Abstract

The wave-based generalized radiance definitions presented in a previous manuscript [J. Opt. Soc. A 18, 902 (2001)] for two-dimensional coherent monochromatic fields in free space are extended here to the three-dimensional case. These new definitions preserve all the properties of their two-dimensional analogs. Notably, they are exactly conserved along rays and well suited for the description of fields traveling in all directions. The different members of this set of functions are seen to correspond to weighted radial projections in momentum of the Wigner function of the field.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.
  2. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.
  3. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.
  4. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 69–78.
  5. For a general overview see, for example, the following two review papers: E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978); A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1978). A collection of the most significant studies on this subject is given in A. T. Friberg (volume editor), Selected Papers on Coherence and Radiometry, Vol. MS69 of SPIE Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1993).
    [CrossRef]
  6. See also Ref. 2, pp. 292–297.
  7. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2000).
    [CrossRef]
  8. 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]
  9. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
    [CrossRef]
  10. N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
    [CrossRef]
  11. H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A. Math. Gen. 25, 5263–5278 (1992). See Eq. (5.17).
    [CrossRef]
  12. G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972). See Eq. (2).
    [CrossRef]
  13. H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887. See Eq. (12).
  14. H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata J. Opt. Soc. Am. A 8, 1518 (1991). See Eq. (6.4).
    [CrossRef]
  15. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993). The missing obliquity factor can be seen from the comparison of Eq. (5.11) of this reference with Eq. (2.9) of the present paper. Whereas the latter involves integration over a solid angle, the former is an integral over the transverse Cartesian coordinates of the direction vector.
    [CrossRef]

2000

1999

1993

1992

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A. Math. Gen. 25, 5263–5278 (1992). See Eq. (5.17).
[CrossRef]

1991

1984

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

1978

1972

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972). See Eq. (2).
[CrossRef]

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Alonso, M. A.

Balasz, N. L.

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

Forbes, G. W.

Jennings, B. K.

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Littlejohn, R. G.

Løkberg, O. J.

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A. Math. Gen. 25, 5263–5278 (1992). See Eq. (5.17).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

Ovchinnikov, G. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972). See Eq. (2).
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A. Math. Gen. 25, 5263–5278 (1992). See Eq. (5.17).
[CrossRef]

H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata J. Opt. Soc. Am. A 8, 1518 (1991). See Eq. (6.4).
[CrossRef]

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887. See Eq. (12).

Tatarskii, V. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972). See Eq. (2).
[CrossRef]

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 69–78.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Winston, R.

Wolf, E.

Wolf, K. B.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A. Math. Gen.

H. M. Pedersen, O. J. Løkberg, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A. Math. Gen. 25, 5263–5278 (1992). See Eq. (5.17).
[CrossRef]

Phys. Rep.

N. L. Balasz, B. K. Jennings, “Wigner’s function and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984).
[CrossRef]

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 740–759 (1932).
[CrossRef]

Radiophys. Quantum Electron.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972). See Eq. (2).
[CrossRef]

Other

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887. See Eq. (12).

See also Ref. 2, pp. 292–297.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 193–199.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 287–307.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, Cambridge, UK, 1995), pp. 69–78.

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Figures (6)

Fig. 1
Fig. 1

Cylindrical coordinates used for the integration of Eq. (3.4).

Fig. 2
Fig. 2

Plots of (k|φ0|)-2M(n) as functions of k|L| for n=0, 1, and 2, for a perfect spherical wave converging to the origin. Note that different numeric scale factors were used for each curve to show the relative reduction of the negative regions for increasing n.

Fig. 3
Fig. 3

Plane (shown in gray) defined by the L1 and L2 axes perpendicular to u, which forms an angle ϕ with the z axis. The L1 axis is chosen to form the largest possible angle with the z axis, and the L2 axis is then perpendicular to both the L1 and z axes. The unit vector w(u, θ) lives in the (L1, L2) plane, and θ is chosen to be the angle from the L1 axis to w.

Fig. 4
Fig. 4

Plots of (k|φ0|)-2M(n) at u=z as functions of k|L| for n=0, 1, and 2, for a spherical wave apertured at infinity, with half-angles ϕ0=π/8, π/4, and π/2.

