Abstract

A new and potentially serious optical-system beam degradation is discussed. The degradation, which we define here as centroid anisoplanatism, deals with the errors and corresponding on-axis intensity reduction that are obtained when centroid or wave-front gradient tracking systems are employed to determine the overall atmospheric-turbulence-induced tilt in short-term imaging and/or laser transmitting systems. The error between overall tilt and centroid measurements becomes more important both at shorter wavelengths and for large-diameter optics. It is also exacerbated by point-ahead limitations and scintillation. Specifically, it is shown that a Strehl ratio of less than 3 × 10−2 results for D/r0 ≳ 100, where D is the optics aperture diameter and r0 is the turbulence-induced lateral coherence length.

© 1985 Optical Society of America

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References

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  1. D. L. Fried, “Varieties of isoplanatism,” Proc. Soc. Photo-Opt. Instrum. Eng. 75, 20 (1976).
  2. D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52 (1982).
    [CrossRef]
  3. G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging systems,” Appl. Opt. 19, 574 (1980).
    [CrossRef] [PubMed]
  4. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251 (1984).
    [CrossRef]
  5. J. W. Hardy, “Active optics: a new technology for the control of light,”IEEE Proc. 66, 651 (1978).
    [CrossRef]
  6. J. Herrmann, “Least squares wavefront errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  7. G. A. Tyler, D. L. Fried, “Wavefront sensing a new approach to wavefront reconstruction,” (Optical Science Company, Placentia, Calif., May1983).
  8. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372 (1966).
    [CrossRef]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Services, Springfield, Va., 1971).
  10. M. T. Tavis, H. T. Yura, “Strong turbulence effects on short wavelength lasers,” (Aerospace Corporation, Los Angeles, Calif., December15, 1979) [note that in Eq. (8) the term θ¯2 should beθ¯].
  11. R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).
  12. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 324.
  13. G. N. Watson, A Treatise of the Theory Bessel Functions (Cambridge U. Press, Cambridge, 1945).

1984 (1)

1982 (1)

1980 (2)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,”IEEE Proc. 66, 651 (1978).
[CrossRef]

1976 (1)

D. L. Fried, “Varieties of isoplanatism,” Proc. Soc. Photo-Opt. Instrum. Eng. 75, 20 (1976).

1966 (1)

Fried, D. L.

D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52 (1982).
[CrossRef]

D. L. Fried, “Varieties of isoplanatism,” Proc. Soc. Photo-Opt. Instrum. Eng. 75, 20 (1976).

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372 (1966).
[CrossRef]

G. A. Tyler, D. L. Fried, “Wavefront sensing a new approach to wavefront reconstruction,” (Optical Science Company, Placentia, Calif., May1983).

Hansen, D. W.

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,”IEEE Proc. 66, 651 (1978).
[CrossRef]

Herrmann, J.

Jones, R. R.

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Julian, A. J.

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Luke, Y. L.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 324.

Rockway, J. W.

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Stotts, L. B.

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Tatarskii, V. I.

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

Tavis, M. T.

M. T. Tavis, H. T. Yura, “Strong turbulence effects on short wavelength lasers,” (Aerospace Corporation, Los Angeles, Calif., December15, 1979) [note that in Eq. (8) the term θ¯2 should beθ¯].

Tyler, G. A.

G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251 (1984).
[CrossRef]

G. A. Tyler, D. L. Fried, “Wavefront sensing a new approach to wavefront reconstruction,” (Optical Science Company, Placentia, Calif., May1983).

Valley, G. C.

Watson, G. N.

G. N. Watson, A Treatise of the Theory Bessel Functions (Cambridge U. Press, Cambridge, 1945).

Yura, H. T.

M. T. Tavis, H. T. Yura, “Strong turbulence effects on short wavelength lasers,” (Aerospace Corporation, Los Angeles, Calif., December15, 1979) [note that in Eq. (8) the term θ¯2 should beθ¯].

Appl. Opt. (1)

IEEE Proc. (1)

J. W. Hardy, “Active optics: a new technology for the control of light,”IEEE Proc. 66, 651 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. L. Fried, “Varieties of isoplanatism,” Proc. Soc. Photo-Opt. Instrum. Eng. 75, 20 (1976).

