Abstract

It is shown that the negative values of the Wigner distribution function in classical optics are a consequence of the phase-space interference among the Gaussian beams into which an arbitrary light distribution (or a superposition of light distributions) can be decomposed. These elementary Gaussian beams partition the phase space in wave optics in adjacent, interacting, finite-area cells, in contrast to geometrical optics, where the phase space is continuous and a light beam can be decomposed into a number of perfectly localized, non-interacting rays.

© 2000 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. V. Buzek, P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics, XXXIV, E. Wolf, ed. (Elsevier, Amsterdam, 1995), pp. 1–158.
  3. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.
  4. K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
    [CrossRef]
  5. H. Konno, P. S. Lomdahl, “The Wigner transform of solitons solutions for the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 63, 3967–3973 (1994).
    [CrossRef]
  6. L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
    [CrossRef]
  7. D. Dragoman, “Wigner-distribution-function representation of the coupling coefficient,” Appl. Opt. 34, 6758–6763 (1995).
    [CrossRef] [PubMed]
  8. D. Dragoman, “Phase space representation of modes in optical waveguides,” J. Mod. Opt. 42, 1815–1823 (1995).
    [CrossRef]
  9. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93(III), 429–457 (1946).
  10. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).
  11. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  12. P. Flandrin, Temps-Fréquence (Hermès, Paris, 1993).
  13. R. L. Hudson, “When is the Wigner quasi-probability density non negative?” Rep. Math. Phys. 6, 249–252 (1974).
    [CrossRef]
  14. A. J. E. M. Janssen, “A note on Hudson’s theorem about functions with non-negative Wigner distributions,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 170–176 (1984).
    [CrossRef]
  15. M. J. Bastiaans, “Gabor’s signal expansion applied to partially coherent light,” Opt. Commun. 86, 14–18 (1991).
    [CrossRef]
  16. S. Qian, D. Chen, “Decomposition of the Wigner–Ville distribution and time-frequency distribution series,” IEEE Trans. Signal Process. 42, 2836–2842 (1994).
    [CrossRef]
  17. F. Pedersen, “The Gabor expansion based positive distribution,” Proc. IEEE 3, 1565–1568 (1998).
  18. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
    [CrossRef]
  19. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
    [CrossRef]

1998

F. Pedersen, “The Gabor expansion based positive distribution,” Proc. IEEE 3, 1565–1568 (1998).

1997

K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
[CrossRef]

1995

D. Dragoman, “Wigner-distribution-function representation of the coupling coefficient,” Appl. Opt. 34, 6758–6763 (1995).
[CrossRef] [PubMed]

D. Dragoman, “Phase space representation of modes in optical waveguides,” J. Mod. Opt. 42, 1815–1823 (1995).
[CrossRef]

1994

H. Konno, P. S. Lomdahl, “The Wigner transform of solitons solutions for the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 63, 3967–3973 (1994).
[CrossRef]

S. Qian, D. Chen, “Decomposition of the Wigner–Ville distribution and time-frequency distribution series,” IEEE Trans. Signal Process. 42, 2836–2842 (1994).
[CrossRef]

1991

M. J. Bastiaans, “Gabor’s signal expansion applied to partially coherent light,” Opt. Commun. 86, 14–18 (1991).
[CrossRef]

1989

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

1984

A. J. E. M. Janssen, “A note on Hudson’s theorem about functions with non-negative Wigner distributions,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 170–176 (1984).
[CrossRef]

1983

1981

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

1980

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

1974

R. L. Hudson, “When is the Wigner quasi-probability density non negative?” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

1946

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93(III), 429–457 (1946).

1932

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

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion applied to partially coherent light,” Opt. Commun. 86, 14–18 (1991).
[CrossRef]

M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

Buzek, V.

V. Buzek, P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics, XXXIV, E. Wolf, ed. (Elsevier, Amsterdam, 1995), pp. 1–158.

Chen, D.

S. Qian, D. Chen, “Decomposition of the Wigner–Ville distribution and time-frequency distribution series,” IEEE Trans. Signal Process. 42, 2836–2842 (1994).
[CrossRef]

Cohen, L.

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Dragoman, D.

D. Dragoman, “Wigner-distribution-function representation of the coupling coefficient,” Appl. Opt. 34, 6758–6763 (1995).
[CrossRef] [PubMed]

D. Dragoman, “Phase space representation of modes in optical waveguides,” J. Mod. Opt. 42, 1815–1823 (1995).
[CrossRef]

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.

Flandrin, P.

P. Flandrin, Temps-Fréquence (Hermès, Paris, 1993).

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93(III), 429–457 (1946).

Hudson, R. L.

R. L. Hudson, “When is the Wigner quasi-probability density non negative?” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen, “A note on Hudson’s theorem about functions with non-negative Wigner distributions,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 170–176 (1984).
[CrossRef]

Knight, P. L.

V. Buzek, P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics, XXXIV, E. Wolf, ed. (Elsevier, Amsterdam, 1995), pp. 1–158.

