Abstract

The photon statistics of a particular photon-imaging detector are studied, and the conditional probability of photon counts in the output given a certain number of counts in an associated reference channel is derived. This result is applied in a variable level discrimination technique which significantly reduces detection errors approaching the ideal limit. These results can be applied to other photon-limited detectors with nonideal pulse height distributions.

© 1984 Optical Society of America

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References

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  1. C. Papaliolios, L. Mertz, Proc. Soc. Photo-Opt. Instrum. Eng. 331, 360 (1982).
  2. C. Papaliolios, personal communication.
  3. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 38 and 39.
  4. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 504.
  5. Ref. 4, p. 505, Eq. (13.1.27).
  6. Ref. 4, p. 504, Eq. (13.1.4).
  7. Ref. 4, p. 556.
  8. Ref. 4, p. 559, Eq. (15.3.3).
  9. Ref. 4, p. 556, Eq. (15.1.20).

1982 (1)

C. Papaliolios, L. Mertz, Proc. Soc. Photo-Opt. Instrum. Eng. 331, 360 (1982).

Mertz, L.

C. Papaliolios, L. Mertz, Proc. Soc. Photo-Opt. Instrum. Eng. 331, 360 (1982).

Papaliolios, C.

C. Papaliolios, L. Mertz, Proc. Soc. Photo-Opt. Instrum. Eng. 331, 360 (1982).

C. Papaliolios, personal communication.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 38 and 39.

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

C. Papaliolios, L. Mertz, Proc. Soc. Photo-Opt. Instrum. Eng. 331, 360 (1982).

Other (8)

C. Papaliolios, personal communication.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 38 and 39.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 504.

Ref. 4, p. 505, Eq. (13.1.27).

Ref. 4, p. 504, Eq. (13.1.4).

Ref. 4, p. 556.

Ref. 4, p. 559, Eq. (15.3.3).

Ref. 4, p. 556, Eq. (15.1.20).

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Figures (6)

Fig. 1
Fig. 1

Four coarsest masks of the STAC would encode the four most significant bits of the digital x coordinate. Note that Gray code is used as opposed to binary to minimize multibit errors.

Fig. 2
Fig. 2

Tm(s,α) plotted over the fundamental range of s for α = 1, 4, and 64.

Fig. 3
Fig. 3

Typical variation of bias B and error E with threshold nt for the mask–spot ratio α = 2.

Fig. 4
Fig. 4

Comparison of the unconditional Bose-Einstein distribution and the conditional or modified Bose-Einstein distribution for r = 20 and Tm = 0.75.

Fig. 5
Fig. 5

Optimum threshold vs reference channel count r for several mask–spot ratios.

Fig. 6
Fig. 6

Comparison of the three discrimination schemes presented in the text.

Equations (38)

