Abstract

Evaluation of photographic optical systems through use of the optical transfer function or sine wave response has become commonplace. The fundamental physical concepts which lie behind the theory are not, however, widely known and are outlined for use by the optical engineer. The significance, limitations, and advantages of this tool for image evaluation are discussed along with some suggestions of direction along which future work may be carried. The discussion is from the point of view of an individual interested in the design, fabrication, evaluation, and use of large systems of very high precision.

© 1963 Optical Society of America

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  1. L. E. Howlett, Can. J. Res. A24, 15 (1946).
  2. F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 151, 239 (1960).
  3. Rayleigh, Phil. Mag. 42, 167 (1896).
  4. H. Frieser, Compt. Rend. Proc. IX, Congr. Intern., Phot. Paris (1935).
  5. E. W. H. Selwyn, Phot. J. 75, 571 (1935).
  6. Symposium on the Evaluation of Optical Imagery 1951, NBS Circular 526 (1954).
  7. R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, R.I.).
  8. P. M. Duffieux, L’Integrale de Fourier et Ses Applications à L’Optique (Besancon, 1946).
  9. O. Schade, RCA Rev. 9 (1948); also, J. Soc. Motion Picture Television Engrs. 56, 137 (1951); J. Soc. Motion Picture Television Engrs. 58, 182 (1952); J. Soc. Motion Picture Television Engrs. 61, 98 (1953); J. Soc. Motion Picture Television Engrs. 64, 593 (1955).
  10. P. Elias, D. S. Grey, D. Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
  11. H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A 208, 263 (1951).
  12. H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 217, 408 (1953).
  13. E. L. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corporation, Lexington, Mass., 1958).
  14. A. B. Porter, Phil. Mag. 11, 154 (1906).
  15. A. A. Michelson, Studies in Optics (Univ. of Chicago Press, Chicago, Ill., 1927).
  16. Recommendation on Nomenclature by ICO Subcommittee for Image Evaluation, Opt. Acta 8, 359 (1961).
  17. R. L. Lamberts, J. Opt. Soc. Am. 49, 425 (1959).
  18. W. J. Bates, Proc. Phys. Soc. (London) 59, 940 (1947).
  19. M. Françon, Opt. Acta 1, 50 (1954).
  20. H. H. Hopkins, Opt. Acta 2, 23 (1955).
  21. F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).
  22. H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 81 (1955).
  23. M. De, Proc. Roy. Soc. (London) Ser. A 233, 91 (1955).
  24. N. S. Bromilow, Proc. Phys. Soc. (London) Ser. B 71, 231 (1958).
  25. K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).
  26. E. H. Linfoot, Opt. Acta 9, 82 (1962).
  27. K. Miyamoto, Progress in Optics (North-Holland, Amsterdam, 1961).
  28. R. E. Hopkins, Notes for Summer Course in Applied Optics (Institute of Optics, Rochester, N.Y., 1962).
  29. R. R. Shannon, S. Perry, J. Opt. Soc. Am. 49, 506 (1959).
  30. G. Black, E. H. Linfoot, Proc. Roy. Soc. (London), Ser. A 239, 522 (1957).
  31. G. Parrent, C. Drane, Opt. Acta 3, 195 (1956).
  32. W. Steel, Rev. Opt. 32, 4, 143, 269 (1953).
  33. R. Barakat, D. Lev, J. Opt. Soc. Am. 53, 324 (1963).
  34. R. Barakat, D. Lev, J. Opt. Soc. Am. 52, 264, 276 (1962).
  35. W. Herriott, J. Opt. Soc. Am. 37, 472 (1947).
  36. D. R. Herriott, J. Opt. Soc. Am. 48, 968 (1958).
  37. P. Lindberg, Opt. Acta 1, 69 (1954).
  38. L. R. Baker, Proc. Phys. Soc. (London) Ser. B 68, 871 (1955).
  39. D. Kelsall, Proc. Phys. Soc. (London) Ser. B 66, 145 (1953).
  40. E. Menzel, Optik 15, 460 (1958).
  41. R. L. Lamberts, J. Opt. Soc. Am. 49, 425 (1959).
  42. Modulation Transfer Data for Kodak Films, Kodak Pamphlet No. P-49.
  43. D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1960); J. Opt. Soc. Am. 51, 319 (1961).
  44. F. Zernicke, Physica 9, 974 (1942).
  45. E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, Oxford, 1955).
  46. J. J. DePalma, E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).
  47. R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).
  48. P. B. Fellgett, E. H. Linfoot, Phil. Trans. Roy. Soc. London Ser. A 247, 369 (1955).
  49. D. H. Kelly, J. Opt. Soc. Am. 50, 1115 (1960).

