Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence

Chunyi Chen; Huamin Yang; Yan Lou; Shoufeng Tong; Rencheng Liu

doi:10.1364/OE.20.007749

1. Introduction

The temporal broadening of optical pulses propagating through atmospheric turbulence is of great interest in many applications, such as remote sensing, lidar operation and high-speed free-space optical communication (FSOC) links. Theoretically the temporal broadening is determined by the width of the temporal averaged pulse intensity [1], and physically it is due to two possible causes: the spreading and the wandering of optical pulses induced by atmospheric turbulence; the effects of the two factors on the temporal broadening were analyzed by Liu and Yeh [1]. Young et al. [2] and Kelly et al. [3] obtained analytical solutions for the near- and far-field temporal broadening of collimated space-time Gaussian (CSTG) pulses passing through weak atmospheric turbulence, respectively. More recently, by making use of the quadratic approximation of the spherical wave structure function, Young [4] derived an analytical formula for the temporal broadening of optical pulses in moderate-to-strong atmospheric turbulence without the near- and far-field limitations. In addition, based on the results given by Ref [2], a Bessel filter used to numerically evaluate the temporal broadening of CSTG pulses propagating through weak atmospheric turbulence was developed by Jurado-Navas et al. [5].

All of the studies mentioned above have considered the situation that optical pulses propagate through Kolmogorov turbulence. Although the Kolmogorov theory [6] for atmospheric turbulence has been widely used, it does not always describe turbulence statistics accurately [7–14]. The non-Kolmogorov turbulence models differ from the Kolmogorov ones in the power-law index of the spatial power spectrum of refractive-index fluctuations, which is generalized and allowed to deviate somewhat from the classic value 11/3 [9–14]. A question arises as to whether the distinction between non-Kolmogorov and Kolmogorov turbulence causes significant differences in the temporal broadening behavior of optical pulses. However, to the best of our knowledge, there has been no formulation concerning the temporal broadening of optical pulses passing through non-Kolmogorov turbulence. Therefore, a deep investigation on this problem is imperative.

In this paper, we first generally formulate the temporal averaged pulse intensity of optical pulses passing in atmospheric turbulence. Then, with the help of these formulations, particular attention is given to studying the temporal broadening behavior of spherical-wave Gaussian (SWG) pulses and CSTG pulses. Taking both non-Kolmogorov and Kolmogorov turbulence into account, an analytical expression for the turbulence-induced temporal half-width of SWG pulses is derived, and an analytical formula for the single-point, two-frequency mutual coherence function (MCF) of collimated Gaussian-beam waves is developed, which provides the necessary groundwork for numerically calculating the temporal broadening of CSTG pulses. Further, the effects of non-Kolmogorov turbulence on the temporal broadening behavior of both SWG and CSTG pulses are analyzed based on numerical calculations.

2. General formulations of temporal averaged pulse intensity

Let us consider a temporal Gaussian pulse originating from a source at z = 0 and propagating predominantly along the positive z axis through atmospheric turbulence to a receiver at z = L, and assume that the pulse p_i(t) = v_i(t)exp(−iω₀t) is a modulated signal with an angular carrier frequency ω₀, where the amplitude v_i(t) = exp(−t²/T₀) represents the pulse shape, and T₀ determines the initial pulse half-width. Following the procedure of Young et al. [2], based on a linear systems approach, the temporal averaged pulse intensity in the receiver plane can be written by

\begin{matrix} 〈 I (r, L; t) 〉 = \frac{T_{0}^{2}}{4 π} \int \int_{- \infty}^{\infty} \exp [- \frac{1}{4} (ω_{1}^{2} + ω_{2}^{2}) T_{0}^{2}] \\ \times Γ_{2} (r, r, L; ω_{0} + ω_{1}, ω_{0} + ω_{2}) \exp [- i (ω_{1} - ω_{2}) t] d ω_{1} d ω_{2}, \end{matrix}

where the angle brackets denote an ensemble average, r is a position vector in the receiver plane, Γ₂(∙) is the single-point, two-frequency MCF defined by

