Abstract

In Part I of this series of two papers, it was explicitly postulated that the internal noise of the detector was large compared with the noise produced by steady radiation falling on the detector (radiation noise). In this Part II both kinds of noise are taken into account. In a sense, therefore, the present paper includes Part I. But in another sense the scope of the present paper is more restricted because the greater complexity with respect to the sources of noise has made it desirable to use throughout the simplifying assumption that was used only in Sec. 3 of Part I.

The emphasis is given not to the information capacity itself, but to the information efficiency, which is the information capacity divided by the mean power of the beam.

A parameter β is introduced that is a measure of the relative amount of the radiation noise and the internal detector noise when the mean signal power is chosen optimally. When β is zero, only radiation noise is present, and when β is infinite, only the internal noise is present. In Sec. 6 the information efficiency I is derived as a function of the parameter β. The two cases represented by the extreme values of β are discussed in Secs. 4 and 5. In Sec. 4 it is found that with symmetrical signal modulation, the maximum possible information efficiency is equal to the detective quantum efficiency of the detector. When, however, nonsymmetrical modulation is permitted, then the maximum possible information efficiency may be greater and is equal to QD multiplied by log2(1/p), where p is the probability that the gate is open.

The information efficiency in bits per photon is calculated in Sec. 8, for five different kinds of detectors: a multiplier phototube, human vision, photographic films, an image orthicon, and heat detectors. An error was made in Part I in the calculation of the information efficiency of the 1P21 phototube; the error is corrected in Sec. 8.

© 1962 Optical Society of America

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References

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  1. R. Clark Jones, J. Opt. Soc. Am. 50, 1166–1170 (1960).
    [CrossRef]
  2. L. Mandel, J. Opt. Soc. Am. 51, 797–798 (1961).
    [CrossRef]
  3. R. Clark Jones, J. Opt. Soc. Am. 50, 883–886 (1960).
    [CrossRef]
  4. R. Clark Jones, Advances in Electronics and Electron Phys. 11, 87–183 (1959), p. 104.
    [CrossRef]
  5. R. Clark Jones, J. Opt. Soc. Am. 52, 493–501 (1962).
    [CrossRef]
  6. P. Görlich, Z. Physik 101, 335–342 (1936).
    [CrossRef]
  7. See reference 4, p. 121.
  8. S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).
  9. E. Baumgardt, Optica Acta 7, 305–316 (1960).
    [CrossRef]
  10. A. Rose, J. Opt. Soc. Am. 38, 196–208 (1948).
    [CrossRef] [PubMed]
  11. A. Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).
  12. R. Clark Jones, J. Wash. Acad. Sci. 47, 100–108 (1957).
  13. R. Clark Jones, J. Opt. Soc. Am. 49, 645–653 (1959).
    [CrossRef]
  14. R. Clark Jones, Phot. Sci. Eng. 2, 198–204 (1958).
  15. R. Clark Jones, Phot. Sci. Eng. 2, 57–65 (1958).
  16. R. Clark Jones, J. Opt. Soc. Am. 51, 1159–1171 (1961).
    [CrossRef]
  17. R. Clark Jones, J. Soc. Motion Picture Television Engrs. 68, 462–466 (1959).
  18. R. Clark Jones, Advances in Electronics 5, 1–96 (1953).
  19. H. Kortum, Jenaer Rundschau 1, 1 (1956).

1962 (1)

1961 (2)

1960 (3)

1959 (3)

R. Clark Jones, Advances in Electronics and Electron Phys. 11, 87–183 (1959), p. 104.
[CrossRef]

R. Clark Jones, J. Soc. Motion Picture Television Engrs. 68, 462–466 (1959).

R. Clark Jones, J. Opt. Soc. Am. 49, 645–653 (1959).
[CrossRef]

1958 (2)

R. Clark Jones, Phot. Sci. Eng. 2, 198–204 (1958).

R. Clark Jones, Phot. Sci. Eng. 2, 57–65 (1958).

1957 (2)

A. Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).

R. Clark Jones, J. Wash. Acad. Sci. 47, 100–108 (1957).

1956 (1)

H. Kortum, Jenaer Rundschau 1, 1 (1956).

1953 (1)

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

1948 (1)

1942 (1)

S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

1936 (1)

P. Görlich, Z. Physik 101, 335–342 (1936).
[CrossRef]

Baumgardt, E.

