Spatiotemporal optical vortices (STOVs) represent a novel class of optical vortices, characterized by transverse orbital angular momentum and distinct phase singularities in the space–time domain.1 These unique wave packets differ significantly from conventional spatial optical vortices and have garnered significant interest due to their analogies with diverse physical systems, such as tropical cyclones,2 plasmonic physics,3 and magnetic nanowires.4 Thus, explorations of STOVs could promote the underlying physics and innovations from optics to other fundamental fields of science. Since their first experimental demonstration in 2016,5 STOVs have become a prominent topic in optics. Recent progress includes advancements in STOV generation techniques,1^,5^–9 studies of their propagation dynamics,1^,10^–13 and exploration of spin-orbital coupling.14^,15 However, due to the inherent asymmetry of spatial diffraction and dispersion during propagation,13^,15 the phase singularities of such beams always exhibit strong spatiotemporal astigmatism compared with conventional spatial-structured light, which greatly hinders the scientific research on light-matter interaction or nonlinear frequency conversion processes involving STOVs.16^–20

One of the most prominent nonlinear effects is the second harmonic generation (SHG), in which a fundamental harmonic of frequency $\omega$ is converted into its second harmonic (SH) at frequency $2\omega$. To date, only two independent research groups have demonstrated the SHG of STOVs, revealing that the space–time topological charge (TC) doubles in the SHG process.16^–18 Notably, the topological properties of the space–time phase swirl may not be fully conserved due to group velocity mismatch (GVM).17 Moreover, as STOVs propagate from the near-field to the far-field, they evolve continuously from a “flying donut” structure to “diagonally separated spatiotemporal lobes.”1 This propagation effect also contributes to the movement of phase singularities in a nonlinear crystal during the SHG process. Consequently, the phase singularity of SH STOVs exhibits distinctive separation and rotation dynamics, which are essential for fostering a more systematic understanding and advancement of the nonlinear conversion of STOVs. However, a comprehensive theoretical description that includes consistent analysis of phase singularity trajectory movement, the influence of dispersion, and propagation property, is highly desired.

To address the above issue, we develop a simple yet powerful theoretical framework to elucidate the effects of spatiotemporal (ST) astigmatism during the SHG, focusing specifically on two dynamical evolutions of the space–time singularities: separation and tilting. Inspired by previous works,16^–18^,21 we demonstrate that the phenomenon of phase singularity separation is primarily governed by GVM in the nonlinear crystal under the nonlinear Maxwell equation. Besides, the separation of phase singularities in STOVs can be accurately predicted under weak ST walk-off conditions via a Simpson-type integral. Furthermore, we show that variations in the spatial phase shift of the fundamental STOVs lead to the directional motion of the SH STOV singularities. In addition, under the combined influence of both GVM and propagation, the phase singularity of the SH STOVs becomes more complex than that of the fundamental STOVs, presenting simultaneous rotation and separation properties. These findings offer deeper insights into the phase singularity evolution in SH STOVs and provide valuable theoretical guidance for applications in structured light manipulation, with implications for microscopy, optical sensing, and nonlinear optics, where precise control over spatiotemporal light characteristics is critical for high-resolution imaging and beam shaping.

