Abstract

A mathematical calculation of the statistical brightness gain of a postulated image intensifier is achieved. Two general cases for the signal-to-noise ratio are formulated: (1) owing to an inevitable nonadditive quantum noise, and (2) owing to the sum of a quantum noise and a postulated additive Poisson noise. The former is found to be somewhat higher than the latter. However, if either the brightness gain of the imaging system or the input photon rate increases, the signal-to-noise ratio of the latter case will asymptotically approach that of the former case. The minimum required scintillation for the two general cases is also determined and is found to be directly proportional to the brightness gain of the imaging system. The illumination of the image screen for the two cases is also calculated. Finally, it is concluded that the brightness gain does not improve the signal-to-noise ratio of the imaging system, which is due to the inevitable quantum noise. However, if the gain of the image illumination could affect the integration time and the perceptive area of the human eye, a substantial change of the signal-to-noise ratio and the required minimum scintillation would be expected.

© 1968 Optical Society of America

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References

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  1. E. Parzen, Modern Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1960), p. 264ff.
  2. J. Adams, B. W. Manley, IEEE Trans. NS-6, 88 (1966).
  3. J. W. Coltman, J. Opt. Soc. Amer. 44, 234 (1954).
    [CrossRef]
  4. A. Rose, J. Opt. Soc. Amer. 38, 196 (1948).
    [CrossRef]
  5. J. W. Coltman, J. Opt. Soc. Amer. 44, 236 (1954).
  6. E. Parzen, Stochastic Processes (Holden Day, Inc., San Francisco, 1962), p. 300ff.

1966

J. Adams, B. W. Manley, IEEE Trans. NS-6, 88 (1966).

1954

J. W. Coltman, J. Opt. Soc. Amer. 44, 234 (1954).
[CrossRef]

J. W. Coltman, J. Opt. Soc. Amer. 44, 236 (1954).

1948

A. Rose, J. Opt. Soc. Amer. 38, 196 (1948).
[CrossRef]

Adams, J.

J. Adams, B. W. Manley, IEEE Trans. NS-6, 88 (1966).

Coltman, J. W.

J. W. Coltman, J. Opt. Soc. Amer. 44, 234 (1954).
[CrossRef]

J. W. Coltman, J. Opt. Soc. Amer. 44, 236 (1954).

Manley, B. W.

J. Adams, B. W. Manley, IEEE Trans. NS-6, 88 (1966).

Parzen, E.

E. Parzen, Stochastic Processes (Holden Day, Inc., San Francisco, 1962), p. 300ff.

E. Parzen, Modern Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1960), p. 264ff.

Rose, A.

A. Rose, J. Opt. Soc. Amer. 38, 196 (1948).
[CrossRef]

IEEE Trans.

J. Adams, B. W. Manley, IEEE Trans. NS-6, 88 (1966).

J. Opt. Soc. Amer.

J. W. Coltman, J. Opt. Soc. Amer. 44, 234 (1954).
[CrossRef]

A. Rose, J. Opt. Soc. Amer. 38, 196 (1948).
[CrossRef]

J. W. Coltman, J. Opt. Soc. Amer. 44, 236 (1954).

Other

E. Parzen, Stochastic Processes (Holden Day, Inc., San Francisco, 1962), p. 300ff.

E. Parzen, Modern Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1960), p. 264ff.

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Figures (2)

Fig. 1
Fig. 1

An idealized image intensifier.

Fig. 2
Fig. 2

Schematic diagram of an electron channel.

Equations (111)

