Abstract

The effect of the turbulent atmosphere on the propagation of an ultrashort pulsed beam is presented in terms of time-domain statistical moment equations. Pulsed beams are localized space–time solutions of the wave equation that propagate unchanged along the ray trajectories. The statistically closed equations that are derived are considerably different from the dispersive multiple-frequency equation that is often used for pulse propagation. It is shown that the coherent intensity of the pulsed beam decays on a scale of the order of 10−3–103 m, depending on the conditions of turbulence. This underscores the importance of the statistical feasibility of propagation in addition to the practical realizability, which is customarily emphasized.

© 1996 Optical Society of America

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  1. J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
    [CrossRef]
  2. R. W. Ziokowski, Phys. Rev. A 39, 2005 (1989).
    [CrossRef]
  3. P. Hillion, J. Appl. Phys. 60, 2981 (1986).
    [CrossRef]
  4. T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
    [CrossRef]
  5. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  6. H. E. Moses, J. Math. Phys. 25, 1905 (1984).
    [CrossRef]
  7. E. Heyman, L. B. Felsen, IEEE Trans. Antennas Propag. AP-34, 1062 (1986).
    [CrossRef]
  8. E. Heyman, IEEE Trans. Antennas Propag. 42, 311 (1994).
    [CrossRef]
  9. I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
    [CrossRef]
  10. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  12. V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  13. J. W. Strohbehn, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 46–70.
  14. M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
    [CrossRef]

1994 (1)

E. Heyman, IEEE Trans. Antennas Propag. 42, 311 (1994).
[CrossRef]

1992 (1)

I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
[CrossRef]

1989 (1)

R. W. Ziokowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef]

1987 (1)

1986 (2)

E. Heyman, L. B. Felsen, IEEE Trans. Antennas Propag. AP-34, 1062 (1986).
[CrossRef]

P. Hillion, J. Appl. Phys. 60, 2981 (1986).
[CrossRef]

1985 (1)

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

1984 (1)

H. E. Moses, J. Math. Phys. 25, 1905 (1984).
[CrossRef]

1983 (1)

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

Besieris, I. M.

I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Brittingham, J. N.

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

Charnotskii, M. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
[CrossRef]

Durnin, J.

Felsen, L. B.

E. Heyman, L. B. Felsen, IEEE Trans. Antennas Propag. AP-34, 1062 (1986).
[CrossRef]

Gozani, J.

M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
[CrossRef]

Heyman, E.

E. Heyman, IEEE Trans. Antennas Propag. 42, 311 (1994).
[CrossRef]

E. Heyman, L. B. Felsen, IEEE Trans. Antennas Propag. AP-34, 1062 (1986).
[CrossRef]

Hillion, P.

P. Hillion, J. Appl. Phys. 60, 2981 (1986).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Moses, H. E.

H. E. Moses, J. Math. Phys. 25, 1905 (1984).
[CrossRef]

Shaarawi, A. M.

I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
[CrossRef]

Strohbehn, J. W.

J. W. Strohbehn, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 46–70.

Tatarski, V. I.

M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wu, T. T.

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

Zavorotny, V. U.

M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
[CrossRef]

Ziokowski, R. W.

I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
[CrossRef]

R. W. Ziokowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

E. Heyman, L. B. Felsen, IEEE Trans. Antennas Propag. AP-34, 1062 (1986).
[CrossRef]

E. Heyman, IEEE Trans. Antennas Propag. 42, 311 (1994).
[CrossRef]

J. Appl. Phys. (3)

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

P. Hillion, J. Appl. Phys. 60, 2981 (1986).
[CrossRef]

T. T. Wu, J. Appl. Phys. 57, 2370 (1985).
[CrossRef]

J. Math. Phys. (2)

H. E. Moses, J. Math. Phys. 25, 1905 (1984).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziokowski, J. Math. Phys. 30, 1254 (1992).
[CrossRef]

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

Phys. Rev. A (1)

R. W. Ziokowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

V. I. Tatarskii, The Effects of Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

J. W. Strohbehn, in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), Chap. 3, pp. 46–70.

M. I. Charnotskii, J. Gozani, V. I. Tatarski, V. U. Zavorotny, in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), p. 203.
[CrossRef]

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Equations (16)

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( Δ x + z 2 v 2 t 2 ) u + = 0 .
u + ( x , z , t ) = 1 2 π 0 d ω exp ( i ω t ) u ˆ r ( x , z , ω ) , Im t < 0 ,
v ( x , z , t ) = v [ 1 + ( x , z , t ) ] 1 / 2 , = 0 , 2 1 ,
u + ( x , z , t ) U + ( x , z , τ ) , τ = t z / c ,
( Δ x + z 2 + 2 c 2 τ 2 2 c z τ ) U + = 0 .
A n ( | x 1 x 2 | , | τ 1 τ 2 | ) d ( z 1 z 2 ) × n ( x 1 , z 1 , t 1 z 1 / c ) n ( x 2 , z 2 , t 2 z 2 / c ) .
z U + U l 0 U λ 2 c τ U + ,
[ z τ c 2 Δ x 1 c n ( x , z , τ ) τ 2 ] U + ( x , z , τ ) = 0 .
z τ Γ 1 + ( z , x , τ ) = c 2 Δ x Γ 1 + ( z , x , τ ) + 1 c n ( x , z , τ ) τ 2 U + ( x , z , τ ) .
n ( x , z , τ ) τ 2 U + ( x , z , τ ) = 1 2 c τ A n ( | 0 | , τ = 0 ) × τ 2 U + ( x , z , τ ) + 1 2 c A n ( | 0 | , 0 ) τ 3 U + ( x , z , τ ) .
z τ Γ 1 + ( z , x , τ ) = [ c 2 Δ x + 1 2 c 2 A n ( | 0 | , 0 ) τ 3 ] × Γ 1 + ( z , x , τ ) ,
U 0 + ( x , z , τ ) = A ( z ) f + [ τ 1 2 x Γ ( z ) x ] , τ = t z / c
u r ( x , z , τ ) = Re U 0 + ( z , x , t ) t ,
z τ σ Γ 2 + ( z , x , y , τ , σ ) = [ c 2 ( Δ x σ + Δ y τ ) + V ( x y , τ , σ ) ] × Γ 2 + ( z , x , y , τ , σ ) ,
V ( x y , τ , σ ) 1 2 c 2 [ A n ( | 0 | , 0 ) ( τ 2 + σ 2 ) + 2 A n ( x y , τ σ ) τ σ ] τ σ ,
Γ 2 + ( z , x , y , τ , 0 ) = 0 d ω 1 2 π d ω 2 2 π exp [ i ( τ ω 1 σ ω 2 ) ] × d 2 x d 2 y U ˆ 0 + ( x , z = 0 , ω 1 ) U ˆ 0 + * ( y , z = 0 , ω 2 ) × 0 , x , y z , x , y D 2 x ( ζ ) D 2 y ( ζ ) × exp { i 2 0 z d ζ [ k 1 | x ˙ ( ζ ) | 2 k 2 | y ˙ ( ζ ) | 2 ] } × exp { 0 z d ζ V ˆ + [ x ( ζ ) y ( ζ ) , ω 1 , ω 2 ] } .

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