Konno, H.

H. Konno, P. S. Lomdahl, “The Wigner transform of solitons solutions for the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 63, 3967–3973 (1994).
[CrossRef]

Lomdahl, P. S.

H. Konno, P. S. Lomdahl, “The Wigner transform of solitons solutions for the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 63, 3967–3973 (1994).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Pedersen, F.

F. Pedersen, “The Gabor expansion based positive distribution,” Proc. IEEE 3, 1565–1568 (1998).

Qian, S.

S. Qian, D. Chen, “Decomposition of the Wigner–Ville distribution and time-frequency distribution series,” IEEE Trans. Signal Process. 42, 2836–2842 (1994).
[CrossRef]

Rivera, A. L.

K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Wigner, E.

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

Wolf, K. B.

K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
[CrossRef]

Appl. Opt.

IEEE Trans. Signal Process.

S. Qian, D. Chen, “Decomposition of the Wigner–Ville distribution and time-frequency distribution series,” IEEE Trans. Signal Process. 42, 2836–2842 (1994).
[CrossRef]

J. Inst. Electr. Eng.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93(III), 429–457 (1946).

J. Mod. Opt.

D. Dragoman, “Phase space representation of modes in optical waveguides,” J. Mod. Opt. 42, 1815–1823 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Soc. Jpn.

H. Konno, P. S. Lomdahl, “The Wigner transform of solitons solutions for the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 63, 3967–3973 (1994).
[CrossRef]

Opt. Acta

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

Opt. Commun.

K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
[CrossRef]

M. J. Bastiaans, “Gabor’s signal expansion applied to partially coherent light,” Opt. Commun. 86, 14–18 (1991).
[CrossRef]

Optik (Stuttgart)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

Phys. Rev.

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

Proc. IEEE

L. Cohen, “Time-frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

F. Pedersen, “The Gabor expansion based positive distribution,” Proc. IEEE 3, 1565–1568 (1998).

Rep. Math. Phys.

R. L. Hudson, “When is the Wigner quasi-probability density non negative?” Rep. Math. Phys. 6, 249–252 (1974).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal.

A. J. E. M. Janssen, “A note on Hudson’s theorem about functions with non-negative Wigner distributions,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 15, 170–176 (1984).
[CrossRef]

Other

V. Buzek, P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” in Progress in Optics, XXXIV, E. Wolf, ed. (Elsevier, Amsterdam, 1995), pp. 1–158.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” in Progress in Optics, XXXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1997), pp. 1–56.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

P. Flandrin, Temps-Fréquence (Hermès, Paris, 1993).

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Figures (5)

Fig. 1
Fig. 1

(a) Field distribution of the first-order Hermite–Gaussian beam and (b) its WDF.

Fig. 2
Fig. 2

(a) Sum of the WDF’s of the Gaussian beams into which the field shown in Fig. 1(a) can be decomposed, and (b) the WDF of the interference term between these Gaussian beams.

Fig. 3
Fig. 3

(a) Total WDF of the sum of the Gaussian beams that approximate the field shown in Fig. 1(a) [the sum of the WDF shown in Figs. 2(a) and 2(b)], and (b) the difference between the sum of Gaussian beams and the original field.

Fig. 4
Fig. 4

(a) Sum of the Gaussian beams that approximate the field shown in Fig. 1(a), obtained according to Gabor decomposition procedure, and (b) its WDF, for x0=2.5xG. See text for details.

Fig. 5
Fig. 5

Same as Fig. 4, but for x0=5xG.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

φ(x)=φ0(x/x0) exp(-πx2/x02),
W(x, p)=φx+x2φ*x-x2exp(ixp)dx,
W(x, p)=x0π2φ022πx2x02+p2x022π-12×exp-2πx2x02-p2x022π.
Wab(x, p)=2πRe a exp(Re b)2Re a-Re c-(x Im a+Im b-p)2Re a-xRe a+Re bRe a2.
φapp(x)=φ1(x)+φ2(x)=φ0 exp(πd2/x02)×{exp[-π(x-d)2/x02]-exp[-π(x+d)2/x02]}φ04πxdx02exp(-πx2/x02)
Wapp(x, p)=W1(x, p)+W2(x, p)+Wint(x, p),
Wi(x, p)=φix+x2φi*x-x2exp(ixp)dx,
i=1, 2,
Wint(x, p)=φ1x+x2φ2*x-x2exp(ixp)dx+φ2x+x2φ1*x-x2×exp(ixp)dx.
φ(x)φapp(x)=n,mamng(x-mxG)exp(inpGx),
amn=φ(x)γ*(x-mxG)exp(-inpGx)dx,
γ(x)=K0-3/2/(21/4π-3/2xG1/2)exp(πx2/xG2)×n+1/2x/xG(-1)n exp[-π(n+1/2)2],
Wmn(x, p)=2 exp-2π(x-mxG)2xG2-xG2(npG+p)22π.

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