The on-axis two-frequency mutual coherence function (MCF) for beam waves propagating along a horizontal path in strong anisotropic atmospheric turbulence is theoretically formulated by making use of the extended Huygens-Fresnel principle. Based on this formulation, a new closed-form expression for the mean square temporal width of Gaussian-beam-wave pulses passing horizontally through strong anisotropic atmospheric turbulence is developed. With the help of this expression, the increments of mean square temporal pulse width due to strong anisotropic atmospheric turbulence under various conditions are further calculated. Results show that the increment of mean square temporal pulse width due to strong anisotropic atmospheric turbulence is basically proportional to the effective anisotropic factor in most situations of interest, with the possible exception of cases in which both the Fresnel ratio and spectral index become relatively small; increasing the effective anisotropic factor can reduce the number of the said exceptions; the turbulence-induced increment of mean square temporal pulse width enlarges as the spectral index increases with a fixed value of the nondimensional turbulence-strength parameter. It is also illustrated that a significant enlargement in the turbulence-induced increment of mean square temporal pulse width occurs by changing the Fresnel ratio from a large to a tiny value if both the effective anisotropic factor and spectral index are relatively small.
© 2015 Optical Society of America
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.  studied the impact that anisotropic atmospheric turbulence has on the mutual dancing of two parallel propagated narrow laser beams. Manning  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  analyzed the degradation of star image caused by stratospheric turbulence, which is usually anisotropic. Kon  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 27], the two-source, two-frequency spherical wave structure function for the case of anisotropic atmospheric turbulence can be developed to give Eq. (2), we have made three assumptions: the first is that refractive-index fluctuations along the propagation direction are almost uncorrelated, meaning that κz ≈ 0 [24,28]; the second is that the propagated optical pulses can be considered narrowband, implying that 1/k1 − 1/k2 ≈ 0 ; the third is that Φn(K) does not change along horizontal propagation paths . In fact, these assumptions have been widely made in the literature. It should be pointed out that rd and sd in the formulae above are assumed to be two-dimensional column vectors, whereas for succinctness the vector operations in Eq. (4) have been written in a manner akin to that used by Andrews and Phillips (, Chap. 5); indeed, only the first two components of K are used to perform the vector operations due to the assumption that κz ≈ 0. We note that, for isotropic atmospheric turbulence, by converting the integrals of K appearing in Eqs. (3) and (4) to polar coordinates and performing the resultant integration over the angle, Eq. (2) reduces to a simpler form which is consistent with Eq. (8) of  under the said narrow-band assumption. In addition, as in , Eq. (2) is expected to be valid for arbitrary turbulence strength.
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  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 , 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 , in the form29], ζe is the degree of anisotropy corresponding to the effective turbulent cell whose size is defined as the turbulent cell size associated with the center of gravity of the curve characterizing how anisotropy varies at different scales between Riso and L0, where Riso represents the scale at which turbulent irregularities become isotropic. For atmospheric turbulence, it is found that 3 < α < 4. At a first glance, there does not seem to be an essential difference in considerations of the degree of anisotropy at various scales between Eq. (5) and the spectrums given by [13,15-17,20-22]. However, this confusion can be removed if we note that ζe is actually determined with the help of the anisotropy distribution function and effective turbulent cell size. By introducing Φn(κx,κy,0) into Eqs. (3) and (4), it follows that (see Appendix A)
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 rd ≡ 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., , where 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 obtainEq. (9), we have assumed that l0/L0 ≪ 1, viz., κ0/κm ≪ 1, which generally holds true in the atmosphere .
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 with 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(km) is an amplitude function depending on the optical wave number km (m = 1, 2), and sm = |sm| (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 (, p. 742, Eq. (23)). Without loss of generality, we specify that 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 . 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  when ζe = 1 and α = 11/3 provided that we exclude the effects due to the introduction of the amplitude function a0(km). By the way, the author of  has noted that Eq. (9) therein agrees with Eq. (50) of  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 nth temporal moment of the pulses can be written by1,3]
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  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 . Furthermore, when ζe = 1 and α = 11/3, it is interesting to find that the sum of the first two terms in Eq. (19), i.e., , is exactly equivalent to the expression for the mean square temporal pulse width developed by Young et al. (, Eq. (20)) based on the Rytov approximation (Notice that κ0 = 1/L0 in ). 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 .
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 appearing in Eq. (5) is in units of m3−α. To avoid the unit choice inconsistency in representation of calculation results pointed out by Charnotskii , here we use the nondimensional parameter as a measure of turbulence strength. In addition, the Fresnel ratio defined in  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 with and , where the dependence of , and on the parameters α, ζe and Λ0 is shown explicitly; note that δ and Δ are related to the parameters α, ζe and Λ0 by Eqs. (15), (16) and (20). It is apparent that 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 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 ζe. Figure 1 illustrates the scaled turbulence-induced increment of mean square temporal pulse width as a function of the effective anisotropic factor ζe 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 ζe. By recalling the definition of , it is easy to find that its first term, i.e., , is proportional to ζe; on the other hand, we note that its second term, viz., , does not vary linearly with ζe. Hence, it can be inferred from the above observations that 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 , which does not depend on Λ0, dominates . Moreover, it is noted from Fig. 1 that with a fixed value of ζe, 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 ζe.
