Page 539 - Sensors and Control Systems in Manufacturing

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T e n
Cha p te r

• Wavelength in microns is converted from the wave number
,
by λ = 10 000 .
σ
• The divergence of a telescope is approximated by θ= Dfs
d
where Dfs is the diameter of the field stop and d the focal
length of the primary.
• The solid angle of a beam at the entrance of a telescope is
Afs
approximated by Ω= where Afs is the area of the field
d 2
stop and d the focal length of the primary.
• The relation between the solid angle and the divergence is
⎛ ⎞ θ
Ω= 2π ⎜1 − cos ⎟. For small angles (e.g., in calculations related
⎝ ⎠ 2
to the FOV of telescopes), the following formula provides a
π
good approximation: Ω= θ .
2
4
• The diameter of an object at a distance d and subtended by a
divergence θ is given by Ds = dθ.
• The throughput between two optical elements is given by
Θ = AΩ, Aas being the area of the aperture stop, and Ω, the
solid angle or divergence, introduced by the field stop. We
can approximate Ω by taking the area of the field stop and
dividing it by the square of the distance between both stops.
The throughput is then expressed by Θ= Afs × Aas .
d 2
• Converting radiance to irradiance is done by multiplying the
radiance by the solid angle of the telescope.
• Converting irradiance to apparent intensity is done by multi-
plying the irradiance by the square of the distance between
the point source and the primary of the telescope.
• Converting apparent intensity-to-intensity is done by correct-
ing the apparent intensity with an atmospheric transmission
spectrum computed by modeling software (Fast Code,
Lowtran, Modtran, and others).

10.5 The SpectRx FFT-NIR Technology Advantage
Spectral information can be obtained using several technical meth-
ods. Most of them rely on the dispersion of light to achieve spectral
separation. This can be done with a prism or a grating. In this case,
the spectral distribution is transformed into a spatial distribution and
requires a scanning mirror in one axis. Circular variable filter radiom-
eters are also currently used, but spatial/spectral smearing effects
add to the other drawbacks of this method.
The fast Fourier transform (FFT) spectrometer does not rely
on spatial dispersion, but rather on time dispersion of the spectral