Modification of the saddle-point equation for strong-field ionization from atomic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> orbitals

The saddle-point approximation (SPA) within the framework of strong-field approximation is extensively applied in strong-field physics because it offers clear physical insight into intense light-matter interactions. In this study, we introduce an $m$-resolved saddle-point approximation ($m$-SPA), where $m$ is the magnetic quantum number, to analyze the ionization dynamics of atoms initially in $p$ orbitals subjected to an intense laser field. Our results reveal that $m$ influences not only the prefactor of the ionization rate but also the saddle-point equation itself, an effect overlooked in prior studies. The accuracy of the $m$-SPA method is validated through comparisons with the backpropagation method and the strong-field approximation, showing its superiority over the conventional SPA approach. By employing the $m$-SPA approach, we are able to extract more accurate distributions of the initial tunneling exit energy and position, thereby allowing for a more precise determination of the asymptotic photoelectron characteristics. Additionally, we extend the conservation law for angular momentum and energy of photoelectrons from $s$ orbitals to $p$ orbitals and from circularly polarized to elliptically polarized laser fields, both at the tunnel exit and in the asymptotic region. This work facilitates future research on strong-field ionization from arbitrary atomic orbitals.