Bicircular twisted Laguerre-Gaussian beams possess a definite torus knot angular momentum (TKAM) $\widehat{J_{\gamma }}=\widehat{L}+\gamma \widehat{S}$ as an alternative a form of angular momentum. TKAM is conserved in nonlinear atomic processes such as high harmonic generation and can be classified by a time delay parameter $\tau$ and a coordination parameter $\gamma$. These parameters are defined by the respective projected orbital angular momentum ${\ell }_i$ and the energy $\hslash p_i{\omega }_0$ of the two superimposed Laguerre-Gaussian beams. We derive a consistent geometric method to determine $\tau$ and $\gamma$ from the driving beam as well as from the high harmonic radiation. The method relates both invariance parameters ($\tau$ and $\gamma$) to a torus knot which can be constructed from the emitted high harmonic radiation. These knots are constructed from the spatiotemporal evolution of the electric field of the respective high harmonic radiation or the driving beam. We demonstrate the classification of the invariance parameters for a planar atomic gas target irradiated by bicircular Laguerre-Gaussian beams explicitly. In addition, we demonstrate that the respective torus knots determined by $\tau$ and $\gamma$ can be mapped onto each other within minor modifications. The numerical calculations are done within the strong-field approximation and the associated quantum orbit approach. Therefore, we also briefly review high harmonic generation by bicircular twisted light beams. This introduced geometric method is a different approach to interpret the invariance parameters $\tau$ and $\gamma$, as well as their underlying relations, compared to a purely formal derivation. The investigations presented in this work are in good agreement with previous findings and provide insight into the dynamical symmetry of TKAM in the context of high harmonic generation induced by bicircular twisted Laguerre-Gaussian beams.

中文翻译：

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高次谐波产生的扭曲阿秒脉冲中的环结角动量

双圆扭曲拉盖尔-高斯光束具有确定的环面结角动量 (TKAM) $\widehat{J_{\gamma }}=\widehat{升}+\gamma \widehat{秒}$作为另一种形式的角动量。TKAM 在非线性原子过程中是守恒的，例如高次谐波产生，并且可以通过时间延迟参数进行分类$\tau$ 和协调参数 $\gamma$. 这些参数由各自的投影轨道角动量定义${\ell }_{\mathrm{一世}}$ 和能量 $\hslash p_{\mathrm{一世}}{\omega }_0$两个叠加的拉盖尔-高斯光束。我们推导出一致的几何方法来确定$\tau$ 和 $\gamma$来自驱动光束以及高次谐波辐射。该方法涉及两个不变性参数（$\tau$ 和 $\gamma$) 到一个环面结，它可以由发射的高次谐波辐射构成。这些结是由各个高次谐波辐射或驱动光束的电场的时空演变构成的。我们明确地展示了由双圆拉盖尔-高斯光束照射的平面原子气体目标的不变性参数的分类。此外，我们证明了各自的环面结由$\tau$ 和 $\gamma$可以在较小的修改内相互映射。数值计算是在强场近似和相关的量子轨道方法中完成的。因此，我们还简要回顾了双圆扭曲光束产生的高次谐波。这种引入的几何方法是解释不变性参数的不同方法$\tau$ 和 $\gamma$，以及它们的潜在关系，与纯粹的形式推导相比。这项工作中提出的研究与先前的发现非常一致，并提供了在双圆扭曲拉盖尔-高斯光束引起的高次谐波产生背景下对 TKAM 动力学对称性的深入了解。