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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 ，Thorlabs）、一个圆柱透镜（ ，焦距为 ，这也决定了元件之间的距离）和一个基于 LC 的 2D 仅相位 SLM（PLUTO-2.1-

b

Fig. 2 SLM phase patterns and experimental setup for generating and characterizing STB vortices. a-d Phase patterns of STB vortices with topological charges of , and 100. e Experimental setup consists of two sections: (1) STB vortex generator consisting of a grating, a cylindrical lens , an aperture, and an SLM; and (2) a time-resolved profile analyser that is realized by a Mach-Zehnder interferometer consisting of two and , a CCD camera, and a motorized translation stage in the reference path. The coordinate in the upper left corner of e marks the orientations in SLM where the phase patterns in a-d are loaded.
图 2 SLM 相位模式及生成和表征 STB 涡旋的实验设置。a-d STB 涡旋的相位模式，拓扑电荷为 和 100。e 实验设置由两个部分组成：(1) STB 涡旋发生器，包括光栅、一个圆柱透镜 、一个光阑和一个 SLM；(2) 通过 Mach-Zehnder 干涉仪实现的时间分辨轮廓分析仪，包括两个 和 、一个 CCD 相机和一个在参考路径中的电动平移台。e 左上角的坐标标记了 SLM 中加载 a-d 相位模式的方向。

NIR-133, Holoeye) with a resolution of , a pixel pitch of , and an active area of (Fig. 2 e ). The frequencies of an ultrashort pulse that comes from a Ti:sapphire laser with a central wavelength of and a pulse duration of are spatially spread by the grating and collimated to the SLM via the , which can be understood as a temporal Fourier transform. We consider the SLM as the plane, where the phase patterns for the generation of STB vortices are loaded. After the SLM, the light field is retroreflected and reconstituted at the same diffraction grating, thereby immediately generating the STB vortices, with no need for a time-delayed SFT. The ST intensities were measured and reconstructed using a Mach-Zehnder interferometer as shown in Fig. 2e (Methods) .
NIR-133，Holoeye）具有 的分辨率， 的像素间距和 的有效区域（图 2 e）。来自中心波长为 、脉冲持续时间为 的钛：蓝宝石激光器的超短脉冲频率通过光栅进行空间扩展，并通过 准直到 SLM，这可以理解为时间傅里叶变换。我们将 SLM 视为 平面，在该平面上加载生成 STB 涡旋的相位模式。在 SLM 之后，光场被反射并在同一衍射光栅处重新构成，从而立即生成 STB 涡旋，无需时间延迟的 SFT。ST 强度通过如图 2e 所示的马赫-曾德干涉仪进行测量和重构（方法） 。

