Abstract

We introduce an analytical model of the influence of paralyzable dead time on registered photon-correlation functions. Distortions of correlation functions in the case of the Poisson point process, the doubly stochastic Poisson point process, and the pairwise point process are calculated. The model permits the analysis of detection systems with constant and random dead times. The results of the analytical model are tested by computer simulation.

© 1995 Optical Society of America

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References

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  1. M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for short-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771 (1981).
    [CrossRef]
  2. B. E. A. Saleh and M. C. Teich, "Multiplication refractoriness in cat's retinal-ganglion-cell discharge at low light levels," Biol. Cybern. 52, 101 (1985).
    [CrossRef]
  3. K. Schatzel, "Dead time correction of photon correlation functions," Appl. Phys. B 41, 95 (1986).
    [CrossRef]
  4. K. Schatzel, "Correction of detection-system dead-time effects on photon correlation functions," J. Opt. Soc. Am. B 6, 937 (1989).
    [CrossRef]
  5. D. J. Cho and G. M. Morris, "Generation and statistical properties of optical dead-time effects," J. Mod. Opt. 35, 667 (1988).
    [CrossRef]
  6. D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
    [CrossRef]
  7. D. J. Daley and D. Ver-Jones, An Introduction to The Theory of Point Processes (Springer-Verlag, New York, 1988).
  8. V. V. Apanasovich, A. A. Kolyada, and A. F. Chernyavski, The Statistical Analysis of Series of Random Events in Physical Experiment (University Press, Minsk, Belarus, 1988).
  9. S. K. Srinivasan, Stochastic Point Processes and Their Application (Griffin, London, 1974), p. 19.
  10. D. R. Cox and V. Isham, Point Processes (Chapman & Hall, London, 1980), p. 71.
  11. B. Saleh, Photoelectron Statistics (Springer, New York, 1978), p. 141.
  12. I. A. Bolshakov and V. S. Rakoshits, Applied Theory of Stochastic Point Processes (Sovradio, Moscow, 1978).
  13. H. Z. Cummins and E. R. Pike, Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).
  14. E. O. Schulz-DuBois, Photon Correlation in Fluid Mechanics (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  15. V. V. Apanasovich and S. V. Paltsev, "Modeling and comparative analysis of efficiency of correlators of series of signals," Electron. Modeling 1, 84 (1994).

1994

V. V. Apanasovich and S. V. Paltsev, "Modeling and comparative analysis of efficiency of correlators of series of signals," Electron. Modeling 1, 84 (1994).

1989

1988

D. J. Cho and G. M. Morris, "Generation and statistical properties of optical dead-time effects," J. Mod. Opt. 35, 667 (1988).
[CrossRef]

1986

K. Schatzel, "Dead time correction of photon correlation functions," Appl. Phys. B 41, 95 (1986).
[CrossRef]

1985

B. E. A. Saleh and M. C. Teich, "Multiplication refractoriness in cat's retinal-ganglion-cell discharge at low light levels," Biol. Cybern. 52, 101 (1985).
[CrossRef]

1981

Apanasovich, V. V.

V. V. Apanasovich and S. V. Paltsev, "Modeling and comparative analysis of efficiency of correlators of series of signals," Electron. Modeling 1, 84 (1994).

V. V. Apanasovich, A. A. Kolyada, and A. F. Chernyavski, The Statistical Analysis of Series of Random Events in Physical Experiment (University Press, Minsk, Belarus, 1988).

Bolshakov, I. A.

I. A. Bolshakov and V. S. Rakoshits, Applied Theory of Stochastic Point Processes (Sovradio, Moscow, 1978).

Chernyavski, A. F.

V. V. Apanasovich, A. A. Kolyada, and A. F. Chernyavski, The Statistical Analysis of Series of Random Events in Physical Experiment (University Press, Minsk, Belarus, 1988).

Cho, D. J.

D. J. Cho and G. M. Morris, "Generation and statistical properties of optical dead-time effects," J. Mod. Opt. 35, 667 (1988).
[CrossRef]

Cox, D. R.

D. R. Cox and V. Isham, Point Processes (Chapman & Hall, London, 1980), p. 71.

Cummins, H. Z.

H. Z. Cummins and E. R. Pike, Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).

Daley, D. J.

