of optical pulses can be examined and analyzed.

The turbulence-induced increment of mean square temporal pulse width has been introduced as a measure of the effects that anisotropic turbulence has on the temporal broadening of optical pulses. Generally speaking, the turbulence-induced increment of mean square temporal pulse width grows with an increasing effective anisotropic factor ζ_e. More specifically, in most cases of interest, the increment of mean square temporal pulse width due to anisotropic atmospheric turbulence is nearly proportional to ζ_e, with the possible exception of situations where both the Fresnel ratio Λ₀ and spectral index α are relatively small. This statement becomes valid over wider ranges of Λ₀ and α as ζ_e grows larger. Moreover, the turbulence-induced increment of mean square temporal pulse width is also largely dependent upon α. With a fixed value of the nondimensional turbulence-strength parameter q, a greater α results in a larger turbulence-induced increment of mean square temporal pulse width. When both α and ζ_e are relatively small, changing Λ₀ from a large to a tiny value may lead to a significant enlargement in the turbulence-induced increment of mean square temporal pulse width.

This work provides revealing insights into the problem concerning how anisotropy of turbulence affects the temporal broadening behavior of optical pulses propagating along a horizontal path in the atmosphere. Our theoretical models are useful for determining the temporal pulse broadening in practical applications, which need to propagate optical pulses along horizontal path in anisotropic atmospheric turbulence.