Abstract

A general phenomenological theory of the static and dynamic behavior of bolometers is presented. The theory assumes as given the fundamental relations between the temperature and resistance of the bolometer, and the past history of the power dissipated within it. From these basic properties are derived a number of the properties of more immediate interest, such as electrical impedance, responsivity as a function of frequency, and the static load curve.

Several equivalent circuits are developed to represent the behavior of the bolometer as a function of frequency at a single operating point. A two-terminal equivalent circuit is described that represents the electrical impedance as a function of frequency. In order to represent the response of the bolometer to incident radiation as a function of frequency, a four-terminal equivalent circuit is described.

An electrical bridge is described that permits one to measure by purely electrical means the electrical response that a bolometer would have to radiation of any given time dependence, including radiation that varies sinusoidally. By purely electrical means and without the need of a radiation source (calibrated or otherwise), the bridge provides a precise measurement of the bolometer’s responsivity (output volts per watt of incident radiation) as a function of frequency. An electrical signal S(t) at the input of the bridge produces the same electrical output as would be produced in the normal use of the bolometer by a radiation signal with the same wave form as S(t).

The presentation is in three parts: static performance; stability; and dynamic performance.

© 1953 Optical Society of America

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References

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  1. Marcel J. E. Golay, Rev. Sci. Instr. 18, 347–356 (1947).
    [Crossref]
  2. Billings, Hyde, and Barr, J. Opt. Soc. Am. 37, 123 (1947).
    [Crossref] [PubMed]

1947 (2)

Marcel J. E. Golay, Rev. Sci. Instr. 18, 347–356 (1947).
[Crossref]

Billings, Hyde, and Barr, J. Opt. Soc. Am. 37, 123 (1947).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

Rev. Sci. Instr. (1)

Marcel J. E. Golay, Rev. Sci. Instr. 18, 347–356 (1947).
[Crossref]

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Figures (14)

F. 1
F. 1

Possible load curves (a) for a positive bolometer (b) for a negative bolometer. The current I flows through the bolometer, and E is the resulting voltage across the bolometer. The arrow on the curves show the direction of increasing temperature rise Δ.

F. 2
F. 2

Possible load curves (a) for a positive bolometer (b) for a negative bolometer, on logarithmic coordinate scales. The diagonal axes correspond to the resistance R and the power P.

F. 3
F. 3

Permitted and forbidden directions of the directed load curve. The permitted direction of the load curve for positive and for negative bolometers does not overlap. The forbidden directions are those in which an increase in the dissipated power would cause the bolometer temperature to decrease, which behavior is impossible. The arrows on the directed load curves show the direction of increasing temperature rise Δ.

F. 4
F. 4

The basic circuit usually employed with a bolometer.

F. 5
F. 5

Demonstration that the operating point P is a conditionally stable operating point.

F. 6
F. 6

This figure is used in the elucidation of the general criteria for the stability of an operating point of a positive bolometer.

F. 7
F. 7

General criterion for the stability of an operating point of a positive bolometer. If the load line intersects the load curve with a slope that lies in the range indicated by u or u′, the operating point represented by the intersection is unstable. Otherwise, it is stable.

F. 8
F. 8

This figure is used in the elucidation of the general criteria for the stability of an operating point of a negative bolometer.

F. 9
F. 9

General criterion for the stability of an operating point of a negative bolometer. If the load line intersects the load curve with a slope that lies in the range indicated by u or u′, the operating point represented by the intersection is unstable. Otherwise, it is stable.

F. 10
F. 10

Showing the way that the magnitude of the electrical impedance depends on the frequency for a positive bolometer, and also showing an electrical circuit that has, for a given operating point, the same electrical impedance at all frequencies as does a positive bolometer with a single time constant.

F. 11
F. 11

Showing the way that the magnitude of the electrical impedance depends on the frequency for a negative bolometer, and also showing an electrical circuit that has, for a given operating point, the same electrical impedance at all frequencies as does a negative bolometer with a single time constant.

F. 12
F. 12

Electrical networks representing the response of a bolometer to radiation signals of arbitrary wave form. At a given operating point, the circuit elements are independent of frequency for bolometers that have single time constants. The radiation signal is represented in (a) by an infinite impedance current generator, and in (b) by a zero impedance voltage generator.

F. 13
F. 13

The basic elements of a bridge circuit that permits one to measure the responsivity of a bolometer by purely electrical means.

F. 14
F. 14

The modified bridge that is necessary if the load impedance of the bolometer is not purely resistive.

Equations (104)