1963

1962

R. Barakat, D. Lev, J. Opt. Soc. Am. 52, 264, 276 (1962).

E. H. Linfoot, Opt. Acta 9, 82 (1962).

J. J. DePalma, E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).

1961

Recommendation on Nomenclature by ICO Subcommittee for Image Evaluation, Opt. Acta 8, 359 (1961).

1960

1959

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

R. L. Lamberts, J. Opt. Soc. Am. 49, 425 (1959).

R. R. Shannon, S. Perry, J. Opt. Soc. Am. 49, 506 (1959).

R. L. Lamberts, J. Opt. Soc. Am. 49, 425 (1959).

1958

E. Menzel, Optik 15, 460 (1958).

D. R. Herriott, J. Opt. Soc. Am. 48, 968 (1958).

N. S. Bromilow, Proc. Phys. Soc. (London) Ser. B 71, 231 (1958).

K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).

1957

G. Black, E. H. Linfoot, Proc. Roy. Soc. (London), Ser. A 239, 522 (1957).

1956

G. Parrent, C. Drane, Opt. Acta 3, 195 (1956).

1955

L. R. Baker, Proc. Phys. Soc. (London) Ser. B 68, 871 (1955).

H. H. Hopkins, Opt. Acta 2, 23 (1955).

F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 81 (1955).

M. De, Proc. Roy. Soc. (London) Ser. A 233, 91 (1955).

P. B. Fellgett, E. H. Linfoot, Phil. Trans. Roy. Soc. London Ser. A 247, 369 (1955).

1954

Symposium on the Evaluation of Optical Imagery 1951, NBS Circular 526 (1954).

P. Lindberg, Opt. Acta 1, 69 (1954).

M. Françon, Opt. Acta 1, 50 (1954).

1953

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

D. Kelsall, Proc. Phys. Soc. (London) Ser. B 66, 145 (1953).

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 217, 408 (1953).

1952

1951

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A 208, 263 (1951).

1948

O. Schade, RCA Rev. 9 (1948); also, J. Soc. Motion Picture Television Engrs. 56, 137 (1951); J. Soc. Motion Picture Television Engrs. 58, 182 (1952); J. Soc. Motion Picture Television Engrs. 61, 98 (1953); J. Soc. Motion Picture Television Engrs. 64, 593 (1955).

1947

W. J. Bates, Proc. Phys. Soc. (London) 59, 940 (1947).

W. Herriott, J. Opt. Soc. Am. 37, 472 (1947).

1946

L. E. Howlett, Can. J. Res. A24, 15 (1946).

1942

F. Zernicke, Physica 9, 974 (1942).

1935

H. Frieser, Compt. Rend. Proc. IX, Congr. Intern., Phot. Paris (1935).

E. W. H. Selwyn, Phot. J. 75, 571 (1935).

1906

A. B. Porter, Phil. Mag. 11, 154 (1906).

1896

Rayleigh, Phil. Mag. 42, 167 (1896).

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. (London) Ser. B 68, 871 (1955).

Barakat, R.

R. Barakat, D. Lev, J. Opt. Soc. Am. 53, 324 (1963).

R. Barakat, D. Lev, J. Opt. Soc. Am. 52, 264, 276 (1962).

Bates, W. J.

W. J. Bates, Proc. Phys. Soc. (London) 59, 940 (1947).

Black, G.

G. Black, E. H. Linfoot, Proc. Roy. Soc. (London), Ser. A 239, 522 (1957).

Bromilow, N. S.

N. S. Bromilow, Proc. Phys. Soc. (London) Ser. B 71, 231 (1958).

De, M.

M. De, Proc. Roy. Soc. (London) Ser. A 233, 91 (1955).

DePalma, J. J.

Drane, C.

G. Parrent, C. Drane, Opt. Acta 3, 195 (1956).

Duffieux, P. M.

P. M. Duffieux, L’Integrale de Fourier et Ses Applications à L’Optique (Besancon, 1946).

Elias, P.

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, Phil. Trans. Roy. Soc. London Ser. A 247, 369 (1955).

Françon, M.

