Gouy Phase-Related Effects in the Free-Space Optical Modulation of Free Electrons

Abstract

Modulating the free-electron wave function with light brings new opportunities to create attosecond electron pulse trains, to probe the quantum coherence of systems with significantly improved spatial resolution, and to generate classical and non-classical states of light with wide tunability. It is therefore crucial to efficiently generate free-electron wave functions that are suitable for these applications. In this study, we theoretically investigate an efficient free-space optical modulation of free electrons with two counter-propagating Gaussian beams. We find that the Gaussian beams' Gouy phase not only plays a crucial role in the interaction, but also enables straight-forward generation of valuable free-electron states, including comb-shape spectra with similar amplitudes, and states with high degree of coherence. We also discuss the feasibility of demonstrating these Gouy phase-related effects with chirped femto-second laser pulses. Our study establishes a theoretical foundation and physical intuition about the role of the Gouy phase. It can provide guidance to efficiently shape the free-electron wave function for a wide range of quantum applications.

Figures (3)

• Figure 1: (a) Schematic of the interaction between the free electron and two counter-propagating Gaussian beams with frequency $\omega_1$ and $\omega_2$. (b) Sketch of the momentum conservation relation, considering the non-zero transverse photon momentum in the Gaussian beams. The blue and green vectors represent the momentum of photons with frequency $\omega_1$ and $\omega_2$, respectively. To satisfy the momentum conservation, a closed triangle should be formed, where two sides are the momentum of the two photons, and the third side is the momentum change of the free electron. (c) Illustration of the total Gouy phase (blue curve) and the additional velocity of the ponderomotive potential (red curve).
• Figure 2: Spatial and spectral evolution with (a-d) and without (e-h) the Gouy phase for the electron with phase-matched velocity ($v_e=c/5$). (a) and (c) show the evolution of the free-electron wave function amplitude in the energy and spatial (in the pseudo-comoving frame) domain, respectively. (b) is the energy-domain wave function amplitude after the interaction. (d) shows $\textrm{DoC}_m$ for $m=1,\,2,\,3,\,4,\,5,\,6$ through the interaction, where the top x-axis indicates $z=v_p t$ normalized by $z_R$. (e-h) are the counterparts for (a-d) when neglecting the Gouy phase.
• Figure 3: (a) Free-electron spectrum after the interaction when changing the field amplitudes of the two Gaussian beams. The green dotted line is the case shown in Fig. \ref{['fig:comparison']}(a-d). The cyan dashed line represents $\delta E_\textrm{Gouy}$. (b-d) Free-electron spectrum during the interaction for $E_1=$ 1.5, 1, 0.4 GV/m, respectively, as indicated by the white dotted lines in (a). (e-g) Normalized $\bar{H}_{int}(z,t)$ (Eq. \ref{['eq:H_int_time_average']}, colored curves) and $|\phi(z,t)|^2$ (gray shaded curves) in the pseudo-comoving frame at $t=[-2,-1,0,1,2,3]\times \frac{z_R}{v_p}$ for $E_1=$ 1.5, 1, 0.4 GV/m, respectively.