Vortices are prevalent in nature in various forms, ranging from quantum vortices to atmospheric vortices, and their existence is not constrained by spatial and temporal scales. In optics, metamaterial and vortex beams with longitudinal orbital angular momentum (OAM) aligned with the wave vector have been extensively studied and applied [1–15]. This optical vortex manifests as a helical phase wavefront featuring a phase singularity at its center and zero intensity. In contrast to longitudinal OAM vortex beams, spatiotemporal optical vortices (STOVs) possess a transverse OAM characteristic that is perpendicular to the wave vector’s direction. This novel class of optical vortex exhibits unique properties across numerous optical phenomena, thereby garnering significant attention [16–24]. In recent years, researchers have achieved remarkable advancements in STOV generation through the utilization of spatial light modulators, subwavelength nanogratings, and photonic crystal plates [25–33]. These developments offer a variety of promising strategies for generating spatiotemporal vortices; however, existing methods either depend on complex time-frequency pulse-shaping devices or face limitations due to challenges in nanostructure fabrication. Consequently, the generation of STOVs within a compact and easily manufacturable platform presents a significant challenge. Specifically, balancing the controllability of the spatiotemporal vortex generating device with the need to maximize the versatility of STOV generation and active tuning characteristics remains a critical issue.

In this paper, we present a tunable transreflective dual-channel and multipolarization terahertz spatiotemporal optical vortices (T-STOV) generation scheme. The reflection and transmission channels can be interconverted by adjusting the temperature of vanadium dioxide (${\mathrm{VO}}_2$). In the reflection channel, we employ a circularly polarized beam incident structure to observe, for the first time, the existence of topological dark lines in parallel tilt. We document the annihilation process of four vortex singularities into two within the wave vector space plane at a single frequency. This observation indicates that our proposed structure can form either a single singularity or two topological singularities with opposite charges in different observation directions, thereby breaking the constraint that topological dark lines are limited to the vertical plane. Consequently, the degree of freedom and flexibility in generating spatiotemporal vortices are significantly enhanced. Moreover, by leveraging the tunable properties of the ${\mathrm{VO}}_2$ phase transition, we successfully observe and plot the trend and direction of topological dark line movement as a function of temperature, further expanding the tunability of our spatiotemporal vortex generation devices. In the transmission channel, we observe independent and topologically robust vortex singularities in the frequency-momentum plane using linearly polarized beams. The phenomenon of singularity migration is analyzed in detail. Through this analysis, we explore a novel direction for achieving tunable frequency and wave vector properties of spatiotemporal vortices. The active tuning scheme for spatiotemporal vortex generation proposed herein greatly enhances the degree of freedom and flexibility required for such generators, broadens the adaptability and application range of the generation device, and provides a novel approach for designing actively regulated spatiotemporal vortex generators.

The photonic crystal unit structure we designed is depicted in Fig. 1(a). The top layer consists of photonic crystals made of ${\mathrm{Si}}_3{\mathrm{N}}_4$, with a thickness denoted as $t_1$, featuring a transverse $\mathrm{U}$-shaped hole at its center. The periodicity of the structure extends in the $x$ and $y$ directions, with a period of $a$. The middle layer is composed of ${\mathrm{SiO}}_2$, with a thickness of $t_2$, while the bottom layer is made of ${\mathrm{VO}}_2$, as illustrated in Fig. 1(d). The angle of inclination for the hole in the photonic crystal plate is denoted as $\theta$.

To fully satisfy our design objectives, we conducted comprehensive screening and optimization of the structural parameters of the metasurface units employed. First, we started with a square structure, and in order to avoid having in-plane $C_2$ symmetry, we created a groove on one side of the square structure. The groove structure has a width of $w$ and a length of $b/2$. We explored the effects of different values for the groove width and length on the reflection topological dark line and the transmission topological singularity. After investigation, we optimized the parameters to be $a=210\mathrm{\mu m}$, $b_1=130\mathrm{\mu m}$, $b_2=150\mathrm{\mu m}$, $w=60\mathrm{\mu m}$, $t_1=52.5\mathrm{\mu m}$, and $t_2=37.5\mathrm{\mu m}$. As for the tilt angle $\theta$, we treated it as a variable and conducted a detailed exploration. We observed its effect on the properties of the topological dark line in the reflection channel and its influence on the frequency and wave vector properties of the topological singularity in the transmission channel.

