Abstract

Rainbow holographic image resolution, primary aberrations, and bandwidth requirements are presented. The results obtained for the rainbow holographic process are rather general, for which the conventional holographic image resolution, aberrations, and bandwidth requirements, can be derived. The conditions for the elimination of the five primary rainbow holographic aberrations are also given. These conditions may be useful for the application of obtaining a high-quality rainbow hologram image. In terms of bandwidth requirements, we have shown that the bandwidth requirement for a rainbow holographic construction is usually several orders lower than that of a conventional holographic process. Therefore, a lower-resolution recording medium can generally be used for most of the rainbow holographic constructions.

© 1983 Optical Society of America

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References

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  1. S. A. Benton, J. Opt. Soc. Am. 59, 1545A (1969).
  2. H. Chen, F. T. S. Yu, Opt. Lett. 2, 85 (1978).
    [CrossRef] [PubMed]
  3. F. T. S. Yu, H. Chen, Opt. Commun. 25, 173 (1978).
    [CrossRef]
  4. A. M. Tai, F. T. S. Yu, H. Chen, Appl. Opt. 18, 61 (1979).
  5. F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
    [CrossRef]
  6. J. C. Wyant, Opt. Lett. 1, 130 (1977).
    [CrossRef] [PubMed]
  7. H. Chen, Appl. Opt. 17, 3290 (1978).
    [CrossRef] [PubMed]
  8. P. N. Tamura, Appl. Opt. 17, 3343 (1978).
    [CrossRef] [PubMed]
  9. S. L. Zhuang, P. H. Ruterbusch, Y. W. Zhang, F. T. S. Yu, Appl. Opt. 20, 872 (1981).
    [CrossRef] [PubMed]
  10. M. Born, E. Wolf, Eds., Principles of Optics (Pergamon, New York, 1964).
  11. R. W. Meier, J. Opt. Soc. Am. 55, 989 (1965).
    [CrossRef]
  12. F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).

1981 (1)

1980 (1)

F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
[CrossRef]

1979 (1)

A. M. Tai, F. T. S. Yu, H. Chen, Appl. Opt. 18, 61 (1979).

1978 (4)

1977 (1)

1969 (1)

S. A. Benton, J. Opt. Soc. Am. 59, 1545A (1969).

1965 (1)

R. W. Meier, J. Opt. Soc. Am. 55, 989 (1965).
[CrossRef]

Benton, S. A.

S. A. Benton, J. Opt. Soc. Am. 59, 1545A (1969).

Chen, H.

F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
[CrossRef]

A. M. Tai, F. T. S. Yu, H. Chen, Appl. Opt. 18, 61 (1979).

F. T. S. Yu, H. Chen, Opt. Commun. 25, 173 (1978).
[CrossRef]

H. Chen, Appl. Opt. 17, 3290 (1978).
[CrossRef] [PubMed]

H. Chen, F. T. S. Yu, Opt. Lett. 2, 85 (1978).
[CrossRef] [PubMed]

Meier, R. W.

R. W. Meier, J. Opt. Soc. Am. 55, 989 (1965).
[CrossRef]

Ruterbusch, P. H.

Tai, A. M.

F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
[CrossRef]

A. M. Tai, F. T. S. Yu, H. Chen, Appl. Opt. 18, 61 (1979).

Tamura, P. N.

Wyant, J. C.

Yu, F. T. S.

S. L. Zhuang, P. H. Ruterbusch, Y. W. Zhang, F. T. S. Yu, Appl. Opt. 20, 872 (1981).
[CrossRef] [PubMed]

F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
[CrossRef]

A. M. Tai, F. T. S. Yu, H. Chen, Appl. Opt. 18, 61 (1979).

F. T. S. Yu, H. Chen, Opt. Commun. 25, 173 (1978).
[CrossRef]

H. Chen, F. T. S. Yu, Opt. Lett. 2, 85 (1978).
[CrossRef] [PubMed]

F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).

Zhang, Y. W.

Zhuang, S. L.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

S. A. Benton, J. Opt. Soc. Am. 59, 1545A (1969).

R. W. Meier, J. Opt. Soc. Am. 55, 989 (1965).
[CrossRef]

Opt. Commun. (1)

F. T. S. Yu, H. Chen, Opt. Commun. 25, 173 (1978).
[CrossRef]

Opt. Eng. (1)

F. T. S. Yu, A. M. Tai, H. Chen. Opt. Eng. 19, 666 (1980).
[CrossRef]

Opt. Lett. (2)

Other (2)

F. T. S. Yu, Introduction to Diffraction Information Processing and Holography (MIT Press, Cambridge, 1973).

M. Born, E. Wolf, Eds., Principles of Optics (Pergamon, New York, 1964).

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Figures (2)

Fig. 1
Fig. 1

A composite rainbow holographic construction and reconstruction process for the evaluation of the hologram image resolution and the primary aberrations: H1, primary hologram; SL, slit aperture; W, slit width; H2, rainbow holographic plate; O, object image point; I, rainbow hologram image point; R, convergent reference point source; C, white-light reconstruction point source.