Γ_{2} (r, r, L; ω, ω') = 〈 U (r, L; ω) U^{*} (r, L; ω') 〉,

where U(∙) is the complex amplitude of a monochromatic wave in the receiver plane, the asterisk represents the complex conjugate. In particular, for a beam wave with a finite cross-section, U(∙) can be written, using the extended Huygens-Fresnel principle [6], by

U (r, L, ω) = - \frac{i ω}{2 π L c} \exp (\frac{i L ω}{c}) \int \int_{- \infty}^{\infty} d^{2} s U_{0} (s, 0, ω) \exp [\frac{i ω | s - r |^{2}}{2 L c} + ψ (s, r, L; ω)],

where U₀(∙) denotes the optical wave field in the source plane, c is the speed of light, ψ(∙) is the random part of the complex phase of a spherical wave with the angular frequency ω propagating in the turbulence from the point (s, 0) to the point (r, L). The substitution of Eq. (3) into Eq. (2) leads to

\begin{matrix} Γ_{2} (r, r, L; ω, ω') = \frac{ω ω'}{4 π^{2} L^{2} c^{2}} \exp [\frac{i L}{c} (ω - ω')] \\ \times \int \int_{- \infty}^{\infty} d^{2} s_{1} \int \int_{- \infty}^{\infty} d^{2} s_{2} U_{0} (s_{1}, 0; ω) U_{0} (s_{2}, 0; ω') \\ \times \exp [\frac{i ω | s_{1} - r |^{2}}{2 L c} - \frac{i ω' | s_{2} - r |^{2}}{2 L c}] M_{2} (r, r, s_{1}, s_{2} L; ω, ω'), \end{matrix}

where M₂(∙) is given by

M_{2} (r, r, s_{1}, s_{2} L; ω, ω') = 〈 \exp [ψ (s_{1}, r, L; ω) + ψ^{*} (s_{2}, r, L; ω')] 〉 .

Assuming that the turbulence is statistically stationary and that ψ(∙) is a complex Gaussian random variable, the following expression is obtained [15]

M_{2} (r, r, s_{1}, s_{2} L; ω, ω') = \exp [- \frac{1}{2} D_{ψ} (r, r, s_{1}, s_{2}, L; ω, ω')],

where D_ψ(∙) is the single-point, two-frequency spherical-wave structure function in the receiver plane. For homogeneous and isotropic turbulence, under the narrow-band assumption, i.e., 1/ω – 1/ω′ ≈0, it follows from Refs [4]. and [16] that

D_{ψ} (r, r, s_{1}, s_{2}, L; ω, ω') = {(2 π / c)}^{2} L \int_{0}^{\infty} d κ κ Φ_{n} (κ) \int_{0}^{1} d ξ [ω^{2} + ω'^{2} - 2 ω ω' J_{0} (κ β)],

where β = ξ∙|s₁ – s₂|, J₀(∙) is a Bessel function of the first kind and zero order, Φ_n(∙) is the general spatial power spectrum of refractive-index fluctuations which can be used to model both non-Kolmogorov and Kolmogorov turbulence and given by [8, 13]

Φ_{n} (κ) = A (α) {\tilde{C}}_{n}^{2} \exp (- κ^{2} / κ_{m}^{2}) {(κ^{2} + κ_{0}^{2})}^{- α / 2}, 3 < α < 5,

where

{\tilde{C}}_{n}^{2}

is a generalized refractive-index structure parameter with units m^3–^α; α is a general spectral index which distinguishes non-Kolmogorov from Kolmogorov turbulence; κ₀ = 2π/L₀, κ_m = c(α)/l₀, c(α) = [2πΓ(5–α/2)A(α)/3]^1/(^α^–5), A(α) = Γ(α–1)cos(απ/2)/(4π²), L₀ and l₀ are the outer and inner scales of turbulence, respectively; Γ(∙) is the Gamma function. In the special case of α = 11/3, Eq. (8) leads to the von Kármán spectrum [6] for Kolmogorov turbulence.

