## Second-order statistics of Gaussian Schell-model pulsed beams propagating through atmospheric turbulence |

Optics Express, Vol. 19, Issue 16, pp. 15196-15204 (2011)

http://dx.doi.org/10.1364/OE.19.015196

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### Abstract

Novel analytical expressions for the cross-spectral density function of a Gaussian Schell-model pulsed (GSMP) beam propagating through atmospheric turbulence are derived. Based on the cross-spectral density function, the average spectral density and the spectral degree of coherence of a GSMP beam in atmospheric turbulence are in turn examined. The dependence of the spectral degree of coherence on the turbulence strength measured by the atmospheric spatial coherence length is calculated numerically and analyzed in depth. The results obtained are useful for applications involving spatially and spectrally partially coherent pulsed beams propagating through atmospheric turbulence.

© 2011 OSA

## 1. Introduction

*et al.*[1

1. G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. **280**(2), 264–270 (2007). [CrossRef]

2. C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. **34**(4), 419–421 (2009). [CrossRef] [PubMed]

3. K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. **50**(2), 025002 (2011). [CrossRef]

5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A **22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

*et al.*[6

6. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(5), 056613 (2003). [CrossRef] [PubMed]

*et al.*[5

5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A **22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

*et al.*[7

7. C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. **10**(9), 095006 (2008). [CrossRef]

## 2. Analytical expressions for the cross-spectral density function

*z*= 0 and propagates into the half-space

*z*> 0, nearly parallel to the positive

*z*direction. The mutual coherence function of a GSMP beam in the source plane is given by [5

5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A **22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

**r**

_{1}and

**r**

_{2}denote two position vectors in the source plane, Γ

_{0}is a constant, and where

*w*

_{0}is the beam width,

*σ*

_{0}is the transverse correlation length,

*r*is the modulus of

_{j}**r**

*(*

_{j}*j*= 1, 2),

*T*

_{0}is the expectation value of the initial pulse width,

*T*is the temporal coherence length,

_{c}*ω*

_{0}is the central angular frequency of the pulses. Using the Fourier transform, the cross-spectral density function of a GSMP beam in the source plane is given by [5

**22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

*ω*(

_{j}*j*= 1, 2) is the angular frequency,

*W*

_{0}= Γ

_{0}

*T*

_{0}/(2πΩ

_{0}) is a coefficient independent of both the position vectors and the angular frequencies, Ω

_{0}= (1/

*T*

_{0}

^{2}+ 2/

*T*

_{c}^{2})

^{1/2}represents the spectral width andwhere Ω

*=*

_{c}*T*Ω

_{c}_{0}/

*T*

_{0}describes the spectral coherence width.

*z*> 0 can be written as

**ρ**

*(*

_{j}*j*= 1, 2) is a position vector in the plane

*z*=

*z*,

_{j}*ψ*(

**r**

*,*

_{j}**ρ**

*,*

_{j}*z*,

_{j}*ω*) represents the random part of the complex phase of a spherical wave with the angular frequency

_{j}*ω*propagating from (

_{j}**r**

*, 0) to (*

_{j}**ρ**

*,*

_{j}*z*) in atmospheric turbulence, the asterisk denotes the complex conjugation,

_{j}*c*is the speed of light, and <∙>

_{m}denotes the average over the ensemble of atmospheric turbulence.

*C*

_{n}^{2}= 0 corresponds to the absence of turbulence in which Γ

_{2}= 1 and

*ρ*

_{0}= ∞; under this condition, Eq. (9) readily proves to be consistent with the formulae obtained by Lajunen

*et al*. (see Eqs. (50)−(60) of [5

**22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

*η*(

*g*

_{1},

*g*

_{2}) has a negative sign before arctan(∙). Indeed, Eq. (60) of [5

**22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

## 3. Average spectral density

**ρ**

_{1}=

**ρ**

_{2}and

*ω*

_{1}=

*ω*

_{2}in Eq. (9), we obtain the average spectral density of a GSMP beam propagating through atmospheric turbulence as follows:where Δ(

*L*,

*ω*), referred to as the spectral expansion coefficient, is written as

**22**(8), 1536–1545 (2005). [CrossRef] [PubMed]

*w*. In fact,

_{c}*S*(

**ρ**,

*L*,

*ω*) is a product of the average spectral density of a GSM beam and the spectrum of a Gaussian pulse. The average spectral density of a GSMP beam decreases as the propagation distance

*L*increases. We can see from Eq. (10) that the reduction in the average spectral density caused by the propagation of a GSMP beam is independent of the spectrally partial coherence of the source. It can be readily found from Eq. (11) that the effect of atmospheric turbulence, associated with

*ρ*

_{0}, on the spectral expansion coefficient resembles the effect of spatially partial coherence of the source, associated with

*σ*

_{0}. In addition, the contribution of

*ρ*

_{0}and

*σ*

_{0}toward the spectral expansion coefficient does not couple together as a GSMP beam propagates through atmospheric turbulence; this manifests that atmospheric turbulence is equivalent to spatially partial coherence of the source in enlarging the spectral expansion coefficient.

