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Thermal imaging system performance estimates
Infrared Physics. 1977, Vol. 17. pp. 415-418. Pergamon Press. Prmted in Great Britain
THERMAL
IMAGING
SYSTEM
PERFORMANCE
ESTIMATES
R. B. EMMONS Lockheed Palo Alto Research Laboratory, Palo Alto, CA 94304, U.S.A. (Received 12 April 1977)
Estimating the performance of a thermal imaging system is complicated by the need to compute the integral of the temperature derivative of a Planck radiation function. When photon detectors are employed in the system, an approximation suggested by Sanderson”’ permits evaluation of this integral’2*3’ in terms of functions found on the radiation slide rule.‘4’ For target and background temperatures near 300K, the approximation is good to =14prn, and covers a great many useful cases. This note shows that the integral in question can be rewritten in terms obtainable from the radiation slide rule without approximation; thus extending the utility of the slide rule to performance estimates for all background temperatures and wavelength intervals. Further, this technique is applicable to systems using thermal detectors as well as to those using photon detectors. The integral
(1) appears in all of the commonly used measures of the performance of thermal imaging systems.‘5T6’The quantities included in the integral are the temperature derivative of the spectral radiant emittance of the object field, aW(A, T)/dT, the spectral specific detectivity of the detector, D*(A), and the transmission of the atmosphere and optical systems, r(A). The wavelength variation of the atmospheric and optical transmission and of the specific detectivity is sufficiently irregular so that for a detailed evaluation of system performance, numerical evaluation of this integral may be required. Reasonable estimates of system performance, and in particular estimates of the best performance possible can be made using a few simplifying assumptions. The thermal image is presumed to arise solely from temperature differences in the object field. The emissivity of both target and background is taken to be wavelength independent, and both are assumed to be Lambertian radiators. The spectral radiant emittance W is therefore given by W=eH
where E is the constant emissivity, and H is Planck’s function, i.e.
H2L
1 As (ec21Ar_ 1)
c1 and c2 are the first and second radiation constants (cr = 2nhc2, c2 = he/k). The spectral photon emittance, which will also be used, is given by
The atmospheric and optical transmission will be assumed constant in the wavelength interval of interest, and zero elsewhere r = to 0
21 < A < 12 elsewhere 415
416
R. B. EMMONS
Finally, the specific detectivity will be assumed proportional below the peak as is appropriate for photon detectors D*(n) = or constant,
as is appropriate
to wavelength
at wavelengths
;D:
for thermal
detectors
D*(n) = 0: With these assumptions,
the integral
of Eq. 1 becomes
s
EQD;
j.2aHi do
I, = __
i-2
i.,
photon
aT ’
I, = Es()D: The temperature
derivative
detectors
thermal
of Planck’s
function
i?H ‘-H
-= aT
(4)
detectors
(5)
is 1
AT2
(1 - e-c2iAT)
Sanderson noted that this derivative is readily obtained from tables of H if the final exponential factor is dropped. This approximation is good when (cJ>.T) >> 1. and when it is made, the integrand on the right side of Eq. 4 becomes
8H i-&Q ?T
1
c21iT
T2
>>1
For a 300K background temperature, the approximate integrand is small by less than 17; at 10 pm, and by less than 4”/, at 14pm. The approximation gets worse at longer wavelengths or at higher temperatures, but remains quite reasonable within 20K of ambient temperatures in the important 8-14pm region. Its substitution in Eq. 4 yields
The integral on the right side, unlike that on the left, is obtainable from a radiation slide rule. The integral averages the error made by approximating the integrand, and yields results in the 280 to 320K temperature region and the 8-14pm wavelength region which are small by less than 2%. Values of the approximate integral obtained from the radiation slide rule are thus quite adequate in the temperature and wavelength interval shown. To obtain a result without approximation (and thus good for all temperatures and wavelength intervals) also in terms of functions obtainable from a radiation slide rule, we first rewrite Eq. 4 in terms of the spectral photon emittance