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R = E / I ,
P = E I .
Δ = T T c .
P = P ( Δ ) ,
R = R ( Δ ) ,
d P ( Δ ) / d Δ > 0 .
α = d log R / d Δ ,
E ( Δ ) = ( P ( Δ ) R ( Δ ) ) 1 2 ,
I ( Δ ) = ( P ( Δ ) / R ( Δ ) ) 1 2 .
P = k Δ ,
R = R c ( 1 + a Δ ) ,
α = a / ( 1 + a Δ ) .
E ( Δ ) = [ k Δ R c ( 1 + α Δ ) ] 1 2 ,
I ( Δ ) = [ k Δ R c ( 1 + α Δ ) ] 1 2 .
I m 2 = k / ( α R c ) .
γ ¯ = R / R c .
E / I m R c = [ γ ¯ ( γ ¯ 1 ) ] 1 2 ,
I / I m = [ ( γ ¯ 1 ) / γ ¯ ] 1 2 ,
P / I m 2 R c = γ ¯ 1 .
E / I m R c = I m I / ( I m 2 I 2 ) .
E = E / ( I m R c )
I = I / I m
1 / E = ( 1 / I ) I .
d P / d Δ = ( d P / d R ) · ( d R / d Δ ) .
Z = d E / d I ,
Z = R ( d log E / d log I ) ,
Z = R d ( log P + log R ) d ( log P log R ) ,
Z = R [ ( k + α P ) / ( k α P ) ] ,
k = d P / d T ,
H = 1 I 2 d P d R = d log P d log R .
H = k / ( α P ) .
Z = R [ ( H + 1 ) / ( H 1 ) ] .
α = k / P = d log P / d T .
α = k / P .
| Z | R α > 0 ,
| Z | R α < 0 .
| d log E / d log I | 1 α > 0 ,
| d log E / d log I | 1 α < 0 .
P = I m 2 ( R R c ) ,
H = I m 2 / I 2 .
Z = R [ ( I m 2 + I m ) / ( I m 2 I 2 ) ]
R = d E / d Q .
R = 1 I · 1 H 1 + ( R / Z L ) ( H + 1 ) ,
R = 1 I · 1 1 + Z / Z L · 1 H 1 .
R = 1 / 2 I H = 1 / 2 I ( d log R / d log P ) .
R = I ( 1 + R / Z L ) I m 2 ( 1 R / Z L ) I 2 .
R R = I / 2 I m 2 = E / 2 I m 2 R .
R = I / ( I m 2 I 2 ) = E / I m 2 R c ,
R R / R = ( I m 2 I 2 ) / 2 I m 2 = 1 / 2 γ ¯ .
R Z = I / [ 2 ( I m 2 I 2 ) ] = E / 2 I m 2 R c .
1 R < 1 Z L < 1 Z .
R < Z L < Z ,
H ( ω ) d log W d log R = 1 I 2 d ( E I + Q ) d ( E / I ) .
H ( ω ) = 1 I 2 d P d R = d log P d log R = d log I d log R + 1 = d log E d log R 1 = ( d log E / d log I ) + 1 ( d log E / d log I ) 1 = Z ( ω ) + R Z ( ω ) R .
H ( ω ) as ω , H ( ω ) = .
G ( ω ) = d W / d Δ .
H ( ω ) = [ ( 1 / I 2 ) ( d Δ / d R ) ] G ( ω ) ,
H ( ω ) = ( 1 / α P ) G ( ω ) .
C ( d Δ / d t ) + k Δ = W .
G ( ω ) = k ( 1 + i ω τ ) ,
τ C / k
H ( ω ) = ( k / α P ) ( 1 + i ω τ ) = H · ( 1 + i ω τ )
Z ( ω ) = R ( d log E / d log I ) ,
Z ( ω ) = R H ( ω ) + 1 H ( ω ) 1 .
H ( ω ) = Z ( ω ) + R Z ( ω ) R .
Z ( ) = R .
Z ( ω ) = R 1 + i ω τ + H 1 1 + i ω τ H 1 .
1 R p = 1 2 R ( k α P 1 ) C = ( 1 / 2 R ) ( k τ / α P ) ,
R n = R 2 ( k | α | P 1 ) , L = R 2 k τ | α | P .
Z ( ω ) = R I m 2 ( 1 + i ω τ ) + I 2 I m 2 ( 1 + i ω τ ) I 2
R ( ω ) d E / d Q .
Z L ( ω ) = d E / d I .
I d E + E d I + d Q d E I E d I I 2 = I 2 H ( ω ) ,
1 R / Z L + 1 / ( I R ( ω ) ) 1 R / Z L = H ( ω ) .
R ( ω ) = 1 I 1 H ( ω ) 1 + ( R / Z L ( ω ) ) ( H ( ω ) + 1 ) .
R ( ω ) = 1 2 I · Z L ( ω ) R · Z ( ω ) R Z ( ω ) + Z L ( ω ) .
R ( ω ) = 1 I 1 1 + Z ( ω ) / Z L ( ω ) 1 H ( ω ) 1 .
R ( ω ) = 1 2 I H ( ω ) , Z L = R ,
R ( ω ) = 1 2 I Z ( ω ) R Z ( ω ) + R , Z L = R .
R ( ω ) = 1 I ( H ( ω ) 1 ) , Z L = ,
R ( ω ) = 1 2 I Z ( ω ) R R , Z L = .
i g = 1 2 Q / E
e g = 1 2 Q / I ,
R ( ω ) = R / ( 1 + i ω τ r ) ,
τ r = H ( R L + R ) R L ( H 1 ) + R ( H + 1 ) τ .
τ r τ as I 0 .
τ r = τ if R = R L .
τ r τ as α R L α R .
R ( ω ) = I I m 2 ( 1 + i ω τ ) I 2 + ( R / Z L ( ω ) ) ( I m 2 ( 1 + i ω τ ) + I 2 ) .
τ r = I m 2 ( R + R L ) I m 2 ( R + R L ) + I 2 ( R R L ) τ .
R 2 / R 1 = R / R L .
Q = K e A ,
K = 2 I R / ( R + R L ) .
R ( ω ) = ( R + R L / 2 I R ) T ( ω ) ,
T ( ω ) e B / e A .
R 2 R 1 + R 2 e A = R R + R L e A .
Z ( ω ) Z ( ω ) + R L e A .
e B = ( Z ( ω ) Z ( ω ) + R L R R + R L ) e A .
T ( ω ) = R L R + R L · Z ( ω ) R Z ( ω ) + R L ,
T ( ω ) = 2 I R R + R L R ( ω ) ,
R L Z L ( 0 ) .
1 Z L ( ω ) = 1 R L + 1 Z s ( ω ) .
Z L ( ω ) R L R L + Z ( ω ) Z L ( ω ) + Z ( ω ) .
T ( ω ) = R R + R L · Z L ( ω ) R Z ( ω ) R Z ( ω ) + Z L ( ω ) ,