Abstract

Formulas for angular, lateral, and longitudinal magnification in reconstructed images by holography (imaging by reconstructed wavefronts) are derived and discussed as a function of object–hologram distance, radii of spherical reference and illuminating wavefronts, wavelength ratio of reconstructing to recording radiation, and scale-change factor of the hologram. Expressions for third-order aberrations in the reconstructed wavefronts of point objects are given and conditions are established under which one or more of the aberration coefficients vanish, taking into account an off-axis angle of both reference and illuminating beam.

© 1965 Optical Society of America

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References

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  1. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  2. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]
  3. L. J. Cutrona and et al., IRE Trans. Inform. Theory IT-6, 386 (1960).
    [Crossref]
  4. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  5. R. S. Longhurst, Geometrical and Physical Optics (John Wiley & Sons, Inc., New York, 1957), p. 288.
  6. G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 313 (1952).
  7. G. W. Stroke and D. G. Falconer, Proceedings of the Symposium on Optical and Electro-optical Information Processing (MIT Press, Cambridge, Massachusetts, 1964).

1963 (1)

1962 (1)

1960 (1)

L. J. Cutrona and et al., IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

1952 (1)

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 313 (1952).

1949 (1)

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

Cutrona, L. J.

L. J. Cutrona and et al., IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

Falconer, D. G.

G. W. Stroke and D. G. Falconer, Proceedings of the Symposium on Optical and Electro-optical Information Processing (MIT Press, Cambridge, Massachusetts, 1964).

Gabor, D.

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

Leith, E. N.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (John Wiley & Sons, Inc., New York, 1957), p. 288.

Rogers, G. L.

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 313 (1952).

Stroke, G. W.

G. W. Stroke and D. G. Falconer, Proceedings of the Symposium on Optical and Electro-optical Information Processing (MIT Press, Cambridge, Massachusetts, 1964).

Upatnieks, J.

IRE Trans. Inform. Theory (1)

L. J. Cutrona and et al., IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

J. Opt. Soc. Am. (2)

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

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

Proc. Roy. Soc. Edinburgh (1)

G. L. Rogers, Proc. Roy. Soc. Edinburgh A63, 313 (1952).

Other (2)

G. W. Stroke and D. G. Falconer, Proceedings of the Symposium on Optical and Electro-optical Information Processing (MIT Press, Cambridge, Massachusetts, 1964).

R. S. Longhurst, Geometrical and Physical Optics (John Wiley & Sons, Inc., New York, 1957), p. 288.

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Figures (2)

F. 1
F. 1

Basic arrangement for recording of the hologram.

F. 2
F. 2

Illustrating the phases of spherical wavefronts in the hologram plane.

Equations (33)

