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Abstract
We propose a method to intelligently design and optimize a multiple-spherical-mirror-based multipass cell (MPC) with Lissajous patterns. The MPC consists of at least three spherical mirrors, which are placed in a rotationally symmetric arrangement. Particle swarm optimization (PSO) is performed to optimize the parameters of the MPC configurations and accelerate the design process. Two Lissajous patterned MPCs with three and five mirrors are built and tested experimentally. We further develop an open-path gas sensor based on a five-mirror-based MPC to detect methane concentrations in ambient laboratory air, and a detection precision of 1.1 ppb with a 123 s averaging time is realized. The PSO algorithm is efficient for optimizing the proposed MPC, which has superior proprieties of symmetry configuration, cost-effectiveness and high detection sensitivity and is well suited for trace gas sensing applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable diode laser absorption spectroscopy (TDLAS) has been widely applied to detect trace gases due to its high selectivity, superior sensitivity, and rapid response [1–7]. The absorbance is proportional to the optical path length (OPL) based on the Beer–Lambert law. As a core component in TDLAS devices, a multiplass cell (MPC) is used to improve the detection sensitivity by increasing the effective OPL. The White cell [8–10] and the Herriott cell [11,12] are two types of classical MPCs, which consist of three spherical mirrors with equal curvature radii and two coaxial spherical mirrors. Reflection spots emerge as a single or double row of spot patterns and elliptical or circular patterns, respectively, which lead to relatively low utilization areas for mirrors. The astigmatic cell [13–15] comprises a pair of astigmatic mirrors and forms Lissajous patterns, which have high spot density and can significantly improve mirror utilization. However, the manufacturing of astigmatic configurations is complex, and extreme-precision tolerance mirrors with exact curvatures have a high cost. Spherical mirrors are affordable with an easily controlled surface quality and are well suited for MPC configurations.
In recent years, dense patterned MPCs with two identical spherical mirrors have been developed for detecting trace gas concentrations using the TDLAS technique [16–28]. The MPC is particularly suitable for applications where the curvature radius of the mirrors is small [16]; however, it is harder to achieve hundreds of reflections between mirrors with small diameters when the curvature radii increase significantly. The newly developed spherical-mirror-based MPC with Lissajous patterns can overcome this drawback. In 2019, Cao et al. proposed a method to design a ring MPC with three spherical mirrors, which forms Lissajous patterns [29]. The calculation model used a paraxial approximation, which resulted in a difference between the matrix model and the actual optical software. In 2021, Hudzikowski et al. used a generic algorithm to design a Lissajous patterned MPC, which consists of multiple spherical mirrors placed in an asymmetric structure [30]. However, the entrance hole is centrally located on the mirror, and a systematic method to find all possible laser beam paths among an arbitrary number of mirrors has not been reported.
In this paper, we propose a method to design and optimize a Lissajous-patterned MPC with multiple cost-effective spherical mirrors, which are placed in a rotationally symmetric arrangement. Optimization algorithms have been successfully performed to accelerate the MPC design process [26–28,30]. With precise optical design, numerous Lissajous patterns can be formed on spherical mirrors with arbitrary curvature radii and diameters. The particle swarm optimization (PSO) algorithm is applied to speed up and optimize the MPC design process intelligently. Two MPCs with three and five spherical mirrors are built to validate the effectiveness of the methods, and the observed patterns are found to be consistent with the simulated results. Wing spot patterns have the properties of regular distributions and easy identification of reflections and OPLs, and we developed a five-mirror-based MPC with wing patterns as an open-path gas sensor. We demonstrated methane (CH$_4$) measurements in ambient laboratory air by using the wavelength modulation spectroscopy (WMS) technique, and a detection precision of 1.1 ppb was achieved with an averaging time of 123 s. The PSO algorithm is efficient for automatically optimizing multiple-mirror-based MPCs under both re-entrant and nonre-entrant conditions. The new type of MPC has superior properties, including high detection sensitivity, high utilization areas for mirrors and symmetry structures, which are suitable for various applications, including atmospheric pollution monitoring, leak detection and medical diagnostics.
