\documentclass[aps,pra,twocolumn,superscriptaddress,showpacs,longbibliography]{revtex4} \usepackage{graphicx} \usepackage{natbib} %\usepackage{biblatex} %\addbibresource{cavity.bib} \usepackage[breaklinks]{hyperref} \usepackage{amsmath} \usepackage{mathtools} \newcommand{\OD}[0]{\mathrm{OD}} \newcommand{\lev}[3]{\textit{#1}$_{#2/#3}$} \newcommand{\hflev}[4]{\textit{#1}$_{#2/#3}$, \textit{F}=#4} \begin{document} %\preprint{} \title{Operating a near-concentric cavity at the last stable resonance} \author{ChiHuan Nguyen} \affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543} \author{Adrian Nugraha Utama} \affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543} \author{Nick Lewty} \affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543} \author{Christian Kurtsiefer} \affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543} \affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542} %\thanks{} \email[]{christian.kurtsiefer@gmail.com} \date{\today} \begin{abstract} %We report on the compact design, construction and stabilization of a Fabry-Perot near-concentric cavity with a length of 11mm. The cavity is designed to obtain strong coupling regime of cavity quantum electrodynamics (CQED) with single neutral atoms, and with large physical space, can be extended to include ions or other collective quantum emitters. We demonstrate the long term stability of the cavity operated at an optical cavity length of sub-wavelength shorter than the concentric point, which corresponds to the stability parameter $g=-0.999962(2)$. %Near-concentric cavities can provide strong interaction of a single atom with multi-modes cavity %We report on the compact design, stabilization, and mode analysis of a Fabry-Perot near-concentric cavity with an optical length of sub-wavelength shorter than the concentric point, which corresponds to the stability parameter $g=-0.999996(28)$. The cavity is designed to obtain strong multimode coupling regime of cavity quantum electrodynamics (CQED) with single neutral atoms, and with large physical space, can be extended to include ions or other collective quantum emitters. With the possibility of reaching a diffraction-limited beam waist, near-concentric cavities can provide technical advantages over the conventional near-planar cavities in applications requiring strong atom-light interaction. However, near-concentric cavities reside on the edge of the stability limit and thus impose challenges on their construction and stabilization. Here, we present an experiment of constructing and stablizing a near-concentric Fabry-Perot optical cavity. We demonstrate that the cavity can be stabilized at the last resonant length with a 780 nm laser before unstable regime while still maintaining the performance in terms of cavity linewidth and transmission. We measure the cavity length to be $207(13)$\,nm shorter than the critical concentric point, which corresponds to a stability parameter $g=-0.999962(2)$ and a cavity beam waist of $2.4\,\mu$m. The presented work also paves the way to explore multimode cavity-based information processing with various quantum systems, including ion traps and other collective quantum emitters. \end{abstract} % insert suggested PACS numbers in braces on next line \pacs{ 32.90.+a, % Other topics in atomic properties and interactions of atoms; 37.30.+i, %Atoms, molecules, and ions in cavities 42.50.Ct % Quantum description of interaction of light and matter; } \maketitle %\textit{I. Introduction} \section{Introduction} Optical cavities are indispensibly used in modern technologies ranging from lasers, gravitational wave detectors, and applications of quantum technologies which require nonlinear atom-light interaction. In particular, atom-cavity systems with ultra-high finesse cavities have been widely used by several groups to demonstrate quantum logic gates, quantum distributed networks, quantum metrology and quantum sensing \citep{Reiserer2015,Ritter2012,Reiserer2014}. % Though displaying remarkable achievements, the intricate high-reflectivity coating of the cavity mirrors used in these experiments poses a challenge on scaling up the system. As a result, there is an increasing trend of exploring new types of optical cavities which can provide additional advantages \citep{Cox2018,Nguyen2017}. % Recently, one of such proposed cavity designs that has been experimentally demonstrated is near-concentric Fabry-Perot cavities \citep{Morin1994,Nguyen2017}. Outside of the atomic physics community, there are also reports and suggestions of using near-unstable cavities such as concentric