Abstract

Exact forward photocounting distributions for a source of optical radiation with arbitrary statistics are obtained in the presence of photodetector dead time. In particular, we examine the counting and pulse-interval distributions that arise from amplitude-stabilized radiation, from chaotic radiation, and from a Van der Pol laser with arbitrary excitation. The exact dead-time-corrected Poisson distribution is graphically compared with a previous approximate result and with the uncorrected Poisson. Plots of the Bose–Einstein distribution clearly indicate the dramatic anti-bunching effects of the dead time in overcoming the inherent bunching of this distribution. A simplified approximate solution is also found. for the Van der Pol laser above threshold; this result is similar to light from an amplitude-stabilized source incident on a photodetector with a gaussian-distributed dead time. Information about photodetector dead-time variation can therefore be obtained either by using an amplitude-stabilized source or by properly choosing system parameters such that irradiance fluctuations are averaged out.

© 1975 Optical Society of America

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References

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  1. L. Mandel, Proc. Phys. Soc. (Lond.) 72, 1037 (1958); Proc. Phys. Soc. (Lond.) 74, 233 (1959).
    [Crossref]
  2. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964); J. Peřina, Coherence of Light (Van Nostrand–Reinhold, New York, 1972).
    [Crossref]
  3. R. J. Glauber, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1965), p. 788.
  4. F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
    [Crossref]
  5. J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North–Holland, Amsterdam, 1967), p. 213.
  6. M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969).
    [Crossref]
  7. P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970); J. Opt. Soc. Am. 60, 1489 (1970).
    [Crossref]
  8. I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).
  9. G. Bédard, Proc. Phys. Soc. (Lond.) 90, 131 (1967).
    [Crossref]
  10. W. Feller, A Volume for the Anniversary of Courant (Wiley–Interscience, New York, 1948), p. 105.
  11. R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).
  12. H. Risken, Z. Physik 186, 85 (1965).
    [Crossref]
  13. V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
    [Crossref]
  14. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Std. (U.S.) Appl. Math. Ser. (U.S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).
  15. M. Lax and M. Zwanziger, Phys. Rev. Lett. 24, 937 (1970); Phys. Rev. A 7, 750 (1973).
    [Crossref]
  16. D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
    [Crossref]

1970 (3)

P. Diament and M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970); J. Opt. Soc. Am. 60, 1489 (1970).
[Crossref]

M. Lax and M. Zwanziger, Phys. Rev. Lett. 24, 937 (1970); Phys. Rev. A 7, 750 (1973).
[Crossref]

D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
[Crossref]

1969 (1)

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969).
[Crossref]

1967 (2)

G. Bédard, Proc. Phys. Soc. (Lond.) 90, 131 (1967).
[Crossref]

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

1966 (2)

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

1965 (1)

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

1964 (2)

I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964); J. Peřina, Coherence of Light (Van Nostrand–Reinhold, New York, 1972).
[Crossref]

1958 (1)

L. Mandel, Proc. Phys. Soc. (Lond.) 72, 1037 (1958); Proc. Phys. Soc. (Lond.) 74, 233 (1959).
[Crossref]

Alley, C. O.

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Armstrong, J. A.

J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North–Holland, Amsterdam, 1967), p. 213.

Arzt, V.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Bédard, G.

G. Bédard, Proc. Phys. Soc. (Lond.) 90, 131 (1967).
[Crossref]

Chang, R. F.

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Davis, W.

D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
[Crossref]

DeLotto, I.

I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).

Detenbeck, R. W.

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Diament, P.

Feller, W.

W. Feller, A Volume for the Anniversary of Courant (Wiley–Interscience, New York, 1948), p. 105.

Glauber, R. J.

R. J. Glauber, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1965), p. 788.

Haken, H.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Hochuli, U.

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Johnson, F. A.

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

Jones, R.

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964); J. Peřina, Coherence of Light (Van Nostrand–Reinhold, New York, 1972).
[Crossref]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964); J. Peřina, Coherence of Light (Van Nostrand–Reinhold, New York, 1972).
[Crossref]

Korenman, V.

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Lax, M.

M. Lax and M. Zwanziger, Phys. Rev. Lett. 24, 937 (1970); Phys. Rev. A 7, 750 (1973).
[Crossref]

Mandel, L.

