Abstract

A filter combination consisting of a lens and two pure phase filters, situated in the two focal planes, is discussed. This element, which we call a tandem component, apparently does not absorb any light; in other words, the tandem component is capable of shaping wave fronts with 100% efficiency independent of the object function. We describe a basic configuration and outline its space-variant system properties. The tandem component can be used for many of the standard applications of computer-generated holograms and possibly for some new types as well in view of its space variance.

© 1984 Optical Society of America

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References

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  1. J. L. Horner, Appl. Opt. 21, 4511 (1982).
    [CrossRef] [PubMed]
  2. H. J. Caulfield, Appl. Opt. 21, 4391 (1982).
    [CrossRef] [PubMed]
  3. H. Kogelnik, in Proceedings, Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 605.
  4. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  5. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  6. W. J. Dallas, in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, New York, 1980).
  7. M. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  8. J. R. Fienup, Opt. Eng. 19, 297 (1980).
    [CrossRef]
  9. B. Braunecker, R. Hauck, W. T. Rhodes, Appl. Opt. 18, 44 (1979).
    [CrossRef] [PubMed]
  10. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]
  11. A. W. Lohmann, D. P. Paris, J. Opt. Soc. Am. 55, 1007 (1965).
    [CrossRef]
  12. R. J. Marks, J. F. Walkup, M. O. Hagler, J. Opt. Soc. Am. 66, 918 (1976).
    [CrossRef]

1982 (2)

1980 (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

1979 (1)

1976 (1)

1974 (1)

1973 (1)

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1965 (1)

Braunecker, B.

Bryngdahl, O.

Caulfield, H. J.

Dallas, W. J.

W. J. Dallas, in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, New York, 1980).

Fienup, J. R.

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Gallagher, M. C.

Hagler, M. O.

Hauck, R.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Horner, J. L.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Kogelnik, H.

H. Kogelnik, in Proceedings, Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 605.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Liu, B.

Lohmann, A. W.

Marks, R. J.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Paris, D. P.

Rhodes, W. T.

Walkup, J. F.

Appl. Opt. (4)

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Eng. (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Other (3)

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

W. J. Dallas, in The Computer in Optical Research, B. R. Frieden, Ed. (Springer, New York, 1980).

H. Kogelnik, in Proceedings, Symposium on Modern Optics, J. Fox, Ed. (Polytechnic Press, Brooklyn, 1967), p. 605.

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Figures (3)

Fig. 1
Fig. 1

Configuration of the tandem component.

Fig. 2
Fig. 2

Processing systems with the tandem component.

Fig. 3
Fig. 3

(A) Original object for a bleached phase filter F1; (B) original object for a bleached phase filter F ˜ 2; (C) reconstruction from the two phase masks F1 and F ˜ 2.

Equations (19)

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η = η s · η o .
F ˜ 1 ( v ) = A ( v ) ,
F ˜ 1 ( v ) = F ˜ 1 ( v ) exp [ i φ 1 ( v ) ] = A ( v ) exp [ i φ 1 ( v ) ] .
F ˜ 2 ( v ) = 1 ,             F ˜ 2 = exp [ i φ 2 ( v ) ] .
u ˜ ( v ) = A ( v ) exp [ i φ ( v ) ] = F ˜ 1 ( v ) · F ˜ 2 ( v ) = A ( v ) exp { i [ φ 1 ( v ) + φ 2 ( v ) ] } .
F 1 ( x ) = 1 , F ˜ 2 ( v ) = 1 , F ˜ 1 ( v ) = A ( v ) , φ 2 ( v ) = φ ( v ) - φ 1 ( v ) ,
u ^ ( x ) = u ( x ) exp [ i π ( x - x ) 2 / λ z ] d x = ( i π x 2 / λ z ) [ u ( x ) exp ( i π x 2 λ z ) ] exp ( - 2 π i x x / λ z ) d x .
F ˜ 1 ( v ) = T ˜ ( v ) [ exp ( i φ 1 ) - exp ( i φ 2 ) ] + δ ( v ) exp ( i φ 2 ) ,
F 1 ( x ) = T ( x ) [ exp ( i φ 1 ) - exp ( i φ 2 ) ] + exp ( i φ 2 ) ;             ( F 1 ( x ) = 1 ) .
F ˜ 1 ( v ) 2 ~ T ˜ ( v ) 2 ,
φ 1 - φ 2 = arccos [ - 1 2 T ˜ ( 0 ) + 1 ] .
F ˜ 2 ( v ) = { exp ( i φ ) if v = 0 , 1 if v 0 ,
φ = - arctan { cos ( φ 1 + φ 2 2 ) cos ( φ 1 - φ 2 2 ) + cos φ 2 sin ( φ 1 + φ 2 2 ) cos ( φ 1 - φ 2 2 ) + sin φ 2 } .
h ( x - x , x ) = S [ δ ( x - x ) ] = P ( x - x ) .
R ( x ) = [ F 1 ( x ) · T ( x ) ] * F 2 ( x ) with F 1 ( x ) = 1 , F ˜ 2 ( v ) = 1 , R ˜ ( v ) = [ F ˜ 1 ( v ) * T ˜ ( v ) ] · F ˜ 2 ( v ) , h ( x - x , x ) = F 1 ( x ) · F 2 ( x - x ) ,
R ( x ) = [ T ( x ) * F 1 ( x ) ] · F 2 ( x ) with F ˜ 1 ( v ) = 1 , F 2 ( x ) = 1 , R ˜ ( v ) = [ T ˜ ( v ) · F ˜ 1 ( v ) ] * F ˜ 2 ( v ) , h ( x - x , x ) = F 1 ( x - x ) F 2 ( x ) .
R ( x ) = F 1 ( x ) · F 2 ( x ) · T ( x ) with F ˜ 1 ( v ) = 1 , F 2 ( x ) = 1 , R ˜ ( v ) = F ˜ 1 ( v ) * F ˜ 2 ( v ) * T ˜ ( v ) , h ( x - x , x ) = δ ( x - x ) [ F 1 ( x ) · F 2 ( x ) ] ,
R ( x ) = [ F 1 ( x ) * F 2 ( x ) ] · T ( x ) with F 1 ( x ) = 1 , F ˜ 2 ( v ) = 1 , R ˜ ( v ) = [ F ˜ 1 ( v ) · F ˜ 2 ( v ) ] * T ˜ ( v ) , h ( x - x , x ) = δ ( x - x ) [ F 1 ( x ) * F 2 ( x ) ] .
R ( x ) = [ F 1 ( x ) · T ( x ) ] * F 2 ( x ) with F ˜ 1 ( v ) = 1 , F ˜ 2 ( v ) = 1 , R ˜ ( v ) = [ F ˜ 1 ( v ) * T ˜ ( v ) ] · F ˜ 2 ( v ) , h ( x - x , x ) = F 1 ( x ) * F 2 ( x - x ) .

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