1 Introduction

Tunable diode laser absorption spectroscopy (TDLS) is an established technique for gas analysis in many fields including environmental and chemical process monitoring, remote detection and combustion analysis [1–3]. Wavelength modulation spectroscopy (WMS) and direct absorption spectroscopy (DAS) are two popular implementations of TDLS. Both methods are based on tuning the emission frequency of a diode laser to let it scan over absorption lines of the gases of interest. WMS uses harmonic lock-in detection with rapid sinusoidal laser modulation and DAS takes the approach of directly detecting the small light absorption on the background of the transmitted laser light. In general, in WMS the background reduction and reduced impact of 1/f intensity noise and thus higher sensitivity are viewed as advantages [1, 2]. DAS is considered to have its strengths in being calibration-free [3] or allowing the straight-forward measurement of line strengths [4]. There are approaches that try to combine advantages of both techniques by recovering the absolute absorption line shape from the signals of a WMS system [5], but this technique is intended for measuring stronger absorptions in process-control with concentration levels in the high percent range and not for trace gas detection.

In DAS and WMS spectrometers, the detector signal is usually digitized and then digitally processed [6, 7]. In this step, one may expect the DAS technique to require a much higher resolution of the analog-to-digital converter (ADC) than WMS, because of the large sloping background masking the tiny absorption signature. WMS, in contrast, allows AC-coupling of the detector signal which eliminates the need for full-scale background digitization. Then the ADC input range can be used more efficiently for the WMS signal that actually carries the absorption information. In this context, the question arises, what the necessary resolution for both techniques is and if quantization noise is likely to be a limiting factor for concentration determination. Recently, this question has been addressed in simulations of DAS and WMS spectrometers [8], but no measurements have been presented. Now, we have implemented a spectrometer that can switch between the two techniques in order to directly compare DAS and WMS performance. While there have been experiments that independently employ both techniques in the same setup [9, 10], they did not aim at a comparison of both techniques.

2 Wavelength modulation spectroscopy and direct absorption measurement principle

2.1 Wavelength modulation spectroscopy

The WMS technique is well described in the literature, for instance in [2, 8, 12]. Figure 1 briefly summarizes the operation principle. A tunable laser diode is modulated by a periodic ramp current to sweep the optical emission frequency ν(t) across the absorption line of the gas under investigation. The ramp frequency f r,wms is usually in the range of several Hz. A harmonic modulation at the frequency f harm of several kHz is superimposed on the ramp modulation (f harm≫f r,wms). Thus, ν(t) oscillates rapidly with the harmonic modulation frequency f harm around the quasistatic instantaneous emission frequency of the ramp scan. Interaction of this modulation of ν(t) with the nonlinear shape of a gas absorption line creates modulation of the transmitted optical power at harmonics of f harm. The transmitted light is detected by a photodiode and an amplifier that outputs the detector voltage U PD(t). This detector voltage is then demodulated by the lock-in amplifier that delivers an output voltage U nf(t) proportional to the respective n th harmonic content. Demodulation at the modulation frequency itself (1f-WMS) or at twice the modulation frequency (2f-WMS) is most common, and the lock-in output sketched in Fig. 1 shows the typical lineshape for second harmonic detection.

Fig. 1
figure 1

General WMS setup and components. The laser diode (DFB-LD) current is ramp-tuned with frequency f r,wms and sinusoidally modulated (f harm). After gas absorption the photodiode (PD) current is amplified by a transimpedance amplifier (TIA) and demodulated by the lock-in amplifier that outputs the 2f measurement signal

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Although the harmonic content in the detector signal decreases with increasing order of the harmonics, higher order detection is helpful in some cases, provided that the absorption is sufficiently strong. Since the lineshape becomes more distinct in higher harmonics, they can be useful in lineshape analysis [11] or to suppress interferences caused by etalons [12], as the interference strength decreases more rapidly as a function of the harmonic than the gas absorption signal itself. Very recent investigations [13] also indicate that a simultaneous evaluation of several harmonics may slightly improve system performance compared to single harmonic evaluations. The scope of this work, however, is restricted to evaluation of the 1f and the 2f WMS-signals, as they are very widely used.