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N e = Q p I p ,
N p = Q e I e ,
p n ( k ; x ) = P n ( N n = k ) ,             k = 0 , 1 , 2 , ,
p n ( k ; x ) x = l ,             k = 0 , 1 , 2 , .
p n ( k ; x + d x ) = p n ( k ; x ) ( 1 - α k d x - β k d x - γ d x ) + p n ( k - 1 ; x ) [ α ( k - 1 ) + γ ] d x + p n ( k + 1 ; x ) β ( k + 1 ) d x .
p n ( k ; x + d x ) - p n ( k ; x ) d x = - [ α k + β k + γ ] p n ( k ; x ) + p n ( k - 1 ; x ) [ x ( k - 1 ) + γ ] + p n ( k + 1 ; x ) β ( k + 1 ) , for k 1.
( / x ) p n ( k ; x ) = - p n ( k ; x ) ( α k + β k + γ ) + p n ( k - 1 ; x ) [ α ( k - 1 ) + γ ] + p n ( k + 1 ; x ) β ( k + 1 ) , for k 1.
( / x ) p n ( k ; x ) = - p n ( k ; x ) ( α k + γ ) + p n ( k - 1 ; x ) [ α ( k - 1 ) + γ ] , for k 1 ,
p n ( k ; o ) = δ n , k ,
δ n , k = { 1 , k = n 0 , k n
p n ( k ; l ) = Γ [ k + ( γ / α ) ] e - α l [ n + ( γ / α ) ] ( 1 - e - α l ) k - n ( k - n ) ! Γ [ n + ( γ / α ) ] ,             for k n .
p n ( k ; l ) = Γ [ k + ( γ / α ) ] ( k - n ) ! Γ [ n + ( γ / α ) ] ( 1 G ) n + ( γ / α ) × ( 1 - 1 G ) k - n ,             for k n .
p n ( k ; l ) ( n / k ) ( k / G ) n e - k / G n ! , for k n , = 0 otherwise .
σ 2 G 2 n
m ¯ = G n .
p ( q ) = λ q e - λ / q ! ,             q = 0 , 1 , 2 , ,
λ = m ¯ a = σ a 2 .
σ N 2 = σ 2 + σ a 2 = G 2 n + λ ,
m ¯ N = G n + λ .
σ N 2 = G 2 n ;
σ N 2 2 G 2 n ;
σ N 2 λ .
N 1 = ( Q G n 1 t A 1 / 2 a ) = 1 2 p t ( A 1 / A 2 ) ,
N 2 = ( Q G n 2 t A 1 / 2 a ) = 1 2 ( 1 - C ) p t ( A 1 / A 2 ) ,
N 1 - N 2 = 1 2 C p t ( A 1 / A 2 ) .
σ 1 = G ( Q n 1 t A 1 / 2 a ) 1 2 = [ 1 2 G p t ( A 1 / A 2 ) ] 1 2 .
σ 2 = G ( Q n 2 t A 1 / 2 a ) 1 2 = [ 1 2 G ( 1 - C ) p t ( A 1 / A 2 ) ] 1 2 .
σ N = ( σ 1 2 + σ 2 2 ) 1 2 = [ 1 2 G ( 2 - C ) p t ( A 1 / A 2 ) ] 1 2 .
R = ( N 1 - N 2 ) / σ N = C [ p t A 1 2 G ( 2 - C ) A 2 ] 1 2 .
N 1 = ( Q G n 1 t + λ t ) ( A 1 / 2 a ) = 1 2 [ p t ( A 1 / A 2 ) + λ t ( A 1 / a ) ]
N 2 = ( Q G n 2 t + λ t ) ( A 1 / 2 a ) = 1 2 [ ( 1 - C ) p t ( A 1 / A 2 ) + λ t ( A 1 / a ) ] ,
σ T 1 2 = ( G 2 Q n 1 t + λ t ) ( A 1 / 2 a ) = 1 2 [ G p t ( A 1 / A 2 ) + ( λ t A 1 / a ) ]
σ T 2 2 = ( G 2 Q n 2 t + λ t ) ( A 1 / 2 a ) = 1 2 [ G ( 1 - C ) p t ( A 1 / A 2 ) + ( λ t A 1 / a ) ] .
σ T N = ( σ T 1 2 + σ T 2 2 ) 1 2 = [ 1 2 G ( 2 - C ) p t ( A 1 / A 2 ) + ( λ t A 1 / a ) ] .
