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Measuring and modeling the frequency response of infrared detectors
INFRAREDPHYSICS &TECHNOLOGY ELSEVIER
InfraredPhysics & Technology38 (1997) 69-74
Measuring and modeling the frequency response of infrared detectors John C. Brasunas
NASA/Goddard SpaceFlight Center, Code 693, Greenbelt, MD 20771, USA Received 20 March 1996
Abstract
It is shown that a light-emitting diode is a simple way of characterizing the frequency response of an infrared detector over several decades of frequency, since the diode has a linear relationship between input current and output radiant power. By normalizing a detector's output voltage to the diode input current, the detector frequency response can be measured as simply as one measures the transfer function of an electrical filter. This is advantageous in studying higher-order effects in a thermal-type detector response, where the detector deviates from the Lorentzian behavior predicted by a simple model of a lumped heat capacity connected to a thermal bath by a lumped thermal conductance path. An example is given of a thermoelectric detector with marked non-Lorentzian behavior. 1. Introduction
The response of an infrared detector is specified in terms of responsivity, the ratio of output volts to input radiant power. Responsivity is typically a function of bias voltage (if any), detector temperature, frequency, wavelength, and degree of thermal isolation (for a thermal-type detector). It is commonly a good assumption that the detector output voltage signal vs is factorable [1], vs( VB, qbs, A a, A, f ) = Vs(VB, ~bs, a d ) v ( A ) v ( f )
(1)
where Vs is the bias voltage, 4's is the input radiant power, A d is the detector area, A is the wavelength and f is the operating frequency. A typical way of measuring v ( f ) is mechanically chopping a blackbody source; a variation of this utilizes not only the chopping frequency itself but also the harmonics, since mechanical chopping is typically nearly square-wave [2]. Another common way of measuring
v ( f ) is electrically modulating the output of a lightemitting diode (LED). We will present results from using an electrically modulated LED, showing that this is an effective way of characterizing the higher-order deviations from the first-order Lorentzian behavior of a thermal-type detector.
2. LED characterization of the frequency response
An LED is an electro-optic device that has a linear relationship between input current and output radiant power (other things such as LED temperature being held fixed). Thus the LED has the virtue of converting a responsivity measurement, typically involving a mixed ratio of detector electrical output and source optical output, into a purely electrical ratio of detector output voltage and source input current. Then measuring responsivity becomes as
1350-4495/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 1350-4495(96)00037-0
J.C. Brasunas / lnfrared Physics & Technology 38 (1997)69-74
70
100.00
1.00 --
0.80 10.00
C3 0.60 oJ
n,"
c
1.00
0.40
o.oo
~ - - = - . . . . . . . . . . . F- ~ - -
"' i 1000000 20000,00 Wavenumbers
0.00
'
r 30000.00
Fig. 1. Normalizedspectrumof the infrared LED. straightforward as measuring the transfer function of an electrical filter or amplifier. When biased above 1.2 V, the LED used (Archer 276-143c) emits in the infrared; see Fig. 1. Peak emission is near 10,800 c m - = (around 920 nm wavelength). The LED is biased with a Stanford Research Systems DS345 function generator (50 fl internal impedance), providing an internal sine wave bias from 2.7 V to 3.7 V; the frequency is swept from 0.1 Hz to a few hundred hertz. A l 0 l'~ resistor is placed in series with the LED to monitor the LED input current. A two-channel spectrum analyzer (HewlettPackard 3582A) directly measures the responsivity by measuring the transfer function, the ratio of the output voltage of the detector under test to the input current to the LED (see Fig. 2). As the LED is a non-thermal device, it is possible that the square-wave chopping can be near ideal to (
m
-
-
IR Stanford Research Systems DS 345
LEE
;;,,o,o,
)
I
pre.amp
[ch. A
tO
HP 3582A spectrum analyzer
Fig. 2. Schematic of the set-up for converting a detector frequency response measurement into a transfer function measurement.
