Abstract

A new method of evaluating the MTF of a Kolmogorov turbulence phase aberration Zernike modal compensating system using an approximation of the correlation of the residual phase aberration and the compensating phase distribution is described. This approach enables the extension of our previous results to cases including high-order terms in the compensating phase function. The Karhunen-Loeve expansion is discussed along with its relation to the Zernike polynomial expansion. Calculation of the optical resolution, or correlation quality, and the relative structure content, shows the Karhunen-Loeve expansion to be noticeably better than the Zernike expansion when terms through third and fifth order are included in the compensating phase distribution.

© 1978 Optical Society of America

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References

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  1. J. Y. Wang, “Optical resolution through a turbulent medium with adaptive phase compensations,” J. Opt. Soc. Am. 67, 383 (1977).
    [Crossref]
  2. 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]
  3. D. L. Fried, “How many pictures do you take to get a good one?” (unpublished); also “Theoretical Study of Non-Standard Imaging Concepts,” , Vols. I and II, Rome Air Development Center, Griffiss Air Force Base, New York (July1975).
  4. The expression of R/Rmax in Eq. (32) of Ref. 1 is slightly in error, Eq. (7) is the correct expression.
  5. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
    [Crossref]
  6. E. H. Linfoot, “Transmission factors and optical design,” J. Opt. Soc. Am. 46, 740 (1956).
    [Crossref]
  7. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 464.
  8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207 (1976).
    [Crossref]
  9. The expression in Eq. (15) for the condition i−i′ even was used in Ref. 8 to represent results for both conditions i− i′ even and i− i′ odd.
  10. D. L. Fried, “Required Number of Degrees-of-Freedom for an Adaptive Optics System,” Optical Science Consultants TR-191 (Oct.1975) (unpublished).
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 461.

1977 (1)

1976 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[Crossref]

1966 (1)

1956 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 464.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 461.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[Crossref]

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]

D. L. Fried, “How many pictures do you take to get a good one?” (unpublished); also “Theoretical Study of Non-Standard Imaging Concepts,” , Vols. I and II, Rome Air Development Center, Griffiss Air Force Base, New York (July1975).

D. L. Fried, “Required Number of Degrees-of-Freedom for an Adaptive Optics System,” Optical Science Consultants TR-191 (Oct.1975) (unpublished).

Linfoot, E. H.

Noll, R. J.

Wang, J. Y.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 461.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 464.

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[Crossref]

Other (6)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 464.

The expression in Eq. (15) for the condition i−i′ even was used in Ref. 8 to represent results for both conditions i− i′ even and i− i′ odd.

D. L. Fried, “Required Number of Degrees-of-Freedom for an Adaptive Optics System,” Optical Science Consultants TR-191 (Oct.1975) (unpublished).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 461.

D. L. Fried, “How many pictures do you take to get a good one?” (unpublished); also “Theoretical Study of Non-Standard Imaging Concepts,” , Vols. I and II, Rome Air Development Center, Griffiss Air Force Base, New York (July1975).

The expression of R/Rmax in Eq. (32) of Ref. 1 is slightly in error, Eq. (7) is the correct expression.

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Figures (6)

FIG. 1
FIG. 1

MTF versus spatial frequency for several values of the normalized lens diameter D/r0 with corrections through coma.

FIG. 2
FIG. 2

MTF versus spatial frequency for several orders of modal corrections. Solid curves—Zernike polynomials; dashed curves—Karhunen-Loeve functions.

FIG. 3
FIG. 3

Dependence of normalized resolution on normalized lens diameter for D ≫ (λz)1/2 (Zernike polynomials).

FIG. 4
FIG. 4

Dependence of relative structure content on normalized lens diameter for D ≫ (λz)1/2. Solid curves—Zernike polynomials; dashed curves—Karhunen-Loeve functions.

FIG. 5
FIG. 5

The first five mode radial functions (a) Zernike polynomials, (b) Karhunen-Loeve functions.

FIG. 6
FIG. 6

Dependence of normalized resolution on normalized lens diameter for D ≫ (λz)1/2 (Karhunen-Loeve functions).

Tables (4)

Tables Icon

TABLE I Relation among Zernike polynomial indices.

Tables Icon

TABLE II(a) Correlations of Zernike polynomial expansion coefficients of largest magnitude.