Other (6)

G. A. Tyler, D. L. Fried, “Wavefront sensing a new approach to wavefront reconstruction,” (Optical Science Company, Placentia, Calif., May1983).

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

M. T. Tavis, H. T. Yura, “Strong turbulence effects on short wavelength lasers,” (Aerospace Corporation, Los Angeles, Calif., December15, 1979) [note that in Eq. (8) the term θ¯2 should beθ¯].

R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, A. J. Julian, “Submarine laser communication evaluation algorithm,” (Naval Ocean Systems Center, San Diego, Calif., May1981).

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 324.

G. N. Watson, A Treatise of the Theory Bessel Functions (Cambridge U. Press, Cambridge, 1945).

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Figures (4)

Fig. 1
Fig. 1

The tilt–centroid correlation coefficient as a function of the obscuration ratio .

Fig. 2
Fig. 2

Strehl ratio as a function of D/r0 for various values of the obscuration ratio.

Fig. 3
Fig. 3

The daytime and nighttime Navy/DARPA models for Cn2 versus altitude.

Fig. 4
Fig. 4

Strehl ratio as a function of D/r0 for various values of the point-ahead angle and .

Tables (3)

Tables Icon

Table 1 R versus

Tables Icon

Table 2 1.8f() versus

Tables Icon

Table 3 1.8fα() versus

Equations (92)