R T = N 1 - N 2 σ T N = C p [ t ( A 1 / A 2 ) 2 G 2 ( 2 - C ) p ( A 1 / A 2 ) + 4 λ ( A 1 / a ) ] ] 1 2 .
R T < R .
R T C [ p t A 1 2 G ( 2 - C ) A 2 ] 1 2 , for large p .
R T C [ p t A 1 2 G ( 2 - C ) A 2 ] 1 2 , for large G .
P min = ( 2 K 2 G A 2 / t A 1 ) [ ( 2 - C ) / C 2 ] = M G [ ( 2 - C ) / C 2 ] ,
M = 2 K 2 A 2 / t A 1 .
P min = M [ ( 2 - C ) / C 2 ] .
( P T ) min = ( K 2 / C 2 ) G ( 2 - C ) + [ ( K 4 / C 4 ) G 2 ( 2 - C ) 2 + 4 λ t ( A 1 / a ) ] 1 2 ( A 1 / A 2 ) t .
( P T ) min M G [ ( 2 - C ) / C 2 ] .
S 1 = Q G n 1 t = m 1 t ,
S 2 = Q G n 2 t = m 2 t ,
σ s 1 2 = G 2 Q n 1 t = G m 1 t ,
σ s 2 2 = G 2 Q n 2 t = G m 2 t ,
K = ( S 1 - S 2 ) / ( σ s 1 2 + σ s 2 2 ) 1 2 ,
( m 1 - m 2 ) t = K [ G t ( m 1 + m 2 ) ] 1 2 .
C = Δ m t / m t ,
C = [ K ( G m t ) 1 2 / m t ] = K ( G / m t ) 1 2 .
m = K 2 G / C 2 t ,
m t = K 2 G E / 2 C 2 t .
L = K 2 G E / 2 ν C 2 t .
I = K 2 G E / 2 ν C 2 t A 2 ,
Q t ( n 1 - n 2 ) = K [ Q t ( n 1 + n 2 ) ] 1 2 ,
C = Δ n t / n t ,
C = K ( 1 / Q n t ) 1 2 .
Q n = K 2 / C 2 t .
( Q n / Q e ) = ( Q / Q e ) ( n 1 + n 2 ) ,
I = K 2 E / 2 ν C 2 t A 2 Q e .
G I = I / I = Q e G .
S 1 = Q G n 1 t + λ t = m 1 t + λ t ,
S 2 = Q G n 2 t + λ t = m 2 t + λ t ,
σ t 1 2 = G 2 Q n 1 t + λ t = G m 1 t + λ t ,
σ t 2 2 = G 2 Q n 2 t + λ t = G m 2 t + λ t ,
K = S 1 - S 2 ( σ t 1 2 + σ t 2 2 ) 1 2 = Δ m t [ G m t + 2 λ t ] 1 2
C = Δ m t / m t ,
C = [ K ( G m t + 2 λ t ) 1 2 ] / m t .
m = K 2 G + K ( K 2 G 2 + 8 C 2 λ t ) 1 2 2 C 2 t .
I T = m E 2 ν A 2 = K 2 G + K ( K 2 G 2 + 8 C 2 λ t ) 1 2 2 C 2 t E 2 ν A 2 .
I T > I .
I T I , for large G .
( / x ) p n ( k ; x ) = - p n ( k ; x ) ( α k + γ ) + p ( k - 1 ; x ) × [ α ( k - 1 ) + γ ] , for k n ,
ψ n ( x , u ) = k p n ( k ; x ) u k .
ψ n ( x , u ) x = k p n ( k ; x ) u k x ,
ψ n ( x , u ) u = k p n ( k ; x ) k u k - 1 .
ψ n ( x , u ) x = - α u ( 1 - u ) ψ n ( x , u ) u - γ ( 1 - u ) ψ n ( x , u ) .
d x = d u / α u ( 1 - u ) ,
d x = - [ 1 / γ ( 1 - u ) ] [ d ψ n ( x , u ) / ψ n ( x , u ) ] .
p n ( k ; x ) x = 0 = δ n , k as ψ n ( x , u ) | x = 0 u = u 0 = p ( k ; x ) u k | x = 0 u = u 0 = u 0 n .
v = 1 / ( 1 - u ) .
d x = d v / α ( v - 1 ) ,
d x = - ( v / γ ) [ d ψ n ( x , v ) / ψ n ( x , v ) ] .
v = exp ( α d x ) [ - α exp ( - α d x ) d x + C ] .
v = 1 + ( v 0 - 1 ) e α x ,
d ψ n ( x , v ) ψ n ( x , v ) = - γ e - α x ( v 0 - 1 ) + e - α x .
