Acetone Absorption Cross-Section in the Near-Infrared of the Methyl Stretch Overtone and Application for Analysis of Human Breath

Abstract

We present an empirical model for the cross-section of low concentration acetone gas in the range of 1671.5–1675 nm that encompasses the absorption band of the methyl stretch overtone. This model is experimentally validated with cavity ring-down spectroscopy (CRDS) measurements performed with a calibration gas and its diluted mixtures with breath samples. Particular attention is paid to accurate wavelength measurements with an interferometric wavemeter. The theoretical framework for analysis of gas mixtures with several absorbing species is presented. We show that the proposed empirical model can be used to accurately determine the concentration of acetone vapor in human breath samples. The comparison of the acetone absorption cross-section with previous results is also presented.

1. Introduction

Spectroscopic detection of acetone is a task that arises in several fields. In the medical domain, it has attracted significant interest due to the potential of acetone as a biomarker for diseases like type-1 diabetes (T1D) and possibly type-2 diabetes (T2D) [1], and heart failure [2,3]. It can also be used for monitoring patients with ketoacidosis [1,4] and those on a ketogenic diet rich with fats [5]. Typical concentrations of acetone in exhaled breath of healthy individuals are from 0.2 ppm to 1.03 ppm, which for T1D patients can be elevated to 1.7–3.8 ppm and even higher [1]. The concentration of acetone in atmospheric air normally is much lower, within 1–10 ppb [6,7], which requires even more sensitive techniques for environmental monitoring. The need for spectroscopic analysis of acetone also appears in astronomical observations [8,9].

Multiple spectroscopic techniques have been employed to accurately detect volatile organic compounds (VOCs) in air and exhaled breath: cavity ring-down spectroscopy (CRDS) [10,11,12], tunable diode laser absorption spectroscopy (TDLAS) [13], wavelength modulation spectroscopy (WMS) [14,15], frequency comb laser absorption spectroscopy (FC-LAS) [16], and the results are summarized in recent reviews [17,18,19]. Another sensitive technique that can be used for detection of VOCs is photoacoustic spectroscopy (PAS) [20,21].

Besides spectroscopic approaches, other sensitive techniques such as gas chromatography-mass spectrometry (GC-MS), proton transfer reaction mass spectrometry (PTR-MS), and selected ion flow tube mass spectrometry (SHIFT-MS) have been used for breath analysis, and some references and results with these techniques can be found in review papers [1,17]. The main advantages of the NIR-CRDS approach presented here are its relative simplicity, low cost, and small footprint, combined with sufficiently high sensitivity and specificity.

For the detection of acetone, electronic transitions in the UV region [22,23] as well as vibrational absorption bands in the near-infrared (NIR) and mid-infrared (MIR) spectral regions were employed. In the UV range, the attenuation due to scattering starts prevailing over the absorption coefficient of acetone when it reaches the value of $2\times {10}^{-6}$ cm⁻¹ [18]. The reflectivity of the mirrors limited at this wavelength by about R = 0.993 also reduces the sensitivity limit to about ${10}^{-6}$ cm⁻¹ [18], which at practical pressures does not make it possible to reach the required sensitivity level of portions of ppm.

In the MIR, several fundamental vibrational modes of acetone are located, which offer strong absorption cross-sections and ultimately a very high sensitivity can be realized with quantum cascade lasers. With WMS near 3367 nm, detection limits as low as 0.12 ppm with 60 s signal averaging time were achieved [15], and with PAS near 8270 nm, the detection limit of 0.25 ppb with the lock-in integration time 10 s was reported [20]. However, compared to NIR, laser sources and detectors in the MIR range are more expensive, and the noise level at room temperature is relatively high, which requires cooling of the detector. For CRDS, the achievable reflectivity of the mirrors in the MIR are typically substantially lower than what is possible in the NIR. Even with the cross-sections in the NIR about 100 times smaller than in the MIR, still NIR CRDS provides sufficient sensitivity for detection of acetone in exhaled breath.

The absorption cross-sections for acetone in the MIR were determined in the 830–1950 cm⁻¹ (5.128–12.05 $\mathsf{\mu }$m) spectral range using a high-resolution FTIR spectrometer [24]. In the NIR, the absorption cross-section of the C–H overtone was determined for isoprene, 1,3-butadiene, and 2,3-dimethyl-1,3-butadiene [23], using continuous wave cavity ring-down spectroscopy (CW-CRDS) and known absorption cross-sections in the UV region. Results on the absorption cross-sections of acetone in the NIR region are scarce, especially measurements performed in a gas mixture. Some previously obtained results are presented in the discussion in Section 5.

In this work, we perform measurements of the absorption cross-section of acetone for the overtone of the C-H stretching mode in the spectral range of 1671.5–1675 nm. For this purpose, NIR-CRDS was used. The technique leverages an optical cavity with highly reflective mirrors, causing light to circulate multiple times through the sample. This extended path length enhances the interaction between the light and the sample, making it possible to detect trace amounts of acetone and other accompanying gases. The high sensitivity of NIR-CRDS allows for detection limits down to the level of tens of ppb, which is sufficient for measurements of breath samples.

NIR-CRDS offers several advantages compared to other spectroscopic techniques. It can target specific overtones and combination bands in the near-infrared region since the laser sources in this spectral range are readily available. The NIR-CRDS setup is relatively simple and can be used in applications such as medical diagnostics or environmental monitoring. Since the technique measures the decay rate of light rather than absolute absorption intensity, it can perform with minimal interference from environmental noise or fluctuations in laser power unlike some other absorption techniques where the light intensity must be carefully monitored for accurate concentration results. The selected wavelength interval provides the advantage of only minor interference of the absorption lines of water vapor and CO₂, which are significantly present in human breath and the atmosphere. NIR-CRDS benefits from the long effective path lengths created by the highly reflective mirrors in the optical cavity. In fact, an interaction length as long as 137 km was reached in our setup.

In the NIR, acetone has a relatively broad absorption band which peaks at a wavelength of 1672.44 nm [25]. The broadband absorption can be seen in the main plot of Figure 1, while the inset shows a zoom of the peak absorption feature. This feature shown in the inset of Figure 1 is particularly interesting because it can be measured with the tuning range of a typical DFB diode laser, and it is this feature that we aim to characterize. The shape and magnitude of this absorption feature of acetone have been measured for pure gas samples and those diluted with laboratory air at low pressure. However, it has yet to be verified if there are distortions in the spectrum shape occurring in the presence of other gas species at breath concentrations and at higher pressures.