As in Ref [7], under the strong fluctuation conditions, replacing J₀(∙) in Eq. (7) by the approximation form J₀(κβ) ≈1 − (κβ)²/4, and after some mathematical manipulations, it follows that

D_{ψ} (r, r, s_{1}, s_{2}, L; ω, ω') \approx {(2 π / c)}^{2} L {(ω - ω')}^{2} Q_{1} + \frac{2}{3} \frac{π^{2}}{c^{2}} ω ω' L {| s_{1} - s_{2} |}^{2} Q_{2},

where

Q_{1} = \int_{0}^{\infty} d κ κ Φ_{n} (κ) = \frac{1}{2} A (α) {\tilde{C}}_{n}^{2} κ_{m}^{2 - α} \exp (κ_{0}^{2} / κ_{m}^{2}) Γ (1 - α / 2, κ_{0}^{2} / κ_{m}^{2}),

\begin{matrix} Q_{2} = \int_{0}^{\infty} d κ κ^{3} Φ_{n} (κ) \\ = \frac{A (α) {\tilde{C}}_{n}^{2}}{2 (α - 2)} [κ_{m}^{2 - α} \exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) (2 κ_{0}^{2} - 2 κ_{m}^{2} + α κ_{m}^{2}) - 2 κ_{0}^{4 - α}], \end{matrix}

where Γ(∙,∙) denotes the incomplete Gamma function.

Note that the substitution of Eq. (9) into Eq. (6) and the specialization of M₂(∙) by letting ω = ω′ lead to Eq. (5) of Ref [7]. Furthermore, in the case of ω = ω′, the results given by Eqs. (4), (6) and (9) are consistent with the expressions presented by Shchepakina et al. (see Eqs. (1) and (2) of Ref [8].) if the two observation points coincide and the initial optical field is completely coherent.

3. Temporal broadening of SWG pulses

In this section, we consider a special case where the temporal Gaussian pulses emanate from a narrow-band spherical-wave source, i.e., a point source, located at z = 0. Under this condition, Eq. (2) can be expressed in the form

Γ_{2} (r, r, L; ω, ω') = U_{s}^{(0)} (r, L; ω) U_{s}^{(0) *} (r, L; ω') M_{2} (r, r, 0, 0, L; ω, ω'),

where U_s⁽⁰⁾(∙) denotes the complex amplitude of a spherical wave in free space at distance L from a unit-amplitude point source and given by [6]

U_{s}^{(0)} (r, L; ω) = \frac{1}{4 π L} \exp (\frac{i ω L}{c} + \frac{i ω r^{2}}{2 L c}),

where r = |r|. For the case in which SWG pulses propagate in free space, i.e., M₂(∙) = 1, substituting Eq. (12) into Eq. (1) and performing the integration lead to

I^{(0)} (r, L; t) = \frac{1}{{(4 π L)}^{2}} \exp [- \frac{2}{T_{0}^{2}} {(t - \frac{L}{c} - \frac{r^{2}}{2 L c})}^{2}],

where the superscript “0” in parenthesis on the left-hand side indicates that it is the temporal pulse intensity in free space. It can be found from Eq. (14) that the received pulse is a Gaussian pulse delayed by L/c + r²/(2Lc) and with a temporal half-width T₀ defined by the e⁻² point of the temporal pulse intensity. Similar free-space propagation behavior of CSTG pulses in the far field has been presented by Ref [6]. In the presence of the turbulence, the temporal averaged pulse intensity becomes

〈 I (r, L; t) 〉 = \frac{1}{{(4 π L)}^{2}} \cdot \frac{T_{0}}{T_{1}} \cdot \exp [- \frac{2}{T_{1}^{2}} {(t - \frac{L}{c} - \frac{r^{2}}{2 L c})}^{2}],

where T₁ = (T₀² + 16π²LQ₁/c²)^1/2 is the turbulence-induced temporal half-width of the pulses defined by the e⁻² point of the temporal averaged pulse intensity, which is due to the combined effects of the spreading and the wandering of pulses caused by the turbulence.

