Abstract

The effect of defocusing on the diffraction images of a general periodic triangular-wave object has been investigated for the case of a narrow rectangular aperture. Results in the form of curves have been given for irradiance distribution and contrast in the images. The aperture has been assumed to be illuminated incoherently. A close similarity has been shown between the sine-wave response and triangular-wave response of a particular object function.

© 1969 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. H. Lnfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).
  2. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1963).
  3. Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press Inc., New York, 1964), p. 484.
  4. E. B. Brown, Modern Optics (Reinhold Publishing Corp., New York, 1965), Ch. 9, p. 479.
  5. K. Muratah in Progress in Optics V, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1966), Ch. IV, p. 201.
  6. J. W. Coltman, J. Opt. Soc. Am. 44, 468 (1954).
    [Crossref]
  7. F. W. Rosberry, J. Res. Natl. Bur. Std. (U.S.) 64C, 57 (1960).
  8. S. H. Emara, J. Res. Natl. Bur. Std. (U.S.) 65A, 465 (1961).
    [Crossref]
  9. A. Maréchal and M. Françon, Diffraction, Structure des Images (Editions de la Revue d’Optique, Paris, 1960).
  10. A. Newman, R. Barakat, and R. Humphreys, Appl. Opt. 5, 670 (1966).
    [Crossref] [PubMed]
  11. F. Kottler and F. H. Perrin, J. Opt. Soc. Am. 56, 377 (1966).
    [Crossref]
  12. K. Singh and A. K. Kavathekar, J. Opt. Soc. Am.,  59, 936 (1969).
    [Crossref]
  13. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1371 (1963).
    [Crossref]
  14. A. W. Lohmann, Optik 14, 510 (1957).
  15. A. W. Lohmann, Appl. Opt. 5, 669 (1966).
    [Crossref] [PubMed]
  16. A. Forstner and H. Köhler, Optik 17, 434 (1960).
  17. G. R. Sapper and H. P. Pulvermacher, Optik 23, 101 (1965).
  18. K. Singh and K. N. Chopra, Appl. Opt. 8, 1695 (1969).
    [Crossref] [PubMed]
  19. P. K. Katti, K. Singh, and A. K. Kavathekar, Opt. Acta,  16, 629 (1969).
    [Crossref]
  20. L. Levi, Applied Optics, A Guide to Modem Optical System Design (John Wiley & Sons, Inc., New York, 1968), p. 505.
  21. K. Rosenhauer, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North–Holland Publishing Co., Amsterdam, 1967).
  22. K. Rosenhauer and K. J. Rosenbruch, Rept. Progr. Phys. 30, 1 (1967).
    [Crossref]
  23. R. M. Bracewell, Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 27.
  24. D. Post, Exper. Mech. 7, 154 (1967).
    [Crossref]
  25. K. Miyamoto, J. Opt. Soc. Am. 48, 567 (1958).For W= 1.0, the sine-wave response is positive at high frequencies, e.g., at ω= 1.8 and not less than zero as given in Ref. 25. This very small error appears to have been made in plotting the portion of the curve near ω= 1.8.
    [Crossref]

1969 (3)

1967 (2)

K. Rosenhauer and K. J. Rosenbruch, Rept. Progr. Phys. 30, 1 (1967).
[Crossref]

D. Post, Exper. Mech. 7, 154 (1967).
[Crossref]

1966 (3)

1965 (1)

G. R. Sapper and H. P. Pulvermacher, Optik 23, 101 (1965).

1963 (1)

1961 (1)

S. H. Emara, J. Res. Natl. Bur. Std. (U.S.) 65A, 465 (1961).
[Crossref]

1960 (2)

A. Forstner and H. Köhler, Optik 17, 434 (1960).

F. W. Rosberry, J. Res. Natl. Bur. Std. (U.S.) 64C, 57 (1960).

1958 (1)

1957 (1)

A. W. Lohmann, Optik 14, 510 (1957).

1954 (1)

Barakat, R.

Bracewell, R. M.

R. M. Bracewell, Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 27.

Brown, E. B.

E. B. Brown, Modern Optics (Reinhold Publishing Corp., New York, 1965), Ch. 9, p. 479.

Chopra, K. N.

Coltman, J. W.

Emara, S. H.

S. H. Emara, J. Res. Natl. Bur. Std. (U.S.) 65A, 465 (1961).
[Crossref]

Forstner, A.

A. Forstner and H. Köhler, Optik 17, 434 (1960).

Françon, M.

A. Maréchal and M. Françon, Diffraction, Structure des Images (Editions de la Revue d’Optique, Paris, 1960).

Houston, A.

Humphreys, R.

Katti, P. K.

P. K. Katti, K. Singh, and A. K. Kavathekar, Opt. Acta,  16, 629 (1969).
[Crossref]

Kavathekar, A. K.

P. K. Katti, K. Singh, and A. K. Kavathekar, Opt. Acta,  16, 629 (1969).
[Crossref]

K. Singh and A. K. Kavathekar, J. Opt. Soc. Am.,  59, 936 (1969).
[Crossref]

Köhler, H.

A. Forstner and H. Köhler, Optik 17, 434 (1960).

Kottler, F.

Levi, L.

L. Levi, Applied Optics, A Guide to Modem Optical System Design (John Wiley & Sons, Inc., New York, 1968), p. 505.

Lnfoot, E. H.

E. H. Lnfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

Lohmann, A. W.

Maréchal, A.

A. Maréchal and M. Françon, Diffraction, Structure des Images (Editions de la Revue d’Optique, Paris, 1960).

Miyamoto, K.

Muratah, K.

