Numerical modeling of laser tunneling ionization in particle-in-cell codes with a laser envelope model

Accurate electron beam phase-space theory for ionization-injection schemes driven by laser pulses

“…The use of reduced envelope models in conjunction with analytical models to correctly mimic the newborn electrons phasespace (e.g. QFluid [18,25] , INF&RNO [26] , ALaDyn [27,28] and Smilei [29,30] ) can therefore be advantageous when long and large grid-size simulations are needed. In this respect, highly accurate analytical predictions of the rms transverse momentum, or even more accurate models for the phasespace distribution of the extracted electrons are needed.…”

“…Once electrons are extracted from the ions, their dynamics follows the prescription for a generic charged particle in an (almost) plane-wave laser pulse. After averaging the momenta during the whole first laser pulse oscillation, we obtain the initial cycle-average normalised 3D momentum u = p/m e c (see [30] and references therein) ūx = −a 0,e sin(ξ e ) , ūy = 0 , ūz = 1 2 a 2 0,e sin 2 (ξ e ) + 1 2 , (1) where ξ e is the ionization pulse phase at the extraction time and a 0,e is the local normalised pulse amplitude at the extraction position. As the electrons slip-back in the laser field, their quivering amplitude decreases, while also the longitudinal ponderomotive force gradually reduces the cycle-averaged longitudinal momentum ūz .…”

“…This is because the longitudinal grid spacing should be small enough to efficiently resolve the extraction process, occurring in a tiny fraction (usually ) of the ionization pulse wavelength. The use of reduced envelope models in conjunction with analytical models to correctly mimic the newborn electrons phase-space (e.g., QFluid [ 18 , 25 ] , INF&RNO [ 26 ] , ALaDyn [ 27 , 28 ] and Smilei [ 29 , 30 ] ) can therefore be advantageous when long and large grid-size simulations are needed. In this respect, highly accurate analytical predictions of the root mean square (rms) transverse momentum or even more accurate models for the phase-space distribution of the extracted electrons are needed.…”

“…After averaging the momenta during the whole first laser pulse oscillation, we obtain the initial cycle-average normalized 3D momentum (see Ref. [30] and references therein): where is the ionization pulse phase at the extraction time and is the local normalized pulse amplitude at the extraction position. As the electrons slip back in the laser field, their quivering amplitude decreases, while also the longitudinal ponderomotive force gradually reduces the cycle-averaged longitudinal momentum .…”

“…After averaging the momenta during the whole first laser pulse oscillation, we obtain the initial cycle-average normalized 3D momentum − → u = − → p /m e c (see Ref. [30] and references therein): u x = −a 0,e sin ξ e , u y = 0, u z = 1 2 a 2 0,e sin 2 ξ e +…”