Next, note length are
that
These partials
the partial
are related
derivatives
of Q with
by
aQ -"Q+Gaf i3T-T
respect
to temperature
and
wave-
Thermal imaging system performance estimates
417
Thus
The second integral on the right hand side may be further reduced with an integration by parts to yield
QdJ + ;[&Q&)
- 4Q@dl
(8)
All of the quantities on the right hand side of this equation may be obtained from a radiation slide rule. Eq. 7 becomes I = lr !!!?L)* 3 P
‘TP
[J
I,
1
12QdA + J-zQ@J- &Q(4)
(9)
This exact result can be evaluated with six references to a radiation slide rule (compared to the three required to evaluate the approximate result of Eq. 6) and one need not be concerned about the range of validity of the Sanderson approximation. Radiation slide rules are generally accurate to 1 or 2%. Should more accuracy be desired or should it be more convenient to use a calculator than the slide rule, the form of Eq. 8 suggests that the approach normally taken to the computation of j’;’ Q dLC4’may be used with s,“’(aQ/ar)dl also. Express the integrand in terms of x = c,/,?T, expand in powers of exp( -x), invert the summation and integration, and integrate by parts to obtain (10) This series converges rapidly if xi = c2/ATis not too small, and its computation is more convenient than a numerical integration. Note also that the first term of the summation is the series for @,/ZJQ(&), and the remaining three are the series for (3/T) ii’ Q d1,(4) confirming Eq. 8. The Sanderson approximation is not helpful when the specific detectivity is independent of wavelength (the assumption appropriate for thermal detectors). Its use leads to the expression
and neither of these functions is on the radiation slide rule. The approach above however works in this case also, and yields the exact expression
suggested
Substitution in Eq. 5 yields iJ I, = ezoD:f 4 H d/? + &H(&) - &H(iJ (12) Lr J.1 1 Again, the expressions on the right side of this equation may be obtained from the radiation slide rule. If more accuracy is desired than is obtainable from the radiation slide rule, the power series
with x1 = cz/& T,may be used. In some older discussions of the sensitivity of thermal imaging systems, it was assumed that the specific detectivity was independent of wavelength (or could be replaced by an
418
R. B. EMMONS
“effective” value outside the integral) whether the detector was a thermal detector or not. (For practical photon detectors near the peak of their response curves, this may still be the appropriate assumption.) But this assumption implied the need for values of the integral ~(~H/ZT)di. in specific wavelength intervals. In the absence of a simple evaluation technique tabular values were often computed and supplied.“,” The exact equations given here for f(ZH/aT)dA and f(3H/JT)idjL in terms of functions which can be evaluated using a radiation slide rule obviate the need for tables of these functions as well as for approximations to them. Values for these integrals. upon which estimates of thermal imaging system performance depend, can be easily, quickly, and accurately obtained directly from a radiation slide rule or from a calculator using the relations derived. REFERENCES 1. 2. 3. 4.
SANDERSON, J. A.. In Guidunce (Edited by A. S. Locke). Van Nostrand, Princeton (1955) Ch. 5. p. 142. DERENIAK, E. L. and F. G. BROWN, Znfrared Phys. 15, 243-248 (1975). LLOYD, J. M., Thermal Imuging Systems. Plenum Press, New York (1975) p. 174. MAKOWSKI, M. W., Rea. Scientific Instr. 20, 876-884 (Dec. 1949). Radiation slide rules are available
commercially from Electra-Optical Industries, Inc., Santa Barbara, Ca., and IRCON. Inc.. Niles, 111. The functions given are H(i.), Sil H(i.)di,, Q(j.), and sal Q(i)dn for any given temperature. H is Planck’s function, and Q = H/hv. 5. LLOYD, J. M. op. cit. Ch. 5. 6. RATCHES, J. A., W. R. LAWSON, L. P. OBERT, R. J. BERGEMANN, T. W. CASSIIIY and J. M. SWENSON. Night Vision Laboratory St&c Performance Model for Thermal I/iea$rzy Systems. Army Electronic Command, Fort Monmouth, N.J., April 1975. Available from the National Technical Information Service (NTIS), U.S. Dept. of Commerce, AD-AOll-212. 7. R. D. HUDSON, Jr., Infrared Systems Engineering. Wiley, New York (1969) p, 431.
