ROTHMAN L S, HARGREAVES R J, HASHEMI R, KARLOVETS E V, SKINNER F M, CONWAY E K, HILL C, KOCHANOV R V, TAN Y, WCISŁO P, FINENKO A A, NELSON K, BERNATH P F, BIRK M, BOUDON V, CAMPARGUE A, CHANCE K V, COUSTENIS A, DROUIN B J, FLAUD J M, GAMACHE R R, HODGES J T, JACQUEMART D, MLAWER E J, NIKITIN A V, PEREVALOV V I, ROTGER M, TENNYSON J, TOON G C, TRAN H, TYUTEREV V G, ADKINS E M, BAKER A, BARBE A, CANè E, CSá SZá R A G, DUDARYONOK A, EGOROV O, FLEISHER A J, FLEURBAEY H, FOLTYNOWICZ A, FURTENBACHER T, HARRISON J J, HARTMANN J M, HORNEMAN V M, HUANG X, KARMAN T, KARNS J, KASSI S, KLEINER I, KOFMAN V Gordon I E Kwabia-Tchana F, Lavrentieva N N

RINSLAND C P, GOLDMAN A, MASSIE S T, EDWARDS D P L S Rothman

First of all, I would like to express my deepest appreciation to the supervisors Prof.

Weidong Chen, and Prof. Lei Dong, for allowing my Ph.D. research in the ULCO and SXU, respectively. Thanks for their guidance and encouragement during my three years' work. My Ph.D. progress would not have been possible without their insight and advice, including their attention to the details of my scientific results. That allowed me to grow academically as well as intellectually. In addition to academic guidance, they also taught me how to be a nice person. Finally, I would like to thank my family for their love and support.

III

Abstract

Over the last few decades, monitoring of trace gas composition of the atmosphere has come to loom large in scientific and public concerns. Such knowledge is crucial to deepen our understanding of the sources/concentrations of key atmospheric species, of their roles in atmospheric chemical processes and of their impacts on local air quality, public health and global climate change.

Laser absorption spectroscopy is an efficient analytical tool for optical metrology of trace gases. Multi-pass cells (MPCs) or optical cavities are usually used for long path absorption spectroscopy to enhance measurement sensitivity. Commercial optical multi-pass cells are bulky, expensive, and not suitable for field monitoring applications requiring cost-effective, compact, and lightweight optical sensors, in particular for network and unmanned aerial vehicle (UAV) observations. In order to further enhance the measurement sensitivity, dielectric mirror-based cavities are used, which allow one to reach some tens of km long optical path length. However, coating on the cavity mirror, used to achieve high-reflectivity, is wavelength-dependent which limits the bandwidth of the mirror's high reflectivity to 5-15 % around the designed central wavelength. As a result, cavities formed by dielectric mirrors are only suitable for measurements in a relatively narrow spectral range.

The present PhD thesis reports on the developments of innovative folded-path dense spot patterned multi-pass cells and a prism-based optical cavity, dedicated to high-sensitivity trace gas sensing. The main works achieved during the PhD work are as follows :

(1) Development of a novel calculation model (patented n°CN109407310A) for the design of folded-path optical absorption cells with various dense spot patterns in a small size. One-circle spot-patterned multi-pass cell (MPC), seven-circle spotpatterned MPC, and single-line spot-patterned MPC were designed and applied for use in optical sensors of CO, CH4, and H2O vapor.

(2) Development of a broadband optical cavity based on prisms used as cavity reflectors. Relying on total internal reflection combined with light incidence at IV Brewster's angle to realize low losses and quasi-wavelength independent high optical finesse over a broad spectral range from the near-UV to the near infrared.

Its optical characteristics and spectroscopic performance are evaluated using absorption spectra of NO2 and NO3 radicals over a broad spectral region in the visible.

Keywords: trace gases, spherical aberration, ABCD matrix, multi-pass cell, prismbased cavity, tunable diode laser absorption spectroscopy, quartz-enhanced photoacoustic spectroscopy, cavity-enhanced absorption spectroscopy V

Ré sumé

Au cours des derniè res dé cennies, la surveillance de la composition chimique de l'atmosphè re est devenue une pré occupation majeure des scientifiques et du public. Ces connaissances sont cruciales pour approfondir notre compré hension des sources/concentrations des principales espè ces atmosphé riques, de leurs rôles dans les processus chimiques atmosphé riques et de leurs impacts sur la qualité de l'air local, la santé publique et le changement climatique mondial.

La spectroscopie d'absorption laser est un outil analytique efficace pour la mé trologie optique des traces de gaz. En gé né ral, des cellules d'absorption à ré flexions multiples (MPC) ou des cavité s optiques sont utilisé es pour la spectroscopie d'absorption à long trajet afin d'amé liorer la sensibilité de mesure. Les cellules multipasses disponibles dans le commerce sont encombrantes, coûteuses et ne conviennent pas aux applications sur le terrain né cessitant des capteurs optiques é conomiques, compacts et lé gers, en particulier pour les applications de mesure en ré seau et sur drones. Afin d'amé liorer davantage la sensibilité de mesure, des cavité s optiques formé es de miroirs dié lectriques sont gé né ralement utilisé es, ce qui permet d'atteindre des longueurs du chemin optique de plusieurs dizaines de km. Cependant, le revê tement sur le miroir de cavité , utilisé pour obtenir une ré flectivité é levé e, est dé pendant de la longueur d'onde, par consé quent la bande passante de la haute ré flectivité du miroir est limité e à 5-15% autour de la longueur d'onde centrale conç ue, ce qui ne permet que d'effectuer des mesures dans une bande spectrale relativement é troite.

Le pré sent manuscrit de thè se dé crit les travaux de recherche sur des dé veloppements de cellules innovantes à chemin optique replié et de cavité s optiques à base de prisme, dé dié es à la dé tection de traces de gaz à haute sensibilité . Les principaux travaux ré alisé s durant la thè se sont les suivants :

(1) Dé veloppement d'un nouveau modè le de calcul (brevet n°CN109407310A) pour la conception de cellules d'absorption optique à chemin replié à diverses modè les de spots denses en petite taille : à un cercle, à sept cercles et en ligne unique. Ces cellules d'absorption à chemin optique replié ont é té conç ues puis appliqué es dans VI des capteurs optiques de monoxyde de carbone (CO), de mé thane (CH4) et de vapeur d'eau (H2O).

(2) Dé veloppement d'une cavité optique large bande basé e sur des prismes utilisé s comme ré flecteurs de cavité . S'appuyant sur la ré flexion interne totale combiné e à l'incidence de la lumiè re à l'angle de Brewster pour ré aliser de faibles pertes et de haute ré flectivité qui est quasi-indé pendante de la longueur d'onde sur une large bande spectrale, cette nouvelle approche offre une cavité optique de haute finesse fonctionnant dans une large ré gion spectrale du proche UV au proche infrarouge.

Ses caracté ristiques optiques et ses performances spectroscopiques sont é valué es à l'aide des spectres d'absorption de NO2, des radicaux NO3, et de vapeur d'eau sur un large domaine spectral dans le visible et dans le proche infrarouge, respectivement.

Mots-clé s: traces de gaz, aberration sphé rique, matrice ABCD, cellule multi-passe, cavité à prisme, Spectroscopie d'absorption par diode laser accordable, spectroscopie photoacoustique à diapason en quartz, spectroscopie d'absorption en cavité

List of Figures

List of Tables

List of Abbreviation

Vital role of trace gas detection

As indicated by scientist Mendeleev, "Any science begins with measurement, and exact science cannot be imagined without measurement", progress in science is closely linked to the advancement of observation and measurement technology. Advanced trace gas detection technology not only enriches the research methodologies of atmospheric science but also plays a vital role in industry, agriculture, public health, and environmental protection.

Trace gases are referred to gases whose mixing ratios are at the levels lower than part per million by volume (ppmv). In environmental sciences, measurements of trace gases in the atmosphere of Earth, including carbon dioxide (CO2), methane (CH4), water vapor (H2O), ozone (O3), nitrogen oxides (NOx), and volatile organic compounds (VOCs) are of significant interest for understanding key atmospheric chemical processes and global climate change trends [1] . In the field of medical diagnostics, noninvasive quantitative analysis of biogenic trace gases, as volatile disease markers, in exhaled breath have gained increasing interest in recent years [2] . These markers can provide important information about the health status of humans [3] . Most of these volatile molecules are present in volume fractions of the order of ppbv or lower.

Gas detection has traditionally relied on laboratory analytical equipment, such as gas chromatographs [4] , mass spectrometry [5] , or low-cost devices, such as semiconductor gas sensors [6] and electrochemical gas sensors [7] . Though gas chromatographs and mass spectrometry offer accurate data, they require sample preparation, and the instruments are very expensive. Semiconductor gas sensors and electrochemical gas sensors, while affordable, are prone to drift and cross-responses to other gases and changing humidity levels [5] . Furthermore the electrochemical gas sensors have limited lifetimes.

Optical sensing of trace gases

Optical sensing methods based on long-path absorption spectroscopy (LPAS) are powerful tools for characterizing trace gas with the following unique advantages : (1) short response time (which can be less than 1 s); (2) high sensitivity and high specificity;

(3) minimal drift; (4) zero cross-response to other gases if the design of the approach is carefully considered [5,10] . When light propagates along an absorption path filled with a target gas, a portion of the light is absorbed due to transitions of the gas from lower to higher energy levels of the atomic or molecular system. Based on the Beer-Lambert law, the target gas concentration can be quantified by its absorption strength [8] . As the intensity transmitted by the sample follows an exponential law as a function of the optical absorption length, the use of long path absorption schemes is a useful way to significantly improve the spectroscopic measurement sensitivity. The improvement in signal-to-noise ratio (SNR) generally scales with the effective pathlength, Leff. For in situ measurements, long path absorption can be achieved by means of : (1) open-path experimental designs provide Leff of ~ km resulting in sub-ppbv detection limits for measurements of integrated trace gas mixing ratios over the entire path length; (2) multiple reflections of probing light in an optical cell (multi-pass cell with Leff up to ~ 200 m leading to ppmv-ppbv detection limits) or in an optical resonant cavity (with Leff up to more than 10 km allowing ppbv-pptv even ppqv detection limit). Long path optical absorption approaches can be operated in open air or they can be enclosed in a vacuum chamber for measurement of static gas samples at a reduced pressure or using controlled gas flows at arbitrary pressures.

Folded-path optical cell-based techniques

In the present thesis, folded-path optical cells are considered through two approaches: multi-pass cells (MPCs, such as Herriott [9,10] , White [START_REF] White | Long Optical Paths of Large Aperture[END_REF] , and Chernin [START_REF] Chernin S M | Optical multipass matrix systems[END_REF] cells) and optical cavities [START_REF] Adler F | Cavity-enhanced direct frequency comb spectroscopy: technology and applications[END_REF] .

Over the last few decades, various spectroscopic techniques involving LPAS have been developed for sensitive gas sensing, such as tunable diode laser absorption spectroscopy (TDLAS) [START_REF] Lackner | Tunable Diode Laser Absorption Spectroscopy (Tdlas) in the Process Industriesa Review[END_REF][START_REF] Cui R | Three-Dimensional Printed Miniature Fiber-Coupled Multipass Cells with Dense Spot Patterns for ppb-Level Methane Detection Using a Near-IR Diode Laser[END_REF][START_REF] Schiff H I | The use of tunable diode laser absorption spectroscopy for atmospheric measurements[END_REF] , long-path (MPC-, cavity-) enhanced photoacoustic spectroscopy (PAS) [START_REF] Bluvshtein | Calibration of a multi-pass photoacoustic spectrometer cell using light-absorbing aerosols[END_REF][START_REF] Nä | Mobile laser spectrometer with novel resonant multipass photoacoustic cell for trace-gas sensing[END_REF][START_REF] Metcalfe G D | Cavity-Enhanced Raman and Helmholtz Resonator Photoacoustic Spectroscopy to Monitor the Mixed Sugar Metabolism of E. coli[END_REF] or quartz-enhanced photoacoustic spectroscopy (QEPAS) [20- 22] , cavity ring-down spectroscopy (CRDS) [START_REF] Maity | Cavity Ring-Down Spectroscopy: Recent Technological Advancements, Techniques, and Applications[END_REF][START_REF] Brown | Absorption spectroscopy in high-finesse cavities for atmospheric studies[END_REF][START_REF] Spence T G | A laser-locked cavity ring-down spectrometer employing an analog detection scheme[END_REF] and cavity enhanced absorption spectroscopy (CEAS) [START_REF]Integrated cavity output analysis of ultra-weak absorption[END_REF][START_REF] Engeln R | Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy[END_REF][START_REF] Zhao W | Wavelength modulated off-axis integrated cavity output spectroscopy in the near infrared[END_REF] .

Folded-path MPC with dense spot pattern based on spherical mirrors has been firstly reported by Stephen So and co-worker in 2012 [29] . They accidentally discovered the dense spot configuration MPCs offers new opportunities to provide longer optical paths in a small volume with minimal beam overlap. They are more compact and smaller in size compared to traditional Herriott MPCs (for instance, a seven-multi-circle spot patterned MPC has a physical length 7 times shorter than a Herriott MPC for the same effective pathlength). While compared to modified Herriott MPCs involving astigmatic mirrors to realize dense spot Lissajous pattern, this kind of MPC employs two simple spherical mirrors to achieve dense spot pattern, resulting in low cost.

A folded-path MPC with a seven-multi-circle spot pattern (Fig. 1.1) was reported by Liu et al. in 2015 using two 2-inch spherical mirrors [START_REF] Liu | Highly sensitive detection of methane by near-infrared laser absorption spectroscopy using a compact dense-pattern multipass cell[END_REF] . Such a folded-path optical cell achieves a 26.4-m optical path length. a seven-multi-circle spot pattern (photographs above the schematic). Figure taken from Ref. [START_REF] Liu | Highly sensitive detection of methane by near-infrared laser absorption spectroscopy using a compact dense-pattern multipass cell[END_REF] .

In 2016, R. Sharma et al. presented the development of a gas analyzer system (Fig. 1.2) based on high-pressure Raman scattering in a folded-path MPC with a line spot pattern and demonstrated its feasibility for real-time natural gas analysis [START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] . [START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] .

All the beams in this folded-path optical cell were manipulated by optical design and focused in the center of the folded-path MPC, leading to sufficient power accumulation for improved detection of Raman signals.

In 2020, Chen et al. designed a high-sensitivity photoacoustic gas detector by employing a miniature folded-path MPC (Herriott cell type) for C2H2 detection [START_REF] Zhang | High-sensitivity photoacoustic gas detector by employing multi-pass cell and fiber-optic microphone[END_REF] . In 2021, Ma et al. designed a line spot pattern folded-path optical cell based on two rightangle prisms for coupling light to a quartz tuning fork (Fig. 1.3), which allowed to further improve the measurement sensitivity by a factor of 3.2 [START_REF] Qiao | Multi-pass quartz-enhanced photoacoustic spectroscopybased trace gas sensing[END_REF] . In all of the reported developments, theoretical guidance for design of dense spot pattern folded-path MPC was lacked which limited the ability to manipulate laser beams in folded-path MPC and to develop other viable folded-path MPC configurations using various dense spot patterns. Hence, it is necessary to develop a folded optical design method that can precisely manipulate ray trajectories in optical cells to produce specific dense spot pattern and hence to provide different types of MPC for specific sensing applications with high sensitivity, small size, low cost, and robustness.

Design of traditional folded-path MPC

Many versatile folded-path optical cells have been designed. One of the earliest examples is the Herriott cell [9] , which was first introduced in 1964. This type of optical cell is composed of two spherical mirrors separated by a distance nearly equal to their common radius of curvature, which can produce a typical one-circle spot pattern on its mirror surface, as shown in Fig. 1.4. In this system, the optical beam is injected through a hole in one spherical mirror and is reflected back and forth many times before exiting from the same hole. This design ensures that the beam remains collimated during transmission and reflection throughout the cell, unlike the White cell, which is formed by three spherical mirrors [START_REF] White | Long Optical Paths of Large Aperture[END_REF] . Herriott cells, owing to their simple design and efficient beam handling, are a popular choice in many optical applications. A further advantage of the Herriott cell is that it is easier to align compared to the White cell, since the direction of the exit beam is not sensitive to changes in the mirror alignment. In the case of spherical mirrors, the beam traces out a one-circular spot pattern on the two mirror surfaces, which cannot give optimum use of the mirror area. The number of passes is determined by the mirror spacing, while the mirror spacing is difficult to adjust due to specific folded-path optical cell design theory. Another folded-path optical cell was later developed by using slightly astigmatic mirrors to create Lissajous spot patterns on the mirror surfaces [START_REF] Herriott D R | Folded Optical Delay Lines[END_REF] , as shown in Fig. 1.5.

As a result, the number of ray reflections in the folded-path optical cell is increased to reach higher utilization efficiency while minimizing the spot overlap [START_REF] Mcmanus | Astigmatic mirror multipass absorption cells for long-path-length spectroscopy[END_REF][START_REF] Gragossian | Astigmatic Herriott cell for optical refrigeration[END_REF] . However, mirrors with high surface accuracy need to be manufactured by grinding and polishing techniques that naturally produce spherical surfaces. The manufacture of aspheric surfaces is more complex, and it is difficult to produce a mirror of sufficient surface accuracy to follow the designed pattern. Therefore, Herriott cells based on spherical mirrors are widely deployed due to the cheaper mirror production and the fact that the quality of the mirror surfaces can be easier controlled [START_REF] Cui R | Calculation model of dense spot pattern multi-pass cells based on a spherical mirror aberration[END_REF] . Metallic mirrors used in MPC approaches can provide quasi wavelength-independent reflectivity of about 95-98 % over a broadband spectral region. However, limited by their relatively low reflectivity, the number of reflections inside the MPC is limited to 100-200, which can typically provide a max Leff of about ~ 200 m (for a physical length of about 1 m). In order to further enhance the measurement sensitivity using longer optical path lengths, dielectric mirror-based cavities can be used. High reflectivity (> 99.9%, up to 99.999%) of the dielectric mirror is achieved via constructive interference of the Fresnel reflection at the interfaces produced by multilayer coatings of alternate high and low refractive index materials [START_REF] Pacheco M Z | Applicability Evaluation of a Laser Light-Mater Interaction Based Computational Tool on Status Identification of Applied Micro-Structured Coatings[END_REF] . Cavities with high reflectivity mirrors enable some tens of km optical path lengths [START_REF] Provencal R | Cavity-enhanced quantum-cascade laser-based instrument for carbon monoxide measurements[END_REF][START_REF] Langridge | Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source[END_REF] with small sample volume. As the coating thickness on the cavity mirror is wavelength-dependent, the bandwidth of the mirror's high reflectivity is limited to 5-15 % around the designed central wavelength. As a result, dielectric mirror cavities are only suitable for measurements in a relatively narrow spectral band.

Optical cavities based on prisms as reflectors can offer high-finesse performance over a broad spectral region from 400 to 2500 nm [START_REF] Meng | Prism-based Broadband Optical Cavity (400 -1600 nm) for High-Sensitivity Trace Gas Sensing by Cavity Enhanced Absorption Spectroscopy[END_REF] . Prism-based cavities provide a versatile spectroscopic measurement tool for high-sensitivity gas sensing.

Objectives of the PhD research

The PhD thesis consists of six chapters as follows:

Chapter 2 overviews the main aspects of optical sensing of trace gases in particular long path absorption spectroscopy based on folded-path optical modules (multi-pass cells and optical cavities).

In ). 3D printing with a printing resolution of 100 µm is adopted to fabricate the MPC frame, which ensures a high level of integration of all components. A NIR DFB laser emitting at 1.65 µm was used as a light source to detect methane at mixing ratios in the ppbv range.

In section 4.3, the development of a single-line spot-patterned MPC is presented.

It was applied to a QEPAS sensor systems [START_REF] Dong L | QEPAS spectrophones: design, optimization, and performance[END_REF][START_REF] Ma | Review of Recent Advances in QEPAS-Based Trace Gas Sensing[END_REF][START_REF] Wojtas | Mid-Infrared Trace Gas Sensor Technology Based on Intracavity Quartz-Enhanced Photoacoustic Spectroscopy[END_REF] . A laser beam is reflected back and forth between two MPC mirrors by passing through a custom-made large prong spacing size quartz tuning fork (QTF) inserted inside the MPC, which produces multiple acoustic point sources along the QTF symmetry axis. As a result, the QTF operating at a fundamental resonance mode is simultaneously excited by acoustic waves from multiple acoustic point sources, the resulting acoustic signal allows one to enhance the measured signal.

Chapter 5 describes recent developments of a novel folded-path optical cell based on prisms as cavity reflectors that offers a high-finesse optical cavity operating in a broadband spectral region from 400 to more than 2500 nm. This novel cavity approach has good potential for high-sensitivity spectroscopic measurements of multi-species or/and broad band absorption of trace gases and aerosols [START_REF] Meng | Prism-based Broadband Optical Cavity (400 -1600 nm) for High-Sensitivity Trace Gas Sensing by Cavity Enhanced Absorption Spectroscopy[END_REF] .

Chapter 6 summarizes the main achievements in this PhD thesis and outlines potential future work, such as the development of a miniature optical sensor for UAVbased observation, and applications of the broadband prism cavity to high-sensitivity frequency-comb spectroscopy.

Chapter 2

Gas sensing by absorption spectroscopic techniques OBJECTIVE Long-path absorption spectroscopy (LPAS) provides an efficient, fundamental, and non-invasive method for identifying and quantifying target molecules or atoms in the gas phase. The basic principle of LPAS relies on the Beer-Lambert law. When the light propagates along a long path through a target gas, the power of the light can be absorbed due to transitions from lower to higher energy levels of the target gases' atoms or molecules. This chapter will overview the principles of laser absorption spectroscopy for gas sensing as well as spectral analysis of the target gas features. The structures of foldedpath optical cells (such as multi-pass cells and optical cavities) will be introduced. Spectroscopic techniques based on multi-pass cells, or optical cavities, will be described in this chapter.

Principle of long-path absorption spectroscopy (LPAS) for gas sensing

The fundamentals of laser absorption spectroscopy and associated instrumental techniques have been discussed widely elsewhere [8,[START_REF] Lackner | Lasers in Chemistry[END_REF][START_REF] B C N A M E | Fundamentals of Molecular Spectroscopy 4th edn[END_REF] . Laser absorption spectroscopy detects and characterizes a medium based on studying the attenuation of the incident light radiation propagating through a sample due to an interaction between the light and molecules, atoms, or any other form of matter in the sample cell. Under specific conditions, the concentration of the absorber in the sample can be derived based on the Beer-Lambert law. According to Fig. 2.1, a proportion of the laser energy can be absorbed after the incident light passes through a medium with an even distribution along the path length L, since the light frequency resonates with the gas absorption line. The attenuated light intensity is frequency-dependent and follows the Beer-Lambert law:

LPAS basic principle

( ) ( ) t0 exp( ) I v I v L  =- (2.1)
Where I0 (v) is the radiation intensity of incident light at frequency v on the sample cell without the absorber, It (v) is the radiation intensity of the light transmitted through the sample cell filled with the absorber, α (in units of cm -1 ) is the absorption coefficient of the sample and L (typically in units of cm) is the optical path-length.

The absorption coefficient α is the product of the sample gas number density c

(molecules•cm -3 ) and the frequency-dependent absorption cross-section σ(v) Where S * is the integrated line intensity of a transition (cm -1 /molecules•cm -2 ), 𝜙 is the frequency-dependent area-normalized absorption line shape function (1/cm -1 ) which depends on the measurement conditions such as sample pressure and temperature. In general, absorption line intensity S (cm -2 atm -1 ) can be expressed as:

(cm
() v S cP  = (2.8)
where c is the number density of the absorber.

For αL ≪ 1, Equation (2.1) is conveniently linear with α, c and L, as follows:

0 ΔI L S cPL I  = (2.9)
Here, ΔI = I0 -I, and ΔI / I0 is the absorbance, which is unitless but often described in "absorbance units" (AU). The attenuation of the radiation is measured by a detector, which can provide the absorption spectrum of a particular transition. An absorption spectrum is a plot of α as a function of wavelength or wavenumber. Public-domain quantified spectra are available from the US National Institute of Standards and Technology (NIST), and Pacific Northwest National Laboratory (PNNL) [START_REF] Sharpe S W | Gas-phase databases for quantitative infrared spectroscopy[END_REF] and may be calculated using the information in the Hitran database [47,48] .

Various chemical species display strong absorption in different regions of the electromagnetic spectrum, such as ultra-violet (UV), near-infrared (NIR), or midinfrared (MIR). These absorption lines or bands enable the identification of each species, forming the basis for LPAS gas sensing. The absorption spectra in different spectral regions exhibit distinct characteristics, as presented in Table 1. In the so-called fingerprint region of the MIR, the absorption spectra of gases exhibit narrow lines resulting from molecular vibrations at discrete energy levels. These can be measured with high resolution, allowing rotational lines to be resolved or at lower resolution, enabling the measurement of the vibrational absorption band. NIR spectra typically represent overtones of fundamental vibrations in the MIR, and hence they can be substantially weaker (e.g., approximately 100 times weaker for methane).

However, the availability of high-quality light sources and detectors, primarily derived from telecommunications applications, can counteract this disadvantage, and the signal-to-noise ratios can be relatively high.

