Abstract

Two pixel-oriented methods for designing Fourier transform holograms—pseudorandom encoding and minimum-distance encoding—usually produce higher-fidelity reconstructions when combined than those produced by each method individually. In previous studies minimum-distance encoding was defined as the mapping from the desired complex value to the closest value produced by the modulator. This method is compared with a new minimum-distance criterion in which the desired complex value is mapped to the closest value that can be realized by pseudorandom encoding. Simulations and experimental measurements using quantized phase and amplitude modulators show that the modified approach to blended encoding produces more faithful reconstructions than those of the previous method.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  25. M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
    [CrossRef] [PubMed]
  26. R. D. Juday, J. Knopp, “HOLOMED—an algorithm for computer generated holograms,” in Optical Pattern Recognition VII, D. P. Casasent, T. Chao, eds., Proc. SPIE2752, 162–172 (1996).
    [CrossRef]
  27. R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
    [CrossRef]
  28. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  31. M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Appl. Opt. 37, 8131–8133 (1998).
    [CrossRef]
  32. The influence that is due to roll-off may at first seem to be surprisingly small, but calculating the effect of this roll-off on the nonuniformity/standard deviation of the ideally uniform spot array gives NU=4.2%. The small influence of the rolloff on NU is further explained by the fact that the simulated values of NU are generally greater than 4% and that standard deviations, rather than being additive, add as the square root of the sum of the squares.

1999

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

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

1998

1997

1996

1995

1994

1993

1992

1991

1989

1984

1973

1971

1969

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

Abushagur, M. A.

Arsenault, H. H.

Bengtsson, J.

Bergeron, A.

Brown, B. R.

Campos, J.

Catino, W. C.

Cho, D. J.

Cohn, R. W.

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

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

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

M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Appl. Opt. 37, 8131–8133 (1998).
[CrossRef]

R. W. Cohn, A. A. Vasiliev, W. Liu, D. L. Hill, “Fully complex diffractive optics via patterned diffuser arrays,” J. Opt. Soc. Am. A 14, 1110–1123 (1997).
[CrossRef]

R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
[CrossRef] [PubMed]

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

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.

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.

Daley, R. C.

Dallas, W. J.

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

Dames, M. P.

Donner, J. T.

Doucet, M.

Dowling, R. J.

Duelli, M.

Farn, M. W.

Gagnon, F.

Gallagher, N. C.

Gauvin, J.

Gianino, P. D.

Gingras, D.

Goodman, J. W.

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.

Hill, D. L.

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.

Johnson, E. G.

Jones, A. 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.

Juvells, I.

Kirk, J. P.

Knopp, J.

R. D. Juday, J. Knopp, “HOLOMED—an algorithm for computer generated holograms,” in Optical Pattern Recognition VII, D. P. Casasent, T. Chao, eds., Proc. SPIE2752, 162–172 (1996).
[CrossRef]

Krackhardt, U.

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–231.

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, B.

Liu, W.

R. W. Cohn, A. A. Vasiliev, W. Liu, D. L. Hill, “Fully complex diffractive optics via patterned diffuser arrays,” J. Opt. Soc. Am. A 14, 1110–1123 (1997).
[CrossRef]

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.

LoCicero, J. L.

Lohmann, A. W.

Mait, J. N.

McKee, P.

Montes-Usategui, M.

Morris, G. M.

Stark, H.

Streibl, N.

Thurman, S. T.

Vasiliev, A. A.

Wood, D.

Appl. Opt.

M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
[CrossRef] [PubMed]

J. Bengtsson, “Kinoform design with an optimal-rotation-angle method,” Appl. Opt. 33, 6879–6884 (1994).
[CrossRef] [PubMed]

N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
[CrossRef] [PubMed]

R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
[CrossRef] [PubMed]

B. R. Brown, A. W. Lohmann, “Complex spatial filter,” Appl. Opt. 5, 967–969 (1966).
[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. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
[CrossRef]

R. D. Juday, “Correlation with a spatial light modulator having phase and amplitude cross coupling,” Appl. Opt. 28, 4865–4869 (1989).
[CrossRef] [PubMed]

M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

A. Bergeron, J. Gauvin, F. Gagnon, D. Gingras, H. H. Arsenault, M. Doucet, “Phase calibration and applications of a liquid-crystal spatial light modulator,” Appl. Opt. 34, 5133–5139 (1995).
[CrossRef] [PubMed]

M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Appl. Opt. 37, 8131–8133 (1998).
[CrossRef]

IBM J. Res. Dev.

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.

J. Opt. Soc. Am. A

Opt. Eng.

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

Opt. Lett.

Opt. Photon. News

J. N. Mait, “Diffractive beauty,” Opt. Photon. News 9, 21–25 (November1998).
[CrossRef]

Other

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–231.

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

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.

