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Abstract
We demonstrate a 13-fold speed improvement in broadband cantilever-enhanced photoacoustic spectroscopy (CEPAS) by combining it with phase-controlled Fourier-transform spectroscopy (PC-FTS) instead of traditional Fourier-transform infrared spectroscopy (FTIR). PC-FTS is a modification of FTIR and capable of fundamentally faster interferogram acquisitions. The speed-improvement is beneficial for CEPAS, which is an especially sensitive version of the background-free photoacoustic spectroscopy technique. We used the PC-FTS-CEPAS technique to measure the absorption spectrum of methane in the mid-infrared region (3.3–3.5 µm) with an optical frequency comb as the light source.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Fourier-transform infrared spectroscopy (FTIR) is one of the most widely used spectroscopic techniques. It is based on the Michelson interferometer, where the input light is split into two arms, back reflected, and combined again [1,2]. The optical field in one of the arms is delayed by scanning the arm length with a translating mirror. This leads to either constructive or destructive interference between the combined beams, depending on the optical path difference, and maps each optical frequency into an easily measurable and unique audio or radio frequency. The resulting time domain signal (the interferogram) is Fourier-transformed to yield the spectrum.
Using rotational instead of translational motion in the delay arm can lead to significantly higher interferogram acquisition rates [3,4]. An especially interesting new technique based on rotational motion is phase-controlled Fourier-transform spectroscopy (PC-FTS) first presented by Hashimoto et al. [5,6]. In PC-FTS, the phase of the optical field in the delay arm is controlled using a grating and a rotating mirror: the input light is dispersed onto the rotating mirror surface, which enhances the delay that can be obtained by tilting the mirror. One can choose which optical frequency is down converted to the zero radio frequency, which leads to efficient use of the detection bandwidth and fundamentally faster interferogram acquisition rates than traditional FTIR. The PC-FTS method has been successfully used in the near-infrared and in the mid-infrared regions for gas-phase spectroscopy and to monitor dynamic mixing of liquids with interferogram acquisition rates as high as 24 kHz [5,6].
We take a different approach and utilize the speed benefit of PC-FTS in cantilever-enhanced photoacoustic spectroscopy (CEPAS). Photoacoustic spectroscopy (PAS) is a highly sensitive spectroscopic technique that is widely used in the analysis of condensed matter [7–12], aerosols [13–15], and molecular gases [16–19]. Here, we mainly focus on trace gas analysis, where PAS has been proven to be one of the most sensitive optical techniques available [20–22]. In PAS, intensity- or wavelength-modulated light enters the sample cell. If absorption occurs, it leads to modulated temperature changes and consequently to modulated pressure changes (i.e., to the generation of an acoustic wave). Conventional capacitor microphones can be used to detect the sound, but the highest sensitivities have been obtained with special techniques, such as quartz-enhanced PAS (QEPAS) [23–25] and cantilever-enhanced PAS (CEPAS) [23,26–28]. In QEPAS, the frequency of the light-field modulation is tuned to coincide with the sharp resonance of a piezoelectric quartz tuning fork that acts as the sound detector. Because the resonance is narrow, simultaneous detection of multiple down-converted optical frequencies is limited to a narrow optical bandwidth [29]. In contrast, CEPAS is based on a silicon cantilever whose movements induced by the pressure changes inside the sample cell are monitored interferometrically. Highly sensitive detection can be performed outside the resonance frequency of the cantilever, allowing simultaneous detection of multiple down-converted optical frequencies – CEPAS is thus compatible with FTIR [30–36].
A drawback of highly sensitive broadband photoacoustic detection (such as CEPAS) is that the detection bandwidth is limited to low audio frequencies (well below 1 kHz), requiring slow scanning velocities with the FTIR instrument. For example, mechanical scanning velocities in the order of 1 mm/s are typically needed in the mid-infrared, which leads to acquisition times ranging from a few seconds to several minutes (depending on the chosen resolution) [33–35]. Here, we demonstrate a 13-fold speed improvement in CEPAS by combining it with the PC-FTS method instead of traditional FTIR. However, this leads to reduced signal-to-noise ratio and also to other trade-offs, which we discuss in detail in this article. The light source we use is a mid-infrared optical frequency comb (MIR OFC) generated using femtosecond pulse-trapped optical parametric generation with continuous wave seeding [37]. As a proof of concept of the PC-FTS-CEPAS technique, we measured the P-branch of the ${\nu _3}$ antisymmetric stretching rovibrational absorption band of methane in the mid-infrared (3.3-3.5 µm) region with a 0.25 Hz scan rate and 10 GHz (0.33 cm-1) typical resolution.
