Wave packets carrying transverse orbital angular momentum are called spatiotemporal vortex pulses (STVPs), which exhibit phase singularities in the coupled space-time dimension and have become a research focus in recent years [1,2]. After about five years of development, people’s interest has shifted from generating [3–5], measuring, and characterizing the STVP [6] to regulating, utilizing spatiotemporal vortices, and exploring more novel structured light fields in higher dimensions, for example, generating vector space-time vortex wave packets [7,8] and space-time skyrmions [9,10], and using the STVP to generate second harmonics [11,12], manipulate particles in the direction of parallel light propagation [13], and accelerate electrons [14].

The fractional Schrödinger equation (FSE) was proposed by Laskin in 2000 when studying Feynman path integrals by replacing Brownian motion with the more general Lévy flight paths [15,16]. The mathematical uniqueness of the FSE is that the Riesz space-fractional derivative replaces the conventional Laplacian [15–17]. Previous theoretical work mainly focused on the propagation dynamics of special beams [18–21], nonlinear soliton solutions [22–29], and the energy bands of PT-symmetric potential [26,30]. In terms of experiments, in 2015, based on optical microcavities, Longhi proposed an experimental plan to realize the spatial FSE and demonstrated an eigensolution (dual-Airy state) of the one-dimensional spatial FSE with a harmonic potential [31]; in 2023, based on frequency control, Liu et al. used a hologram to act as the optical Lévy et al. waveguide with fractional group-velocity dispersion and observed the diverse propagation dynamics of pulse wave packets in one-dimensional temporal FSE [32]. However, little attention has been paid to the propagation of STVPs in the FSE. Since the fractional Laplacian differential is a non-local operator, it has always been a complex problem to solve the propagation of wave packets in fractional systems analytically [18,33–36].

In this paper, we investigated the propagation dynamics of Bessel-type spatiotemporal vortex pulses (BSTVPs) in the spatial fractional wave equation. We analyzed the effects of pulse linewidth, linear chirp modulation, and vortex topological charge on beam propagation. We revealed the phenomenon of orbital angular momentum backflow resulting from fractional differentiation. Our research results can be readily generalized to other types of space-time vortex pulses (such as Laguerre–Gaussian-type space-time vortex pulse).

The wave equation for light in vacuum can be written as $$\frac{{\partial }^{2}E}{\partial x^2}+\frac{{\partial }^{2}E}{\partial y^2}+\frac{{\partial }^{2}E}{\partial z^2}=\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}.$$(1)

Assuming that the second derivative of the electric field in the $y$ dimension is much smaller than that in the $x$ and $z$ dimensions, the corresponding $x$ dimensional spatial fractional wave equation (FWE) can be written as $$D_{\alpha }\frac{{\partial }^{\alpha }E}{\partial x^{\alpha }}+\frac{{\partial }^{2}E}{\partial z^2}=\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}.$$(2)

Here $\alpha$ is the Lévy index ($1\le \alpha \le 2$), and $D_{\alpha }$ is the quantum diffusion constant with dimension $[D_{\alpha }]={\mathrm{erg}}^{1-\alpha }{\mathrm{cm}}^{\alpha }{\sec }^{-\alpha }$ [16,19]. Corresponding to the optics, its dimension is $[D_{\alpha }]={\mathrm{cm}}^{\alpha -2}$ [31]. Under the harmonic approximation $E(x,z,t)=E(x,z)\exp (-i\omega t)$, there is $$D_{\alpha }\frac{{\partial }^{\alpha }E}{\partial x^{\alpha }}+\frac{{\partial }^{2}E}{\partial z^2}=-\frac{{\omega }^2}{c^2}E.$$(3)

Here $\omega$ is the angular frequency, and using the definition of the Riesz derivative [19], the propagation dynamics of a beam can be expressed more simply in the momentum space: $$\begin{gathered} {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }(D_{\alpha }{|k_x|}^{\alpha }+k_z^2-\frac{{\omega }^2}{c^2})\times S(k_x,k_z) \\ \times \exp (i{k}_{x}x+i{k}_{z}z)\mathrm{d}{k}_x\mathrm{d}{k}_z=0. \end{gathered}$$(4)

From Eq. (4), the dispersion relationship in one-dimensional FWE can be obtained as $$(D_{\alpha }{|k_x|}^{\alpha }+k_z^2)c^2={\omega }^2.$$(5)

From Eq. (5), it is evident that the dispersion relationship in the fractional system differs from that in the integer case ($\alpha =2$) [7], where rotational symmetry is lost in the $k_x-k_z$ plane. This article focuses on the limiting case $\alpha =1$, widely regarded as the most interesting [18,26,30,31]. The corresponding dispersion relationship is illustrated in Fig. 1(a).