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I = | o + r | 2 = | o | 2 + | r | 2 + o * r + o r * .
H = c ( | o | 2 + | r | 2 + o * r + o r * ) .
H R = c o * r and H V = c o r * .
φ 0 ( x , y ) = ( 2 π / λ 0 ) d = ( 2 π / λ 0 ) ( P Q P 0 ) = ( 2 π / λ 0 ) { [ ( x x 0 ) 2 + ( y y 0 ) 2 + z 0 2 ] 1 2 ( x 0 2 + y 0 2 + z 0 2 ) 1 2 } = ( 2 π / λ 0 ) z 0 ( { 1 + [ ( x x 0 ) 2 + ( y y 0 ) 2 ] / z 0 2 } 1 2 [ 1 + ( x 0 2 + y 0 2 ) / z 0 2 ] 1 2 ) .
φ 0 ( x , y ) = ( 2 π / λ 0 ) [ 1 2 z 0 1 ( x 2 + y 2 2 x x 0 2 y y 0 ) 1 8 z 0 3 ( x 4 + y 4 + 2 x 2 y 2 4 x 3 x 0 4 y 3 y 0 4 x 2 y y 0 4 x y 2 x 0 + 6 x 2 x 0 2 + 6 y 2 y 0 2 + 2 x 2 y 0 2 + 2 y 2 x 0 2 + 8 x y x 0 y 0 4 x x 0 3 4 y y 0 3 4 x x 0 y 0 2 4 x y 0 2 y 0 ) + higher - order terms ] .
Φ V = φ c + φ 0 φ r and Φ R = φ c φ 0 + φ r .
Φ v ( 1 ) = 2 π λ c 1 2 x 2 + y 2 2 x x c 2 y y c z c + 2 π λ 0 1 2 x 2 + y 2 2 x x 0 2 y y 0 z 0 2 π λ 0 1 2 x 2 + y 2 2 x x r 2 y y r z r .
Φ v ( 1 ) = 2 π λ c 1 2 { ( x 2 + y 2 ) ( 1 z c + μ m 2 z 0 μ m 2 z r ) 2 x ( x c z c + μ x 0 m z 0 μ x r m z r ) 2 y ( y c z c + μ y 0 m z 0 μ y r m z r ) } .
Φ v ( 1 ) = 2 π λ 0 1 2 x 2 + y 2 2 x a v 2 y b v Z v ,
Z v = m 2 z c z 0 z r m 2 z 0 z r + μ z c z r μ z c z 0 ; a v = m 2 x c z 0 z r + μ m x 0 z c z r μ m x r z c z 0 m 2 z 0 z r + μ z c z r μ z c z 0 .
Z R = m 2 z c z 0 z r m 2 z 0 z r μ z c z r + μ z c z 0 ; a R = m 2 x c z 0 z r μ m x 0 z c z r + μ m x r z c z 0 m 2 z 0 z r μ z c z r + μ z c z 0 .
M ang = d ( a / 2 ) / d ( x 0 / z 0 ) = ± μ / m .
M lat = d a / d x 0 ,
M V lat = m / [ 1 + ( m 2 z 0 / μ z c ) ( z 0 / z r ) ]
M R lat = m / [ 1 ( m 2 z 0 / μ z c ) ( z 0 / z r ) ] .
1 z 0 + 1 Z R = 1 f R = 1 z 0 + 1 z c μ m 2 z 0 + μ m 2 z r .
M long = d Z R d z 0 = m 2 μ d d z 0 z 0 1 z 0 [ ( m 2 / μ z c ) + ( 1 / z r ) ] = 1 μ m 2 { 1 z 0 [ ( m 2 / μ z c ) + ( 1 / z r ) ] } 2 = 1 μ M 2 lat .
Δ Z = ± ( Δ a / tan α 0 ) M lat ,
Δ Z = ± ( Δ a / tan α 0 ) ( 1 / μ ) M lat ,
Φ ( 3 ) = ( 2 π / λ c ) [ 1 8 Z 3 ( x 4 + y 4 + 2 x 2 y 2 4 x 3 a 4 y 3 b 4 x y 2 a 4 x 2 y b + 6 x 2 a 2 + 6 y 2 b 2 + 2 x 2 b 2 + 2 y 2 a 2 + 8 x y a b 4 x a 3 4 y b 3 4 x a b 2 4 y a 2 b ) ] .
W = ( 2 π / λ c ) [ 1 8 p 4 S spherical aberration ; + 1 2 ρ 3 ( cos θ C x + sin θ C y ) coma ; 1 2 ρ 2 ( cos 2 θ A x + sin 2 θ A y + 2 cos θ sin θ A x y ) astigmatsm ; 1 4 ρ 2 F field curvature ; + 1 2 ρ ( cos θ D x + sin θ D y ) ] distortion .
S = 1 z c 3 μ m 4 z 0 3 + μ m 4 z r 3 1 z R 3 = μ m 4 [ ( μ 2 m 2 1 ) ( 1 z 0 3 1 z r 3 ) 3 μ z c ( 1 z 0 2 + 1 z r 2 ) + 3 ( m 2 z c 2 μ m 2 z 0 z r ) ( 1 z 0 1 z r ) + 6 μ z 0 z r z c ] .
S = ( μ / m 4 ) [ ( μ 2 / m 2 ) 1 ] z 0 3
C x = x c z c 3 μ x 0 m 3 z 0 3 + μ x r m 3 z r 3 a R z R 3 = x c z c [ 1 z c 2 ( 1 z c μ m 2 z 0 + μ m 2 z r ) 2 ] x 0 μ z 0 m [ 1 m 2 z 0 2 ( 1 z c μ m 2 z 0 + μ m 2 z r ) 2 ] + x r z r μ m [ 1 m 2 z r 2 ( 1 z c μ m 2 z 0 + μ m 2 z r ) 2 ] .
C x = μ m 3 z 0 2 [ x c z c μ m x 0 z 0 ( μ 2 m 2 1 ) + x r z r μ 2 m 2 ] .
C x = μ m [ x 0 z 0 ( 1 m 2 z 0 2 1 z c 2 ) + x r z r ( 1 m 2 z 0 2 1 z c 2 ) ] .
A x = x c 2 z c 3 μ x 0 2 m 2 z 0 3 + μ x r 2 m 2 z r 3 a R 2 Z R 3 = x c 2 z c 3 μ x 0 2 m 2 z 0 3 + μ x r 2 m 2 z r 3 ( x c z c μ x 0 m z 0 + μ x r m z r ) 2 ( 1 z c μ m 2 z 0 + μ m 2 z r ) .
A x = μ m 2 z 0 [ x 0 2 z 0 2 ( μ 2 m 2 1 ) 2 μ m x 0 z 0 ( x c z c + μ x r m z r ) + ( x c z c + μ x r m z r ) 2 ] .
A x = μ m [ x 0 2 z 0 2 ( 1 m z 0 + μ m z c ) + 2 z c x 0 z 0 ( x c z c + μ x r m z r ) + x r 2 z r 2 ( 1 m z 0 μ m z c ) 2 z c x c x r z c z r ] .
A x = μ m 2 [ x 0 2 z 0 2 ( 1 z 0 + μ z c ) + x r 2 z r 2 ( 1 z 0 + μ z c ) ] .
F = x c 2 + y c 2 z c 3 μ ( x 0 2 + y 0 2 ) m 2 z 0 3 + μ ( x r 2 + y r 2 ) m 2 z r 3 a R 2 + b R 2 z R 3 .
D x = x c 3 + x c y c 2 z c 3 μ ( x 0 3 + x 0 y 0 2 ) m z 0 3 + μ ( x r 3 + x r y r 2 ) m z r 3 a R 3 + a R b R 2 Z R 3 .
D x = μ m [ x 0 3 z 0 3 ( μ 2 m 2 1 ) 3 μ m x 0 2 z 0 2 ( x c z c + μ m x r z r ) + 3 x 0 z 0 ( x c z c + μ m x r z r ) 2 x r 3 z r 3 ( μ 2 m 2 1 ) 3 x c x r z c z r ( x c z c + μ m x r z r ) 2 + x 0 y 0 2 z 0 3 ( μ 2 m 2 1 ) ] .