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I rel = I ( 0 ) I D L ( 0 ) .
I rel = d 2 r exp [ - ½ D ( r ) ] K ( r ) ,
K ( r ) = d 2 R W ( R + ½ r ) W ( R - ½ r ) d 2 R W ( R ) 2 ,
W ( r ) = { 1 for D 2 r D 2 0 otherwise .
K ( x ) = 8 π 2 D 2 [ cos - 1 x - x ( 1 - x 2 ) 1 / 2 ] ,
D ( r 1 - r 2 ) = [ δ ϕ ( r 1 , t ) - δ ϕ ( r 2 , t ) ] 2 ,
δ ϕ ( r ) = ϕ ( r ; p ) - ϕ B ( r ; p ) ,
ϕ ( r ) = k r · θ t + δ ϕ ( r )
ϕ B ( r ) = k r · θ ^ t + δ ϕ ^ ( r ) .
ϕ ( r ) = p = 1 a p Z p ( r ) ,
δ ϕ ^ ( r ) = δ ϕ ( r ) .
D ( r ) = k 2 r 2 2 ( θ t - θ ^ t ) 2 ,
D ( r ) = k 2 r 2 2 ( θ t 2 + θ ^ t 2 - 2 θ t · θ t ) ,
I rel = d 2 r K ( r ) exp ( - ½ k 2 σ ϕ 2 r 2 ) ,
σ ϕ 2 = θ t 2 γ ,
γ = ½ [ 1 + θ ^ t 2 θ t 2 ] - ( θ ^ t 2 θ t 2 ) 1 / 2 R ,
R = θ t · θ ^ t ( θ t 2 ) 1 / 2 ( θ ^ t 2 ) 1 / 2
I rel 2 π K ( 0 ) k 2 σ ϕ 2 ,             σ ϕ 2 .
I rel 1 / 1 + [ k 2 σ ϕ 2 / 2 π K ( 0 ) ] .
I rel 1 / 1 + π 2 2 σ ϕ 2 ( λ / D ) 2 ( 1 - 2 ) .
θ t = r ϕ ( r ) W ( r ) d 2 r ½ k r 2 W ( r ) d 2 r .
θ t 2 = θ t 0 2 ( 1 - 4 ) 2 [ ( 1 + 23 / 3 ) - 4 Γ ( ¹⁷ / ) Γ ( ²⁹ / ) Γ ( ¹⁴ / ) F 2 1 ( , - ¹¹ / , 3 , 2 ) ] ,
θ t 0 2 = 14.35 ( D r 0 ) 5 / 3 ( 1 k D ) 2
θ ^ t = θ c ,
θ c = I ( θ ) θ d 2 θ I ( θ ) d 2 θ ,
θ c = ϕ ( r ) W ( r ) 2 d 2 r k W ( r ) 2 d 2 r .
θ c 2 = θ c 0 2 ( 1 - 2 ) 2 [ ( 1 + 11 / 3 - 2 2 Γ ( ¹⁷ / ) Γ ( ¹¹ / ) Γ ( 8 / 3 ) F 2 1 ( , - 5 / 6 , 2 , 2 ) ] ,
θ c 0 2 = 13.39 ( D r 0 ) 5 / 3 ( 1 k D ) 2 .
R = θ t · θ c ( θ t 2 θ c 2 ) 1 / 2 .
R = ( 11 / 6 ) 1 / 2 A 1 / ( A 2 A 3 ) 1 / 2 ,
A 1 = Γ ( ¹¹ / ) ( 1 + 17 / 3 ) Γ ( ¹⁷ / ) Γ ( ²³ / ) - 4 2 F 2 1 ( - , , 3 , 2 ) - 6 2 11 F 2 1 ( - ¹¹ / , , 2 , 2 ) ,
A 2 = Γ ( ¹⁴ / ) Γ ( ¹⁷ / ) Γ ( ²⁹ / ) ( 1 + 23 / 3 ) - 4 F 2 1 ( , - ¹¹ / , 3 , 2 ) ,
A 3 = Γ ( / ) ( 1 + 11 / 3 ) Γ ( ¹⁷ / ) Γ ( ¹¹ / ) - 2 2 F 2 1 ( , - , 2 , 2 ) ,
R 0 = 0 d x x - 11 / 3 J 1 ( x ) J 2 ( x ) [ 0 d x x - 14 / 3 J 2 2 ( x ) 0 d x x - 8 / 3 J 1 2 ( x ) ] 1 / 2 = ( 8 × 23 11 × 17 ) 1 / 2 = 0.9919.
I rel = 0 1 d x x M ( x ) exp [ - k 2 D 2 σ ϕ 2 2 x 2 ] ,
σ ϕ 2 = θ t 2 γ ,
γ = 1 / 2 ( 1 + θ c 2 θ t 2 ) - ( θ c 2 θ t 2 ) 1 / 2 R ,
M ( x ) = 16 π ( 1 - 2 ) 2 ( cos - 1 x - x ( 1 - x 2 ) 1 / 2 - [ 2 cos - 1 ( x / ) - x ( 2 - x 2 ) 1 / 2 ] H ( - x ) - π 2 H ( 1 + 2 - x ) + { α 2 - θ ¯ + 4 x 2 sin 2 α [ cot ( α - θ ¯ ) - cot α ] } × H ( x - 1 - 2 ) H ( 1 + 2 - x ) ) ,
