1 Introduction

For decades, sensitive tunable diode laser (TDL) absorption measurements have been performed with wavelength modulation spectroscopy (WMS) in a wide variety of practical applications [1–5]. With its better noise-rejection characteristics through laser wavelength modulation strategies, WMS has long been recognized as the method of choice for sensitive measurements of small values of absorption, and thus is favored for trace species detection [6–8], measurements with significant noise [9], or both [10]. Injection-current-tuned TDL-WMS is attractive in systems with significant non-absorption attenuation of the optical transmission because it is possible to account for changes in laser intensity by normalizing the signals at even harmonics of the modulation by the first harmonic signal [6–11].

Most early TDL-WMS applications were either in an atmospheric pressure open path [6, 12, 13] or in a relatively low-pressure (P < 2 atm) vessel [14–16], where the optimal modulation depths for achieving the strongest WMS signal were small and accessible within the tuning range of typical TDLs. However, to achieve optimal WMS conditions for higher pressure gas sensing, a larger modulation depth is required [17, 18] because the collisional width of the targeted transition is broadened. Based on Kluczynski and Axner’s model describing the WMS-nf signals by Fourier analysis [19–21], Li et al. [22] demonstrated TDL-WMS measurements with 2f detection at pressures to 10 atm using a large modulation depth. In the same paper, the concept of 1f-normalized WMS-2f considering the non-ideal modulation characteristics of injection-current-tuned TDLs was discussed. Later Reiker et al. [23] developed a general protocol for 1f-normalized TDL-WMS measurements. The normalization scheme makes the WMS measurements independent of the laser intensity, and thus allows a calibration-free WMS measurement without the need to acquire the zero-absorption baseline during the absorption measurements [24]. This benefit is important, since at pressures > 10 atm where the transitions are strongly blended, it is generally difficult to find a near zero-absorption region near the targeted transition for the measurement of a direct absorption (DA) baseline. In addition, laboratory bench measurements indicate that WMS measurements have smaller uncertainties than DA for non-Lorentzian lineshape effects at high pressures, such as the breakdown of the impact approximation [25] and line-mixing [26]. All these advantages demonstrate the feasibility of using 1f-normalized TDL-WMS for high-pressure gas sensing.

Among all harmonics of WMS, the second harmonic of the transmitted laser intensity, called WMS-2f, is the most commonly employed for gas sensing, owing to its relatively large magnitude. However, at high pressures >10 atm, when an optimal modulation depth for WMS-2f is obtained with a typical injection-current-tuned diode laser, the 2f background signal can be significant. The 2f background signals have two primary contributions: (1) the simultaneous non-linear intensity modulation of the injection-current-tuned laser and (2) optics in the line-of-sight (LOS) whose transmissions are wavelength dependent [20]. Since the background 2f signal must be vector subtracted from the measured 2f signal to infer the absolute molecular absorption, an inaccurate measurement of the background signal can generate dramatic errors in the gas mole fraction determined from the measurements. In a practical absorption measurement, with interference patterns generated from reflections between parallel surfaces (etalons), the background signal can also drift over time due to thermal expansion or other movements of the optical windows [27, 28], temperature change of the laser components [29], and slight variation in the optical alignment [30]. Even under quiet laboratory conditions in an evacuated gas cell with the temperature stabilized at 296 ± 2 K, when a relatively large modulation depth of 1.52 cm−1 (Nanoplus TDL, near 2,325 nm) was used, a 33 % change in the 2f background was observed over ~ 15 h as shown in Fig. 1. As simultaneous monitoring of the WMS background is not possible for practical TDL absorption sensor applications, schemes are desired to reduce the impact of WMS background variations.

Fig. 1
figure 1

Measured 2f background magnitude versus time, normalized by the laser intensity (Measured with a 2.3 μm Nanoplus TDL, traveling through an evacuated, 17.3 cm cell with wedged windows. Modulation depth = 1.52 cm−1, modulation frequency = 1 kHz. No external intervention presented during measurement)

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The advantages of the larger signal-to-background ratio (SBR) of WMS higher harmonics (e.g., 4f or 6f) have been discussed previously [19–21, 28]. Kluczynski et al. [21] noted that the 4f background signal was about two orders-of-magnitude smaller than that for 2f when the background signals were predominantly caused by etalons in the LOS. Gustafson et al. [28] used WMS in graphite furnaces, where they found more than an order-of-magnitude improvement in the detection limit for Rb absorption using 4f and 6f instead of 2f. In this paper, the extension to TDL-WMS with higher harmonics for high-pressure gas sensing with a large modulation depth is presented. This discussion includes the first use of WMS-1f to normalize the higher harmonic signals and the contributions to WMS signals from the characteristics of injection current tuning of the laser as well as reflective interference induced in the transmitted laser intensity by other optical components. More specifically, this paper seeks to make the following five contributions:

  1. 1.