The main advantage of the 2f-WMS scheme (and all higher harmonic schemes) is an ideally background-free measurement with zero voltage at the lock-in output for zero absorption. In practice, the background is often not completely eliminated due to nonlinearities in the laser current-power characteristics that lead to a residual amplitude modulation (RAM) [14].

By second harmonic detection, the detection frequency is shifted to 2f harm in the high kHz range, where the 1/f-noise of the laser intensity is much lower than in the baseband. The lock-in amplifier acts as a narrow-band filter around the detection frequency with the detection bandwidth being determined by the lock-in time constant τ. Higher time constants result in better noise suppression but slower response time. Usually, the time constant is chosen small enough to not distort the 2f-waveform and large enough to achieve reasonable noise suppression. For the detection at the modulation harmonics, the TIA may be AC-coupled to make better use of the dynamic range of the lock-in amplifier. To become independent of possible fluctuations of the average detector power, the 2f-signal may then be normalized to the 1f-component, instead of normalization to DC voltage [6].

For the evaluation of the gas concentration, the 2f lineshape is usually compared to a reference spectrum that has been measured or calculated for a known concentration at the same ambient conditions. To take background signals and offsets into account, which are due to RAM, a linear regression model [15] can be used for the concentration evaluation from the reference and measurement spectra.

2.2 Direct absorption

Compared to the WMS technique, DAS gives up the advantage of background free measurements in favor of the simpler setup shown in Fig. 2. No lock-in amplifier is required and the laser diode is tuned by a ramp current only. The small absorption signature of the gas must be detected on the large background of laser intensity that is caused by the ramp modulation. To extract the absorption information, the DC-coupled detector signal U PD(t) is digitized by an analog-to-digital converter (ADC) and then processed by a computer. No lock-in scheme is used to escape 1/f intensity noise but several scans are averaged instead to reduce noise. As described previously [8], this averaging process acts as a filter with a comb-like transmission spectrum with passbands around harmonics of f r,das. Increasing the number of averaged scans corresponds to narrowing these passbands. The ramp frequency f r,das is typically in the range of several 10 Hz up to the low kHz range, in order to be able to average many scans in a limited measurement time. After scan averaging, a baseline fit is applied to the detected signal to remove the sloping baseline. From the measured power transmission T(t), the time-dependent absorbance is given as α(t)=−ln(T(t)). The recovered absorbance spectrum then has to be rescaled to a wavenumber-dependent version α(ν). Here, the generally nonlinear temporal tuning behavior ∂ν/∂t of the laser diode, which must be characterized in advance [3, 16], needs to be considered. Finally, a Voigt line fit can be applied to α(ν) to evaluate one (or more) of the parameters gas concentration, temperature, or pressure. A more comprehensive description of DAS is given in [1, 3, 17].

Fig. 2
figure 2

Setup and components in a spectrometer using the direct absorption technique. The laser diode (DFB-LD) is tuned by ramp modulation only (frequency f r,das) and the transmitted power is detected by the photodiode (PD) and transimpedance amplifier (TIA). The detector voltage U PD(t) including DC is digitized by an analog-to-digital converter (ADC). Further signal processing and baseline removal yields the absorbance spectrum

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3 Experimental detail

3.1 Setup of the WMS/DAS-switchable spectrometer

Our goal was to compare both the WMS and DAS scheme without variation in hardware components and the spectrometer setup. For that, we used the spectrometer setup shown in Fig. 3, which has been used for both techniques.