Absorption line profile

Strictly speaking, the frequency is not monochromatic at a single absorption one line. The absorption line profile function reflects the relative change of the absorption coefficient with respect to frequency. The line profile function, represented by 𝜙, is a normalized function, meaning that its integration over the entire frequency domain is equal to 1 as described in Equation (2.10). An absorption line has a pressure-broadened Lorentzian profile [8] as described by Equation (2.11):

22 0 ( ( ) ) cS       = +- (2.11)
Where S is the line intensity (cm -1 /molecule cm -2 ), γ is the line halfwidth at half maximum (HWHM, cm -1 ), ν is the wavenumber (cm -1 ) and ν0 is the position of the line center. Fig. Accurate measurement of linewidth is critical in high-resolution measurement schemes, and it is important to consider the variations in linewidth at different pressures.

The background matrix for the gas can also affect the linewidth. To account for these effects, self-broadening and air-broadening parameters are commonly used. Selfbroadening refers to the linewidth for a gas surrounded by itself, whereas airbroadening refers to the linewidth for small quantities of the gas in air. These parameters are well-known for commonly detected gases, and the Hitran database provides known halfwidths for air-broadened and self-broadened lines (designated as γair and γself, respectively).

At atmospheric pressure, the halfwidths are dependent on both pressure and temperature. Therefore, it is important to consider the pressure and temperature of the gas when calculating the value of γ. Specifically, for gas at pressure P (atm), temperature T (K), and a partial pressure Ppartical (atm), the value of γ is given by [47] :

( )

air partical self partical P P P    = - + (2.12)
When dealing with significantly different matrices, it may be necessary to measure the linewidth, particularly for gases that exhibit strong interactions with their neighbors. This is the case with water vapor, for example, which can form hydrogen bonds.

Nonetheless, it is important to note that, regardless of the specifics of the molecular environment, the integrated area under the curve is always proportional to the number of molecules present.

Structures of folded-path optical cells

There are two ways to improve the sensitivity of LPAS: by increasing the absorption signal or by reducing the noise on the signal. The value of σ is restricted by the transitions accessible to the laser, which should be free from complications arising from interference with absorption from other species. According to Equation (2.1), the only other adjustable parameter that can be changed is the optical path length (OPL), L, since the concentration of the target gas is the independent variable. The method by which the path length is increased varies on the application and environment.

The sensitivity of most LPAS sensing systems is limited not only by laser or detector noise, but also by optical fringes superimposed on the measured spectrum [START_REF] Lackner | Lasers in Chemistry[END_REF][START_REF] Werle | Laser excess noise and interferometric effects in frequencymodulated diode-laser spectrometers[END_REF] .

Multi-pass cells (MPCs)

To increase the OPL within a limited space, one solution is to utilize an MPC.

Such folded-path optical cells achieve long OPLs by using mirrors to fold the optical path, giving typically ~ 100 pass counts for a 50 cm base-length optical cell [START_REF] Lackner | Lasers in Chemistry[END_REF] . Design criteria for MPCs involve a trade-off between total path length (OPL), volume (V), pass counts (N) of the beam in the MPC, and the quality of the exit beam. One of the earliest designs of such a system was constructed by White in 1942, which consisted of three spherical mirrors. A single large spherical mirror is placed opposite a pair of smaller spherical mirrors in such a way that a beam directed past the edge of the single mirror onto one of the pair will bounce back and forth several times before exiting past the opposite edge [START_REF] White | Long Optical Paths of Large Aperture[END_REF] . Small adjustments to the mirrors allow easy tunability of the number of passes, and thus the path length, within the sample volume. A typical cell of this type would have a 50 cm mirror spacing, a volume of the MPC is 10 L, and an effective optical path-length of 50 m [START_REF] Lackner | Lasers in Chemistry[END_REF] . Since in this MPC, the spots on the mirror surface can overlap after every 4th pass count of the beam, as shown in Fig. 2.3, there will be a standing wave, producing a fringe system with a period of 0.005 cm -1 or 150 MHz. The extent of this fringe system is comparable to the absorption linewidth, so it is difficult to distinguish. In 1965, a similar system was designed by using only two spherical mirrors that were separated by nearly their radius of curvature, which is called a Herriott cell [9] . In this system, the optical beam is injected through a hole in one spherical mirror and is reflected back and forth several times before exiting from the same hole. The ray trajectories of the Herriott cell are described in Fig. 2.3. Unlike the White cell, the beam remains collimated during transmission and reflection throughout the cell. One of the advantages of the Herriott cell is its ease of alignment compared to the White cell. This is because changes in the mirror alignment do not significantly affect the direction of the exit beam. However, in the case of spherical mirrors, the beam traces out a circular spot pattern on the two mirror surfaces, which does not take advantage of the full mirror area. Although the number of passes is determined by the mirror spacing, adjusting the mirror spacing is difficult due to the specific MPC design theory. To overcome this issue, another MPC was developed using slightly astigmatic mirrors typically employed to modify the Herriott cell's arrangement [START_REF] Herriott D R | Folded Optical Delay Lines[END_REF][START_REF] Mcmanus | Astigmatic mirror multipass absorption cells for long-path-length spectroscopy[END_REF] . As a result, this MPC creates Lissajous spot patterns, increasing the number of ray reflections while minimizing spot overlap [36] . Astigmatic mirrors designed for MPCs are known to possess higher utilization efficiency of the mirrors. However, high-precision mirrors are typically produced using grinding and polishing techniques, resulting in naturally occurring spherical surfaces.

The manufacturing process for aspheric surfaces is more complex, and it is challenging to achieve the necessary surface accuracy required to conform to the design specifications. Consequently, spherical mirrors are widely used in MPCs due to their simplicity, ease of surface quality control, and lower cost [START_REF] Cui R | Calculation model of dense spot pattern multi-pass cells based on a spherical mirror aberration[END_REF] .

The thesis proposes a novel model for the design of MPCs, which employs two spherical mirrors. A theoretical description of the MPC and its stability conditions is provided. Design considerations leading to practical MPCs with a dense spot pattern are discussed in Chapter 2. Furthermore, Chapter 3 details the implementation of various gas sensing methods utilizing dense spot-pattern MPCs.

Optical cavities

Another method to increase OPL in a limited space is using an optical cavity. An optical cavity is a set of mirrors coated with a highly reflective dielectric coating, resulting in extremely high reflectivity at a specific wavelength [START_REF] Hood C | Characterization of high-finesse mirrors: Loss, phase shifts, and mode structure in an optical cavity[END_REF] . The use of highreflectivity dielectric mirrors to construct a high-finesse optical cavity enables one to achieve long OPLs of up to several kilometers for high-sensitivity spectroscopy applications [START_REF] Brown | Absorption spectroscopy in high-finesse cavities for atmospheric studies[END_REF][START_REF] Nakagawa | Highly sensitive detection of molecular absorption using a high finesse optical cavity[END_REF] . The most common type of optical cavities consists of two opposing high-reflectivity mirrors separated by a distance L with an equal reflectivity R(λ) (>99.9 % at wavelength λ). The effective optical path length, Leff = L/(1-R) can be several thousand times the cavity base length L (~0.5-1 m), reaching several kilometers.

Multilayer dielectric mirror coatings are specifically selected for design wavelengths.

The spectral coverage of the highest reflectivity mirrors is limited to only a few percent of the central design wavelength. For simultaneous multi-species detection using a single optical cavity, broadband high reflectivity mirrors are required. Although, with the proper choice of materials and thickness for the layers, dielectric mirrors cannot be designed over a wide range of wavelengths from the near-UV to the infrared. These mirrors achieve high reflectivity by constructive interference of the Fresnel reflection of many interfaces produced by multilayer coatings of alternate high and low-index materials. The interference conditions limit the wavelength coverage of the highest reflectivity mirrors. Losses will approximately double for a shift in wavelength of ± 5% from the central design wavelength and increase rapidly thereafter [START_REF] Birch | Introduction to Modern Optics[END_REF][START_REF] Lehmann | Recent Advances in Cavity Ring-Down Spectroscopy[END_REF] .

Recently, prism reflector-based cavities were developed. They rely on total internal reflection combined with light incidence at Brewster's angle to realize low losses and quasi-wavelength independent high reflectivity over a broadband from the UV to the near-infrared [START_REF] Langridge | Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source[END_REF][START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF][START_REF] Lee | A high-finesse broadband optical cavity using calcium fluoride prism retroreflectors[END_REF][START_REF] Moosmüller | Brewster's Angle Porro Prism: A Different Use for a Pellin-Broca Prism[END_REF][58][START_REF] Engel | Ring down spectroscopy with a Brewster's angle prism resonator[END_REF] . Theoretical bandwidth of the high reflectivity is limited by critical angle conditions for total internal reflection at the internal surface of the prism. Here optical cavities based on Brewster-angle prisms are more suitable for broadband cavity-enhanced absorption spectroscopy. The development of folded-path optical cells based on two prism retroreflectors, as an optical cavity, is detailed in Chapter 5.

Advanced LPAS techniques

Many researchers have been focusing on developments of advanced LPAS (such as "Tunable diode laser absorption spectroscopy (TDLAS)", "photoacoustic spectroscopy (PAS)", "Quartz-enhanced photoacoustic spectroscopy (QEPAS)", "Cavity ringdown spectroscopy (CRDS)" and "Cavity enhanced absorption spectroscopy (CEAS)") based on folded-path optical cells for gas sensing, which have unique and excellent properties, such as long-term stability, inherent self-monitoring, and lowest possible cross-sensitivity to other gases when the laser source is suitable. In this section, we will detail some of the current advanced LPAS technology methods for trace gas detection. TDLAS, pioneered in 1960, is a spectroscopic method for trace gas sensing [START_REF] Werle | A review of recent advances in semiconductor laser based gas monitors[END_REF] .

Tunable Diode Laser Absorption Spectroscopy (TDLAS)

TDLAS is a direct absorption technique and is usually carried out in a long-path absorption cell such as a Herriot cell [START_REF] Li | A review of signal enhancement and noise reduction techniques for tunable diode laser absorption spectroscopy[END_REF][START_REF] Jiang | Multi-gas detection in power transformer oil based on tunable diode laser absorption spectrum[END_REF][START_REF] Jiang | TDLAS-based detection of dissolved methane in power transformer oil and field application[END_REF] , with a tunable light source. To further improve the sensitivity of the TDLAS gas sensing system, a 2f-wavelength modulation is an effective method for reducing the excessive laser 1/f noise [START_REF] Li | Compact TDLAS based optical sensor for ppb-level ethane detection by use of a 3.34 μm room-temperature CW interband cascade laser[END_REF][START_REF] Köhring | QCL-based TDLAS sensor for detection of NO toward emission measurements from ovarian cancer cells[END_REF][START_REF] Bomse D S | Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser[END_REF] . The modulation technique in TDLAS provides an advantage because the amplitude of the difference signal is proportional to the target gas concentration [START_REF] Werle | A review of recent advances in semiconductor laser based gas monitors[END_REF][START_REF] Shemshad | A review of developments in near infrared methane detection based on tunable diode laser[END_REF][START_REF] Sun | Highly sensitive acetylene detection based on a compact multi-pass gas cell and optimized wavelength modulation technique[END_REF] , and the signal is detected at harmonics of the modulating frequency, where the laser noise is significantly reduced [69,70] .

In modulation techniques, the signal to be measured is modulated which transforms it from a highly noisy low-frequency region to a less noisy high-frequency region. The noise components associated with the signal are then removed by filtering.

This noise-free signal is again transformed back to the original frequency region by the process of demodulation.

The optical probe beam with a sinusoidal modulation having modulation index m, modulation frequency ωm, and optical carrier frequency ω0 is given by:

  00 00 ( ) exp( )exp ( ) exp( ) ( ) exp( ) mm nm E t E i t i t msin t E i t J m jn t     =+ = (2.13)
Written as summation of n th -order Bessel functions, Jn, and the co-efficient of Jn characterizes frequency components of the modulated light spectrum [START_REF] Gupta | Multipass Optical Cavity Based Technique for Trace Level Detection of Heavy Water[END_REF] . Depending on values of modulation index m and frequency ωm, modulation spectroscopy is divided into two categories: Frequency Modulation Spectroscopy (FMS) and Wavelength-Modulation Spectroscopy (WMS) [START_REF] Bomse D S | Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser[END_REF][START_REF] Silver | Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods[END_REF][START_REF] Rieker G B | Calibration-free wavelengthmodulation spectroscopy for measurements of gas temperature and concentration in harsh environments[END_REF] . In FMS, ωm is greater than the half-width (Γ) of the probed absorption feature (100 MHz to several GHz range) and the modulation index m << 1. In WMS, ωm is less than Γ (kHz to several MHz range) with m >> 1.

Recent advancements in MPC, particularly a dense spot pattern-MPC, have enabled the development of compact sensors that use TDLAS combined with WMS and demonstrate excellent sensitivity. In Chapter 4, an MPC-based TDLAS gas sensing system will be detailed.

Photoacoustic Spectroscopy (PAS)

Photoacoustic spectroscopy (PAS) differs from direct absorption spectroscopy like TDLAS or CEAS (in Section 2.3.6) in the way that the absorbed light is detected on basis of the resulting pressure wave in a surrounding medium [START_REF] Rosencwaig | Photoacoustic spectroscopy. New tool for investigation of solids[END_REF][START_REF] West G A | Photoacoustic spectroscopy[END_REF][START_REF] Schmid | Photoacoustic spectroscopy for process analysis[END_REF] . In TDLAS or CEAS sensing systems, measuring the transmitted light intensity in the presence and absence of target gas absorption respectively is a basic method to derive the level of absorbed light according to Equation (2.1). In the PAS sensing system, the absorbed light is indirectly measured without monitoring the intensity of the light transmitted by the sample, as illustrated in Fig. 2.5. Light energy is absorbed by the target gas and then converted into heat. This generally occurs via molecular collision-induced non-radiative relaxation of excited states in the target molecule [START_REF] Patimisco | Quartz enhanced photoacoustic sensors for trace gas detection in the IR and THz spectral range; proceedings of the THz and Security Applications: Detectors, Sources and Associated Electronics for THz Applications[END_REF] . The temperature increase causes the analyte and surrounding medium (gas, liquid or solid) to expand. If the light is chopped or modulated, the expansion produces pressure waves that can be detected using a microphone [START_REF] Dumitras | Laser photoacoustic spectroscopy: principles, instrumentation, and characterization[END_REF] .

In the linear regime of the Beer-Lambert law (Equation (2.1)), the signal SPAS from a resonant photoacoustic cell is equal to [START_REF] Mordmüller | An electrically and optically cooperated QEPAS device for highly integrated gas sensors[END_REF] :

oo PAS L cI Q S fV   = (2.14)
where I0 is the optical power in W, Q is the quality factor of the resonant PAS cell, V is the cell volume, L0 (a few cm in length) is the length of the PAS cell, f is the modulation frequency and η is a system efficiency factor that considers the microphone transfer function and other potential loss factors. In the PAS cell, if a MPC were used, ideally, a mirror reflectivity of 100 % is assumed, the signal SPAS should be equal to:

0 11 1 n ii A oo PS i L cI Q NL cI Q NcI Q S fV fV fA       -- = =  =  (2.15)
where N is the pass count in the MPC, and A is the cross-sectional area of the cell. It can be seen from Equation (2.15) that a smaller cross-sectional area and larger pass counts in an MPC-based PAS gas sensing system are desired to increase the PAS signal [START_REF] Nä | Mobile laser spectrometer with novel resonant multipass photoacoustic cell for trace-gas sensing[END_REF][START_REF] Zhao | Ultra-high sensitive photoacoustic gas detector based on differential multi-pass cell[END_REF][START_REF] Rey | Investigation and optimisation of a multipass resonant photoacoustic cell at high absorption levels[END_REF] . However, the microphone's wide response frequency makes it susceptible to ambient noise, which may affect the accuracy of measurements. Furthermore, the large volume of the photoacoustic cell limits its suitability for measuring gases in small samples.

Quartz-enhanced PAS (QEPAS)

The quartz-enhanced PAS (QEPAS) system of Rice University was designed with acoustic noise suppression in mind [START_REF] Kosterev | QEPAS detector for rapid spectral measurements[END_REF] . The sensors module with a small sample volume (a few mL in volume) is very compact. The quartz tuning fork (QTF) at the heart of the sensor is a quadrupole, which is immune to externally generated noise, because the environmental noises rend the prongs of the tuning fork moving in the same direction which is less likely to induce quadrupole vibrations [82] . An example of such a QTF is shown in Fig. 2.6. The piezoelectric current generated by the QTF is detected using a trans-impedance amplifier, giving a normalized αmin in the range 10 -8 -10 -9 cm -1 excitation laser, an MPC or an optical cavity can be coupled to a QEPAS sensing system for improving the QEPAS signal [START_REF] Qiao | Multi-pass quartz-enhanced photoacoustic spectroscopybased trace gas sensing[END_REF][START_REF] Borri | Intracavity quartz-enhanced photoacoustic sensor[END_REF] . In Chapter 4, an MPC-QEPAS gas sensing system will be detailed.

Fig. 2.6

Detection of the photoacoustic effect in a traditional QEPAS gas sensing system.

Cavity ring down spectroscopy (CRDS) Fig. 2.7 Idea of CRDS (top) and CEAS (bottom) operation

Cavity-enhanced spectroscopic techniques utilize an optical cavity comprised of two or more highly reflective mirrors as mentioned in Section 2.2. The first cavityenhanced technique is cavity ring-down spectroscopy (CRDS), which was used to measure an absorption spectrum of oxygen in open air [START_REF] Deacon D A G | Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources[END_REF] . Developed by O'Keefe and Deacon in 1988, it's based on the measurement of the exponential decay of light intensity leaking from a high-finesse cavity. The ring-down time (τ) is used to calculate the concentration of the absorbing species present in the cavity without any secondary calibration. The decay time of an optical cavity is given by:

1 [(1 ) ] c R L L   -+ = (2.16)
When the cavity is empty (α=0), the determining decay time is given by the reflectivity R of the mirrors in this optical cavity and the cavity length or mirror spacing L, the decay time τ0 is derived from:

1 (1 ) o cR L  - = (2.17)
Hence, the absorption coefficient is derived from the Equation (2.15) and (2.16) and given by:

11 1 o c    =-  (2.18)
These techniques also offer a good temporal resolution, compact instrumentation, and the ability to conduct quantitative concentration analysis from the acquired spectra.

The CRDS technique has been widely applied to measure ppbv (10 -9 ) mixing ratios of atmospheric trace gases [START_REF] Kosterev A A | Cavity ringdown spectroscopic detection of nitric oxide with a continuouswave quantum-cascade laser[END_REF] .

However, CRDS gas sensing has a drawback in that only a few photons enter or escape the optical cavity, which results in low-power being incident on the detector for analysis. Additionally, the performance of CRDS depends on various factors such as laser beam quality, optical alignment, mechanical and thermal stabilities, aerosols, Rayleigh scattering, and the contamination of mirrors [START_REF] Mazurenka | 4 Cavity ring-down and cavity enhanced spectroscopy using diode lasers[END_REF] .

Cavity-enhanced absorption spectroscopy (CEAS)

CRDS and CEAS are two measurement techniques used in LPAS. CRDS is a timedomain method that measures the absorption of light by analyzing the ringdown times or phase shifts. In contrast, CEAS measures the intensity of light that leaks out from the cavity and integrates it over a long period [START_REF] Mazurenka | 4 Cavity ring-down and cavity enhanced spectroscopy using diode lasers[END_REF] , making use of simpler electronics while still retaining the long pathlengths of high-finesse cavities [START_REF] Thorpe M | Cavity-enhanced direct frequency comb spectroscopy[END_REF] .

The detection bandwidth required for CEAS is typically lower than that of CRDS, ranging from 10 to 200 kHz depending on the finesse of the cavity [START_REF] Alquaity A B | Detection of shock-heated hydrogen peroxide (H2O2) by off-axis cavity-enhanced absorption spectroscopy (OA-CEAS)[END_REF] . CRDS requires a detection bandwidth of over 1 MHz. The technique was first demonstrated in two papers: one by O'Keefe, referred to as integrating cavity output spectroscopy (ICOS) [26,89] , and the other by Engeln et al., referred to as cavity-enhanced absorption spectroscopy (CEAS) [START_REF] Engeln R | Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy[END_REF][START_REF] Fiedler S E | Incoherent broad-band cavity-enhanced absorption spectroscopy[END_REF] .

Today, the term ICOS is generally used to refer to techniques that involve integrating over many cavity modes by dithering the cavity length and / or a laser wavelength. On the other hand, CEAS is used to describe alternative variations. This convention is adopted here to differentiate between the two techniques.

In ICOS, many closely spaced cavity modes are excited, and the amplitude of the resonant peaks is reduced due to absorption losses by the molecules inside the cavity.

The change in the amplitude of these resonance peaks is proportional to the absorption coefficient of the sample. Thus, the absorption spectrum is obtained at discrete points when the laser frequency is tuned around the absorption peak. Cavity-length scanning is required to shift the resonant peaks to record a continuous spectral feature. To avoid undesirable saturation effects associated with the build-up of light in the cavity (as in CRDS), the cavity must also be dithered on a timescale faster than the ringdown / buildup time [START_REF] Romanini | Diode laser cavity ring down spectroscopy[END_REF] . Alternatively, the laser emission wavelength can be dithered to capture more cavity modes with an amplitude that still permits the resolution of the gas absorption line. Limitations of ICOS include the statistical residual uncertainty caused by intentional coupling to a finite number of cavity modes and the low mean optical throughput. Averaging over many such scans is necessary to eliminate the mode spectrum, which remains the main source of noise limiting the sensitivity of the ICOS technique.

Summary

As discussed above, LPAS can be used for identifying and quantifying molecules in the gas phase. According to the Beer-Lambert law, there are two ways to improve the detectability of an analytical instrument: enhancing the absorption signal and decreasing the noise. A more significant absorption signal can be obtained by selecting a different transition with a larger cross-section or line intensity and increasing the optical absorption path length. The long interaction path length between light and absorbing medium can be realized by using either a multi-pass cell or an optical cavity.

Among LPAS techniques, different spectroscopic techniques have their special advantages since their detection method, employing a suitable folded-path optical cell with a specific structure is an easy way to enhance the sensitivity of the LPAS techniques.

Chapter 3

Novel model for design of folded-path optical cells

OBJECTIVE

The focus of this chapter is on the multi-pass cell (MPC), a folded-path optical cell based on two spherical mirrors. In this chapter, a novel calculation model for the design of MPCs based on spherical aberration will be presented.

Additionally, the parameters of a practical MPC with a dense spot pattern and a small size will be discussed, including the location and angle of the incident light, the focal length of the spherical mirror, and mirror spacing.

Calculation model of dense spot-pattern MPCs based on spherical mirror aberration

Introduction

Multi-pass cells (MPCs) with a long effective optical path length constructed with highly reflective mirrors are widely used in optical absorption spectroscopy and gasphase optical delay lines for the detection of trace amounts of gas molecules [START_REF] Krzempek K | CW DFB RT diode laser-based sensor for trace-gas detection of ethane using a novel compact multipass gas absorption cell[END_REF][START_REF] Zheng | A near-infrared C2H2/CH4 dual-gas sensor system combining offaxis integrated-cavity output spectroscopy and frequency-divisionmultiplexing-based wavelength modulation spectroscopy[END_REF][START_REF] Wei | Palm-sized methane TDLAS sensor based on a mini-multi-pass cell and a quartz tuning fork as a thermal detector[END_REF] . Early spherical MPCs developed by White [START_REF] White | Long Optical Paths of Large Aperture[END_REF] and Herriott [9] are still in use in laser-based spectroscopic trace gas sensors due to their simplicity, reliability robustness, and operability. To improve the utilization efficiency of mirror surfaces and hence obtain a longer optical path length, Herriot described astigmatic MPCs in which the x-z and yz planes of a spherical mirror have different focal lengths, thus creating Lissajous spot patterns [START_REF] Herriott D R | Folded Optical Delay Lines[END_REF] . As a result, this increases the number of ray reflections in an MPC, while minimizing the spot overlap. Recently, many variants based on the astigmatic MPCs were reported with a similar high fill factor pattern, in which at least one spherical mirror was replaced with a cylindrical mirror [START_REF] Silver | Simple dense-pattern optical multipass cells[END_REF][START_REF] Hao L-Y | Cylindrical mirror multipass Lissajous system for laser photoacoustic spectroscopy[END_REF] . In 2017, S. Ozharar et al. designed an aspherical mirror whose focal length varies inversely to the ray height from the optical axis [START_REF] Ozharar | Mirrors with designed spherical aberration for multi-pass cavities[END_REF] . With an appropriate incident angle and position of a ray, a rich set of dense patterns was achieved.

However, so far, the densest pattern MPCs are based on aspherical mirrors. In practice, mirrors with high surface accuracy are manufactured by grinding and polishing techniques that naturally produce spherical surfaces. The manufacture of aspheric surfaces is more complex and it is difficult to produce a mirror of sufficient surface accuracy to follow the designed pattern. Therefore, spherical mirrors are required in these MPCs due to their simplicity, more easily controlled surface quality, and more importantly lower cost.

