## Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence |

Optics Express, Vol. 20, Issue 7, pp. 7749-7757 (2012)

http://dx.doi.org/10.1364/OE.20.007749

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### 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

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. 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]

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. 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]

## 2. General formulations of temporal averaged pulse intensity

*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(−

*iω*

_{0}

*t*) is a modulated signal with an angular carrier frequency

*ω*

_{0}, where the amplitude

*v*(

_{i}*t*) = exp(−

*t*

^{2}/

*T*

_{0}) represents the pulse shape, and

*T*

_{0}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]

**r**is a position vector in the receiver plane, Γ

_{2}(∙) is the single-point, two-frequency MCF defined bywhere

*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], bywhere

*U*

_{0}(∙) 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

*M*

_{2}(∙) is given by

*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]

*D*(∙) 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/

_{ψ}*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]

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]

*β*=

*ξ*∙|

**s**

_{1}–

**s**

_{2}|,

*J*

_{0}(∙) is a Bessel function of the first kind and zero order, Φ

*(∙) 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*

_{n}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. 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]

^{3–}

*;*

^{α}*α*is a general spectral index which distinguishes non-Kolmogorov from Kolmogorov turbulence;

*κ*

_{0}= 2π/

*L*

_{0},

*κ*=

_{m}*c*(

*α*)/

*l*

_{0},

*c*(

*α*) = [2πΓ(5–

*α*/2)

*A*(

*α*)/3]

^{1/(}

^{α}^{–5)},

*A*(

*α*) = Γ(

*α*–1)cos(

*α*π/2)/(4π

^{2}),

*L*

_{0}and

*l*

_{0}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.

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]

*J*

_{0}(∙) in Eq. (7) by the approximation form

*J*

_{0}(

*κβ*) ≈1 − (

*κβ*)

^{2}/4, and after some mathematical manipulations, it follows thatwhere where Γ(∙,∙) denotes the incomplete Gamma function.

*M*

_{2}(∙) 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]

*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]

## 3. Temporal broadening of SWG pulses

*z*= 0. Under this condition, Eq. (2) can be expressed in the formwhere

*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]where

*r*= |

**r**|. For the case in which SWG pulses propagate in free space, i.e.,

*M*

_{2}(∙) = 1, substituting Eq. (12) into Eq. (1) and performing the integration lead towhere 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*+

*r*

^{2}/(2

*Lc*) and with a temporal half-width

*T*

_{0}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 becomeswhere

*T*

_{1}= (

*T*

_{0}

^{2}+ 16π

^{2}

*LQ*

_{1}/

*c*

^{2})

^{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.

^{2}

*LQ*

_{1}/

*c*

^{2}in

*T*

_{1}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]

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. Temporal broadening of CSTG pulses

*s*= |

**s**|, and

*W*

_{0}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]and performing the integration, after a long but straightforward calculation, the following result is foundwhere

*T*

_{1}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

*Q*

_{1}=

*Q*

_{2}= 0 leads to the temporal half-width

*T*

_{0}of GSTG pulses propagating in free space.

## 5. Numerical calculations and analysis

*ε*=

*T*

_{1}/

*T*

_{0}≥ 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

*T*

_{0}and

*L*are fixed, the RTBC of SWG pulses passing in atmospheric turbulence depends only on

*Q*

_{1}, and achieves its maximum with the value of

*α*which maximizes

*Q*

_{1}. Further, it can be seen from Eq. (10) that

*Q*

_{1}is uniquely determined by

*L*

_{0}and

*l*

_{0}, and independent of

*T*

_{0}. As a result, in Fig. 1 the value of

*α*which maximizes the RTBC of SWG pulses with various

*T*

_{0}is the same.

*T*

_{0}; 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

*T*

_{0}, 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

*T*

_{0}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

*T*

_{0}, 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.

*α*

_{max}as the value of

*α*that maximizes the RTBC of pulses with a given combination of

*T*

_{0},

*L*,

*L*

_{0}and

*l*

_{0}. Figure 2 shows the dependence of

*α*

_{max}on the turbulence parameters

*L*

_{0}and

*l*

_{0}. It can be found from Fig. 2 that although

*α*

_{max}is a strong function of

*L*

_{0}, it varies very little with the change of

*l*

_{0}.

*T*

_{1}of CSTG pulses on the radial distance

*r*= |

**r**|. To facilitate the analysis, we compute the normalized temporal half-width

*T*defined by the ratio of the turbulence-induced temporal half-width

_{n}*T*

_{1}with an observation point at

**r**to that on the axis, i.e.,

*T*(

_{n}*r*) =

*T*

_{1}(

*r*)/

*T*

_{1}(

*r*= 0), where the content in parentheses explicitly indicates the radial dependency of

*T*

_{1}. Needless to say,

*T*

_{1}has a strong radial dependency if the deviation of

*T*from 1 varies significantly with the radial distance

_{n}*r*.

*T*in terms of the scaled radial distance

_{n}*r*/ [

*W*

_{0}

^{2}(1 +

*L*

^{2}/

*z*

_{0}

^{2})]

^{1/2}, where the term [

*W*

_{0}

^{2}(1 +

*L*

^{2}/

*z*

_{0}

^{2})]

^{1/2}denotes the free-space beam radius in the receiver plane, and

*z*

_{0}= π

*W*

_{0}

^{2}/

*λ*

_{0}. It can be observed from Fig. 3(a) and 3(b) that

*T*rises as the scaled radial distance increases with a propagation distance

_{n}*L*of both 1km and 5 km, and the smaller

*T*

_{0}is, the faster

*T*increases. On other hand, it is clear from Fig. 3 that all values of

_{n}*T*are very close to 1, and the differences in

_{n}*T*with various scaled radial distances are indeed really tiny. Hence, the dependence of the turbulence-induced temporal half-width

_{n}*T*

_{1}of CSTG pulses on the radial distance can be generally neglected.

## 6. Conclusions

*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

*L*

_{0}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

*L*

_{0}and the inner scale

*l*

_{0}, that of the pulses in non-Kolmogorov turbulence is in terms of

*L*

_{0},

*l*

_{0}and

*α*.

## Acknowledgments

## References and links

1. | C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. |

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. |

3. | D. E. T. T. S. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulses,” Waves Random Media |

4. | C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE |

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. |

6. | L. C. Andrews and R. L. Phillips, |

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. |

8. | E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express |

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. |

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. |

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 |

12. | W. Du, L. Tan, J. Ma, and Y. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express |

13. | O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. |

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 |

15. | R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE |

16. | Y. Baykal and M. A. Plonus, “Two-source, two-frequency spherical wave structure functions in atmospheric turbulence,” J. Opt. Soc. Am. |

17. | I. S. Gradshteyn and I. M. Ryzhik, |

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 |

**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

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### References

- C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979). [CrossRef]
- 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]
- 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]
- C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]
- 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]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 3, 4, 7 and 18.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- O. Korotkova, E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef] [PubMed]
- 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]
- R. L. Fante, “Electromagnetic beam propagation in turbulent media: an update,” Proc. IEEE 68(11), 1424–1443 (1980). [CrossRef]
- 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]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.
- 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]

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