Abstract

The Wigner distribution function is used to analyze moiré patterns that originate from a superposition of nonperiodic masks. For patterns with well-defined local frequencies, the concept of the Wigner distribution function allows one to extend the description of the moiré effect in terms of vector sums. How this picture can be applied to design moiré patterns and to analyze their information content is also discussed.

© 2000 Optical Society of America

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References

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  1. A. W. Lohmann, S. Sinzinger, “Moiré effect as a tool for image processing,” J. Opt. Soc. Am. A 10, 65–68 (1993).
    [CrossRef]
  2. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).
  3. G. Oster, M. Waserman, C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
    [CrossRef]
  4. O. Bryngdahl, “Moiré: formation and interpretation,” J. Opt. Soc. Am. 64, 1287–1294 (1974).
    [CrossRef]
  5. A. W. Lohmann, Optical Information Processing, 3rd ed. (A. W. Lohmann, Uttenreuth, Germany, 1986), Chaps. 3 and 5.
  6. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]
  7. T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution function—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).
  8. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  9. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  10. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  11. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  12. O. Bryngdahl, “Moiré and higher grating harmonics,” J. Opt. Soc. Am. 65, 685–694 (1975).
    [CrossRef]
  13. J. M. Burch, D. C. Williams, “Varifocal moiré plates for straightness measurements,” Appl. Opt. 16, 2445–2450 (1977).
    [CrossRef] [PubMed]
  14. P. P. Huang, “Holographic anti-counterfeit method and device with encoded pattern,” in Diffractive/Holographic Technologies, Systems, and Spatial Light Modulators, VI, I. Cindrich, S. H. Lee, R. L. Sutherland, eds., Proc. SPIE3633, 61–67 (1999).
    [CrossRef]
  15. A. Kolodziejczyk, Z. Jaroszewicz, “Diffractive elements of variable optical power and high diffraction efficiency,” Appl. Opt. 32, 4317–4322 (1993).
    [CrossRef] [PubMed]

1996 (1)

1993 (2)

1982 (1)

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980 (1)

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution function—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

1979 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1977 (1)

1975 (1)

1974 (1)

1967 (1)

1964 (1)

Bastiaans, M. J.

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

Bryngdahl, O.

Burch, J. M.

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution function—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Dorsch, R. G.

Ferreira, C.

Huang, P. P.

P. P. Huang, “Holographic anti-counterfeit method and device with encoded pattern,” in Diffractive/Holographic Technologies, Systems, and Spatial Light Modulators, VI, I. Cindrich, S. H. Lee, R. L. Sutherland, eds., Proc. SPIE3633, 61–67 (1999).
[CrossRef]

Jaroszewicz, Z.

Kolodziejczyk, A.

Lohmann, A. W.

Mecklenbräuker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution function—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Mendlovic, D.

Oster, G.

Paris, D. P.

Patorski, K.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

Sinzinger, S.

Waserman, M.

Williams, D. C.

Zalevsky, Z.

Zwerling, C.

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Philips J. Res. (1)

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution function—a tool for time-frequency signal analysis. Part I: Continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Other (3)

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993).

A. W. Lohmann, Optical Information Processing, 3rd ed. (A. W. Lohmann, Uttenreuth, Germany, 1986), Chaps. 3 and 5.

P. P. Huang, “Holographic anti-counterfeit method and device with encoded pattern,” in Diffractive/Holographic Technologies, Systems, and Spatial Light Modulators, VI, I. Cindrich, S. H. Lee, R. L. Sutherland, eds., Proc. SPIE3633, 61–67 (1999).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Formation of Schuster fringes displayed in the Wigner domain: (a) The WDF’s of both masks (dotted lines and dashed lines, respectively) drawn in a single Wigner chart. (b) Additional terms generated by a multiplicative superposition: The difference terms (solid lines) result in a constant frequency that corresponds to the Schuster fringes. The sum terms (dotted–dashed lines) represent a Fresnel zone pattern.

Fig. 2
Fig. 2

Self-moiré of phase function Φ=(x3/a)+(y2/b): (a) Fringe pattern of the mask function, (b) moiré obtained from simultaneous shifting in x and y; the Schuster fringes generated by the lateral shift in y are superposed with the Fresnel zone pattern of the x displacement, forming a set of parabolic fringes.

Fig. 3
Fig. 3

Self-moiré obtained by mutual rotation of two masks: (a) mask, (b) moiré pattern of the phase function Φ(r, ϕ)=rϕ/a.

Fig. 4
Fig. 4

Multiplexing two patterns as phase modulations of different carrier frequencies: (a) WDF chart illustrating the SBP occupied by two encoded signals, (b) transmission function encoding the two patterns, (c) reconstruction of the one pattern by superposition with the respective carrier frequency, (d) reconstruction of the second pattern by superposition with the carrier shifted by half a period.

Fig. 5
Fig. 5

Multiplexing two patterns as phase modulations of random carriers: (a) WDF chart of two signals modulating random carriers, (b) transmission function encoding two patterns, (c) reconstruction of the one pattern by use of the random carrier for demodulation, (d) reconstruction based on a superposition with a cosinusoidal pattern.

Fig. 6
Fig. 6

Multiplexing two patterns as phase modulations of chirped carriers: (a) WDF chart of two signals modulating chirped randomized carrier without overlap, (b) reconstruction of one of the two multiplexed patterns, (c) WDF chart of two signals modulating chirped carriers with overlap, and (d) reconstruction containing cross talk.

Equations (20)

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TM(x)=T1(x)T2(x)
W(x, ν)=-u(x+x/2)u*(x-x/2)×exp(-i2πνx)dx,
W(x, ν)=W1(x, ν)**W2(x, ν)=-W1(x, ν)W2(x, ν-ν)dν,
W(x, ν)=δν-12πΦx.
Tn(x)=12+12cos[Φn(x)],
Tn(x)=12+14exp[iΦn(x)]+14exp[-iΦn(x)],
Wn(x, μ,)=14δ(ν)+14δν-12πΦnx+14δν+12πΦnx+Wcross(x, ν),
Wm(x, ν)=W1(x, ν)**W2(x, ν)=14W1(x, ν)+14W2(x, ν)+Wsum(x, ν)+Wdiff(x, ν)+Wcross(x, ν),
Wsum(x, ν)=1162δν-12πΦ1x-12πΦ2x+1162δν+12πΦ1x+12πΦ2x,
Wdiff(x, ν)=1162δν-12πΦ1x+12πΦ2x+1162δν+12πΦ1x-12πΦ2x.
Φsum(x)x=±Φ2(x)x+Φ1(x)x,
Φdiff(x)x=±Φ2(x)x-Φ1(x)x.
Φdiff(x, y)=±|Φ2(x, y)-Φ1(x, y)|.
Φfz1(x, y)=xa3+yb,
Φfz2(x, y)=x2ya3,
Φfz1/2(x-x0, y-y0)x=P1(x, y)+P2(x0, y0)+αxf(x0, y0),
Φ(r, ϕ)=(2rϕ/a, r/a).
Tc(x)=12+12Nn=1N cos{2πνn[x+mnPn(x)]},
Tc(x)=12+12Nn=1N cos[2πνnx+Rn(x)+2πνnmnPn(x)],
Tc(x)=12+12Nn=1N cos{2πνn[αx2+x+mnPn(x)]},

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