Based on the nonlinear conversion law of STOVs, to generate SH STOVs with a doubled topological charge, the fundamental STOVs are initially produced in a “flying donut” shape with a TC of $l=1$,1^,6 which can be expressed as $$u_f(x,\tau )=(\frac{\tau }{{\tau }_s}+i\frac{x}{x_s})u_0(x,\tau ),$$(1)$$u_0(x,\tau )=\exp (-\frac{x^2}{x_s^2}-\frac{{\tau }^2}{{\tau }_s^2}),$$(2)where $u_f(x,\tau )$ is the envelope of the fundamental STOVs, $u_0(x,\tau )$ is the STOV-free envelope assuming a Gaussian shape, and $x_s$ and ${\tau }_s$ are the spatial and temporal half-widths of the fundamental STOVs, respectively. Here, we ignore the influence in the $y$-direction because it does not carry any vortex phase and substitute the local time coordinate $\tau =t-z/c$ for the time domain $t$ of the pulse. As shown in Ref. 17, the splitting of phase singularities is primarily driven by GVM, whereas higher-order dispersions such as group delay dispersion reduce the degree of splitting. The slowly varying amplitude approximation assumes that the envelope of a forward-traveling wave pulse varies slowly in time and space compared with the period or wavelength of the carrier wave, which is widely used in beam propagation and nonlinear processes.22 It should be pointed out that such an approximation is independent of the beam transverse distribution, making it applicable to transverse OAM beams, which has been confirmed in previous studies.21^,23 To exclusively study the phase singularity separation dynamics of STOVs, we consider the nonlinear Maxwell equation governing the SHG process with GVM as the sole factor under the assumption of slowly varying amplitude approximation and small signal transmission, which can be rewritten as $$\frac{\partial u_f}{\partial z}=\frac{\eta }{2}\frac{\partial u_f}{\partial t},$$(3)$$\frac{\partial u_s}{\partial z}=-\frac{\eta }{2}\frac{\partial u_s}{\partial t}-(\varrho +i\mathrm{\Delta }k)u_s-i{\rho }_2{u}_f^2,$$(4)where $u_f$ represents the envelope of the fundamental pulse, $u_s$ represents the envelope of the SH pulse, $\eta$ is the GVM in the nonlinear crystal, $\varrho$ is the linear loss, $\mathrm{\Delta }k$ is the phase mismatch, and ${\rho }_2$ is the nonlinear coupling coefficient. The phase-matching bandwidth during the SHG refers to the range of wavelengths over which the phase-matching condition can be effectively maintained, allowing efficient conversion of the fundamental frequency to the second harmonic frequency.24 Through the Fourier transform relations, constraints in the spectral domain inherently impact the temporal domain. Nevertheless, we observe that the spectral bandwidth of the generated STOV pulses is constrained, primarily due to the bandwidth limitations of the phase mask or spatial light modulator. By carefully selecting an appropriate nonlinear crystal, it is possible to cover the full phase-matching bandwidth of the fundamental waves. Therefore, we assume perfect phase-matching conditions, which can cover the full pulse bandwidth. Besides, ${\rho }_2$ is the nonlinear coupling coefficient, whose magnitude depends on the phase-matching conditions and properties of the nonlinear medium. In terms of the perfect phase-matching conditions and the slowly varying amplitude approximation, ${\rho }_2$ can be treated as a constant value. Moreover, our theoretical framework builds upon the experimental studies presented in Refs. 17 and 18, which demonstrated that the SHG could be observed when fundamental pulses propagate directly through a nonlinear crystal. In these studies, the fundamental pulses passed directly through a nonlinear crystal without any focusing, yet SH STOVs were still obtained. Until now, the studies of SH STOVs are still scarce. Consequently, the diffraction effects in the generation of SH STOVs are typically neglected due to their unbalanced spatial and temporal diffraction. Under these assumptions, the envelope of the SH STOVs can be expressed as $$u_S(x,\tau )={\int }_0^{1}d\zeta u_f^2[x,\tau +\eta L(\zeta -\frac{1}{2}\left)\right]e^{\varrho L\zeta },$$(5)where $L$ is the length of the nonlinear crystal, and $\zeta$ is the longitudinal coordinate along the propagation direction in the nonlinear crystal. Simpson’s rule is a numerical method used to approximate the definite integral of a function. It works by approximating the integrand with a quadratic polynomial that passes through three equally spaced points within the integration interval.25 By employing the Simpson-type integral and assuming $\eta L/{\tau }_s\ll 1$, we can simplify Eq. (5) as $$\begin{gathered} u_{\mathrm{SS.}}(x,\tau )\approx [\frac{\tau }{{\tau }_s}+i(\frac{x}{x_s}-\frac{\alpha }{2\sqrt{3}}\left)\right][\frac{\tau }{{\tau }_s}+i(\frac{x}{x_s}+\frac{\alpha }{2\sqrt{3}}\left)\right]u_0(x,\tau {)}^2 \\ =[(\frac{\tau }{{\tau }_s}+i\frac{x}{x_s}{)}^2+{\left(\frac{\alpha }{2\sqrt{3}}\right)}^2]u_0(x,\tau {)}^2, \end{gathered}$$(6)where $u_{\mathrm{SS.}}$ represents the envelope of the SH STOVs under Simpson-type integral approximation. In Eq. (6), we introduce a normalized length-scale parameter $\alpha =L/L_T$, named as the normalized temporal walk-off (NTWO), where $L_T={\tau }_s/\eta$ is the temporal walk-off length. It is important to note that the NTWO is a parameter that encapsulates both the crystal length and the temporal walk-off length, whereas the pulse parameters, such as wavelength and pulse duration, still play a significant role in phase singularity decomposition. The condition $u_{\mathrm{SS.}}(x,\tau )=0$ in Eq. (6) determines the positions of space–time singularities, which can be expressed as $$(x_0,{\tau }_0)=(\pm \frac{\alpha }{2\sqrt{3}}{x}_s,0).$$(7)