E. Baumgardt, Optica Acta 7, 305–316 (1960).
[CrossRef]

Clark Jones, R.

R. Clark Jones, J. Opt. Soc. Am. 52, 493–501 (1962).
[CrossRef]

R. Clark Jones, J. Opt. Soc. Am. 51, 1159–1171 (1961).
[CrossRef]

R. Clark Jones, J. Opt. Soc. Am. 50, 883–886 (1960).
[CrossRef]

R. Clark Jones, J. Opt. Soc. Am. 50, 1166–1170 (1960).
[CrossRef]

R. Clark Jones, J. Soc. Motion Picture Television Engrs. 68, 462–466 (1959).

R. Clark Jones, Advances in Electronics and Electron Phys. 11, 87–183 (1959), p. 104.
[CrossRef]

R. Clark Jones, J. Opt. Soc. Am. 49, 645–653 (1959).
[CrossRef]

R. Clark Jones, Phot. Sci. Eng. 2, 198–204 (1958).

R. Clark Jones, Phot. Sci. Eng. 2, 57–65 (1958).

R. Clark Jones, J. Wash. Acad. Sci. 47, 100–108 (1957).

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

Görlich, P.

P. Görlich, Z. Physik 101, 335–342 (1936).
[CrossRef]

Hecht, S.

S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Kortum, H.

H. Kortum, Jenaer Rundschau 1, 1 (1956).

Mandel, L.

Pirenne, M. H.

S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Rose, A.

A. Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).

A. Rose, J. Opt. Soc. Am. 38, 196–208 (1948).
[CrossRef] [PubMed]

Schlaer, S.

S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

Advances in Biol. and Med. Phys. (1)

A. Rose, Advances in Biol. and Med. Phys. 5, 211–242 (1957).

Advances in Electronics (1)

R. Clark Jones, Advances in Electronics 5, 1–96 (1953).

Advances in Electronics and Electron Phys. (1)

R. Clark Jones, Advances in Electronics and Electron Phys. 11, 87–183 (1959), p. 104.
[CrossRef]

J. Gen. Physiol. (1)

S. Hecht, S. Schlaer, and M. H. Pirenne, J. Gen. Physiol. 25, 819–840 (1942).

J. Opt. Soc. Am. (7)

J. Soc. Motion Picture Television Engrs. (1)

R. Clark Jones, J. Soc. Motion Picture Television Engrs. 68, 462–466 (1959).

J. Wash. Acad. Sci. (1)

R. Clark Jones, J. Wash. Acad. Sci. 47, 100–108 (1957).

Jenaer Rundschau (1)

H. Kortum, Jenaer Rundschau 1, 1 (1956).

Optica Acta (1)

E. Baumgardt, Optica Acta 7, 305–316 (1960).
[CrossRef]

Phot. Sci. Eng. (2)

R. Clark Jones, Phot. Sci. Eng. 2, 198–204 (1958).

R. Clark Jones, Phot. Sci. Eng. 2, 57–65 (1958).

Z. Physik (1)

P. Görlich, Z. Physik 101, 335–342 (1936).
[CrossRef]

Other (1)

See reference 4, p. 121.

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Tables (3)

Tables Icon

Table I A summary of the results calculated in Sec. 8. All of the results are calculated for a wavelength near 0.45 μ.

Tables Icon

Table II A summary of the results for the four Kodak films at a wavelength of 430 mμ. The films are fully identified, and the developing conditions are described in reference 15.

Tables Icon

Table III Exact maximization of Eq. (6.1).

Equations (70)

Equations on this page are rendered with MathJax. Learn more.

