Abstract

Images from high aspect ratio annulus apertures are spatially filtered in a coherent optical processor for equalization of the modulation transfer function. The imagery is relatively insensitive to spherical aberration, field curvature, and longitudinal color; and the depth of focus is substantially greater than that of conventional imagery with comparable resolution.

© 1971 Optical Society of America

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References

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  1. P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 3, Chap. 2.
  2. W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953).
  3. E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956).
    [CrossRef]
  4. J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).
  5. D. J. McLean, Proc. Roy. Soc. (London) A263, 545 (1961).
  6. E. G. Bowen, Nature 195, 649 (1962).
    [CrossRef]
  7. A. W. L. Carter, J. P. Wild, Proc. Roy. Soc. (London) 282, 252 (1964).
    [CrossRef]
  8. J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).
  9. J. P. Wild, Phys. Today19, 28 (July1966).
    [CrossRef]
  10. A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [CrossRef]
  11. B. R. Brown, A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  12. A. Vander Lugt, Opt. Acta 15, 1 (1968).
    [CrossRef]
  13. G. W. Stroke, R. G. Zech, Phys. Lett. 25A, 89 (1967).
  14. A. W. Lohmann, Appl. Opt. 7, 561 (1968).
    [CrossRef]
  15. K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 54.
    [CrossRef]

1968 (2)

A. Vander Lugt, Opt. Acta 15, 1 (1968).
[CrossRef]

A. W. Lohmann, Appl. Opt. 7, 561 (1968).
[CrossRef]

1967 (1)

G. W. Stroke, R. G. Zech, Phys. Lett. 25A, 89 (1967).

1966 (1)

1965 (1)

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

1964 (2)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

A. W. L. Carter, J. P. Wild, Proc. Roy. Soc. (London) 282, 252 (1964).
[CrossRef]

1962 (1)

E. G. Bowen, Nature 195, 649 (1962).
[CrossRef]

1961 (2)

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

D. J. McLean, Proc. Roy. Soc. (London) A263, 545 (1961).

1956 (1)

1953 (1)

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953).

Bowen, E. G.

E. G. Bowen, Nature 195, 649 (1962).
[CrossRef]

Brown, B. R.

Carter, A. W. L.

A. W. L. Carter, J. P. Wild, Proc. Roy. Soc. (London) 282, 252 (1964).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 3, Chap. 2.

Lohmann, A. W.

McLean, D. J.

D. J. McLean, Proc. Roy. Soc. (London) A263, 545 (1961).

Miyamoto, K.

K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 54.
[CrossRef]

O’Neill, E. L.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 3, Chap. 2.

Steel, W. H.

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953).

Stroke, G. W.

G. W. Stroke, R. G. Zech, Phys. Lett. 25A, 89 (1967).

Vander Lugt, A.

A. Vander Lugt, Opt. Acta 15, 1 (1968).
[CrossRef]

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

Wild, J. P.

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

A. W. L. Carter, J. P. Wild, Proc. Roy. Soc. (London) 282, 252 (1964).
[CrossRef]

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

J. P. Wild, Phys. Today19, 28 (July1966).
[CrossRef]

Zech, R. G.

G. W. Stroke, R. G. Zech, Phys. Lett. 25A, 89 (1967).

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (1)

E. G. Bowen, Nature 195, 649 (1962).
[CrossRef]

Opt. Acta (1)

A. Vander Lugt, Opt. Acta 15, 1 (1968).
[CrossRef]

Phys. Lett. (1)

G. W. Stroke, R. G. Zech, Phys. Lett. 25A, 89 (1967).

Proc. Roy. Soc. (London) (4)

A. W. L. Carter, J. P. Wild, Proc. Roy. Soc. (London) 282, 252 (1964).
[CrossRef]

J. P. Wild, Proc. Roy. Soc. (London) A286, 499 (1965).

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

D. J. McLean, Proc. Roy. Soc. (London) A263, 545 (1961).

Rev. Opt. (1)

W. H. Steel, Rev. Opt. 32, 4, 143, 269 (1953).

Other (3)

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 3, Chap. 2.

J. P. Wild, Phys. Today19, 28 (July1966).
[CrossRef]

K. Miyamoto, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1961), Vol. 1, p. 54.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Coherent processor for correcting transparencies obtained with annulus aperture.

Fig. 2
Fig. 2

MTF of in-focus annulus of width 1/25× outer radius.

Fig. 3
Fig. 3

Geometry for calculating misfocused MTF. Shaded areas contribute to autocorrelation of annulus vs relative displacement 2σ.

Fig. 4
Fig. 4

Processed image of radial line resolution chart. Best measured resolution indicated by outside edges of straight black lines near center. Original transparency taken in focus.

Fig. 5
Fig. 5

Same as Fig. 4, but original transparency taken out of focus by 230 mm. We tried to mark the black lines indicating resolution more conservatively here.