In this investigation, we show that the absorption cross-section of acetone in the range of 1671.5–1675 nm has a consistent shape that is independent of pressure in the range of 30–500 Torr and is not affected by the high concentrations of water vapor present in human breath. Additionally, we characterize this absorption feature by providing an empirical formula for the cross-section valid to high accuracy, which we demonstrate is useful for providing an accurate determination of acetone concentration in human breath samples. The developed NIR CRDS setup, despite its simplicity, is robust and efficient in performing breath analysis in the desired sensitivity range down to tens of ppb level for acetone.

2. Theoretical Basis

2.1. Description of Spectral Measurement with CRDS

The method of CW-CRDS [27] allows for calibration-free measurement of the absorption coefficient of weakly absorbing species present in the mode volume of a high finesse optical cavity. The presence of this absorbing species modifies the finesse of the optical cavity, and this change can be sensitively observed through the ring-down time of the cavity [28].

The ring-down time of the optical cavity is measured with CW-CRDS by coupling laser light into the cavity and rapidly shutting off this laser with an external modulator. The light intensity, which is transmitted through one of the cavity mirrors, can then be observed to undergo and exponential decay, which is well modeled by the function

$$y\left(t\right)=A{e}^{-t/\tau }+B$$

(1)

where $\tau$ is the ring-down time, and A, B are two fitting constants.

In the absence of any absorbing species, the optical cavity will have some baseline ring-down time which, in terms of the speed of light c and the round-trip length of the cavity L, is

$${\tau }_0\left(\lambda \right)={\displaystyle \frac{L/c}{1-R^2\left(\lambda \right)+X}}$$

(2)

which is in general dependent on wavelength. Here, $R\left(\lambda \right)$ is the wavelength-dependent reflectivity of the cavity mirrors, which we assume is the same for both mirrors. Additional losses in the cavity (diffraction and scattering) are taken into account by the X term. This baseline can be measured by evacuating the chamber which houses the optical cavity to remove any gaseous species.

In the presence of a species having absorption coefficient $\alpha$ which is weakly absorbing ($\alpha L<<1$), the ring-down time is given for a two-mirror cavity of round-trip length L by [28]

$$\tau \left(\lambda \right)={\displaystyle \frac{L/c}{1-R^2\left(\lambda \right)+X+\alpha \left(\lambda \right)L}}$$

(3)

In the limit of small absorptions ($L/c\tau <<1$), the absorption coefficient can be extracted in a calibration-free manner by the use of

$$\alpha \left(\lambda \right)={\displaystyle \frac{1}{c}}\left({\displaystyle \frac{1}{\tau \left(\lambda \right)}}-{\displaystyle \frac{1}{{\tau }_0\left(\lambda \right)}}\right)$$

(4)

By scanning the laser wavelength over the interval of interest, we obtain a measurement of the absorption spectrum $\alpha \left(\lambda \right)$. Since the quantities (2) and (3) cannot be measured simultaneously for the optical cavity, we must conduct two separate scans of the baseline and filled chamber. To evaluate ${\tau }_0\left(\lambda \right)$ at some wavelength, we evaluate the straight line fit to the baseline measurement. Our experimental results show that this is a valid approximation for the behavior of the baseline. The behavior of the baseline is fairly stable as long as the cavity mirrors are not contaminated.

2.2. Retrieving Concentrations from Absorption Measurements: Analysis of a Gas Mixture

For a gas mixture having m different absorbing species with mole fractions $y_j$ (which we will refer to as concentrations) and absorption cross-sections ${\sigma }_j\left(\lambda \right)$, we can write the absorption coefficient as

$$\alpha \left(\lambda \right)={\displaystyle \frac{P}{k_{b}T}}\left(y_1{\sigma }_1\left(\lambda \right)+y_2{\sigma }_2\left(\lambda \right)+\dots +y_m{\sigma }_m\left(\lambda \right)\right)$$

(5)

In general, the cross-sections ${\sigma }_j$ depend on the presence of other gas species [29] and the temperature T and pressure P of the mixture [30].

$${\sigma }_j\left(\lambda \right)={\sigma }_j(\lambda ;T,P,y_1,\dots ,y_m)$$

(6)

We give the analytic solutions for the concentrations $y_j$ in Equations (8) and (A29) which are computed under the assumption that the cross-sections do not depend explicitly on $y_i$. For the gas species measured in this work, which are cataloged in the HITRAN database [30], only very small corrections [29] to the lineshape functions are known to be arising from the interaction with other gas species. The NIR absorption of acetone is not well cataloged in the HITRAN database, but it was also found in Section 4.1 that the acetone cross-section displayed neither pressure dependence nor distortion in the presence of other species. Therefore, we maintain the assumption that the cross-sections do not depend on $y_j$.

In the case where the cross-section has explicit dependence on $y_j$, iteration can be used. Using an initial guess for $y_j$ to compute the cross-section, the analytic solution of either (8) or (A29) is used to compute a new set of concentrations which can then be used to update the cross-section ${\sigma }_j(\lambda ;T,P,y_1,\dots ,y_m)$. By iterating this process only a few times, a solution is obtained which includes the effect of other gas species on the cross-section. For the rest of this work, we will only consider cross-sections which have dependence on the total temperature and pressure ${\sigma }_j\left(\lambda \right)={\sigma }_j(\lambda ;T,P)$.

