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.

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  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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2011 (3)

2010 (4)

2009 (1)

2008 (2)

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]

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]

2002 (1)

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

1999 (1)

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]

1998 (1)

1980 (2)

1979 (1)

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

Andrews, L. C.

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]

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]

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]

Bai, X.

Baykal, Y.

Cao, L.

Cao, X.

Cao, X. G.

Castillo-Vázquez, M.

Chen, C.

Cui, L.

Cui, L. Y.

Dang, A.

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]

Dong, J. K.

Du, W.

Fante, R. L.

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

Ferrero, V.

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]

Garrido-Balsells, J. M.

Golbraikh, E.

Guo, H.

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]

Ishimaru, A.

Jiang, Y.

Jurado-Navas, A.

Kelly, D. E. T. T. S.

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]

Kopeika, N. S.

Korotkova, O.

Liu, C. H.

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

Lou, Y.

Luo, B.

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]

Ma, J.

Phillips, R. L.

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]

Plonus, M. A.

Puerta-Notario, A.

Shchepakina, E.

Tan, L.

Tong, S.

Toselli, I.

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]

Wang, J. N.

Wu, G.

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]

Xue, B.

Xue, B. D.

Xue, W.

Yang, H.

Yeh, K. C.

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

Young, C. Y.

Yu, S.

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]

Zhao, T.

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]

Zheng, S.

Zhou, F.

Zilberman, A.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

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]

Opt. Express (5)

Opt. Lett. (2)

Optik-Int. J. Light Electron. (1)

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]

Proc. IEEE (1)

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

Proc. SPIE (1)

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

Radio Sci. (1)

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

Waves Random Media (1)

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]

Other (2)

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

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

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Figures (3)

Fig. 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.

Fig. 2
Fig. 2

Dependence of αmax associated with the maximum RTBC of pulses on the turbulence parameters C ˜ n 2 , L0 and l0.

Fig. 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.

Equations (19)

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I ( r , L ; t ) = T 0 2 4 π exp [ 1 4 ( ω 1 2 + ω 2 2 ) T 0 2 ] × 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 ,
Γ 2 ( r , r , L ; ω , ω ' ) = U ( r , L ; ω ) U * ( r , L ; ω ' ) ,
U ( r , L , ω ) = i ω 2 π L c exp ( i L ω c ) d 2 s U 0 ( s , 0 , ω ) exp [ i ω | s r | 2 2 L c + ψ ( s , r , L ; ω ) ] ,
Γ 2 ( r , r , L ; ω , ω ' ) = ω ω ' 4 π 2 L 2 c 2 exp [ i L c ( ω ω ' ) ] × d 2 s 1 d 2 s 2 U 0 ( s 1 , 0 ; ω ) U 0 ( s 2 , 0 ; ω ' ) × exp [ i ω | s 1 r | 2 2 L c i ω ' | s 2 r | 2 2 L c ] M 2 ( r , r , s 1 , s 2 L ; ω , ω ' ) ,
M 2 ( r , r , s 1 , s 2 L ; ω , ω ' ) = exp [ ψ ( s 1 , r , L ; ω ) + ψ * ( s 2 , r , L ; ω ' ) ] .
M 2 ( r , r , s 1 , s 2 L ; ω , ω ' ) = exp [ 1 2 D ψ ( r , r , s 1 , s 2 , L ; ω , ω ' ) ] ,
D ψ ( r , r , s 1 , s 2 , L ; ω , ω ' ) = ( 2 π / c ) 2 L 0 d κ κ Φ n ( κ ) 0 1 d ξ [ ω 2 + ω ' 2 2 ω ω ' J 0 ( κ β ) ] ,
Φ n ( κ ) = A ( α ) C ˜ n 2 exp ( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α / 2 ,   3 < α < 5 ,
D ψ ( r , r , s 1 , s 2 , L ; ω , ω ' ) ( 2 π / c ) 2 L ( ω ω ' ) 2 Q 1 + 2 3 π 2 c 2 ω ω ' L | s 1 s 2 | 2 Q 2 ,
Q 1 = 0 d κ κ Φ n ( κ ) = 1 2 A ( α ) C ˜ n 2 κ m 2 α exp ( κ 0 2 / κ m 2 ) Γ ( 1 α / 2 , κ 0 2 / κ m 2 ) ,
Q 2 = 0 d κ κ 3 Φ n ( κ ) = A ( α ) C ˜ n 2 2 ( α 2 ) [ κ m 2 α exp ( κ 0 2 κ m 2 ) Γ ( 2 α 2 , κ 0 2 κ m 2 ) ( 2 κ 0 2 2 κ m 2 + α κ m 2 ) 2 κ 0 4 α ] ,
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 ' ) ,
U s ( 0 ) ( r , L ; ω ) = 1 4 π L exp ( i ω L c + i ω r 2 2 L c ) ,
I ( 0 ) ( r , L ; t ) = 1 ( 4 π L ) 2 exp [ 2 T 0 2 ( t L c r 2 2 L c ) 2 ] ,
I ( r , L ; t ) = 1 ( 4 π L ) 2 T 0 T 1 exp [ 2 T 1 2 ( t L c r 2 2 L c ) 2 ] ,
U 0 ( s , 0 , ω ) = exp ( s 2 W 0 2 ) ,
exp ( p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) π p ,   ( Re { p 2 } > 0 ) ,
Γ 2 ( r , r , L ; ω , ω ' ) = 9 c 2 4 L 2 ω ω ' 9 c 4 a 1 a 2 ( π 2 L ω ω ' Q 2 ) 2 × exp [ 2 π 2 L c 2 ( ω ω ' ) 2 Q 1 ] exp [ i ( ω ω ' ) L c ] × exp [ i ( ω ω ' ) r 2 2 L c ] exp ( ω ' 2 r 2 4 L 2 c 2 a 2 ) × exp [ 9 4 ( 1 L c π 2 ω ' 2 Q 2 3 c 3 a 2 ) 2 c 4 a 2 ω 2 r 2 9 c 4 a 1 a 2 ( π 2 L ω ω ' Q 2 ) 2 ] ,
a 1 = 1 W 0 2 + π 2 L ω ω ' Q 2 3 c 2 i ω 2 L c , a 2 = 1 W 0 2 + π 2 L ω ω ' Q 2 3 c 2 + i ω ' 2 L c .

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