0,01
0.10
~ '";J~'[ 1.00
~ ~ ~'~l 10.00 Frequency (Hz)
i
~ =~l,~, I
,
100.00
Fig. 3. Amplitude frequency response of the thermopile (thick line), and a fit based on a single Lorentzian with 88.0 ms time constant (thin line).
high frequencies. This was verified by first measuring the frequency response of a silicon photo-diode detector (Newport 818-SL), which should have about a 2 /~s rise time and thus a very flat frequency response. A flat response was seen; compared with the near-dc response, the 200 Hz response was down by 1%. Thus the electrically modulated LED can be used directly to measure the frequency response to a few hundred Hz, without the need to normalize the results to a reference detector. The frequency response was next measured for a thin-film thermopile detector (Oriel 7109); this was obtained with a metal oxide coating which has a nominal time constant between 50-100 ms. Amplitude frequency response data from 0.4 to 250 Hz are shown in Fig. 3 (thick line): these data were obtained by running the spectrum analyzer three times, from dc to 10 Hz, from dc to 100 Hz, and from dc to 250 Hz, each time discarding the 5 lowest-frequency points. The thin line in Fig. 3 is a fit using a Lorentzian response with an 88.0 ms time constant; the fit is good at low frequencies, but the roll-off appears to be faster than a single Lorentzian at frequencies approaching and exceeding 100 Hz. The phase frequency response data from 0.4 to 250 Hz are shown in Fig. 4 (thick line), and a product-oftwo-Lorentzians fit with a second time constant of
J.C. Brasunas / Infrared Physics & Technology 38 (1997) 69-74
0.00l
71
mensions of the detector element (absorber and thermometer), or the dimensions of the thermal conductance path (connecting wires for a bolometer, the thermoelectric material itself for a thermopile detector) are not small compared with the thermal diffusion length [3] l o, where
i -4O.OO
Io = 2~--//oJ
(4)
where the diffusivity a is defined as a = K/c
p
(5)
-120.00
0~o
1.0o
Frequency(Hz) 1ooo
Ioo.oo
Fig. 4. Phase frequency response of the thermopile (thick line), and a fit based on the product of two Lorentzians, with time constants of 88.0 ms and 0.7 ms (thin line).
0.7 ms is shown with the thin line. The double Lorentzian fit is quite good; each Lorentzian produces an asymptotic, high frequency phase delay of 90 ° .
3. Interpretation A thermal detector such as a thermopile should have a frequency response that is approximately Lorentzian [3],
at=
wo V'G~ + co2C 2
(2)
where AT is the amplitude of the detector temperature response, Wo is the amplitude of input heat radiation, oJ = 2~rf, G is the thermal conductance of the detector element to its surroundings, and C is the thermal or heat capacity of the detector element. The time constant r is commonly defined as
z= C//G.
(3)
To be precisely Lorentzian, the detector has to behave as a lumped heat capacity, coupled to a thermal bath with a lumped thermal conductance path. Deviations from Lorentzian behavior occur when the di-
where K is the thermal conductivity, Cp is the specific heat and p is the density. Near-Lorentzian behavior requires that l o be longer than detector length scales; in this case, G and C are approximately independent of oJ and the response of Eq. (2) is a Lorentzian. If the length of the thermal conductance path exceeds 1o/2, the thermal conductance G becomes frequency dependent [3] and the detector response of Eq. (2) is no longer a Lorentzian with a single time constant. If the dimensions of the absorber and thermometer are not small compared with l o, the lumped-element approach is also no longer correct, and deviations occur from Lorentzian behavior. Deviations are likely at high frequencies, because l o is a monotonically decreasing function of frequency. Deviations are also likely at non-cryogenic temperatures, since then the specific heat is at a maximum described by the law of Dulong and Petit. A thermal model more detailed than a lumped heat capacity and a lumped thermal conductance path is suggested by the observed frequency responses of Figs. 3 and 4. The secondary time constant is more than two orders of magnitude shorter than the primary time constant. Then by Eq. (2) the dominant contributor in the denominator at frequencies much greater than the reciprocal of the first time constant is oJC, and thus the secondary time constant cannot be explained by appealing to thermal diffusion changing the effective thermal isolation. Thermal diffusion within the thermal conductance path does change the effective contribution of the thermal conductance path to the total heat capacity of the detector [3], but in the wrong direction. Thermal diffusion lessens the contribution of the thermal con-
