Abstract

The transfer function of an annular aperture in the presence of spherical aberration and defocusing is evaluated. The technique employed is the sampling method developed in a previous paper. The Maréchal aberration-balancing theory is extended to annular apertures. Representative numerical results are discussed.

© 1965 Optical Society of America

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References

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  1. W. H. Steel, Rev. Opt. 32, 4 (1953).
  2. E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956).
    [Crossref]
  3. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
    [Crossref]
  4. Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).
  5. C. G. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).
  6. T. Asakura and R. Barakat, Oyo Butsuri 30, 728 (1961).
  7. Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902), Vol. 3, Article 148.
  8. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
    [Crossref]
  9. A. Maréchal, Rev. Opt. 26, 257 (1947).

1964 (1)

1963 (1)

1961 (1)

T. Asakura and R. Barakat, Oyo Butsuri 30, 728 (1961).

1956 (1)

1953 (1)

W. H. Steel, Rev. Opt. 32, 4 (1953).

1947 (1)

A. Maréchal, Rev. Opt. 26, 257 (1947).

Asakura, T.

T. Asakura and R. Barakat, Oyo Butsuri 30, 728 (1961).

Barakat, R.

Houston, A.

Kopal, Z.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

O’Neill, E. L.

Rayleigh,

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902), Vol. 3, Article 148.

Steel, W. H.

W. H. Steel, Rev. Opt. 32, 4 (1953).

Steward, C. G.

C. G. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).

J. Opt. Soc. Am. (3)

Oyo Butsuri (1)

T. Asakura and R. Barakat, Oyo Butsuri 30, 728 (1961).

Rev. Opt. (2)

W. H. Steel, Rev. Opt. 32, 4 (1953).

A. Maréchal, Rev. Opt. 26, 257 (1947).

Other (3)

Rayleigh, Collected Papers (Cambridge University Press, Cambridge, England, 1902), Vol. 3, Article 148.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

C. G. Steward, The Symmetrical Optical System (Cambridge University Press, Cambridge, England, 1928).

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Figures (4)

Fig. 1
Fig. 1

Transfer function for aberration-free annular aperture with obscuration =0.4 for amounts of defocusing corresponding to: (A) W2=0, (B) W2=0.25, (C) W2=0.50, (D) W2=0.75.

Fig. 2
Fig. 2

Transfer function for aberration-free annular aperture with obscuration =0.6 for amounts of defocusing corresponding to: (A) W2=0, (B) W2=0.25, (C) W2=0.50, (D) W2=0.75.

Fig. 3
Fig. 3

Transfer function for optimum-balanced third-order spherical aberration of amount W4=1.0 for obscurations: (A) =0.4, (B) =0.5, (C) =0.6, (D) =0.7, (E) =0.8.

Fig. 4
Fig. 4

Transfer function for optimum-balanced fifth-order spherical aberration of amount W6=3.0 for obscurations: (A) =0, (B) =0.3, (C) =0.5, (D) =0.7.

Tables (3)

Tables Icon

Table I Wavefront coefficient ratios for optimum-balanced fifth-order spherical aberration as a function of .

Tables Icon

Table II Strehl criterion of an annular aperture possessing third-order spherical aberration for: (A) optimum-balanced wavefront with W4 = 1.0, (B) regular wavefront with W4 = 1.0.

Tables Icon

Table III Strehl criterion of an annular aperture possessing fifth-order spherical aberration for: (A) optimum-balanced wavefront with W6 = 3.0, (B) regular wavefront with W6 = 3.0.

Equations (17)

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T ( ω ) = n = 1 1 J 1 2 ( α n ) t ( α n 2 ) J 0 ( α n ω 2 ) ,
t ( v ) = 4 ( 1 - 2 ) 2 | 1 e i k W ( ρ ) J 0 ( v ρ ) ρ d ρ | 2 ,
W ( ρ ) = W 2 ρ 2 + W 4 ρ 4 + W 6 ρ 6 + .
t ( v ) = J 0 2 ( v )
J 0 ( v ) ~ ( 2 / π v ) 1 2 cos [ v - ( π / 4 ) ] ,
t ( v ) ~ O ( v - 2 ) .
t ( v ) ~ O ( v - 2 + ) ,
E 0 = 2 ( 1 - 2 ) 1 [ W ( ρ ) ] 2 ρ d ρ - 4 ( 1 - 2 ) 2 [ 1 W ( ρ ) ρ d ρ ] 2 .
W = W 2 ρ 2 + W 4 ρ 4 + W 6 ρ 6 .
( 1 - 2 ) 2 E 0 F = [ ( 6 ) 6 - ( 4 ) 2 8 ( 2 ) ] W 2 2 + [ ( 10 ) 10 - ( 6 ) 2 18 ( 2 ) ] W 4 2 + [ ( 14 ) 14 - ( 8 ) 2 32 ( 2 ) ] W 6 2 + [ ( 12 ) 6 - ( 6 ) ( 8 ) 12 ( 2 ) ] W 4 W 6 + [ ( 8 ) 4 - ( 4 ) ( 6 ) 6 ( 2 ) ] W 2 W 4 + [ ( 10 ) 5 - ( 4 ) ( 8 ) 8 ( 2 ) ] W 2 W 6 ,
( n ) = 1 - n .
F / W 2 j = 0 ,             ( j = 1 , 2 , , n - 1 ) ,
F W 2 = [ ( 6 ) 3 - ( 4 ) 2 4 ( 2 ) ] W 2 + [ ( 8 ) 4 - ( 4 ) ( 6 ) 6 ( 2 ) ] W 4 + [ ( 10 ) 5 - ( 4 ) ( 8 ) 8 ( 2 ) ] W 6 = 0 , F W 4 = [ ( 8 ) 4 - ( 4 ) ( 6 ) 6 ( 2 ) ] W 2 + [ ( 10 ) 5 - ( 6 ) 2 9 ( 2 ) ] W 4 + [ ( 12 ) 6 - ( 6 ) ( 8 ) 12 ( 2 ) ] W 6 = 0.
W 2 W 4 = - [ ( 8 ) 4 - ( 4 ) ( 6 ) 6 ( 2 ) ] / [ ( 6 ) 3 - ( 4 ) ( 4 ) 4 ( 2 ) ] .
W 2 = - W 4 .
W ( ρ ) = W 4 ( ρ 4 - ρ 2 ) ,
W ( ρ ) = W 6 ( ρ 6 - 3 2 ρ 4 + ρ 2 ) .