THERMAL
IMAGING
SYSTEM
PERFORMANCE
ESTIMATES
R. B. EMMONS Lockheed Palo Alto Research Laboratory, Palo Alto, CA 94304, U.S.A. (Received 12 April 1977)
Estimating the performance of a thermal imaging system is complicated by the need to compute the integral of the temperature derivative of a Planck radiation function. When photon detectors are employed in the system, an approximation suggested by Sanderson”’ permits evaluation of this integral’2*3’ in terms of functions found on the radiation slide rule.‘4’ For target and background temperatures near 300K, the approximation is good to =14prn, and covers a great many useful cases. This note shows that the integral in question can be rewritten in terms obtainable from the radiation slide rule without approximation; thus extending the utility of the slide rule to performance estimates for all background temperatures and wavelength intervals. Further, this technique is applicable to systems using thermal detectors as well as to those using photon detectors. The integral
(1) appears in all of the commonly used measures of the performance of thermal imaging systems.‘5T6’The quantities included in the integral are the temperature derivative of the spectral radiant emittance of the object field, aW(A, T)/dT, the spectral specific detectivity of the detector, D*(A), and the transmission of the atmosphere and optical systems, r(A). The wavelength variation of the atmospheric and optical transmission and of the specific detectivity is sufficiently irregular so that for a detailed evaluation of system performance, numerical evaluation of this integral may be required. Reasonable estimates of system performance, and in particular estimates of the best performance possible can be made using a few simplifying assumptions. The thermal image is presumed to arise solely from temperature differences in the object field. The emissivity of both target and background is taken to be wavelength independent, and both are assumed to be Lambertian radiators. The spectral radiant emittance W is therefore given by W=eH
where E is the constant emissivity, and H is Planck’s function, i.e.
H2L
1 As (ec21Ar_ 1)
c1 and c2 are the first and second radiation constants (cr = 2nhc2, c2 = he/k). The spectral photon emittance, which will also be used, is given by
The atmospheric and optical transmission will be assumed constant in the wavelength interval of interest, and zero elsewhere r = to 0
21 < A < 12 elsewhere 415
416
R. B. EMMONS
Finally, the specific detectivity will be assumed proportional below the peak as is appropriate for photon detectors D*(n) = or constant,
as is appropriate
to wavelength
at wavelengths
;D:
for thermal
detectors
D*(n) = 0: With these assumptions,
the integral
of Eq. 1 becomes
s
EQD;
j.2aHi do
I, = __
i-2
i.,
photon
aT ’
I, = Es()D: The temperature
derivative
detectors
thermal
of Planck’s
function
i?H ‘-H
-= aT
(4)
detectors
(5)
is 1
AT2
(1 - e-c2iAT)
Sanderson noted that this derivative is readily obtained from tables of H if the final exponential factor is dropped. This approximation is good when (cJ>.T) >> 1. and when it is made, the integrand on the right side of Eq. 4 becomes
8H i-&Q ?T
1
c21iT
T2
>>1
For a 300K background temperature, the approximate integrand is small by less than 17; at 10 pm, and by less than 4”/, at 14pm. The approximation gets worse at longer wavelengths or at higher temperatures, but remains quite reasonable within 20K of ambient temperatures in the important 8-14pm region. Its substitution in Eq. 4 yields
The integral on the right side, unlike that on the left, is obtainable from a radiation slide rule. The integral averages the error made by approximating the integrand, and yields results in the 280 to 320K temperature region and the 8-14pm wavelength region which are small by less than 2%. Values of the approximate integral obtained from the radiation slide rule are thus quite adequate in the temperature and wavelength interval shown. To obtain a result without approximation (and thus good for all temperatures and wavelength intervals) also in terms of functions obtainable from a radiation slide rule, we first rewrite Eq. 4 in terms of the spectral photon emittance
Next, note length are
