Abstract

Pseudorandom encoding is a method of statistically approximating desired complex values with those values that are achievable with a given spatial light modulator. Originally developed for phase-only modulators, pseudorandom encoding is extended to modulators for which amplitude is a function of phase. This is accomplished by transforming the phase statistics to compensate for the amplitude coupling. Example encoding formulas are derived, evaluated, and compared with a noncompensating pseudorandom-encoding algorithm. Compensating algorithms encode a smaller area of the complex plane and can produce more noise than is possible for arbitrary pseudorandom algorithms. However, the encoding formulas have greatly simplified numerical implementations.

© 1998 Optical Society of America

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References

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    [CrossRef]
  3. L. G. Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid crystal television,” Appl. Opt. 34, 1944–1950 (1995).
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    [CrossRef] [PubMed]
  7. R. W. Cohn, A. A. Vasiliev, W. Liu, D. L. Hill, “Fully complex diffractive optics by means of patterned diffuser arrays,” J. Opt. Soc. Am. A 14, 1110–1123 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. R. W. Cohn, W. Liu, “Pseudorandom encoding of fully complex modulation to bi-amplitude phase modulators,” in Diffractive Optics and Microoptics, 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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  15. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [CrossRef]

1997

1996

1995

1994

1993

1989

1974

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

1972

1971

1966

1949

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Brown, B. R.

Chu, D. C.

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

D. C. Chu, J. W. Goodman, “Spectrum shaping with parity sequences,” Appl. Opt. 11, 1716–1724 (1972).
[CrossRef] [PubMed]

Cohn, R. W.

Daley, R. C.

Fienup, J. R.

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

Goodman, J. W.

Hassebrook, L. G.

Hill, D. L.

Jones, A. L.

Juday, R. D.

Kirk, J. P.

Lhamon, M. E.

Liang, M.

Liu, W.

R. W. Cohn, A. A. Vasiliev, W. Liu, D. L. Hill, “Fully complex diffractive optics by means of 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 Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 237–240.

Lohmann, A. W.

Monroe, S. E.

C. Soutar, S. E. Monroe, “Selection of operating curves of twisted-nematic liquid crystal televisions,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 280–291 (1994).
[CrossRef]

Neto, L. G.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 5, pp. 101–102 and Chap. 8, pp. 226–229.

Roberge, D.

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Sheng, Y.

Soutar, C.

C. Soutar, S. E. Monroe, “Selection of operating curves of twisted-nematic liquid crystal televisions,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 280–291 (1994).
[CrossRef]

Vasiliev, A. A.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

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

Opt. Lett.

Proc. IRE

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Other

C. Soutar, S. E. Monroe, “Selection of operating curves of twisted-nematic liquid crystal televisions,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 280–291 (1994).
[CrossRef]

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

A. Papoulis, Probability, Random Variables, and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), Chap. 5, pp. 101–102 and Chap. 8, pp. 226–229.

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Figures (10)

Fig. 1
Fig. 1

Amplitude-coupled phase modulation characteristics illustrated in (a) rectangular and (b) polar plots. The characteristics have continuous phase ranges of 4π (solid curves) and 2π (dotted curves). Although amplitude coupling is drawn as a linear function, it is not necessary that it be linear for example 3 in Section 4.

Fig. 2
Fig. 2

Probability density functions (pdf’s) for (a) uniform and (b) binomial random distributions of phase. In Section 4 effective pdf’s of these same forms are sought.

Fig. 3
Fig. 3

Effective amplitudes for examples 1 and 2. The amplitude coupling used is identical to the solid curves in Fig. 1. The curve without a legend (dashed–dotted–dotted curve) is the effective amplitude for ψ0=-π with the maximum effective amplitude normalized to unity. The two other curves even more closely match the sinc (phase-only) curve if they are normalized similarly.

Fig. 4
Fig. 4

Effective amplitude curves for example 3. The amplitude coupling is identical to the dotted curve in Fig. 1(b). The effective amplitudes for phase-only encoding with use of the pdf from Fig. 2(b) are included for comparison.

Fig. 5
Fig. 5

Map of the effective complex amplitudes that can be pseudorandom encoded (clear region) and those values that cannot be pseudorandom encoded (striped region) on the unit disk by the encoding method from example 3. The amplitude coupling used is the same as the dotted curve in Fig. 1(b), and it is replotted in this illustration. Note that for effective phases between 0 and π/4 the effective amplitude can be larger than the amplitude of the coupling function. For effective phases between -π/2 and π/2, the effective-amplitude curves (Fig. 4) have jump discontinuities. For effective phases between 0 and π/2, the values on each side of the discontinuity are plotted as dotted curves. For effective phases between -π/2 and 0, the discontinuities form the boundary of the portion of the unrealizable region that has radii less than the amplitude-coupling function.