In the following, we first explain the experimental setup, after which we focus on the benefits and drawbacks of the PC-FTS method itself in more detail. We then present the results of the broadband methane measurements and compare the performance of the background-free CEPAS detection to a conventional transmission spectroscopy approach. The Supplement 1 Notes contain additional information about the group- and phase-delay corrections required in PC-FTS, the MIR OFC light source, and other important details of our experimental setup.
2. Experimental setup
The PC-FTS-CEPAS setup is depicted in Fig. 1. The PC-FTS part is basically a Michelson interferometer similar to that used in traditional FTIR. However, instead of a translating mirror, the delay arm has a rotating mirror, a focusing optic, and a grating in 4f geometry [5]. The 4f geometry ensures back-reflection despite the deflection caused by the mirror tilt. The grating disperses the input light onto the rotating mirror surface, creating a larger phase-delay difference between adjacent frequencies than in traditional FTIR where all the frequencies would strike the same spot on the mirror. This results in a large group delay and therefore a high resolution with only a small tilt of the mirror. The rotating mirror we used is a simple 1-inch (25.4 mm) square silver mirror (Thorlabs ME1S-P01) mounted on a galvanometric scanner (GSI VM500+). The optics in the reference arm mimic the geometry of the delay arm. This ensures similar beam propagation in the two arms and efficient overlap and interference of the combined beams on the detector side. The most relevant components for PC-FTS (such as the grating groove density and the focal length in the 4f geometry) are discussed in Section 3 and listed in Table 1.
Fig. 1. Experimental setup. M: plane mirror. The mirror materials are given in Supplement 1, Note 4, to estimate the overall losses; HW: half-wave plate; L: uncoated CaF2 lens; BR: back-reflecting plane mirrors. The lengths of the reference and delay arms are both approx. 83 cm (distance from the BS through the 4f geometry to the BR). Note that the light beams are vertically deflected in both arms such that the BRs reside below the incident beams; CM: 150 mm focal length (${l_\textrm{f}}$) curved mirror. The distance from the grating to CM1 is also ${l_\textrm{f}}$; SM: scanning mirror; $\mathrm{\Delta }\theta $: maximum mechanical scan angle; W: CEPAS cell windows (uncoated CaF2); P: pressure sensor; Note that, for simplicity, only a single frequency light beam is drawn to traverse the experimental setup in the schematic. The grating will disperse a broadband light source onto the scanning mirror surface as illustrated in Fig. 1a of Ref. [5].
Table 1. Specifications of the PC-FTS setup used in this work and in Ref. [6] (two different resolutions)
The half-wave plate in Fig. 1 was used to adjust the polarization of the input light to be perpendicular to the grating grooves (i.e., parallel to the plane of the optical table, the plane of paper). This minimized losses at the grating and resulted in the strongest interference signal (Supplement 1, Note 4). For the light sources, we used a continuous wave (CW) difference frequency generation (DFG) source [38] (for the group- and phase-delay corrections discussed in Section 4) and the MIR OFC [37] (for broadband spectroscopy). The MIR OFC is described in more detail in Supplement 1, Note 6, and its spectrum is shown in Fig. 2 (measured with a spectrum analyzer; Bristol 771B). Note that the resolution of the spectrum analyzer or the PC-FTS instrument described here is not enough to resolve the comb lines. Typically, 1–2 mW of the CW DFG power and 24 mW (approx. 0.11 mW/cm-1 spectral power density) of the MIR OFC power entered the PC-FTS system. However, only a small fraction of these input powers actually arrives at the CEPAS cell (2% and 7% from the delay and reference arms, respectively). High losses are inherent to PC-FTS (see Supplement 1, Note 4), which is problematic for photoacoustic spectroscopy, where the signal strength is proportional to the optical power entering the sample cell.
