Photons can carry spin angular momentum (SAM) related to their polarization state and orbital angular momentum (OAM) related to their spiral phase. In recent years, spatiotemporal optical vortices (STOVs) that carry transverse OAM have gained rapidly growing interest in the field of optics due to their capacity to introduce new degrees of freedom for regulating the optical field. Current research is primarily concentrated on low-order STOVs, with limited attention given to the generation and propagation characteristics of high-order STOVs. We mainly investigate the generation of high-order strongly focused STOVs based on incident wavepacket preconditioning using mode conversion theory. Additionally, we also study the propagation characteristics of highly localized STOVs both before and after the focal plane.

Research has shown that when STOVs are focused by a high numerical aperture (NA) objective lens, they will experience the so-called “spatiotemporal astigmatism” effect similar to the focusing effect of a cylindrical lens on the Laguerre-Gaussian (LG) light field. Thus, STOVs will collapse and lose their spatiotemporal spiral phase on the focal plane of the high NA objective lens. With the understanding of such a spatiotemporal astigmatism, we preconditioned the incident wavepacket by a linear superposition of LG spatiotemporal wavepackets and then employed a high NA objective lens to focus the preconditioned incident wavepacket. The high-order STOVs on the focal plane or propagating away from the focal plane were calculated based on the Richards Wolf diffraction theory. In the simulations, it was assumed that the NA of the objective lens was 0.9, and the waist radius was set to be 0.5. Meanwhile, spatiotemporal coupling was ignored, and thus each temporal slice of the incident wavepacket was focused onto its corresponding temporal slice within the focal volume. The spatial sizes of the incident wavepackets were normalized to the pupil of the objective lens.

Although the third-order STOV exhibits transverse OAM (Fig. 1), its helical phase disappears when strongly focused by a high NA objective lens (Fig. 3). Based on mode conversion theory, we construct a third-order diagonal Hermitian-Gaussian (HG) wavepacket as the incident wavepacket by linearly superimposing third-order LG wavepackets. The preconditioned incident wavepacket is split, but the corresponding tightly focused wavepacket returns to a doughnut shape and regains the spatiotemporal spiral phase (Fig. 4). The phase of the focused wavepacket undergoes three continuous changes from -π to π in the clockwise direction, indicating that the strongly focused wavepacket is a high-order STOV with a purely transverse OAM with a topological charge of +3. Tightly focused STOVs with purely transverse OAM with a higher topological charge can also be generated based on mode conversion theory. For example, the fourth-order preconditioned incident wavepacket is strongly focused on realizing a fourth-order highly confined STOV (Fig. 5). We can find that the intensity distribution of the focused wavepacket exhibits a doughnut shape in the x-t plane. Additionally, the phase distribution changes continuously four times in the clockwise direction from -π to π in the x-tplane. The results once again demonstrate the feasibility of the presented method.

To study the propagation characteristics of highly localized STOVs both before and after the focal plane, we calculate the focused wavepackets at zf=λ and zf=-λ, respectively (Fig. 6). λ is the center wavelength of the wavepacket. From them, we can learn that the propagated focused three-order STOV degrades into three first-order vortices distributed diagonally in the x-tplane, and each vortex carries a transverse OAM of topological charge of +1. However, when the focused wavepacket propagates from zf=-λ to zf=λ, the intensity distribution of the focused wavepacket will rotate 90°. The total topological charge of the focused wavepacket in each case is 3. The results show that the total topological charge remains unchanged. The intensity distribution rotates, and the third-order phase singularity degrades into three first-order phase singularities for the propagated highly confined STOVs.

In the paper, we study the generation of high-order STOVs and their propagation characteristics. Due to the spatiotemporal astigmatism effect of the objective lens, the LG STOV will collapse after being strongly focused by a high NA objective lens, which will lead to the disappearance of the spatiotemporal spiral phase. To solve the problem, we linearly superimpose the LG wavepackets into diagonal HG wavepackets based on the mode conversion theory. Using the preconditioned wavepackets as the incident wavepackets of the focusing system, we successfully generate STOVs with topological charges of +3 and +4 on the focal plane of a high NA objective lens. When highly confined STOV propagates in the vicinity of the focal plane, the topological charge remains unchanged, and the intensity distribution rotates. The high-order spatiotemporal phase singularity will degrade into several first-order spatiotemporal phase singularities, which can be employed to generate spatiotemporal vortex arrays.