Abstract

The process of producing a hologram of an object that is transmitting or reflecting diffuse, partially coherent, quasimonochromatic light is described mathematically. The discussion shows how the degree of coherence between the reference beam and the beam illuminating the object affects the reconstruction. The types of image degradation resulting from the use of partially coherent light are outlined. The application of holography to the measurement of second-order spatial coherence is suggested and a possible experiment is described.

© 1966 Optical Society of America

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References

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  1. D. Gabor, Nature 161, 777 (1948).
    [Crossref]
  2. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  3. E. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  4. For a discussion of the degree of coherence see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), in particular 2nd ed., Sec. 10.3.
  5. Strictly speaking, this requires a positive plate with γ= 2 since, for a positive exposed to an irradiance I(P) for a time t0, logtransmitted fluxincident flux=γ logI(P)t0. For γ= 2, [transmitted fluxincident flux]12=I(P)t0 so transmitted amplitudeincident amplitude=I(P)t0. However, holograms can be made with other γ’s and are, in fact, usually made with negative plates developed to γ’s of 6 or more.
  6. U(P,t)− reconstructs the conjugate real image. See Ref. 3, p. 1300.
  7. This discussion has been restricted to plane waves for simplicity. It is easy to show that for an undistorted reconstruction, it is essential that the reference beam have the same form as the reconstructing beam, but this form can be as complicated as desired. For an interesting example of this, see H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
    [Crossref]
  8. See, for example, Ref. 4, p. 508 ff.

1965 (1)

This discussion has been restricted to plane waves for simplicity. It is easy to show that for an undistorted reconstruction, it is essential that the reference beam have the same form as the reconstructing beam, but this form can be as complicated as desired. For an interesting example of this, see H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
[Crossref]

1964 (1)

1962 (1)

1948 (1)

D. Gabor, Nature 161, 777 (1948).
[Crossref]

Born, M.

For a discussion of the degree of coherence see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), in particular 2nd ed., Sec. 10.3.

Gabor, D.

D. Gabor, Nature 161, 777 (1948).
[Crossref]

Kogelnik, H.

This discussion has been restricted to plane waves for simplicity. It is easy to show that for an undistorted reconstruction, it is essential that the reference beam have the same form as the reconstructing beam, but this form can be as complicated as desired. For an interesting example of this, see H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
[Crossref]

Leith, E.

Upatnieks, J.

Wolf, E.

For a discussion of the degree of coherence see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), in particular 2nd ed., Sec. 10.3.

Bell System Tech. J. (1)

This discussion has been restricted to plane waves for simplicity. It is easy to show that for an undistorted reconstruction, it is essential that the reference beam have the same form as the reconstructing beam, but this form can be as complicated as desired. For an interesting example of this, see H. Kogelnik, Bell System Tech. J. 44, 2451 (1965).
[Crossref]

J. Opt. Soc. Am. (2)

Nature (1)

D. Gabor, Nature 161, 777 (1948).
[Crossref]

Other (4)

See, for example, Ref. 4, p. 508 ff.

For a discussion of the degree of coherence see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), in particular 2nd ed., Sec. 10.3.

Strictly speaking, this requires a positive plate with γ= 2 since, for a positive exposed to an irradiance I(P) for a time t0, logtransmitted fluxincident flux=γ logI(P)t0. For γ= 2, [transmitted fluxincident flux]12=I(P)t0 so transmitted amplitudeincident amplitude=I(P)t0. However, holograms can be made with other γ’s and are, in fact, usually made with negative plates developed to γ’s of 6 or more.

U(P,t)− reconstructs the conjugate real image. See Ref. 3, p. 1300.

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Figures (3)

Fig. 1
Fig. 1

A system for producing a hologram in the plane H. P is any point in that plane. R is a plane in the plane-wave reference beam. O is an illuminated, diffusely reflecting object. s and smP are the distances indicated.

Fig. 2
Fig. 2

W(P,t) is the total wave due to the illuminated object and crossing the plane H.

Fig. 3
Fig. 3

A system to record, holographically, the coherence of the light reaching the object O. O is a piece of ground glass or other uniform, diffusing object. M is a very small mirror centered at the point m1 in the object. S is a small, circular, incoherent source. L1 and L2 are lenses to form a plane wave reference beam that is directed to the hologram plate H by the lower mirror shown. R is defined as in Fig. 1.

Equations (18)

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I ( P ) = V P ( t ) V P * ( t ) ,
V P ( t ) = V R P ( t ) + m V m P ( t ) .
I ( P ) = m V m P ( t ) V m P * ( t ) + m n m V m P ( t ) V n P * ( t ) + V R P ( t ) V R P * ( t ) + m 2 Re V m P ( t ) V R P * ( t ) .
I ( P ) = I o ( P ) + I B ( P ) + 2 m Re V m P ( t ) V R P * ( t ) ,
V m P ( t ) = A m ( t - s m P / c ) e j ( ω t - s m P k + ϕ m ) ,             k = ω / c
V R P ( t ) = A R P ( t - s / c ) e j ( ω t - s k ) ,
Re V m P ( t ) V R P * ( t ) = Re ( 1 / s m P ) A m ( t - s m P / c ) A R P ( t - s / c ) * × exp { j [ ω ( s - s m P ) / c + ϕ m ] } = Re { 1 / s m P ) A m ( t ) A R P ( t - τ m P ) * e j ( ω τ m P + ϕ m ) } , τ m P = ( s - s m P ) / c .
γ m R P ( τ m P ) = A m ( t ) A R P ( t - τ m P ) * e j ( ω τ m P + ϕ m ) [ s m P 2 I m ( P ) I R ( P ) ] 1 2
I R ( P ) = A R P ( t - τ m P ) A R P ( t - τ m P ) *
exp j [ arg A m ( t ) - arg A R P ( t - τ m P ) ] = M e j θ m R P ,
A m ( t ) A R P ( t - τ m P ) * = A m A R P M e j θ m R P .
Re V m P ( t ) V R P * ( t ) = Re { [ I m ( P ) I R ( P ) ] 1 2 γ m R P ( τ m P ) } = [ I m ( P ) I R ( P ) ] 1 2 γ m R P ( τ m P ) × cos ( ω τ m P + ϕ m + θ m R P ) .
I ( P ) = I 0 ( P ) + I B ( P ) + 2 [ I R ( P ) ] 1 2 m [ I m ( P ) ] 1 2 × γ m R P ( τ m P ) cos ( ω τ m P + ϕ m + θ m R P ) .
m [ I R ( P ) I m ( P ) ] 1 2 γ m R P ( τ m P ) × cos ( ω t ± ω τ m P ± ϕ m ± θ m R P ) = m [ I R ( P ) ] 1 2 s m P A m γ m R P ( τ m P ) × cos ( ω t s m P k ± [ ϕ m + s k + θ m R P ] ) = U ( P , t ) ± ,
I m ( P ) = ( 1 / s m P 2 ) A m ( t ) A m ( t ) * = ( 1 / s m P 2 ) A m 2
[ I m ( P ) ] 1 2 = ( 1 / s m P ) A m .
W ( P , t ) = m 1 s m P A m cos ( ω t - s m P k + ϕ m ) ,
γ m R P ( τ m P ) γ m R P ( O ) γ m m 1 ( O ) ,