1. Introduction

Since its first description by Bjorklund in 1980, laser-based frequency modulation spectroscopy (FMS) has proven to be a powerful tool for the detection of molecular species in both spectroscopic and reaction kinetics studies [1,2]. FM techniques combine sub-microsecond time resolution with shot-noise limited sensitivity and are therefore particularly suitable for investigating short-lived species in the gas phase. In contrast to conventional absorption-based methods, FM spectroscopy is a derivative technique that is based on unequal absorption or dispersion of the sidebands in the frequency-modulated spectrum of the detection laser. In theory, intensity fluctuations of the light source and other forms of sideband-balanced effects such as broadband background absorption do not yield FM signal. An impressive example highlighting the combined advantages of the FM detection approach has been reported by Crofton and Peterson [3]. They were able to detect transient SiH$_2$ radicals (i.e., weak absorptions of a short-lived species) during the formation of particles (i.e., strong interfering and time-dependent background absorption) in a shock tube experiment (i.e., an experimental setup known to be prone to acoustic noise contributions).

Earlier, Sears and coworkers demonstrated the high sensitivity and high time resolution of FM spectroscopy for the detection of NH$_2$ and HCCl radicals in supersonic jets [4] and CN radicals in a low-pressure flow reactor [5]. Very recently, they used an FM detection scheme for detecting OH radicals in the near-infrared in a kinetic experiment [6]. Another example exploiting the excellent temporal response of FM spectroscopy is a study by Alagappan et al. [7], who investigated the collision dynamics of CN radicals with a time resolution of about $20$ ns applying a $400$ MHz modulation scheme.

The research groups of Wagner and Hanson pioneered the use of FM spectroscopy as a detection technique for shock tube studies. Such high-temperature studies are especially demanding with respect to quantitative determination of reactant concentrations, which are often in the range of a few ppm that need to be detected with microsecond time resolution under harsh conditions (e.g., beam steering and pressure-induced birefringence in shock tube windows) in a real-time single-shot experiment [8–10]. Among the first species investigated with FM spectroscopy in shock tube experiments were NH$_2$, $^1$CH$_2$, and HCO [11–14], more recently HNO has been detected as well [10].

So far, FM-based applications have mostly been limited to the visible and NIR regions, with more recent advances into the UV and VUV [15,16]. The expansion of FM spectroscopy into the mid-infrared (MIR) region promises new detection options for many molecular species. Instead of relying on often weak absorption lines of overtone or combination bands observed in the NIR, detection on stronger fundamental rovibrational transitions in the MIR offers an increase in sensitivity. Our own interest lies in detection schemes for HO$_2$, CH$_3$, and HCN. However, the limited bandwidth of MIR detectors and the low modulation efficiency of electro-optical modulators in the MIR (i.e., the high half-wave voltage, which is proportional to the detection wavelength) have made the widespread use of MIR-FM spectroscopy with external modulation largely unfeasible as yet.

Moreover, the availability of reliable and robust cw-MIR laser sources was limited for a long time. Today, quantum cascade lasers (QCLs) and systems based on either difference frequency generation (DFG) or optical parametric oscillators (OPOs) are the most common MIR-light sources. Yet, there is a number of studies reporting the implementation of MIR-FM spectroscopy with approaches involving both external and direct modulation.

In the late nineties, Taatjes and coworkers introduced MIR-FM spectroscopy with external modulation for the detection of HCl in kinetic studies that were concerned with H-abstraction reactions [17,18]. The low bandwidth of mid-infrared detectors available at the time required them to use the two-tone FM approach [19,20], which limits the maximum achievable sensitivity [21] and makes quantitative measurements more difficult. In 2006, Borri et al. reported the implementation of single-tone MIR-FM spectroscopy with a current-modulated quantum cascade laser [22]. Shortly afterwards, Maddaloni et al. described a two-tone FM spectrometer based on DFG with a modulated diode laser achieving a signal-to-noise ratio improvement by a factor of 100 compared with the direct absorption setup [23]. Similarly, Lindsay et al. demonstrated a single-tone MIR-FM application by recording FM spectra of CH$_4$ at $3175$ cm$^{-1}$ using a cw-OPO-based laser system directly modulated with $153$ MHz [24]. All three approaches employ direct modulation of the light source, resulting in undesirable amplitude modulation (AM) and reduced sensitivity. Chen et al. introduced the concept of modulating the frequency of a cw-quantum cascade laser by illuminating the front facet with 100 fs Ti:sapphire laser pulses, which also caused high levels of amplitude modulation [25]. This issue was later addressed by both all-optical and advanced electronic control schemes for AM suppression [26,27], which holds the potential for future QCL-based FM spectroscopic applications.