Fig. 5
Fig. 5

Plots of 4π(1-ϕ/ϕ0)-1(k|ϕ0|)-2M(0) as functions of kL1 and kL2 for (a) ϕ=0, (b) π/8, and (c) 3π/16, for a uniform spherical wave apertured at infinity with half-angle ϕ0=π/4.

Fig. 6
Fig. 6

Plot of 8π(k|φ0|)-2M(0) at L=0, as a function of ϕ, for a uniform spherical wave apertured at infinity with half-angle ϕ0=π/4.

Equations (77)

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B(r, u)u·ndσdΩ=d2Φ.
B(r, u)0.
u·B(r, u)=0.
(2+k2)U(r)=0.
U(r)=k2π1/22πφ(θ)exp[ikr·u(θ)]dθ,
B(n)[r, u(θ)]=M(n)[r·u(θ), θ],
M(n)(l, θ)=k2π-ππφθ+α2φ*θ-α2×exp2ikl sinα2cosnα2dα,
2πB(0)[r, u(θ)]dθ=S(r)  |U(r)|2,
2πB(2)(r, θ)dθ=H(r)  12[|U(r)|2+k-2|U(r)|2],
2πB(1)[r, u(θ)]u(θ)dθ=F(r)  12ik[U*(r)U(r)-U(r)U*(r)],
RB(n)[lu(θ), u(θ)]dl=J[u(θ)]  |φ(θ)|2,
W(r, p)=k2πNRUr+r2U*r-r2×exp(-ikr·p)dNr,
RW(r, p)dNp=|U(r)|2=S(r).
RW(r, p)pdNp=RUr+r2×U*r-r2ikδ(r)dNr=12ik[U*(r)U(r)-U(r)U*(r)]=F(r),
RW(r, p)p·pdNp=-RUr+r2×U*r-r21k22δ(r)dNr=12[U*(r)U(r)+k-2U(r)·U*(r)]=H(r),
B(0)(r, u)0W(r, pu)pN-1dp.
2(N-1)πB(0)(r, u)dN-1α=S(r),
B(n)(r, u)0W(r, pu)pN+n-1dp.
2(N-1)πB(2)(r, u)dN-1α=H(r),
2(N-1)πB(1)(r, u)udN-1α=F(r).
U(r)=k2π4πφ(u)exp(iku·r)dΩ
=k2πRφ(p)exp(ikp·r)δ(|p|-1)d3p,
W(r, p)
=k2π5RRRφ(p)δ(|p|-1)×φ*(p)δ(|p|-1)
×expikp·r+r2-p·r-r2-r·p
×d3pd3pd3r
=k2π2RRφ(p)δ(|p|-1)φ*(p)δ(|p|-1)×exp[ikr·(p-p)]δ3p+p2-p×d3pd3p.
s=p+p2,v=p-p,
W(r, p)=k2π2RRφs+v2δs+v2-1×φ*s-v2δs-v2-1×exp(ikr·v)δ3(s-p)d3sd3v=k2π2Rφp+v2φ*p-v2×δp+v2-1δp-v2-1×exp(ikr·v)d3v.
w(u, θ)·u=0,w(u, θ)·w(u, θ)=cos(θ-θ),
v=2τu+2γw(u, θ),d3ν=8γdτdγdθ,
δp±v2-1=δ{[(p±τ)2+γ2]1/2-1}.
δ{[(p+τ)2+γ2]1/2-1}δ{[(p-τ)2+γ2]1/2-1}=δ(τ)δ[p-(1+γ2)1/2]+δ(p)δ(τ-(1+γ2)1/2)2|τ2-p2|.
B(n)(r, u)=4k2π22π01φ[(1-γ2)1/2u+γw(u, θ)]×φ*[(1-γ2)1/2u-γw(u, θ)]×exp[2ikγr·w(u, θ)]γ(1-γ2)n/2dγdθ.
B(n)(r, u)=k2π22π0πφcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)×exp2ikr·w(u, θ)sinα2sin α×cosnα2dαdθ.
u·B(n)(r, u)=0.
AnB(n)(r, u)dσ=|φ(u)|2n·u=J(u)n·u.
B(r, u)=M(r×u, u).
r=u×L+ξu
M(n)(L, u)  B(n)(u×L+ξu, u)
=k2π22π0πφcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)×exp2iku×L·w(u, θ)sinα2×sin α cosnα2dαdθ
