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Abstract
Using hollow waveguide hybrid optical integration, a miniaturized mid-infrared laser absorption spectrometer for 13CO2/12CO2 isotopologue ratio analysis is presented. The laser analyzer described focuses on applications where samples contain a few percent of CO2, such as breath analysis and characterization of geo-carbon fluxes, where miniaturization facilitates deployment. As part of the spectrometer design, hollow waveguide mode coupling and propagation is analyzed to inform the arrangement of the integrated optical system. The encapsulated optical system of the spectrometer occupies a volume of 158 × 60 × 30 mm3 and requires a low sample volume (56 µL) for analysis, while integrating a quantum cascade laser, coupling lens, hollow waveguide cell and optical detector into a single copper alloy substrate. The isotopic analyzer performance is characterized through robust error propagation analysis, from spectral inversion to calibration errors. The analyzer achieves a precision of 0.2‰ in 500 s integration. A stability time greater than 500 s was established to allow two-point calibration. The accuracy achieved is 1.5‰, including a contribution of 0.7‰ from calibrant gases that can be addressed with improved calibration mixtures.
Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
The analysis of stable isotopologues and their ratios provides, though fractionation, information about the chemical production pathways of molecules. In the case of carbon dioxide (CO2), the measurement of δ13CO2, defined as the relative difference of the 13CO2/12CO2 isotopic ratio of a sample with that of a reference standard ratio, provides insight into production processes and/or production sources. Through fractionation process modelling, isotopologue information also contributes to our understanding of the carbon budget, sources and sinks [1,2]. The field of application of 13CO2/12CO2 analysis is vast and encompasses many different scientific disciplines. The instrument development presented in this work focuses on applications relevant to samples with high concentrations of CO2 (few %), requiring the ability for real-time, in situ sensing of isotopic ratio, and miniaturized robust analyzers to facilitate field applications, and/or to enable deployment from small autonomous vehicles. Characterization of geological carbon reservoirs and fluxes [3] and breath analysis [4,5] are the primary target applications.
In geochemistry, an accuracy of between 1 and 0.5‰ is required to interpret carbon reservoirs and understand the formation and mechanisms of volcanic systems, as well as their fluxes [6]. In breath analysis, the requirements are more stringent. Whilst the 13C urea breath test for Helicobacter pylori gives a positive diagnosis for an increase in delta of 3.0‰ over a patient’s baseline measurement [7], in other cases, such as the investigation of 13CO2/12CO2 as a biomarker for sepsis [8] or diagnosis of small intestine bacterial overgrowth [9], 0.2 to 0.3‰ is required. Therefore, an accuracy target of 0.3‰ seems appropriate for a miniaturized analyzer.
13CO2 and 12CO2 can be separated and measured with an isotope-ratio mass spectrometer (IRMS), to very high precision (0.01‰) [10], but IRMS suffers from lack of portability and temporal resolution, and requires extensive sample preparation. High-resolution laser spectroscopy, particularly in the mid–infrared (MIR) part of the spectrum, is an alternative approach that can obviate the IRMS drawbacks, at the cost of losing the large isotopologue flexibility IRMS offers [11].
Optical techniques using laser absorption spectrometry in the near-infrared (NIR) have been investigated for 13CO2/12CO2 ratio measurements, and include wavelength modulation spectroscopy [12] and cavity ring-down spectroscopy [13]. Moving to the MIR and the fundamental υ3 band of CO2 centered at 4.3 µm, molecular transition intensities are more than four orders of magnitude greater than in the NIR telecom bands (≈1.5 µm). This avoids the necessity of using resonant optical cavities, which require stringent alignment and stability, and high reflectivity mirrors that can be subject to degradation. In the υ3 band, for trace levels of CO2 as found in environmental applications, a multi-pass cell can be used instead [14,15]. But for isotopologue analysis of high CO2 concentration fumaroles [16], or in exhaled breath [17], absorption path length of a few centimeters is sufficient.
We propose and demonstrate spectrometer ruggedization and miniaturization based on hollow waveguide (HW) hybrid integration. Hollow core channels and fibers have been used as gas cells for high-resolution MIR laser absorption spectroscopy [18] and isotopic analysis [19], benefitting from ultra-low sample volume requirements. The use of ceramic HW for MIR laser light was pioneered as part of the development of CO2 waveguide lasers [20]. This led to the idea of fully encapsulated laser-based optical instruments that maintain single-mode EH11 propagation through a HW circuit, together with precisely positioned, encapsulated, optical components within the HW substrate [21]. An important point to note is that even though the HW can support a very large numbers of modes, optimized field coupling promotes single fundamental mode propagation, unlike a light pipe. In contrast to hollow core fibers, the rigidity of the HW structure provides high mode fidelity and is not subject to bends.
The manuscript is organized as follows: the first section describes the rationale behind the choice of isotopologue transitions, from which operating pressure and pathlength are derived, in addition to the theoretical HW concepts, which together determine the design of the integrated spectrometer. The second section describes the design and fabrication of the miniature spectrometer, providing detail of integral subsystems. The third section deals with characterization of both the laser source and the HW coupling. The final section before conclusion presents demonstration of the HW integrated spectrometer for 13CO2/12CO2 analysis, describing and assessing the measurement methodology and associated error budget.