Figure 1(a) shows the calculated distributions of the SH STOVs with $\alpha$ ranging from 0 to 1.5 using Eq. (5). We set the parameters according to Ref. 18 as follows: $x_s=400\mu \mathrm{m}$ and ${\tau }_s=250\mathrm{fs}$. Our theory generating $u_S(x,\tau )$ shows that in the case of extremely small $\alpha$, the SH STOVs with $l=2$ do not break up and the envelope can be considered as the doubling of the fundamental STOVs $u_S(x,\tau )=u_f(x,\tau {)}^2$, indicating that the single-phase singularity rotates anticlockwise with a $4\pi$ azimuthal spiral phase in space–time. The introduction of the NTWO causes the SH STOVs with a TC $l=2$ to decompose into two STOVs with TC $l=1$, which are located symmetrically around the spatial axis. The two singularities, which rotate anticlockwise with $2\pi$ independently, give rise to $4\pi$. This phenomenon highlights the fact that although $\alpha$ occurs in the temporal domain, the singularity splitting arises in the spatial domain. Intriguingly, theoretical calculations reveal that the temporal parameter can induce spatial modal decomposition, confirmed by previous experimental results.26^,27 This underscores the significant coupling between the temporal and spatial domains, highlighting the complex dynamics of STOVs during nonlinear interactions. However, as the NTWO increases, the singularity separation of the SH STOVs intensifies, and the pulse broadening induced by dispersion becomes increasingly pronounced. To compare with Fig. 1(a), we omit the temporal broadening in Fig. 1(b) using an integral approximation [Eq. (6)], resulting in a discrepancy between the calculated and Simpson-type integral results.

To determine the conditions under which the approximation is valid, we compare the degree of singularity splitting between the calculated and approximate results, see Fig. 2. Figure 2(a) illustrates that as $\alpha$ increases, the distance of phase singularities by Simpson-type integral (red line) gradually deviates from the calculated results (blue line). This discrepancy may be because of the temporal broadening during the SHG process. Figure 2(b) shows the zoomed-in view of Fig. 2(a) with $\alpha$ ranging from 0 to 0.25. At this point, the relative error, defined as the ratio of the difference between the calculated and the Simpson-type integral results to the Simpson-type integral result [(Simpson. − Calculated.)/Simpson.], is less than 0.05. The convergence of the calculated and Simpson-type integral curves further demonstrates the accuracy of Eq. (6). Therefore, similar to the weak walk-off condition caused by the spatial walk-off angle in optical vortices, we can still define the weak ST walk-off condition.26^–28

In Sec. 2.1, we derived an approximate solution related to the decomposition degree of the SH STOVs and NTWO induced by the SHG process under the weak ST walk-off condition. However, beyond this approximation, when the NTWO approaches infinity, the situation of decomposed SH STOVs is still unclear. Therefore, we pose a question: is there a limit to the modal decomposition of SH STOVs in an SHG process dominated only by GVM? In this section, inspired by an earlier study on the SHG of optical vortices,29 we theoretically calculate the maximum separation of singularities caused solely by GVM. Now, we proceed to solve Eq. (5) to see if we can derive an explicit analytical expression in singularity positions for the SH STOVs. Substituting Eq. (1) into Eq. (5) yields $$u_S(x,\tau )=e^{-2{\xi }_x^2}{\int }_0^{1}d\zeta {[{\xi }_{\tau }-\alpha (\zeta -\frac{1}{2})+i{\xi }_x]}^2{e}^{-2[{\xi }_{\tau }-\alpha (\zeta -\frac{1}{2}){]}^2},$$(8)with $$\{\begin{array}{c}{\xi }_x=\frac{x}{x_s}\\ {\xi }_{\tau }=\frac{\tau }{{\tau }_s}\\ \alpha =\frac{\eta L}{{\tau }_s}\end{array}.$$(9)

To simplify the equation, we choose ${{\xi }_{\tau }}^{\prime }={\xi }_{\tau }-\alpha (\zeta -\frac{1}{2})$, and Eq. (8) can be divided into two parts and then set to zero to determine the exact locations of the singularities $$R(x,\tau )=\frac{e^{-2{\xi }_x^2}}{\alpha }{\int }_{{\xi }_{\tau }-\frac{\alpha }{2}}^{{\xi }_{\tau }+\frac{\alpha }{2}}\mathrm{d}{{\xi }_{\tau }}^{\prime }({{\xi }_{\tau }}^{\prime 2}-{\xi }_x^2)e^{-2\xi {\tau }^{\prime 2}}=0,$$(10)$$I(x,\tau )=\frac{e^{-2{\xi }_x^2}}{\alpha }{\int }_{{\xi }_{\tau }-\frac{\alpha }{2}}^{{\xi }_{\tau }+\frac{\alpha }{2}}\mathrm{d}{{\xi }_{\tau }}^{\prime }2i{{\xi }_{\tau }}^{\prime }{\xi }_x{e}^{-2{\xi }_{\tau }{\prime }^2}=0,$$(11)where $R(x,\tau )$ and $I(x,\tau )$ represent the real and image parts of $u_S(x,\tau )$, respectively. One can directly solve Eq. (11) to obtain Eq. (12) $$e^{-2({\xi }_{\tau }+\frac{\alpha }{2}{)}^2}-e^{-2({\xi }_{\tau }-\frac{\alpha }{2}{)}^2}=0.$$(12)

