1. Introduction

Optical fringing due to Fabry-Pérot etalons is an important concern for laser spectrometers [1]. If the scattering/reflecting optics that directly contribute to the formation of an etalon can be identified, in some cases improved alignment can help, and the optical components can also be modified to change the etalon free spectral range such that it minimally affects the absorption signal [2]. In cases where etalons are unavoidable, many techniques have been developed to mitigate optical fringing over the past decades. For example, fast mechanical dithering of etalon cavity length allows fringes to be incoherently averaged without affecting the signal [3,4]. Similarly, “fringe scrambling” through thermal cycling has also been attempted [5]. However, these methods require compatible systems and can be power consuming for field applications. Many numerical techniques have also been developed [6–10], demonstrating in most cases suppression of slowly drifting etalons or fringe patterns with relatively few etalon components. Another fringe suppression approach leverages etalon information obtained from a reference detection channel through balanced detection, which may not be applicable depending on the sensing platform [11,12]. Techniques targeting specific fringes through carefully chosen laser modulation parameters have also demonstrated immunity from one or two etalons [13–15]. Apart from the complexity of implementation, these approaches become inefficient in systems that exhibit complex fringe structures resulting from multiple etalons. While numerous approaches discussed above are effective in specific systems where the necessary criteria are met, we present a software-based fringe suppression methodology that is generally applicable to tunable laser absorption spectroscopy.

We demonstrate the proposed fringe suppression method on a silicon photonic waveguide absorption sensor [16]. This integrated sensing platform is an emerging technology as a scalable sensing solution for applications such as pipeline methane leak detection, which can help reduce emissions from natural gas distribution [17]. However, distributed backscattering from waveguide sidewall roughness along with spurious facet reflections result in background spectral structures due to numerous etalons along the waveguide [18–20], which is also the case for our test sensor. Drifts in the background fringe spectrum limit the long-term stability of the waveguide sensor when ambient conditions such as temperature cannot be precisely controlled. As hardware modifications are preferably avoided due to its integrated on-chip infrastructure, computational approaches are low-cost alternatives.

Unlike the dynamic etalon fitting-routine that has been proposed to mitigate time-varying drifts of etalons whose dynamics can be approximated collectively [7], this work is a generalized method based on spectral fringe decomposition that can be applied to more complex fringe structures. The key to this approach lies in proper fringe background spectrum decomposition into etalon groups that can be approximated as drifting in unison. Then, an adaptive algorithm based on least mean squares (LMS) fitting is applied to track the change of individual etalon groups and to dynamically reconstruct the spectroscopic baseline.

2. Adaptive fringe correction method

In tunable laser absorption spectroscopy, the presence of a gas analyte results in the attenuation of incident light as described by the Beer-Lambert Law. The resulting transmitted spectrum can be modeled using the lineshape profile of the molecule (Voigt, Lorentzian, Gaussian, etc.) along with spectral baseline structures such as the laser’s L-I curve. In the presence of Fabry-Pérot etalons, the transmission spectra of etalons also lead to variations in the transmitted optical power as a function of laser frequency (hereafter referred to as the fringe background). The transmitted spectrum generally consists of a multiplication of the fringe background, the above-mentioned gas absorption profile, and other forms of baseline structures. If the fringe background is static, a zero-gas (no absorption) spectrum can be recorded during the calibration stage, then directly accounted for in the subsequent measurements in the process known as baseline correction. However, etalon drift causes the fringe background to change over time, resulting in incomplete fringe background suppression by conventional baseline correction approaches. The contaminated baseline then causes the fitting algorithm to mistake spectral distortion due to fringes as the analyte absorption signal, which leads to inaccurate concentration retrieval. If an optical system contains a dominant etalon or a set of etalons that exhibit the same characteristics (i.e. drifting at the same rate), the fringe background can be heuristically approximated by performing nonlinear transformation on the overall zero-gas background [7]. However, in the case of multiple etalons drifting at varying rates, uniform correction of the fringe background is insufficient.

The waveguide sensor under study exhibits such a situation where numerous etalons are simultaneously present and exhibit different drift rates [21]. The spectroscopic setup is shown in Fig. 1(a). A fiber coupled diode laser is used to target the methane R4 line near 6037.1 cm^-1 with sawtooth laser frequency scanning. A fiber pigtailed waveguide sensor is temperature regulated with an analog PID controller (Arroyo 5305). The sensor is enclosed in a chamber that allows a steady flow of the analyte, which in this case is a methane gas balanced by nitrogen at fixed pressure. A data acquisition card (NI USB-6251) continuously acquires transmission spectra as the laser frequency scans across the target methane transition. To study the complex fringe behavior on this sensor, the waveguide temperature is varied from 19 °C to 23 °C at increments of 0.001°C. The transmission spectra in the absence of methane were collected after each step of temperature change and are vertically cascaded in Fig. 1(b). As the arrows marked in Fig. 1(b) indicate, etalons with different free spectral range (FSR) drift at a different rate under the same temperature variation.