that
These partials
the partial
are related
derivatives
of Q with
by
aQ -"Q+Gaf i3T-T
respect
to temperature
and
wave-
Thermal imaging system performance estimates
417
Thus
The second integral on the right hand side may be further reduced with an integration by parts to yield
QdJ + ;[&Q&)
- 4Q@dl
(8)
All of the quantities on the right hand side of this equation may be obtained from a radiation slide rule. Eq. 7 becomes I = lr !!!?L)* 3 P
‘TP
[J
I,
1
12QdA + J-zQ@J- &Q(4)
(9)
This exact result can be evaluated with six references to a radiation slide rule (compared to the three required to evaluate the approximate result of Eq. 6) and one need not be concerned about the range of validity of the Sanderson approximation. Radiation slide rules are generally accurate to 1 or 2%. Should more accuracy be desired or should it be more convenient to use a calculator than the slide rule, the form of Eq. 8 suggests that the approach normally taken to the computation of j’;’ Q dLC4’may be used with s,“’(aQ/ar)dl also. Express the integrand in terms of x = c,/,?T, expand in powers of exp( -x), invert the summation and integration, and integrate by parts to obtain (10) This series converges rapidly if xi = c2/ATis not too small, and its computation is more convenient than a numerical integration. Note also that the first term of the summation is the series for @,/ZJQ(&), and the remaining three are the series for (3/T) ii’ Q d1,(4) confirming Eq. 8. The Sanderson approximation is not helpful when the specific detectivity is independent of wavelength (the assumption appropriate for thermal detectors). Its use leads to the expression
and neither of these functions is on the radiation slide rule. The approach above however works in this case also, and yields the exact expression
suggested
Substitution in Eq. 5 yields iJ I, = ezoD:f 4 H d/? + &H(&) - &H(iJ (12) Lr J.1 1 Again, the expressions on the right side of this equation may be obtained from the radiation slide rule. If more accuracy is desired than is obtainable from the radiation slide rule, the power series
with x1 = cz/& T,may be used. In some older discussions of the sensitivity of thermal imaging systems, it was assumed that the specific detectivity was independent of wavelength (or could be replaced by an
418
R. B. EMMONS
“effective” value outside the integral) whether the detector was a thermal detector or not. (For practical photon detectors near the peak of their response curves, this may still be the appropriate assumption.) But this assumption implied the need for values of the integral ~(~H/ZT)di. in specific wavelength intervals. In the absence of a simple evaluation technique tabular values were often computed and supplied.“,” The exact equations given here for f(ZH/aT)dA and f(3H/JT)idjL in terms of functions which can be evaluated using a radiation slide rule obviate the need for tables of these functions as well as for approximations to them. Values for these integrals. upon which estimates of thermal imaging system performance depend, can be easily, quickly, and accurately obtained directly from a radiation slide rule or from a calculator using the relations derived. REFERENCES 1. 2. 3. 4.
SANDERSON, J. A.. In Guidunce (Edited by A. S. Locke). Van Nostrand, Princeton (1955) Ch. 5. p. 142. DERENIAK, E. L. and F. G. BROWN, Znfrared Phys. 15, 243-248 (1975). LLOYD, J. M., Thermal Imuging Systems. Plenum Press, New York (1975) p. 174. MAKOWSKI, M. W., Rea. Scientific Instr. 20, 876-884 (Dec. 1949). Radiation slide rules are available
commercially from Electra-Optical Industries, Inc., Santa Barbara, Ca., and IRCON. Inc.. Niles, 111. The functions given are H(i.), Sil H(i.)di,, Q(j.), and sal Q(i)dn for any given temperature. H is Planck’s function, and Q = H/hv. 5. LLOYD, J. M. op. cit. Ch. 5. 6. RATCHES, J. A., W. R. LAWSON, L. P. OBERT, R. J. BERGEMANN, T. W. CASSIIIY and J. M. SWENSON. Night Vision Laboratory St&c Performance Model for Thermal I/iea$rzy Systems. Army Electronic Command, Fort Monmouth, N.J., April 1975. Available from the National Technical Information Service (NTIS), U.S. Dept. of Commerce, AD-AOll-212. 7. R. D. HUDSON, Jr., Infrared Systems Engineering. Wiley, New York (1969) p, 431.