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Abstract
We proposed a method to intelligently design two-spherical-mirror-based multipass cells (MPCs) and optimize multiple objectives simultaneously. By integrating the K-means algorithm into the particle swarm optimization (PSO) algorithm, an efficient method is developed to optimize MPC configurations possessing characteristics of both long optical path lengths (OPLs) and circle patterns. We built and tested an MPC with four concentric circle patterns, which achieved an OPL of 54.1 m in a volume of 273.1 cm3. We demonstrated the stability and detection precision of the developed gas sensor. Continuous measurement of methane in ambient laboratory air was realized, with a detection precision of 8 ppb and an averaging time of 13 s. The combination of K-means and PSO algorithms is effective in optimizing MPCs with multiple objectives, which makes it suitable for designing versatile MPCs satisfying various requirements of field applications, including pollution and greenhouse gas emission monitoring and high-sensitivity measurements of other trace gases.
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1. Introduction
Tunable diode laser absorption spectroscopy (TDLAS) has been widely used for trace gas detection in many applications due to its advantages of fast response, high selectivity and sensitivity [1–7]. Based on the Beer–Lambert law, the absorbance is proportional to the optical path length (OPL). Multipass cells (MPCs) play a key role in TDLAS, which can be used to improve the instrument detection sensitivity by increasing the effective OPL. Typical MPCs include the White cell [8–10] and the Herriott cell [11,12]. The White cell consists of three mirrors and a relatively large volume, which limits its application in miniaturized gas sensors. The Herriott cell has been commonly used in industry due to its simple configuration and superior opto-mechanical stability. However, the Herriott cell only forms one circle or ellipse pattern on mirrors, and the effective utilization area of mirrors is low. In recent years, new types of MPCs have been designed and developed for detecting trace gas concentrations [13–19]. However, conventional methods used to design suitable MPCs are usually complex and time-consuming [19–26], as a number of patterns can be formed by varying design parameters, including radii of the mirror curvature, mirror sizes, mirror distances, and the position and angles of the incident beam.
There has been increasing interest in applying optimization algorithms to design MPC configurations and accelerate the design process [18,24–26]. In 2018, Hudzikowski et al. presented a custom-made tool for MPC and optical cavity design and implemented a genetic algorithm to automate the design process [24]. In 2021, Liu et al. used Monte Carlo and Nelder–Mead simplex algorithms to design two-mirror-based MPCs and calculated the characteristic distances between mirrors for independent circle-patterned MPCs [25]. In 2021, kong et al. applied random walk and gradient descent algorithms to optimize two types of MPCs with dense patterns, which consist of two and four spherical mirrors, respectively [26]. In 2021, Hudzikowski et al. used a genetic algorithm to design a MPC, which combines cost-effective spherical mirrors and forms Lissajous patterns [18]. Based on actual applications, desired MPCs usually consist of multiple objectives that need to be optimized, and it is necessary and valuable to design MPC configurations satisfying multiple measurement criteria and goals simultaneously, including long effective OPLs, desired spot patterns, high utilization rates of mirrors and so on.
In this paper, we present an efficient and intelligent way to optimize a two-mirror-based MPC enabled by decomposition-based multiobjective optimization. The MPC consists of two cost-effective spherical mirrors in a coaxial arrangement and forms dense patterns on each mirror. MPCs with circle patterns have characteristics of perfect regularity and easy identification of reflections and OPLs, which are useful in practice. We use the K-means algorithm to recognize independent circle and concentric circle patterns, and calculate the mean square deviation of the radius on each circle automatically. By integrating the K-means algorithm into the particle swarm optimization (PSO) algorithm, the multiple objectives for designing MPCs with both long OPLs and desired patterns are optimized in a cooperative and intelligent manner. The quality of the output beam is further optimized and implemented to evaluate the MPC designs. A compact MPC with four concentric- circle patterns was built and tested experimentally, which achieved an OPL of 54.1 m in a volume of 273.1 cm$^3$. To evaluate the performance of the developed MPC, a continuous measurement of methane (CH$_4$) was implemented in ambient laboratory air by using a 1653 nm distributed feedback laser and an uncooled InGaAs photodetector, and a detection precision of 8 ppb was achieved with an averaging time of 13 s. The PSO algorithm combined with the K-means algorithm is effective in automatically optimizing MPCs with multiple objectives, and the optimized MPCs have the advantages of high detection sensitivities, simple configuration, compactness and cost-effectiveness, which are suitable for various applications, including leak detection, industry safety inspections, and air pollution monitoring.