The experimental and simulated (Methods) results of STB vortices with , and 100 are shown in Fig. 3. The reconstructed intensities are plotted in Fig. 3a-d, and simulations are drawn with respective illustrations accordingly. Notably, the spatial and temporal bandwidths were set to (by the phase patterns) and (by the onedimensional slit with a width of (Fig. 2e)) for temporal and spatial scale consistency, i.e., the standard STB vortices should be circularly symmetric pulses with equal width and length (when we project axis to axis) to make the integral OAM value is per photon, otherwise, the OAM value will become larger . The corresponding is . Despite the slight distortion, the experimental results are in good agreement with the simulations. To ensure that SLM does not affect the polarization, we adjusted the incident light to be linearly polarized in the direction, which means that the polarization of the generated STB vortices is perpendicular to the plane. According to the theoretical analysis in ref. , this out-of-plane polarization will result in a zero intensity at the centre of STB vortices (verified by Fig. 3a-d). In contrast, the in-plane polarization will lead to an observable nonzero intensity in the centre of STB vortices due to transverse spin-orbit interaction .
STB 涡旋的实验和模拟（方法）结果如图 3 所示， 和 100。重建的强度绘制在图 3a-d 中，模拟结果相应地绘制了相应的插图。值得注意的是，空间和时间带宽分别设置为 （通过相位模式）和 （通过宽度为 的单维狭缝（图 2e）），以保持时间和空间尺度的一致性，即标准 STB 涡旋应为具有相等宽度和长度的圆对称脉冲（当我们将 轴投影到 轴时），以使每个光子的积分 OAM 值为 ，否则，OAM 值将变得更大 。相应的 为 。尽管存在轻微的失真，实验结果与模拟结果非常一致。为了确保 SLM 不影响偏振，我们将入射光调整为在 方向上线性偏振，这意味着生成的 STB 涡旋的偏振与 平面垂直。根据参考文献中的理论分析。 ，这种面外极化将在 STB 涡旋中心导致零强度（由图 3a-d 验证）。相比之下，面内极化将由于横向自旋-轨道相互作用在 STB 涡旋中心导致可观察的非零强度 。
Besides, the phase reconstruction of high-order modes is difficult due to the instability of the interferometer caused by vibrations such as air disturbance (Methods). Nevertheless, the simulated phases (Fig. 3e-h) verify the spiral phases of the corresponding topological charges, implying the carried transverse OAM (see also the reconstructed phases with , and 25 shown in Supplementary Fig. 3). The reconstructed three- dimensional intensity iso-surface profiles of these beams from measured data are shown in Fig. 3i-l (Methods). Furthermore, we calculated the spatial and temporal diameters of these beams from Eq. (2). The theoretical, simulated, and experimental results as a function of topological charges are shown in Fig. 3m, n, indicating that their diameters are linearly related to the topological charge, except for . Additionally, the widths of the STB vortices on the and axes are inversely proportional to the spatial and temporal spectral bandwidths (see also Supplementary Fig. 3), respectively, which are consistent with conventional spatial beams and short pulses. Enabled by the proposed modulation, our results provide the experimental observation of ultrahigh transverse OAM even beyond , which is two orders of magnitude higher than the results reported thus far.
此外，由于干涉仪因空气扰动等振动引起的不稳定性，高阶模式的相位重建较为困难（方法）。尽管如此，模拟相位（图 3e-h）验证了相应拓扑电荷的螺旋相位，暗示了携带的横向 OAM（另见补充图 3 中重建的相位 和 25）。从测量数据得到的这些光束的三维强度等值面轮廓如图 3i-l 所示（方法）。此外，我们根据公式（2）计算了这些光束的空间和时间直径。理论、模拟和实验结果作为拓扑电荷的函数如图 3m、n 所示，表明它们的直径与拓扑电荷呈线性关系，除了 。此外，STB 涡旋在 和 轴上的宽度与空间和时间光谱带宽成反比（另见补充图 3），这与传统空间光束和短脉冲一致。 通过所提议的 调制，我们的结果提供了超高横向 OAM 的实验观察，甚至超出了 ，其值比迄今为止报告的结果高出两个数量级。

Quantifying ST coupling by an intrinsic dispersion factor . To investigate the propagation dynamics, we captured two STB vortices with opposite topological charges of and -100 at different positions along . The intensities with at , and 150 mm are shown in Fig. 4a. We marked the position of a standard mode (described in Eq. (2)) as and observed a time-symmetrical evolution on the axis. Remarkably, unlike the longitudinal OAM beam, for which the sign of the topological charge is difficult to judge without extra devices, such as the cylindrical lens, the results with are distinguishable because they look like the mirror version of (Fig. 4b). Since the modulation-induced degradation is circumvented, we immediately realized that such unique evolution could be explained by the time diffraction, which describes the accumulated phase difference in STB vortices between waves with different frequencies . In this model, STB vortices could be considered as the coherent superposition of a series of plane waves ,
量化 ST 耦合的内在色散因子 。为了研究传播动态，我们在 的不同位置捕获了两个具有相反拓扑电荷 和-100 的 STB 涡旋。图 4a 显示了在 、 和 150 毫米处的强度。我们将标准模式的位置（在方程(2)中描述）标记为 ，并观察到在 轴上的时间对称演化。值得注意的是，与纵向 OAM 光束不同，后者的拓扑电荷符号在没有额外设备（如圆柱透镜）的情况下难以判断，结果 是可区分的，因为它们看起来像 的镜像版本（图 4b）。由于避免了调制引起的降解，我们立即意识到这种独特的演化可以通过时间衍射来解释，该衍射描述了不同频率波之间在 STB 涡旋中的累积相位差 。在这个模型中，STB 涡旋可以被视为一系列平面波的相干叠加 。

where is the speed of light in a vacuum, is the spatial bandwidths, and is the electric field. As shown in Fig. 4c, the results by calculating intensities with the same parameters in our experiment verified our observation of pure time diffraction.
其中 是真空中的光速， 是空间带宽， 是电场。如图 4c 所示，通过计算在我们实验中使用相同参数的强度 的结果验证了我们对纯时间衍射的观察。

