Abstract

By using Fourier optics a theoretical analysis of the rainbow holographic process is made. The monochromatic and the polychromatic point spread functions, the monochromatic and the polychromatic transfer functions are derived. From these functions the real resolution (including monochromatic resolution and color blur) of the rainbow holographic image for different limiting apertures and the cutoff spatial frequency of the transfer function are calculated. These calculations are less laborious in contrast to the tedious evaluation of the diffraction theory. The formulas are general and applicable to the conventional hologram. The imaging plane hologram is first explained as a specific case of the rainbow hologram in this paper.

© 1984 Optical Society of America

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References

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  1. Y. W. Zhang, W. G. Zhu, F. T. S. Yu, “Rainbow Holographic Aberrations and the Bandwidth Requirements,” Appl. Opt. 22, 164 (1983).
    [Crossref] [PubMed]
  2. J. C. Wyant, “Image Blur for Rainbow Holograms,” Opt. Lett. 1, 130 (1977).
    [Crossref] [PubMed]
  3. H. Chen, “Color Blur of the Rainbow Hologram,” Appl. Opt. 17, 3290 (1978).
    [Crossref] [PubMed]
  4. P. N. Tamura, “One-Step Rainbow Holography with a Field Lens,” Appl. Opt. 17, 3343 (1978).
    [Crossref] [PubMed]
  5. S. L. Zhuang, P. H. Ruterbusch, Y. W. Zhang, F. T. S. Yu, “Resolution and Color Blur of the One-Step Rainbow Hologram,” Appl. Opt. 20, 872 (1981).
    [Crossref] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1983 (1)

1981 (1)

1978 (2)

1977 (1)

Chen, H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Ruterbusch, P. H.

Tamura, P. N.

Wyant, J. C.

Yu, F. T. S.

Zhang, Y. W.

Zhu, W. G.

Zhuang, S. L.

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Figures (6)

Fig. 1
Fig. 1

Rainbow holographic process: A, limiting aperture; MH, primary hologram; RH, rainbow hologram; O, object point; I, image point; R, convergent reference beam; C, divergent illuminating beam.

Fig. 2
Fig. 2

Geometrical representation of Fig. 1.

Fig. 3
Fig. 3

Rectangular slit aperture.

Fig. 4
Fig. 4

Cross slit aperture.

Fig. 5
Fig. 5

Circular aperture.

Fig. 6
Fig. 6

Polychromatic point spread function of wavelength spread: (a) Δλ = 50 Å;(b) Δλ = 97 Å;(c) Δλ = 150 Å.

Equations (47)