J.C. Brasunas / lnfiared Physics & Technology 38 (1997) 69-74
72
ductance path to the total heat capacity of the detector; this would produce a roll-off less rapid than Lorentzian - - the opposite of what is observed. Thus the non-Lorentzian behavior is not explained by appealing to thermal diffusion within the thermal conductance path between the thermometer and the thermal bath. Another possible explanation is that as the heat power is absorbed at the top of the absorber, it is damped within the absorber material before reaching the thermometer region (the thermoelectric material). This is analogous to the harmonic temperature variations induced by diurnal or seasonal variations in solar heating being exponentially damped within the interior of the Earth. The temperature variations at a depth x within a semi-infinite absorber, compared with the variations on the surface, satisfy the e x p o n e n t i a l attenuation [4] exp( 2 7r x / 2 ~ r l o) = exp( X / l o ) . While appeal to this mechanism does provide the leverage for suppressing the detector response below Lorentzian, this simple model would not explain the apparent good fit of a double Lorentzian. Perhaps a geometrically more detailed model [5] would. Rather than considering continuously distributed elements within the absorber-thermometer model, let us consider discretely distributed elements (Fig. 5). Incoming radiant heat W ( t ) flows into the absorber with heat capacity C 2. The absorber is coupled with thermal resistance R E to the thermometer with heat capacity C l, which is coupled with thermal resistance R¿ to the heat bath at temperature To. Let us -
-
compute the steady-state solution in response to W ( t ) = W e j°''. T h e steady-state solution is the thermometer temperature T~(t) = To + ~ T 1 e jc°t and the absorber temperature T2(t) = To + 6T~ e j~' + 6 T2 e j,o,. Now by elementary thermodynamics adding heat AQ to a body with heat capacity C causes a temperature rise AT, where AQ= CAT.
(6)
And in equilibrium, the rate of flow of heat W through a thermal resistance R (equals 1 / G ) in response to a temperature difference AT satisfies the relation W = - d(AQ)/dt=
l/RAT.
(7)
Defining the heat flow from the absorber to the thermometer to be W2(t)= W2eJO't, and the heat flow from the thermometer to the thermal bath to be Wl(t) = W l e i°'t, then by Eq. (7) 6T t = RiW l
(8)
and 6 T 2 = REW 2.
(9)
Also, by Eq. (6) W2(t ) - W , ( t ) = d ( A Q , ) / d t = C n d ( T l ( t ) ) / d t (10) or
W2 - W l = Cj jto~ T l .
w absorber C2 ( ~ W2 i ~ thermometer C1 W1
(11)
Also, by Eq. (6)
$ To+ dT1+ dT2
W ( t) - W2(t ) = d( A a 2 ) / d t = C2d(T2( t) ) / d t
(12)
R2 To+ dT1 RI
or
W-
W 2 = C 2 j t o ( ~ T , + 8T2).
(13)
Then combining Eqs. (8), (9), (11) and (13) to eliminate W t and W2, Fig. 5. A thermal model of the detector, with separately lumped absorber and thermometer,
t$ T 2 / R 2 - 6 T n / R t = C l jto6 T n
(14)
J.C. Brasunas / Infrared Physics & Technology 38 (1997) 69-74 and W-
8 T 2 / R 2 = C2joJ( a T I + 8T2).
(15)
Combining Eqs. (3), (14) and (15) to solve for 6T~, we have R~W aT I =
73
Table 1 Retrieved values for the model parameters from three runs of data, based on the criterion of minimizing the mean-square-difference between the phases of the data and of the model Test conditions r'1 (ms) r~ (ms) msd (o) 1st run, low amplitude 2nd run, low amplitude 3rd run, high amplitude
85.5 85.5 86
0.7l 0.71 0.715
4.817 3.62 3.022
j w r l ( c z / C ,) + ( 1 + j w r l) ( 1 + j w r 2) " (16)
Defining O = C 2 / C ~, then the detector response may be re-expressed as the product of two Lorentzians, R~W 6T, =
(1 + j t o r ' , ) ( 1 +jtor'2)
(17)
where r'j and r'2 are the effective time constants defined by r',r'2 = % %
5. Conclusions
(18)
and r', + r [ = % ( 1 + O )
creases msd by 0.4%; increasing r~ by 1% increases msd by 0.4%. Using the criterion that msd not be increased by more than 0.4%, these procedures appear to be capable of estimating r'~ to 1%, and r': to 1%, which is also the level of consistency among the three runs.