Fig. 3
figure 3

Measurement setup used for both WMS and DAS. The respective modulation voltage U mod(t) for the laser diode controller (LDC) is provided by the analog output (ao) of the data acquisition card (DAQ). The amplifier (TIA) is AC-coupled for WMS and DC-coupled for DAS using the digital output (do) of the DAQ. The detector voltage U PD(t) is digitized by the analog input (ai) of the DAQ and stored on hard disk for further signal processing

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The fiber coupled DFB-laser module (Mitsubishi FU-68PDF) is driven by a Thorlabs ITC 502 diode laser controller (LDC). The required modulation voltage U mod(t) for the LDC modulation input is provided by the analog output of a NI 6281 data acquisition card (DAQ). The modulation voltage is converted to a current modulation by the LDC with a transconductance of 20 mA/V. A fused fiber coupler splits off 10% of the laser power for the absorption measurements to avoid saturation of the photodiode. The light is collimated and directed through a 50 cm gas tube filled with different concentrations of CO2 at a pressure of p=100 mbar. The transmitted light is focused on the detector photodiode and the output voltage U PD(t) of a low noise transimpedance amplifier (Femto DLPCA 200) is digitized with a sample rate of F S =500 kHz by the DAQ at a constant resolution of 18 bit. The analog bandwidth has been limited to 200 kHz by an antialias filter. The DAQ card is connected via USB to a standard computer that runs a Matlab control software for data storage, processing, and evaluation. For WMS, the computer takes the role of a software lock-in amplifier and for DAS the averaging and baseline removal procedures are carried out. The respective data processing steps are discussed in detail in Sect. 3.2 for WMS and in Sect. 3.3 for DAS. Depending on whether a WMS or DAS measurement is made, the computer generates a modulation voltage via the analog output (ao) of the DAQ card that takes the form

(1)

for the WMS technique, or

$$ U_\mathrm{mod,das}(t) = U_\mathrm{r,das}\,\mathrm{tri}(2\pi f_\mathrm{r,das}\,t)$$
(2)

for the DAS version, respectively. In (1) and (2), U r,wms and U harm are the ramp and harmonic modulation amplitudes for WMS and f r,wms and f harm the respective frequencies. U r,das and f r,das are the amplitude and frequency of the ramp for DAS measurements. The tri-function denotes a triangular, zero-mean, 50%-duty cycle ramp that takes the same arguments as the sin-function. The respective values used in the WMS and DAS measurements are listed in Table 1 along with the resulting peak-to-peak wavenumber tuning Δν. The value of U harm has been optimized for a maximum 2f-peak-to-peak signal for the given absorption linewidth of Δν line = 0.020 cm−1 (FWHM) at p=100 mbar.

Table 1 Modulation parameters of the WMS and DAS implementation
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Since for the WMS measurements, the DC part of the detector voltage U PD(t) may be cut off, the transimpedance amplifier is switched to AC-coupling for WMS measurements and to DC-coupling for the DAS technique using the digital output of the DAQ board. By setting the modulation voltage output to either U mod,wms(t) or U mod,das(t) and by adjusting the ADC-range of the DAQ analog input by the measurement software, it is possible to switch between the two techniques. The software allows to either make a given number of measurements with one of the two techniques followed by a series with the other technique or to make alternate measurements, one with WMS and one with DAS for a selectable number of cycles.

3.2 WMS signal processing

Figure 4 gives an overview over the signal processing steps after the digitization of the detector voltage U PD(t) for WMS. All signal processing is carried out by the Matlab program that controls the spectrometer. U PD(t) is demodulated in two orthogonal channels for 1f- and 2f-detection. This is done by multiplication with the respective software-generated sine and cosine reference, followed by digital lowpass filtering. The reference phase is adjusted for maximum signals in the in-phase channels, which are then used for further processing.

Fig. 4
figure 4

Signal processing steps for the WMS mode

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The lowpass filter bandwidth for lock-in amplifiers is traditionally expressed by the integration time constant τ. For the digital lock-in detection, it is adjustable via the characteristics of the digital filter. The lowpass filtering is based on a Hamming window FIR (finite impulse response) digital filter. Two different filters have been used with the two −3 dB cutoff frequencies f c1=328 Hz, corresponding to a small time constant, and f c2=34 Hz, corresponding to a larger time constant. The equivalent noise bandwidths of the filters were ENBW 1=682 Hz and ENBW 2=71 Hz, respectively.