Fig. 2. The MIR OFC spectrum measured with a spectrum analyzer (blue trace). Note that the resolution of the spectrum analyzer is not enough to resolve the comb lines. The magenta line indicates the optical frequency striking the pivot point of the scanning mirror; the red line indicates the edge of the mirror. The green lines indicate the locations of the methane absorption lines P11 and P2 that were typically used for the phase- and group-delay corrections (Section 4). The orange trace is a simulated methane absorption spectrum (without instrumental broadening) according to the HITRAN database [39,40].
The irises before and after the PC-FTS system were used to ensure that the beams from all the light sources travel the same path in the system. Flip mirrors were used to switch between the light sources. The flip mirror shown in Fig. 1 was used to guide the CW light to a wavelength meter (EXFO WA-1500) to perform the group- and phase-delay corrections. The combined beams exiting the interferometer were focused through the 10 cm long CEPAS cell (Gasera Ltd. PA201). The CEPAS cell was equipped with uncoated CaF2 windows and contained either 1% methane in nitrogen for the absorption measurements, or lab air for background measurements. The measurements were performed at 1000 mbar pressure (measured with a pressure sensor at the outlet of the cell; Honeywell HSCMRNN015PAAA5) and 295 K temperature (lab temperature; temperature of the cell was not actively stabilized). Flushing and gas exchange were performed using a vacuum pump (KNF NMP830KNDC). The cell inlet and outlet were closed during the measurements.
In addition to CEPAS detection, interferograms were simultaneously measured with a MIR detector (Vigo UM-I-10.6) after the CEPAS cell. This way we could obtain the MIR OFC power spectrum for normalizing the CEPAS spectra. We also wanted to compare the absorption spectra measured with these two detection schemes. The interferograms were measured by driving the galvanometric scanner with a ramp signal from a waveform generator (Agilent 33220A). The Transistor-Transistor Logic (TTL) signal of the waveform generator was used to trigger a DAQ (GaGe CompuScope 14200) for interferogram digitization.
The interferograms were digitally high-pass (50 Hz) and low-pass (1 kHz) filtered and further processed in MATLAB. This data processing included averaging of interferograms or spectra, the phase- and group-delay corrections, rectangular apodization of the interferograms to ensure symmetric trimming about the centerbursts, zero-padding, Fourier-transforming the interferograms into the corresponding spectra, and redefining the frequency axes according to the PC-FTS theory. Note that all the interferograms were double-sided and that the spectra were calculated as the modulus of the complex Fourier-transform. Further processing of the raw spectra depends on the detection method (CEPAS vs. MIR detector) and is explained in Section 5.
3. PC-FTS
In PC-FTS, the optical frequencies are down-converted into corresponding radio frequencies according to Eq. (1). See supplementary information of the original paper by Hashimoto and Ideguchi [5] for the derivation. To keep the nomenclature simple, throughout the following text, we will use the term “radio frequency” (RF) when referring to the down-converted counterpart of an optical frequency. However, note that for CEPAS detection the down-converted frequencies are below 1 kHz and thus in the audio frequency domain.
where the down-conversion factorHere, ${l_\textrm{f}}$ is the focal length of the focusing optic in the 4f geometry, N is the groove density of the grating, ${\nu _0}$ is the optical frequency that strikes the pivot point of the scanning mirror and $\omega $ is the angular speed of the scanning mirror. We have expressed the angular speed as $\omega = \mathrm{\Delta \theta }/\mathrm{\Delta }T = 2\mathrm{\Delta }\theta {f_{\textrm{scan}}}$, where $\mathrm{\Delta }\theta $ is the maximum mechanical scan angle, $\mathrm{\Delta }T$ is the length of the interferogram in seconds, and ${f_{\textrm{scan}}}$ is the scan frequency of the rotating mirror. Note that with this definition, two interferograms are obtained during $1/{f_{\textrm{scan}}}$ but in different scan directions (hence the factor of two). For simplicity, we usually omitted the other scan direction.