 

Fig. 1. a) Schematic of the setup for performing line-scanned absorption spectroscopy with a fiber-pigtailed waveguide sensor. The fiber isolator (ISO) is installed after the laser to prevent optical feedback, and the polarization controller (PC) allows operation in TE polarization for maximum waveguide transmission. The data acquisition board (DAQ) synchronously provides the line-scanning sawtooth waveform while acquiring spectral signal from the detector. An optical microscope image of the waveguide sensor is also shown. b) Fringe background spectral cascade under waveguide temperature tuning.

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Motivated by these experimental observations, we propose to address complicated optical fringing by separating the fringes into different sub-groups that exhibit similar characteristics so that they can be corrected separately. The proposed fringe correction procedure is illustrated in Fig. 2 in two stages: background fringe decomposition and dynamic fringe correction. First, a spectrum under zero-gas condition is measured as the fringe calibration background in step 1. Then, by taking the Fourier transform of the calibration background the various etalons are arranged according to their FSR. In the Fourier domain, etalons can be separated through bandpass filtering, which is shown in step 2 of Fig. 2. Note that the sectioning of the fringe background needs to divide etalons of disparate behaviors into different partitions, while the number of divisions should be minimized to avoid overfitting. Optimal etalon partitioning can be experimentally determined by performing etalon drift experiments (e.g. via temperature scanning) such as shown in Fig. 1(b) and observing the behavior of specific etalons. In this demonstration, the calibration background is separated into 5 etalon groups (a-e), each of which is inverse Fourier transformed to construct a set of background etalon basis shown in step 3.

 

Fig. 2. Illustration of the fringe correction procedure. Background fringe decomposition is performed in the Fourier domain, followed by nonlinear regression that adaptively tracks the fringe movement.

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Having obtained a set of etalon basis, subsequent spectral measurements can be corrected through a multi-group etalon fitting algorithm. Step 4 in Fig. 2 shows the newly acquired spectrum (black trace acquired a few minutes after the calibration background while system was operated with no temperature regulation), which has clearly changed from the calibration background (gray trace). Direct baseline correction without properly accounting for the updated fringe background would result in large residual transmission fringe noise. Based on the etalon basis, the current fringe background spectrum can be estimated through least mean squares (LMS) fit. The regression model is a product of the approximated fringe background, the absorption profile for the gas of interest, and a low-order polynomial baseline representing the laser L-I characteristic. Each background fringe component in the etalon basis is assigned two free variables to allow for constrained frequency drift and amplitude change. Step 5 in Fig. 2 shows the fitted result of the current fringe components’ positions (red traces) with respect to the calibration etalon basis (gray traces). The overall fringe background approximation considering the changes of individual fringe components is shown in step 6, where close agreement is obtained between the estimation (red) and the actual (black) fringe background. This multi-group etalon fit allows for more accurate baseline correction, effectively accounting for complex etalon drift during post-processing. It is noted that due to the increased number of free parameters in the LMS fit, the computation time increased by a factor of 5 when performing 5-group etalon fit in comparison to concentration retrieval without etalon correction. Even with the 5-group etalon fit the current computation time remains less than 300 ms for each spectral fit. For applications where shorter computation time is required, fewer etalon groups should be considered, and the LMS algorithm can be optimized (e.g. tighter fitting parameter constraints and more accurate initialization) or dedicated computational hardware (such as FPGAs) could also be considered. A detailed procedure of implementing background fringe decomposition and dynamic fringe approximation is provided in the Supplement 1.

3. Application to a waveguide methane sensor

The efficacy of the multi-group etalon correction is examined through the concentration retrieval performed using the silicon photonic spectrometer. A gas dilution system (Environics) is used to mix pure methane with nitrogen, and a steady flow of the gas mixture is delivered to the sensing chamber while maintaining 700 Torr chamber pressure. In the experiment presented in Fig. 3(a), nitrogen flows in the first 170 seconds of measurement to provide a zero-gas background, followed by the addition of methane in increments of 3% concentration every 80 seconds. After 3 methane concentration steps, we return to zero-gas nitrogen flow starting at 410 s. Due to the varying temperature of the gas mixture waveguide temperature fluctuations on the order of 10s of millikelvin are measured by a thermistor located on the surface of the sensor chip. As a result of temperature instability, fringe drift is observed throughout the measurement.

 

Fig. 3. a) Methane concentration retrieval using three different methods during an experiment with varying methane concentrations. b) Zero-gas methane concentration Allan deviation analysis. c-e) The background removed spectrum acquired at 220s for each retrieval method with fitted methane profile.