Equation (12) indicates that the $t$-axis coordinate of the singularities is ${\xi }_{\tau }=0$. By substituting the result into Eq. (10), the equation can be modified as $${\int }_{-\frac{\alpha }{2}}^{\frac{\alpha }{2}}\mathrm{d}{{\xi }_{\tau }}^{\prime }({{\xi }_{\tau }}^{\prime 2}-{\xi }_x^2)e^{-2{{\xi }_{\tau }}^{\prime 2}}=0.$$(13)

Equation (13) is a complex expression carrying a hard-to-solve Gaussian error function. For $\alpha \to \infty$, we can get the following solutions: $$\{\begin{array}{l}{\xi }_x=\pm \frac{1}{2}\\ {\xi }_{\tau }=0\end{array}.$$(14)

It is worth mentioning that after the SHG, the frequency of the SH pulse is doubled, and the width becomes $1/\sqrt{2}$ times the fundamental pulse,24^,30 so Eq. (14) needs to be corrected as $$\{\begin{array}{l}{\xi }_x^{\mathrm{SH}}=\pm \frac{\sqrt{2}}{2}\\ {\xi }_{\tau }^{\mathrm{SH}}=0\end{array}.$$(15)

Therefore, as the NTWO approaches infinity, the positions of the two singularities of the SH STOVs can be explicitly determined: $(x_{\max },{\tau }_{\max })=(\pm \frac{\sqrt{2}}{2}{x}_s,0)$, confirming the maximum distance of the singularity is $\mathrm{\Delta }x=\sqrt{2}{x}_s$. We suppose that the incomplete extension of the separation to the singularities of the STOVs stems from constraints imposed by energy redistribution dynamics. The nonlinear interaction between the fundamental and SH waves inherently balances the spatial and temporal degrees of freedom. Although GVM drives singularity separation in the spatial domain, the temporal broadening of the pulse (due to dispersion) counteracts further spatial splitting. This spatiotemporal coupling ensures that energy remains localized within the original spatial envelope of the STOVs, preventing the singularities from moving to the edge. Besides, it should be emphasized that ${\tau }_s$ progressively broadens as the NTWO increases. To derive a more concise expression for SH STOVs and to facilitate a more intuitive understanding of the propagation process to be discussed in Sec. 3, we scale the nonlinear process associated with the infinite parameter to a linear regime. Therefore, Eq. (5) can be clearly expressed as $$u_S(x,\tau )=\left[\right(\frac{\tau }{{\tau }_s}+i\frac{x}{x_s}{)}^2+f(\alpha )]u_0(x,\tau {)}^2,$$(16)where ${\tau }_s$ is a variable temporal half-width that shows a positive correlation with the NTWO, and $f(\alpha )=(\mathrm{\Delta }x/2x_s{)}^2$ is a variable related to the NTWO $\alpha$ with a range from 0 to 0.5. Under weak walk-off conditions, i.e., Eq. (6) (Simpson-type integral), $f(\alpha )$ can be expressed as $(\frac{\alpha }{2\sqrt{3}}{)}^2$. Figure 3 illustrates the simulated distribution as $f(\alpha )$ increases. The intensity and phase distributions of the SH STOVs are scaled by $1/x_s$ and $1/{\tau }_s$ for better visualization. It is obvious that $f(\alpha )$ does not have a specific analytical expression. Nevertheless, we can still present different modal decomposition scenarios by varying the values of $f(\alpha )$. Although the theoretical predictions successfully describe the relationship between the pulse and NTWO, the case of $f(\alpha )=0.5$ is unachievable in practical experiments, because the NTWO is unlikely to approach infinity. In Sec. 3.1, we will discuss the influence of propagation dynamics in fundamental and SH STOVs.

