Abstract

Spatial filtering techniques used to detect thin phase structures can be represented by a unique transfer function (phase-contrast function, PhCF). The PhCF is here formulated in terms of mathematical symmetry to define, analyze, and classify schlieren techniques (ST). It is shown that all ST can be identified by essentially the same PhCF. This latter function performs a passband Hilbert transformation under coherent illumination. It can also act as a differential filter provided that the illumination is noncoherent.

© 1979 Optical Society of America

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References

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  1. E. Menzel, Optik (Stuttgart) 15, 460 (1958).
  2. E. Menzel, Optik (Stuttgart) 39, 170 (1973).
  3. H. Hausser, Opt. Acta 9, 121 (1962).
    [Crossref]
  4. K. J. Hanszen, Advances in Optical and Electron Microscopy (Academic Press, London, 1971), Vol. 4, p. 43.
  5. H. Pulvermacher, Optik (Stuttgart) 45, 111 (1976).
  6. M. G. van Heel, Optik (Stuttgart) 49, 389 (1978).
  7. J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).
  8. J. Ojeda-Castaneda, Bol. Inst. Tonantzintla 2, 293 (1978).
  9. J. Ojeda-Castaneda, Appl. Opt.18, in press (1979).
    [PubMed]
  10. H. H. Hopkins, J. Opt. Soc. Am. 69, 5 (1979).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 111.
  12. R. V. Shack, Proc. SPIE 46, 39 (1974).
    [Crossref]
  13. H. Wolter, Ann. Phys. (Leipzig) 7, 341 (1950).
  14. H. H. Hopkins, Proc. R. Soc. London Ser. A: 208, 263 (1951).
    [Crossref]
  15. H. H. Hopkins, Proc. R. Soc. London Ser. A: 217, 408 (1953).
    [Crossref]
  16. J. Ojeda-Castaneda, Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 251.
  17. J. Ojeda-Castaneda, Opt. Acta (Revised manuscript submitted for publication).
  18. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 267.
  19. Ref. 11, p. 193.
  20. R. Barakat, J. Opt. Soc. Am. 59, 1432 (1969).
    [Crossref]
  21. S. Lowenthal, Y. Belvaux, Appl. Phys. Lett. 11, 49 (1967).
    [Crossref]
  22. J. E. A. Landgrave, M. Sc. Report, Imperial College of Science and Technology, U.K. (1974).
  23. J. Ojeda-Castaneda, ICO-11 Proceedings, J. Bescos et al., Eds. (1978) p. 288.
  24. J. B. De Velis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), p. 168.
  25. J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
    [Crossref]
  26. J. Ojeda-Castaneda, Ph.D. thesis, University of Reading, U.K. (1976), pp. 7–23.

1979 (1)

H. H. Hopkins, J. Opt. Soc. Am. 69, 5 (1979).

1978 (2)

M. G. van Heel, Optik (Stuttgart) 49, 389 (1978).

J. Ojeda-Castaneda, Bol. Inst. Tonantzintla 2, 293 (1978).

1977 (1)

J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).

1976 (1)

H. Pulvermacher, Optik (Stuttgart) 45, 111 (1976).

1974 (1)

R. V. Shack, Proc. SPIE 46, 39 (1974).
[Crossref]

1973 (1)

E. Menzel, Optik (Stuttgart) 39, 170 (1973).

1969 (1)

1967 (1)

S. Lowenthal, Y. Belvaux, Appl. Phys. Lett. 11, 49 (1967).
[Crossref]

1965 (1)

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

1962 (1)

H. Hausser, Opt. Acta 9, 121 (1962).
[Crossref]

1958 (1)

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

1953 (1)

H. H. Hopkins, Proc. R. Soc. London Ser. A: 217, 408 (1953).
[Crossref]

1951 (1)

H. H. Hopkins, Proc. R. Soc. London Ser. A: 208, 263 (1951).
[Crossref]

1950 (1)

H. Wolter, Ann. Phys. (Leipzig) 7, 341 (1950).

Barakat, R.

Belvaux, Y.

S. Lowenthal, Y. Belvaux, Appl. Phys. Lett. 11, 49 (1967).
[Crossref]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 267.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

De Velis, J. B.

J. B. De Velis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), p. 168.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 111.

Hanszen, K. J.

K. J. Hanszen, Advances in Optical and Electron Microscopy (Academic Press, London, 1971), Vol. 4, p. 43.

Hausser, H.

H. Hausser, Opt. Acta 9, 121 (1962).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, J. Opt. Soc. Am. 69, 5 (1979).

H. H. Hopkins, Proc. R. Soc. London Ser. A: 217, 408 (1953).
[Crossref]

H. H. Hopkins, Proc. R. Soc. London Ser. A: 208, 263 (1951).
[Crossref]

Landgrave, J. E. A.

J. E. A. Landgrave, M. Sc. Report, Imperial College of Science and Technology, U.K. (1974).

Lowenthal, S.

S. Lowenthal, Y. Belvaux, Appl. Phys. Lett. 11, 49 (1967).
[Crossref]

Menzel, E.

E. Menzel, Optik (Stuttgart) 39, 170 (1973).

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

Moya, J.

J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, Bol. Inst. Tonantzintla 2, 293 (1978).

J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).

J. Ojeda-Castaneda, Appl. Opt.18, in press (1979).
[PubMed]

J. Ojeda-Castaneda, Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 251.

J. Ojeda-Castaneda, Opt. Acta (Revised manuscript submitted for publication).

J. Ojeda-Castaneda, ICO-11 Proceedings, J. Bescos et al., Eds. (1978) p. 288.

J. Ojeda-Castaneda, Ph.D. thesis, University of Reading, U.K. (1976), pp. 7–23.

Pulvermacher, H.

