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Vortex structures that carry orbital angular momentum (OAM) have been the subject of extensive research and applied to diverse areas, including optics , acoustics , and electronics , both in classical and quantum regions. The best-known optical vortices with OAM can be generated by introducing a spiral phase into the two-dimensional transverse plane (i.e., the plane) of light fields. Note that such OAM is generally longitudinal, which means the OAM vector is parallel to the propagation direction . Using a similar manner, one can conveniently generate longitudinal OAM beams in other physical systems.
携带轨道角动量（OAM）的涡旋结构一直是广泛研究的主题，并应用于包括光学 、声学 和电子学 等多个领域，涵盖经典和量子区域。最著名的具有 OAM 的光学涡旋可以通过在光场的二维横向平面（即 平面）中引入螺旋相位来生成。请注意，这种 OAM 通常是纵向的，这意味着 OAM 向量与传播方向 平行。以类似的方式，可以方便地在其他物理系统中生成纵向 OAM 光束。
Recently, there is rapidly growing interest in transverse OAM where the OAM vector is orthogonal to the propagation direction. Such transverse vortices are not uncommon in nature and science. As early as 1770, Captain Cook discovered a similar phenomenon in the intriguing movement of the boomerang used by the First Nations Australian peoples. The tropical cyclone, a transversely moving vortex flow, has always been a research hotspot in meteorology and even economics . Similar vortex flow has also been observed in human left ventricular blood flow, which can be used to diagnose and assess ventricular contractility . In magnetic nanowires, controlling the formation and movement of vortex domain walls at the nanoscale is a fertile ground to explore emergent phenomena and their technological prospects, such as microelectronic devices and robust memory system . In optics, such transverse OAM was first observed in femtosecond filaments in air and was subsequently realized in polychromatic wave packets in free space , known as spatiotemporal (ST) optical vortices. This transverse OAM beam shows great potential to extend the applications of longitudinal OAM and to promote optics or other basic physics. However, previous studies still have some limitations. For instance, although a few studies suggested a strong ST coupling in ST vortices compared with their longitudinal counterparts, a precise description and accurate analysis of this nontrivial coupling mechanism are still lacking. Most importantly, existing experiments were limited in low transverse OAM, e.g., (where is the topological charges), in which a 2 -order ST vortex rapidly degrades into two first-order ST vortices during propagation or nonlinear interaction . Particularly, such degradation leads to instability of integral OAM value , limiting greatly the scientific research for and the real-world application of transverse OAM. Till now, most experimental efforts attributed this mode degradation to ST astigmatism effect, i.e., the mismatch between spatial diffraction and medium dispersion . In contrast, a few theoretical studies introduced a novel effect-termed time diffraction-via a plane-wave expansion method, providing a new perspective to investigate ST vortices and their degradation . Notably, ST astigmatism and temporal diffraction appear to be distinct because the former depends on the medium, while the latter is intrinsic, similar to spatial diffraction of monochromatic beams.
最近，对横向 OAM 的兴趣迅速增长，其中 OAM 矢量与传播方向正交。这种横向涡旋在自然和科学中并不少见。早在 1770 年，库克船长就发现了类似现象，体现在第一民族澳大利亚人民使用的回旋镖的有趣运动中。热带气旋，作为一种横向移动的涡旋流，一直是气象学甚至经济学的研究热点。类似的涡旋流也在人类左心室血流中被观察到，这可以用于诊断和评估心室收缩性。在磁性纳米线中，控制涡旋畴壁在纳米尺度上的形成和运动是探索新兴现象及其技术前景的肥沃土壤，例如微电子设备和稳健的存储系统。在光学中，这种横向 OAM 首次在空气中的飞秒光柱中被观察到，随后在自由空间中的多色波包中实现，被称为时空（ST）光涡旋。 这种横向 OAM 光束显示出极大的潜力，可以扩展纵向 OAM 的应用，并促进光学或其他基础物理。然而，以前的研究仍然存在一些局限性。例如，尽管一些研究表明，与其纵向对应物相比，ST 涡旋中存在强 ST 耦合，但对这种非平凡耦合机制的精确描述和准确分析仍然缺乏。最重要的是，现有实验在低横向 OAM 方面受到限制，例如 （其中 是拓扑电荷），在传播 或非线性相互作用 过程中，二阶 ST 涡旋迅速降解为两个一阶 ST 涡旋。特别是，这种降解导致整体 OAM 值的不稳定 ，极大地限制了横向 OAM 的科学研究和实际应用。到目前为止，大多数实验努力将这种模式降解归因于 ST 散光效应，即空间衍射与介质色散之间的不匹配 。 相反，一些理论研究通过平面波展开方法引入了一种新颖的效应——时间衍射，为研究 ST 涡旋及其降解提供了新的视角 。值得注意的是，ST 散光和时间衍射似乎是不同的，因为前者依赖于介质，而后者是内在的，类似于单色光束的空间衍射。
Here, based on wavevector analysis, the nontrivial features and inherent ST coupling of transverse OAM are uncovered. We show that the time-delayed modulation relies on a spatial Fourier transform (SFT) in current experiments actually made the most contributions to the observed mode degradation, which could be circumvented by an immediate modulation via the inverse design of phase. This allows us to generate theoretically equivalent spatiotemporal Bessel (STB) vortices and observe ultrahigh transverse OAM even beyond . We also show that STB vortices behave an opposite time-symmetrical evolution with respect to signs of due to therein inherent ST coupling. By circumventing the modulation-induced degradation, we confirm that such a never-seen-before phenomenon can be explained perfectly by time diffraction, with no need for considering the ST astigmatism. Beyond, we further show that such nontrivial ST coupling can be quantified as an intrinsic dispersion factor, and thus, theoretically, can be compensated by the medium dispersion thereby obtaining time diffraction-free STB vortices. Our work paves the way for further research and application of this unique OAM .
在这里，基于波矢分析，揭示了横向 OAM 的非平凡特征和固有的 ST 耦合。我们表明，时间延迟的 调制依赖于当前实验中的空间傅里叶变换 (SFT)，实际上对观察到的模式退化贡献最大，这可以通过相位的逆设计实现即时 调制来规避。这使我们能够理论上生成等效的时空贝塞尔 (STB) 涡旋，并观察到超高的横向 OAM，甚至超过 。我们还表明，STB 涡旋在 符号的时间对称演化中表现出相反的行为，这是由于其固有的 ST 耦合。通过规避调制引起的退化，我们确认这种前所未见的现象可以通过时间衍射完美解释，无需考虑 ST 像散。此外，我们进一步表明，这种非平凡的 ST 耦合可以量化为一个内在的色散因子，因此，从理论上讲，可以通过介质色散进行补偿，从而获得无时间衍射的 STB 涡旋。 我们的工作为进一步研究和应用这种独特的 OAM 铺平了道路。

