Abstract

A unified mathematical formulation for designing and analyzing even the most general optical processor is presented. It exploits the Wigner distribution function to characterize the illumination, the input, the inherent filter, and the output results. To characterize the propagation of the light through the optical processor setup, we exploit the Wigner matrix formalism, which is appealing because it allows simple geometric analysis. The Wigner distribution function was extended to include illumination of arbitrary coherence so that processors using either coherent light or partially coherent light can be designed and analyzed with the same Wigner formalism. The basic principles, design, and analysis of the imaging and Fourier-transform operations and use of the Wigner formalism to evaluate the performance and tolerances of optical processors are presented.

© 2001 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. T. Alieva, M. J. Bastiaans, “Self-affinity in phase space,” J. Opt. Soc. Am. A 17, 756–761 (2000).
    [CrossRef]
  3. D. Peris, V. C. Georgopoulos, “Wigner distribution representation and analysis of audio signals: an illustrated tutorial review,” J. Audio Eng. Soc. 47, 1043–1053 (1999).
  4. K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998).
    [CrossRef]
  5. T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  7. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  8. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
    [CrossRef]
  9. G. A. Deschamps, “Ray techniques in electromagnetism,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  10. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  11. A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).
  12. R. H. Katyl, “Compensating optical systems. III. Achromatic Fourier transformation,” Appl. Opt. 11, 1255–1260 (1972).
    [CrossRef] [PubMed]
  13. S. Leon, E. N. Leith, “Optical processing and holography with polychromatic point source illumination,” Appl. Opt. 24, 3638–3642 (1985).
    [CrossRef] [PubMed]
  14. P. Andres, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992).
    [CrossRef] [PubMed]
  15. E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
    [CrossRef]
  16. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
    [CrossRef] [PubMed]
  17. P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, A. W. Lohmann, “All-incoherent dispersion-compensated optical correlator,” Opt. Lett. 24, 1331–1333 (1999).
    [CrossRef]
  18. A. Pe’er, D. Wang, A. W. Lohmann, A. A. Friesem, “Optical correlation with totally incoherent light,” Opt. Lett. 24, 1469–1471 (1999).
    [CrossRef]
  19. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  20. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  21. Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
    [CrossRef] [PubMed]

2000

1999

1998

K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

1996

1995

1994

1992

1986

1985

1984

1981

1980

T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

1979

1972

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Alieva, T.

Andres, P.

E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

P. Andres, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992).
[CrossRef] [PubMed]

Andrés, P.

Banaszek, K.

K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998).
[CrossRef]

Bastiaans, M. J.

Bitran, Y.

Classen, T. A. C. M.

T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Climent, V.

P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, A. W. Lohmann, “All-incoherent dispersion-compensated optical correlator,” Opt. Lett. 24, 1331–1333 (1999).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetism,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Dorsch, R. G.

Friesem, A. A.

A. Pe’er, D. Wang, A. W. Lohmann, A. A. Friesem, “Optical correlation with totally incoherent light,” Opt. Lett. 24, 1469–1471 (1999).
[CrossRef]

A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).

Furlan, W. D.

Georgopoulos, V. C.

D. Peris, V. C. Georgopoulos, “Wigner distribution representation and analysis of audio signals: an illustrated tutorial review,” J. Audio Eng. Soc. 47, 1043–1053 (1999).

Katyl, R. H.

Lancis, J.

Leith, E. N.

Leon, S.

Lohmann, A. W.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

Mecklenbrauker, W. F. G.

T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Mendlovic, D.

Mínguez-Vega, G.

Morris, G. M.

Ozaktas, H. M.

Pe’er, A.

A. Pe’er, D. Wang, A. W. Lohmann, A. A. Friesem, “Optical correlation with totally incoherent light,” Opt. Lett. 24, 1469–1471 (1999).
[CrossRef]

A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).

Peris, D.

D. Peris, V. C. Georgopoulos, “Wigner distribution representation and analysis of audio signals: an illustrated tutorial review,” J. Audio Eng. Soc. 47, 1043–1053 (1999).

Tajahuerce, E.

P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, A. W. Lohmann, “All-incoherent dispersion-compensated optical correlator,” Opt. Lett. 24, 1331–1333 (1999).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Wang, D.

