The optical orbital angular momentum (OAM) can exist either as longitudinal OAM in the spatial vortex beam or transverse OAM in the spatiotemporal optical vortices. In contrast to the amount of research focused on longitudinal OAM, very few pay attention to optical fields with transverse OAM. Unlike longitudinal OAM which is only affected by diffraction, transverse OAM can be affected by both diffractive effect and dispersive effect. One of the biggest challenges in utilizing optical fields carrying transverse OAM is to overcome diffraction and dispersion as the optical field propagates. Diffraction and dispersion will cause the fields to spread in space and time, which limits the applications of the optical field with OAM. We introduce a class of three-dimensional (3D) spatiotemporal localized wave packets with transverse optical OAM. The combination of the transverse OAM and the localized waves enables it to be immune to both dispersion and diffraction as the wave packet propagates. 3D spatiotemporal localized wave packets carrying transverse OAM provide a new opportunity for the utilization of transverse OAM and are expected to be applied in optical communication, quantum optics, and other fields in the future.

In previous studies, the vortex phase is placed in the spatial x-y plane and the resulting localized wave packet carries longitudinal OAM. In this study, we rotate the polar axis by 90°, so that it is now aligned in the y-direction. Therefore, the vortex phase term eim? locates in the x-t plane. Two spatiotemporal localized wave packets carrying two types of OAM: longitudinal OAM and transverse OAM are plotted (Fig. 1). Then, the theoretical derivation [Eqs. (4)-(6)] proves that the transverse OAM possessed by each photon is m?. In Fig. 2, 3D spatiotemporal localized wave packets described by Eq. (7) with different orders are presented. From the basic-order to higher-order 3D spatiotemporal localized wave packets with transverse OAM, a kind of 3D spatiotemporal localized wave packets in abnormal medium is proposed.

To investigate the localized property, we choose one of the family of 3D localized wave packets and simulate its propagation in a virtual medium BK7 with negative material dispersion (β2=-25.26 fs² mm^-1) at the central wavelength of 1550 nm. As a comparison, we filter out the central lobe of the wave packet and propagate it in the same medium. Due to the condition that the effects of diffraction and dispersion are equalized, a proper pulse duration and beam size of the filtered wave packet is 112.25 fs and 0.30 mm at L=0 mm, respectively. Hence, we have diffractive length and dispersive length around Ldiff=Ldis=180 mm. As shown in Figs. 3 and 4, the spatiotemporal localized wave packet keeps its intensity shape without any distorts during propagation. It is noted that the central lobe wave packet experiences dramatic change and is magnified proportionally in intensity profile compared with the spatiotemporal localized wave packet. The propagation invariability of spatiotemporal localized wave packets has been presented. The ability of the wave packet to propagate free of diffraction/dispersion is only valid when the diffraction effect and the dispersion effect are balanced with each other. In other words, the wave packets propagate unstably in the unbalanced diffraction and dispersion. In addition, the localized capacity cannot be continued permanently due to finite energy in practice. However, the limited invariantly propagated length is longer than the length of the filtered wave packets. On the other side, self-healing is also often used to characterize non-spreading wave packets, leading to a wavefront reconstruction after an electromagnetic absorption obstacle. To verify the self-healing of the spatiotemporal localized waves, we numerically simulate that a rectangular plate (around widths of 600 μm) perfectly absorbing electromagnetic fields is placed in the central part of the spatiotemporal localized wave packet and propagate the blocked wave in BK7 (β2=-25.26 fs² mm^-1). 3D iso-intensity profile of the blocked wave packet in anomalous medium at different propagated lengths (0, 230, 320, and 500 mm) is shown in Fig. 5. We can see that the up blocked area and the down area are split into two rings and move towards to the central part in Fig. 5(d). In the end, the wave packets can be recovered to their original spatiotemporal localized wave packets. The linear momentum density and intensity distribution of the blocked wave packet at different propagated distances in y-t plane are shown in Figs. 5(e)-5(h). The direction of linear momentum density is labelled by arrows and points to the blocked areas visually indicating the reason why self-healing can happen in the spatiotemporal localized wave packets.

In summary, we present a new class of 3D spatiotemporal localized wave packets carrying transverse optical OAM. These wave packets exist in abnormal dispersion and can propagate invariantly when the diffractive effect and the dispersive effect are equal. To investigate the non-spreading nature of these wave packets, we simulate a wave packet (l,m)=(2,1) propagating in a proper and real medium BK7 glass. The results show that the wave packet propagates over several Rayleigh lengths while keeping its structure invariant. The wave packet can be recovered to its origin even when passing through a blocked obstacle. This kind of wave packets may provide new applications related to transverse OAM in the fields such as quantum optics and optical communications.