Abstract

The joint probability of finding m2 photons in detection interval 2 and m1 photons in detection interval 1 was previously determined by the author [ J. Mod. Opt. 34, 91 ( 1987)] in terms of a series expansion of the time-integrated intensity correlation function with coefficients given by a class of orthogonal polynomials in the discrete variables m2 and m1. This expansion is now used to obtain an expression for the average of two general functions υ(m2) and u(m1), 〈υ(m2)u(m1)〉. These functions are specialized to cover two important situations: singly and doubly clipped covariances for essentially arbitrary counting times. In the doubly clipped case, different clipping levels are permitted in the two channels in order to study the influence of clipping mismatch. Numerical calculations, in the form of graphs, are used to illustrate typical behavior.

© 1988 Optical Society of America

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References

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  1. J. H. Van Vleck, D. M. Middleton, “The spectrum of clipped noise,” Proc. IEEE 54, 2–19 (1966) [Originally Rep. 51 of the Radio Research Laboratory (Harvard University, Cambridge, Mass., July21, 1943)].
    [CrossRef]
  2. E. Jakeman, E. R. Pike, “Spectrum of clipped photon-counting fluctuations of Gaussian light,”J. Phys. A 2, 411–412 (1969).
    [CrossRef]
  3. R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
    [CrossRef] [PubMed]
  4. V. Degiorgio, J. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2049 (1971).
    [CrossRef]
  5. H. Kelly, “A comparison of the information gathering capacities of photon-correlation devices,” IEEE J. Quantum Electron. QE-7, 541–550 (1971).
    [CrossRef]
  6. S. H. Chen, P. Tartaglia, “Light scattering from N noninteracting particles,” Opt. Commun. 6, 119–124 (1972).
    [CrossRef]
  7. E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon counting fluctuations,”J. Phys. A 3, 201–215 (1970).
    [CrossRef]
  8. D. E. Koppel, “Analysis of Gaussian light by clipped photocount autocorrelation: the effect of finite sampling times and incomplete coherence,” J. Appl. Phys. 42, 3216–3225 (1971).
    [CrossRef]
  9. M. Singh, “Higher order photon counting statistics of Gaussian–Lorentz light,” Opt. Acta 31, 1293–1305 (1984).
    [CrossRef]
  10. M. Singh, “Singly-clipped correlation and the Bell polynomials,” Phys. Lett. A 121, 159–163 (1987).
    [CrossRef]
  11. J. Blake, R. Barakat, “Two-fold photoelectron counting statistics: the clipped correlation function,”J. Phys. A 6, 1196–1210 (1973).
    [CrossRef]
  12. R. Barakat, “Second order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
    [CrossRef]
  13. B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
    [CrossRef]
  14. R. Barakat, J. Blake, “Theory of photoelectron counting statistics: an essay,” Phys. Rep. 60, 226–340 (1980).
    [CrossRef]
  15. R. Barakat, “First order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 901–912 (1973).
    [CrossRef]
  16. E. Feldheim, “On a system of orthogonal polynomials associated with a distribution of Stieltjes type,”C. R. Dokl. Akad. Sci. URSS 31, 528–533 (1941).
  17. G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Chap. 3.
  18. R. Barakat, “Clipped correlation functions of aperture integrated laser speckle,” Appl. Opt. 25, 3885–3888 (1986).
    [CrossRef] [PubMed]

1987 (2)

R. Barakat, “Second order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

M. Singh, “Singly-clipped correlation and the Bell polynomials,” Phys. Lett. A 121, 159–163 (1987).
[CrossRef]

1986 (1)

1984 (1)

M. Singh, “Higher order photon counting statistics of Gaussian–Lorentz light,” Opt. Acta 31, 1293–1305 (1984).
[CrossRef]

1980 (1)

R. Barakat, J. Blake, “Theory of photoelectron counting statistics: an essay,” Phys. Rep. 60, 226–340 (1980).
[CrossRef]

1973 (2)

R. Barakat, “First order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 901–912 (1973).
[CrossRef]

J. Blake, R. Barakat, “Two-fold photoelectron counting statistics: the clipped correlation function,”J. Phys. A 6, 1196–1210 (1973).
[CrossRef]

1972 (1)

S. H. Chen, P. Tartaglia, “Light scattering from N noninteracting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

1971 (3)