As shown in Fig. 3.1, the incident ray propagates between two mirrors (M1 and M2) and then reflects on the M2 surface to complete a pass count in the simple geometry of a two-spherical-mirror MPC. With conventional calculations of ray tracing analysis, the paraxial theory was applied based on two assumptions: (i) the optical path length of any rays between two mirrors is dn, where n is the pass counts; the distance D between two mirrors is constant; (ii) all the rays make small angles to the optic axial of the system, so that three important angle approximations are valid, i.e., sinθ ≈ θ, tanθ ≈ θ and cosθ ≈ 1. However, with more highly curved surfaces, particularly marginal rays, and more pass counts, the paraxial theory generates increasingly large deviations from the actual performance due to spherical aberrations. For example, at an angle of 10 o , the paraxial approximation of sinθ ≈ θ has an error of 0.5 %, where these errors are not accounted for in the standard ABCD matrix. With an increasing pass count, these errors will be accumulated and amplified, and distort the real pattern. The paraxial approximation and the lack of equation-based designs that match the actual ray trajectories in MPC limit the ability to develop other spot patterns that may provide other viable MPC arrangements. In particular, the lack of consideration of spherical aberration effects, which are not easily simulated in conventional matrix-based approaches, is a major limitation of these design endeavors. In this part, a novel dense pattern MPC design using two common identical spherical mirrors is reported. An accurate ABCD matrix without the paraxial approximation was developed to describe the propagation of both the paraxial and marginal rays between the two spherical mirrors and the reflection of these rays on the mirror surface. A rich set of spot patterns on the mirrors can be created by numerical simulations due to the presence of spherical aberrations. These novel spot distributions from a simpler mirror surface curvature and with a lower cost, have similar patterns to those obtained by S. Ozharar et al., who used an aspherical mirror designed with varying focal lengths [START_REF] Ozharar | Mirrors with designed spherical aberration for multi-pass cavities[END_REF] .

Calculation principle of a dense spot patterned MPC based on aberration

The reported MPC was assumed to consist of two identical spherical mirrors. In the paraxial analysis of a two-spherical-mirror MPC, the ABCD matrix describing one pass count, consists of a standard transmission matrix and a standard reflection matrix:

1 0 1 D . 2 R 1 0 1 AB CD       =       -       (3.1)
In order to remove the paraxial approximation, we define two operators S and L, which are Sφ = sinφ and Lφ = -2arcsin(φ/R), respectively, where φ is an arbitrary rational number. Therefore, the new ABCD matrix describing one pass count without the paraxial approximation can be expressed as:

1 0 1 . 1 0 1 A B d S n C D L       =             (3.2)
An iterative scheme is used to calculate the ray parameters for each pass count since the modified ABCD matrix varies with each pass count. Using a Cartesian coordinate system, a ray before the transmission and reflection between the (n-1) th and the nth pass count is described by the coordinates (xn-1, yn-1) of the point where it is located on the mirror surface of M1 or M2, and for which the ray has the inclination angles (x'n-1, y'n-1), as depicted in Fig. 3.1. The corresponding ray parameters after n th pass count are calculated using the new ABCD matrix as follows:

-

1 -1 -1 ;' R sin ' 2 arcsin ' , n n n n n n x n x x x d x x = +  = - + -1 -1 -1 ;' R sin ' 2 arcsin ' . n n n n n n y n y y y d y y = +  = - + (3.3)
The above iterative equations can be carried out as long as dn for each pass count is obtained.

If an incident ray enters the MPC from M1 with the initial location (x0, y0) on the M1 surface and inclination angles (x'0, y'0), the location z0 on the M1 surface and the rest inclination angle z'0 can be calculated using the spherical equation for M1:

2 2 2 0 0 0 , z R R x y = - -- 22 0 0 0 1 ( ) ( ) . z arcsin sinx siny    = - - (3.4)
Hence, zn can be expressed by solving the line-sphere intersections:

( )

2 2 1 1 ( 1) , 22 n nn n n n n bb z sinz a c  - -    =  -+ -  -     (3.5)
where an, bn and cn are given by:

2 1 , () nn a sinz  - - = ' 1 1 1 1 1 ' '' 2 11 1 sin sin 1 1 sin (sin ) 2, 1 ( 1)
( 1) 2

n n n n n nn n n n x x y y z zz b DR - - - - - -- -    +  + -      =   +-  -  + -     22 '' 11 1 1 1 1 '' 11 2 1 2 sin sin sin sin . 1 ( 1) ( 1) 2 nn n n n n nn n n n xy x z y z zz c D R R -- -  - - -- -       -  + -           =    +- +  + - -     (3.6)
The optical path length for each pass count can be expressed as:

( )

1 ' 1 . nn n n zz d sinz - - - = (3.7)
Thus z'n can be calculated using the results from Equation (3.4):

( ) ( ) for aspherical mirrors, the four plots in Fig. 3.3 depicted similar spot patterns as in Fig. 3 of Ref. [START_REF] Ozharar | Mirrors with designed spherical aberration for multi-pass cavities[END_REF] where an aspherical mirror was employed. All spots on the two mirrors were projected onto x-y plane. The initial inclination angles and the focal length of the spherical mirrors are taken as (6.56 o , 6.56 o ) and 25 mm, respectively. In all these patterns, the only different parameters for each pattern are the initial entry locations on M1, the mirror spacing, and the pass count under re-entrant condition, which are given in Table 2. With a dense spot pattern, the two-spherical-mirror MPC achieves a long total optical path length, but in a small size. A long total optical path length can improve the detection sensitivity, while a small size can not only enhance the rate of gas exchange and hence reduce the response time, but also realize a compact MPC. The total optical path length L and the volume V of each MPC are listed in Table 2. Here the distance from the farthest spot to the original point for each pattern is selected as their mirror radius. The volume is defined as the product of the mirror area and the mirror spacing.

The longest optical path length is the seven nonintersecting-circle spot pattern with a value of 12.92 m in Fig. 3.3 (c), while the smallest volume is the three nested-circle spot pattern. In fact, a ratio of the total optical path length to the volume can better characterize the space utilization of the ray pattern in an MPC. According to column 7

in Table 2, the most efficient space utilization is the three nested-circle spot pattern in The deficiency of the paraxial approximation for these unusual spot patterns results mainly from the large initial inclination angle of the entry ray, which plays an important role in spot pattern evolution caused by spherical aberration. When the initial inclination angle remains unchanged, each spot pattern in Fig. 3. In other words, the incident ray can continue to retrace the same spot pattern again and again after every ν pass counts. In this simulation, with a fixed inclination angle (6.56 o , 6.56 o ), the mirror spacing D and initial location (x0, y0) to produce this pattern were adjusted manually in order to make the final (xν, yν) and (x'ν, y'ν) as well as the initial

Summary

In conclusion, we developed a new ABCD matrix without the paraxial approximation, which allows the investigation of a two-spherical-mirror MPC in the presence of spherical aberration. The results show that the spherical aberration was accumulated and amplified for each transmission and reflection, producing an abundant set of intricate spot patterns, which are very different from the standard Herriot circle spot pattern. The excellent ratio of the total optical path length to the volume can be used for the construction of a highly sensitive and compact gas sensor, with low costs due to the use of a pair of common spherical mirrors. Compact or portable MPCs such as the one described in this manuscript have many uses in defense, atmospheric monitoring, and medical diagnostics. Further topics of interest include the fabrication and testing of the two-spherical-mirror MPC, as well as the investigation of the effect of beam interference in the MPC of this type.

Two-spherical-mirror MPCs with dense multi-circle spot patterns 3.2.1 Introduction

The developed calculation model did not take into account the variation of spot shapes, due to the usage of an ideal single ray. The deformation of spot shapes generated by the spherical mirror aberrations may lead to overlaps between adjacent spots and hence cause an etalon background. A study of the evolution of spot shapes is necessary to obtain practical MPCs. This section focuses on multi-circle spot patterned MPCs. In the developed calculation model, an ideal single ray was replaced with a geometric ray set as an incident light beam, so that the evolution of the spot shape can be clearly observed via direct tracing of each separate ray forming the incident light beam. Based on this modified calculation model, a set of practical multi-circle patterns without spot overlaps was identified and verified.

Modified calculation model of a dense spot patterned MPC based on aberration

The used light beam consists of a plurality of separate rays. These rays are in parallel and encircle a center ray, as shown in Fig. 3.6. The circular outline of the cross section of this ray set defines the initial spot shape of the incident light beam. Each ray in this ray set is independently tracked in a three-dimensional coordinate system using the single ray calculation mode [START_REF] Cui R | Calculation model of dense spot pattern multi-pass cells based on a spherical mirror aberration[END_REF] , as described in Equation (3.2). The ABCD matrix consists of a modified transmission matrix and a modified reflection matrix in which L and S represent two operators. The operators are defined as Sφ = sinφ and Lφ = -2arcsin(φ/R), respectively, where R is the radius of curvature of the spherical mirrors and φ is an arbitrary rational number. The parameter dn is the optical path length for the nth transmission between two spherical mirrors. The obtained outlines of this ray beam incident on two spherical mirror surfaces are recorded. The spherical mirror aberrations change an initial circular spot into a crescent spot due to its spherical nature, as shown in Fig. 3.6. Therefore, the observation of the spot shape evolution is important, which can effectively avoid overlaps between two adjacent spots on each mirror. Using the multi-ray calculation model in this section, the influence from spherical aberration, astigmatism and coma can be considered.

Simulation results and discussion

In order to identify practical multi-circle spot patterns, a large number of numerical calculations were carried out using Equation (3.2). Two 2-inch diameter spherical mirrors were selected with focal lengths of 50 mm. The beam diameter was selected to be 1.0 mm, which is an easily obtained value for a near-infrared laser beam. The diameters of both entrance and exit holes were set to 2 mm. The inclination angle of the incident beam is fixed to (-7.4 o , 3.4 o ) since such an inclination angle was found to decrease the spot divergence. Due to the issue of spot overlap, not all multi-circle spot patterns can be adopted in practice. Hence three criteria are followed: 1) circles do not intersect; 2) spots are evenly distributed on two mirror surfaces; 3) spots do not overlap and the spot size is < 2 mm before the ν th reflections, where ν is as large as possible.

With the above fixed parameters, the production of a multi-circle spot pattern strongly depends on other parameters, such as the initial entry location (x0, y0), the mirrors spacing D and the pass count ν. The process of searching for a practical spot pattern is as follows: the mirror spacing is increased from small to large until a multi-circle spot pattern appears; the initial entry location (x0, y0) is adjusted to make the spot pattern zoom in and out around the center point of the mirrors, so that the spots can be evenly distributed on two mirror surfaces; and an appropriate pass count ν is determined according to the condition of the overlap of the spots and the size of the exit hole.

Practical three-, four-, five-, seven-, eight-, nine-multi-circle spot patterns are shown in Fig. 3.7. All spot patterns were projected onto the x-y plane of the exit mirror, respectively. All corresponding parameters for multi-circle spot patterns are listed in Table 3. The absence of a six-multi-circle spot pattern is since a set of appropriate parameters was not found to evenly distribute the spots of the six-multi-circle pattern on the mirror surface.

When the evolution of the spot shapes is not considered and the incident beam is an ideal ray, the incident ray is able to continue to retrace the same spot pattern.

However, the deformation of spot shapes resulting from spherical mirror aberration can cause the spots to overlap before a complete cycle, in which the beam diameter plays an important role. With the 1-mm beam diameter, the five-, eight-and nine-multi-circle spot patterns in Figs. 3.7 (c)(e)(f) can complete a cycle without a spot overlap. The reason why the last few spots were removed in Fig. 3.7 (e) and (f) is that their spot sizes exceed 2 mm. These large size spots cannot exit from a 2-mm exit hole. The effective pass counts ν is listed in Table 3. In contrast, the spot overlap occurs in the case of a 1mm beam for the three-, four-and seven-circle spot patterns after the v th effective pass counts. These overlap spots were removed from Fig. 3 The initial entry location (x0, y0) of the beam can be used to control the spot distribution on the two mirror surfaces. Based on Table 3, the values of the entry location do not change much for the different spot patterns and are distributed in an annular area with a diameter of 6.4 mm. To assess the space utilization of the different spot patterns, a ratio of the total optical path length to the volume, RLV was used as shown in column 7 in Table 3. As the number of circles increases, the RLV increases monotonically except for the eight-multi-circle spot pattern. The total optical path length L equals the product of the effective pass counts ν and mirror spacing D, while volume V equals the product of the mirror area and mirror spacing D. Since the mirror area is a constant in our case, the RLV depends on the effective pass counts ν, which means that a higher effective pass count offers a larger RLV. Therefore, the eight-multicircle spot pattern has the largest RLV of all the patterns outlined in Table 3. To verify the validity of the calculation in Fig. 3.7, the eight-and nine-multi-circle spot pattern in Fig. 3.7 (e) and (f) were demonstrated experimentally as shown in Fig. 3.8 (a) and (b), respectively. Two 2-inch spherical mirrors with a curvature radius of 100 mm were employed and the mirror reflectivity of each mirror was 98%. Based on the acquired 2f spectrum in Fig. 3.9, a signal-to-noise ratio of ~93 was obtained.

Summary

In conclusion, we modified the single-ray calculation model for two spherical mirror MPCs into a multi-ray calculation model in which the evolution of the MPC spot shapes on the spherical mirror can be accurately simulated. A set of practical multicircle spot patterns were identified. These exotic multi-circle spot patterns can offer small-sized MPCs with a high fill factor and long absorption length, thus paving the way towards a sensitive, low-cost, compact trace gas sensor suitable for the large-scale deployment of distributed sensor networks and for handheld mobile devices.

One-circle patterned MPC for CO sensing by TDLAS at 2.33 μm

Introduction

Sulfur hexafluoride (SF6) has been widely used as an insulating medium in electric power transmission equipment of a >110 kV high-voltage power grid, such as gasinsulated switchgear (GIS) and gas circuit breakers (GCBs). The monitoring of the operating condition of electric power transmission equipment is vital to ensure failurefree operation of the electric power transmission lines. The online detection for SF6 decomposition is an effective means for fault early warning of electric power transmission equipment, which is based on the fact that SF6 can be decomposed in the case of superheating and a partial discharge (PD) caused by latent inner insulation defects. In this process, various sulfur-containing, fluorine-containing compounds, such as SF4, HF, SO2F2, SOF2, SO2, CO2, CO, and SF4, are generated via a reaction between water, oxygen, metal vapor, or other impurities [START_REF] Kurte R | Application of infrared spectroscopy to monitoring gas insulated high-voltage equipment: electrode material-dependent SF6 decomposition[END_REF][START_REF] Duffour | Molecular dynamic simulations of the collision between copper ions, SF6 molecules and a polyethylene surface: a study of decomposition products and an evaluation of the self-diffusion coefficients[END_REF][START_REF] Luo | Research on the detection of SF6 decomposition products based on nonresonant photoacoustic spectroscopy[END_REF][START_REF] Shinkai | A study on condition assessment method of gas-insulated switchgear. part II. influence of moisture in the SF6, detection of a partial discharge on a spacer, repetition discharge and overheating by incomplete contact[END_REF][START_REF] Yin | Highly sensitive SO2 photoacoustic sensor for SF6 decomposition detection using a compact mW-level diode-pumped solid-state laser emitting at 303 nm[END_REF][START_REF] Khan Q | Partial discharge detection and diagnosis in gas insulated switchgear: State of the art[END_REF] . These decomposition gases can be employed as indicator gases to locate and identify fault types and their locations.

Carbon monoxide (CO) is one of the indicator gases indicating equipment overheating. When the concentration level of sulfur-containing compounds is low, an increase in CO content suggests that low-temperature overheating may have occurred inside the equipment. This is because organic insulating materials, such as polyester ethylene, insulating paper, and insulating paint, can release CO gas slowly at lowtemperature overheating due to their low thermal decomposition temperature.

Therefore, the CO content can identify the low-temperature overheating fault of the equipment. Moreover, the concentration level of CO increases with cumulative equipment operation time increasing. The operation time of the equipment can be estimated. For example, for 1,100 kV GIS equipment, the concentration levels are < 20 ppmv under normal operating conditions [START_REF] Khan Q | Partial discharge detection and diagnosis in gas insulated switchgear: State of the art[END_REF] . Therefore, there is considerable interest in developing an online, cost-effective, and ppmv-level CO sensor for SF6 decomposition analysis.

Trace gas detection based on tunable diode laser absorption spectroscopy (TDLAS) is a versatile technique for the analysis of gas compositions since it offers narrow spectral resolution, high sensitivity, and fast response time, which permits online monitoring of target gas concentrations [START_REF] Ren | Sensitive detection of formaldehyde using an interband cascade laser near 3.6 μm[END_REF][START_REF] Wang | Measurements on co concentration and gas temperature at 1.58 um with tunable diode laser absorption spectroscopy[END_REF] . The performance of TDLAS-based sensors is dependent on the availability of suitable laser sources, such as single-mode laser diodes in the near-infrared (near-IR) wavelength up to ~3 μm, quantum cascade lasers (QCLs) in the mid-infrared (mid-IR) region beyond ~3.7 μm and GaSb-based interband cascade lasers (ICLs) emitting in the 3-4 μm wavelength. F. Wang et al. used the TDLAS technique and a 1.58-μm VCSEL diode laser as a radiation source to monitor CO concentration levels [START_REF] Wang | Measurements on co concentration and gas temperature at 1.58 um with tunable diode laser absorption spectroscopy[END_REF] . A detection sensitivity of ~200 ppmv was achieved. In 2009, X. Chao et al. developed a TDLAS-based sensor for CO detection by use of a distributed feedback (DFB) diode laser operating at 2.3 μm, which resulted in a detection sensitivity of 3 ppmv [START_REF] Chao | Absorption sensor for CO in combustion gases using 2.3 µm tunable diode lasers[END_REF] . In 2017, J. Dang et al. employed a high-power, continuous wave (CW), distributed feedback (DFB) QCL emitting at 4.7 μm and a 1.6m multi-pass gas cell (MPC) to detect CO [START_REF] Dang | A CO trace gas detection system based on continuous wave DFB-QCL[END_REF] , which resulted in a minimum detection limit of 26 ppbv. Furthermore, R. Ghorbani et al. reported a compact sensor for CO detection by employing a 4-m circular MPC with an ICL emitting at 4.69 μm [START_REF] Ghorbani R | ICL-based TDLAS sensor for real-time breath gas analysis of carbon monoxide isotopes[END_REF] . The detection limit obtained was 9 ppbv. Although a ppb-level CO sensor was achieved by the use of the TDLAS technique with a mid-IR QCL or ICL targeting the strongest fundamental absorption band, these gas sensors were designed to operate in a nitrogen (N2) buffer gas environment. Hence, they are unsuitable for use in electric power transmission equipment since high SF6 concentration levels have abundant absorption spectra in the IR region.

In this section, a TDLAS-based CO sensor based on a CW-DFB diode laser with low power consumption emitting at 2.33 μm for SF6 decomposition detection is described. A careful investigation of the line selection was carried out and a compromise is made in terms of cross-response, cost-effectiveness, and detection sensitivity. 2f wavelength modulation spectroscopy (2f-WMS) and a 14.5-m MPC with a one-circle spot pattern were employed to achieve a high detection sensitivity. The optical sensor system using the selected target spectrum results in a ppmv-level CO detection limit in an SF6 buffer gas environment, which can be applied to the online monitoring of a low-temperature overheating fault inside electric power transmission equipment.

Line selection

The selection of a gas absorption line is crucial for the detection of gas concentration levels using narrow linewidth lasers, which affects both the resolution and sensitivity of a detection system. In general, the selection of a gas absorption line is mainly determined by two main aspects. First, the wavelength where the selected absorption line is located should be covered by commercially available laser sources.

Secondly, the selected absorption line must have sufficient line strength to achieve the required sensitivity and simultaneously avoid interference with other existing gases.

The absorption band of CO, which has been employed to determine the CO concentration levels, includes the fundamental vibration band near 4.6 μm [START_REF] Ma Y | QEPAS based ppb-level detection of CO and N2O using a high power CW DFB-QCL[END_REF][START_REF] Dong L | Ultrasensitive carbon monoxide detection by using EC-QCL based quartz-enhanced photoacoustic spectroscopy[END_REF] , the first overtone band near 2.3 μm [START_REF] Ma | Sensitive detection of carbon monoxide based on a QEPAS sensor with a 2.3 μm fiber-coupled antimonide diode laser[END_REF][START_REF] Li | Improved evanescent-wave quartzenhanced photoacoustic CO sensor using an optical fiber taper[END_REF] , and the second overtone band near 1.55 μm [START_REF] Yin | Impact of humidity on quartz-enhanced photoacoustic spectroscopy based CO detection using a near-IR telecommunication diode laser[END_REF] .

The fundamental vibration band near 4.6 μm possesses the largest absorption crosssection and their wavelength can be reached by lead-salt lasers [START_REF] Bomse D S | Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser[END_REF] and quantum cascade lasers (QCLs) [START_REF] Li | Real-time measurements of atmospheric CO using a continuous-wave room temperature quantum cascade laser based spectrometer[END_REF] with a ppbv-level detection sensitivity. Both the first and second overtone bands near 2.3 μm and 1.55 μm can be covered by cost-efficient DFB diode lasers, whose detection sensitivities, however, at 2.3 and 1.55 μm the absorption crosssections of the corresponding overtones are two and four orders of magnitude smaller than that of the fundamental vibration band, respectively.

To select an interference-free CO absorption line, the absorption spectrum of SF6 has to be investigated, since a TDLAS-based CO sensor for SF6 decomposition detection is required to operate in the environment of an SF6 buffer gas. The HITRAN database provides the line parameters regarding the strong ν3 band of SF6 at 10.5 μm.

However, the concentration of SF6 in an electric power system is usually > 99.8 %. As a result, some weaker unidentified absorption bands of SF6 between 2.5 μm and 10 μm cannot be disregarded in the case of high SF6 concentration levels. A Fourier transform infrared (FTIR) spectrometer (Thermofisher Nicolet IS50) equipped with a special multi-pass gas cell with a 10 m path length was employed to investigate the interference between the absorption lines of CO and SF6. Pure SF6 (99.99 %) was introduced into the gas cell first at atmospheric pressure. The dashed navy line in Fig. 4.1 depicts the absorption spectra of pure SF6 between 2 and 5.5 μm. The pure SF6 spectra show an absorption band at a wavelength of > 2.5 μm. Next, a mixture of 1000 ppmv CO in N2 was filled into the gas cell after flushing the gas cell with a flow of pure N2 gas. The red line in Fig. 4.1 represents the fundamental vibration and first overtone bands of 1000 ppmv CO located at 4.6 μm and 2.33 μm, respectively. The absorption band near 4.9 μm is due to pentacarbonyl-iron (Fe(CO)5) which is produced by the reaction between CO and the gas cylinder material iron. The fundamental vibration band of CO near 4.6 μm overlaps strongly with SF6 absorption features. Since the CO spectra are 3A). The wavelength of the diode laser can be tuned between 4282 cm -1 and 4290 cm -1 , as shown in Fig. 4.2 (a). The wavelength tuning range covers the selected R(6) line plotted as a black line in Fig. 4.2 (a). The current and temperature tuning coefficients of the diode laser are -0.049 cm -1 /mA and -0.403cm -1 /°C, respectively. An operating current and temperature of 85 mA and 31.3 °C were selected. An operating temperature higher than room temperature is helpful to decrease the power consumption of the laser temperature controller since a heating process has a higher efficiency than a cooling process. The measured L-I-V (light-current-voltage) curve of the laser at an operating temperature of 31.3 °C is shown in Fig. 4.2 (b). The output optical power is 2.2 mW with a forward voltage of 1.1 V when the diode laser was operated at a current of 85 mA. In this condition, the power consumption of the diode laser is only 90 mW.

Sensor design

The schematic of the CO sensor system is depicted in Fig. 4.3. The CO sensor system consisted of three parts: an optical unit, a control unit, and a gas handling unit.

In the optical unit, the CW-DFB diode laser operating at 2.3 μm was collimated to a parallel beam with a diameter of 7.3 mm by a collimator (Thorlabs, F810APC-2000). In the control unit, a current driver (Newport, ILX-3220) and a commercial temperature controller (Thorlabs, model TED 200C) were employed to tune the central wavelength of the DFB laser to 4285.01 cm -1 . 2f based wavelength modulated spectroscopy (2f-WMS) was used to obtain a high detection sensitivity. Function generator 1 generated a saw tooth waveform to scan the laser wavelength, while function generator 2 generated a sinusoidal waveform with a small amplitude to dither the laser wavelength. The two waveforms from function generators 1 and 2 were added by a custom-made electrical adder and then sent to the modulation port of the current driver. The output signal from the InGaAs detector (Thorlabs, PDA10D-EC) was sent to a lock-in amplifier (Stanford, SR830). The lock-in amplifier demodulated the signal in the 2f mode to the sync signal from function generator 2. An acquisition system consisting of a DAQ card, and a laptop was triggered by function generator 1 to acquire the 2f spectra from the InGaAs detector.

The gas handling unit consisted of standard gas cylinders, a gas dilution system, a pressure controller, a mass flow meter, and a vacuum pump. The gas dilution system diluted a certified 500 ppmv CO/SF6 mixture gas (Air Products) with a 99.99% pure SF6 gas (Air Products) to produce the CO/SF6 mixture gases with different concentration levels. The 99.99 % pure SF6 gas contains an unavoidable impurity of 3 ppmv CO, which was verified by the supplier and was taken into account in all gas dilution processes. A needle valve (NV) was placed upstream to the MPC to adjust the gas flow, while the pressure controller and vacuum pump were placed downstream to the MPC to maintain a constant pressure in the MPC. A mass flow meter was inserted between the pressure controller and vacuum pump to monitor the gas flow.