Highly efficient electro-optical modulators with high-Q resonators, suitable for external modulation of MIR laser beams with sufficient modulation depth, have recently become commercially available. In addition, fast MIR detectors with up to $1$ GHz bandwidth have entered the market, which enables the practical realization of single-tone MIR-FM spectroscopy. This work presents the first measurements conducted with a newly established single-tone MIR-FM spectrometer taking advantage of external EOM modulation at $500$ MHz. The results include FM spectra of CH$_4$ and concentration-time profiles of the stable species HCl and transient OH radicals. A sensitivity analysis of the detection scheme based on Allan variance was performed.

2. Theoretical background

A brief outline of the principle behind FM spectroscopic methods is given here, whereas more detailed theoretical descriptions can be found elsewhere [1,2,21]. Frequency modulation of a monochromatic laser beam with the modulation frequency $f_{\textrm {m}}$ leads to the emergence of discrete sidebands around the carrier frequency $f_{\textrm {c}}$ (see upper part of Fig. 1), where the modulation index $M$ is a measure for the modulation depth that determines the relative intensities of the carrier and the sidebands. $M$ is defined as the ratio of the maximum induced frequency deviation to the modulation frequency $f_{\textrm {m}}$. The modulation of the laser frequency can be achieved either by direct modulation of the light source or by external modulation using an electro-optical modulator.

 

Fig. 1. Schematic of the MIR-FM setup. OPO: optical parametric oscillator; EOM: electro-optical modulator; AC/DC: alternating/direct current; RF: radio frequency; LO: local oscillator; IF: intermediate frequency; DSO: digital storage oscilloscope.

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If a frequency-modulated beam passes through a sample that causes unequal absorption or dispersion of the opposing sidebands, an amplitude modulation of the electric field oscillating at the frequency $f_{\textrm {m}}$ is induced. More generally, the effect of absorption and dispersion on the electric field is described by applying the complex transmission function $T_n(\omega ) = \textrm {exp}(-{\delta _n}-i{\phi _n})$ to the electric field, with $\delta _n$ and $\phi _n$ as the electric field coefficients for absorption and dispersion of the $n$th order sideband, respectively. Using the angular frequencies $\omega _{\textrm {c}}~=~2 \pi f_{\textrm {c}}$ and $\omega _{\textrm {m}}~=~2 \pi f_{\textrm {m}}$, the electric field $E_T(t)$ of the transmitted amplitude-modulated light can be written as

The FM approach transfers absorption and dispersion information to a high frequency, which leads to a significant reduction of $1/f$ noise and other forms of low-frequency noise. In principle, FM spectroscopic methods can accomplish shot-noise limited detection even on short timescales. In practice, however, the shot-noise limit is often not reached due to residual amplitude modulation (RAM). Generally, all contributions to amplitude modulation that do not stem from absorption or dispersion induced by the sample are referred to as RAM. Etalon effects, photorefractive scattering, and the mismatch of the polarization axes due to temperature fluctuations in the modulator crystal are the main factors contributing to RAM noise in applications with external frequency modulation [30]. Possible strategies for reducing RAM include tilting the EOM relative to the beam path, photorefractive erasure techniques, and active thermal stabilization of the modulator [31–33].

