3D+1 Crank-Nicolson algorithm implementation for solving nonlinear ultrafast pulse propagation in a Multi-pass cell

P.RUEDA SUESCUN; F. FURCH; F. VIDELA; G.A. TORCHIA

Conferencia; Ultrafast Optics UFO XIII-2023; 2023

A new routine to model the Homogeneous Non-Linear Schrodinger partial differential equation in three spatial and one-time coordinates was implemented by a 4D Tensor, Crank-Nicolson, second order ADI (Alternating Direction Implicit) and Adam-Bashforth algorithms in Python. This model includes self-focusing and plasma phenomena to study spectralbroadening in light propagation.Laser pulse compression is a central topic for isolated attosecond pulse generation considering that typically few cycle pulses are required to drive the nonlinear process of high harmonic generation. Nonlinear pulse compressionpermits reducing the laser pulse duration from tens or hundreds of optical cycles to only a few optical cycles and even reaching the single-cycle limit. This process is based on the generation of new frequencies driven by theself-phase modulation effect which is a third-order phenomenon that depends directly on the input pulse intensity and its interaction with a non-linear media. Then, the pulse compression is realized by spectral phase flattening,typically utilizing a set of chirped mirrors that introduce negative second-order dispersion. In a recent work, we demonstrated pulse compression from 45 fs long pulses at 800 nm and 1kHz repetition rate to around 8 fs [1]. Inthis case, spectral broadening of the input pulses was realized during 5.5 round trips in a multi-pass cell enclosed in a chamber filled with Argon gas at 1.5 atm of pressure. . The current work introduces a model to describe thebehavior of intense light pulses propagating in a cavity considering the experimental conditions in our recent experiment.In order to recreate the experimental conditions described above, we use a 3D+1D homogeneous non-linear Schrödinger partial differential equation (NLSE) to simulate nonlinear pulse propagation in the cavity and model the spectral broadening process [2]. In our model, the following linear and nonlinear terms are considered: diffraction, first and second-order dispersion, Kerr effect, self-focusing, self-phase modulation, and self-steepening. Normally, the conditions during pulse propagation in the multi-pass cell are adjusted to avoid plasma generation. Our computational model describes weak ionization effects around the focal plane of the cell. In the literature, there exist several numerical methods for solving such equations. In this work, we use the Crank-Nicolson splitting method. Split step Fourier method (SSFM) is perfect for solving nonlinear effects, but, when dealing with larges spectral broadening becomes time-consuming for spectrum broaden than 10 THz.Additionally, FFT operations in a large-size array, may lead to an overall computation time measured in days.In this case, the Crank-Nicolson performance would be a better solution because the spectral bandwidth of the propagating pulse result greater than 100THz, and because of its versatility in the sense that we could treat our NLSE in either time or spectral domain.In addition, we have implemented a second-order ADI algorithm to operate a 3D tensor in which we can distinguish two spatial coordinates representing the cross sections of our Gaussian beam, and a third coordinate for time. The last coordinate is the propagation along the whole round-trip in the multi-pass cell cavity. Finally, in our numerical code, we have implemented a routine called second-order Adam Bashforth method which involves non-linearterms such as self-phase, modulation, self-steepening, and plasma effects [3,4].References[1] P. Rueda, F. Videla, T. Witting, G. Torchia, and F. Furch, ?8-fs laser pulses from a compact gas-filled multi-passcell," Opt. Express 29, 27004-27013 (2021).[2] Couairon, A., Brambilla, E., Corti, T. et al. Practitioner?s guide to laser pulse propagation models andsimulation. Eur. Phys. J. Spec. Top. 199, 5?76 (2011).[3] Smith, G. D., & Smith, G. D , Numerical solution of partial differential equations: finite difference methods.Oxford University Press (1985).[4] Segarra, J. Métodos numéricos Runge-Kutta y Adams Bashforth-Moulton en Mathematica. Revista Ingeniería,Matemáticas y Ciencias de la Información, 7(14), 13-32 (2020).