D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
[Crossref]

L. Mandel, Proc. Phys. Soc. (Lond.) 72, 1037 (1958); Proc. Phys. Soc. (Lond.) 74, 233 (1959).
[Crossref]

Manfredi, P. F.

I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).

McLean, T. P.

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

Meltzer, D.

D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
[Crossref]

Pike, E. R.

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

Principi, P.

I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).

Risken, H.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

Sauerman, H.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Schmid, C.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Smith, A. W.

J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North–Holland, Amsterdam, 1967), p. 213.

Teich, M. C.

Weidlich, W.

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Zwanziger, M.

M. Lax and M. Zwanziger, Phys. Rev. Lett. 24, 937 (1970); Phys. Rev. A 7, 750 (1973).
[Crossref]

Appl. Phys. Lett. (1)

D. Meltzer, W. Davis, and L. Mandel, Appl. Phys. Lett. 17, 242 (1970).
[Crossref]

Energia Nucleare (Milan) (1)

I. DeLotto, P. F. Manfredi, and P. Principi, Energia Nucleare (Milan) 11, 557 (1964); Energia Nucleare (Milan) 11, 599 (1964).

J. Appl. Phys. (1)

M. C. Teich and P. Diament, J. Appl. Phys. 40, 625 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Lett. (1)

R. F. Chang, R. W. Detenbeck, V. Korenman, C. O. Alley, and U. Hochuli, Phys. Lett. 25A, 272 (1967).

Phys. Rev. (1)

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964); J. Peřina, Coherence of Light (Van Nostrand–Reinhold, New York, 1972).
[Crossref]

Phys. Rev. Lett. (2)

F. A. Johnson, R. Jones, T. P. McLean, and E. R. Pike, Phys. Rev. Lett. 16, 589 (1966).
[Crossref]

M. Lax and M. Zwanziger, Phys. Rev. Lett. 24, 937 (1970); Phys. Rev. A 7, 750 (1973).
[Crossref]

Proc. Phys. Soc. (Lond.) (2)

L. Mandel, Proc. Phys. Soc. (Lond.) 72, 1037 (1958); Proc. Phys. Soc. (Lond.) 74, 233 (1959).
[Crossref]

G. Bédard, Proc. Phys. Soc. (Lond.) 90, 131 (1967).
[Crossref]

Z. Physik (2)

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

V. Arzt, H. Haken, H. Risken, H. Sauerman, C. Schmid, and W. Weidlich, Z. Physik 197, 207 (1966).
[Crossref]

Other (4)

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun. Natl. Bur. Std. (U.S.) Appl. Math. Ser. (U.S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

W. Feller, A Volume for the Anniversary of Courant (Wiley–Interscience, New York, 1948), p. 105.

J. A. Armstrong and A. W. Smith, in Progress in Optics, VI, edited by E. Wolf (North–Holland, Amsterdam, 1967), p. 213.

R. J. Glauber, in Physics of Quantum Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York, 1965), p. 788.

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Figures (4)

FIG. 1
FIG. 1

The Poisson distribution with no dead time is represented by the solid curve (mean = variance = 25.0). The effects of dead time produce the distribution shown by the dotted curve (mean = 16.3, variance = 5.6). Note the reduction of the mean (shown by arrows) due to the elimination of output pulses. The dead-time-corrected distribution normalized to approximately the same mean as the uncorrected Poisson distribution is shown by the dashed curve (mean = 24.6, variance = 3.6); this clearly demonstrates the decrease of variance brought about by dead-time effects. The ratio of dead time to sampling time τ/T was 0.025 for all curves.

FIG. 2
FIG. 2

Photocounting distributions corresponding to exact (solid curve) and approximate (dashed curve) corrections for dead time. The mean value for both curves (shown by arrows) is 25.3. These curves are normalized to the same irradiance and correspond to a value τ/T = 0.005.

FIG. 3
FIG. 3

Dead-time-corrected photocounting distributions for amplitude-stabilized (dashed curve) and Van der Pol laser (solid curve) radiation. The mean irradiance of the Van der Pol laser is equal to the irradiance of the amplitude-stabilized source to first order. The excitation parameter w of thee laser is taken to be 12.5. As in Fig. 1, τ/T = 0.025.