Equation (1) reveals that by adjusting the location of the pivot point of the scanning mirror one can choose which optical frequency is mapped into the zero radio frequency. As pointed out by Hashimoto and Ideguchi [5], this freedom of choice makes PC-FTS resemble dual-comb spectroscopy [41] and enables fundamentally faster acquisitions compared to traditional FTIR, where it is the zero optical frequency that is inevitably mapped to the zero radio frequency. As PC-FTS down-converts the optical bandwidth of interest to efficiently fill the whole available RF bandwidth, a lower resolution in the down-converted spectrum (i.e., a shorter measurement time) suffices to resolve the spectral features as illustrated in Fig. 3. In fact (Supplement 1, Note 1), when the scan frequency and amplitude of the rotating mirror in PC-FTS are matched to the translational speed of the moving mirror in traditional FTIR such that the maximum RFs in the down-converted spectra are the same, the time needed to measure an interferogram with given resolution is reduced by a factor of ${\nu _{\textrm{max}}}/\mathrm{\Delta }{\nu _\textrm{g}}$, where $\mathrm{\Delta }{\nu _\textrm{g}} = {\nu _{\textrm{max}}} - {\nu _0}$ is the optical bandwidth down-converted by the PC-FTS method. Note that Fig. 3 describes the case where the maximum down-converted frequency is limited by the detection bandwidth set by the detector response and not by the maximum available sampling frequency. In this case, the method of undersampling [1,42] cannot be used to speed up the traditional FTIR measurement as the scanning speed cannot be further increased.
Fig. 3. Illustration of the difference between the down-conversion principles of PC-FTS and traditional FTIR. The PC-FTS approach down-converts the optical bandwidth of interest ($\mathrm{\Delta }{\nu _\textrm{g}}$) to efficiently fill the whole detection bandwidth ($0\; \textrm{Hz} - {f_{\textrm{RF},\textrm{max}}}$), which leads to fundamentally faster acquisitions, as lower resolution in the down-converted spectrum (i.e., shorter interferogram) suffices to resolve the optical components. See Table 1 for the explanation of the symbols in the PC-FTS down-conversion factor ${c_\textrm{g}};$ the symbol u in the FTIR down-conversion factor ${c_{\textrm{FTIR}}}$ denotes the mechanical scan velocity of a traditional FTIR instrument. The symbol c denotes the speed of light. Note that a frequency comb light source is not a prerequisite for PC-FTS, but for simplicity the optical spectrum is drawn as discrete lines to illustrate the optical resolution.
As an example of the speed-benefit in PC-FTS, our rotating mirror supports an optical bandwidth $\mathrm{\Delta }{\nu _\textrm{g}}$ = 6.9 THz (230 cm-1) with ${\nu _{\textrm{max}}}$ = 90.7 THz (3025.5 cm-1). The speed improvement in our case is thus 13-fold compared to traditional FTIR. Note, however, that the speed benefit leads to reduced signal-to-noise ratio (SNR), because, as in any FTIR scheme [1,6,43], the SNR is proportional to the square root of measurement time. In addition, the speed benefit is a trade-off with the optical bandwidth; Supplement 1, Note 2 discusses how to predict the optical bandwidth with chosen scanning mirror width and 4f geometry components. Figure 2 illustrates the excellent accommodation of the MIR OFC spectrum onto the wide scanning mirror we used.
We define optical resolution as the full width at half maximum (FWHM) of the instrument lineshape function (ILS) when measuring a double-sided interferogram and using rectangular apodization (in which case the ILS is a sinc function). The optical resolution is then
The lengths of the longest interferograms we measured were $\mathrm{\Delta }T$ = 1.82 s and a typical down-conversion factor was ${c_\textrm{g}}$ = 9.1505 × 10−11, which down-converted the optical bandwidth with the chosen 4f geometry components (Table 1) below approximately 630 Hz. This corresponds to $\delta \nu $ = 7.2 GHz (0.24 cm-1), which is the highest (double-sided) optical resolution obtainable with our setup. For comparison, in traditional FTIR such resolution would require a 2.5 cm mechanical scan length. Because the FTIR down-conversion factor is $2u/c$, where u is the mechanical scan velocity and c the speed of light [1], $u$ = 1.04 mm/s would be required to down-convert the maximum optical frequency of interest (${\nu _{\textrm{max}}}$ = 90.7 THz or 3025.5 cm-1) to the 630 Hz value. This corresponds to a recording time of 2.5 cm/u = 24 s, which agrees with the speed-benefit discussion above.