Fig. 2
Fig. 2

A composite rainbow holographic construction and reconstruction process for determining the bandwidth requirements: H1, primary hologram; SL, slit aperture; W, slit width; H2, rainbow holographic plate; O1O2, extended hologram object image; R, convergent reference point source.

Tables (2)

Tables Icon

Table I Rainbow Holographic Aberrations and the Conditions for Elimination

Tables Icon

Table II Rainbow Holographic Bandwidth Requirements

Equations (36)

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P 11 = ( 1 d g ) x 0 + d w 2 g ,
P 12 = ( 1 d g ) x 0 d w 2 g ,
U 1 ( x ; k 1 ) = A 1 exp { i k 1 [ d 2 + ( x x 0 ) 2 ] 1 / 2 } ,
U 2 ( x ; k 1 ) = A 2 exp { i k 1 [ L 1 2 + ( x x r ) 2 ] 1 / 2 } ,
U 3 ( x ; k 2 ) = A 3 exp { i k 2 [ L 2 2 + ( x x c ) 2 ] 1 / 2 } ,
h l ( α x ; k 2 ) = A 4 exp { i k 2 [ l 2 + ( α x ) 2 ] 1 / 2 } ,
B i ( α ; k 2 ) = C 1 P 12 P 11 U 1 U * 2 U 3 h l d x = C 2 sinc [ d w λ 2 g l i ( α + α i ) ] ,
l i = d L 1 L 2 μ L 1 L 2 + d L 1 μ d L 2 ,
α i = l i ( μ x 0 d x c L 2 + μ x r L 1 ) ,
M i = α i x 0 = μ l i d = ( 1 d L 1 + d μ L 2 ) 1 ,
Δ H i = λ 2 l i ( 1 + s d ) W = λ 2 L 1 L 2 ( d + s ) ( μ L 1 L 2 + L 1 d μ L 2 d ) W ,
Δ H i = λ 1 ( d + s ) W .
φ ( x , k ) = k [ L 2 + ( x a ) 2 ] 1 / 2 = k [ 1 + ( x a ) 2 2 L ( x a ) 4 8 L 3 + ] .
φ i = k 2 { μ [ L 1 2 + ( x x r ) 2 + ( y y r ) 2 ] 1 / 2 μ [ d 2 + ( x x 0 ) 2 + ( y y 0 ) 2 ] 1 / 2 [ L 2 2 + ( x x c ) 2 + ( y y c ) 2 ] 1 / 2 + [ l i 2 + ( α i x ) 2 + ( β i y ) 2 ] 1 / 2 } .
φ i = φ c + φ p + φ t + φ h ,
φ t = k 2 8 { μ d 3 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 2 μ L 1 3 [ ( x x r ) 2 + ( y y r ) 2 ] 2 + 1 L 2 3 [ ( x x c ) 2 + ( y y c ) 2 ] 2 1 l i 3 [ ( α i x ) 2 + ( β i y ) 2 ] 2 } .
φ t = k 2 { 1 8 ( μ d 3 μ L 1 3 + 1 L 2 3 1 l i 3 ) r 4 1 2 [ ( μ d 3 x 0 μ L 1 3 x r + x c L 2 3 α i l i 3 ) cos θ + ( μ d 3 y 0 μ L 1 3 y r + y c L 2 3 β i l i 3 ) sin θ ] r 3 + 1 4 [ μ d 3 ( x 0 2 + y 0 2 ) μ L 1 3 ( x r 2 + y r 2 ) + 1 L 2 3 ( x c 2 + y c 2 ) 1 l i 3 ( α i 2 + β i 2 ) ] r 2 + 1 2 [ ( μ d 3 x 0 2 μ L 1 3 x r 2 + x c 2 L 2 3 α i 2 l i 2 ) cos 2 θ + ( μ d 3 y 0 2 μ L 1 3 y r 2 + y c 2 L 2 3 β i 2 l i 3 ) sin 2 θ + ( μ d 3 x 0 y 0 μ L 1 3 x r y r + 1 L 2 3 x c y c 1 l i 3 α i β i ) sin θ cos θ ] r 2 1 2 [ ( μ d 3 x 0 3 μ L 1 3 x r 3 + 1 L 2 3 x c 3 1 l i 3 α i 3 ) cos θ + ( μ d 3 y 0 3 μ L 1 3 y r 3 + 1 L 2 3 y c 3 1 l i 3 β i 3 ) sin θ ] r } .
