Redstone™ Optical Spectrum Analyzer Chapter 16: Appendix
Rev C, January 21, 2022 Page 78
Chapter 16 Appendix
16.1. Calculating Absorption Cross Sections
As described in Section 8.10, the OSA software calculates absorption cross sections from the HITRAN line-by-
line references by separating each line into a line strength and a line shape:
Here,
is the temperature-dependent strength of the line, is a function describing the shape of the line,
is the center wavenumber of the line, and is the shift in pressure.
The line strength,
, is dependent on temperature because the temperature determines how strongly the
upper and lower energy levels are populated.
Here,
is the energy of the lower energy level, represents the temperature dependence of the partition
function, is the given temperature and
is the reference temperature.
The line shape is affected by two effects: Doppler broadening – caused by random molecular motion – and
pressure broadening – caused by collisions between molecules. Doppler broadening leads to a Gaussian line
shape with half-width at half-max given by:
where is the molecular mass. In contrast, pressure broadening leads to a Lorentzian line shape with half-
width at half-max given by:
where
is the broadening caused by collisions with other species,
is the broadening caused by
molecules of the same species, and is the mixing ratio of the gas. The exponent is transition-dependent and
given by the line-by-line reference.
The combination of Doppler broadening and pressure broadening can be calculated by a convolution between
the Gaussian and Lorentzian line shapes, and produces a profile known as the Voigt line shape.
In many cases, pressure broadening has a significantly greater effect than Doppler broadening. If the half-width
at half-max of the Gaussian line shape is less than 1% of the half-width at half-max of the Lorentzian line shape,
the software approximates the line shape by using only the Lorentzian. Otherwise, the full convolution is
calculated.
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