Experimental results and discussions

(1) Modulation depth and gas pressure optimization With the 2f-WMS-based CO sensor, the pressure-dependent absorption linewidth must match the modulation depth of the laser, since the maximum peak of the 2f signal occurs at the optimal modulation depth and gas pressure [START_REF] Tittel F K | Quantum cascade laser-based sensor system for nitric oxide detection[END_REF][START_REF] Wang | Quartz-enhanced photoacoustic detection of ethylene using a 10.5 μm quantum cascade laser[END_REF] . A 20-ppmv CO/SF6 mixture gas produced by the gas dilution system was filled into the MPC. The 2f signal amplitudes with a modulation depth between 0.014 cm -1 and 0.19 cm -1 and a pressure range of 200 -700 Torr were acquired and are shown Fig. 4.4. The 2f signal amplitudes vary significantly with modulation depth and gas pressure. The 2f signal amplitudes from the different pressures linearly increase together with the modulation depth increasing in the range of 0.014-0.07 cm -1 . When the modulation depth is > 0.07 cm -1 , the 2f signal amplitudes for the different gas pressures reach a maximum value and in turn decrease gradually. The neighboring curve interval of the 2f signal amplitude decreases with increasing gas pressure. The amplitude curves at different pressures do not intersect except for two curves at 600 and 700 Torr. This is due to the fact that in Doppler broadening (at low pressures), the spectral amplitude increases with increasing gas pressure, while in the pressure broadening regime (at high pressures), the spectral amplitude is insensitive to the gas pressure. Hence an operating pressure of 600-700 Torr is optimum based on Fig. 4.4. The SF6 gas pressure in an electric power system is ~ 4,400 Torr, which is too high for the CO sensor outlined here due to spectral broadening. Therefore, the gas pressure has to be reduced before spectral measurements can be carried out. The Taiyuan area, China, has an atmospheric pressure of ~700 Torr as a result of its 784-m altitude. A straightforward way to deduce the pressure in field applications is to set an automatic reducing valve into the MPC, which can reduce the pressure to the outside atmospheric pressure. Therefore, the optimal modulation depth of 0.138 cm -1 at a pressure of 700 Torr according to the results in Fig. 4.4 in the following experiments was selected, which corresponds to a modulation index of 2.3.

(2) Spectral data processing method For an online sensor system, a fast data output rate is desirable. Therefore, a 6 Hz saw tooth waveform from function generator 1 was used to scan the 2.33 μm diode laser wavelength from 4284.9 cm -1 to 4285.2 cm -1 , while a 2 kHz sinusoidal waveform from function generator 2 was employed to dither the laser wavelength with an optimal modulation depth of 10 mA, to simplify the hardware construction and reduce the workload of the acquisition system. The lock-in amplifier was set with a time constant of 3 ms and filter slope of 24 dB, corresponding to an equivalent noise bandwidth (ENBW) Δf = 26.04 Hz, according to its operating manual. It was verified that with such integration parameters, the observed spectral lines were not only distorted by the lock-in detection but also had a lower noise level. The sampling rate of the acquisition system triggered by the sync signal from function generator 1 was set to 12 kHz. concentration of CO/SF6 gas mixture. Then pure SF6 was flown through the MPC at different flow rates. The CO mixing ratios were monitored during this process. The SF6 molecules have a molecular mass of 146 a.u., which is considerably larger than the molecular mass of air, at 29 a.u.. As a result, the maximum flow rate was limited to 200 standard cubic centimeters per minute (SCCM) since the mass flow controller was designed for air or nitrogen. (4) Sensor linearity, precision, and stability

The linearity of the sensor system was first evaluated. The different concentration levels of CO, 40 ppmv, 30 ppmv, 20 ppmv, 10 ppmv, and 6 ppmv, were generated by diluting the certificated 500 ppmv CO/SF6 gas mixture using the gas dilution system and were directed into the sensor system in turn. 200 points were measured continuously for each concentration level. Then the SF6 buffer gas and high-purity N2 were introduced into the sensor system for a small signal and zero baseline measurements. The stability of the CO sensor system, characterized by the Allen-Werle variance, was investigated. Figure 4.9 shows the Allan-Werle variance obtained from the same time series measurements used in the histogram plot. When the averaging time is <48 s, the Allan-Werle variance follows a 1/τ dependence during the duration of the measurement series, which indicates that the thermal noise of the sensor system remains the dominant noise source on such a time scale. This yields an optimum averaging time of 48 s with a precision of ~6 ppbv. In field applications, a faster flow rate of >200 SCCM can be set to match an optimum averaging time to achieve the optimum working parameters of the sensor system. wastewater treatment plants and sewer networks. Furthermore, CH4 is the main component of natural gas used widely in industry and urban as an industrial fuel and domestic fuel, respectively. The boom in shale-gas production based on "fracking" technology raises concern about the environmental impact of shale-gas drilling.

Therefore, reducing CH4 fugitive emissions is a key requirement for natural gas harvesting as a clean energy alternative. It is of great significance to study and monitor the change of CH4 concentrations in terms of climate change and industrial monitoring, CH4 production, and emission from sewers. Ideally, large-scale deployment of sensors in a network configuration will provide both spatial and temporal resolution for CH4 emission localization and quantification, respectively. Benchtop TDLAS sensor systems have been developed for online gas analysis1 and higher sensitivities have been achieved using MPCs with long interaction lengths [9,[START_REF] Dong L | Compact TDLAS based sensor design using interband cascade lasers for mid-IR trace gas sensing[END_REF] . However, widespread use of these TDLAS sensor systems for distributed sensor networks (DSNs) or handheld mobile devices is limited by structure complexity, extremely large size and the tight optical alignment tolerances of these systems. A desired TDLAS sensor system based on a simple structure with a small footprint, but high sensitivity, is preferred.

To achieve a high detection sensitivity with TDLAS, the most effective method is to shift the target absorption line of the molecular compound from the NIR overtone spectral region to the MIR "fingerprint" spectral region [5] and select the strongest MIR interference-free absorption line at the wavelength region that the employed MIR laser source covers. For example, an inter-band cascade laser (ICL) operating at room temperature combined with a 54.6-m MPC, results in a CH4 sensor with a detection sensitivity of ~10 ppbv [START_REF] Dong L | Compact TDLAS based sensor design using interband cascade lasers for mid-IR trace gas sensing[END_REF] . However, MIR lasers and detectors are costly and nondurable compared to robust diode lasers operating in the NIR region. Furthermore, the MIR optical fiber coupling technique is not yet mature and not readily available for implementation of a cost-effective compact, and robust MIR TDLAS sensor design.

An alternative approach to obtaining high detection sensitivity for TDLAS is to increase the optical absorption path length by using a commercial or custom-made MPC, which is constructed with highly reflective mirrors. A variety of MPC-based TDLAS sensors were demonstrated for numerous spectroscopic applications. However, the traditional Herriott cell [START_REF] Cui R | Highly sensitive and selective CO sensor using a 2.33 mum diode laser and wavelength modulation spectroscopy[END_REF] based on two spherical mirrors yields a standard circular / elliptical spot pattern with a low utilization efficiency of the mirror surfaces, which is difficult to realize in a small handheld sensor system. Many variants based on the Herriott cell, such as astigmatic mirror cells can produce dense dot patterns to take advantage of the mirror surfaces, thus realizing a compact size with a long optical path length. But the manufacture of aspheric surfaces is more complex, and it is difficult to produce a mirror of sufficient surface accuracy to follow the designed curvature.

Therefore, spherical mirrors are required in MPCs due to low requirements for mirror processing technology, more easily controlled surface quality, and more importantly lower cost.

In this section, a mini-MPC with a 4.2-m optical absorption path length is designed and fabricated which to the best of our knowledge is currently the smallest MPC in terms of the same optical path-length. The MPC employs two spherical mirrors and yields a seven-nonintersecting-circle spot pattern by means of its own spherical mirror aberration. A fiber-coupled collimator and an uncooled InGaAs photodetector with a preamplifier circuit board are integrated into this MPC. The 3D printed technique with a printing resolution of 100 μm was used to fabricate the MPC frame, which ensures a high integration of all components. As a result, the mini-MPC is lightweight, costeffective, free from active optical alignment and easy to fabricate and has a competitive optical path length. A NIR DFB laser emitting at 1.65 μm was used as a light source to detect methane at ppbv levels. A demonstration of continuous CH4 monitoring was carried out near sewers and in ambient air for 13 and 11 days, respectively. Our results indicate the potential for a highly integrated mini-MPC in a small handheld sensor system and in a large-scale deployment of DSN providing high detection sensitivity and selectivity for leak detection and monitoring atmospheric pollutants [START_REF] Cui R | Three-Dimensional Printed Miniature Fiber-Coupled Multipass Cells with Dense Spot Patterns for ppb-Level Methane Detection Using a Near-IR Diode Laser[END_REF] .

Experimental section

This section focused the research on a nonintersecting-circle spot pattern due to a higher ratio of the total optical path length to the volume. A large number of numerical calculations were carried out using Equation (3.2) with a variety of initial parameters to determine an optimal dense spot pattern with a longer optical path length and a smaller volume as well as without spot overlap. An incident beam diameter of 0.6 mm was adopted. The diameter and the focal length of the spherical mirror was fixed to 25. In practice, the reflectivity of a spherical mirror cannot be 100 %. With a specific mirror reflectivity, the SNR of an MPC is not a monotonically increasing function of the pass count. To achieve high SNR spectra with a long optical path length, the pass count must be appropriately selected to avoid an excessive loss of laser power.

According to the Lambert-Beer law, the power absorbed by the target gas can be expressed after the n th pass as: mm. With the improvement of mirror reflectivity, the achievable SNR and the optimal pass counts increase rapidly. When the reflectivity increases from 95 % to 99.06 %, the optimal optical path length and the achievable SNR are improved by a factor of ~ 5.6 and 4.9, respectively. Hence, a high mirror reflectivity is required in the mini-MPC. To integrate the fiber-coupled collimator and the optical photodetector into the mini-MPC, an MPC frame of the cage system was designed and fabricated using a photosensitive resin and the SLA 3D printing technique with a printing resolution of 100 μm. The 3D-printed cage system consists of two cage plates as mirror mounts. The In this work, the main objective was to demonstrate the powerful capacities of a cost-effective, highly integrated mini-MPC for CH4 gas detection. To test the mini-MPC response to different CH4 concentration levels, the closed mini-MPC was used.

Different concentration levels of CH4 were generated by diluting the certified 30 ppmv CH4:N2 gas mixture using a commercial gas dilution system and then were directed into with an R-square value of >0.999 between the measured and nominal concentrations confirms the excellent linearity of the sensor response to CH4. of 1.47×10 -5 Hz -1/2 was achieved, which is one order magnitude higher than the conventional TDLAS system [START_REF] Bartlome R | Vapor-phase infrared laser spectroscopy: from gas sensing to forensic urinalysis[END_REF] . This work [START_REF] Cui R | Three-Dimensional Printed Miniature Fiber-Coupled Multipass Cells with Dense Spot Patterns for ppb-Level Methane Detection Using a Near-IR Diode Laser[END_REF] Therefore, the mini-MPC-based sensor system can realize highly sensitive gas detection in a small gas sampling volume. The fill factor reflects the space utilization of the ray pattern in an MPC, while the NEA reflects the minimum detection limit of a sensor, which is normalized to the absorption-line strength and the detection bandwidth.

Here we define a new performance indicator, the ratio of NEA to fill factor (RNF), which allows MPC-based sensing techniques to be compared without reference to the specific target gas and at the same time considering the space utilization of the ray pattern in an MPC. The mini-MPC used here has a fill factor of 21 cm -2 , achieving an The CH4 concentration varied from 1.98 to 9.28 ppmv, with a mean value of 2.93 ± 0.19 ppmv. As the measured concentration data near sewers depend on the CH4 content in the atmosphere, the CH4 random emissions from these sewage wells, as well as weather conditions such as wind, humidity and temperature, a clearly regular behavior are not observed in Fig. 4.19 (a). To determine the periodicity of CH4 emission, the diurnal profile of Fig. 4.19 (a) was calculated and plotted in Fig. 4.19 (b), which reveals an increase of the hourly mean values in the CH4 concentration from dusk to the early morning with a subsequent gradual decrease after ~8:00 SCT. However, the distribution of hourly concentrations for different days was large.

To measure CH4 concentrations in the atmosphere, the sensor system was moved the sampling period are also in agreement with previous studies [START_REF] Bamberger | Spatial variability of methane: attributing atmospheric concentrations to emissions[END_REF] .

Summary

In conclusion, an integrated mini-MPC with a 4.2-m optical path length was developed using two common spherical mirrors and was demonstrated its effective capacity for trace gas sensing in the NIR spectral region. Conventional MPC-based sensors usually employ discrete optical elements [START_REF] Weibring | Ultrahigh-precision mid-IR spectrometer II: system description and spectroscopic performance[END_REF] with tight alignment tolerances and the MPC takes up most of the space, resulting in a large size and high cost limiting large-scale deployment and handheld applications. In contrast, the mini-MPC is configured with a fiber-coupled collimator and a photodetector, the accurate relative positions between these optical elements were through a high-resolution 3D printing technique that eliminated the requirement of an active optical alignment. Therefore, the highly integrated mini-MPC is a suitable candidate for large-scale deployment of the DSNs to monitor pollutants in the atmosphere or in the workplace. Its small dimensions of 4×4×6 cm 3 (open cavity) can offer a handheld mobile sensor device for safety inspections, leak detection, or medical diagnostics. A future mini-MPC upgrade will employ the macromolecule polymer as a mirror substrate instead of glass mirrors. In this manner, the mirror mounts and the mirrors can be 3D printed together, which results in a further significant weight reduction of the sensor system.

Single-line spot patterned MPC for MPC enhanced QEPAS sensing 4.3.1 Introduction

Several laser-based methods, such as tunable diode laser absorption spectroscopy (TDLAS), quartz-enhanced photoacoustic spectroscopy (QEPAS), have been developed for trace gas detection, leading to the advent of reliable and robust sensors.

In these gas detection techniques, QEPAS is an attractive approach characterized by high cost-effectiveness, high sensitivity, and small footprint [START_REF] Dong L | QEPAS spectrophones: design, optimization, and performance[END_REF][START_REF] Kosterev | QEPAS detector for rapid spectral measurements[END_REF] , since a high Q-factor, low-cost quartz tuning fork (QTF) is used as the acoustic detector. In QEPAS, a modulated laser beam is focused through the gap of a QTF. When the laser beam interacts with the target analyte around the QTF, an acoustic point source is generated between two QTF prongs, forcing the QTF prongs to oscillate back and forth. In this way, the in-plane anti-symmetrical vibration of the QTF is excited [START_REF] Petra N | Theoretical analysis of a quartz-enhanced photoacoustic spectroscopy sensor[END_REF] . As a result, the QTF can be treated as an acoustic quadrupole, providing some immunity to external (outside between two prongs) environmental sound [START_REF] Ma Y | In-plane quartz-enhanced photoacoustic spectroscopy[END_REF][START_REF] Wu | Beat frequency quartz-enhanced photoacoustic spectroscopy for fast and calibration-free continuous trace-gas monitoring[END_REF] . An outgoing cylindrically symmetric pressure wave is produced along all the laser transmission paths, not just between the two prongs. But the thickness of an ordinary QTF is < 0.4 mm, which means that only the sound wave from the QTF-thickness-long sound source is effectively picked up. If more sound waves along the beam path can be collected, the QEPAS signal can be significantly improved.

As the QTF thickness is limited by its geometric design and manufacturing process, an alternative approach is to fold the laser beam, making it pass through the QTF repeatedly and thus generate multiple sound sources between two QTF prongs. All the sound waves from these sound sources together drive the two QTF prongs to vibrate, hence enhancing signal amplitude. In 2017, a quartz-enhanced photoacoustic spectrophone driven by two acoustic point sources was demonstrated, in which a laser beam passed through the QTF two times utilizing a mirror and an optical circulator, producing two acoustic point sources along the QTF symmetry axis. The system successfully exited the two resonance antinodes of the QTF operating at a 1st overtone resonance mode [START_REF] Zheng | Double antinode excited quartz-enhanced photoacoustic spectrophone[END_REF] . Finally, a signal gain factor of 3 was achieved. However, the phases of the laser beams at the two antinodes must meet a specific phase relationship, which requires an additional phase control system, resulting in more complexity of the equipment.

A multi-pass cell (MPC) composed of two high-reflectivity spherical or aspherical mirrors is a powerful tool to increase the number of beams reflections. But a big challenge is to make all these back-and-forth laser beams between two mirrors fall on a plane so that all of the beams can pass through the narrow QTF gap between the two Compared to a conventional QEPAS measurements, the prism-enhanced QEPAS provided an enhancement of the signal level of ~3.2 times [START_REF] Qiao | Multi-pass quartz-enhanced photoacoustic spectroscopybased trace gas sensing[END_REF] . However, the number of reflections in such approach is low. In 2020, a spherical-aberration-based theoretical model of two-spherical-mirror MPCs was developed by our group, which can generate exotic spot patterns under a careful design, thus enabling the combination of MPCs with QEPAS.

In this part, multiple-sound-source-excitation quartz-enhanced photoacoustic spectroscopy (MSSE-QEPAS) is reported for the first time, in which a photoacoustic combination of a QTF and an MPC is demonstrated. The MPC design included spherical aberration, which yielded a single-line spot pattern on the two MPC mirrors, which implies that all laser beams between the two mirrors fall on one plane. A custombuilt QTF with a large prong spacing is inserted into the MPC. The laser beam passes through the QTF's gap repeatedly, producing multiple acoustic point sources along the QTF symmetry axis. As a result, the QTF operating at a fundamental resonance mode is simultaneously excited by the multiple acoustic point sources and effectively collects more sound waves to enhance the signal output.

Experimental section

In MSSE-QEPAS, the design of the MPC should follow the below rules: 1) all laser beams must fall on one plane in order to pass through a QTF gap, 2) all light spots on the mirrors should be concentrated within a short single-line segment due to the limited prong length of a QTF, 3) the more beam reflections are desirable to increase the number of sound sources between the two prongs, 4) spot overlaps are allowed which can increase the radiation intensity of the sound sources generated by the overlapping spots. In order to obtain an optimal MPC with a single-line spot pattern, a large number of numerical calculations was carried out using the theoretical model of spherical aberration with a variety of initial parameters. The diameter and the radius of curvature of the MPC's spherical mirrors were fixed to 10 mm and 100 mm, respectively.

The incident beam diameter was set as 0.6 mm. When the mirror spacing was 36. The schematic diagram of a MSSE-QEPAS sensor system is depicted in Fig. 4.21.

A custom QTF with a resonance frequency of f0 = 7.2057 kHz was employed, which had a prong length and spacing of 10 mm and 800 μm, respectively. Such a prong spacing ensures all the ray trajectories in this MPC to fall within the QTF's gap without the light-induced photoacoustic thermal noise [START_REF] Spagnolo | Mid-infrared fiber-coupled QCL-QEPAS sensor[END_REF] . The custom QTF was inserted into

Results and discussion

The performance of the MSSE-QEPAS sensor system was optimized by detecting the fixed concentration of water vapor at atmospheric pressure and room temperature.

The selected H2O target line is located at 7181 cm -1 . Since the QEPAS signal strongly depends on the position of the laser focusing point along the QTF axis [START_REF] Wu | Quartz enhanced photoacoustic H2S gas sensor based on a fiber-amplifier source and a custom tuning fork with large prong spacing[END_REF] , the depth of the QTF into the MPC has to be optimized according to the output signal. The signal gain factor of the MSSE-QEPAS sensor system at the optimal moving distance of 9 mm was assessed. The 2f signal peak was measured to be 2.14 mV with an 8500-ppmv H2O/N2 at room temperature and normal pressure, as shown in Fig. 4.23.

In order to compare with traditional QEPAS, the MPC was removed from the sensor system, enabling the laser beam to pass through the QTF once. The QEPAS signal amplitudes were scanned from the top (point a) to the bottom (point b) along the QTF symmetry axis under the same conditions, as recorded in Fig. 4.24 (a). The highest signal amplitude was 112.5 μV, which was observed at ~1.2 mm. The optimal position is consistent with the previous result [START_REF] Wu | Quartz enhanced photoacoustic H2S gas sensor based on a fiber-amplifier source and a custom tuning fork with large prong spacing[END_REF] . Therefore, the signal gain factor of MSSE-QEPAS is 19 with respect to traditional QEPAS, as shown in Fig. 4.23. The used MPC provides 60 passes through the QTF. An anticipated signal gain factor should be ~60 if a linear enhancement mechanism was taken into account.

However, the actual signal gain factor is 3 times less than the anticipated value. The process of a QTF inserting into an MPC can be described as a convolution operation:

[ ] [ ] [ ] [ ] [ ] i F m S m G m S i G m i   =- =  = -  (4.

Summary

In conclusion, here realized a QEPAS sensor excited by 60 sound sources by use of a single-line pattern MPC and a QTF operating at the fundamental resonance mode.

Unlike the 1st overtone resonance mode reported previously, all sound waves from the 60 sound sources contributed to the in-phase prong deflections of the QTF without any phase shift, producing a signal gain factor of ~20. Highly sensitive QEPAS sensors based on MSSE-QEPAS described in this letter have many uses in atmospheric monitoring, industry process control, and medical diagnostics. Further topics of interest include reducing the mirror size according to the single-line spot pattern and increasing the pass count of the light in the MPC, as well as concentrating all the beams within the highly sensitive top part of a QTF.

Introduction

For ppbv-ppmv level gas sensing, MPC-based LAS techniques are suitable [START_REF] Cui R | Three-Dimensional Printed Miniature Fiber-Coupled Multipass Cells with Dense Spot Patterns for ppb-Level Methane Detection Using a Near-IR Diode Laser[END_REF][START_REF] Ghorbani R | ICL-based TDLAS sensor for real-time breath gas analysis of carbon monoxide isotopes[END_REF] .

For mixing ratios at the ppbv-pptv level or even lower, optical cavity-based LAS technique is necessary to further increase the optical absorption path length up to several kilometers allowing to significantly enhance the needed measurement sensitivity. The most common type of optical cavities consists of two opposing highreflectivity mirrors separated by a distance L with an equal reflectivity R(λ) (typically >99.9%). Effective optical path length, Leff = L/(1-R), can be thus several thousand times the cavity base length L (typically ~0.5-1 m), reaching several kilometers [START_REF] Deacon D A G | Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources[END_REF] . Various cavity-enhanced absorption spectroscopy (CEAS) approaches have been developed and applied to trace gas sensing [START_REF] Provencal R | Cavity-enhanced quantum-cascade laser-based instrument for carbon monoxide measurements[END_REF][START_REF] Morville | Fast, low-noise, mode-by-mode, cavity-enhanced absorption spectroscopy by diode-laser selflocking[END_REF][START_REF] Varma R M | Light extinction by secondary organic aerosol: an intercomparison of three broadband cavity spectrometers[END_REF][START_REF] Dixneuf | The near infrared cavity-enhanced absorption spectrum of methyl cyanide [J][END_REF] , such as CRDS (Cavity Ring-Down Spectroscopy) [START_REF] Lehmann | Recent Advances in Cavity Ring-Down Spectroscopy[END_REF][START_REF] Mazurenka | 4 Cavity ring-down and cavity enhanced spectroscopy using diode lasers[END_REF] , OA-ICOS (Off-Axis Integrated Cavity Output Spectroscopy) [START_REF] Zhao W | Wavelength modulated off-axis integrated cavity output spectroscopy in the near infrared[END_REF][START_REF] Baer D S | Sensitive absorption measurements in the near-infrared region using off-axis integrated-cavityoutput spectroscopy[END_REF][START_REF] Chen W | H2S trace concentration measurements using off-axis integrated cavity output spectroscopy in the near-infrared[END_REF] , IBBCEAS (Incoherent Broadband Cavity Enhanced Absorption Spectroscopy) [START_REF] Ruth A A | Broadband Cavity-Enhanced Absorption Spectroscopy with Incoherent Light[END_REF][START_REF] Wu | Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode[END_REF][START_REF] Yi | Sensing atmospheric reactive species using light emitting diode by incoherent broadband cavity enhanced absorption spectroscopy[END_REF][START_REF] Chen W | 3 -Broadband optical cavity methods[END_REF] , and OF-CEAS (Optical Feedback Cavity-Enhanced Absorption Spectroscopy) [START_REF] Maisons | Optical-feedback cavity-enhanced absorption spectroscopy with a quantum cascade laser[END_REF] . In almost all the reported works, dielectric mirrors with a high reflectivity of 99.99% (low loss with 100 parts per million (ppm) [START_REF] Rempe | Measurement of ultralow losses in an optical interferometer[END_REF] ) have been employed to build high-finesse optical cavities [START_REF] Chen L | Fabrication and design of an integrable subwavelength ultrabroadband dielectric mirror[END_REF] . Dielectric mirror-based optical cavities are easy to align and simple for use in field applications. They have been largely used for high-sensitivity gas sensing. For a mirror at normal incidence, the thickness d of each layers is designed equal to one quarter of the wavelength : di = /(4ni) [START_REF] Macleod H A | Thin-film optical filters[END_REF] . Combined with a phase change of 180° due to light reflection from the surface of a medium with higher refractive index, the resulting reflected light waves from each interface of the multiple dielectric layers constructively interfere (in phase) with each other, leading to a high reflectivity of light at a specific wavelength range.

