Abstract
Parrentâ€™s criterion for the complete coherence of an optical wave field is reexamined and a new proof is obtained using FourierStieltjes integrals.
Citation
RICHARD BARAKAT, "Theorem in Coherence Theory," J. Opt. Soc. Am. 56, 739740 (1966)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa566739
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References

G. Parrent, Opt. Acta 6, 285 (1959). See also: M. Beran and G. Parrent, Theory of Partial Coherence (PrenticeHall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).

The author prefers the use of realvalued functions, although the subsequent analysis can also be effectuated using complexvalued functions.

The braces { } denote ensemble properties.

M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.

Actually there is a third component termed the singular continuous component; this component is continuous but is without a spectral density at every point. Such functions do not have any physical significance and are always omitted.

H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).

This theorem was first stated by Kampe de Feriet (in an almost completely unknown paper) for the case of an autocorrelation function; the extension to crosscorrelation functions presented here is a straightforward extension of his proof. See: M. J. Kampe de Feriet, Ann. Soc. Sci. Bruxelles, Ser. 1 59, 145 (1939).

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.
Balachandran, A. P.

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).
Bartlett, M. S.

M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.
Bohr, H.

H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).
de Feriet, M. J. Kampe

This theorem was first stated by Kampe de Feriet (in an almost completely unknown paper) for the case of an autocorrelation function; the extension to crosscorrelation functions presented here is a straightforward extension of his proof. See: M. J. Kampe de Feriet, Ann. Soc. Sci. Bruxelles, Ser. 1 59, 145 (1939).
Edwards, S. F.

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).
Metha, C. L.

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).
Parrent, G.

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

G. Parrent, Opt. Acta 6, 285 (1959). See also: M. Beran and G. Parrent, Theory of Partial Coherence (PrenticeHall, Inc., Englewood Cliffs, N. J., 1964), p. 47.
Watson, G. N.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.
Whittaker, E. T.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.
Wolf, E.

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).
Other

G. Parrent, Opt. Acta 6, 285 (1959). See also: M. Beran and G. Parrent, Theory of Partial Coherence (PrenticeHall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).

The author prefers the use of realvalued functions, although the subsequent analysis can also be effectuated using complexvalued functions.

The braces { } denote ensemble properties.

M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.

Actually there is a third component termed the singular continuous component; this component is continuous but is without a spectral density at every point. Such functions do not have any physical significance and are always omitted.

H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).

This theorem was first stated by Kampe de Feriet (in an almost completely unknown paper) for the case of an autocorrelation function; the extension to crosscorrelation functions presented here is a straightforward extension of his proof. See: M. J. Kampe de Feriet, Ann. Soc. Sci. Bruxelles, Ser. 1 59, 145 (1939).

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.
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