Spatiotemporal Torquing of Light

We demonstrate the controlled spatiotemporal transfer of transverse orbital angular momentum (OAM) to electromagnetic waves: the spatiotemporal torquing of light. This is a radically different situation from OAM transfer to longitudinal, spatially defined OAM light by stationary or slowly varying refractive-index structures such as phase plates or air turbulence. We show that net transverse OAM per photon can be spatiotemporally imparted to a light pulse only if (1) a transient phase perturbation is well overlapped with the pulse in spacetime, or (2) the pulse initially has nonzero transverse OAM density, and the perturbation removes energy from it. Physical insight is provided by the mechanical analogy of torquing a wheel or removing mass as it spins. Our OAM theory for spatiotemporal optical vortex (STOV) pulses [S. W. Hancock et al.,Phys. Rev. Lett. 127, 193901 (2021)] correctly quantifies the light-matter interaction of our experiments and provides a spatiotemporal-torque-based explanation for the first measurement of STOVs [N. Jhajj et al., Phys. Rev. X 6, 031037 (2016)].

Popular Summary

In earlier work, we reported a surprising fact: Light pulses can have orbital angular momentum (OAM) pointing transverse to the propagation direction. We dubbed these electromagnetic structures spatiotemporal optical vortex (STOV) pulses, one form of which resembles a rolling doughnut. Knowing how much transverse OAM is carried per photon in a STOV pulse is fundamental to understanding light-matter interactions with these structures, and hence to all their potential applications. Here, we experimentally verify our theory: The transverse OAM per photon in a STOV pulse is one-half of that in a pulse with longitudinal OAM.

To perturb transverse OAM, we introduce the notion of spatiotemporal torque—a spacetime twist applied to a laser pulse. Guided by this idea, we use a second ultrashort laser pulse to generate a narrow, ultrafast rise-time plasma in the air to introduce a phase perturbation precisely positioned in space and time, defining a spatiotemporal torque lever arm. For each location of the lever arm, we capture the full complex electric field of the perturbed pulse midflight using a specially developed imager. This yields the torque-dependent change in OAM, which we then compare with theory.

Besides addressing the nature of transverse OAM, our experiments and theory demonstrate that spatiotemporally induced changes to transverse OAM require ultrafast perturbations that rarely occur in the environment. This points to the robustness of transverse OAM states and potential applications.

Figure 1

Illustrative figure of STOV pulse propagation for spacetime asymmetry ratio $\alpha =w_{0\xi }/w_{0x}=0.24$, with $z_{0x}=\frac{1}{2}{k}_0{w}_{0x}^2=4.5\mathrm{cm}$ (a) Top two rows: modal theory [22] plots of spatiotemporal intensity and phase of $l=1$ STOV pulse propagating right to left through its beam waist from $z/z_{0x}=-0.41$ to $z/z_{0x}=0.24$ (and right to left within each panel). Bottom two rows: experimental intensity and phase plots extracted by TG-SSSI. (b) Lineouts along $(0,\xi )$ and $(x,0)$ of the experimental intensity profile at $z/z_{0x}=0.02$ (solid lines) and fits to the modal theory curves (dashed lines).

Figure 2

Plots of analytic solutions [Eqs. (6a) and (6b)] of $\mathrm{\Delta }⟨L_y⟩$ vs $(x_0,{\xi }_0)$ for the spatiotemporal phase shift $\mathrm{\Delta }{\varphi }_p(x,\xi )$ of Eq. (4) applied to (a) a Gaussian pulse $A_{\mathrm{G}}(x,\xi )$ [Eq. (5a)] and to (b) an $l=1$ STOV pulse ${\mathrm{A}}_{\mathrm{STOV}}(x,\xi )$ [Eq. (5b)]; $x_0$ and ${\xi }_0$ are the central space location and turn-on time of the perturbation. In Eq. (4), we choose $\mathrm{\Delta }{\varphi }_{p0}=-0.5$ to model the plasma generated by optical field ionization (OFI) of air.