    Extend the 1f-normalized WMS-2f model for injection-current-tuned TDLs to a generalized 1f-normalized WMS-nf model for the large modulation depths needed for high-pressure gas sensing. The presentation follows the general theory of Kluczynski and Axner. The result accounts for the performance characteristics of the complete optical system including the injection-current-tuned TDL and the LOS optics.

  2. 2.

    Validate the TDL-WMS-nf model by measurements of the 1f-normalized WMS-nf lineshapes of the R(11) transition in the 1st overtone band of CO molecule near 2.3 μm at 4 different pressures ranging from 5 to 20 atm with a range of CO mole fractions from 0.21 to 2.8 %, all at room temperature.

  3. 3.

    Develop a sensor for CO mole fraction measurements in high-pressure environment using 1f-normalized WMS-nf.

  4. 4.

    Investigate background signals for WMS-nf and demonstrate the advantages of WMS-nf (n > 2) over WMS-2f at elevated pressures.

  5. 5.

    Discuss the potential for WMS-nf (n > 2) to reduce the interference absorption from neighboring transitions.

2 Generalized WMS-nf model considering non-ideal diode laser performance

2.1 WMS-nf model

The Li et al. [22] model for WMS-2f including the performance of injection-current-tuned diode lasers is extended to WMS-nf including higher harmonic terms that were ignored in most past WMS-2f studies.

Typically, for a WMS absorption measurement, the injection current of a TDL is modulated by a sinusoidal function at frequency f, resulting in simultaneous modulation of the laser wavelength (or frequency) and intensity,

$$ \nu (t) = \bar{\nu } + a\cos (2\pi ft + \psi ), $$
(1)
$$ I_{0} (t) = \bar{I}_{0} \left[ {1 + \sum\limits_{m = 1}^{\infty } {i_{m} \cos (m \cdot 2\pi ft + \psi_{m} )} } \right], , $$
(2)

where \( \bar{\nu } \) (cm−1) is the center wavelength (or frequency) of the laser without modulation, a (cm−1) is the modulation depth, \( \psi_{{}} \)is the initial phase of the wavelength modulation, \( \bar{I}_{0} \) is the incident laser intensity without modulation, \( i_{m} \) is the mth Fourier coefficient of the laser intensity, and \( \psi_{m} \) is the initial phase of the mth order intensity modulation.

Most previous WMS-2f studies truncate the series in Eq. (2) to include only the first-order, linear intensity modulation term \( i_{1} \), and the second-order, non-linear intensity modulation amplitude \( i_{2} \); all other higher order Fourier components are ignored. This assumption is accurate when the laser intensity response to the injection current modulation has small nonlinearities and transmissions of the optical components in the system, which are not significantly wavelength dependent over the wavelength region of concern. These assumptions are not always valid, for example an external cavity TDL produces a significant interference (etalon) pattern in the output intensity, resulting in non-negligible \( i_{3} \) and \( i_{4} \) terms even with small modulation depths (<0.1 cm−1) [31]. Therefore, the model developed here includes all the Fourier components to represent a more general case. The highest order needed will depend on the specific laser architecture and modulation depth.

In the model, the choice of time-zero is arbitrary. To be consistent with the theory derived by Kluczynski et al. [19] and Li et al. [22], the time-zero is defined as \( \psi /2\pi f \), which eliminates the initial phase of the wavelength modulation; Eqs. (1) and (2) are then rewritten:

$$ \nu (t) = \bar{\nu } + a\cos (2\pi ft), $$
(3)
$$ I_{0} (t) = \bar{I}_{0} \left[ {1 + \sum\limits_{m = 1}^{\infty } {i_{m} \cos (m \times 2\pi ft + \psi_{m} - m \times \psi )} } \right]. $$
(4)