P ¯ = P ( t ) Av .
( t ) = P ( t ) - P ¯
P 2 = 2 ( t ) Av .
F ( f ) = - 1 2 T 1 2 T ( t ) e - 2 π i f t d t ,
W P ( f ) = 2 T - 1 F ( f ) 2 Ens . Av .
P 2 = 0 W P ( f ) d f .
m = 1 / ( e E / k T s - 1 ) ,
λ T s 14 400 micron-degrees ,
α P / P ¯ .
D ( f ) = R ( f ) / [ W N ( f ) ] 1 2 ,
Δ m = 2 [ 0 [ D ( f ) ] 2 d f ] 1 2 .
Δ m = 2 f m 1 2 D ,
D max 2 = 1 / 2 E P ¯ .
Q D = 2 E P ¯ D 2 .
Q D = 2 E P ¯ D r 2 ,
1 / D 2 = 1 / D i 2 + 1 / D r 2 .
1 / D 2 = 1 / D i 2 + 2 E P ¯ / Q D .
β α Q D / ( E f m 1 2 D i ) = 2 α Q D / ( E Δ m i ) ,
γ Q D P ¯ / ( 2 f m E )
C = 0 log 2 [ 1 + D ( f ) 2 W P ( f ) ] d f
I C / P ¯ .
I = α I I ,
I = f m P ¯ log 2 [ 1 + α 2 P ¯ 2 f m ( 1 D i 2 + 2 E P ¯ Q D ) - 1 ] .
I 1 ( f ) = P ¯ - 1 log 2 [ 1 + α 2 D i ( f ) 2 P ¯ 2 ] ,
I = f m P ¯ log 2 [ 1 + α 2 Q D P ¯ 2 f m E ] .
I = I E = ( f m E / P ¯ ) log 2 [ 1 + α 2 γ ] ,
I = 1 2 [ log 2 e ] α 2 Q D = 0.72 α 2 Q D
I = Q D ,             α = 1 ,             γ 1.
I = Q D log 2 ( 1 / p ) ,
I = f m P ¯ log 2 [ 1 + α 2 P ¯ 2 D i 2 f m ] .
I = α f m 1 2 D i = 1 2 α Δ m i
α 2 P ¯ 2 D i 2 / f m = 1.
4 α 4 γ 2 / β = 1.
I = 1 2 Δ m i ,
I 1 ( f ) = α D i ( f )
α 2 P ¯ 2 D i 2 ( f ) = 1.
I 1 ( f ) = D i ( f ) ,
I = f m P ¯ log 2 [ 1 + β 2 P ¯ 2 ρ 2 + 2 ρ P ¯ ] ,
ρ Q D / ( E D i 2 ) .
I = f m / P ¯ ,
ρ 2 + 2 ρ P ¯ - β 2 P ¯ 2 = 0
P ¯ = ρ [ 1 + ( 1 + β 2 ) 1 2 ] / β 2 .
I = ( 1 2 α 2 Q D ) · 2 / [ 1 + ( 1 + β 2 ) 1 2 ]
I = ( 1 2 α Δ m i ) · β / [ 1 + ( 1 + β 2 ) 1 2 ] ,
I = 1 2 α 2 Q D
I = 1 2 α Δ m i .
I = 1 2 Q D .
β 2 Q D / Δ m i ,
ρ 4 f m Q D / Δ m i .
I = ( 1 2 Δ m i ) β / [ 1 + ( 1 + β 2 ) 1 2 ] ,
I = Q D / [ 1 + ( 1 + β 2 ) 1 2 ] .
I = 1 / ( 2 / Q D + 2 / Δ m i )
I = 1 / ( 1 / Q D + 2 / Δ m i ) .
P ¯ = 2 f m ( 1 / Δ m i + 1 / Q D ) .
Ē = E N + 1 / Q D .
I = 0.1 bit / photon .
I = 7.2 × 10 - 3 bit / photon .
Δ m i = 3.99 × 10 - 5 / photon .
I = 2.0 × 10 - 5 bit / photon .
Δ m i = 0.833 × 10 11 / joule ,
Δ m i = 3.31 × 10 - 8 / photon
I = 1.65 × 10 - 8 bit / photon .
( ρ / f m ) I = x log 2 S ,
x ρ / P ¯ ,
S 1 + β 2 / ( x 2 + 2 x ) .
L S log e S / [ 2 ( S - 1 ) ] = ( x + 1 ) / ( x + 2 ) ,
x = ( 2 L - 1 ) / ( 1 - L )
β 2 = ( x 2 + 2 x ) ( S - 1 ) .
x 2 = β 2 / [ 1 + ( 1 + β 2 ) 1 2 ] .
I exact / I S = 2 = ( x log S ) / x 2