The maximum reflectivity of a dielectric mirror is closely related to its designed central wavelength, the reflectivity gradually decreases then on either side. The HR bandwidth limits the applications of such dielectric mirror-based optical cavities [START_REF] Morville | Trace gas detection with DFB lasers and cavity ring-down spectroscopy[END_REF][START_REF] Yang | Mid-infrared cavity-enhanced absorption sensor for ppb-level N2O detection using an injection-currentmodulated quantum cascade laser[END_REF] . For instance, measurement of wavelength-dependent optical properties of aerosols over broad spectral regions of the solar radiation (300 -2000 nm) are important for both atmospheric science/climate change research and environmental applications [START_REF] Marenco | Optical properties of tropospheric aerosols determined by lidar and spectrophotometric measurements (Photochemical Activity and Solar Ultraviolet Radiation campaign)[END_REF][START_REF] Xu | Three-wavelength cavity-enhanced albedometer for measuring wavelength-dependent optical properties and single-scattering albedo of aerosols[END_REF] . The corresponding measurements require broadband optical cavities to ensure a high-sensitivity.

In this chapter, a novel optical cavity using two Brewster-angle prism retroreflectors made of fused silica is described.

Cavity modes

Due to the effects of interference, a cavity (also called resonator) allows only certain patterns and frequencies of light being reproduced on every round-trip through the cavity, which ensures light bouncing back and forth thousands of times between cavity mirrors to realize a very long light-matter interaction path length. These specific patterns and frequencies are the eigenmodes, known as the modes of the cavity. Cavity modes are classified as longitudinal modes, differing in frequency from each other;

and transverse modes, differing in both frequency and the intensity pattern of the light. The longitudinal modes, describing a "comb-like" transmission pattern, result from frequency-dependent cavity resonances (Fig. 5.3).

Thus, the center frequency vq of the q th cavity transmission mode is:

2 q qc v nL = (5.1)
Where n is the refractive index inside the cavity.

The spacing of the modes, called the free spectral range (FSR), is inversely proportional to the cavity length, i.e., the mirror separation (L), which is equal to:

1 2 qq c FSR v v nL + = -= (5.2)
The FSR of an empty cavity with a length L in the range of 30 -50 cm is in the 300 -500 MHz range. Axial modes have Lorentzian profiles with a resonance width (FWHM : full width at half maximum) of , due to perturbation on either the light radiation frequency or the cavity length. F is called the cavity finesse, a useful FSR F = 93 specific parameter for characterizing the optical cavities performance. It depends solely on the mirror reflectivity, and is usually expressed as: (5.3) where R is the mirror reflectivity.

The performance of an optical cavity is usually characterized with the Q factor, or quality factor. Using the definition of the FWHM, the Q factor can be expressed as follows:

Q  =  ()
where  is the resonant frequency.

Brewster-angle prism retroreflectors

Broadband optical cavities based on prism reflectors rely on total internal reflection in combination with light being incident on the prism surfaces at Brewster's angle to realize low-losses and quasi-wavelength independent high-reflectivity over a broad spectral band from the UV to the near-infrared [START_REF] Langridge | Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source[END_REF][START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF][START_REF] Lee | A high-finesse broadband optical cavity using calcium fluoride prism retroreflectors[END_REF][START_REF] Moosmüller | Brewster's Angle Porro Prism: A Different Use for a Pellin-Broca Prism[END_REF][58][START_REF] Engel | Ring down spectroscopy with a Brewster's angle prism resonator[END_REF] .

Brewster's angle depends on the refractive indices of the prism material (n1), and that of the optically thin medium (n0). It can be calculated as follows: https://en.wikipedia.org/wiki/Brewster%27s_angle

(1 )

FSR F R R  = = - 
of the prism. In this work, the prism used as reflector is made of fused silica. The refractive index n of fused silica can be expressed using Equation (5.6) [START_REF] Malitson | Interspecimen Comparison of the Refractive Index of Fused Silica*, † [J][END_REF] : The data is taken from Ref. [START_REF] Malitson | Interspecimen Comparison of the Refractive Index of Fused Silica*, † [J][END_REF] . The light is then injected into the prism without suffering from Fresnel reflection losses. The injected light ray strikes the surface AB at R2 with an incident angle of 45º (> TIR), which ensures the light ray will undergo TIR. The light ray after the first TIR will then propagate toward the surface BC and reach it at R3 with an angle of incidence of 45º as , which ensures that the light ray undergoes a second TIR. After these two TIRs, the light ray will follow the path from R3 to R4 which is parallel in the horizontal plane to the incident ray path from R1 to R2. The light ray will exit the prism at the Brewster angle into air through the surface AD at R4. The exiting ray is parallel to the incident ray.

By the law of sines, the light ray will be centered on the AB side if the length AR1 is given by [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] :

sin( ) 1 4 sin 2 B AR AB   = (5.9)
Hence, the length of AR1 can be derived from Equation (5.9): The propagation distance |R2R3| is equal to . The propagation distance |R3R4| is given by Equation (5.12):

sin( ) 2 3 3 4 1 2 2 3 (1 ) sin 22 B B AB R R R R R R n    - = +  = + (5.12)
Thus, the total pathlength Lp inside a prism can be thus derived as below:

1 1 2 2 3 3 4 2 (1 ) p L R R R R R R AB n - = + + = + (5.13)
In this work, the dimensions of the fused silica prism are as follows : AB=16.00 mm and a total path inside the prism is Lp=38.23 mm. This is an important specific Prism II has identical dimensions except for surface EF which has a radius of curvature of 6.0 m. This radius of curvature is necessary for getting a stable optical cavity [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] .

P-polarized light propagating inside the prism cavity forms a closed loop passing through the following points (one round trip in the cavity) :

R4-R5-R6 (TIR)-R7 (TIR)-R8-R1-R2 (TIR)-R3 (TIR)-R4

The incident light starts at point R4. The incident ray is p-polarized, i.e. it is polarized in the plane of the incoming and the reflected ray. If the angle of incidence is perfectly aligned at the Brewster angle, no light is reflected toward R5 to follow the pathway R4-R5 due to the zero reflection for the p-polarized beam at the Brewster incidence angle. The incoming light is deliberately tuned away from the Brewster angle by δθ = ±1° to ensure the injection of the incident light into the cavity, which then follows the pathway R4→R5→R6→R7→R8→R1→R2→R3→R4. By rotating prism I the fractional reflectivity at the surface AD can be controlled.

After reflection at R4, the injected p-polarized light is reflected toward R5. The light at R5 enters prism II at the Brewster angle without Fresnel reflection losses. The injected p-polarized light undergoes two total internal reflections at points R6 and R7, it is then leaving prism II at point R8 at the Brewster angle with respect to the surface normal. After propagation through the cavity, the light reaches prism I at R1 at the same angle of incidence, B ± δθ, as the input light. A small fraction of light (Light-out) at R1 exits the cavity for detection. The vast majority of the light enter prism I and undergoes two total internal reflections to arrive at R4. A light loop is thus formed between the two prisms and light bounces back and forth thousands of times between the two prisms to achieve a several kilometers long effective optical path-length.

Ray transfer matrix

To ensure that the injected light is trapped inside the prism-based optical cavity, the ray trajectories should be described and analyzed by using the ABCD matrix. The Gaussian beam transfers through an optical system once, the ray trajectory is described in Fig. 5.12. The matrix M is the so-called ray transfer ABCD matrix. Equation (5.16) is known as the ABCD law for the radius of curvature r. It relates the input and out radius of curvature for an optical system described by its ABCD matrix.

Accordingly, the transport of a beam through a number of optical systems can be calculated by multiplying the individual transfer matrices that given by:

3 2 1 1 N N x M M M M x → → =    (5.17)
For a Gaussian beam as shown in Fig. 5.13, it is possible to define a complex radius q describing both the curvature of the wavefront and transversal size of the beam.

The nature of this radius of curvature is complex values. It is given by: ω 0 : beam waist, is the radius at which the intensity is 1/e 2 of its value on the axis; Θ: total angular spread; r(z): radius of curvature of the beam's wavefronts at z; ω(z): radius of the spot size at z.

Numerical modeling of a prism-cavity using the ABCD matrix method

The ABCD matrix for a prism cavity describing one round-trip can be obtained by multiplying the translation matrices and refraction matrices of all relevant elements in the cavity. For R6→R7 in prims II, the ray undergoes first total internal reflection at R6 located at the surface with a radius of curvature of Rc=6 m. The internal reflection angle is set to 45º . MR6(T) and MR6(s) represent reflection matrices for reflections on the tangential direction and sagittal planes, respectively [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] . The MR6 matrix can be given by: After the first TIR, the ray propagates to R7 over a distance of LR6-R7. The transmission matrix MR6-R7 is given by: For R7→R8, the ray will undergo a second TIR at R7 located at a flat surface and then travels a distance of LR7-R8 to R8, the corresponding matrix MR7-R8 is given by: For R8→R1, the ray passes through prism II in air at the Brewster angle and subsequently toward R1 through a distance of LR8-R1, the corresponding matrices MR8-R1 for the tangential and sagittal plane are given by:

103 81 8 1( ) 2 1 0 1 01 0 RR R R T L n M n - -    =     (5.28) 81 8 1( ) 1 1 0 0 1 0 RR R R S L M n - -    =      
(5.29)

For R1→R2, the ray enters prism I at R1 and travels to R2 through a distance of LR1-R2, the corresponding matrices MR1-R2 are given by: For R2→R3 in prims I, the ray undergoes at first TIR at R2. Thereafter the ray propagates the ray transmits to R3 through a distance of LR2-R3. This matrix MR2-R3 can be obtained by: For R3→R4 in prism I, the ray undergoes a second TIR at R3. Then the ray propagates to R4 over a distance of LR3-R4. This matrix MR3-R4 is given by: For R4→R4 in prims I, the ray is from prism to air with a complementary Brewster angle (90º -θB). The matrix MR4-R4 can be obtained by: Based on these matrices, the one round-trip matrix of this prism-based optical cavity for the tangential plane and sagittal plane can be obtained by multiplication of the matrices one-by-one in the direction opposite to the beam propagation direction as shown below:

11 T R R T R R R R R R T TT overall T R R R R TT tL n t L n L L t n f n nf n AB M CD nn n L L n ff - - - - --    - - + + +        ==       - - + +    
(5.38) (1 )

1 s R R R R s s R R R R RR S S S overall S RR S S t nL L t t L L L AB f n f n nf M CD L nf nf - - - - - - +   - - + - +      ==     --     (5.39)
Where

12 2 45 4 p R R T p R R LL t n L L n - - - = + + 56 81 2 p R R s R R LL t nL n - - - =+
LR5-R6 = LR1-R2, LR8-R1 = LR4-R5. 1 ,

Stability criteria

T R R R R TT nn A t D n L L n n f f -- = - = - +
+ can be obtained from Equation (5.38) for the tangential direction.

Since is half of , astigmatic behaviors will be most considerable for the tangential direction [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] . To simplify the calculation, we consider only the astigmatism of the beam propagation in tangential direction. In order to form a stable optical cavity in the tangential direction, using Eq. (5.46) and given LR1-R2=9.56

overall T AB M CD     ==     -     (5.48)
Based on the ABCD matrix analysis (5.48), radius r of the Gaussian beam at the point of R4 is 3531.257 cm derived by Equations (5.44). The size of the Gaussian beam (z) at R4 is obtained from Equation (5.45) at R4 located at prism I and was found to be 404 µm.

For one round-trip starting from R6 (R6-R7-R8-R1-R2-R3-R4-R5-R6), the matrix of the full prism cavity system can be written as follows:

( ) 5
( ) overall T R R T R R T R R T R R T R R T R R T R R T R R T R R T R T M M M M M M M M M M M - - - - - - - - - =          6
(5.49)

This ABCD matrix can be determined as follows:

() 0.82487 266.9742 0.00684

1 overall T AB M CD -     ==     -     (5.50)
Based on this ABCD matrix (5.50), the radius of curvature r of the Gaussian beam at point R6 was found to be 292.596 cm using Equations (5.44). This calculated radius of curvature is in good agreement with Rc(T) = 2fT = 292.6 cm of the EF surface in prism II with an (z) = 476.5 µm (Eq. 5.45). The mode matching condition is thus met with sufficient accuracy.

Optical losses

Though the reflectivity of the prism-retroreflector relies in principle on the material losses, other factors also affect the real effective reflectivity, such as surface scattering due to imperfections of the prism material and dirt on the prism surfaces, bulk absorption and scattering losses in the prism substrates, residual birefringence (concerting p to s polarization) due to either strain or misalignment of the optic axis of the prism substrate material, and small deviations from Brewster's angle. Optical losses of the prism retroreflector can be divided into intrinsic and extrinsic losses.

The intrinsic losses are subdivided into light absorption losses and light scattering losses. It is well-known that alkalis and transition metal impurities have strong absorption bands in the UV, visible, and near IR [START_REF] Sigel | Vacuum ultraviolet absorption in alkali doped fused silica and silicate glasses [J][END_REF][START_REF] Schultz P C | Optical Absorption of the Transition Elements in Vitreous Silica [J][END_REF] . They are not a concern for fused silica prisms because of their exceptionally low impurity levels [START_REF] Moore L A | Fused silica as an optical material[END_REF] . For visible and shorter wavelengths, the intrinsic material losses are dominated by Rayleigh scattering, which is caused by fluctuations related to density or compositional inhomogeneities in the glass structure that are smaller than the light wavelength [START_REF] Moore L A | Fused silica as an optical material[END_REF] .

Rayleigh scattering has a 1/λ 4 wavelength-dependence and is, therefore stronger for ultraviolet light than for infrared light. Due to the wavelength dependence, this loss becomes unacceptably large for wavelengths shorter than 400-500 nm, which correspond to scattering loss of 350-150 ppm/pass, respectively [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] .

The extrinsic losses are a consequence of the p-polarized light being injected into the cavity at an angle that slightly deviates from Brewster's angle. The Fresnel loss (FL) due to the angle deviation per surface is given by [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] :

( ) ( ) ( ) 2 4 2 6 1 4 n FL n    - =  (5.51)
where δθ is the small angular difference to the Brewster angle, and n is the refraction index of fused silica. For instance, if the detuning angle δθ is 1˚ from the Brewster's angle, the resulting Fresnel loss can be calculated as a function of wavelength from Equation (5.51). The result is plotted in Fig. 5.16 for a wavelength range of 400-2000 nm for a Brewster's angle fixed for =1064 nm. Another source of extrinsic losses is scattering losses (SLT) [START_REF] Elson | Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties[END_REF] resulting from total internal reflection, expressed as:

( ) ( ) 2 4 cos n SLT        =     (5.52)
where θ is the incident angle. σ is the root mean squared (RMS) surface roughness with a pretty good value of 0.1 nm [START_REF] Lehmann K K | Brewster angle prism retroreflectors for cavity enhanced spectroscopy[END_REF] . The theoretical optical loss generated by scattering at the surface of the total internal reflection inside the prims is shown in Fig. 5.17. The scattering losses at prism air-prism surface (SLS) is an intrinsic loss, which can be calculated by Equation (5.53) [START_REF] Elson | Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties[END_REF] and plotted in Fig. 5.18.

( ) ( ) Using a super polished surface with RMS roughness of ~ 0.1 nm, the surface scattering can be estimated as 0.14 and 0.94 ppm at λ=1064 nm for a transmission at Brewster's angle on the prism air-prism surface and one total internal reflection from the surface inside the prim, respectively. For one prism, total scattering losses at prism surface and TIR are calculated using Equation (5.52) and (5.53); the estimated loss is 2.16 ppm due to two TIRs and two passes in one prism.

2 2 2 1 2 1 n SLS n       -   =   +   (

Spectroscopic characterization of the prism-based cavity

An optical setup using a prism cavity was built as shown in Fig. 5.19 for the spectroscopic characterization of its performance. The prism-cavity spectrometer mainly includes three components that are : (1) light source module, (2) fused silica prism cavity, and (3) photodetector or CCD spectrometer. A broadband supercontinuum light source working at 400-2000 nm is used to validate the broadband performances of the prism-cavity in the region from the visible to the near-infrared. As the broadband light can't excite the cavity resonance mode while simultaneously tuning the cavity length, a tunable diode laser operating at 1074 nm is employed to characterize the optical cavity in the near-infrared.

Two lenses (f1 and f2) are used to inject the light beam into the cavity to realize mode matching. The injected light radiation is linearly polarized. A polarization rotator is used to vary the polarization state of the injected beam to p-or s-polarized in the plane of incidence which is very helpful for optical alignment.

A 632 nm He-Ne laser is used for optical alignment. S-polarized light beam is used to perform cavity optical alignment. Once it is done, polarization of the light beam can be changed from s-polarization to p-polarization to realize cavity modes excitation by p-polarized light.

Two iris (iris 1 and iris 2) are used for optical alignment with the help of four plane deflection mirrors M1-M4. A polarizer is used to generate p-polarized light in the plane of incidence before the injection to prism I. The light exiting from the prism cavity is focused, through lens f3 with a focal length of 50 mm, on a photodetector (PDA10CS, Thorlabs) for detection of narrow-band laser radiation or on a spectrometer (Model Ocean Optics QE65000) for measurements of broad-band light for spectral analysis.

Results and discussions

Study of prism cavity mode using an external cavity laser

To characterize the prism cavity, a tunable external cavity laser (New focus, Model 6321) was used. The output power was 9.0 mW at =1.05 nm when powered with a current of 66 mA at 20 °C. A lens with a focal length of 6.24 mm (Thorlabs Inc, Model A110TM-C) and a second lens with a focal length of 50 mm were both used to collimate the laser output beam and then focus the laser beam at Prism I with a beam diameter of less than 1 mm to ensure mode matching. The injected laser power is around 2 mW after passing through the two lenses, the steering mirrors (M1, M2, M3, M4) and two irises (the distance between the laser output and input of the prism I is ~75 cm). As the laser beam was linearly polarized, the polarizer shown in Fig. 5.19 was not useful and was removed. The theoretical FSR can be expressed as follows:

Broadband absorption spectrum measurement in the 500-675 nm region

In order to demonstrate the broadband operation performance, broadband cavity enhanced absorption spectroscopy approach (BBCEAS) was realized by coupling a supercontinuum light source (SC-Pro-M, YSL Photonics) to a fused silica prism cavity.

Real time optical monitoring of NO3 radical production resulting from NO2+O3→NO3 reaction in the 500-675 nm spectral region was performed using this BBCEAS setup.

A CCD spectrometer (Ocean Optics) with a spectral resolution of 0.42 nm was used for spectral measurements. The prism cavity was installed in a box for this experimental demonstration. The cavity was initially purged with zero air to provide background spectra I0(λ). After that, 2 ppmv NO2 was injected into the cavity, the NO2 concentration was simultaneously measured with the BBCEAS setup and a chemiluminescence NOx

Ré sumé

Comme l'a indiqué le scientifique Mendeleï ev, "Toute science commence par la mesure, et une science exacte ne peut ê tre imaginé e sans mesure", le progrè s en science est é troitement lié à l'avancement de la technologie d'observation et de mesure. La

technologie avancé e de dé tection des gaz traces enrichit non seulement les mé thodologies de recherche en science de l'atmosphè re, mais joue é galement un rôle vital dans l'industrie, l'agriculture, la santé publique et la protection de l'environnement.

On entend par gaz traces des gaz dont les rapports de mé lange sont infé rieurs à une partie par million en volume (ppmv). En sciences de l'environnement, les mesures des gaz traces dans l'atmosphè re terrestre, y compris le dioxyde de carbone (CO2), le mé thane (CH4), la vapeur d'eau (H2O), l'ozone (O3), les oxydes d'azote (NOx) et les composé s organiques volatils (COV), revê tent une importance significative pour la compré hension des principaux processus chimiques atmosphé riques et des tendances de changement climatique mondial [1] . Dans le domaine du diagnostic mé dical, l'analyse quantitative non invasive des gaz traces biogè nes, en tant que marqueurs volatils de maladies, dans le souffle exhalé a suscité un inté rê t croissant ces derniè res anné es [2] .

Ces marqueurs peuvent fournir des informations importantes sur l'é tat de santé des humains [3] . La plupart de ces molé cules volatiles sont pré sentes en fractions volumiques de l'ordre de ppbv ou moins.

La dé tection de gaz s'appuie traditionnellement sur des é quipements analytiques de laboratoire, tels que les chromatographes en phase gazeuse [4] , la spectromé trie de masse [5] , ou des dispositifs peu coûteux, tels que les capteurs de gaz à semi-conducteurs [6] et les capteurs de gaz é lectrochimiques [7] . Bien que les chromatographes en phase gazeuse et la spectromé trie de masse offrent des donné es pré cises, ils né cessitent une pré paration d'é chantillon, et les instruments sont trè s coûteux. Les capteurs de gaz à semi-conducteurs et les capteurs de gaz é lectrochimiques, bien qu'abordables, sont sujets à la dé rive et aux ré ponses croisé es à d'autres gaz et aux variations du taux d'humidité [5] . De plus, les capteurs de gaz é lectrochimiques ont une duré e de vie limité e. ê tre quantifié e par sa puissance d'absorption [8] . Comme l'intensité transmise par Dans la pré sente thè se, les cellules optiques à chemin plié sont considé ré es à travers deux approches : les cellules à passages multiples (MPC, telles que les cellules Herriott [9,10] , White [START_REF] White | Long Optical Paths of Large Aperture[END_REF] , et Chernin [START_REF] Chernin S M | Optical multipass matrix systems[END_REF] ) et les cavité s optiques [START_REF] Adler F | Cavity-enhanced direct frequency comb spectroscopy: technology and applications[END_REF] .

Au cours des derniè res dé cennies, diverses techniques spectroscopiques impliquant la LPAS ont é té dé veloppé es pour la dé tection sensible des gaz, telles que la spectroscopie d'absorption par laser à diode ré glable (TDLAS) [START_REF] Lackner | Tunable Diode Laser Absorption Spectroscopy (Tdlas) in the Process Industriesa Review[END_REF][START_REF] Cui R | Three-Dimensional Printed Miniature Fiber-Coupled Multipass Cells with Dense Spot Patterns for ppb-Level Methane Detection Using a Near-IR Diode Laser[END_REF][START_REF] Schiff H I | The use of tunable diode laser absorption spectroscopy for atmospheric measurements[END_REF] , la spectroscopie photoacoustique amé lioré e à chemin long (MPC, cavité ) (PAS) [START_REF] Bluvshtein | Calibration of a multi-pass photoacoustic spectrometer cell using light-absorbing aerosols[END_REF][START_REF] Nä | Mobile laser spectrometer with novel resonant multipass photoacoustic cell for trace-gas sensing[END_REF][START_REF] Metcalfe G D | Cavity-Enhanced Raman and Helmholtz Resonator Photoacoustic Spectroscopy to Monitor the Mixed Sugar Metabolism of E. coli[END_REF] ou la spectroscopie photoacoustique à quartz amé lioré e (QEPAS) [START_REF] Kosterev A A | Applications of quartz tuning forks in spectroscopic gas sensing[END_REF][START_REF] Lewicki R | Carbon dioxide and ammonia detection using 2 μm diode laser based quartz-enhanced photoacoustic spectroscopy[END_REF][START_REF] Dong L | QEPAS spectrophones: design, optimization, and performance[END_REF] , la spectroscopie d'extinction en anneau de cavité (CRDS) [START_REF] Maity | Cavity Ring-Down Spectroscopy: Recent Technological Advancements, Techniques, and Applications[END_REF][START_REF] Brown | Absorption spectroscopy in high-finesse cavities for atmospheric studies[END_REF][START_REF] Spence T G | A laser-locked cavity ring-down spectrometer employing an analog detection scheme[END_REF] et la spectroscopie d'absorption amé lioré e en cavité (CEAS) [START_REF]Integrated cavity output analysis of ultra-weak absorption[END_REF][START_REF] Engeln R | Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy[END_REF][START_REF] Zhao W | Wavelength modulated off-axis integrated cavity output spectroscopy in the near infrared[END_REF] .

Le MPC à chemin plié avec un motif de taches denses basé sur des miroirs sphé riques a é té rapporté pour la premiè re fois par Stephen So et ses collè gues en 2012 3 Un MPC à chemin plié avec un motif de sept cercles multiples (Fig. 1.1) a é té rapporté par Liu et al. en 2015 en utilisant deux miroirs sphé riques de 2 pouces [START_REF] Liu | Highly sensitive detection of methane by near-infrared laser absorption spectroscopy using a compact dense-pattern multipass cell[END_REF] .

Une telle cellule optique à chemin plié atteint une longueur de trajet optique de 26.4 mè tres. a seven-multi-circle spot pattern (photographs above the schematic). Figure taken from Ref. [START_REF] Liu | Highly sensitive detection of methane by near-infrared laser absorption spectroscopy using a compact dense-pattern multipass cell[END_REF] .

En temps ré el du gaz naturel [START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] . [START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] .

Tous les faisceaux de cette cellule optique à chemin plié ont é té manipulé s par une conception optique et focalisé s au centre du MPC à chemin plié , ce qui a permis une accumulation de puissance suffisante pour amé liorer la dé tection des signaux Raman.

