Abstract

Liquid-crystal light valves can have intensity-dependent resolution. We find for a nematic liquid-crystal light valve that this effect is well modeled as a phase that has been blurred by a linear space-invariant filter. The phase point-spread function is measured and is used in simulations to demonstrate that it introduces intermodulation products to the diffraction patterns of computer-generated Fourier transform holograms. Also, the influence of phase blurring on a pseudorandom-encoding algorithm is evaluated in closed form. This analysis applied to a spot array generator design indicates that nonlinear effects are negligible only if the diameter of the point-spread function is a small fraction of the pixel spacing.

© 2000 Optical Society of America

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References

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  1. D. Casasent, “Performance evaluation of spatial light modulators,” Appl. Opt. 18, 2445–2453 (1979).
    [CrossRef] [PubMed]
  2. A. D. Fisher, J. N. Lee, “The current status of two-dimensional spatial light modulator technology,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. SPIE634, 352–371 (1986).
    [CrossRef]
  3. T. D. Hudson, D. A. Gregory, “Optically addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
    [CrossRef]
  4. D. V. Wick, T. Martinez, M. V. Wood, J. M. Wilkes, M. T. Gruneisen, V. L. Berenberg, M. V. Vasil’ev, A. P. Onokhov, L. A. Beresnev, “Deformed-helix ferroelectric liquid-crystal spatial light modulator that demonstrates high diffraction efficiency and 370-line pairs/mm resolution,” Appl. Opt. 38, 3798–3803 (1999).
    [CrossRef]
  5. R. W. Cohn, A. A. Vasiliev, W. Liu, D. L. Hill, “Fully complex diffractive optics by means of patterned diffuser arrays: encoding concept and implications for fabrication,” J. Opt. Soc. Am. A 14, 1110–1123 (1997).
    [CrossRef]
  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.
  7. 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.
  8. R. W. Cohn, M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. 35, 2488–2497 (1996).
    [CrossRef] [PubMed]
  9. R. W. Cohn, M. Duelli, “Ternary pseudorandom encoding of Fourier transform holograms,” J. Opt. Soc. Am. A 16, 71–84; “errata,” 16, 1089–1090 (1999).
  10. 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]
  11. R. W. Cohn, M. Liang, “Approximating fully complex spatial modulation with pseudorandom phase-only modulation,” Appl. Opt. 33, 4406–4415 (1994).
    [CrossRef] [PubMed]
  12. G. Paul-Hus, Y. Sheng, “Optical on-axis real-time phase-dominant correlator using liquid crystal television,” Opt. Eng. 32, 2165–2172 (1993).
    [CrossRef]
  13. R. W. Cohn, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. 33, 5432–5439 (1994).
    [CrossRef] [PubMed]
  14. E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).
  15. 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]
  16. R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
    [CrossRef]
  17. M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Appl. Opt. 37, 8131–8133 (1998).
    [CrossRef]
  18. R. R. Read, J. L. Shanks, S. Treitel, “Two dimensional recursive filtering,” in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, Berlin, 1979), pp. 131–176.
    [CrossRef]
  19. J. S. Lim, “Image restoration,” in Two-Dimensional Signal- and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 524–588.
  20. A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 530.

1999 (3)

1998 (1)

1997 (1)

1996 (2)

1994 (2)

1993 (1)

G. Paul-Hus, Y. Sheng, “Optical on-axis real-time phase-dominant correlator using liquid crystal television,” Opt. Eng. 32, 2165–2172 (1993).
[CrossRef]

1991 (1)

T. D. Hudson, D. A. Gregory, “Optically addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

1988 (1)

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

1979 (1)

Berenberg, V. L.

Beresnev, L. A.

Bernstein, H.

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

Casasent, D.

Cohn, R. W.

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

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 by means of patterned diffuser arrays: encoding concept and implications for fabrication,” 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, 2488–2497 (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, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. 33, 5432–5439 (1994).
[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, M. Duelli, “Ternary pseudorandom encoding of Fourier transform holograms,” J. Opt. Soc. Am. A 16, 71–84; “errata,” 16, 1089–1090 (1999).

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.

Duelli, M.

Fisher, A. D.

A. D. Fisher, J. N. Lee, “The current status of two-dimensional spatial light modulator technology,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. SPIE634, 352–371 (1986).
[CrossRef]

Friesem, A. A.

