Accelerating and Decelerating Space-Time Optical Wave Packets in Free Space

Although a plethora of techniques are now available for controlling the group velocity of an optical wave packet, there are very few options for creating accelerating or decelerating wave packets whose group velocity varies controllably along the propagation axis. Here we show that “space-time” wave packets in which each wavelength is associated with a prescribed spatial bandwidth enable the realization of optical acceleration and deceleration in free space. Endowing the field with precise spatiotemporal structure leads to group-velocity changes as high as $\sim c$ observed over a distance of $\sim 20\mathrm{mm}$ in free space, which represents a boost of at least $\sim 4$ orders of magnitude over $X$ waves and Airy pulses. The acceleration implemented is, in principle, independent of the initial group velocity, and we have verified this effect in both the subluminal and superluminal regimes.

Figure 1

Setup for the synthesis and characterization of accelerating ST wave packets. BS, beam splitter; $G$, diffraction grating; SLM, spatial light modulator; DL, delay line; $L$, lens.

Figure 2

(a) Representation of the spectral support domain of ST wave packets on the surface of the light cone. For propagation-invariant ST wave packets, the support domain is a conic section at the intersection of the light cone with a tilted spectral plane (here ${\theta }_1=45.2°$ and ${\theta }_2=47°$). For an accelerating ST wave packet, the support domain is the 2D area at the intersection of the light cone with a wedge-shaped volume bounded by two tilted spectral planes (${\theta }_1$ and ${\theta }_2$). (b) Phase distribution $\mathrm{\Phi }$ imparted by the SLM in Fig. 1 to the spectrally resolved wave front to produce ST wave packets of group velocity ${\tilde{v}}_1=c\tan {\theta }_1$, group velocity ${\tilde{v}}_2=c\tan {\theta }_2$, and acceleration ${\tilde{v}}_1\to {\tilde{v}}_2$. Insets show the phases (modulo $2\pi$) for a fixed temporal frequency (identified by the green box): ${\mathrm{\Phi }}_1(x)$ and ${\mathrm{\Phi }}_2(x)$ are linear phase distributions, each corresponding to a particular spatial frequency; ${\mathrm{\Phi }}_3(x)$ is chirped between ${\mathrm{\Phi }}_1(x)$ and ${\mathrm{\Phi }}_2(x)$, thus corresponding to a finite spatial bandwidth. (c) Spectral projections onto the $(k_x,\omega /c)$ and $(k_z,\omega /c)$ planes for the ST wave packets in (a) and (b). The projections are 1D curves for constant-$\tilde{v}$ wave packets and 2D domains for accelerating wave packets. The dashed green line is the light line $k_z=\omega /c$.

Figure 3

(a) Measured axial evolution of the time-averaged intensity $I(x,z)$ for two constant-$\tilde{v}$ ST wave packets (${\tilde{v}}_1=c\tan {\theta }_1$ and ${\tilde{v}}_2=c\tan {\theta }_2$) and a decelerating ST wave packet ${\tilde{v}}_1\to {\tilde{v}}_2$ (${\tilde{v}}_2<{\tilde{v}}_1$); ${\theta }_1=60°$ and ${\theta }_2=50°$. The white line represents the normalized on-axis intensity evolution in $z$. (b) Measured spatiotemporal intensity profiles $I(x,z;\tau )$ of the ST wave packets at $z_1=5\mathrm{mm}$ and $z_2=25\mathrm{mm}$, corresponding to the marked axial planes in (a). (c) Measured spatiotemporal spectral projections onto the $(k_x,\omega )$ and $(k_z,\omega /c)$ planes. The dotted red curves are theoretical predictions for the constant-$\tilde{v}$ ST wave packets, the dashed green line is the light line $k_z=\omega /c$, and ${\lambda }_o\approx 800\mathrm{nm}$ (see Supplemental Material [43] for theoretical predictions).

Figure 4

(a) Measured group delays (left) with respect to a pulse traveling at the initial group velocity ${\tilde{v}}_1$ at $z=0$ and group velocities (right). We plot data for accelerating and decelerating subluminal ST wave packets over a propagation distance of 30 mm and also plot for comparison data for a constant-$\tilde{v}$ ST wave packet at the initial group velocity ${\tilde{v}}_1$. (b) The same as (a) for superluminal ST wave packets. (c) The same as (b) for superluminal ST wave packets with large acceleration and deceleration. Points are the measured data, and the dotted curves are theoretical fits (quadratic for the delays, linear for the group velocities).