Abstract

Probabilities of detection of coherent optical signals in thermal background noise of a Lorentzian spectrum are calculated and compared for two cases: counting the number of photoelectrons within a fixed time T and measuring time intervals between photoelectrons within the same T. Although a comparison between the two methods is not always possible, it is shown that in some cases the counting method leads to better performance.

© 1983 Optical Society of America

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References

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  1. C. W. Helstrom, “Photoelectric detection of coherent light in filtered background light,” IEEE Trans. Aerospace Electron. Syst. AES-7, 210–213 (1971).
    [Crossref]
  2. D. L. Snyder, Random Point Processes (Wiley, New York, 1975), pp. 310–316.
  3. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), p. 342.
  4. C. Bendjaballah, “Analyse de champs optiques par les méthodes de comptage et coincidence de photons,” , Thesis (Université de Paris-Sud, Orsay, France, 1972).
  5. J. A. McFadden, “On the lengths of intervals in a stationary point process,” J. R. Stat. Soc. B 24, 364–382 (1962).
  6. S. R. Laxpati and G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
    [Crossref]
  7. H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part 1, p. 43.
  8. C. W. Helstrom, “Performance of an ideal quantum receiver of a coherent signal of random phase,” IEEE Trans. Aerospace Electron. Syst. AES-5, 562–564 (1969).
    [Crossref]
  9. F. Davidson and J. Amoss, “Sequential photon-counting statistics and maximum-likelihood estimation techniques for Gaussian optical field,” J. Opt. Soc. Am. 63, 30–37 (1973).
    [Crossref]

1973 (1)

1972 (1)

S. R. Laxpati and G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[Crossref]

1971 (1)

C. W. Helstrom, “Photoelectric detection of coherent light in filtered background light,” IEEE Trans. Aerospace Electron. Syst. AES-7, 210–213 (1971).
[Crossref]

1969 (1)

C. W. Helstrom, “Performance of an ideal quantum receiver of a coherent signal of random phase,” IEEE Trans. Aerospace Electron. Syst. AES-5, 562–564 (1969).
[Crossref]

1962 (1)

J. A. McFadden, “On the lengths of intervals in a stationary point process,” J. R. Stat. Soc. B 24, 364–382 (1962).

Amoss, J.

Bendjaballah, C.

C. Bendjaballah, “Analyse de champs optiques par les méthodes de comptage et coincidence de photons,” , Thesis (Université de Paris-Sud, Orsay, France, 1972).

Davidson, F.

Helstrom, C. W.

C. W. Helstrom, “Photoelectric detection of coherent light in filtered background light,” IEEE Trans. Aerospace Electron. Syst. AES-7, 210–213 (1971).
[Crossref]

C. W. Helstrom, “Performance of an ideal quantum receiver of a coherent signal of random phase,” IEEE Trans. Aerospace Electron. Syst. AES-5, 562–564 (1969).
[Crossref]

Lachs, G.

S. R. Laxpati and G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[Crossref]

Laxpati, S. R.

S. R. Laxpati and G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[Crossref]

McFadden, J. A.

J. A. McFadden, “On the lengths of intervals in a stationary point process,” J. R. Stat. Soc. B 24, 364–382 (1962).

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), p. 342.

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), pp. 310–316.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part 1, p. 43.

IEEE Trans. Aerospace Electron. Syst. (2)

C. W. Helstrom, “Photoelectric detection of coherent light in filtered background light,” IEEE Trans. Aerospace Electron. Syst. AES-7, 210–213 (1971).
[Crossref]

C. W. Helstrom, “Performance of an ideal quantum receiver of a coherent signal of random phase,” IEEE Trans. Aerospace Electron. Syst. AES-5, 562–564 (1969).
[Crossref]

J. Appl. Phys. (1)

S. R. Laxpati and G. Lachs, “Closed-form solutions for the photocount statistics of superposed coherent and chaotic radiation,” J. Appl. Phys. 43, 4773–4776 (1972).
[Crossref]

J. Opt. Soc. Am. (1)

J. R. Stat. Soc. B (1)

J. A. McFadden, “On the lengths of intervals in a stationary point process,” J. R. Stat. Soc. B 24, 364–382 (1962).

Other (4)

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part 1, p. 43.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), pp. 310–316.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978), p. 342.

C. Bendjaballah, “Analyse de champs optiques par les méthodes de comptage et coincidence de photons,” , Thesis (Université de Paris-Sud, Orsay, France, 1972).

