Editing Tunable diode laser absorption spectroscopy (section)

==Basic principles==

=== Concentration measurement ===
The basic principle behind the TDLAS technique is simple. The focus here is on a single absorption line in the absorption spectrum of a particular species of interest. To start, the wavelength of a [[laser diode|diode laser]] is tuned over a particular absorption line of interest and the intensity of the transmitted radiation is measured. The transmitted intensity can be related to the concentration of the species present by the [[Beer-Lambert law]], which states that when a radiation of [[wavenumber]] <math> (\tilde{\nu}) </math> passes through an absorbing medium, the intensity variation along the path of the beam is given by,<ref>See Bernath, Peter F. (2005), C7§6 p.272–274.</ref>

:<math>I(\tilde{\nu}) = I_{0}(\tilde{\nu}) \exp(-\alpha(\tilde{\nu})L) = I_{0}(\tilde{\nu}) \exp(-\sigma(\tilde{\nu})NL)</math>
where,
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:<math> I(\tilde{\nu}) </math> is the transmitted intensity of the radiation after it has traversed a distance <math> L </math> through the medium,
:<math> I_{0}(\tilde{\nu}) </math> is the initial intensity of the radiation,
:<math> \alpha(\tilde{\nu}) = \sigma(\tilde{\nu})N = S(T)\phi(\tilde{\nu}- \tilde{\nu}_{0}) </math> is the absorbance of the medium,
:<math> \sigma(\tilde{\nu}) </math> is the absorption cross-section of the absorbing species,
:<math> N \!</math> is the [[number density]] of the absorbing species,
:<math> S(T) \!</math> is the line strength (i.e. the total absorption per molecule) of the absorbing species at temperature <math> T </math>,
:<math> \phi(\tilde{\nu}- \tilde{\nu}_{0}) </math> is the lineshape function for the particular absorption line. Sometimes also represented by <math> g(\tilde{\nu}- \tilde{\nu}_{0}) </math>,
:<math>\tilde{\nu}_{0}</math> is the center frequency of the spectrum.

=== Temperature measurement ===
The above relation requires that the temperature <math> T \!</math> of the absorbing species is known. However, it is possible to overcome this difficulty and measure the temperature simultaneously. There are number of ways to measure the temperature. A widely applied method, which can measure the temperature simultaneously, uses the fact that the line strength <math> S(T) \! </math> is a function of temperature alone. Here two different absorption lines for the same species are probed while sweeping the laser across the absorption spectrum, the ratio of the integrated absorbance, is then a function of temperature alone.

:<math> R =\left( \frac{S_{1}}{S_{2}}\right)_{T} = \left(\frac{S_{1}}{S_{2}} \right)_{T_0} \exp\left[-\frac{hc(E_{1}-E_{2})}{k}\left(\frac{1}{T}-\frac{1}{T_{0}} \right) \right] </math>
where,
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:<math>T_{0} \!</math> is some reference temperature at which the line strengths are known,
:<math>\Delta E = (E_{1} - E_{2}) \! </math> is the difference in the lower [[energy levels]] involved in the transitions for the lines being probed.

Another way to measure the temperature is by relating the [[Full width at half maximum|FWHM]] of the probed absorption line to the [[Doppler broadening|Doppler line width]] of the species at that temperature. This is given by,

:<math>FWHM (\Delta\tilde{\nu}_{D}) = \tilde{\nu}_{0} \sqrt{\frac{8kT\ln 2}{mc^{2}}} = \tilde{\nu}_{0} (7.1623\mbox{x}10^{-7}) \sqrt{\frac{T}{M}} </math>
where,
:<math> m</math> is the weight of one molecule of the species, and
:<math>M</math> is the [[Mole (unit)#Proposed future definition|molecular weight]] of the species.
Note: In the last expression, <math>T</math> is in kelvins and <math>M</math> is in g/mol.
However, this method can be used, only when the gas pressure is low (of the order of few [[mbar]]). At higher pressures (tens of millibars or more), [[Spectral line broadening#Spectral line broadening and shift|pressure or collisional broadening]] becomes important and the lineshape is no longer a function of temperature alone.

=== Velocity measurement ===
The effect of a mean flow of the gas in the path of the laser beam can be seen as a shift in the absorption spectrum, also known as [[Doppler effect|Doppler shift]]. The shift in the frequency spectrum is related to the mean flow velocity by,

:<math>\Delta\tilde{\nu}_{D} = \frac{V}{c}\tilde{\nu}_{0}\cos\theta </math>
where,
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:<math>\theta</math> is the angle between the flow direction and the laser beam direction.
Note : <math>\Delta\tilde{\nu}_{D}</math> is not the same as the one mentioned before where it refers to the width of the spectrum. The shift is usually very small (3×10<sup>−5</sup> cm<sup>−1</sup> ms<sup>−1</sup> for near-IR diode laser) and the shift-to-width ratio is of the order of 10<sup>−4</sup>.