The influence that is due to roll-off may at first seem to be surprisingly small, but calculating the effect of this roll-off on the nonuniformity/standard deviation of the ideally uniform spot array gives NU=4.2%. The small influence of the rolloff on NU is further explained by the fact that the simulated values of NU are generally greater than 4% and that standard deviations, rather than being additive, add as the square root of the sum of the squares.

R. D. Juday, J. Knopp, “HOLOMED—an algorithm for computer generated holograms,” in Optical Pattern Recognition VII, D. P. Casasent, T. Chao, eds., Proc. SPIE2752, 162–172 (1996).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Modulation characteristics for which the minimum-distance mapping to the modulation characteristic and to the encoding range (striped regions) of the PRE algorithm are (a) identical and (b) different. In (a) the modulation characteristic is a circle, and in (b) the modulation characteristic is the three dots, one at each apex of the triangle.

Fig. 2
Fig. 2

Comparison of (a) the MDE algorithm with (b) the PRE algorithm for a tri-phase phase-only modulation characteristic.

Fig. 3
Fig. 3

Illustration of the encoding range and the fully complex encoding range, and their relationship to the scaling parameter γ for a quad-phase modulation characteristic.

Fig. 4
Fig. 4

Classification of the various modulation characteristics considered in this paper.

Fig. 5
Fig. 5

Illustration of the individual MDE and PRE algorithms together with their blending for continuous modulation characteristics.

Fig. 6
Fig. 6

Illustration of the individual MDE and PRE algorithms together with their minimum-distance and modified-minimum-distance blendings for [(a)–(d)] tri-phase and [(e)–(h)] quad-phase phase-only modulation characteristics. Parts (a) and (e) identify the decision regions for MDE. Parts (b) and (f) show the encoding ranges for the PRE algorithms together with the fully complex ranges, which are bounded by each dashed circle. Part (f) also indicates that there are two regions. Each triangular region is encoded by Eqs. (7) and (8) with use of the three modulation values at the corners of the corresponding regions.

Fig. 7
Fig. 7

Illustration of the individual MDE and PRE algorithms for bi-amplitude modulation characteristics: (a) MDE and (b) PRE for tri-phase SLM's; (c) MDE and (d) PRE for quad-phase SLM's. Parts (a) and (c) show the decision regions for the MDE algorithms. Parts (b) and (d) show the individual subregions that are each encoded by using ternary PRE. The striped areas of region IV in (a) are outside the encoding range for PRE in (b). Therefore the MD-PRE blending of (a) and (b) requires that values in the striped areas be mapped to zero according to the MDE algorithm.

Fig. 8
Fig. 8

Simulated performance of blended algorithms as a function of the blending parameter for phase-only and bi-amplitude phase modulation characteristics.

Fig. 9
Fig. 9

Simulated far-field intensity patterns of the tri-phase phase-only SLM for (a) PRE, (b) mMD-PRE, (c) MD-PRE, and (d) MDE. The images show intensity with a linear gray scale. To bring out the background noise, the maximum gray-scale value (full white) is 30% of the average intensity of the 49 spots.

Fig. 10
Fig. 10

Cross sections of the far-field intensity patterns of the tri-phase phase-only SLM from Fig. 9. The cross section is taken across the diagonal of the 7×7 spot array and through the optical axis (indicated by the dashed vertical line). The traces are normalized so that the average intensity of each spot array is of identical vertical length on each plot. 

Fig. 11
Fig. 11

Simulated performance of blended algorithms as a function of the blending parameter for quantized-phase phase-only modulation characteristics.

Fig. 12
Fig. 12

Simulated performance of blended algorithms as a function of the blending parameter for quantized-phase bi-amplitude phase modulation characteristics.

Fig. 13
Fig. 13

Comparison between theory (thick solid curve) and experiment (thin solid curve and dots) of the performance of mMD-PRE on quad-phase phase-only SLM's. The dashed curve on the plots of NU shows the theory with the frequency response roll-off (which is to the aperture of the SLM pixels) taken into account.

Tables (3)

Tables Icon

Table 1 Best Performance of Encoding Algorithms for Continuous SLM's

Tables Icon

Table 2 Best Performance of Encoding Algorithms for Phase-Only SLM's

Tables Icon

Table 3 Best Performance of Encoding Algorithms for Bi-Amplitude SLM's

Equations (9)

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

pi=(1+|aci|)/2,
ai=exp[j arg(aci)]if0si<pi-exp[j arg(aci)]ifpisi1,
pi=|aci|,
ai=exp[j arg(aci)]if0si<pi0ifpisi1.
ai=am1if0si<piam2ifpisi<1-riam3if1-risi1,
Re(aci)Im(aci)1=Re(am1)Re(am2)Re(am3)Im(am1)Im(am2)Im(am3)111piqiri,
ai=1if0si<pi±jifpisi<1-ri-1if1-risi1,
Re(aci)Im(aci)1=10-10±10111piqiri,
ac(x, y)=k=17 exp(jθk)exp(j2πkx)×l=17 exp(jθl)exp(j2πly),

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