After the demodulation the 2f signal is normalized to the 1f signal and the concentration is determined by a linear regression algorithm [15] that uses a reference measurement made at a concentration of 10% CO2, an offset and a linear slope as model. For a more detailed description of linear regression applied to WMS, see [18], for instance.

To investigate the impact of ADC resolution to the determined concentrations, the raw detector data U PD(t) is stored on the computer’s hard disk for each measurement cycle, before demodulation and evaluation. Thus, the same data can be used to assess the effect of different resolutions, eliminating the influence of accidental scan-to-scan variations. The raw detector data is sampled with 18 bit resolution by the DAQ board. In all measurements, attention has been paid that more than half of the input range was used when sampling the raw data, to avoid loss of resolution in this step. The resolution is then artificially reduced by software, before the concentration evaluation proceeds. The new input range is assumed to be bipolar [−U max,+U max], as it is the case for many data acquisition devices. The value of U max is ideally matched to the maximum of U PD(t). The quantization steps of the new resolution are then evenly spread over that interval and the original 18 bit measurement data are rounded to the respective nearest quantization steps. Possible nonidealities of the ADC are not considered. To validate that this software reduction of the resolution delivers results that are comparable to real analog-to-digital conversion anyway; we performed a 2f-WMS measurement using the DAQ card with software-reduced resolution and simultaneously used an 8 bit digital oscilloscope (LeCroy 9354) for data acquisition.

Figure 5 shows WMS spectra measured with 7 bit and 8 bit software-reduced resolution and compares them to the 8 bit oscilloscope measurement (middle trace). It can be recognized that the noise caused by quantization effects for the oscilloscope is smaller than for the 7 bit and slightly larger than for the 8 bit DAQ board measurement. Given that the oscilloscope measurement includes the nonidealities of the real ADC, we find that the software reduction of resolution emulates a real ADC reasonably well.

Fig. 5
figure 5

A WMS measurement for 10% CO2 taken with resolutions artificially reduced from 18 bit to 7 bit (top trace) and 8 bit (bottom trace) compared to the acquisition with an 8 bit oscilloscope (middle trace)

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When reduced resolutions are used in our WMS measurements, also the reference spectrum for the concentration evaluation has been requantized, assuming that the reference will usually be measured with the same spectrometer that is used for measurements on unknown samples. In this work, resolutions between 16 bit and 7 bit have been investigated.

3.3 DAS signal processing

For the direct absorption measurements, the DC-coupled ramp signal is digitized by the ADC. In the same way, as for the WMS setup, the U PD(t) is saved to disk prior to the signal processing steps that are summarized in Fig. 6. The ADC resolution reduction is carried out in the same manner as for the WMS method.

Fig. 6
figure 6

Signal processing steps for the DAS mode

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For each concentration determination, 500 ramp scans are averaged, resulting in 1 s total measurement time per cycle, the same as for the WMS setup. Although the averaging process partially reduces noise, optional FIR filtering is provided by the software to evaluate its interplay with the resolution reduction. The FIR filter used in the presented results is specified by a −3 dB cutoff frequency f c=65 kHz and a noise bandwidth of ENBW=130 kHz. These specifications have been chosen as to not distort the absorption lineshape by too tight filtering.

In the next step, a robust low-order polynomial baseline fit to the rising ramp slope is performed to yield the baseline U 0(t). The time-dependent absorbance is then calculated by α(t)=−ln(U PD(t)/U 0(t)). For rescaling α(t) to a wavenumber-dependent absorbance α(ν), the laser diode dynamic scanning behavior has been characterized in a separate measurement with the help of a solid silicon etalon (FSR=0.0119 cm−1) by evaluating the distance of the etalon fringes during the scan.

Finally, a Voigt line fit is applied to α(ν) using a nonlinear least square algorithm with boundary conditions. The numerical calculation of the Voigt function is realized as evaluation of the real part of the complex error function following [19], as described previously [17]. For the fit, the temperature and the pressure are fixed to T=296 K and p=100 mbar and the CO2 concentration c(CO2) is extracted from scaling the normalized Voigt profile to the measured absorbance spectrum with the help of the absorption line parameters listed in Sect. 3.4.

