Abstract

To improve the imaging properties of a defocused optical system, the use of shaded apertures is studied theoretically and experimentally. The study is based on the optical transfer function (OTF). The two shaded apertures studied are the type in which the amplitude transmittance decreases gradually from the center of the pupil toward its rim, TA, and the type in which the amplitude transmittance decreases from its rim toward the center, TB. For comparison, the effects achieved with a clear aperture, TC, are included. The results of the calculations show that near focus the OTF for TA has higher values in the low frequency region than has either TB or TC. When the system is defocused, the shaded aperture of the type TA yields an improved defocused image that is faithful to the outline of the object. The quality of the defocused image obtained with TB is worsened. When the OTF is used as a means for judging the quality of the defocused image, the two necessary conditions on the functions appear to be that the OTF (1) must be a monotonically decreasing function and (2) must be nonnegative. These conditions are confirmed by experiment. Since the transmittance variation of the shaded apertures is achieved by absorption, the effects due to the resultant decreases in light level are also considered.

© 1971 Optical Society of America

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References

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  1. J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 4, p. 133.
    [CrossRef]
  2. P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, p. 31.
    [CrossRef]
  3. H. H. Hopkins, Proc. Phys. Soc. (London) 62, 22 (1949).
    [CrossRef]
  4. H. F. A. Tschunko, J. Opt. Soc. Am. 55, 1 (1965).
    [CrossRef]
  5. T. Asakura, “Report of the Institute of Industrial Science,” The University of Tokyo, Ser. 10917, 51 (Dec.1966).
  6. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  7. H. Osterberg, J. E. Wilkins, J. Opt. Soc. Am. 39, 553 (1949).
    [CrossRef]
  8. E. L. O’Neill, J. Opt. Soc. Am. 46, 285 (1956).
    [CrossRef]

1965 (1)

1956 (1)

1955 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

1949 (2)

H. Osterberg, J. E. Wilkins, J. Opt. Soc. Am. 39, 553 (1949).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) 62, 22 (1949).
[CrossRef]

Asakura, T.

T. Asakura, “Report of the Institute of Industrial Science,” The University of Tokyo, Ser. 10917, 51 (Dec.1966).

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

H. H. Hopkins, Proc. Phys. Soc. (London) 62, 22 (1949).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, p. 31.
[CrossRef]

O’Neill, E. L.

Osterberg, H.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, p. 31.
[CrossRef]

Tschunko, H. F. A.

Tsujiuchi, J.

J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 4, p. 133.
[CrossRef]

Wilkins, J. E.

J. Opt. Soc. Am. (3)

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

H. H. Hopkins, Proc. Phys. Soc. (London) 62, 22 (1949).
[CrossRef]

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).

Other (3)

T. Asakura, “Report of the Institute of Industrial Science,” The University of Tokyo, Ser. 10917, 51 (Dec.1966).

J. Tsujiuchi, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1963), Vol. 2, Chap. 4, p. 133.
[CrossRef]

P. Jacquinot, B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, p. 31.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Amplitude transmittance for the circular aperture to be studied. TA and TB represent TA(r) = 1 − r2 and TB(r) = r2, respectively, where r2 = x2 + y2.

Fig. 2
Fig. 2

OTF’s of the defocused optical system with shaded and clear apertures. The abscissa represents spatial frequency s. Curves labeled TA, TB, and TC correspond to the OTF’s for the apertures of TA(x,y) = 1 − (x2 + y2), TB(x,y) = x2 + y2, and TC(x,y) = 1, respectively. The defocus coefficient given by Eq. (3) is ω20. (a) ω20 = 0, (b) ω20 = λ/π, (c) ω20 = 2λ/π, (d) ω20 = 3λ/π, (e) ω20 = 5λ/π, (f) ω20 = 10λ/π, and (g) ω20 = 30λ/π.

Fig. 3
Fig. 3

Defocused images of Siemens star chart at the defocus coefficient ω20 ≈ 40λ. Type of aperture: (a) TA(x,y) = 1 − (x2 + y2), (b) TB(x,y) = x2 + y2, (c) TC(x,y) = 1.

Fig. 4
Fig. 4

Comparison between diffraction OTF’s and geometrical OTF’s for the defocused optical system with shaded and clear apertures. The abscissa represents spatial frequency s. Solid curves give the diffraction OTF’s, dashed curves the geometrical OTF’s. (a) ω20 = 20λ/π, (b) ω20 = 60λ/π.

Fig. 5
Fig. 5

OTF’s of equal light level, where aperture diameter of clear aperture aperture diameter of shaded apertures = 1 3 .The condition of defocus and the value of abscissa are indicated, respectively, following ω20 and s of the shaded apertures TA and TB. (a) ω20 = 1.5λ/π, (b) ω20 = 6λ/π, and (c) ω20 = 45λ/π.

Fig. 6
Fig. 6

Photographs taken by (a) the lens of F = 22 with clear aperture, (b) the lens of F = 2.8 with the shaded aperture of TA, (c) the lens of F = 2.8 with clear aperture, and (d) the lens of F = 4.7 with clear aperture.