+ r z.
(19)
The solutions for r] and r [ are real. For example, if r, = r = g 2 = 1, the effective time constants are (3 + ~ ' - 1 / 2 .
4. Estimation of ~"1 and ~'[ We estimate r'~ and r~ by calculating the phase of the response function of Eq. (17) for an array of values of r ' l and r~, spaced by 0.5 ms for r ' 1 and 0.005 ms for r~. For each r ' l / r ' z pair, the meansquare difference (msd) is calculated between the theoretical curve and the phase of the measured data; the solution is the r'l/r'9 pair that minimizes msd. We took the detector data a second time to look for non-repeatability, and a third time to look for systematic effects by doubling the modulation amplitude of the LED, looking for artifacts due to detector non-linearity or to LED self-heating. The data as shown in Table 1 are quite reproducible for parameter retrieval. The criterion msd is quite sensitive to r', and to r~. Compared with the best fit for the high LED modulation case, increasing r'~ by 1% in-
It has been shown that electrical modulation of an LED is a simple way of characterizing the frequency response of an infrared detector over a wide frequency range. It is possible to retrieve not only the first-order Lorentzian response of a thermal detector, but also higher-order deviations from the Lorentzian. In particular the thermopile under study can be modeled with a double Lorentzian, indicative of a discretely distributed absorber and thermometer, and lumped thermal conductance paths between the absorber and thermometer and between the thermometer and the thermal bath. Thus the LED measurements are useful for the study of the thermal structure of the detector, in addition to the precise characterization of the frequency response itself.
Acknowledgements I thank Larry Herath of SSAI of Lanham, MD for suggesting I consider the phase response.
References [1] W.L. Wolfe and G.J. Zissis (eds.), The Infrared Handbook, revised Ed. (Office of Naval Research, Washington, DC, 1985).
74
J.C. Brasunas / lnfrared Physics & Technology 38 (1997) 69-74
[2] J.C. Brasunas and J.C. Balleza, Appl. Opt. 29 (1990) 14. [3] R.A. Smith, F.E. Jones and R.P. Chasmar, The Detection and Measurement of Infrared Radiation, 2nd Ed. (Oxford, London, 1968).
[4] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd Ed. (Oxford, Oxford, 1959). [5] S. Bauer and B. Ploss, J. Appl. Phys. 68 (1990) 6361.
InfraredPhysics & Technology38 (1997) 69-74
Measuring and modeling the frequency response of infrared detectors John C. Brasunas
NASA/Goddard SpaceFlight Center, Code 693, Greenbelt, MD 20771, USA Received 20 March 1996
Abstract
It is shown that a light-emitting diode is a simple way of characterizing the frequency response of an infrared detector over several decades of frequency, since the diode has a linear relationship between input current and output radiant power. By normalizing a detector's output voltage to the diode input current, the detector frequency response can be measured as simply as one measures the transfer function of an electrical filter. This is advantageous in studying higher-order effects in a thermal-type detector response, where the detector deviates from the Lorentzian behavior predicted by a simple model of a lumped heat capacity connected to a thermal bath by a lumped thermal conductance path. An example is given of a thermoelectric detector with marked non-Lorentzian behavior. 1. Introduction
The response of an infrared detector is specified in terms of responsivity, the ratio of output volts to input radiant power. Responsivity is typically a function of bias voltage (if any), detector temperature, frequency, wavelength, and degree of thermal isolation (for a thermal-type detector). It is commonly a good assumption that the detector output voltage signal vs is factorable [1], vs( VB, qbs, A a, A, f ) = Vs(VB, ~bs, a d ) v ( A ) v ( f )
(1)
where Vs is the bias voltage, 4's is the input radiant power, A d is the detector area, A is the wavelength and f is the operating frequency. A typical way of measuring v ( f ) is mechanically chopping a blackbody source; a variation of this utilizes not only the chopping frequency itself but also the harmonics, since mechanical chopping is typically nearly square-wave [2]. Another common way of measuring
v ( f ) is electrically modulating the output of a lightemitting diode (LED). We will present results from using an electrically modulated LED, showing that this is an effective way of characterizing the higher-order deviations from the first-order Lorentzian behavior of a thermal-type detector.