Equation (15) shows an important result, obtained by this paper, with respect to the temporal broadening of SWG pulses propagating through either non-Kolmogorov or Kolmogorov turbulence under the strong fluctuation conditions. Obviously, it is the term 16π²LQ₁/c² in T₁ that provides the turbulence-induced contribution towards the temporal broadening. Hence, it can be readily found that the temporal broadening of SWG pulses caused by either non-Kolmogorov or Kolmogorov turbulence is independent of the optical frequency. Note that the temporal broadening behavior of CSTG pulses propagating through weak Kolmogorov turbulence in the near and far field, obtained by Refs [2]. and [3], respectively, exhibits the similar independency on the optical frequency.

4. Temporal broadening of CSTG pulses

The optical wave field of a collimated unit-amplitude Gaussian beam in the source plane can be written as [6]

U_{0} (s, 0, ω) = \exp (- \frac{s^{2}}{W_{0}^{2}}),

where s = |s|, and W₀ is the beam radius at which the field amplitude falls to e⁻¹ of that on the beam axis. Substituting Eq. (16) into Eq. (4), making use of the identity [17]

\int_{- \infty}^{\infty} \exp (- p^{2} x^{2} \pm q x) d x = \exp (\frac{q^{2}}{4 p^{2}}) \frac{\sqrt{π}}{p}, (Re {p^{2}} > 0),

and performing the integration, after a long but straightforward calculation, the following result is found

\begin{matrix} Γ_{2} (r, r, L; ω, ω') = \frac{9 c^{2}}{4 L^{2}} \cdot \frac{ω ω'}{9 c^{4} a_{1} a_{2} - {(π^{2} L ω ω' Q_{2})}^{2}} \\ \times \exp [- \frac{2 π^{2} L}{c^{2}} {(ω - ω')}^{2} Q_{1}] \cdot \exp [\frac{i (ω - ω') L}{c}] \\ \times \exp [\frac{i (ω - ω') r^{2}}{2 L c}] \cdot \exp (- \frac{ω'^{2} r^{2}}{4 L^{2} c^{2} a_{2}}) \\ \times \exp [- \frac{9}{4} {(\frac{1}{L c} - \frac{π^{2} ω'^{2} Q_{2}}{3 c^{3} a_{2}})}^{2} \cdot \frac{c^{4} a_{2} ω^{2} r^{2}}{9 c^{4} a_{1} a_{2} - {(π^{2} L ω ω' Q_{2})}^{2}}], \end{matrix}

where

\begin{array}{l} a_{1} = \frac{1}{W_{0}^{2}} + \frac{π^{2} L ω ω' Q_{2}}{3 c^{2}} - i \frac{ω}{2 L c}, \\ a_{2} = \frac{1}{W_{0}^{2}} + \frac{π^{2} L ω ω' Q_{2}}{3 c^{2}} + i \frac{ω'}{2 L c} . \end{array}

Equation (18), which is another important result derived by this paper, is an analytical expression for the single-point, two-frequency MCF of collimated Gaussian-beam waves propagating through either non-Kolmogorov or Kolmogorov turbulence under the strong fluctuation conditions. Different from the case of SWG pulses, an analytical formula for the temporal averaged pulse intensity of CSTG pulses is not directly available by inserting Eq. (18) into Eq. (1). However, based on Eq. (18) numerical methods can be used to calculate the temporal broadening of CSTG pulses. The turbulence-induced temporal half-width T₁ of GSTG pulses propagating through atmospheric turbulence can be computed by evaluating the temporal averaged pulse intensity numerically and finding the corresponding e⁻² point, and the specialization of this procedure by letting Q₁ = Q₂ = 0 leads to the temporal half-width T₀ of GSTG pulses propagating in free space.