3.4 Scope of the investigation

Concentration determinations with DAS and WMS have been carried out on different test concentrations of CO2. Table 2 gives an overview over the parameters of the absorption line according to [20] and the test concentrations used. Absorptions at the line center have been calculated assuming pure air broadening. We compared the performance of DAS and WMS for different ADC resolutions at three stages. Firstly, we observed the influence of resolution reduction on the lineshapes recorded for either technique. The respective results are discussed in Sect. 4.1. In a second stage, we consider short-term series of 20 concentration evaluations with both techniques for the three concentration levels of Table 2 to investigate how the influence of ADC resolution manifests after the signal processing for concentration evaluation. Here, we used the concentration mean μ c and the standard deviation σ c of those 20 measured concentrations to assess the influence of different resolutions. Results are presented and discussed in Sect. 4.2. Finally, we performed a long-term series of concentration evaluations with WMS and DAS over several hours at the concentration of 1.65% CO2 with a twofold aim. On the one hand, we compare the two techniques, WMS and DAS, with respect to their performance in the presence of drift with full ADC-resolution. On the other hand, we can observe the impact of resolution reduction to the long-term behavior in either technique. The results are presented in the Allan plots of Sect. 4.3.

Table 2 Concentrations and line data used in the measurements
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4 Results and discussion

4.1 Impact on lineshapes

Figure 7 shows an example of the 2f-WMS lineshapes for the 10% CO2 measurement for 16 bit, 8 bit and 7 bit resolution, each evaluated for the two different cutoff frequencies of the lowpass filter. At 16 bit resolution, the line is smooth with no noticeable noise with either filter. At a reduced resolution of 8 bit, the lineshape begins to degrade. The quantization becomes visible as increased noise for the larger cutoff frequency f c1 (see insets in Fig. 7). However, the noise can be suppressed, if tighter filtering with the cutoff frequency f c2 is used and one can expect correct concentration evaluations. For a further resolution reduction to 7 bit the effect becomes more pronounced and even with tighter filtering, the original lineshape is not maintained, so that wrong concentrations are to be expected here. Indeed, the evaluated concentrations are considerably too small as will be shown in Sect. 4.2. This is due to the fact that the reference is recorded with the same resolution and the different realizations of quantization noise in the measurement and the reference spectrum lead to a too low correlation between the two spectra.

Fig. 7
figure 7

Examples for measured 2f-WMS lineshapes for 10% CO2 quantized with different ADC resolutions. Each graph shows the demodulated 2f-signal for the larger cutoff frequency (blue) and the smaller cutoff frequency (red) of the lowpass filter

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The results of the corresponding investigation of the lineshape for DAS are shown in Fig. 8. The measured absorbance spectrum is shown for resolutions of 16 bit, 10 bit and 9 bit, each evaluated with and without the additional FIR filtering. Also, here a clean lineshape can be observed for 16 bit resolution, irrespective of FIR filtering. For 10 bit resolution, a distinct triangular pattern becomes visible. The triangular shape is plausible, given that the quantization of the ramp signal U PD(t) will result in a staircase-like shape, which is then normalized to the sloping baseline, resulting in the visible triangles. The amplitude of this modulation increases for larger wavenumbers (smaller wavelengths) because the laser intensity is lower at this point of the scan and thus, during the baseline fit, the staircase is referenced to a smaller value. For a further resolution reduction to 9 bit, the steps of the staircase become wider and higher, resulting in the shape of the bottom graph of Fig. 8. Filtering significantly alleviates this effect, but for 9 bit resolution, considerable deformation of the lineshape remains.

