Abstract

Error diffusion (ED) and pseudorandom encoding (PRE) methods of designing Fourier transform holograms are compared in terms of their properties and the optical performance of the resulting far-field diffraction patterns. Although both methods produce a diffuse noise pattern due to the error between the desired fully complex pattern and the encoded modulation, the PRE errors reconstruct uniformly over the nonredundant bandwidth of the discrete-pixel spatial light modulator, while the ED errors reconstruct outside the window of the designed diffraction pattern. Combining the two encoding methods produces higher-fidelity diffraction patterns than either method produces individually. For some designs the fidelity of the ED–PRE algorithm is even higher over the entire nonredundant bandwidth than for the previously reported [J. Opt. Soc. Am. A 16, 2425 (1999)] minimum-distance-PRE algorithm.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. R. W. Cohn, W. Liu, “Pseudorandom encoding of fully complex modulation to bi-amplitude phase modulators,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 237–240.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  25. To maintain as much consistency as possible in comparing all the curves and tables and to avoid excessive computation, we have calculated and reported all performance met-rics for values of γ=1, 1.1, 1.2,… .This results in adequately smooth and sampled curves except in one case. For the SPRmcurve of ED–PRE in Fig. 5, finer sampling led to a significant increase in SPRm,from 53 at γ=1.2and 1.3 to 60 at γ=1.26.This additional point is included in the plot in Fig. 5. We also checked the maxima of other SPR and SPRmperformance curves, using finer sampling increments. However, since the change in appearance is minimal and the maximum values of the curves would change by no more than a few tenths, these additional findings are omitted.

1999 (4)

1998 (1)

1996 (2)

1994 (2)

1993 (1)

1992 (1)

1991 (1)

1989 (1)

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

1988 (1)

1984 (2)

1976 (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

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]

1966 (1)

Barnard, E.

Brown, B. R.

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]

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

Cohn, R. W.

R. W. Cohn, M. Duelli, “Ternary pseudorandom encoding of Fourier transform holograms,” J. Opt. Soc. Am. A 16, 71–84 (1999); Errata, 1089–1090 (1999).
[CrossRef]

M. Duelli, M. Reece, R. W. Cohn, “A modified minimum-distance criterion for blended random and nonrandom encoding,” J. Opt. Soc. Am. A 16, 2425–2438 (1999).
[CrossRef]

R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
[CrossRef]

M. Duelli, R. W. Cohn, “Pseudorandom encoding for real-valued ternary spatial light modulators,” Appl. Opt. 38, 3804–3809 (1999).
[CrossRef]

R. W. Cohn, “Pseudorandom encoding of complex valued functions onto amplitude coupled phase modulators,” J. Opt. Soc. Am. A 15, 868–883 (1998).
[CrossRef]

L. G. Hassebrook, M. E. Lhamon, R. C. Daley, R. W. Cohn, M. Liang, “Random phase encoding of composite fully complex filters,” Opt. Lett. 21, 272–274 (1996).
[CrossRef] [PubMed]

R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2497 (1996).
[CrossRef] [PubMed]

R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudorandom phase-only modulation,” Appl. Opt. 33, 4406–4415 (1994).
[CrossRef] [PubMed]

R. W. Cohn, L. G. Hassebrook, “Representations of fully complex functions on real-time spatial light modulators,” in Optical Information Processing, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 15, pp. 396–432.

R. W. Cohn, W. Liu, “Pseudorandom encoding of fully complex modulation to bi-amplitude phase modulators,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 237–240.

Cottrell, D. M.

Daley, R. C.

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 291–366.

Davis, J. A.

Duelli, M.

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Gianino, P. D.

Goodman, J. M.

J. M. Goodman, “Effects of film nonlinearities,” in Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Sec. 8-6, pp. 230–241.

Hall, T. J.

A. G. Kirk, A. K. Powell, T. J. Hall, “The design of quasi-periodic Fourier plane array generators,” in Optical Information Technology, S. D. Smith, R. F. Neale, eds. (Springer-Verlag, Berlin, 1993), pp. 47–56.

Hassebrook, L. G.

L. G. Hassebrook, M. E. Lhamon, R. C. Daley, R. W. Cohn, M. Liang, “Random phase encoding of composite fully complex filters,” Opt. Lett. 21, 272–274 (1996).
[CrossRef] [PubMed]

R. W. Cohn, L. G. Hassebrook, “Representations of fully complex functions on real-time spatial light modulators,” in Optical Information Processing, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 15, pp. 396–432.

Hauk, 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]

Horner, J. L.

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]

Juday, R. D.

Kirk, A. G.

A. G. Kirk, A. K. Powell, T. J. Hall, “The design of quasi-periodic Fourier plane array generators,” in Optical Information Technology, S. D. Smith, R. F. Neale, eds. (Springer-Verlag, Berlin, 1993), pp. 47–56.

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]

Lhamon, M. E.

Liang, M.

Liu, W.

R. W. Cohn, W. Liu, “Pseudorandom encoding of fully complex modulation to bi-amplitude phase modulators,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 237–240.

Lohmann, A. W.

Powell, A. K.

A. G. Kirk, A. K. Powell, T. J. Hall, “The design of quasi-periodic Fourier plane array generators,” in Optical Information Technology, S. D. Smith, R. F. Neale, eds. (Springer-Verlag, Berlin, 1993), pp. 47–56.