cavities to reduce mirror coating thermal noise \citep{Wang2018}. Among all geometrical configurations of Fabry-Perot cavities, concentric cavities where the length of the cavity is twice the radius of curvature of cavity mirrors ($l_{cav}=2R_{C}$) exhibit the tightest focusing mode at the waist of the cavity. This strong focusing mode, which in ideal conditions can reach a diffraction-limited waist, effectively concentrates the cavity field at the location of the trapped atoms. % Therefore, despite being operated at milimeter cavity length, the near-concentric cavity can provide a small effective mode volume comparable to state-of-the-art cavities of micrometer lengths. The milimeter cavity length allows narrow cavity linewidths with relatively low finesse mirrors, more optical access and space to prepare and manipulate quantum matters inside the cavity. % Furthermore, the near-degenacy in resonant frequencies of transverse modes of near-concentric cavities is an intriguing feature to explore the physics of multi-mode strong coupling in cavity quantum electrodynamics \citep{Ballantine2017}. Despite these advantages, near-concentric cavities have not been widely explored yet mainly because of technical hurdles of constructing and stabilizing them. It is due to the fact that the concentric cavity is close to the margin of the stability limit and hence sensitive to the misalignment in the transverse positions of the two cavity mirrors. %Though shortly mentioned in some textbooks, to the extent of our knowledge, there has not been a report or any technical account to describe the construction, stabilization and analysis of concentric or near-concentric cavities. Here, we report on a compact design of a near-concentric cavity with a length of 11 mm, stabilized at sub-wavelength shorter than concentric point. In addition, we demonstrate the long-term stability of the cavity and show that properties of the cavity's fundamental mode is well preserved at the last resonant length with a 780 nm laser before the unstable regime. The design is particularly useful to study atom-light interaction but can be easily adapted to a wider range of experiments. %The paper is organized as follows. In Section I, the cavity design and the alignment procedure are described. In Section III, we discuss about the transverse modes spacing measurement to characterize the length of the cavity. We also address the issue of cavity stabilization in transverse directions in Section IV. \begin{figure} \includegraphics[width=0.8\columnwidth]{figures/fig1/fig1.pdf} \caption{\label{fig:cms} Schematic of a near-concentric cavity assembly. Arrows indicate the moving directions of piezo segments. %The footprint of the design is 25 } \end{figure} %\textit{II. Design and optical setup} \section{Design and optical setup} %\textit{1. Cavity Design} \subsection{Cavity design} We first describe the design of our cavity system. %The overview schematic of the design is shown in Fig.~\ref{fig:cms}. Central to our system are spherical cavity mirrors with a radius of curvature $R_{C}= 5.5$\,mm, and a nominal reflectivity of $R=99.5 \%$ at a wavelength of 780\,nm. For effective mode-matching, the input surface of the mirror has an aspherical shape instead of the usual planar surface, which is introduced to transform the collimated Gaussian beam into the desired cavity mode. The details of characterization and abberation analysis of the mirrors can be found in \citep{Durak2014}. To align the cavity and correct for misalignment due to thermal drifting, we placed one of the cavity mirrors on a shear piezo stack (PI P-153.6710H) with a capability of travelling in three orthorgonal directions. In each direction, the piezo is able to move $\pm \,5\, \mu$m with sub-nanometer resolution. % The cavity mounting system is designed to be compact and fit into a cuvette which can provide convenient optical access to the experiment. The system essentially consists of the following parts: a piezo holder, cavity mirror shields, and a moveable mirror holder (see Fig.