2. LED characterization of the frequency response
An LED is an electro-optic device that has a linear relationship between input current and output radiant power (other things such as LED temperature being held fixed). Thus the LED has the virtue of converting a responsivity measurement, typically involving a mixed ratio of detector electrical output and source optical output, into a purely electrical ratio of detector output voltage and source input current. Then measuring responsivity becomes as
1350-4495/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 1350-4495(96)00037-0
J.C. Brasunas / lnfrared Physics & Technology 38 (1997)69-74
70
100.00
1.00 --
0.80 10.00
C3 0.60 oJ
n,"
c
1.00
0.40
o.oo
~ - - = - . . . . . . . . . . . F- ~ - -
"' i 1000000 20000,00 Wavenumbers
0.00
'
r 30000.00
Fig. 1. Normalizedspectrumof the infrared LED. straightforward as measuring the transfer function of an electrical filter or amplifier. When biased above 1.2 V, the LED used (Archer 276-143c) emits in the infrared; see Fig. 1. Peak emission is near 10,800 c m - = (around 920 nm wavelength). The LED is biased with a Stanford Research Systems DS345 function generator (50 fl internal impedance), providing an internal sine wave bias from 2.7 V to 3.7 V; the frequency is swept from 0.1 Hz to a few hundred hertz. A l 0 l'~ resistor is placed in series with the LED to monitor the LED input current. A two-channel spectrum analyzer (HewlettPackard 3582A) directly measures the responsivity by measuring the transfer function, the ratio of the output voltage of the detector under test to the input current to the LED (see Fig. 2). As the LED is a non-thermal device, it is possible that the square-wave chopping can be near ideal to (
m
-
-
IR Stanford Research Systems DS 345
LEE
;;,,o,o,
)
I
pre.amp
[ch. A
tO
HP 3582A spectrum analyzer
Fig. 2. Schematic of the set-up for converting a detector frequency response measurement into a transfer function measurement.
0,01
0.10
~ '";J~'[ 1.00
~ ~ ~'~l 10.00 Frequency (Hz)
i
~ =~l,~, I
,
100.00
Fig. 3. Amplitude frequency response of the thermopile (thick line), and a fit based on a single Lorentzian with 88.0 ms time constant (thin line).
high frequencies. This was verified by first measuring the frequency response of a silicon photo-diode detector (Newport 818-SL), which should have about a 2 /~s rise time and thus a very flat frequency response. A flat response was seen; compared with the near-dc response, the 200 Hz response was down by 1%. Thus the electrically modulated LED can be used directly to measure the frequency response to a few hundred Hz, without the need to normalize the results to a reference detector. The frequency response was next measured for a thin-film thermopile detector (Oriel 7109); this was obtained with a metal oxide coating which has a nominal time constant between 50-100 ms. Amplitude frequency response data from 0.4 to 250 Hz are shown in Fig. 3 (thick line): these data were obtained by running the spectrum analyzer three times, from dc to 10 Hz, from dc to 100 Hz, and from dc to 250 Hz, each time discarding the 5 lowest-frequency points. The thin line in Fig. 3 is a fit using a Lorentzian response with an 88.0 ms time constant; the fit is good at low frequencies, but the roll-off appears to be faster than a single Lorentzian at frequencies approaching and exceeding 100 Hz. The phase frequency response data from 0.4 to 250 Hz are shown in Fig. 4 (thick line), and a product-oftwo-Lorentzians fit with a second time constant of
J.C. Brasunas / Infrared Physics & Technology 38 (1997) 69-74
0.00l
71
mensions of the detector element (absorber and thermometer), or the dimensions of the thermal conductance path (connecting wires for a bolometer, the thermoelectric material itself for a thermopile detector) are not small compared with the thermal diffusion length [3] l o, where
i -4O.OO
Io = 2~--//oJ
(4)
where the diffusivity a is defined as a = K/c
p
(5)
-120.00
0~o
1.0o
Frequency(Hz) 1ooo
Ioo.oo
Fig. 4. Phase frequency response of the thermopile (thick line), and a fit based on the product of two Lorentzians, with time constants of 88.0 ms and 0.7 ms (thin line).
