Measurement of the laser pulse phase velocity in plasma channel for DLA optimization

We demonstrate a novel, direct method for measuring the phase velocity $v_{\phi}$ of an intense laser pulse within a plasma channel - the crucial parameter that controls the resonance condition in direct laser acceleration (DLA). The technique exploits the second harmonic (SH) radiation generated at the channel sheath - a phenomenon previously observed in laser-wakefield acceleration experiments. The SH emission angle is governed by a phase-matching condition that directly depends on $v_{\phi}$. Experimental measurements performed using a 1 TW, 10 Hz Ti:Sa laser system yield phase velocities in the range $v_{\phi}=(1.010-1.030)c$ for plasma electron densities in the range $n_e=(0.01-0.06)n_{cr}$. The diagnostic is validated through quasi-3D particle-in-cell (PIC) simulations that reproduce the experimental conditions. This work provides a way to optimize DLA schemes by enabling in-situ measurement of the laser pulse phase velocity in plasma channels.