~\ref{fig:cms}). Except for the cavity mirror shields, all the mechanical parts are made from Titanium to reduce the structural change of the mounting system due to thermal fluctuation. %The piezo holder is the main body of the cavity mounting system and used to accommodates the stack piezo. The holder is machined on a computer numerical controlled (CNC) milling machine from a block of Ti6Al4V Grade-5 titanium. The piezo holder has a protruding arm with a spherical opening to hold one of the cavity mirrors. The four columns at the opening provide anchor points to glue the cavity mirror. The other mirror would be mounted inside a similar structure located on the L-shaped block. The diameter of the spherical openings is $ \sim 100 \mu$m larger than the diameter of the mirror shields. The gap between the shield and the four columns allows adjustment of the mirror's position during the alignment process. Larger size of the gap, though allows for more tolerance in the machining process, requires more glues to apply and hence causes more drifting of cavity mirrors' position during the curing process. %\textit{2. Alignment procedure} \subsection{Alignment procedure} %We proceed to describe the cavity mirror alignment procedure. %Contrary to other cavity configurations, the relative transverse positions of the two mirrors of the concentric cavity is critical. The underlying principle of the alignment is that optical axes of the two cavity mirrors must be coincident. %Figure \ref{fig:cas} shows the schematic of our alignment setup. A laser beam between two fiber-couplers defines a reference line for the alignment of the cavity mirrors. The right cavity mirror is pre-assembled in the moveable mirror holder. The left cavity mirror, contained in a separate external holder (not part of the final assembled cavity system), is gradually moved into the cavity holder. % Throughout the alignment process, the reflected beams from the two cavity mirrors are monitored and ensured to couple back to the fiber couplers. This reduces the tilting misalignment and provides a coarse translational alignment between the two cavity mirrors. The fine adjustment is carried out by an external piezo system that controls the position of the left mirror via the external holder. The left mirror is secured in the aligned positions inside the cavity holder by vacuum-compatible epoxy (Torrseal). %In other words, the optical axis of two cavity mirrors will be aligned to be coincident with the reference laser beam. %This provides a coarse alignment of cavity and finer adjustment will be done later by maximizing the transmission of the fundamental mode of the cavity. %The cavity mounting system with one glued cavity mirror is stationed on a xyz precision translation stage (Thorlabs PT3/M) with a tip-tilt mirror mount (KM100-E02). This arrangement allows adjustment of full degrees of freedom in translation and two rotational degrees of freedom. %\begin{figure} %[ht] %\centering % \includegraphics[width=\columnwidth]{figures/fig2/fig2.pdf} % \caption{\label{fig:cas} % Schematic of cavity alignment setup. Two single-mode fiber couplers (FC) coupled to each other define a reference line for the optical axis of concentric cavity (CC). External holder (EH) is placed on the left translation stage with an external piezo system (EPS). The cavity mounting system is on the right translation stage (TS). After being glued, the left cavity mirror is released from the delivery holder. Rotational degrees of freedom are provided by tip-tilt mounts (TM). M: highly reflective mirrors. %} %\end{figure} %The next step is to align the translation stage such that the laser beam hit the centers of the cavity mirrors. This can be ensured by observing the symmetry of the reflected beam from the mirror on a camera. The adjustment of the tip-tilt mirror is used to make the reflected beam from the cavity mirror back to the fiber couplers. %On the left side, the other mirror (that we call left mirror) is clamped to a delivery holder, which is moved around by a translation stage. %The mirror can be released from the holder by untightening the screws on the top of the delivery holder after the gluing process. %Following the same procedure, the left mirror optical axis is aligned along the reference beam. Here, beside the coarse movement of the micrometer knob, the left mirror can be moved with nm resolution step provided by the Jena piezo system. %When the two cavity mirrors are well aligned with the reference beam,the left mirror is slowly moved into the holder through the spherical opening of the cavity holder. %In order for the left mirror to avoid touching the cavity holder which can alter the alignment, we monitor the electrical continuity between the two holders with a multimeter. In addition, the optical power of the reflected beam into the fiber couplers was made sure to be relatively constant during the transport process. When the cavity mirror is totally inside the holder, fine adjustment of the transverse position is done with the Jena piezo system. The transmission of the cavity detected on photodiode is used as a