D. J. Daley and D. Ver-Jones, An Introduction to The Theory of Point Processes (Springer-Verlag, New York, 1988).

Isham, V.

D. R. Cox and V. Isham, Point Processes (Chapman & Hall, London, 1980), p. 71.

Kolyada, A. A.

V. V. Apanasovich, A. A. Kolyada, and A. F. Chernyavski, The Statistical Analysis of Series of Random Events in Physical Experiment (University Press, Minsk, Belarus, 1988).

Miller, M. I.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

Morris, G. M.

D. J. Cho and G. M. Morris, "Generation and statistical properties of optical dead-time effects," J. Mod. Opt. 35, 667 (1988).
[CrossRef]

Paltsev, S. V.

V. V. Apanasovich and S. V. Paltsev, "Modeling and comparative analysis of efficiency of correlators of series of signals," Electron. Modeling 1, 84 (1994).

Pike, E. R.

H. Z. Cummins and E. R. Pike, Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).

Rakoshits, V. S.

I. A. Bolshakov and V. S. Rakoshits, Applied Theory of Stochastic Point Processes (Sovradio, Moscow, 1978).

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, New York, 1978), p. 141.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, "Multiplication refractoriness in cat's retinal-ganglion-cell discharge at low light levels," Biol. Cybern. 52, 101 (1985).
[CrossRef]

M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for short-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771 (1981).
[CrossRef]

Schatzel, K.

Schulz-DuBois, E. O.

E. O. Schulz-DuBois, Photon Correlation in Fluid Mechanics (Springer-Verlag, Berlin, 1983).
[CrossRef]

Snyder, D. L.

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

Srinivasan, S. K.

S. K. Srinivasan, Stochastic Point Processes and Their Application (Griffin, London, 1974), p. 19.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, "Multiplication refractoriness in cat's retinal-ganglion-cell discharge at low light levels," Biol. Cybern. 52, 101 (1985).
[CrossRef]

M. C. Teich and B. E. A. Saleh, "Interevent-time statistics for short-noise-driven self-exciting point processes in photon detection," J. Opt. Soc. Am. 71, 771 (1981).
[CrossRef]

Ver-Jones, D.

D. J. Daley and D. Ver-Jones, An Introduction to The Theory of Point Processes (Springer-Verlag, New York, 1988).

Appl. Phys. B

K. Schatzel, "Dead time correction of photon correlation functions," Appl. Phys. B 41, 95 (1986).
[CrossRef]

Biol. Cybern.

B. E. A. Saleh and M. C. Teich, "Multiplication refractoriness in cat's retinal-ganglion-cell discharge at low light levels," Biol. Cybern. 52, 101 (1985).
[CrossRef]

Electron. Modeling

V. V. Apanasovich and S. V. Paltsev, "Modeling and comparative analysis of efficiency of correlators of series of signals," Electron. Modeling 1, 84 (1994).

J. Mod. Opt.

D. J. Cho and G. M. Morris, "Generation and statistical properties of optical dead-time effects," J. Mod. Opt. 35, 667 (1988).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Other

D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

D. J. Daley and D. Ver-Jones, An Introduction to The Theory of Point Processes (Springer-Verlag, New York, 1988).

V. V. Apanasovich, A. A. Kolyada, and A. F. Chernyavski, The Statistical Analysis of Series of Random Events in Physical Experiment (University Press, Minsk, Belarus, 1988).

S. K. Srinivasan, Stochastic Point Processes and Their Application (Griffin, London, 1974), p. 19.

D. R. Cox and V. Isham, Point Processes (Chapman & Hall, London, 1980), p. 71.

B. Saleh, Photoelectron Statistics (Springer, New York, 1978), p. 141.

I. A. Bolshakov and V. S. Rakoshits, Applied Theory of Stochastic Point Processes (Sovradio, Moscow, 1978).

H. Z. Cummins and E. R. Pike, Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1974).

E. O. Schulz-DuBois, Photon Correlation in Fluid Mechanics (Springer-Verlag, Berlin, 1983).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Correlation function f(i) = f2(τ) distorted by a detection system with constant paralyzable dead time Tm. The results of the simulation experiments are marked by asterisks. The calculated data are represented by solid curves. The width of channel is h = 0.02, f1(t) = 4, g2(τ) = 6 exp(−4τ), f2(τ) = 16 + 6 exp(−4τ), and τ = ih.