2. Spectrometer design rationale
2.1 Isotopologue transition selection
The sensing performance of any tunable laser absorption spectrometer is partly determined by the choice of the molecular, and in this case isotopologue, transitions selected to carry out spectroscopic measurements. When distributed feedback (DFB) quantum cascade lasers (QCLs) are the spectroscopic source, a continuously tunable spectral window of ≈1 cm−1 is typically accessible when current tuning from threshold to roll-over. Within this narrow window, molecular parameters such as transition strength, lineshape overlap, absorption interference, and transition temperature sensitivity must be considered and optimized.
The isotopologue transition selection analysis is also dependent on the target application, and many cases have been considered in previous work [14,15,17,22–26]. Here, given the requirement for miniaturization, a single laser source is considered. The expected CO2 concentration of the sample is set to 5% in air. Following the methodology for 13CO2/12CO2 transition selection previously described [22], the criteria are:
- (1) the strengths of the 12CO2 and 13CO2 transitions must both be sufficient to enable an absorption path length within acceptable dimensions of a miniature instrument, ≈140 mm or less. Note that following HITRAN’s practice, the line strength is weighted by the natural abundance;
- (2) the peak absorptions from the 12CO2 and 13CO2 transitions must be similar to minimize the dynamic range of absorption prone to introducing non-linearities. Maintaining no more than 50% transmission difference, Δτ, is desirable.
- (3) the temperature sensitivity of δ13CO2 must be smaller than 200‰/K to minimize drifts originating from temperature uncertainties. This condition relates to the transitions’ ground state energies as described by Eq. (1) [27], in which ΔE is the difference in ground state energies, T the molecular gas temperature (20°C), kB the Boltzmann constant, and Δδ/ΔT the temperature sensitivity of the measured delta value;
- (4) the transitions must not overlap, but must lie within the tuning range of the QCL. A spectrum where the 12CO2 and 13CO2 lines have the same frequency provides no information about δ13CO2. The minimum separation at which the lines are just resolved can be defined by the Sparrow criterion [28]. If the line shape is approximately Gaussian, the Sparrow criterion occurs when the line separation, $|{{\sigma_{13}} - {\sigma_{12}}} |= 0.85 \cdot FWHM$, where FWHM is the full width at half maximum of the Gaussian. Using linewidth data from HITRAN 2016 [29], and a pressure of 100 Torr, this criterion gives a minimum line separation of 0.016 cm−1;
- (5) interference from other molecules in the mixture, for example H2O, must be minimized; we choose that these must have absorption less than 10%. No contributing water absorption lines were found within the analysis band. In application to real samples, other compounds that may be present in breath may also need to be considered [30].
The analysis was run in the υ3 band of CO2 (2200 to 2400 cm−1), which contains the strongest transitions. The line-pair candidates returned all lie in the region of overlap between the υ3 band of 12CO2 and the υ3 band of 13CO2 (2290 to 2320 cm−1). The outcome for the best four pairs is given in Table 1. Finally, taking into consideration technical parameters related to the QCL operation (operating temperature and power), line pairs LP2 and LP3 were retained.
Table 1. Candidate line pairs for the HW integrated laser spectrometer that meet the selection criteria. Transition frequencies and line strengths are reported, with uncertainties <0.001 cm−1 and <2%, respectively [29].
Once the transitions are selected, using the instrument and error propagation model described in section 5, the optimum absorption length and operating pressure can be determined for a 5% CO2 mixture in air. A pressure of 100 ± 50 Torr and a path length of 10 cm were chosen, ensuring peak absorption ≈80%.
2.2 Hollow waveguide characteristics
Key to the miniaturization and ruggedization approach is the use of a HW for both beam transport and gas sample cell, in combination with hybrid optical integration. HWs in this context are rigid square channels fabricated in bulk dielectric or metal, with a separate ‘lid’ in the same material completing the fourth wall of the channel [31]. Square HW with defined input coupling conditions in suitable materials have proved low loss at MIR wavelengths; losses are comparable to circular HW when propagation is in the fundamental mode. HWs benefit from ease of manufacture (5-axis precision milling), and the rigidity of a channel structure within a bulk material prevents mode sensitivity to bends. The hybrid integration approach combines HW beam transport, HW gas cell, and integrated laser, detector and ancillary optical components, to complete a fully encapsulated laser isotope spectrometer.
HW dimensions, material, surface properties, and operating wavelength determine the coupling efficiency, the propagation losses, and the coupling tolerances of a HW structure. The coupling properties relate to every integrated optical component and therefore constrain their positioning tolerance into the structure. These HW coupling and propagation characteristics will be considered in turn.
For wavelengths within 3–12 µm, optimum HW dimensions are on the scale of several 100’s of micron. As such, a HW supports many thousands of higher order linear EHpq hybrid modes, where p and q are non-zero. For lowest loss and control of the propagating modes, coupling between the incoming beam and the fundamental EH11 mode of the HW should be optimized. Figure 1(a) shows the variation of intensity coupling coefficient (overlap integral) between an input TEM00 Gaussian beam and the lowest-order modes of the HW; ω0/a is the ratio of the 1/e2 Gaussian beam waist, ω0, to the half-width of the HW channel, a. The theoretical approach follows the analytic method presented by Laakmann and Steier [32]; the data presented here were modelled numerically using FIMMPROP for convenience.
