OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7749–7757
« Show journal navigation

Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence

Chunyi Chen, Huamin Yang, Yan Lou, Shoufeng Tong, and Rencheng Liu  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7749-7757 (2012)
http://dx.doi.org/10.1364/OE.20.007749


View Full Text Article

Acrobat PDF (752 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

General formulations of the temporal averaged pulse intensity for optical pulses propagating through either non-Kolmogorov or Kolmogorov turbulence are deduced under the strong fluctuation conditions and the narrow-band assumption. Based on these formulations, an analytical formula for the turbulence-induced temporal half-width of spherical-wave Gaussian (SWG) pulses is derived, and the single-point, two-frequency mutual coherence function (MCF) of collimated Gaussian-beam waves in atmospheric turbulence is formulated analytically, by which the temporal averaged pulse intensity of collimated space-time Gaussian (CSTG) pulses can be calculated numerically. Calculation results show that the temporal broadening of both SWG and CSTG pulses in atmospheric turbulence depends heavily on the general spectral index of the spatial power spectrum of refractive-index fluctuations, and the temporal broadening of SWG pulses can be used to approximate that of CSTG pulses on the axis with the same turbulence parameters and propagation distances. It is also illustrated by numerical calculations that the variation in the turbulence-induced temporal half-width of CSTG pulses with the radial distance is really tiny.

© 2012 OSA

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

1. C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]

], 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

1. C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]

]. Young et al. [2

2. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]

] and Kelly et al. [3

3. D. E. T. T. S. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999). [CrossRef]

] 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

4. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]

] 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

2. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]

], 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

5. A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “Numerical model for the temporal broadening of optical pulses propagating through weak atmospheric turbulence,” Opt. Lett. 34(23), 3662–3664 (2009). [CrossRef] [PubMed]

].

All of the studies mentioned above have considered the situation that optical pulses propagate through Kolmogorov turbulence. Although the Kolmogorov theory [6

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

] for atmospheric turbulence has been widely used, it does not always describe turbulence statistics accurately [7

7. G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Spreading of partially coherent Hermite-Gaussian beams through a non-Kolmogorov turbulence,” Optik-Int. J. Light Electron. 122(22), 2029–2033 (2011). [CrossRef]

14

14. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011). [CrossRef] [PubMed]

]. 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

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]

14

14. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011). [CrossRef] [PubMed]

]. 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 pi(t) = vi(t)exp(−0t) is a modulated signal with an angular carrier frequency ω0, where the amplitude vi(t) = exp(−t2/T0) represents the pulse shape, and T0 determines the initial pulse half-width. Following the procedure of Young et al. [2

2. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]

], based on a linear systems approach, the temporal averaged pulse intensity in the receiver plane can be written by
I(r,L;t)=T024πexp[14(ω12+ω22)T02]×Γ2(r,r,L;ω0+ω1,ω0+ω2)exp[i(ω1ω2)t]dω1dω2,
(1)
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
Γ2(r,r,L;ω,ω')=U(r,L;ω)U*(r,L;ω'),
(2)
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

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

], by
U(r,L,ω)=iω2πLcexp(iLωc)d2sU0(s,0,ω)exp[iω|sr|22Lc+ψ(s,r,L;ω)],
(3)
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
Γ2(r,r,L;ω,ω')=ωω'4π2L2c2exp[iLc(ωω')]×d2s1d2s2U0(s1,0;ω)U0(s2,0;ω')×exp[iω|s1r|22Lciω'|s2r|22Lc]M2(r,r,s1,s2L;ω,ω'),
(4)
where M2(∙) is given by

M2(r,r,s1,s2L;ω,ω')=exp[ψ(s1,r,L;ω)+ψ*(s2,r,L;ω')].
(5)

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

15. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68(11), 1424–1443 (1980). [CrossRef]

]
M2(r,r,s1,s2L;ω,ω')=exp[12Dψ(r,r,s1,s2,L;ω,ω')],
(6)
where Dψ(∙) 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

4. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]

]. and [16

16. Y. Baykal and M. A. Plonus, “Two-source, two-frequency spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 70(10), 1278–1279 (1980). [CrossRef]

] that
Dψ(r,r,s1,s2,L;ω,ω')=(2π/c)2L0dκκΦn(κ)01dξ[ω2+ω'22ωω'J0(κβ)],
(7)
where β = ξ∙|s1s2|, J0(∙) is a Bessel function of the first kind and zero order, Φn(∙) 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

8. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

, 13

13. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]

]
Φn(κ)=A(α)C˜n2exp(κ2/κm2)(κ2+κ02)α/2, 3<α<5,
(8)
where C˜n2 is a generalized refractive-index structure parameter with units m3–α; α is a general spectral index which distinguishes non-Kolmogorov from Kolmogorov turbulence; κ0 = 2π/L0, κm = 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

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

] for Kolmogorov turbulence.

As in Ref [7

7. G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Spreading of partially coherent Hermite-Gaussian beams through a non-Kolmogorov turbulence,” Optik-Int. J. Light Electron. 122(22), 2029–2033 (2011). [CrossRef]

], 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
Dψ(r,r,s1,s2,L;ω,ω')(2π/c)2L(ωω')2Q1+23π2c2ωω'L|s1s2|2Q2,
(9)
where
Q1=0dκκΦn(κ)=12A(α)C˜n2κm2αexp(κ02/κm2)Γ(1α/2,κ02/κm2),
(10)
Q2=0dκκ3Φn(κ)=A(α)C˜n22(α2)[κm2αexp(κ02κm2)Γ(2α2,κ02κm2)(2κ022κm2+ακm2)2κ04α],
(11)
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

7. G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Spreading of partially coherent Hermite-Gaussian beams through a non-Kolmogorov turbulence,” Optik-Int. J. Light Electron. 122(22), 2029–2033 (2011). [CrossRef]

]. 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

8. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

].) 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
Γ2(r,r,L;ω,ω')=Us(0)(r,L;ω)Us(0)*(r,L;ω')M2(r,r,0,0,L;ω,ω'),
(12)
where Us(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

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

]
Us(0)(r,L;ω)=14πLexp(iωLc+iωr22Lc),
(13)
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)=1(4πL)2exp[2T02(tLcr22Lc)2],
(14)
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

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

]. In the presence of the turbulence, the temporal averaged pulse intensity becomes
I(r,L;t)=1(4πL)2T0T1exp[2T12(tLcr22Lc)2],
(15)
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

2. C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]

]. and [3

3. D. E. T. T. S. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999). [CrossRef]

], 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

6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

]
U0(s,0,ω)=exp(s2W02),
(16)
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

17. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

]
exp(p2x2±qx)dx=exp(q24p2)πp, (Re{p2}>0),
(17)
and performing the integration, after a long but straightforward calculation, the following result is found
Γ2(r,r,L;ω,ω')=9c24L2ωω'9c4a1a2(π2Lωω'Q2)2×exp[2π2Lc2(ωω')2Q1]exp[i(ωω')Lc]×exp[i(ωω')r22Lc]exp(ω'2r24L2c2a2)×exp[94(1Lcπ2ω'2Q23c3a2)2c4a2ω2r29c4a1a2(π2Lωω'Q2)2],
(18)
where

a1=1W02+π2Lωω'Q23c2iω2Lc,a2=1W02+π2Lωω'Q23c2+iω'2Lc.

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

Figure 1
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.
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 C˜n2, 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.

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 αmax as the value of α that maximizes the RTBC of pulses with a given combination of T0, L, C˜n2, L0 and l0. Figure 2
Fig. 2 Dependence of αmax associated with the maximum RTBC of pulses on the turbulence parameters C˜n2, L0 and l0.
shows the dependence of αmax on the turbulence parameters C˜n2, L0 and l0. It can be found from Fig. 2 that although αmax is a strong function of L0, it varies very little with the change of C˜n2 and (or) l0.