D. E. Koppel, “Analysis of Gaussian light by clipped photocount autocorrelation: the effect of finite sampling times and incomplete coherence,” J. Appl. Phys. 42, 3216–3225 (1971).
[CrossRef]

V. Degiorgio, J. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2049 (1971).
[CrossRef]

H. Kelly, “A comparison of the information gathering capacities of photon-correlation devices,” IEEE J. Quantum Electron. QE-7, 541–550 (1971).
[CrossRef]

1970 (2)

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon counting fluctuations,”J. Phys. A 3, 201–215 (1970).
[CrossRef]

1969 (1)

E. Jakeman, E. R. Pike, “Spectrum of clipped photon-counting fluctuations of Gaussian light,”J. Phys. A 2, 411–412 (1969).
[CrossRef]

1966 (1)

J. H. Van Vleck, D. M. Middleton, “The spectrum of clipped noise,” Proc. IEEE 54, 2–19 (1966) [Originally Rep. 51 of the Radio Research Laboratory (Harvard University, Cambridge, Mass., July21, 1943)].
[CrossRef]

1941 (1)

E. Feldheim, “On a system of orthogonal polynomials associated with a distribution of Stieltjes type,”C. R. Dokl. Akad. Sci. URSS 31, 528–533 (1941).

Barakat, R.

R. Barakat, “Second order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

R. Barakat, “Clipped correlation functions of aperture integrated laser speckle,” Appl. Opt. 25, 3885–3888 (1986).
[CrossRef] [PubMed]

R. Barakat, J. Blake, “Theory of photoelectron counting statistics: an essay,” Phys. Rep. 60, 226–340 (1980).
[CrossRef]

J. Blake, R. Barakat, “Two-fold photoelectron counting statistics: the clipped correlation function,”J. Phys. A 6, 1196–1210 (1973).
[CrossRef]

R. Barakat, “First order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 901–912 (1973).
[CrossRef]

Blagrove, R.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Blake, J.

R. Barakat, J. Blake, “Theory of photoelectron counting statistics: an essay,” Phys. Rep. 60, 226–340 (1980).
[CrossRef]

J. Blake, R. Barakat, “Two-fold photoelectron counting statistics: the clipped correlation function,”J. Phys. A 6, 1196–1210 (1973).
[CrossRef]

Chen, S. H.

S. H. Chen, P. Tartaglia, “Light scattering from N noninteracting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

Degiorgio, V.

V. Degiorgio, J. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2049 (1971).
[CrossRef]

Feldheim, E.

E. Feldheim, “On a system of orthogonal polynomials associated with a distribution of Stieltjes type,”C. R. Dokl. Akad. Sci. URSS 31, 528–533 (1941).

Foord, R.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Jakeman, E.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon counting fluctuations,”J. Phys. A 3, 201–215 (1970).
[CrossRef]

E. Jakeman, E. R. Pike, “Spectrum of clipped photon-counting fluctuations of Gaussian light,”J. Phys. A 2, 411–412 (1969).
[CrossRef]

Kelly, H.

H. Kelly, “A comparison of the information gathering capacities of photon-correlation devices,” IEEE J. Quantum Electron. QE-7, 541–550 (1971).
[CrossRef]

Koppel, D. E.

D. E. Koppel, “Analysis of Gaussian light by clipped photocount autocorrelation: the effect of finite sampling times and incomplete coherence,” J. Appl. Phys. 42, 3216–3225 (1971).
[CrossRef]

Lastovka, J.

V. Degiorgio, J. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2049 (1971).
[CrossRef]

Middleton, D. M.

J. H. Van Vleck, D. M. Middleton, “The spectrum of clipped noise,” Proc. IEEE 54, 2–19 (1966) [Originally Rep. 51 of the Radio Research Laboratory (Harvard University, Cambridge, Mass., July21, 1943)].
[CrossRef]

Oliver, C.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Peacocke, A.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Pike, E. R.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

E. Jakeman, E. R. Pike, “Spectrum of clipped photon-counting fluctuations of Gaussian light,”J. Phys. A 2, 411–412 (1969).
[CrossRef]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
[CrossRef]

Singh, M.

M. Singh, “Singly-clipped correlation and the Bell polynomials,” Phys. Lett. A 121, 159–163 (1987).
[CrossRef]

M. Singh, “Higher order photon counting statistics of Gaussian–Lorentz light,” Opt. Acta 31, 1293–1305 (1984).
[CrossRef]

Szego, G.