Fig. 8
figure 8

Examples for measured DAS-lineshapes for 10% CO2 quantized with different ADC resolutions. Each graph shows the result without the optional FIR filtering (blue) and with the filter (red)

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4.2 Mean concentration and standard deviation in short-term measurements

While the lineshapes can give a first indication of the impact of resolution, it is more significant whether the correct concentration is recovered by the signal processing or not. Three series of 20 concentration evaluations each have been taken with both WMS and DAS for the concentrations listed in Table 2. Each series has been evaluated using different ADC resolutions.

Figure 9 shows the results for the WMS technique with the larger cutoff frequency f c1 (left column) and for the tighter filtering with cutoff frequency f c2 (right column). When looking at the results for 10% CO2 in the top row of the figure, it is visible that for both lowpass filters, resolutions between 9 bit and 16 bit deliver comparable results in terms of the average concentration μ c and its standard deviation σ c . For 8 bit resolution, the measured concentration is too low (mean μ c8=9.69% CO2) for the larger cutoff frequency but improves to μ c8=9.96% CO2 when using tighter filtering. Although the average concentration improves, the standard deviation σ c remains unchanged by the filtering. Figure 10 shows a plot of σ c over the ADC resolution for the three concentration levels and the two different lowpass filter cutoff frequencies. For resolutions between 10 bit and 16 bit, the standard deviation remains constant. It begins to increase for resolutions of 9 bit or lower. In general the standard deviation does not seem to depend on the lowpass filter cutoff frequency, except for the resolution of 7 bit, which delivers the wrong mean concentration μ c for the measurement of 10% CO2 and is therefore not considered in the further discussion.

Fig. 9
figure 9

Measured concentrations of three WMS series measured for 10% CO2 (top), 1.65% CO2 (middle), and 0% CO2 (bottom). The original raw data has been processed for different resolutions between 16 bit and 7 bit and with two different lock-in lowpass filters (left/right)

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Fig. 10
figure 10

Standard deviation σ c over ADC resolution for the concentration values determined in the measurements of Fig. 9. An increase of σ c is observable for resolutions smaller than 9 bit

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The results for the measurements of 1.65% CO2 (middle row of Fig. 9) show a similar behavior with the correct average value for all resolutions, but an increased standard deviation for resolutions below 10 bit (green traces of Fig. 10).

The bottom row of Fig. 9 shows the concentration results of measurements performed for 0% CO2. For any spectrometer, only concentrations will be detectable that are significantly above the standard deviation of measurements without the target gas present. Except for the 7 bit resolution, the mean of the 20 measurements is within ±0.012% CO2. The standard deviation is σ c16=0.015% CO2 for 16 bit, corresponding to a relative absorption of 2.4⋅10−6, and begins to increase for resolutions smaller than 10 bit.

In summary for the WMS measurements at the three different concentrations, there is no significant performance loss for resolutions down to 10 bit in terms of mean value μ c or standard deviation σ c . When using the tighter lowpass filtering with the smaller cutoff frequency, the mean values for 9 bit and 8 bit become acceptable, but at the cost of a considerable increase of the standard deviation. The standard deviation shows no significant dependence on the lowpass filter cutoff frequency. This can be understood against the background that the linear regression, which is used for concentration determination, is tolerant to noise and acts as a filter itself.

An equivalent set of concentration evaluations has been carried out for the DAS technique. The results are shown in Fig. 11 and allow similar observations, as in the WMS technique. However, the degradation of performance already begins at resolutions that are higher compared to the WMS case. Without FIR filtering (left column of Fig. 11), an increase of the standard deviation and a wrong mean value are obvious for resolutions of 11 bit or smaller. When using additional FIR filtering (right column) before the baseline fit, the performance improves. The measurements with 11 bit resolution now deliver a correct result for all three concentration levels. Although it is hardly visible from Fig. 11, the standard deviations σ c in that case are already slightly increased, compared to the 16 bit measurements. For the 1.65% CO2 measurements, for instance, it increases from σ c16=0.012% CO2 for 16 bit resolution to σ c11=0.021% CO2 for 11 bit. Further evaluations for 12 bit and 13 bit, which are not shown in Fig. 11, indicate that the deterioration begins for resolutions of 12 bit, but that 13 bit resolution results in unchanged performance. The measurements at 0% CO2 (bottom row of Fig. 11) yield mean values μ c <0.011% CO2 and a standard deviation σ c <0.03% CO2 for resolutions between 12 and 16 bit, a value twice as high as in the respective WMS measurements. A side note should be made about the relation of the relative absorptions involved and the ADC resolutions in DAS measurements. For the measurement of 1.65% CO2, the relative absorption in the line center amounts to 2.63⋅10−4 while for a resolution of 11 bit, the significance of the least significant bit (LSB) relative to the bipolar full scale ADC input range is 2/211=9.77⋅10−4, which is more than three times larger than the peak absorption in the line center. The measurement of absorptions much below the value of the LSB becomes possible only because of the averaging of many ramp scans, which can be interpreted as a dithering-averaging scheme, which leads to an enhanced resolution [8].