Fig. 1. (a) Modelled intensity mode coupling coefficients for square HWs. Solid lines are the coupling coefficients of a Gaussian beam into the defined mode; while the dashed line indicates the total intensity coupling coefficient of all modes shown, plus their equivalents (i.e., EH31, EH51, EH53). When p or q is even, the intensity coupling coefficient is zero. (b) Theoretical propagation loss of l = 10 cm HW at 4.36 µm. Total propagation loss (solid lines) of Macor (black) and Cu (red: EH11 mode; blue: EH31 mode) are shown. Power loss (dashed lines) contribution to the total loss is shown (these are coincident with the total propagation loss lines (solid) for both modes in Cu). Scattering loss (dotted) has a greater contribution to the total loss of the fundamental Macor HW mode than either Cu mode. (c) Transmission of modes in a 10 cm HW for Macor and Cu at 4.36 µm, combining the mode intensity coupling coefficient and the total propagation loss of the mode. Macor (black) and Cu (red: EH11 mode; blue: EH31 mode). (d) Lateral and angular waveguide coupling tolerance with changing waveguide width, to defined 99% power coupled metric. For λ=4.36 µm, Cu waveguide. Dashed lines indicate values for 750 µm waveguide.
Optimization of coupling into the EH11 mode occurs when ω0/a = 0.703 (bold, black); more than 98% of input intensity is coupled to the EH11 mode. Behavior of these waveguides as a ‘light pipe’, where higher order modes are excited, is seen to the left of this maximum (dashed, black); the total power coupled to the waveguide is close to 1 despite lower contribution of the fundamental mode. However, it should be noted that this plot shows only coupling losses, and these higher order modes have intrinsically higher propagation losses than the fundamental mode, leading to lower overall power transmission through the waveguide within this ‘light pipe’ region.
Propagation losses comprise two contributions: 1) losses inherent to attenuation into the material (power loss), 2) scattering losses related to the quality of the HW inner surfaces. The first contribution is given in Eq. (2), following Laakmann and Steier [32], where n–ik is the complex refractive index of the material host. The second contribution is given by Eq. (3), following Danilov et al. [33], where σs is the average statistical height (Ra) of the roughnesses, and S is the average peak-to-peak distance between roughnesses (Sm). Equation (3) was obtained from the case of the cylindrical HW where 2πS/λ>>1, to apply to the case of the square HW.
At 4.36 µm, for Cu, n = 1.4961 and k = 31.349. For Macor, a direct value at 4.36 µm is not known; we take the refractive index of the greatest constituent part by mass, SiO2 (46%), n = 1.3742 and k = 0. The calculated power loss, αpowerl, for the EH11 mode is shown in Fig. 1(b) (dashed), for both materials. In addition, to highlight the point made above regarding increased loss of higher order modes, the power loss of the EH31 mode in Cu is also shown. To calculate the scattering losses, αscatterl, the σs and S values were derived from surface roughness measurements of representative machined Macor and Cu HW surfaces obtained using a diamond-tipped surface profiler; for Macor, σs=0.597 µm and S≈33 µm, and for Cu, σs=0.166 µm and S≈40 µm. The scattering loss is shown in Fig. 1(b) (dotted). The plot shows the lesser contribution of scattering loss to the total loss for Cu compared to Macor. Additionally, Fig. 1(b) shows the total propagation loss of the mode (solid); this is the sum of the power loss and the scattering loss calculated above. These solid lines obscure the dashed ‘power loss’ lines for both Cu HW modes, indicating the small contribution of scatter to the total propagation loss of the mode.
Figure 1(c) shows the normalized transmission of the HW at 4.36 µm, comprising the intensity mode coupling coefficient (for ω0/a = 0.703) and the total propagation loss of the waveguide, following Eq. (4):
The coupling loss is the dominant loss mechanism. The very small difference in total HW EH11 mode transmission between Macor and Cu is ≈0.004 for 2a = 750 µm.The final aspect to consider is the coupling tolerance when injecting coherent light into the HW. For ω0/a = 0.703, we define our tolerance to be 99% of input intensity coupled to the EH11 mode (99% of the maximum value of the calculated intensity coupling coefficient) when the incoming beam is coupled with a lateral or angular offset relative to the optical axis of the waveguide. Lateral tolerance is directly proportional to the HW width, whereas angular tolerance is proportional to the ratio of the operating wavelength to HW width [35]. Optimizing coupling into the EH11 mode sets the input beam waist to HW width ratio (ω0/a = 0.703). The propagation loss analysis indicates that, irrespective of the material used, a HW width >0.5 mm is required to keep the throughput losses below 1% over 10 cm. The coupling tolerance analysis in Fig. 1(d) shows that the angular tolerance is far more critical than the lateral, and narrow HW widths are preferred to relax this constraint. We chose to maintain the angular tolerance >0.5 mrad as this is realistic for a physical system. As a result, the HW cross-section dimension, 2a, was chosen to be 750 µm. This choice also minimizes dead volume within the gas handling components, as it best matches the inner diameter of the miniature pipework feeding gas into the HW. The lateral and angular coupling tolerances are δt = 26 µm, and δθ=0.527 mrad, respectively.
3. HW spectrometer design and fabrication
3.1 Optical system
A copper alloy was chosen to form the substrate in which the HW structure is machined. Its thermal properties and machinability outperform Macor, as temperature control and stability is critical to achieve the desired measurement precision.
The design for the HW spectrometer requires the laser source, optics, HW cell and detector to be integrated into a single structure, with positioning requirements that maintain fundamental EH11 mode propagation throughout the optical path. A schematic of the HW integrated system is shown in Fig. 2(a). The reference datum is chosen as the intersection of the HW input port plane with the HW optical axis. Design and metrology are referenced against this datum.
