Abstract

Beam wander of a finite optical beam propagating in a turbulent medium is investigated theoretically. Using the optical analog of Ehrenfest’s theorem, it is shown that the centroid of a finite beam propagates as a paraxial ray in a certain effective refractive index that depends on the irradiance profile of the beam. Ray statistics in the effective refractive index are studied for arbitrary irradiance profiles and new results are obtained for the variance of spot displacement and beam angle of arrival. These results are then applied to the particular cases of focused and collimated gaussian beams in atmospheric turbulence with a modified Von Karman power spectrum to yield the functional dependence of spot dancing and angle-of-arrival statistics on the inner and outer scales of turbulence and on the Fresnel number for focused gaussian beams.

© 1975 Optical Society of America

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References

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  1. L. I. Schiff, Quantum Mechanics, 2nd ed. (McGraw–Hill, New York, 1955), p. 25.
  2. P. Beckmann, Radio Sci. 69D, 629 (1965).
  3. L. A. Chernow, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw–Hill, New York, 1960), Ch. 2.
  4. J. B. Keller, Proc. Symp. Appl. Math. 13, 226 (1962).
  5. T. Chiba, Appl. Opt. 10, 2456 (1971).
    [Crossref] [PubMed]

1971 (1)

1965 (1)

P. Beckmann, Radio Sci. 69D, 629 (1965).

1962 (1)

J. B. Keller, Proc. Symp. Appl. Math. 13, 226 (1962).

Beckmann, P.

P. Beckmann, Radio Sci. 69D, 629 (1965).

Chernow, L. A.

L. A. Chernow, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw–Hill, New York, 1960), Ch. 2.

Chiba, T.

Keller, J. B.

J. B. Keller, Proc. Symp. Appl. Math. 13, 226 (1962).

Schiff, L. I.

L. I. Schiff, Quantum Mechanics, 2nd ed. (McGraw–Hill, New York, 1955), p. 25.

Appl. Opt. (1)

Proc. Symp. Appl. Math. (1)

J. B. Keller, Proc. Symp. Appl. Math. 13, 226 (1962).

Radio Sci. (1)

P. Beckmann, Radio Sci. 69D, 629 (1965).

Other (2)

L. A. Chernow, Wave Propagation in a Random Medium, translated by R. A. Silverman (McGraw–Hill, New York, 1960), Ch. 2.

L. I. Schiff, Quantum Mechanics, 2nd ed. (McGraw–Hill, New York, 1955), p. 25.

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Figures (3)

FIG. 1
FIG. 1

Diffusion function (solid line), two-term approximation to the diffusion function (curve 2), and four-term approximation to the diffusion function (curve 4).

FIG. 2
FIG. 2

Beam angle-of-arrival coefficient A(N, Δ) versus Fresnel number N = π ω 0 2 / λ f at several values of the inner-scale parameter Δ = l 0 / λ f.

FIG. 3
FIG. 3

Beam-wander coefficient B(N, Δ) versus Fresnel number N = π ω 0 2 / λ f at several values of the inner-scale parameter Δ = l 0 / λ f.

Equations (67)