feedback for the alignment. Note that during the alignment procedure, the alignment of laser beams are maintained fixed, only the positions of two cavity mirrors are adjusted to get the optimized cavity transmission. %When the alignment is done, vacuum-compatible epoxy (Torrseal) are applied at the contacts between the cavity mirror shield and the four columns at the spherical opening of the holder. Torrseal is chosen because of its relatively low shrinkage, ultrahigh vacuum compatibility and high elasticity modulus. We observe that the curing process can change volume of glues and hence pushing the cavity mirror out of the alignment, so it is critical to apply an equal amount of glues at the four corners to balance the drifting during the curing process. We observe that it takes about two hours at room temperature for the Torrseal to be cured and hardened. During the curing process, the cavity mirrors position needs to be constantly monitored and adjusted to maintain the cavity alignment. After the mirror's position is secured by the cured Torrseal, the mirror is releashed from the delivery holder and the holder is withdrawn out, leaving the cavity mirrors assembled and aligned inside the cavity holder. \begin{figure*} [ht!] \centering \includegraphics[width=\textwidth]{figures/opticalsetup/optical_setup.pdf} \caption{\label{fig:opticalsetup} Locking scheme of the near-concentric cavity setup. Red and orange lines indicate the beams from 780 nm probe laser and 810 nm lock laser, respectively. The frequency of the probe laser is stabilized to a D2 transition of \textsuperscript{87}Rb by modulation transfer spectroscopy. The lock laser's sideband is locked to resonance of the transfer cavity, which in turn is stabilized to the probe laser. % The frequency of the lock laser can be tuned by adjusting the sideband's frequency. The near-concentric cavity is stabilized to the lock laser. All cavity locking schemes use the standard Pound-Drever-Hall technique with 20 MHz phase modulation. The cavity transmission of probe and lock lasers are seperated by a dichroic mirror (DM). A camera with linear response (C) and a photodetector (PD1) are placed at the cavity transmission's 780 nm arm to observe the resonant modes. PBS: polarization beam splitter. BS: beam splitter. SMF: single-mode fibers. } \end{figure*} %\textit{3. Optical setup} \subsection{Optical setup} The optical layout is presented in Fig.~\ref{fig:opticalsetup}. We couple two lasers at wavelengths of 780 nm and 810 nm into the near-concentric cavity, which we label as the probe and the lock laser, respectively. Our probe laser is referenced to a D2 transition of \textsuperscript{87}Rb via a modulation transfer spectroscopy~\cite{McCarron2008}. The stability of the probe laser is passed to the lock laser at 810 nm wavelength via a transfer cavity. In particular, one of phase-modulated sidebands of the 810\,nm laser, which is generated by an electro-optical-modulator (EOM), is locked to the resonance of the transfer cavity. By tuning the frequency of the sideband, the frequency of the lock laser can be adjusted. The frequency of the sideband is chosen such that the probe and lock laser are simultaneously resonant with the near-concentric cavity. %We combine the probe and the lock laser at a dichroic mirror and coupled them to the cavity. The cavity transmission of the lock laser is used to derive the feedback signal to lock the near-concentric cavity. All locking with the cavities are carried out by using the standard Pound-Drever-Hall technique~\citep{Drever1983}. %The reason why there is an additional lock lasers instead of locking the cavity directly with the probe laser is that the locking scheme bears a similarity to the previous scheme employed for atomic CQED setups, in which the lock laser plays an additional role of the intra-cavity dipole trap. %\textit{III. Cavity length measurement} \section{Cavity length measurement} Laguerre-Gaussian (LG) functions form a complete basis to solutions of the paraxial wave equation, and thus can be used to describe the eigenmodes of spherical optical resonator. We denote cavity modes as $\textrm{LG}_{nlp}$ where $n,l,p$ are integer numbers. Modes of different $n$ are known as the longitudal modes, while the indices $(l,p)$ indicate spatial dependences of the cavity modes on transverse coordinates, hence known as transverse modes. % The resonance frequencies of the cavity modes are determined by the condition that the round-trip phase shift in the cavity must be an integer multiple of $2 \pi$. % As the cavity length approaches