Interference in $\omega -2\omega $ atomic ionization within the strong-field approximation: Beyond the perturbative regime

We analyze interference processes in atomic ionization induced by a two-color laser with fundamental frequency $\omega$ and its second harmonic $2\omega$. The interplay between inter- and intracycle interference processes give rise to multiphoton peaks which can be named as main or ATI peaks and sidebands, in analogy to the well-known RABBIT (reconstruction of attosecond harmonic beating by interference of two-photon transitions). We use the saddle point approximation (SPA) to extract the complex ionization times of the interfering electron trajectories. Changing the relative phase between the two colors, the doubly differential momentum distribution of emitted electrons can be controlled. We study the dependence of the electron emission as a function of the relative phase between the $\omega$ and $$ fields within the strong field approximation (SFA) but beyond the perturbative regime. We focus on the extraction of the phase delays accounting the electron forward emission in the direction of the polarized electric fields. We characterize the time delays in the emission of electrons for visible frequency of the pump and its first harmonic as a probe [Ti:Sapphire laser ($800$ nm) together with the first harmonic ($400$ nm)] for a typical $\omega -2\omega$ configuration for argon ionization. We find excellent agreement between our SPA results and the corresponding SFA (without any further approximation) and also with previous perturbative theories.