${\mathrm{VO}}_2$ is a unique phase change material; when it transitions from an insulating state to a metallic state, its conductivity increases dramatically from $\sigma =20\mathrm{S}/\mathrm{m}$ to $\sigma =\mathrm{200,000}\mathrm{S}/\mathrm{m}$. This conversion can be achieved by varying the temperature, allowing ${\mathrm{VO}}_2$ to revert from the metallic state back to the insulating state. In the terahertz frequency range, the relative dielectric constant of ${\mathrm{VO}}_2$ can be characterized using the Drude–Lorentz model [34]: $${\epsilon }_{{\mathrm{VO}}_2}(\omega )={\epsilon }_{\infty }-\frac{{\omega }_p^2({\sigma }_{{\mathrm{VO}}_2})}{{\omega }^2+i\gamma \omega },$$(1)where ${\epsilon }_{\infty }=12$ is the relative dielectric constant at the limit frequency, $\gamma =5.75\times {10}^{13}\mathrm{rad}/\mathrm{s}$ is the damped frequency, and the plasma frequency can be approximately described by Eq. (2) [35], where ${\sigma }_0=3\times {10}^5\mathrm{S}/\mathrm{m}$ and ${\omega }_p({\sigma }_0)=1.4\times {10}^{15}\mathrm{rad}/\mathrm{s}$: $${\omega }_p^2({\sigma }_{{\mathrm{VO}}_2})=\frac{\sigma }{{\sigma }_0}{\omega }_p^2({\sigma }_0).$$(2)

We employ temporal coupled-mode theory (TCMT) to analyze the T-STOV phenomena of the system in transmission and reflection modes. First, when the temperature of ${\mathrm{VO}}_2$ exceeds 68°C, vanadium dioxide enters a fully metallic state, and the system operates in reflective mode, as illustrated in Fig. 2(a). Our proposed structure exhibits longitudinal mirror symmetry; however, it lacks in-plane $C_2$ rotational symmetry. This characteristic enables the conversion of incident circularly polarized light into its orthogonal counterpart, leading to the formation of topological dark singularities. We have provided relevant explanations in Appendix A.

At the resonant frequency, the additional phase shift of $\pi$ in the resonant reflection within the resonant scattering channel causes the photonic crystal plate to function as a half-wave plate. This configuration converts incident left circularly polarized light into right circularly polarized light while simultaneously converting incident right circularly polarized light into left circularly polarized light. This process corresponds to the transition of one pole to the other on the Poincaré sphere, as illustrated in Fig. 2(c).

As the temperature increases, ${\mathrm{VO}}_2$ undergoes a phase transition from an insulator to a metal, resulting in a significant increase in its conductivity. This phase transition does not occur abruptly at a single temperature point; rather, it progresses gradually within a specific temperature range. The characteristics of this diffuse phase transition enable ${\mathrm{VO}}_2$ and its applications to exhibit smoother temperature control and adjustment capabilities in practical use [36]. This phenomenon provides substantial support for the active tuning of the properties of space-time vortices. The relationship between temperature and conductivity is detailed in Appendix B. Since the conductivity, dielectric constant, and loss of ${\mathrm{VO}}_2$ vary with temperature, the time-reversal symmetry condition and the resonant frequency of the system are altered as the state of ${\mathrm{VO}}_2$ changes.