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

Gregory, D. A.

T. D. Hudson, D. A. Gregory, “Optically addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

Grubel, H.

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

Gruneisen, M. T.

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.

Horner, J. L.

Hudson, T. D.

T. D. Hudson, D. A. Gregory, “Optically addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

Lee, J. N.

A. D. Fisher, J. N. Lee, “The current status of two-dimensional spatial light modulator technology,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. SPIE634, 352–371 (1986).
[CrossRef]

Lhamon, M. E.

Liang, M.

Lim, J. S.

J. S. Lim, “Image restoration,” in Two-Dimensional Signal- and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 524–588.

Liu, W.

Martinez, T.

Onokhov, A. P.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 530.

Paul-Hus, G.

G. Paul-Hus, Y. Sheng, “Optical on-axis real-time phase-dominant correlator using liquid crystal television,” Opt. Eng. 32, 2165–2172 (1993).
[CrossRef]

Read, R. R.

R. R. Read, J. L. Shanks, S. Treitel, “Two dimensional recursive filtering,” in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, Berlin, 1979), pp. 131–176.
[CrossRef]

Reece, M.

Shafer, R. W.

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 530.

Shafir, E.

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

Shanks, J. L.

R. R. Read, J. L. Shanks, S. Treitel, “Two dimensional recursive filtering,” in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, Berlin, 1979), pp. 131–176.
[CrossRef]

Sheng, Y.

G. Paul-Hus, Y. Sheng, “Optical on-axis real-time phase-dominant correlator using liquid crystal television,” Opt. Eng. 32, 2165–2172 (1993).
[CrossRef]

Treitel, S.

R. R. Read, J. L. Shanks, S. Treitel, “Two dimensional recursive filtering,” in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, Berlin, 1979), pp. 131–176.
[CrossRef]

Vasil’ev, M. V.

Vasiliev, A. A.

Wick, D. V.

Wilkes, J. M.

Wood, M. V.

Appl. Opt. (6)

J. Opt. Soc. Am. A (3)

Opt. Eng. (3)

G. Paul-Hus, Y. Sheng, “Optical on-axis real-time phase-dominant correlator using liquid crystal television,” Opt. Eng. 32, 2165–2172 (1993).
[CrossRef]

E. Shafir, H. Bernstein, A. A. Friesem, H. Grubel, “Method for measuring the spatial frequency response of phase-modulating spatial light modulators,” Opt. Eng. 27, 71–74 (1988).

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

Opt. Laser Technol. (1)

T. D. Hudson, D. A. Gregory, “Optically addressed spatial light modulators,” Opt. Laser Technol. 23, 297–302 (1991).
[CrossRef]

Opt. Lett. (1)

Other (6)

A. D. Fisher, J. N. Lee, “The current status of two-dimensional spatial light modulator technology,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. SPIE634, 352–371 (1986).
[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.

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.

R. R. Read, J. L. Shanks, S. Treitel, “Two dimensional recursive filtering,” in Picture Processing and Digital Filtering, Vol. 6 of Topics in Applied Physics, T. S. Huang, ed. (Springer-Verlag, Berlin, 1979), pp. 131–176.
[CrossRef]

J. S. Lim, “Image restoration,” in Two-Dimensional Signal- and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 524–588.

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), p. 530.

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Figures (6)

Fig. 1
Fig. 1

Far-field diffraction patterns from (a) a DOE, (b) a LCLV, and (c) a simulation of (b) that includes a linear shift-invariant phase-blurring model. Maximum white in the gray-scale images correspond in (a) and (c) to 30% and in (b) to 20% of the average of the peak intensities of the spots in the desired spot array.

Fig. 2
Fig. 2

Simulated effect of phase blurring on performance for phase PSF’s of various diameters. The vertical bars indicate the value of ΔPSF/Δ corresponding to the phase PSF measured for the actual LCLV. The dotted lines indicate the performance values for ΔPSF/Δ=0.15.