Nontrivial features and inherent ST coupling of transverse OAM. To better understand why the transverse OAM is less trivial than the longitudinal case, let us re-exam the difference between these two situations. Generally, the angular momentum density of optical fields can be written as , where is position vector and is linear momentum density associated with a local wavevector . For a vortex beam, the carried OAM should remain stable during propagation. This means that the field with the same position vector should move as a whole, and thus they share the same propagation vector and the linear momentum density with the same amplitude , i.e., a local wavevector .
横向 OAM 的非平凡特征和固有的 ST 耦合。为了更好地理解为什么横向 OAM 比纵向情况更复杂，让我们重新审视这两种情况之间的差异。一般来说，光场的角动量密度可以写成 ，其中 是位置向量， 是与局部波矢量 相关的线动量密度。对于涡旋光束，携带的 OAM 在传播过程中应保持稳定。这意味着具有相同位置向量 的场应作为一个整体移动，因此它们共享相同的传播向量 和具有相同幅度 的线动量密度，即局部波矢量 。

For the longitudinal OAM beam, the local wavevector lies on the plane and is always perpendicular to during propagation (Fig. 1a), i.e., . One sees immediately that, a monochromatic light field can naturally meet the demand, that is, const, where is the whole wavevector. While for the transverse OAM beam, the plane is now rotated by 90 degrees to follow the rotation of the OAM vector, becoming the plane (Fig. 1b). Note that is no longer perpendicular to . If the monochromaticity is still forcibly retained, another wavevector varying along the ring is required to compensate for the wavevector mismatch, that is, . Obviously, such conditions are difficult to satisfy, at least experimentally. Nevertheless, if we turn to the polychromatic field, can be easily compensated by a broadened time spectrum , i.e., . This explains why transverse OAM was first found in femtosecond filaments, and polychromatic wave packets become the possible solution for carrying transverse OAM. Most importantly, the strong interaction between and leads to an inherent ST coupling in transverse OAM beams, indicating their nontrivial time-varying features. This coupling is also reflected in the specific ST frequency-frequency relationship inside an ST vortex, as we will demonstrate below .
对于纵向 OAM 光束，局部波矢 位于 平面上，并且在传播过程中始终与 垂直（图 1a），即 。人们立刻可以看到，单色光场自然可以满足这一要求，即 常数，其中 是整个波矢。而对于横向 OAM 光束， 平面现在旋转 90 度以跟随 OAM 矢量的旋转，变为 平面（图 1b）。注意， 不再与 垂直。如果仍然强行保持单色性，则需要另一个沿环变化的波矢 来补偿波矢不匹配，即 。显然，这种条件很难满足，至少在实验上是如此。然而，如果我们转向多色场， 可以通过扩展的时间谱 轻松补偿，即 。这解释了为什么横向 OAM 最初是在飞秒细丝中发现的，而多色波包成为携带横向 OAM 的可能解决方案。 最重要的是， 和 之间的强相互作用导致横向 OAM 光束中固有的 ST 耦合，表明它们具有非平凡的时间变化特征。这种耦合也反映在 ST 涡旋内部的特定 ST 频率-频率关系中，正如我们将在下面 中演示的。

