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.

© 1973 Optical Society of America

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References

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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.
    [Crossref]
  2. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  3. G. Bedard, Phys. Rev. 151, 1038 (1966); G. Bedard, 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).
    [Crossref]
  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).
    [Crossref]
  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 Sphéroĩdales (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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  12. C. L. Mehta, Ref. 1, Appendix B.
  13. We are thankful to J. Raynal for help on this point.

1966 (1)

G. Bedard, Phys. Rev. 151, 1038 (1966); G. Bedard, 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).
[Crossref]

1965 (3)

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

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).
[Crossref]

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).
[Crossref]

1957 (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.
[Crossref]

1944 (1)

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

Bedard, G.

G. Bedard, Phys. Rev. 151, 1038 (1966); G. Bedard, 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).
[Crossref]

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.

Hanbury Brown, R.

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.
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).

Rice, S. O.

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

Robin, L.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroĩdales (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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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.
[Crossref]

Wolf, E.

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

Bell Syst. Tech. J. (2)

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

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).
[Crossref]

J. Math. Phys. (1)

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).
[Crossref]

Phys. Rev. (1)

G. Bedard, Phys. Rev. 151, 1038 (1966); G. Bedard, 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).
[Crossref]

Proc. R. Soc. A (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.
[Crossref]

Rev. Mod. Phys. (1)

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

Other (7)

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.

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

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).

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 Sphéroĩdales (Gauthier–Villars, Paris, 1959), Tome III, p. 250, formule (255).

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

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

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Figures (9)

Fig. 1
Fig. 1

E(n,T), the probability that a randomly chosen time interval of duration T will contain exactly n counts, as a function of 1 2 π σ T for various values of n and for αg(0) = 1. This figure as well as Figs. 2 and 5 refer to completely polarized thermal light having a rectangular spectral profile.

Fig. 2
Fig. 2

E(n,T) for αg(0) = 5 and for various values of n. The spectral profile is rectangular.

Fig. 3
Fig. 3

E(n,T) for αg(0) = 9 and for various values of n. The spectral profile is rectangular.

Fig. 4
Fig. 4

Characteristic function C(ih) as a function of 1 2 π σ T for hg(0) = 5, 9, 20, 36, 50, and 90. The graphs for Rice’s approximation CR(ih) [Eq. (21)] are practically indistinguishable from the exact ones [Eq. (15)] except in the range 0.5 < 1 2 π σ T < 2, where they are denoted by dotted lines. The spectral profile is rectangular.

Fig. 5
Fig. 5

Probability density P(W/〈W〉) for various values of σT. As noted in the text, these graphs do not depend on the spectral irradiance g(0), because W is measured in units of its average value 〈W〉. The spectral profile is rectangular.

Fig. 6
Fig. 6

Same as Fig. 1, except that the spectral profile is lorentzian. The constant A in Eq. (2) is chosen to be 2g(0)σ2/π. Thus, the width σ and the total area Γ(0) are the same in the two cases.

Fig. 7
Fig. 7

Same as Fig. 2, except that the spectral profile is lorentzian.

Fig. 8
Fig. 8

Same as Fig. 3, except that the spectral profile is lorentzian.

Fig. 9
Fig. 9

Same as Fig. 4, except that the spectral profile is lorentzian.

Equations (38)

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Γ ( τ ) = V * ( t ) V ( t + τ ) ,
g ( ν ) = A { 4 ν 2 + σ 2 } 1 .
g ( ν ) = g ( 0 ) = π A 2 σ 2 , | ν | 1 2 σ g ( ν ) = 0 , | ν | > 1 2 σ
E ( n , T ) = 0 ( 1 / n ! ) ( α W ) n e α W P ( W ) d W ,
W = t t + T I ( t ) d t ,
I ( t ) = V * ( t ) V ( t ) .
Γ ( τ ) = V * ( t ) V ( t + τ ) ,
Γ ( τ ) = + g ( ν ) e 2 π i ν τ d ν ,
Γ i j ( τ ) = V i * ( t ) V j ( t + τ ) , { i = x , y j = x , y
λ k φ k ( t 1 ) = T / 2 + T / 2 Γ ( t 1 t 2 ) φ k ( t 2 ) d t 2 .
T / 2 + T / 2 φ j * ( t ) φ k ( t ) d t = δ j k .
V ( t ) = k a k φ k ( t )
a j * a k = λ j δ j k
P ( { a k } ) = ( π λ k ) 1 exp { k | a k | 2 / λ k } .
W = T / 2 + T / 2 V * ( t ) V ( t ) d t = k | a k | 2
C ( h ) = e i h W = k ( 1 i h λ k ) 1 ;
P ( W ) = 1 2 π + e i h W C ( h ) d h
= j 1 λ j e W / λ j k ( j ) λ j ( λ j λ k ) 1 .
E ( 0 , T ) = k ( 1 + α λ k ) 1
E ( n , T ) = E ( 0 , T ) ( k ) j = 1 n α λ k j ( 1 + α λ k j ) 1 ,
E ( n , T ) = α n 2 π + C ( h ) ( α + i h ) n + 1 d h = ( i α ) n n ! [ d n d h n C ( h ) ] h = + i α .
n 2 n 2 = j α λ j ( 1 + α λ j ) .
Γ ( τ ) = g ( 0 ) π τ sin ( π σ τ ) .
W = k | a k | 2 = k λ k = Γ ( 0 ) T .
C R ( i h ) = ( 1 + h W / N ) N ,
N = T 2 { T / 2 + T / 2 | Γ ( t t ) / Γ ( 0 ) | 2 d t d t } 1 = ( λ k ) 2 / λ k 2 .
sin 2 z z 2 = n = 0 ( 1 ) n 2 2 n + 1 ( 2 n + 2 ) ! z 2 n ,
Γ 2 T / 2 + T / 2 | Γ ( t t ) / Γ ( 0 ) | 2 d t d t = 2 ( π σ T ) 2 0 π σ T ( π σ T z ) sin 2 z z 2 d z = n = 0 ( 1 ) n 2 2 n + 2 ( 2 n + 2 ) ! ( π σ T ) 2 n ( 2 n + 1 ) ( 2 n + 2 ) .
P ( W / W ) = exp { W / W } .
C ( h ) = e π σ T { cosh z + 1 2 sinh z ( z π σ T + π σ T z ) } 1 ,
z = π σ T ( 1 2 i h π g ( 0 ) ) 1 2 = π σ T ( 1 i h A σ 2 ) 1 2 .
1 + i x i + i j x i x j + i j k x i x j x k + = i ( 1 x i ) 1 ,
i x i + 2 i j x i x j + 3 i j k x i x j x k + = i ( 1 x i ) 1 j x j ( 1 x j ) 1 ,
i j x i x j + 3 i j k x i x j x k + 6 i j k l x i x j x k x l + = i ( 1 x i ) 1 j k x j x k ( 1 x j ) 1 ( 1 x k ) 1 .
i x i + 2 2 i j x i x j + 3 2 i j k x i x j x k + = i ( 1 x i ) 1 × { j k 2 x j x k ( 1 x j ) 1 ( 1 x k ) 1 + j x j ( 1 x i ) 1 } .
x j = α λ j / ( 1 + α λ j ) ,
n n n E ( n , T ) = j α λ j = α Γ ( 0 ) T
n 2 n n 2 E ( n , T ) = j k 2 α λ j α λ k + j α λ j = ( j α λ j ) 2 + j α λ j ( 1 + α λ j ) .