## 4. Spectral degree of coherence

_{2}and exp(−|

**ρ**

_{1}−

**ρ**

_{2}|

^{2}/

*ρ*

_{0}

^{2}) in Eq. (13) are two turbulence-induced attenuation factors associated with the angular frequency separation |

*ω*

_{1}−

*ω*

_{2}| and the position separation |

**ρ**

_{1}−

**ρ**

_{2}|, respectively, which have an influence on the modulus of the spectral degree of coherence. Furthermore, the argument of the spectral degree of coherence is the same as the cross-spectral density function.

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| and the wavelength

*λ*with different combinations of

*ρ*

_{0}and

*σ*

_{0}. Note that the quantity |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| actually describes the frequency-varying degree of transverse spatial coherence of the beam, and indeed, |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| does not depend on

*T*

_{0}and

*T*. It can be seen that |

_{c}*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| ≡ 1 if a spatially completely coherent pulsed beam propagates through free space without atmospheric turbulence, i.e.,

*ρ*

_{0}= ∞ and

*σ*

_{0}= ∞; this result agrees with what one might expect. For other combinations of

*ρ*

_{0}and

*σ*

_{0}, |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| shows a very slow monotonic growth with the increasing wavelength; the maximum values of |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)|

_{λ}_{= 1.61μm}− |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)|

_{λ}_{= 1.49μm}are 5.9 × 10

^{−3}and 1.1 × 10

^{−2}for Figs. 1(a) and 1(b), respectively. This variation behavior of |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| with the wavelength is also consistent with the known theory for a GSM beam (see Eqs. (5.6-84) and (5.6-107) of [4]) when

*ρ*

_{0}= ∞. Based on these numerical results, one can say that in the presence of spatially partial coherence of the source and (or) atmospheric turbulence, the degree of transverse spatial coherence in an observation plane

*z*> 0 is slightly smaller for a spectral component of a GSMP beam with a higher frequency than one with a lower frequency, that is, as opposed to the source plane in which the transverse correlation length for all spectral components is the same, the degree of transverse spatial coherence in the observation plane for a given spectral component of a GSMP beam is dependent on the frequency. Moreover, it can be observed from Fig. 1 that |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*,

*ω*)| reduces as the values of

*ρ*

_{0}and (or)

*σ*

_{0}decrease with the same wavelength

*λ*.

*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| as a function of the wavelength separation

*δ*associated with the two spectral components of a GSMP beam, where the central angular frequency

_{λ}*ω*

_{0}= 2π

*c*/

*λ*

_{0}is fixed and the angular frequency

*ω*

_{1}= 2π

*c*/(

*λ*

_{0}+

*δ*) varies with

_{λ}*δ*. It should be noted that the quantity |

_{λ}*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| is a measure of spectral coherence at the position

**ρ**for the frequencies of

*ω*

_{0}and

*ω*

_{1}. Figure 2(a) corresponds to a spatially and spectrally completely coherent pulsed beam, and Fig. 2(b) represents a spatially and spectrally partially coherent pulsed beam. We can see from Figs. 2(a) and 2(b) that |

*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| reduces monotonically as

*ρ*

_{0}decreases with the same

*δ*; this variation behavior of |

_{λ}*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| with

*ρ*

_{0}manifests the fact that atmospheric turbulence induces the decrease of spectral coherence of a pulsed beam, which is just the reason for the temporal pulse broadening. It needs to be pointed out that the curves of |

*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| are actually not rigorously symmetric with respect to

*δ*= 0; this fact proves that |

_{λ}*μ*(

**ρ**,

**ρ**,

*L*,

*ω*

_{0},

*ω*

_{1})| is determined exactly by a given combination of

*ω*

_{0}and

*ω*

_{1}instead of the frequency separation |

*ω*

_{0}−

*ω*

_{1}|.