FIG. 4
FIG. 4

The Bose–Einstein distribution with no dead time is represented by the solid curve (mean = 25.0, variance = 1275.0). The effects of dead time produce the distributions shown by the dashed curve (τ/T = 0.025, mean = 12.7, variance 54.5) and by the dotted curve (τ/T = 0.1, mean = 6.5, variance = 4.7). Note the reduction of the mean (shown by arrows) due to the elimination of output pulses. The dotted curve labeled P represents the dead-time-corrected Poisson distribution (τ/T = 0.1, mean = 6.7, variance = 0.7), clearly demonstrating the similarity of diverse counting distributions in the dead-time-limited domain.

Equations (49)

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p ( n , λ ) = ( λ T ) n exp ( - λ T ) / n ! ,
λ = α I .
p 0 ( n , λ , τ ) = k = 0 n { λ k ( T - n τ ) k / k ! } exp { - λ ( T - n τ ) } - k = 0 n - 1 { λ k [ T - ( n - 1 ) τ ] k / k ! } exp { - λ [ T - ( n - 1 ) τ ] } .
n ¯ = λ T ( 1 + λ τ ) - 1 + 1 2 ( λ τ ) 2 ( 1 + λ τ ) - 2
σ 2 = λ T ( 1 + λ τ ) - 3 ,
p k ( n , λ ) = { λ k ( T - n τ ) k / k ! } exp { - λ ( T - n τ ) } ,
p 0 ( n , λ , τ ) = k = 0 n p k ( n , λ ) - k = 0 n - 1 p k ( n - 1 , λ ) .
p ( n , λ , τ ) = p 0 ( n , λ , τ ) M ,
M = α t t + T I ( t ) d t .
p ( n , λ , τ ) = k = 0 n p k ( n , λ ) M - k = 0 n - 1 p k ( n - 1 , λ ) M .
P ( I ) = 2 π - 1 / 2 [ I 1 ( 1 + erf w ) ] - 1 exp { - [ ( I / I 1 ) - w ] 2 } .
p k ( n , λ ) M = [ exp ( w 2 ) erfc ( - w ) ] - 1 × { ( ν t ) k exp [ ( ν t / 2 ) - w ] 2 i k erfc [ ( ν t / 2 ) - w ] } ,
I = I 1 [ w + π - 1 / 2 ( 1 + erf w ) - 1 exp ( - w 2 ) ] .
f ( t ) = { ν π - 1 / 2 [ exp ( w 2 ) erfc ( - w ) ] - 1 { 1 - π 1 / 2 [ ( ν / 2 ) ( t - τ ) - w ] exp [ ( ν / 2 ) ( t - τ ) - w ] 2 erfc [ ( ν / 2 ) ( t - τ ) - w ] } ,             t τ 0 ,             t < τ .
λ = α I ( t ) ,
P ( λ ) = ( 2 π ) - 1 / 2 σ - 1 exp { - 1 2 [ ( λ - λ ¯ ) / σ ] 2 } ,
λ ¯ = α I 1 w
σ = α I 1 / ( 2 1 / 2 ) .
p ( n , σ , λ ¯ ) = 0 p 0 ( n , λ , τ ) P ( λ ) d λ ,
p ( n , σ , λ ¯ ) = 0 ( k = 0 n { λ k ( T - n τ ) k / k ! } exp { - λ ( T - n τ ) } - k = 0 n - 1 { λ k [ T - ( n - 1 ) τ ] k / k ! } exp { - λ [ T - ( n - 1 ) τ ] } ) × P ( λ ) d λ .
p k , n = 0 { λ k ( T - n τ ) k / k ! } exp { - λ ( T - n τ ) } P ( λ ) d λ .
f ( λ ) = ln [ p k ( n , λ ) ] - ( λ - λ ¯ ) 2 / 2 σ 2
q m ( n , k , λ ) = m ln p k ( n , λ ) / λ m .
f ( λ 0 ) = 0