Fig. 2. (a) Schematic of the HW spectrometer. The integrated HW section is enclosed in the dashed box; QCL, L, W1, W2, and PD are the quantum cascade laser, lens, window 1, window 2, and photodetector, respectively. The remainder of the diagram is the fluidic system. Blue solid line is 1/4-in. piping, while black is 1/16-in. Two sample gas cylinders, G1 and G2, are connected to a three-way ball valve (BV), via metering valves (MV1 and MV2), air filters (F1 and F2) and reducers (R1 and R2). After BV, the pressure is controlled and the flow monitored before gas input to the HW cell. At the cell output, a reducer (R3), valve (V), metering valve (MV3) and diaphragm pump control the flow. (b) Photograph of the completed spectrometer, including laser electrical and cooling connectors (left), gas conduits (top) and detector mounting (right). (c) Photograph of hollow waveguide channel machined in copper. (d) Photograph of end facet of copper hollow waveguide.
Integrating the QCL into the structure requires the laser facet coordinates with respect to the laser chip submount to be known to ~2 micron. To that end, metrology of the laser chip and its positioning on the submount is carried out using an OGP SmartScope ZIP 300. Once the QCL metrology is known, a QCL positioning system based on dowels, spring loaded positioners and holding clamp is designed within the HW structure to ensure QCL facet positioning with respect to the datum within 4 µm and, ideally, alignment of the QCL cavity with the optical axis within 0.5 mrad. Epi-down mounted QCLs do not always allow this alignment procedure to be followed, due to inability to determine the position of the rear facet of the laser cavity. The clamp also provides electrical contact to the QCL.
The light from the single mode QCL waveguide is highly divergent and requires a fast lens to couple to the HW. To that end, a 2 mm diameter Ge28Sb12Se60 aspheric lens (effective focal length of 0.495 mm at 4.3 µm, NA = 0.71, antireflection coated for 3–5 µm) is used. The positioning of the lens inside the HW structure is made fairly loose (±50 µm) so that it can be manually adjusted to optimize the QCL coupling into the HW and compensate for small errors in the QCL positioning. A nanopositioner (XYZ translation) is used to manually position the lens and optimize the coupling based on the spatial mode profile and power outputting the HW. The lens is then glued in position using UV curing epoxy.
The part of the HW structure supporting the QCL is thermally decoupled from the rest of the system using fiberglass insulating bridges. Temperature control and stability requirements for precise measurement warrants accurate control of the QCL frequency and its stability. To ensure few mK control and stability, a thermoelectric cooler (TEC) is installed beneath the Cu structure supporting the QCL. The TEC is connected to a linear PID controller, whose control signal is provided by a thermistor located close to the laser submount.
The 10 cm long HW forming the absorption cell is hermetically sealed by gluing the lid of the waveguide, and gluing CaF2 windows on the input and output ports. Gas input and output connections are installed on the lid using 1/16-in. (1 in. = 2.54 cm) tubing and corresponding Swagelok fittings. A thermistor is embedded into the structure to monitor the temperature, taken to be the representative temperature of gas flowing through the HW cell.
The final part of the structure is an uncooled MIR detector. The detector mounting is electrically insulated from the HW structure, and holds the detector to capture the light from the output port of the HW.
All surfaces with the potential to produce optical feedback have been tilted by a small angle (10°–15°) to prevent the formation of standing waves.
A photograph of the optical part of the HW spectrometer is shown in Fig. 2(b).
3.2 Fluidics
As previously mentioned, the fluidics at the HW interface is formed of 1/16-in. stainless steel tubing. The fluidics schematic is shown in Fig. 2(a). Gas input to the HW spectrometer can be selected by a manual three-way ball valve. Then, a PID pressure controller ensures accurate active control of the upstream pressure. Typically, 100 Torr was used. A flow meter measures the flow rate, between 0–10 mL/min. Typically, 8 mL/min was observed; there was no specific requirement of temporal resolution for the HW spectrometer demonstration. The flow enters the HW where it interacts with the laser light. At the gas output port of the system a set of valves allows flow adjustment or isolation of the HW channel. A small diaphragm pump provides the pressure differential required to the fluidics system.
3.3 Electronics and acquisition
The InAsSb detector (Hamamatsu) has a 0.7 × 0.7 mm2 active area, a sensitivity of 4.5 mA/W and a bandwidth of ≈350 kHz. The detectivity (D*) is 1 × 109 cmHz1/2/W, giving a noise equivalent power (NEP) of 7.0 × 10−11 W/Hz1/2. The detector current signal is amplified by a transimpedance amplifier (106 V/A gain, 200 kHz bandwidth (3 dB)), directed to a data acquisition board (USB-6259 16-bit digitizer, National Instruments) with 1.7 MHz bandwidth and read with software coded in LabVIEW (National Instruments). The bandwidth of the complete acquisition system is 200 kHz, limited by the amplifier. Each spectrum acquired consisted of 4000 datapoints, acquired at 400 kSample/s, and synchronized to the QCL injection current ramp. Due to buffer limitation and disc writing speed, spectral data were written to the hard drive only at a rate of ≈8.4 spectra/s, effectively decimating the dataset.