The resolution of our PC-FTS setup is limited by the width of the grating. As the angle of the rotating mirror is scanned, resolution improves until the beam misses the optical components in the 4f geometry (Supplement 1, Note 3). We also chose to use a slightly poorer resolution (lower scan amplitude) than the highest obtainable value for the broadband measurements and used rectangular apodization to ensure symmetrical trimming about the centerbursts of the interferograms. The expected resolution was then reduced to 8 GHz. However, the experimentally observed resolution was typically somewhat poorer than this due to self-apodizing effects. We characterized the resolution by measuring the spectrum of our monochromatic but wavelength-tunable CW DFG light source at different optical frequencies. The FWHMs of these spectra ranged from 8.7–9.4 GHz (9.0 GHz mean) with the MIR detector and 9.5–11.7 GHz (10.4 GHz mean) with CEPAS detection. In particular, the CEPAS interferograms suffered from self-apodizing effects due to irregular mirror scanning: the response of the CEPAS detector is not constant with respect to the down-converted frequency, for which reason nonlinear scan velocity couples to the signal strength. The experimental characterization of the instrument lineshape function is discussed in length in Supplement 1, Note 8.
Table 1 lists the specifications of the 4f geometry components we used and the PC-FTS performance we obtained. For comparison, we have also included in Table 1 the specifications of the PC-FTS setup Hashimoto et al. used for MIR gas phase spectroscopy with two different spectral resolutions [6]. The main difference between their setup and ours is the different interferogram acquisition rates: they demonstrated high-speed measurements at 12 kHz scan rate whereas we require lower scan rates for our application. In their case, the resolution was not limited by the widths of the gratings they used (52 mm) [6]. For this reason, the grating width and the scan angles are left empty in Table 1. The change from low resolution (3.0 × 0.603 cm-1) to high (0.29 × 0.603 cm-1) required them to limit the optical bandwidth from 317 cm-1 to approximately 63 cm-1. We can accommodate high optical bandwidth simultaneously with high resolution due to the wide rotating mirror we used.
We have further collected lookup tables in Supplement 1, Note 10. They list typical components (such as N and ${l_\textrm{f}}$) available for different spectral ranges (such as MIR and NIR) and the performance expected with them (such as speed benefit and resolution). We hope these tables are useful for anyone wishing to implement PC-FTS for their application.
4. Phase- and group-delay correction
Generally, a measured interferogram should be sampled linearly with respect to the optical path difference or group delay before Fourier-transforming it into the corresponding spectrum. In traditional FTIR, one way of ensuring this is to have the instrument equipped with a continuous wave laser that may also serve as an inner frequency reference [1,44,45]. The interferogram of the laser is monitored simultaneously with the broadband interferogram and the sampling of the latter is referenced to the zero crossings of the former. A similar kind of group-delay (GD) correction is required also in PC-FTS. In addition to GD correction, a special type of phase-delay (PD) correction is required because the scanning mirror is not infinitely thin (the mirror surface usually does not lie on the pivot point). It follows that even the frequency ${\nu _0}$ experiences phase delay when the mirror is scanned. This must be corrected to recover the spectra accurately [5].
Note that this phase-delay correction is not the same as the phase correction [1,2,46] that is typically used in FTIR to combat noise linearities and to accurately recover the spectra as the real part of the complex Fourier transform without instrument lineshape function distortion. Here, we do not phase correct the spectra but measure two-sided interferograms and present the spectra as the modulus of the complex Fourier-transform. Even though the background-free CEPAS measurements are emissive in their nature, and thus the phase spectrum is not well defined where the signal intensity is zero, phase-correction would be possible, for example, as explained in Ref. [46].
The GD and PD corrections are described in detail in Supplement 1, Note 7. Briefly, the CW DFG light source is used to measure CW interferograms at two different optical frequencies. These CW interferograms are used to calculate the group-delay curve that is then used to resample the interferogram at constant intervals of the group delay. The CW interferograms are also used to calculate the phase-delay curve for a frequency that is the least delayed during scan (i.e., for ${\nu _0}$). This phase-delay curve is then used to induce a phase shift prior to resampling the interferogram with respect to the linearized group delay and performing the Fourier transform. Note that we define the down-conversion factor as the slope of the linearized group delay. The down-conversion factor and the ${\nu _0}$ value can be used to switch between the optical and down-converted frequency axes according to Eq. (1).
