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Errors in minimum resolvable temperature difference charts
Infrared Physics, 1977, Vol. 17. pp. 375-379. Pergamon Press. Printed in Great Britain
ERRORS
IN MINIMUM RESOLVABLE TEMPERATURE DIFFERENCE CHARTS WILLIAML. WOLFE
Optical
Sciences
Center,
University (Rrceiwd
of Arizona, 14 March
Tucson,
Arizona
85721. U.S.A.
1977)
INTRODUCTION The test chart or target usually employed in assessing the minimum resolvable temperature difference (MRTD) of infrared devices as a function of frequency consists of several groups of four bars. Each group has a different fundamental spatial frequency. The bars have an aspect ratio of 7 : 1. The test target is generally made from two plates operated at slightly different temperatures and located one behind the other. The bars are slots in the front plate. Figure 1 is a schematic diagram of such a device. The sensor then views alternately the front plate and the bars. The output signal-to-noise ratio is recorded as a function of the temperature difference between the rear plate and the front plate. The usual assumption is that the front and rear plates have the same emissivity. For many reasons this assumption is never quite true. For example, there is a reasonable cavity effect causing an enhancement of the effective emissivity of the rear plate, and as a consequence, the bars. We have analyzed the errors that accrue from the fact that these two emissivities are different. The results are functions of the plate temperature, the bar temperature, the background temperature, the two emissivities and the spectral band of operation. THE PHYSICAL MODEL We assume that the front plate has a temperature T and emissivity E; the bar has a temperature T + dT, emissivity E + de; and we assume that the background is uniform in temperature with unity emissivity. The analysis is done first for a system that senses
60
0.009
50 0.007
40
0.005
30 20
0.003
IO 0.001 0 -10 -20 -30 -40 -50 -60 -70 -60 -90 -100 200
250
350
309
Temperature,
400
K
Fig. 1. The relative error in bar charts for the full spectrum of power as a function of temperature for different values of dr/dT and a background temperature of 310K. 375
376
WILLI4M
L.
WOLFI
radiation power of all wavelengths equally. This shows the basic principles. A similar expression for a detector that responds equally to photons of all wavelengths is easily obtained from this. Next the more complicated analysis for monochromatic radiation is performed. Interestingly the error for the monochromatic case is independent of whether a power distribution or photon distribution is used. Finally finite-spectrum and non-uniform background cases are considered. FULL
For full-spectrum
quantities
SPECTRlJM
ANALYSIS
one can write L= 71 ‘CUT4 + (I - E)7r_‘ITT;:
where: L= radiance; 0 = Stefan Boltzmann constant; T = target E = target plate emissivity; T, = background temperature. The difference in radiance between bar and plate then is given by:
a quantity
dT defined
by:
4eoT3 dT,, Then
dT,,, is the apparent
error
= 4oT”dT
temperature
+ aT4dc
- aT,4dc
and it is given by:
difference
= dT + a(T4 - T,“) dQ&T3
dT,, The relative
temperature;
+ nT4 de - UT, de
ndL = 4EaT3dT Consider
plate
in temperature
is therefore:
@IT,, - dT)/dT
=
i’
; -;; -Tm [1 - (TB/T)4] dc!dTj
RE = D(T/4)[1
- (T,/T)‘]
where D is the quotient of the two changes and E is assumed to be one. This shows the error is zero when the target and background are the same temperature or when there is no difference in emissivity. Further, the error will get larger as the difference between T and T, gets larger. Figure 2 is a curve for the relative error for different values of dc/dT as a function of T when T, = 310K. It seems obvious that for a photon distribution the error is given by:
;f
, 300
310
,
,
320
330
Temperature,
, 340
1 350
K
Fig. 2. The relative error in bar charts for the full spectrum of photon flux rate for the region 300K to 350K for different values of dc/dT and background temperature of 3IOK.
Errors
in MRTD
charts
311
The error is zero for de = 0 and for T = TB. The first of these results is just satisfaction of the basic assumption. The second is the situation for which the background irradiation exactly makes up for the emissivity difference. The error is larger for larger de and smaller for larger dT. The quantity D might be made smaller by making the effective emissivity of the back plate more nearly equal to that of the front plate. MONOCHROMATIC
In this case the fundamental
ANALYSIS
equation is: L, = LfB + (1 - E) Ly
The change is: dL = La’ xeX(eX - 1)-r dT/T + (LYE - La) de where: x = cJAT;
c2 = 14388 cm K.