Tables Icon

TABLE II(b) Behavior of the correlation of Zernike polynomial expansion coefficients as the order differential increases for n = 1.

Tables Icon

TABLE III List of expansion coefficients of Karhunen-Loeve functions in terms of Zernike functions for the first 20 modes.

Equations (64)

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τ ( ρ ) = G ( r + ½ ρ ) G * ( r - ½ ρ ) × W ( r + ½ ρ ) W ( r - ½ ρ ) d 2 r ,
W ( r ) = { 1 r D / 2 0 r > D / 2.
G ( r + ½ ρ ) G * ( r - ½ ρ ) = exp { l ( r + ½ ρ ) + j [ φ ( r + ½ ρ ) - Φ ( r + ½ ρ ) ] + l ( r - ½ ρ ) - j [ φ ( r - ½ ρ ) - Φ ( r - ½ ρ ) ] } ,
G ( r + ½ ρ ) G * ( r - ½ ρ ) = exp { - ½ D ( ρ ) - ½ [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] 2 + [ φ ( r + ½ ρ ) - φ ( r - ½ ρ ) ] × [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] } ,
D ( ρ ) = 6.88 ( ρ / r 0 ) 5 / 3 ,
r 0 = 1.68 ( C n 2 z k 2 ) - 3 / 5 ,
R R max = 8 ( D r 0 ) 2 0 1 τ ( ρ ) ρ d ρ .
T = τ ( ρ ) 2 d 2 ρ τ 0 ( ρ ) 2 d 2 ρ ,
F ( r ) even i = [ 8 ( n + 1 ) ] 1 / 2 R n m ( r ) cos m θ , F ( r ) odd i = [ 8 ( n + 1 ) ] 1 / 2 R n m ( r ) sin m θ , ]             m 0 F ( r ) i = [ 4 ( n + 1 ) ] 1 / 2 R n 0 ( r ) ,             m = 0
R n m ( r ) = s = 0 ( n - m ) / 2 × ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! ( 2 r D ) n - 2 s .
φ ( r ) = i = 1 a i F i ( r ) ,
a i = d 2 r W ( r ) φ ( r ) F i ( r ) .
d 2 r W ( r ) F i ( r ) F i ( r ) = π D 2 δ i i ,
Φ ( r ) = i = 2 N a i F i ( r ) .
a i a i = { 0.0072 ( D r 0 ) 5 / 3 ( - 1 ) ( n + n - 2 m ) / 2 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 π 8 / 3 δ m m × Γ ( 14 / 3 ) Γ [ ( n + n - 5 / 3 ) / 2 ] Γ [ ( n - n + 17 / 3 ) / 2 ] Γ [ ( n - n + 17 / 3 ) / 2 ] Γ [ ( n + n + 23 / 3 ) / 2 ] , for i - i even 0 , for i - i odd
C ( ρ , r ) [ φ ( r + ½ ρ ) - φ ( r - ½ ρ ) ] × [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] = ½ i = 2 N [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] d 2 x W ( x ) F i ( x ) × [ D φ ( r - ½ ρ - x ) - D φ ( r + ½ ρ - x ) ] ,
D φ ( ρ ) = [ φ ( r + ½ ρ ) - φ ( r - ½ ρ ) ] 2 .
P i ± ( ρ ; r ) ½ [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] × d 2 x F i ( x ) D φ ( r ± ½ ρ - x ) ,
τ ( ρ ) = 4 D 2 exp [ - ½ D ( ρ ) ] 0 π / 2 d θ 0 L ( θ ) r d r × exp [ - ½ Q ( ρ ; r ) + i = 2 N P i - ( ρ ; r ) - i = 2 N P i + ( ρ ; r ) ] ,