A. Pe’er, D. Wang, A. W. Lohmann, A. A. Friesem, “Optical correlation with totally incoherent light,” Opt. Lett. 24, 1469–1471 (1999).
[CrossRef]

A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wodkiewicz, K.

K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998).
[CrossRef]

Zalevsky, Z.

Appl. Opt.

J. Audio Eng. Soc.

D. Peris, V. C. Georgopoulos, “Wigner distribution representation and analysis of audio signals: an illustrated tutorial review,” J. Audio Eng. Soc. 47, 1043–1053 (1999).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998).
[CrossRef]

Opt. Lett.

Philips J. Res.

T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A

K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998).
[CrossRef]

Proc. IEEE

G. A. Deschamps, “Ray techniques in electromagnetism,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Other

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).

A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).

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Figures (4)

Fig. 1
Fig. 1

Effect of lens and free space on WDF: (a) input WDF, (b) WDF x sheared after free-space propagation, and (c) WDF ν sheared after lens action.

Fig. 2
Fig. 2

Effect of imaging and Fourier transformation on the WDF. (a) Input WDF, (b) perfect imaging: a π-rad rotation, (c) imperfect imaging: a π-rad rotation and frequency shear, (d) perfect Fourier: a π/2-rad rotation, (e) imperfect Fourier: a π/2-rad rotation and frequency shear, and (f) perfect fractional Fourier transform of order p: a rotation by an angle φ = pπ/2.

Fig. 3
Fig. 3

Effect of degree of coherence on WDF and Fourier plane intensity distribution. (a) The double rectangular input function U(x), (b) the coherent WDF and (c) the corresponding Fourier plane intensity P(ν), (d) the WDF with a high degree of coherence and (e) the corresponding Fourier plane intensity, and (f) WDF with a low degree of coherence and (g) the corresponding Fourier plane intensity.

Fig. 4
Fig. 4

Geometric analysis of tolerances: (a) schematic WDF at the input plane and (b) schematic WDF at the filter plane.

Equations (36)

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Wx, ν-dxUx-x2U*x+x2exp2πiνx,
Ix=|Ux|2=-dνWx, ν,Pν=|Ũν|2=-dxWx, ν,
Sz=1λz01,  Lf=10-1/λf1,
xoutνout=Axinνin=abcdxinνin,
xoutνout=a0c1/axinνin.
δxout=aλ-a0¯xin+b¯Δνin,
xoutνout=0b-1/bdxinνin.
δxout=āΔxin+bλ-b0¯νin,
xoutνout=a0 cos φb0 sin φ-1/b0sin φ1/a0cos φxinνin,
Wpcx, νWcx, ν=-dxΓx-x2, x+x2exp2πiνx,
W+x, ν=-dxU+x-x2U+*x+x2×exp2πiνx=-dxU-x-x2U-*x+x2×fx-x2f*x+x2exp2πiνx.
W+x, ν=-dxU-x-x2U-*x+x2×fx-x2f*x+x2exp2πiνx.
W+x, ν=-dνW-x, νWfx, ν-ν=W-x, νν*Wfx, ν,
Γx-x/2, x+x/2=Ixδx.
W+x, ν=I-x|fx|2=I+x,
T=m0c1/m=10c/m1m001/m=LM,
T=BA,
B=LMA-1.
W+xν=-dνW-xνWFxν-ν,
WoutBxν=-dνWinA-1xνWFxν-ν.
Woutdx-bν-cx+aν=-dνWindx-bν-cx+aνWFxν-ν.
xoutνout=A-1xν=dx-bν-cx+aν,xin=dx-bν-xout,
Woutxoutνout=-1b-dxinWinxin-xoutab xin+νoutWFaxout+bνoutxinb.
Woutxoutνout=1b-dxinWinxin-xoutνoutWFbνoutxinb.
Ioutxout=-dνoutWoutxoutνout=-1b-dxinIinxout-xinĨFxinb,
δxF=|a|Δxin,
δxF=|b|δνin.
|a|Δxin<|b|δνin.
|a|<|b0|δνinΔxin.
δνF=δxin|b|.
EνF=Δxinb0-Δxinb<δxin|b|.
=b-b0b0.
||<δxinΔxin.
|aλ|<|b0| δνinΔxin,  |λ|<δxinΔxin.
a2λ<|b0| δνinΔxin,  2λ<δxinΔxin,
a2λ= dλa2λSλ dλSλ.

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