Figure 3

Effect of a non-energy-conserving pure amplitude perturbation $\mathrm{\Gamma }(x,\xi )=1-\exp [-{(x/h)}^8]$ on pulses with and without transverse OAM. (a) Preperturbation $l=1$ STOV and Gaussian pulse intensities $|E_{\mathrm{s}}(x,\xi ){|}^2$ at $z=0^{-}$, followed by the pulse intensity evolution $|E_{\mathrm{sp}}(x,y=0,\xi ;z){|}^2$ from $z=0$ to $z=2z_{0x}$ determined by $\mathbf{E}$ and $\mathbf{H}$ field propagation computed by yappe (Appendix pp2). Here, $2h=100\mathrm{\mu }\mathrm{m}$ and $w_{0x}=w_{0\xi }=100\mathrm{\mu }\mathrm{m}$. (b) Change in transverse OAM per photon vs $z$ ($\mathrm{\Delta }{⟨L_y⟩}_z$) for the Gaussian and STOV pulses calculated directly from the fields using Eq. (1b) (points) and calculated using Eq. (1a) (solid lines).

Figure 4

Configuration for measuring the effect of a transient phase perturbation on field $E_{\mathrm{s}}$ (from a $4f$ -pulse shaper) imposed by the ultrafast OFI plasma induced by field $E_{\mathrm{TW}}$. This OFI plasma is the transient wire. The perturbed pulse $E_{\mathrm{sp}}$ and unperturbed pulse $E_{\mathrm{s}}$ ($E_{\mathrm{TW}}$ off) are measured by TG-SSSI [17]. The angle between the beams is $\theta =18.5°$. A detailed experimental diagram is shown in Appendix pp3.

Figure 5

(a) TG-SSSI measured $I_{\mathrm{s}}=|{\mathbf{E}}_{\mathrm{s}}(x,\tau ){|}^2$ (transient wire off). (b) and (c) TG-SSSI measured $|{\mathbf{E}}_{\mathrm{sp}}(x,\tau ){|}^2$ and $\mathrm{\Delta }{\varphi }_p(x,\tau )$ (in rad) (transient wire on). (a′)–(c′) corresponding simulated $I_{\mathrm{s}}^{\mathrm{sim}}=|{\mathbf{E}}_{\mathrm{s}}^{\mathrm{sim}}(x,\tau ){|}^2$, $I_{\mathrm{sp}}^{\mathrm{sim}}=|{\mathbf{E}}_{\mathrm{sp}}^{\mathrm{sim}}(x,\tau ){|}^2$, and $\mathrm{\Delta }{\varphi }_p^{\mathrm{sim}}(x,\tau )$. (ci) Lineout of (c) along dashed red line (solid red) and fit (dotted green). (cii) Lineout of (c) along dashed blue line (solid blue) and fit (dotted pink). The fit curve neglects the oscillations on the right, which are due to the imaging plane being 3 mm past the interaction (see text). The fits in (ci) and (cii) are to $\mathrm{\Delta }{\varphi }_p^{\mathrm{sim}}(x,\tau )=\frac{1}{2}\mathrm{\Delta }{\varphi }_{p0}(1+\mathrm{erf}[\sqrt{2}(\tau -{\tau }_0)/h_{\tau }])\exp \{-{[(x-x_0)/h_x]}^8\}$, giving $h_{\tau }=h_{\xi }/v_g=44\mathrm{fs}$ and $h_x=40\mathrm{\mu }\mathrm{m}$. In (ci) $x=120\mathrm{\mu }\mathrm{m}(=x_0)$, and in (cii) $\tau =100\mathrm{fs}$.