3. Experimental

Figure 1 shows a schematic of the experimental setup. Standard electronic radio frequency (RF) components (Mini Circuits) have been used in the modulation-demodulation circuitry. A combination of a fiber-coupled DFB pump laser (Koheras AdjustiK Y10 PM, NKT Photonics, 15 mW), a Yb-doped fiber amplifier (YAR-10K-1064-LP-SF, IPG Photonics, 10 W), and a continuous-wave optical parametric oscillator (cw-OPO, Argos SF-10, Lockheed Martin, modules B and C, >1.2 W idler power) was used as the narrow-bandwidth light source (<60 kHz linewidth). Only the idler output beam of the OPO module was used for detection and its wavelength was monitored with a wavemeter (Model 621, Bristol Instruments). For kinetic measurements, the frequency of the detection laser was kept fixed, shifted by $f_{\textrm {m}} = 500$ MHz relative to the center of the respective absorption line in order to maximize the FM factor $\Delta f$. Spectral measurements were performed by applying a triangle-shaped voltage profile to the piezoelectric element in the pump laser that controls the emission wavelength.

A resonant LiTaO$_3$ electro-optical modulator (EO-T500T3-MIR, Qubig) was used for the modulation of the detection beam with $f_{\textrm {m}} = 500$ MHz. According to the specified half-wave voltage of about $40$ V/rad at $\lambda = 4.5~\mu$m, a modulation index of $M\approx 0.5$ should be achievable at the maximum RF power rating of $P_{\textrm {max}} = 5$ W. Since scanning etalons for the mid-infrared are not readily commercially available, the modulation index was initially estimated from the ratio of the first and second-order peaks in FM spectra measured at the highest RF power on the modulator. Later, a newly established self-made scanning etalon with custom-coated YAG mirrors ($R\approx 98.5$%, $2500~$cm$^{-1} \leq \tilde {\nu } \leq 5000~$cm$^{-1}$, Layertec) and a piezo actuator (P-401-10, piezosystem jena) was used. Two YVO$_4$ polarizers, one in front of the EOM and one behind it, ensured that the beam polarization matched the required input polarization of the EOM and that the absolute phase angle was set to pure absorption ($\theta = 0^\circ$ or $180^\circ$; for details on the issue of setting the demodulation phase angle we refer to [13]).

The output of an RF source (Signal Generator 2023, Marconi Instruments) was split by a directional coupler (ZFDC-10-5-S+) and the resulting coupled and main signals were guided to the modulation line and the demodulation circuit, respectively. To set the power on the EOM and the local oscillator line of the demodulation circuit to appropriate levels, the RF source output power was adjusted and a set of attenuators (SAT series) as well as two amplifiers (ZHL-20W-13+, ZRl-700+) were used. After passing through an absorbing gas sample, the amplitude-modulated detection beam was focused onto a two-stage thermoeletrically cooled high-bandwidth GaAs detector (PVI-2TE-5, Vigo System) offering a fast time response ($t_{\textrm {r}} \leq 0.7~$ns) and high detectivity ($D_{\textrm {opt}}^{\star } \geq 6 \cdot 10^{10}~$cm Hz$^{1/2}$ W$^{-1}$). Within the detector module, the measured signal was pre-amplified and split into its AC and DC components with a bias tee. The AC signal was demodulated using a frequency mixer (ZFM-2000+), a phase shifter (JSPHS-661+), and a low-pass filter (SLP-1.9), after which the resulting FM signal was amplified and low-pass filtered to 1 MHz bandwidth using an additional amplifier (Model SR560, Stanford Research Systems). Both the FM signal and the DC signal were recorded with a digital storage oscilloscope (MSO 8104A, Agilent Technologies). The gain factor of the demodulation circuit $G$ was determined by comparing the FM signal with the conventional absorption signal of a $\textrm {CH}_4$ sample measured at a known FM factor (see Eq. (2)), which yielded a value consistent with the total gain of the components in the demodulation circuit specified by the manufacturers.

