Abstract

The image degradation due to Kolmogorov atmospheric turbulence is considered in terms of the time-averaged optical transfer function, point spread function, Strehl ratio, and encircled energy for imaging systems with annular pupils. The applications include imaging with mirror telescopes and propagation of obscured laser beams through turbulence. Numerical results are given for obscuration ratios of 0, 0.25, 0.50, and 0.75.

© 1981 Optical Society of America

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References

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  1. I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
    [CrossRef]
  2. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  3. D. M. Chase, J. Opt. Soc. Am. 56, 33 (1966).
    [CrossRef]
  4. D. L. Fried, Appl. Opt. 13, 2620 (1974).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.5.2.
  6. R. E. Hufnagel, N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [CrossRef]
  7. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966). Here the notation R∞/Rmax is used in place of η.
    [CrossRef]
  8. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Eq. (8.20).
  9. Ref. 5, Sec. 10.4.1.
  10. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971) (see Ref. 14 also).
    [CrossRef] [PubMed]
  11. V. N. Mahajan, Appl. Opt. 17, 3329 (1978).
    [CrossRef] [PubMed]
  12. H. F. Willis, Philos. Mag. 39455 (1948), Eq. (e), p. 457.
  13. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  14. H. T. Yura, AppL Opt. 10, 2771 (1971).
    [CrossRef] [PubMed]

1978 (1)

1974 (1)

1971 (2)

1967 (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

1966 (2)

1965 (2)

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
[CrossRef]

1964 (1)

1948 (1)

H. F. Willis, Philos. Mag. 39455 (1948), Eq. (e), p. 457.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.5.2.

Chabot, A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Chase, D. M.

Fried, D. L.

Goldstein, I.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Hufnagel, R. E.

Lutomirski, R. F.

Mahajan, V. N.

Miles, P. A.

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Eq. (8.20).

Willis, H. F.

H. F. Willis, Philos. Mag. 39455 (1948), Eq. (e), p. 457.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.5.2.

Yura, H. T.

AppL Opt. (1)

H. T. Yura, AppL Opt. 10, 2771 (1971).
[CrossRef] [PubMed]

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Philos. Mag. (1)

H. F. Willis, Philos. Mag. 39455 (1948), Eq. (e), p. 457.

Proc. IEEE (2)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

I. Goldstein, P. A. Miles, A. Chabot, Proc. IEEE 53, 1172 (1965).
[CrossRef]

Other (3)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Eq. (8.20).

Ref. 5, Sec. 10.4.1.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.5.2.

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Figures (4)

Fig. 1
Fig. 1

Variation of η(D/r0;) with D/r0.

Fig. 2
Fig. 2

Variation of Strehl ratio with D/r0. ΔS() is given by Eq. (31).

Fig. 3
Fig. 3

Encircled energy for various values of D/r0: (a) = 0; (b) = 0.25; (c) = 0.50; (d) = 0.75; v0 is measured in units of λR/D.

Fig. 4
Fig. 4

Energy in an Airy disk (v0 = 1.22) as a function of D/r0.

Tables (1)

Tables Icon

Table I Strehl Ratio for Various Values of and D/r0

Equations (33)

