Abstract

A procedure to calculate a highly quantized, blazed phase structure is presented. Characteristics that are concentrated on are a high diffraction efficiency and a large signal-to-noise ratio. The calculation techniques are based on iterative Fourier-transform algorithms. Stagnation problems are discussed, and methods to overcome them are described.

© 1990 Optical Society of America

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References

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  1. R. Magnusson, T. K. Gaylord, “Diffraction efficiencies of thin phase gratings with arbitrary grating shape,” J. Opt. Soc. Am. 68, 806–809 (1978).
    [CrossRef]
  2. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  3. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
    [CrossRef]
  4. S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
    [CrossRef]
  5. T. Yatagai, R. Sugawara, H. Hashizume, M. Suki, “Phase-only computer-generated hologram produced by an ion-exchange technique,” Opt. Lett. 13, 952–954 (1988).
    [CrossRef] [PubMed]
  6. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  7. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  8. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  9. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object/dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).
  10. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  11. F. Wyrowski, “Considerations on the bandwidth of phase factors,” to be submitted to Opt. Commun.
  12. F. Wyrowski, “Diffraction efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
    [CrossRef]
  13. H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef] [PubMed]
  14. L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [CrossRef]
  15. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  16. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18, 3661–3669 (1979).
    [CrossRef] [PubMed]
  17. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  18. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  19. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  20. H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).
  21. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]
  22. H. Bartelt, J. Horner, “Improving binary phase correlation filters using iterative techniques,” Appl. Opt. 24, 2894–2897 (1985).
    [CrossRef] [PubMed]
  23. M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1988).
    [CrossRef]

1990 (1)

1989 (2)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

1986 (1)

1985 (2)

H. Bartelt, J. Horner, “Improving binary phase correlation filters using iterative techniques,” Appl. Opt. 24, 2894–2897 (1985).
[CrossRef] [PubMed]

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

1984 (1)

1982 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1979 (1)

1978 (1)

1973 (1)

1970 (2)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

1967 (1)

Akahori, H.

Allebach, J. P.

Arnold, S. M.

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Bartelt, H.

Broja, M.

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1988).
[CrossRef]

Bryngdahl, O.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1988).
[CrossRef]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

Dammann, H.

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Gallagher, N. C.

Gaylord, T. K.

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Hashizume, H.

Hauck, R.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object/dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Horner, J.

Jordan, J. A.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object/dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Jordon, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Lee, W.-H.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object/dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Liu, B.

Lohmann, A. W.

Magnusson, R.

Paris, D. P.

Seldowitz, M. A.

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Sugawara, R.

Suki, M.

Sweeney, D. W.

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Wyrowski, F.

F. Wyrowski, “Diffraction efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
[CrossRef]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1988).
[CrossRef]

F. Wyrowski, “Considerations on the bandwidth of phase factors,” to be submitted to Opt. Commun.

Yatagai, T.

Appl. Opt. (8)

IBM J. Res. Dev. (2)

L. B. Lesem, P. M. Hirsch, J. A. Jordon, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital halftoning by iterative procedure,” Opt. Commun. 69, 205–210 (1988).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Opt. Lett. (1)

Optik (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Other (4)

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object/dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

F. Wyrowski, “Considerations on the bandwidth of phase factors,” to be submitted to Opt. Commun.

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Figures (12)

Fig. 1
Fig. 1

Illustration of the relation between g(x) and f(x): the complex amplitude g(x) is given within the total field and the signal f(x) within the signal window F.

Fig. 2
Fig. 2

Realization of a linear phase as a blazed grating with sawtoothed profile.

Fig. 3
Fig. 3

Diagram of an iterative Fourier-transform algorithm.

Fig. 4
Fig. 4

Illustration of a stagnation of the iterative procedure: (a) initial distribution |g0(x)|2, resulting distributions (b) |20(x)|2 and (c) |1000(x)|2 in case of exclusive use of the phase freedom and (d) |40(x)|2 when a combination of amplitude and phase freedom is utilized.

Fig. 5
Fig. 5

a, Magnified part of an iteratively calculated analog blazed structure; b, calculated modulus of Fraunhofer pattern.