The relative error can then be written: RE = (1 - L:/LfB) (xeX(eX - 1)-l)-’
For the Wien approximation
DT
this reduces to: RE = [l - exp(x - xs)] x-l DT
The relative error for the region in which the Planck function must be used can be written: RE = [l - (e, - 1) (eXB- 1)-r (1 - ewX)]x-l DT
These relationships are shown in Figs. 3 and 4 for two values of de/dT and for several wavelengths. Figure 5 shows the same information for temperatures near the background temperature. FINITE
BAND
ANALYSIS
(GRAY
BODIES)
Over any finite band one can write: L,A = L;f
+ (1 - E) Lf;,
The same manipulations lead to the same basic equation for the relative error. This is true because the temperature and emissivity differentiation are independent of wave-
Temperature,
K
Fig. 3. The relative error in bar charts based on the Planck Law for de/dT = 0.001 and for different wavelengths and a background temperature of 310K.
WILLIAM L. WOLFE
378
1
I
I 310
-‘00300
I
320 Temperature,
Fig. 4. The relative
I
I 340
330
3x,
K
error in bar charts based on the Planck Law for dr/dT wavelengths and a background temperature of 31OK.
= 0.01 for different
length for graybodies. Then : 1-
RE = (PI-/x)
I
;:-qex - I)-” di,/ : J-JA
3‘3i.
xe”X”(e”
- f)-‘di. /
DISCUSSION
Most measurements are made with the front plate near ambient temperature (300K-320K). The rear plate usually varies in temperature from a few degrees above the front plate to 10 or 20 degrees above the front plate. fn this region the errors can be reduced by measuring the effective emissivities of the bar chart, so the errof would then depend upon the measLIring instrument. Part of the reason that there is any error at all arises from the fact that most infrared instruments measure differences of irradiance on their entrance pupils rather than diEerences in either temperature or emissivity. An additional contribution to the error is
7
6
304
306
308
Temperature, Fig. 5. Detail
of the relative
error
1
310
1 -2-r-5k A
~
320
K
near the background
tcmperat~lrc
of 31OK
Errors
in MRTD
charts
379
the use of the bar chart as a primary standard. These calculations seem to underline the fact that the charts should be calibrated. Perhaps we are also using the wrong specifications for the performance. Only the apparent MRTD should be specified in terms of the variations of the charts as measured by an appropriate radiometer. The measurement procedure and specifications must be described very carefully. Some charts of this type use the contrast between sky or other ambient radiation and the front plate. These need to be calibrated by their very design, so no additional analysis of the type carried out here is necessary.
ERRORS
IN MINIMUM RESOLVABLE TEMPERATURE DIFFERENCE CHARTS WILLIAML. WOLFE
Optical
Sciences
Center,
University (Rrceiwd
of Arizona, 14 March
Tucson,
Arizona
85721. U.S.A.
1977)
INTRODUCTION The test chart or target usually employed in assessing the minimum resolvable temperature difference (MRTD) of infrared devices as a function of frequency consists of several groups of four bars. Each group has a different fundamental spatial frequency. The bars have an aspect ratio of 7 : 1. The test target is generally made from two plates operated at slightly different temperatures and located one behind the other. The bars are slots in the front plate. Figure 1 is a schematic diagram of such a device. The sensor then views alternately the front plate and the bars. The output signal-to-noise ratio is recorded as a function of the temperature difference between the rear plate and the front plate. The usual assumption is that the front and rear plates have the same emissivity. For many reasons this assumption is never quite true. For example, there is a reasonable cavity effect causing an enhancement of the effective emissivity of the rear plate, and as a consequence, the bars. We have analyzed the errors that accrue from the fact that these two emissivities are different. The results are functions of the plate temperature, the bar temperature, the background temperature, the two emissivities and the spectral band of operation. THE PHYSICAL MODEL We assume that the front plate has a temperature T and emissivity E; the bar has a temperature T + dT, emissivity E + de; and we assume that the background is uniform in temperature with unity emissivity. The analysis is done first for a system that senses
60
0.009
50 0.007
40
0.005
30 20
0.003
IO 0.001 0 -10 -20 -30 -40 -50 -60 -70 -60 -90 -100 200
250
350
309
Temperature,
400
K
Fig. 1. The relative error in bar charts for the full spectrum of power as a function of temperature for different values of dr/dT and a background temperature of 310K. 375
376
WILLI4M
L.