When the laser beam travels through the absorbing medium, the wavelength-dependent transmission \( \tau \) is described by the Beer-Lambert law:

$$ \tau = \left( {\frac{{I_{t} }}{{I_{0} }}} \right) = \exp \left( { - \sum\limits_{j} {S_{j} \times \phi_{j} \times P \times x_{i} \times L} } \right), $$
(5)

where \( S_{j} \) and \( \phi_{j} \) are the line strength and line shape function of transition j, P is the total pressure of the gas, x i is the mole fraction of absorber i and L is the path length. Since the laser wavelength is modulated at frequency f, the transmission also becomes a periodic function and can be expanded into Fourier series:

$$ \tau (t) = \sum\limits_{k = 0}^{\infty } {H_{k} \cos (k \times 2\pi ft)} , $$
(6)

where H k is the kth order Fourier coefficient defined as:

$$ H_{k} = \frac{1}{{(1 + \delta_{k0} )\pi }}\int\limits_{ - \pi }^{\pi } {\tau (\bar{\nu } + a\cos \theta )\cos k\theta d\theta } . $$
(7)

Note that no assumption has been made about the optical depth and this model is valid for arbitrary absorbance.

The transmitted intensity with the presence of absorbers becomes:

$$ \begin{aligned} I_{t} (t) = & I_{0} (t) \times \tau (\bar{\nu } + a\cos (2\pi ft)) \\ = & \bar{I}_{0} \left[ {1 + \sum\limits_{m = 1}^{\infty } {i_{m} \cos (m \times 2\pi ft + \psi_{m} - m \cdot \psi )} } \right] \times \left[ {\sum\limits_{k = 0}^{\infty } {H_{k} \cos (k \times 2\pi ft)} } \right]. \\ \end{aligned} $$
(8)

The harmonics at the modulation frequency of the transmitted laser intensity are extracted by lock-in amplifiers with a bandwidth determined by the low-pass filter. The X-component [detector signal × cos(n × 2πft)] and Y-component [detector signal × sin(n × 2πft)] of the nf signal are written as:

$$ X_{nf} = \frac{1}{2}G\bar{I}_{0} \left[ {H_{n} + \frac{1}{2}\sum\limits_{k = 1}^{\infty } {(H_{n + k} + (1 + \delta_{nk} )H_{{\left| {n - k} \right|}} )i_{k} \cos (\psi_{k} - k \times \psi )} } \right], $$
(9)
$$ Y_{nf} = \frac{1}{2}G\bar{I}_{0} \left[ {\frac{1}{2}\sum\limits_{k = 1}^{\infty } {(H_{n + k} - (1 + \delta_{nk} )H_{{\left| {n - k} \right|}} )i_{k} \sin (\psi_{k} - k \times \psi )} } \right], $$
(10)

where G is the electro-optical gain of the detector. For cases where non-linear intensity response is insignificant (laser without large etalons in the intensity tuning), the following approximations can be used:

$$ X_{nf} \approx \frac{1}{2}G\bar{I}_{0} \left[ {H_{n} + \frac{1}{2}(H_{n + 1} + (1 + \delta_{n1} )H_{{\left| {n - 1} \right|}} )i_{1} \cos (\psi_{1} - \psi ) + \frac{1}{2}(H_{n + 2} + (1 + \delta_{n2} )H_{{\left| {n - 2} \right|}} )i_{2} \cos (\psi_{2} - 2 \times \psi )} \right] $$
(11)
$$ Y_{nf} \approx \frac{1}{2}G\bar{I}_{0} \left[ {\frac{1}{2}(H_{n + 1} - (1 + \delta_{n1} )H_{{\left| {n - 1} \right|}} )i_{1} \sin (\psi_{1} - \psi ) + \frac{1}{2}(H_{n + 2} - (1 + \delta_{n2} )H_{{\left| {n - 2} \right|}} )i_{2} \sin (\psi_{2} - 2 \times \psi )} \right]. $$
(12)

The WMS-nf signal magnitude can be written:

$$ R_{nf} = \sqrt {X_{nf}^{2} + Y_{nf}^{2} } . $$
(13)

