Abstract

Pseudorandom encoding is a statistical method for designing Fourier transform holograms by mapping ideal complex-valued modulations onto spatial light modulators that are not fully complex. These algorithms are notable because their computational overhead is low and because the space–bandwidth product of the encoded signal is identical to the number of modulator pixels. All previous pseudorandom-encoding algorithms were developed for analog modulators. A less restrictive algorithm for quantized modulators is derived that permits fully complex ranges to be encoded with as few as three noncollinear modulation values that are separated by more than 180° on the complex plane.

© 1999 Optical Society of America

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Corrections

Robert W. Cohn and Markus Duelli, "Ternary pseudorandom encoding of Fourier transform holograms: errata," J. Opt. Soc. Am. A 16, 1089-1090 (1999)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-16-5-1089

References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  16. A. Papoulis, Probability, Random Variables and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), pp. 43–56.
  17. R. W. Cohn, M. Liang, “Spot array generator designed by the method of pseudorandom encoding,” presented at the Annual Meeting of the Optical Society of America, Rochester, New York, October 20–24, 1996.
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1998 (2)

M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Eng. Lab. Notes 9, 3–5 (1998).

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

1997 (2)

1996 (1)

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1973 (1)

1970 (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)

Brown, B. R.

Burckhardt, C. B.

Cohn, R. W.

R. W. Cohn, “Pseudorandom encoding of complex-valued 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,” Eng. Lab. Notes 9, 3–5 (1998).

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, 2488–2498 (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, M. Liang, “Spot array generator designed by the method of pseudorandom encoding,” presented at the Annual Meeting of the Optical Society of America, Rochester, New York, October 20–24, 1996.

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.

R. W. Cohn, “Analyzing the encoding range of amplitude–phase coupled spatial light modulators,” in Spatial Light Modulators, R. L. Sutherland, ed., Proc. SPIE3297, 122–128 (1998).
[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.

Duelli, M.

M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Eng. Lab. Notes 9, 3–5 (1998).

Gallagher, N. C.

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.

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]

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.

Kettunen, V.

Krackhardt, U.

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.

Liu, W.

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, 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.

Mait, J. N.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), pp. 53–55 and 211–212.

A. Papoulis, Probability, Random Variables and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), pp. 43–56.

Streibl, N.

Turunen, J.

Vahimaa, P.

Vasiliev, A. A.

Wyrowski, F.

Appl. Opt. (7)

Eng. Lab. Notes (1)

M. Duelli, D. L. Hill, R. W. Cohn, “Frequency swept measurements of coherent diffraction patterns,” Eng. Lab. Notes 9, 3–5 (1998).

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 (3)

Opt. Lett. (1)

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.

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), pp. 53–55 and 211–212.

R. W. Cohn, “Analyzing the encoding range of amplitude–phase coupled spatial light modulators,” in Spatial Light Modulators, R. L. Sutherland, ed., Proc. SPIE3297, 122–128 (1998).
[CrossRef]

A. Papoulis, Probability, Random Variables and Stochastic Process, 3rd ed. (McGraw-Hill, New York, 1991), pp. 43–56.

R. W. Cohn, M. Liang, “Spot array generator designed by the method of pseudorandom encoding,” presented at the Annual Meeting of the Optical Society of America, Rochester, New York, October 20–24, 1996.

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Figures (8)

Fig. 1
Fig. 1

Systems definition of complex-valued encoding. (a) Systems viewpoint of the Fourier transform computer-generated hologram. A desired complex-valued signal is encoded into a realizable SLM modulation. This signal is decoded through diffraction into a spectrum that approximates the desired complex-valued spectrum. (b) Systems viewpoint specialized for pseudorandom encoding. The desired complex-valued signal ac(x) is pseudorandom encoded (PRE) to produce the realizable modulation a(x). The observed intensity diffraction pattern I(f), which is the squared magnitude of the Fourier transform of a(x), approximates the desired intensity diffraction Ic(f) in a statistical sense. Specifically, the expected value of I(f), i.e., I(f), is the desired diffraction pattern Ic(f) on a background of white noise.

Fig. 2
Fig. 2

Distinctions between (a) Burckhardt’s ternary encoding method and (b) pseudorandom ternary encoding. In Burckhardt’s method the magnitudes of the available complex amplitudes a1, a2, and a3 (their loci indicated by the three connected arrows) can be continuously varied between 0 and 1. In pseudorandom encoding, the available complex amplitudes are constant, but the probabilities p, q, and r of selecting a1, a2, and a3 can be varied continuously between 0 and 1. The constraint that p+q+r=1 leads to pseudorandom encoding having a different encodable/realizable range from that of Burckhardt’s method. The fully complex range is the maximum-diameter circular region that surrounds the origin of the complex plane and that does not exceed the extremal encoding range. The fully complex range is drawn for the specific case that the origin is the center of the circular region.

Fig. 3
Fig. 3

Geometry of pseudorandom biamplitude encoding. Any desired value ac between the two available modulation values a1 and a2 can be encoded by pseudorandom encoding. The product of the lengths of the line segments that connect a1 and a2 to ac is the encoding error ϵ=l1l2.

