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Object View Spectral Radiation
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Figure 10.7 shows the radiation from an ideal blackbody source at various temperatures between 100 C and 1600 C on a logarithmic
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FIGURE 10.6
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Ten
5.26 0.01 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 Wavenumber (cm 1) Source Temperature (K) 373 W/(cm2 sr cm 1) 573 0.0001 773 973 1173 0.00001 1373 1573 0.000001 5 4.76 4.54 4.35 4.17 4 3.85 3.7 3.57 3.45 Wavelength (um)
FIGURE 10.7 Object view spectral radiance versus temperature.
scale. It can be seen that the dynamic range requirement of the system is quite large, spanning almost four orders of magnitude at 3.5 m. Spectral radiance from the object view is given by Planck s equation: L( ) = C1 3 C exp 2 1 T (10.1)
where C1 = 1.191 10 12 W/(cm2 sr [cm 1]4) is the first radiation constant C2 = 1.439 K cm is the second radiation constant = wave number in cm 1 T = temperature of the object view in K L( ) = radiated power density or spectral radiance in W/(cm2 sr cm 1). This version of Planck s equation expressed in wave-numbers is different from the usual equation expressed in wavelengths: M( ) = C1 5 C exp 2 1 T (10.2)
The difference comes from the fact that Planck s function is a power density. Because = 1/ , the infinitesimal wave-number element d is equal to d / 2. Thus:
L( ) d =
M( ) d
(10.3)
S p e c t R x N I R Te c h n o l o g y
and the apparent disagreement between Eqs. (10.1) and (10.2) disappears.
10.7.2 Transmission of the Spectroradiometer
Transmission of the spectroradiometer is the ratio of the radiant signal at the detector to that at the input port, taking into account the reflectivity of mirrors and transmission through windows, beamsplitters, and lenses. The description and transmission coefficient of these elements is shown in Table 10.1. The system transmission T( ) (no units) is given by: T ( ) = i ( )
i=1 n
(10.4)
where i( ) is the spectral transmission efficiency of element i (no units). For the SpectRx system, the standard beam-splitter is ZnSe, a nonhygroscopic material well adapted to field applications. The optical components of the SpectRx systems along with their transmittance is presented in Table 10.1. The total estimated system transmission is about 30 percent. This transmission can be improved to about 50 percent by using an anti-reflection coating on the protective windows; however, this will have a detrimental effect on the spectral range coverage. Another way to improve the transmission is to use different window materials. Unfortunately, most acceptable infrared window materials (Kbr, Kcl, etc.) are hygroscopic and require more care than ZnSe.
10.7.3 Throughput
The throughput of the SpectRx system is 0.004 cm2. The system can be used without having to reduce the throughput to a spectral resolution below 1 cm 1.
Power at the Detector
Pdetector ( ) = L( ) T ( ) 2 (10.5)
The power at the detector (W/cm 1) is given by:
where T( ) = spectral instrument transmission efficiency (no units) L( ) = spectral radiance from the object view (W/cm2 cm 1 sr) = instrument throughput (cm2 sr) The SpectRx system is equipped with a single input output port and dual output ports. The evaluation of the power at the detector takes into account that the energy from a single input port is divided
Ten
equally between the two output ports, thus representing a factor of 2 in Eq. (10.5).
Detector Current
The detector s responsivity is the detector output current for a given input illumination power. The responsivity is directly proportional to the wavelength multiplied by the quantum efficiency ( ). This is because shorter wavelength photons have more energy per photon; therefore, there are fewer photons per watt of flux of either the signal or the background. The responsivity is thus dependent on the spectral frequency.
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