K. Muratah in Progress in Optics V, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1966), Ch. IV, p. 201.

Newman, A.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1963).

Perrin, F. H.

Post, D.

D. Post, Exper. Mech. 7, 154 (1967).
[Crossref]

Pulvermacher, H. P.

G. R. Sapper and H. P. Pulvermacher, Optik 23, 101 (1965).

Rosberry, F. W.

F. W. Rosberry, J. Res. Natl. Bur. Std. (U.S.) 64C, 57 (1960).

Rosenbruch, K. J.

K. Rosenhauer and K. J. Rosenbruch, Rept. Progr. Phys. 30, 1 (1967).
[Crossref]

Rosenhauer, K.

K. Rosenhauer and K. J. Rosenbruch, Rept. Progr. Phys. 30, 1 (1967).
[Crossref]

K. Rosenhauer, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North–Holland Publishing Co., Amsterdam, 1967).

Sapper, G. R.

G. R. Sapper and H. P. Pulvermacher, Optik 23, 101 (1965).

Singh, K.

Appl. Opt. (3)

Exper. Mech. (1)

D. Post, Exper. Mech. 7, 154 (1967).
[Crossref]

J. Opt. Soc. Am. (5)

J. Res. Natl. Bur. Std. (U.S.) (2)

F. W. Rosberry, J. Res. Natl. Bur. Std. (U.S.) 64C, 57 (1960).

S. H. Emara, J. Res. Natl. Bur. Std. (U.S.) 65A, 465 (1961).
[Crossref]

Opt. Acta (1)

P. K. Katti, K. Singh, and A. K. Kavathekar, Opt. Acta,  16, 629 (1969).
[Crossref]

Optik (3)

A. W. Lohmann, Optik 14, 510 (1957).

A. Forstner and H. Köhler, Optik 17, 434 (1960).

G. R. Sapper and H. P. Pulvermacher, Optik 23, 101 (1965).

Rept. Progr. Phys. (1)

K. Rosenhauer and K. J. Rosenbruch, Rept. Progr. Phys. 30, 1 (1967).
[Crossref]

Other (9)

R. M. Bracewell, Fourier Transform and its Applications (McGraw–Hill Book Co., New York, 1965), p. 27.

L. Levi, Applied Optics, A Guide to Modem Optical System Design (John Wiley & Sons, Inc., New York, 1968), p. 505.

K. Rosenhauer, in Advanced Optical Techniques, A. C. S. van Heel, Ed. (North–Holland Publishing Co., Amsterdam, 1967).

A. Maréchal and M. Françon, Diffraction, Structure des Images (Editions de la Revue d’Optique, Paris, 1960).

E. H. Lnfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Inc., Reading, Mass., 1963).

Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press Inc., New York, 1964), p. 484.

E. B. Brown, Modern Optics (Reinhold Publishing Corp., New York, 1965), Ch. 9, p. 479.

K. Muratah in Progress in Optics V, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1966), Ch. IV, p. 201.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

F. 1(a)
F. 1(a)

A general periodic object of triangular-wave form. Width of bright portion at the mean value of irradiance is αp where α 50. a is the mean background irradiance and b is the modulation. (b) Sharpened moiré fringes with profile as in figure 1(a). (c) Moiré fringes with triangular-wave profile (α = 0.50).

F. 2
F. 2

Far-field irradiance distribution in the images of triangular-wave object. α = 0. The spatial frequencies are ω = 0.20, 0.60, and 1.0. ωz′ is the reduced distance in image plane.

F. 3
F. 3

Same as Fig. 2. α = 0.10. Broken lines with dots (— · — · — ·) show the edges of the geometrical image.

F. 4
F. 4

Same as Fig. 3. α = 0.25.

F. 5
F. 5

Same as Fig. 3. α = 0.50.

F. 6
F. 6

Comparison of normalized irradiance distribution in the images formed by a narrow rectangular aperture for W = 0.25 and W = 0.50. (a) ω = 0.60, (b) ω = 1.40. α = 0.10, 0.25, and 0.50 in each case. Solid and dotted lines are for W = 0.25 and 0.50, respectively.

F. 7
F. 7

Triangular-wave response for slit aperture with W = 0. α = 0.10, 0.25, and 0.50. Curve marked 5 represents the sine-wave response for the same value of W. Broken lines with dots (— · — · — ·) represent the ideal case of uniform illumination.

F. 8
F. 8

Same as Fig. 7. W = 0.25.

F. 9
F. 9

Same as Fig. 7. W = 0.50.

F. 10
F. 10

Same as Fig. 7. W = 10.

F. 11
F. 11

Comparison of triangular-wave response (α =0.50) of slit aperture for various values of aberration parameter (W = 0.25, 0.50, and 1.0). Broken lines with dots (— · — · — ·) correspond to the ideal case.

F. 12
F. 12

Comparison of triangular-wave response of slit (W = 0.50) for α = 0.45 and sine-wave response (S) of slit.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

H ( z ) = F ( z z ) G ( z ) d z ,
G ( z ) = a b + 2 α b + 4 b π 2 α n = 1 sin 2 n π α n 2 cos n ω z ,
H ( z ) = a b + 2 α b + 4 b π 2 α n = 1 n f ( n ω ) sin 2 n π α n 2 cos n ω z ,
f ( ω ) = sin [ 4 π W ω ( 1 | ω / 2 | ) ] 4 π W ω when | ω | 2.0 = 0 when | ω | > 2.0 ,
G ( z ) 1 + 2 n = 1 cos n ω z .
H ( z ) 1 + 2 n = 1 n f ( n ω ) cos n ω z .