Fig. 2
Fig. 2

Correlation function f(i) = f2(τ) distorted by a detection system with random paralyzable dead time. The distribution function B(τ) = 1 − exp(−). The results of the simulation experiments are marked by asterisks. The calculated data are represented by solid curves. The width of channel h = 0.02, f1(t) = 4, g2(τ) = 6 exp(−4τ), f2(τ) = 16 + 6 exp(−4τ), and τ = ih.

Equations (30)

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L [ u , G ] = j = 1 v [ 1 + u ( ζ j ) ] ζ , v ,
L [ u ; G ] = k = 0 1 k ! G π k ( t 1 , , t k ; G ) × j = 1 k [ 1 + u ( t j ) ] d t 1 d t k .
δ L [ u ; G ] δ u ( t 1 ) = { δ δ λ L [ u ( · ) + λ δ ( · - t 1 ) ; G ] } | λ = 0 ,
δ L i [ u ; G ] δ u ( t 1 ) δ u ( t i ) = δ δ u ( t i ) { δ L i - 1 [ u ; G ] u ( t 1 ) δ u ( t i - 1 ) } ,
δ L i [ u ; G ] δ u ( t 1 ) δ u ( t i ) = k = i k ( k - 1 ) ( k - i + 1 ) k ! × G π k ( t 1 , , t i , t i + 1 , , t k ; G ) × j = i + 1 k [ 1 + u ( t j ) ] d t i + 1 d t k .
π i ( t 1 , , t i ) = δ i L [ u ; G ] δ u ( t 1 ) δ u ( t i ) | u ( · ) = - 1 ,             i = 1 , 2 , .
f i ( t 1 , , t i ) = δ i L [ u ; G ] δ u ( t 1 ) δ u ( t i ) | u ( · ) = 0 ,             i = 1 , 2 , ,
L [ v ; G ; u ; G ; w ; X ] = k = 0 1 k ! G k X k π k ( τ 1 , , τ k ; G ) j = 1 k [ 1 + v ( τ j ) ] [ 1 + w ( x j ) ] × L 2 k C [ u ; G τ 1 , , τ k , x 1 , , x k ] d B ( x 1 - τ 1 ) d B ( x k - τ k ) d τ 1 d τ k ,
G = m = 1 k ( τ m , x m ] .
L 2 k C [ u ; G τ 1 , , τ k , x 1 , , x k ] = j = 1 k [ 1 + u ( τ j ) ] u ( τ ) = 0 , τ G .
L C [ u ; G ] = k = 0 1 k ! G k X k π k ( τ 1 , , τ k ; G ) × j = 1 k { 1 + u ( τ j ) m = 1 m j k [ θ ( τ m - τ j ) + θ ( τ j - x m ) ] } d B ( x 1 - τ 1 ) d B ( x k - τ k ) d τ 1 d τ k , θ ( x ) = { 0 x < 0 1 x 0 .
δ L C [ u ; G ] δ u ( t 1 ) = k = 1 k k ! G k - 1 X k π k ( τ 1 , , τ k - 1 , t 1 ; G ) × j = 1 k - 1 { 1 + u ( τ j ) m = 1 m j k - 1 [ θ ( τ m - τ j ) + θ ( τ j - x m ) ] [ θ ( t 1 - τ j ) + θ ( τ j - x k ) ] } × m = 1 k - 1 [ θ ( τ m - t ) + θ ( t 1 - x m ) ] d B ( x 1 - τ 1 ) d B ( x k - 1 - τ k - 1 ) d B ( x k - t 1 ) d τ 1 d τ k - 1 .