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τ ( ρ ) = τ l ( ρ ) τ a ( ρ ) ,
τ a ( ρ ) = exp [ ψ ( u ) + ψ * ( u - λ R ρ ) ] ,
ψ ( u ) = i Φ ( u ) + χ ( u ) .
τ a ( ρ ) = exp [ - 1 / 2 D ψ ( λ R ρ ) ] ,
D ψ ( u 1 - u 2 ) = ψ ( u 1 ) - ψ ( u 2 ) 2
D ψ ( r ) = 2.914 k 2 r 5 / 3 sec θ h 1 h 2 C n 2 ( h ) [ ( h 2 - h ) / ( h 2 - h 1 ) ] 5 / 3 d h ,
D ψ ( r ) = 6.88 ( r / r 0 ) 5 / 3 ,
r 0 = 0.1847 λ 1.2 { sec θ h 1 h 2 C n 2 ( h ) [ ( h 2 - h ) / ( h 2 - h 1 ) ] 5 / 3 d h } - 0.6
τ ( ρ ) = τ l ( ρ ) exp [ - 3.44 ( λ R ρ / r 0 ) 5 / 3 ] .
I ( v ) = τ ( ρ ) exp ( - 2 π i v · ρ ) d ρ .
I ( v ) = 2 π τ ( ρ ) J 0 ( 2 π v ρ ) ρ d ρ ,
τ ( ρ ) = τ l ( ρ ) exp [ - 3.44 ( λ R ρ / r 0 ) 5 / 3 ] .
E ( v 0 ) = 0 v 0 I ( v ) v d v / 0 I ( v ) v d v .
E ( v 0 ) = 2 π v 0 τ ( ρ ) J 1 ( 2 π v 0 ρ ) d ρ ,
τ ( ρ ; ) = τ l ( ρ ; ) τ a ( ρ ) ,
τ l ( ρ ; ) = τ l ( ρ ) + 2 τ l ( ρ / ) - τ 12 ( ρ ; ) ,
τ l ( ρ ) = 2 π ( 1 - 2 ) [ cos - 1 ρ - ρ ( 1 - ρ 2 ) 1 / 2 ] , 0 ρ 1 , = 0 , otherwise ,
τ 12 ( ρ , ) = 2 2 1 - 2 , 0 ρ 1 - 2 , = 2 2 1 - 2 [ 1 - 1 + 2 2 π 2 β - 1 π sin β + 1 - 2 π 2 tan - 1 ( 1 + 1 - tan β 2 ) ] , 1 - 2 ρ 1 + 2 , = 0 , otherwise ,
β = cos - 1 ( 1 + 2 - 4 ρ 2 2 ) .
τ l ( 0 ; ) = 1 ,
0 τ l ( ρ ; ) ρ d ρ = ( 1 - 2 ) / 8.
τ a ( ρ ) = exp [ - 3.44 ( ρ D / r 0 ) 5 / 3 ] .
I ( v ; ; D / r 0 ) = I ( 0 ; ) I n ( v ; ; D / r 0 )
I ( 0 ; ) = π ( 1 - 2 ) P D 2 / 4 λ 2 R 2
I n ( v ; ; D / r 0 ) = [ 8 / ( 1 - 2 ) ] 0 τ ( ρ ; ; D / r 0 ) J 0 ( 2 π v ρ ) ρ d ρ
τ ( ρ ; ; D / r 0 ) = τ l ( ρ ; ) exp [ - 3.44 ( ρ D / r 0 ) 5 / 3 ] .
E ( v 0 ; ; D / r 0 ) = 2 π v 0 τ ( ρ ; ; D / r 0 ) J 1 ( 2 π v 0 ρ ) d ρ .
η ( D / r 0 ; ) = ( 1 - 2 ) ( D / r 0 ) 2 I n ( 0 ; ; D / r 0 ) = 8 ( D / r 0 ) 2 0 τ ( ρ ; ; D / r 0 ) ρ d ρ
I ( 0 ; ; D / r 0 ) = ( π P r 0 2 / 4 λ 2 R 2 ) η ( D / r 0 ; ) .
I ( 0 ; ; D / r 0 ) max = π P r 0 2 / 4 λ 2 R 2 .
S ( D / r 0 ; ) = I n ( 0 ; ; D / r 0 ) = η ( D / r 0 ; ) / η ( 0 , )
= [ 8 / ( 1 - 2 ) ] 0 τ ( ρ ; ; D / r 0 ) ρ d ρ .
Δ S ( ) = 10 [ S ( D / r 0 ; 0 ) - S ( D / r 0 ; ) ]

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