Abstract

Two- and three-element combinations of hologram optical elements are considered, and analytical conditions are obtained for absence of longitudinal dispersion. Solutions of the conditional equations are compared with another author's computed design results. A simple doublet design is considered for an eyepiece, and its performance is described.

© 1976 Optical Society of America

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References

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  1. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  2. E. N. Leith, J. Upatnieks, K. A. Haines, J. Opt. Soc. Am. 55, 981 (1965).
    [CrossRef]
  3. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  4. M. R. B. Forshaw, Optica Acta 20, 669 (1973).
    [CrossRef]
  5. R. H. Katyl, Appl. Opt. 11, 1241 (1972).
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  6. J. N. Latta, Appl. Opt. 11, 1686 (1972).
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  7. J. N. Latta, Appl. Opt. 10, 599 (1971).
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  8. J. N. Latta, Appl. Opt. 10, 609 (1971).
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1973 (1)

M. R. B. Forshaw, Optica Acta 20, 669 (1973).
[CrossRef]

1972 (2)

1971 (2)

1967 (1)

1965 (2)

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Figures (4)

Fig. 1
Fig. 1

(a) Recording geometry and (b) reconstruction geometry for a hologram element.

Fig. 2
Fig. 2

Achromatic combination of two hologram elements.

Fig. 3
Fig. 3

Galilean telescope with −20 diopter hologram doublet eyepiece.

Fig. 4
Fig. 4

Variation of wavefront curvature (I/R12) with μ for (a) single element; (b) doublet F1 = +12 · 45 diopters; F2 = −43 · 20 diopters; (c) doublet F1 = −80 · 45 diopters; F2 = +23 · 17 diopters.

Tables (1)

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Table I Values of μ for Broadband Two- and Three-Element Combinations

Equations (39)

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F = 1 / f = ( 1 / R I ) ( 1 / R c ) = ± μ [ ( 1 / R o ) ( 1 / R R ) ] ,
Δ F = ± ( Δ λ C / λ R ) [ ( 1 / R o ) ( 1 / R R ) ] .
D = Δ F / F = Δ λ C / λ C .
V = 1 / D = λ C / Δ λ C .
F = F 1 + F 2 ,
Δ F = Δ F 1 + Δ F 2 .
D F = D 1 F 1 + D 2 F 2 .
D 1 F 1 + D 2 F 2 = 0 ,
F 1 + F 2 = 0.
F = F 1 + F 2 A F 1 F 2 = ± ( λ C / λ R 1 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] ± ( λ C / λ R 2 ) [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] A ( λ C 2 / λ R 1 λ R 2 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] ,
F / λ C = ± ( 1 / λ R 1 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] ± ( 1 / λ R 2 ) [ 1 / R O 2 ) ( 1 / R R 2 ) ] ( 2 A λ C / λ R 1 λ R 2 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] .
A = 1 2 ( f 1 + f 2 ) .
1 / R I 1 = ( 1 / R C 1 ) ± μ 1 [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] ,
1 / R I 2 = ( 1 / R C 2 ) ± μ 2 [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] ,
R C 2 = R I 1 A .
( / λ C ) ( 1 / R I 1 ) = ± ( 1 / λ R 1 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] = F 1 / λ C ,
( / λ C ) ( 1 / R I 2 ) = ( / λ C ) ( 1 / R C 2 ) ± ( 1 / λ R 2 ) [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] = ( / λ C ) ( 1 / R C 2 ) + ( F 2 / λ C ) ,
R C 2 / λ C = R I 1 / λ C .
( / λ C ) ( 1 / R I 1 ) = ( 1 / R I 1 2 ) ( R I 1 / λ C ) .
R I 1 / λ C = R I 1 2 ( F 1 / λ C ) .
( / λ C ) ( 1 / R I 2 ) = ( 1 / R C 2 2 ) ( R C 2 / λ C ) + ( F 2 / λ C ) .
( / λ C ) ( 1 / R I 2 ) = ( R I 1 2 / R C 2 2 ) ( F 1 / λ C ) + ( F 2 / λ C ) .
R I 1 2 F 1 = R C 2 2 F 2 .
R I 1 > R C 2 .
R C 2 < 1 / F 2 .
R I 1 < 1 / F 1 .
1 / R I 1 = ( 1 / R C 1 ) ± μ 1 [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] ,
1 / R I 2 = ( 1 / R C 2 ) ± μ 2 [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] ,
1 / R I 3 = ( 1 / R C 3 ) ± μ 3 [ ( 1 / R O 3 ) ( 1 / R R 3 ) ] ,
R C 2 = R I 1 A 12 ,
R C 3 = R I 2 A 23 ,
( / λ C ) ( 1 / R I 1 ) = ± ( 1 / λ R 1 ) [ ( 1 / R O 1 ) ( 1 / R R 1 ) ] = F 1 / λ C ,
( / λ C ) ( 1 / R I 2 ) = ( / λ C ) ( 1 / R C 2 ) ± ( 1 / λ R 2 ) [ ( 1 / R O 2 ) ( 1 / R R 2 ) ] = ( / λ C ) ( 1 / R C 2 ) + ( F 2 / λ C ) ,
( / λ C ) ( 1 / R I 3 ) = ( / λ C ) ( 1 / R C 3 ) ± ( 1 / λ R 3 ) [ ( 1 / R O 3 ) ( 1 / R R 3 ) ] = ( / λ C ) ( 1 / R C 3 ) + ( F 3 / λ C ) ,
R C 2 / λ C = R I 1 / λ C ,
R C 3 / λ C = R I 2 / λ C ,
( / λ C ) ( 1 / R I 3 ) = ( R I 2 2 / R C 3 2 ) ( R I 1 2 / R C 2 2 ) ( F 1 / λ C ) + ( R I 2 2 / R C 3 2 ) ( F 2 / λ C ) + ( F 3 / λ C ) ,
R I 2 2 R I 1 2 F 1 + R I 2 2 R C 2 2 F 2 + R C 2 2 R C 3 2 F 3 = 0.
0.0004 F 2 3 0.012 F 2 2 0.8 F 2 + 20 = 0.

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