A single-beam photothermal interferometer for in situ measurements of aerosol light absorption

Abstract. We have developed a novel single-beam photothermal interferometer and present here its application for the measurement of aerosol light absorption. The use of only a single laser beam allows for a compact optical set-up and significantly easier alignment compared to standard
dual-beam photothermal interferometers, making it ideal for field measurements. Due to a unique configuration of the reference interferometer arm, light absorption by aerosols can be determined directly – even in the presence of light-absorbing gases. The instrument can be calibrated directly with light-absorbing gases, such as NO2, and can be used to calibrate other light absorption instruments. The detection limits (1σ) for absorption for 10 and 60 s averaging times were determined to be 14.6 and 7.4 Mm−1, respectively, which for a mass absorption cross section of 10 m2 g−1 leads to equivalent black carbon concentration detection limits of 1460 and 740 ng m−3, respectively. The detection limit could be reduced further by improvements to the isolation of the instrument and the signal detection and processing schemes employed.



Adaptation of the PTI equation for a focused laser beam with Gaussian beam profile
Here, we calculate the phase shift in our modulated single-beam PTI (MSPTI) for a fundamental (or TEM 00 ) transverse Gaussian mode.This calculation can be expanded to other beam profiles, if the profile is known.Figure S1 shows the intensity profile along the direction of propagation z for a focused beam with a TEM 00 mode profile.The focal point of the beam is arbitrarily set to z = 0, which we assume to be located in the centre of the measurement chamber of length 2a.
The intensity  (units of W m -2 ) in the chamber is given by where  is the radial distance from the central axis of the beam and z is the propagation axis with the focal point at  = 0.  0 = ( = 0) is the waist of the laser beam in the focal point, and () is given by and depends on the Rayleigh distance The total laser power (units of W) is In analogy to the calculation presented in Sedlacek (2006), the induced phase change is calculated in an infinitesimally small toroidal volume  = 2 •  • .For an absorption coefficient   the absorbed power within  is During the heating time ∆, the induced temperature change within  is where ,   and  are the infinitesimal mass, heat capacity and density of the air, respectively.The induced phase change from  is For calculating the total phase change ∆, one needs to account for the z dependence of the intensity in the chamber and weigh ∆ with a weighting factor before integrating over the entire volume.This weighting factor is the normalized intensity The integration over the measurement chamber volume yields Combining S8 with S1-S4 and using the identities In the following paragraph we present an example calculation for a 10 cm long (a = 5 cm) sample chamber.Figure S1a shows for a constant laser power of  = 100 mW the Illuminance in the beam centre (r = 0) as a function of the chamber length position z, where the laser beam is focused in the centre of the chamber (z = 0), for a number of different beam waists  0 .
For a heating time of ∆ = 5.5 ms (91 Hz modulation), and absorption coefficient   = 10 −4 m -1 and air at standard conditions, Figure S1b presents the corresponding spatial contribution to the measured phase shift, i.e.
This example illustrates that focussing at the correct position is important to centre and focus the sensitive region into a well-confined range and to lower signal background contributions from the chamber windows.

Description of signal processing methodology
The rate of change of phase during the heating cycle is determined by two processes.Firstly, the linear heating process arising from the absorption of light within the beam volume and secondly, by the exponential cooling of the beam volume driven by the temperature difference to the surroundings.The effect of these two competing processes is that the beam volume reaches an equilibrium temperature.The phase change during the heating cycle for the PTI developed in this work is found to be empirically best approximated with an exponential function of the form: where the parameter  is the limit of the phase change and is defined as the phase shift at which the heating rate due to absorption and cooling rate due to heat loss out of the beam volume are equal,  is the mean lifetime of the cooling process and  is the offset from phase quadrature.The parameter  is characteristic for the system and is dependent on the pump beam focusing and the pump/probe beam geometry, and the thermal diffusivity of the sample gas (Monson et al., 1989).
An example of a 1-second average of experimental data for a soot concentration of approximately 100 µg m -³ is shown in Figure S2.The experimental data has been offset corrected by firstly subtracting  and then ∆ measured in a pure argon atmosphere to remove the PTI contributions from the interferometer optics, leaving just the absorption dynamics of the sample.
The heating curves were analysed for a range of different concentrations of electrical discharge generated BC and the results of the fits are shown in Figure S3.By comparing the  and  values from the fits for a range of concentrations it was found that  remained constant within experimental error, and that  showed a linear relationship with soot concentration.This can also be understood directly from the definitions of these values:  is dependent on the laser beam geometry and thus the time required for heat from light absorption to leave the beam volume.The beam geometry did not change between measurements and so  is observed to remain virtually constant.The parameter  is directly related to the heat increase of the beam volume due to light absorption and is therefore expected to vary linearly with the absorption coefficient, i.e. the concentration of the absorbing species.
At  ~ 0 the heat losses out of the beam volume are negligible and a near linear increase of ∆ is observed with time.
The rate of increase of ∆ for  ~ 0 is given by In Figure S4, the exponential fit from Figure S2 is plotted together with fitted  and a line of the form: The best linear fits to the heating curves were also determined.A comparison of the results obtained from the exponential and linear fits to the heating curves in terms of with the rate of increase of the exponential fits a factor of approximately 2.7 higher than that of the linear fits.As ∆  is calibrated to   via a gas of known absorption (NO 2 in this case), the absolute value of the fitted rate of increase is unimportant, so long as the value determined for ∆  increases linearly with   .This linear relationship has been demonstrated for both the exponential and linear fitting methods, thus validating the use of the linear fitting methodology.

Physical layout of the single-beam photothermal interferometer
See Figure S6 for a photographic representation of the MSPTI.

Raw data for calculation of standard deviation and drift
See Figure S7 for the best least-squares linear fit to the raw data for the subtraction of the linear drift.
Figure S5.The results of the exponential fits are presented as   as per Equation S12.Both sets of fits show a clear linear relationship with eBC concentration,

Figure S1 :Figure S3 - 105 Figure S4 -
Figure S1: The optical intensity of the laser beam along the propagation axis z is shown in a).The position of the focus has been set to z = 0.In b) the spatial contribution to the MSPTI signal is plotted as a function of z.The localisation of the signal contribution around the focal point as the beam waist decreases shows the importance of the positioning of the sample chamber.Note the log scale on the y-axis.

Figure S7 - 125 Figure S8 -
Figure S7 -Raw data for the determination of the baseline drift.The data points are 1 second averages and the black line is the best linear fit to the data.The drift was subtracted from the raw data before calculation of the detection limits for the instrument.