G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Chap. 3.

Tartaglia, P.

S. H. Chen, P. Tartaglia, “Light scattering from N noninteracting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

Van Vleck, J. H.

J. H. Van Vleck, D. M. Middleton, “The spectrum of clipped noise,” Proc. IEEE 54, 2–19 (1966) [Originally Rep. 51 of the Radio Research Laboratory (Harvard University, Cambridge, Mass., July21, 1943)].
[CrossRef]

Wood, E.

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Appl. Opt. (1)

C. R. Dokl. Akad. Sci. URSS (1)

E. Feldheim, “On a system of orthogonal polynomials associated with a distribution of Stieltjes type,”C. R. Dokl. Akad. Sci. URSS 31, 528–533 (1941).

IEEE J. Quantum Electron. (1)

H. Kelly, “A comparison of the information gathering capacities of photon-correlation devices,” IEEE J. Quantum Electron. QE-7, 541–550 (1971).
[CrossRef]

J. Appl. Phys. (1)

D. E. Koppel, “Analysis of Gaussian light by clipped photocount autocorrelation: the effect of finite sampling times and incomplete coherence,” J. Appl. Phys. 42, 3216–3225 (1971).
[CrossRef]

J. Mod. Opt. (1)

R. Barakat, “Second order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

J. Phys. A (3)

J. Blake, R. Barakat, “Two-fold photoelectron counting statistics: the clipped correlation function,”J. Phys. A 6, 1196–1210 (1973).
[CrossRef]

E. Jakeman, “Theory of optical spectroscopy by digital autocorrelation of photon counting fluctuations,”J. Phys. A 3, 201–215 (1970).
[CrossRef]

E. Jakeman, E. R. Pike, “Spectrum of clipped photon-counting fluctuations of Gaussian light,”J. Phys. A 2, 411–412 (1969).
[CrossRef]

Nature (1)

R. Foord, E. Jakeman, C. Oliver, E. R. Pike, R. Blagrove, E. Wood, A. Peacocke, “Determination of diffusion coefficients of haemocyanin at low concentration by intensity fluctuation spectroscopy of scattered laser light,” Nature 227, 242–245 (1970).
[CrossRef] [PubMed]

Opt. Acta (2)

M. Singh, “Higher order photon counting statistics of Gaussian–Lorentz light,” Opt. Acta 31, 1293–1305 (1984).
[CrossRef]

R. Barakat, “First order probability densities of laser speckle patterns observed through finite-size scanning apertures,” Opt. Acta 20, 901–912 (1973).
[CrossRef]

Opt. Commun. (1)

S. H. Chen, P. Tartaglia, “Light scattering from N noninteracting particles,” Opt. Commun. 6, 119–124 (1972).
[CrossRef]

Phys. Lett. A (1)

M. Singh, “Singly-clipped correlation and the Bell polynomials,” Phys. Lett. A 121, 159–163 (1987).
[CrossRef]

Phys. Rep. (1)

R. Barakat, J. Blake, “Theory of photoelectron counting statistics: an essay,” Phys. Rep. 60, 226–340 (1980).
[CrossRef]

Phys. Rev. A (1)

V. Degiorgio, J. Lastovka, “Intensity-correlation spectroscopy,” Phys. Rev. A 4, 2033–2049 (1971).
[CrossRef]

Proc. IEEE (1)

J. H. Van Vleck, D. M. Middleton, “The spectrum of clipped noise,” Proc. IEEE 54, 2–19 (1966) [Originally Rep. 51 of the Radio Research Laboratory (Harvard University, Cambridge, Mass., July21, 1943)].
[CrossRef]

Other (2)

B. Saleh, Photoelectron Statistics (Springer-Verlag, New York, 1978).
[CrossRef]

G. Szego, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939), Chap. 3.

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Figures (5)

Fig. 1
Fig. 1

Normalized singly clipped covariance function for k = 1, μ = 0.1: —, α = 1; ·····, α = 2; −··, α = 3; −−, α = 5.

Fig. 2
Fig. 2

Normalized doubly clipped covariance for α = 1, k2 = k1 = 1: −−, μ = 0.1; —, μ = 1; −·−, μ = 2.

Fig. 3
Fig. 3

Normalized doubly clipped covariance function for α = 1, k2 = k1 = 2: −·−, μ = 0.1; —, μ = 1; — —; μ = 2.