All spectroscopic and kinetic experiments were conducted at room temperature in a 42 cm (CH$_4$ and HCl detection) or $18.5~$cm (OH detection) long slow-flow gas cell equipped with quartz windows. An ArF excimer laser (RD-EXC-200, Radiant Dyes) was used as a photolysis light source for the generation of Cl atoms and OH radicals by $193~$nm photolysis of oxalyl chloride, (COCl)$_2$, and H$_2$O$_2$, respectively. The output beam of the excimer laser was overlapped with and divided from the collinear detection laser beam using two dichroic mirrors, resulting in pulse energies of $20-39~$mJ/pulse in the reactor. The chemicals used for the measurements were pure CH$_4$ (Air Liquide, 99.5%) and freshly distilled oxalyl chloride (Sigma-Aldrich, > 99%) in argon (Air Liquide, 99.999%). Gas mixtures of about 1% H$_2$O$_2$ in argon were prepared by passing the carrier gas through a heated trap filled with solid adduct of H$_2$O$_2$ and urea (Sigma-Aldrich, 97%) at a gas flow rate of $4$ sccm, $T = 50^{\circ }$C, and $p \leq 8.6$ mbar [34].

4. Results

In order to demonstrate the capabilities of the newly established MIR-FM spectrometer, we recorded (i) FM spectra of CH$_4$ at different modulation indices, (ii) time profiles of HCl formed by the reaction Cl + CH$_4$ following UV photolysis of oxalyl chloride, and (iii) OH transients produced by UV photolysis of H$_2$O$_2$. Moreover, in order to assess the sensitivity of the setup and to reveal dominating noise sources, an Allan deviation analysis was conducted using FM signal noise recorded at different light power levels on the photodetector.

1 Spectroscopic examples

Figure 2(b) shows FM spectra of the weak and spectroscopically isolated $0200~1E \leftarrow 0000~1A_{1}$ transition of CH$_4$ at $\tilde {\nu }_{0} = 3071.39$ cm$^{-1}$ [35]. The individual spectra have been recorded with different modulation depths ($0.20 < M < 0.69$) and the demodulation phase set to pure absorption. The scan rate and the number of scans used for averaging were $0.77~$cm$^{-1}$s$^{-1}$ and $16$, respectively. For comparison, the directly measured absorption line, corresponding to a peak absorption of $A_{\textrm {c}}= 3.9 \cdot 10^{-2}$, is included in Fig. 2(b) as a black curve. A typical etalon trace of the frequency-modulated light at $M =~$0.60 is shown in Fig. 2(a). Here, the vertical bars represent the exact positions of the carrier and sideband frequencies, which are not fully resolved by the scanning etalon setup. Note that the spectra in Fig. 2(b) have already been baseline corrected using previously recorded baseline signals. This was necessary due to recurrent undulations of the FM signal during wavelength scans, which we mostly attribute to minor polarization instabilities of the OPO laser system. An example trace of an uncorrected signal is shown in Fig. 2(c). Pronounced baseline effects only occurred when the detection laser wavelength was scanned, while for constant wavelengths the FM signal background exhibited only minor long-term drifts. After baseline correction, high-quality derivative-like signals have been observed. Actually, as the high modulation frequency of $500~$MHz already exceeded the width of the CH$_4$ absorption line ($\Delta \nu _{\textrm {HWHM}} = 140~$MHz), instead of resembling a simple first derivative lineshape, the absorption peak is reproduced twice with positive and negative maxima at $|\nu - \nu _{0}| \approx f_{\textrm {m}} = 500~$MHz. It is well-known from the literature [2,21] and also directly follows from Eq. (4) that for $f_{\textrm {m}}/\Delta \nu _{\textrm {HWHM}} > 1$ the FM signal originates from the direct interaction of a single sideband with the absorption feature, where the different signs of the signal reflect the relative phases of the opposing sidebands. For the experiment at the highest modulation index of $M = 0.69$ in Fig. 2(b), a second-order peak at $|\nu - \nu _{0}| \approx 1~$GHz emerges due to interaction of the second-order sideband with the spectral line.

 

Fig. 2. (a) Frequency-spectrum of the modulated light at $M =~$0.60. (b) FM spectra of CH$_4$ at different modulation indices and corresponding absorption spectra in units of the FM factor $\Delta f$ ($p = 1.5~$mbar, $\theta = 180^{\circ }$, $\tilde {\nu }_{0} = 3071.39$ cm$^{-1}$, $A_{\textrm {c}}= 3.9 \cdot 10^{-2}$). (c) Typical uncorrected FM signal (blue curve) and corresponding baseline (black curve). (d) Experimental (solid curves) and simulated (dashed curves) FM spectra of four closely spaced CH$_4$ lines ($p = 2.0~$mbar, $M = 0.32$, $A_{\textrm {c}}= 5.2 \cdot 10^{-2}$ for the highest peak) as well as simulated direct absorption spectrum (dashed black curve).

