Abstract

The results of previous work on the theory of holography with partially coherent light are applied to the measurement of spatial coherence by means of Fourier-transform holograms. It is shown that if a Fourier-transform hologram is made using a coherent reference beam, the amplitude at any point in the reconstructed image will be multiplied by the magnitude of the normalized degree of coherence between the reference beam and the light from the corresponding point on the object. Thus, the reconstructed image can be used to measure coherence. An example of such a measurement is discussed in detail. The usefulness of the partial-coherence approach in interpreting several known holographic phenomena is mentioned.

© 1968 Optical Society of America

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References

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  1. D. J. de Bitetto, Appl. Phys. Letters 8, 78 (1966).
    [Crossref]
  2. G. W. Stroke and R. C. Restrick, Appl. Phys. Letters 7, 229 (1965).
    [Crossref]
  3. M. Lurie, J. Opt. Soc. Am. 56, 1369 (1966).
    [Crossref]
  4. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  5. G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
    [Crossref]
  6. J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
    [Crossref]
  7. M. Lurie, dissertation, Newark College of Engineering (June1967).
  8. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Sec. 10.3.1.
  9. See, for example, M. Born and E. Wolf, Ref. 8, Sec. 10.4.2.
  10. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Engelwood Cliffs, N. J., 1964), p. 28.
  11. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [Crossref]
  12. Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
    [Crossref]
  13. M. Lurie, J. Opt. Soc. Am. 57, 573A (1967).

1967 (2)

Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
[Crossref]

M. Lurie, J. Opt. Soc. Am. 57, 573A (1967).

1966 (2)

D. J. de Bitetto, Appl. Phys. Letters 8, 78 (1966).
[Crossref]

M. Lurie, J. Opt. Soc. Am. 56, 1369 (1966).
[Crossref]

1965 (4)

G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
[Crossref]

R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
[Crossref]

G. W. Stroke and R. C. Restrick, Appl. Phys. Letters 7, 229 (1965).
[Crossref]

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

1964 (1)

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Engelwood Cliffs, N. J., 1964), p. 28.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Sec. 10.3.1.

Brumm, D.

Caulfield, H. J.

Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
[Crossref]

de Bitetto, D. J.

D. J. de Bitetto, Appl. Phys. Letters 8, 78 (1966).
[Crossref]

Funkhouser, A.

Hermstreet, H. W.

Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
[Crossref]

Leith, E.

Lu, Sun

Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
[Crossref]

Lurie, M.

M. Lurie, J. Opt. Soc. Am. 57, 573A (1967).

M. Lurie, J. Opt. Soc. Am. 56, 1369 (1966).
[Crossref]

M. Lurie, dissertation, Newark College of Engineering (June1967).

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Engelwood Cliffs, N. J., 1964), p. 28.

Powell, R. L.

Restrick, R. C.

G. W. Stroke and R. C. Restrick, Appl. Phys. Letters 7, 229 (1965).
[Crossref]

Stetson, K. A.

Stroke, G. W.

G. W. Stroke and R. C. Restrick, Appl. Phys. Letters 7, 229 (1965).
[Crossref]

G. W. Stroke, D. Brumm, and A. Funkhouser, J. Opt. Soc. Am. 55, 1327 (1965).
[Crossref]

Upatnieks, J.

Winthrop, J. T.

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Sec. 10.3.1.

Worthington, C. R.

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Appl. Phys. Letters (2)

D. J. de Bitetto, Appl. Phys. Letters 8, 78 (1966).
[Crossref]

G. W. Stroke and R. C. Restrick, Appl. Phys. Letters 7, 229 (1965).
[Crossref]

J. Opt. Soc. Am. (5)

Phys. Letters (2)

Sun Lu, H. W. Hermstreet, and H. J. Caulfield, Phys. Letters 25A, 294 (1967).
[Crossref]

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Other (4)

M. Lurie, dissertation, Newark College of Engineering (June1967).

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), Sec. 10.3.1.

See, for example, M. Born and E. Wolf, Ref. 8, Sec. 10.4.2.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Engelwood Cliffs, N. J., 1964), p. 28.

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Figures (9)

Fig. 1
Fig. 1

An arrangement for producing a Fourier-transform hologram of an object O. The illumination of the object can have any degree of spatial coherence, but the reference beam, coming from the very small, quasimonochromatic source S, must be highly coherent.

Fig. 2
Fig. 2

An arrangement for reconstructing a real image from a hologram produced as in Fig. 1. The hologram is illuminated with spherical waves converging to the point S.

Fig. 3
Fig. 3

A system for illuminating an object with partially coherent, quasimonochromatic light and exposing a hologram of the object. The rotating ground glass G destroys the spatial coherence of the output of the laser L. Therefore, the coherence of the light reaching the object O from the small circular aperture A is easily calculated. A portion of the light near the optical axis passes through another aperture near the object and then through a 3-cm focal-length lens L and is used for the reference beam. M is a microscope objective, C is a condensing lens, and F is the film to record the hologram.

Fig. 4
Fig. 4

A drawing of the object O which is a 36 × 46 mm piece of ground glass with radial lines originating on the optical axis and marked in centimeters. The 2-mm-diam aperture centered on the axis, near the edge of the object, allows a highly coherent wave to pass undiffused. This is spread by a 3-cm-focal-length lens and forms the reference beam.

Fig. 5
Fig. 5

A photograph of the apparatus described in the text. The components can be identified with those in the schematic drawing in Fig. 3.

Fig. 6
Fig. 6

A print of a hologram made with the apparatus shown in Fig. 5. The pattern of rings is the Fresnel diffraction pattern of the coherent reference beam.

Fig. 7
Fig. 7

(a) A conventional photograph of the object drawn in Fig. 4. The coherence of the illumination varies over the object but this is not shown by the photograph. (b). A holographic reconstruction of the same object, illuminated precisely as it was in (a). Regions of low coherence are darker than those of high coherence. The pattern displayed agrees with the calculated variation of coherence over the object. A mirror image can be seen on the left. It is out of focus because the hologram is not a true Fourier-transform hologram, as explained in the text.

Fig. 8
Fig. 8

An example of holographic measurement of spatial coherence | γ r ( 0 ) | 2, showing the agreement with calculated values (solid curve). The curve was fitted to the data at the circled point. The minimum at r = 14.9 is nearly obscured by noise.

Fig. 9
Fig. 9

(a). A conventional photograph of the object used in Figs. 5 and 8, illuminated through stationary ground glass. The illuminating wavefront has a random distribution of amplitude and phase over the object, but is coherent. (b). A holographic reconstruction of the same object, illuminated in the same way. Because the illumination is coherent, the reconstruction is an accurate representation of the object as it appeared to the eye or to a conventional camera as in (a). Once again, the out-of-focus mirror image can be seen on the left.

Equations (6)

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B V ( m ) = B 0 ( m ) | γ Rm | 2 ,
B R ( m ) = B 0 ( m ) | γ Rm | 2 ,
| γ 12 | = | 2 J 1 ( υ ) / υ | = | γ r | ,
ρ = 0.0077 mm D = 30 cm ω / c = 2 π / λ = 2 π / 633 nm ,
γ = J 0 ( C Δ ) ,
A = A J 0 ( C Δ ) ,