En 2020, Chen et al. ont conç u un dé tecteur de gaz photoacoustique à haute sensibilité en utilisant un CPM miniature à chemin plié (de type cellule Herriott) pour la dé tection de C2H2 [START_REF] Zhang | High-sensitivity photoacoustic gas detector by employing multi-pass cell and fiber-optic microphone[END_REF] . En 2021, Ma et al. ont conç u une cellule optique à chemin plié à motif de tache liné aire basé e sur deux prismes à angle droit pour coupler la lumiè re à une fourche d'accord en quartz (Fig. 1.3), ce qui a permis d'amé liorer encore la sensibilité de mesure d'un facteur de 3.2 [START_REF] Qiao | Multi-pass quartz-enhanced photoacoustic spectroscopybased trace gas sensing[END_REF] . De nombreuses cellules optiques polyvalentes à chemin plié ont é té conç ues. L'un des premiers exemples est la cellule Herriott [9] , qui a é té introduite pour la premiè re fois en 1964. Ce type de cellule optique est composé de deux miroirs sphé riques sé paré s par une distance presque é gale à leur rayon de courbure commun, ce qui peut produire un motif de tache circulaire typique sur sa surface de miroir, comme illustré dans la Fig. est formé e par trois miroirs sphé riques [START_REF] White | Long Optical Paths of Large Aperture[END_REF] . Les cellules Herriott, en raison de leur conception simple et de leur manipulation efficace du faisceau, sont un choix populaire dans de nombreuses applications optiques. Un autre avantage de la cellule Herriott est qu'elle est plus facile à aligner par rapport à la cellule White, car la direction du faisceau de sortie n'est pas sensible aux changements dans l'alignement du miroir. Dans le cas des miroirs sphé riques, le faisceau trace un motif de tache circulaire sur les deux surfaces du miroir, ce qui ne permet pas une utilisation optimale de la surface du miroir.

Le nombre de passages est dé terminé par l'espacement des miroirs, tandis que l'espacement des miroirs est difficile à ajuster en raison de la thé orie spé cifique de la conception de la cellule optique à chemin plié . Un autre type de cellule optique à chemin plié a é té ulté rieurement dé veloppé en utilisant des miroirs lé gè rement astigmatiques pour cré er des motifs de taches de Lissajous sur les surfaces des miroirs [START_REF] Herriott D R | Folded Optical Delay Lines[END_REF] , comme illustré dans la Fig. 1.5. Par consé quent, le nombre de ré flexions des rayons dans la cellule optique à chemin plié est augmenté pour atteindre une efficacité d'utilisation plus é levé e tout en minimisant le chevauchement des taches [START_REF] Mcmanus | Astigmatic mirror multipass absorption cells for long-path-length spectroscopy[END_REF][START_REF] Gragossian | Astigmatic Herriott cell for optical refrigeration[END_REF] . Cependant, les miroirs pré sentant une grande pré cision de surface doivent ê tre fabriqué s par des techniques de meulage et de polissage qui produisent naturellement des surfaces sphé riques. La fabrication de surfaces asphé riques est plus complexe, et il est difficile de produire un miroir avec une pré cision de surface suffisante pour suivre le motif conç u. Par consé quent, les cellules Herriott basé es sur des miroirs sphé riques sont largement dé ployé es en raison de la production moins coûteuse des miroirs et du fait que la qualité des surfaces des miroirs peut ê tre plus facilement contrôlé e [START_REF] Cui R | Calculation model of dense spot pattern multi-pass cells based on a spherical mirror aberration[END_REF] . Les miroirs mé talliques utilisé s dans les approches des MPC peuvent fournir une ré flectivité quasi indé pendante de la longueur Pour amé liorer davantage la sensibilité de mesure en utilisant des longueurs de trajet optique plus longues, des cavité s à base de miroirs dié lectriques peuvent ê tre utilisé es. Une ré flectivité é levé e (> 99,9 %, jusqu'à 99,999 %) du miroir dié lectrique est obtenue via l'interfé rence constructive de la ré flexion de Fresnel aux interfaces produite par des revê tements multicouches de maté riaux à indice de ré fraction é levé et faible en alternance [START_REF] Pacheco M Z | Applicability Evaluation of a Laser Light-Mater Interaction Based Computational Tool on Status Identification of Applied Micro-Structured Coatings[END_REF] . Les cavité s avec des miroirs à haute ré flectivité permettent des longueurs de trajet optique de quelques dizaines de kilomè tres [START_REF] Provencal R | Cavity-enhanced quantum-cascade laser-based instrument for carbon monoxide measurements[END_REF][START_REF] Langridge | Cavity enhanced absorption spectroscopy of multiple trace gas species using a supercontinuum radiation source[END_REF] Les cavité s optiques basé es sur des prismes en tant que ré flecteurs peuvent offrir des performances à haute finesse sur une large plage spectrale de 400 à 2500 nm [START_REF] Meng | Prism-based Broadband Optical Cavity (400 -1600 nm) for High-Sensitivity Trace Gas Sensing by Cavity Enhanced Absorption Spectroscopy[END_REF] .

Les cavité s à base de prismes fournissent un outil de mesure spectroscopique polyvalent pour la dé tection de gaz à haute sensibilité .

La thè se de doctorat se compose de six chapitres comme suit : Dans la section 4.3, le dé veloppement d'un MPC à motif de tache unique est pré senté . Il a é té appliqué à des systè mes de capteurs QEPAS [START_REF] Dong L | QEPAS spectrophones: design, optimization, and performance[END_REF][START_REF] Ma | Review of Recent Advances in QEPAS-Based Trace Gas Sensing[END_REF][START_REF] Wojtas | Mid-Infrared Trace Gas Sensor Technology Based on Intracavity Quartz-Enhanced Photoacoustic Spectroscopy[END_REF] . Un faisceau laser est ré flé chi en avant et en arriè re entre deux miroirs de CPM en passant à travers une fourche d'accord en quartz (QTF) de taille d'espacement de dents personnalisé e insé ré e à l'inté rieur du MPC, ce qui produit de multiples sources ponctuelles acoustiques le long de l'axe de symé trie du QTF. En consé quence, le QTF fonctionnant en mode de ré sonance fondamentale est simultané ment excité par des ondes acoustiques provenant de multiples sources ponctuelles acoustiques, le signal acoustique ré sultant permet d'amé liorer le signal mesuré .

Le chapitre 5 dé crit les ré cents dé veloppements d'une nouvelle cellule optique à chemin plié basé e sur des prismes en tant que ré flecteurs de cavité qui offre une cavité optique à haute finesse fonctionnant dans une plage spectrale large de 400 à plus de 2500 nm. Cette nouvelle approche de cavité pré sente un bon potentiel pour des mesures spectroscopiques à haute sensibilité de multi-espè ces ou/et d'absorption à large bande de gaz traces et d'aé rosols [START_REF] Meng | Prism-based Broadband Optical Cavity (400 -1600 nm) for High-Sensitivity Trace Gas Sensing by Cavity Enhanced Absorption Spectroscopy[END_REF] .

Le chapitre 6 ré sume les principales ré alisations de cette thè se de doctorat et trace les grandes lignes des travaux futurs potentiels, tels que le dé veloppement d'un capteur optique miniature pour l'observation basé e sur un UAV, et les applications de la cavité à prismes à large bande à la spectroscopie à peigne de fré quence à haute sensibilité .

I

  would like to thank Prof. Hongpeng Wu and Dr. Tong Nguyen-Ba for their support and patience in my Ph.D. works. I would also like to give my sincere acknowledgment to the members of the jury, Prof. Christa Fittschen, Prof. Yanxia Cui, Prof. Andy Ruth and Prof. Kun Liu, for their excellent evaluation works on my PhD research. I would like to extend my sincere thanks to Dr. Hongbing Chen and Dr. Gaoxuan Wang for their support and discussions on the development of prism cavity. Thanks to Dr. Minh Nhut NGO and Dr. Eric Fertein for their help during my work in the ULCO. I would like to express my particular gratitude to Prof. Azer Yalin at Colorado State University (USA) for the fruitful collaboration in the development of a broadband prism-cavity in the ULCO in France. Owing to this collaboration, this work can be performed with great success. I would like to acknowledge Prof. Kun Liu at Anhui Institute of Optics and Fine Mechanics, China for discussing the design of dense spot pattern multi-pass cells. Thanks to my team members, including Minh Nhut Ngo, Ghoufrane Abichou, Marie-Hé lè ne Mammez, Sama Molaie, Anastasia Pienkeina, Zhen Liu at LPCA, and Huandan Zheng, Xukun Yin, Shangzhi Li, Tingting Wei, Biao Li, Bo Sun, Yufeng Pan, Zhijin Shang, Yongyong Hu, Chaofan Feng, Gang Wang, Xiaoli Liu, Yong Wang, II Zheng Chen, Jiapeng Wang, et.al. in SXU. Thanks to my friends, including Bao-Anh Phung-Ngoc, Bouthayna Alrifai, Fatima Al Ali, and Ghoufrane Abichou in LPCA and Yueting Zhou, Shuangping Han, Zhuo Cai in SXU.

Fig. 1 . 1 18 Fig. 1 . 2 18 Fig. 1 . 3 19 Fig. 1 . 4 20 Fig. 1 . 5 2 Fig. 2 . 1 25 Fig. 2 . 2 28 Fig. 2 . 3 30 Fig. 2 . 4 33 Fig. 2 . 5 34 Fig. 2 . 6 36 Fig. 2 . 7 3 Fig. 3 . 1

 111812181319142015221252228233024332534263627331 Fig. 1.1 Schematic diagram of a CH 4 sensor (bottom) involving a compact folded-path optical cell with a seven-multi-circle spot pattern (photographs above the schematic). Figure taken from Ref. [30]. ..... 18 Fig. 1.2 Schematic of the folded-path optical cell based-Raman natural gas (CH 4 ) analyzer system. Figure taken from Ref. [31]. .............................................................................................................................. 18 Fig. 1.3 Folded-path optical cell-based QEPAS system for water vapor detection. Figure taken from Ref. [33]. ........................................................................................................................................................ 19 Fig. 1.4 One-circle spot pattern on the mirror surface of Herriott cell using a visible laser. .................. 20 Fig. 1.5 Nonsynchronously sampled Lissajous pattern on the mirror surface. Figure taken from Ref. [34]. ................................................................................................................................................................ 21

Fig. 3 . 2 45 Fig. 3 . 3 46 Fig. 3 . 4 46 Fig. 3 . 5 Fig. 3 . 6 50 Fig. 3 . 7 52 Fig. 3 . 8 53 Fig. 3 . 9 54 Fig. 3 . 4 Fig. 4 . 1 60 Fig. 4 . 2 61 Fig. 4 . 3 62 Fig. 4 . 4 64 Fig. 4 . 65 Fig. 4 . 6

 32453346344635365037523853395434416042614362446446546 Fig. 3.2 Standard Herriott spot patterns produced by a conventional ABCD matrix with large (a) and small (c) angle incident rays. The corresponding spot patterns calculated by the modified ABCD matrix with large (b) and small (d) angle incident rays. Gray lines are the trajectories between two neighbor spots (similarly hereinafter). ................................................................................................................... 45 Fig. 3.3 Four exotic spot patterns generated by two-spherical-mirror MPC. (a) morning-glory-like spot pattern; (b) three nested-circle spot pattern; (c) seven nonintersecting-circle spot pattern; (d) sunflowerlike spot pattern. The initial ray parameters for each pattern are listed in Table 2. ................................ 46 Fig. 3.4 Simulations of morning-glory-like spot pattern (a) and seven nonintersecting-circle spot pattern (b) on M2 by use of TracePro. Spot size stands for irradiance. .............................................................. 46 Fig. 3.5 Manipulation of the spot density (a) and pattern size (b) for a sunflower-like spot pattern with a fixed ray inclination angle. ..................................................................................................................... 48 Fig. 3.6 Reflection of incident ray set on a spherical mirror. ................................................................. 50 Fig. 3.7 Six exotic spot patterns generated by a two-spherical-mirror MPC. (a) three-, (b) four-, (c) five-, (d) seven-, (e) eight-, and (f) nine-multi-circle spot patterns. The plots represent the spot projection of the exit mirror, black spots represent the exit positions of beam, green rings mark black spots. ........... 52 Fig. 3.8 Photographs of the real (a) eight-and (b) nine-multi circle spot pattern on the exit spherical mirror. ..................................................................................................................................................... 53 Fig. 3.9 Ambient CH 4 2f absorption spectrum at normal atmospheric pressure. .................................... 54 Fig. 3.10 Evolutionary process of spot patterns on a two-spherical-mirror MPC for increasing distance, D, between the mirrors. .......................................................................................................................... 55

Fig. 4 . 7 67 Fig. 4 . 8 68 Fig. 4 . 9 68 Fig. 4 . 72 Fig. 4 . 73 Fig. 4 . 74 Fig. 4 . 75 Fig. 4 . 76 Fig. 4 . 77 Fig. 4 . 78 Fig. 4 . 78 Fig. 4 . 79 Fig. 4 . 80 Fig. 4 . 84 Fig. 4 . 85 Fig. 4 . 86 Fig. 4 . 87 Fig. 4 . 5 Fig. 5 . 1 90 Fig. 5 . 2 91 Fig. 5 . 3 92 Fig. 5 . 4 93 Fig. 5 . 5 94 Fig. 5 . 6 94 Fig. 5 . 7 95 Fig. 5 . 8 95 Fig. 5 . 9 96 Fig. 5 . 10 96 Fig. 5 . 98 Fig. 5 . 12 99 Fig. 5 . 13 100 Fig. 5 . 14 101 Fig. 5 . 104 Fig. 5 . 108 Fig. 5 . 17 108 Fig. 5 . 18 109 Fig. 5 . 19 110 Fig. 5 . 111 Fig. 5 . 112 Fig. 5 . 22 112 Fig. 5 . 24

 4767486849684724734744754764774784784794804844854864874551905291539254935594569457955895599651096598512995131005141015104510851710851810951911051115112522112524 Fig. 4.7 (a) CO concentration response of the sensor system; (b) Linear dependence of the measured concentrations as a function of the five CO nominal concentrations. .................................................... 67 Fig. 4.8 Percent deviation of the measured concentration compared with the calculated value. Histogram plot of the deviation. The red line corresponds to a Gaussian profile. ................................................... 68 Fig. 4.9 Allan variance plot for the same time series measurements with the histogram plot. ............... 68 Fig. 4.10 Calculated seven-nonintersecting circle spot pattern on the (a) entrance and (b) exit spherical mirror surfaces; photographs of the real seven-nonintersecting circle spot pattern on the (c) entrance and (d) exit spherical mirror surfaces. A red visible diode laser beam was used to visualize the spot pattern. ................................................................................................................................................................ 72 Fig. 4.11 Signal-to-noise ratio as a function of pass counts in a MPC for different reflectivities. ......... 73 Fig. 4.12 (a) Photograph of a custom-made pre-amplifier; (b) Photograph of an InGaAs photodiode detector on the custom-made amplifier circuit board. ............................................................................ 74 Fig. 4.13 (a) Photograph of an open mini-MPC with a seven-nonintersecting circle spot pattern shown on the exit spherical mirror surface; (b) Photograph of the ray trajectories between the two spherical mirrors; (c) Photograph of a closed mini-MPC with a gas inlet and an outlet; (d) Top view of the closed mini-MPC without the cap; (e) Schematic diagram of the CH 4 sensor. ................................................. 75 Fig. 4.14 CH 4 concentration response of the TDLAS sensor system. .................................................... 76 Fig. 4.15 Rise time and fall time of the mini-MPC sensor system. Rise time: the time interval between initial response and 90% of the final response. Fall time: the time interval between initial response to a step decrease in concentration and 90% of the final response. ............................................................... 77 Fig. 4.16 2 ppmv CH 4 Allan-deviation stability of the mini-MPC based TDLAS sensor. ..................... 78 Fig. 4.17 Deviation of the measured concentration compared with the calculated value. Histogram plot of the deviation. The black line depicts a Gaussian profile. ................................................................... 78 Fig. 4.18 2f absorption spectrum of 2 ppmv CH 4 at normal atmospheric pressure. ............................... 79 Fig. 4.19 Continuous monitoring of CH 4 concentrations (a) measured from Apr. 6, 2019 to Apr. 18, 2019 near sewers and (c) measured from Apr. 28, 2019 to May 8, 2019 far away from sewers on the Shanxi University campus, China; Diurnal variations of CH 4 mixing ratio (b) near sewers and (d) far away from sewers. Bottom whisker, bottom box line, and top whisker in (b) and (d) indicate the 10th, 25th, 75th, and 90th percentiles, respectively. Lines inside the boxes and continuous solid lines in (b) and (d) represent the hourly median and mean of the data, respectively. ........................................................... 80 Fig. 4.20 (a) Single line spot pattern generated by a theoretical model of a two-spherical-mirror MPC based on spherical aberration. The plot represents the spot projection of the entrance mirror. Black spot represents the coincident entrance and exit positions of the beam. (b) Verification of single-line spot pattern on the entrance mirror using the TracePro. Spot size represents the irradiance. ........................ 84

FWHM

  address the major scientific issues related to environmental pollution and climate change, knowledge about the exact gas compositions and their concentrations in the atmosphere is crucial. This thesis presents studies, developments, and applications of folded-path optical cells dedicated to gas sensing by laser absorption spectroscopy. The developed folded-path optical cells during the present PhD work are not commercially available. They are very suitable for specific field monitoring that require cost-effective, compact, and lightweight optical sensors, such as applications in network and UAV observations. This chapter describes the motivation of the PhD research project, accompanied with a brief historical overview of folded-path optical cells and the current development states. The main content of each chapter of the PhD thesis is introduced at the end of this chapter.

Fig. 1 . 1

 11 Fig. 1.1 Schematic diagram of a CH 4 sensor (bottom) involving a compact folded-path optical cell with

Fig. 1 . 2

 12 Fig. 1.2 Schematic of the folded-path optical cell based-Raman natural gas (CH 4 ) analyzer system. Figure taken from Ref.[START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] .

Fig. 1 . 3

 13 Fig. 1.3 Folded-path optical cell-based QEPAS system for water vapor detection. Figure taken from Ref [33].

Fig. 1 . 4

 14 Fig. 1.4 One-circle spot pattern on the mirror surface of Herriott cell using a visible laser.

Fig. 1 . 5

 15 Fig. 1.5 Nonsynchronously sampled Lissajous pattern on the mirror surface. Figure taken from Ref. [34].

Fig. 2 . 1

 21 Fig. 2.1 Illustration of the parameters involved in the Beer-Lambert law.

  ensures that: the area under the line profile function represents the probability of a photon being absorbed in an arbitrarily small frequency interval, and the line profile function can be compared across different absorption lines and experiments if they have been normalized in the same way. As such, the line profile function is an essential tool in spectroscopy, allowing researchers to quantify the absorption of light by different materials and gain insights into their properties.

Fig. 2 . 3

 23 Fig. 2.3 White cell (left) and Herriott cell (right). M 1 , M 2 , M 3 are spherical mirrors. Red lines are ray trajectories.

Fig. 2 . 4

 24 Fig. 2.4 Conventional configuration for TDLAS employing a laser diode, showing detected signals as a function of laser drive current for (a) direct scanning of the gas line, and (b) 2nd harmonic wavelength modulation spectroscopy (2 f-WMS).

Fig. 2 . 5

 25 Fig. 2.5 Detection of the photoacoustic effect in a closed cell.

Fig. 3 . 1

 31 Fig. 3.1 Transmission of rays between two identical spherical mirrors, M1 and M2, and reflection on the M2 surface, in the x-z plane of a Cartesian coordinate system. R: radius of curvature of the spherical mirrors; D: mirror spacing; O: origin of coordinates; d n : optical path length for the n th transmission between M1 and M2; (x n-1 , x' n-1 ): spot location on M1 and initial ray inclination angle; (x n , x' n ): spot location on M2 and ray inclination angle after n th transmitting between M1 and M2 and reflected by M2.

3

 3 Numerical solutions of a dense spot patterned MPC based on aberration Comparisons were made between the calculated difference from the standard ABCD matrix Equation (3.1) and the modified ABCD matrix without the paraxial approximation of Equation (3.2). Two identical spherical mirrors with 25-mm focal length and 35.5-mm mirror spacing were employed. The initial location and inclination angles are (-8.5 mm, 4.1 mm) and (16.1 o , 8.1 o ), respectively. The result from Equation (3.1) shows a standard 2×2 cm 2 Herriot pattern as depicted in Fig. 3.2 (a), while the result from Equation (3.2) displays a spot pattern consisting of five arcs, as shown in Fig. 3.2 (b). Spherical aberration makes the symmetrical Herriot distribution spots from a paraxial calculation yield five separated arcs. Therefore, when a marginal ray with a large inclination angle enters an MPC comprising two spherical mirrors, the aberration must be taken into account. The modified ABCD matrix describes the gradual evolution of the spot patterns on the mirror surface, which is a practical design tool. Subsequently, the parameters of the spherical mirrors were changed to 200-mm focal length and 266.3-mm mirror spacing. The small inclination angles (-3.4 o , 0 o ) and the initial location (8.1 mm, -10.8 mm) were employed. The result from Equation (3.2) in Fig. 3.2 (d) is in excellent agreement with that from Equation (3.1) in Fig. 3.2 (c), which verifies that the modified ABCD matrix can return to the standard ABCD matrix with a small angle incident ray.

Fig. 3 . 2

 32 Fig. 3.2 Standard Herriott spot patterns produced by a conventional ABCD matrix with large (a) and small (c) angle incident rays. The corresponding spot patterns calculated by the modified ABCD matrix with large (b) and small (d) angle incident rays. Gray lines are the trajectories between two neighbor spots (similarly hereinafter).

  Figure 3.3 (a) gives a morning-glory-like spot pattern in which the 'petals' face to the 'flower heart', while Fig. 3.3 (d) exhibits a sunflower-like spot pattern in which the 'petals' face to the outside. More concise spot patterns were obtained by adjusting the mirror spacing and the entry location of the incident ray on M1. Figure 3.3 (b) shows three nested-circle spot patterns, while Fig. 3.3 (c) exhibits a seven nonintersecting-circle spot pattern. For these spot patterns, an angle tolerance of ±0.3 o is allowed with 2-mm entry and exit holes of light. To validate the calculation model, the spot patterns in Fig. 3.3 (a) and (c) were simulated using TracePro. The results were shown in Fig. 3.4. The consistency between the calculated and simulated results verifies the validity of the modified ABCD matrix.

Fig. 3 . 3

 33 Fig. 3.3 Four exotic spot patterns generated by two-spherical-mirror MPC. (a) morning-glory-like spot pattern; (b) three nested-circle spot pattern; (c) seven nonintersecting-circle spot pattern; (d) sunflowerlike spot pattern. The initial ray parameters for each pattern are listed in Table2.

Fig. 3 . 4

 34 Fig. 3.4 Simulations of morning-glory-like spot pattern (a) and seven nonintersecting-circle spot pattern (b) on M2 by use of TracePro. Spot size stands for irradiance.

Fig. 3 .

 3 Fig. 3.3 (b), and the second best in the list is the seven nonintersecting-circle spot pattern in Fig. 3.3 (c).

  3 can be manipulated to increase the spot density or scale up or down. To increase the spot density, the initial location on M1 (x0, y0) of the entry ray and the inclination angle (x'0, y'0) remain unchanged. The focal length f and mirror spacing D multiply the intended gain factor.The pass count n will increase or decrease by the gain factor. For example, the pass count n in Fig.3.3 (d) can be two times more (412 times) for the following parameters: f = 50 mm, D = 122.79 mm, as shown in Fig.3.5 (a). It should be noted that the increase in the spot density does not occur on the original sunflower-like spot pattern. The new sunflower-like spot pattern in Fig.3.5 (a) becomes wide and grows two times larger although the entry location and angle remained unchanged. Moreover, the new and original distance D do have not the exact double relationship, but 1.99 times. In other words, after f and D multiply the gain factor, the D requires fine-tuning in order to obtain a new sunflower-like spot pattern, since otherwise the spot pattern is distorted.A tentative explanation is that there occurs nonlinear calculation processing in Equation(3.3).When the inclination angle (x'0, y'0) of the entry ray remains unchanged, and the initial entry location (x0, y0), the focal length f, and mirror spacing D multiply the same scaling factor simultaneously, the spot patterns can be scaled up or down, as can be seen in Equation(3.3). For example, if (x0, y0), f, and D in Fig.3.3 (d) are increased by a factor of 2, i.e. (6.28 mm, -11.26 mm), f = 50 mm and D = 123.52 mm, a two times larger similar pattern without any distortions can be obtained with the same pass count n, as shown in Fig. 3.5 (b).

Fig. 3 . 5

 35 Fig. 3.5 Manipulation of the spot density (a) and pattern size (b) for a sunflower-like spot pattern with a fixed ray inclination angle.

  y0) and (x'0, y'0) overlap. The spot patterns in Fig.3.3 are those that have been adjusted manually. The last two columns in Table2list the spot locations on M1 and ray inclination angles after the νth pass count for the four patterns in Fig.3.3. Based on these values, the re-entrant conditions can be met for all four patterns. To understand the re-entrant behavior, a periodic function sinφ should be paid attention to in the expression of xn and yn. If x'n and y'n are linear, a re-entrant trajectory can be achieved for any initial beam parameters. However, a nonlinear function arcsinφ occurs in x'n and y'n. The mutual interaction and restriction between arcsinφ and sinφ generate periodic patterns and meet the re-entrant condition with some specific initial beam parameters.

Fig. 3 . 6

 36 Fig. 3.6 Reflection of incident ray set on a spherical mirror.

  .7 (a), (b), and (d). However, when the beam diameters decrease to 0.6 mm for Fig. 3.7 (a), 0.7 mm for Fig. 3.7 (b), and 0.8 mm for Fig. 3.7 (d), these multi-circle spot patterns can be closed.

Fig. 3 . 7

 37 Fig. 3.7 Six exotic spot patterns generated by a two-spherical-mirror MPC. (a) three-, (b) four-, (c) five-, (d) seven-, (e) eight-, and (f) nine-multi-circle spot patterns. The plots represent the spot projection of the exit mirror, black spots represent the exit positions of beam, green rings mark black spots.

