Measure CO2 – Part 4 Differential Beam

Goal

Determine in-situ atmospheric gas concentration using low-frequency differential continuous-wave laser measurements.

Theoretical Basis

Atmospheric gases absorb laser light of very specific “colors” or wavelengths. If two laser beams are transmitted – one the ON-line wavelength for absorption by a species of interest; one slightly different – the OFF-line wavelength – the intensity of the ON-line wavelength is attenuated to a greater degree than the intensity of the OFF-line wavelength yet both beams have the same path noise and errors. The amount of target gas along the laser path may be determined by comparing the two beams after they travel through the atmosphere.

The photodetector has a much wider response range than desired; optical filtering will be required. As an example – and not intended as an endorsement, ThorLabs has filter – FB1570-12 – which has center wavelength of 1570 ± 2 nm with FWHM of 12 ± 2.4 nm available for $95. For a bit more (much more -$ 1000s), filters with tighter specs are available.

Methodology

The desired measurement is the optical conductivity – aka optical depth – of a specific gas species through a path in the atmosphere. Using techniques similar to electrical ohm-meters, the measurements of the laser transmission of slightly different wavelengths can be compared resulting in a measure of the absorption of the atmospheric species of interest.

One beam is of an absorption wavelength for the desired species; the other beam wavelength is slightly different. Both beams are transmitted through the same optical path allowing a direct comparison in differential intensity. In electrical terms, I measure the differential component while rejecting the common-mode component. The electronic circuit analogy would be a differential pair.

Basic Theory

The fundamental expression for optical conductivity may be expressed as:

$\vartheta \; = \; \int\limits_0^Z n\text{\footnotesize{(z)}} \, \sigma \text{\footnotesize{(z)}} \; \text{d}z$

where $\vartheta$ is the relative optical conductivity. n is the number of optical attenuating elements within the optical path and σ is the cross-sectional area of the attenuating elements; both taken as functions of path length z.

When the integration is carried out, the resulting expression has the form:

$\text{RX} \; = \; \text{TX} \; exp[-z \, \sigma \, (n_a + n_s)]$

where RX is the received intensity and TX the transmitted intensity. Attenuation factor n has been expanded to (na + ns) where na represents the number of absorbing elements and ns the number of scattering elements $^\textbf{\normalsize 1}$ .

The sum of na and ns is the total number of attenuation elements in optical path cross-section σ. It is assumed the cross-sectional area of the absorbing and scattering elements are approximately equal. It is more probable that the scattering elements will be both larger and smaller but I’m assuming an average size here.

The individual expressions for the on-wavelength(absorbing) and off-wavelength (non-absorbing) signals are defined as:

$\text{RX}_{ON} \; = \; \text{TX}_{ON} \, exp[-z \, \sigma \, n_{ON}] \; = \; \text{TX}_{ON} \, exp[-z \, \sigma \, (n_a + n_s)]$

$\text{RX}_{OFF} \; = \; \text{TX}_{OFF} \, exp[-z \, \sigma \, n_{OFF}] \; = \; \text{TX}_{OFF} \, exp[-z \, \sigma \, n_s]$

$\text{TX}_{ON}$ and $\text{TX}_{OFF}$ are the known transmitted intensities of the ON and OFF channels; $\text{RX}_{ON}$ and $\text{RX}_{OFF}$ are the received intensities. Distance z will also be a known quantity.

Define:

$\sigma \, \text{n}_s \; \equiv \; \alpha_s \qquad \qquad \sigma \, ( \text{n}_a \, + \, \text{n}_s ) \; \equiv \; \alpha_a \, + \, \alpha_s$

where $\alpha_a$ represents a loss of intensity due to absorption and $\alpha_s$ represents all other intensity losses (the causes of other losses may be of interest, but I’m only interested in received photons … which might include some scattered elements. But that’s measurement interpretation, not the measurement.)

$\text{RX}_{ON} \; = \; \text{TX}_{ON} \, exp[-z \, (\alpha_a + \alpha_s)]$

$\text{RX}_{OFF} \; = \; \text{TX}_{OFF} \, exp[-z \, \alpha_s]$

It may be demonstrated that in situations in which beam divergence angle is considered, the basic expression becomes dependent on optical path volume ν which itself is a function of z. The optical path has the form of a cone rather than cylinder.

That’s good for now.

$^\textbf{\normalsize 1}$ Scattering factor ns represents the common-mode loss to both channels while absorption factor na is the differential-mode loss which is the parameter of interest.