Abstract

We investigate a general form of the Wigner function for wave fields that satisfy the Helmholtz equation in two-dimensional free space. The momentum moment of this Wigner function is shown to correspond to the flux of the wave field. For a forward-propagating wave field, the negative regions of the Wigner function are seen to be associated with small regions of backward flux in the field. We also study different projections of the Wigner function, each corresponding to a distribution in a reduced phase space that fully characterizes the wave field. One of these projections is the standard Wigner function of the field at a screen. Another projection introduced by us has the added property of being conserved along rays and is better suited to the description of nonparaxial wave fields.

© 1999 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);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  3. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  4. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  5. Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.
  6. The existence of an invertible Fourier transform [Eq. (3)] of a (twice-differentiable) Helmholtz wave field Ψ(q) is ensured by the conditions of the Fourier integral theorem only when it is absolutely integrable and of bounded variation. Plane waves are in this sense limit functions for which Fourier analysis can be extended as usual with Dirac δ’s in the p plane and on the unit circle [Eq. (4)]. However, evanescent solutions (i.e., of exponential growth in the plane) do not possess a Fourier transform.
  7. P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995).
    [CrossRef]
  8. For other references about flux vortices see, for example, M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A Nucl. Math. Gen. 27, L391–L398 (1994).
    [CrossRef]
  9. A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).
  10. K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997).
    [CrossRef]
  11. M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978);“Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979);“The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
    [CrossRef]
  12. S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
    [CrossRef]
  13. L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
    [CrossRef]

1998

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

1997

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

1995

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995).
[CrossRef]

1994

For other references about flux vortices see, for example, M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A Nucl. Math. Gen. 27, L391–L398 (1994).
[CrossRef]

1993

1981

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

1980

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

1978

M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978);“Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979);“The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

1973

1968

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Atakishiyev, N. M.

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

Bastiaans, M.

M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978);“Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979);“The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

Berry, M. V.

For other references about flux vortices see, for example, M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A Nucl. Math. Gen. 27, L391–L398 (1994).
[CrossRef]

Chumakov, S. M.

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

González-Casanova, P.

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995).
[CrossRef]

Kravtsov, Yu. A.

Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.

Littlejohn, R. G.

Lohmann, A.

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

Nieto, L. M.

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

Orlov, Yu. A.

Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.

Rivera, A. L.

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

Steinberg, S.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Winston, R.

Wolf, K. B.

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

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

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995).
[CrossRef]

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

J. Math. Phys.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A Nucl. Math. Gen.

For other references about flux vortices see, for example, M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A Nucl. Math. Gen. 27, L391–L398 (1994).
[CrossRef]

L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998).
[CrossRef]

Num. Meth. Partial Diff. Eqs.

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995).
[CrossRef]

Opt. Commun.

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

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

M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978);“Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979);“The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Other

Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.

The existence of an invertible Fourier transform [Eq. (3)] of a (twice-differentiable) Helmholtz wave field Ψ(q) is ensured by the conditions of the Fourier integral theorem only when it is absolutely integrable and of bounded variation. Plane waves are in this sense limit functions for which Fourier analysis can be extended as usual with Dirac δ’s in the p plane and on the unit circle [Eq. (4)]. However, evanescent solutions (i.e., of exponential growth in the plane) do not possess a Fourier transform.

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Figures (12)

Fig. 1
Fig. 1

Support of the integrand of Eq. (8) consists of two points, where the momentum arguments of the two functions, the vectors p-(1/2)s and p+(1/2)s, meet the unit circle.

Fig. 2
Fig. 2

(a) Intensity and flux-vector field of a forward rectangle beam with spectral function ψ(θ)=Rectπ/2(θ). Note that beyond the field waist there are flux vortices, and saddle points along the x axis. The flux-vector field for the region inside the dotted box is shown in detail in (b).  

Fig. 3
Fig. 3

Gray area represents support in momentum for the Wigner function of a forward-propagating field.

Fig. 4
Fig. 4

p1-p2W(Ψ|p, q) for a forward-rectangle beam, evaluated at (a) kq=(0, 0) and (b) kq=(2.7, 0). The shade of gray in the background of both figures corresponds to zero, and lighter (darker) shades of gray correspond to higher (lower) values.

Fig. 5
Fig. 5

For a rectangle beam of width w=π/16, as defined by Eq. (20): (a) spectral function on the circle, (b) intensity and wave fronts of the wave field on the q plane, (c) screen marginal at qz=0, and (d) screen marginal at kqz=50, where qx=qx+pxqz/1-px2.

