Abstract

Two-dimensional Fourier transform kinoforms can be calculated by use of a discrete Fourier transform. It is well known that the off-axis reconstruction has lower reconstruction error than the on-axis one. Here we make what to our knowledge is a new analysis on the effect of phase quantization in the Fourier domain. We find that the kinoform reconstruction error changes periodically according to the position of the desired image when a large dummy area is added. The error dependence of quantized kinoform reconstruction is simulated on the position of the desired image by use of the iterative dummy area method and the iterative interlacing technique.

© 2000 Optical Society of America

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References

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  1. L. B. Lesem, P. M. Hirsch, T. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [CrossRef]
  2. T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).
  3. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  4. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  5. G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
    [CrossRef]
  6. H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef] [PubMed]
  7. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structure,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  8. S. Yang, T. Shimomura, “Interlacing technique approach for the synthesis of kinoforms,” Appl. Opt. 35, 6983–6989 (1996).
    [CrossRef] [PubMed]
  9. J. W. Goodman, A. M. Silvestri, “Some effect of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]
  10. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  11. S. Yang, T. Shimomura, “Error reduction of quantized kinoforms by means of increasing the kinoform size,” Appl. Opt. 37, 6931–6936 (1998).
    [CrossRef]

1998 (1)

1996 (1)

1994 (1)

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

1990 (1)

1989 (1)

1986 (1)

1980 (1)

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

1975 (1)

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

1973 (1)

1970 (1)

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

1969 (1)

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

Akahori, H.

Fienup, J. R.

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

Gallagher, N. C.

Goodman, J. W.

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

Hirsch, P. M.

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

Jordan, T. A.

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

Lesem, L. B.

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

Liu, B.

Lu, G.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

Shimomura, T.

Silvestri, A. M.

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

Takeda, M.

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

Tanone, A.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

Wyrowski, F.

Yang, S.

Yatagai, T.

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

Yu, F. T. S.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

Zhang, Z.

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

Appl. Opt. (5)

IBM J. Res. Dev. (2)

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

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

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

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

G. Lu, Z. Zhang, F. T. S. Yu, A. Tanone, “Pendulum iterative algorithm for phase retrieval from modulus data,” Opt. Eng. 33, 548–555 (1994).
[CrossRef]

Optik (Stuttgart) (1)

T. Yatagai, M. Takeda, “Effect of phase nonlinearity in kinoform,” Optik (Stuttgart) 43, 337–352 (1975).

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Figures (8)

Fig. 1
Fig. 1

Configuration of a dummy area added to the desired image.

Fig. 2
Fig. 2

Desired image.

Fig. 3
Fig. 3

Error dependence of the IDA method on the position of the desired image with 4 × 4 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Fig. 4
Fig. 4

Error dependence of the IDA method on the position of the desired image with 8 × 8 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Fig. 5
Fig. 5

Error dependence of the IDASQ method on the position of the desired image with 4 × 4 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Fig. 6
Fig. 6

Error dependence of the IDASQ method on the position of the desired image with 8 × 8 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Fig. 7
Fig. 7

Error dependence of the IT method on the position of the desired image with 4 × 4 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Fig. 8
Fig. 8

Error dependence of the IT method on the position of the desired image with 8 × 8 times the size of the desired image enlarged kinoform. (a) Four-level kinoform, (b) eight-level kinoform.

Equations (19)

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Gu, v=|Gu, v|expjΦu, v=x=-PN/2PN/2-1y=-QN/2QN/2-1 gx, y|exp-j2π×xuPN+yvQN,
WLu, v=|A|expjΦLu, v,
gLx, y=1PQN2u=-PN/2PN/2-1v=-QN/2QN/2-1 WLu, v×expj2πxuPN+y, vQN.
GΔx0,Δy0u, v=|GΔx0,Δy0u, v|expjΦΔx0,Δy0u, v=x=-PN/2PN/2-1y=-QN/2QN/2-1 gx-Δx0, y-Δy0|exp-j2πxuPN+yvQN=|Gu, v|expjΦu, v-2πΔx0uPN+Δy0vQN.
WL,Δx0,Δy0u, v=WLu, vexp-j2π×Δx0uPN+Δy0vQN.
gL,Δx0,Δy0x, y=gLx-Δx0, y-Δy0.
WLu, v=|A| m=-sincm+1/LexpjmL+1Φu, v=|A| m=-sincm+1/LexpjmL+1Ψu, vexp-j2πmL+1×x0uPN+y0vQN=m=0sincm+1/L|A|1-|mL+1|×exp-j2πmL+1x0uPN+y0vQN×W0mL+1u, v+m=--1sincm+1/L×|A|1-|mL+1|×exp-j2πmL+1x0uPN+y0vQN×W0|mL+1|u, v*,
gLx, y=m=-sincm+1/L|A|1-|mL+1|×fmx-mL+1x0, y-mL+1y0,
fmx, y=1PQN2u=-PN/2PN/2-1v=-QN/2QN/2-1 W0mL+1u, v×expj2πxuPN+yvQN  for m0,  =1PQN2u=-PN/2PN/2-1v=-QN/2QN/2-1W0|mL+1|u, v*×expj2πxuPN+yvQN  for m<0.
gkx, y=|fx-x0, y-y0|expj arg gk-1x, yx  -N/2+x0, N/2-1+x0y  -N/2+y0, N/2-1+y0αkgk-1x, yotherwise,
gk-1x, y=1PQN2u=-PN/2PN/2-1v=-QN/2QN/2-1 Wk-1u, v×expj2πxuPN+yvQN.
gk,Δx0,Δy0x, y=gkx-Δx0, y-Δy0.
Wˆiu, v=|A|arg Wiu, v<0.5Δ i  |A|expjlΔlΔ-0.5Δ iarg Wiu, v<lΔ+0.5Δ i  |A|2π-0.5Δ iarg Wiu, vWiu, votherwise,
gp,qx, y=λp,q|fx, y| expjηp,q-1x, y-gp,q-1,Lx, yexp-j 2πNpxP+qyQ×exp-j 2πNpx0P+qy0Q,
gp,q,Lx, y=1PQN2u=-N/2N/2-1v=-N/2N/2-1 WLu, v×expj 2πNxu+yvexpj 2πNpxP+qyQexpj 2πNpx0P+qy0Q,
gp,q,L,Δx0,Δy0x, y=gp,q,Lx, y.
EMSE=10 log×x=-N/2+x0N/2-1+x0y=-N/2+y0N/2-1+y0 I0x, y-αIrx, y2x=-N/2+x0N/2-1+x0y=-N/2+y0N/2-1+y0 I0x, y2,
I0x, y=|fx, y|2,
Irx, y=|gx, y|2.

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