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*

_{0},

*ω*

_{1})| in terms of the relative atmospheric spatial coherence length

*ρ*

_{0}/[

*w*

_{0}

^{2}∙Δ(

*L*,

*ω*

_{0})]

^{1/2}which is defined as the ratio of the atmospheric spatial coherence length to the beam width. In Fig. 3(a), the frequency separation |

*ω*

_{0}−

*ω*

_{1}| is fixed and the position separation

*d*= |

**ρ**

_{1}−

**ρ**

_{2}| is specified as 0cm, 2cm, 4cm and 6cm, respectively. In Fig. 3 (b), the position separation |

**ρ**

_{1}−

**ρ**

_{2}| = 2cm is fixed and the frequency separation |

*ω*

_{0}−

*ω*

_{1}| is specified by various values of the wavelength separation

*δ*. We can see from Figs. 3(a) and 3 (b) that |

_{λ}*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*

_{0},

*ω*

_{1})| rises quickly as the relative atmospheric spatial coherence length increases. In Fig. 3(a), for the case of |

**ρ**

_{1}−

**ρ**

_{2}| = 0, |

*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*

_{0},

*ω*

_{1})| approaches to saturation as the relative atmospheric spatial coherence length increases past 1, and for other cases, with the increase of |

**ρ**

_{1}−

**ρ**

_{2}|, the saturation point moves gradually toward the right side. However, in Fig. 3(b), we can find that almost for all values of the wavelength separation

*δ*, |

_{λ}*μ*(

**ρ**

_{1},

**ρ**

_{2},

*L*,

*ω*

_{0},

*ω*

_{1})| approaches to saturation as the relative atmospheric spatial coherence length increases beyond 1.

## 5. Conclusions

*z*> 0 for a given spectral component of a GSMP beam depends on the frequency, and the higher the frequency is, the slightly smaller the degree of transverse spatial coherence becomes; the spectral degree of coherence is determined exactly by a given combination of the two frequencies instead of the corresponding frequency separation; the modulus of the spectral degree of coherence rises quickly with the increasing relative atmospheric spatial coherence length and then approaches to saturation as the relative atmospheric spatial coherence length increases beyond a given value determined by the parameters of the positions and the frequencies.

## Acknowledgments

## References and links

1. | G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. |

2. | C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. |

3. | K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. |

4. | L. Mandel and E. Wolf, |

5. | H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A |

6. | L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

7. | C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. |

8. | R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. |

9. | C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. |

10. | C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. |

11. | H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express |

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

13. | Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. |

14. | G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. |

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

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

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 1, 2011

Revised Manuscript: July 2, 2011

Manuscript Accepted: July 7, 2011

Published: July 22, 2011

**Citation**

Chunyi Chen, Huamin Yang, Yan Lou, and Shoufeng Tong, "Second-order statistics of Gaussian Schell-model pulsed beams propagating through atmospheric turbulence," Opt. Express **19**, 15196-15204 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-16-15196

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### References

- G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280(2), 264–270 (2007). [CrossRef]
- C. Chen, H. Yang, X. Feng, and H. Wang, “Optimization criterion for initial coherence degree of lasers in free-space optical links through atmospheric turbulence,” Opt. Lett. 34(4), 419–421 (2009). [CrossRef] [PubMed]
- K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50(2), 025002 (2011). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chaps. 4 and 5.
- H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005). [CrossRef] [PubMed]
- L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003). [CrossRef] [PubMed]
- C. Ding and B. Lü, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008). [CrossRef]
- R. L. Fante, “Two-position, two-frequency mutual-coherence function in turbulence,” J. Opt. Soc. Am. 71(12), 1446–1451 (1981). [CrossRef]
- C. Ding, L. Pan, and B. Lü, “Effect of turbulence on the spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in atmospheric turbulence,” J. Opt. A, Pure Appl. Opt. 11(10), 105404 (2009). [CrossRef]
- C. Ding, Z. Zhao, X. Li, L. Pan, and X. Yuan, “Influence of turbulent atmosphere on polarization properties of stochastic electromagnetic pulsed beams,” Chin. Phys. Lett. 28(2), 024214 (2011). [CrossRef]
- H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010). [CrossRef] [PubMed]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005), Chaps. 4 and 7.
- Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010). [CrossRef]
- G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13(3), 035706 (2011). [CrossRef]
- C. Y. Young, “Broadening of ultra-short optical pulses in moderate to strong turbulence,” Proc. SPIE 4821, 74–81 (2002). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, 2007), Chap. 3.

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