concentric point, the shift of the transverse mode frequencies approaches the free spectral range. Therefore all transverse modes become co-resonant in the concentric regime. %While this feature also happens for confocal cavities, in confocal cavities, the degeneracy splitted into groups of odd and even mode number, while for concentric cavities, all the transverse mode frequencies are identical. Hence, we can say that concentric cavity has a higher degree of de Making use of this property, we determine the cavity length by measuring the spacing of resonant frequencies between the fundamental mode $\textrm{LG}_{00}$ and the transverse mode $\textrm{LG}_{10}$. Under paraxial approximation, the resonance frequencies of the cavity with identical spherical mirrors are: \begin{equation}\label{Eq:res_freq} \nu_{n,l,p}=n\frac{c}{2l_\textrm{cav}}+ \left(1+|l|+2p\right)\frac{c}{2l_\textrm{cav}} \frac{\Delta \psi}{\pi}, \end{equation} where $c$ is the speed of light, $\Delta \psi=2\tan^{-1}\left(l_{cav}/{2z_{0}}\right)$ the Gouy phase difference after one round trip of $\textrm{LG}_{00}$, and $z_{0}$ the Rayleigh range of the cavity \citep{Saleh2001}. \begin{figure} %[ht] \centering \includegraphics[width=\columnwidth]{figures/fig3/fig3.pdf} \caption{\label{fig:tms} Transverse-mode frequency spacing ($\Delta\nu_{tr}$) at different critical distance ($d$) of cavity lengths that are resonant with 780\,nm laser. The solid line is the fit based on Eq.~\ref{eq:tms}. Error bars show the standard deviation of the measurement. The inset shows a typical cavity transmission spectrum and the derived $\Delta\nu_{tr}$. } \end{figure} From Eq.~\ref{Eq:res_freq}, we derive the expression for frequency spacing of $\textrm{LG}_{00}$ and $\textrm{LG}_{10}$ in terms of $l_{cav}$ and $R_{C}$: \begin{equation} \Delta \nu_\textrm{tr} = \nu_{n00}-\nu_{n10} =\frac{c}{2l_\textrm{cav}} \left(1- \frac{\cos^{-1}g}{\pi} \right),\label{eq:tms} \end{equation} where $g=1-l_{cav}/R_{C}$ is the stability parameter. We obtain the cavity transmission spectra by scanning the cavity length within a free spectral range. We record the spectra at different resonant cavity lengths and use a peak-detection algorithm to determine the resonant frequencies. The intensity distribution of cavity transmission on a camera helps us to distinguish different transverse modes. To obtain a frequency marker, we modulate the probe laser by an electro-optical phase modulator (EOM). The two sidebands emerged in the cavity transmission as a result of phase modulation of the laser are used as a frequency reference for the peak-detection algorithm. Figure~\ref{fig:tms} shows the transverse mode frequency spacing at different cavity lengths which are resonant with the 780 nm laser. We define the critical distance as $d=2R_{C}-l_{cav}$. %From a least-squares fit of experimental data points to Eq.~\ref{eq:tms}, we determine $R_C=5.49946(4)$\,mm, which agrees well with the design value. From a least squares fitting of experimental data points to Eq.~\ref{eq:tms}, we determine $d= 207(13)$ nm at the data point b, which corresponds to the stability parameter $g=-0.99996(2)$. As a result, point b is the last geometrically stable length of the cavity that is resonant with the 780\,nm laser. This is consistent with our observation that when increasing the cavity length by another half wavelength beyond point b, the cavity enters the unstable regime and exhibits lossy cavity modes (see Fig.~\ref{fig:cavity_trans} and the next section). The good agreement between the experimental data, including the last resonant point, and the fit based on the paraxial equation prompts us to discuss the validity of the paraxial approximation in our near-concentric cavity. The complex amplitude of the electric field distribution that propagates in the $z$ direction can described as $E\left(x,y,z\right)=u\left(x,y,z\right)e^{-ikz}$, where $k$ is the longitudinal wave vector component. To be valid for paraxial approximation, it requires: \begin{equation} \left| \frac{\partial^2 u}{\partial^2 z} \right| \ll \left|2k \frac{\partial u}{\partial z}\right|.