Fig. 4
Fig. 4

Normalized doubly clipped covariance function for α = 1, μ = 2: —, k2 = k1 = 3; −−, k2 = 3, k1 = 2; ·····, k2 = 3, k1 = 4.

Fig. 5
Fig. 5

Normalized doubly clipped covariance function for α = 1, μ = 5: —, k2 = k1 = 6; −−, k2 = 6, k1 = 5; ·····, k2 = 6, k1 = 7.

Equations (51)

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Ω = 0 T E ( ) ( t ) E ( + ) ( t ) d t ,
0 T g ( t t ) ψ l ( t ) d t = T Ω m l ψ l ( t ) .
g ( t t ) = E ( t ) E * ( t ) | E ( t ) | 2 .
W ( Ω ) = l = 0 W l exp ( Ω / b l ) ,
W ( Ω ) = 1 Ω exp ( Ω / Ω ) ,
W ( Ω ) = 1 Γ ( α ) ( α Ω ) α Ω α 1 exp ( α Ω / Ω ) ,
α = Ω 2 Ω 2 Ω 2 Ω 2 var ( Ω ) .
P ( m ) = 0 W ( Ω ) ( Ω m e Ω m ! ) d Ω .
P ( m ) = P BE ( m ) m m ( 1 + m ) m + 1 ,
P ( m ) = P NB ( m ) ( m + α 1 α 1 ) ( m α ) m ( 1 + m α ) m + α ,
Ω j = T j E ( ) ( t ) E ( + ) ( t ) d t ( j = 1 , 2 ) .
T 1 = ( 0 , T ) , T 2 = ( τ , τ + T ) ,
( λ 1 T 1 + λ 2 T 2 ) g ( t t ) ψ l ( t ) d t = T Ω m l ψ l ( t ) ,
Q ( λ 2 , λ 1 ) = exp ( λ 2 Ω 2 λ 1 Ω 1 ) = 0 W ( Ω 2 , Ω 1 ) exp ( λ 2 Ω 2 λ 1 Ω 1 ) d Ω 2 d Ω 1 .
P ( m 2 , m 1 ) = 0 W ( Ω 2 , Ω 1 ) Ω 2 m 2 exp ( Ω 2 ) m 2 ! × Ω 1 m 1 exp ( Ω 1 ) m 1 ! d Ω 2 d Ω 1 .
W ( Ω 2 , Ω 1 ) = W g ( Ω 2 ) W g ( Ω 1 ) n = 0 [ r Ω ( τ ) ] n ( n + α 1 n ) × L n ( α 1 ) ( α Ω 2 μ ) L n ( α 1 ) ( α Ω 1 μ ) ,
r Ω ( τ ) = Ω 2 Ω 1 Ω 2 Ω 2 Ω 2 , 0 r Ω ( τ ) 1.
r Ω ( τ ) = 1 T 2 0 T r ( t t + τ ) d t d t .
r ( τ ) = e 2 τ ,
r Ω ( τ ) = e 2 τ ( sinh T T ) 2 = 1 4 T 2 { exp [ 2 ( T + τ ) ] + exp [ 2 ( T τ ) ] 2 e 2 T + 4 ( T τ ) } ,
W ( Ω 2 , Ω 1 ) = W g ( Ω 2 ) δ ( Ω 2 Ω 1 ) .
P ( m 2 , m 1 ) = P NB ( m 2 ) P NB ( m 1 ) × n = 0 [ r Ω ( τ ) ] n ( n + α 1 n ) Q n ( α ) ( m 2 ) Q n ( α ) ( m 1 ) ,
P ( m 2 , m 1 ) = Γ ( m 2 + m 1 + α ) m 2 ! m 1 ! Γ ( α ) ( μ α ) m 2 + m 1 ( 1 + 2 μ α ) m 2 + m 1 + α ,
P ( m 2 , m 1 ) = P p ( m 2 ) P p ( m 1 ) ,
P p ( m ) = μ m e u m ! .
Q 0 ( α ) ( m ) = 1 , Q 0 ( α ) ( m ) = l = 0 n ( m + α 1 + l m + α 1 ) ( n + α 1 n l ) ( 1 ) l ( 1 + μ α ) l .