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Three ways of retrieving methane concentration are shown in Fig. 3(a) for comparison. In the first case, no etalon correction is applied, and the first 10 seconds of the zero-gas spectra are averaged and used directly as the baseline for consecutive measurement. Methane concentration is then retrieved by fitting the R4 transition model to the conventionally baseline-corrected transmission data. The retrieved concentration in Fig. 3(a) shows a continuous drift below 0%, demonstrating the detrimental impact of drifting fringe contamination on the methane concentration retrieval. In the second case, uniform etalon drift correction is applied to the acquired spectra. As discussed earlier, this method assumes all the etalons in the system drift at the same rate, which is an inaccurate assumption as demonstrated in Fig. 1(b). As Fig. 3 shows, uniform etalon drift correction dramatically improves the concentration retrieval accuracy in comparison to the retrieval with simple pre-recorded baseline correction. While positive and increasing methane concentrations are obtained, residual etalon drift causes a slow drift in concentration retrieval to persist throughout the measurement in addition to 2.5% false methane concentration retrieval during the zero-gas acquisition at the end of the experiment. Finally, the proposed multi-group etalon correction is applied to the measured spectra. The first 10 seconds of zero-gas spectra are averaged and decomposed to obtain the calibration etalon basis as explained in Fig. 2. As shown in Fig. 3(a), the retrieval results with multi-group etalon correction closely agrees with the expected methane concentration. Allan deviation [22] analysis adopted to spectroscopic system stability evaluation by Werle et al. [23] is performed for the zero-gas acquisition; the sensitivity at 10-second integration time improved from 4300 ppmv with conventional baseline correction, to 1400 ppmv with uniform etalon correction, to 630 ppmv with multi-group etalon correction.

A closer look at the effectiveness of the fringe noise suppression using various correction methods helps explain the retrieval results. The baseline corrected spectrum acquired at 220 s along with the fitted methane model are shown for each retrieval method in Figs. 3(c)-(e). Evidently, concentration retrieval is completely unreliable without proper etalon suppression as the spectrum is dominated by fringe noise in Fig. 3(c). The standard deviation of the fit residual for the uniformly etalon corrected spectrum in Fig. 3(d) is improved by a factor of 11 compared to the residual obtained from fitting without fringe correction, and further residual improvement by a factor of 2 is observed using the multi-group etalon correction.

It should be mentioned that Fourier domain partitioning of the fringe background does not guarantee complete separation of etalons. Fundamentally, an etalon spectrum can be modeled using the Airy distribution, whose Fourier transform contains infinite harmonics. In systems where multiple etalons are simultaneously present, the various harmonics of one etalon may overlap with those of another. However, the amplitude of a higher etalon harmonic is reduced by a factor of ${R^N}$, where R is the Fabry-Pérot cavity reflectivity and N represents the harmonic number [24]. For R < < 1, which is the case in most carefully designed optical systems, the amplitude of the 2^nd harmonic can already be negligible, causing little interference amongst different etalon harmonics. A more serious concern is that when multiple etalons are present, their spectral responses are multiplicative, and therefore the Fourier transform of the resulting fringe background is a convolution of different etalon components. In addition, given the relatively small laser frequency scanning range of single mode diode lasers, the Fourier transform resolution may not be sufficient to resolve etalons with close FSRs. Therefore, multi-group etalon fitting merely provides a close approximation of the actual fringes, and its efficacy depends on the decoupling of fringe structures with different behaviors. As there are limitations to approximations, recalibration would eventually be required depending on the rate of etalon drift and severity of fringe noise, and the calibration etalon basis needs to be updated to reflect the actual fringe background. In our experiment, the laser optical frequency is scanned over 1.4 cm^-1, leading to the Fourier domain axis resolution of ∼0.71 cm. Experiments show that this resolution is sufficient to achieve validity of initial calibration throughout the 500-second acquisition and the multi-group etalon fitting successfully captures the evolving fringe background in this setup. In sensors where etalons with similar FSRs have very different dynamics, wider optical scan range would be needed to improve the Fourier domain resolution in order to better separate these etalons. For systems where zero-gas calibration cannot be performed, active temperature stabilization can be performed as detailed in a related work [21].

4. Conclusion

A fringe suppression method based on fringe spectral decomposition is developed to mitigate the effects of etalon drift. We show that fast and effective separation of a complex fringe spectrum into subsets of etalon groups can be performed in the Fourier domain, and a regression model allowing for the correction of individual fringe components improves fringe background approximation. The proposed multi-group etalon correction algorithm is demonstrated on a silicon photonic waveguide absorption sensor. Zero-gas Allan deviation analysis shows better stability of the system with improved precision by a factor of 20 after 80 s of integration, and more accurate methane retrieval is also demonstrated in a gas flow experiment where agreement between measured and expected methane concentrations is obtained with multi-group etalon correction. As the proposed method does not require any hardware modification, it is easily adaptable, cost-effective, and generally applicable to tunable laser absorption systems.