Abstract

Tracing rays through a hologram is treated as a generalized case of tracing rays through a diffraction grating. The local “diffracting power” at a point of the hologram is determined by the geometry and wavelength of the beams whose recorded interference pattern constitutes the hologram. The resulting ray-tracing equations are valid for zone plates. An example of the analysis of the aberrations of a holographic system and a description of a modified system, in which these aberrations are corrected, are given.

© 1966 Optical Society of America

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References

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  1. G. R. Rosendahl, J. Opt. Soc. Am. 51, 1 (1961).
    [Crossref]
  2. G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 672 (1962).
    [Crossref]
  3. G. Toraldo di Francia, NBS Circ. 526, 165 (1954).
  4. D. Gabor, Nature 161, 777 (1948).
    [Crossref]
  5. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
    [Crossref]
  6. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  7. E. N. Leith and J. Upatnieks, J. Opt. Am. Soc.,  53, 1377 (1963).
    [Crossref]
  8. R. Barakat and A. Houston, Optica Acta 13, 1 (1966).
    [Crossref]
  9. A. S. Hoffman, J. G. Doidge, and D. G. Mooney, J. Opt. Soc. Am. 55, 1559L (1965).
    [Crossref]

1966 (1)

R. Barakat and A. Houston, Optica Acta 13, 1 (1966).
[Crossref]

1965 (1)

A. S. Hoffman, J. G. Doidge, and D. G. Mooney, J. Opt. Soc. Am. 55, 1559L (1965).
[Crossref]

1963 (1)

E. N. Leith and J. Upatnieks, J. Opt. Am. Soc.,  53, 1377 (1963).
[Crossref]

1962 (2)

1961 (1)

1954 (1)

G. Toraldo di Francia, NBS Circ. 526, 165 (1954).

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref]

1948 (1)

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

Barakat, R.

R. Barakat and A. Houston, Optica Acta 13, 1 (1966).
[Crossref]

Doidge, J. G.

A. S. Hoffman, J. G. Doidge, and D. G. Mooney, J. Opt. Soc. Am. 55, 1559L (1965).
[Crossref]

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref]

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

Hoffman, A. S.

A. S. Hoffman, J. G. Doidge, and D. G. Mooney, J. Opt. Soc. Am. 55, 1559L (1965).
[Crossref]

Houston, A.

R. Barakat and A. Houston, Optica Acta 13, 1 (1966).
[Crossref]

Leith, E. N.

E. N. Leith and J. Upatnieks, J. Opt. Am. Soc.,  53, 1377 (1963).
[Crossref]

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
[Crossref]

Mooney, D. G.

A. S. Hoffman, J. G. Doidge, and D. G. Mooney, J. Opt. Soc. Am. 55, 1559L (1965).
[Crossref]

Murty, M. V. R. K.

Rosendahl, G. R.

Spencer, G. H.

Toraldo di Francia, G.

G. Toraldo di Francia, NBS Circ. 526, 165 (1954).

Upatnieks, J.

E. N. Leith and J. Upatnieks, J. Opt. Am. Soc.,  53, 1377 (1963).
[Crossref]

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
[Crossref]

J. Opt. Am. Soc. (1)

E. N. Leith and J. Upatnieks, J. Opt. Am. Soc.,  53, 1377 (1963).
[Crossref]

J. Opt. Soc. Am. (4)

Nature (1)

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

NBS Circ. 526 (1)

G. Toraldo di Francia, NBS Circ. 526, 165 (1954).

Optica Acta (1)

R. Barakat and A. Houston, Optica Acta 13, 1 (1966).
[Crossref]

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Coordinate system used to describe a diffraction grating.

Fig. 2
Fig. 2

Geometry for recording a hologram. The hologram is recorded in the y, z plane with the center of the object wave at (−L, Hy, Hz) and the center of the reference wave at (−R cosβ, −R sinβ, 0).

Fig. 3
Fig. 3

Off-axis imagery of hologram made with axial object point. The hologram located in the y, z plane is illuminated by a source in the Hy, Hz plane. A virtual image is formed in the Hy, Hz plane.

Fig. 4
Fig. 4

Arrangement for recording a hologram which is a component of a corrected system.

Equations (20)

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m = m + p f λ
n = n .
m m = p f λ cos θ ,
n n = p f λ sin θ .
φ m = f cos θ = ( m m ) / p λ ,
φ n = f sin θ = ( n n ) / p λ .
m r = m r + p r λ r φ m ,
n = n r + p r λ r φ n .
l = [ 1 + 2 y sin β / R + ( y 2 + z 2 ) / R 2 ] 1 2 cos β
m = l ( sin β + y / R ) / cos β
n = l z / R cos β .
l = [ 1 + ( y H y ) 2 / L 2 + ( Z H z ) 2 / L 2 ] 1 2
m = l ( y H y ) / L
n = l ( Z H z ) / L .
φ m ( y , z , L , H y , H z ) = ( m m ) / λ
φ n ( y , z , L , H y , H z ) = ( n n ) / λ .
H y H y / cos 4 ° = 6.5 × 10 4 ( H y + H z ) + 9.26 × 10 5 H y Y + 4.63 × 10 5 H z Z
H z H z = 4.63 × 10 5 H z y
φ m ( y , z , L ) = y / λ ( L 2 + y 2 + z 2 ) 1 2
φ n ( y , z , L ) = z / λ ( L 2 + y 2 + z 2 ) 1 2 .