For a measured spectrum consisting of N values of the absorption coefficient ${\alpha }_i$ at wavelength ${\lambda }_i$, we have for m different absorbing species having each cross-sections ${\sigma }_j$

$$\left(\begin{array}{c}{\alpha }_1\\ {\alpha }_2\\ {\alpha }_3\\ ⋮\\ {\alpha }_N\end{array}\right)={\displaystyle \frac{P}{k_{B}T}}\left(\begin{array}{cccc}{\sigma }_1\left({\lambda }_1\right)& {\sigma }_2\left({\lambda }_1\right)& \cdots & {\sigma }_m\left({\lambda }_1\right)\\ {\sigma }_1\left({\lambda }_2\right)& {\sigma }_2\left({\lambda }_2\right)& \cdots & {\sigma }_m\left({\lambda }_2\right)\\ {\sigma }_1\left({\lambda }_3\right)& {\sigma }_2\left({\lambda }_3\right)& \cdots & {\sigma }_m\left({\lambda }_3\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\sigma }_1\left({\lambda }_N\right)& {\sigma }_2\left({\lambda }_N\right)& \cdots & {\sigma }_m\left({\lambda }_N\right)\end{array}\right)\left(\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_m\end{array}\right)$$

(7)

Here, ${\alpha }_i=(1/{\tau }_i-1/{\tau }_0)/c$ is the absorption coefficient determined from the measured ring down time ${\tau }_i$ conducted at wavelength ${\lambda }_i$. We denote the N by m matrix in Equation (7) as A, the length N column vector on the left as $\overrightarrow{\alpha }$, and the length m vector on the right as $\overrightarrow{y}$. For $N>m$, the over-determined system (7) can then be written as $\overrightarrow{\alpha }=A\overrightarrow{y}$, and the least squares solution is given by

$$\overrightarrow{y}={\left(A^{T}A\right)}^{-1}{A}^T\overrightarrow{\alpha }$$

(8)

The above expression gives values of the concentrations $y_i$ which provide the best fit to the data after they have been converted to absorption. This simple analytic solution can be written since ${\alpha }_{\mathrm{model}}$ is linear in the gas concentrations $y_j$. The use of the weighting function can give reduced squared deviation from the data to the fitting curve in spaces other than absorption, as described in Appendix B. We also show in Appendix C that the weighting function can improve the estimates for the concentrations recovered from the spectrum fitting. It is possible to account for baseline instability by the inclusion of additional terms in the fitting model such as the ${\alpha }_B\left(\lambda \right)$ models described and analyzed in Appendix C.

3. Experimental Setup and Methods

3.1. Experimental Apparatus

The schematic of the experimental setup is shown in Figure 2. The laser source is a commercially available distributed feedback diode laser (DFB) (Eblana Photonics EP1673-7-DM-B01-FM). It is available as a 14-pin butterfly package. The output of this fiber pigtailed laser is fed through a fiber-coupled isolator (ISO) and then collimated using a fiber collimator (FC) package. In order to use this CW laser source, it was necessary to employ a fast optical modulator [27]. This is accomplished by using the first diffraction order output of an acousto-optic modulator (AOM) operated in the Bragg regime, as this allows for a high extinction ratio and fast switch-off times of the first-order deflected beam. The optical high finesse optical cavity is of hemispherical type, consisting of a planar and concave high reflectivity mirrors separated by a distance of 1.72 m. The input coupler is a flat mirror, and the output coupler has a concave shape with 2 m radius of curvature. The output coupler is mounted to a ring-shaped piezoelectric transducer (PZT), which slightly alters the length of the cavity. The output beam passes through the middle of the PZT, out of the chamber using a wedged AR-coated window, and is focused onto an amplified photodiode (Thorlabs PDA10CS) with an off-axis parabolic mirror.

The first-order deflected beam is mode-matched to the optical cavity with a mode matching telescope consisting of two planoconvex lenses. The alignment and mode-matching of the cavity were verified by eliminating the presence of higher-order transverse modes. These could be seen by sweeping the length of the cavity using the PZT and monitoring the cavity transmission using the photodiode.

The cavity resonance excitation is performed by applying a high voltage triangular wave (shown as HV ramp) to the PZT, which modulates the optical frequency of the cavity resonances. The amplitude of the triangle wave is chosen such that a single cavity resonance always sweeps over the frequency of the laser. Also shown is the laser diode controller (LDC) and GPIB connection to the computer. The acquisition system consists of a single board computer (Red Pitaya STEMlab 125-14, Solkan, Slovenia) which has onboard 14-bit analog-to-digital converter (ADC) and a field-programmable gate array (FPGA), which is used to provide a low-latency switching signal to the AOM driver, allowing the laser to shutoff when resonant excitation of the cavity is detected on the photodiode (PD) by a rapid signal rise to a pre-defined threshold. This single board computer also performs the fitting of the exponentially decaying signal of the transmitted through the cavity light with the model Equation (1). The amplitude term A captures variations in the coupling between the laser and cavity.

The analog ring-down signal is processed using a low-noise pre-amplifier (Stanford Research Systems SR560, Sunnyvale, CA, USA) which features a 1 MHz single-pole low-pass filter. This signal is acquired at 125 MHz sampling rate by the acquisition system, which is down-sampled to 125/64 MHz. Additional digital filtering is performed by the FPGA prior to down-sampling in order to reduce noise aliased to low frequency [31]. The down-sampled signal is stored in the RAM of the single-board computer and passed to the operating system, which is then fit using an optimized fitting routine written in C. The initial portion of the recording is trimmed to remove the cavity buildup and corresponding transients [31].

The signal is recorded using 4000 data points and the fit is conducted with 20 iterations of the Levenberg–Marquardt algorithm. This fitting is performed on the same single board computer which performs the measurement to avoid additional latency from transferring the analog trace. An initial guess of the parameters is calculated using the Fourier transform method of [32] using the FPGA of the acquisition system to perform the Fourier integrals in real time.

From the fit of the model Equation (1) to the recorded signal, we obtain the measured cavity ring-down time $\tau$. The amplitude term A captures variations in the coupling between the laser and cavity. The B term allows for the small offset that is output by the photodetector electronics.

For precise control of the gas mixture measured by the cavity ring-down technique, the optical cavity is constructed inside of a vacuum chamber. The pressure of the chamber is monitored by a pressure sensor (PS). Two separate pressure transducers are used; one is a diaphragm-type pressure gauge (Vacuum Research 902321, Pittsburgh, PA, USA), providing a measure of the total pressure inside the chamber which is independent of gas composition. The other is a thermocouple gauge (Varian 531) used to verify that a pressure of <${10}^{-3}$ Torr is obtained when evacuating the chamber using a fore-pump (Leybold, Scrollvac SC15D, Cologne, Germany).