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2 U - n 2 c 2 2 U t 2 = 0 ,
U = u ( x , y , z ) exp ( i k z - i ω t ) ,
2 u + 2 i k u z + k 2 ( n 2 - 1 ) u = 0.
i u z = - 1 2 k T 2 u - 1 2 k ( n 2 - 1 ) u ,
n ( r ) = 1 + n 1 ( r ) ,
i u z = - 1 2 k T 2 u - k n 1 u .
i Ψ t = - 2 2 m T 2 Ψ + V Ψ ,
t z ,             1 Ψ u ,             m k V ( x , y , t ) - k n 1 ( x , y , z ) .
m d 2 ρ d t 2 = - T V ,
ρ = - ρ Ψ * Ψ d x d y
T V = - T V Ψ * Ψ d x d y ,
d 2 R d z 2 = - T u 1 u * u d x d y = - T n 1 I d x d y ,
R = - ρ I d x d y .
- Ψ * Ψ d x d y = 1 ,
- u * u d x d y = 1.
I ( x , y , z ) = I 0 ( x - , y - η , z ) .
d 2 R d z 2 = - T n 1 I 0 ( x - , y - η , z ) d x d y .
d 2 d z 2 = - n 1 ( x , y , z ) x I 0 ( x - , y - η , z ) d x d y = - n 1 ( x + , y + η , z ) x I 0 ( x , y , z ) d x d y = - n 1 ( x + , y + η , z ) I 0 ( x , y , z ) d x d y = - n 1 ( x + , y + η , z ) I 0 ( x , y , z ) d x d y = - n 1 ( x , y , z ) I 0 ( x - , y - η , z ) d x d y .
μ ( , η ) = - n 1 ( x , y , z ) I 0 ( x - , y - η , z ) d x d y ,
d 2 R d z 2 = T μ .
n 1 ( x 1 , y 1 , z 1 ) n 1 ( x 2 , y 2 , z 2 ) = B n ( ( x 2 + y 2 + z 2 ) 1 / 2 ) ,
B μ = μ ( 1 , η 1 , z 1 ) μ ( 2 , η 2 , z 2 ) = - d x 1 d y 1 d x 2 d y 2 I 0 ( x 1 - 1 , y 1 - η 1 , z 1 ) × I 0 ( x 2 - 2 , y 2 - η 2 , z 2 ) n 1 ( x 1 , y 1 , z 1 ) n 1 ( x 2 , y 2 , z 2 ) .
B μ ( , η , z ; z 1 ) = - T ( - x , η - y , z ; z 1 ) B n ( x , y , z ) d x d y ,
T ( , η , z ; z 1 ) = - I 0 ( x , y , z 1 ) I 0 ( x - , y - η , z + z 1 ) d x d y
B μ ( , η , z ) = - T ( - x , η - y ) B n ( x , y , z ) d x d y ,
T ( , η ) = - I 0 ( x , y ) I 0 ( x - , y - η ) d x d y .
I 0 ( x , y ) = 2 π ω 2 exp { - 2 ( x 2 + y 2 ) ω 2 } .
T ( , η ) = 1 π ω 2 exp { - 2 + η 2 ω 2 } .
B n ( x , y , z ) = n 1 2 exp { - ( x 2 + y 2 + z 2 ) / a 2 } ,
B μ ( , η , z ) = n 1 2 1 + ω 2 / a 2 exp { - ( 2 + η 2 a 2 [ 1 + ω 2 / a 2 ] + z 2 a 2 ) } .
T ( , η , z ; z 1 ) = 2 π ( ω 2 ( z 1 ) + ω 2 ( z + z 1 ) ) × exp { - 2 ( 2 + η 2 ) ω 2 ( z 1 ) + ω 2 ( z + z 1 ) } .
T ( , η , z ; z 1 ) = 1 π ω 2 ( z 1 ) exp { - 2 + η 2 ω 2 ( z 1 ) }