Equations (21)

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A ( ω ) 1 , ρ 1 k / d ω ρ 2 k / d , A ( ω ) 0 , otherwise ,
i ( r ) = - exp ( - i ω · r ) I ( ω ) d ω ,
F ( ω ) = - exp ( i ω · r ) f ( r ) d r .
MTF ~ ( Δ ρ / 2 π ρ ¯ ) ( ω ¯ / ω ) ( 1 - ω 2 / ω ¯ 2 ) - 1 2 ,
s ( r - r 0 ) = | c ρ 1 ρ J 0 ( k ρ r - r 0 / d ) ρ d ρ | 2 ,
s ( r - r 0 ) c ( 2 ρ ¯ d ) 1 2 ( π k r - r 0 ) - 1 2 × ρ 1 ρ 2 cos ( k ρ r - r 0 / d - π / 4 ) d ρ 2 = ( 8 c 2 ρ ¯ d 3 ) ( π k 3 r - r 0 3 ) - 1 cos 2 ( ω ¯ r - r 0 / 2 - π / 4 ) sin 2 ( Δ ω r - r 0 / 4 ) .
s ( r - r 0 ) ~ ( 2 c 2 ρ ¯ d Δ ρ 2 ) ( π k r - r 0 ) - 1 × cos 2 ( ω ¯ r - r 0 / 2 - π / 4 )
Φ f = K ρ 2 2 / 2 ,
Φ a = K ρ ¯ Δ ρ ,
K = k ( 1 / d - 1 / d ) k ( d - d ) / d 2 for ( d - d ) d .
ρ 1 2 ( x - d ) 2 + y 2 ρ 2 2 , ρ 1 2 ( x + d ) 2 + y 2 ρ 2 2 .
Δ y = [ ( ρ ¯ + Δ ρ / 2 ) 2 - ( x + σ ) 2 ] 1 2 - [ ( ρ ¯ - Δ ρ / 2 ) 2 - ( x - σ ) 2 ] 1 2
Φ + = K [ ( x + σ ) 2 + y 2 ] / 2 , Φ - = K [ ( x - σ ) 2 + y 2 ] / 2 , Φ + - Φ - = 2 K x σ .
MTF = 4 ( 2 π ρ ¯ Δ ρ ) - 1 0 ρ 0 ( ρ ¯ 2 - σ 2 ) - 1 2 ( ρ ¯ Δ ρ - 2 x σ ) cos ( 2 K σ x ) d x = ( Δ ρ / 2 π ρ ¯ ) ( ρ ¯ / σ ) ( 1 - σ 2 / ρ ¯ 2 ) - 1 2 ( sin K ρ ¯ Δ ρ / 2 ) 2 / ( K ρ ¯ Δ ρ / 2 ) 2 .
K a / K f = ( 0.668 ) 2 ρ 2 2 / π ρ ¯ Δ ρ = 0.425 ( 1 + 0.5 Δ ρ / ρ 2 ) ρ 2 / Δ ρ .
Φ + - Φ - = S [ ( x + σ ) 2 + y 2 ] 2 - S [ ( x - σ ) 2 + y 2 ] 2 = 8 S [ x 2 + ( y - y 0 ) 2 + 2 y 0 ( y - y 0 ) + ρ ¯ 2 ] x σ ,
( k / d ) ρ 0 = ( k / d ) 2 ( ρ 2 2 - ρ 1 2 ) / 2 ω , ( k / d ) ρ m = ( k / d ) ρ 2 ( ρ 2 2 + ω 2 - ρ 1 2 ) / 2 ρ 2 ω - ω / 2 ,
g ( a , x ) = 2 x ( a 2 - x 2 ) 1 2 + 2 a 2 sin - 1 ( x / a ) ,
1 / K = π ( ρ 2 k / d ) 2 - π ( ρ 1 k / d ) 2 ,
A A * ( ω ) = - K g ( ρ 2 k / d , ω / 2 ) - K g ( ρ 1 k / d , ρ 1 k / d ) + K g ( ρ 1 k / d , - ω / 2 ) + K g ( ρ 2 k / d , ρ 2 k / d ) }             for 0 ω < ρ 2 k / d - ρ 1 k / d = K g ( ρ 2 k / d , ω / 2 + ρ 0 k / d ) - K g ( ρ 2 k / d , ω / 2 ) + K g ( ρ 1 k / d , - ω / 2 ) - K g ( ρ 1 k / d , ρ 0 k / d - ω / 2 ) }             for ρ 2 k / d - ρ 1 k / d ω < 2 ρ 1 k / d = K g ( ρ 2 k / d , ω / 2 + ρ 0 k / d ) - K g ( ρ 2 k / d , ω / 2 ) - K g ( ρ 1 k / d , ρ 1 k / d ) + K g ( ρ 1 k / d , ω / 2 - ρ 0 k / d ) }             for 2 ρ 1 k / d ω < ρ 1 k / d + ρ 2 k / d = K g ( ρ 2 k / d , ρ 2 k / d ) - K g ( ρ 2 k / d , ω / 2 ) }             for ρ 1 k / d + ρ 2 k / d ω 2 ρ 2 k / d = 0 for 2 ρ 2 k / d < ω ;
A A ( ω ) * = - K g ( ρ 2 k / d , ω / 2 ) - K g ( ρ 1 k / d , ρ 1 k / d ) + K g ( ρ 1 k / d , - ω / 2 ) + K g ( ρ 2 k / d , ρ 2 k / d ) }             for 0 ω < 2 ρ 1 k / d = K g ( ρ 2 k / d , ρ 2 k / d ) - K g ( ρ 2 k / d , ω / 2 ) - 2 π K ( ρ 1 k / d ) 2 }             for 2 ρ 1 k / d ω < ρ 2 k / d - ρ 1 k / d = K g ( ρ 2 k / d , ω / 2 + ρ m k / d ) - K g ( ρ 2 k / d , ω / 2 ) - K g ( ρ 1 k / d , ρ 1 k / d ) + K g ( ρ 1 k / d , ω / 2 - ρ m k / d ) }             for ρ 2 k / d - ρ 1 k / d ω < ρ 2 k / d + ρ 1 k / d = K g ( ρ 2 k / d , ρ 2 k / d ) - K g ( ρ 2 k / d , ω / 2 ) }             for ρ 2 k / d + ρ 1 k / d ω 2 ρ 2 k / d = 0 for 2 ρ 2 k / d < ω .

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