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p ( N ) = N N N ! exp ( - N ) ,
p ( N ) = N N [ 1 + N ] N + 1 .
p ( N N ) = ( N N ) T N ( 1 - T ) N - N ,
N 5.0 × 10 4 , T 0 1.1 × 10 - 3 , η 0.15.
r = η T 0 N 8.3.
T m = k = - [ 1 π 2 k w ( 2 k + 1 ) w exp [ - ( x - x 0 ) 2 / 2 σ 2 ] d x ] ,
α = w 2 σ and s = x 0 w .
T m ( s , α ) = ½ k = - erf [ ( 2 k - s + 1 ) α ] - erf [ ( 2 k - s ) α ] .
E ( s ) = { p ( n n t ) , for - 1 / 2 s < 0 ; p ( n < n t ) , for 0 s 1 / 2.
E 0 = p ( n n t ) s , for s < 0 ; E 1 = p ( n < n t ) s for s 0 ; E = 1 2 ( E 0 + E 1 ) .
B = E 1 - E 0 .
n = T m r .
p ( r N ) = { ( N r ) ( η T 0 ) r ( 1 - η T 0 ) n - r , for r N ; 0 , for r > N .
p ( N r ) = p ( r N ) p ( N ) p ( r ) .
p ( n r ) = N = 0 p ( n N ) p ( N r )
p ( N r ) = { [ ( 1 - η T 0 ) N ] N - r ( N - r ) ! exp ( r - N ) , for r R , 0 , for r > N .
μ = ( 1 - η T 0 ) N = N - r ,
p ( N r ) = { μ N - r ( N - r ) ! exp ( - μ ) , for N r , 0 , for N < r .
p ( n N ) = { ( N n ) β n ( 1 - β ) N - n , for n N , 0 , for n > N ,
β = η T 0 T m .
p ( n r ) = N = N 0 ( N n ) β n ( 1 - β ) N - n μ N - r ( N - r ) ! exp ( - μ )
N 0 = max ( n , r ) .
p ( n r ) = { ( r n ) ( 1 - β ) r - n β n exp ( - μ ) M [ r + 1 ; r - n + 1 ; μ ( 1 - β ) ] n r ; μ n - r ( n - r ) ! β n exp ( - μ ) M [ n + 1 ; n - r + 1 ; μ ( 1 - β ) ] n r .
M ( a ; b ; z ) = exp ( z ) M ( b - a ; b ; - z ) ,
p ( n r ) = { ( r n ) β n ( 1 - β ) r - n exp ( - μ β ) M [ - n ; r - n + 1 ; μ ( β - 1 ) ] , n r ; ( μ β ) n - r ( n - r ) ! exp ( - μ β ) β r M [ - r ; n - r + 1 ; μ ( β - 1 ) ] , n r .
μ 1 , β 1 , and μ ( 1 - β ) μ 1.
M ( a ; b ; z ) = Γ ( b ) Γ ( a ) exp ( z ) z a - b [ 1 + O ( z - 1 ) ] ,
p ( n r ) n n n ! exp ( - n ) ,
p ( N r ) = { ( N r ) ( 1 + r ) r + 1 ( 1 + N ) N + 1 μ N - r , for N r , 0 , for N < r ,
p ( n r ) = N = N 0 ( N n ) ( N r ) β n ( 1 - β ) N - n ( 1 + r ) r + 1 ( 1 + N ) N + 1 μ N - r ,
p ( n r ) = ( r n ) β n ( 1 - β ) r - n [ ( 1 + r ) ( 1 + N ) ] r + 1 × F [ r + 1 , r + 1 ; r - n + 1 ; μ ( 1 - β ) 1 + N ] ,             n r = ( n r ) β r ( μ β ) n - r ( 1 + r ) r + 1 ( 1 + N ) n + 1 × F [ n + 1 , n + 1 ; n - r + 1 ; μ ( 1 - β ) 1 + N ] ,             n r .
F ( a , b ; c ; z ) = ( 1 - z ) c - a - b F ( c - a , c - b ; c ; z ) ,
p ( n r ) = ( r n ) β n ( 1 - β ) r - n ( 1 - r ) r + 1 ( 1 + N ) n ( 1 + r + μ β ) r + n + 1 × F [ - n , - n ; r - n + 1 ; μ ( 1 - β ) 1 + N ] ,             n r = ( n r ) β r ( μ β ) n - r ( 1 - r ) r + 1 ( 1 + N ) r ( 1 + r + μ β ) r + n + 1 × F [ - r , - r ; n - r + 1 ; μ ( 1 - β ) 1 + N ] ,             n r .
μ ( 1 - β ) 1 + N 1
F ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) ,
p ( n r ) ( r + n n ) ( 1 - r ) r + 1 n n ( 1 + r + n ) r + n + 1 ,
r = 0 p ( n r ) p ( r ) = n n ( 1 + n ) n + 1 ,
MBE : σ 2 n r = 1 + n - ( r 1 + r ) n , Bose - Einstein : σ 2 n = 1 + n , Poisson : σ 2 n = 1 ,

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