d 2 R d z 2 = T μ ( R , z ) ,
R = R 0 + R 1 + 2 R 2 + .
d 2 R 0 d z 2 + d 2 R 1 d z 2 + = T μ ( R 0 + R 1 + , z ) .
d 2 R 0 d z 2 = 0.
d 2 R 1 d z 2 = T μ ( R 0 , z ) .
d 2 R 1 d z 2 = T μ ( 0 , 0 , z ) .
d R 1 d z = 0 z T μ ( 0 , 0 , t ) d t ,
R 1 = 0 z ( z - t ) T μ ( 0 , 0 , t ) d t .
ρ 2 = R 1 2 = 0 z d t 1 ( z - t 1 ) 0 z d t 2 ( z - t 2 ) × T 1 μ ( 0 , 0 , t 1 ) · T 2 μ ( 0 , 0 , t 2 )
α 2 = 0 z 0 z d t 1 d t 2 T 1 μ ( 0 , 0 , t 1 ) · T 2 μ ( 0 , 0 , t 2 ) ,
μ ( ρ 1 , z 1 ) μ ( ρ 2 , z 2 ) = B μ ( ρ 2 - ρ 1 , z ; z 1 ) ,
T 1 μ ( ρ 1 , z 1 ) · T 2 μ ( ρ 2 , z 2 ) = - T 2 B μ ( ρ 2 - ρ 1 , z ; z 1 ) .
α 2 = - 0 z d t 1 - t 1 z - t 1 T 2 B μ ( 0 , t ; t 1 ) d t .
α 2 = 4 0 z D μ ( t ) d t ,
D μ ( z ) = - 1 4 - T 2 B μ ( 0 , t ; z ) d t
ρ 2 = 4 0 z ( z - t ) 2 D μ ( t ) d t .
ρ 2 = 4 3 D μ z 3 ,
α 2 = 4 D μ z .
ρ 2 = 2 S D - 1 D 2 ( 4 9 - e - 2 D S 2 + e - 6 D S 18 ) ,
cos α = e - 2 D S ,
B n ( r ) = - Φ n ( k ) exp { - i k · r } d 3 k ,
Φ n ( k ) = 0.033 C n 2 exp { - ( k l 0 ) 2 } [ k 2 + L 0 - 2 ] 11 / 6 ,
B μ ( , η , z ; z 1 ) = 1 π ω 2 - exp { - ( - x ) 2 + ( η - y ) 2 ω 2 } B n ( x , y , z ) d x d y .
D μ ( z ) = - 1 4 ( 2 2 + 2 η 2 ) - B μ ( , η , t , z ) d t | = 0 , η = 0
D μ ( z ) = 0.033 π 2 C n 2 0 exp { - ( ω 2 ( z ) + 4 l 0 2 ) k 2 / 4 } k 3 d k [ k 2 + L 0 - 2 ] 11 / 6 .
x 2 = ( ω 2 + 4 l 0 2 ) / L 0 2 ,
D μ = 0.033 π 2 C n 2 L 0 1 / 3 F ( x ) ,
F ( x ) = x - 1 / 3 0 exp { - u 2 / 4 } [ u 2 + x 2 ] 11 / 6 u 3 d u .
F ( x ) 3.2 1 / 3 Γ ( 7 8 ) x - 1 / 3 - 3.60 + 33 20 · 2 1 / 3 Γ ( 7 6 ) x 5 / 3 - 54 35 x 2 = 3.51 x - 1 / 3 - 3.60 + 1.93 x 5 / 3 - 1.54 x 2 .
ω 2 ( z ) = ω 0 2 f 2 [ z 2 N 2 + ( z - f ) 2 ] ,
D μ ( z ) = 1.143 C n 2 ω 0 1 / 3 [ 1 N 2 ( z f ) 2 + ( z f - 1 ) 2 + 4 l 0 ω 0 2 ] - 1 / 6 - 1.173 C n 2 L 0 1 / 3 .
α 2 = 6.85 C n 2 f ω 0 1 / 3 A ( N , Δ ) - 4.69 C n 2 f L 0 1 / 3 ,
ρ 2 = 1.71 C n 2 f 3 ω 0 1 / 3 B ( N , Δ ) - 1.56 C n 2 f 3 L 0 1 / 3 ,
A ( N , Δ ) = 2 3 0 1 [ ( u - 1 ) 2 + ( u / N ) 2 + 4 π Δ 2 / N ] - 1 / 6 d u ,
B ( N , Δ ) = 8 3 0 1 [ u - 1 ] 2 [ ( u - 1 ) 2 + ( u / N ) 2 + 4 π Δ 2 / N ] - 1 / 6 d u .