Reece, M.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Weissbach, S.

S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[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]

Wyrowski, F.

Appl. Opt. (7)

IBM J. Res. Dev. (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]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

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. (1)

R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
[CrossRef]

Opt. Lett. (3)

Proc. Soc. Inf. Disp. (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Other (6)

R. W. Cohn, L. G. Hassebrook, “Representations of fully complex functions on real-time spatial light modulators,” in Optical Information Processing, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 15, pp. 396–432.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), Chap. 6, pp. 291–366.

A. G. Kirk, A. K. Powell, T. J. Hall, “The design of quasi-periodic Fourier plane array generators,” in Optical Information Technology, S. D. Smith, R. F. Neale, eds. (Springer-Verlag, Berlin, 1993), pp. 47–56.

J. M. Goodman, “Effects of film nonlinearities,” in Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Sec. 8-6, pp. 230–241.

To maintain as much consistency as possible in comparing all the curves and tables and to avoid excessive computation, we have calculated and reported all performance met-rics for values of γ=1, 1.1, 1.2,… .This results in adequately smooth and sampled curves except in one case. For the SPRmcurve of ED–PRE in Fig. 5, finer sampling led to a significant increase in SPRm,from 53 at γ=1.2and 1.3 to 60 at γ=1.26.This additional point is included in the plot in Fig. 5. We also checked the maxima of other SPR and SPRmperformance curves, using finer sampling increments. However, since the change in appearance is minimal and the maximum values of the curves would change by no more than a few tenths, these additional findings are omitted.

R. W. Cohn, W. Liu, “Pseudorandom encoding of fully complex modulation to bi-amplitude phase modulators,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 237–240.

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Figures (10)

Fig. 1
Fig. 1

Illustration of the encoding methods: (a) MDE, (b) ED, (c) PRE, (d) MD–PRE, (e) ED–PRE for bij outside the unit circle, (f ) ED–PRE for bij inside the unit circle. The error diffused forward from previously encoded pixels is represented in the illustration by δij(εi-1, j+εi, j-1)/2. In (f ) the encoding error εε/χ, where ε is the amount of the encoding error that is diffused forward with use of Eq. (4).

Fig. 2
Fig. 2

Flow chart for the ED–PRE algorithm.

Fig. 3
Fig. 3

On-axis diffraction patterns for encoding by (a)–(d) and (f) ED and (e) MDE. The value of γ used for ED is (a) 1.0, (b) 1.2, (c) 1.3, (d) 1.6, and (f ) 5.5. The dotted square encloses the area used to calculate SPRm. The gray-scale intensities are scaled so that full white corresponds to 3% of the average peak intensities of the 49 desired spots.

Fig. 4
Fig. 4

(a)–(c) On-axis and (d)–(f ) off-axis diffraction patterns for the maximum SPR design by (a) ED, γ=1.5; (b) MD–PRE, γ=1.4; (c) ED–PRE, γ=1.3, χ=0.6; (d) ED, γ=1.5; (e) MD–PRE, γ=1.4; (f ) ED–PRE, γ=1.3, χ=1. The gray-scale normalization and the dotted square are identical to those used in Fig. 3.

Fig. 5
Fig. 5

Performance curves of the various encoding algorithms as a function of γ for the on-axis test function.25 For ED–PRE the specific curve shown for NU is for χ=0.9, for SPR is for χ=0.6, and for SPRm is for χ=0.7. These curves achieve the best performance for ED–PRE as reported in Table 1.

Fig. 6
Fig. 6

Sensitivity of fidelity metrics of ED–PRE to the free parameter χ[0.5, 1] for the on-axis test function.

Fig. 7
Fig. 7

Statistical variations of the fidelity metrics of (a) ED–PRE and (b) MD–PRE for an ensemble of encodings of the on-axis function. Shaded regions are bounded by the maximum and minimum values found for 21 random trials of each encoding algorithm. The respective curves from Fig. 5 are reproduced for comparison.

Fig. 8
Fig. 8

Performance curves of the various encoding algorithms as a function of γ for the off-axis test function. For ED–PRE the specific curve shown for NU is for χ=0.7, for SPR is for χ=1, and for SPRm is for χ=0.6. These curves achieve the best performance for ED–PRE as reported in Table 2.

Fig. 9
Fig. 9

Sensitivity of fidelity metrics of ED–PRE to the free parameter χ[0.5,1] for the off-axis test function.

Fig. 10
Fig. 10

Statistical variations of the fidelity metrics of (a) ED–PRE and (b) MD–PRE for an ensemble of encodings of the off-axis test function. Shaded regions are bounded by the maximum and minimum values found for 21 random trials of each encoding algorithm. The respective curves from Fig. 8 are reproduced for comparison.

Tables (2)

Tables Icon

Table 1 Best Encoding Performance for the On-Axis Function

Tables Icon

Table 2 Best Encoding Performance for the Off-Axis Function

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

aij=exp[j arg(ac ij)],
eij=ac ij-aij.
aij=exp[j arg(bij)],
bij=ac ij+(εi-1,j+εi,j-1)/2,
εij=bij-aij
aij=cos(νij /2)exp(ψc ij).
νij=2 arccos(|ac ij |).
εij=χ(bij-aij).

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