Fig. 3
Fig. 3

Far-field diffraction patterns resulting from the identical PRE design of a spot array generator: (a) and (d) without phase blurring, (b) and (e) with phase blurring α = 0.5, and (c) and (f) with phase blurring of a predistorted phase for α = 0.5. (a) and (b) are simulated, (c) is the average of ten simulations each using a different random sequence for PRE, (d) and (e) are as measured for the BNS SLM, and (f) is the expected far-field intensity pattern as calculated with Eq. (15). Maximum white in the gray-scale images corresponds in (a) and (d) to 30%, in (b) and (e) to 50%, and in (c) and (f) to 1% of the average peak intensity of the desired spot array.

Fig. 4
Fig. 4

Performance of PRE as a function of α, the degree of phase blurring. Solid curves report the computer-simulated results for the effect of phase blurring. Dots show the experimentally measured values obtained with the BNS SLM. Thin curves report the simulated performance if the phase is first predistorted through the inverse filter of Eq. (17). The inset plots E1/E-1 over an extended range that shows the simulated performance for phase blurring and for phase blurring of the predistorted phase.

Fig. 5
Fig. 5

Illustration of the distortion of a modded phase ramp of 2π range due to the phase PSF of Eq. (1).

Fig. 6
Fig. 6

Far-field diffraction patterns resulting from the identical PRE design of a spot array generator: (a) and (d) without phase blurring, (b) and (e) with phase blurring α = 0.3, and (c) and (f) with phase blurring α = 0.5. (a)–(c) are simulated, and (d)–(f) are as measured for the BNS SLM. Maximum white in the gray-scale images corresponds to 15% of the peak intensity level of the desired spot array.

Tables (4)

Tables Icon

Table 1 Performance of PRE Implemented on a DOE

Tables Icon

Table 2 Performance of PRE Implemented on a LCLV

Tables Icon

Table 3 Diffraction-Order Intensities for a Modded Phase Ramp That Is Phase Blurred a

Tables Icon

Table 4 Diffraction-Order Intensities That Result from Phase Blurring of PRE a

Equations (24)

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h(x)=(1-α)δ(x)+αδ(x-Δ),
 exp[jϕ(x) * h(x)]=exp[jϕ(x)]×exp{jα[ϕ(x-Δ)-ϕ(x)]},
exp[jϕ(x) * h(x)]=exp[j(1-α)ϕ(x)]exp[jαϕ(x-Δ)].
aci=api(a)/daai,
aci=exp(jϕ)pi(ϕ)dϕ=exp(ϕ)i.
ϕi=ψi-νi/2,0si<1/2ψi+νi/2,1/2<si1,
pi(ϕ)=12{δ [ϕ-(ψi-νi/2)]+δ [ϕ-(ψi+νi/2)]},
ai=cos(νi/2)exp(jψi).
νi/2=arccos(|aci|),
A(fx)i=1Naiexp(-j2πiΔfx),
Ac(fx)=A(fx)=i=1Naciexp(-j2π iΔfx),
|A(fx)|2=i=1Nk=1Naiak*exp[-j2π(i-k)Δfx]= |Ac( fx)|2+i=1N(1-|aci|2).
biexp{j[(1-α)ϕi+αϕi-1]}=exp[j(1-α)ϕi]exp(jαϕi-1)=cos[(1-α)νi/2]cos(ανi-1/2)×exp{j[(1-α)ψi+αψi-1]}.
|A(fx)|2
= |B(fx)|2+i=1N(1-|bi|2)+2 i=1N-1Re({exp[j(1-2α)ϕi]exp[-j(1-α)ϕi+1]×exp(jαϕi-1)-bibi+1*}exp(j2πΔfx)).
exp[j(1-2α)ϕi]exp[j(1-α)ϕi]exp[-jαϕi]
|A(fx)|2
= |B(fx)|2+i=1N(1-|bi|2)+2 Rei=1Ntan[(1-α)νi/2]×tan(ανi/2)bibi+1*exp(j2πΔfx).
|A( fx)|2= |B( fx)|2+C1+C2cos(2πΔfx+Φ),
hi-1=11-α-α1-αi;i0.
ψb(x)=2π[(x-αΔ)f0+α]if0x<Δ2π(x-αΔ)f0ifΔx<1/f0.
dk=f001/f0exp[jψb(x)]exp(-j2πkf0x)dx,
|dk|2
=(1-Δf0)2+(Δf0)2+2(Δf0)(1-Δf0)cos(2πα)ifk=1|[2/π(1-k)]sin[πΔf0(1-k)]sin(πα)|2ifk1.

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