4. System characterization
4.1 QCL
Following the isotopic line selection, a QCL fulfilling the requirements was acquired (DQ4-M945R-NS, Maxion). The laser can be tuned between 2293.4 cm−1 and 2300.0 cm−1, when operated between −10°C and + 12°C with injection current in the range 810 to 1080 mA. At −10°C, the QCL typically delivers up to 3.2 mW of optical power. Given this tuning range, we expect to be able to observe LP2 and LP3. The linear approximation of the current tuning rate was found to be −16.3 cm−1/A, and similarly the temperature tuning rate was −0.15 cm−1/K.
Example spectra at two laser temperatures are shown in Fig. 3(a), along with a calculated transmission spectrum in Fig. 3(b). The laser was operating at a DC current of 1 A, on top of which a 100 Hz sawtooth current signal of ± 0.1 A peak-to-peak amplitude was added. A 5% CO2 mixture in ultra-zero grade air was flown through the HW channel. Each spectrum was acquired in 10 ms.
Fig. 3. (a) Measured spectra of 5% CO2 with δ=−22.7‰ at −1.0°C (red) and −7.5°C (blue). In both spectra, the laser threshold is marked ‘th’. (b) Three transmission spectra simulated from the HITRAN2016 database: 5.0000% 12CO2 (orange), 0.0546% 13CO2 (purple, for δ13CO2=−22.7‰), and 0.02003% 16O12C18O (green). The shaded regions, LP2 and LP3, denote the line pairs selected for δ measurement.
The gas sample used was given to be δ=−22.7‰. In Fig. 3(a), at −1.0°C laser operating temperature (red line), the HW channel temperature and pressure were 16.887°C and 87.5 Torr. At −7.5°C laser operating temperature (blue line), the channel temperature was 16.932°C and 87.5 Torr. The channel temperature was measured by the embedded uncalibrated thermistor, and the pressure was measured by the sensor of the pressure controller and corrected following a spectroscopic pressure measurement.
The relative frequency tuning properties of the QCL were calibrated by temporarily inserting a 76.42 mm Germanium etalon between the output port of the HW and the optical detector. The relative frequency tuning can be cross-validated using the CO2 lines appearing in the spectrum, whose frequencies are known to a high level of accuracy from the HITRAN database. However, pressure shift contribution must be added to the nominal line frequencies.
The QCL was found to operate as desired and was able to resolve the chosen line pairs LP2 and LP3. These pairs are shown within the shaded areas in Fig. 3.
4.2 HW coupling
To characterize the QCL to HW coupling performance, far-field output profiles from the HW were used. Figures 4(a) and 4(b) show the measured and modelled profiles before lens gluing, while Figs. 4(c) and 4(d) show the measured and modelled profiles after lens gluing. The measured beam profiles in Figs. 4(a) and 4(c) were obtained using a Pyrocam (Ophir), positioned as close as possible to the output of the waveguide, in place of the detector. Evidence of higher-order mode coupling is observed in these data. This is due to the properties of the aspheric lens; using this lens, it was not possible to achieve the correct beam waist dimension to meet the ω0/a = 0.703 criterion. Gaussian beam propagation indicated that for an image distance of 10.5 mm, as implemented here, ω0/a = 0.24.
Fig. 4. Measured and modelled far-field beam profiles, for a 750 µm cross-section copper waveguide, and ω0/a = 0.24. For measured profiles, the scales are the absolute position on the camera array. (a) Measured and (b) modelled beam profiles prior to gluing the lens. The misalignments introduced to form the modelled profile in (b) were: 1 mrad angular misalignment in both transverse planes, and −10 µm on the vertical scale of the plot. (c) Measured and (d) modelled beam profiles after gluing the lens. The misalignments introduced to form the modelled profile in (d) were: 2 mrad angular misalignment in both transverse planes, and −10 µm on the vertical scale of the plot.
The modelled output profile in Fig. 4(b) applies this ω0/a ratio. In addition, the model includes angular and lateral misalignments, as could be expected experimentally due to errors in the laser placement or metrology. A 1 mrad misalignment of the input beam from the optical axis was introduced to the model, in both transverse planes, in addition to a 10 µm vertical offset of the incoming beam from the optical axis. While the lateral offset falls within the lateral tolerance defined above, the angular deviation is twice as large as the quoted tolerance value for EH11 mode propagation. This is due to a combination of factors. Firstly, the tolerances above are for ω0/a = 0.703, not ω0/a = 0.24 as in the real system. Figure 1(a) shows the increased rate of change of the coupling coefficient of the EH11 mode at ω0/a = 0.24, resulting in much tighter tolerances than physically possible (<0.5 mrad) in this HW system. Secondly, due to the epi-down arrangement of the laser chip on the submount, it was not possible to obtain quantitative measurement of the angle of the QCL waveguide with respect to the submount, as the rear facet of the QCL chip could not be accessed by optical metrology. Thus, no correction was made to account for the angle of the QCL waveguide; this is very likely to be non-zero. Qualitatively, good agreement is observed between the “misalignment” model output shown in Fig. 4(b), and the observed mode profile in Fig. 4(a).
On gluing the lens in position, the mode profile shown in Fig. 4(c) was observed. This indicates additional misalignment of the beam waist at the waveguide input; the modelled expectation of this is shown in Fig. 4(d). An increased angular misalignment, up to 2 mrad, was introduced to the model, once again in both transverse planes; the lateral misalignment remained as 10 µm. Again, reasonably good qualitative agreement is found between the measurement and model; clearly, the higher order mode structure is becoming more pronounced with this relatively small increased misalignment. The physical process behind this misalignment is expected shrinkage of the glue on curing, by approximately 2 µm.