From time-delayed modulation to immediate modulation. The above discussion also implies that the way to generate an ST vortex must be immediate. In contrast, any timedelayed modulation inevitably resonates with the inherent ST coupling within an ST vortex, resulting in strong mode degradation. We note that all recent experiments have been realized by directly loading a spiral phase via a conventional pulse shaper that contains a phase device between two gratings in a system. In this arrangement, the phase-loaded plane is regarded as plane and the ST vortex generally corresponds to a "patch" on the light-cone (Fig. 1c). Naturally, such a scheme is inspired by the generation of longitudinal OAM vortices. Because the -shaper is only used to realize temporal modulation, an additional SFT is required to convert the spiral phase onto plane, which was realized by a cylindrical lens or free-space transmission. Its principle is, essentially, to use the transmission to achieve the SFT; obviously, such approach is time-delayed which works well around a certain position-typically, the Fourier plane of the cylindrical lens or the far field-and the ST vortices inevitably
从时间延迟的 调制到即时的 调制。上述讨论还暗示，生成 ST 涡旋的方式必须是即时的。相比之下，任何时间延迟的调制不可避免地与 ST 涡旋内在的 ST 耦合产生共振，导致强模式退化。我们注意到，所有最近的实验都是通过直接加载一个螺旋相位实现的，使用的是一个包含两个光栅之间相位设备的常规脉冲整形器 ，在一个 系统中。在这种安排中，加载相位的平面被视为 平面，而 ST 涡旋通常对应于光锥上的一个“补丁”（图 1c）。自然，这种方案受到生成纵向 OAM 涡旋的启发。由于 整形器仅用于实现时间调制，因此需要额外的 SFT 将螺旋相位转换到 平面，这通过一个圆柱透镜或自由空间传输实现。其原理基本上是利用传输来实现 SFT；显然，这种方法是有时间延迟的，通常在某个位置有效——典型地是圆柱透镜的傅里叶平面或远场——而 ST 涡旋不可避免地
a

C

Fig. 1 Wavevector analysis of longitudinal and transverse OAM and two schemes for transverse OAM beams generation. a For a monochromatic longitudinal OAM beam, the local wavevector is always perpendicular to the propagation vector For a transverse OAM beam, the monochromaticity is broken due to the interaction between the local wavevector and the propagation vector , resulting in the generation of polychromatic ST vortex with inherent ST coupling. Note that spiral and cycloid curve in represent the linear momentum vectors respectively, which donate the OAM by . c An Gaussian-like ST vortex with I corresponds to a "patch" with a spiral phase on the light cone . Notably, the current scheme cannot effectively produce such a spiral phase on the plane due to the time-delayed modulation, and the 5 -order ST vortex degrades into five first-order ST vortices rapidly. Inverse design of the phase makes it possible to immediately produce an impulse ring with a spiral phase on the plane, leading to the generation of a degradation-free 5 -order STB vortex.
图 1 纵向和横向 OAM 的波矢分析以及横向 OAM 光束生成的两种方案。a 对于单色纵向 OAM 光束，局部波矢 始终与传播矢量 垂直。对于横向 OAM 光束，由于局部波矢 与传播矢量 之间的相互作用，单色性被打破，导致生成具有固有 ST 耦合的多色 ST 涡旋。注意， 中的螺旋和摆线曲线分别表示线动量矢量 ，通过 提供 OAM。c 一个类似高斯的 ST 涡旋与 I 对应于光锥 上具有螺旋相位的“补丁”。值得注意的是，当前方案无法有效地在 平面上产生这样的螺旋相位，由于时间延迟调制，5 阶 ST 涡旋迅速降解为五个一阶 ST 涡旋。 相位的逆设计使得在 平面上立即产生具有螺旋相位的脉冲环成为可能，从而生成无降解的 5 阶 STB 涡旋。
degrade at other positions (Fig. 1c). This explains the observed strong degradation in recent experiments, especially for highorder vortices.
在其他位置降解（图 1c）。这解释了最近实验中观察到的强降解，特别是对于高阶涡旋。
In this work, we show that according to the inverse design of the spiral phase, the time-delayed SFT is no longer required, thereby realizing a degradation-free ST vortex directly from immediate modulation, where "immediate" means that the SFT is pre-processed on the phase pattern, with no need for extra propagation. Therefore, one can use the conventional liquidcrystal (LC) based spatial light modulators (SLM) to accomplish such modulation, although their response is not fast. Such operation corresponds to an impulse ring with a spiral phase on the plane (as shown in Fig. 1d), i.e.,
在这项工作中，我们展示了根据螺旋相位的逆设计，时间延迟的 SFT 不再是必需的，从而实现了直接从即时 调制中获得无降级的 ST 涡旋，其中“即时”意味着 SFT 在相位模式上进行了预处理，无需额外传播。因此，可以使用传统的基于液晶（LC）的空间光调制器（SLM）来完成这种调制，尽管它们的响应速度并不快。这种操作对应于在 平面上具有螺旋相位的脉冲环（如图 1d 所示），即，