The ring-down time of the cavity, which is in general wavelength-dependent, gives a sensitive measure of the intracavity losses including the contribution from absorbing gas species present between the cavity mirrors. The zero-order beam emerging from the AOM is coupled into a single mode fiber and sent to the wavemeter, which is described in the next section.

3.2. Scan of the Wavelength

The laser wavelength is tuned by changing the temperature and injection current of the laser diode, both of which are controlled by GPIB communication between the computer and the laser diode controller (LDC in Figure 2). Fine control of the laser wavelength is established by adjusting the laser diode injection current. The laser current is stabilized within 25 ppm with a resolution of 0.01 mA using a Newport LCM-39420 module in an LDC-3900 laser diode controller. For course wavelength adjustment, the temperature of the diode is actively stabilized by the LCM-39420 module. The set point of this stabilization can be adjusted to a precision of 0.1 °C.

The scan of the laser wavelength is conducted by first scanning the laser current from 110.00 to 109.00 mA. After finishing the scan of laser current, the temperature is changed by 0.2 °C and the current scan is reset. This scan order is preferred to minimize the frequency of temperature changes of the laser diode since there is a longer time required for the laser to settle into a stable wavelength with temperature variations compared to changes in the current.

At each point in the scan, the AOM trigger is disabled and the wavelength is measured from the zero-order output beam of the AOM. This is achieved by fiber coupling this beam and then sending the output of the fiber to a home-made precision wavemeter (Appendix A). The AOM trigger is disabled so that the intensity changes caused by the AOM on the zero-order beam do not affect the wavelength measurement. The AOM creates a small shift equal to our driving frequency of 80 MHz between the zero- and first-order AOM beams. This discrepancy is much smaller than any of the spectrum features in this investigation and we will not be concerned with it.

3.3. Acetone NIR Absorption Characterization

Calibrated gas mixtures of acetone were obtained from GasCo at 5 ppm and 30 ppm. The calibration of the mixtures are NIST-traceable, and the calibration certificates of the mixtures claim 10.02 ppm ± 2% and 27.33 ppm ± 5%, respectively. We investigate the shape of the NIR absorption spectrum of acetone by measuring the sample of 27.33 ppm calibration gas under various pressures and conditions of dilution, including conditions of human breath samples.

The chamber was first evacuated and the baseline ring-down time measured. A linear fit of the baseline ring-down time against wavelength is later used to evaluate ${\tau }_0\left(\lambda \right)$ when the absorption is calculated from (4). The evacuated chamber was filled with an initial pressure measured on the diaphragm gauge, and the wavelength dependent ring-down time of the cavity was determined by taking the average of 100 separate ring-down measurements conducted at 1000 different wavelengths. The duration of the scans was in the range of 1–2 h, with spectra containing larger absorptions taking longer due to the decreased light being transmitted from the cavity.

The calibration gas was first measured with minimal dilution to establish that the acetone absorption cross-section does not show any detectable pressure dependence in the range of 30–141 Torr. Using the same sample of calibration gas, we also investigated the acetone absorption in the range of 141–500 Torr by diluting with ambient air. No distortions of the acetone cross-section were detectable at these increased pressures.

During this investigation, it was found that the amount of time spent vacuuming the chamber had an effect on the long-term stability of the acetone absorption measurement. This effect is investigated in more detail in Section 4.2. In light of this drift, an optimized procedure was used to minimize the effect and to report the parameters for the acetone cross-section in Section 4.3.

In Section 4.4, we demonstrate the utility of the acetone cross-section model by measuring the acetone concentration in samples of human breath. We further demonstrate the accuracy of this process by diluting these samples with a known amount of calibration gas and showing that the subsequently measured spectrum is consistent with the amount of acetone gas added and the initial measurement.

The ring-down time measurement noise is characterized in Section 4.5, and this information is used in Appendix C to simulate the effect of this noise on the spectrum fitting process and quantify the error which is propagated to the final gas concentration determination.

4. Results

4.1. Pressure Dependence of the NIR Acetone Absorption

Cavity ring-down spectroscopy provides an absolute measurement of the absorption coefficient $\alpha$ [28], which is defined as the absorption cross-section $\sigma$ times the number density of molecules n,

$$\alpha \left(\lambda \right)=n\sigma \left(\lambda \right)$$

(9)

The cross-section can, in general, be dependent on the pressure, temperature, and presence of other gas species. The linearity of the acetone NIR absorption with number density has been established for pure acetone samples in [33] from $2\times {10}^{-3}$ to $9\times {10}^{-2}$ Torr, indicating that the NIR cross-section for acetone must, to some degree, be independent of pressure. In this paper, we extend the range of conditions under which the NIR absorption of acetone is known to be linear.

We monitored the broadband shape of the NIR acetone absorption feature in the range of 1671.5–1675 nm by constructing an empirical model of the peak shape. We give a normalized peak shape ${\tilde{\sigma }}_{\mathrm{ac}}$ in the form

$${\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)={\displaystyle \frac{{\Gamma }^2(\lambda -p_2)}{{(\lambda -p_2)}^2+{\Gamma }^2(\lambda -p_2)}}$$

(10)

where the width parameter $\Gamma \left(\lambda \right)$ is given by a linear dependence on wavelength

$$\Gamma \left(\lambda \right)=p_0+p_1\lambda$$

(11)

The $p_1$ term characterizes the asymmetry of the peak, and for $p_1=0$ the above function reduces to a symmetric Lorentzian peak. We scale ${\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)$ to an amplitude $c_0$ and offset by a constant $c_1$ to give the model acetone cross-section

$${\sigma }_{\mathrm{ac}}\left(\lambda \right)=c_0{\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)+c_1$$

(12)

The peak of this model cross-section occurs at $\lambda =p_2$ with a magnitude $c_0+c_1$.

The proposed shape of the NIR acetone absorption cross-section (12) fits well with the experimental data in the wavelength range of this investigation, but results in difficulties when this function must be extrapolated to fit extended wavelength ranges in future investigations.The peak shape function ${\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)$ approaches a constant at large values of $\left|\lambda -p_2\right|$ and has a zero at $\Gamma (\lambda -p_2)=0$. To amend this, the function $\Gamma \left(\lambda \right)$ can be clamped such that a linear behavior is displayed in this wavelength range but then rolls off to a constant value outside of it before encountering the zero of $\Gamma$. These considerations could be used to construct a modified model which extrapolates well.