Fig. 6
Fig. 6

Illustration of the effect of the quantization operator for Z = 3. The dashed lines indicate the thresholds.

Fig. 7
Fig. 7

Calculated |(x)| for different offsets x0 to show the effect of quantization noise for Z = 3: a, x0 = 0; b, off-axis signal.

Fig. 8
Fig. 8

Illustration of stagnative iteration: stagnation occurs if the values of Gj+1(u) [hatched parts in (b)] do not change enough to cross the threshold levels (dashed lines).

Fig. 9
Fig. 9

Illustration of the effect of the operator Q3p.

Fig. 10
Fig. 10

Illustration of a stepwise introduction of the quantization operator Q3 during iteration from (a) the analog to the (d) quantized distribution.

Fig. 11
Fig. 11

a, Iteratively calculated trilevel blazed structure; b, calculated intensity of diffracted light.

Fig. 12
Fig. 12

a, Iteratively calculated four-level blazed structure; b, calculated intensity of diffracted light.

Tables (2)

Tables Icon

Table 1 Signal-to-Noise Ratios of Noniteratively and Iteratively Quantized, Blazed Structures

Tables Icon

Table 2 Diffraction Efficiency of Blazed Structures Quantized in Z Levels

Equations (24)

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f ( x ) = FT { exp [ i Φ ( u ) ] } ,
| g ( x ) | 2 | f ( x x 0 ) | 2 , x F ,
g ( x ) = f ( x x 0 ) ,
G ( u ) = exp [ i 2 π u x 0 ] ,
g ( x ) | f ( x x 0 ) | exp [ i φ ( x x 0 ) ] + c ( x )
c ( x ) = 0 , x F
X _ [ j ( x ) ] = g j + 1 ( x ) = { c j | f ( x x 0 ) | exp [ i γ ¯ j ( x ) ] , x F 0 , x F
U _ [ G j ( u ) ] = j ( u ) = exp [ i Γ j ( u ) ] .
c j = F | f ( x x 0 ) | | j ( x ) | d x d y D ( f ) | f ( x ) | 2 d x d y ,
X _ [ j ( x ) ] = g j + 1 ( x ) = { c j | f ( x x 0 ) | exp [ i γ ¯ j ( x ) ] , x F j ( x ) , x F .
f ( m f ) = | f ( m f ) | exp [ i φ ( m f ) ] = f ( x ) comb ( x , δ x ) ,
comb ( x , δ x ) α , β δ ( x α δ x ) δ ( y β δ y ) ,
g 0 ( m ) = f ( m m 0 ) ,
Γ ¯ ( u ) { π , π + Δ , , π Δ } ,
QZ _ [ G ( u ) ] = ( u ) = { exp ( i π ) , Γ ( u ) + π 0.5 Δ exp [ i ( π + z Δ ) ] , ( z 0.5 ) Δ < Γ ( u ) + π ( z + 0.5 ) Δ , exp ( i π ) , ( Z 0.5 ) Δ < Γ ( u ) + π
( u ) = G ( u ) = Q ( u ) .
( x ) = g ( x ) + q ( x ) f ( x x 0 ) + c ( x ) + q ( x ) .
q ( x ) = 0 , x F
| f ( x x 0 ) + q ( x ) | 2 | f ( x x 0 ) | 2 , x F .
q ¯ j ( x ) = g j + 1 ( x ) j ( x ) ,
G j + 1 ( u ) = j ( u ) + Q ¯ j ( u ) .
QZ _ p [ G j ( u ) ] = j ( u ) = { exp [ i π ] , Γ ( u ) + π 0.5 Δ · ( p ) exp [ i ( π + z Δ ) ] , ( z 0.5 ( p ) ) Δ < Γ ( u ) + π ( z + 0.5 ( p ) ) Δ exp [ i π ] , ( z 0.5 ( p ) ) Δ < Γ ( u ) + π G j ( u ) , otherwise ,
0 < ( 1 ) < ( 2 ) < ( p ) < ( P ) = 1
η ¯ ( sinc 1 / Z ) 2 · η ,

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