In experiments, the ST asymmetric property of the STOVs introduces a phase shift term (Gouy phase shift) $\exp (-i{\varphi }_{\mathrm{FW}})$ into the spatial domain under space dispersionless propagation, resulting in the fundamental STOVs evolving from a pure “flying donut” type to the “space–time separated lobes,” where $u_f^{\text{farfield}}=(\frac{\tau }{{\tau }_s}+\frac{x}{x_s})u_0$, as shown in Fig. 4.13 Notably, the transverse diffractive spreading results in spatial widening during propagation. As a result, we scale the distributions of the STOVs by factors $1/x_s$ and $1/{\tau }_s$ to account for this effect. It is worth noting that the propagation distance $z$ can never reach infinity, and ${\varphi }_{\mathrm{FW}}(z)$ can only approximate ${\varphi }_{\mathrm{FW}}(\infty )=0.5\pi$ asymptotically.31^–33 Moreover, the evolution of the spatial term of the fundamental STOVs shifts from the real part to the imaginary part. This numerical transition represents the gradual evolution of the fundamental STOVs, including the gradual disappearance of their phase.

Naturally, this envelope shape of the fundamental STOVs also affects the space–time distribution of the SH STOVs. When fundamental STOVs are introduced into nonlinear crystal exhibiting NTWO, the splitting singularities are no longer only confined to the $x$-axis. Compared with the SHG involving pure STOVs, there will be a certain degree of angular displacement, which is related to the shifted phase ${\varphi }_{\mathrm{FW}}$, and can be expressed by modifying Eq. (16) as $$\begin{gathered} u_S^{\prime }(x,\tau )=\{\frac{\tau }{{\tau }_s}+\frac{x}{x_s}\sin ({\varphi }_{\mathrm{FW}})+i[\frac{x}{x_s}\cos ({\varphi }_{\mathrm{FW}})-\sqrt{f(\alpha )}\left]\right\} \\ \{\frac{\tau }{{\tau }_s}+\frac{x}{x_s}\sin ({\varphi }_{\mathrm{FW}})+i[\frac{x}{x_s}\cos ({\varphi }_{\mathrm{FW}})+\sqrt{f(\alpha )}\left]\right\}{u}_0(x,\tau {)}^2\mathrm{.} \end{gathered}$$(17)

Therefore, the degree of decomposition can also be modified as $$(x_0^{\prime },{\tau }_0^{\prime })=(\pm \sqrt{f(\alpha )}\frac{x_s}{\cos ({\varphi }_{\mathrm{FW}})},\mp \sqrt{f(\alpha )}\tan ({\varphi }_{\mathrm{FW}})).$$(18)

Figure 5(a) illustrates the SHG result of the fundamental STOVs with varying fundamental spatial shifted phases. Here, we set the NTWO parameter $\alpha \approx 0.75$ according to Ref. 18 to simulate this process as follows: the BBO crystal length $L=1\mathrm{mm}$ and GVM $\eta =1/v_g^{(2\omega )}-1/v_g^{(\omega )}=193.3\mathrm{fs}/\mathrm{mm}$ at a fundamental center wavelength $\lambda =800\mathrm{nm}$. The results suggest that the singularity positions of the SH STOVs generated by the fundamental STOVs with different spatial shifts are rotating, effectively illustrating the influence of the fundamental phase shift on the ST singularities in the SH pulses. Even a slight fundamental phase shift (e.g., ${\varphi }_{\mathrm{FW}}=0.1$, $z=0.1z_R$ in Fig. 4) can significantly affect the motion of the singularities (e.g., ${\varphi }_{\mathrm{FW}}=0.1$, $z=0.1z_R$ in Fig. 5), which correspond to the experimental results presented earlier.18 To quantitatively analyze the relationship between the fundamental phase shift and singularity trajectory, we define two parameters to investigate the motion trajectory of the singularities in detail: the degree of rotation $\theta$ and the distance $S$ of singularities, which can be expressed as $$\theta ({\varphi }_{\mathrm{FW}})=\mathrm{arc}\tan \left[\frac{({\tau }_{\text{top}}-{\tau }_{\mathrm{bot}})/{\tau }_s}{(x_{\text{top}}-x_{\mathrm{bot}})/x_s}\right],$$(19)$$S({\varphi }_{\mathrm{FW}})=\sqrt{(\frac{{\tau }_{\text{top}}-{\tau }_{\mathrm{bot}}}{{\tau }_s}{)}^2+(\frac{x_{\text{top}}-x_{\mathrm{bot}}}{x_s}{)}^2}.$$(20)