Figure 6

Effect of transient wire onset time ${\tau }_0$ on changing the transverse OAM of STOV and Gaussian pulses. Onset time is scanned between $-200$ and 800 fs in $\mathrm{\Delta }{\tau }_0=66\mathrm{fs}$ steps. (a) $l=1$ STOV pulse, with transient wire centered at $x_0=+120\mathrm{\mu }\mathrm{m}$; (b) $l=1$ STOV pulse, with transient wire centered at $x_0=-120\mathrm{\mu }\mathrm{m}$; (c) Gaussian pulse ($l=0$), with transient wire centered at $x_0=60\mathrm{\mu }\mathrm{m}$. The top two rows in (a)–(c) are $I_{\mathrm{sp}}(x,\tau )=|E_{\mathrm{sp}}(x,\tau ){|}^2$ and $\mathrm{\Delta }{\varphi }_p(x,\tau )=\arg (E_{\mathrm{sp}})-\arg (E_{\mathrm{s}})$ (in rad), where $E_{\mathrm{s}}(x,\tau )$ and $E_{\mathrm{sp}}(x,\tau )$ are the unperturbed and perturbed complex fields extracted from TG-SSSI measurements. The spatial location of the perturbation is indicated by the red dotted lines. The bottom panels in (a)–(c) plot $\mathrm{\Delta }{⟨L_y⟩}_{1a}$ and $\mathrm{\Delta }{⟨L_y⟩}_{1b}$, which are the change in spatiotemporal OAM per photon computed by inserting the measured $E_{\mathrm{s}}$ and $E_{\mathrm{sp}}$ into Eq. (1a), and the $\mathbf{E}$ and $\mathbf{H}$ fields computed from them into Eq. (1b). The error bars are the $\pm$ standard deviation over 500–750 shots of extracted data. Overlaid in (a)–(c) is $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$ calculated using Eqs. (6a) and (6b), in which we use measured and fit quantities. The FWHM of $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$ in (a) and (b) is 280 fs. For $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$ in panel (a) $\mathrm{\Delta }{\varphi }_{p0}=-0.45$, $w_{0x}=120\mathrm{\mu }\mathrm{m}$, $w_{0\xi }=56\mathrm{\mu }\mathrm{m}$, $h=40\mathrm{\mu }\mathrm{m}$, $x_0=120\mathrm{\mu }\mathrm{m}$). For $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$ in panel (b) $\mathrm{\Delta }{\varphi }_{p0}=-0.21$, $w_{0x}=110\mathrm{\mu }\mathrm{m}$, $w_{0\xi }=56\mathrm{\mu }\mathrm{m}$, $h=40\mathrm{\mu }\mathrm{m}$, $x_0=-120\mathrm{\mu }\mathrm{m}$. For $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$ in panel (c) $\mathrm{\Delta }{\varphi }_{p0}=-0.31$, $w_{0x}=100\mathrm{\mu }\mathrm{m}$, $w_{0\xi }=61\mathrm{\mu }\mathrm{m}$, $h=40\mathrm{\mu }\mathrm{m}$, $x_0=60\mathrm{\mu }\mathrm{m}$. The “$-\infty$” mark on the time axes refers to the ${\xi }_0\to -\infty$ limit of $\mathrm{\Delta }{⟨L_y⟩}_{\text{theory}}$.

Figure 7

(a) Pulse with longitudinal OAM propagating at center-of-energy velocity ${\mathbf{v}}_{\mathrm{CE}}$. Monochromatic, “infinitely long” pulses can be viewed as this pulse stretched in the $\pm {\mathbf{v}}_{\mathrm{CE}}$ directions. In that case, the calculation of intrinsic OAM is done using ${\mathbf{L}}_{\mathrm{int}}={(4\pi c)}^{-1}\int d^3\mathbf{r}(\mathbf{r}-{\mathbf{r}}_{\mathrm{CE}})\times (\mathbf{E}\times \mathbf{H})$ with no reference to ${\mathbf{v}}_{\mathrm{CE}}$. (b) STOV pulse (with transverse OAM) propagating at center-of-energy velocity ${\mathbf{v}}_{\mathrm{CE}}$. All coordinates in (a) and (b) are in the lab frame.

Figure 8

Propagation evolution of $\mathrm{\Delta }⟨L_y⟩=u_{\mathrm{sp}}^{-1}⟨A_{\mathrm{sp}}|L_y|A_{\mathrm{sp}}{⟩}_z$ (red curve) and $\mathrm{\Delta }⟨{\pounds }_y⟩=u_{\mathrm{sp}}^{-1}⟨A_{\mathrm{sp}}|{\pounds }_y|A_{\mathrm{sp}}{⟩}_z$ (green curve), where the $z$ evolution of $A_{\mathrm{sp}}(x,y,\xi ;z)$ is nonparaxially computed using our code yappe (Appendix pp2). The perturbation [given by Eq. (7), with $\mathrm{\Delta }{\varphi }_{p0}=-0.5$, $x_0=-100\mathrm{\mu }\mathrm{m}$, ${\xi }_0=0,h_x=50\mathrm{\mu }\mathrm{m}$, and $h_{\xi }=50\mathrm{\mu }\mathrm{m}$] is applied to the Gaussian pulse of Eq. (5a), with $w_{ox}=100\mathrm{\mu }\mathrm{m}$ and $w_{0\xi }=100\mathrm{\mu }\mathrm{m}$. The immediate postperturbation OAM changes are $\mathrm{\Delta }{⟨L_y⟩}_{z=0}=0.012$ and $\mathrm{\Delta }{⟨{\pounds }_y⟩}_{z=0}=-0.10$. Also plotted are large blue points $\mathrm{\Delta }{L}_y^S$ computed using Eq. (1b), with the $\mathbf{E}$ and $\mathbf{H}$ fields propagated nonparaxially with yappe, and small black points computed as $\mathrm{\Delta }{⟨{\pounds }_y⟩}_z=\mathrm{\Delta }{⟨{\pounds }_y⟩}_{z=0}+k_0^{-1}z{⟨{\partial }^2/\partial x\partial \xi ⟩}_{z=0}$ from integration of Eq. (a10), where $⟨{\partial }^2/\partial x\partial \xi {⟩}_{z=0}=⟨A_{\mathrm{sp}}|{\partial }^2/\partial x\partial \xi |A_{\mathrm{sp}}⟩$ for $A_{\mathrm{sp}}=A_{\mathrm{sp}}(x,y,\xi ;z=0)$.