To model the experimentally observed spectra, we include the effect of water vapor contamination and an imperfect absorption baseline. The calibration gas mixtures used for this measurement were not perfectly pure and contained some additional absorption primarily due to water vapor. This contamination was small in comparison to the desorption of water from the aluminum chamber walls. Due to limitations of the chamber construction, the setup could not be heated to remove this outgassing. It was therefore necessary to separate the contributions from acetone and water vapor absorptions. The need for baseline corrections results from the drift which is discussed in Section 4.2.

In order to fit the experimentally measured spectra of the calibration gas at different pressure values, the following model is used:

$$\alpha \left(\lambda \right)=a_0{\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)+a_1+a_2(\lambda -{\lambda }_0)+a_3{\sigma }_{{\mathrm{H}}_2\mathrm{O}}\left(\lambda \right)$$

(13)

The $a_{0,1}$ are related to the $c_{0,1}$ cross-section parameters by the ideal gas number density $P/k_{B}T$ and the concentration of acetone $y_{\mathrm{ace}}$, and similarly, the $a_3$ is related to the concentration of water vapor $y_3$ by

$$a_{0,1}=\left(y_{\mathrm{ace}}{\displaystyle \frac{P}{k_{B}T}}\right)c_{0,1}{a}_3={\displaystyle \frac{P}{k_{B}T}}{y}_3$$

(14)

The $a_2$ term is included to reduce the effect of drift which tends to tilt the observed spectra when the acetone absorption is slightly drifting. This was necessary for some of the calibration gas measurements since a long-term drift in the acetone absorption was observed (see Section 4.2). The use of this term corresponds to the $B=1$ fitting model discussed in Appendix C, and offers improved immunity to spectrum tilt at the expense of degraded $a_{0,1}$ determination. It was also found that the use of the $a_2$ baseline slope parameter introduced a slight amount of crosstalk with the $p_1$ parameter and hence was not used in the final determination of the cross-section parameters. The ${\lambda }_0$ parameter does not affect the final expanded form of the baseline polynomial $a_1+a_2(\lambda -{\lambda }_0)$, but choosing ${\lambda }_0=1673.5$ nm in the center of the scan reduced correlation between $a_1$ and $a_2$.

The absorption cross-section of water ${\sigma }_{{\mathrm{H}}_2\mathrm{O}}\left(\lambda \right)$ was calculated using the HAPI Python module [34], which can simulate the absorption cross-sections of many atmospheric molecules using parameters in the HITRAN database [30]. For these calculations, measurements of the total chamber pressure and temperature are needed. The broadening parameters were initially calculated under the assumption of a dry air mixture, with this assumption iteratively relaxed (Section 2.2).

The fit of the model was performed using Levenberg–Marquardt algorithm to vary the nonlinear model parameters $p_0$, $p_1$, $p_2$, and the wavelength correction factor $p_{\lambda }$ from Equation (A2). For each trial value of these parameters, the linear coefficients $a_{0-3}$ of the model absorption were determined using the unweighted linear decomposition procedure (8).

The fit of Equation (13) to the observed absorption spectrum yields the concentration of water vapor from the values of $a_3$ using the relation of Equation (14) and the relative contributions of the offset term $a_1$ and the peak amplitude $a_0$. The baseline slope term $a_2$ includes the contribution from a tilt of the spectrum which was attributed to the long-term drift (see Section 4.2) occurring during the course of the acquisition.

The $p_{0-2}$ define the shape of the acetone absorption, and changes in these values are indicative of a changing cross-section. In this investigation, we confirmed that the values of $p_{0-2}$ stayed relatively constant and the values of $a_0$ and $a_1$ were linear with acetone number density over a range of 30–500 Torr, indicating that the cross-section showed no dependence on pressure. Additionally, we confirmed this behavior in the presence of laboratory air in the range of 141–500 Torr.

Absorption Linearity in the Range of 30–500 Torr

For pressures in the range of 30–141 Torr, we investigated the absorption pressure dependence of a sample of dry air mixture 27.33 ppm calibration gas. The sample was used as provided from the supplier, preventing dilution by flushing the gas lines several times and evacuating the measurement chamber before filling. The linear behavior of the absorption magnitude is seen from $a_{0,1}$ when plotted against pressure, indicating that there is very little effect of pressure and self-broadening on the absorption cross-section.

Shown in Figure 3 are two sequences of acetone calibration gas measurements along with their fits to the model absorption of Equation (13). The pressure was changed by partially evacuating the chamber using the scroll pump. For the measurements shown, the chamber was filled at 120 Torr and 50 Torr. A slight upward tilt of the measured absorption spectrum can be seen on the larger wavelength side of these two measurements, which was later attributed to a long-term drift of the acetone absorption (Section 4.2). The narrow absorptions of water can be seen superposed on the acetone absorption. The values of the coefficients obtained from the fit to the absorption curve are shown as by the points of Figure 4 plotted against pressure.

Very little distortion of the peak shape was revealed by the near constant values of $p_0$ and $p_2$ resulting from the fitting, as seen in Figure 4. The relatively large variance of $p_1$ is attributed to a coupling with the $a_2$ baseline slope term. Due to the presence of drift, we could not remove the interfering term without detrimentally affecting the fit. Because of this, we did not use the data presented in Figure 4 for the final resolution of the coefficients $p_{0-2}$. This is discussed further in Section 4.3. Additionally, this drift resulted in poor control of the calibrated mixture, so we did not use the slope of the fit line (as was done in [33]) to resolve the final value for the cross-section magnitude.

In the range of pressures 141–500 Torr, we further investigated the linearity of the absorption by diluting the same sample of acetone calibration gas using ambient laboratory air in several increments. In this case, diluting the sample does not appreciably change the number density n, but only changes the pressure of the measurement environment and introduces contaminants. We therefore expect the absorption magnitudes $a_0$ and $a_1$ of the resulting fit to remain approximately constant as the dilution is performed for a linear absorption.

Indeed, $a_0$ and $a_1$ did not show any effect of the dilution and only showed a long-term dependence which is shown in the left panel of Figure 5 and resembles the long-term drift behavior further investigated in Section 4.2. A similar drift behavior can be seen in Figure 6a. With the exception of a few outliers, the values for $p_{0-2}$ shown in the right panel of Figure 5 showed stable behavior, indicating that the shape of the acetone peak was unchanging.