f 1 C ( t 1 ) = k = 1 k k ! G k - 1 π k ( τ 1 , , τ k - 1 , t 1 ; G ) × X k m = 1 k - 1 [ θ ( τ m - t 1 ) + θ ( t 1 - x m ) ] d B ( x 1 - τ 1 ) d B ( x k - 1 - τ k - 1 ) d B ( x k - t 1 ) d τ 1 d τ k - 1 = k = 1 k k ! G k - 1 π k ( τ 1 , , τ k - 1 , t 1 ; G ) m = 1 k - 1 [ θ ( τ m - t 1 ) + B ( t 1 - τ m ) ] d τ 1 d τ k - 1 .
f 1 C ( t 1 ) = δ L [ v ; G ] δ v ( t 1 ) | v ( · ) = θ ( · - t 1 ) + B ( t 1 - · ) - 1 .
δ 2 L C [ u ; G ] δ u ( t 1 ) δ u ( t 2 ) = k = 2 k ( k - 1 ) k ! G k - 2 X k π k ( τ 1 , , τ k - 2 , t 1 , t 2 ; G ) × j = 1 k - 2 { 1 + u ( τ j ) m = 1 m j k - 2 [ θ ( τ m - τ j ) + θ ( τ j - x m ) ] [ θ ( t 1 - τ j ) + θ ( τ j - x k - 1 ) ] [ θ ( t 2 - τ j ) + θ ( τ j - x k ) ] } × m = 1 k - 2 [ θ ( τ m - t 1 ) + θ ( t 1 - x m ) ] [ θ ( t 2 - t 1 ) + θ ( t 1 - x k ) ] m = 1 k - 2 [ θ ( τ m - t 2 ) + θ ( t 2 - x m ) ] × [ θ ( t 1 - t 2 ) + θ ( t 2 - x k - 1 ) ] d B ( x 1 - τ 1 ) d B ( x k - 2 - τ k - 2 ) d B ( x k - 1 - t 1 ) d B ( x k - t 2 ) d τ 1 d τ k - 2 .
f 2 C ( t 1 , t 2 ) = δ 2 L C [ u ; G ] δ u ( t 1 ) δ u ( t 2 ) | u ( · ) = 0 = k = 2 k ( k - 1 ) k ! G k - 2 π k ( τ 1 , , τ k - 2 , t 1 , t 2 ; G ) × m = 1 k - 2 { X [ θ ( τ m - t 1 ) + θ ( t 1 - x m ) ] × [ θ ( τ m - t 2 ) + θ ( t 2 - x m ) ] d B ( x m - τ m ) } × [ θ ( t 2 - t 1 ) + B ( t 1 - t 2 ) ] × [ θ ( t 1 - t 2 ) + B ( t 2 - t 1 ) ] d τ 1 d τ k - 2 .
f 2 C ( t 1 , t 2 ) = B ( t 2 - t 1 ) k = 2 k ( k - 1 ) k ! × G k - 2 π k ( τ 1 , , τ k - 2 , t 1 , t 2 ; G ) × m = 1 k - 2 [ θ ( τ m - t 2 ) + θ ( τ m - t 1 ) B ( t 2 - τ m ) + B ( t 1 - x m ) ] d τ 1 d τ k - 2 .
f 2 C ( t 1 , t 2 ) = B ( t 2 - t 1 ) δ 2 L [ v ; G ] δ v ( t 1 ) δ v ( t 2 ) | u ( · ) = θ ( · - t 2 ) + θ ( · - t 1 ) B ( t 2 - · ) + B ( t 1 - · ) - 1.
L [ v , G ] = exp G ξ ( τ ) v ( τ ) d τ ξ ,
δ 2 L [ v ; G ] δ v ( t 1 ) δ v ( t 2 ) = ξ ( t 1 ) ξ ( t 2 ) exp G ξ ( τ ) v ( τ ) d τ ξ .
f 2 C ( t 1 , t 2 ) = B ( t 2 - t 1 ) ξ ( t 1 ) ξ ( t 2 ) exp G ξ ( τ ) [ B ( t 1 - τ ) + B ( t 2 - τ ) θ ( τ - t 1 ) + θ ( τ - t 2 ) - 1 ] d τ ξ .
f 2 C ( τ ) = B ( τ ) ξ ( 0 ) ξ ( τ ) exp { - - 0 ξ ( t ) [ 1 - B ( - t ) ] d t - 0 τ ξ ( t ) [ 1 - B ( τ - t ) ] d t } ξ .
f 2 C ( t 1 , t 2 ) = f 1 ( t 1 ) f 1 ( t 2 ) B ( t 2 - t 1 ) × exp { - T 1 t 1 f 1 ( τ ) [ 1 - B ( t 1 - τ ) ] d t } × exp { - t 1 t 2 f 1 ( τ ) [ 1 - B ( t 2 - τ ) ] d t } .
