Abstract

This paper discusses some general properties of Zernike polynomials, such as their Fourier transforms, integral representations, and derivatives. A Zernike representation of the Kolmogoroff spectrum of turbulence is given that provides a complete analytical description of the number of independent corrections required in a wave-front compensation system.

© 1976 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 9.2.
  2. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [Crossref]
  3. L. C. Bradley and J. Herrmann, Appl. Opt. 13, 331 (1974).
    [Crossref] [PubMed]
  4. S. N. Bezdid’ko, Sov. J. Opt. Tech. 41, 425 (1974).
  5. Defining the aperture weight function W(r) as shown allows the aperture weighted variance, σ2, of a phase function, ϕ, to be written as σ2=∫d2rW(r)ϕ2(r).
  6. Although Eq. (29) has not been proven to be an asymptotic form of Eq. (28), a graph of Eq. (28) yields a linear log plot for large J.Equation (29) represents a fit to such a plot.
  7. L. I. Golden, R. V. Shack, and P. N. Slater, NASA Final Report, (1974).
  8. D. L. Fried (private communication).

1974 (2)

L. C. Bradley and J. Herrmann, Appl. Opt. 13, 331 (1974).
[Crossref] [PubMed]

S. N. Bezdid’ko, Sov. J. Opt. Tech. 41, 425 (1974).

1965 (1)

Bezdid’ko, S. N.

S. N. Bezdid’ko, Sov. J. Opt. Tech. 41, 425 (1974).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 9.2.

Bradley, L. C.

Fried, D. L.

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

D. L. Fried (private communication).

Golden, L. I.

L. I. Golden, R. V. Shack, and P. N. Slater, NASA Final Report, (1974).

Herrmann, J.

Shack, R. V.

L. I. Golden, R. V. Shack, and P. N. Slater, NASA Final Report, (1974).

Slater, P. N.

L. I. Golden, R. V. Shack, and P. N. Slater, NASA Final Report, (1974).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 9.2.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Sov. J. Opt. Tech. (1)

S. N. Bezdid’ko, Sov. J. Opt. Tech. 41, 425 (1974).

Other (5)

Defining the aperture weight function W(r) as shown allows the aperture weighted variance, σ2, of a phase function, ϕ, to be written as σ2=∫d2rW(r)ϕ2(r).

Although Eq. (29) has not been proven to be an asymptotic form of Eq. (28), a graph of Eq. (28) yields a linear log plot for large J.Equation (29) represents a fit to such a plot.

L. I. Golden, R. V. Shack, and P. N. Slater, NASA Final Report, (1974).

D. L. Fried (private communication).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sec. 9.2.

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Tables (4)

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TABLE I Zernike polynomials. The modes, Zj, are ordered such that even j corresponds to the symmetric modes defined by cos, while odd j corresponds to the antisymmetric modes given by sin. For a given n, modes with a lower value of m are ordered first.

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TABLE II Zernike polynomial derivative matrix: γ j j x.

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TABLE III Zernike polynomial derivative matrix: γ j j y.

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TABLE IV Zernike-Kolmogoroff residual errors (ΔJ). (D is the aperture diameter.)

Equations (37)