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Figures (2)

Fig. 1
Fig. 1

Probability of detection Q1, versus S/N ratio for thermal background of Lorentzian spectrum with Q0 = 0.01, m = 3, and N = 1. The curves are labeled with the values of WT. Curve P is obtained from counting strategy with Poisson distribution.

Fig. 2
Fig. 2

Probability of detection Q1 versus S/N ratio for thermal background of Lorentzian spectrum of WT = 1, with Q0 = 0.01, N = 1, and m ranging from 1 to 7. Curve C is obtained from counting strategy with relevant photocounting distribution.

Equations (34)

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V ( t ) = B ( t ) ,             I ( t ) = I 0 ( t ) ,             p 0 ( n , T ) ,
V ( t ) = S ( t ) + B ( t ) ,             I ( t ) = I 1 ( t ) ,             p 1 ( n , T ) ,
n H 0 H 1 λ .
Q i = 1 + f p i ( λ , T ) - k = 0 λ p i ( k , T )             ( i = 0 , 1 ) ,
r i , m ( θ ) = E { I i ( θ ) [ J i ( θ ) ] m - 1 exp [ - J i ( θ ) ] / ( m - 1 ) ! } ,
J i ( θ ) = 0 θ I i ( u ) d u .
H 0 :             intensity I 0 ( t ) ( mean value I 0 ) ;             θ m ( T ) with r 0 , m ( θ ) ,
H 1 :             intensity I 1 ( t ) ( mean value I 1 ) ;             θ m ( T ) with r 1 , m ( θ ) .
Λ m ( θ ) = r 1 , m ( θ ) / r 0 , m ( θ )
Λ m ( θ ) H 0 H 1 η .
θ H 0 H 1 μ .
Λ m ( θ ) = R 0 , m ( T ) R 1 , m ( T ) ( I 1 I 0 ) m exp [ - ( I 1 - I 0 ) θ ] ,
R i , m ( T ) = 0 T r i , m ( θ ) d θ ,
r i , m ( θ ) = I i ( I i θ ) m - 1 exp ( - I i θ ) / ( m - 1 ) ! .
ρ H 0 H 1 ( I 1 - I 0 ) θ ,
Λ m ( θ ) = R 0 , m ( T ) ( I 1 ) m ( 1 + I 0 θ ) m + 1 R 1 , m ( T ) ( I 0 ) m ( 1 + I 1 θ ) m + 1 ,
r i , m ( θ ) = m I i m θ m - 1 ( 1 + I i θ ) - ( m + 1 ) .
Q i , m ( μ ) = [ R i , m ( T ) ] - 1 0 μ r i , m ( θ ) d θ ,
p i ( k , θ ) = E { [ J i ( θ ) ] k exp [ - J i ( θ ) ] / k ! }
r i , m ( θ ) = - k = 0 m - 1 θ p i ( k , θ ) .
Q i , m ( μ ) = [ 1 - k = 0 m - 1 p i ( k , μ ) ] / R i , m ( T ) ,
R i , m ( T ) = 1 - k = 0 m - 1 p i ( k , T ) .
Q i , m ( μ ) = 1 - I i exp ( - I i μ ) k = 0 m - 1 ( I i μ ) k / k ! 1 - I i exp ( - I i T ) k = 0 m - 1 ( I i T ) k / k ! .
p 1 ( k , θ ) = p 0 ( k , θ ) exp [ - I S / ( 1 + I 0 θ ) ] × L k { - I S / [ I 0 ( 1 + I 0 θ ) ] } ,
p 0 ( k , θ ) = ( I 0 θ ) k ( 1 + I 0 θ ) - ( k + 1 ) .
r i , m ( θ ) = m θ - 1 p i ( m , θ )
k = 0 m - 1 P 0 ( k , μ ) > k = 0 m - 1 P 0 ( k , T ) ,
M ~ α log n ( 1 / Q 0 ) + β log n ( S ) ,
T γ ( t - τ ) ϕ i ( τ ) d τ = χ i ϕ i ( t ) ,
γ ( x ) = γ ( 0 ) exp ( - W x ) .
α i = T ( t ) ϕ i ( t ) d t ,
p ( k + 1 , T ) = j = 0 k p ( j , T ) F ( k , j ) / ( k + 1 ) ,
F ( k , j ) = i = 1 ( χ i n T 1 + χ i n T ) k + 1 - j + ( k + 1 - j ) i = 1 α i n c ( χ i n T ) k - j ( 1 + χ i n T ) k + 2 - j ,
p ( O , T ) = i exp [ - α i 2 n c × ( 1 + χ i n T ) ] / ( 1 + χ i n T ) .