The plane wave spectrum of a scalar ($y$-direction polarized) ideal Bessel-type spatiotemporal vortex pulse (BSTVP) is [6,37] $$S(k_x,k_z)=\delta [\sqrt{{(k_z-k_0)}^2+k_x^2}-\mathrm{\Delta }k]\exp (il\varphi ).$$(6)

Here $\mathrm{\Delta }k$ represents spatial bandwidth and determines the degree of paraxiality and monochromaticity of the BSTVP, $l$ is the topological charge, $\varphi$ is the azimuth angle, and the schematic diagram of the corresponding plane wave spectrum of the BSTVP is shown in Fig. 1(b). The energy of an ideal BSTVP is infinite, so it is impossible to be realized in physics [37,38]. In previous experiments, the achieved Bessel beams were all truncated. So, when studying the BSTVP’s transmission characteristics in fractional systems, we will truncate it by taking the number of rings as 15. The studied BSTVP can be expressed as follows [6]: $$\begin{gathered} E(x,z,t=0)=J_l(\mathrm{\Delta }k·r)\exp [i(k_{0}z+l\phi )]P(r), \\ P=\{\begin{array}{cc}1& r\le r_{15}^l\\ 0& r>r_{15}^l\end{array}. \end{gathered}$$(7)

The propagation of the spatiotemporal vortex beam can be numerically simulated using the following angular spectrum method: $$E(x,z,t)=F*[F[E(x,z,t=0)]\exp (-i\omega t)].$$(8)

Among them, $F$ represents the Fourier transform, and $F*$ represents the inverse Fourier transform. By using Eqs. (5) and (8), the transmission of any spatiotemporal vortex pulse in the FWE can be quickly calculated. According to the dispersion relationship in the fractional system and the plane wave spectrum of the BSTVP, a propagation constant $t_R=k_0/(c·\mathrm{\Delta }{k}^{\alpha })$ and two dimensionless variables $z_c=(z-ct)\mathrm{\Delta }k$ and $x_c=x·\mathrm{\Delta }k$ can be defined. The following simulations have been conducted in units of $t_R$, $z_c$, $x_c$.

Assuming the central wavelength of the BSTVP is ${\lambda }_0=1064\mathrm{nm}$, using Eqs. (5) and (8), the BSTVP’s transmission characteristics in narrowband conditions ($\mathrm{\Delta }k\ll k_0$) are first studied. When BSTVP carries transverse orbital angular momentum (vortex phase), its electric field complex amplitude no longer has spatial rotational symmetry in the $x–z$ plane [1,2], so it is not like a Gaussian beam symmetrically split into two identical beams during propagation in the fractional system [18,20]. Instead, due to breaking the symmetry, it will gradually split into two identical but opposite-direction opening C-shaped beams (see Fig. 2), and the larger $l$, the larger the opening. In addition, the intensity distribution of the C-shaped beam remains almost unchanged with the increase of propagation distance, except for the linear position shift. Therefore, it can be considered that in the narrowband case, BSTVPs have similar diffraction-free characteristics to monochromatic waves in fractional system transmission.

Figures 3(a) and 3(b) depict the intensity phase diagrams of the BSTVP without and with transverse chirp phase modulation $\exp (ivx)$ for $l=3$, respectively. The center wavenumber of the BSTVP is $k_0$, which carries a linear chirp phase in the $z$-direction [$\exp (i{k}_{0}z)$, see Eq. (7)]. Therefore, the BSTVP without $\exp (ivx)$ exhibits a linear phase distribution along the $z$-axis. When the BSTVP is modulated by $\exp (ivx)$, it displays a linear phase distribution along an inclined direction, where the inclined angle is $\arctan (v/k_0)$.

Applying linear chirp modulation [$\exp (ivx)$] to the incident monochromatic beam shifts its spectrum to one side of $k_x$, thereby eliminating beam splitting in the $x$ dimension and enabling propagation in a single direction without diffraction [18,20]. Similarly, when linear chirp modulation [$\exp (ivx)$] is applied to the BSTVP, its spectrum shifts to one side of $k_x$ [see Fig. 3(c)], leading to similar non-diffraction effects. As depicted in Fig. 4, after applying linear chirp modulation [$\exp (2i·\mathrm{\Delta }k·x)$] to the narrowband BSTVP, the beam propagates entirely along one direction in the $x–z$ plane without diffraction.

The increase in pulse linewidth in momentum space results in increased time diffraction in real space, which primarily degrades the spatiotemporal vortex pulses (STVPs) [6,37]. When the narrowband condition is not met, pulse transmission in fractional systems also suffers significant time diffraction compared to integer systems. Figure 5 illustrates the transmission of BSTVPs in the fractional wave equation (FWE) under broadband conditions ($\mathrm{\Delta }k\propto k_0$). It is observed that BSTVPs with low-order vortex topological charges split into two nearly symmetric pulses along the $z$-axis, losing their characteristic C-shaped openings [see Figs. 5(a1)–5(a4), 5(b1)–5(b4)]. Conversely, BSTVPs with high-order vortex topology maintain C-shaped openings in the split beams, but the opening asymmetry increases with propagation distance [see Figs. 5(c1)–5(c4), 5(d1)–5(d4)]. Comparing the propagation of the BSTVP with the high-order and low-order vortex, it is evident that increasing the vortex topological charge has a certain inhibitory effect on time diffraction. After applying linear modulation, a more intuitive observation of the suppression of diffraction can be obtained by comparing the transmission of BSTVPs with low-order and high-order topologies (see Fig. 6).