We selected states of ${\mathrm{VO}}_2$ corresponding to conductivities of 20,000, 30,000, 50,000, 70,000, 100,000, and 200,000 S/m and plotted the 2D energy band distributions of the photonic crystal system for these different states, as illustrated in Fig. 2(d). The figure demonstrates that the conductivity of ${\mathrm{VO}}_2$ increases with rising temperature, leading to an observed increase in the resonance frequency. Notably, when the state of ${\mathrm{VO}}_2$ approaches a complete transition to the metallic state, the 2D energy bands nearly coincide.

When ${\mathrm{VO}}_2$ is at room temperature, the system operates in transmission mode, allowing it to adapt to the incidence of a linearly polarized beam and generate isolated singularities in the frequency-momentum plane. This process results in the formation of spatiotemporal vortices, as illustrated in Fig. 3(a). We also employed the double-resonance four-port time-coupled mode theory (TCMT) model, as illustrated in Fig. 3(b). The relevant aspects of the time-coupled mode theory concerning the system’s transmission modes and the conditions for singularity generation are further elaborated in Appendix A.

In the fully metallic state of ${\mathrm{VO}}_2$, characterized by a conductivity of 200,000 S/m, a left circularly polarized light is incident on a photonic crystal structure with a frequency of 1.108 THz. The copolarized reflectivity and copolarized reflection phase are plotted in the wave vector plane, as illustrated in Fig. 4. We observe the presence of four singularities in the wave vector plane, symmetric about the $k_x=0$ plane, indicated as red labeled dots in the figure. The existence of these singularities markedly contrasts with the conventional single nodal and topological dark lines.

Figures 4(a) and 4(c) present the copolarized reflection phase and copolarized reflectivity maps in the vector space at an inclination angle of $\theta =8°$ for the photonic crystal dug hole, where four vortex singularities with two opposite topological charges are observable. These vortex singularities represent the intersections of two topological dark lines with the wave vector plane in 3D frequency-momentum space. Figures 4(b) and 4(d) illustrate the copolarized reflection phase and reflectivity maps in the vector space for $\theta =15°$. It is evident that variations in the inclination angle $\theta$ result in differing positions of the four singularities within the wave vector plane. This phenomenon corresponds to the increased curvature of the dark line with the inclination angle, which arises from the $C_2$ symmetry of the structure and the degree of broken $z$-mirror symmetry that intensifies with an increase in $\theta$.

The inclination angle $\theta$ significantly influences the copolarized reflectivity and conversion efficiency of the structure; thus, we selected a moderate angle of $\theta =15°$. The bottom-left inset of Fig. 5 illustrates the topological dark lines present in the 3D frequency-momentum space of the system at 1.1045 to 1.108 THz. We generated copolarized reflectivity and reflection phase maps in the wave vector plane at various frequencies, connecting the vortex singularities in the wave vector plane with black curves. This resulted in two tilted curves in the 3D frequency-momentum space that are symmetric about the $k_x=0$ plane. In the wave vector space across different frequencies, we identified four independent singularities with two opposite topological charges. By fixing the value of $k_{x}a/(2\pi )$ at $-0.028$ and plotting the copolarized reflection phase cuts in 3D frequency-momentum space, we observed an isolated vortex singularity. Additionally, when we fixed the values of $k_{y}a/(2\pi )$ at 0.02 and $-0.02$, we detected a pair of vortex singularities with opposite topological charges, respectively. This observation is attributed to the fact that each topological dark line does not align routinely parallel to either the frequency-transverse wave vector plane or the frequency-longitudinal wave vector plane but rather exists in a tilt-symmetric manner. Notably, these two pairs of topological singularities are distinctly positioned in the figure, and their locations in the frequency-transverse wave vector plane do not coincide. This phenomenon aligns with the position and state of the topological dark lines depicted in the lower-left inset.

