Despite recent progress in nonlinear optics in wavelength-scale resonators, there are still open questions on the possibility of parametric oscillation in such resonators. We present a general approach to predict the behavior and estimate the oscillation threshold of multi-mode subwavelength and wavelength-scale optical parametric oscillators (OPOs). As an example, we propose an OPO based on Mie-type multipolar resonances, and we demonstrate that due to the low-$Q$ nature of multipolar modes in wavelength-scale resonators, there is a nonlinear interaction between these modes. As a result, the OPO threshold, compared to the single-mode case, can be reduced by a factor that is significantly larger than the number of interacting modes. The multi-mode interaction can also lead to a phase transition manifested through a sudden change in the parametric gain as well as the oscillation threshold, which can be utilized for enhanced sensing. We establish an explicit connection between the second-harmonic generation efficiency and the OPO threshold. This allows us to estimate the OPO threshold based on measured or simulated second-harmonic generation in different classes of resonators, such as bound states in the continuum and inversely designed resonators. Our approach for analyzing and modeling miniaturized OPOs can open unprecedented opportunities for classical and quantum nonlinear photonics.

中文翻译：

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波长尺度光参量振荡器

尽管最近在波长尺度谐振器中的非线性光学方面取得了进展，但是关于这种谐振器中的参数振荡的可能性仍然存在未解决的问题。我们提出了一种通用的方法来预测行为，并估计多模子波长和波长尺度光学参量振荡器（OPO）的振荡阈值。例如，我们提出了一种基于Mie型多极共振的OPO，并且我们证明了由于$ Q $低在波长尺度谐振器的多极模式的本质上，这些模式之间存在非线性相互作用。结果，与单模式情况相比，可以将OPO阈值降低一个明显大于交互模式数量的因子。多模式相互作用还可以导致通过参数增益以及振荡阈值的突然变化而表现出的相变，该相变可以用于增强感测。我们在二次谐波产生效率和OPO阈值之间建立了明确的联系。这使我们能够基于在不同类别的谐振器中测量或模拟的二次谐波生成来估计OPO阈值，例如连续体中的约束状态和反向设计的谐振器。