A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.

[CrossRef]

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).

[CrossRef]

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).

[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).

[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). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.

[CrossRef]

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).

[CrossRef]

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).

[CrossRef]

This function was previously proposed in Ref. 16. A similar function was defined in E. Marchand, E. Wolf, “Angular correlation in the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972). However, that function corresponds to the double Fourier transform in the transverse spatial variables of the cross-spectral density at a plane of fixed z. It therefore differs from the form used here by an obliquity factor and a mapping, and it assumes forward propagation.

[CrossRef]

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).

[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).

[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).

[CrossRef]

M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).

[CrossRef]

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]

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), Chap. X, pp. 554–632.

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

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).

[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).

[CrossRef]

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).

[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).

[CrossRef]

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).

[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).

[CrossRef]

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).

[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

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).

[CrossRef]

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).

[CrossRef]

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

[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, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.

[CrossRef]

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.

[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). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.

[CrossRef]

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).

[CrossRef]

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).

[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).

[CrossRef]

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).

[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).

[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).

[CrossRef]

This function was previously proposed in Ref. 16. A similar function was defined in E. Marchand, E. Wolf, “Angular correlation in the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972). However, that function corresponds to the double Fourier transform in the transverse spatial variables of the cross-spectral density at a plane of fixed z. It therefore differs from the form used here by an obliquity factor and a mapping, and it assumes forward propagation.

[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

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), Chap. X, pp. 554–632.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

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). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.

[CrossRef]

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).

[CrossRef]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).

[CrossRef]

This function was previously proposed in Ref. 16. A similar function was defined in E. Marchand, E. Wolf, “Angular correlation in the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972). However, that function corresponds to the double Fourier transform in the transverse spatial variables of the cross-spectral density at a plane of fixed z. It therefore differs from the form used here by an obliquity factor and a mapping, and it assumes forward propagation.

[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).

[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).

[CrossRef]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).

[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).

[CrossRef]

M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).

[CrossRef]

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

[CrossRef]

M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).

[CrossRef]

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]

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).

[CrossRef]

R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).

[CrossRef]

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).

[CrossRef]

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).

[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).

[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1987).

[CrossRef]

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.

[CrossRef]

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).

[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.

Walther’s two generalized radiances, as well as the more general family to which they belong to, are asymptotically conserved along rays in the limit of small wavelength and for planar quasi-homogeneous sources, as shown in Refs. 9-12. Definitions of generalized radiance that are explicitly conserved along rays are given in Refs. 26-30below.

Some of the most relevant papers in the subject are compiled in A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. MS69 of Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1993).

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), Chap. X, pp. 554–632.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

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

See Ref. 5, p. 610, and Ref. 6, pp. 162–163.

See Ref. 5, p. 567, and Ref. 6, p. 170. The cross-spectral density is often denoted by the letter W. However, in order to avoid confusion with the Wigner function used here, we denote the cross-spectral density by G, as in Ref. 5.

See Ref. 6, p. 170.

These properties are found in Ref. 6, pp. 183 and 215.

See Ref. 6, pp. 214–216. Notice that the orthogonality relation there differs from the one to be given here: Whereas Eq. (4.7–12) of Ref. 6 involves integration over all space, Eq. (2.17) of this paper involves integration over all directions.

See Ref. 6, p. 290.

This condition for the region of interest in explained in Ref. 14.

See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

Remember that the denominator is given by the square of the total power defined in Eq. (4.2). The fact that this quantity diverges, however, does not mean that the radiation inside a blackbody cavity is infinite, because the field does not extend over all space but only inside the cavity.