The illustrations in the lower-left corner of Fig. 5 for frequencies above 1.1045 THz all contain points with zero copolarization reflectivity. Near 1.1045 THz, dark lines with opposite topological charges converge and subsequently annihilate. Additionally, Fig. 5 reveals two distinct directions of copolarized reflection phases, illustrated in the frequency-transverse wave vector tangent and frequency-longitudinal wave vector tangent of copolarization. These different orientations of the reflection phase cuts demonstrate variations in the number and position of frequency-momentum vortex singularities, indicative of changes in the number of vortex singularities within the space-time domain when observed from different directions. This observation first confirms the existence of two tilted, juxtaposed symmetric topological dark lines in the 3D frequency-momentum space for our proposed photonic crystal structure. Further, it supports the realization of differential changes in the number and nature of spatiotemporal vortex singularities across various directions in the spatiotemporal domain.

${\mathrm{VO}}_2$ functions as a temperature-sensitive material, with its complex dielectric constant and conductivity exhibiting temperature-dependent variations. To qualitatively assess the movement direction and the trend of change of topological dark lines as the temperature is decreased, we systematically altered the state of ${\mathrm{VO}}_2$, facilitating the transition of ${\mathrm{VO}}_2$ from a fully metallic state. Figures 6(a)–6(h) present the reflectance phase and reflectivity plots in a 3D frequency-wave vector spatial section at $k_{x}a/(2\pi )=-0.02$. As the conductivity of ${\mathrm{VO}}_2$ is sequentially varied to 200,000, 150,000, 100,000, and 70,000 S/m, the movement direction and trend of the vortex singularity in the phase cross-section are as illustrated in Figs. 6(a), 6(b), 6(e), and 6(f), while Figs. 6(c), 6(d), 6(g), and 6(h) depict the corresponding reflectivities. Notably, the range of singularity dispersion increases slightly as conductivity decreases. Our findings indicate that, when a reduction in temperature leads to a significant decrease in the conductivity of ${\mathrm{VO}}_2$, while still remaining above or equal to 70,000 S/m, the ${\mathrm{VO}}_2$ gradually transitions away from the fully metallic state. At this juncture, the vortex singularity in the cross-section progressively shifts upward along the path of maximum phase change rather than moving vertically.

However, this does not provide a clear indication of the movement direction of the two topological dark lines. To address this, we plotted the reflection phase diagrams in the wave vector plane at 1.105 THz, as illustrated in Figs. 6(i) and 6(j). In these figures, the red markers indicate the intersections of the topological dark lines with the wave vector plane. Figure 6(i) depicts the frequency of the incident circularly polarized light on ${\mathrm{VO}}_2$ at a conductivity of 200,000 S/m, with the copolarized reflection phase diagram fixed at 1.105 THz. At this frequency, both topological dark lines approach the annihilation frequency, resulting in the red dots in the figure being close to the plane defined by $k_{y}a/(2\pi )=0$. In contrast, Fig. 6(j) presents the reflection phase diagram at the same frequency but with the conductivity of ${\mathrm{VO}}_2$ set to 100,000 S/m. Although the four vortex singularities exhibit a similar outward shift, this shift is attributed to changes in the conductivity and complex permittivity of ${\mathrm{VO}}_2$, which fundamentally differs from the outward shift of the singularities caused by the increased curvature due to changes in the aperture angle $\theta$ of the photonic crystals, as shown in Fig. 4, as well as the frequency-induced shift observed in the inset of Fig. 5. This phenomenon is linked to the downward shift of the topological dark line, consistent with the trend observed in Fig. 2(d), which shows that the structure-guided resonance frequency decreases with decreasing conductivity. The trend of phase singularity position shifts in response to decreasing ${\mathrm{VO}}_2$ conductivity is also evident from the positions of the dark points in Figs. 6(c), 6(d), 6(g), and 6(h). Finally, Figs. 6(k) and 6(l) display the copolarized reflection amplitudes corresponding to Figs. 6(i) and 6(j), clearly illustrating the four singularities where the two topological dark lines intersect the wave vector plane.

The observed phenomenon indicates that the topological dark line shifts in response to changes in the state of ${\mathrm{VO}}_2$. We posit that this shift is primarily attributed to the alteration in the substrate loss of ${\mathrm{VO}}_2$. Consequently, we conclude that, as the external temperature decreases and ${\mathrm{VO}}_2$ transitions gradually from a fully metallic state, the motion of the two topological dark lines in 3D frequency-momentum space trends downward and inward, as illustrated in Fig. 7.