Q ( ρ ; r ) = [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] 2 = i = 2 N i = 2 N a i a j [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] × [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ]
L ( θ ) = - 1 2 ( ρ D ) cos ( θ - ϕ ) + 1 2 ( ρ D ) [ ( ρ D ) - 2 - sin 2 ( θ - ϕ ) ] 1 / 2 .
E ( ρ ; r ) - ½ D ( ρ ) - ½ [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] 2 + [ φ ( r + ½ ρ ) - φ ( r - ½ ρ ) ] [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] = - ½ D ( ρ ) + ½ [ Φ ( r + ½ ρ ) - Φ ( r - ½ ρ ) ] 2 + i = 2 N i = N + 1 a i a i [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] × [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] .
τ ( ρ ) = 4 D 2 exp [ - ½ D ( ρ ) ] 0 π / 2 d θ 0 L ( θ ) r d r × exp { ½ Q ( ρ ; r ) + i = 2 N i = N + 1 i max a i a i [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] } .
ϕ ( r ) = φ ( r ) - φ ( r ) ,
φ ( r ) = ( π D 2 4 ) - 1 W ( r ) φ ( r ) d 2 r .
ϕ ( r ) = i β i G i ( r ) ,
β i β i * = Λ i 2 δ i i .
d 2 r W ( r ) G i * ( r ) G i ( r ) = π D 2 δ i i ,
β i = d 2 r W ( r ) G i * ( r ) ϕ ( r ) .
d 2 r W ( r ) G i ( r ) ϕ ( r ) ϕ ( r ) = Λ i 2 G i ( r ) .
C ϕ ( r - r ) = [ φ ( r ) - φ ] [ φ ( r ) - φ ] = - ½ D φ ( r - r ) + 1 2 ( π D 2 4 ) - 1 d 2 r 1 W ( r 1 ) × D φ ( r - r 1 ) + 1 2 ( π D 2 4 ) - 1 d 2 r 2 W ( r 2 ) D φ ( r - r 2 ) - 1 2 ( π D 2 4 ) - 2 d 2 r 1 d 2 r 2 W ( r 1 ) W ( r 2 ) × D φ ( r 1 - r 2 ) ,
G i ( r ) K p q ( r ) e j q θ ,
K p q ( r ) e j q θ Λ i 2 = 0 D / 2 K p q ( r ) r d r × 0 2 π d θ C ϕ ( r , r , θ - θ ) e j q θ = e j q θ 0 D / 2 K p q ( r ) r d r 0 2 π d θ C ϕ ( r , r , θ ) e j q θ ,
R q ( r , r ) r 0 2 π d θ C ϕ ( r , r , θ ) e j q θ ,
0 D / 2 R q ( r , r ) K p q ( r ) d r = Λ p q 2 K p q ( r ) .
Λ p q 2 ( D r 0 ) 5 / 3 λ p q 2 .
0 1 / 2 R q ( x , x ) K p q ( x ) d x = λ p q 2 K p q ( x ) ,
R 0 ( x , x ) = - x 0 2 π d θ B 0 ( [ x 2 + x 2 - 2 x x cos θ ] 1 / 2 ) + 2 π x [ B 1 ( x ) + B 1 ( x ) - B 2 ] ,
R q ( x , x ) = - x 0 2 π d θ B 0 ( [ x 2 + x 2 - 2 x x cos θ ] 1 / 2 ) × cos ( q θ ) , q = ± 1 , ± 2 ,
B 0 ( x ) = 3.44 x 5 / 3 ,
B 1 ( x ) = 3.44 ( 4 π ) 0 1 / 2 x d x 0 2 π d θ ( x 2 + x 2 - 2 x x cos θ ) 5 / 6 ,
B 2 = 8 0 1 / 2 x d x B 1 ( x ) .
0 1 / 2 K p q ( x ) K p q ( x ) x d x = { 1 if q 0 and p = p 1 / 2 if q = 0 and p = p 0 if p p .
K p ± q ( x ) = n = m C n m R n m ( x ) ,