The only indication that there is some distortion of the peak shape can be seen in the bottom left of Figure 5, where there is a slight fluctuation of $a_0/a_1$. This ratio characterizes the height of the acetone absorption peak to the local offset and is independent of acetone concentration in our model. We attribute this small variation over time to contaminants which result in an apparent broadband and featureless absorption that increases the value of $a_0$ in the fit. This could result from increased scattering as an increasing amount of atmospheric gas is let into the chamber.

4.2. Drift of the Acetone Absorption

It was determined that the main limiting factor in our spectrum measurements was the apparent drift of the acetone absorption over a span of many hours. Since scans of the spectrum took a period of 1–2, h this drift was accounted for by using the $a_2$ slope term in the fit Equation (13). Coupling of this term with the $p_1$ resulted in a degraded ability to resolve the value of this coefficient.

To quantify the nature of the spectrum drift, we choose five wavelengths which reside on the acetone feature between the sharp water features, and a sixth wavelength was chosen to monitor an isolated water peak. The results of two different long-term drift measurements is shown in Figure 6. It was found that the time dependence of these absorption values was well modeled by a double exponential function plus a linear term

$$\alpha \left(t\right)=mt+b+A_1{e}^{-t/T_1}+A_2{e}^{-t/T_2}$$

(15)

In Figure 6, two different drift measurements are shown in panes (a) and (b) and compared in pane (c). The preparation of the sample was nearly identical in both cases except for the amount of time spent by the chamber under low vacuum prior to acetone calibration gas introduction. In (a), the chamber had been continuously pumped for six hours after the thermocouple gauge had reached its minimum reading of 0.001 Torr, and in (b) the chamber was only pumped down to 0.1 Torr. A much greater defect between the initial absorption value and that obtained roughly 20 h later is observed in the conditions of (a). It should be noted that the values of peak acetone absorption obtained in other studies agree most closely with this initial absorption value. In light of this finding, we decided to conduct the measurements for the coefficients in Table 1 under the conditions of (b), where the defect is smallest.

For each wavelength in the drift measurement, we fit a curve to the absorption coefficient over time using Equation (15). The water absorption was fit using a simple linear equation. The fit equations are shown as black curves in panes (a) and (b) of Figure 6. The evaluation of these curves was used to check for distortions of the acetone absorption profile.

We additionally used the model drift Equation (15) to evaluate five different absorption values from which a fit of $p_1$, $a_0$, and $a_1$ was obtained. This was done to eliminate the considerable time required for the laser to fully stabilize to a new wavelength, as evaluating the temporal fit equation allows us to know the absorption at several wavelengths simultaneously. To recover the shape of the acetone absorption, $p_0$ and $p_2$ were held fixed and $a_2$ was held at zero. This three-parameter model was used to fit to the five different acetone probe points to monitor for changes of the peak shape during the time evolution of the spectrum. It was found that the value of $p_1$ extracted from these fits then became stable and all of the spectrum evolution was contained in $a_0$ and $a_1$, which seemed to vary together keeping the ratio $a_0/a_1$ nearly constant. This indicates that the acetone absorption evolved as a scale factor, and the drift is most likely due to a changing number density of acetone, and not a distortion of the NIR peak.

Under the assumption that the change in the acetone absorption profile is due to a changing acetone concentration, Figure 6c was constructed from the measurements shown in panes (a) and (b). For a set of chosen times in the measurement interval, the fit equations to the changing absorption were evaluated to yield six absorption measurements at six known wavelengths. These were used to set up the system of Equation (7) to be established from which the concentrations of acetone and water vapor could be solved using (8). By repeating this procedure for many selected times in the measurement interval, we construct the curves shown in Figure 6c.

From Figure 6c, we see that, in the first 20 h, the drift is much more severe for the case wherein the chamber walls were dried with vacuum desiccation prior to sample introduction. Due to the long time scale, we attribute the drift to an interaction of acetone with the aluminum walls of the chamber, but this requires further investigation. The chamber in our case was rather large, having the shape of a rectangular prism of approximate internal dimensions 9 × 13 × 76 cm.

4.3. Determination of Model Coefficients

Determination of the cross-section coefficients was performed by fitting the model

$$\alpha \left(\lambda \right)=a_0{\tilde{\sigma }}_{\mathrm{ac}}\left(\lambda \right)+a_1+a_2{\sigma }_{{\mathrm{H}}_2\mathrm{O}}\left(\lambda \right)$$

(16)

to the absorption measurements conducted at the relatively stable laboratory temperature of 21.8 °C, as measured by an AccuRite Iris weather station located in the laboratory. It was found that the slope term contained in (13) was not necessary for achieving a good fit to the data when the drift was minimized. The above model gave improved stability of the $p_1$ coefficient contained in the ${\tilde{\sigma }}_{\mathrm{ac}}$ term and is equivalent to the $B=0$ model simulated in Appendix C. The resulting fit parameters allow the offset $a_1$ and peak amplitude $a_0$ to be determined independently, which correspond to $c_{0,1}$ through (14).

In determining the values in Table 1, care was taken to evaluate only spectra which had been taken in the minimum of the temporal variation observed in Figure 6. This point is marked by the black circle marked A in pane (b). At this point, the tilt of the spectrum induced by the drift of the acetone absorption is minimized and we can uniquely determine the spectrum shape. Additionally, we analyzed data collected in the linear portions of the drift with the wavelength scan conducted in a back-and-forth motion, allowing the relatively slow drift to make roughly equal and opposite contributions across the scan. Such a point is marked B in Figure 6b.

The shape of the spectrum was not found to show any dependence on pressure or the presence of other species under the conditions that were investigated. Since the coefficients of this model resulting from the fitting procedure gave stable values, we simply took an average of the results to report the numbers in Table 1.

Table 1. Values of the coefficients resulting from the fit of the acetone cross-section at $T=21.8$ °C. The peak of the model acetone cross-section is given by $a_0+a_1\sim 1.22\pm 0.01{\mathrm{cm}}^2$.

