Abstract

An exact geometrical-optics theory of holography is worked out. A simple derivation of the holographic ray-tracing equations is given; it is used to determine the principal points of a hologram. The well-known paraxial conjugate equations of holography are shown to be exact relations, if the distance and the angles are measured in an appropriate manner. The intersection of all the principal rays determine the position of the aberration-free image.

© 1975 Optical Society of America

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References

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  1. C. W. Helstrom, J. Opt. Soc. Am. 56, 433 (1966).
    [Crossref]
  2. A. Offner, J. Opt. Soc. Am. 56, 1509 (1966).
    [Crossref]
  3. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [Crossref]
  4. I. A. Abramowitz and J. M. Ballantyne, J. Opt. Soc. Am. 57, 1522 (1967).
    [Crossref]
  5. J. N. Latta, Appl. Opt. 10, 2698 (1971).
    [Crossref] [PubMed]

1971 (1)

1967 (2)

1966 (2)

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Figures (4)

FIG. 1
FIG. 1

Some of the principal points of a hologram. The plane that contains the reference source S, the reconstruction source R, and the object O is the meridian plane. The intersection of the meridian plane with the hologram is the meridian line. The line SO is the principal axis of the hologram and the intersection of the axis SO with the hologram is the vertex V′. O′ and S′ are the virtual object and the virtual reference source, respectively. Not shown in the figure is the sagittal plane, which is a plane perpendicular to the meridian plane that contains the meridian line. Also not shown here are the mirror images in the sagittal plane of the object and reference sources, which are useful for constructing certain principal rays.

Fig. 2
Fig. 2

Three of the principal rays for the direct images of a hologram. The intersection of the axis SO with the meridian line is the vertex of the hologram. The three principal rays that radiate from the reconstruction source R all intersect at the direct image Id, after being deviated at the hologram in the manner described in the text.

Fig. 3
Fig. 3

Three of the principal rays for the conjugate image of a hologram. The notation is the same as in Fig. 1. The three principal rays that radiate from the reconstruction source R all meet at the conjugate image point Ic.

Fig. 4
Fig. 4

Geometrical construction for the derivation of the conjugate equation for the direct image.

Equations (29)

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1 ρ i = 1 ρ r ± ( 1 ρ s - 1 ρ 0 ) ,
θ i - θ 0 = θ r - θ s ,
F s ( M , t ) = A s exp { i [ k s · ( r - r s ) - ω t ] } ,
F 0 ( M , t ) = A 0 exp { i [ k 0 · ( r - r 0 ) - ω t ] } ,
L ( M ) = F s ( M , t ) + F 0 ( M , t ) 2 , = A 0 2 + A s 2 + A s A 0 exp { i [ ± ( k s - k 0 ) · r ± ( k s · r s - k 0 · r 0 ) ] } .
F r ( M , t ) = A r exp { i [ k r · ( r - r r ) - ω t ] } ,
F ( M ) = ( A 0 + A s ) 2 A r exp { i [ k r · ( r r - r ) ] } + A s A 0 A r exp { i k r [ ± ( k s - k 0 ) ] · r - k r · r r ± ( k s · r s - k 0 · r 0 ) } .
k · r = [ k r ± ( k s - k 0 ) ] · r , k r cos θ = k r r cos θ r ± ( k s r cos θ s - k 0 r cos θ 0 ) ,
ψ = k r r r ± ( k s r s - k 0 r 0 ) .
cos θ d = cos θ r - ( cos θ s - cos θ 0 ) ,
cos θ c = cos θ r + ( cos θ s - cos θ 0 ) .
θ d = θ 0 , ψ d = k 0 r 0 ,
cos θ d = cos θ 0 .
cos θ d = cos θ r .
θ d = θ s ± π .
y R x - x R y = 0 , y 0 x - ( x 0 - x R y s - x s y R y s - y R ) y - ( x R y s - x s y R ) y 0 y s - y R = 0 , y s x + ( x s - x 0 y R + x R y 0 y R + y 0 ) y - x s y s = 0.
y s x - ( x s + x R y 0 - x 0 y R y R + y 0 ) y - x R y 0 - x 0 y R y R + y 0 y s = 0.
R P I Q = P V Q V ,
R P B V = P V + V O V O ,
I Q B V = Q V + V S V S .
R P I Q = ( P V + V O ) V S ( Q V + V S ) V O .
P V Q V = ( P V + V O ) V S ( Q V + V S ) V .
1 ρ i d = 1 ρ r - 1 ρ s + 1 ρ o ,
θ o - θ i = θ s - θ r .
1 ρ i c = 1 ρ r + 1 ρ s - 1 ρ 0 ,
θ i c - θ 0 = θ r - θ s ,
1 ρ i 1 ρ 0 ,
1 ρ i = 1 ρ r ± 1 ρ 0 .
1 F = ± ( 1 ρ s - 1 ρ 0 ) .