## Inverse problem with quasi-homogeneous sources

JOSA A, Vol. 2, Issue 11, pp. 1994-2000 (1985)

http://dx.doi.org/10.1364/JOSAA.2.001994

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### Abstract

The problem is discussed of determining the distribution of the intensity and of the degree of spectral coherence of a planar, secondary quasi-homogeneous source from the cross-spectral density function of the field measured over any plane that is parallel to the source. A solution to this problem is presented under the assumption that the degree of spectral coherence *g*(*ρ*_{1} − *ρ*_{2}, *ω*) of the quasi-homogeneous source does not vary appreciably across the source over distances |*ρ*_{1} − *ρ*_{2}| that are of the order of or less than the wavelength λ corresponding to the frequency to. The results are illustrated by computational reconstruction of Gaussian-correlated quasi-homogeneous sources.

© 1985 Optical Society of America

**History**

Original Manuscript: February 28, 1985

Manuscript Accepted: May 30, 1985

Published: November 1, 1985

**Citation**

William H. Garter and Emil Wolf, "Inverse problem with quasi-homogeneous sources," J. Opt. Soc. Am. A **2**, 1994-2000 (1985)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-2-11-1994

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### References

- We are aware of only two papers regarding problems of this kind, viz., A. J. Devaney, “The inverse problem for random sources,” J. Math. Phys. 20, 1687–1691 (1979);I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985). [CrossRef]
- W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977). [CrossRef]
- J. C. Dainty, “An introduction to Gaussian speckle,” Proc. Soc. Photo-Opt. Instrum. Eng. 243, 2–8 (1980).
- See, for example, E. Collett, E. Wolf, “Is a complete spatial coherence necessary for the generation of highly directional beams?” Opt. Lett. 2, 27–29 (1978);P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979);J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980). [CrossRef]
- M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.2.
- L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976), Eq. (2.10). [CrossRef]
- E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).There is an error in Eq. (4.6) of this paper. The constant k should be omitted in the exponent of the integral in that equation. This error does not affect the subsequent equations. [CrossRef]
- To simplify the notation we do not display from now on the dependence of the various quantities on the frequency ω.
- The cross-spectral density may be determined by taking the temporal Fourier transform of the mutual coherence function Γ(ρ1, ρ2, z, τ). The mutual coherence function may be determined from Young’s interference experiments, with pinholes at the points P1(ρ1, z) and P2(ρ2, z) (Ref. 5, Sec. 10.3). Alternatively, the cross-spectral density may be determined more directly from such experiments, if narrow-band filters, with passbands centered at the frequency ω, are placed in front of the pinholes [cf. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983)].
- Since strictly speaking the preceding analysis applies to fields produced by finite sources, it would be more appropriate to assume that Eq. (4.1) holds when ρ≤ a and that I(0)(ρ) = 0 when ρ> a, where a is the radius of the source. However, when a ≫ σI, as we will assume from now on, no significant error is introduced by taking I(0)(ρ) to be given by Eq. (4.1) for all values of ρ.
- E. Wolf, “Completeness of coherent-mode eigenfunctions of Schell-model sources,” Opt. Lett. 9, 387–389 (1984), Eqs. (2.9) and (2.15). [CrossRef] [PubMed]
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944), p. 20, Eq. (5) (with an obvious substitution).

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