θ ¯ = cos - 1 ( 1 + 4 x 2 - 2 4 x ) ,
α = cos - 1 ( 1 - 4 x 2 - 2 4 x ) ,
H ( x ) = { 1 , x 0 0 , x < 0 .
I rel 1 / 1 + 1.8 ( D / r 0 ) 5 / 3 f ( ) ,
f ( ) = γ ( 1 - 2 ) 0 d x x - 14 / 3 [ J 2 ( x ) - 2 J 2 ( x ) ] 2 ( 1 - 4 ) 2 0 d x x - 14 / 3 J 2 2 ( x ) ,
I rel 1 / 1 + 0.0150 ( D / r 0 ) 5 / 3 .
D ( x , y ) = k 2 { x 2 [ θ t ( p ) - θ c ( p ) ] x 2 + y 2 [ θ t ( p ) - θ c ( p ) ] y 2 } .
I rel = d x d y K ( x , y ) exp [ - ( 1 / 2 ) k 2 ( σ ϕ x 2 x 2 + σ ϕ y 2 y 2 ) ] ,
σ ϕ x , y 2 = θ t 2 γ x , y ,
γ x , y = 1 / 2 ( 1 + θ c 2 θ t 2 ) - ( θ c 2 θ t 2 ) 1 / 2 R x , y ,
R x , y = θ t · θ c x , y ( θ t 2 x , y ) 1 / 2 ( θ c 2 x , y ) 1 / 2
I rel = 0 1 d x x M ( x ) exp [ - k 2 D 2 4 θ t 2 ( γ x + γ y ) x 2 ] × I 0 [ k 2 D 2 4 θ t 2 ( γ x - γ y ) x 2 ] ,
R x , y ( α ) = R { 0 d x x - 11 / 3 F x , y ( x ) [ J 1 ( x ) - J 1 ( x ) ] [ J 2 ( x ) - 2 J 2 ( x ) ] I 0 d x x - 11 / 3 [ J 1 ( x ) - J 1 ( x ) ] [ J 2 ( x ) - 2 J 2 ( x ) ] } ,
F x , y = path [ J 0 ( 2 x α z D ) J 2 ( 2 x α z D ) ] C n 2 ( z ) d z ,
I = path C n 2 ( z ) d z .
I rel 1 / 1 + 1.8 ( D / r 0 ) 5 / 3 f ( α ) ,
f ( α ) = [ γ x ( α ) γ y ( α ) ] 1 / 2 ( 1 - 2 ) × 0 d x x - 14 / 3 [ J 2 ( x ) - 2 J 2 ( x ) ] 2 ( 1 - 4 ) 2 0 d x x - 14 / 3 J 2 2 ( x )
θ t 2 = A t - 2 d 2 r 1 d 2 r 2 r 1 · r 2 B ϕ ( r 1 - r 2 ) W ( r 1 ) W ( r 2 ) ,
B ϕ ( r 1 - r 2 ) = ϕ ( r 1 ) ϕ ( r 2 )
B ϕ ( r ) = d 2 q ( 2 π ) 2 P ϕ ( q ) exp ( i q · r )
r W ( r ) = d 2 q ( 2 π ) 2 H t ( q ) exp ( i q · r ) ,
P ϕ ( q ) = d 2 r B ϕ ( r ) exp ( - i q · r )
H t ( q ) = d 2 r r W ( r ) exp ( - i q · r ) ,
θ t 2 = A t - 2 d 2 q ( 2 π ) 2 P ϕ ( - q ) H t ( q ) 2 .
H t ( q ) = - i π D 2 u ^ 2 q [ J 2 ( q D / 2 ) - 2 J 2 ( q D / 2 ) ] ,
u ^ = q / q
P ϕ ( q ) = 8.186 k 2 I q - 11 / 3 ,
I = path C n 2 ( z ) d z
r 0 = ( 0.423 k 2 I ) - 3 / 5 ,
θ t 2 = θ t 0 2 ( 1 - 4 ) 2 0 d x x - 14 / 3 [ J 2 ( x ) - 2 J 2 ( x ) ] 2 0 d x x - 14 / 3 J 2 2 ( x ) ,
0 t - λ J μ ( a t ) J ν ( b t ) d t = ( b / a ) ν ( a / 2 ) λ - 1 Γ ( μ + ν - λ + 1 2 ) 2 Γ ( ν + 1 ) Γ ( μ - ν + λ + 1 2 ) × F 2 1 ( μ + ν - λ + 1 2 , ν - μ - λ + 1 2 , ν + 1 , b 2 a 2 ) , R ( μ + ν - λ + 1 ) > 0 ,             R ( λ ) > - 1 ,             0 < b < a ,