0.7 ms is shown with the thin line. The double Lorentzian fit is quite good; each Lorentzian produces an asymptotic, high frequency phase delay of 90 ° .
3. Interpretation A thermal detector such as a thermopile should have a frequency response that is approximately Lorentzian [3],
at=
wo V'G~ + co2C 2
(2)
where AT is the amplitude of the detector temperature response, Wo is the amplitude of input heat radiation, oJ = 2~rf, G is the thermal conductance of the detector element to its surroundings, and C is the thermal or heat capacity of the detector element. The time constant r is commonly defined as
z= C//G.
(3)
To be precisely Lorentzian, the detector has to behave as a lumped heat capacity, coupled to a thermal bath with a lumped thermal conductance path. Deviations from Lorentzian behavior occur when the di-
where K is the thermal conductivity, Cp is the specific heat and p is the density. Near-Lorentzian behavior requires that l o be longer than detector length scales; in this case, G and C are approximately independent of oJ and the response of Eq. (2) is a Lorentzian. If the length of the thermal conductance path exceeds 1o/2, the thermal conductance G becomes frequency dependent [3] and the detector response of Eq. (2) is no longer a Lorentzian with a single time constant. If the dimensions of the absorber and thermometer are not small compared with l o, the lumped-element approach is also no longer correct, and deviations occur from Lorentzian behavior. Deviations are likely at high frequencies, because l o is a monotonically decreasing function of frequency. Deviations are also likely at non-cryogenic temperatures, since then the specific heat is at a maximum described by the law of Dulong and Petit. A thermal model more detailed than a lumped heat capacity and a lumped thermal conductance path is suggested by the observed frequency responses of Figs. 3 and 4. The secondary time constant is more than two orders of magnitude shorter than the primary time constant. Then by Eq. (2) the dominant contributor in the denominator at frequencies much greater than the reciprocal of the first time constant is oJC, and thus the secondary time constant cannot be explained by appealing to thermal diffusion changing the effective thermal isolation. Thermal diffusion within the thermal conductance path does change the effective contribution of the thermal conductance path to the total heat capacity of the detector [3], but in the wrong direction. Thermal diffusion lessens the contribution of the thermal con-
J.C. Brasunas / lnfiared Physics & Technology 38 (1997) 69-74
72
ductance path to the total heat capacity of the detector; this would produce a roll-off less rapid than Lorentzian - - the opposite of what is observed. Thus the non-Lorentzian behavior is not explained by appealing to thermal diffusion within the thermal conductance path between the thermometer and the thermal bath. Another possible explanation is that as the heat power is absorbed at the top of the absorber, it is damped within the absorber material before reaching the thermometer region (the thermoelectric material). This is analogous to the harmonic temperature variations induced by diurnal or seasonal variations in solar heating being exponentially damped within the interior of the Earth. The temperature variations at a depth x within a semi-infinite absorber, compared with the variations on the surface, satisfy the e x p o n e n t i a l attenuation [4] exp( 2 7r x / 2 ~ r l o) = exp( X / l o ) . While appeal to this mechanism does provide the leverage for suppressing the detector response below Lorentzian, this simple model would not explain the apparent good fit of a double Lorentzian. Perhaps a geometrically more detailed model [5] would. Rather than considering continuously distributed elements within the absorber-thermometer model, let us consider discretely distributed elements (Fig. 5). Incoming radiant heat W ( t ) flows into the absorber with heat capacity C 2. The absorber is coupled with thermal resistance R E to the thermometer with heat capacity C l, which is coupled with thermal resistance R¿ to the heat bath at temperature To. Let us -
-
compute the steady-state solution in response to W ( t ) = W e j°''. T h e steady-state solution is the thermometer temperature T~(t) = To + ~ T 1 e jc°t and the absorber temperature T2(t) = To + 6T~ e j~' + 6 T2 e j,o,. Now by elementary thermodynamics adding heat AQ to a body with heat capacity C causes a temperature rise AT, where AQ= CAT.
(6)
And in equilibrium, the rate of flow of heat W through a thermal resistance R (equals 1 / G ) in response to a temperature difference AT satisfies the relation W = - d(AQ)/dt=
l/RAT.