The scaled turbulence-induced increment of mean square temporal pulse width in terms of the Fresnel ratio Λ0 with various combinations of ζe 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 ζe results in an increase in . However, it is also seen from Fig. 2 that although the variation in 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 ζe; 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 if both α and ζe 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
Equation (21) is indeed the on-axis two-frequency MCF shown by Eq. (10) with the term removed; based on it, we can calculate the wave-number coherence bandwidth of turbulent atmosphere. In accordance with Fante (, p. 1449), the wave-number coherence bandwidth can be defined according to the value of k12 at which |Γturb(k12)| is equal to e−1 of its value at k12 = 0. Figure 3 shows |Γturb(k12)|/Γturb (0) in terms of k12 with various combinations of ζe, Λ0 and α. Examination of Fig. 3(a) reveals that with a fixed ζe, the wave-number coherence bandwidth decreases as Λ0 changes from 1 to 10−3, and a smaller ζe 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 ζe 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 , which is independent of T0.
Figure 4 illustrates the ratio of to in terms of the spectral index α with various Λ0. Note that, denotes the increment of mean square temporal pulse width caused by isotropic atmospheric turbulence; consequently, for the specified values of α, ζe and Λ0, the ratio of to actually quantifies the effects of anisotropy on the temporal pulse broadening; in other words, a value of 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), is basically always equal to ζe times 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 ζe times its isotropic counterpart. The result in this case resembles that obtained by Kon (, 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 , we recognize that and the Fresnel ratio Λ0 affects the value of only through the term . The above discussions imply once again that is dominated by when either Λ0 or α is relatively large, whereas the picture changes when both Λ0 and α become relatively small.
At this point, it is interesting to investigate which one of the two terms, that is, and , 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 to ; it is noted that if this ratio is much smaller than 1, is dominated by and hence can be ignored without significant error; conversely, becomes negligible when this ratio is much larger than 1. Figure 5 shows the contours of as functions of α and Λ0, where the regions shaded using white roughly represent , those using cyan approximately denote , and those using yellow and green indicate the transition regions. It is found from Figs. 5(a) – 5(d) that holds true in most situations of interest, whereas the case of occurs only in a narrow region. As stated previously, when , one can ignore the contribution of to the turbulence-induced increment of mean square temporal pulse width without significant error; under this condition, it is easy to find that , which has actually been confirmed in the analysis of Fig. 4. Similar reasoning can lead to another statement that if , is related to ζe 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 ζe results in wider regions where is dominated by . The reason for this fact is the following: if ζe grows so large that and , the quantity Δ shown by Eq. (20) nearly does not change with a further increase in ζe; in this case, keeping δ ∝ ζe in mind, it can be readily concluded that will dominate the turbulence-induced increment of mean square temporal pulse width as ζe becomes large enough. Finally, we examine in two limiting cases. In the limit of Λ0 = 0, i.e., w0 → ∞ with and L fixed, 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 , where 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 , and , which in turn lead to , where in this case. Hence, in the limit of Λ0 → ∞, i.e., w0 = 0 with and L fixed, it follows that , meaning that the last term in Eq. (19) vanishes completely. This result is indeed consistent with that presented in  about the temporal broadening of spherical-wave pulses if we let ζe = 1 and note that κ0/κm ≪ 1.
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 ζe in the previous calculations. Determining practical values of ζe 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 ζe is determined. Furthermore, it should be noted that the spatial power spectrum of refractive-index fluctuations for anisotropic atmospheric turbulence proposed in , 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.
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 ζe. More specifically, in most cases of interest, the increment of mean square temporal pulse width due to anisotropic atmospheric turbulence is nearly proportional to ζe, 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 ζe 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 ζe 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.
Appendix AEqs. (2) and (5), it follows that
We let and . Then, Eq. (A3) can be written asEq. (8). Following the approach used by Yura (, Eqs. (19) and (20)), we find 28], Sec. 6.4.1), the integration on in Eq. (A5) can be evaluated and the result is shown by Eq. (6).
The integrals appearing in Eqs. (B3) – (B5) cannot be analytically evaluated in a rigorous manner to yield closed-form solutions. However, it is observed that the exponential functions in terms of occurring therein have an appreciable value only in the range of . Recall that the propagated pulses are narrowband relative to the carrier frequency; accordingly, the peak of the said exponential functions centered at is very sharp. On the other hand, actually does not appreciably change across the range of in situations of practical interest. For the reasons above, we may replace in Eqs. (B3) – (B5) by (m = 1, 2, 3). Then, the result shown by Eq. (19) can be readily obtained.
This work was supported by the National Natural Science Foundation of China (Nos. 61007046, 61275080 and 61475025), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20132216110002).
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