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 Tn 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., Tn(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 Tn from 1 varies significantly with the radial distance r.

Figure 3
Fig. 3 Normalized temporal half-width Tn 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.
shows the normalized temporal half-width Tn 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 Tn 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 Tn increases. On other hand, it is clear from Fig. 3 that all values of Tn are very close to 1, and the differences in Tn 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.

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 C˜n2, the outer scale L0 and the inner scale l0, that of the pulses in non-Kolmogorov turbulence is in terms of C˜n2, 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.

References and links

1.

C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]

2.

C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]

3.

D. E. T. T. S. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999). [CrossRef]

4.

C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]

5.

A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “Numerical model for the temporal broadening of optical pulses propagating through weak atmospheric turbulence,” Opt. Lett. 34(23), 3662–3664 (2009). [CrossRef] [PubMed]

6.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.

7.

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Spreading of partially coherent Hermite-Gaussian beams through a non-Kolmogorov turbulence,” Optik-Int. J. Light Electron. 122(22), 2029–2033 (2011). [CrossRef]

8.

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]

9.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]

10.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]

11.

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]

12.

W. Du, L. Tan, J. Ma, and Y. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(6), 5763–5775 (2010). [CrossRef] [PubMed]

13.

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]

14.

L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011). [CrossRef] [PubMed]

15.

R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68(11), 1424–1443 (1980). [CrossRef]

16.

Y. Baykal and M. A. Plonus, “Two-source, two-frequency spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 70(10), 1278–1279 (1980). [CrossRef]

17.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

18.

C. Chen, H. Yang, Y. Lou, and S. Tong, “Second-order statistics of Gaussian Schell-model pulsed beams propagating through atmospheric turbulence,” Opt. Express 19(16), 15196–15204 (2011). [CrossRef] [PubMed]

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(320.5550) Ultrafast optics : Pulses

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: January 26, 2012
Revised Manuscript: February 26, 2012
Manuscript Accepted: February 27, 2012
Published: March 20, 2012

Citation
Chunyi Chen, Huamin Yang, Yan Lou, Shoufeng Tong, and Rencheng Liu, "Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence," Opt. Express 20, 7749-7757 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7749


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]
  2. C. Y. Young, L. C. Andrews, A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998). [CrossRef] [PubMed]
  3. D. E. T. T. S. Kelly, L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media 9(3), 307–325 (1999). [CrossRef]
  4. C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]
  5. A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, A. Puerta-Notario, “Numerical model for the temporal broadening of optical pulses propagating through weak atmospheric turbulence,” Opt. Lett. 34(23), 3662–3664 (2009). [CrossRef] [PubMed]
  6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.
  7. G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, H. Guo, “Spreading of partially coherent Hermite-Gaussian beams through a non-Kolmogorov turbulence,” Optik-Int. J. Light Electron. 122(22), 2029–2033 (2011). [CrossRef]
  8. E. Shchepakina, O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef] [PubMed]
  9. I. Toselli, L. C. Andrews, R. L. Phillips, V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]
  10. A. Zilberman, E. Golbraikh, N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]
  11. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010). [CrossRef] [PubMed]
  12. W. Du, L. Tan, J. Ma, Y. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(6), 5763–5775 (2010). [CrossRef] [PubMed]
  13. O. Korotkova, E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]
  14. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011). [CrossRef] [PubMed]
  15. R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68(11), 1424–1443 (1980). [CrossRef]
  16. Y. Baykal, M. A. Plonus, “Two-source, two-frequency spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. 70(10), 1278–1279 (1980). [CrossRef]
  17. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.
  18. C. Chen, H. Yang, Y. Lou, S. Tong, “Second-order statistics of Gaussian Schell-model pulsed beams propagating through atmospheric turbulence,” Opt. Express 19(16), 15196–15204 (2011). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited