Measure CO2 – Part 2 Considerations of Measurement

Optical methods. Aircraft-based instrument platform.

The Platform

Keep it basic. Aircraft travels 200 m/s (447 mph) in x-direction at an altitude of 10 km (32,800 ft). This is roughly the top of the troposphere – the “atmosphere” for us ground-dwellers. I’m going to assume no y-axis motion and cylindrical symmetry about the z-axis so I have a two-dimensional problem to work in the x-z plane.

Rough approximations such as this can assume constant velocities:

$\nu \; = \; \{200, \, 0, \, 0 \} \; \text{m/s} \qquad \qquad \nabla \nu \; = \; 0$

Light travels roughly 1 km in 3.3 μs. Assuming an altitude of 10 km, the time delay between transmission and reception is about 66 μs. In a platform moving with a ground speed of about 200 m/s, the platform moves an insignificant lateral distance (13.2 mm) during a measurement frame.

For all intents and purposes, the transmit and receive optical paths are identical. The molecular content of the optical path may be assumed constant. IR vibrational relaxation times are in the fsec – psec range … light travels 300 μm in 1 ps … so that the “same” molecule in the optical path could absorb a photon from both the incident beam and the reflected beam. This suggests the estimation for concentration could be off by as much as a factor of two if each molecule absorbs two photons.

Quantum Absorption

Electromagnetic radiation consists of photons. For any given frequency of radiation λ, each photon has only a single value of quantum energy.

$\mathcal{E} \; = \; \frac{\, \mathpzc{h} \, c \, }{\lambda}$

where Planck Constant $\mathpzc{h}$ is equal to 6.626070 $e\text{-}34$ J · s and c is the speed of light (299,792,458 m/s) (The approximation of 300 $e6$ m/s is about 0.07 % high).

Energy levels of atoms and molecules exist only in quantized states specific to the properties of the material. Transitions between states occur during one of three photon processes: absorption, emission, or stimulated emission. Each of these can only occur if the photon energy is equal to the energy separation between states.

Total energy of the atom or molecule increases with absorption and decreases with emission.

Absorption can only occur when the photon energy hν is equal to the difference in state energies $E_1$ and $E_2$ where are specific to the material. Although the absorption state is of interest here, molecular level vibration and rotation may also be affected by electromagnetic energy. Infrared energy generally causes energy transitions due to absorption, microwave energy generally causes rotational transitions.

Infrared radiation covers a wide bandwidth of frequencies with wavelengths extending from just below the red portion of visible light to near the upper communications frequencies. The range nearest the visible spectrum is “near” infrared (with wavelengths longer than about 750 nm); those closer to the communications bands are “far” infrared (with wavelengths shorter than about 1 mm). Infrared interactions with matter generally increase the vibration frequency of the material (“heat”).

Infrared absorption and re-radiation is considered the basis of the “greenhouse” effect. When a photon is absorbed, the molecular bond changes to a higher state, then relaxes back to a nominal state by emitting a lower frequency photon – usually in the form of heat. In addition to water vapor, primary atmospheric gases of concern are carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O).

The atmosphere has “windows” through which various electromagnetic radiation of specific wavelengths can pass – the most obvious being visible light (where “red” has the longest wavelength and “blue” the shortest). For purposes of this discussion, the wavelengths of interest are those on the order of 1-2 μm – wavelengths just longer than “red” (hence, “near infrared”).

Absorption cross-section is a parameter describing a molecule’s ability to absorb a photon of a specific wavelength. The units are of area but the parameter is a measure of probability depending on the density and state of the target molecule.

$\sigma \; = \; \frac{\, \mu \, }{\rho} \times \frac{m}{\, N_A \, }$

where μ is the attenuation coefficient, ρ is the mass density, m is the atomic molar mass (in g/mol), and $N_A$ is Avogadro’s number.

Photoelectric Effect – Quantum Electrodynamics

The thermoelectric effect is the emission of free electrons over a potential energy barrier caused by thermal energy. This is the basis of operation of vacuum tubes: a cathode is heated to sufficient temperature which causes the release of electrons. These electrons are drawn to the tube’s plate which is positively charged. The effect was noted in 1873 and re-discovered by Thomas Edison in 1880. The effect is unipolar, electrons are not attracted to a negatively charged plate. This is the basis for diodes.

The photoelectric effect is similar: free electron charge carriers are produced when light strikes a material. The free electron is produced when the photon imparts sufficient energy to kick a valence band electron into the conduction band. The effect is dependent on the frequency of the photon, not the intensity of the light.

Einstein proposed in 1905 that light exists as discrete particles (photons). The effect was confirmed by Millikan in 1914. While the effect may occur in many materials, it is most prevalent in metals. As it turned out, all electromagnetic phenomena is explained by the principles of what is now known as quantum electrodynamics. The difference between the thermoelectric and photoelectric effect is the energy content of the photon. Thermal radiation is long wavelength infrared (8 – 15 μm).

This discussion will focus on near infrared. For atmospheric measurements, absorption by water vapor is a significant and usually undesired effect – an “open” wavelength band that exists between 1530 and 1570 nm is feasible for such spectral measurements.

The intensity of an optical beam as it passes through a finite medium is described by the Beer-Lambert law:

$\Phi \; = \; \Phi_o \, exp \left[ \, \sum\limits_{i=1}^N \, \int\limits_0^Z \rho_i \, \delta_i \; \text{d} z \, \right]$

where $\Phi$ is intensity of the beam, i is the species index, ρ is the density, δ is the absorption/scattering cross section, and Z is the optical path length.

The expression may be simplified:
Assume only a single species is of interest (by the use of a monochromatic laser as an optical source)
Assume the cross-section area is constant

$\Phi \; = \; \Phi_o \, exp \left[ \, \delta \, \int\limits_0^Z \rho_i \; \text{d} z \, \right]$

I’m interested in the near-IR absorption characteristics.

The Beer-Lambert Law relates the intensity of light entering an absorbing media to that leaving it. The source intensity Io is projected along the z-axis over path length Z. The photon flux crosses through an incremental length dz of an area A normal to the direction of light. The amount of light absorbed within the incremental area is a function of both the light entering the area and the probability a photon will be absorbed.

The probability of absorption is determined by the ratio of the effective absorbing cross-sectional area of the molecule to the effective cross-sectional area of the photon flux (the absorption cross-section $\sigma$ is different than the physical cross-section; it encompasses the region within which a photon will be captured). The probability of absorption is therefore:

$p \; = \; N \, \frac{\sigma}{\, A \, }$

where N is the number of molecules within the flux cross-section.

It is more convenient to work with molecular concentration – molecules per volume – such that:

$\mathbb{c} \; = \; \frac{N}{\, V \, } \; = \; \frac{N}{\, A \, \text{d}z \, }$

A bit of shifting of variables gives:

$p \; = \; \sigma \, \mathbb{c} \, \text{d}z$

Substituting:

$\text{d}I\text{\footnotesize{(z)}} \; = \; - \sigma \, \mathbb{c} \, I\text{\footnotesize{(z)}} \, \text{d}z \; \; \Rightarrow \; \; \frac{ \, \text{d}I\text{\footnotesize{(z)}} \, }{I\text{\footnotesize{(z)}}} \; = \; - \sigma \, \mathbb{c} \, \text{d}z$

$\int\limits_{\Phi_o}^\Phi \frac{ \, \text{d}\Phi\text{\footnotesize{(z)}} \, }{\Phi\text{\footnotesize{(z)}}} \; \; \equiv \; \; \int\limits_0^Z - \sigma \, \mathbb{c} \, \text{d}z \quad \Rightarrow \quad -ln\left(\frac{\Phi}{\, \Phi_o \, } \right) \; = \; \sigma \, \mathbb{c} \, Z$

which is the Beer-Lambert expression.

Concentration is best described as moles per litre rather than molecules per volume. This necessitates a change of parameters from absorption cross-section to molar absorptivity. This modifies Beer-Lambert to:

$-ln\left(\frac{\Phi}{\, \Phi_o \, } \right) \; = \; \epsilon \, \mathbb{c} \, Z$

Given the species molar absorptivity coefficient and optical path length, the concentration of a species may be determined by optical means.

$\mathbb{c} \; = \; (\sigma \, Z)^{-1} \, ln\left(\frac{\Phi}{\, \Phi_o \, } \right)$

where $\Phi_o$ , $\sigma$ , and Z are known quantities.

Goal: Measure the received intensity.

Transverse Electromagnetic Propagation

However, a laser is a directed monochromatic beam; the energy is not isotropic, but is rather more of the form of a transverse electromagnetic field along a transmission line which, for the electric field component, is defined as:

$E_r \; = \; E_o \, \mathbb{e}^{- \jmath \, k \cdot r}$

where $E_r$ is the received electric field after passing through a material characterized by wavenumber $k$ and position vector $r$ .

In a single dimension, the wavenumber k may be defined in terms of material properties:

$E_r \; = \; E_o \, \mathbb{e}^{- \alpha \, z} \, \mathbb{e}^{- \jmath \, \beta \, z} \, \mathbb{e}^{- \jmath \, \omega \, t}$

Term α is real and represents field attenuation. As distance z increases, the term tends to zero. “Skin depth” is defined as the distance in which the magnitude has decreased by a factor of $\mathbb{e}^{-1}$ related to the initial magnitude. Note that skin depth is a dimensionless measure of attenuation but is not a direct indicator of detectability.

Term β represents a wave varying sinusoidally with z ( $\mathbb{e}^{- \jmath \, \beta \, z}$ = $cos \beta z - \jmath sin \beta z$ ). For this discussion, β may be assumed zero.

Term ω represents a wave varying sinusoidally with t … but the wavelength of the photons is such that the flux is essentially constant. $\omega \, t$ goes to 0

Geophysical Model and Reflection Coefficient

I note that the Beer-Lambert expression is of identical form to that of electromagnetic field propagation:

$\mathcal{E} \; = \; E_o \, exp^{- \jmath \, k \, z} \, + \; E_r \, exp^{+ \jmath \, k \, z}$

where wavenumber k in free space is:

$k \; = \; \omega \, \sqrt{ \, \varepsilon \, \mu \; }$

The reflection coefficient is expressed in terms of material properties as:

$\mathbb{r} \; = \; \frac{E_r}{\, E_o \,} \; = \; \frac{\eta_1 \, - \, \eta_2}{\, \eta_1 \, + \, \eta_2 \,}$

where material impedance $\eta$ is:

$\eta \; = \; \sqrt{ \, \frac{ \, \mu \, }{\varepsilon} \, }$

where $\mu$ is $\mu_o$ , the reflection coefficient becomes:

$\mathbb{r} \; = \; \frac{\sqrt{ \, \varepsilon_1 \, } \, - \, \sqrt{ \, \varepsilon_2 \, }}{\sqrt{ \, \varepsilon_1 \, } \, + \, \sqrt{ \, \varepsilon_2 \, }}$

This should not be surprising in that photons are the basic particles of electromagnetism.

Assuming one media is air, then:

$\mathbb{r} \; = \; \frac{1 \, - \, \sqrt{ \, \varepsilon_{r2} \, }}\, 1 \, + \, \sqrt{ \, \varepsilon_{r2} \, }}$

(The dielectric permittivity of air is not quite that of free-space but the value of 1.00059 is typically rounded off to 1)

Reflection coefficients from earth surface materials can range from near-zero (basalts, dark soils) to almost 1 (snow, ice) although the actual reflection coefficient is wavelength-dependent.

The project expression is based on the Beer-Lambert relationship, but there are photon losses due to the effects of atmospheric scattering, ground scattering, and ground absorption. I’ll neglect ground absorption as those effects are an indirect part of the reflection coefficient.

The reflection coefficient is the most significant – and least characterizable – factor in the optical path. The coefficient is so variable that even a small percentage uncertainty will restrict the possible precision of the measure interpretation (the measure itself – of number of photons received – may be precise and accurate … but the interpretation of concentration depends on the reflection coefficient.)

Returning to the core function, the intensity at the ground only requires a modest change in representation:

$\mathcal{F}_1 \; = \; \frac{1}{\, \sigma \, z_1 \,} \, ln\left(\frac{\Phi_g}{\, \Phi_o \, } \right)$

The photon flux at the ground is either reflected back to the receiver, scattered away, or absorbed by the reflecting surface. Since I’m only interested in the detectable return signal, I won’t consider scattering or absorption losses – those are accounted for by the effective reflection coefficient (more losses, lower coefficient).

The reflected intensity is simply $I_g \times \mathbb{R}$ . The received intensity may now be expressed as:

$\mathcal{F}_r \; = \; \frac{1}{\, \sigma \, z_2 \,} \, ln\left(\frac{\Phi_r}{\, \Phi_g \, \mathbb{R} \,} \right)$

I’ve assumed $\sigma$ to be constant – I’ve not yet considered the effects of altitude, temperature, and pressure, nor have I considered atmospheric particle scattering.

It’s beginning to appear as if there are so many uncharacterizable variables that the measurement data will not be able to be properly interpreted …

As with EM fields, the effect of the total path is now:

$\mathcal{F} \; = \; \mathcal{F}_1 \, + \, \mathcal{F}_2 \; = \; \frac{1}{\, \sigma \, z_1 \,} \, ln\left(\frac{\Phi_g}{\, \Phi_o \, } \right) \; + \; \frac{1}{\, \sigma \, z_2 \,} \, ln\left(\frac{\Phi_r}{\, \Phi_g \, \mathbb{R} \,} \right)$

or, after wiggling the equation around a bit,

$\mathbb{c} = \; \frac{1}{\, \sigma \, Z \,} \, ln\left(\frac{\Phi_r}{\, \Phi_o \, } \, \frac{1}{\, \mathbb{R} \,} \right)$

The Reflection Coefficient

This has such a major effect on the signal, it should be discussed in and of itself.

And I know I worked on this someplace. I just need to find the file or re-do it

The Source

A continuous wave laser will be transmitted through the atmosphere. For the purposes of electronic interfacing, the laser beam may be considered analogous to a piece of wire: it is simply a means of collecting and transmitting the information desired[1].

In simplistic terms, photons having specific energy will interact with certain molecules. The energy of the photon is absorbed by the molecule; the molecule undergoes an increase in temperature. The energy of a photon is related to the wavelength as:

$\mathcal{E}_\gamma \; = \; \frac{\, \mathcal{h} \, \text{c} \, }{\lambda}$

where $\mathcal{h}$ is Planck’s Constant, c is photon velocity (speed-of-light in air), and $\lambda$ is the photon wavelength.

Here’s where the rubber meets the road … the photon wavelength must be at an absorption frequency of the species desired. The wavelength selected also has to be detectable – an appropriate photodetector must be available. For this experiment, I’ll use a photodetector with suitable responsivity in the near-infrared between 1300 and 1650 nm.

Carbon dioxide (CO2) and methane (CH4) are two gases of strong interest. Using SpectralCalc, I see the spectra appear as:

where RED is methane; BLK is CO2. This, however, is a log scale – the instrument measures a linear scale (although log processing is a possibility and should be considered considering the signal is of a log/exp nature).

The following shows the response on a linear scale. C2 is quite prominent in three regions … but I haven’t examined other atmospheric constituents such as water vapour. Not going to right now either – I could spend an article discussing wavelength selection.

For this experiment, I’ll pick a wavelength in the neighbourhood of 1570 nm. The exact wavelength will be more precise than this. The photodetector is not capable of the desired discrimination; to prevent overlap, an optical filter will be used – these can provide bandpass filtering in the neighbourhood of ±20 pm.

The photon/molecular interaction will result in a linear loss of beam intensity.

The Photodiode

Photons are converted to electrons via the photoelectric effect. In a manner of speaking, this is similar to molecular absorption.

For now, assume a 1:1 correspondence between photon energy and charge creation with 100% efficiency in the photodiode having a transductance factor of hÂP = i where h is dimensionless conversion efficiency, Â is responsivity in A/W, P is optical power (energy in units of joules per photon times intensity in photons per second), and i is photodiode output current (a “typical” value of Â might be on the order of 10A/W)

The detector will require a minimum optical power to provide a usable measurement; it also has a maximum limit of optical power beyond which the detector will possibly be destroyed. While there are potential techniques to extract certain types of signals from below the noise floor (at a cost of increased complexity), the upper limit is defined by the risk of physical damage. These limits define the detector dynamic range; the laser transmit power and expected intensity losses need to be matched with a detector to maximize a measurement dynamic range.

All else is secondary.

6666666666666666666666
If an electron within a material (such as a semiconductor lattice) absorbs the energy of a photon and acquires more energy than the electron binding energy of the material, it breaks free of the lattice structure and forms a free charge carrier. If the photon energy is too low, the electron remains bonded to the lattice.

The release of an electron is dependent on the energy of an individual photon via the optical wavelength rather than intensity – an increase of intensity of low-energy photons simply increases the number of low-energy photons; a change in intensity does not increase the energy of any single photon. The creation of emitted electrons depends only on the energy of an individual photons. This energy transfer occurs between the incident photon and the outermost valence electrons.

Electrons will absorb energy from photons but in an all-or-nothing manner: all of the energy from one photon must be absorbed and used to liberate one electron from atomic binding. If not, the energy is re-emitted. If the photon energy is absorbed, some of the energy liberates the electron from the atom, and the rest contributes to the electron’s kinetic energy as a free particle.

The detection of an optical beam is a fundamentally continuous time process for optical intensities of interest. Optical intensity (power per unit area in J/s/m $^2$ ) and optical power (energy per second in J/s) are numerically identical quantities if the beam intensity is normalized to the detector area. A photodiode converts photons to charge-carrying electrons. (Electrical current has units of charge/sec). Measuring intensity with a photodiode implies measuring charge over some period of time.

The photodetector used as an example is sensitive to optical wavelengths in the vicinity of 1550 nm. As such, each photon has energy:

$\mathcal{E}_\gamma \; = \; \frac{\, h \, \text{c} \, }{\lambda} \, \frac{\, \text{J} \cdot \text{s} \times \text{m/s} \, }{\text{m}} \; \; \Rightarrow \; \; \mathcal{E}_\gamma \; = \; \frac{\, \oldstylenums{6.626}e\text{-34} \times \oldstylenums{300}e\text{6} \, }{\oldstylenums{1550}e\text{-9}} \; = \; \oldstylenums{128}e\text{-21}} \; \text{J}$

Power is energy per sec: 1 nW corresponds to:

$\text{P}_{\text{opt}} \; = \; 1 \times 10^{-9} \; \text{W = J/s} \; = \; n \, \mathcal{E}_\gamma \; = \; n \, \oldstylenums{128}e\text{-21}} \; \text{W = J/s} \; \; \Rightarrow \; \; \oldstylenums{7.81}e\text{9}} \; \gamma\text{/s}$

The basic photodiode is modelled as a reverse-biased semiconductor pn-junction. Semiconductor physics suggests that the balance of charge at a pn-junction forms a depletion zone – a region empty of free charge carriers – at the junction. The width of this depletion region increases as reverse-biased potential is increased. This forms a potential-dependent capacitance across the device. High frequency measurements tend to reverse-bias the diode at high voltages to reduce the shunt capacitance; that is not necessary for lower frequency measures such as those encountered with this project.

When a photon of sufficient energy strikes the lattice in the depletion region, it is absorbed into the structure by expending its energy into the release of an electron (a charge hole is also formed, but is not significant in this discussion). The electron is drawn to the device terminal; the movement of charge forms a current.

Noise in the photodiode primarily arises from one of two sources. A semiconductor junction is a delicately balanced structure of charge energy. Additional thermal energy can upset this balance and cause the release of an electron from the lattice. The aggregate of this energy transfer forms thermal current – commonly referred to as “white noise” which has a Gaussian probability distribution. The other primary noise source is created by the drift of charge carriers across the potential of a pn-junction. This is referred to as “shot noise”. This noise also has a Gaussian probability distribution.

“Flicker noise” is neither Gaussian nor stationary; it has a probability distribution inversely proportional to frequency and is sometimes referred to as “1 over f” noise. This is a low frequency phenomena; as the frequency of event increases, the magnitude decreases.

Thermal noise is often the most significant noise factor – this noise is reduced by a factor of the square root of the decrease in temperature. A reduction in temperature from “room” (300 K) to “LN77” (liquid nitrogen: 77 K) temperature will reduce theoretical thermal RMS noise by a factor of about ½.

$\frac{\text{Noise}_{77,rms}}{\, \text{Noise}_{300,rms} \, } \; \propto \; \sqrt{ \, \frac{77}{\, 300 \, } \, } \; = \, 0.5066$

This does not include additional noise sources related to cooling the detector. Shot and flicker noise are not temperature dependent (to 1st-order). The question to cool or not cool is deferred; this project assumes no cooling. In any case, it is more important to stabilize temperature than maintain a specific value; ΔT is more critical than T. Shot noise is usually not significant at moderate current levels but becomes more so at low signal magnitudes. Shot noise has a Poisson probability distribution with rms magnitude of 18 fA/rtHz at a DC current of 1 nA.

A noiseless photodiode has two primary parameters: dark current and responsivity. The dark current represents a “leakage” current within the structure – this current flows without optical stimulation. It has a Gaussian probability distribution consisting of both thermal and shot elements. Empirical measurements are more effective than analysis for determining dark current uncertainty. This dark current noise is usually less significant than other system noise sources but does establish the minimum detectable signal level.

To 1st-order, dark current is considered constant and represents the minimum device current. Optical stimulation adds to the dark current to form the device current such that the output is expressed as:

$\text{I}_{pd} \; = \; \text{I}_{dark} \, + \, \text{I}_{sig}$

In determining signal dynamic range, the dark current forms the lower signal bound. If the signal current is equal to the dark current, the dynamic range – at that signal magnitude – may be defined as 0 dB. This output current has 100% error.

Both dark and signal current flow in the same direction – this being a diode. The photodiode is reverse or zero biased; these currents are analogous to leakage current in a traditional diode. Electrical current is the movement of charge. Each electron carries a quanta of charge, 1.69E-19 coulomb; 1 C consists of 6.24E18 electrons.

The responsivity $\mathbb{R}$ in units of amperes/watt describes the relationship between optical power on the detector and the generated signal current. A responsivity value of 1 A/W is used in this discussion (the actual photodiode has $\mathbb{R}$ = 0.95 A/W). In terms of actual response, units of nA/nW are more realistic.

The system counts electrons by measuring charge for some defined period of time and reports the results. The number of electrons is related to the number of photons by the photodetector responsivity:

$\frac{\; \text{number of electrons} \;}{\text{time}} \; \; = \; \; \mathbb{R} \, \frac{\; \text{number of photons} \; }{\text{time}}$

Expanding:

$\mathbb{R} \; \equiv \; \frac{\; \text{amperes} \;}{\text{watt}} \; \; = \; \; \frac{\; \frac{ \; \text{coulombs} \; }{\text{sec}} \; }{\frac{ \; \text{energy} \; }{\text{sec}} } \; = \; \frac{\; \text{coulombs} \;}{\text{energy}} \; = \; \text{number of electrons} \;}{\text{number of photons}}$

The responsivity parameter defines the relationship between optical power and electrical current. Power and intensity have the same units if intensity is normalized to the detection area of the photodiode.

The selected photodetector has responsivity of 0.95 A/W

$\text{I}_{\text{pd}} \; = \; \frac{\, dq \, }{\text{dt}} \; = \; \text{P}_{\text{opt}} \, \mathbb{R} \; \; \Rightarrow \; \; 1 \times 10^{-9} \, \times \, 0.95 \; = \; 0.95 \; \text{nA}$

This corresponds to:

$7.81 \times 10^9 \; \gamma / \text{s} \; \; = \; \; 6.24 \times 10^9 \; \text{q/s}$

If a clocked system has frequency of 100 MHz, each incremental period converts:

$78.1 \; \gamma \; \; \equiv \; \; 62.4 \; \text{q}$

Charge
So perhaps considering the information as charge rather than current:

$\mathbb{R} \, \frac{\, \text{d} \gamma \, }{\text{dt}} \; = \; \frac{\, \text{dq} \, }{\text{dt}} \; \; \Rightarrow \; \; \frac{\, \text{d} \gamma \, }{\text{dt}} \; = \; \left. \frac{\, \text{dq} \, }{\text{dt}} \right|_{\mathbb{R}=1}$

While photodetectors may vary in material and wavelength response, they are all at heart simply diodes – pn-junctions – and once the photon-electron conversion has occurred, the signal from all types is simply a current. Common electronic processing procedures should be adaptable to a variety of photodiode types.

Although the match between number of photons and number of electrons is not quite 1:1, it is close enough to use in approximations such as for this discussion.

1 nW ( 1E-9 J/s ) of monochromatic optical energy will produce 1 nA ( 1E-9 C/s ) of current. Rather than relate watts to amps, consider number of photons to number of electrons. Monochromatic photons all have identical energy; electrons all carry identical charge.

The amount of charge accumulated in a given time is proportional to the number of impinging photons in that given time: the specific time period is not a factor. This allows time to be a controlling variable – integrate over time from 0 to T.

A capacitor is an electronic component which accumulates charge. As charge accumulates on the capacitor, the sum of the individual charges cause a voltage potential to develop across the capacitor.