Substituting Eq. (18) to Eqs. (19) and (20) yields $$\theta ({\varphi }_{\mathrm{FW}})=-{\tan }^{-1}[\sin ({\varphi }_{\mathrm{FW}})],$$(21)$$S({\varphi }_{\mathrm{FW}})=\sqrt{f(\alpha )}\frac{\sqrt{1+{\sin }^2({\varphi }_{\mathrm{FW}})}}{\cos ({\varphi }_{\mathrm{FW}})}.$$(22)

Equations (21) and (22) are used to plot the relationship among ${\varphi }_G$, $\theta$, and $S$. It is evident that as the two singularities rotate counterclockwise, their distance also increases with ${\varphi }_{\mathrm{FW}}$. As shown in Fig. 5(b), $\theta$ changes monotonically between 0 and $-0.25\pi$, suggesting the rotation of the spatial shifted phase. Conversely, Fig. 5(c) shows that as ${\varphi }_{\mathrm{FW}}$ approaches $0.5\pi$, $S$ increases more rapidly. When ${\varphi }_{\mathrm{FW}}=0.5\pi$, the singularity vanishes and the phase difference along the diagonal is $\pi$ ($0.5\pi$ at the upper right corner and $-0.5\pi$ at the lower left corner). This envelope evolution is the same as the mode conversion from Laguerre–Gaussian modes to Hermite–Gaussian modes in traditional vortex beams. The STOVs with “diagonally aligned lobes,” which carry $0.5\pi$ phase shift, double due to the SHG process, exhibit a disappearance of the phase difference along the diagonal, as can be seen in the rightmost column of Fig. 5(a), which is first theoretically calculated in Ref. 16. The influence of the fundamental spatial phase on the frequency-doubling process provides a solid explanation for previous results, particularly the ST splitting of the singularities of SH STOVs. However, the impact on STOVs is not limited to the aforementioned discussion; the evolutionary propagation of second-order harmonic light should also be taken into account. In Sec. 3.2, we will conduct a systematic discussion on the propagation process of SH STOVs and present their impact on the evolution of singularities.

Here, we start with the paraxial wave equation for the slowly varying amplitude $u(x,\tau ,z)$13^,34$$\frac{1}{k_0}\frac{{\partial }^{2}u}{\partial x^2}-{k_0}^{″}\frac{{\partial }^{2}u}{\partial {\tau }^2}+2i\frac{\partial u}{\partial z}=0,$$(23)where $k_0$ is the wave number at the center wavelength $\lambda$, and ${k_0}^{″}$ is the group velocity dispersion. On the basis of previous works,1^,13^,35 we assume that the various components of $x$ and $\tau$ can be separated $u(x,\tau ,z)=u^x(x,z)u^{\tau }(\tau ,z)$. Under this assumption, the most general Hermite–Gaussian mode solution of Eq. (23) is $$u_m^x(x,z)=\frac{C_m}{\sqrt{x_s}}{H}_m\left(\frac{\sqrt{2}x}{x_s}\right)e^{-\frac{x^2}{x_s^2}}{e}^{i\frac{k_0{x}^2}{2R_x}}{e}^{-i(m+\frac{1}{2}){\varphi }_x},$$(24)$$u_n^{\tau }(\tau ,z)=\frac{C_n}{\sqrt{{\tau }_s}}{H}_n\left(\frac{\sqrt{2}\tau }{{\tau }_s}\right)e^{-\frac{{\tau }^2}{{\tau }_s^2}}{e}^{-i\frac{{\tau }^2}{2k_0^{″}{R}_{\tau }}}{e}^{i(n+\frac{1}{2}){\varphi }_t},$$(25)where $H$ is a Hermite polynomial of order $m$; $x_s$ and $x_{\tau }$ are the spatial and temporal radii of the pulse in the $x$ and $t$ domains [$x_s=x_{s0}\sqrt{1+(z/z_R{)}^2}$ and ${\tau }_s={\tau }_{s0}\sqrt{1+(z/z_R{)}^2}$], respectively; $R_x$ and $R_{\tau }$ are the spatial and temporal radii of the curvature, respectively; ${\varphi }_x$ and ${\varphi }_t$ are the spatial and temporal Gouy phase shifts [${\varphi }_x={\tan }^{-1}(z/z_R)$ and ${\varphi }_t=v_g^2{k}_0{k_0}^{″}{\tan }^{-1}(z/z_R)$], respectively. In a vacuum or a very dilute medium (air), ${k_0}^{″}=0$ and $v_g=c$. The pulse propagation equation must satisfy Eq. (23), so we can superpose different Hermite–Gaussian modes to obtain the desired ST envelope, such that STOVs during the SHG process with an extremely small NTWO can be expressed as13^,36$$\begin{gathered} u_s^0(x,\tau ,z)=u_0^x(x,z)u_2^{\tau }(\tau ,z)+u_1^x(x,z)u_1^{\tau }(\tau ,z)+u_2^x(x,z)u_0^{\tau }(\tau ,z) \\ \begin{array}{c}=\left\{\right(\frac{\tau }{{\tau }_s}{)}^2-\frac{1}{4}+2i\frac{x\tau }{x_s{\tau }_s}{e}^{-i{\varphi }_{\mathrm{SH}}}-\left[\right(\frac{x}{x_s}{)}^2-\frac{1}{4}\left]e^{-2i{\varphi }_{\mathrm{SH}}}\right\}{u}_0\end{array}, \end{gathered}$$(26)where ${\varphi }_{\mathrm{SH}}={\varphi }_x={\tan }^{-1}(z/z_R)$ is the additional phase shift of the second harmonic pulse during propagation. If Eq. (22) is assumed to be the near-field expression of the SH STOVs (at $z=0$), the introduction of NTWO will lead to the introduction of a constant term $u_0^x(x,z)u_0^{\tau }(\tau ,z)$ in the propagation equation. Equation (26) can be modified as $$\begin{gathered} u_s(x,\tau ,z,L)=u_s^0(x,\tau ,z)+f(\alpha )u_0^x(x,z)u_0^{\tau }(\tau ,z) \\ =\left\{\right(\frac{\tau }{{\tau }_s}{)}^2-\frac{1}{4}+2i\frac{x\tau }{x_s{\tau }_s}{e}^{-i{\varphi }_{\mathrm{SH}}}-\left[\right(\frac{x}{x_s}{)}^2-\frac{1}{4}]e^{-2i{\varphi }_{\mathrm{SH}}}+f(\alpha )\}{u}_0. \end{gathered}$$(27)