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More complicated FM spectra arise for the case of closely spaced absorption lines causing partial overlap of the corresponding FM signal contributions. An illustrative example with four CH$_4$ absorption lines between $3070.42$ cm$^{-1}$ and $3070.66$ cm$^{-1}$ is shown in Fig. 2(d). The 16 times averaged FM spectra have been recorded with a scan rate of $1.10~$cm$^{-1}$s$^{-1}$ at a total pressure of $p = 2.0~$mbar and a modulation index of $M = 0.32$ by setting the demodulation phase angle $\theta$ to either absorption (solid blue curve) or dispersion (solid red curve). Simulations of the absorption and dispersion FM spectra (dashed colored curves) based on Eqs. (2) and (3) as well as the conventional absorption spectrum (dashed black curve) were added for comparison. Apart from minor deviations, most notably around $3070.63$ cm$^{-1}$, which can be attributed to small baseline issues and uncertainties of $\theta$ and $M$, the simulated FM spectra are in excellent agreement with the experimental ones, demonstrating the reliable acquisition of FM spectra.

2 Reaction kinetics examples

Practical use of the new MIR-FM detection scheme for reaction kinetics studies has been demonstrated by the detection of a stable reaction product as well as a reactive transient radical.

The formation of HCl from the reaction

 

Fig. 3. (a) HCl profile (blue curve) and single exponential fit (red curve) in units of absorbance and mole fraction at $2925.90$ cm$^{-1}$ ($M = 0.48$, $\Delta f_{\textrm {max}} = 0.45$, $p = 31.0~$mbar, $\Delta f_{\textrm {BW}} = 1~$MHz, $P_0 = 0.6~$mW). The inset shows the noise level on the recorded signal. (b) OH profile (blue curve) in units of absorbance and mole fraction at $3447.90$ cm$^{-1}$ ($M = 0.48$, $\Delta f_{\textrm {max}} = 0.45$, $p = 8.6~$mbar, $\Delta f_{\textrm {BW}} = 1~$MHz, $10$ point moving average, $P_0 = 1.0~$mW) and kinetic simulation (red curve) of the experimental profile based on rate constants adopted from Atkinson et al. [37] and Baulch et al. [38].

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Following photolysis at $t = 0$, Fig. 3(a) reveals the direct formation of HCl from the reaction Cl + CH$_4$. The small and short negative signal occurring over an interval of about $1.5~\mu$s is a typical measurement artifact caused by the intense excimer laser pulse. The inset in Fig. 3(a) highlights the noise level of the recorded profile, which is well below $1 \cdot 10^{-3}$ for this experiment. The experimental data can be nicely represented by first-order kinetics (red curve in Fig. 3(a)), indicating pseudo-first order reaction conditions for the reaction Cl + CH$_4$. A kinetic pseudo-first order analysis of the experiments performed at varying mole fractions of the excess species CH$_4$ yielded the expected linear trend and a bimolecular rate constant $k_{\textrm {exp}} = (1.07 \pm 0.02) \cdot 10^{-13}~$cm$^3$molec$^{-1}$s$^{-1}$, which is in quantitative agreement with the recommended literature value of $k_{\textrm {lit}} = (1.0 \pm 0.1) \cdot 10^{-13}~$cm$^3$molec$^{-1}$s$^{-1}$ [39].