Coefficient	Value	Uncertainty	Unit
$c_0$	$3.92\times {10}^{-22}$	$0.05\times {10}^{-22}$	${\mathrm{cm}}^2$
$c_1$	$8.3\times {10}^{-22}$	$0.1\times {10}^{-22}$	${\mathrm{cm}}^2$
$p_0$	$0.704$	0.002	nm
$p_1$	$0.174$	0.002	unitless
$p_2$	$1672.442$	$0.001$	nm

Table 1. Values of the coefficients resulting from the fit of the acetone cross-section at $T=21.8$ °C. The peak of the model acetone cross-section is given by $a_0+a_1\sim 1.22\pm 0.01{\mathrm{cm}}^2$.

Coefficient	Value	Uncertainty	Unit
$c_0$	$3.92\times {10}^{-22}$	$0.05\times {10}^{-22}$	${\mathrm{cm}}^2$
$c_1$	$8.3\times {10}^{-22}$	$0.1\times {10}^{-22}$	${\mathrm{cm}}^2$
$p_0$	$0.704$	0.002	nm
$p_1$	$0.174$	0.002	unitless
$p_2$	$1672.442$	$0.001$	nm

4.4. Measurements with Diluted Breath Samples and Breath Dilution Procedure

We demonstrate that the concentration of acetone can be recovered from the NIR spectrum of a breath sample using the acetone model cross-section (12) with coefficient values from Table 1. We first measure the absorption spectrum of a breath sample, dilute this sample with a known amount of acetone calibration gas, and then measure the absorption spectrum again. If the concentration of acetone is accurately recovered from the two breath samples, then the concentrations obtained from the two spectra should be consistent with the known dilution.

A pressure around 70 Torr was found to be the optimum to measure breath samples in our apparatus in this wavelength range. A higher sample pressure would cause the water peaks to saturate the measurement such that $\tau \to 0$, resulting in the cavity producing nearly zero transmission signal. Operating just before this point allows for the highest signal-to-noise ratio of the observed signal while still fully capturing the large water absorptions.

We first filled the chamber (after the baseline ${\tau }_0\left(\lambda \right)$ is characterized) with a breath sample to 70 Torr and measured the spectrum. The cavity was then partially evacuated to 65 Torr and then diluted with calibration gas to the original pressure of 70 Torr. To transfer the breath sample to the measurement chamber, the volunteer is asked to fill a latex balloon, which is then temporarily sealed with a binder clip. This balloon is placed in a refrigerator for approximately 15 min to quickly reduce the temperature of the sample. This is done to prevent condensation on the inside of the chamber and gas lines, since fresh breath gas is warmer than the 21.8 °C laboratory environment. The final transfer to the evacuated measurement chamber was performed by attaching the balloon to a 1″ KF connector and opening a needle valve in a controlled manner to avoid disturbing the optics. The chamber filling rate can be monitored with high sensitivity by monitoring the positions of the cavity resonances in the scan of the PZT (shown in Figure 2), since the refractive index of the gas changes monotonically with pressure. The final filling pressure is determined using the diaphragm gauge.

The measurements of both the diluted and un-diluted spectra were conducted at 1000 different wavelengths with 100 averages each. This number of averages is smaller than the optimum (see Section 4.5) but this smaller number of averages enabled a faster scan, limiting the effect of the drift mentioned in Section 4.2. This smaller number of averages also places our measurement in the regime of Gaussian white noise (the linear portion of the Allan plot in Figure 8b), which enables a more direct comparison with the simulation results of Appendix C.

The concentrations of the various gas species recovered from the fitting procedure are shown in

Table 2. Results for breath samples diluted by calibration gas. The breath was intially measured at 70 Torr resulting in the pre-dilution concentrations shown in the second column. The breath sample was then reduced in pressure to 65 Torr using a vacuum pump, and then diluted with 27.33 ppm calibration gas until the total pressure again reached 70 Torr. After this final dilution, the spectrum was measured again, resulting in the post-dilution concentrations in the third column. The fourth column shows the concentrations which are expected of the post-dilution if the initial concentrations are accurate. The percentages of fractional change from these expected numbers are shown in the last column. The final diluted breath spectrum is shown in Figure 7.

Gas	Pre-Dil.	Post-Dil.	Expected	Deviation
Species	(ppm)	(ppm)		(%)
CH4	7.34	6.74	6.32	6.57%
${(}^{13}\mathrm{C}){\mathrm{H}}_4$	0.092	0.088	0.080	11.6%
H2O	21,730	20,940	18,740	11.7%
CO2	31,400	28,600	27,000	5.8%
${(}^{13}\mathrm{C}){\mathrm{O}}_2$	344	300	296	2.7%
$\mathrm{CO}{(}^{18}\mathrm{O})$	128.0	119.5	110.3	8.27%
$\mathrm{CO}{(}^{17}\mathrm{O})$	24	24	21	17%
${\mathrm{H}}_2{(}^{18}\mathrm{O})$	32	36	28	29%
Acetone	0.34	2.31	2.29	0.74%

for the sample shown in Figure 7 before and after the dilution. The calibration gas used for the dilution was measured to contain only a trace amount <0.01 ppm of methane and <100 ppm water. The pre-dilution concentrations recovered from the fit are used to compute the expected concentrations resulting from a dilution with pure acetone calibration gas and these are shown in the fourth column of Table 2 and compared to the values resulting from the spectrum fit of the diluted gas.

4.5. Detection Limit

We can evaluate the detection limit of our system by the fluctuations in the measured ring-down time [35] near the baseline. For a weak absorption, the measured ring-down time $\tau$ is nearly that of the empty cavity baseline ${\tau }_0$, and must be distinguished from the intrinsic fluctuations of the measurements. For a single ring-down measurement, the detection limit is defined as the absorption which produces a change in ring-down time equal to the standard deviation of the baseline ring-down time $\Delta \tau$. This gives the detection limit of a single ring-down measurement as

$$\Delta \alpha ={\displaystyle \frac{1}{c{\tau }_0^2}}\Delta \tau$$

(17)

When the measurement is conducted, many averages of the ring-down time are taken at each wavelength to effectively reduce the deviation $\Delta \tau$ of the measurements. However, this cannot be continued indefinitely due to drift. The modified Allan variance [35] can be used to determine the optimum averaging time and the minimum possible value of $\Delta \tau$ that can be obtained through averaging. This allows the minimum detectable absorption to be determined. Equivalently, we can calculate the Allan variance directly from measurements of the absorption coefficient to give directly the minimum variance of absorption coefficient $\Delta \alpha$. This is shown in Figure 8, where the minimum of the modified Allan deviation of the absorption coefficient gives the (1$\sigma$) minimum detectable absorption. For 15.8 s of averaging time, we obtained a detection limit of $1.1\times {10}^{-10}$ cm⁻¹. This corresponds to an acetone concentration of 44 ppb at 70 Torr when the absolute value of the peak absorption cross-section is considered, and 134 ppb when just the height of the characteristic peak is considered.