5. Numerical calculations and analysis

In this section, we evaluate the temporal broadening of both SWG and CSTG pulses propagating through either non-Kolmogorov or Kolmogorov turbulence and analyze its dependence on the general spectral index α. Here, we consider optical pulses of which the central wavelength λ₀, related to the angular carrier frequency by ω₀ = 2πc/λ₀ [18], is 1.06μm. The values of turbulence and Gaussian-beam parameters used in the following calculations are as follows: ${\tilde{C}}_{n}^{2}$ = 10⁻¹³m^3–^α, L₀ = 1m, l₀ = 1mm, W₀ = 1.8cm, unless other values are specified in the figures.

Figure 1 illustrates the relative temporal broadening coefficient (RTBC), defined by ε = T₁/T₀ ≥ 1, of optical pulses propagating through atmospheric turbulence as a function of the general spectral index α; the propagation distance L is 1km and 5km for Fig. 1(a) and 1(b), respectively, where the solid curves are for SWG pulses and the dotted curves for CSTG pulses with |r| = 0, i.e., an on-axis observation point. Note that the greater the RTBC is, the more significant the turbulence-induced temporal broadening of optical pulses becomes, and the turbulence does not contribute to the temporal broadening if the RTBC equals 1. Indeed, it is easy to prove that if the values of both T₀ and L are fixed, the RTBC of SWG pulses passing in atmospheric turbulence depends only on Q₁, and achieves its maximum with the value of α which maximizes Q₁. Further, it can be seen from Eq. (10) that Q₁ is uniquely determined by ${\tilde{C}}_{n}^{2}$ , L₀ and l₀, and independent of T₀. As a result, in Fig. 1 the value of α which maximizes the RTBC of SWG pulses with various T₀ is the same.

Fig. 1 Relative temporal broadening coefficient (RTBC) ε as a function of the general spectral index α with various values of T₀ for both SWG and CSTG pulses propagating through atmospheric turbulence.

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It can be observed from both Fig. 1(a) and 1(b) that there exist few differences in the RTBC between SWG and CSTG pulses with all values of T₀; therefore, the on-axis temporal broadening of CSTG pulses can be approximated properly by that computed based on the expressions for SWG pulses with the same turbulence parameters and propagation distances. As shown in Fig. 1, with the same T₀, in the range of 3 < α < 5, the RTBC first rises almost from 1 with increasing values of α, then achieves its maximum at α = 3.45, and finally reduces almost to 1 as α continues to increase. In addition, it can be seen that with the same α, the larger T₀ is, the closer the RTBC becomes to 1. Indeed, the effect of the turbulence-induced temporal broadening of pulses on practical applications can be ignored properly, provided that the RTBC approaches 1. Comparisons between Fig. 1(a) and 1(b) show that with the same T₀, a longer propagation distance leads to a larger value of the RTBC; this fact is what one might expect because a longer propagation distance means stronger effects of the turbulence.

Let us define α_max as the value of α that maximizes the RTBC of pulses with a given combination of T₀, L, ${\tilde{C}}_{n}^{2}$ , L₀ and l₀. Figure 2 shows the dependence of α_max on the turbulence parameters ${\tilde{C}}_{n}^{2}$ , L₀ and l₀. It can be found from Fig. 2 that although α_max is a strong function of L₀, it varies very little with the change of ${\tilde{C}}_{n}^{2}$ and (or) l₀.

Fig. 2 Dependence of α_max associated with the maximum RTBC of pulses on the turbulence parameters ${\tilde{C}}_{n}^{2}$ , L₀ and l₀.