In principle, any two frequencies can be used for the corrections, but we found that a large wavenumber separation is preferable. In addition, to measure CW interferograms with CEPAS, an absorbing sample is required inside the sample cell and the CW light source wavenumber must be tuned onto an absorption line. We typically used the large wavenumber separation of 92 cm-1 between the methane absorption lines P11 and P2 as indicated in Fig. 2. In addition, we used separate GD and PD corrections for the interferograms measured with the MIR detector and with CEPAS detection (correction curves determined for the MIR detector could not be readily used for CEPAS and vice versa). It is noteworthy that in Ref. [6] the corrections were performed using a single CW laser, which is possible if the rotating mirror is placed on a translational stage and the missing wavenumber reference is taken from an absorption feature in a measured broadband spectrum (see the supplementary information of Ref. [6]).
As the PD and GD corrections required us to measure CW interferograms at two distinct optical frequencies, they were not performed in real time. This means that the GD and PD correction curves are determined based on a separate measurement, and the curves are stored, and used in future measurements to correct the interferograms. It may be preferable to perform the GD and PD correction measurements on the same day as the actual broadband measurements, but there was no conclusive inaccuracy if correction curves of a separate day were used. However, the lack of real-time correction did pose problems with residual phase noise as discussed in Supplement 1, Note 8. The phase noise distorts the broadband spectra as discussed in the following section.
5. Broadband absorption spectroscopy
Figure 4 shows typical group-delay corrected broadband interferograms from a background (lab air) measurement and a methane measurement using the MIR detector (left panel), and from a methane measurement using the CEPAS detector (right panel). Each interferogram is an average of 10 interferograms. Note the fundamental differences between the interferograms measured with the two detection methods. The centerburst of an interferogram measured with the MIR detector mainly describes the envelope of the light source, and any structure in the spectrum (such as methane absorption) is encoded into the sidebursts further away from the centerburst. In contrast, PAS is a background-free technique, which means that ideally all structure in the interferogram is due to absorption. This leads to efficient use of the dynamic range of the detector and foreshadows higher signal-to-noise ratios (SNRs) in the PAS spectra than in the MIR detector spectra [1].
Fig. 4. Typical interferograms measured with the MIR detector (left panel) and CEPAS (right panel). The lengths of the time windows are 1.77 s and 1.78 s for the MIR detector interferograms and the CEPAS interferogram, respectively. The corresponding down-conversion factors are 8.5027 × 10−11 and 8.4887 × 10−11, respectively. These values differ due to the different phase- and group-delay corrections used for the two detection methods.
Four interferograms like the ones shown in Fig. 4 (MIR detector background, MIR detector methane, or CEPAS methane; each an average of 10 interferograms) were processed into a spectrum and averaged. Without averaging, the envelopes of subsequently measured spectra varied from spectrum to spectrum and prevented the determination of a reliable transmission spectrum. The variation from spectrum to spectrum is presumably due to the jitter discussed further in Supplement 1, Note 8; the mirror scans were not perfectly reproducible from scan to scan, which leads to inaccurate phase- and group-delay corrections.
Figure 5 shows the averaged spectra based on the MIR detector measurements. To yield the absorption spectrum based on these raw spectra, the methane spectrum was divided by the background spectrum, after which the Beer–Lambert law [47] was applied to obtain the absorption coefficient $\alpha ={-} \ln ({I/{I_0}} )/L$, where I is the methane spectrum, ${I_0}$ the background spectrum, and L the absorption path length. Division by the background spectrum left a residual sloping background, which was corrected using Savitzky–Golay filtering. In addition, a constant offset of 1.9 GHz was applied to the frequency axis. The resulting final absorption spectrum based on the MIR detector measurements is shown in Fig. 6. Only the methane lines P14–P3 are shown, as lines beyond this range were masked by noise.
Fig. 5. Raw methane and background spectra measured with the MIR detector. The horizontal axis spans the full bandwidth from the pivot point (${\tilde{\nu }_0} = $ 2795.372 cm-1) to the edge of the scanning mirror (3025.5 cm-1). The artefacts at the down-converted frequencies approx. 45 Hz and 145 Hz are due to the MIR detector fans. The etalon structure is presumably due to the uncoated CaF2 lenses, the CEPAS cell windows, or both. Also note how intensity leaks below 50 Hz despite the high-pass filtering. This leaking is due to the imperfect phase- and group-delay corrections (residual phase noise) caused by the unreproducible mirror scans (discussed further in Supplement 1, Note 8).