2. Design and optimization of multipass cells with two spherical mirrors
2.1 Modeling and principles of two-spherical-mirror-based multipass cells
The MPC consists of two cost-effective spherical mirrors of equal curvature $R$, which are placed in a coaxial arrangement and separated by a distance $d$. The geometry of the MPC and incident angles $\theta$ and $\varphi$ are illustrated in Fig. 1. The incident beam enters the cell through the entrance hole, reflects $N$ times and leaves through the exit hole at mirror M1 or M2. Under re-entrance conditions, the laser beam leaves the cell through the same entrance hole. $P_0^{(i)}$ , $P_C^{(i)}$, $r^{(i)}$ and $r_N^{(i)}$ represent the position of the incident point, the center of curvature, the direction vector of the incident beam, and the normal vector to the sphere, respectively (the subscript $i$ denotes the $i^{th}$ reflection, where $i=1,2\cdots, N$). Line-sphere equations and successive reflections between two mirrors are described below:
Based on iteration equations, a MATLAB program was used for ray tracing and guide the parametric design of the MPC, which has been detailed in previously reported work [23]. Patterns can be scaled up or down by keeping the direction of the incident beam unchanged and multiplying the parameters $(d, R, \theta, \varphi )$ with the same scaling factor [21]. To simplify the calculations, the radius of curvature R and the x-coordinate value of the incident beam were set to 100 mm and 0 mm, respectively. Therefore, the design parameters for the MPC were simplified to four variables: $d$, $y$, $\theta$, and $\varphi$.
2.2 Decomposition-based multiobjective optimization for multipass cells
Based on actual applications, multiple objectives need to be considered, and it is valuable to design MPCs satisfying various criteria and requirements. As numerous MPC configurations with dense patterns can be formed by varying design parameters, optimization algorithms were performed to accelerate and automate the MPC design process. We combined K-means and PSO algorithms to implement decomposition-based multiobjective optimization, and the MPC configurations with both circle patterns and long OPLs were optimized simultaneously and efficiently.
2.2.1 Automatic recognition of circle patterns
The K-means algorithm is an unsupervised learning algorithm that can be used to solve clustering problems, requires precise numbers in determining the number of clusters [27] and is suitable for identifying and recognizing desired patterns. MPCs with circle patterns have the characteristics of regular and even distributions of spots, easy identification of reflections and OPLs and are widely used in various applications [14–17]. Hence, we focus on circle patterns, including independent and concentric circle patterns, and use the K-means algorithm to recognize and classify the above desired patterns automatically.
The number of circles and spots on mirror M2 are k and n, respectively. Three steps are involved in the clustering procedure for optimizing circle patterns. Step 1: initialize k cluster centroids and set the maximum number of iterations. For independent circle patterns, k positions of spots are selected randomly as the initial k cluster centroids. For concentric circle patterns, the radius r is defined as the distance between each spot and the center of the mirror. We randomly choose k numbers of radius $r_p~(p=1,2\cdots,k)$ as initial cluster centroids, where the subscript p represents the spot located on the $p_{th}$ circle. Step 2: assign each spot to its closest centroid. For independent patterns, the assignment relies on the minimum Euclidean distance between each spot to be classified and cluster centroids. For concentric patterns, we calculate the radius of each spot on the mirror and subtract $r_p$ successively, and the spot with the minimum difference $\vert \Delta min_p\vert$ is reassigned to the corresponding $p_{th}$ cluster. Step 3: recompute cluster centroids. The cluster centroids are recalculated and return to Step 2 until convergence, that is, the spot number in each cluster remains unchanged or the iteration achieves the maximum limit.