When $z=0$ (i.e., in the near-field), the SH STOVs maintain a “flying donut shape,” with a splitting singularity distance of $\mathrm{\Delta }x/x_s=2\sqrt{f(\alpha )}$. The envelope shape of the second-order STOVs undergoes spatial singularity splitting because the NTWO continuously evolves during propagation. As the propagation distance approaches infinity (i.e., ${\varphi }_{\mathrm{SH}}=0.5\pi$, in the far-field), the envelope evolves into lobes that are diagonally separated in space–time, attributed to the ST asymmetry. Figure 6 illustrates the propagation evolution of the SH STOVs with varying $f(\alpha )$ at different propagation positions. For the convenience of observing the entire propagation process, we replace the propagation position with an additional phase shift ${\varphi }_{\mathrm{SH}}$.

Figure 6 clearly indicates that the influence of different NTWO on SH STOVs is profound and not just limited to the spatial separation of singularities in the shape of a flying donut STOV in the near-field. The introduced NTWO during the frequency doubling process influences the intensity distribution of the SH STOVs in the far-field. When ${\varphi }_{\mathrm{SH}}=0.5\pi$, as $f(\alpha )$ increases, the far-field intensity distribution of the SH STOVs evolves from three ST diagonally aligned lobes to two lobes, with the middle lobe gradually disappearing. At this point, the mode of the SH STOVs can be considered as a complete decomposition, as the intensity distribution of the two diagonally aligned lobes in space–time is similar to the far-field distribution of the fundamental beam (see Fig. 3), where $u_s=(\frac{\tau }{{\tau }_s}+\frac{x}{x_s}{)}^2{u}_0=(u_f^{\text{farfield}}{)}^2$.

We have systematically investigated the generation, propagation, and decomposition dynamics of SH STOVs during the SHG process. Our analysis reveals that the GVM leads to a distinct spatial splitting of the SH STOVs singularities, with the separation increasing with the NTWO. Under weak ST walk-off conditions, the dynamics of singularity separation can be accurately predicted, providing crucial insights into the nonlinear interactions governing the STOV behavior. The influence of the spatial phase shift of the fundamental STOVs is found to be significant in determining the singularity dynamics of the SH STOVs, inducing directional movement and rotation of the singularities. In addition, we provided an in-depth exploration of the propagation dynamics of the SH STOVs, demonstrating their evolution from a “flying donut” structure in the near-field to diagonally separated lobes in the far-field. This transition is indicative of the unique ST coupling inherent to STOVs, driven by the asymmetry introduced during spatial dispersionless propagation. When the NTWO increases, the far-field SH STOVs exhibit a distinct transformation in intensity distribution from three diagonally aligned lobes to two lobes, signifying a complete modal decomposition. This behavior underscores the critical role of nonlinear crystal parameters in shaping the ST evolution of the STOVs, which has been further validated through rigorous numerical simulations.