ψ n ( x , v ) = [ 1 - ( 1 / v 0 ) ] n [ 1 - ( e α x - 1 ) / v 0 e α x ] ( γ / α ) .
ψ n ( x , u ) = u n exp - α x [ n + ( γ / α ) ] × [ 1 - u ( 1 - e - α x ) ] - [ n + ( γ / α ) ] .
[ 1 - u ( 1 - e - α x ] - [ n + ( γ / α ) ] = 1 + [ n + ( γ / α ) 1 ! ( 1 - e - α x ) u + [ n + ( γ / α ) ] [ n + ( γ / α ) + 1 ] ( 1 - e - α x ) 2 u 2 m ! + + [ n + ( γ / α ) ] [ n + ( γ / α ) + 1 ] [ n + ( γ / α ) + m - 1 ] × ( 1 - e - α x ) m u m m ! = m = 0 [ n + ( γ / α ) ] [ n + ( γ / α ) + 1 ] [ n + ( γ / α ) + m - 1 ] × ( 1 - e - α x ) m u m m ! ,
ψ n ( x , u ) = m = 0 u m + n [ n + ( γ / α ) ] [ n + ( γ / α ) + 1 ] × [ n + ( γ / α ) + m - 1 m × e - α x [ n + ( γ / α ) ] ( 1 - e - α x ) m .
ψ n ( x , u ) = k n u k [ ( γ / α ) + k - 1 ] [ ( γ / α ) + k - 2 ] × [ n + ( γ / α ) ] ( k - n ) ! × e - α x [ n + ( γ / α ) ] ( 1 - e - α x ) k - n .
ψ n ( x , u ) = k n u k Γ [ k + ( γ / α ) ] ( k - n ) ! Γ [ n + ( γ / α ) ] e - α x [ n + ( γ / α ) ] × ( 1 - e - α x ) k - n .
p n ( k ; x ) = Γ [ k + ( γ / α ) ( k - n ) ! Γ [ n + ( γ / α ) ] exp { - α x [ n + ( γ / α ) ] × ( 1 - e - α x ) k - n , for k n = 0 , otherwise .
p n ( k ; x ) x = l = Γ [ k + ( γ / α ) ] ( k - n ) ! Γ [ n + ( γ / α ) ] ( 1 G ) n + ( γ / α ) × ( 1 - 1 G ) k - n , for k n = 0 , otherwise .
( 1 - 1 G ) k - n 1 - k G + 1 2 ! ( k G ) 2 - 1 3 ! ( k G ) 3 + = e ( - k / G )
Γ [ k + ( γ / α ) ] ( k - n ) ! = [ k + ( γ / α ) - 1 ] [ k + ( γ / α ) - 2 ] × [ k + ( γ / α ) - n ] ( γ / α ) Γ ( γ / α ) ( k - n ) ! k n + ( γ / α ) - 1 .
p n ( k ; l ) k n + ( γ / α ) - 1 e ( - k / G ) G n + ( γ / α ) Γ [ n + ( γ / α ) ] , for k 0 = 0 , otherwise .
Γ [ n + ( γ / α ) ] Γ ( n ) = ( n - 1 ) ! .
p n ( k ; l ) n k ( k / G ) n e ( - k / G ) n ! , for k n = 0 , otherwise .
m ¯ = ψ n ( l , u ) u | u = 1 = k k p n ( k ; l ) u k - 1 u = 1.
ψ n ( l , u ) = u n e - n α x [ 1 - u ( 1 - e - α x ) ] - n .
m ¯ = [ ψ n ( l , u ) / u ] u = 1 = n e α l
m ¯ = n G .
2 ψ n ( l , u ) 2 u = k k ( k - 1 ) p n ( k ; l ) u k - 2 .
[ 2 ψ n ( l , u ) ] / ( 2 u ) u = 1 = m 2 ¯ - m ¯ .
[ 2 ψ n ( l , u ) ] / ( 2 u ) u = 1 = n [ n e α l - e α l + n e - 2 α l + e 2 α l ] .
[ 2 ψ n ( l , u ) ] / ( 2 u ) u = 1 = n 2 G - n G + n 2 G 2 + n G 2 .
m 2 ¯ = 2 ψ n ( l , u ) 2 u | u = 1 + m ¯ = n 2 G + n 2 G 2 + n G 2 .
σ 2 = m 2 ¯ - m ¯ 2 = n 2 G + n G 2 .
σ 2 n G 2 .

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