φ t = k 2 [ r 4 8 S + r 3 2 ( C x cos θ + C y sin θ ) r 2 4 F r 2 2 ( A x cos 2 θ + 2 A x y sin θ cos θ + A y sin 2 θ ) + r 2 ( D x cos θ + D y sin θ ) ] ,
Δ S = ( 1 Δ λ 2 λ 2 ) [ S Δ λ 2 λ 1 ( 1 L 1 3 1 d 3 ) + λ 2 ( 1 l i 3 ) Δ λ 2 ] S Δ λ 2 λ 2 S + Δ λ 2 λ 1 [ ( 1 L 1 3 1 d 3 ) + L 2 ( L 1 d ) ( μ L 1 L 2 + L 1 d μ L 2 d ) 2 ( d L 1 L 2 ) 3 ] ,
Δ C x = ( 1 Δ λ 2 λ 2 ) { C x ( x 0 d 3 x r L 1 3 ) Δ λ 2 λ 1 Δ λ 2 λ 1 [ 1 l i 2 ( x 0 d x r L 1 ) + 2 L 2 ( L 1 d ) l i 2 ( μ L 1 L 2 + L 1 d μ L 2 d ) ( μ x 0 d + x c L 2 μ x r L 1 ) ] } C x Δ λ 2 λ 2 C x + Δ λ 2 λ 1 { ( x r L 1 3 x 0 d 3 ) 1 l i 2 [ ( x 0 d x r L 1 ) + 2 L 2 ( L 1 d ) ( μ L 1 L 2 + L 1 d μ L 2 d ) ( μ x 0 d + x c L 2 μ x r L 1 ) ] } ,
Δ F = Δ A x = ( 1 Δ λ 2 λ 2 ) { F + ( x r 2 L 1 3 x 0 2 d 3 ) Δ λ 2 λ 1 + Δ λ 2 λ 1 d L 1 L 2 [ L 2 ( L 1 d ) ( μ x 0 d + x c L 2 μ x r L 1 ) 2 + 2 ( μ L 1 L 2 + L 1 d μ L 2 d ) ( μ x 0 d + x c L 2 μ x r L 1 ) ( x 0 d x r L 1 ) ] } F Δ λ 2 λ 2 F + Δ λ 2 λ 1 { ( x r 2 L 1 3 x 0 2 d 3 ) + ( μ x 0 d + x c L 2 μ x r L 1 ) d L 1 L 2 [ L 2 ( L 1 d ) ( μ x 0 d + x c L 2 μ x r L 1 ) + 2 ( x 0 d x r L 1 ) ( μ L 1 L 2 + L 1 d μ L 2 d ) ] } ,
Δ D x = ( 1 Δ λ 2 λ 2 ) { D x + [ ( x r 3 L 1 3 x 0 3 d 3 ) 3 ( μ x 0 d + x c L 2 μ x r L 1 ) 2 ( x 0 d x r L 1 ) ] Δ λ 2 λ 1 } D x Δ λ 2 λ 2 D x + Δ λ 2 λ 1 ( x r L 1 x 0 d ) [ ( x 0 2 d 2 + x 0 x r d L 1 + x r 2 L 1 2 ) + 3 ( μ x 0 d + x c L 2 μ x r L 1 ) 2 ] .
P 11 = ( 1 d g ) x 01 + d 2 g W , P 12 = ( 1 d g ) x 01 d 2 g W , P 21 = ( 1 d g ) x 02 + d 2 g W , P 22 = ( 1 d g ) x 02 d 2 g W .
U 1 U * 3 + U 3 U * 1 = 2 | A 1 | | A 3 | cos { k 1 [ L 1 d ( x x 01 ) 2 2 d + ( x x r ) 2 2 L 1 ] + θ 1 } ,
U 2 U * 3 + U 3 U * 2 = 2 | A 2 | | A 3 | cos { k 1 [ L 1 d ( x x 02 ) 2 2 d + ( x x r ) 2 2 L 1 ] + θ 2 } ,
φ 31 = k 1 [ L 1 d ( x x 01 ) 2 2 d + ( x x r ) 2 2 L 1 ] + θ 1 ,
φ 32 = k 1 [ L 1 d ( x x 02 ) 2 2 d + ( x x r ) 2 2 L 1 ] + θ 2 ,
ν 31 = 1 2 π d φ 31 d x = 1 λ 1 [ ( 1 L 1 1 d ) x + x 01 d x r L 1 ] ,
ν 32 = 1 2 π d φ 32 d x = 1 λ 1 [ ( 1 L 1 1 d ) x + x 02 d x r L 1 ] ,
Δ ν = ν 32 | x = P 22 ν 31 | x = P 11 = 1 λ 1 L 1 g [ ( L 1 + s ) Δ x 0 + ( L 1 d ) W ] ,
Δ ν 1 = 1 λ 1 g ( 1 + s L 1 ) Δ x 0 ,
Δ ν 2 = 1 λ 1 g ( 1 d L 1 ) W ,
Δ ν = Δ ν 1 + Δ ν 2 ,
Δ ν = 1 λ 1 g [ Δ x 0 + W + ( s Δ x 0 d W ) 1 L 1 ] .
Δ ν x = 1 λ 1 g [ | 1 + s L 1 | Δ x 0 + | 1 d L 1 | W ] , Δ ν y = 1 λ 1 g [ | 1 + s L 1 | Δ y 0 + | 1 d L 1 | L ] ,
Δ ν x = 1 λ 1 d [ Δ x 0 + | 1 d L 1 | L x ] , Δ ν y = 1 λ 1 d [ Δ y 0 + | 1 d L 1 | L y ] .

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