\label{eq:field} \end{equation} Equation~\ref{eq:field} implies that optical beams can be focused with an angle up to approximately 30 degrees before the paraxial approximation breaks down~\cite{siegman86}. For the near-concentric cavities, the beam divergence for the fundamental mode is $\theta = {\lambda}/{\pi w_{0}}$, where $\lambda$ is the wavelength of the resonant mode taken to be 780 \,nm, and $w_{0}$ is the cavity beam waist. Taking the beam divergence as our focusing angle, we approximate the condition for the paraxial approximation to be invalid for our cavity parameters when $w_{0} \leq 496$\, nm, or equivalently when $d \leq 0.5 $\,nm. As a result, the paraxial approximation is still valid to describe our near-concentric cavity. %\textit{IV. Cavity mode analysis} \section{Cavity mode analysis} It has been reported before that the cavity finesse reduces significantly as the cavity is pushed toward the geometrical instability~\cite{Haase2006}. In this section, we demonstrate that our near-concentric cavity can maintain the transmission and linewidth at the last two resonant cavity lengths before the unstable regime. The typical cavity transmission spectra are displayed in Fig.~\ref{fig:cavity_trans}. The transmission of the cavity with two modes can be modeled by a summation of two Lorentzian functions: % \begin{equation} T(\nu)=\frac{T_{1}}{4(\nu-\nu_{1})^2/\gamma_{1}^2+1}+\frac{T_{2}}{4(\nu-\nu_{2})^2/\gamma_{2}^2+1},\label{eq:transmission} \end{equation} % where $T_{1(2)}$, $\nu_{1(2)}$, and $\gamma_{1(2)}$ are transmission coefficients, resonant frequencies, and linewidths of cavity modes $\textrm{LG}_{00}$ and $\textrm{LG}_{10}$ respectively. From a fit of cavity transmission spectra with Eq.~\ref{eq:transmission}, we can determine the cavity parameters at multiple cavity lengths. % \begin{figure} [ht!] \centering \includegraphics[width=\columnwidth]{figures/fig4/fig4.pdf} \caption{\label{fig:cavity_trans} Cavity transmission spectra measured by detuning the cavity length. (a) $d= 597\,$\,nm. The dashed line is the fit based on a summation of two Lorentzian functions, corresponding to two resonant peaks. (b) $d= 207\,$\,nm. Transverse modes become degenerate and form a long tail extending out to the lower frequencies. (c) $d=-183\,$\,nm. The cavity is in the unstable regime. The insets show the transverse mode profiles. } \end{figure} At $d=207$\,nm, we observe that the cavity fundamental mode maintains the similar cavity linewidths and transmissions. In particular, the observed linewidth of the fundamental mode $\textrm{LG}_{00}$ agrees well with the nominal value of 21.7 MHz, determined from the cavity mirror's design reflectivity of 0.995 at 780\,nm wavelength. However, at the last resonant length, as the transverse modes start to overlap and the probe laser simultanously couples to multiple cavity modes, the second cavity mode becomes difficult to be identified, and has a broadened effective linewidth, which is determined from the fit to be 98(2) MHz. When increasing the cavity length by another half wavelength, there is a sudden decrease in the cavity transmission and an sudden increase in the cavity linewidth, which indicate the signal of unstable cavity modes. Besides the scattering and absorption loss, due to the finite mirrors' aperture, the cavity can exhibit additional geometrical losses if there is misalignment between the two optical axes of the cavity mirrors. This diffraction loss due to mislaignment becomes more critical for near-unstable cavities. Hence, it is important to assess the degree of misalignment in our cavity based on the observed variation of cavity linewidths across the cavity lengths. Here, we assume that the misalignment is entirely due to the tilting of the mirrors, as one of the mirrors can be aligned translationally by the piezo. Under that assumption, the misalignment loss per round trip is given by ~\citep{Hauck1980}: \begin{equation} \alpha=\theta^2 \frac{1+g^2}{(1-g^2)^{3/2}} \frac{\pi l_{cav}}{\lambda}\frac{(a/w_{m})^2}{\textrm{exp}[2(a/w_{m})^2]-1}, \end{equation} where $\theta$ is the misalignment angle, $a$ the radii of cavity mirror's aperture, and $w_{m}$ the beam waist on the mirrors. From the observation that the cavity linewidth is comparable to the nominal value and assuming that all the cavity loss is due to the tilting misalignment, we can estimate the tilting angle between the two cavity mirrors to be better than 0.5 degrees in the near-concentric regime. This estimation agrees with what can be guaranteed in the alignment procedure, as the reflected laser beams from the cavity mirrors are ensured to couple back to the fiber couplers.