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Input port B can be set to look at a cold source to reduce the photon noise.
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NESR
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The system s noise equivalent spectral radiance (NESR) (W/cm2 sr cm 1) is given by:
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NESR = (( I photon )2 + ( I quantization )2 + ( I darknoise )2 + ( I Johnson )2 + ( I jitter )2 + ( I scan.inst )2 ) t R( ) F
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(10.30) where I photon = noise current due to photon noise ( A / Hz ) I = noise current due to quantization noise ( A / Hz ) I Johnson = noise current due to the Johnson noise of the resistance of the first amplification stage ( A / Hz ) I darknoise = noise current due to dark noise ( A / Hz ) I jitter = noise current due to the jitter of the sampling ( A / Hz ) I scan. inst. = noise current due to scanning speed variations in the spectroradiometer, coupled with the analog filter slope and the delay mismatch ( A / Hz ) t = acquisition time (s) = spectral interval (cm 1)
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= instrument efficiency (no units) R( ) = responsivity of the system (A/W) = instrument throughput (cm2 sr) F = noise reduction apodization factor (no units); equal to 1 if no apodization is used Being incoherent, the various sources of noise are added in an RSS manner in Eq. (10.11).
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SNR( ) = P( ) NESR (10.31)
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Finally, the signal-to-noise ratio SNR is given by:
Instrument Line Shape and Spectral Resolution
The ILS (instrument line shape) is the response of the spectrometer to a monochromatic spectral stimulus. This is illustrated schematically in Fig. 10.10. A distinct ILS spanning from = to + exists for each wave-number , although it changes from one to the next in a slow and progressive manner. The ILS at a given wave number represents the noiseless spectrum that would be obtained if the spectrometer were submitted to monochromatic radiation for example, that of a laser at = 1/ . The ILS function ILS ( ) is thus characterized by two indices, and , as illustrated in Fig. 10.10. An observed spectrum Sobserved( ) is the convolution of the ILS functions with the spectral distribution of photon incidence B( ).
Sobserved ( ) =
B( ) ILS ( ) d
(10.32)
If the spectral features or variations of B( ) are much wider than the width of the ILS peak, then Sobserved( ) will be very similar to B( ).
IR Stimulus
ILS ( ) Spectrometer
FIGURE 10.10 Monochromatic IR stimulus and instrumental response function (ILS).
S p e c t R x N I R Te c h n o l o g y
On the other hand, if B( ) has features much narrower than the ILS, Sobserved( ) will be greatly affected by the response of the system. This is often referred to as under-resolving. In this case, the narrow lines from B( ) would all have widths very similar to that of the ILS. In the limiting case of an isolated monochromatic line, the measured spectrum is the ILS itself, as illustrated in Fig. 10.10. With respect to the instrument structure, many parameters influence the ILS, such as the finite mirror scan length, the finite divergence in the interferometer, the IR alignment, and so on.
Spectral Resolution
In the present system, the width of the ILS is mainly governed by the extent of the finite length of the interferogram. For a system with negligible divergence that is equipped with a round detector, the normalized ILS is given by: ILS( , ) = Sin (2 MPD) 2 MPD (10.33)
The ILS function due to the finite interferogram length is independent of (i.e., it is the same for all wave numbers). This ILS function is illustrated in Fig. 10.10. The line shape seen in Fig. 10.11 is very similar to the actual simulated ILSs displayed in Fig. 10.10. The FWHM of the ILS given by Eq. (10.15), FWHMboxcar, can be derived analytically and is: FWHM boxcar = 1 . 207 2 MPD (10.34)
1 0.8 Boxcar ILS amplitude 0.6 0.4 0.2 0 40 0.2 0.4 Wavenumbers (cm 1)
FIGURE 10.11 Plot of sin(2 L)/(2 L) (where L = 0.122 cm).
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