Fig. 4
Fig. 4

Geometric relationships for ternary pseudorandom encoding. (a) The three vectors (thick lines) correspond to the three terms in Eq. (14). Each term corresponds to a vector that has a3 as its origin. (b) Geometry for ternary encoding when the probability r=0. For this condition ternary encoding reduces to biamplitude encoding between a1 and a2. This construction also identifies two triangles that are identical except for scaling by p and q. The thick lines indicate the two vectors that add together to produce the desired complex value ac. (c) Geometry for ternary encoding when the probability r is 0<r<1. This construction shows that there are three triangles that are identical except for scaling by p,  q, and r. The thick lines indicate the six line segment lengths that are used in Eq. (22) to calculate the encoding error. The products of the lengths of the three pairs of collinear segments are added together to give the encoding error.

Fig. 5
Fig. 5

Relative encoding errors for various pseudorandom-encoding algorithms. The desired magnitude ac is normalized by γ, the maximum radius for the fully complex encoding range, and the relative encoding error is ϵrel=ϵ/γ2. The striped region gives all possible encoding errors for the m-ary 2 algorithm. The phase-only curve also represents the relative encoding error for pseudorandom encoding with M phase-only values that are uniformly spaced in angle. In this case the plotted phase-only curve is offset by tan2(π/M) [see Eq. (24)].

Fig. 6
Fig. 6

[(a)–(f)] Simulated and [(g)–(i)] experimental gray-scale images of the diffraction pattern intensity resulting from various encoding algorithms. All encodings are pseudorandom except (a), which is nonrandom phase-only. The simulated pseudorandom encodings are (b) biamplitude phase, (c) m-ary 2, (d) phase-only, (e) m-ary 1, and (f) ternary. The experimental pseudorandom encodings are (g) phase-only, (h) m-ary 1, and (i) ternary.

Fig. 7
Fig. 7

Simulated cross sections of the diffraction pattern intensity resulting from various encoding algorithms: (a) pseudorandom encodings, (b) nonrandom encodings. Each cross section is along a diagonal that contains the (7,7) order (leftmost spot), the (1,1) order (rightmost spot), and the optical axis (the rightmost side of the curve). For all the pseudorandom curves in (a) and the nonrandom phase-only curve in (b), the cross sections are the intensity values from the diagonal (from upper left to lower right) of the corresponding gray-scale image in Fig. 6. The four other nonrandom encodings in (b) are plotted over an identical range.

Fig. 8
Fig. 8

Delineation of nonlinear effects on encoding: (a) simulated and (b) experimental diffraction pattern intensity for nonrandom ternary encoding, (c) experimental diffraction pattern for pseudorandom ternary encoding. These patterns show a larger view of the diffraction pattern than those in Figs. 6 and 7. Each intensity cross section is along the diagonal of the corresponding gray-scale image. In (a) and (b) the nonrandom ternary encoding produces mixing products, as evident in the lower left corner of each gray-scale image. Although speckle noise is evident in this same region for pseudorandom ternary encoding [(c)], it is much lower in intensity than the mixing products for (b). The saturated spot (centered on the optical axis) in (b) and (c) is primarily a result of the SLM cover glass not being antireflection coated. The most severe effect of the SLM’s limited resolution is the appearance, to the lower left of the optical axis, of a duplicate 7×7 spot array in (b) and (c).

Tables (3)

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Table 1 Defining Parameters and Metrics for Various Pseudorandom-Encoding Algorithms

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Table 2 Performance Measures of the Pseudorandomly Encoded Desired Function

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Table 3 Performance Measures of the Nonrandomly Encoded Desired Function

Equations (31)

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a=ap(a)da
Ic(f)=i Aci2=Fi aci2,
I(f)=Ic(f)+i (|Ai|2-|Aci|2),
ϵ=|a|2-|ac|2,
p(a)=pδ(a-a1)+qδ(a-a2),
a=pa1+(1-p)a2.
ac=l2la1+l1la2,
ϵ=l1l2=pql2,
a=a1if0spa2ifp<s1,
p(a)=pδ(a-a1)+qδ(a-a2)+rδ(a-a3),
p+q+r=1.
aca=pa1+qa2+ra3.
acraci1=a1ra2ra3ra1ia2ia3i111pqr,
ac-a3=p(a1-a3)+q(a2-a3).
0lc3=l13 sin θ13-l23 sin θ23l13 cos θ13-l23 cos θ23pq,
p=lc3l13sin θ23sin(θ13+θ23),q=lc3l23sin θ13sin(θ13+θ23).
a=a1if0spa2ifp<sp+qa3ifp+q<s1,
ϵ=p|a1|2+q|a2|2+r|a3|2-|pa1+qa2+ra3|2,
ϵ=(p-p2)a12+(q-q2)a22+(r-r2)a32-2pqa1a2 cos ϕ12-2pra1a3 cos ϕ13-2qra2a3 cos ϕ23,
li, j|ai-aj|2=ai2+aj2-2ai aj cos ϕij
p-p2=pq+pr,
q-q2=pq+qr,
r-r2=pr+qr,
ϵ=pql122+prl132+qrl232.
γM=cos(π/M).
ϵrelϵM/γM2=1+tan2(π/M)-(ac/γM)2.
SNR=NBaca2ϵa.
η=1Ni=1Naci2aca2.
SNR=NBη1-η.
ac(x, y)=k=17 exp(jθk)exp(j2πkx)×l=17 exp(jθl)exp(j2πly)
a(x, y)=exp{j arg[ac(x, y)]}.

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