0 t - λ J μ ( a t ) J ν ( b t ) d t = ( a / b ) μ ( b / 2 ) λ - 1 Γ ( μ + ν - λ + 1 2 ) 2 Γ ( μ + 1 ) Γ ( ν - μ + λ + 1 2 ) × F 2 1 ( μ + ν - λ + 1 2 , μ - ν - λ + 1 2 , μ + 1 , a 2 b 2 ) , R ( μ + ν - λ + 1 ) > 0 ,             R ( λ ) > - 1 ,             0 < a < b ,
0 t - λ J μ ( a t ) J ν ( a t ) d t = ( a / 2 ) λ - 1 Γ ( λ ) Γ ( μ + ν - λ + 1 2 ) 2 Γ Γ ( ν - μ + λ + 1 2 ) Γ ( ν + μ + λ + 1 2 ) Γ ( μ - ν + λ + 1 2 ) , R ( μ + ν + 1 ) > R ( λ ) > 0 ,             a > 0.
θ c 2 = A c - 2 d 2 r 1 d 2 r 2 1 · 2 B ϕ ( r 1 - r 2 ) W ( r 1 ) W ( r 2 ) ,
θ c 2 = A c - 2 d 2 q ( 2 π ) 2 q 2 P ϕ ( - q ) H c ( q ) 2 ,
H c ( q ) = d 2 r W ( r ) 2 exp ( - i q · r ) ,
H c ( q ) = π D q [ J 1 ( q D / 2 ) - J 1 ( q D / 2 ) ] .
θ c 2 = θ c 0 2 ( 1 - 2 ) 2 { 0 d x x - 8 / 3 [ J 1 ( x ) - J 1 ( x ) ] 2 0 d x x - 8 / 3 J 1 2 ( x ) } ,
θ t · θ c = ( A t A c ) - 1 d 2 r 1 d 2 r 2 W ( r 1 ) W ( r 2 ) 2 × r 1 · 2 [ B ϕ ( r 1 - r 2 ) ] .
θ t · θ c = ( A t A c ) - 1 d 2 q ( 2 π ) 2 P ϕ ( - q ) q · H t ( q ) H c ( q ) .
R = d 2 q P ϕ ( - q ) q · H t ( q ) H c ( q ) [ d 2 q P ϕ ( - q ) q 2 H c ( q ) 2 d 2 q P ϕ ( - q ) H t ( q ) 2 ] 1 / 2 .
R = R 0 ( 0 d x x - 11 / 3 [ J 1 ( x ) - J 1 ( x ) ] [ J 2 ( x ) - 2 J 2 ( x ) ] { 0 d x x - 14 / 3 [ J 2 ( x ) - 2 J 2 ( x ) ] 2 0 d x x - 8 / 3 [ J 1 ( x ) - J 1 ( x ) ] 2 } ) .
θ t · θ c = ( A t A c ) - 1 d 2 r 1 d 2 r 2 W ( r 1 ) W ( r 2 ) 2 × r 1 · 2 [ B ϕ ( r 1 - r 2 ; p - p ) ] ,
P ϕ ( q ; α ) = 8.186 k 2 q - 11 / 3 path d z C n 2 ( z ) exp ( i z q · α ) .
θ t · θ c x , y = constant d z C n 2 ( z ) G ( z ) x , y ,
G ( z ) x , y = 0 d q q - 11 / 3 J 1 ( D q / 2 ) J 2 ( D q / 2 ) × 0 2 π d ϕ ( cos 2 ϕ sin 2 ϕ ) exp ( i z q α cos ϕ ) = π d q q - 11 / 3 J 1 ( D q / 2 ) J 2 ( D q / 2 ) × [ J 0 ( q α z ) J 2 ( q α z ) ] ,
J 0 ( x ) + J 2 ( x ) = 2 J 1 ( x ) / x ,
J 0 ( x ) - J 2 ( x ) = 2 J 1 ( x ) x - 2 J 2 ( x ) ,
J ν ( a x ) J ν ( b x ) x ν = [ a b 2 ] ν Γ ( ν + 1 / 2 ) Γ ( 1 / 2 ) 0 π J ν ( ω x ) sin 2 ν ϕ d ϕ ω ν ,
ω = ( a 2 + b 2 - 2 a b cos ϕ ) 1 / 2 .
R x , y ( α ) = R 0 C n 2 ( h ) F x , y ( h ) d h I 0 1 ρ 8 / 3 M ( ρ ) d ρ ,
F x ( α ) = 2 0 1 ρ M ( ρ ) { ρ 2 ( ρ + η ) 1 / 3 [ F 2 1 ( , ½ , 1 , z ) - ½ F 2 1 ( , ³ / , 3 , z ) ] - ( ρ η ) 2 6 ( ρ + η ) 1 / 3 } d ρ ,
F y ( α ) = 0 1 ρ 3 M ( ρ ) ( ρ + η ) 1 / 3 F 2 1 ( , ³ / , 3 , z ) d ρ ,
η = h α D , z = ρ η ( ρ + η ) 2 ,

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