Figure 3(b) shows a typical kinetic profile of the transient OH radical. Following fast formation of OH from photodissociation of H$_2$O$_2$, the OH radical decay takes place on a timescale of several hundred microseconds. Next to loss of OH from the reaction OH + H$_2$O$_2$, a full kinetic analysis has to account for other reactions such as OH + HO$_2$ as well. A numerical simulation of the profile (red curve in Fig. 3(b)) is in full agreement with the experimental trace. From the $1\sigma$ standard deviation of the noise, one can estimate a minimum detectable single-shot absorbance of $A_{\textrm {min}}= 1 \cdot 10^{-4}$ ($\Delta f_{\textrm {BW}} = 1~$MHz, $10$ point moving average). Based on the known absorption cross section of $\sigma _{\textrm {OH}} = 5.2 \cdot 10^{-18}~$cm$^{2}$molec$^{-1}$ ($T = 298~$K, $p = 8.6~$mbar) [35], this yields a OH radical detection limit of $\sim 4~$ppm for the specified conditions. Of course, the detection limit could be further improved to sub-ppm levels by signal-averaging procedures. Note that the OH detection presented in this work serves as an illustrative example for the detection of a transient species. In case of OH, due to more than two orders of magnitude higher absorption cross sections of its rovibrational $A^2\Sigma ^+ - X^2\Pi$ transitions at wavelengths around 308 nm, the detection limit achieved by sensitive UV laser absorption methods is even in the ppb range [15,40].

3 Allan sensitivity analysis

In this work, the preferred measure of sensitivity is the Allan deviation of the FM signal expressed in units of absorbance. Allan deviation is based on the two-sample variance of adjacent intervals of averaged data and was originally developed in order to investigate the frequency fluctuations of atomic clocks [41,42].

Consider a data set of an arbitrary quantity $y$ consisting of $N$ data points measured with a sampling frequency $f_{\textrm {s}}$, so that individual points of data are evenly spaced by the time $\tau _{\textrm {s}} = 1/f_{\textrm {s}}$ without any dead time. One can divide the data into $K$ intervals of adjacent points, with $k = N/K$ as the number of points in each interval and $\tau = k \cdot \tau _{\textrm {s}}$ as the corresponding interval time period. The difference between the average $\bar {y}_{i}$ for the $i$th interval and the average $\bar {y}_{i+1}$ for the next interval is dependent on the selected averaging period $\tau$ as well as the noise level affecting the quantity $y$. The Allan variance $\sigma _{\textrm {A}}^{2}(\tau )$ is defined as the expectation value of the infinite time average of the sample variance regarding two adjacent intervals for a given value of $\tau$:

An example of FM signal noise recorded at $3071$ cm$^{-1}$, preamplified with $\Delta f_{\textrm {BW}} = 1$ MHz (corresponding to a 10%-to-90% rise time of $0.35~\mu$s), and sampled with the oscilloscope in intervals of $\tau _{\textrm {s}} = 1~\mu$s is shown in Fig. 4(a). Note that the signal is displayed on five different timescales to better illustrate the frequency dependence of the noise, where the individual segments of the signal noise with $n > 0$ were offset from the zero line for better visualization.

 

Fig. 4. (a) Measured FM signal noise on five different timescales ($M = 0.20$, $P_0 = 480~\mu$W, $\tilde {\nu } = 3071$ cm$^{-1}$). The individual segments of FM noise were offset with $20~$mV relative to each other for better visualization. (b) Corresponding Allan plot regarding $I_{\textrm {FM}}$ with estimated errors.

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On a small timescale ($n\geq 3$), the signal noise appears to be statistical white noise, whereas on longer timescales ($n \leq 1$) low-frequency signal undulations occur. The corresponding Allan plot in Fig. 4(b) shows a linear decrease of the Allan deviation $\sigma _{\textrm {A}}$ with $\sqrt {\tau }$ up to averaging periods of about $40~\mu$s, which is the expected $\tau$-dependence for white noise. For longer averaging periods $\sigma _{\textrm {A}}$ starts to increase, with a maximum in the millisecond range. Since the noise on this timescale was very sensitive to tilting of the EOM relative to the beam path, it can most likely be attributed to RAM noise contributions. In contrast, we attribute the small shoulder at $\tau \approx 10^{-4}$ s to parasitic noise from the detection electronics.

