Abstract

A simple formalism is found for the measurement of wave fields that satisfy the Helmholtz equation in free space. This formalism turns out to be analogous to the well-known theory of measurements for quantum-mechanical wave functions: A measurement corresponds to the squared magnitude of the inner product (in a suitable Hilbert space) of the wave field and a field that is associated with the detector. The measurement can also be expressed as an overlap in phase space of a special form of the Wigner function that is tailored for Helmholtz wave fields.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.
  2. E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  3. H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  4. See, for example, S. Stenholm, “The Wigner function. I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980).
    [CrossRef]
  5. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  6. P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 11, 77–91 (1995).
    [CrossRef]
  7. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.
  8. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed (Cambridge U. Press, New York, 1995), pp. 128–133.
  9. 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]
  10. See Ref. 1, p. 216.
  11. Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.
  12. Notice that, for example, the spectral function for the first measurement in Eq. (18b) can be written as μ(θ)=2 sin θρ(r, θ)=i2limL→0 ρ(x+L/2, z, θ)-ρ(x-L/2, z, θ)kL.The corresponding measurement is then achieved by a phase-locked pair of point detectors, aligned in the z direction with a separation much smaller than the wavelength, and with a phase mismatch of π. Notice that these two detectors are coupled (i.e., their relative weights including phase are fixed) and therefore do not belong to the class of composite detectors presented in Section 8. The remaining measurements in Eqs. (18b) and (18c) are achieved with similar detectors.
  13. See Ref. 1, pp. 221.
  14. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  15. M. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  16. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
    [CrossRef]
  17. See Ref. 8, p. 169.

1999 (1)

1995 (2)

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 11, 77–91 (1995).
[CrossRef]

1981 (1)

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 (2)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
[CrossRef]

See, for example, S. Stenholm, “The Wigner function. I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980).
[CrossRef]

1979 (1)

1968 (1)

1932 (1)

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

Alonso, M. A.

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
[CrossRef]

Bastiaans, M.

Brenner, K. H.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.

Forbes, G. W.

González-Casanova, P.

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 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.

Laloë, F.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.

Lee, H. W.

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Lohmann, A. W.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed (Cambridge U. Press, New York, 1995), pp. 128–133.

Orlov, Yu. A.

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

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]

Stenholm, S.

See, for example, S. Stenholm, “The Wigner function. I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980).
[CrossRef]

Walther, A.

Wigner, E.

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

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed (Cambridge U. Press, New York, 1995), pp. 128–133.

Wolf, K. B.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 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]

Eur. J. Phys. (1)

See, for example, S. Stenholm, “The Wigner function. I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980).
[CrossRef]

J. Math. Phys. (1)

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. (2)

J. Opt. Soc. Am. A (1)

Numer. Methods Partial Differential Equations (1)

P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 11, 77–91 (1995).
[CrossRef]

Opt. Commun. (1)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980).
[CrossRef]

Phys. Rep. (1)

H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Phys. Rev. (1)

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

Other (8)

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed (Cambridge U. Press, New York, 1995), pp. 128–133.

See Ref. 1, p. 216.

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

Notice that, for example, the spectral function for the first measurement in Eq. (18b) can be written as μ(θ)=2 sin θρ(r, θ)=i2limL→0 ρ(x+L/2, z, θ)-ρ(x-L/2, z, θ)kL.The corresponding measurement is then achieved by a phase-locked pair of point detectors, aligned in the z direction with a separation much smaller than the wavelength, and with a phase mismatch of π. Notice that these two detectors are coupled (i.e., their relative weights including phase are fixed) and therefore do not belong to the class of composite detectors presented in Section 8. The remaining measurements in Eqs. (18b) and (18c) are achieved with similar detectors.

See Ref. 1, pp. 221.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.

See Ref. 8, p. 169.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

A ray is fully characterized by its direction given by the angle θ from the z axis and by its impact parameter l.

Fig. 2
Fig. 2

When the coordinate origin is translated to a point r0, the cylinder over which the angle-impact Wigner function is defined undergoes a linear shear as described by Eq. (24). The magnitude of the maximum shift in l caused by this shear equals the magnitude of r0 and corresponds to directions of propagation that are perpendicular to r0.

Fig. 3
Fig. 3

The integral of M is nonnegative over any contour that corresponds to the intersection of the cylinder and any plane that contains the center of the cylinder at l=0.

Fig. 4
Fig. 4

Plot of the nonlocal factor T(l), defined in Eq. (32).

Fig. 5
Fig. 5

Plots of (a) Mμ and (b) Mμ¯ for a detector with a constant spectral function. The values represented by the different levels of gray follow the charts below the figures.

Fig. 6
Fig. 6

Plots of MμC¯ for the composite detector described by the mutual spectrum given in Eq. (50), for (a) kL=0.5 and (b) kL=8.654.