Figure 9

Comparison of the predictions of Refs. [33, 34] against our experimental results. The above panels replot the curves of Fig. 6 and overlay $\mathrm{\Delta }⟨{\pounds }_y⟩$, where the latter is computed using Eq. (1a), with the complex fields provided by the TG-SSSI measurements.

Figure 10

(a) Change in transverse angular momentum density $\mathrm{\Delta }{M}_y(x,\xi )=A_{\mathrm{sp}}^{*}{L}_y{A}_{\mathrm{sp}}$ of a Gaussian pulse $A_{\mathrm{G}}(x,\xi )$ using step-function perturbation, Eq. (4). (b) Same as (a) except using smoothed perturbation, Eq. (7). In both panels, $\mathrm{\Delta }{M}_y(x,\xi )$ is normalized by the maximum value $|\mathrm{\Delta }{M}_y(x,\xi ){|}_{\max }$.

Figure 11

Plots of analytic solutions [45] for change in transverse OAM per photon, $\mathrm{\Delta }{⟨L_y⟩}_{x_0,{\xi }_0}$ imparted to an optical pulse as a function of $(x_0,{\xi }_0)$ by a spatiotemporal phase perturbation $\mathrm{\Gamma }(x,\xi )=\exp [i\mathrm{\Delta }{\varphi }_p(x,\xi )]$. Here, $\mathrm{\Delta }{\varphi }_p(x,\xi )=\mathrm{\Delta }{\varphi }_{p0}\exp [-{(x-x_0)}^2/h_x^2-{(\xi -{\xi }_0)}^2/h_{\xi }^2$], $\mathrm{\Delta }{\varphi }_{p0}=1$, ${\beta }_2=0$, and $h_x/w_{0x}=0.25$. (a) Perturbation $\mathrm{\Gamma }(x,\xi )$ applied to an $l=1$ STOV pulse $A_{\mathrm{STOV}}(x,\xi )=(\xi /w_{0\xi }+ix/w_{0x})A_{\mathrm{G}}(x,\xi )$ for $h_{\xi }/w_{0\xi }=0.25$ and $\alpha =w_{0\xi }/w_{0x}=0.5$, 1, and 2. The red dashed circle in the center panel is the contour of peak intensity of $|A_{\mathrm{STOV}}{|}^2$. (b) $\mathrm{\Gamma }(x,\xi )$ applied to $A_{\mathrm{STOV}}(x,\xi )$ as in (a), here with $\alpha =1$ and transient width $h_{\xi }/w_{0\xi }=0.5$, 1, and 10. The curves immediately below plot the maximum absolute change in OAM ($|\mathrm{\Delta }⟨L_y{⟩}_{x_0,{\xi }_0}{|}_{\max }$) vs transient width $h_{\xi }$, for the cases of fixed peak phase shift $\mathrm{\Delta }{\varphi }_{p0}=1$ and constant integrated phase shift ${(h_x{h}_{\xi })}^{-1}\int dxd\xi \mathrm{\Delta }{\varphi }_p(x,\xi )=1$. The overlaid dashed red circle follows the maximum intensity contour of $A_{\mathrm{STOV}}$.

Figure 12

Detailed experimental configuration. Compressor 1 adjusts the pulse for the $4f$-pulse shaper, TG-SSSI spatial interferometry pulse ${\mathcal{E}}_i$, and transient wire pulse $E_{\mathrm{TW}}$. Compressor 2 adjusts the pulse for the TG-SSSI probe and reference supercontinuum pulses [17]. CM, concave mirror; BP, bandpass filter 3-nm FWHM, 800-nm center; AS, adjustable slit; SLM, spatial light modulator; WP, $\lambda /2$ wave plate; SP, short-pass filter transmits below 750 nm; VR, vertical retroreflector; RM, pump and interferometric reference rejection mirror; G, grating; L, lens.