Fig. 3 . 8

 38 Fig. 3.8 Photographs of the real (a) eight-and (b) nine-multi circle spot pattern on the exit spherical mirror.

Fig. 3 . 9

 39 Fig. 3.9 Ambient CH 4 2f absorption spectrum at normal atmospheric pressure.

Fig. 3 .

 3 Fig. 3.10 Evolutionary process of spot patterns on a two-spherical-mirror MPC for increasing distance, D, between the mirrors.

  for trace gas sensing OBJECTIVE In this chapter, to validate the ability of the calculation model based on spherical aberration, the one-circle spot patterned MPC, seven-circle spot patterned MPC, and single-line spot patterned MPC were designed and experimentally implemented. These MPCs have different structures, which can be used for different LPAS techniques. To evaluate the performances of these MPCs, three MPC-based gas sensing systems were built, and the corresponding experimental results are discussed in this section.

Fig. 4 . 2

 42 Fig. 4.2 (a) Diode laser wavelength as a function of current tuning at five different operating temperatures. (b) L-I-V curve of the 2.33 μm CW, DFB diode laser.

A

  semiconductor laser diode (alignment laser) emitting 635 nm visible light is used as the guide beam for the invisible infrared (IR) light beam. The light paths of the visible and IR laser beams were overlapped utilizing a flip mirror. Two plane mirrors, Mirror 1 (M1) and Mirror 2 (M2) were used to steer the beam into a Herriott MPC. A Herriott MPC was used to enhance the detection sensitivity, which has an invar enclosure with a typical mean coefficient of thermal expansion of 10 -6 /°C. An effective optical path length of 14.5 m was obtained after 52 beam passes in the Herriott MPC. The IR beam leaving the MPC was subsequently received by an InGaAs detector (Thorlabs, PDA 10D-EC). To couple the invisible IR into the MPC, the flip mirror first reflected the visible alignment beam into the MPC. A circle pattern with a designed 14.5 m light path can be achieved by adjusting the horizontal and vertical angles of mirrors, M1 and M2. Then the flip mirror was moved into a position so that the invisible IR beam was guided into the MPC. In this way, the invisible IR beam has the same light path as the visible guide beam once the two light paths overlapped.

Fig. 4 . 3

 43 Fig. 4.3 Schematic of CO gas sensor system for SF 6 decomposition analysis.

Fig. 4 . 4

 44 Fig. 4.4 Measured 2f signal amplitude as a function of the modulation depths for a 20-ppmv CO : SF 6 gas mixture as a function of operating gas pressure.

Figure 4 .

 4 6 depicts the time-dependent mixing ratios normalized to the initial value at the start of the measurement as a function of time for different flushing rates of the MPC. The flushing time decreases with an increased gas flow rate because a larger flow rate increases the gas exchange rate in the MPC. The parameter τ in the exponential decay fit of e -t/τ was defined to the normalized concentration signal curve as the flow response time. The response time of the sensor system was estimated to be 3 min, 4.2 min, and 5.8 min for a fixed flow rate of 200, 100, and 30 SCCM, respectively. If a larger flow rate is provided, a faster response time of the sensor system can be achieved.

Fig. 4 . 6

 46 Fig. 4.6 Flow response time of the sensor system with pure SF 6 flushing at three different gas flow rates.

  Figure 4.7 (a) depicts the experimental results. The sensor exhibits

Fig. 4 . 8

 48 Fig. 4.8 Percent deviation of the measured concentration compared with the calculated value. Histogram plot of the deviation. The red line corresponds to a Gaussian profile.

Fig. 4 . 9

 49 Fig. 4.9 Allan variance plot for the same time series measurements with the histogram plot.

4 mm and 25

 425 mm, respectively. When the mirror spacing was 39.4 mm and the incident angle of the initial ray combination was ~ 8°, a seven-nonintersecting circle spot pattern was realized, as shown in Fig.4.10 (a) and (b). At such an angle, the paraxial approximation of sinθ ≈ θ has an error of 0.3 %. This error was eliminated with the new modified ABCD matrix. In Fig.4.10 (a) and (b), the spot pattern can meet the re-entrant condition after every 168 times pass counts ignoring the spot size. Based on all the determined parameters (such as entry location of incident beam, incident angle of incident beam, mirror spacing, and radius of curvature of spherical mirrors), an MPC consisting of two identical spherical mirrors was set up to verify the designed dot pattern.

Figure 4 .

 4 10 (c) and (d) show the actual spot patterns on the entrance and exit mirrors, respectively, which are in excellent agreement with those shown in Fig. 4.10 (a) and (b).

Fig. 4 .

 4 Fig. 4.10 Calculated seven-nonintersecting circle spot pattern on the (a) entrance and (b) exit spherical mirror surfaces; photographs of the real seven-nonintersecting circle spot pattern on the (c) entrance and (d) exit spherical mirror surfaces. A red visible diode laser beam was used to visualize the spot pattern.

  I0 represents the initial intensity, R is the effective reflectivity of the spherical mirror surfaces, n is the pass count, and k and c are the absorption coefficient and the target gas number density, respectively. L is defined as an effective optical path length of the beam interacting with gas molecules. Inside the MPC, L = D • n, where D represents the mirror spacing. For weak absorptions, the first-order approximation can be used to simplify Equation (4.1) as Equation (4initial laser intensity, a specific mirror spacing, a selected absorption line, and a specific target gas concentration, the intensity absorbed by target gas, I is a function of R and n. The condition for the maximum I, Rn = 1/e can be obtained by setting the derivative of I/Iokc with respect to n to zero, which implies that the optimal optical path length is reached when the intensity is reduced to e -1 ≈ 0.368 of its initial value via reflection losses. The SNR as a function of the pass counts and the reflectivity of the spherical mirrors are shown in Fig.4.11. The mirror spacing was chosen as 39.4

Fig. 4 .

 4 Fig. 4.11 Signal-to-noise ratio as a function of pass counts in a MPC for different reflectivities.

two 25 .

 25 4 mm spherical mirrors were installed into the two cage plates. The two cage plates were connected using four 6-mm diameter steel rods, which ensures an adequate degree of mechanical stability. The two cage plates can slide along the steel rods to adjust the mirror spacing and four screws were mounted to fix the position of the two cage plates. The first cage plate served as the incident end face of the mini-MPC and included an 8° hole for a fiber-coupled collimator, whose position was at the entrance hole on the front mirror. The collimator can produce a ~600 μm beam waist at a working distance of 80 mm and can be easily inserted into the first cage plate and fixed, thus avoiding the need for optical calibration. An InGaAs photodiode detector and a custommade amplifier circuit (see Fig.4.12) were mounted in a small, enclosed cell on the second cage plate.

Fig. 4 .

 4 Fig. 4.12 (a) Photograph of a custom-made pre-amplifier; (b) Photograph of an InGaAs photodiode detector on the custom-made amplifier circuit board.

Fig. 4 .

 4 Fig. 4.14 CH 4 concentration response of the TDLAS sensor system.

  the sensor system sequentially. The linearity of the CH4 sensor is validated by measuring the 2f signal amplitude for CH4 concentrations ranging from 2 ppmv to 30 ppmv in Fig. 4.14. Data for each CH4 concentration step are averaged and plotted against the CH4 nominal concentrations in the inset of Fig. 4.14. A linear relationship

Fig. 4 .

 4 Fig. 4.15 Rise time and fall time of the mini-MPC sensor system. Rise time: the time interval between initial response and 90% of the final response. Fall time: the time interval between initial response to a step decrease in concentration and 90% of the final response.

Fig. 4 .

 4 Fig. 4.16 2 ppmv CH 4 Allan-deviation stability of the mini-MPC based TDLAS sensor.

Fig. 4 . 17

 417 Fig. 4.17 Deviation of the measured concentration compared with the calculated value. Histogram plot of the deviation. The black line depicts a Gaussian profile.

Fig. 4 .

 4 Fig. 4.18 2f absorption spectrum of 2 ppmv CH 4 at normal atmospheric pressure.

  -7 cm 2 •Hz -1/2 . A side-by-side comparison of RNFs for the mini-MPC-based CH4 sensor and those current state-of-the-art MPC-based CH4 sensors are shown in Table 4. This mini-MPC-based CH4 sensor has the best RNF value due to the large value of the fill factor and the small value of the NEA. Continuous monitoring of CH4 concentrations near sewers and in the atmosphere was performed to demonstrate the online performance of our mini-MPC-based CH4 sensor. The sensor system shown in Fig. 4.13 (e) was first placed on the first floor of the Shaw Amenities Building on the Shanxi University campus in Taiyuan, China, with GPS coordinates: 37°47'58.1"N, 112°35'14.2"E. There is a sewage well 3 m away from the front of the window and four sewage wells 8 m away from the left-hand side of the window. A 2-m long °φ6-mm PTFE tube was used to sample air from outside the laboratory window. A 3 μm micropore PTFE filter membrane was used at the front end of the sampling tube as an air filter to avoid the contamination of the MPC by dust or soot particles. A small diaphragm pump was placed downstream to provide a pressure difference. The concentration data near the sewers for thirteen days (6-18 Apr 2019) are shown in Fig. 4.19 (a). High fluctuations along with numerous spikes in concentration levels of CH4 were observed due to the presence of the sewage wells.

Fig. 4 .

 4 Fig. 4.19 Continuous monitoring of CH 4 concentrations (a) measured from Apr. 6, 2019 to Apr. 18, 2019 near sewers and (c) measured from Apr. 28, 2019 to May 8, 2019 far away from sewers on the Shanxi University campus, China; Diurnal variations of CH 4 mixing ratio (b) near sewers and (d) far away from sewers. Bottom whisker, bottom box line, and top whisker in (b) and (d) indicate the 10th, 25th, 75th, and 90th percentiles, respectively. Lines inside the boxes and continuous solid lines in (b) and (d) represent the hourly median and mean of the data, respectively.

  from the first floor to the third floor and was placed on the other side of the Shaw Amenities Building which is far away from the sewage wells. The concentration data of continuous monitoring for eleven days (Apr 28 to May 8, 2019) are shown in Fig.4.19 (c).The concentration curve as a function of time during that period turned out to be smooth without any spikes. The CH4 concentration varied from 1.55 to 3.24 ppmv with a mean value of 2.26 ± 0.11 ppmv. Several high-concentration days, April 28, May 1, 3, and 7 were observed in Fig.4.19 (c). A reasonable explanation for this observation is that the CH4 fugitive emission from a natural gas station located three kilometers southeast of the campus was blown to the test site since the wind direction was from the southeast for these four days. The diurnal profile of Fig. 4.19 (d) displays a variation trend of the CH4 concentration which is consistent with Fig. 4.19 (b) but has a concentrated distribution. The early morning peak and diurnal variation trends during

  prongs. In 2021, Y. Ma et al. designed a line spot pattern MPC based on two right-angle prisms for coupling light to a quartz tuning fork, which improved the development of quartz-enhanced photoacoustic spectroscopy (QEPAS) based trace gas sensing.

Fig. 4 .

 4 Fig. 4.20 (a) Single line spot pattern generated by a theoretical model of a two-spherical-mirror MPC based on spherical aberration. The plot represents the spot projection of the entrance mirror. Black spot represents the coincident entrance and exit positions of the beam. (b) Verification of single-line spot pattern on the entrance mirror using the TracePro. Spot size represents the irradiance.

Fig. 4 .

 4 Fig. 4.21 MSSE-QEPAS system configuration. Red lines: transmission of the laser beams between the two identical spherical mirrors, M1 and M2; QTF: quartz tuning fork; CCB: control circuit board. Inset: Photograph of the single-line spot pattern on the M1 surface and the QTF.

  the designed MPC. A photograph was shown in the inset of Fig. 4.21. A continuous wave (CW) fiber-coupled distributed feedback (DFB) laser with a wavelength of 1.395 μm and output power of 16 mW was used as a probe light source. A custom laser control circuit board (CCB) was used to control the driving current and laser temperature by means of a laptop. To improve the performance of the MSSE-QEPAS sensor, wavelength modulation technique was implemented. A ramp signal generated from the laptop scans the laser wavelength across the absorption line. A sine wave signal modulates the laser wavelength at a frequency of f0/2. The DFB laser was connected to a fiber collimator with a ~600 μm beam waist at a working distance of 80 mm. The laser beam was coupled into the designed MPC. The output electrical signal from the custom QTF was connected to a low noise transimpedance amplifier and then transferred to a commercial lock-in amplifier (Stanford Research Systems, Model SR830). The lock-in amplifier demodulated the signal in the 2f mode with respect to the sync signal. The lock-in amplifier was set to a 12 dB/oct filter slope and a time constant τ = 1 s corresponding to a detection bandwidth of 0.25 Hz. The demodulation signal of 2f spectra was delivered to the laptop for analysis and display.

A 3 -

 3 dimensional linear stage with a typical step size of 1 μm was used to move the MPC from the top (point a) to the bottom (point b) along the QTF symmetry axis, as shown in Fig. 4.22.

Fig. 4 .

 4 Fig. 4.22 Normalized MSSE-QEPAS signal amplitudes as a function of the vertical moving distance of the MPC.

Fig. 4 .

 4 Fig. 4.23 2f signals of traditional QEPAS (blue line) and MSSE-QEPAS (red line).

  Here, a theoretical model to understand the behavior is proposed. S[i] is defined as the traditional QEPAS signals as shown in Fig.4.24 (a), the maximum of which at ~1.2 mm was normalized to unit 1. Variable i is the QTF symmetry axis. It is discretized for the convenience of the numerical calculation and is an integer, one of which stands for a typical 1-μm step size of the 3-dimensional translation linear stage. G[i] is defined as the laser beam intensity, which is equal to Rn(i) [0≤n≤59] or 0 for a laser beam being present at position i or not, respectively. Here R=0.98 is the mirror reflectivity of the MPC, and n (i) is the number of beam reflections at i. The first laser beam intensity without any reflections (n=0) is unit 1. The 60 laser beam intensities between two QTF prongs as a function of i are shown in Fig.4.24 (b).

3 )

 3 where m is the moving distance and F[m] is the signal amplitudes of MSSE-QEPAS at different QTF insertion depths. The calculation process and the result of the convolution are shown in Fig. 4.24 (c) and Fig. 4.24 (d), respectively. The ladder pattern of F[m] is attributed to the distribution of the discrete laser intensity G[i]. The maximum signal amplitude of 19.8 was achieved at m = 8988, which corresponds to the moving depth of ~9 mm. Considering the normalized QEPAS signal amplitude in Fig.4.24 (a), the calculated signal gain factor is 19.8, which is in excellent agreement with the experimental results.

Fig. 4 .

 4 Fig. 4.24 (a) Normalized traditional QEPAS signals measured by changing the position of the excitation light from the top to the bottom of the QTF. (b) 60 laser beams normalized intensities and distribution along i. (c) Calculation process of the convolution model. (d) Signal amplitudes of MSSE-QEPAS at different QTF insertion depths.

A

  dielectric mirror consists of multiple layers of dielectric materials with alternating refractive indices. The layers are deposited onto a substrate, usually glass or a transparent polymer, as shown in Fig. 5.1. The refractive index of each layer is represented by "ni" and the thickness of each layer is represented by "di".

Fig. 5 . 1

 51 Fig. 5.1 Schematic of a dielectric high reflectivity (HR) coating exploiting constructive interference of Fresnel reflections. n 1 is a high index, n 2 is a low index material, and n 1 d 1 =/4 with  being the design center wavelength. https://www.edmundoptics.com.tw/knowledge-center/applicationnotes/optics/highly-reflective-coatings/

  The highreflectivity (HR) bandwidth is typically about 5-15 % of the center wavelength. The reflectivity depends on the center wavelength, coating design and manufacturing process. For instance, reflectance spectra of quarter-wave stacks consisting of 15 pairs of Ta2O5/SiO2 and TiO2/SiO2 for  = 800 nm (n1/n2 = 2.1/1.46 and 2.35/1.46, respectively) are shown in Fig. 5.2. The high-reflectivity bandwidth is about 15 %.

Fig. 5 . 2

 52 Fig. 5.2 Reflectance spectra of quarter-wave stacks consisting of 15 pairs of Ta 2 O 5 /SiO 2 and TiO 2 /SiO 2 .https://www.layertec.de/files/catalog/2020/LAYERTEC_Optics_And_Coatings_Optical_Coatings.pdf.

Fig. 5 . 3

 53 Fig. 5.3 An example of cavity transmission consisting of equally spaced axial modes.

Fig. 5 . 4

 54 Fig. 5.4 Illustration of a polarized light ray incident on a separation surface at Brewster's angle (θ B ). The angle between the reflected ray and refracted ray is 90º .

  the vacuum wavelength in micrometers. Over a wide wavelength range from 400 nm to 2000 nm, the fused silica refractive index n varies from 1.467 -1.438, as shown in Fig.5.5. This makes the corresponding Brewster-angles range from 55.79º to 55.18º (see Fig.5.8), almost near-achromatic in such a broadband. This is essential to realize a broadband high-finesse optical cavity.

Fig. 5 . 5

 55 Fig. 5.5 Refractive index vs.wavelength for fused silica. The data is taken from Ref.[START_REF] Malitson | Interspecimen Comparison of the Refractive Index of Fused Silica*, † [J][END_REF] .

Fig. 5 . 6

 56 Fig. 5.6 Reflectance vs. angle of incidence of a polarized beams on a fused silica prism, where  B = 55.4° is calculated for a wavelength of =1064 nm.

Figure 5 .Fig. 5 . 7 Fig. 5 . 8

 55758 Figure 5.6 shows light reflectance versus incidence angle on the interface between air and fused silica. When the incident angle is equal to the Brewster angle B (=arctan(n)), the reflectance is virtually zero for p-polarized light, hence there is no Fresnel reflection loss on the incident surface for injection of p-polarized light into the prism.

96 The

 96 Brewster angle (B = arctan (n)) and the critical angle (TIR = arcsin (n)) are plotted as a function of wavelength in Fig. 5.8.

Figure 5 .Fig. 5 . 9

 559 Figure 5.9 shows the Fresnel loss-limited SiO2 prism reflectivity (calculated using Equation (5.8)) over a broadband spectral range of 300-2000 nm with a Brewster angle fixed for =1000 nm. Material loss-limited reflectivity of more than 99.99% can be obtained.

Fig. 5 .

 5 Fig. 5.10 Prism retroreflectors associated with total internal reflection (TIR) inside. For fused silica, Brewster angle (θ B ) is 55.4º at a wavelength of 1.064 µm.

ABFig. 5 .

 5 Fig. 5.11 Schematic of fused silica prism-based optical cavity. θ B : Brewster angle.

Fig. 5 .

 5 Fig.5.12 The optical system is represented by the ABCD matrix. The input and output rays are characterized by their locations and angles with respect to the optical axis as (x 1 , θ 1 ) and (x 2 , θ 2 ), respectively. The radius of curvature, r 1 and r 2 , are related to the distances between the intersection of the rays with the optical axis and the input plane and the output plane of the optical system, respectively.

  z) is a radius of phase front curvature of the Gaussian beam; ω(z) is the beam radius at which the optical intensity drops to 1/e 2 (≈ 13.5%) of the value on the beam axis.

Fig. 5 .

 5 Fig. 5.13 Gaussian beam width ω (z) as a function of the distance z along the beam propagation direction.

  of the prism retroreflector are found to be :

Fig. 5 .

 5 Fig. 5.15 Illustration of the q parameter of a Gaussian beam inside a cavity after one round-trip being self-consistent. R4 represents the point on the surface of prism I.

  distance between the two prisms LR4-R5 (and LR8-R1) should satisfy the -R5 should be less than 137.9 cm.For  = 1.064 µm , n = 1.45, LR1-R2 = 9.56 mm, Rc = 6 m, Lp = 38.23 mm, and by setting the distance between the two prisms to LR4-R5 = 43 cm, the matrix for one roundtrip (R4-R5-R6-R7-R8-R1-R2-R3-R4) can be obtained from Equation (5.38):

Fig. 5 . 16

 516 Fig.5.16 Fresnel loss per surface in parts per million (ppm) for fused silica for a Brewster angle fixed for =1064 nm.

Fig. 5 . 17

 517 Fig. 5.17 Scattering losses at the surface of the total internal reflection in parts per million for fused silica for a Brewster angle fixed at 1064 nm.

5 . 53 )Fig. 5 .

 5535 Fig. 5.18 Scattering losses at fused silica prism surface in parts per million for a Brewster angle fixed at 1064 nm.

Fig. 5 .

 5 Fig. 5.19 Schematic of prism cavity-based spectrometer. f 1 , f 2 , and f 3 are lenses with focal lengths of 6.24 mm, 50 mm, and 50 mm, respectively. Mi (i=1...5) are flat mirrors. PR: polarization rotator. PD: photodetector.

A 20

 20 Hz sine-wave signal with an amplitude ranging from -50 mV to +50 mV was generated with a function generator (Agilent, 33220A Function), which was applied to the piezo electric transducer (PZT) of the laser to finely scan the external cavity laser frequency. Sweeping the frequency modulation input with an amplitude of ± 50 mV could lead to a laser frequency scans the laser over ± 500 MHz. A photodetector (Model PDA10CS, Thorlabs Inc.) equipped with an amplifier was used to measure the laser intensity leaking from the prism cavity. Cavity modes were observed with an oscilloscope (LeCroy, Model 104MXs) as shown in Fig. 5.20 (cavity modes generated by s-polarized light excitation) and in Fig. 5.21 (p-polarized light excited cavity modes).

Fig. 5 .

 5 Fig. 5.20 Cavity modes generated by s-polarized light (red line).

Fig. 5 .

 5 Fig. 5.21 Cavity modes generated by p-polarized light (red line).

  .2 mm is the path-length inside the prism with a refractive index of n=1.45. 2L = 86 cm is the double of the cavity lengths in air between the two prisms. The calculated value of FSR is 310 MHz. The observed discrepancy between FSRexp (364 MHz) and FSRcal (310 MHz) can be attributed to the nonlinear relationship between the laser frequency and the sweeping voltage.

Fig. 5 .

 5 Fig. 5.22 Cavity mode of p-polarized light (blue line) accompanied with a Gauss Fit (red line).

  analyzer (Model 42i, Thermo Inc). At the same time, O3 generated from an O3 generator (Monitor Europe ML 9841B, Thermo Corp.) was introduced into the prism cavity box to produce NO3 via the reaction: NO2 + O3 → NO3 + O2. Measurements of NO2 and NO3 concentrations were simultaneously carried out. A typical BBCEAS spectrum is shown in Fig. 5.24.

Fig. 5 .

 5 Fig. 5.23 Measured absorption coefficient (black line) of 116 ppbv NO 2 and 3.5 ppbv NO 3 by the prismbased BBCEAS spectrometer and its fit (red line). Residual (gray line) between the measurement and the fit with a stand deviation of ~4.010 -8 cm -1 [58] .

Figure 5 .

 5 Figure 5.24 presents a typical experimental broadband absorption spectrum (black line) of 116 ppbv NO2 and 3.5 ppbv NO3 over 175 nm and the corresponding fit (red line). A detection sensitivity of ~ 4.010 -8 cm -1 was estimated based on the residual (gray line).

  Les mé thodes de dé tection optique basé es sur la spectroscopie d'absorption à chemin optique long (LPAS) sont des outils puissants pour caracté riser les gaz traces, avec les avantages uniques suivants : (1) temps de ré ponse court (pouvant ê tre infé rieur à 1 s) ; (2) haute sensibilité et haute spé cificité ; (3) dé rive minimale ; (4) absence de 2 ré ponse croisé e à d'autres gaz si la conception de l'approche est soigneusement é tudié e [5,10] . Lorsque la lumiè re se propage le long d'un trajet d'absorption rempli d'un gaz cible, une partie de la lumiè re est absorbé e en raison des transitions du gaz des niveaux d'é nergie infé rieurs aux niveaux d'é nergie supé rieurs du systè me atomique ou molé culaire. En se basant sur la loi de Beer-Lambert, la concentration du gaz cible peut

  l'é chantillon suit une loi exponentielle en fonction de la longueur d'absorption optique, l'utilisation de sché mas d'absorption à chemin optique long est un moyen utile pour amé liorer considé rablement la sensibilité de la mesure spectroscopique. L'amé lioration du rapport signal sur bruit est gé né ralement proportionnelle à la longueur de chemin efficace, Leff. Pour les mesures in situ, l'absorption à chemin optique long peut ê tre obtenue par : (1) des conceptions expé rimentales à chemin ouvert offrant un Leff d'environ 1 km, ce qui permet d'obtenir des limites de dé tection infé rieures à la ppbv pour les mesures de rapports de mé lange de gaz trace inté gré s sur toute la longueur du trajet ; (2) de multiples ré flexions de la lumiè re de sonde dans une cellule optique (cellule multipass avec Leff allant jusqu'à 200 m, permettant des limites de dé tection de l'ordre de ppmv-ppbv) ou dans une cavité ré sonante optique (avec Leff de plus de 10 km, permettant des limites de dé tection de l'ordre de ppbv-pptv, voire ppqv). Les approches d'absorption optique à chemin long peuvent ê tre utilisé es à l'air libre ou enfermé es dans une chambre à vide pour la mesure d'é chantillons de gaz statiques à une pression ré duite ou en utilisant des flux de gaz contrôlé s à une pression arbitraire.

.

