Fund Project:Project supported by the National Natural Science Foundation of China (Grants Nos. 11674032, 11774035, 91950108)
Received Date:31 May 2020
Accepted Date:28 June 2020
Available Online:09 November 2020
Published Online:20 November 2020
Abstract:Optical cavity is a fundamental device of modern optics and has a wide range of applications in the fields of laser generation, nonlinear optical conversion, and optical sensors. A major aspect of the properties of optical cavity is the stability analysis. According to different geometric losses, these optical cavities can be divided into three types: stable cavity, critical cavity, and unstable cavity. The determination of the stability of the optical cavity is the basic problem of a classic system, but the research and analysis of this point have been much insufficient in the past. In this paper, by extending the definition domain of the inverse trigonometric function, the propagation matrices of the symmetric confocal cavity and the asymmetric confocal cavity are solved. The sudden change of stability with the change of geometric parameters is explained by algebraic analysis and optical ray topology.The mathematical analysis shows that the stability catastrophe of confocal cavity is due to the sudden change in the value of inverse cosine function at the critical point of the traditional domain of definition. From the perspective of geometric topology, we define the topological charge of the cavities according to the geometric propagation path of light in the cavity. Only the cavities with zero topological charge are found to be stable, and the change of topological charge is quantized, which explains the sudden change of confocal cavity stability. Finally, we build a coupled stable cavity consisting of two unstable cavities with the same parameters. The quality factors of the coupled stable cavity and the unstable cavity are analyzed by the finite difference time domain method, which further verifies the origin of the sudden change in the stability of the confocal cavity. We propose that the coupled unstable dual cavities with opposite topological charges are able to be stable, and we also find that there are new modes in the coupled cavities which are not found in the corresponding single cavity. These findings suggest a new method for controlling microcavity loss, which has a certain value for studying the new micro-nano lasers, on-chip nonlinear devices, and non-Hermitian optical sensors. Keywords:confocal cavity/ stability analysis/ topological structure
式中, $ {R}_{1} $和$ {R}_{2} $分别表示两腔镜的半径, $ L $表示腔长, 如图1(a)所示. 若将光线表示为列向量的形式, 则在谐振腔内往返$ n $次的光线与入射光线满足如 (3) 式: 图 1 腔的稳定性分析 (a)谐振腔的示意图及参数, O1和O2代表腔镜的球心, R1和R2分别表示两腔镜的半径, L表示腔长; (b)稳定图, 白色的区域表示稳定腔, 坐标轴和双曲线上的点表示临界腔, 灰色的区域表示非稳腔 Figure1. Stability analysis of the cavity: (a) Schematic diagram and parameters of the resonator cavity: O1 and O2 represent the spherical center of the cavity mirror, R1 and R2 represent the radius of the two mirrors, L represents the cavity length; (b) stability diagram, the white area represents the stable cavity, the coordinate axis and the hyperbola point represents the critical cavity, and the gray area represents the unstable cavity.
将$ \Delta L $分别取0, 0.15L和–0.25L, 即${R}_{1}+{R}_{2}= 2 L$, $ {R}_{1}+{R}_{2}=2.3 L $和$ {R}_{1}+{R}_{2}=1.5 L $三种腔在稳定图上对应的曲线画出, 如图2所示. 图中实线表示共焦腔, 但线上的$ {g}_{1}=1 $且$ {g}_{2}=1 $的点表示平行平面腔, 不符合共焦腔和非共焦腔的概念, 在图中用红色圆圈除去. 虚线表示满足$ {R}_{1}+{R}_{2}=2.3 L $的非共焦腔, 点划线表示满足$ {R}_{1}+{R}_{2}=1.5 L $的非共焦腔. 图 2 共焦腔和几种非共焦腔对应的曲线, 实线表示共焦腔, 虚线表示满足$ {R}_{1}+{R}_{2}=2.3 L $的非共焦腔, 点划线表示满足$ {R}_{1}+{R}_{2}=1.5 L $的非共焦腔(红色圆圈除外) Figure2. Stability curves corresponding to the confocal cavity and non-confocal cavities. The solid line represents the confocal cavity, the dotted line represents the non-confocal cavity satisfying $ {R}_{1}+{R}_{2}=2.3 L $, and the dash dot line represents the non-confocal cavity satisfying ${R}_{1}+ {R}_{2}=1.5 L$.(except for the red circle).
表2对称、非对称共焦腔及耦合腔内模式的Q值 Table2.Q factor of modes in symmetric, asymmetric and coupled confocal cavities.
通过模拟对比对称共焦腔及耦合腔内的模式, 发现除了两种腔都支持的四个模式外, 耦合腔在模式1和模式4旁边还出现了新的模式1-1及4-1. 为进一步分析新模式出现的原因, 分别探究了对称共焦腔及耦合腔内的相位分布. 图5所示为模式1和模式4的相位, 其中模式1为两种腔都支持的模式, 模式1-1为耦合腔内出现的新模式, 如图4(b)中的蓝色箭头所示. 通过对比磁场的相位分布, 发现对称共焦腔所支持的四种模式的相位分布都是关于结构反对称分布, 而耦合腔内出现的新模式其相位分布为对称分布. 根据相位分布可见, 耦合腔内出现的新模式是由两个非对称共焦腔之间的耦合形成的, 这证明了耦合的非对称共焦腔拥有比单个腔更多的调节自由度. 图 5 模式1和模式4的相位 (a)对称共焦腔内模式1的相位分布; (b)耦合腔内模式1的相位分; (c)耦合腔内模式1-1的相位分布 Figure5. Phase distribution of mode 1 and mode 4: (a) Phase distribution of mode 1 in a coupled cavity; (b) phase distribution of mode 1 in a coupled cavity; (c) phase distribution of mode 1-1 in a coupled cavity.