Fig. 6
Fig. 6

For a rectangle beam of width w=π/4, as defined by Eq. (20): (a) spectral function on the circle, (b) intensity and wave fronts of the wave field on the q plane, (c) screen marginal at qz=0, and (d) screen marginal at kqz=50, where qx=qx+pxqz/1-px2.

Fig. 7
Fig. 7

(a) Spectral function on the circle, (b) intensity, (c) angle-position marginal at qz=0, and (d) angle-impact marginal, for the Bessel monopole field defined by Eq. (40).

Fig. 8
Fig. 8

Angle-impact marginal for rectangle beams of width w=π/16 and w=π/4, as defined by Eq. (20).

Fig. 9
Fig. 9

(a) Spectral function on the circle, (b) intensity, and (c) angle-impact marginal, for the Bessel dipole field defined by Eq. (41).

Fig. 10
Fig. 10

(a) Spectral function on the circle, (b) intensity and wave fronts of the wave field on the q plane, and (c) angle-impact marginal, for the Gaussian wave field defined by Eq. (42) of waist w2=0.1.

Fig. 11
Fig. 11

(a) Spectral function on the circle, (b) intensity and wave fronts of the wave field on the q plane, and (c) angle-impact marginal, for the superposition of two Gaussian beams displaced in position along the qz=0 line.

Fig. 12
Fig. 12

(a) Spectral function on the circle, (b) intensity and wave fronts of the wave field on the q plane, and (c) angle-impact marginal, for the superposition of two Gaussian beams displaced in angle, centered at θ=(1/2)π.

Equations (54)

