Two-lens treatment of a gaussian beam with high harmonic frequency (1/N in wavelength), via Transfer-Matrix (q-vector, only rational part) treatment.
High harmonic generation e.g. via laser beams is developed for high photon coherent sources with sub-fs pulse durations. Commonly a laser with a fundamental wavelength is down focused into / onto a medium (gas or solid) where in the focus by the interaction high harmonic radiation with mulitple frequencies of the fundamental freuqency is emitted.
For the special case of solid dense - laser high harmonic creation, the interaction is based on the Relativistic Mirror Model, where a sufficient strong laser creates a plasma on the solid surface with which the rest of the laser pulse interacts. Electrons in the density gradient of the plasma surface are accelerated by the laser and form an relativistic oscillating dense surface upon which the laser is reflected. By the doppler-shift the reflected light is once per laser cycle upshifted to a higher photon energy (increase of the reflected pulse bandwidth to a quasi continuum like distribution). This creates a pulse train over the driving laser pulse, where each pulse train member has a significant smaller pulse duration (< 0.5 laser cycle). On a detector with a integration time that is much longer the driving laser pulse duration (electronic switching can NOT resolve fs or as!!!) each pulse of the pulse train is integrated, that by the uncertainty relation leads to an interference on the detector (cf. SPIDER measurment method) - where the interference can be described as modulation with the laser cycle duration (once per cycle) that in the spectrum is found as high harmonics of the fundamental frequency (or 1/N in wavelength).
This raytracing script tries to model high-harmonic gaussian beam creation close and in the laser focal region. It focuses only on the spatial characteristics of such a created beam. The created HHG are diverging in space after creation proportional to the driving laser frequency *1/N (theoretical model). In case of a plasma surface denting, the source of creation changes the HHG divergence like a curved mirror - in other words focus the beam. The mathematical treatment uses as a two lens - ray transfer matrix, where lens 1 is the initial driving laser focal lens and lens 2 is the curved plasma surface, that can be either dented outwards or inwards. The model treats the case for a displacement of the plasma surface (target) from the laser focuse, that leads to a bigger spotsize on the target as well as a reduction of the intensity. The denting depth is depending on both parameters scaling differently with either defocusing or additionaly scaling the intensity by a pulse duration enlargement (cf. chirped pulses). The model is based on Vincenti et. al, NatComm. 2014, but enlarged his model by the defocusing range.
The script focuses on real experimental parameters, which are known for the highfield laser community: M^2, Rayleigh-Length, hole boring velocity, w0 for gaussian beams, Denting depth of a plasma surface, scaling laws.
Python 3.6 ... mathematical details will follow sooon