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Now we determine the dependence of the turbulence-induced temporal half-width T₁ of CSTG pulses on the radial distance r = |r|. To facilitate the analysis, we compute the normalized temporal half-width T_n defined by the ratio of the turbulence-induced temporal half-width T₁ with an observation point at r to that on the axis, i.e., T_n(r) = T₁(r)/T₁(r = 0), where the content in parentheses explicitly indicates the radial dependency of T₁. Needless to say, T₁ has a strong radial dependency if the deviation of T_n from 1 varies significantly with the radial distance r.

Figure 3 shows the normalized temporal half-width T_n in terms of the scaled radial distance r / [W₀²(1 + L²/z₀²)]^1/2, where the term [W₀²(1 + L²/z₀²)]^1/2 denotes the free-space beam radius in the receiver plane, and z₀ = πW₀²/λ₀. It can be observed from Fig. 3(a) and 3(b) that T_n rises as the scaled radial distance increases with a propagation distance L of both 1km and 5 km, and the smaller T₀ is, the faster T_n increases. On other hand, it is clear from Fig. 3 that all values of T_n are very close to 1, and the differences in T_n with various scaled radial distances are indeed really tiny. Hence, the dependence of the turbulence-induced temporal half-width T₁ of CSTG pulses on the radial distance can be generally neglected.

Fig. 3 Normalized temporal half-width T_n in terms of the scaled radial distance r / [W₀²(1 + L²/z₀²)]^1/2 with various values of T₀ for CSTG pulses propagating through non-Kolmogorov turbulence.

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6. Conclusions

In this paper, general formulations of the temporal averaged pulse intensity for optical pulses passing in either non-Kolmogorov or Kolmogorov turbulence under the strong fluctuation conditions and the narrow-band assumption have been obtained by using the extended Huygens-Fresnel principle. Two specializations of these formulations for SWG and CSTG pulses, respectively, were considered further. An analytical formula for the turbulence-induced temporal half-width of SWG pulses was derived, and the single-point, two-frequency MCF of collimated Gaussian-beam waves in either non-Kolmogorov or Kolmogorov turbulence, based on which the temporal averaged pulse intensity of CSTG pulses can be calculated numerically, was formulated analytically. It has been shown from the numerical calculations that the on-axis temporal broadening of CSTG pulses approximates very well to that calculated by the formula for SWG pulses with the same turbulence parameters and propagation distances; the temporal broadening of both SWG and CSTG pulses is heavily dependent on the general spectral index α; in the range of 3 < α < 5 (α = 11/3 for Kolmogorov turbulence), the RTBC first rises almost from 1 with increasing values of α until it arrives at its peak value, and then it reduces almost to 1 as α continues to increase; the turbulence-induced temporal half-width of CSTG pulses depends very little on the radial distance. Based on the analysis in this paper, it has been found that the value of α which maximizes the RTBC of the pulses with a given combination of the initial pulse half-width and the propagation distance is determined mainly by the outer scale L₀ of the turbulence; in contrast to the temporal broadening of the pulses in Kolmogorov turbulence, which is a function of the generalized refractive-index structure parameter ${\tilde{C}}_{n}^{2}$ , the outer scale L₀ and the inner scale l₀, that of the pulses in non-Kolmogorov turbulence is in terms of ${\tilde{C}}_{n}^{2}$ , L₀, l₀ and α.

The work reported here provides an insight into the problem that whether there are significant differences in the temporal broadening behavior between optical pulses propagating through non-Kolmogorov and Kolmogorov turbulence. Our results are useful for applications involving the propagation of optical pulses in non-Kolmogorov turbulence.

Acknowledgments

The authors are very grateful to the reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China (NSFC) under grant 61007046 and the Jilin Provincial Development Programs of Science & Technology of China under grant 201101096.

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Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence

Abstract

1. Introduction

2. General formulations of temporal averaged pulse intensity

3. Temporal broadening of SWG pulses

4. Temporal broadening of CSTG pulses

5. Numerical calculations and analysis

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (19)

Optics Express