Fig. 6. Final absorption spectrum of methane lines P14–P3 measured with the MIR detector (blue trace). Note that especially lines P4 and P3 have poor SNR and uncertainty in the background subtraction. The black dotted trace is a simulation of the expected absorption spectrum. It is the result of simulating the interferogram of a HITRAN transmission spectrum, multiplying it with the experimental apodization function, and reprocessing the result into an absorption spectrum. The orange trace is a HITRAN reference that yields the best match with the measured spectrum. It is the result of convoluting a HITRAN transmission spectrum with a Lorentz function whose FWHM was fixed to 9 GHz. The residual in the lower panel is calculated against the Lorentz reference.
Figure 7 shows the raw spectrum based on the CEPAS measurements. To yield the absorption spectrum, the raw spectrum was normalized with the MIR detector background spectrum, as the PAS signal at each optical frequency is proportional to the optical power at that frequency. In addition, the CEPAS spectrum was normalized with the CEPAS response plotted as the yellow line in Fig. 7 (the response was determined experimentally, see Supplement 1, Note 9). Furthermore, CEPAS requires calibration with respect to the sample concentration. This was considered by fitting a scaling parameter to yield a good match between the measured spectrum and the chosen HITRAN reference discussed below. Finally, a residual background was subtracted using Savitzky–Golay filtering, and a constant shift of 1.8 GHz was applied to the frequency axis. The resulting final absorption spectrum based on the CEPAS measurements is shown in Fig. 8.
Fig. 7. Raw methane spectrum measured with the CEPAS detector (before normalizing with the MIR detector background spectrum in Fig. 5, or with the CEPAS response plotted as the yellow trace in this figure). The horizontal axis spans the full bandwidth from the pivot point (${\tilde{\nu }_0} = $ 2795.340 cm-1) to the edge of the scanning mirror (3025.5 cm-1). Note the increasing background signal above approx. 3000 cm-1 due to noise close to the CEPAS detector resonance (570 Hz). Also note that the P2 line is clearly visible with CEPAS detection but it is masked by noise in the MIR detector measurements (Fig. 5).
The black dotted line in the final MIR detector spectrum in Fig. 6 (CEPAS spectrum in Fig. 8) is a simulation of the methane spectrum via simulating the interferogram of a HITRAN transmission (absorption) spectrum, multiplying it with the experimental apodization function and by reprocessing the interferogram into an absorption spectrum. The experimental apodization function was determined from the average envelope of a few CW interferograms measured at lines P11, P6, and P2 (see discussion in Supplement 1, Note 8). It can be seen that the measured methane lines are much flatter and wider than expected. The orange line in Fig. 6 (Fig. 8) is a convolution of a HITRAN transmission (absorption) spectrum with a Lorentzian function with its FWHM fixed to 9 GHz (10.5 GHz) [39,40]. This provided the best match with the measured spectrum compared to a few different convolution functions we tried. The HITRAN references assume air-broadened lines even though the methane measurements were performed in nitrogen. In addition, speed-dependent effects on the lineshapes and line-mixing are known to be important for methane [48]. However, inclusion of such effects [48] and changing the air broadenings to nitrogen broadenings [48,49] had negligible effect on the residuals plotted in the lower panels of Figs. 6 and 8. The found discrepancies between the measured and the expected spectra are presumably due to the phase noise (jitter) that the experimental ILS function ignores. This hypothesis was supported by tests where we introduced jitter to simulated methane interferograms. Other reasons for the discrepancies may include different alignment of the MIR OFC light source through the system compared with the CW light source, unaccounted for aberration effects [6,50], the lack of phase correction [46], or detector nonlinearity, which is also known to distort spectra in a similar manner [51]. More rigorous analysis of the ILS could be performed based on measurements of isolated absorption lines [52] instead of relying on separate CW measurements as we have done.
The standard deviations of the MIR detector spectrum residuals (Fig. 6) and the CEPAS spectrum residuals (Fig. 8) are both approximately 4% from the respective maximum values in the spectra (the line P6 for both). As the residuals under the peaks mainly describe the match of the measured spectrum with the assumed reference spectrum, we define SNR as the line P6 height divided by the standard deviation of the residual with the residuals under the absorption peaks and data above approximately 2970 cm-1 excluded. This approach yields an SNR of 57 for the MIR detector spectrum and of 116 for the CEPAS spectrum. For the MIR detector spectrum, the SNR corresponds to a noise-equivalent limit of detection (NELOD) of 180 ppm.
