Optical vortices (OVs), characterized by a twist wavefront of the form *e^ilθ*, exhibit a hollow intensity profile due to on-axis phase singularities. The integer *l*, the azimuthal quantum number (topological charge), determines the longitudinal orbital angular momentum (OAM) carried by spatial OVs, such as Laguerre-Gaussian beams. This OAM has found applications in diverse fields, including light-matter interaction, quantum optics, telecommunications, and holographic data storage. Recent advancements in ultrafast optics have led to the exploration of self-torque beams with time-varying longitudinal OAM. The discovery of transverse OAM, perpendicular to the propagation direction, in spatiotemporal optical vortices (STOVs) represents a significant breakthrough. STOVs exhibit singularities in space-time, and their generation through spiral phase modulation in the spatial frequency-frequency domain of pulsed beams opens new avenues for research in various areas such as photonic topology, structured ultrafast wavepackets, nonlinear optics, and nanophotonics. However, most research has focused on STOVs based on Gaussian-type wavepackets. Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) modes form complete orthonormal bases for light's spatial structure, with LG modes characterized by azimuthal (*l*) and radial (*p*) indices. The azimuthal number is linked to the longitudinal OAM, while the radial number (*p*) governs the beam's hyperbolic momentum. Controlling both indices in LG-OVs is advantageous in various applications. This study aims to explore the relatively unexplored spatiotemporal LG modes with both radial and azimuthal quantum numbers, opening a new frontier in this field.

Extensive research has been conducted on optical vortices and their applications. Allen et al. (1992) discovered the orbital angular momentum of light and its relation to Laguerre-Gaussian modes. Subsequent research has highlighted the use of spatial OVs in various fields such as manipulating particles, quantum information processing, and increasing communication channel capacity (Yao & Padgett, 2011; Padgett & Bowman, 2011; Yang et al., 2021; Mafu et al., 2013; Nicolas et al., 2014; Forbes & Nape, 2019; Bozinovic et al., 2013; Wang et al., 2012; Fang et al., 2020; Kong et al., 2023). The development of ultrafast optics has enabled the investigation of self-torque beams with time-varying longitudinal OAM (Rego et al., 2019; Zhao et al., 2020; Cruz-Delgado et al., 2022). The existence of transverse OAM in spatiotemporal optical vortices (STOVs) has been both theoretically and experimentally confirmed (Sukhorukov & Yangirova, 2005; Bliokh & Nori, 2012, 2015; Jhajj et al., 2016; Chong et al., 2020; Hancock et al., 2019). Studies on STOVs have also explored various applications in areas such as photonic topology, structured ultrafast wavepackets, nonlinear optics, and nanophotonics (Wan et al., 2022; Zdagkas et al., 2022; Wan et al., 2022; Zhong et al., 2023; Hancock et al., 2021; Cao et al., 2022; Chen et al., 2022; Chen et al., 2022; Lin et al., 2023; Piccardo et al., 2023; Hancock et al., 2021; Gui et al., 2021; Fang et al., 2021; Huang et al., 2023; Wang et al., 2021; Huang et al., 2022; Ge et al., 2023; Zhang et al., 2023; Bliokh, 2023; Wan et al., 2023). The use of Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) modes in spatial light has been extensively studied, with their ability to carry orbital angular momentum being significant (Plick et al., 2013; Plick & Krenn, 2015; Zhao et al., 2015; Krenn et al., 2016; Forbes et al., 2021; Mazanov et al., 2021; Fontaine et al., 2019; Zhou et al., 2017; Krenn et al., 2017; Salakhutdinov et al., 2012; Harris et al., 2015; Zia et al., 2023; Li et al., 2011; Li et al., 2023; Huang et al., 2022; Huo et al., 2024).