n = m ( C n m ) 2 = 1.
τ ( ρ ) = d 2 r W ( r + ½ ρ ) W ( r - ½ ρ ) exp { - ½ D ( ρ ) + ½ i β i 2 [ G i ( r + ½ ρ ) - G i ( r - ½ ρ ) ] 2 } .
ϕ ( r ) = φ ( r ) - φ ( r ) - α · r ,
α = ( π D 4 64 ) - 1 d 2 r W ( r ) r φ ( r ) .
S N = 2 π ( η e ) A ¯ s 2 D 2 d 2 ρ d 2 r W ( r + ½ ρ ) W ( r - ½ ρ ) × exp [ - 1 / 2 D ( ρ ) ] ,
S N = 2 π ( η e ) A ¯ s 2 D 2 d 2 ρ τ ( ρ ) ,
S / N SNR max = 8 ( D r 0 ) 2 0 1 τ ( ρ ) ρ d ρ ,
S . R . = 0 1 τ ( ρ ) ρ d ρ 0 1 τ 0 ( ρ ) ρ d ρ ,
S . R . = R / R max 8 ( D / r 0 ) 2 0 1 τ 0 ( ρ ) ρ d ρ .
P i ± ( ρ ; r ) ½ [ F i ( r + ½ ρ ) - F i ( r - ½ ρ ) ] d 2 x × W ( x ) F i ( x ) D φ ( r + ½ ρ - x ) .
y ± - r ½ ρ + x = ξ cos δ i ˆ + ξ sin δ j ˆ .
P 7 ± ( ρ ; r ) = 1024 A π ( { 3 [ ( r D ) 2 + 1 4 ( ρ D ) 2 ] - 1 2 } sin ϕ + 6 ( r D ) 2 cos ( θ - ϕ ) sin θ ) × 0 2 π d δ ( { [ 3 ( r D ) 2 + 3 4 ( ρ D ) 2 ± 3 ( r D ) ( ρ D ) cos ( θ - ϕ ) - 1 2 ] ( r D ) sin θ + [ 3 2 ( ρ D ) ( r D ) cos ( θ - ϕ ) ± 3 2 ( r D ) 2 ± 3 8 ( ρ D ) 2 1 4 ] ( ρ D ) sin ϕ } H 1 ± + { [ 6 ( r D ) cos ( θ - δ ) ± 3 ( ρ D ) cos ( ϕ - δ ) ] ( r D ) sin θ + [ 3 2 ( ρ D ) cos ( ϕ - δ ) ± 3 ( r D ) cos ( θ - δ ) ] ( ρ D ) sin ϕ + [ 3 ( r D ) 2 + 3 4 ( ρ D ) 2 ± 3 ( r D ) ( ρ D ) - 1 2 ] sin δ } H 2 ± + { 3 ( r D ) sin θ ± 3 2 ( ρ D ) sin ϕ + [ 6 ( r D ) cos ( θ - δ ) ± 3 ( ρ D ) cos ( ϕ - δ ) ] sin δ } H 3 ± + 3 sin δ H 4 ± ) ,
P 8 ± ( ρ ; r ) = same as P 7 ± ( ρ ; r ) except replacing sin θ by cos θ , sin ϕ by cos ϕ , and sin δ by cos δ ,
P 9 ± ( ρ ; r ) = 1024 A π [ - 3 ( r D ) 2 sin ( 2 θ + ϕ ) - 1 4 ( ρ D ) 2 sin 3 ϕ ] × 0 2 π d δ { [ - ( r D ) 3 sin 3 θ 1 8 ( ρ D ) 3 sin 3 ϕ 3 2 ( r D ) 2 ( ρ D ) sin ( 2 θ + ϕ ) - 3 4 ( r D ) ( ρ D ) 2 sin ( 2 ϕ + θ ) ] H 1 ± + [ - 3 ( r D ) 2 sin ( 2 θ + δ ) - 3 4 ( ρ D ) 2 × sin ( 2 ϕ + δ * ) 3 ( r D ) ( ρ D ) sin ( θ + ϕ + δ ) ] H 2 ± + [ - 3 ( r D ) sin ( 2 δ + θ ) 3 2 ( ρ D ) 2 sin ( 2 δ + ϕ ) ] H 3 ± - sin 3 δ H 4 ± } ,
P 10 ± ( ρ ; r ) = same as P 9 ± ( ρ ; r ) except replacing sin ( sum of 3 angles ) by cos ( same sum of 3 angles ) ,
A = 6.88 ( D r 0 ) 5 / 3 ( ρ D ) ,
H n = ( η ± / D ) 8 / 3 + n 8 / 3 + n ,             n = 1 , 2 , 3 , 4 ,
η ± = - Δ ± cos ( δ - δ 0 ± ) + [ ¼ - ( Δ ± ) 2 sin 2 ( δ - δ 0 ± ) ] 1 / 2 ,
Δ ± = [ r 2 + ¼ ρ 2 ± ρ r cos ( θ - ϕ ) ] 1 / 2 ,
δ 0 ± = tan - 1 ( r sin θ ± ½ ρ sin ϕ r cos θ ± ½ ρ cos ϕ ) .