Abstract

The primary chromatic aberrations of a reconstructed holographic image can be compensated by a dispersive lens placed at the quasi-achromatic (QA) point of a hologram. The conditions for obtaining this compensation are derived. Three experimental systems that employ this compensation are discussed. Two of these are in line and use either a zone plate lens or flint glass lens as the dispersive element centered at the QA point. In the third system, an off-axis hologram is followed by an off-axis zone plate lens positioned at the offset QA point.

© 1972 Optical Society of America

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References

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  1. R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966); J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  2. E. B. Champagne, J. Opt. Soc. Am. 57, 51 (1967).
    [CrossRef]
  3. J. N. Latta, Appl. Opt. 10, 609 (1971).
    [CrossRef] [PubMed]
  4. J. N. Latta, Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  5. C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966); see also R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 501ff.
  6. D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
    [CrossRef]
  7. H. W. Rose, J. Opt. Soc. Am. 61, 667A (1971).
  8. J. N. Latta, J. Opt. Soc. Am. 61, 667A (1971).
  9. O. Bryngdahl, A. Lohmann, J. Opt. Soc. Am. 60, 281 (1970).
    [CrossRef]
  10. R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 51.
  11. G. L. Fillmore, R. F. Tynan, J. Opt. Soc. Am. 61, 199 (1971).
    [CrossRef]
  12. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, Appl. Opt. 5, 1303 (1966).
    [CrossRef] [PubMed]

1971 (5)

1970 (1)

1967 (1)

1966 (4)

R. W. Meier, J. Opt. Soc. Am. 56, 219 (1966); J. Opt. Soc. Am. 55, 987 (1965).
[CrossRef]

E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, Appl. Opt. 5, 1303 (1966).
[CrossRef] [PubMed]

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966); see also R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 501ff.

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Bryngdahl, O.

Burckhardt, C. B.

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966); see also R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 501ff.

Champagne, E. B.

DeBitetto, D. J.

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Fillmore, G. L.

Kozma, A.

Latta, J. N.

Leith, E. N.

Lohmann, A.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 51.

Marks, J.

Massey, N.

Meier, R. W.

Rose, H. W.

H. W. Rose, J. Opt. Soc. Am. 61, 667A (1971).

Tynan, R. F.

Upatnieks, J.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

D. J. DeBitetto, Appl. Phys. Lett. 9, 417 (1966).
[CrossRef]

Bell Syst. Tech. J. (1)

C. B. Burckhardt, Bell Syst. Tech. J. 45, 1841 (1966); see also R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), p. 501ff.

J. Opt. Soc. Am. (6)

Other (1)

R. S. Longhurst, Geometrical and Physical Optics (Longmans, Green and Co., London, 1960), p. 51.

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Figures (8)

Fig. 1
Fig. 1

Geometry of experimental systems. Dispersive relaying lens centered at QA point is used to correct primary chromatic aberrations of reconstructed image. Systems A and B are in line and use compound holographic elements. System C is offset type system. Holographic parameter fh for elements used as shown: A:a/2, B: 0.44a, C: −2a/3. Focal length of dispersive lenses at 4880 Å: A: 2a, B: a/2, C: 3a/2.

Fig. 2
Fig. 2

Experimental construction parameters. Reference beam angles were 14° for systems A and B, 14° for HC, and 9° for L2C. In using these elements in the system of Fig. 1, the following elements were used in conjugate order: L1A, L1B, HB, HC. The separation distance of hologram to lens, a, was: A: 19 cm, B: 24 cm, C: 10 cm. Hologram and zone plate lens apertures were 20 mm in all systems.

Fig. 3
Fig. 3

Coordinate system used and example of off-axis quasi-achromatic point rc. r01 and r02 are object points; small crosses are corresponding reconstructed image points of differing wavelength.

Fig. 4
Fig. 4

Plot of reciprocal focal length vs reciprocal normalized wavelength for (1) flint glass lens, (2) zone plate lens, (3) compound lens made up of (1) and (2). Dotted lines indicate required functional behavior of lens for exact correction of holographic chromatic aberrations.

Fig. 5
Fig. 5

Normalized image plane shift with wavelength for systems A, B, and C, computed from Eq. (6).

Fig. 6
Fig. 6

System A. (a) Image of holographic reconstruction with 37-cm achromatic lens at QA point. (b) Corrected image with zone plate lens at QA point.

Fig. 7
Fig. 7

System B. (a) Image of holographic reconstruction with 12-cm achromatic lens at QA point. (b) Corrected image using flint glass lens.

Fig. 8
Fig. 8

Corrected image of system C.

Equations (20)

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x i = z i [ x c z c + μ ( x 0 z 0 x r z r ) ] ,
y i = z i ( μ y 0 / z 0 ) ,
( 1 / z i ) = ( 1 / z c ) + ( μ / f h ) ,
x i = x z m x z i = z c θ c m x ( z i z c ) ,
y i = y z m y z i = m y ( z i z c ) ,
x z = f h ( θ 0 x θ r ) , y z = f h θ 0 y , m x = ( f h / z c ) ( θ 0 x θ r ) θ c , m y = ( f h / z c ) θ 0 y , θ r = ( x r / z r ) ; θ c = ( x c / z c ) ; θ 0 x = ( x 0 / z 0 ) ; θ 0 y = ( y 0 / z 0 ) .
r N = 1 z r μ z c [ ( x c z r μ x r z c ) , 0 , z r z c ( 1 μ ) ] .
Φ T = C g O g * R g R h * O h ( true image ) , Φ c = C g O g R g * R h O h * ( conjugate image ) ,
( 1 / S ) = ( 1 / f ) [ 1 / ( a z i ] .
( 1 / f ) = [ ( 1 / S ) + ( 1 / a ) ] + ( f h / μ a 2 ) .
D = λ f f λ = f μ ( 1 / f ) ( 1 / μ ) = f f h μ 0 a 2 .
f 0 f h = μ 0 2 a 2 ( zone plate lens ) ,
f f h = 3.46 μ 0 a 2 υ D ( glass lens ) .
1 S = 1 S 0 + J = 2 1 J ! d J ( 1 / f ) d β J ( β β 0 ) J ,
1 S = 1 S 0 + 1 f 0 Δ μ 2 μ 0 ,
S S 0 ~ ( S 0 2 / f 0 ) ( Δ μ 2 / μ 0 ) .
1 S 1 S 0 = 1 2 2 ( 1 / S ) β 2 Δ β 2 + 2 ( 1 / S ) β z 0 Δ β Δ z 0 ,
Δ S S 0 = 1 2 ( S 0 f 0 ) G ( Δ μ 2 μ 0 ) ( Δ μ μ 0 ) ( M Δ z 0 S 0 ) ,
( Δ μ / μ 0 ) = ( 2 d s p / M Δ z ) ( S 0 / d )
( θ r l / θ r h ) = a / f 0 .

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