WOLFI
radiation power of all wavelengths equally. This shows the basic principles. A similar expression for a detector that responds equally to photons of all wavelengths is easily obtained from this. Next the more complicated analysis for monochromatic radiation is performed. Interestingly the error for the monochromatic case is independent of whether a power distribution or photon distribution is used. Finally finite-spectrum and non-uniform background cases are considered. FULL
For full-spectrum
quantities
SPECTRlJM
ANALYSIS
one can write L= 71 ‘CUT4 + (I - E)7r_‘ITT;:
where: L= radiance; 0 = Stefan Boltzmann constant; T = target E = target plate emissivity; T, = background temperature. The difference in radiance between bar and plate then is given by:
a quantity
dT defined
by:
4eoT3 dT,, Then
dT,,, is the apparent
error
= 4oT”dT
temperature
+ aT4dc
- aT,4dc
and it is given by:
difference
= dT + a(T4 - T,“) dQ&T3
dT,, The relative
temperature;
+ nT4 de - UT, de
ndL = 4EaT3dT Consider
plate
in temperature
is therefore:
@IT,, - dT)/dT
=
i’
; -;; -Tm [1 - (TB/T)4] dc!dTj
RE = D(T/4)[1
- (T,/T)‘]
where D is the quotient of the two changes and E is assumed to be one. This shows the error is zero when the target and background are the same temperature or when there is no difference in emissivity. Further, the error will get larger as the difference between T and T, gets larger. Figure 2 is a curve for the relative error for different values of dc/dT as a function of T when T, = 310K. It seems obvious that for a photon distribution the error is given by:
;f
, 300
310
,
,
320
330
Temperature,
, 340
1 350
K
Fig. 2. The relative error in bar charts for the full spectrum of photon flux rate for the region 300K to 350K for different values of dc/dT and background temperature of 3IOK.
Errors
in MRTD
charts
311
The error is zero for de = 0 and for T = TB. The first of these results is just satisfaction of the basic assumption. The second is the situation for which the background irradiation exactly makes up for the emissivity difference. The error is larger for larger de and smaller for larger dT. The quantity D might be made smaller by making the effective emissivity of the back plate more nearly equal to that of the front plate. MONOCHROMATIC
In this case the fundamental
ANALYSIS
equation is: L, = LfB + (1 - E) Ly
The change is: dL = La’ xeX(eX - 1)-r dT/T + (LYE - La) de where: x = cJAT;
c2 = 14388 cm K.
The relative error can then be written: RE = (1 - L:/LfB) (xeX(eX - 1)-l)-’
For the Wien approximation
DT
this reduces to: RE = [l - exp(x - xs)] x-l DT
The relative error for the region in which the Planck function must be used can be written: RE = [l - (e, - 1) (eXB- 1)-r (1 - ewX)]x-l DT
These relationships are shown in Figs. 3 and 4 for two values of de/dT and for several wavelengths. Figure 5 shows the same information for temperatures near the background temperature. FINITE
BAND
ANALYSIS
(GRAY
BODIES)
Over any finite band one can write: L,A = L;f
+ (1 - E) Lf;,
The same manipulations lead to the same basic equation for the relative error. This is true because the temperature and emissivity differentiation are independent of wave-
Temperature,
K
Fig. 3. The relative error in bar charts based on the Planck Law for de/dT = 0.001 and for different wavelengths and a background temperature of 310K.
WILLIAM L. WOLFE
378
1
I
I 310
-‘00300
I
320 Temperature,
Fig. 4. The relative
I
I 340
330
3x,
K
error in bar charts based on the Planck Law for dr/dT wavelengths and a background temperature of 31OK.
= 0.01 for different
length for graybodies. Then : 1-
RE = (PI-/x)
I
;:-qex - I)-” di,/ : J-JA
3‘3i.
xe”X”(e”
- f)-‘di. /
DISCUSSION
Most measurements are made with the front plate near ambient temperature (300K-320K). The rear plate usually varies in temperature from a few degrees above the front plate to 10 or 20 degrees above the front plate. fn this region the errors can be reduced by measuring the effective emissivities of the bar chart, so the errof would then depend upon the measLIring instrument. Part of the reason that there is any error at all arises from the fact that most infrared instruments measure differences of irradiance on their entrance pupils rather than diEerences in either temperature or emissivity. An additional contribution to the error is
7
6
304
306
308
Temperature, Fig. 5. Detail
of the relative
error
1
310
1 -2-r-5k A
~
320
K
near the background
tcmperat~lrc
of 31OK
Errors
in MRTD
charts
379
the use of the bar chart as a primary standard. These calculations seem to underline the fact that the charts should be calibrated. Perhaps we are also using the wrong specifications for the performance. Only the apparent MRTD should be specified in terms of the variations of the charts as measured by an appropriate radiometer. The measurement procedure and specifications must be described very carefully. Some charts of this type use the contrast between sky or other ambient radiation and the front plate. These need to be calibrated by their very design, so no additional analysis of the type carried out here is necessary.