5. Discussion

The overtone absorption bands, as the C-H stretch band that we investigate in this paper, appear due to anharmonicity [36]. The overtones with relatively high transition frequencies, observed in the NIR region, involve primarily small mass hydrogen atoms coupled with larger atoms [37].

The profiles of the overtone absorption bands often show similarity with the profiles of the corresponding bands of fundamental wavelength. Indeed, the fundamental band [38] around ${\nu }_f$ = 3320 cm⁻¹ and the overtone absorption band that we are investigating around ${\nu }_0=5979$ cm⁻¹ both show a central peak on an elevated base. The anharmonicity is reflected in the fact that ${\nu }_0<2{\nu }_f$ and leads to the frequency of the overtone being smaller than the corresponding multiple of the fundamental frequency. This was used for the determination of the molecular structure from spectroscopy of overtones [39]. The assignment of the observed band to the first overtone 2${\nu }_a$ of the asymmetric stretching vibration of the methyl group CH₃ of acetone with certainty follows from previous research [33,40,41].

Our determination of the absorption cross-section is based on the formula of Equation (8) and is not relying on the known cross-section of acetone for a different absorption band (in UV) as in ref. [23]. Strictly speaking, Equation (13) determines the total attenuation cross-section, which also includes losses due to scattering. To the scattering losses contribute all gas components that are present in the optical cavity. However, the scattering cross-section at the used wavelengths is many orders of magnitude smaller than the absorption cross-section [42] and can be neglected. We note that the absorption from hot band transitions [43] at our temperature is also negligible.

We performed measurements with a mixture of gases and used the well-known absorption cross-sections of methane, water, and carbon dioxide in the NIR region. The relatively high concentration of acetone and methane compared to other VOCs and high spectral resolution made it possible to accurately determine their concentrations in gas mixtures of exhaled breath. The peak value of the acetone NIR cross-section and position of the peak absorption for the investigated absorption band are shown in

Table 3. Cross-sections and positions of the peak absorption for the stretching 2${\nu }_a$ overtone of CH3 group in acetone.

Source	Cross-Section	Peak Absorption
	$\mathbf{\sigma }$  (cm2)	Wavelength  $\mathbf{\lambda }$  (nm)
This Work	$1.22\times {10}^{-21}$	1672.442
[40]	$1.2\times {10}^{-21}$	1672.15
[33]	$1.2\times {10}^{-21}$	1677.8
[25]	$1.4\times {10}^{-21}$	1672.44
[44]	$1.3\times {10}^{-21}$	1672.48

in comparison to the values obtained in previous research. The obtained value of the peak absorption of $1.22\times {10}^{-21}{\mathrm{cm}}^2$ is in good agreement with refs. [33,40,44], and the position of the peak of 1672.4 nm is also in good agreement with refs. [25,40].

Some variations in the observed absorption and consequently in the retrieved acetone concentration as determined from spectroscopic measurements can be related to the equilibration process after the admission of the gaseous sample into the chamber, partial condensation, and interactions in the mixture of acetone and water molecules. Such mixtures can show azeotropic behavior [45], adsorption effects on condensed fraction and on the walls of the chamber [46,47], and interactions due to hydrogen bonds [48,49,50]. These effects can be especially important before the equilibrium is achieved. Indeed, in our measurements, the initially observed absorption gradually reduced by about 10% on a time scale of hours, as seen in Figure 6.

Our measurements with diluted samples of the calibration gas show that the achieved detection limit for acetone of 44 ppb at 70 Torr total pressure and 15.8 s integration time is low enough for the analysis of human breath samples, since the typical range of concentrations even for healthy individuals is from 0.2 ppm to 1.03 ppm [1]. This is comparable in sensitivity with the MIR sensor employing a 7.32 µm quantum cascade laser and a multi-pass cell [51], which demonstrated the detection limit of 21.4 ppb with a 78 s integration time. Note that our sensor does not require any preliminary acetone enrichment or separation as is the case for acetone detection with a chemoresistive sensor combined with a catalytic filter and separation column [52]. Despite a relatively narrow measured spectral interval (as compared, for instance, to the frequency comb spectroscopy [19]), it also enables the detection of methane as is shown in Figure 7 and Table 2, which often requires a specialized sensor [53]. The high spectral resolution allowed for the identification of spectral lines contributed by different isotopes, as presented in Table 2. Of particular interest are carbon isotopes for breath analysis [54] and for monitoring atmospheric environments [55].

6. Conclusions

We performed cavity ring-down spectroscopy of the C-H stretch overtone absorption band in the NIR spectral range of 1671.5–1675.0 nm. The absorption cross-section was determined from the measurements with calibration gas and its mixtures with breath samples. We have shown that a simple empirical model gives good agreement with the absorption cross-section of dilute acetone gas in the measured wavelength range. The peak absorption value of $1.22\times {10}^{-21}{\mathrm{cm}}^2$ positioned at 1672.442 nm in the investigated spectral range is in agreement with some of the previous measurements. The absorption peak of acetone is distinct enough from those of other molecules present in human breath to allow for quantitative analysis of acetone in human breath samples. These measurements required multispecies analysis of gas mixtures that was developed and demonstrated on human breath samples. Some variations in the observed absorption of acetone due to interactions with water and the walls of the measuring chamber were observed and discussed. The detection limit down to 44 ppb at 70 Torr total pressure in the chamber with 15.8 s integration time was demonstrated as sufficient for sensitive analysis of exhaled breath. Thus, the measurements of the C-H stretch overtone absorption band within the tuning range of a diode laser present a viable and inexpensive approach to measure acetone content in exhaled breath.