L [ v , G ] = exp [ G f 1 ( τ ) v ( τ ) d τ + 1 2 G 2 g 2 ( τ 1 , τ 2 ) v ( τ 1 ) v ( τ 2 ) d τ 1 d τ 2 ] ,
δ 2 L [ v ; G ] δ v ( t 1 ) δ v ( t 2 ) = exp [ G f 1 ( τ ) v ( τ ) d τ + 1 2 G 2 g 2 ( τ 1 , τ 2 ) v ( τ 1 ) v ( τ 2 ) d τ 1 d τ 2 ] × { [ f 1 ( t 1 ) + G g 2 ( t 1 , τ ) v ( τ ) d τ ] × [ f 1 ( t 2 ) + G g 2 ( t 2 , τ ) v ( τ ) d τ ] + g 2 ( t 1 , t 2 ) } .
f 2 C ( t 1 , t 2 ) = B ( t 2 - t 1 ) exp { T 1 t 1 f 1 ( τ ) [ B ( t 1 - τ ) - 1 ] d τ + t 1 t 2 f 1 ( τ ) [ B ( t 2 - τ ) - 1 ] d τ + T 1 t 1 τ 1 t 1 g 2 ( τ 1 , τ 2 ) × [ B ( t 1 - τ 1 ) - 1 ] [ B ( t 1 - τ 2 ) - 1 ] d τ 1 d τ 2 + T 1 t 1 t 1 t 2 g 2 ( τ 1 , τ 2 ) [ B ( t 1 - τ 1 ) - 1 ] [ B ( t 2 - τ 2 ) - 1 ] d τ 1 d τ 2 + t 1 t 2 τ 1 t 2 g 2 ( τ 1 , τ 2 ) [ B ( t 2 - τ 1 ) - 1 ] [ B ( t 2 - τ 2 ) - 1 ] d τ 1 d τ 2 } ( { f 1 ( t 1 ) + T 1 t 1 g 2 ( t 1 , τ ) [ B ( t 1 - τ ) - 1 ] d τ + t 1 t 2 g 2 ( t 1 , τ ) [ B ( t 2 - τ ) - 1 ] d τ } { f 1 ( t 2 ) + T 1 t 1 g 2 ( t 2 , τ ) [ B ( t 1 - τ ) - 1 ] d τ + t 1 t 2 g 2 ( t 2 , τ ) [ B ( t 2 - τ ) - 1 ] d τ } + g 2 ( t 1 , t 2 ) ) .
B ( τ ) = { 1 τ T m 0 τ < T m ,
f 2 C ( τ ) = 1 ( τ - T m ) exp { - 2 C T m + 2 0 T m ( T m - z ) g 2 ( z ) d z + 0 T m z [ g 2 ( z + τ - T m ) + g 2 ( z + τ ) ] d z } × { [ C + 0 T m g 2 ( z ) d z + 0 T m g 2 ( z - τ ) d z ] × [ C + 0 T m g 2 ( z ) d z + 0 T m g 2 ( z + τ ) d z ] + g 2 ( τ ) } .
f 2 C ( τ ) = 1 ( τ - T m ) exp { - 2 C T m + 2 X Y 2 [ exp ( - Y T m ) + Y T m - 1 ] - X T m Y [ exp ( - Y τ ) + exp ( - Y τ - Y T m ) ] + X Y 2 [ exp ( - Y τ + Y T m ) - exp ( - Y τ - Y T m ) ] } ( { C + X Y [ exp ( - Y T m ) - 1 ] [ 1 - exp ( - Y T m ) ] } × { C + X Y [ exp ( - Y T m ) - 1 ] [ 1 + exp ( - Y τ ) ] } + X exp ( - Y τ ) ) .
f 2 C ( τ ) = [ 1 - exp ( - A τ ) ] exp { C A [ exp ( - A τ ) - 2 ] + X Y 2 - A 2 [ exp ( - A τ ) - exp ( - Y τ ) ] + X Y - A 1 2 A [ 2 - exp ( - 2 A τ ) ] + 1 Y + A [ exp ( - A τ - Y τ ) - 2 ] } ( { C - X Y + A + X Y - A [ exp ( - Y τ ) - exp ( - A τ ) ] } × { C - X Y + A [ 1 + exp ( - Y τ ) - exp ( - A τ - Y τ ) ] } + X exp ( - Y τ ) ) .

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