When no absorbers are present, \( H_{k} = \delta_{0k} \), the absorption-free WMS-nf background signals in the system are given as:

$$ X_{nf}^{0} = \frac{1}{2}G\bar{I}_{0} i_{n} \cos (\psi_{n} - n \times \psi ), $$
(14)
$$ Y_{nf}^{0} = - \frac{1}{2}G\bar{I}_{0} i_{n} \sin (\psi_{n} - n \times \psi ), $$
(15)
$$ R_{nf}^{0} = \sqrt {(X_{nf}^{0} )^{2} + (Y_{nf}^{0} )^{2} } = \frac{1}{2}G\bar{I}_{0} i_{n} , $$
(16)
$$ R_{0f}^{0} = G\bar{I}_{0} . $$
(17)

After vector-subtracting the background, the WMS-nf signal becomes:

$$ S_{nf} = [(X_{nf} - X_{nf}^{0} )^{2} + (Y_{nf} - Y_{nf}^{0} )^{2} ]. $$
(18)

Note that all harmonics of the WMS-nf signal are proportional to laser intensity and electro-optical gain, thus the WMS-1f signal can be used to normalize all harmonics of the WMS signals, similar to the 1f-normalized WMS-2f scheme developed previously [24]:

$$ S_{{nf\;{\text{normalized}}}} = \sqrt {\left[ {\left( {\frac{{X_{nf} }}{{R_{1f} }}} \right) - \left( {\frac{{X_{nf}^{0} }}{{R_{{_{1f} }}^{0} }}} \right)} \right]^{2} + \left[ {\left( {\frac{{Y_{nf} }}{{R_{1f} }}} \right) - \left( {\frac{{Y_{{_{nf} }}^{0} }}{{R_{{_{1f} }}^{0} }}} \right)} \right]^{2} } . $$
(19)

The normalization accounts for the laser intensity and electro-optical gain of the detector, making the normalized measurement independent of the laser intensity impinging on the detector and thus suitable for environments involving large non-absorption transmission losses or variations.

2.2 Laser characterization

To obtain an accurate WMS simulation, the absorption is described by the spectroscopic parameters including line strength and collisional broadening coefficients, and the parameters that describe the injection current tuning behavior of the specific TDL. These parameters include a (cm−1), the modulation depth; and ψ, the initial phase of the wavelength modulation. The method to evaluate the parameters that quantify the injection current tuning characteristics of the TDLs has been discussed previously, and the reader is referred to the literature [22].

2.3 Optical system characterization

The transmission can vary as a function of laser wavelength due to the optical components along the LOS. Most often such variation is due to interference from reflection by parallel optical surfaces, commonly called “etalon” effects. For example, many IR-detectors have flat windows in front of the active area for protections, which can result in reflective interference when the laser is tuned. As this interference is wavelength dependent, it can produce background signals at the harmonics of the laser modulation frequency. Even larger interference can be produced by other optical components such as measurement volume windows, optical filters, or optical amplifiers. As a result, it will be more accurate to evaluate the intensity modulation characteristics (\( i_{m} \) and \( \psi_{m} \)) of the entire optical system rather than simply the laser.

\( i_{m} \) and \( \psi_{m} \) are extracted from the WMS background signals for a measurement with no absorbers in the LOS using Eqs. (14) and (15). During the measurements, time-zero is determined by the data acquisition and laser modulation triggering, and it is practical to use Eqs. (1) and (2) (instead of Eqs. (3) and (4)) to define the laser intensity modulation; Eqs. (14) and (15) then become:

$$ X_{{_{nf} }}^{0} = \frac{1}{2}G\bar{I}_{0} i_{n} \cos \psi_{n} , $$
(20)
$$ Y_{{_{nf} }}^{0} = - \frac{1}{2}G\bar{I}_{0} i_{n} \sin \psi_{n} . $$
(21)

Then \( \psi_{m} \) can be obtained as:

$$ \psi_{m} = \left\{\begin{array}{ll} \tan^{ - 1} ( - Y_{mf}^{0} /X_{mf}^{0} ) &{\text{for}}\quad Y_{mf}^{0} \le 0 \hfill \\ \tan^{ - 1} ( - Y_{mf}^{0} /X_{mf}^{0} ) + \pi & {\text{for}}\quad Y_{mf}^{0} > 0 \hfill \\ \end{array} \right. $$
(22)

and \( i_{m} \) can be obtained with Eqs. (16) and (17) as:

$$ i_{m} = 2R_{mf}^{0} /R_{0f}^{0} . $$
(23)

This illustrates that the parameters (\( i_{m} \) and \( \psi_{m} \)) of the intensity modulation used for WMS lineshape simulations can be obtained from the WMS zero-absorption background signals that combine the performance of the LOS optics and the TDL. For practical measurements, when some external interference (e.g., etalons) cannot be easily eliminated, or if the system includes optical components that are spectrally sensitive, characterizing the overall system will be more relevant than characterizing only the laser.

3 WMS-nf model validation by high-pressure CO WMS spectra measurements

3.1 WMS-nf model validation

The WMS-nf model presented was validated by measurements of CO WMS lineshapes at different pressures and absorber mole fractions. The probed CO transition was R(11) of the 1st overtone band near 4,300.7 cm−1. The measurements included gas pressures ranging from 5 to 20 atm and CO mole fractions ranging from 0.21 to 2.8 % diluted in N2, all at room temperature.

The average pressure investigated was 12.5 atm. At this pressure, the optimal modulation depth (maximum signal magnitude) for WMS-2f is 1.56 cm−1 (modulation index, m = a/Δν ~ 2.2, where the transition half-width at half-maximum (HWHM) Δν is 0.71 cm−1 at 12.5 atm). The optimal modulation depths for WMS-nf signals (n > 2) are larger than WMS-2f [28]; for this case, 2.93 cm−1 for 4f and 4.33 cm−1 for 6f. However, the maximum modulation depths achievable by the available l TDLs are limited by their injection current limits. Although the modulation frequency can be decreased to obtain a larger modulation depth, this will reduce the time resolution of the measurement, and make WMS less effective in rejecting low-frequency noise. For this study, a modulation depth of 1.52 cm−1 and a modulation frequency of 1 kHz were selected. The selected modulation depth is very close to the optimal modulation depth for WMS-2f, but smaller than the optimal for WMS-nf signals (n > 2), especially when the pressures are larger than 10 atm.

The WMS lineshapes were measured at room temperature (296 K) in a 100.5-cm-long cell (Fig. 2), equipped with wedged sapphire windows. A DFB laser (center wavelength 4,300 cm−1 from Nanoplus) was used with output coupled into a single-mode fiber (corning, SMF-28). The collimated-laser beam was directed through the cell and focused (focal length 5 cm) onto a New Focus extended-λ InGaAs photo-receiver of 700 kHz bandwidth with 1 mm2 active area. The detector voltage signal was then sampled using a 12-bit National Instrument PCI-6110 DAQ card, at a sampling rate of 2.5 MHz. Gas mixtures (CO in N2) were prepared in a mixing tank and allowed to diffusively mix overnight before the measurements.

Fig. 2
figure 2

Experimental setup for high-pressure CO gas sensing

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The injection current of the laser was modulated with a 1 kHz sine wave. The center frequency of the modulated laser was scanned by stepping the laser temperature in 0.1 °C steps resulting in 0.04 cm−1 laser frequency resolution. After each step, a 15 s delay was given for the laser temperature to stabilize. The WMS lineshape was collected for wavelength tuning over the range from 4,298 to 4,304 cm−1.

The background signals were measured before introducing the CO absorber into the cell to obtain \( i_{m} \) and \( \psi_{m} \)as described in Sect. 2.3. The WMS-nf signals simulations used line strength and broadening coefficients from Chao et al. [10], and the pressure-induced line center-shift coefficients from HITEMP 2010 [32].

Figures 3 and 4 compare the measured background subtracted WMS-nf spectra with simulations for the R(11) transition with 1.59 % CO in N2 at pressures from 5 to 20 atm. Although at these pressures, a closed form Lorentzian lineshape could have been used in the simulations without introducing significant errors, the more general Voigt lineshape function was used to facilitate validation of the measurements over a wider range of pressure (both high and low-pressure applications). At all pressures, the measured WMS-2f, 3f, and 4f lineshapes agreed very well with the simulations based on the model described in Sect. 2.1, with an average deviation of less than 2 %. An overall measurement uncertainty was estimated to be about 2.5 %, including 0.5 % in the gas mixture mole fraction, 1 % in the spectroscopic parameters, 0.5 % in laser characterization parameters, and 0.5 % in pressure. Thus, the observed deviation between the measurements and simulations for WMS-2f, 3f and 4f lineshapes were all less than the system uncertainty. However, for WMS-5f and 6f lineshapes, at pressures above 10 atm, obvious discrepancies between the measurements and simulations were observed. We speculate that these discrepancies were the results of three factors: (1) The modulation depth was much smaller than optimal for WMS-5f and WMS-6f for P > 10 atm; (2) Potential contributions from non-linearity in the wavelength (or frequency) modulation, which is not included in the present WMS model. Since these higher harmonics were under-modulated for measurements above 10 atm, their signals can be more sensitive to such non-linearities; (3) The absolute magnitudes of the higher harmonic signals were small, exacerbated by the use of modulation depths below the optimal values for higher harmonics at very high-pressure conditions (P > 15 atm). Similar results were observed in measurements with other CO mole fractions ranging from 0.21 to 2.8 %.

