Abstract

An upper limit is obtained for the sensitivity of any radiation detector which satisfies Lambert’s law and whose effective area is independent of the wave-length of the radiation. Expressions for the minimum detectable power are given by Eqs. (3.8) and (5.4).

The treatment given is based on thermodynamics, and makes no use of statistical mechanics. In the special case in which the fluctuations in the output of the detector are due entirely to fluctuations in the radiation, the results obtained agree with those obtained from a statistical mechanical treatment of the fluctuations in the radiation.

The treatment of the first five sections is confined to detectors, called “temperature detectors”, which operate by means of the change in their temperature brought about by the radiation. Such detectors include the thermocouple, the bolometer, and the radiometer. In Sections VI and VIII it is shown that the results hold for any detector satisfying the conditions in the first paragraph, provided that the emissivities are interpreted as quantum efficiencies. Section VII indicates that if the condition regarding Lambert’s law is given up, then there exists no lower limit on the minimum detectable power. Section IX is devoted to numerical examples and a comparison with the reported sensitivities of bolometers.

© 1947 Optical Society of America

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Corrections

R. Clark Jones, "Erratum: The Ultimate Sensitivity of Radiation Detectors," J. Opt. Soc. Am. 39, 343-343 (1949)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-39-5-343

References

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  1. Paul S. Epstein, Textbook of Thermodynamics (John Wiley and Sons, Inc., New York, 1937), pp. 389–390.
  2. J. K. Roberts, Heat and Thermodynamics (Blackie and Son, Ltd., London, 1933), second edition, Chapter 19.
  3. E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Teddington, England, 1935), fourth edition, Chapter 13.
  4. S. O. Rice, Bell. Syst. Tech. J. 23, 282 (1944).
    [Crossref]
  5. R. J. Havens, J. Opt. Soc. Am. 36, 355A (1946).
  6. Ralph W. Engstrom, J. Opt. Soc. Am. 37, 420 (1947).
    [Crossref]
  7. J. M. W. Milatz and H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369 (1943).
    [Crossref]

1947 (1)

1946 (1)

R. J. Havens, J. Opt. Soc. Am. 36, 355A (1946).

1944 (1)

S. O. Rice, Bell. Syst. Tech. J. 23, 282 (1944).
[Crossref]

1943 (1)

J. M. W. Milatz and H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369 (1943).
[Crossref]

Engstrom, Ralph W.

Epstein, Paul S.

Paul S. Epstein, Textbook of Thermodynamics (John Wiley and Sons, Inc., New York, 1937), pp. 389–390.

Havens, R. J.

R. J. Havens, J. Opt. Soc. Am. 36, 355A (1946).

Milatz, J. M. W.

J. M. W. Milatz and H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369 (1943).
[Crossref]

Rice, S. O.

S. O. Rice, Bell. Syst. Tech. J. 23, 282 (1944).
[Crossref]

Roberts, J. K.

J. K. Roberts, Heat and Thermodynamics (Blackie and Son, Ltd., London, 1933), second edition, Chapter 19.

van der Velden, H. A.

J. M. W. Milatz and H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369 (1943).
[Crossref]

Watson, G. N.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Teddington, England, 1935), fourth edition, Chapter 13.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Teddington, England, 1935), fourth edition, Chapter 13.

Bell. Syst. Tech. J. (1)

S. O. Rice, Bell. Syst. Tech. J. 23, 282 (1944).
[Crossref]

J. Opt. Soc. Am. (2)

R. J. Havens, J. Opt. Soc. Am. 36, 355A (1946).

Ralph W. Engstrom, J. Opt. Soc. Am. 37, 420 (1947).
[Crossref]

Physica (1)

J. M. W. Milatz and H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369 (1943).
[Crossref]

Other (3)

Paul S. Epstein, Textbook of Thermodynamics (John Wiley and Sons, Inc., New York, 1937), pp. 389–390.

J. K. Roberts, Heat and Thermodynamics (Blackie and Son, Ltd., London, 1933), second edition, Chapter 19.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Teddington, England, 1935), fourth edition, Chapter 13.