M. Françon, Opt. Acta 1, 50 (1954).

Frieser, H.

H. Frieser, Compt. Rend. Proc. IX, Congr. Intern., Phot. Paris (1935).

Grey, D. S.

Herriott, D. R.

Herriott, W.

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 2, 23 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 81 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 217, 408 (1953).

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A 208, 263 (1951).

Hopkins, R. E.

R. E. Hopkins, Notes for Summer Course in Applied Optics (Institute of Optics, Rochester, N.Y., 1962).

Howlett, L. E.

L. E. Howlett, Can. J. Res. A24, 15 (1946).

Kelly, D. H.

Kelsall, D.

D. Kelsall, Proc. Phys. Soc. (London) Ser. B 66, 145 (1953).

Lamberts, R. L.

Lev, D.

R. Barakat, D. Lev, J. Opt. Soc. Am. 53, 324 (1963).

R. Barakat, D. Lev, J. Opt. Soc. Am. 52, 264, 276 (1962).

Lindberg, P.

P. Lindberg, Opt. Acta 1, 69 (1954).

Linfoot, E. H.

E. H. Linfoot, Opt. Acta 9, 82 (1962).

G. Black, E. H. Linfoot, Proc. Roy. Soc. (London), Ser. A 239, 522 (1957).

P. B. Fellgett, E. H. Linfoot, Phil. Trans. Roy. Soc. London Ser. A 247, 369 (1955).

E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, Oxford, 1955).

Lowry, E. M.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, R.I.).

Menzel, E.

E. Menzel, Optik 15, 460 (1958).

Michelson, A. A.

A. A. Michelson, Studies in Optics (Univ. of Chicago Press, Chicago, Ill., 1927).

Miyamoto, K.

K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).

K. Miyamoto, Progress in Optics (North-Holland, Amsterdam, 1961).

O’Neill, E. L.

E. L. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corporation, Lexington, Mass., 1958).

Parrent, G.

G. Parrent, C. Drane, Opt. Acta 3, 195 (1956).

Perrin, F. H.

F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 151, 239 (1960).

Perry, S.

R. R. Shannon, S. Perry, J. Opt. Soc. Am. 49, 506 (1959).

Porter, A. B.

A. B. Porter, Phil. Mag. 11, 154 (1906).

Rayleigh,

Rayleigh, Phil. Mag. 42, 167 (1896).

Robinson, D. Z.

Schade, O.

O. Schade, RCA Rev. 9 (1948); also, J. Soc. Motion Picture Television Engrs. 56, 137 (1951); J. Soc. Motion Picture Television Engrs. 58, 182 (1952); J. Soc. Motion Picture Television Engrs. 61, 98 (1953); J. Soc. Motion Picture Television Engrs. 64, 593 (1955).

Scott, R. M.

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

Selwyn, E. W. H.

E. W. H. Selwyn, Phot. J. 75, 571 (1935).

Shannon, R. R.

R. R. Shannon, S. Perry, J. Opt. Soc. Am. 49, 506 (1959).

Smith, F. D.

F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).

Steel, W.

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

Zernicke, F.

F. Zernicke, Physica 9, 974 (1942).

Can. J. Res.

L. E. Howlett, Can. J. Res. A24, 15 (1946).

Compt. Rend. Proc. IX, Congr. Intern., Phot. Paris

H. Frieser, Compt. Rend. Proc. IX, Congr. Intern., Phot. Paris (1935).

J. Opt. Soc. Am.

J. Soc. Motion Picture Television Engrs.

F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 151, 239 (1960).

Opt. Acta

Recommendation on Nomenclature by ICO Subcommittee for Image Evaluation, Opt. Acta 8, 359 (1961).

P. Lindberg, Opt. Acta 1, 69 (1954).

G. Parrent, C. Drane, Opt. Acta 3, 195 (1956).

E. H. Linfoot, Opt. Acta 9, 82 (1962).

M. Françon, Opt. Acta 1, 50 (1954).

H. H. Hopkins, Opt. Acta 2, 23 (1955).

Optik

E. Menzel, Optik 15, 460 (1958).

Phil. Mag.

Rayleigh, Phil. Mag. 42, 167 (1896).

A. B. Porter, Phil. Mag. 11, 154 (1906).

Phil. Trans. Roy. Soc. London Ser. A

P. B. Fellgett, E. H. Linfoot, Phil. Trans. Roy. Soc. London Ser. A 247, 369 (1955).

Phot. J.