Fig. 11
figure 11

Measurement results of three DAS series measured for 10% CO2 (top), 1.65% CO2 (middle), and 0% CO2 (bottom). The raw data has been processed for resolutions between 16 bit and 8 bit. The left column shows the results without the optional FIR filter, the right column with additional filtering

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4.3 Long-term behavior and Allan deviation

For a characterization of long-term behavior, the considerations made above are not sufficient, since they cannot assess the influence of slow drifts, for instance. To investigate the long-term stability of the spectrometer using WMS and DAS, we collected alternate measurements. For DAS, the additional FIR filter was activated and for WMS, the smaller cutoff frequency f c2 was used. Altogether, 8,000 concentration measurements were taken, 4,000 with each technique. Because the raw data of each measurement cycle needed to be saved to hard disk and because the modulation output of the DAQ board U mod(t) had to be reinitialized for every measurement, the time between two concentration determinations was 1.95 s for a data acquisition time of 1 s. As WMS and DAS were used in turns, the resulting interval between two measurement cycles with the same technique was Δt=3.91 s. The advantage of taking alternate measurements is that both systems experience the same drift, and thus become more comparable than in two independent measurement series.

The resulting plots of the Allan or “two-sample” deviation are shown in Fig. 12. Its calculation methods and the interpretation of the resulting plots are discussed in detail in [21, 22]. The diagrams in Fig. 12 have been calculated using the τ-overlap method of [22].

Fig. 12
figure 12

Plots of the Allan (two-sample) deviation for the measurements with WMS (left) and DAS (right). Different curves are for different resolutions used for quantization of the raw data

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Before looking at reduced resolutions, it is worth noting that without resolution reduction, i.e., for 16 bit resolution, both techniques deliver very similar results (purple traces). The optimum subgroup size k opt, for that the minimum value of the two-sample-deviation is achieved, in both cases is in the range of k opt≈60. The lowest achieved two-sample deviation is slightly but not significantly lower for the WMS measurement and is in the range of 5⋅10−7 absorbance for both cases. In our opinion, this similar performance is not self-evident. Given the rather different nature of the measurement principles and especially of the evaluation procedures of the two methods, different robustness of the concentration evaluation against drift would be conceivable.

Compared to the direct absorption signal, the 2f-signal has a reduced amplitude. The fact that both systems perform comparable anyway, may be understood as an indication that even with scan averaging the DAS system is considerably affected by 1/f intensity noise while the impact to the WMS detection at 2f harm=60 kHz is much lower. The comb-like filter spectrum of the DAS averaging scheme (see Sect. 2.2) has some of its passbands in the spectral region where 1/f intensity noise is strong, specifically around DC and around the lower order harmonics of the scanning frequency f r,das. This idea agrees with the simulation results in [8] that suggested that for a white spectral noise distribution (meaning a suppression of 1/f-noise, e.g., by balanced detection), the advantage of larger signal amplitudes in DAS does result in better performance. One important result of this investigation certainly is the indication that none of the two approaches is intrinsically superior. Nevertheless, we do not dispute that the performance of both approaches can probably be enhanced by additional efforts such as noise canceling, more sophisticated signal processing, or multiharmonic detection.

