Experimental confirmation of electron figure-8 motion in a strong laser field

We measure polarization-resolved fundamental, second, and third harmonic nonlinear Thomson scattering out the side of a laser focus with ${10}^{18}\mathrm{W}/{\mathrm{cm}}^2$. The separate measured polarization components are each associated with a distinct dimension of predicted electron figure-8 motion. Taken together, the measured angular emission patterns for the two polarizations unambiguously confirm the figure-8 motion. Electrons are donated from low-density helium (${10}^{-3}$ to 1 Torr) ionized early during the laser pulse. Time-resolved single-photon detection is used to distinguish signal from noise.

Figure 1

(a) Electron trajectory computed in the average center-of-mass frame of a uniform plane wave ($1\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$, $\alpha =0.7$, 800 nm). (b) Electron trajectory computed in the laboratory frame for a plane-wave pulse ($\tau =32$ fs).

Figure 2

Experimental setup inside vacuum. An off-axis parabola focuses the laser pulses. A half wave plate (not shown) rotates the linear polarization of the beam before focusing. Scattered photons traverse a wire-grid polarizer and are imaged onto the end of an optical fiber. The inset depicts an alternate viewpoint where instead the laser polarization remains fixed along the $x$ axis while the detector rotates.

Figure 3

Far-field angular emission patterns for first, second, and third harmonics in the plane perpendicular laser propagation (i.e., $\theta ={90}^{\circ }$), resolved by polarization along $\widehat{\mathit{\varphi }}$ (blue, dark gray) and $\widehat{\mathit{\theta }}$ (green, light gray). Column (a) shows the number of (redshifted) photons per steradian emitted during a single laser cycle of a plane wave ($I=1\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$) according to Eq. (8). Column (b) shows the number of photons per steradian emitted during an entire laser pulse ($I_{\text{peak}}=1\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$, $\tau =32$ fs) from an electron initially at the focal center ($w_0=3.2\mu \mathrm{m}$), computed according to Eq. (10). Column (c) gives the predicted number of photons emitted during 1000 laser shots from an ensemble of electrons distributed randomly throughout the laser focus ($\rho <4w_0$ and $\left|z\right|<z_0$), with experimentally relevant electron density, collection/detection efficiency and spectral windowing applied to harmonics (see text). Column (d) shows the number of photons measured at various angles for 1000 laser shots.

Figure 4

Cumulative histogram showing arrival times for all photons measured for the second-harmonic data set plotted in Fig. 3. The inset repeats Fig. 3 column (d) for second-harmonic using a 200 ns time window.

Figure 5

Far-field emission pattern ($\widehat{\mathit{\varphi }}$ polarized) computed without the $\mathbf{B}$-field (blue, dark gray) and with the $z$ component of the electron trajectory artificially set to zero (orange, light gray). Compare with column (b) of Fig. 3. The $\widehat{\mathit{\theta }}$ polarization component vanishes.