E. W. H. Selwyn, Phot. J. 75, 571 (1935).

Phot. Sci. Eng.

R. M. Scott, Phot. Sci. Eng. 3, 201 (1959).

Physica

F. Zernicke, Physica 9, 974 (1942).

Proc. Phys. Soc. (London)

W. J. Bates, Proc. Phys. Soc. (London) 59, 940 (1947).

Proc. Phys. Soc. (London) Ser. B

L. R. Baker, Proc. Phys. Soc. (London) Ser. B 68, 871 (1955).

D. Kelsall, Proc. Phys. Soc. (London) Ser. B 66, 145 (1953).

N. S. Bromilow, Proc. Phys. Soc. (London) Ser. B 71, 231 (1958).

Proc. Roy. Soc. (London)

G. Black, E. H. Linfoot, Proc. Roy. Soc. (London), Ser. A 239, 522 (1957).

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A 208, 263 (1951).

Proc. Roy. Soc. (London) Ser. A

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 217, 408 (1953).

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 81 (1955).

M. De, Proc. Roy. Soc. (London) Ser. A 233, 91 (1955).

RCA Rev. 9

O. Schade, RCA Rev. 9 (1948); also, J. Soc. Motion Picture Television Engrs. 56, 137 (1951); J. Soc. Motion Picture Television Engrs. 58, 182 (1952); J. Soc. Motion Picture Television Engrs. 61, 98 (1953); J. Soc. Motion Picture Television Engrs. 64, 593 (1955).

Rev. Opt.

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

Symposium on the Evaluation of Optical Imagery 1951

Symposium on the Evaluation of Optical Imagery 1951, NBS Circular 526 (1954).

Other

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, R.I.).

P. M. Duffieux, L’Integrale de Fourier et Ses Applications à L’Optique (Besancon, 1946).

E. L. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corporation, Lexington, Mass., 1958).

A. A. Michelson, Studies in Optics (Univ. of Chicago Press, Chicago, Ill., 1927).

K. Miyamoto, Progress in Optics (North-Holland, Amsterdam, 1961).

R. E. Hopkins, Notes for Summer Course in Applied Optics (Institute of Optics, Rochester, N.Y., 1962).

Modulation Transfer Data for Kodak Films, Kodak Pamphlet No. P-49.

E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, Oxford, 1955).

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Figures (17)

Fig. 1
Fig. 1

A sine wave imaged on a moderately fine grained film. The apparent contrast as a function of line length depends both on granularity and the inherent behavior of the visual system.

Fig. 2
Fig. 2

Image formation from a plane wave.

Fig. 3
Fig. 3

Sinusoidal distribution of luminance.

Fig. 4
Fig. 4

Notation for image formation.

Fig. 5
Fig. 5

Graphical representation of the real and imaginary portions of T(ω).

Fig. 6
Fig. 6

Transfer function for aberration-free optical system with circular aperture.

Fig. 7
Fig. 7

Transfer function for hypothetical divided aperture.

Fig. 8
Fig. 8

Region of integration for diffraction calculation of T.

Fig. 9
Fig. 9

Transfer function from a given annular aperture (a) for symmetrical case (b) off-axis with vignetting.

Fig. 10
Fig. 10

Phase shift between wavefronts in a shearing interferometer for a relative displacement ξ0.

Fig. 11
Fig. 11

Simulated shearing interferograms obtained by moiré interference between grids for a circular aperture at shifts of 0.25, 0.5, and 0.75 the diameter of the aperture.

Fig. 12
Fig. 12

Transfer function for one wavelength of third-order coma obtained from patterns like those in Fig. 11.

Fig. 13
Fig. 13

Computed modulation transfer function for the axial image of a well-corrected doublet lens of focal length 66 in. (168 cm) at several focal settings with (a) all spherical surfaces and (b) one aspheric surface (after Barakat).

Fig. 14
Fig. 14

Transfer function analysis of system with secondary spectrum as displayed on computer cathode-ray tube by R. R. Shannon. Upper portion shows the transfer function vs wavelength for spatial frequencies (beginning at the top curve) of 0.1, 0.2, 0.3, …, 1.0. The lower portion shows the over-all transfer function which results from integrating over all wavelengths.