Fig. 3
figure 3

Measured (square, red) and simulated (solid line, black) 1f-normalized WMS-2f, 3f, and 4f lineshape signals at different pressures. Gas mixture: 1.59 % CO in N2; T = 296 K; optical pathlength L = 100.5 cm

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Fig. 4
figure 4

Measured (square, red) and simulated (solid line, black) 1f-normalized WMS-5f and 6f lineshape signals at different pressures. Gas mixture: 1.59 % CO in N2; T = 296 K; optical pathlength L = 100.5 cm

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3.2 High-pressure CO sensor design based on WMS-nf detection

After the WMS-nf model was validated by measuring wavelength-scanned WMS-nf lineshapes, a sensor adopting fixed-wavelength WMS was evaluated for practical sensing of gas mole fraction at elevated pressures. This fixed-wavelength scheme can have much faster time resolution than the slow wavelength scanning using temperature tuning. A modulation frequency of 1 kHz and a lowpass filter with bandwidth of 100 Hz was used to give 20 ms temporal resolution for CO mole fraction measurements. To validate the sensor, mixtures with CO mole fractions ranging from 0.21 to 2.8 % were used and the experiment conditions were same as the previous section. The measured CO mole fractions were determined from the peak values of the 1f-normalized WMS-nf spectra as shown in Fig. 5 for n = 2–6. The results were compared with the calibrated CO mixture mole fractions. The averaged deviations between measurements of WMS-2f, 3f, 4f and signals simulated for known gas mixtures were all less than the measurement uncertainty analyzed in section A. However, the measured WMS-5f and 6f signals had larger deviations due to the limited modulation depth.

Fig. 5
figure 5

Measured CO mole fractions by 1f-normalized WMS-nf technique and comparison with calibrated CO mole fractions (dashed line), bath gas: N2; T = 296 K; optical pathlength L = 100.5 cm

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Similar to the WMS lineshape results in Sect. 3.1, the fixed-frequency WMS results show that the lower harmonics (for n = 2–4) have the advantage of larger signal magnitude in gas mole fraction measurements at high pressures. This explains why the 2f signal is favored for most applications, especially trace species measurements in a simple optical system without parallel surfaces (windows, mirrors, etc.), where etalon interferences are small compared with those often present in practical high-pressure applications (e.g., coal gasifiers, gas turbine or internal combustion engines). The measurement time was only a few minutes and the WMS background signals were measured just before the absorption measurements. The accuracy of the CO mole fraction measurements indicates that the background drifts were sufficiently small for this short time period and that no significant errors were incurred. However, for applications where the measurement period is long (e.g., more than 1 day), or when large variations in measurement conditions are present, or when background measurements are not feasible, significant measurement error due to the WMS background can occur, which would potentially increase the measurement uncertainty and affect the accuracy of the absorption measurements. For these cases when the drift of the WMS background cannot be regularly measured during an absorption measurement, WMS higher harmonic signals can have the advantage of smaller background drifts, which will be discussed in detail in the following section.

4 Influence of the background drift to WMS-nf measurements

At high pressures, the magnitude of the absorption-induced 2f signal (background subtracted) can be comparable to its absorption-free background signal. When the pressure is higher than 10 atm and a relatively large modulation depth is used, such as one close to the optimal modulation depth, the 2f signal can even be smaller than its background signal for small absorbance (<0.1) cases. Figure 6 shows the measured 1f-normalized WMS-2f and 4f signals (background subtracted) as well as their background signals for 0.21 % CO in N2 at 10 atm in a 100.5-cm-long cell (peak absorbance ~0.085). Note that at these experiment conditions, the 2f signal peak is smaller than its background signal at the same wavelength, while the 4f signal peak is much larger than its corresponding background signal. To quantify the relative magnitudes of the background signal, we here define a signal-to-background ratio (SBR) as the absorption-induced signal to the absorption-free background signal at the same wavelength. Therefore, in the above measurement, the 2f and 4f signal peaks have SBRs of 0.7 and 35, respectively.