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Figures (1)

F. 1
F. 1

The solid line shows the theoretical, minimum detectable power for a derector whose quantum efficiency is unity and whose effective area is one square millimeter at all wave-lengths, as predicted by Eqs. (3.8) and (5.4). The detector is assumed to have a temperature of 300°F, and to obey Lambert’s law. The dashed line shows the minimum detectable power, according to an engineering estimate made by R. J. Havens of what might be done in constructing bolometers with current materials and techniques. The four circles show the minimum detectable powers, computed on the basis specified in the Appendix from performance data supplied by their designers, of four bolometers constructed during or since the last war.

Equations (49)

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C ( d θ / d t ) + κ θ = 0 .
θ T T 0 .
C ( d θ / d t ) + κ θ = H ( t ) .
H = H 0 cos 2 π f t .
θ = H 0 / κ [ 1 + ( 2 π f τ ) 2 ] 1 2 ,
τ C / κ .
τ ( d θ / d t ) + θ = ( H ( t ) / κ ) .
( l l ¯ ) 2 Av = k / 2 S l 2 ,
θ r.m.s . 2 = θ 2 Av = k / 2 S T 2 .
S T = ( 1 T ) ( Q T ) = C T .
θ r.m.s . 2 = k T 2 / C .
θ r.m.s . 2 = k T 2 / ( κ τ ) .
H m = κ θ r.m.s . / s .
H m 2 = κ k T 2 / ( s 2 τ ) .
κ = d H / d T ,
H = A r σ T 4 ,
κ = A σ d ( r T 4 ) d T ,
κ = 4 A u σ T 3 ,
u = 1 4 T 3 d ( r T 4 ) d T = d ( r T 4 ) d T 4 .
H m 2 = 4 A σ T 4 k T u / ( s 2 τ ) .
s = d A H s ( ν , r ) ( ν , r ) d ν d A H s ( ν , r ) d ν = d A H s ( ν , r ) ( ν , r ) d ν H m .
r = d A J ( ν , T ) ( ν , r ) d ν d A J ( ν , T ) d ν = d A J ( ν , T ) ( ν , r ) d ν A σ T 4 ,
κ = d A J ( ν , T ) T ( ν , r ) d ν ,
u = d A J ( ν , T ) T ( ν , r ) d ν 4 A σ T 3 .
J ( ν , T ) = 2 π h ν 3 c 2 ( e h ν / k T 1 ) ,
J ( ν , T ) T = 2 π h 2 ν 4 c 2 k T 2 e h ν / k T ( e h ν / k T 1 ) 2 .
ξ h ν / k T .
κ = 2 π k c 2 ( k T h ) 3 d A 0 ξ 4 e ξ ( e ξ 1 ) 2 ( ξ , r ) d ξ .
u = 15 4 π 4 A d A 0 ξ 4 e ξ ( e ξ 1 ) 2 ( ξ , r ) d ξ ,
σ 2 π 5 15 k 4 c 2 h 3 .
0 ξ 4 e ξ ( e ξ 1 ) 2 d ξ = 24 ξ ( 4 ) = 4 π 4 15 ,
1 / τ 4 0 | Z ( f ) | 2 d f .
Z ( f ) = 1 1 + 2 π i f τ ,
H m 2 = 4 κ k T 2 s 2 0 | Z ( f ) | 2 d f | Z ( f ) | 2 .
H m 2 = 16 A σ T 4 k T u s 2 0 | Z ( f ) | 2 d f | Z ( f ) | 2 .
16 A σ T 4 κ T .
H m = 2.76 × 10 12 watt τ 1 2 ,
σ T 4 = 4.64 × 10 2 watt / cm 2 , k T = 4.11 × 10 21 joule , 4 A = 0.04 cm 2 .
H m = 3.0 × 10 12 watt τ ,
u s 2 = 100 15 4 π 2 ξ 0 ξ 4 e ξ ( 1 e ξ ) 2 d ξ
ξ 0 h ν 0 / k T ,
ξ 0 = 48 .
u / s 2 = 3.85 ξ 0 4 e ξ 0 3 × 10 14 .
H m 5 × 10 19 watt τ 1 2
f = 1 / 2 π τ .
A = 3.6 mm 2 , τ = 50 milliseconds .
A = 0.2 mm 2 , τ = 4.1 milliseconds .
A = 0.6 mm 2 , τ = 3 milliseconds .
A = 6 mm 2 , τ = 4 milliseconds .