Comparison of the transmission properties of the waveguide was made for these different ω0/a values to determine the effect of the non-ideal beam profile on the measurement quality. Analysis was made using the data presented in Figs. 1(a) and 1(b), following Eq. (4) above; the transmission was calculated for each individual mode, and the results summed to obtain the overall transmission. Only modes where the coupling efficiency was greater than 1% were considered for brevity. Despite the obvious presence of higher order modes in the observed beam profiles, for a 10 cm channel length, the theoretical overall transmission of the waveguide changes very little, from 98.3% for ω0/a = 0.703, to 95.5% for ω0/a = 0.24. The laser used provided ample power to minimize the effect of this HW misalignment loss on the absorption measurement.
4.3 Spectrometer
Ideally, the spectrometer performance should be driven by the detector performance (detector-limited operation). The dark signal of the detector was measured to be 17.3 mV with a standard deviation of 0.3 mV over 12 minutes (the measurement limited by the digitization voltage resolution). Spectrum processing will require this background to be removed, which calls for a measurement protocol that includes recording the detector signal at currents below the laser current threshold to capture the dark signal relevant to each spectrum.
Given the NEP, the responsivity, and the acquisition bandwidth of the detector, the expected detector noise should be 0.14 mV. The noise on the dark signal is close to the expected value.
The noise on the spectra shown in Fig. 3(a) was estimated by taking the standard deviation of the signal in spectral regions away from molecular absorption. Noise values of 3.2 mV and 3.7 mV were estimated for the −1°C and the −7.5°C spectra, respectively. Measured noise levels are double the estimated detector noise value, indicating additional excess noise which exhibits dependence on the total laser power incident on the detector. The maximum signal-to-noise ratio on these spectra is therefore 1100 and 1900 for the −1°C and the −7.5°C spectra, respectively. These values will be used to carry out the error propagation from spectral noise to isotopologue concentrations.
Having quantified the random high frequency noise, the residual fringing was investigated. This is the baseline modulation observed on the −7.5°C spectrum shown in Fig. 3(a), and is thought to be due to optical fringing. To investigate this feature, the HW channel was flushed with N2, and baseline spectra were acquired. After removal of the linear trend in the baseline trace, a sinusoidal-like pattern with a free spectral range (FSR) of 1.64 cm−1 was observed. This FSR corresponds to an optical cavity of ≈3 mm. The aspheric lens is suspected to be the cause of this cavity, together with the output facet of the QCL. The lens has a center thickness of 1 mm and a refractive index of 2.6197; adding the 120 µm working distance, this would correspond to a cavity length of 2.74 mm.
The effect of this modulation on the measurement will be limited since the two isotopic lines selected cover a range of only ≈0.5 cm−1. However, a higher-order polynomial baseline is required to model the power modulation.
5. Isotopologue measurements
5.1 From spectra to δ13CO2 values
The HW spectrometer output consists of a spectrum from which the 12CO2 and 13CO2 concentrations needs to be obtained, among other parameters. To do this, the optimum estimation method (OEM) is used; this approach has been described previously [36,37]. The method relies on building a forward model that describes the instrument output as accurately as possible. In this case, the measurement vector y is a frequency-calibrated spectrum, and the forward model F relates y to the unknown parameters x (the ‘state vector’ comprising the isotopologue concentrations), and a set of known parameters b that describe the physics of the model. The model takes into account the measurement noise in the form of an additional vector ɛ. The forward model is then expressed by Eq. (5).
Determining the unknown state vector x requires inverting the forward model F to determine the function R that maps measurement vectors into the state vectors, as described by Eq. (6). The inversion provides an estimate $\hat{{\boldsymbol x}}$ of the state vector (in comparison to the true state vector x), and most often the physical parameters b are also estimates, defined as $\hat{{\boldsymbol b}}$. Vector c contains all the additional parameters required to condition the inversion algorithm.Once locally linearized, the inversion problem can be expressed in term of linear algebra using matrix and vector notation and arithmetic. The forward model is then expressed by Eq. (7), where K is the Jacobian or weighting function matrix. This is an important quantity that describes quantitatively the sensitivity of the measurement y to a change of the true state x. The local linearization implies that iterative convergence towards the solution will be necessary to obviate the non-linearity of the problem.
Bayes’ theorem is used to estimate the optimum solution to Eq. (7), and is stated in Eq. (8).Making the fundamental assumption that errors are described by multivariate Gaussian probability distribution functions, maximizing the a posteriori as a function of x is equivalent to minimizing the scalar cost function, J(x), given by Eq. (9). It is beyond the scope of this manuscript to go into further detail on the inversion methods; Brasseur and Jacob [38] provides the full description. In Eq. (9), the a priori state vector xa and the covariance matrix Sa characterize the prior knowledge we have of the solution, and should be informed by physical arguments. Sɛ is the covariance matrix describing the measurement noise.
When applying these concepts to process the spectra of the HW integrated isotopic spectrometer, the following conditions apply:
- - The forward model F consists of a line-by-line molecular transmission model to which a quadratic power modulation baseline is added.
- - The state vector x comprises the 12CO2, 13CO2, 16O12C18O concentrations, the three baseline coefficients describing the laser power modulation, an absolute frequency shift parameter, and a relative frequency scaling parameter.
