## Radiometry and wide-angle wave fields III: partial coherence

JOSA A, Vol. 18, Issue 10, pp. 2502-2511 (2001)

http://dx.doi.org/10.1364/JOSAA.18.002502

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### Abstract

The analogs of the generalized radiances introduced in two previous manuscripts [J. Opt. Soc. Am. A 18, 902 (2001) and J. Opt. Soc. Am. A 18, 910 (2001)] for fully coherent fields in two- and three-dimensional free space are given here for the case of partial coherence. These functions are exactly conserved along rays and are suitable for the description of fields with components propagating in any direction. Also defined here is a global measure of coherence, which can be expressed in terms of the new functions. The cases of radiation in a blackbody cavity and partially coherent focused waves are considered as examples.

© 2001 Optical Society of America

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.5620) Coherence and statistical optics : Radiative transfer

(030.5630) Coherence and statistical optics : Radiometry

**History**

Original Manuscript: February 1, 2001

Revised Manuscript: April 5, 2001

Manuscript Accepted: March 27, 2001

Published: October 1, 2001

**Citation**

M. A. Alonso, "Radiometry and wide-angle wave fields III: partial coherence," J. Opt. Soc. Am. A **18**, 2502-2511 (2001)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-10-2502

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### References

- L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]
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- 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).
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- 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]
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- 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.
- 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]
- 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.
- 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]
- 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]
- 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]
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- 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.
- 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, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992). [CrossRef]
- R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993). [CrossRef]
- 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.
- 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]
- 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.
- 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]

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