Fig. 3 Experimental and simulated results of STB vortices with topological charges of , 50, and 100. a-d Reconstructed intensities.
图 3 具有拓扑电荷 、50 和 100 的 STB 涡旋的实验和模拟结果。a-d 重建强度。
Simulated phases. i-I Reconstructed 3D profiles. Spatial or temporal diameter dependence of STB vortices. The white dotted squares (circles) in a-d (e-h) show the simulated intensities (spiral phases) of each STB vortex. The corresponding topological charges are marked at the top of each column. Exp: experimental results; Simu: simulated results.
模拟相位。i-I 重建的 3D 轮廓。 STB 涡旋的空间或时间直径依赖性。a-d（e-h）中的白色虚线方框（圆圈）显示了每个 STB 涡旋的模拟强度（螺旋相位）。相应的拓扑电荷标记在每列的顶部。Exp: 实验结果；Simu: 模拟结果。

We note that the time diffraction actually acts on STB vortices in a similar manner as the medium dispersion acts on optical pulses . Indeed, by setting a pre-chirp to , we observed consistent results in the simulation (Fig. 4d), which means the pre-chirp is completely compensated after a short distance of propagation from (the elliptical chirped STB vortex) to (the standard non-chirped STB vortex). At first, we attributed this to the positive dispersion of the air, however, the corresponding dispersion factor is times the second-order dispersion of air-which implies that this dispersion should be intrinsic. Inspired by the inherent wavevector interaction within transverse OAM beams, we re-examine the ST spectra of STB vortices as described in Eq. (1), where the spatial frequencies and temporal frequencies actually hold a one-to-one correspondence. By combining Eq. (1) with the light cone equation , we can obtain the relation between the propagation constant and . Formally, an intrinsic dispersion factor could be obtained by Taylor series expansion of , i.e., . In this manner, the dispersion factor could be written as
我们注意到，时间衍射实际上对 STB 涡旋的作用方式与介质色散对光脉冲的作用相似 。确实，通过设置预啁啾为 ，我们在模拟中观察到了一致的结果（图 4d），这意味着在从 （椭圆啁啾 STB 涡旋）到 （标准非啁啾 STB 涡旋）的短距离传播后，预啁啾被完全补偿。起初，我们将此归因于空气的正色散，然而，相应的色散因子 是空气的二阶色散的 倍——这意味着这种色散应该是内在的。受到横向 OAM 光束内在波矢相互作用的启发，我们重新审视了 STB 涡旋的 ST 谱，如公式（1）所述，其中空间频率 和时间频率 实际上存在一一对应关系。通过将公式（1）与光锥方程 结合，我们可以获得传播常数 与 之间的关系。形式上，可以通过对 进行泰勒级数展开来获得内在色散因子 ，即 。以这种方式，分散因子可以写成
(Supplementary Note 5) （补充说明 5）

Using Eq. (4), we directly obtain , which is perfectly consistent with the dispersion factor estimated by our simulation. This makes it clear that the inherent ST coupling in STB vortices can be quantified as an intrinsic dispersion factor . Base on this, we propose a group dispersion delay (GDD) model to provide a theory-accurately and engineering-friendly description of STB vortices. For simplicity, we only consider the second-order dispersion and ignore the higher-order terms. In this model, the ST spectra of STB vortices could be rewritten as
使用公式（4），我们直接得到 ，这与我们模拟估计的色散因子完全一致。这清楚地表明，STB 涡旋中的固有 ST 耦合可以量化为一个内在的色散因子 。基于此，我们提出了一种群色散延迟（GDD）模型，以提供一个理论准确且工程友好的 STB 涡旋描述。为简化起见，我们仅考虑二阶色散，忽略高阶项。在该模型中，STB 涡旋的 ST 光谱可以重写为

donates the GDD. Therefore, the fields on the plane become
捐赠了 GDD。因此， 平面上的字段变为

As expected, by solving Eq. (5) numerically with , the results are perfectly consistent with the
如预期，通过使用 数值求解方程 (5)，结果与

