Abstract

Pseudorandom encoding (PRE) is a statistics-based procedure in which a pure-phase spatial light modulator (SLM) can yield, on the average, the prescribed diffraction pattern specified by the user. We seek to combine PRE with the optimization of an aperture-based target function. The target function is a fully complex input transmittance, unrealizable by a phase-only SLM, that generates a prescribed light intensity. The optimization is done to increase the diffraction efficiency of the overall process. We compare three optimization methods—Monte Carlo simulation, a genetic algorithm, and a gradient search—for maximizing the diffraction efficiency of a spot-array generator. Calculated solutions are then encoded by PRE, and the resulting diffraction patterns are computer simulated. Details on the complexity of each procedure are furnished, as well as comparisons on the quality, such as uniformity of the output spot array.

© 2000 Optical Society of America

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    [CrossRef]
  2. J. N. Mait, “Fourier array generators,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 293–323.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  7. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  8. J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
    [CrossRef]
  9. H. Stark, W. C. Catino, J. L. LoCicero, “Design of phase gratings by generalized projections,” J. Opt. Soc. Am. A 8, 566–571 (1991).
    [CrossRef]
  10. H. Stark, Y. Yang, D. Gurkan, “Factors affect convergence in the design of diffractive optics by iterative vector space methods,” J. Opt. Soc. Am. A 16, 149–159 (1999).
    [CrossRef]
  11. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  12. 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]
  13. J. Bengtsson, “Kinoform design with an optimal-rotation-angle method,” Appl. Opt. 33, 6879–6884 (1994).
    [CrossRef] [PubMed]
  14. B. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–970 (1966).
    [CrossRef] [PubMed]
  15. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1978), Vol. 16, pp. 119–231.
  16. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer, Berlin, 1980), Chap. 6, pp. 291–366.
  17. 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]
  18. D. C. Chu, J. R. Fienup, “Recent approaches to computer-generated holograms,” Opt. Eng. 13, 189–195 (1974).
    [CrossRef]
  19. D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (1986).
    [CrossRef] [PubMed]
  20. D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
    [CrossRef]
  21. M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Computer and Optically Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 34–42 (1991).
    [CrossRef]
  22. J. D. Stack, M. R. Feldman, “Recursive mean-squared-error algorithm for iterative discrete on-axis encoded holograms,” Appl. Opt. 31, 4839–4846 (1992).
    [CrossRef] [PubMed]
  23. 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.
  24. R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudo-random phase-only modulation,” Appl. Opt. 33, 4406–4415 (1994).
    [CrossRef] [PubMed]
  25. R. W. Cohn, M. Liang, “Pseudo-random phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2498 (1996).
    [CrossRef] [PubMed]
  26. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  27. S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef]
  28. H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).

1999 (1)

1996 (1)

1995 (1)

1994 (2)

1992 (3)

1991 (3)

1990 (1)

1989 (1)

1988 (1)

1986 (1)

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1983 (1)

S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

1974 (1)

D. C. Chu, J. R. Fienup, “Recent approaches to computer-generated holograms,” Opt. Eng. 13, 189–195 (1974).
[CrossRef]

1973 (1)

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)

Allebach, J. P.

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Bengtsson, J.

Brown, B. R.

Bryngdahl, O.

Casasent, D.

Catino, W. C.

Chu, D. C.

D. C. Chu, J. R. Fienup, “Recent approaches to computer-generated holograms,” Opt. Eng. 13, 189–195 (1974).
[CrossRef]

Cohn, R. W.

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

R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudo-random 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.

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.

Dandliker, R.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
[CrossRef]

Dowling, R. J.

Farn, M. W.

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Computer and Optically Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 34–42 (1991).
[CrossRef]

Feldman, M. R.

Fienup, J. R.

D. C. Chu, J. R. Fienup, “Recent approaches to computer-generated holograms,” Opt. Eng. 13, 189–195 (1974).
[CrossRef]

Gale, M. T.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
[CrossRef]

Gallagher, N. C.

Gelatt, C. D.

S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gurkan, D.

Hassebrook, L. G.

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.

Herzig, H. P.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
[CrossRef]

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]

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]

Kirpatrick, S.

S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[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]

Liang, M.

Liu, B.

LoCicero, J. L.

Lohmann, A. W.

Mait, J. N.

McKee, P.

Prongue, D.

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
[CrossRef]

Rozzi, W. A.

Stack, J. D.

Stark, H.

Streibl, N.

Sweeney, D. W.

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Vecchi, M. P.

S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Wood, D.

Wyrowski, F.

Yang, Y.

H. Stark, Y. Yang, D. Gurkan, “Factors affect convergence in the design of diffractive optics by iterative vector space methods,” J. Opt. Soc. Am. A 16, 149–159 (1999).
[CrossRef]

H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).

Appl. Opt. (11)

D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5707–5711 (1992).
[CrossRef]

B. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–970 (1966).
[CrossRef] [PubMed]

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

D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (1986).
[CrossRef] [PubMed]

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

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. D. Stack, M. R. Feldman, “Recursive mean-squared-error algorithm for iterative discrete on-axis encoded holograms,” Appl. Opt. 31, 4839–4846 (1992).
[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]

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

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

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

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]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

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

Opt. Eng. (1)

D. C. Chu, J. R. Fienup, “Recent approaches to computer-generated holograms,” Opt. Eng. 13, 189–195 (1974).
[CrossRef]

Opt. Lett. (1)

Science (1)

S. Kirpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef]

Other (7)

H. Stark, Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, New York, 1998).

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Computer and Optically Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 34–42 (1991).
[CrossRef]

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.