- - The fixed parameters vector, b, consists of the HW channel pressure, temperature, and length. The first two parameters are measured during acquisition.
- - The measurement vector, y, is the spectrum delivered by the instrument, corrected from the dark signal and calibrated in frequency. The measurement noise covariance matrix is diagonal and constant; the constant value is the variance in the spectrum estimated from a subset of data points away from molecular absorption features.
- - The a priori state vector parameters were taken to be 5.00% 12CO2 and 0.05% 13CO2. 0.01% 16O12C18O was also included in the a priori state vector to avoid model errors due to the influence of these weak lines. The errors on the a priori concentrations were 50%. Initial baseline parameters were determined by fitting a polynomial to six points in the spectrum away from absorption lines. The a priori errors on these parameters were set to 100%, even though they can be further constrained from long-term baseline variability measurements. Initial frequency offset, σs, and scaling, kσ, were set to 0.00002 ± 0.02 cm−1, and 0.005 ± 0.005, respectively.
Fig. 5. (a) Part of the measured spectrum (red line) of a calibrant gas with known δ=−22.7‰, measured at a laser temperature of −1°C, showing the spectral region around LP2. The fitted model and fitted baseline are also shown. (b) The residual of the spectrum and model shown in (a). (c) Part of the measured spectrum of the same gas at a lower laser temperature (−7.5 °C) showing LP3. (d) Residual of (c).
The fitted models agree well with the measurements, as witnessed by the residuals. For the spectrum covering LP2, the residual in spectral regions away from the absorption features has random intensity noise of 3.6 mV. On the flanks of the stronger 13CO2 absorption feature, the noise increases to ≈20 mV (standard deviation). This suggests that laser phase noise is converted to intensity noise. Given the maximum local slope on the flanks of the absorption line is 115 V/cm−1, this corresponds to a frequency noise of 5.2 MHz over a timescale of 30 µs.
The OEM approach enables diagnostics to be run to ascertain the quality of the inversion. First, the cross-correlation between the measured parameters is scrutinized, with particular emphasis on the 12CO2 and 13CO2 concentrations. The worst (largest) relative cross-correlation for parameters in Table 2 is 4.5 × 10−5, between the concentration of 12CO2 and σs for LP2. Secondly, a well-conditioned inversion should provide J(x)=m, where m is the number of points of the spectrum. For the case of Figs. 5(a) and 5(b), J(x)/m was 8 and 37, respectively. This indicates the measurement errors were slightly underestimated; this is not surprising as the phase to intensity noise conversion effect was not included in the noise model. As a result, propagated errors to 12CO2 and 13CO2 concentrations are also underestimated.
Table 2. Optimum solution for the state vector and associated errors for the inversions shown in Fig. 5.
The inversions presented in Figs. 5(a) and 5(b) yield δ=−61.4‰ and δ=−78.9‰, respectively; these are far from the nominal value of the calibrant at −22.7‰. The propagated δ-value errors from the random measurement noise are 1.0‰ for LP2, and 0.5‰ for LP3. The isotopic ratios are given using the Vienna Pee Dee Belemnite reference ratio, Rref = 0.0111802 [39]. The main source of bias identified thus far originates from the uncertainty in determining the actual temperature of the molecular gas. Assuming the error observed is solely due to the temperature bias between the HW block thermistor and the actual molecular gas temperature, the molecular gas temperature is underestimated by −2.8 K and −3.2 K for LP2 and LP3, respectively, using the intrinsic δ-value bias dependence with temperature from Table 1. This suggests that the gas temperature has not equilibrated to the lower-than-ambient block temperature during the measurement, and a temperature correction is required.
5.2 Stability, precision, and repeatability
The highest precision of the isotopic HW spectrometer can be achieved by averaging signals until the long-term drift dominates the noise statistics of the system. To that end, an Allan variance analysis is run [40]. The instrument measured a sample from the same cylinder for ≈15 min, with all the settings on the instrument kept identical. Figure 6 shows the corresponding temporal record of 12CO2 and 13CO2 concentrations and the calculated δ-values, with the Allan deviation of each temporal record shown alongside, for line pairs LP2 in Fig. 6(a), and LP3 in Fig. 6(b). Consistently, the precision of δ13CO2 is 0.2‰ after 500 s, and the target precision of 0.3‰ is reached at ≈220 s. The roll-over point of δ13CO2, at which the long term drift starts to appear, is not reached before 500 s. In both cases, the system appears to be dominated by white noise. Interestingly, whilst an upward trend to equilibrium can be seen on the concentration values, which limits the stability of the concentration measurements, δ13CO2 remains unaffected. The limit of detection of 12CO2 (resp. 13CO2) is inferred as the minimum of the Allan variance plots shown in Fig. 6, and is 39 ppm (resp. 448 ppb) for LP2, and 61 ppm (resp. 484 ppb) for LP3.
Fig. 6. (a) The concentrations of 12CO2, 13CO2, and calculated δ13CO2 over 15 min, derived from model inversion of 7000 spectra (gray) of δ=−22.7‰ gas, acquired for LP2 near 2294.4 cm−1. The data have been smoothed (red line) with a 200 point (≈26 s) wide window. The Allan deviation of each is shown on the right. The white noise dependence is shown as a dashed black line. (b) Corresponding measurements acquired for LP3 near 2296.0 cm−1, with the same gas.