Equations (67)

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

2-1c2 2t2E(r, t)=0.
(2+k2)U(r)=0,
U(x, z)=k2π1/2Sϕ(θ)exp[ik(x sin θ+z cos θ)]dθ,
ϕ(θ)=sign(cos θ)2 k2π1/2×U(x, 0)cos θ-ik Uz (x, 0)×exp(-ikx sin θ)dx,
ϕin(θ)  S^in(θ)U  limr expikr-i π4×r2 1+ik rU(-r sin θ,-r cos θ),
ϕout(θ)  S^out(θ)U  limr exp-ikr+i π4×r2 1-ik rU(r sin θ, r cos θ),
Φin=S|ϕin(θ)|2dθ,
Φout=S|ϕout(θ)|2dθ.
UT(x, z)=U(x, z)-cUμ(x, z),
S^in(θ)Uμ0.
μ(θ)  S^out(θ)Uμ.
S|μ(θ)|2dθ=1.
ΔΦ=Φin-Φout=S|ϕ(θ)|2dθ-S|ϕ(θ)-cμ(θ)|2dθ=cSϕ*(θ)μ(θ)dθ+c*Sμ*(θ)ϕ(θ)dθ-cc*,
c=Sμ*(θ)ϕ(θ)dθ=(Uu|U),
ΔΦ=|(Uμ|U)|2=Sμ*(θ)ϕ(θ)dθ2.
U(r)=(r|U)=kSρ*(r, θ)ϕ(θ)dθ,
ρ(r, θ)  12π exp[-ik(x sin θ+z cos θ)].
I(r)  |U(r)|2=|(r|U)|2.
J(r)  12ik [U(r)U*(r)-U*(r)U(r)].
J(r)-k2 Sρ*(r, θ)ϕ(θ)dθS(sin θ, cos θ)×ρ(r, θ)ϕ*(θ)dθ+S(sin θ, cos θ)ρ*(r, θ)ϕ(θ)dθ×Sρ(r, θ)ϕ*(θ)dθ.
I(r)=kSρ*(r, θ)ϕ(θ)dθ2,
j0,xz(r)=kSsin θcos θρ*(r, θ)ϕ(θ)dθ2,
j0, xz(r)=kS1+sin θcos θρ*(r, θ)ϕ(θ)dθ2,
Jxz(r)=12 [I(r)+j0,xz(r)-j1,xz(r).
W(x, Px; z)  k2π U(x+x2, z)U*x-x2, z×exp(-ikxpx)dx,
W(x, px; z)W(x-zpx, px; 0).
M(l, θ)  k2π -ππϕθ+α2×ϕ*θ-α2exp2ikl sin α2dα,
l=x cos θ-z sin θ.
l=l-x0 cos θ+z0 sin θ.
SM(x cos θ-z sin θ, θ)dθ=|U(x, z)|2=I(x, z).
M(l, θ)dl=|ϕ(θ)|2.
SM(l, θ)dldθ=S|ϕ(θ)|2dθ=(U|U).
ϕ(θ)ϕ*(θ)=cosθ+θ2Ml, θ+θ2×exp-2ikl sinθ-θ2dl.
Mμ(l, θ)  k2π -ππμθ+α2μ*θ-α2×exp2ikl sin α2dα.
μ(θ)μ*(θ)=cosθ+θ2Mμl, θ+θ2×exp-2ikl sinθ-θ2dl.
ΔΦ=SM(l, θ)Mμ(l, θ)T(l-1)dldldθ,
T(l)-ππ cos2 β2 exp-2ikl sin β2dβ=πJ1(2kl)kl.
ΔΦSM(l, θ)M¯u(l, θ)dldθ,
M¯μ(l, θ)Mμ(l, θ)T(l-l)dl
=-ππμθ+α2μ*θ-α2×exp2ikl sin α2cos α2 dα.
μ(θ)=12π.
Mμ(l, θ)=k2π J0(2kl),
M¯(l, θ)=sin(2kl)πkl.
ΔΦ=S|ϕ(θ)|2dθ-Sϕ(θ)-ncnμn(θ)2dθ
=n(cnan*+cn*an)-n,ncnAn,ncn
=ca*+c*a-c*Ac,
anSμn*(θ)ϕ(θ)dθ=(Uμn|U),
An,n Sμn*(θ)μn(θ)dθ=(Uμn|Uμn).
Ac=a.
ΔΦ=a*A-1a=SSϕ*(θ)μC(θ, θ)ϕ(θ)dθdθ,
μC(θ, θ)n,nμn(θ)(A-1)n,nμn*(θ).
MμC(l, θ)  k2π -ππμCθ+α2, θ-α2×exp2ikl sin α2dα,
M¯μC(l, θ) -ππμCθ+α2, θ-α2×exp2ikl sin α2cos α2 dα,
ΔΦ=SM(l, θ)MμC(l, θ)T(l-l)dldldθ=SM(l, θ)M¯μC(l, θ)dldθ.
μ1(θ)=12π exp-ik L2 sin θ,
μ2(θ)=12π exp+ik L2 sin θ,
A=1J0(kL)J0(kL)1.
A-1=11-J02(kL) 1-J0(kL)-J0(kL)1.
μC(θ, θ)=12π[1-J02(kL)] coskL sin θ-sin θ2-J0(kL)coskL sin θ+sin θ2.
MμC(l, θ)=k2π {J0[k(2l-L cos θ)]+J0[k(2l+L cos θ)]},
M¯μC(l, θ)=2 sin[k(2l-L cos θ)]πk(2l-L cos θ)+2 sin[k(2l+L cos θ)]πk(2l+L cos θ).
IPC(r, r)=U(r)U*(r),
ϕPC(θ, θ)=ϕ(θ)ϕ*(θ).
IPC(r,r)=k2π SSϕPC(θ, θ)×exp[ik(x sin θ+z cos θ-x sin θ-z cos θ)]dθdθ.
ΔΦ=SSϕPC(θ, θ)μC(θ, θ)dθdθ,
MPC(l, θ)  k2π -ππϕPCθ+α2, θ-α2×exp2ikl sin α2dα.
μ(θ)=2 sin θρ(r, θ)=i2limL0 ρ(x+L/2, z, θ)-ρ(x-L/2, z, θ)kL.

Metrics