Abstract

Pinhole imagery is discussed by using modulation transfer functions. The optimum condition between the size of the pinhole and the focus position is understood as the balance of the diffraction caused by the aperture size and the geometrical-optical aberration given by the aperture size and focus position. Field characteristics, chromatic characteristics and relations with receptor systems are discussed easily in the spatial-frequency domain.

© 1967 Optical Society of America

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References

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  1. H. Gernsheim, The History of Photography (Oxford University Press, New York, 1955), p. 1.
  2. J. Petzval, Wien Ber. XXVI, 33 (1857);Phil. Mag. XVII, 1 (1859).
  3. Lord Rayleigh, Phil Mag. XXXI, 87 (1891);Sci. Paper Vol. I (Dover Publications, New York, 1965), p. 513.
  4. E. W. H. Selwyn, Phot. Z. 90B, 47 (1950).
  5. For example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Book Hill Company, Inc., New York, 1932), p. 124.
  6. For example, R. W. Wood, Physical Optics (Macmillan Company, New York, 1934), 3rd ed., p. 270.
  7. For example, J. M. Stone, Radiation and Optics (McGraw–Book Hill Company, Inc., New York, 1963), p. 204.
  8. For example, R. Kingslake, Lenses in Photography (A. S. Barnes and Company, Inc., New York, 1963), Rev. ed., p. 60.
  9. For example, J. Flügge, Das Photographische Objektiv (Springer– Verlag, Wien, 1955), p. 8.
  10. W. Merte, R. Richter, and M. V. Rohr, Das Photographische Objektiv (Verlag von Julius Springer, Wien, 1932), p. 17.
    [Crossref]
  11. A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
    [Crossref] [PubMed]
  12. H. H. Hopkins, Proc. Plays. Soc. (London) A231, 91 (1955).
    [Crossref]
  13. W. H. Steel, Opt. Acta 3, 65 (1956).
    [Crossref]
  14. K. Sayanagi, Oyo Butsuri 25, 193 (1956).
  15. K. Strehl, Z. Instrumentenk 15, 362 (1895).
  16. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., p. 440 and 441.
  17. H. H. Hopkins, Wave Theory of Aberration (Clarendon Press, Oxford, 1950), p. 16.
  18. Assuming that the spread function is the uniform disk.
  19. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 91 (1955).
  20. H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).
  21. H. J. Zweig, G. C. Higgins, and D. L. MacAdam, J. Opt. Soc. Am. 48, 926 (1958).
    [Crossref]
  22. R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
    [Crossref]

1965 (1)

A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
[Crossref] [PubMed]

1961 (1)

1958 (1)

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

1956 (2)

W. H. Steel, Opt. Acta 3, 65 (1956).
[Crossref]

K. Sayanagi, Oyo Butsuri 25, 193 (1956).

1955 (2)

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

H. H. Hopkins, Proc. Plays. Soc. (London) A231, 91 (1955).
[Crossref]

1950 (1)

E. W. H. Selwyn, Phot. Z. 90B, 47 (1950).

1895 (1)

K. Strehl, Z. Instrumentenk 15, 362 (1895).

1891 (1)

Lord Rayleigh, Phil Mag. XXXI, 87 (1891);Sci. Paper Vol. I (Dover Publications, New York, 1965), p. 513.

1857 (1)

J. Petzval, Wien Ber. XXVI, 33 (1857);Phil. Mag. XVII, 1 (1859).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., p. 440 and 441.

Flügge, J.

For example, J. Flügge, Das Photographische Objektiv (Springer– Verlag, Wien, 1955), p. 8.

Gallas, A. H.

A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
[Crossref] [PubMed]

Gernsheim, H.

H. Gernsheim, The History of Photography (Oxford University Press, New York, 1955), p. 1.

Gilbert, C. A.

A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
[Crossref] [PubMed]

Hardy, A. C.

For example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Book Hill Company, Inc., New York, 1932), p. 124.

Higgins, G. C.

Hitterdal, A. B.

A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
[Crossref] [PubMed]

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

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

H. H. Hopkins, Proc. Plays. Soc. (London) A231, 91 (1955).
[Crossref]

H. H. Hopkins, Wave Theory of Aberration (Clarendon Press, Oxford, 1950), p. 16.

Jones, R. C.

Kingslake, R.

For example, R. Kingslake, Lenses in Photography (A. S. Barnes and Company, Inc., New York, 1963), Rev. ed., p. 60.

MacAdam, D. L.

Merte, W.

W. Merte, R. Richter, and M. V. Rohr, Das Photographische Objektiv (Verlag von Julius Springer, Wien, 1932), p. 17.
[Crossref]

Perrin, F. H.

For example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Book Hill Company, Inc., New York, 1932), p. 124.

Petzval, J.

J. Petzval, Wien Ber. XXVI, 33 (1857);Phil. Mag. XVII, 1 (1859).

Rayleigh, Lord

Lord Rayleigh, Phil Mag. XXXI, 87 (1891);Sci. Paper Vol. I (Dover Publications, New York, 1965), p. 513.