2. Modeling and principle of multiple-spherical-mirror-based MPCs
The MPC consists of multiple (at least three) identical spherical mirrors, which are placed in a rotationally symmetric arrangement. The configuration of an MPC with three mirrors is shown in Fig. 1(a), and the top view is exhibited in Fig. 1(b). We set the center of all mirrors as the origin O of the coordinate system, and the variable d is defined as the distance between the origin O and the center C of each mirror. The number, curvature radius and diameter of the mirrors are M, R and D, respectively. A laser beam enters through the entrance hole, reflects N times and leaves the MPC through the exit hole. The beam can leave though the exit hole on arbitrary mirrors, and exits the cell through the same entrance hole under the re-entrant condition. The following line-sphere equations are used to calculate the chief ray tracing and simulate successive reflections between M mirrors:
The ray tracing is simulated accurately based on line-sphere equations, and the following criteria should be satisfied: The laser beam is reflected within the boundary of each mirror and creates long effective OPLs. The spots avoid overlapping to reduce the interference effect. Only the incident beam enters the MPC through the entrance hole, the emergent beam leaves through the exit hole, and the other reflected beams cannot leave through the entrance hole or exit hole. With precise design parameters, the multiple spherical mirrors in a rotationally symmetrical configuration generate Lissajous patterns, which are usually formed by a pair of expensive aspherical mirrors. The proposed multiple-spherical-mirror-based MPC has the advantages of a symmetrical structure, affordable cost and high mirror utilization area, which are well suited for trace gas detection in field applications.
3. Optimizing the MPC design with the PSO algorithm
For MPCs with three and four spherical mirrors, the laser beam reflects on the neighboring mirrors successively and forms one laser beam route. The beam routes are increased by adding a number M of mirrors. Given the number M of spherical mirrors, we plan the laser beam’s route, which should reflect on each mirror once, return finally to the starting mirror and generate a polygon with M edges. As the spherical mirrors are placed in rotation symmetrically, the actual beam paths have the shape of regular polygons. The Schläfli symbol ${p/q}$ [31] and the one-stroke drawing method [32] are effective to find all possible laser beam paths for an M-mirror-based MPC. The exemplary beam routes for MPCs with three to thirteen spherical mirrors are demonstrated in Fig. 2. The black circles represent the positions of the spherical mirrors, which are placed rotation-symmetrically on a circle shape (marked by gray dotted line), and the red lines show laser beam routes within the MPC. In addition, an abundant set of Lissajous patterns are generated by adjusting the design parameters, and it is a complex and time-consuming task to design the desired MPC configurations. The PSO algorithm is applied to optimize the desired MPC configurations in an intelligent and very fast way.
Fig. 2. Laser beam routes (marked in red lines) for MPCs with three to thirteen spherical mirrors; the black circles represent the positions of the spherical mirrors.
PSO is an intelligent heuristic algorithm inspired by the social behavior of bird flocks and has proprieties of strong global search ability, simple implementation, high efficiency and superior accuracy [33]. We implement the PSO algorithm to speed up and optimize the design of multiple-mirror-based MPCs. The particles and positions are represented by MPC configurations and multidimensional design parameters, and the fitness function is defined as $F(d,x^{(0)},y^{(0)},x^{(1)},y^{(1)})=1/L$ , where $L=\sum \nolimits _{i=1}^Nt^{(i)}$. $L$ and $t^{(i)}$ represent the total OPL of an MPC and the OPL of the $i^{th}$ reflection. In the PSO-aided optical design, the initial particles are generated randomly, and the individual and global solutions are selected and updated at each iteration. By minimizing the fitness value, the MPC configuration is finally optimized in the five-dimensional searching space. We predefined the diameter of the mirrors as 50.8 mm and set the number of particles and iterations to 50 and 500, respectively. The MATLAB-coded program was implemented using an Intel Core-i7 with Windows 10, and the optimization process only required several minutes. Compared to the grid search method [22] and the Monte Carlo algorithm [26], which takes several days and several hours, the PSO algorithm is quite effective for speeding up MPC design.
