# Stabilization of Transverse Modes for a High Finesse Near-Unstable Cavity
## 1 Introduction  
In the development of narrow-band entangled photon-pair sources for applications such as quantum storage and quantum information processing [1], high-finesse optical cavities have been used to enhance the spontaneous parametric down-conversion (SPDC) [2; 3; 4; 5; 6; 7; 8]. This method was proposed by [PERSON] et al. in 1991 [2], and generates entangled photon pairs using a second-order nonlinear effect and narrows the linewidth of single photon pairs by putting the nonlinear crystal inside the optical cavity. The narrow spectral linewidth of SPDC has been achieved in various types of cavities, for example, bow-tie cavity [6; 7; 8; 9; 10; 11] and the Fabry-Perot (FP) cavity [4; 12; 13; 14; 15; 16]. The bow-tie cavity consists of four mirrors with a long cavity length for a narrow spectral linewidth. It is expensive and is sensitive to the environmental perturbation due to the long cavity length. The FP cavity usually consists of two components, a mirror and a nonlinear crystal with a highly reflective coating on the output surface with a curvature. The FP cavity and periodically poled crystals have been successfully used for narrow-band polarization-entangled photon pair generation [4]. However, the performance of this kind of FP cavity is limited by the coating technology for a high-finesse because most nonlinear crystals, e.g., Beta-Barium Borate crystal (BBO) and Potassium Dihydrogen Phosphate crystal (KDP), cannot be coated with highly reflective films. Three-component cavity solves this problem by putting the nonlinear crystal in the cavity for SPDC. In a three-component FP cavity, it is easy to replace the nonlinear crystal and the cavity is suitable for both type-I and type-II cavity-enhanced SPDC, where the type-I and type-II refer to the phase-matching conditions of the nonlinear crystal. In type-I, the down-converted photons are either in extraordinary rays (e rays) or ordinary rays (o rays), and theirpolarizations are orthogonal to the pump polarization [3]. While in type-II, the polarizations of the down-converted photons are orthogonal [4]. The selection of FP cavity type is crucial in generating entangled photon pairs. For example, confocal cavity is a kind of widely used cavity since it is highly stable and easy to align. However, it has a relatively low threshold when using a nonlinear crystal as the gain media for SPDC. With the gain media, the cavity is called optical parametric oscillation (OPO) cavity. The narrow-band entangled photons can only be generated in a condition of far-below threshold, and the OPO cavity will be lasing above the threshold. Therefore, working at far-below threshold is crucial for cavity-enhanced SPDC [2]. In this paper, we designed a kind of near-unstable FP cavity, which is away from the confocal region but close to the stability boundary of cavity. Similar near-unstable cavities were recently proposed for gravitational wave detection [17]. In their case, a larger beam spot size on the mirrors is required to reduce the mirror coating thermal noise. We intentionally selected the near-unstable cavity due to its relatively high threshold when using it in OPO, therefore the far-below threshold condition can be easily achieved. The FP near-unstable cavity consisted of a plane mirror and a concave mirror with a large radius. It is difficult to lock this kind of cavity with an ultra-high-finesse for an extremely narrow linewidth. We used the Pound-Drever-Hall (PDH) technique to lock the designed FP cavity in both the fundamental mode and the high-order transverse modes and dynamically linked a high-finesse external FP cavity (F = 7330) to the dye laser cavity at the same wavelength of sub-megahertz narrow linewidth.