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Z even j = n + 1 R n m ( r ) 2 cos m θ Z odd j = n + 1 R n m ( r ) 2 sin m θ } m 0 Z j = n + 1 R n o ( r ) , m = 0
R n m ( r ) = s = 0 ( n - m ) ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! r n - 2 s .
d 2 r W ( r ) Z j Z j = δ j j ,
W ( r ) = 1 / π r 1 = 0 r > 1.
ϕ ( R ρ , θ ) = j a j Z j ( ρ , θ ) ,
a j = d 2 ρ W ( ρ ) ϕ ( R ρ , θ ) Z j ( ρ , θ )
a j = ( 1 / R 2 ) d 2 r W ( r / R ) ϕ ( r , θ ) Z j ( r / R , θ ) .
W ( ρ ) Z j ( ρ , θ ) = d 2 k Q j ( k , ϕ ) e - 2 π i k · ρ .
Q even j ( k , ϕ ) = Q odd j ( k , ϕ ) = Q j ( k , ϕ ) = } n + 1 J n + 1 ( 2 π k ) π k { ( - 1 ) ( n - m ) / 2 i m 2 cos m ϕ , ( - 1 ) ( n - m ) / 2 i m 2 sin m ϕ , ( - 1 ) n / 2 ,             ( m = 0 )
R n m ( ρ ) = 2 π ( - 1 ) ( n - m ) / 2 0 d k J n + 1 ( 2 π k ) J m ( 2 π k ρ ) .
( d / d ρ ) R n m = 2 π ( - 1 ) ( n - m ) / 2 × 0 d k J n + 1 ( 2 π k ) d J m ( 2 π k ρ ) d ρ .
d J l / d x = 1 2 [ J l - 1 ( x ) - J l + 1 ( x ) ]
x J l + 1 ( x ) = 2 l J l ( x ) - x J l - 1 ( x ) ,
( d / d ρ ) R n m = n [ R n - 1 m - 1 + R n - 1 m + 1 ] + ( d / d ρ ) R n - 2 m .
Z j = j γ j j , Z j .
γ j j x = d 2 ρ Z j d Z j d x
γ j j y = d 2 ρ Z j d Z j d y .
D ϕ ( r ) = 2 [ ϕ 2 ( r 1 ) - ϕ ( r 1 ) ϕ ( r 1 + r ) ] .
D ϕ ( r ) = 6.88 ( r / r 0 ) 5 / 3 .
D ϕ ( r ) = 2 d k Φ ( k ) [ 1 - cos ( 2 π k · r ) ] .
0 x - P [ 1 - J 0 ( b x ) ] d x = π b P - 1 2 P [ Γ ( P + 1 ) / 2 ] 2 sin [ π ( P - 1 ) / 2 ] ,
Φ ( k ) = ( 0.023 / r 0 5 / 3 ) k - 11 / 3 ,
a j * a j = d ρ d ρ W ( ρ ) W ( ρ ) Z j ( ρ , θ ) × C ( R ρ , R ρ ) Z j ( ρ , θ ) ,
C ( R ρ , R ρ ) = ϕ ( R ρ ) ϕ ( R ρ ) .
a j * a j = d k d k Q j * ( k ) Φ ( k / R , k / R ) Q j ( k ) ,
Φ ( k / R , k / R ) = 0.023 ( R / r 0 ) 5 / 3 k - 11 / 3 δ ( k - k ) .
a j * a j = ( 0.046 / π ) ( R / r 0 ) 5 / 3 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 × ( - 1 ) ( n + n - 2 n ) / 2 δ m m × d k k - 8 / 3 J n + 1 ( 2 π k ) J n + 1 ( 2 π k ) k 2 ,
ϕ C = j = 1 J a j Z j .
Δ = d ρ W ( ρ ) [ ϕ ( R ρ ) - ϕ C ( R ρ ) ] 2 .
Δ J = ϕ 2 - j = 1 J a j 2 ,
Δ J 0.2944 J - 3 / 2 ( D / r 0 ) 5 / 3 [ rad 2 ] .
I n n = 0 d k k - 8 / 3 J n + 1 ( k ) J n + 1 ( k ) k 2 .
I n n = Γ ( 14 3 ) Γ [ ( n + n - 14 3 + 3 ) / 2 ] 2 14 / 3 Γ [ ( - n + n + 14 3 + 1 ) / 2 ] Γ [ ( n - n + 14 3 + 1 ) / 2 ] Γ [ ( n + n + 14 3 + 3 ) / 2 ]
0 { 1 - [ 4 J 1 2 ( x ) ] / x 2 } x - P d x
0 d x J 1 ( x ) 2 x q = Γ [ ( 3 - q ) / 2 ] Γ ( q ) 2 q Γ [ ( q + 1 ) / 2 ] Γ [ ( q + 1 ) / 2 ] Γ [ ( q + 3 ) / 2 ] = F 1 ( q )             0 < q < 3.
0 ( 1 - 4 J 1 2 ( x ) x 2 ) x - P d x = π Γ ( P + 2 ) 2 P { Γ [ ( P + 3 ) / 2 ] } 2 Γ [ ( P + 5 ) / 2 ] Γ [ ( 1 + P / 2 ) ] sin [ ( π / 2 ) ( P - 1 ) ] .
σ2=d2rW(r)ϕ2(r).