The analysis of the orbital angular momentum of polychromatic waves is a challenging problem, as most theoretical methods are developed for monochromatic waves [6]. In addition, under broadband conditions, the vortex topological charge of the STVP is unstable. It quickly degrades into $l$ first-order vortices [37], making it difficult to define the orbital angular momentum of the pulse beam. Fortunately, under narrowband conditions, the intrinsic angular momentum of a pulse beam can be uniformly defined.

The intrinsic orbital angular momentum density of spatiotemporal vortex pulse beams is defined as follows [1,6,39]: $$L_y^{\mathrm{in}}={\omega }_0^{-1}[\mathrm{Im}(E^{*}·\frac{\partial }{\partial \phi }E)+k_0·x·I].$$(9)

Here $\phi =\arg [x+i(z-ct)]$ is the azimuth angle. So the number of intrinsic orbital angular momenta contained in a single photon is $$N_L^y=\frac{L_y^{\mathrm{in}}}{n\hslash }=\frac{\partial }{\partial \phi }\arg [E·\exp (-i{k}_{0}z)].$$(10)

Under narrowband conditions ($\mathrm{\Delta }k\ll k_0$), an approximation can be done: $\frac{\omega }{c}=\sqrt{D_{\alpha }{|k_x|}^{\alpha }+k_z^2}\approx k_z+\frac{D_{\alpha }{|k_x|}^{\alpha }}{2k_0}$. Using this approximation a corresponding analytical solution for the propagation of ideal BSTVPs in the FWE can be provided as follows: $$\begin{gathered} E(x,z,t)=F^{*}[S(k_x,k_z)\exp (-i\omega t)] \\ =E_{+}(x-\frac{ct{D}_{\alpha }}{2k_0},z-ct)+E_{-}(x+\frac{ct{D}_{\alpha }}{2k_0},z-ct), \end{gathered}$$(11)$$\begin{gathered} E_{+}(x-\frac{ct{D}_{\alpha }}{2k_0},z-ct)={\int }_0^{\pi }\exp [i(k_{0}z+\mathrm{\Delta }k\cos \varphi z+\mathrm{\Delta }k\sin \varphi x+l\varphi )]\mathrm{d}\varphi , \\ {E}_{-}(x+\frac{ct{D}_{\alpha }}{2k_0},z-ct)={\int }_{\pi }^{2\pi }\exp [i(k_{0}z+\mathrm{\Delta }k\cos \varphi z+\mathrm{\Delta }k\sin \varphi x+l\varphi )]\mathrm{d}\varphi . \end{gathered}$$(12)

According to Eqs. (11) and (12), in the narrowband case, the propagation of the BSTVP in a fractional system is equivalent to the mapping of its half-plane wave spectrum to the entire real space, and as the propagation distance increases, the two split pulses only undergo a simple linear shift, which is consistent with the numerical simulation in Fig. 2. In Fig. 7, we demonstrate this mapping. From the perspective of half-edge mapping, the propagation of the BSTVP in the FWE can be seen as the translation of two half-Bessel spatiotemporal vortex pulses (half-BSTVPs). Subsequently, using Eqs. (10) and (12), we studied the $N_L^y$ of a half-BSTVP.

As shown in Figs. 8(a3), 8(b3), 8(c3), and 8(d3), the photon’s $N_L^y$ is not spatially uniform. It exhibits oscillatory behavior with the changes in the azimuth angle ($\phi =\arg [(x-\frac{ct{D}_{\alpha }}{2k_0})+i(z-ct)]$), and the oscillation frequency increases with the increase of $l$. Furthermore, it should be noted that the value of $N_L^y$ at the C-junction of the half-BSTVP is the smallest and can be negative [see Figs. 8(a3) and 8(b3)]. This negative $N_L^y$ reveals a phenomenon of photon backflow. The concept of photon backflow is derived from quantum backflow, and known optical methods include vortex superposition [40], superoscillation [41,42], and suboscillation [43]. The difference from the previous method is that our approach to achieving optical backflow is to change the spatial dispersion relationship of the propagation system. In addition, we also numerically calculated the average $N_L^y$ of the half-BSTVP and found that its value is equal to the vortex topological charge (${\overline{N}}_L^y=l$). So when the BSTVPs propagate in the FWE, their angular momentum is conserved.

In summary, in this paper, we studied the propagation dynamics of the BSTVP in the FWE and analyzed the effects of pulse bandwidth, vortex topological charge ($l$), and linear chirp modulation on propagation. The increase of $l$ in broadband conditions has a certain suppression effect on the degradation caused by time diffraction. In narrowband cases, the propagation of the BSTVP in the FWE is approximately equal to the coherent superposition of two half-BSTVPs. For half-BSTVP, the distribution of $N_L^y$ is extremely uneven and may be accompanied by orbital angular momentum backflow. Our research results can be straightforwardly generalized to other types of space-time vortices.