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2qx2+2qz2Ψ(q)=-k2Ψ(q).
Ψk(q)exp i(kxqx+kzqz).
Ψ˜(p)=k2π R2d2qΨ(q)exp(-ikpq),
(p2-1)Ψ˜(p)=0Ψ˜[p]=2πk δ(p-1)ψ(θ),
Ψ(qx, qz)=k2π Sdθψ(θ)exp ik(qx sin θ+qz cos θ).
ψ(θ)=σz(θ)2 k2π Rdqx[ΨS(qx)cos θ-ik-1ΨzS(qx)]exp(-ikqx sin θ).
W(Ψ|p, q)=k2π2R2d2rΨ(q-12r)*×exp(-ikpr)Ψ(q+12r)
=k2π2R2d2sΨ˜(p-12s)*×exp(+ikqs)Ψ˜(p+12s).
R2d2pR2d2qW(Ψ|p, q)=(Ψ, Ψ)L2(R2)=R2d2qΨ(q)*Ψ(q)=R2d2pΨ˜(p)*Ψ˜(p),
W(Ψ|p, q)=k2π SdθSdθψ(θ)*ψ(θ)×δ[px-12(sin θ+sin θ)]×δ[pz-12(cos θ+cos θ)]×exp{-ik[qx(sin θ-sin θ)+qz(cos θ-cos θ)]}=k2π Sdα-ππdβψ(α-12β)*ψ(α+12β)×δ(px-cos 12β sin α)×δ(pz-cos 12β cos α)×exp[2ik(qx cos α-qz sin α)sin 12β].
W(Ψ|p, q)=kπp1-p2 {ψ(θ-12ϖ)*ψ(θ+12ϖ)×exp[2ik1-p2(qx cos θ-qz sin θ)]+ψ(θ+12ϖ)*ψ(θ-12ϖ)×exp[-2ik1-p2×(qx cos θ-qz sin θ)]},
ϖ=2 arccos p(0, π).
W(Ψ|p, qx, qz)=W(Ψ|p, qx-qz tan θ, 0).
Tu : Ψ(q)=Ψ(q+u),
Tu : ψ(θ)=exp[ik(ux sin θ+uz cos θ)]ψ(θ),
Rβ : Ψ(q)=Ψ[R(β)q],Rβ : ψ(θ)=ψ(θ+β),
W(TuΨ|p, q)=W(Ψ|p, q+u),
W(RβΨ|p, q)=W(Ψ|R(β)p, R(β)q).
I(Ψ|q)=R2d2pW(Ψ|p, q)=|Ψ(q)|2.
J(Ψ|q)=R2d2ppW(Ψ|p, q)=ik k2π2R4d2pd2rΨ(q-12r)*×[r exp(-ikpr)]Ψ(q+12r)=i2k k2π2R4d2pd2r[-Ψ(q+12r)×qΨ(q-12r)*+Ψ(q-12r)*qΨ(q+12r)]exp(-ikpr)=12ik [Ψ(q)qΨ(q)*-Ψ(q)*qΨ(q)]=1k Im[Ψ(q)qΨ(q)*].
ρ(θ; ω)=(2ω)-1 Rectω(θ)=1/2ω|θ|<ω,0otherwise.
Ψ(qx, 0)=k2π -π/2π/2dθ exp(ikqx sin θ)=k2π J0(kqx),
Ψqx(qx, 0)=ikk2π -π/2π/2dθ sin θ exp(ikqx sin θ)=-kπk2π J1(kqx),
Ψqz(qx, 0)=ikk2π -π/2π/2dθ cos θ exp(ikqx sin θ)=2ikk2πsin kqxqx,
Jz(Ψ|qx, 0)=kπ J0(kqx) sin kqxqx.
K(Ψ|px, qx; qz)=RdpzW(Ψ|p, q)=k2π RdrxΨ(qx-12rx, qz)*×exp(-ikpxrx)Ψ(qx+12rx, qz).
K(Ψ|px, qx; qz)K(Ψ|px, qx-pxqz, 0).
K(Ψ|px, qx; 0)
=k2π Sdα-ππdϖψ(α-12ϖ)*×ψ(α+12ϖ)δ(px-cos 12ϖ sin α)×exp[2ik(qx cos α-qz sin α)sin 12ϖ],
K(Ψ|0, qx; 0)=k2π -ππ dϖcos 12ϖ [ψ(-12ϖ)*ψ(12ϖ)×exp(2ikqx sin 12ϖ)+ψ(π-12ϖ)*ψ(π+12ϖ)×exp(-2ikqx sin 12ϖ)].
RdpxRdqxK(Ψ|px, qx; qz)
=RdqxΨ(qx, qz)*Ψ(qx, qz)
=Sdθ |ψ(θ)|2|cos θ|.
L(Ψ|θ, qx; qz)=01pdpW(Ψ|p, q)=k2π -ππdαψ(θ-12α)*ψ(θ+12α)×exp[2ik(qx cos θ-qz sin θ)sin 12α].
L(Ψ|θ, qx; qz)=L(Ψ|θ, qx-qz tan θ; 0).
K(Ψ|sin θ, qx; qz)L(Ψ|θ, qx; qz).
M(Ψ|θ, l)=k2π -ππdαψ(θ-12α)*ψ(θ+12α)×exp(2ikl sin 12α)
L(Ψ|θ, qx; qz)=M(Ψ|θ, qx cos θ-qz sin θ).
RdqxL(Ψ|θ, qx, 0)exp(-2ikqx cos θ sin 12α)
=ψ(θ-12α)*ψ(θ+12α)|cos θ cos12α|.
ψ(θ)*ψ(θ)=|cos θ+cos θ|2×RdqxL[Ψ|12(θ+θ), qx; 0]×exp[ikqx(sin θ-sin θ)]=|cos θ+cos θ|2 Rdqx×M[Ψ|12(θ+θ), qx cos 12(θ+θ)]×exp[ikqx(sin θ-sin θ)].
SdθL(Ψ|θ, qx; qz)=SdθM(Ψ|θ, qx cos θ-qz sin θ)=|Ψ(q)|2,
RdlM(Ψ|θ, l)=|ψ(θ)|2.
SdθRdlM(Ψ|θ, l)=Sdθψ(θ)*ψ(θ)=(Ψ, Ψ)L2(S),
ψ(θ)=1,i.e.,Ψ(q)J0(k|q|).
ψ(θ)=cos θ,i.e.,Ψ(q)J1(k|q|)qz/|q|.
ψ(θ)=γ(θ; w)=Σm exp(-m2w2+imθ).
(Φ, Ψ)L2(S)=Sdθϕ(θ)*ψ(θ).
(Φ, Ψ)L2(S)=RdqxRdqxΦ(qx)Φz(qx)×Mk(qx-qx)Ψ(qx)Ψz(qx),
Mk(q)=18π Sdθk cos2 θ-i cos θi cos θ1/k×exp(ikq sin θ).
Mk(q)=14 diag[J1(kq)/q, J0(kq)/k].
(Φ, IzΨ)L2(S)=RdqxRdqxΦ(qx)Φz(qx)×Ik(qx-qx)Ψ(qx)Ψz(qx),
Ik(q)=i2k 1π sin kqq 0-110.
(Φ, IzΨ)L2(S)=12ik Rdqx[Φz(qx)*×Ψ(qx)-Φ(qx)*Ψz(qx)].

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