Notably, as $\alpha =\frac{\eta L}{{\tau }_s}$ suggests, achieving an infinite NTWO ($\alpha \to \infty$) would require one of the following conditions: (1) a vanishingly small pulse width (${\tau }_s\to 0$) and (2) an infinitely large GVM ($\eta \to \infty$) or an infinitely large crystal length ($L\to \infty$). However, all of these conditions are experimentally unattainable. Fortunately, the evolutionary behavior of the STOVs during propagation provides an alternative. The transformation from the “flying donut” type to the “spatiotemporal diagonally aligned lobes” type enables the maximal separation of singularities without requiring an infinite NTWO. Given that a lens can be regarded as a spatial Fourier transform system, the ST diagonally separated lobe-type STOVs can be achieved with NTWO $\alpha =0$ at the Fourier plane, which is equivalent to that of direct SHG with an infinite NTWO. As lenses are generally used in the SHG to increase power density and thereby scale up the output power, our theoretical framework offers robust support for the optimization and analysis of such systems. Herein, we continue to assume that the fundamental STOV is a “flying donut” with a TC = 1. After the SHG, the SH STOVs can be expressed as $$u_s=F_{\text{space}}^{-1}\{[F_{\text{space}}(u_f){]}^2\},$$(28)where $F_{\text{space}}$ and $F_{\text{space}}^{-1}$ are the forward and inverse spatial Fourier transforms, respectively.

Beyond the simplified scenario, the SHG process is further complicated by additional factors besides GVM, such as group dispersion delay,37 phase mismatch,38^,39 spatial walk-off,27 and pump depletion.40 A rigorous theoretical study of the envelope shape of SH STOVs must consider the factors mentioned above, which represent an entirely new research topic. However, this is beyond the scope of the current work.

Besides, the pulse energy is set to be 540 nJ in our calculations, and current research focuses more on characterizing STOV behavior than on optimizing efficiency during nonlinear processes. Although practical experimental implementation may require higher incident energies due to coating losses or crystal damage thresholds, laser systems deliver outputs of 100 nJ and even to mJ level are commercially available. Besides, the SH beam pattern can be observed even at relatively low energy levels using a high-sensitivity charge-coupled device (CCD). As a result, our theoretical framework provides crucial guidance for experimental design, enabling the exploration of STOV dynamics in the SHG while considering practical constraints.

In addition, our theory is based on two key assumptions: perfect phase-matching and slowly varying amplitude approximation. In high harmonic generation, however, the driving pulse duration is significantly shorter, which may invalidate the slowly varying amplitude approximation. Moreover, high harmonic generation involves a more complex phase-matching process that requires consideration of additional factors, such as the spatial chirp as discussed in Ref. 20 and the inhomogeneity parameters $\eta$ as discussed in Ref. 41. Although our work focuses on the impact of STOV propagation properties during the SHG, it provides a solid foundation for extending this approach to higher-order nonlinear processes.

In summary, the findings presented in this work offer a comprehensive understanding of the nonlinear generation and propagation mechanisms of STOVs, paving the way for advancement in ST light manipulation techniques. These insights establish a theoretical foundation for future exploration of STOV singularity dynamics and hold significant potential for applications in advanced microscopy, precision optical sensing, and the broader study of singularity evolution in complex light fields.

Xuechen Gao is currently pursuing his PhD at the School of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Tianjin, China. His current research interests include nonlinear optics and spatiotemporal light field.

Jintao Fan is an associate professor at the School of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Tianjin, China. He received his PhD in 2019, conducted postdoctoral research at the Hannover University (2019–2021), and joined the Tianjin University in 2021. Recipient of Marie Curie Optics Fellowship, Wang Daheng Optics Award, and Jin Guofan Scholarship, his research focuses on ultrafast optical parametric dynamics and structured light field frequency conversion.

Wei Chen is an assistant professor at the College of Engineering and Applied Sciences, Nanjing University, Nanjing, China. He received his BS degree in optoelectronics in 2012 and his PhD in optical engineering in 2018, both from the Tianjin University, Tianjin, China. His current research interests focus primarily on structured light and nonlinear optical microstructures.

Minglie Hu is a professor at the School of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Tianjin, China. He obtained his PhD in 2005, then held postdoctoral/visiting positions at the Tianjin University, the Jena University (Germany), and the CUHK (2005–2007). A recipient of the National Excellent Young Scientist Fund, he was selected for the Ministry of Educations New Century Talent Program and awarded the National 100 Outstanding Doctoral Dissertations. His research focuses on ultrafast pulsed laser technology and novel high-power fiber femtosecond lasers.

Biographies of the other authors are not available.