H. Pulvermacher, Optik (Stuttgart) 45, 111 (1976).

Reynolds, G. O.

J. B. De Velis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), p. 168.

Rodríguez, G.

J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).

Shack, R. V.

R. V. Shack, Proc. SPIE 46, 39 (1974).
[Crossref]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

van Heel, M. G.

M. G. van Heel, Optik (Stuttgart) 49, 389 (1978).

Wolter, H.

H. Wolter, Ann. Phys. (Leipzig) 7, 341 (1950).

Ann. Phys. (Leipzig) (1)

H. Wolter, Ann. Phys. (Leipzig) 7, 341 (1950).

Appl. Phys. Lett. (1)

S. Lowenthal, Y. Belvaux, Appl. Phys. Lett. 11, 49 (1967).
[Crossref]

Bol. Inst. Tonantzintla (2)

J. Ojeda-Castaneda, J. Moya, G. Rodríguez, Bol. Inst. Tonantzintla 2, 209 (1977).

J. Ojeda-Castaneda, Bol. Inst. Tonantzintla 2, 293 (1978).

J. Opt. Soc. Am. (2)

H. H. Hopkins, J. Opt. Soc. Am. 69, 5 (1979).

R. Barakat, J. Opt. Soc. Am. 59, 1432 (1969).
[Crossref]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[Crossref]

Opt. Acta (1)

H. Hausser, Opt. Acta 9, 121 (1962).
[Crossref]

Optik (Stuttgart) (4)

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

E. Menzel, Optik (Stuttgart) 39, 170 (1973).

H. Pulvermacher, Optik (Stuttgart) 45, 111 (1976).

M. G. van Heel, Optik (Stuttgart) 49, 389 (1978).

Proc. R. Soc. London Ser. A (2)

H. H. Hopkins, Proc. R. Soc. London Ser. A: 208, 263 (1951).
[Crossref]

H. H. Hopkins, Proc. R. Soc. London Ser. A: 217, 408 (1953).
[Crossref]

Proc. SPIE (1)

R. V. Shack, Proc. SPIE 46, 39 (1974).
[Crossref]

Other (11)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 111.

J. Ojeda-Castaneda, Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 251.

J. Ojeda-Castaneda, Opt. Acta (Revised manuscript submitted for publication).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 267.

Ref. 11, p. 193.

J. Ojeda-Castaneda, Appl. Opt.18, in press (1979).
[PubMed]

K. J. Hanszen, Advances in Optical and Electron Microscopy (Academic Press, London, 1971), Vol. 4, p. 43.

J. Ojeda-Castaneda, Ph.D. thesis, University of Reading, U.K. (1976), pp. 7–23.

J. E. A. Landgrave, M. Sc. Report, Imperial College of Science and Technology, U.K. (1974).

J. Ojeda-Castaneda, ICO-11 Proceedings, J. Bescos et al., Eds. (1978) p. 288.

J. B. De Velis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967), p. 168.

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Figures (4)

Fig. 1
Fig. 1

(a) Normalized coordinates employed to represent the image-formation process. (b) Schematic representation of the image-formation process of a thin phase grating employing the PhCF formalism.

Fig. 2
Fig. 2

Coherent and phase-contrast transfer functions (for coherent and noncoherent illuminaton) of the following ST: (a) Foucault’s test, (b) Wolter’s Minimumschlierenverfahren, (c) modulation contrast, and (d) modified modulation contrast.

Fig. 3
Fig. 3

(a) Filtered Fourier spectrum of thin phase bars. For the dashed line the width of the bar is u0 = 0.25, and for the solid line the width is u0 = 2.0. (b) Image irradiance variations about the irradiance background due to the above-mentioned thin phase bars when the PhCF is that shown in the third column of Fig. 2(c). The scale along the B′ axis is in units of α, where α is the phase difference produced by the thin phase bars.

Fig. 4
Fig. 4

Coherent and phase-contrast transfer functions associated with two one-sided methods. (a) Variable passband phase contrast, (b) Minimumschlierenverfahren.

Equations (12)

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A ( u , v ) = 1 + i Φ ( u , v ) ,
A ( u , v ) = f ( x = 0 , y = 0 ) + i - f ( x , y ) ϕ ( x , y ) × exp [ i 2 π ( u x + v y ) ] d x d y ,
B ( u , v ) = A ( u , v ) 2 = 1 + - p ( x , y ) ϕ ( x , y ) exp [ i 2 π ( u x + v y ) ] d x d y ,
p ( x , y ) = i [ f * ( o , o ) f ( x , y ) - f ( o , o ) × f * ( - x , - y ) ] / f ( o , o ) 2 .
A ( u , v ) = 1 + i α cos 2 π s u ,
B ( u , v ) = 1 + α p ( x = s , y = o ) × cos [ 2 π s u + β ( x = s , y = o ) ] ,
p ( s , o ) = ( p ( s , o ) exp [ i β ( s , o ) ] ,
p ( s , o ) = R ( s , o ) + i I ( s , o ) ,
B ( u , v ) = 1 + α R ( s , o ) cos 2 π s u , - α I ( s , o ) sin 2 π s u ,
p ( s , o ) = Even [ p ( s , o ) ] + i odd [ p ( s , o ) ] .
R ( s , o ) = Even [ p ( s , o ) ] ,
i I ( s , o ) = Odd [ p ( s , o ) ] .

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