Abstract

Spheroidal functions appearing in many other contexts are used here to analyze the photoelectron-counting statistics of a polarized thermal light beam of rectangular spectral profile.

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  1. R. Hanbury Brown and R. Q. Twiss, Proc. R. Soc. A 242, 300 (1957); Proc. R. Soc. A 243, 29 (1957); J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North—Holland, Amsterdam, 1967), p. 211; J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), Ch. 2; C. L. Mehta, in Progress in Optics, VIII, edited by E. Wolf (North—Holland, Amsterdam, 1971), p. 374.
  2. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
  3. G. Bédard, Phys. Rev. 151, 1038 (1966); G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967); E. Jakeman and E. R. Pike, J. Phys. A 1, 128 (1968); A. K. Jaiswal and C. L. Mehta, Phys. Rev. 186, 1355 (1969).
  4. Note that as far as the counting statistics are concerned, only the shape of the spectrum g (ν) is important and not its relative position. We therefore take the center of the spectrum to be situated at the origin.
  5. S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944); Bell Syst. Tech. J. 24, 46 (1945).
  6. See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), Vol. 1, § 4.8. We are thankful to J. des Cloizeaux for indicating this method.
  7. L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroīdales (Gauthier—Villars, Paris, 1959), Tome III, p. 250, formule (255).
  8. D. Slepian, J. Math. Phys. 44, 99 (1965); see this article for other references; M. Gaudin, Nucl. Phys. 25, 447 (1961); J. des Cloizeaux and M. L. Mehta, J. Math. Phys. 13, 1745 (1972).
  9. J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spheroidfunktionen (Springer, Berlin, 1954); C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).
  10. J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).
  11. D. Slepian and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965); M. L. Mehta and J. des Cloizeaux, Indian J. Pure Appl. Math 3, 329 (1972).
  12. C. L. Mehta, Ref. 1, Appendix B.
  13. We are thankful to J. Raynal for help on this point.

Bédard, G.

G. Bédard, Phys. Rev. 151, 1038 (1966); G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967); E. Jakeman and E. R. Pike, J. Phys. A 1, 128 (1968); A. K. Jaiswal and C. L. Mehta, Phys. Rev. 186, 1355 (1969).

Brown, R. Hanbury

R. Hanbury Brown and R. Q. Twiss, Proc. R. Soc. A 242, 300 (1957); Proc. R. Soc. A 243, 29 (1957); J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North—Holland, Amsterdam, 1967), p. 211; J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), Ch. 2; C. L. Mehta, in Progress in Optics, VIII, edited by E. Wolf (North—Holland, Amsterdam, 1971), p. 374.

Chu, L. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

Corbatto, P. J.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

Feshbach, H.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), Vol. 1, § 4.8. We are thankful to J. des Cloizeaux for indicating this method.

Little, J. D. C.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

Mehta, C. L.

C. L. Mehta, Ref. 1, Appendix B.

Meixner, J.

J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spheroidfunktionen (Springer, Berlin, 1954); C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).

Morse, P. M.

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), Vol. 1, § 4.8. We are thankful to J. des Cloizeaux for indicating this method.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

Raynal, J.

We are thankful to J. Raynal for help on this point.

Rice, S. O.

S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944); Bell Syst. Tech. J. 24, 46 (1945).

Robin, L.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroīdales (Gauthier—Villars, Paris, 1959), Tome III, p. 250, formule (255).

Schafke, F. W.

J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spheroidfunktionen (Springer, Berlin, 1954); C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).

Slepian, D.

D. Slepian and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965); M. L. Mehta and J. des Cloizeaux, Indian J. Pure Appl. Math 3, 329 (1972).

D. Slepian, J. Math. Phys. 44, 99 (1965); see this article for other references; M. Gaudin, Nucl. Phys. 25, 447 (1961); J. des Cloizeaux and M. L. Mehta, J. Math. Phys. 13, 1745 (1972).

Sonnenblick, E.

D. Slepian and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965); M. L. Mehta and J. des Cloizeaux, Indian J. Pure Appl. Math 3, 329 (1972).

Stratton, J. A.

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Proc. R. Soc. A 242, 300 (1957); Proc. R. Soc. A 243, 29 (1957); J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North—Holland, Amsterdam, 1967), p. 211; J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), Ch. 2; C. L. Mehta, in Progress in Optics, VIII, edited by E. Wolf (North—Holland, Amsterdam, 1971), p. 374.

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

Other (13)

R. Hanbury Brown and R. Q. Twiss, Proc. R. Soc. A 242, 300 (1957); Proc. R. Soc. A 243, 29 (1957); J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North—Holland, Amsterdam, 1967), p. 211; J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968), Ch. 2; C. L. Mehta, in Progress in Optics, VIII, edited by E. Wolf (North—Holland, Amsterdam, 1971), p. 374.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).

G. Bédard, Phys. Rev. 151, 1038 (1966); G. Bédard, J. C. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967); E. Jakeman and E. R. Pike, J. Phys. A 1, 128 (1968); A. K. Jaiswal and C. L. Mehta, Phys. Rev. 186, 1355 (1969).

Note that as far as the counting statistics are concerned, only the shape of the spectrum g (ν) is important and not its relative position. We therefore take the center of the spectrum to be situated at the origin.

S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944); Bell Syst. Tech. J. 24, 46 (1945).

See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw—Hill, New York, 1953), Vol. 1, § 4.8. We are thankful to J. des Cloizeaux for indicating this method.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroīdales (Gauthier—Villars, Paris, 1959), Tome III, p. 250, formule (255).

D. Slepian, J. Math. Phys. 44, 99 (1965); see this article for other references; M. Gaudin, Nucl. Phys. 25, 447 (1961); J. des Cloizeaux and M. L. Mehta, J. Math. Phys. 13, 1745 (1972).

J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spheroidfunktionen (Springer, Berlin, 1954); C. Flammer, Spheroidal Wave Functions (Stanford U.P., Stanford, Calif., 1957).

J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and P. J. Corbatto, Spheroidal Wave Functions (MIT Press, Cambridge, Mass., 1956); B. J. King and A. L. Van Buren, A Fortran Computer Program for Calculating the Prolate and Oblate Spheroidal Functions of the First Kind and Their First and Second Derivatives, NRL Report No. 7161 (U.S. Government Printing Office, Washington, D.C., 1970).

D. Slepian and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965); M. L. Mehta and J. des Cloizeaux, Indian J. Pure Appl. Math 3, 329 (1972).

C. L. Mehta, Ref. 1, Appendix B.

We are thankful to J. Raynal for help on this point.

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