J. N. Mait, “Fourier array generators,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 293–323.

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

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.

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Figures (8)

Fig. 1
Fig. 1

Diffraction intensity produced by an ideal generating function [Eq. (25)] with no phase optimization.

Fig. 2
Fig. 2

Diffraction intensity produced by the NPRE algorithm.

Fig. 3
Fig. 3

Diffraction intensity realized by PRE after phase optimization by MCS1.

Fig. 4
Fig. 4

Diffraction intensity realized by PRE after phase optimization by MCS2.

Fig. 5
Fig. 5

Diffraction intensity after phase optimization by the genetic algorithm followed by PRE.

Fig. 6
Fig. 6

Diffraction intensity after phase optimization by the gradient algorithm followed by PRE.

Fig. 7
Fig. 7

Phase-only diffraction intensity produced by a genetic-algorithm-optimized generating transmittance.

Fig. 8
Fig. 8

Phase-only diffraction intensity produced by a gradient-search-optimized generating transmittance.

Tables (2)

Tables Icon

Table 1 Optimum Phase Anglesa

Tables Icon

Table 2 Performance of Optimization Routinesa

Equations (51)

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ηin=1L-L/2L/2|g(x)|2 dx(continuous)1Ni=1N|g(iΔ)|2(discrete),
g1(x)=Ki=1N exp[jϕ(iΔ)]rectx-iΔ-Δ/2Δ,
G1(u)=Δn=1N exp[jϕ(nΔ)]exp(-j2πunΔ)
G1(u)¯=Δn=1Ng(nΔ)exp(-j2πunΔ),
σ12(u)=|G1(u)|2¯-[G1(u)¯]2=N-n=1N|g(nΔ)|2=N(1-ηin).
SNR1=[G1(u)¯]2σ12=[G1(u)¯]2N[1-ηin].
G2(u)=n=12N exp[jϕ(nΔ/2)]exp(-j2πunΔ/2)
=n=1N{exp[jϕ((2n-1)Δ/2)]×exp[-j2πu(2n-1)Δ/2]+exp[jϕ(2nΔ/2)]exp(-j2πu2nΔ/2)},
G2(u)¯=n=1N[g((2n-1)Δ/2)exp(-j2πunΔ)×exp(jπuΔ)+g(nΔ)exp(-j2πunΔ)].
G2(u)¯n=1Ng((n-1/2)Δ))exp(-j2πunΔ)+n=1Ng(nΔ)exp(-j2πunΔ)2G1(u)¯.
σ22(u)=|G2(u)|2¯-[G2(u)¯]2=2N-n=12N|g(nΔ/2)|2=2σ12(u).
SNR2=[G2(u)¯]2σ222×SNR1.
g(x, ϕ)=f(x, ϕ)maxx|f(x, ϕ)|,
f(x, ϕ)=k=1M exp[j(2πukx+ϕk)],
ηin(ϕ)=α2(ϕ)M,
α(ϕ)1maxx|f(x, ϕ)|.
ϕ*=arg[maxϕ α2(ϕ)].
e(β, ϕ)k=1N[β-|f(kΔ, ϕ)|2]2
(β, ϕ)k+1=(β, ϕ)k-γe(β, ϕ),
e(β, ϕ)=eβeϕ1eϕMT,
Tϕ+Tmax+Tηin,
TT(MC)=C(Tϕ+Tmax+Tηin).
TT(GA)=25Q(Tϕ+Tmax+Tηin+Ts+Tco+Tmu)25Q(Tϕ+Tmax+Tηin),
ηin(ϕ)=k=1Nα2|f(kΔ, ϕ)|2=k=1Nα2l=1M i=1Mexp{j[2π(ul-ui)kΔ+ϕl-ϕi]}=k=1Nα2[M+2Q(kΔ)],
Q(kΔ)=l=1Mi=1+1M cos[2π(ul-ui)kΔ+ϕl-ϕi].
e(β, ϕ)=k=1N[β-M-2Q(kΔ)]2,
eβ=k=1N[2β-2M-4Q(kΔ)],
eϕl=k=1N[2β-2M-4Q(kΔ)][2Sl(kΔ)],
l=1,, M,
Sl(kΔ)=i=1ilM sin[2π(ul-ui)kΔ+ϕl-ϕi],
l=1,, M.
TT(GR)=PMTηin.
ηout=energyindesireddiffractionpatterntotalenergyinfrequencyplane=RI(u)du-I(u)du,
ηout=i=110I(ui)n=0512I(nΔu).
ADCi=Imax,p(i)-Imin,p(i)Imax,p(i)+Imin,p(i),
ADC¯=110i=110ADCi,
ADC*=maxi ADCi.
σN¯=110i=110σN(i),
σN(i)=n=110[Ipeak,n(i)-I¯peak(i)]21/2I¯peak(i),
I¯peak(i)=110n=110Ipeak,n(i).
g=- exp(jα)fθ(α)dα
ϕθ(α; c, w)=1wrectα-cw,
g=exp(jc)sin(w/2)w/2=exp(jc)sinc(w/2π).
c(x):c=arg[g(x)],
w(x):sinc(w/2π)|=|g(x)|.
pi=ηin(i)i=125ηin(i),qk=i=1kpi,
ηin(i)*=αηin(i)+β,
α=Cηmax-ηavgηmax-ηavg,
ηmax=maxi ηin(i),
ηavg=125i=125ηin(i),
β=1-α,

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