To assess the repeatability of the measurement, the gas input was switched repeatedly between two gases using the three-way valve: 1) ≈5% CO2 calibrant gas with δ=−7.2‰, and 2) nitrogen (N2). The valve was operated every 60 s. The spectra were fitted using the model inversion of LP2. The resulting record of concentrations of 12CO2 and 13CO2 are shown in Figs. 7(a) and 7(b), respectively, along with derived δ13CO2 values in Fig. 7(c). The response time, defined as the time between switching the input from N2 to CO2 and reaching a stable δ13CO2, was 4.3 s. An expanded view of the transient is shown in Figs. 7(d)–7(f). The volume of the HW channel is 0.05625 mL. With a gas flow rate of 8 ml/min, the response time should be 0.45 s. However, the fluidic system is not yet optimized and there is a relatively large reservoir of gas within the controllers and pipework, which explains the extended response time.
Fig. 7. (a) The concentration of 12CO2 from the model inversion measured over time. The gas input was switched between 5% CO2 (δ=−7.2‰) and N2 every 60 s. (b) The concentration of 13CO2 from the same model inversion. (c) The δ values calculated from the concentrations in (a) and (b). The black lines indicate the mean for each measurement. The measurement results (1–6) are shown in Table 3. (d–f) Expanded view of the transient behaviour (240–250 s) leading to measurement 3, defined in (c). The dashed line indicates the end of the transient period, 4.3 s after the switch from N2 to CO2.
Table 3. Results of six repeated measurements of δ, from the data in Fig. 7(c). All units are ‰.
The red regions in Fig. 7 show the stable periods, after switching the gas and waiting for the response time. The δ values for each of these 6 periods were averaged to produce a set of 6 δ measurements, shown in Table 3. The standard deviation (σ) is also shown, along with the standard error $\sigma /\sqrt N$, where N is the number of δ-values averaged. The standard deviation of the 6 δ measurements gives the repeatability as 0.35‰, and the mean of the standard errors is 0.57‰.
5.3 Calibration and associated errors
Whilst the precision is demonstrated to be 0.2‰ for a 500 s measurement, owing to biases, obtaining the accuracy requires calibration against known isotopic mixtures. An initial gain and offset two-point calibration is preferred. The calibration must occur within the stability time, therefore a sample measurement is immediately preceded and succeeded by measurement of two different calibrant gases. Measurement of gas C1, with nominal δ1=−22.7 ± 0.2‰, precedes the sample measurement, while measurement of gas C2, with nominal δ2=−7.2 ± 0.4‰, succeeds it. Each gas is measured for 120 s.
The two-point linear calibration will require additional error propagation to the final uncertainty of δ13CO2. The calibrated measurement δx is derived from the direct measurement δmx using Eq. (10), where the slope s and the intercept i, corresponding to the gain and offset, respectively, are determined from the two-point calibration. A direct sum of squares error propagation gives Eq. (11), where Δ indicates the uncertainty of a physical quantity.
Table 4. Summary of the values involved in the calibration, and associated errors.
Despite the temperature correction, measurement biases are still present. Propagating the errors including calibration yields −38.4 ± 1.5‰. The significant uncertainties on the δ-values of calibrant gases (0.7‰) contribute almost half of the final uncertainty, and are an obvious source of improvement.
6. Conclusion
The development of a highly miniaturized (16 × 6×3 cm3) hollow waveguide integrated mid-infrared (4.36 µm) spectrometer for 13CO2/12CO2 analysis has been thoroughly described, including the underpinning hollow waveguide concepts. The spectrometer was demonstrated to encapsulate the quantum cascade laser, coupling optics, hollow waveguide channel, and the detector within a monolithic copper alloy block. The performance of the system was evaluated in the laboratory using calibrated mixtures of 5% CO2. The measurement methodology, including the spectra inversion method and associated error propagation, was described to demonstrate a precision of 0.2‰ for a 500 s measurement. Measurements were found to be affected by a large bias due to the uncertainty in the knowledge of the molecular gas temperature. This emphasizes the need for efficient sample gas equilibration within the hollow waveguide structure. The overall accuracy achieved using two-point calibration is 1.5‰, which includes a contribution of 0.7‰ from the large uncertainty in the reference calibration mixtures, that can be addressed by using improved calibrants.
The fully encapsulated isotope analyzer is highly robust with no moving parts. The hollow waveguide approach, with mode propagation control, ensures optimization of light coupling and thus minimization of loss into the waveguide circuit. Benefits include a small sample volume and improved temperature homogeneity and control, compared to previous free-space optical instruments. This demonstration enables the development of miniaturized isotopic analyzers for both bedside breath analysis and geochemical flux analysis. The latter application benefits from this miniaturized and robust system for field deployment using unmanned autonomous vehicles.
Current developments include addressing the sample gas temperature equilibration issue highlighted above, via an offline gas conduit equilibrated with the hollow waveguide block. In addition, hollow waveguide integrated multi-pass structures are being tested to enable the analysis of lower concentration species, whilst maintaining a small form factor and a low sample volume. Finally, on-board calibration techniques are being developed to facilitate long term deployment to areas and fields not well suited for complex instrumentation.
Funding
Science and Technology Facilities Council.
Acknowledgments
Instrument manufacture was carried out by Wayne Robins from RAL Space Precision Development Facility. Metrology was carried out by Dave Wilsher from the RAL Metrology Facility.
Disclosures
DW: Mirico Ltd (I, E, P). Remaining authors declare no conflicts of interest.
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