Based on the K-means algorithm, spots are classified, and target patterns with concentric and independent circles are recognized in an automatic and accurate way. We calculate the mean square deviation of the radius on each circle and compute the total mean square deviation $\sigma _j (j=1,2,\ldots,k)$ . The optimal target pattern is formulated as solving the minimum $\overline {\sigma }=(\sum _{j=1}^{k}\sigma _j)/k$ . The clustering results for two representative patterns with seven independent circles and two concentric circles are shown in Figs. 2(a) and 2(b), respectively. The K-means algorithm is successfully implemented to recognize desired patterns automatically and calculate the mean square deviation of radii on each circle, which is one of the subobjectives for MPC optimization.
Fig. 2. Pattern recognition by using the K-means algorithm, and the clusters are marked with colors and Roman numerals; (a) seven independent circle patterns; (b) two concentric circle patterns.
2.2.2 Multiobjective optimization and analysis of multipass cells
PSO is an intelligent heuristic algorithm that uses the wisdom of swarms and adopts personal and global learning strategies to solve complex optimization problems [28]. In PSO, the positions of initial particles are produced randomly, and particle velocities are continuously updated by the individual and global optimal locations [28]. After a number of iterations, the particles will find the optimal solution in the search domain. In PSO-aided optical design, the particles and positions are represented by MPC configurations and corresponding design parameters $(d, y, \theta, \varphi )$, respectively. The performance of each particle is assessed by using a predefined fitness function, which can be adjusted based on practical requirements.
The PSO algorithm has the advantages of a strong global search ability and simplicity of implementation. By combining the K-means and PSO algorithms, we applied decomposition-based multiobjective optimization to intelligently design and optimize MPCs. To optimize MPCs with both long OPLs and circle patterns, we decomposed the multiobjective optimization into finding the optimality of two objectives with a set of weight vectors $[\lambda _1, \lambda _2]$. Optimizing the desired MPCs is transferred to solve the minimum of the fitness function $F(d, y, \theta, \varphi )=\lambda _1(1/OPL)+\lambda _2\overline {\sigma }$, where the value $\overline {\sigma }$ is calculated by the K-means algorithm. The first subobjective to achieve long OPLs and the second subobjective for evolving circle patterns are simultaneously optimized in a collaborative manner. The above method is effective and efficient in optimizing MPCs with multiple objectives and can be extended to design versatile MPCs satisfying the diversified demands of various field applications.
By combining K-means into the PSO algorithm, we intelligently optimized MPC configurations with both desired patterns and long OPLs. Furthermore, the quality of the output beam was optimized and used to evaluate the desired MPC configurations. Multiple rays are evenly placed on a circle outline with a diameter $D_{in}$ and are converged at one point $P_f$ on the line $P_0^{(1)}P_0^{(2)}$. The diameter $D_{in}$ can be adjusted in practical applications, and we set the diameter from 1 mm to 3 mm with an interval of 0.1 mm. Spots are deformed into oval shapes due to spherical aberrations, the length of the longer axis is expressed as $D_e$, and the beam size magnification $\alpha$ is written as the ratio of the length $D_e$ to the diameter $D_{in}$. By adjusting the point $P_f$, the quality of the output beam is optimized with a minimal magnification $\alpha$.
In the MATLAB-coded program, we predefined the number of particles and iterations in PSO as 500 and 50 and set the maximum reflections, the diameter of mirrors and the diameter of the entrance hole to 400, 50.8 mm and 3 mm, respectively. The following criteria should be considered when designing MPC configurations: spots are reflected within the boundaries of the mirrors, and the minimum distance of the spots is set as 1 mm to avoid spot overlap. To increase the diversity of solutions, the program is ran fifty times and the calculated non-dominated solutions are stored as the Pareto-optimal set for each pattern model, including two, three, four and five concentric circle patterns and independent circle patterns from three to nine. Among the Pareto-optimal solutions, the MPCs with an OPL of > 50 m were selected to confirm high detection sensitivities. For each pattern model, the quality of the output beam was further optimized, and the MPC with the best beam quality was targeted. Based on the above requirements, circle patterns with optimal beam qualities are exhibited in Fig. 3, and the corresponding parameters are listed in Table 1.
Fig. 3. Computed circle patterns with optimal quality of the output beam; the OPLs are all larger than 50 m.
Table 1. Parameters for the circle patterns shown in Fig. 3.
