Abstract

There are many ways in which the paths of two waves through turbulence can become separated, thereby leading to anisoplanatic effects. Among these are a parallel path separation, an angular separation, one caused by a time delay, and one that is due to differential refraction at two wavelengths. All these effects can be treated in the same manner. Gegenbauer polynomials are used to obtain an approximation for the Strehl ratio for these anisoplanatic effects, yielding a greater range of applicability than the Maréchal approximation.

© 1992 Optical Society of America

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References

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  1. J. Belsher, D. Fried, “Chromatic refraction induced pseudo anisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1981).
  2. B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Co., Placentia, Calif., 1984).
  3. D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
    [CrossRef]
  4. D. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  5. D. Korff, G. Druden, R. P. Leavitt, “Isoplanicity: the translation invariance of the atmospheric Green’s function,” J. Opt. Soc. Am. 65, 1321–1330 (1975).
    [CrossRef]
  6. J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65–68 (1975).
    [CrossRef]
  7. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
    [CrossRef]
  8. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  9. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971).
  10. R. E. Hufnagel, Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).
  11. J. L. Bufton, P. O. Minott, M. W. Fitzmaurice, P. J. Titterton, “Measurements of turbulence profiles in the troposphere,” J. Opt. Soc. Am. 62, 1068–1070 (1972).
    [CrossRef]
  12. G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging system,” Appl. Opt. 19, 574–577 (1980).
    [CrossRef] [PubMed]
  13. M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” RADC-TR-79-131 (Rome Air Development Center, Griffiss Air Force Base, N.Y, 1979).
  14. D. P. Greenwood, “Bandwidth specifications for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–393 (1977).
    [CrossRef]
  15. D. L. Fried, “Time-delay-induced mean-square error in adaptive optics,” J. Opt. Soc. Am. A 7, 1224–1225 (1990).
    [CrossRef]
  16. C. W. Allen, Astrophysical Quantities (Athlone, London, 1963).
  17. A. E. Cole, A. Court, A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).
  18. G. Tyler, J. Belsher, D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1982).

1990 (1)

1984 (1)

1982 (1)

1980 (1)

1977 (1)

1975 (3)

1972 (1)

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Athlone, London, 1963).

Belsher, J.

G. Tyler, J. Belsher, D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1982).

J. Belsher, D. Fried, “Chromatic refraction induced pseudo anisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1981).

Bufton, J. L.

Cole, A. E.

A. E. Cole, A. Court, A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).

Court, A.

A. E. Cole, A. Court, A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).

Druden, G.

Ellerbroek, B. L.

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Co., Placentia, Calif., 1984).

Fitzmaurice, M. W.

Fried, D.

D. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

G. Tyler, J. Belsher, D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1982).

J. Belsher, D. Fried, “Chromatic refraction induced pseudo anisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1981).

Fried, D. L.

D. L. Fried, “Time-delay-induced mean-square error in adaptive optics,” J. Opt. Soc. Am. A 7, 1224–1225 (1990).
[CrossRef]

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Greenwood, D. P.

Hufnagel, R. E.

R. E. Hufnagel, Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Kantor, A. J.

A. E. Cole, A. Court, A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).

Korff, D.

Leavitt, R. P.

Miller, M. G.

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” RADC-TR-79-131 (Rome Air Development Center, Griffiss Air Force Base, N.Y, 1979).

Minott, P. O.

Roberts, P. H.

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Co., Placentia, Calif., 1984).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Shapiro, J. H.

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971).

Titterton, P. J.

Tyler, G.

G. Tyler, J. Belsher, D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1982).

Tyler, G. A.

Valley, G. C.

Zieske, P. L.

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” RADC-TR-79-131 (Rome Air Development Center, Griffiss Air Force Base, N.Y, 1979).

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Radio Sci. (1)

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[CrossRef]

Other (9)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971).

R. E. Hufnagel, Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

M. G. Miller, P. L. Zieske, “Turbulence environmental characterization,” RADC-TR-79-131 (Rome Air Development Center, Griffiss Air Force Base, N.Y, 1979).

C. W. Allen, Astrophysical Quantities (Athlone, London, 1963).

A. E. Cole, A. Court, A. J. Kantor, Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965).

G. Tyler, J. Belsher, D. Fried, “Amelioration of chromatic refraction induced pseudoanisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1982).

J. Belsher, D. Fried, “Chromatic refraction induced pseudo anisoplanatism,” (Optical Sciences Co., Placentia, Calif., 1981).

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Co., Placentia, Calif., 1984).

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Figures (12)

Fig. 1
Fig. 1

Comparison of the Maréchal and the two- to six-term approximations with the exact value of the Strehl ratio, for an anisoplanatic displacement, for D/r0 equal to 1.

Fig. 2
Fig. 2

Comparison of the Maréchal and the five- and six-term approximations with the exact value of the Strehl ratio, with an anisoplanatic displacement that is constant along the propagation path, for D/r0 equal to 5.

