Abstract

An analytic expression is derived for the long-term temporal broadening (fluctuations of arrival time) of a collimated space-time Gaussian pulse propagating along a horizontal path through weak optical turbulence. General results are presented for nominal parameter values characterizing laser communication through the atmosphere. Specific examples are calculated for both upper-atmosphere and ground-level cross links. It is shown that, for upper-atmosphere cross links, pulses shorter than 100 fs have considerable broadening, whereas at ground level, broadening is predicted in pulses as long as 1 ps.

© 1998 Optical Society of America

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References

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  1. C. Y. Young, A. Ishimaru, L. C. Andrews, “Two-frequency mutual coherence function of a Gaussian beam pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 35, 6522–6526 (1996).
    [CrossRef] [PubMed]
  2. S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
    [CrossRef]
  3. I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
    [CrossRef]
  4. I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
    [CrossRef] [PubMed]
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).
  6. R. Ziolkowski, J. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  7. C. H. Liu, K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain, or fog,” J. Opt. Soc. Am. 67, 1261–1266 (1977).
    [CrossRef]
  8. C. H. Liu, K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14, 925–931 (1979).
    [CrossRef]
  9. C. E. Coulman, J. Vernin, Y. Coqueugniot, J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
    [CrossRef] [PubMed]

1996 (1)

1992 (1)

1988 (1)

1979 (2)

1977 (1)

1976 (2)

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Andrews, L. C.

Caccia, J. L.

Coqueugniot, Y.

Coulman, C. E.

Hong, S. T.

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Ishimaru, A.

C. Y. Young, A. Ishimaru, L. C. Andrews, “Two-frequency mutual coherence function of a Gaussian beam pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 35, 6522–6526 (1996).
[CrossRef] [PubMed]

I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).

Judkins, J.

Liu, C. H.

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

C. H. Liu, K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain, or fog,” J. Opt. Soc. Am. 67, 1261–1266 (1977).
[CrossRef]

Sreenivasiah, I.

I. Sreenivasiah, A. Ishimaru, “Beam wave two-frequency mutual coherence function and pulse propagation in random media: an analytic solution,” Appl. Opt. 18, 1613–1618 (1979).
[CrossRef] [PubMed]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Vernin, J.

Yeh, K. C.

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

C. H. Liu, K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain, or fog,” J. Opt. Soc. Am. 67, 1261–1266 (1977).
[CrossRef]

Young, C. Y.

Ziolkowski, R.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Radio Sci. (3)

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

S. T. Hong, A. Ishimaru, “Two-frequency mutual coherence function, coherence bandwidth, and coherence time of millimeter and optical waves in rain, fog, and turbulence,” Radio Sci. 11, 551–559 (1976).
[CrossRef]

I. Sreenivasiah, A. Ishimaru, S. T. Hong, “Two-frequency mutual coherence function and pulse propagation in random media: an analytic solution to the plane wave case,” Radio Sci. 11, 775–778 (1976).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, New York, 1997).

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Figures (2)

Fig. 1
Fig. 1

Expected pulse broadening for an upper-atmosphere cross link.

Fig. 2
Fig. 2

Expected pulse broadening for a ground-level cross link.

Equations (44)