where relates to the detuning frequency, is the central frequency, is the reduction coefficient for temporal and spatial scale consistency, and are the polar coordinates on the plane, and is the modulated radius. Equation (1) also describes the strong correlation between spatial frequency and temporal frequency inside this beam, implying that changing the spectral properties also changes its spatial properties. The field on the plane can be calculated by a two-dimensional Fourier transform (Supplementary Note 1):
其中 与失谐频率相关， 是中心频率， 是时间和空间尺度一致性的缩减系数， 和 是 平面上的极坐标， 是调制半径。方程 (1) 还描述了该光束内空间频率 和时间频率 之间的强相关性，这意味着改变光谱特性也会改变其空间特性。可以通过二维傅里叶变换（补充说明 1）计算 平面上的场。

where and are the polar coordinates on the - plane, is the -order Bessel function of the first kind, and is the retarded time in pulse frame where is the group velocity. Equation (2) shows that the generated fields are strictly equivalent to the optical STB vortices, the exact solutions of the paraxial wave equation described by Dallaire et al. in 2009. Actually, it was not until 2012 that Bliokh and Nori theoretically pointed out that these beams carry transverse OAM. Their very recent work theoretically proposed an observable spin-orbit interaction between the transverse OAM and spin in such beams, providing an intersection of these two hot topics. However, due to the similar issue of modulation method, these beams lack experimental observation so far. We note that a strategy to synthesize STB vortices via superimposing a spiral phase with a conical phase has been proposed very recently . Still, due to its reliance on time-delayed modulation, obvious separation of topological charges even for has been observed.
在 - 平面上， 和 是极坐标， 是 阶第一类贝塞尔函数， 是脉冲框架中的滞后时间，其中 是群速度。方程（2）表明，生成的场严格等同于光学 STB 涡旋，这是 Dallaire 等人在 2009 年描述的平面波方程的精确解。实际上，直到 2012 年，Bliokh 和 Nori 理论上指出这些光束携带横向角动量（OAM）。他们最近的工作理论上提出了在这些光束中横向 OAM 与自旋之间可观察的自旋-轨道相互作用，提供了这两个热门话题的交集。然而，由于调制方法的类似问题，这些光束迄今为止缺乏实验观察。我们注意到，最近提出了一种通过叠加螺旋相位与锥相位合成 STB 涡旋的策略 。尽管如此，由于其依赖于时间延迟调制，即使对于 也观察到了拓扑电荷的明显分离。

To achieve such STB vortices via phase modulation, the spiral phases are inversely designed, inspired by the fact that the space coordinate and the spatial frequency are essentially a Fourier transform pair. Therefore, one can project a single point in the spatial frequency domain onto a location-shifted grating in the axis, wherein the period and shifted displacement of the grating are and , respectively (Supplementary Fig. 1). The phases for generating STB vortices with topological charges of , and 100 are shown in Fig. 2a-d, in which topological charges are manifested as the amount of dislocation between the left and right main lobes in the phase diagram.
通过 相位调制实现这样的 STB 涡旋，螺旋相位是反向设计的，灵感来源于空间坐标 和空间频率 本质上是一对傅里叶变换对。因此，可以将空间频率域中的单个点 投影到 轴上位置偏移的光栅 ，其中光栅的周期和偏移位移分别为 和 （补充图 1）。生成拓扑电荷为 和 100 的 STB 涡旋的相位如图 2a-d 所示，其中拓扑电荷表现为相位图中左右主瓣之间的位错量。

STB vortices with ultrahigh transverse OAM even beyond . To generate STB vortices, we used a custom -pulse shaper consisting of a diffraction grating ( 1800 lines , Thorlabs), a cylindrical lens ( with a focal length of , which also determines the distances between the elements), and an LC-based 2D phase-only SLM (PLUTO-2.1-
STB 涡旋具有超高的横向 OAM，甚至超过 。为了生成 STB 涡旋，我们使用了一个定制的 脉冲整形器，包含一个衍射光栅（1800 条