Fig. 3
Fig. 3

Comparison of the Maréchal and six-term approximations with the exact value of the Strehl ratio, with an anisoplanatic displacement, for D/r0 equal to 10.

Fig. 4
Fig. 4

Strehl ratio for angular anisoplanatic error at zenith, for various turbulence models, versus separation angle for a 0.6-m system. Upper-altitude turbulence has a strong effect on the Strehl ratio.

Fig. 5
Fig. 5

Strehl ratio for angular anisoplanatism at 30° from zenith for a 0.6-m system.

Fig. 6
Fig. 6

Strehl ratio versus time delay at zenith for a 0.6-m system.

Fig. 7
Fig. 7

Strehl ratio versus time delay for a 0.6-m system at 30° zenith angle.

Fig. 8
Fig. 8

Difference (×106) in refractive index between 0.5 μm and other wavelengths.

Fig. 9
Fig. 9

Normalized air density versus altitude.

Fig. 10
Fig. 10

Comparison of the Strehl ratios at infinity and in the 300-km range versus beacon wavelength. The two results are close in value. This permits a simpler calculation for finite-range problems.

Fig. 11
Fig. 11

Strehl ratio for SLCSAT-Day turbulence with the scoring beam at 0.5 μm for a 0.6-m system and zenith angles between 10° and 45°. Note that above the 45° elevation angle the Strehl ratio exceeds 0.7 for the first beam at 0.5 μm and the second at any longer wavelength.

Fig. 12
Fig. 12

Strehl ratios for displacement, angle, and time-delay anisoplanatism presented separately and combined for SLCSAT-Day turbulence model. The displacement is constant along the propagation path. The angle offset in microradihns is 850 times the displacement in meters. The time delay in milliseconds is 130 times the displacement in meters. The combined effects are for all displacements in line and at 120° to one another.

Tables (1)

Tables Icon

Table 1 Values of T2 and T5/3 to Solve for the Chromatic Displacement for Various Turbulence Models for a Wavelength of 0.5 μma

Equations (61)