As the temperature continues to decrease, ${\mathrm{VO}}_2$ gradually transitions from a fully metallic state to an insulating state, at which point the conductivity decreases to 20 S/m, resulting in a shift in the system’s mode of operation to a transmission mode. Building on our previous discussion, we utilized linearly polarized light incident on the photonic crystal along the $y$ direction and plotted the copolarized transmission phase in the frequency-wave vector plane. In the frequency-transverse wave vector plane with $k_y=0$, we observe isolated vortex singularities, as illustrated in Figs. 8(a) and 8(e).

To better reflect the real situation, we take into account the nonradiative losses of the material in our design. Simulations indicate that the singularity remains present. We further examine the evolution of this singularity in the frequency-transverse wave vector space for various inclination angles. In this analysis, we assume that the nonradiative loss remains constant with respect to $\theta$. As $\theta$ increases gradually, the vortex singularity approaches the $\mathrm{\Gamma }$ point; however, the singularity and the vortex phase persist and continue to shift with the inclination angle $\theta$.

As the inclination angle $\theta$ increases, the $C_2$ symmetry and $z$-mirror symmetry of the system are progressively broken. The location where the second condition of Eq. (A13) is satisfied approaches the $\mathrm{\Gamma }$ point. At other locations, while the first and third conditions of Eq. (A13) are met, the second condition remains unsatisfied. With an increasing inclination angle $\theta$, the singularity appears at a specific position in the frequency-wave vector plane, as illustrated in Fig. 8(i). Notably, even with a significant increase in the inclination angle $\theta$, the singularity position does not fully coincide with the $\mathrm{\Gamma }$ point. This deviation is attributed to nonradiative losses induced by the photonic crystal structure and the ${\mathrm{VO}}_2$ substrate within the system, which causes the topological singularity to shift in frequency-momentum space. This results in a change in the time-reversal symmetry condition of the system, causing topological singularities to deviate in frequency-momentum space. We verify this in Appendix C, which opens up new ideas and directions for the tunable properties of spatiotemporal vortices. Nevertheless, the singularity and the phase helix remain unaffected, further demonstrating that the singularity at this point is topologically robust within the 2D parameter space, as anticipated.

In this study, we investigate the generation and tunability of vortex singularities in the frequency-wave vector plane within the reflection and transmission channels, utilizing the phase-change tunable properties of ${\mathrm{VO}}_2$. Our proposed structure employs a circularly polarized beam to irradiate the photonic crystal structure based on topological dark point theory. In the 3D frequency-wave vector space of the reflection channel, we observe a pair of topological dark lines composed of topological dark points. Two phase vortices with opposite topological charges ($\pm 1$) are identified in the frequency-transverse wave vector section, while a single independent phase vortex is noted in the frequency-longitudinal wave vector section. By tuning the conductivity and complex permittivity of ${\mathrm{VO}}_2$ through temperature adjustments, we successfully observe the moving trajectories of the topological dark lines and achieve tuning of the center frequency of the spatiotemporal vortex. When ${\mathrm{VO}}_2$ transitions to the transmissive mode due to temperature reduction, we generate independent phase vortices in the frequency-wave vector plane using line-polarized light incident on the photonic crystal structure. The position of the singularity approaches the $\mathrm{\Gamma }$ point as the inclination angle $\theta$ increases, and the singularity demonstrates topological robustness within the parameter space, allowing for conversion to STOV via Fourier optics. The photonic crystal structure presented in this study exhibits exceptional performance in generating spatiotemporal vortices and tuning functions, significantly enhancing the tunability and degrees of freedom in the generation of spatiotemporal vortices using a photonic crystal system. This advancement offers a novel pathway for the design of actively tunable spatiotemporal vortex generation devices.