Few-Attosecond electron pulse trains with tunable periods produced by two counter-propagating lasers

Attosecond electron pulses enable real-time probing of ultrafast matter dynamics, yet conventional modulation schemes suffer from drastically shortened longitudinal focal lengths when targeting sub-attosecond durations. To address this bottleneck, we propose and demonstrate a compact scheme utilizing two counter-propagating lasers that reveals a previously unidentified stable modulation regime. Contrary to established models of ponderomotive forces and stochastic acceleration in dual-laser fields, we show that a specific parametric resonance condition permits the electron beam to be stably modulated into highly periodic attosecond trains with rapid energy gain. Using a sub-relativistic electron beam, simulations confirm the generation of ~1 as pulses with a Lorentz factor up to 15 and a relative energy spread below 0.02%, extending the focal length by three orders of magnitude compared with conventional approaches. This work identifies the critical transition from ordered modulation to stochastic acceleration, offering a viable route to overcoming the focal-length barrier in attosecond electron-pulse applications.