Fig. 8. Final absorption spectrum of methane lines P14–P3 measured with the CEPAS detector (blue trace). The black dotted trace is a simulation of the expected absorption spectrum. It is the result of simulating the interferogram of a HITRAN absorption spectrum, multiplying it with the experimental apodization function and Fourier-transforming the result back into an absorption spectrum. The orange trace is a HITRAN reference that yields the best match with the measured spectrum. It is the result of convoluting a HITRAN transmission spectrum with a Lorentz function whose FWHM was fixed to 10.5 GHz. The residual in the lower panel is calculated against the Lorentz reference.
The NELOD for the CEPAS spectrum should be determined in a different way than for the MIR detector spectrum. Mikkonen et al. [35] have shown that PAS detection leads to noise reduction in FTIR compared with traditional detection schemes, as only the spectral components absorbed by the sample contribute to the observed noise. Similarly, it is clear that the phase-noise contribution by a spectral component (Supplement 1, Note 8) scales with the absorbed power. This means that as the sample concentration is decreased, the absorption peaks and the noise decrease with constant SNR until the background noise of the detector starts to dominate. The NELOD should then be determined from a separate CEPAS spectrum measurement where light enters the sample cell but no methane is present. We measured such a spectrum (a single spectrum from an average of 10 interferograms), calculated the standard deviation of the noise between 2870 cm-1 and 2970 cm-1, divided the P6 peak height of the raw methane spectrum in Fig. 7 with the calculated standard deviation, and arrived at a NELOD of 7 ppm. This is a factor of 25 better than the NELOD obtained with the MIR detector. The NELOD for CEPAS is of the same order or in some cases a factor of ten poorer than values reported in previous CEPAS-FTS demonstrations [28,33].
6. Conclusions
We have demonstrated a 13-fold speed improvement in broadband cantilever-enhanced photoacoustic spectroscopy (CEPAS) by using the phase-controlled Fourier-transform spectroscopy (PC-FTS) instead of traditional Fourier-transform infrared spectroscopy (FTIR). Indeed, the typical 1.8 s long interferograms measured with PC-FTS would require up to 24 seconds of scanning to yield the expected optical resolution of 7.2 GHz with traditional FTIR. In practice, the resolution was degraded due to self-apodization effects caused by non-constant scan velocity of the rotating mirror. In addition, unreproducible scanning resulted in residual phase noise, which distorted the broadband spectra. These problems could potentially be avoided by using a more stable scanner, implementing the phase- and group-delay corrections in real time [53,54], or both. The optical resolution could then be improved by using a wider grating, unless aberration effects prevent this [6,50]. It is noteworthy that the speed benefit of PC-FTS is a trade-off with the optical bandwidth. The scanning mirror we used accommodated a maximum bandwidth of 6.9 THz. However, we were able to consider only the P-branch of methane due to the limited tuning of the MIR OFC light source. Despite the inherently high optical losses of PC-FTS and the power scaling of CEPAS signal, the CEPAS detection was shown to be at least 25 times more sensitive than the traditional transmission spectroscopy approach. Even though averaging was required to yield reasonable limits of detection compared to previous CEPAS-FTS demonstrations, the speed-benefit of PC-FTS may be useful in applications where the SNR is not the limiting factor.
Funding
Academy of Finland (326444); Alfred Kordelin Foundation; University of Helsinki.
Acknowledgments
We thank Dr. Kazuki Hashimoto, Prof. Takuro Ideguchi and Dr. Markus Metsälä for their insights and constructive feedback on the project. M. Roiz acknowledges financial support from the Alfred Kordelin Foundation. S. Larnimaa acknowledges financial support from the CHEMS doctoral program of the University of Helsinki. Open access funded by Helsinki University Library. Author contributions include S. Larnimaa constructed the experimental setup (not including the mid-infrared optical frequency comb light source), performed the measurements, analyzed the data, and wrote the manuscript; M. Roiz constructed the mid-infrared optical frequency comb light source, assisted with its use, and reviewed the manuscript; M. Vainio conceptualized and supervised the project, and reviewed the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data and MATLAB codes underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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