Voltage and charge are related as:

$\text{Q} \; = \; \text{C V}$

A change in charge produces a change in voltage:

$\text{d}Q \; = \; C \, \text{d}V$

Related to time:

$text{d}Q \, }{\text{d}t} \; = \; C \, text{d}V \, }{\text{d}t}$

Noting that

$\frac{\, \text{d}q \, }{\text{d}t} \; = \; \text{I}$

Gives:

$\text{I} \; = \; C \, \frac{\, \text{d}v \, }{\text{d}t}$

Integrating:

$\int\limits_T I \, \text{d}t \; = \; \int\limits_T \frac{\, \text{d}q \,}{\text{d}t} \, \text{d}t \; = \; C \, \int\limits_T \frac{\, \text{d}v \,}{\text{d}t} \, \text{d}t$

$\int\limits_T I \, \text{d}t \; = \; \text{Q} \; = \; \text{C V}$

A rather roundabout way of stating that by integrating the photodiode current over some time period T, the accumulated charge – directly related to optical intensity – may be inferred from the voltage developed across a known capacitance. A simple switch is used to “reset” the capacitor after time T.

666666666666666666

The Method

Two source lasers of similar but barely different wavelengths. Close enough to use the same photodetector as a receiver. One laser will be tuned to an absorption frequency of a molecule of interest; the other tuned just a squeech off – just far enough to avoid absorption. Both beams have the same optical path.

The desired information is extracted from:

$\text{N}_A \; = \; - \frac{1}{\, 2 \, \text{L} \, \Delta \sigma \, } \, ln \left[ \, \frac{R_{ON}}{\, R_{OFF} \, } \, \frac{\, S_{OFF} \, }{S_{ON}} \, \right]$

where I should know the source intensities S $_{ON}$ and S $_{OFF}$ for the ON and OFF channels, and I measure the received intensities R $_{ON}$ and R $_{OFF}$ .

Sensitivity

In general, the sensitivity of function y to a variation in parameter x is defined as:

$\mathcal{S}_x^y \; = \; \frac{\, x \, }{y} \, \frac{\, \partial y \, }{\partial x}$

It can be demonstrated that the absorbance function has identical sensitivity to both distance and concentration.

… where it is assumed that absorbance is a constant material property (since I assume a random spatial molecular orientation in statistically significant numbers, any possible tensor properties are assumed negated at a macro scale.

Performance of the reference channel will be critical to the success of the measurement.

The first part of the foundation for measurement variations is the sensitivity of the measurement to variations in the function parameters.

In general, the sensitivity of function y to a variation in parameter x is defined as:

$\mathbb{S}_x^y \; = \; \frac{x}{\, y \, } \, \frac{\, \partial y \, }{\partial x}$

Not Measured

It is important to note what is not measured.

The term “scattering” is not intended to mean the same thing as the term “reflection” although scattering is a form of reflection.

Scattering is intended to be the random loss of photons due to non-absorption interactions of photons with atmospheric particles – particles on the order of the photon wavelength will cause the photon to change direction to random directions away from the beam path. If the two beams are of similar wavelength and travel along identical paths, the variations in intensity loss will be statistically identical over a path length significantly longer than the dimension of the scattering agent. The assumption of common-mode scattering will be considered valid during the initial design phase

The term “reflection” is intended to refer to action of reversing the beam direction – generally the earth surface. Although the base assumption of common path is used, timing issues may suggest this assumption is not valid. Changes in the reflecting material and/or angle will possibly change between the two beams depending on the horizontal velocity, the horizontal integration time, and changes in topography and material at the surface. Small changes in reflectivity can have significant changes in the intensity of the received signal. This will become more apparent during discussions of numerical values of parameters and timing. This assumption is dubious and needs to be considered during the initial instrument design phase.

The instrument only measures the beam intensity at the transmit and receive ends. The instrument only counts electrons which have been generated by photon interaction within the detector. Given sufficiently close laser wavelengths, the detector by itself will not differentiate between the two beams.

The instrument does not differentiate segments of distance; it simply measures the end result of the integration over the path length. A thin layer of high concentration will not be differentiated from a wide layer of lesser concentration.

That’s good for now.

[1] The mechanics of generating the laser beam and the physics of the photon-to-electron transformation in the photodiode are of interest … at that level. For the purposes of extracting the desired information, the photodetector produces a current having a linear relationship to beam intensity. Ideally, 1 photon produces 1 electron. All else is scaling.