The PSO algorithm was successfully used to design and optimize MPCs with arbitrary numbers of spherical mirrors and laser beam routes. We first set the curvature radius R as 100 mm, and the representative Lissajous patterns are shown in Figs. 3(a)–3(c). For MPCs with three spherical mirrors, diamond spot patterns and cross-style patterns are formed, as shown in Figs. 3(a) and 3(b), respectively; for a five-mirror-based MPC with star laser beam routes, wing spot patterns are achieved and exhibited in Fig. 3(c). Subsequently, we calculate the three-mirror-based MPC with curvature radii of 50 mm, 400 mm and 1000 mm, and the exemplary Lissajous patterns are depicted in Figs. 3(d)–3(f), respectively. For the MPC with a curvature radius of 50 mm, the distance d is 92.3 mm, and the footprint is relatively low. When the curvature R is increased to 1000 mm, an OPL of 318.4 m is achieved after the laser beam is reflected 330 times in the cell. The PSO algorithm is successfully performed to design and optimize MPCs with arbitrary numbers of spherical mirrors and laser beam routes.
Fig. 3. Computed Lissajous patterns formed by multiple-spherical-mirror-based MPCs; the mirrors are rotated to make the centers locate on the z-axis.
To visualize the patterns clearly, we rotate the mirrors to ensure that the centers are located on the z-axis. All spots are projected onto an x-y plane, and the corresponding parameters are listed in Table 1. The PSO-aided method is generalized for designing MPCs with both centrally located holes and off-axis holes. Usually, less spots are reflected on the center areas of mirrors, which can effectively avoid spot overlap near the center hole; off-axis holes provides more choices for the locations of the input beam and the output beam. By analyzing the results calculated by the PSO algorithm, the multiple-spherical-mirror-based MPC can be generally used for all sizes of mirrors with any curvature and diameter. The MPC with small curvature radii has a compact structure and a relatively high OPL to volume ratio. By increasing the curvature radius, MPCs with an OPL of hundreds of meters can be easily achieved to improve the detection sensitivity, which is particularly suitable for open-path detention of trace gases. Based on actual applications, a required MPC with suitable mirror sizes and design parameters can be designed and optimized in a fast and intelligent way.
Table 1. Parameters of the Lissajous patterns show in Fig. 3. The coordinates of the incident spot and the first reflected spot are rotated and projected onto the x-y plane.
Two MPCs with three and five spherical mirrors are built and tested experimentally, and the laser beam paths and observed spot patterns are exhibited in Figs. 4(a)–4(d). For the three-mirror-based MPC, a laser beam is reflected onto neighboring mirrors successively. By adjusting the suitable locations for the incident point and first reflected point, cross-style patterns are formed with a distance d of 81.8 mm, as shown in Fig. 4(c). The OPL is 48.5 m after the beam passes 342 times between the three spherical mirrors. As exhibited in Figs. 4(c) and 4(d), the star laser beam route and the wing spot patterns are formed, and an effective OPL of 38.1 m is achieved after the beam is reflected 260 times. The simulated patterns are exhibited in Figs. 3(b) and 3(c), which are consistent with observed patterns and validate the effectiveness of design methods aided by the PSO algorithm.
Fig. 4. The laser beam propagation and observed patterns on mirrors; (a)(c) three-mirror-based MPC with cross style patterns, (b)(d) five-mirror-based MPC with wing spot patterns.
4. CH$_4$ detection by using an open-path multipass cell
We constructed and developed a five-mirror-based MPC with wing spot patterns as an open-path gas sensor and detected the methane concentrations in ambient laboratory air. The wavelength modulation spectroscopy (WMS) technique was applied to enhance detection sensitivities, and an open-path TDLAS experimental system is shown in Fig. 5. The wing patterns have the advantages of easy identification of reflections and OPLs, and the observed patterns are exhibited in Fig. 5. The curvature radius, diameter and reflectivity of the spherical mirrors are 200 mm, 50.8 mm and $\sim$99$\%$, respectively. With a specific mirror reflectivity, the signal-to-noise ratio (SNR) of an MPC is not a monotonic increasing function of the number of reflections [24], and the optimal number of reflections for the spherical mirrors is about 95. In addition, multiple off-axis reflections from concave mirrors make spot shapes diffuse due to the spherical aberration [22,26]. The deformation of spots can be effectively reduced by strategically choosing design parameters, including the curvature radius, the number of reflections, and the position and incident angles of the input beam [26]. The laser beam passes through a fiber collimator FC and enters the designed MPC with an effective OPL of 28 m. After reflecting 95 times, the beam exits through the same entrance hole and is collected by an InGaAs photodetector (Thorlabs, PDA10CS-EC).
