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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 66, Iss. 6 — Jun. 1, 1976
  • pp: 529–535

Spectral coherence and the concept of cross-spectral purity

L. Mandel and E. Wolf  »View Author Affiliations

JOSA, Vol. 66, Issue 6, pp. 529-535 (1976)

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A new measure of correlations in optical fields, introduced in recent investigations on radiometry with partially coherent sources, is studied and applied to the analysis of interference experiments. This measure, which we call the complex degree of spectral coherence, or the spectral correlation coefficient, characterizes the correlations that exist between the spectral components at a given frequency in the light oscillations at two points in a stationary optical field. A relation between this degree of correlation and the usual degree of coherence is obtained and the role that the complex degree of spectral coherence plays in the spectral structure of a two-beam interference pattern is examined. It is also shown that the complex degree of spectral coherence provides a clear insight into the physical significance of cross-spectral purity. When the optical field at two points is cross-spectrally pure, the absolute value of the complex degree of spectral coherence at these points is found to be the same for every frequency component of the light. This fact is reflected in the visibility of the spectral components of the interference fringes formed by light from these points.

© 1976 Optical Society of America

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529-535 (1976)

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975). It should be noted that the mutual coherence function in this book differs by interchange of the points r1, r2 from the definition (1. 2a).
  2. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
  3. E. Wolf and W. H. Carter, Opt. Commun. 13, 205 (1975).
  4. W. H. Carter and E. Wolf, J. Opt. Soc. Am. 65, 1067 (1975).
  5. E. Wolf and W. H. Carter, Opt. Commun. 16, 297 (1976).
  6. The concept of cross-spectral purity was introduced by L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
  7. For reasons well known in the theory of stationary random processes, the Fourier representation (2. 1) does not exist in the sense of ordinary function theory, and must be understood in terms of the theory of generalized functions [cf. Y. L. Lumley, Stochastic Tools in Turbulence (Academic, New York, 1970)]. One may avoid the use of generalized functions by other mathematical refinements, for example with the help of a certain truncation procedure (cf. Ref. 1, Secs. 10.2 and 10. 3), or by the use of the Fourier-Stieltjes integral [cf. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions (Prentice-Hall, Englewood Cliffs, N.J., 1962)].
  8. We are dealing here with a cross-spectral density function associated with a stationary random process V(r, t) labeled by a continuous (vector)parameter r. For a finite-dimensional process labeled by a discrete parameter, a similar non-negative definiteness condition on the cross-spectral density function follows from a theorem due to H. Cramér, Ann. Math. 41, 215 (1940).
  9. When light of frequency v is incident on the plane of the pinholes along or close to the normal direction, and the angles that the diffracted directions P1P and P2P make with the normal to are also small, then Kj ~ (/csjAj (j = 1, 2), where δAj is the area of the pinhole at rj (ef. Ref. 1, Secs. 8.2 and 8.3).
  10. L. Mandel, J. Opt. Soc. Am. 52, 1335 (1962).
  11. W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).

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