Equations (14)

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f ( x , y ) = T ( x , y ) exp [ i k ω 20 ( x 2 + y 2 ) ] ( x 2 + y 2 1 ) , = 0 ( x 2 + y 2 > 1 ) ,
ω 20 = ( 1 / 2 ) n δ z sin 2 ϕ ,
ω 20 = ( δ z ) / ( 8 F 2 ) .
T A ( x , y ) = 1 - ( x 2 + y 2 ) ( x 2 + y 2 1 ) , = 0 ( x 2 + y 2 > 1 ) ,
T B ( x , y ) = x 2 + y 2 ( x 2 + y 2 1 ) , = 0 ( x 2 + y 2 > 1 ) .
τ ( s ) = - - f ( x + s / 2 , y ) f * ( x - s / 2 , y ) d x d y - - f ( x , y ) f * ( x , y ) d x d y ,
s = 2 F λ N .
τ ( s ) = 3 π - [ 1 - ( s / 2 ) 2 ] 1 2 [ 1 - ( s / 2 ) 2 ] 1 2 - [ ( 1 - y 2 ) 1 2 - s / 2 ] ( 1 - y 2 ) 1 2 - s / 2 [ 1 - ( x + s / 2 ) 2 - y 2 ] × [ 1 - ( x - s / 2 ) 2 - y 2 ] exp ( i a x ) d x d y ,
x 2 m cos a x d x             ( m = 0 , 1 , 2 ) ,
τ ( a ) = 6 π { J 0 ( a ) ( q 2 β + q 3 sin β + q 2 sin 2 β 2 + q 1 sin 3 β 3 ) + J 1 ( a ) [ p 3 β + 3 p 2 sin β + ( p 3 + p 1 ) sin 2 β 2 + p 2 sin 3 β 3 + p 1 sin 4 β 4 ] - J 2 ( a ) [ q 2 β + ( q 3 + q 1 ) sin β + 2 q 2 sin 2 β 2 + q 3 sin 3 β 3 + q 2 sin 4 β 4 + q 1 sin 5 β 5 ] - J 3 ( a ) ( p 1 β + p 2 sin β + p 3 sin 2 β 2 + 2 p 2 sin 3 β 3 + p 3 sin 4 β 4 + p 2 sin 5 β 5 + p 1 sin 6 β 6 ) + n = 1 ( - 1 ) n + 1 J 2 n + 2 ( a ) [ q 1 sin ( 2 n - 1 ) β 2 n - 1 + q 2 sin 2 n β 2 n + q 3 sin ( 2 n + 1 ) β 2 n + 1 + 2 q 2 sin ( 2 n + 2 ) β 2 n + 2 + q 3 sin ( 2 n + 3 ) β 2 n + 3 + q 2 sin ( 2 n + 4 ) β 2 n + 4 + q 1 sin ( 2 n + 5 ) β 2 n + 5 ] + n = 1 ( - 1 ) n + 1 J 2 n + 3 ( a ) [ p 1 sin 2 n β 2 n + p 2 sin ( 2 n + 1 ) β 2 n + 1 + p 3 sin ( 2 n + 2 ) β 2 n + 2 + 2 p 2 sin ( 2 n + 3 ) β 2 n + 3 + p 3 sin ( 2 n + 4 ) β 2 n + 4 + p 2 sin ( 2 n + 5 ) β 2 n + 5 + p 1 sin ( 2 n + 6 ) β 2 n + 6 ] } ,
p 1 = - 4 a 3 cos a s 2 - 2 s a 2 sin a s 2 , p 2 = 12 s a 3 cos a s 2 + ( 2 s a 2 - 24 a 4 ) sin a s 2 , p 3 = ( - 4 s 2 + 12 a 3 + 48 a 5 ) cos a s 2 + ( - 6 s a 2 + 24 s a 4 ) sin a s 2 , q 1 = - 2 s a 2 cos a s 2 + 4 a 3 sin a s 2 , q 2 = ( 2 s a 2 - 24 a 4 ) cos a s 2 - 12 s a 3 sin a s 2 , q 3 = ( - 6 s a 2 + 24 s a 4 ) cos a s 2 - ( - 4 s 2 + 12 a 3 + 48 a 5 ) sin a s 2 , β = cos - 1 ( s / 2 ) , a = 2 k ω 20 s ,
p 1 = - 4 a 3 cos a s 2 - 2 s a 2 sin a s 2 , p 2 = ( - 2 s a + 12 s a 3 ) cos a s 2 + ( 2 s 2 + 4 a 2 - 24 a 4 ) sin a s 2 , p 3 = ( 2 s 2 + 2 a - 4 s 2 + 20 a 3 + 48 a 5 ) cos a s 2 + ( - 10 s a 2 + 24 s a 4 ) sin a s 2 , q 1 = - 2 s a 2 cos a s 2 + 4 a 3 sin a s 2 , q 2 = ( 2 s 2 + 4 a 2 - 24 a 4 ) cos a s 2 - ( - 2 s a + 12 s a 3 ) sin a s 2 , q 3 = ( - 10 s a 2 + 24 s a 4 ) cos a s 2 - ( 2 s 2 + 2 a - 4 s 2 + 20 a 3 + 48 a 5 ) sin a s 2 .
E = 1 π - [ T ( x , y ) ] 2 d x d y ,
aperture diameter of clear aperture aperture diameter of shaded apertures = 1 3 .

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