The study experimentally synthesized spatiotemporal Laguerre-Gaussian (STLG) wavepackets using a complex-amplitude modulation technique integrated with a 2D pulse shaper. A theoretical analysis, employing the paraxial wave equation and considering a slowly varying envelope approximation, was used to describe the evolution of the wavepackets in a dispersive medium. The electric field of a generic pulsed beam was expressed as a product of the carrier and the slowly varying envelope. Equations were derived to describe the spatiotemporal Laguerre-Gaussian wavepacket and the spatial-spectral coupled polychrome Laguerre-Gaussian beam. Numerical simulations were performed to model the generation of STLG wavepackets at different diffraction distances in free space, showing the generation of a multi-layered doughnut-shaped topological structure. Mode conversion from STLG to spatiotemporal Hermite-Gaussian (STHG) wavepackets was simulated by introducing spatiotemporal astigmatism. Experimentally, a mode-locked fiber laser provided a pulsed beam, split into probe and object pulses. The object pulse was shaped using a 2D ultrafast holographic pulse shaper comprising a grating, cylindrical lens, and a spatial light modulator (SLM). The SLM imprinted a phase pattern encoding the complex field amplitude of the polychrome beam. The reconstituted pulse after the shaper resulted in the STLG wavepacket. A 3D diagnostic measurement technique, using off-axis interference between the object pulse and a de-chirped probe pulse, was used to characterize the spatiotemporal profile of the STLG wavepacket. Time-delay scanning allowed for reconstruction of the 3D intensity and phase distributions. Mode conversion to STHG was achieved by applying a spatiotemporal astigmatism on the SLM. The purity of the synthesized wavepackets was analyzed using the orthonormality of LG modes. The experimental setup included a pulse compressor, a 2D ultrafast holographic pulse shaper, and a time-delay line system to fully reconstruct the 3D profile of the generated ST wavepacket.

The research successfully generated spatiotemporal Laguerre-Gaussian (STLG) wavepackets with well-defined radial (*p*) and azimuthal (*l*) quantum numbers. Experimental results showed high-purity 3D STLG wavepackets exhibiting multi-layered doughnut-shaped topological structures. These wavepackets possess screw singularities and edge dislocations, resulting in an intrinsic transverse OAM composed of two directionally opposite components. Mode conversion from STLG to spatiotemporal Hermite-Gaussian (STHG) wavepackets was experimentally demonstrated by applying controlled spatiotemporal astigmatism. The converted STHG wavepackets showed a decoupled intensity distribution in space and time, facilitating efficient and accurate recognition of ultrafast STLG wavepackets with various *p* and *l* values. The experimental results showed good agreement with numerical simulations. Specifically, the experimentally reconstructed 3D iso-intensity surfaces of STLG and STHG wavepackets matched the simulated profiles. The spatial and temporal widths of the generated STLG wavepackets were controllable by manipulating the diffraction and dispersion phases on the SLM. The mode conversion process showed a gradual deformation of the STLG wavepacket under increasing spatiotemporal astigmatism, culminating in a distinct STHG profile. Analysis of the spatial projections of converted STHG wavepackets revealed a clear relationship between the number of dark gaps in the spatial intensity profiles and the radial and azimuthal quantum numbers, providing a method for recognizing the quantum numbers of STLG wavepackets.

The successful generation of STLG wavepackets with controlled radial and azimuthal indices addresses the need for creating complex spatiotemporal light fields with tailored properties. The intrinsic transverse OAM and the multi-ring topology of these wavepackets open exciting possibilities in various fields. The demonstrated mode conversion to STHG wavepackets, with its decoupled spatial and temporal intensity distribution, offers a practical approach for characterizing and manipulating these wavepackets. The ability to control and distinguish these high-order modes is particularly important for applications in high-dimensional quantum information processing and high-capacity optical communication. This research extends the capabilities of spatiotemporal light field engineering, providing new tools for controlling and manipulating light at the ultrafast scale. The results pave the way for utilizing STLG wavepackets in advanced optical technologies and potentially in other wave phenomena such as acoustics and electron waves.

This work demonstrated the experimental generation and characterization of high-purity spatiotemporal Laguerre-Gaussian (STLG) wavepackets with controlled radial and azimuthal quantum numbers. The unique features of these wavepackets, including their intrinsic transverse orbital angular momentum and multi-ring topology, were experimentally verified. Mode conversion to STHG wavepackets was achieved using spatiotemporal astigmatism, facilitating efficient identification of STLG wavepacket parameters. This study advances the field of structured light and opens new possibilities for applications in high-dimensional quantum information, photonic topology, nonlinear optics, and other wave phenomena. Future research could explore the generation and control of even higher-order STLG wavepackets and the integration of STLG wavepacket manipulation with nanophotonic devices.

The current experimental setup, using a phase-only spatial light modulator, involves both amplitude and phase modulation, which leads to some energy loss. The spatial and temporal widths of the synthesized STLG wavepackets are determined by the time-dispersion and space-diffraction phases. While this allows for control, it also sets limitations on the achievable spatial and temporal resolutions. The off-axis interference scheme for spatiotemporal reconstruction can be affected by the instability of the interferometer. Future research could focus on improving the efficiency of the complex-amplitude modulation technique and enhancing the stability of the measurement system for higher fidelity characterization.