Richter, R.

W. Merte, R. Richter, and M. V. Rohr, Das Photographische Objektiv (Verlag von Julius Springer, Wien, 1932), p. 17.
[Crossref]

Rohr, M. V.

W. Merte, R. Richter, and M. V. Rohr, Das Photographische Objektiv (Verlag von Julius Springer, Wien, 1932), p. 17.
[Crossref]

Sayanagi, K.

K. Sayanagi, Oyo Butsuri 25, 193 (1956).

Selwyn, E. W. H.

E. W. H. Selwyn, Phot. Z. 90B, 47 (1950).

Steel, W. H.

W. H. Steel, Opt. Acta 3, 65 (1956).
[Crossref]

Stone, J. M.

For example, J. M. Stone, Radiation and Optics (McGraw–Book Hill Company, Inc., New York, 1963), p. 204.

Strehl, K.

K. Strehl, Z. Instrumentenk 15, 362 (1895).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., p. 440 and 441.

Wood, R. W.

For example, R. W. Wood, Physical Optics (Macmillan Company, New York, 1934), 3rd ed., p. 270.

Zweig, H. J.

J. Opt. Soc. Am. (2)

J. Soc. Motion Picture Television Engrs. (1)

A. H. Gallas, C. A. Gilbert, and A. B. Hitterdal, J. Soc. Motion Picture Television Engrs. 74, 321 (1965).P. A. Newman and V. E. Rible, Appl. Opt. 5, 1225 (1966).
[Crossref] [PubMed]

Opt. Acta (1)

W. H. Steel, Opt. Acta 3, 65 (1956).
[Crossref]

Oyo Butsuri (1)

K. Sayanagi, Oyo Butsuri 25, 193 (1956).

Phil Mag. (1)

Lord Rayleigh, Phil Mag. XXXI, 87 (1891);Sci. Paper Vol. I (Dover Publications, New York, 1965), p. 513.

Phot. Z. (1)

E. W. H. Selwyn, Phot. Z. 90B, 47 (1950).

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

H. H. Hopkins, Proc. Phys. Soc. (London) B70, 449 (1957).

Proc. Plays. Soc. (London) (1)

H. H. Hopkins, Proc. Plays. Soc. (London) A231, 91 (1955).
[Crossref]

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

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

Wien Ber. (1)

J. Petzval, Wien Ber. XXVI, 33 (1857);Phil. Mag. XVII, 1 (1859).

Z. Instrumentenk (1)

K. Strehl, Z. Instrumentenk 15, 362 (1895).

Other (10)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1965), 3rd ed., p. 440 and 441.

H. H. Hopkins, Wave Theory of Aberration (Clarendon Press, Oxford, 1950), p. 16.

Assuming that the spread function is the uniform disk.

H. Gernsheim, The History of Photography (Oxford University Press, New York, 1955), p. 1.

For example, A. C. Hardy and F. H. Perrin, The Principles of Optics (McGraw–Book Hill Company, Inc., New York, 1932), p. 124.

For example, R. W. Wood, Physical Optics (Macmillan Company, New York, 1934), 3rd ed., p. 270.

For example, J. M. Stone, Radiation and Optics (McGraw–Book Hill Company, Inc., New York, 1963), p. 204.

For example, R. Kingslake, Lenses in Photography (A. S. Barnes and Company, Inc., New York, 1963), Rev. ed., p. 60.

For example, J. Flügge, Das Photographische Objektiv (Springer– Verlag, Wien, 1955), p. 8.

W. Merte, R. Richter, and M. V. Rohr, Das Photographische Objektiv (Verlag von Julius Springer, Wien, 1932), p. 17.
[Crossref]

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Figures (10)

F. 1
F. 1

Geometrical scheme of pinhole.

F. 2
F. 2

Modulation transfer functions for varying pinhole. Diameter l = 20.9 mm, λ = 500 mμ and μ = 1l = 0.12 mm20.1630.2040.23.

F. 3
F. 3

Modulation transfer functions for varying focus. Position d = 0.20 mm. λ = 500 mμ and μ = 1l = 62.8 mm231.4320.9415.7512.6610.5.

F. 4
F. 4

Field characteristics of optimum focus position; ——— radial image plane, – – – – – tangential image plane.

F. 5
F. 5

Resolving power for diameter change. Focus position 20.9 mm, λ = 500 mμ.

F. 6
F. 6

Resolving power for focus position change. Diameter 0.2 mm, λ = 500 mμ.

F. 7
F. 7

Modulation transfer function at radial and tangential optimum image plane. μ = 3.

F. 8
F. 8

Diffraction pattern at radial and tangential optimum image plane. μ = 3.

F. 9
F. 9

Off-axial modulation transfer function for the image plane of μ = 3 at on-axis.

F. 10
F. 10

Chromatic difference of modulation transfer function when μ = 3 for 480 mμ.