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SR = 1 2 π d α K ( α ) exp [ - D ( α ) 2 ] .
K ( α ) = 16 π [ cos - 1 ( α ) - α ( 1 - α 2 ) 1 / 2 ] U ( 1 - α ) ,
U ( x ) = 1 for x 0 , U ( x ) = 0 for x < 0.
D ( α D ) = 2 ( 2.91 ) k 0 2 0 d z C n 2 ( z ) [ ( α D ) 5 / 3 + d 5 / 3 ( z ) - ½ α D + d ( z ) 5 / 3 - ½ α D - d ( z ) 5 / 3 ] ,
α D ± d ( z ) 5 / 3 = [ ( α D ) 2 ± 2 α D d ( z ) cos ( φ ) + d 2 ( z ) ] 5 / 6 ,
( 1 - 2 α x + a 2 ) - λ = p = 0 C p λ ( x ) a p .
( 1 - 2 a x + a 2 ) - 1 / 2 = p = 0 P p ( x ) a p .
C p λ [ cos ( φ ) ] = m = 0 p Γ [ λ + m ] Γ [ λ + p - m ] cos [ ( p - 2 m ) φ ] m ! ( p - m ) ! ( Γ [ λ ] ) 2 ,
C 2 - 5 / 6 [ cos ( φ ) ] = / [ 1 - cos 2 ( φ ) ] .
D ( α D ) = 2 ( 2.91 ) k 0 2 0 d z C n 2 ( z ) × { d 5 / 3 ( z ) - ( α D ) 5 / 3 p = 1 C 2 p - 5 / 6 [ cos ( φ ) ] [ d ( z ) α D ] 2 p } .
d m 2.91 k 0 2 0 d z C n 2 ( z ) d m ( z )
σ φ 2 = d 5 / 3 .
D ( α D ) = 2 σ φ 2 - 2 x ,
x = d 2 [ 1 - cos 2 ( φ ) ] / ( α D ) - 1 / 3 .
SR exp ( - σ φ 2 ) 2 π d α K ( α ) × ( 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 ) .
Φ ( n ) = 1 2 π 0 2 π d φ [ 1 - cos 2 ( φ ) ] n = 1 2 π m = 0 n ( n n - m ) 3 - m 0 2 π d φ cos 2 m ( φ ) ,
( n n - m ) = n ! ( n - m ) ! m ! .
0 π / 2 d φ cos 2 m ( φ ) = π ( 2 m - 1 ) ! ! 2 ( 2 m ) ! ! ,
( 2 m - 1 ) ! ! = ( 2 m - 1 ) ( 2 m - 3 ) ( 3 ) ( 1 ) ,
( 2 m ) ! ! = ( 2 m ) ( 2 m - 2 ) ( 4 ) ( 2 ) .
Φ ( n ) = 1 - m = 1 n ( n n - m ) 3 - m ( 2 m - 1 ) ! ! ( 2 m ) ! ! .
Y ( n ) = 0 1 d α α 1 - n / 3 K ( α ) .
Y ( n ) = 8 ( 2 - n / 3 ) π Γ [ - n / 6 + ³ / - n / 6 + 3 ]             for n < 6.
SR ( 1 + 0.9736 E + 0.5133 E 2 + 0.2009 E 3 + 0.0697 E 4 + 0.02744 E 5 ) exp ( - σ φ 2 ) ,
E = d 2 D 1 / 3 .
μ m = 0 C n 2 ( z ) z m d z = sec m + 1 ( ξ ) 0 C n 2 ( h ) h m d h ,
C n 2 ( h ) = 0.00594 ( W 27 ) 2 ( 10 - 5 h ) 10 exp ( - h 1000 ) + 2.7 × 10 - 16 exp ( - h 1500 ) + A exp ( - h 100 ) ,
C n 2 ( H ) = 3.96 × 10 - 13 / H 1.05 , 18.5 H 232 = 1.3 × 10 - 15 , 232 H 880 = 8.87 × 10 - 7 H - 3 , 880 H 7220 = 2.0 × 10 - 16 H - 0.5 , 7220 H 20 , 500 .
d ( z ) = d ,
d 2 = 2.91 k 0 2 μ 0 d 2 ,
E = 6.88 ( d D ) 2 ( D r 0 ) 5 / 3 ,
σ φ 2 = 2.91 k 0 2 μ 0 d 5 / 3 = 6.88 ( d r 0 ) 5 / 3 .
d ( z ) = θ z ,
d 2 = 2.91 k 0 2 μ 2 θ 2 ,
E = 6.88 μ 2 μ 0 ( θ D ) 2 ( D r 0 ) 5 / 3 ,
σ φ 2 = 2.91 k 0 2 θ 5 / 3 0 L d z C n 2 ( z ) z 5 / 3 = ( θ / θ 0 ) 5 / 3 ,
θ 0 = ( 2.19 k 0 2 μ 5 / 3 ) - 3 / 5 .
d ( z ) = ν ( z ) τ ,
d 2 = 2.91 k 0 2 0 L d z C n 2 ( z ) ν 2 ( z ) τ 2 = ( τ / τ 2 ) 2 ,
E = τ 2 τ 2 2 D 1 / 3 ,
σ φ 2 = 2.91 k 0 2 0 L d z C n 2 ( z ) ν 5 / 3 ( z ) τ 5 / 3 = ( τ / τ 5 / 3 ) 5 / 3 ,
1 / τ m 5 / 3 = 2.91 k 0 2 0 L d z C n 2 ( z ) ν m ( z ) .
f G τ 5 / 3 = [ 0.0175 n sin ( 5 π / 6 n ) ] 3 / 5 ,
f G = 0.254 k 0 6 / 5 ν 5 / 3 3 / 5 ,
ν ( h ) = ν g + 30 exp [ - ( h - 9400 4800 ) 2 ] ,
Δ ξ ( h ) = - Δ n ( h ) tan [ ξ ( h ) ] .
d c ( z ) = 0 z d z Δ ξ ( h ) = - ξ sin ( ξ ) Δ n 0 ξ cos 2 ( ξ ) 0 z d z α ( z ) ,
Δ n 0 = ( λ 1 2 - λ 2 2 ) [ 29 498.1 ( 146 λ 2 2 - 1 ) ( 146 λ 1 2 - 1 ) + 255.4 ( 41 λ 2 2 - 1 ) ( 41 λ 1 2 - 1 ] 10 - 6 .
α ( h ) = exp ( - 1.11 × 10 - 4 h ) , h < 10 km , α ( h ) = 1.6 exp ( - 1.57 × 10 - 4 h ) , h > 10 km .
d c ( z ) = - ξ sin ( ξ ) Δ n 0 ξ cos 2 ( ξ ) [ 0 z d z α ( z ) - z L 0 L d z α ( z ) ] .
I ( z ) = 0 z d z α ( z ) .
I ( z ) = 9010 [ 1 - exp ( - 1.11 × 10 - 4 z ) ] for z < 10 km , I ( z ) = 8161 - 10 , 190 exp ( - 1.57 × 10 - 4 z ) for z > 10 km .
d m = [ sin ( ξ ) Δ n 0 cos 2 ( ξ ) ] m T m ,
T m = 2.91 k 0 2 sec ( ξ ) 0 H d h C n 2 ( h ) [ I ( h ) - h sec ( ξ ) L I ( L ) ] m .
T m = 2.91 k 0 2 sec ( ξ ) 0 H d h C n 2 ( h ) I m ( h ) .
E = [ sin ( ξ ) Δ n 0 cos 2 ( ξ ) ] 2 T 2 D 1 / 3 ,
σ φ 2 = [ sin ( ξ ) Δ n 0 cos 2 ( ξ ) ] 5 / 3 T 5 / 3 .
d t ( z ) = d + θ z + ν ( z ) τ + d c ( z ) ,
E = d 2 D 1 / 3 ,
σ φ 2 = d 5 / 3 ,
d m = 2.91 k 0 2 0 d z C n 2 ( z ) d t ( z ) m .

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