Coming back to the impact of quantization, it can be found that the WMS measurement is not affected by reduced resolutions down to 11 bit (Fig. 12, left). For 10 bit resolution, the two-sample deviation begins to increase, an effect that has not been notable that clearly in the much shorter series of Sect. 4.1. For 9 bit resolution, the two-sample deviation is increased by more than an order of magnitude, meaning that the measurement time must be more than 10 times larger to achieve the same sensitivity as with 16 bit resolution. The increased quantization noise becomes evident as an additional white noise contribution that dominates the influence of the drift. In the plot, this becomes obvious by an increase of both the optimum subgroup length, and the lowest possible two-sample deviation. Using a longer integration time then improves the result, because the white noise contribution is now increased relative to the drift. However, the two-sample deviation of higher resolutions can never be reached.

For the DAS technique (Fig. 12, right) the increase in deviation begins for resolutions of 12 bit. Evaluations for 13 bits result in virtually no deterioration compared to 16 bit resolution. Compared to the WMS system this means a requirement of a resolution that is two bits higher for DAS to achieve comparable results.

This limit of 13 bit resolution for DAS, for which the performance is not yet affected by resolution issues is not a universal or arbitrary number, but is related to the noise present in the system. As briefly addressed at the end of Sect. 4.2, the DAS technique relies on a resolution enhancement through the averaging of many scans. For this procedure to work, there must be noise present in the analog signal that is strong enough to toggle the least significant bit (LSB) in different scans that later will be averaged. If, for instance, in one instant of a first scan the LSB is set to 1 by noise and, in a second scan it takes the value of 0, averaging the two scans will result in a value between the 0 and the 1 state of the LSB thus leading to increased resolution. This enhancement will work only if the noise is strong enough compared to the LSB. In our DAS measurements, we found a correlation of this condition to the 13 bit limit. When looking at the noise that is superimposed to the same sample of all the 500 scans that were averaged in DAS, we found that the standard deviation of the noise corresponds to 2.9 LSB for 14 bit resolution, 1.4 LSB for 13 bit resolution, and 0.7 LSB for 12 bit resolution. That means that for 14 bit and 13 bit, the LSB is constantly toggled by the noise, making the dithering work fine. For 12 bit the LSB is toggled less often, as the noise standard deviation is lower than the value of the LSB and thus the dithering-averaging scheme works less efficiently. Thus, the resolution necessary for DAS measurements is not given by a fundamental limit, but is a result of the interplay of averaging, the noise present in the system and the number of scans being averaged. An according guideline for a general estimation of necessary resolution for DAS would be to measure the fluctuations of the same sample on the different ramps that are included in the averaging cycle and choose the minimum resolution so that the value of the LSB is in the range of the measured noise standard deviation.

5 Conclusion

Using a DAQ card and a standard computer, we realized a spectrometer that can easily switch between the 2f-WMS and the DAS technique based on software selection. Using the capability of the presented system to take alternate measurements, the two methods could directly be compared under the same hardware and environmental conditions. Results show that with full ADC resolution, both methods achieve quite similar performance in terms of optimum integration time and lowest achieved Allan deviation.

The presented system has further been used to experimentally investigate the role of ADC resolution and its impact to concentration evaluation for the two techniques. WMS has generally lower requirements to DAQ resolution than DAS. Our WMS system achieves a performance comparable to DAS with resolutions that are 2 bit lower. For the long term investigation, WMS maintained its performance down to 11 bit, while the corresponding limit in DAS was found to be 13 bit. Both systems still deliver the correct average concentration for resolutions that are 2 bit lower than these values, but at the cost of an increased standard deviation of the measured concentrations, so that longer averaging is needed. Though the requirements for DAS are higher, averaging and filtering allow to measure relative absorptions lower than the value of the least significant bit. Hence, for both systems, readily available analog-to-digital converters with resolutions of 14 bit or 16 bit can be considered well suited. For measurement tasks that are not too demanding, the WMS technique might even be used with an 8 bit resolution ADC.