Fig. 5. Schematic diagram of the developed CH$_4$ sensor (FC: Fiber Collimator, PD: Photodetector) and the observed pattern.
The CH$_4$ transition located at 6046.95 cm$^{-1}$ was targeted. A 1653 nm DFB diode laser (NLK1U5FAAA, NEL) was driven by a commercial laser controller (ILX Lightwave, LDC-3724C), and the temperature and laser drive current were set at 34.42 $^{\circ }$C and 69.98 mA, respectively. A LabVIEW program and a multifunctional DAQ card (NI-USB-6363) were used to realize the laser scanning and modulation, data processing and lock-in amplification detection. A sinewave signal with a frequency and amplitude of 20 kHz and 0.65 V, respectively, was superimposed onto a 1 V, 10 Hz saw tooth wave. To calibrate the open-path sensor, a high-sensitivity instrument reported in Ref. [28] was applied to simultaneously measure the methane concentration in ambient air, and the open-path sensor was calibrated based on the corresponding detected methane value.
To evaluate the performance of the developed sensor, continuous measurements of CH$_4$ in ambient laboratory air were implemented for forty minutes. The measurements were performed by averaging ten scans with a scan rate of 10 Hz. A total of 2400 data points were obtained, and the corresponding concentrations are plotted in Fig. 6(a). By analyzing the Allan deviation, a detection precision of 11.3 ppb and 1.1 ppb was achieved with averaging times of 1 s and 123 s, respectively. The detection precision is comparable with previously reported experiments [18,24,28,34]. The histogram plot for the measured concentrations is exhibited in Fig. 6(b). The data follow a Gaussian profile distribution, and the standard deviation value ($\sigma$) and half width at half maximum (HWHM) are 12.0 ppb and 14.2 ppb, respectively. The experiments show that the proposed method has the ability to achieve highly sensitive gas detection and promote the development of open-path gas sensors for air quality monitoring and diverse field applications.
Fig. 6. Continuous measurements of CH$_4$ in ambient laboratory air (top), and Allan deviation analysis for the developed open-path gas sensor (bottom), (b) histogram plot for the measured concentrations.
5. Conclusions
In this work, we propose an intelligent method to design and optimize an MPC with multiple spherical mirrors, which are placed in a rotationally symmetric arrangement. By adjusting suitable design parameters, Lissajous patterns can be formed on mirrors with arbitrary sizes and curvatures. The small-diameter-based MPC has a compact structure with a small curvature radius and can be used to realize an effective OPL of hundreds of meters by increasing the curvature radii. We implemented the PSO algorithm to determine the parameters of the MPC configurations and accelerate the design process. The whole optimization process takes only several minutes, which represents a significant reduction in computational power and calculation time.
The PSO algorithm was successfully performed to optimize the proposed MPCs in a fast and automatic way. Two exemplary MPCs with three and five spherical mirrors were built and tested, and the observed cross-style patterns and wing spot patterns were found to be consistent with the simulated results. A five-mirror-based MPC with wing-like patterns was constructed and developed as an open-path-based gas sensor. Continuous CH$_4$ measurements were carried out in ambient laboratory air, and a detection precision of 1.1 ppb was achieved with an averaging time of 123 s. This high performance validates the effectiveness of the proposed method. This new type of MPC has the advantages of high detection sensitivity, symmetrical structure and cost effectiveness, which provides a useful tool for trace gas sensing, especially in open-path-based TDLAS systems.
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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