  Ils ont dé couvert accidentellement que les MPC à configuration de taches denses offrent de nouvelles opportunité s pour fournir des chemins optiques plus longs dans un petit volume avec un chevauchement minimal des faisceaux. Ils sont plus compacts et de taille plus ré duite par rapport aux MPC Herriott traditionnels (par exemple, un MPC à motif de taches de sept cercles multiples a une longueur physique 7 fois plus courte qu'un MPC Herriott pour la mê me longueur de chemin efficace). Par rapport aux MPC Herriott modifié s impliquant des miroirs astigmatiques pour ré aliser un motif de Lissajous à taches denses, ce type de MPC utilise deux simples miroirs sphé riques pour obtenir un motif de taches denses, ce qui le rend peu coûteux.

Fig. 1 . 1

 11 Fig. 1.1 Schematic diagram of a CH4 sensor (bottom) involving a compact folded-path optical cell with

  2016, R. Sharma et al. ont pré senté le dé veloppement d'un systè me d'analyse de gaz (Fig. 1.2) basé sur la diffusion Raman à haute pression dans un MPC à chemin plié avec un motif de tache liné aire et en ont dé montré la faisabilité pour l'analyse en

Fig. 1 . 2

 12 Fig. 1.2 Schematic of the folded-path optical cell based-Raman natural gas (CH4) analyzer system. Figure taken from Ref.[START_REF] Sharma R | Raman analyzer for sensitive natural gas composition analysis[END_REF] .

Fig. 1 . 3

 13 Fig. 1.3 Folded-path optical cell-based QEPAS system for water vapor detection. Figure taken from Ref [33].

1. 4 .

 4 Dans ce systè me, le faisceau optique est injecté à travers un trou dans un miroir sphé rique et est ré flé chi en avant et en arriè re de nombreuses fois avant de sortir par le mê me trou. Cette conception garantit que le faisceau reste collimaté pendant la transmission et la ré flexion dans toute la cellule, contrairement à la cellule White, qui

Fig. 1 . 4

 14 Fig. 1.4 One-circle spot pattern on the mirror surface of Herriott cell using a visible laser.

  d'onde d'environ 95 à 98 % sur une large ré gion spectrale. Cependant, limité s par leur ré flectivité relativement faible, le nombre de ré flexions à l'inté rieur du MPC est limité à 100-200, ce qui peut gé né ralement fournir un Leff maximal d'environ ~ 200 m (pour une longueur physique d'environ 1 m).

Fig. 1 . 5

 15 Fig. 1.5 Nonsynchronously sampled Lissajous pattern on the mirror surface. Figure taken from Ref. [34].

  avec un petit volume d'é chantillon. Comme l'é paisseur du revê tement sur le miroir de la cavité dé pend de la longueur d'onde, la largeur de bande de la haute ré flectivité du miroir est limité e à 5-15 % autour de la longueur d'onde centrale conç ue. Par consé quent, les cavité s à miroirs dié lectriques ne conviennent que pour des mesures dans une bande spectrale relativement é troite.

Le chapitre 2

 2 donne un aperç u des principaux aspects de la dé tection optique des gaz traces, en particulier la spectroscopie d'absorption à chemin long basé e sur des modules optiques à chemin plié (cellules à passages multiples et cavité s optiques).Au chapitre 3, un nouveau modè le de calcul basé sur l'aberration sphé rique est pré senté dans la section 3.1. Par rapport à la mé thode de conception pour la cellule Herriott traditionnelle basé e sur une approximation paraxiale, le modè le de calcul pré senté peut dé crire pré cisé ment la propagation des rayons entre deux miroirs sphé riques et la ré flexion sur les surfaces du miroir en utilisant la thé orie de l'aberration.Pour prendre en compte l'é volution et la dé formation des formes des taches causé es par les aberrations, qui peuvent produire un effet d'é talon ré sultant de chevauchements de taches, ce modè le de calcul est optimisé dans la section 3.2. De nombreuses simulations du modè le de conception optique plié e sont effectué es pour obtenir divers CPM à motif de taches, chacun ayant une structure diffé rente, pour une application à la LPAS basé e sur le MPC. Les paramè tres (emplacement et angle d'un faisceau ou d'un rayon incident, distance focale du miroir sphé rique, espacement des miroirs) d'une sé rie de MPC pratiques à motif de taches de cercles multiples peuvent ê tre dé terminé s, ce qui peut ouvrir la voie au dé veloppement de MPC miniatures à faible bruit avec un motif de taches dense à l'é tape suivante. Au chapitre 4, les procé dures de conception d'un MPC à motif de tache circulaire, d'un MPC à motif de taches à sept cercles et d'un MPC à motif de tache unique sont discuté es, ce qui valide le modè le de calcul basé sur l'aberration sphé rique. Trois systè mes de dé tection de gaz basé s sur des MPC sont dé crits, et les ré sultats expé rimentaux correspondants sont discuté s pour é valuer les performances de ces MPC. Un TDLAS basé sur un MPC à motif de tache circulaire pour la dé tection de CO est dé crit dans la section 4.1. Une enquê te minutieuse sur la sé lection de la ligne a é té mené e et un compromis est ré alisé en termes de ré ponse croisé e, de rentabilité et de sensibilité de dé tection. La spectroscopie de modulation de longueur d'onde 2f (2f-WMS) combiné e à une cellule Herriott de 14.5 m a é té utilisé e pour atteindre une limite de dé tection minimale de CO au niveau ppmv dans un environnement de gaz tampon SF6. La configuration a é té appliqué e à la surveillance en ligne d'un dé faut de surchauffe à basse tempé rature à l'inté rieur d'é quipements de transmission d'é nergie é lectrique. La section 4.2 pré sente les dé tails de conception d'un capteur CH4 dé veloppé basé sur le TDLAS utilisant un MPC à motif de taches à sept cercles. Un mini-MPC avec une longueur de trajet d'absorption optique de 4.2 m et un volume d'é chantillonnage de 20 cm 3 a é té dé veloppé . À notre connaissance, il s'agit actuellement du MPC le plus petit en termes de la mê me longueur de trajet optique (~ 4,2 m). L'impression 3D avec une ré solution d'impression de 100 µm est adopté e pour fabriquer le cadre du MPC, ce qui garantit un haut niveau d'inté gration de tous les composants. Un DFB NIR laser é mettant à 1.65 µm a é té utilisé comme source lumineuse pour dé tecter le mé thane à des rapports de mé lange dans la plage de ppbv.

Table 1 .

 1 Origin of laser absorption spectra in different regions of the electromagnetic spectrum. ........ 27

Table 2 .

 2 Initial Entry Location (x 0 , y 0 ) on M1, Mirror Spacing (D) and Pass Count under Re-entrant Condition (v) Used to Obtain the Spot Patterns Depicted in Figs. 3.3 (a)-(d); Total Optical Path Length (L), Volume of MPC (V) and Ratio of L to V (RLV); Spot Location (x ν , y ν ) on M1 and Ray Inclination Angle (x' ν , y' ν ) After ν th Pass Count. ...................................................................................................... 47

Table 3 .

 3 Initial Entry Location (x 0 , y 0 ) on Entry Mirror, Mirror Spacing (D) and Pass Count (ν) Used to Obtain the Spot Patterns Depicted in Figs. 3.7; Total Optical Path Length (L), Volume of MPC (V) and Ratio of L to V (RLV); Spot Location (x ν , y ν ) on Exit Mirror and Ray Inclination Angle (x' ν , y' ν ) After ν th Pass Count. .......................... 53

Table 4 .

 4 Performance

indicators of methane sensors based on multi-pass cells (MPCs). Noise Equivalent Absorption (NEA), Total Optical Path Length of MPC (L), Volume of MPC (V), Fill Factor (FF), Ratio of NEA to FF (RNF) and Reference (Ref). ............................................................................................ 79

  2

Table 1 .

 1 Origin of laser absorption spectra in different regions of the electromagnetic spectrum.

	Spectral region	Cause of absorption
	UV (200 -400 nm)	Electronic transitions
	NIR (700 nm-2.5 µm)	Molecular vibration-rotation, first
		harmonic
	MIR (2.5 -14 µm)	Molecular vibration-rotation, fundamental
	THz (0.03 -3 mm)	Molecular rotation

Table 2 .

 2 Initial

	Pattern	(x0, y0) (mm)	D (mm)	v	L (m)
	Fig. 3.3 (a)	(8.56, -5.35)	61.16	130	7.95	75.65	10.51	(8.52, -5.51)	(6.55, 6.45)
	Fig. 3.3 (b)	(5.60, -6.18)	24.57	306	7.52	6.44	116.77	(5.69, -6.40)	(6.40, 6.37)
	Fig. 3.3 (c)	(3.17, -5.54)	61.54	210	12.92	26.42	48.90	(3.19, -5.58)	(6.51, 6.50)
	Fig. 3.3 (d)	(3.14, -5.63)	61.76	206	12.72	26.53	47.95	(3.10, -5.55)	(6.63, 6.63)

Entry Location (x 0 , y 0 ) on M1, Mirror Spacing (D) and Pass Count under Re-entrant Condition (v) Used to Obtain the Spot Patterns Depicted in Figs. 3.3 (a)-(d); Total Optical Path Length (L), Volume of MPC (V) and Ratio of L to V (RLV); Spot Location (x ν , y ν ) on M1 and Ray Inclination Angle (x' ν , y' ν ) After ν th Pass Count. V (mL) RLV (cm -2 ) (xν, yν) (mm) (x'ν, y'ν) ( o )

Table 3 .

 3 Initial Entry Location (x 0 , y 0 ) on Entry Mirror, Mirror Spacing (D) and Pass Count (ν) Used to Obtain the Spot Patterns Depicted in Figs. 3.7; Total Optical Path Length (L), Volume of MPC (V) and Ratio of L to V (RLV); Spot Location (x ν , y ν ) on Exit Mirror and Ray Inclination Angle (x' ν , y' ν ) After ν th Pass Count.

	Pattern	(x0, y0) (mm)	D (mm)	ν	L (m) V (mL) RLV (cm -2 )	(xν, yν) (mm)	(x'ν, y'ν) ( o )
	Fig. 3.7 (a) (17.58, 10.51)	50.60	107	5.41	102.56	5.27	(7.43, -17.22)	(6.79, 2.87)
	Fig. 3.7 (b) (17.38, 11.23)	29.54	133	3.93	59.87	6.56	(-8.66, -19.47)	(1.09, -14.16)
	Fig. 3.7 (c) (17.81, 11.46)	69.91	159	11.12 141.70	7.85	(1.79, -9.12)	(14.95, 6.30)

Fig. 3.7 (d) (16.23, 6.94)

123.06 203 24.98 249.42 10.02 (13.36, 6.33) (8.99, 12.35) Fig. 3.7 (e) (16.30, 5.83) 138.99 235 32.66 281.71 11.59 (-12.53, 1.79) (-11.15, -6.32) Fig. 3.7 (f) (14.91, 6.72) 118.16 223 26.35 239.49 11.00 (11.80, 0.26) (9.79, 8.80)

Table 4 .

 4 Performance indicators of methane sensors based on multi-pass cells (MPCs). Noise Equivalent Absorption (NEA), Total Optical Path Length of MPC (L), Volume of MPC (V), Fill Factor (FF), Ratio of NEA to FF (RNF) and Reference (Ref).

	Wavelength (μm)	NEA (Hz -1/2 )	L (m)	V (mL)	FF (cm -2 )	RNF (cm 2 • Hz -1/2 )	Ref.
	1.648 µm	4-8×10 -5	74	6967	1.06	3.77-7.55×10 -5	[126]
	1.654 µm	Not stated	252	3600	7.00	Not stated	[127]
	3.3 µm	3.5×10 -4	6.8	262	2.60	1.35×10 -4	[128]
	8.03 µm	4.6×10 -5	76	500	15.20	3.03×10 -6	[129]
	1.653 µm	9.5×10 -6	26.4	249	11	8.64×10 -7	[30]
	3.3 µm	7.3×10 -4	54.6	213	25.60	2.85×10 -5	[130]
	1.65 µm	1.47×10 -5	4.2	20	21	7×10 -7

ppmv CH 4 Concentration [ppmv]Linear fit

5 1 2

Acknowledgments

This work was realized at the Laboratoire de Physico-Chimie de l'Atmosphère (LPCA), Université du Littoral Côte d'Opale (ULCO), France and Laser Spectroscopy Institute, Shanxi University (SXU), China, in the framework of a co-advisorship (Cotutelle) PhD program between France and China.

Remerciements

from 1000 ppmv CO/N2 mixture gas, it is a challenge to identify the variation of a 1 ppmv concentration level in the presence of a large background interference from SF6.

Therefore, the fundamental vibration band is not suitable for CO detection in electric power systems. The first overtone band of CO between 2.3 and 2.4 μm is 100 times weaker than the fundamental vibration band. No SF6 absorption features are observed in this spectral region as shown in the inset of Fig. 4.1. The first overtone band of CO is interference-free from SF6 spectra. We selected the R(6) transitions located at 4285.01 cm -1 in the first overtone band of CO as the target detection line since it has a large stronger absorption line strength of 8.32 × 10 -2 cm -2 /atm. Thus, a ppmv-level detection sensitivity can be expected based on the experience of the CO sensor operating with an N2 buffer gas in this wavelength region, which can meet the requirements for an SF6 decomposition analysis. A CW-DFB antimonide diode laser from Nanoplus Nanosystems and Technologies GmbH (http://www.nanoplus.com/) was employed as the light source.

The single mode laser emitting single-mode radiation at a center wavelength of 2333.7 nm. The output wavelength and power of the diode laser were characterized at different laser currents using the FTIR spectrometer and a power meter (Ophir Optironica, model

The acquired CO spectrum was processed using a three-step algorithm. In the first step, a consecutive acquisition of 6,000 data points was carried out after the acquisition system was triggered, which included 3 complete CO spectra. In the second step, the 3 CO spectra were separated and averaged. In the third step, a standard CO 2f spectrum, which was calculated and stored in the computer memory in advance, was fitted to the averaged 2f spectra using a Levenberg-Marquardt nonlinear least-squares fit procedure [117] . The fitting routine was written in Labview@ code, which can output a scale factor between the standard and acquired CO spectra. From the scale factor, CO mixing ratios can be retrieved since the standard spectrum is based on a calibrated 40-ppmv mixing ratio of CO. The acquisition and signal processing time take 0.5 s and 0.4 s, respectively. A queue operation was adopted to implement the data acquisition and processing in parallel, resulting in a final data updating rate of the concentration value is 0.6 s/point.

A fitting example of the 20-ppmv CO 2f-WMS spectrum is shown in Fig. 4.5. The obtained scale factor is 0.499, which yields a measured concentration value of 19.97 ppmv.

(3) Flow response time

The flow response time is an important parameter, especially for online sensor systems, which is a function of flow, MPC volume, and surface adsorption-desorption rate of the target gas molecule. The volume of the MPC is 0.9 L. The response time was investigated as a function of the flow rate. The MPC was initially filled with a constant excellent performance in terms of signal stability for each constant concentration. The signal amplitude of the SF6 buffer gas is 3.04 ppmv which is in good agreement with the information from the SF6 supplier. To check the real zero baseline level, the 'signal' from the high-purity N2 measurement with an amplitude of 1 ppmv was also plotted in Fig. 4.7 (a). This can be explained by the fact that an etalon effect occurs in the sensor system. When there is no CO gas present, the Levenberg-Marquardt nonlinear leastsquares fit procedure tries to fit an etalon pattern using the standard CO 2f spectrum and then returns a mixing ratio, which reflects the etalon level of the sensor system, i.e., the detection sensitivity is limited at 1 ppmv. Data for each step are averaged and plotted against CO nominal concentrations in Fig. 4.7 (b). A linear relationship with an Rsquare value of 0.9994 between the measured and nominal concentrations confirms the excellent linearity of the sensor response to CO concentrations. To further examine the capabilities of the sensor system in terms of measurement precision and stability, time series measurements of 20 ppmv CO/SF6 gas mixture were carried out for 85 mins, which includes 8,500 concentration points. A histogram plot of measured concentrations showing an approximately normal distribution around the mean value is plotted in Fig. 4.8, which can be used to assess the measurement precision.

A Gaussian profile is fitted to the distribution histogram with an R-square value of 0.993, resulting in a half-width at half-maximum (HWHM) of 39 ppbv and sigma value of 33 ppbv with a concentration output rate of 0.6 s. This verifies that the sensor measurement precision is better than 40 ppbv.

Summary

In this work, we developed a ppmv-level CO sensor system for an SF6 decomposition analysis by use of a 2.33 μm DFB diode laser and a 14.5-m MPC. The challenge of quantifying CO concentration levels in SF6 is to avoid the strong spectral interference from SF6 absorption. The selection of the CO first overtone band for CO detection, where no SF6 absorption behavior is observed and whose absorption coefficients are stronger than the CO second overtone band at 1.55 μm, is a compromise between the detection sensitivity, the spectral interference, and cost efficiency. The identification of the optimal detection wavelength results in a minimum detection sensitivity of 1 ppmv, which is limited by etalon effects from the MPC. Such detection sensitivity is sufficient for online monitoring of electric power systems. A measurement precision of 40 ppbv can be obtained with a concentration update rate of 0.6 s, while the precision can be improved to 6 ppbv with an average time of 48 s. This is attributed to the use of a Levenberg-Marquardt nonlinear least-squares fit algorithm, in which the information from all data points based on the 2f-WMS spectrum is fully utilized. The flow rate effect was tested in the range of 30 SCCM to 200 SCCM due to the performance limitation of the gas dilution system. However, a higher flow rate matching the optimal averaging time of 48 s can make the sensor operating at optimized operation conditions for field work. Deployment of the sensor system in gas-insulated switchgear to further evaluate its performance for CO concentration monitoring in the field will be conducted soon.

Seven-circle patterned MPC for ppbv-level CH 4 sensing

Introduction

Tunable diode laser absorption spectroscopy (TDLAS) based on multi-pass cell (MPC) is a powerful analytical tool and is widely applied to air quality monitoring, industrial process control, and medical diagnostics [START_REF] Wu | Atmospheric CH4 measurement near a landfill using an ICL-based QEPAS sensor with V-T relaxation selfcalibration[END_REF][START_REF] Shao | Detection of nitric oxide at low ppb•m concentrations by differential absorption spectrometry using a fully diode-laser-based ultraviolet laser system [J][END_REF] . Particularly, an important application involves greenhouse-gas sensing, which has recently undergone significant study and development on account of increasing emissions control regulations. For example, the emissions of methane (CH4) contribute to the increasing greenhouse effect on the earth's surface [START_REF] Nisbet E G | Atmospheric science. Methane on the rise--again[END_REF][START_REF] Feldman D R | Observationally derived rise in methane surface forcing mediated by water vapour trends[END_REF] . Some recent studies have shown significant CH4 production in sewers [START_REF] Liu | Online monitoring of methane in sewer air[END_REF] . Water utilities are committed to reducing CH4 emissions from The ray trajectories between the two spherical mirrors can be observed when a plume of smoke is introduced between two spherical mirrors as shown in Fig. 4

Results and discussion

To demonstrate the potential of the highly integrated mini-MPC, a CH4 sensor system was set up with the mini-MPC as the core component. Fig. 4.13 (e) shows the basic components of the CH4 gas sensor system. In the present experiment, the probe light source is a CW fiber-coupled 1.65 μm DFB laser, which was directly connected to the fiber input port of the mini MPC. The laser source can be tuned to the R3 rotational branch of the 2ν3 band in CH4 located at 6046.95 cm -1 , which is an interference-free absorption from other species present in ambient air. At this wavenumber, the DFB laser delivers single-mode radiation of 6.3 mW when the DFB laser current and temperature were set to 60 mA and 21.36°C, respectively. The laser Chapter 5

Broadband high finesse optical cavity based on prism OBJECTIVE

In general, an optical cavity is built using high-reflectivity dielectric mirrors, which allows one to achieve long optical path-lengths of up to several kilometers. High reflectivity is achieved using multilayer coatings on the dielectric mirror. Whereas the thickness of each multilayer is wavelengthdependent, as a result high reflectivity is specifically realized for a target wavelength range with a bandwidth of about 5-15 % of the center wavelength.

Broadband optical cavity is highly desired for high-sensitivity simultaneous measurements of multiple species or wide band absorbers.

In this Chapter, the development of a novel optical cavity based on prisms used as cavity reflectors is described. This approach offers a high-finesse optical cavity operating in a broadband spectral region from the blue to the near infrared spectral region. The principle of the prism-based cavity and its characterization are presented and discussed.

In optical design systems, the vertical plane is generally called the "tangential plane", the "sagittal plane" is the plane perpendicular to the tangential plane; it contains the principal ray. Astigmatism results from different focal lengths in the tangential and sagittal plane. Since this the Brewster-angle prism cavity acts differently in the tangential and the sagittal planes. The transfer matrices of the tilted plane in the plane are described by M(T) and M(S), respectively. The composition of the ABCD matrix of the prism-based cavity is illustrated graphically in Fig. 5.14 and further developed in detail below: For R4→R5, the incoming ray is first reflected at the Brewster angle and then travels from R4 to R5 over a distance of LR4-R5 in air (nair=1). This trajectory can be described by a translation matrix MR4-R5(S, T): For R5→R6, the ray travels into prism II with an angle of incidence equal to θB at R5 and then reaches R6 after traveling a distance of LR5-R6 in fused silica (SiO2) nSiO2=n, the matrix MR5-R6 of the tilted plane in the two planes as described by M R5-R6(T) and M R5-R6(S), respectively: ( )

(5.37) Equations (5.36) and (5.37) lead to the ABCD matrix for one round-trip (R4→R5→R6→R7→R8→R1→R2→R3→R4) as follows:

( ) ( )

() 73 44 To calculate the eigenmodes of a cavity, the q parameter of a Gaussian beam must be self-consistent, that means it must meet the round-trip condition.

The q2 parameter of the Gaussian beam after one round-trip in the optical cavity should be equal to the initial q1 parameter of the beam, which implies that : Therefore, after one round-trip, the q parameter can be derived from the ABCD matrix method as [START_REF] Kogelnik | Laser beams and resonators[END_REF] :

From a comparison of coefficients with Equation (5.18) the radius of curvature r and spot size radius ω at the beam's equiphase surface can be found: The usual stability condition for the case of the prism cavity is given by [START_REF] Kogelnik | Laser beams and resonators[END_REF] :

where ( )

A cavity mode excited by p-polarized light is plotted in Fig. 5.22. The full width at half maximum (FWHM) of the cavity mode was found to be 0.44 MHz by fitting a Gaussian profile to the measured spectrum. A finesse (F) of this prism cavity is then derived to be 827.27 using Equation (5.3), which results in a reflectivity R of the prisms of ~ 99.8% (Eq. 5.3) at λ=1.05 µm, the corresponding Q factor is determined to be 6.487×10 8 using Equation (5.4).

ICOS by prism cavity

Continuous wave integrated cavity output spectroscopy (ICOS) at 1050 nm was used to characterize the prism cavity. The ICOS method [START_REF] Scherer J J | CW integrated cavity output spectroscopy[END_REF][START_REF] Silva | Integrated cavity output spectroscopy measurements of NO levels in breath with a pulsed room-temperature QCL[END_REF] is based on systematically disrupting the cavity resonant mode to integrate frequency-dependent light intensity transmitted through the cavity. This method provides a far simpler and lower-cost alternative to CRDS (cavity ringdown spectroscopy) [START_REF] Deacon D A G | Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources[END_REF][START_REF] Mazurenka | 4 Cavity ring-down and cavity enhanced spectroscopy using diode lasers[END_REF] . The ICOS method relies on measuring the light intensity transmitted through a high finesse optical cavity as a function of wavelength. The light intensity (It) leaking out the optical cavity can be written as below [START_REF] Fiedler S E | Incoherent broad-band cavity-enhanced absorption spectroscopy[END_REF] :

where I0 is the light intensity leaking out of the prism cavity without absorber (with nitrogen flushing the cavity), R is the equivalent reflectivity of the cavity prism, D is the distance between the two prisms, and α is the gas absorption coefficient. For small absorption coefficients and high reflectivity mirrors, the absorption coefficient (α) can be approximated:

From Equation (5.56), the equivalent mirror reflectivity R can be deduced from the measured light intensity and known absorption coefficient α of a calibrated standard species:

Summary

A fused silica prism-based optical cavity was developed. Its broadband performance was evaluated through the measurements of NO2 and NO3 absorption at 500-675 nm over a broadband region of 175 nm. The preliminary results in spectroscopic applications are promising. Using the broadband prism cavity together with a broadband frequency comb source is planned for the future.

An equivalent prism reflectivity of ~ 99.9 % was experimentally determined. The theoretical value based on Fresnel losses would result in a higher reflectivity of up to 99.99%. Further investigation on additional optical losses should be conducted in order to further improve the performance of the prism cavity setup.

Chapter 6

General conclusion and perspective

The main subject of this PhD thesis is the development, investigation, and improvement of folded-path dense spot pattern optical cells for trace gas detection. A (1) A one-circle spot patterned MPC with an Leff = 14.5 m using a 2.33 µm DFB diode laser. This MPC was developed for sensing ppmv-levels of CO from SF6 decomposition.

(2) A seven-circle spot patterned mini-MPC in a small volume of 20 cm 3 with a Leff of 4.2 m, the smallest mini-MPC for the same effective optical path length. It is configured for optical alignment-free operation based on fiber coupling for both lightinjection and light-detection. Owing to its performance, this integrated mini-MPC is a suitable candidate for large-scale deployment in specific field applications.

( Enfin, je tiens à remercier ma famille pour leur amour et leur soutien