λ 0 = λ ¯ + σ 2 q 1 ( n , k , λ 0 ) .
f ( λ ) = f ( λ 0 ) + f ( λ 0 ) x + f ( λ 0 ) x 2 / 2 ! + R ( x )
p k , n [ 1 - σ 2 q 2 ( n , k , λ 0 ) ] - 1 / 2 { p k ( n , λ 0 ) exp [ - 1 2 ( λ 0 - λ ¯ ) / σ 2 ] } .
q 1 ( n , k , λ ) = k / λ - ( T - n τ ) ,
q 2 ( n , k , λ ) = - k / λ 2 ,
q 3 ( n , k , λ ) = 2 k / λ 3 ,
λ 0 = λ ¯ + σ 2 k / λ 0 - σ 2 ( T - n τ )
p k , n ( 1 + σ 2 k / λ 0 2 ) - 1 / 2 { p k ( n , λ 0 ) exp [ - 1 2 ( λ 0 - λ ¯ ) / σ 2 ] } ;
p ( n , σ , λ ¯ ) = k = 0 n p k , n - k = 0 n - 1 p k , n - 1 .
p k ( n , λ ) M = { λ [ T - n τ ] } k / { 1 + λ [ T - n τ ] } k + 1 ,
p ( n , λ , τ ) = k = 0 n { λ [ T - n τ ] } k / { 1 + λ [ T - n τ ] } k + 1 - k = 0 n - 1 { λ [ T - ( n - 1 ) τ ] } k / { 1 + λ [ T - ( n - 1 ) τ ] } k + 1 .
P n * ( 0 , t ) = 0 t [ p n * ( 0 , t ) - p n + 1 * ( 0 , t ) ] d t ,
P n * ( s ) = s - 1 [ p n * ( s ) - p n + 1 * ( s ) ] .
p n * ( s ) = p 1 * ( s ) p n - 1 ( s ) = p 1 * ( s ) [ p 1 ( s ) ] n - 1 ,
P n * ( s ) = s - 1 p 1 * ( s ) [ 1 - p 1 ( s ) ] [ p 1 ( s ) ] n - 1 ,
p 1 * ( s ) = p 1 ( s ) = λ ( s + λ ) - 1 .
p 1 , u * ( s ) = λ ( s + λ ) - 1 and p 1 , u ( s ) = λ ( s + λ ) - 1 exp ( - s τ ) ,
P n , u * ( s ) = s - 1 λ ( s + λ ) - 1 [ 1 - λ ( s + λ ) - 1 exp ( - s τ ) ] × [ λ ( s + λ ) - 1 exp ( - s τ ) ] n - 1 ,
P n , u * ( s ) = s - 1 λ n ( s + λ ) - n exp [ - s τ ( n - 1 ) ] - s - 1 λ n + 1 ( s + λ ) - ( n + 1 ) exp [ - ( n s τ ) ] .
s - 1 ( s + λ ) - n = K 1 s - 1 + K 2 , n ( s + λ ) - n + K 2 , n - 1 ( s + λ ) - ( n - 1 ) + + K 2 , 2 ( s + λ ) - 2 + K 2 , 1 ( s + λ ) - 1 .
K 1 = λ - n and K 2 , n - m = - λ - ( m + 1 ) ,             m = 0 , 1 , 2 , , n - 1.
L - 1 { s - 1 ( s + λ ) - n } = u ( t ) λ - n - m = 0 n - 1 λ - ( m + 1 ) L - 1 [ ( s + λ ) - ( n - m ) ] = u ( t ) λ - n - m = 0 n - 1 λ - ( m + 1 ) [ t ( n - m - 1 ) e - λ t / ( n - m - 1 ) ! ] ,
L - 1 { λ n exp [ - s τ ( n - 1 ) ] s - 1 ( s + λ ) - n } = u [ t - ( n - 1 ) τ ] - m = 0 n - 1 λ n - ( m + 1 ) { [ t - ( n - 1 ) τ ] ( n - m - 1 ) / ( n - m - 1 ) ! } × exp { - λ [ t - ( n - 1 ) τ ] } .
L - 1 { λ n + 1 exp [ - ( n s τ ) ] s - 1 ( s + λ ) - ( n + 1 ) } = u ( t - n τ ) - m = 0 n λ n - m { [ t - n τ ] n - m / ( n - m ) ! } exp [ - λ ( t - n τ ) ] .
p 0 ( n , λ , τ ) = P n , u * ( 0 , T ) = k = 0 n { λ k ( T - n τ ) k / k ! } exp { - λ ( T - n τ ) } - k = 0 n - 1 { λ k [ T - ( n - 1 ) τ ] k / k ! } exp { - λ [ T - ( n - 1 ) τ ] } ,