Fig. 4 Experimental, theorical, and simulated results of propagation dynamics of two STB vortices with topological charges of and . a Reconstructed intensities of an STB vortex with the topological charge of at , and 150 mm , where the position of the standard mode (described in Eq. (2)) is marked as . b Same as , but with a topological charge of . Theorical calculated results of using the time diffraction model. Simulated results of with a pre-chirp of . Theorical calculated results of using the GDD model with an intrinsic dispersion factor . The corresponding eccentricities are given in the lower right corner of each figure in a-e. The positions are marked at the top of each column. Time diff. model: time diffraction model.
图 4 两个拓扑电荷为 和 的 STB 涡旋的传播动态的实验、理论和模拟结果。a 在 和 150 mm 处，拓扑电荷为 的 STB 涡旋的重建强度，其中标准模式（在方程（2）中描述）的位置标记为 。b 同 ，但拓扑电荷为 。 使用时间衍射模型计算的理论结果。 带有 预啁啾的 的模拟结果。 使用具有内在色散因子 的 GDD 模型计算的理论结果。每个图形 a-e 右下角给出了相应的偏心率。位置标记在每列的顶部。时间衍射模型：时间衍射模型。
experiment, time diffraction model, and the simulation, as shown in Fig. .
实验、时间衍射模型和模拟，如图 所示。

Additionally, we calculated the eccentricity of these two beams at different positions to quantify and compare the mode distortions caused by the time diffraction, i.e., , where and are the major and minor axes of these beams on the plane. The farther the value of is from 0 -corresponding to a standard circle-the larger the distortion of the mode is. The evolution of eccentricity further verifies the time-symmetrical evolution, despite the minor deviation in the experiment owing to the slight beam divergence caused by incomplete collimation.
此外，我们计算了这两束光在不同位置的离心率，以量化和比较时间衍射引起的模式畸变，即 ，其中 和 是这两束光在 平面上的大轴和小轴。 的值越远离 0（对应于标准圆），模式的畸变就越大。离心率的演变进一步验证了时间对称演变，尽管由于不完全准直引起的轻微光束发散，实验中存在轻微偏差。

Towards time diffraction-free STB vortices. We next try to eliminate the time diffraction as much as possible. One way that has been theoretically proposed is to produce the so-called Lorentz-boosted STB vortices , however, such beams are difficult to generate experimentally because of the need to give a monochromatic Bessel vortex an extra speed along the direction. Another rough method is just to simultaneously reduce the time and space bandwidth, i.e., and , akin to the fact that pulses with narrower bandwidths are less sensitive to the dispersion. Here, based on the GDD model, we propose that by engineering the dispersion of media, the time diffraction can be completely suppressed, thereby achieving time diffraction-free STB vortices.
朝着无时间衍射的 STB 涡旋。接下来，我们尝试尽可能消除时间衍射。理论上提出的一种方法是产生所谓的洛伦兹增强 STB 涡旋 ，然而，由于需要给单色贝塞尔涡旋在 方向上施加额外的速度，这种光束在实验上难以生成。另一种粗略的方法是同时减少时间和空间带宽，即 和 ，类似于带宽更窄的脉冲对色散的敏感性较低的事实。在这里，基于 GDD 模型，我们提出通过工程化介质的色散，可以完全抑制时间衍射，从而实现无时间衍射的 STB 涡旋。
The simulated evolutions of an STB vortex with propagating in virtual media with the second-order dispersion of and are shown in Fig. 5a, b, respectively, where the time diffraction is suppressed to varying degrees. We summarize the calculated integral OAM values (Methods) and eccentricities at different propagating distances with media dispersions varying from to (Fig. 5c). Naturally, one can obtain STB vortices propagating in a stable and invariant manner if the high-order dispersion terms are further compensated. Additionally, we find that integral OAM values in Fig. 5 c can be approximately expressed as , where is the OAM value of the standard STB vortex, is a dimensionless constant related to where is the media dispersion, and is the propagation distance from the standard location. The time-varying transverse may be considered as a special physical quantity to define ST vortices because it indicates that their OAM fundamentally depends on the intrinsic dispersion factor It would be interesting and meaningful to derive a more precise analytical expression of and find other unique physical quantities of the ST vortices in the future.
STB 涡旋的模拟演化如图 5a、b 所示， 在具有 和 的二阶色散的虚拟介质中传播，其中时间衍射被抑制到不同程度。我们总结了在不同传播距离下计算的积分 OAM 值（方法）和偏心率，介质色散从 变化到 （图 5c）。自然地，如果进一步补偿高阶色散项，可以获得以稳定和不变的方式传播的 STB 涡旋。此外，我们发现图 5c 中的积分 OAM 值可以近似表示为 ，其中 是标准 STB 涡旋的 OAM 值， 是与 相关的无量纲常数，其中 是介质色散， 是从标准位置的传播距离。 时间变化的横向 可以被视为定义 ST 涡旋的特殊物理量，因为它表明它们的 OAM 在根本上依赖于内在的色散因子 。推导出更精确的 的解析表达式并在未来找到 ST 涡旋的其他独特物理量将是有趣且有意义的。