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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 64, Iss. 9 — Sep. 1, 1974
  • pp: 1219–1226

Radiometry with sources of any state of coherence

E. W. Marchand and E. Wolf  »View Author Affiliations

JOSA, Vol. 64, Issue 9, pp. 1219-1226 (1974)

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The basic laws of radiometry are generalized to fields generated by a two-dimensional stationary source of any state of coherence. Important in this analysis is the concept of the generalized radiance function, introduced by Walther in 1968. The concepts of generalized radiant emittance and of generalized radiant intensity are introduced and it is shown how all these quantities may be expressed in terms of coherence functions of the source. Both the generalized radiance of Walther and the generalized radiant emittance may take on negative values, indicating that these quantities have, in general, a less-direct physical meaning than have the corresponding quantities of traditional radiometry (which presumably represents the incoherent limit of the present theory). The generalized radiant intensity is, however, always found to be non-negative and, just as in the incoherent limit, represents the angular distribution of the energy flux in the far zone.

E. W. Marchand and E. Wolf, "Radiometry with sources of any state of coherence," J. Opt. Soc. Am. 64, 1219-1226 (1974)

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  1. M. Planck, The Theory of Heat Radiation, translation from the Second Edition (Dover, New York, 1959).
  2. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford and New York, 1970), §10.4.2.
  3. A preliminary account of our main results was published in Opt. Commun. 6, 305 (1972) and was also presented at a meeting of the Optical Society of America held in Rochester, N. Y., 9–12 October 1973 [J. Opt. Soc. Am. 63, 1285A (1973)]. In Eq. (12) of the published paper, a factor (2π)-2 should be omitted.
  4. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968). In a recent paper [J. Opt. Soc. Am. 63, 1622 (1973)] Walther modified his original definition of the generalized radiance after asserting that it depends on the choice of the coordinate system. This assertion, however, is misleading in the context of his earlier paper relating to radiation from planar sources. For, as we show in a Letter on p. 1273 in the present issue, the generalized radiance, as originally defined by Walther, is, in fact, invariant with respect to an arbitrary displacement of the origin of coordinates in the source plane and is also invariant with respect to rotation of axes about the normal to the plane of the source. In any case, as will be clear from the discussion in the present paper, it is not the generalized radiance, but rather the generalized radiant intensity that has a direct physical significance.
  5. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 62, 379 (1972). In Eq. (10) and in some of the subsequent equations of this reference there is an error: m2 should be replaced by its complex conjugate m2*. This error does not, however, affect the main results.
  6. E. Wigner, Phys. Rev. 40, 749 (1932). For a good discussion of some of the properties of the Wigner distribution function, see K. Imre, E. Ozizmir, M. Rosenbaum, and P. F. Zweifel, J. Math. Phys. 8, 1907 (1967).
  7. See, for example, G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. P., 1922), p. 20, Eq. (5) (with an obvious substitution).
  8. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965), p. 688, formula 1 of §6.567, with v = 0, µ = ½.
  9. Alternative representations of spatially incoherent sources are discussed in a paper by M. Beran and G. Parrent, Nuovo Cimento 27, 1049 (1963).
  10. (a) C. L. Mehta, Nuovo Cimento 28, 401 (1963); (b) R. C. Bourret, Nuovo Cimento 18, 347 (1960); (c) C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964); Phys. Rev. 134, A1149 (1964).
  11. J. Peřina, Coherence of Light (Van Nostrand, London, 1972), §4.2.

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