%, which are located around 20 cm away from the cavity. \\ \begin{figure} \centering \includegraphics[width=\columnwidth]{figures/fig5/2D_T_profile_2.eps} \caption{\label{fig:figure5} Sensitivity of cavity transmission coupled to a single-mode fiber as a function of transverse displacements. The cavity mirror is displaced in x and y directions while the cavity length is stabilized. } \end{figure} %\textit{V. Transverse stabilization of near-concentric cavities} \section{Transverse stabilization of near-concentric cavities} % Stabilization of optical cavities mostly involves keeping the cavity length in resonance. %We demonstrate the long-term stability of the cavity when operated at the last resonant length and describe the algorithm to compensate for the drifting in transverse directions. The alignment of near-concentric cavities is sensitive to the transverse positions of the cavity mirrors. To quantify this effect, we measure the cavity resonant transmission coupled to a single-mode fiber as we displace one of the cavity mirrors in x and y directions (see Fig.~\ref{fig:figure5}). Throughout the measurement, the cavity length is locked to the frequency stabilized 810\,nm laser. The transmission profile in Fig.~\ref{fig:figure5} shows a FWHM of 59(3)\,nm in both transverse directions. Therefore, the change of temperature of the cavity on the order of 100 mK is enough to reduce the transmission of the fundamental mode by 10\%. % \begin{figure} \centering \includegraphics[width=\columnwidth]{figures/fig6/transverse_lock.eps} \caption{\label{fig:figure6} Long-term stability of the near-concentric cavity at $d=207$ nm. The cavity length is locked during the measurement. The slow drift of cavity transmission on the order of minutes is due to the transverse misalignment which is caused by temperature change. %The transverse compensation algorithm is activated when the cavity transmission drops below the threshold value which is chosen to be 10\% of the maximum transmission. Vertical arrows indicate the activation of the stabilization algorithm. The cavity transmission recovers to the maximum value after the successful implementation of the algorithm. } \end{figure} % To actively compensate for such drifting in alignment, we implement an algorithm based on gradient-search method to maximize the cavity transmission. %The algorithm starts when the cavity transmission drops below a threshold value. The cavity mirror is transversely scanned in incremental steps surrounding the initial position to find the direction of steepest ascent of the cavity transmission. %We reduce the size of the scanning step as the cavity transmission increase to avoid the looping of the algorithm jumping between two points. The algorithm repeats the iteration until the cavity transmission reaches the chosen value within a predefined range of tolerance. Figure~\ref{fig:figure6} shows a typical record of cavity transmission at the last resonant length when the stabilization algorithm is activated. We attribute the slow drift on the order of minutes to the temperature change of the cavity. %, while the fast fluctuation of the maximum transmission is due to the vibration of the cavity length. %The reduction of cavity transmission below a chosen threshold will trigger the algorithm. %The algorithm is triggered when the cavity transmission drops below a chosen threshold. %90\%, which is chosen for illustration purpose of displaying the slow drift of the cavity transmission. %In general, the algorithm performs well for any threshold value larger than longitudinal fluctuation of cavity transmission. %The threshold can be as high as 97\% to have a high duty cycle. The average search time to recover the maximum cavity transmission is on the order of seconds. With a combination of both temperature stabilization and our active transverse stabilization algorithm, the near-concentric cavity can remain aligned for the course of few hours. %\textit{VI. Conclusion} \section{Conclusion} We presented a compact design, alignment procedure and stablization methods of a Fabry-Perot near-concentric optical cavity. We experimentally demonstrated that the cavity design can preserve cavity linewidth and cavity transmission when being operated at 207(13) nm shorter than the concentric point. The constructed cavity can be employed to explore multi-mode strong coupling CQED with various types of quantum emitters. \begin{acknowledgments}This work was supported by the Ministry of Education in Singapore (AcRF Tier 1) and the National Research Foundation, Prime Minister’s office (partly under grant no NRF-CRP12-2013-03). \end{acknowledgments} \bibliographystyle{apsrev4-1} \bibliography{cavity} %merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked %Control: key (0) %Control: author (72) initials jnrlst %Control: editor formatted (1) identically to author %Control: production of article title (-1) disabled %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled \end{document}