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p i t = ν i t exp - i ω 0 t ,
p 0 t = ν 0 t exp - i ω 0 t .
P i ω = -   p i t exp i ω t d t = - ν i t exp i ω - ω 0 t d t , = V i ω - ω 0 ,
ν 0 r ,   z ;   t = 1 2 π -   V i ω U r ,   z ;   ω + ω 0 exp - i ω t d ω ,
B ν r 1 ,   r 2 ,   z ;   t 1 ,   t 2 = ν 0 r 1 ,   z ;   t 1 ν 0 * r 2 ,   z ;   t 2 = 1 2 π 2 -   V i ω 1 V i * ω 2 × Γ 2 r 1 ,   r 2 ,   z ;   ω 1 ,   ω 2 exp - i ω 1 t 1 + i ω 2 t 2 d ω 1 d ω 2 ,
Γ 2 r 1 ,   r 2 ,   z ;   ω 1 ,   ω 2 = U r 1 ,   z ;   ω 1 + ω 0 × U * r 2 ,   z ;   ω 2 + ω 0 .
ν i t = exp - t 2 / T 0 2 ,
V i ω = - exp - t 2 / T 0 2 exp i ω t d t = π T 0   exp - 1 4   ω 2 T 0 2 ,
I r ,   z ;   t = T 0 2 4 π - exp - 1 2   ω c 2 T 0 2 exp - 1 8   ω d 2 T 0 2 × Γ 2 r ,   r ,   z ;   ω c + ω 0 ,   ω d × exp - i ω d t d ω c d ω d ,
ω c = 1 2 ω 1 + ω 2 ,   ω d = ω 1 - ω 2 .
Γ 2 r ,   r ,   z ,   ω c + ω 0 ,   ω d = exp - 2 r 2 W 0 2 × exp - α ω d 2 + iz c   ω d ,
α = 0.3908 C n 2 zL 0 5 / 3 c 2 ,
I 0 r ,   t ,   z = exp - 2 r 2 W 0 2 exp - 2 t - z / c 2 T 0 2 ,
I r ,   t ,   z = T 0 T 2 exp -   2 r 2 W 0 2 exp - 2 t - z / c 2 T 2 2 ,
T 2 = T 0 2 + 8 α 1 / 2 ,
t a = M 1 M 0 ,   t a 2 = M 2 M 0 ,   σ ta 2 = t a 2 - t a 2 ,
M n = -   t n ν 0 r ,   z ;   t ν 0 * r ,   z ;   t d t .
-   t n   exp i ω d t d t = 2 π - i n δ n ω d , -   δ n ω d f ω d d ω d = - 1 n n ω d n   f ω d | ω d = 0 ,
M n = i n 2 π - n ω d n V i ω c + ω d 2 V i * ω c - ω d 2 × Γ 2 r ,   r ,   z ,   ω c + ω 0 ,   ω d ω d = 0 d ω c .
t a = z / c ,   σ ta 2 = T 0 2 / 4 + 2 α .
4 σ ta 2 = T 2 2 = T 0 2 + 8 α .
σ 1 2     1 , weak   optical   turbulence , T 0   >   20   fs , narrow   band , Ω 2     1 , near   field ,
σ 1 2 = 1.23   C n 2 k 7 / 6 z 11 / 6 ,   Ω = 2 z kW 0 2 .
T 2 = T 0 2 + 8 α 1 / 2 ,
α = 0.322   σ 1 2 Q 0 - 5 / 6 ω 0 2 ,
σ 1 2 = 1.23 C n 2 k 7 / 6 z 11 / 6 ,   Q 0 = z kL 0 2 ,   ω 0 = kc .
σ 1 2 = 0.2 ,   Q 0 = 2.47 × 10 - 4 ,   ω 0 = 1.2 × 10 15 ,
T 2 = 27.4   fs ,
σ 1 2 = 0.33 ,   Q 0 = 4.22 × 10 - 7 ,   ω 0 = 3.6 × 10 15 ,
T 2 = 155   fs ,
Γ 2 r 1 ,   k 1 ,   r 2 ,   k 2 ,   z = U r 1 ,   k 1 ,   z U * r 2 ,   k 2 ,   z ,
Φ n κ = 0.033 C n 2 exp - κ 2 / κ m 2 κ 2 + κ 0 2 11 / 6 ,
Γ 2 r 1 ,   k 1 ,   r 2 ,   k 2 ,   z = U 0 r 1 ,   k 1 ,   z U 0 * r 2 ,   k 2 ,   z M 2 ,
U 0 r 1 ,   k 1 ,   z = Θ 1 - i Λ 1 exp ik 1 z × exp ik 1 2 z Θ ˜ 1 + i Λ 1 r 1 2 , U 0 * r 2 ,   k 2 ,   z = Θ 2 + i Λ 2 exp - ik 2 z × exp - ik 2 2 z Θ ˜ 2 - i Λ 2 r 2 2 ,
M 2 = exp - 2 π 2 k 1 2 + k 2 2 z   0   κ Φ n κ d κ + 4 π 2 k 1 k 2 0 1 0   κ Φ n κ J 0 κ | r | γ 1 - γ 2 * exp - i κ 2 z ξ 2 γ 1 k 1 - γ 2 * k 2 d κ d ξ ,
γ 1 = 1 - Θ ˜ 1 + i Λ 1 ξ ,   γ 2 * = 1 - Θ ˜ 2 - i Λ 2 ξ ,
Θ 1 = Ω 0 Ω 0 2 + Ω 1 2 , Λ 1 = Ω 1 Ω 0 2 + Ω 1 2 , Θ ˜ 1 = 1 - Θ 1 , Θ 2 = Ω 0 Ω 0 2 + Ω 2 2 , Λ 2 = Ω 2 Ω 0 2 + Ω 2 2 , Θ ˜ 2 = 1 - Θ 2 , Ω 0 = 1 - z R 0 , Ω 1 = 2 z k 1 W 0 2 , Ω 2 = 2 z k 2 W 0 2 .
Θ 1,2 = 1 ,   Θ ˜ 1,2 = 0 ,   Λ 1,2 = Ω 1,2 ,   γ 1 = γ 2 * = 1 ,
Γ 2 r ,   r ,   k 1 ,   k 2 ,   z = exp iz k 1 - k 2 exp - 2 r 2 W 0 2 M 2 ,
M 2 = exp - 2 π 2 k 1 2 + k 2 2 z   0   κ Φ n κ d κ + 4 π 2 k 1 k 2 × 0 1 0   κ Φ n κ exp - i κ 2 z ξ 2 1 k 1 - 1 k 2 d κ d ξ .
M 2 = exp - 2 π 2 k 1 2 + k 2 2 - 2 k 1 k 2 z   0   κ Φ n κ d κ .
ω d = k d c = k 1 - k 2 c ,
Γ 2 r ,   r ,   z ,   ω c + ω 0 ,   ω d = exp - 2 r 2 W 0 2 × exp - α ω d 2 + iz c   ω d ,
α = 0.3908 C n 2 zL 0 5 / 3 c 2 .

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