3. CH$_4$ detection using the optimized multipass cell
To demonstrate that the MPC optimized by intelligence algorithms can be effectively applied in gas sensors, continuous measurements of CH$_4$ were performed. A MPC with four concentric circle patterns was built and tested experimentally, and the observed pattern on mirror M2 is shown in Fig. 4. The wavelength modulation spectroscopy (WMS) technique can be used to significantly reduce 1/f noise by shifting detection to higher frequencies, and was applied to improve the detection sensitivity. The experimental setup is shown in Fig. 4. A fiber-coupled DFB diode laser (NLK1U5FAAA, NEL) was controlled by a commercial laser current and temperature controller (ILX Lightwave, LDC-3724C). To target the CH$_4$ transition at 6046.95 cm$^{-1}$, the DFB laser drive current and temperature were set to 69.98 mA and 34.42 $^{\circ }$C, respectively. A saw tooth signal with a frequency and amplitude of 10 Hz and 1 V was used to tune the wavelength, and a 0.65 V, 20 kHz sine wave signal was superimposed onto the saw tooth wave. The scan and modulation signal used to drive the DFB laser were generated by a LabVIEW program.
Fig. 4. Schematic diagram of the developed CH$_4$ sensor (FC: Fiber Collimator, PD: Photodetector) and the observed pattern on mirror M2.
The laser beam passed through an optical collimator and was then coupled to the proposed MPC with an OPL of 54.1 m. After reflecting 392 times, the beam exited the cell through the same entrance hole and was amplified by a detector (Thorlabs, PDA10CS-EC) with a gain of 30 dB. The lock-in amplification was realized by LabVIEW code and a multifunctional DAQ card (NI-USB-6211). The designed gas sensor was calibrated with standard gas consisting of 10 ppm CH$_4$ at atmospheric pressure.
To evaluate the stability and the detection precision of the designed gas sensor, CH$_4$ in ambient laboratory air was measured for thirty minutes. The measurements were performed by averaging ten scans at a rate of 10 Hz, and the sampling interval (temporal resolution) was 1 s. A total of 1800 data points were obtained, and the corresponding CH$_4$ concentrations are shown in Fig. 5(a). The Allan deviation analysis is presented in Fig. 5(a), the detection precision was 21.5 ppb at an integration time of 1 s and improved to 8 ppb with an averaging time of 13 s. The detection precision is comparable with previously reported experiments [14,17,19]. The histogram of the measured CH$_4$ concentrations was shown in Fig. 5(b), and the data distribution follows a Gaussian profile with a standard deviation value ($\sigma$) and half width at half maximum (HWHM) of 29 ppb and 34 ppb, respectively. The performed experiments show that the proposed method can be used as a powerful tool to improve detection sensitivities and promote the development of miniature gas sensors.
Fig. 5. (a) Time series measurements of CH$_4$ in ambient laboratory air (top), and Allan deviation analysis for the developed gas sensor (bottom), (b) histogram plot for the CH$_4$ concentrations.
4. Conclusion
In this work, we proposed an automatic and intelligent method to optimize MPCs based on decomposition-based multiobjective optimization. The optimization process aimed to obtain MPCs with both long OPLs and circular patterns. The K-means algorithm was used to recognize circle patterns automatically and calculate the mean square deviation of the radius on each circle. By integrating the K-means algorithm into the PSO algorithm, an effective and efficient method was developed to optimize MPCs with multiple subobjectives simultaneously. Based on the results calculated by the PSO algorithm, the quality of the output beam was further optimized and used to evaluate MPC arrangements.
We designed and built an MPC with four- concentric- circle patterns, and demonstrated its effective capacity for trace gas sensing in the NIR spectral region. The compact MPC achieved an OPL of 54.1 m, and the observed patterns fit well with the theoretical calculations. Continuous measurements of CH$_4$ in ambient laboratory air were performed, and a detection precision of 8 ppb was obtained with an integration time of 13 s. The high performance validates the effectiveness of the proposed method, which efficiently optimizes MPCs with multiple objectives that can potentially accelerate the development of trace gas sensors in versatile applications, including air quality monitoring, industry safety inspections and medical diagnostics.
Funding
Beijing Normal University (10100-312232102).
Acknowledgments
This work was supported by the Scientific Research Foundation of the High Level Scholars of Beijing Normal University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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