Figure 5(a) shows the results of another typical Allan analysis of FM signal noise measured at $\tilde {\nu } = 3447$ cm$^{-1}$ with $M = 0.48$ at four different light power levels $P_0$. Here, the Allan deviation is expressed in units of absorption, where the $\tau$-dependent Allan deviation in units of voltage was converted to the Allan deviation in terms of FM-equivalent absorption using Eq. (2). Overall, the Allan deviation decreases with increasing $P_0$ and, for small values of $\tau$, decreases linearly with increasing $\sqrt {\tau }$, although there is a slight curvature in all four Allan plots below $10~\mu$s. For the detection of transient species with $\tau = 1~\mu$s and at the highest value of $P_0$, the minimal detectable absorption is $A_{\textrm {min}}= 4 \cdot 10^{-4}$. This is considerably lower than for conventional absorption-based techniques with dual-beam setups and multi-pass cells, which are limited to a noise-equivalent absorption of $\sim 5 \cdot 10^{-6}$ Hz$^{-0.5}$ under ideal conditions, corresponding to a minimal detectable absorption of $\sim 5 \cdot 10^{-3}$ at a comparable detection bandwidth of $\Delta f_{\textrm {BW}} = 1~$MHz [43,44]. More advanced methods using direct modulation schemes such as 2$f$ wavelength modulation absorption spectroscopy with DFG [45,46] and interband cascade lasers [47] achieve minimal detectable absorption levels of $>1 \cdot 10^{-3}$ for the same bandwidth. Note that for the specific data shown in Fig. 5(a), the RAM noise at $\tau \approx 3$ ms was much more pronounced than for the FM signal shown in Fig. 4(b). Clearly, careful optimization of the optical setup is necessary to reduce RAM noise components for sensitive detection of transients with concentration changes taking place on millisecond timescales.

 

Fig. 5. (a) Minimal detectable absorption $A_{\textrm {min}}$ derived from Allan analysis of FM signal noise at different light power levels $P_0$ as function of the averaging period $\tau$ ($M = 0.48$, $\Delta f_{\textrm {max}} = 0.45$, $\tilde {\nu } = 3447$ cm$^{-1}$). (b) log-log plot of experimental $A_{\textrm {min}}$ values at $\tau = 1~\mu$s versus light power $P_0$ ($M = 0.48$, $\Delta f_{\textrm {max}} = 0.45$, $\tilde {\nu } = 3447$ cm$^{-1}$). The theoretical shot-noise limit is shown for comparison ($\eta = 0.52$, $\Delta f_{\textrm {BW}} = 1~$MHz).

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Finally, in order to compare the achieved sensitivity with the theoretical shot-noise limit, Fig. 5(b) shows a log-log plot of $A_{\textrm {min}}$ values for $\tau = 1~\mu$s as a function of the incident light power $P_0$ (diamond symbols). The expected shot-noise limit $A_{\textrm {min}}^{\textrm {SN}}$ [2] according to

5. Conclusion

The present study reports the first implementation of a single-tone MIR-FM detection scheme with external modulation. MIR-FMS holds high potential for sensitive detection of molecular species and is particularly useful for reaction kinetics applications as it allows to quantitatively measure concentration-time profiles of reactive species with microsecond time resolution. FM spectra and transient profiles of CH$_4$, HCl, and OH radicals detected on selected transitions in the respective X-H stretch-vibrational bands around $3~\mu$m served as illustrative examples. A detection limit of $A_{\textrm {min}}= 4 \cdot 10^{-4}$ for $\Delta f_{\textrm {BW}} = 1~$MHz was achieved using a resonant EOM with a modulation depth of $M = 0.48$. This detection limit is only about a factor of 4 higher than the shot-noise limit. Further improvements to the spectrometer are possible. A thermoelectrically cooled EOM in double-pass configuration is currently being implemented to further increase the modulation depth and to reduce RAM noise, which currently sets the sensitivity limit for measurements on longer timescales. Moreover, the photodetector will be replaced by a tailored model with multi-stage thermoelectrical cooling and a higher optical saturation limit. By combining higher modulation depth, reduced thermal noise, and higher detectable optical power, we expect to increase the overall sensitivity of the spectrometer by at least a factor of 5 and to approach the shot-noise limit. We also plan to demonstrate the use of MIR-FM spectroscopy for the detection of transient species under harsh shock tube conditions.