Fig. 6
figure 6

Measured 1f-normalized WMS-2f and 4f signals as well as their background signals. (0.21 % CO in N2, P = 10 atm, T = 296 K, L = 100.5 cm; a = 1.52 cm−1, f = 1 kHz)

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Such measurements were carried out for various pressures, and the 1f-normalized WMS-nf peak-to-background ratios are plotted versus pressure in Fig. 7. Although the absolute WMS-2f signal was larger than all other harmonic signals, the ratio of the 2f signal to its background signal was at least one order-of-magnitude smaller than for the other harmonics. The result also indicates that for P ≥ 10 atm, the absorption-induced 2f signal peak was smaller than its background signal for the condition studied here. The small SBRs will not result in significant errors in mole fraction measurements as long as the background signal is well determined and does not change significantly during the measurement period as for measurements shown in Fig. 5. However, if the background signal drifts over time due to changes in the beam path, e.g., signals associated with optical interferences and reflections, this may lead to significant error in the mole fraction determination if the SBR is small. WMS higher harmonic signals have much larger SBRs than WMS-2f and potentially can thus have less relative error in absorption measurements from the background drift. In addition, since the peak signal of the higher harmonics (n > 2) is much larger than their background signal, knowledge or accurate evaluation of the background signals may not be required. This advantage is useful for practical sensing applications such as that in a continuously running coal-gasifier where opportunities to monitor for zero-absorption background signal are difficult to achieve.

Fig. 7
figure 7

Measured ratio of 1f-normalized WMS-nf signal to its 1f-normalized background signal (a = 1.52 cm−1, f = 1 kHz, T = 296 K, L = 100.5 cm)

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To quantify the effect of the background drift, the background signal of the optical system used in the previous measurement was monitored overnight and Fig. 8 shows the ratio of the WMS-nf signal peak to the change in background, the latter defined as the standard deviation of the background signal over the measurement period (~10 h). The result is consistent with the SBR results in Fig. 7. It shows that the background drift has less effects on higher harmonics measurements, which is consistent with Kluczynski’s study in analyzing the background signals of different WMS harmonics with a thin etalon [29, 33]. In this study, the potential source of thin etalon interference includes the laser cavity and the thin window right of the detector.

Fig. 8
figure 8

Measured ratios of 1f-normalized WMS-nf signal to the drift magnitude of its 1f-normalized background signal. (a = 1.52 cm−1, f = 1 kHz, T = 296 K, L = 100.5 cm)

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For mixtures where the change of the absorber concentration does not significantly change the transition broadening width and the peak absorbance is small (<0.1), the 1f-normalized WMS-nf signal peak is linearly proportional to the absorber concentration. This proportionality lets the detection limit be estimated from the measured concentration and the percentage drift of the background signal with respect to the 1f-normalized WMS-nf peak magnitudes. Figure 9 shows the detection limits for 2,100 ppm CO concentration measurements estimated from the drift in the TDL-WMS background. The detection limits are much smaller for WMS-nf signals with n > 2 than for WMS-2f. For 20 atm pressure, the estimated detection limit using 2f was 62 ppm, corresponding to about 3 % for the measurement of 0.21 % (2,100 ppm) CO, which was larger than the measurement uncertainty analyzed in Sect. 3 for short term use of the sensor. This detection limit will be increased when the pressure is higher. If a 50 ppm detection limit in mole fraction measurements is required, the highest pressure condition measurable by WMS-2f will be less than 16 atm, whereas for WMS-nf with n > 2, pressures up to 30 atm will be feasible. These results are representative for laboratory conditions, whereas larger background drifts and higher detection limits are anticipated for practical field conditions. Work is underway to study the benefits of using higher harmonics for high-pressure gas sensing in harsh environments where temperatures of the cell or windows are not stable.