Tables (2)

Tables Icon

Table I k value of the optimum condition d2 = kλl by different authors. (k = 4μ/π)

Tables Icon

Table II k value of the optimum condition d2 = kλl for different conditions. (k = 4μ/π)

Equations (69)

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D = d + 2 l λ / d ,
D / d = 1 2 l λ / d 2 ,
d p 2 = 2 l λ ,
d R 2 = 3.6 l λ .
d s 2 = 3 l λ .
W ( ξ , η ) = a 20 ( ξ 2 + η 2 ) ,
a 20 = d 2 / 8 l ,
a 20 = d 2 ( a + l ) / 8 a l ,
μ = π d 2 / 4 λ l ,
l = π d 2 / 4 λ μ ,
d 2 = ( 4 μ / π ) λ l .
F = l / d = π d / 4 λ μ .
u c = 2 sin α / λ = 4 μ / π d = d / λ l ,
Q I I = T ( u , v ) dudv ,
Q I x = T ( u , o ) d u , Q I y = T ( o , v ) d v , }
Q I I = t ( 0 , 0 ) ,
Q I x = h x ( 0 ) Q I y = h y ( 0 ) } ,
t ( x , y ) = T ( u , v ) exp { 2 π i ( u x + v y ) } dudv ,
h x ( x ) = t ( x , y ) d y , h y ( y ) = t ( x , y ) d x } .
I = ( π d 2 A 4 λ f 2 ) 2 [ sin ( U / 4 ) U / 4 ] 2 ,
U = ( 2 π / λ ) ( d / 2 f ) 2 Z ,
a 20 = 1 2 ( d / 2 f ) 2 Z
I = 4 A 2 / l 2 [ sin ( π d 2 / 8 λ l ) ] 2 .
t ( x , y ) dxdy = 1 .
( π d 2 / 4 ) A 2 = 1 ,
A 2 = 4 / π d 2 .
Q I I = 16 / π d 2 l 2 [ sin ( π d 2 / 8 λ l ) ] 2 .
Q I I d = 8 π d 2 l 2 ( sin π d 2 8 λ l ) { π d λ l cos π d 2 8 λ l 4 d sin π d 2 8 λ l } = 0 ,
tan ( π d 2 / 8 λ l ) = π d 2 / 4 λ l ,
μ 0.742 π 2.331 .
d Q I I 2 = 2.968 λ l .
μ 2.2 ,
d Q I 2 = 2.7 λ l .
Q I I l = 32 π d 2 l 3 ( sin π d 2 8 λ l ) { sin π d 2 8 λ l + π d 2 8 λ l cos π d 2 8 λ l } = 0 ;
tan ( π d 2 / 8 λ l ) = π d 2 / 8 λ l ,
μ 1.292 π 4.056 .
d 2 = 5.168 λ l Q I I ,
l Q I I = d 2 / 5.168 λ .
μ 3.0 ,
d 2 = 3.8 λ l Q I ,
l Q I = d 2 / 3.8 λ .
d 2 = k λ l ,
k = 4 μ / π .
T ( s ) = ( 1 / A ) p exp [ iksW ( ξ , η ; s ) ] d ξ d η
W ( ξ , η ; s ) = ( 1 / s ) [ W ( ξ + 1 2 s , η ) W ( ξ 1 2 s , η ) ]
W ( ξ , η ; s ) = 2 a 20 ξ ,
T ( t ) = ( 1 / A ) p exp [ iktW ( ξ , η ; t ) ] d ξ d η ,
W ( ξ , η ; t ) = ( 1 / t ) [ W ( ξ , η + 1 2 t ) W ( ξ ; η 1 2 t ) ] ,
W ( ξ , η ; t ) = 2 a 20 η .
and d r = d , d t = d cos θ }
l r = ( π d r 2 / 4 λ μ ) cos θ = ( π d 2 / 4 λ μ ) cos θ , l t = ( π d t 2 / 4 λ μ ) cos θ = ( π d 2 / 4 λ μ ) cos 3 θ .
u c r = 4 μ / π d r = 4 μ / π d
u c t = 4 μ / π d t = 4 μ / π d cos θ .
u c t = ( 4 μ / π d cos θ ) cos θ = 4 μ / π d .
μ = π d 2 / 4 λ l
u c = d / λ l
Q I I / λ = ( 4 / λ 2 l 3 ) sin ( π d 2 / 8 λ l ) cos ( π d 2 / 8 λ l ) = 0
π d 2 / 8 λ l = π / 2 .
μ c I I = π d 2 / 4 λ l = π
d 2 = 4 λ l .
μ c I = 3.0
d 2 = 3.8 λ l .
μ = π d / 4 λ l = c μ c ,
μ c = π d 2 / 4 ( c λ ) l = π d 2 / 4 λ l ,
u c = u c · l = d / λ .
( exposure × resolution ) film = const .
illuminance of the image 1 / F = 16 λ 2 μ 2 / π 2 d 2 ,
resolution in the image ( 1 / μ c 2 ) = π 2 d 2 / 16 μ 2 .
( illuminance × resolution ) λ 2 = const .