Abstract

A theoretical study is made of the possibilities of using an annular aperture to increase the focal depth of a photographic objective. It is shown that for a given gain in focal depth the loss in speed is the same for both annular apertures and conventional stopping down. For images of isolated point objects, the definition is improved by using an annular stop. The gain in focal depth is less for off-axis points, but it is found that, for example, a factor of 2.7 in focal depth gained by means of an annular aperture is barely affected at a field angle of 30°.

© 1960 Optical Society of America

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References

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  1. Rayleigh, Monthly Notices Roy. Astron. Soc. 33, 59 (1872), reprinted in Sci. Papers, 1, 163.
  2. G. C. Steward, The Symmetrical Optical System (Cambridge University Press, New York, 1928), Chap. VII.
  3. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).
  4. W. H. Steel, Rev. opt. 32, 4 (1953).
  5. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944).

1953 (2)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

W. H. Steel, Rev. opt. 32, 4 (1953).

1872 (1)

Rayleigh, Monthly Notices Roy. Astron. Soc. 33, 59 (1872), reprinted in Sci. Papers, 1, 163.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Rayleigh,

Rayleigh, Monthly Notices Roy. Astron. Soc. 33, 59 (1872), reprinted in Sci. Papers, 1, 163.

Steel, W. H.

W. H. Steel, Rev. opt. 32, 4 (1953).

Steward, G. C.

G. C. Steward, The Symmetrical Optical System (Cambridge University Press, New York, 1928), Chap. VII.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944).

Wolf, E.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Monthly Notices Roy. Astron. Soc. (1)

Rayleigh, Monthly Notices Roy. Astron. Soc. 33, 59 (1872), reprinted in Sci. Papers, 1, 163.

Proc. Phys. Soc. (London) (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Rev. opt. (1)

W. H. Steel, Rev. opt. 32, 4 (1953).

Other (2)

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944).

G. C. Steward, The Symmetrical Optical System (Cambridge University Press, New York, 1928), Chap. VII.

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Figures (6)

Fig. 1
Fig. 1

Notation for annular aperture formulas.

Fig. 2
Fig. 2

Airy patterns for annular and unobstructed apertures. 1, Unobstructed aperture; 2, annular aperture with obstruction ratio 0.8; 3, annular aperture with obstruction ratio tending to unity; 4, unobstructed aperture stopped down to give same focal depth as for 2.

Fig. 3
Fig. 3

Notation for off-axis annular aperture formulas.

Fig. 4
Fig. 4

Strehl intensity for annular apertures off-axis. Ordinate: field angle; abscissa: defocusing coordinate. The graphs are loci of constant Strehl intensity 0.8, 0.6, 0.4, 0.2, as indicated.

Fig. 5
Fig. 5

Proportion of light flux U (v) within radius v of center of image for obstruction ratio . The broken line curves A, B, and C are for conventional stopping down to give the same increase of focal depth as for =0.7, 0.8, and 0.9, respectively.

Fig. 6
Fig. 6

Ideal wide-angle lenses to be used with annular aperture.

Equations (14)

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I ( 0 , ρ ) = a 4 λ 2 R 2 [ 2 J 1 ( v ) v - 2 J 1 ( v ) v ] 2 ,
v = 2 π a ρ / λ R
I ( z , 0 ) = a 4 λ 2 R 2 { sin [ 1 2 p ( 1 - 2 ) ] 1 2 p } 2 ,
p = π a 2 z / λ R 2 .
A = exp ( i p r 2 ) r d r d ϕ ,
{ σ cos θ = r cos ϕ σ sin θ cos β = r sin ϕ .
A = cos β 1 0 2 π exp [ i p σ 2 ( 1 - sin 2 θ sin 2 β ) ] σ d σ d θ .
A = 2 π cos β 1 exp [ i p σ 2 ( 1 - 1 2 sin 2 β ) ] × J 0 ( 1 2 p σ 2 sin 2 β ) σ d σ .
A = [ π cos β exp [ i p σ 2 ( 1 - 1 2 sin 2 β ) ] i p ( 1 - 1 2 sin 2 β ) × n = 0 ( i sin 2 β 2 - sin 2 β ) n J 0 ( n ) ( 1 2 p σ 2 sin 2 β ) ] 1 ,
I = | [ exp [ i p σ 2 ( 1 - 1 2 sin 2 β ) ] i p ( 1 - 1 2 sin 2 β ) ( 1 - 2 ) × n = 0 ( i sin 2 β 2 - sin 2 β ) n J 0 ( n ) ( 1 2 p σ 2 sin 2 β ) ] 1 | 2 ,
I = 4 sin 2 [ 1 2 p ( 1 - 2 ) ] / p 2 ( 1 - 2 ) 2 .
U ( v ) = 2 1 - 2 0 v [ J 1 ( v ) - J 1 ( v ) ] 2 d v v
= 2 1 - 2 0 v { J 1 2 ( v ) v + 2 J 1 2 ( v ) v - 2 v J 1 ( v ) J 1 ( v ) } d v .
U ( v ) = 1 1 - 2 { J 1 2 ( v ) - 2 J 1 2 ( v ) + 2 J 0 2 ( v ) - 2 J 0 2 ( v ) } + - 2 1 - 2 n = 2 [ ( 1 - 2 ) n v n 2 n n ! ( n - 1 ) { J 0 ( v ) J n ( v ) - J 1 ( v ) J n - 1 ( v ) } ] .