Fig. 15
Fig. 15

Sine-wave response of the following Kodak films developed in the following Kodak developers: (1) Tri-X, DK-50, 5 min; (2) Panatomic-X, D-19, 3 min; (3) Micro-File, D-11, 4 min; (4) Experimental document-recording, DK-50 (1:10), 10 min. Exposed through Kodak Wratten No. 61 filter (after Lamberts). Figure reproduced by permission of the Editor J. Opt. Soc. Am.

Fig. 16
Fig. 16

Schematic layout for a coherent optical system with point source.

Fig. 17
Fig. 17

Bessel function. Appearance of J0 spatial pattern proposed as a stimulus for visual research (after Kelly).

Equations (35)

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Δ / ξ = x / f .
A ( x ) = - A ( ξ ) e 2 π i ξ x d ξ ,
A ( ξ ) = - A ( x ) e - 2 π i ξ x d x .
C = B max - B min B max + B min .
ω = 2 π / p .
T ( ω ) = Contrast of the image at frequency ω Contrast of the object at frequency ω .
O ( x ) = 1 + C 0 cos ω x ,
i ( x ) = - A ( x - x ) 2 ( 1 + C 0 cos ω x ) d x .
x - x = X , i ( x ) = - A ( x ) 2 [ 1 + C 0 cos ω ( x - X ) ] d X = - A ( X ) 2 d X + C 0 cos ω x - A ( x ) 2 cos ω X d X + C 0 sin ω x - A ( x ) 2 sin ω X d X .
i ( x ) = a 0 + a 1 C 0 cos θ cos ω x + a 1 C 0 sin θ sin ω x .
i ( x ) = a 0 [ 1 + a 1 C 0 a 0 cos ( ω x - θ ) ] ,
i ( x ) = a 0 [ 1 + a 1 C 0 a 0 cos ω x ] T ( ω ) = a 1 C 0 a 0 / C 0 ,
T ( ω ) = - A ( X ) 2 cos ω x d X .
T ( ω ) = K - A ( X ) 2 e i ω X d X ,
A ( ξ ) = A ( ξ ) e 2 π i Δ ( ξ ) .
A ( X ) = - A ( ξ ) e 2 π i ξ X d ξ ,
A ( ξ ) = - A ( X ) e - 2 π i ξ X d X ,
T ( ω ) = K - A ( X ) A * ( X ) e i ω X d X = - A ( X ) [ - A ( ξ ) e - 2 π i ξ X d ξ ] e i ω X d X = - A ( ξ ) [ - A ( X ) e 2 π i ( ξ - ω / 2 π ) X d X ] d ξ ,
T ( ω ) = K - A ( ξ ) A * ( ξ - ξ 0 ) d ξ ,
ξ 0 = ω 2 π = 1 p .
T ( ω x ω y ) = K - - A ( ξ , η ) A * ( ξ - ξ 0 , η - η 0 ) d ξ d η .
ξ 0 = d λ = 1 λ ( F . No . ) .
O ( ω ) = - o ( x ) e i ω x d x = - o ( x ) cos ω x d x + i - o ( x ) sin ω x d x .
i ( x ) = - A ( X ) 2 o ( x ) d x .
- i ( x ) e i ω x d x = - [ - A ( X ) 2 o ( x ) d x ] e i ω x d x = - A ( x ) 2 e i ω X d X - o ( x ) e i ω x d x .
I ( ω ) = T ( ω ) O ( ω ) .
I ( ω x , ω y ) = T ( ω x , ω y ) O ( ω x , ω y ) .
Δ ( ξ , η ) - Δ ( ξ - ξ 0 , η ) .
cos ( δ 1 - δ 2 ) 2 = 1 2 [ cos ( δ 1 - δ 2 ) - 1 ] .
T ( ω ) = e i δ 1 e - i δ 2 d ξ d η = cos ( δ 1 - δ 2 ) d ξ d η + i sin ( δ 1 - δ 2 ) d ξ d η .
δ 1 - δ 2 - 1 2 π ,
1 2 [ cos ( δ 1 - δ 2 - π ) - 1 ] = 1 2 [ sin ( δ 1 - δ 2 ) - 1 ] .
T g ( ω x , ω y ) = lim λ 0 T ( ω x ω y ) = K - - A ( ξ , η ) 2 × exp [ 2 π i ( ω x Δ ξ + ω y Δ η ) d ξ d η
E max - E min E max + E min
γ ( δ x ) = - Γ ( μ ) e i μ δ x d μ ,

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