Fig. 6
Fig. 6

Map of the effective complex amplitudes that can be pseudorandom encoded (clear region) and those values that cannot be pseudorandom encoded (striped region) on the unit disk by the encoding method from example 2. Additionally, the minimum value of effective amplitude has been calculated for phases between π and 2π by using Eq. (21) (dashed curve). This has increased greatly the number of complex values that can be encoded. The amplitude coupling used is the same as the solid curve in Fig. 1(b), and it is replotted in this illustration.

Fig. 7
Fig. 7

Map of the effective complex amplitudes that can be pseudorandom encoded (clear region) and those values that cannot be pseudorandom encoded (striped region) on the unit disk by use of all possible binomial distributions. The amplitude coupling (dashed curve) is the same as the dotted curve in Fig. 1 (b). For a given pair of samples on the amplitude-coupling curve, any effective complex amplitude can be realized on the line segment connecting the two points. This is shown for four pairs of samples. Any of the three thin lines could be used to produce the same effective complex amplitude at their common intersection. The thick line segment and the amplitude-coupling curve bound a convex set of all complex values that can be realized by pseudorandom encoding for the particular modulator characteristic.

Fig. 8
Fig. 8

Plots of the desired fully complex function ac used in the simulation study: (a) desired magnitudes ac as they would appear on the 128×128-pixel SLM, (b) desired values ac shown on the complex plane. Each value shown occurs numerous times.

Fig. 9
Fig. 9

Gray-scale plots of the intensity of the diffraction patterns resulting from (a) the desired fully complex function and (b)–(f) the various algorithms b–f. The algorithms are described in Section 8. Each image shows the central 65×512 samples of the simulated 512×512-sample diffraction pattern. In each image a fully white gray scale corresponds to an intensity that is 15% of the maximum spot intensity plus the minimum spot intensity divided by 2 [(max+min)/2].

Fig. 10
Fig. 10

Cross sections of the diffraction patterns resulting from (a) the desired function and (b)–(f) the various algorithms b–f. Each cross section is the central 1×512 samples of the simulated 512×512-sample diffraction pattern. Each trace is normalized so that the maximum spot intensity plus the minimum spot intensity divided by 2 [(max+min)/2] is of identical length on the vertical scale of each plot.

Tables (1)

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Table 1 Encoding Errors for Minimum-Error, Compensated, Maximum-Error, and Phase-only Pseudorandom Encoding for Various ac

Equations (32)

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a=ap(a)da
Ic(fx)=iAci2=Fiaci2,
I(fx)=Ic(fx)+i(|Ai|2-|Aci|2),
a=a(ψ)p(ψ)exp(jψ)dψa0 exp(jψ0),
γ=maxi(aci)
η=1Ni=1N|aci|2.
p(ψ; ψ, ν)=1νrectψ-ψν,
a=sinc(ν/2π)exp(jψ)a0 exp(jψ0).
a0=|sinc(ν/2π)|,
ψ0=ψ,sinc(ν/2π)>0ψ+π,sinc(ν/2π)<0,
a=sinc(ν/2π)exp(jψ),0ν2π.
ν=2π sinc-1(ac).
ψ=ψ+ν[ran(s)-1/2].
ψeff=ψpeff(ψ)dψ,
P(ψ)=-ψ p(ϕ)dϕ.
ψ=P-1(s).
a(ψ)=mψ+b,ψ[-2π, 2π],
peff(ψ)rect[(ψ-ψ0)/ν].
p(ψ)=1ψ+b/m×lnψ0+ν/2+b/mψ0-ν/2+b/m-1 rectψ-ψ0ν.
ψ=(ψ0+ν/2+b/m)s(ψ0-ν/2+b/m)s-1-b/m.
a=mνlnψ0+ν/2+b/mψ0-ν/2+b/m-1 sincν2πexp(jψ0).
peff(ψ)δ(ψ-ψ0+ν/2)+δ(ψ-ψ0-ν/2),
p(ψ)=a(ψ0+ν/2)δ(ψ-ψ0+ν/2)+a(ψ0-ν/2)δ(ψ-ψ0-ν/2)a(ψ0-ν/2)+a(ψ0+ν/2)
ac=a=2a(ψ0+ν/2)a(ψ0-ν/2)a(ψ0+ν/2)+a(ψ0-ν/2)cos(ν/2)exp(jψ0).
ψ=ψ0-ν/2ifsa(ψ0+ν/2)a(ψ0-ν/2)+a(ψ0+ν/2)ψ0+ν/2otherwise.
a=dau+(1-d)al,
a=[dau+(1-d)al]exp(jψ0),
p(ψ; d)=dpu(ψ; ψ0, νu)+(1-d)pl(ψ; ψ0, νl),
a=da1+(1-d)a2,
=a2(ψ)p(ψ)dψ-|ac|2.
=d(1-d)[a12+a22-2a1a2 cos(ψ1-ψ2)]subjecttoac=da1+(1-d)a2,
=|ac-a1||ac-a2|.

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