(7)
Defining the heat flow from the absorber to the thermometer to be W2(t)= W2eJO't, and the heat flow from the thermometer to the thermal bath to be Wl(t) = W l e i°'t, then by Eq. (7) 6T t = RiW l
(8)
and 6 T 2 = REW 2.
(9)
Also, by Eq. (6) W2(t ) - W , ( t ) = d ( A Q , ) / d t = C n d ( T l ( t ) ) / d t (10) or
W2 - W l = Cj jto~ T l .
w absorber C2 ( ~ W2 i ~ thermometer C1 W1
(11)
Also, by Eq. (6)
$ To+ dT1+ dT2
W ( t) - W2(t ) = d( A a 2 ) / d t = C2d(T2( t) ) / d t
(12)
R2 To+ dT1 RI
or
W-
W 2 = C 2 j t o ( ~ T , + 8T2).
(13)
Then combining Eqs. (8), (9), (11) and (13) to eliminate W t and W2, Fig. 5. A thermal model of the detector, with separately lumped absorber and thermometer,
t$ T 2 / R 2 - 6 T n / R t = C l jto6 T n
(14)
J.C. Brasunas / Infrared Physics & Technology 38 (1997) 69-74 and W-
8 T 2 / R 2 = C2joJ( a T I + 8T2).
(15)
Combining Eqs. (3), (14) and (15) to solve for 6T~, we have R~W aT I =
73
Table 1 Retrieved values for the model parameters from three runs of data, based on the criterion of minimizing the mean-square-difference between the phases of the data and of the model Test conditions r'1 (ms) r~ (ms) msd (o) 1st run, low amplitude 2nd run, low amplitude 3rd run, high amplitude
85.5 85.5 86
0.7l 0.71 0.715
4.817 3.62 3.022
j w r l ( c z / C ,) + ( 1 + j w r l) ( 1 + j w r 2) " (16)
Defining O = C 2 / C ~, then the detector response may be re-expressed as the product of two Lorentzians, R~W 6T, =
(1 + j t o r ' , ) ( 1 +jtor'2)
(17)
where r'j and r'2 are the effective time constants defined by r',r'2 = % %
5. Conclusions
(18)
and r', + r [ = % ( 1 + O )
creases msd by 0.4%; increasing r~ by 1% increases msd by 0.4%. Using the criterion that msd not be increased by more than 0.4%, these procedures appear to be capable of estimating r'~ to 1%, and r': to 1%, which is also the level of consistency among the three runs.
+ r z.
(19)
The solutions for r] and r [ are real. For example, if r, = r = g 2 = 1, the effective time constants are (3 + ~ ' - 1 / 2 .
4. Estimation of ~"1 and ~'[ We estimate r'~ and r~ by calculating the phase of the response function of Eq. (17) for an array of values of r ' l and r~, spaced by 0.5 ms for r ' 1 and 0.005 ms for r~. For each r ' l / r ' z pair, the meansquare difference (msd) is calculated between the theoretical curve and the phase of the measured data; the solution is the r'l/r'9 pair that minimizes msd. We took the detector data a second time to look for non-repeatability, and a third time to look for systematic effects by doubling the modulation amplitude of the LED, looking for artifacts due to detector non-linearity or to LED self-heating. The data as shown in Table 1 are quite reproducible for parameter retrieval. The criterion msd is quite sensitive to r', and to r~. Compared with the best fit for the high LED modulation case, increasing r'~ by 1% in-
It has been shown that electrical modulation of an LED is a simple way of characterizing the frequency response of an infrared detector over a wide frequency range. It is possible to retrieve not only the first-order Lorentzian response of a thermal detector, but also higher-order deviations from the Lorentzian. In particular the thermopile under study can be modeled with a double Lorentzian, indicative of a discretely distributed absorber and thermometer, and lumped thermal conductance paths between the absorber and thermometer and between the thermometer and the thermal bath. Thus the LED measurements are useful for the study of the thermal structure of the detector, in addition to the precise characterization of the frequency response itself.
Acknowledgements I thank Larry Herath of SSAI of Lanham, MD for suggesting I consider the phase response.
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