=k2π22π0πφcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)×exp2ikL·wu, θ-π2×sinα2sin α cosnα2dαdθ.
RM(n)(L, u)d2L=|φ(u)|2=J(u),
4πRM(n)(L, u)d2LdΩ=Φ,
φ(x sin ϕ cos β+y sin ϕ sin β+z cos ϕ)=φ*(z)cosnϕ2-1RR×M(n)(x sin β-y cos β)L1+x cosϕ2cos β+y cosϕ2sin β-z sinϕ2×L2,x sinϕ2cos β+y sinϕ2sin β+z cosϕ2×exp-2ikL1 sinϕ2dL1dL2,
φ(u)=φ0.
M(n)(L, u)
=kφ02π22π0πexp2ikL·wu, θ-π2sinα2×sin α cosnα2dαdθ
=kφ02π20π2πexp2ik|L|sin θ sinα2×dθ sin α cosnα2dα
=k2|φ0|22π0πJ02k|L|sinα2sin α cosnα2dα.
M(0)(L, u)=k2|φ0|2πJ1(2k|L|)k|L|,
M(1)(L, u)=2k2|φ0|2πsin(2k|L|)-2k|L|cos(2k|L|)(2k|L|)3,
M(2)(L, u)=k2|φ0|2πk|L|J1(2k|L|)-J2(2k|L|)+k|L|J3(2k|L|)(k|L|)2.
φ(u)=f(uz)=f(u·z).
z·w(u, θ)=-sin ϕ cos θ,
L·wu, θ-π2=L1 sin θ-L2 cos θ,
M(n)(L, u)=k2π22π0πfcosα2cos ϕ-sinα2sin ϕ cos θ×f*cosα2cos ϕ+sinα2sin ϕ cos θ×exp2ik(L1 sin θ-L2 cos θ)sinα2×sin α cosnα2dαdθ.
f(cos ϕ)=ϕ0,cos ϕcos ϕ00,cos ϕ<cos ϕ0,
r=r1m1+r2m2+rnn,
m1=β-1(u-n·un),m2=β-1n×u,
β=[1-(n·u)2]1/2.
r·w(u, θ)=β-1[r1w(u, θ)·n×u-r2w(u, θ)·nn·u]+rnw(u, θ)·n.
n=n·uu+βw(u, θ0),
n×u=βw(u, θ0)×u=βwu, θ0-π2.
r·w(u, θ)=-r1 sin(θ-θ0)-r2n·ucos(θ-θ0)+rnβ cos(θ-θ0).
λnB(n)(r, u)dσ=4k2π22π01φ[(1-γ2)1/2u+γw(u, θ)]×φ*[(1-γ2)1/2u-γw(u, θ)]×RRexp{-2ikγ[r1 sin(θ-θ0)+r2n·u×cos(θ-θ0)]}dr1dr2×exp[2ikγrnβ cos(θ-θ0)]γ(1-γ2)n/2dγdθ=1n·u2π01φ[(1-γ2)1/2u+γw(u, θ)]×φ*[(1-γ2)1/2u-γw(u, θ)]×exp[2ikγrnβ cos(θ-θ0)](1-γ2)n/2×δ[γ sin(θ-θ0)]δ[γ cos(θ-θ0)]γdγdθ.
L=L1w(u, 0)+L2wu, π2.
L·wu, θ-π2=L1 sin θ-L2 cos θ.
I(n)=RRM(n)(L, u)exp-2ikL·wu, θ-π2×sinα2cosnα2dL1dL2,
I(n)=2π0πφcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)cosnα2×δ2 sin θ sinα2-2 sin θ sinα2×δ2 cos θ sinα2-2 cos θ sinα2sin αdαdθ
=2π0πφcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)cosnα2×δ(θ-θ)δ(α-α)dαdθ
=φcosα2u+sinα2w(u, θ)×φ*cosα2u-sinα2w(u, θ)cosnα2.
cosα2u-sinα2w(u, θ)=z,
cosα2u+sinα2w(u, θ)=x sin ϕ cos β+y sin ϕ sin β+z cos ϕ,
u=x sinϕ2cos β+y sinϕ2sin β+z cosϕ2.
w(u, θ)=x cosϕ2cos β+y cosϕ2sin β-z sinϕ2,
wu, θ-π2=x sin β-y cos β.

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