Fig. 9
figure 9

Background drift induced WMS-nf detection limits at different pressures (CO in N2, T = 296 K, L = 100.5 cm; a = 1.52 cm−1, f = 1 kHz)

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5 Advantage of WMS-nf (n > 2) in reducing the interference from neighbors

One advantage of WMS-2f over DA is the higher sensitivity to the absorption lineshape curvature, giving larger signals near the transition line center than in the wings. This is attractive for elevated pressure measurements where the transitions are blended by collisional broadening. For WMS-2f measurements, the signal peak usually appears close to the line center, where sensitivity is the highest; therefore, interference from the wings of neighbor transitions will be minimized. It follows that this advantage can be even more pronounced for higher harmonics measurements, e.g., WMS-4f. The WMS-2f signal is dominated by the Fourier components from \( H_{0} \) to\( H_{4} \), whereas WMS-nf (n > 2) is dominated by higher Fourier components \( H_{k} \)as shown in Eqs. (11) and (12). For example, WMS-4f signal can be written as

$$ X_{4f} \approx \frac{1}{2}G\bar{I}_{0} \left[ {H_{4} + \frac{1}{2}(H_{5} + H_{3} )i_{1} \cos (\psi_{1} - \psi ) + \frac{1}{2}(H_{6} + H_{2} )i_{2} \cos (\psi_{2} - 2 \cdot \psi )} \right], $$
(24)
$$ Y_{4f} \approx \frac{1}{2}G\bar{I}_{0} \left[ {\frac{1}{2}(H_{5} - H_{3} )i_{1} \sin (\psi_{1} - \psi ) + \frac{1}{2}(H_{6} - H_{2} )i_{2} \sin (\psi_{2} - 2 \cdot \psi )} \right]. $$
(25)

Figure 10 shows the simulated\( H_{k} \)‘s for an individual, isolated CO transition near 4,300.7 cm−1. The higher the order of the Fourier component, the faster its magnitude decays to zero as the wavelength deviates from the line center. As WMS higher harmonics include contributions mostly from higher order \( H_{k} \)‘s, transitions detected with WMS higher harmonics will have less interference from neighboring transitions.

Fig. 10
figure 10

Simulated H k for targeted transition near 4,300.7 cm−1 at 20 atm (a = 1.52 cm−1, f = 1 kHz, 0.21 % CO in N2, T = 296, K = 100.5 cm)

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When the 1f-normalized WMS-nf detection method is used to make the absorption measurement, this advantage of WMS higher harmonics will be degraded to a certain extent, since the 1f signal used for normalization includes contributions from mainly low order \( H_{k} \). This is especially the case for large absorbance cases (> 0.5) where the 1f signal will be more dependent on the absorption. However, in the optically thin case (absorbance < 0.1), where the 1f signal is mainly dominated by the variation of laser power from the injection current tuning, this advantage will remain significant.

6 Conclusion

A generalized model for 1f-normalized WMS-nf detection with an injection-current-tuned diode laser was presented that accounts for performance by the laser and etalon interference by the optical components in the LOS. This model was validated using measurements of the CO transition of R (11) in the 1st overtone band near 2.3 μm at room temperature for a range of CO mole fractions (0.21–2.8 %) and pressures (5–20 atm). For high-pressure gas sensing, wavelength modulation spectroscopy with higher order harmonic detection (WMS-nf, n > 2) was found to have less influence from the WMS background signals when the selected modulation depth was near the optimal modulation depth for the WMS-2f signal. Similar levels of accuracy in CO mole fraction measurements were observed by 1f-normalized WMS-2f, 3f and 4f techniques, but WMS-3f and WMS-4f detection showed better accuracy than WMS-2f when uncertainty in the WMS background signal was significant (i.e., from the long term drift of the background signals). The large signal-to-background ratio of WMS higher harmonics (WMS-nf, n > 2) potentially offers the advantage that knowledge or accurate evaluation of the background signals may not be required. This advantage is useful for practical sensing applications such as that in a continuously running coal-gasifier where opportunities to monitor the zero-absorption background signal are difficult to achieve. In optically thin cases, absorption measurements of the targeted transition had less interference from neighboring transitions by using detection of 1f-normalized WMS higher harmonics. However, WMS-nf (n > 2) detection required a larger modulation depth than WMS-2f for the optimal signal-to-noise ratio, which may be achieved at a lower modulation frequency, but can in turn result in decreasing the measurement time resolution. In addition, as the absolute amplitudes of the higher harmonic signals are smaller than those for 2f, using higher harmonics may lower the signal-to-noise ratio.