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E i ( α , β ; λ ) = c 1 P 1 P 2 Q 1 Q 2 × exp { - j k 1 l i [ ( α - α 2 ) x + ( β - β i ) y ] } d x d y .
l i = λ 1 L 1 L 2 d λ 1 L 1 L 2 - λ d L 2 + λ 1 d L 1 ,
α i = λ x 0 L 1 L 2 - λ x r d L 2 + λ 1 x r d L 1 λ L 1 L 2 - λ d L 2 + λ 1 d L 1 ,
β i = λ y 0 L 1 L 2 - λ y r d L 2 + λ 1 y c d L 1 λ L 1 L 2 - λ d L 2 + λ 1 d L 1 .
x = d g ξ + ( 1 - d g ) x 0 ,
y = d g η + ( 1 - d g ) y 0 .
E i ( α , β ; λ ) = c 2 - w / 2 w / 2 - L / 2 L / 2 × exp { - j k d g l i [ ( α - α i ) ξ + ( β - β i ) η } d ξ d η .
E i ( α , β ; λ ) = c 2 - A ( ξ , η ) × exp { - j 2 π d λ g l i [ ( α - α i ) ξ + ( β - β i ) η ] } d ξ d η = F { A ( ξ , η ) } f x = d / λ g l i ( α - α i ) , f y = d / λ g l i ( β - β i ) ,
A ( ξ , η ) = rect ( ξ W ) rect ( η L ) ,
E i m ( α , β ) = λ - Δ λ / 2 λ + Δ λ / 2 C ( λ ) E i 2 ( α , β ; λ ) d λ .
C ( λ ) = S ( λ ) F ( λ ) V ( λ ) ,
E i m ( α , β ) = c λ - Δ λ / 2 λ + Δ λ / 2 E i 2 ( α , β ; λ ) d λ .
A ( ξ , η ) = rect ( ξ W ) rect ( η L ) .
E i ( α , β ; λ ) = W L sinc ( W f x sinc ( L f y ) f x = d / λ g l i ( α - α i ) , f y = d / λ g l i ( β - β i ) = W L sinc [ W d λ g l i ( α - α i ) ] sinc [ L d λ g l i ( β - β i ) ] .
Δ R x = λ g l i W d ,
Δ R y = λ g l i L d .
Δ R x = λ 1 λ L 1 g ( λ L 1 + λ 1 d - λ d ) W = 7.78 × 10 - 2 mm .
Δ R x = λ l i L x ,
Δ R y = λ l i L y .
A ( ξ , η ) = rect ( ξ W ) rect ( η L ) + rect ( ξ L ) rect ( η W ) - rect ( ξ W ) rect ( η W ) .
E i ( α , β ; λ ) = L W sinc [ W d λ g l i ( α - α i ) ] sinc [ L d λ g l i ( β - β i ) ] + L W sinc [ L d λ g l i ( α - α i ) ] sinc [ W d λ g l i ( β - β i ) ] - W 2 sinc [ W d λ g l i ( α - α i ) ] × sinc [ W d λ g l i ( β - β i ) ] .
Δ R x = λ g l i W d .
Δ R y = λ g l i W d .
Δ R x = Δ R x = λ g l i W d ,
Δ R y = Δ R y = λ g l i W d .
A ( ξ , η ) = circ ( r r 0 ) ,
E i ( α , β ; λ ) = r 0 J 1 ( 2 π r 0 ρ ) ρ ,
ρ = ( f x 2 + f y 2 ) 1 / 2 = d λ g l i [ ( α - α i ) 2 + ( β - β i ) 2 ] 1 / 2 ,
Δ R r = [ ( α - α i ) 2 + ( β - β i ) 2 ] 1 / 2 = 0.61 λ g l i r 0 d .
Δ R r = 0.61 λ 1 λ g L 1 L 2 r 0 ( λ L 1 L 2 - λ d L 2 + λ 1 d L 1 ) .
Δ R r = 0.61 λ 1 S r 0 .
E i m ( α ) = c λ - Δ λ / 2 λ + Δ λ / 2 W 2 sinc 2 [ W d λ g l i ( α - α i ) ] d λ .
x ˜ = d λ g l i ζ ,             y ¯ = d λ g l i η ;             α ˜ = α - α i ,             β ˜ = β - β i .
E i ( α ˜ , β ˜ ; λ ) = ( λ g l i d ) 2 - A ( λ g l i d x ˜ , λ g l i d y ¯ ) × exp [ - j 2 π ( α ˜ x ˜ + β ˜ y ˜ ) ] d x ˜ d y ˜ = N F [ A ( λ g l i d x ˜ , λ g l i d y ˜ ) ] ,
N = ( λ g l i / d ) 2 .
H ( f x , f y ; λ ) = - E i ( α ˜ , β ˜ ; λ ) exp [ - j 2 π ( f x α ˜ + f y β ˜ ) ] d α ˜ d β ˜ = F [ E i ( α ˜ , β ˜ ; λ ) ] = N F { F [ A ( λ g l i d x ˜ , λ g l i d y ˜ ) ] } = N A ( - λ g l i d f x , - λ g l i d f y ) .
H ( f x , f y ; λ ) = N A ( λ g l i d f x , λ g l i d f y ) .
H ( f x , f y ; λ ) = N rect ( λ g l i f x W d ) rect ( λ g l i f x L d ) .
f x , max = W d 2 λ g l i ,
f y , max = L d 2 λ g l i .
f x , max = L x 2 λ l i ,
f y , max = L y 2 λ l i .
H m ( f x f y ) = - E i m ( α ˜ , β ˜ ) exp [ - j 2 π ( f x α ˜ + f y β ˜ ) ] d α ˜ d β ˜ / - E i m ( α ˜ , β ˜ ) d α ¯ d β ˜ = λ - Δ λ / 2 λ + Δ λ / 2 - E i 2 ( α ˜ , β ˜ ; λ ) exp [ - j 2 π ( f x α ˜ + f y β ˜ ) ] d α ˜ d β ˜ d λ / λ - Δ λ / 2 λ + Δ λ / 2 - E i 2 ( α ˜ , β ˜ ; λ ) d α ˜ d β ˜ d λ = λ - Δ λ / 2 λ + Δ λ / 2 - H ( ξ - f x 2 , η - f y 2 ; λ ) × H * ( ξ + f x 2 , η + f y 2 ; λ ) d ξ d η d λ / λ - Δ λ / 2 λ + Δ λ / 2 - H 2 ( ξ , η ; λ ) d ξ d η d λ .
H m ( f x , f y ) = λ - Δ λ / 2 λ + Δ λ / 2 - A ( ξ - λ g l i 2 d f x , η - λ g l i 2 d f y ; λ ) A * ( ξ + λ g l i 2 d f x , η + λ g l i 2 d f y ; λ ) d ξ d η d λ / λ - Δ λ / 2 λ + Δ λ / 2 - A 2 ( ξ , η ; λ ) d ξ d η d λ .
E ( x i - x ˜ 0 , y i - y ˜ 0 ) = - A ( λ d i x ˜ , λ d i y ˜ ) exp { - j 2 π [ ( x i - x ˜ 0 ) x ˜ + ( y i - y ˜ 0 ) y ˜ ] } d x ¯ d y ¯ .
H ( f x , f y ; λ ) = A ( λ d i f x , λ d i f y ) .
E i ( α ) = F { A ( M λ 1 g x ˜ ) } = F [ A ( λ g l i d x ˜ ) ] .

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