Q n ( α ) ( m ) = ( n + α 1 n ) F 2 [ n , m + α , α , ( 1 1 + μ α ) ] .
m x , n n , α α + 1 , ( 1 + μ / α ) 1 1 e λ .
m = 0 Q j ( α ) ( m ) Q k ( α ) ( m ) P NB ( m ) = ( j + α 1 j ) ( μ α 1 + μ α ) j δ k j ,
υ ( m 2 ) u ( m 1 ) = n = 0 r Ω n ( n + α 1 n ) × υ ( m 2 ) Q n ( α ) ( m 2 ) u ( m 1 ) Q n ( α ) ( m 1 ) ,
w ( m ) Q n ( α ) ( m ) m = 0 w ( m ) Ω n ( α ) ( m ) P NB ( m ) .
υ ( m 2 ) u ( m 1 ) = υ ( m 2 ) u ( m 1 ) .
υ ( m 2 ) u ( m 1 ) = m 2 = 0 m 1 = 0 υ ( m 2 ) u ( m 1 ) Γ ( m 2 + m 1 + α ) m 2 ! m 1 ! Γ ( α ) × ( μ α ) m 2 + m 1 ( 1 + 2 μ α ) m 2 + m 1 + α .
R k ( r Ω | α ) g k ( m 2 ) m 1 ,
g k ( m ) = 1 if m k = 0 if m < k
R k ( r Ω | α ) = n = 0 r Ω n ( n + α 1 n ) g k ( m 2 ) Q n ( α ) ( m 2 ) × m 1 Q n ( α ) ( m 1 ) .
g k ( m 2 ) Q n ( α ) ( m 2 ) = m 2 = k Q 0 ( α ) ( m 2 ) Q n ( α ) ( m 2 ) P NB ( m 2 ) ,
m 2 = k Q 0 ( α ) ( m 2 ) Q n ( α ) ( m 2 ) P NB ( m 2 ) = δ n 0 m 2 = 0 k 1 Q n ( α ) × ( m 2 ) P NB ( m 2 )
g k ( m 2 ) Q n ( α ) ( m 2 ) = δ n 0 m 2 = 0 k 1 Q n ( α ) ( m 2 ) P NB ( m 2 ) ,
m 1 Q 0 ( α ) ( m 1 ) = m 1 μ .
m 1 = μ Q 0 ( α ) ( m 1 ) ( 1 + μ α ) Q 1 ( α ) ( m 1 ) .
m 1 Q 1 ( α ) ( m 1 ) = μ ,
R k ( r Ω | α ) = μ [ 1 m 2 = 0 k 1 P NB ( m 2 ) ] + ( μ α ) r Ω m 2 = 0 k 1 Q 1 ( α ) ( m 2 ) P NB ( m 2 ) .
R k ( r | 1 ) = μ ( μ 1 + μ ) k [ 1 + ( k 1 + μ ) r ] .
R 1 ( r Ω | α ) = μ ( 1 + μ α ) α { [ ( 1 + μ α ) α 1 ] + ( μ / α 1 + μ / α ) r Ω } .
R k 2 k 1 ( r Ω | α ) g k 2 ( m 2 ) g k 1 ( m 1 ) ,
R k 2 k 1 ( r Ω | α ) = n = 0 r Ω n ( n + α 1 n ) g k 2 ( m 2 ) Q n ( α ) ( m 2 ) × g k 1 ( m 1 ) Q m ( α ) ( m 1 ) .
R k 2 k 1 ( r Ω | α ) = n = 0 r Ω n ( n + α 1 n ) [ δ n 0 m 2 = 0 k 2 1 Q n ( α ) ( m 2 ) P NB ( m 2 ) ] × [ δ n 0 m 1 = 0 k 1 1 Q n ( α ) ( m 1 ) P NB ( m 1 ) ] ,
R 00 ( r Ω | α ) = 1.
P BE ( 0 ) = 1 ( 1 + μ ) , Q n ( 1 ) ( 0 ) = 2 F 1 [ n , 1,1 , ( 1 + μ ) 1 ] = ( μ 1 + μ ) n .
R 11 ( r | 1 ) = 1 2 ( 1 + μ ) + 1 ( 1 + μ ) 2 n = 0 ( r 1 / 2 μ 1 + μ ) 2 n = ( μ 1 μ + 1 ) + 1 [ 1 + 2 μ + μ 2 ( 1 r ) ] .

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