Tools for numerical modelling of nonlinear propagation in hollow capillary fibres and their application

Abstract

The development of new coherent and ultrashort light sources is of great relevance for exploring fundamental processes and different applications in science. The most successful technique for generating ultrashort laser pulses, in terms of energy and pulse duration, is using hollow capillary fibre compressors. The different strategies to further increase the pulse energy and to achieve shorter pulses at non-conventional wavelengths, lead to continuous improvement of this technique. In this work, we present the theoretical framework of the nonlinear propagation in hollow capillary fibres through the propagation equation and the spatio-temporal effects that appear in the nonlinear dynamics. To numerically study the nonlinear propagation of the pulse in the hollow capillary fibre, we present different numerical models considering only the spatial effects, (1+1)D model, the full spatio-temporal dynamics and ionization, (2+1)D model, or the case with lack of cylindrical symmetry, (3+1)D model. To show the performance of some of these models in a particular case, we study the generation of ultrashort and energetic dispersive waves inside the hollow capillary fibre. We show that the emission of a dispersive wave at a fixed wavelength for different pump wavelengths is possible by parameter scaling.

Robust Isolated Attosecond Pulse Generation with Self-Compressed Subcycle Drivers from Hollow Capillary Fibers

“…In addition, to first explore different selfcompression scenarios and perform systematic parameter scans, we reduce the (2 + 1)D computationally demanding model to a time-dependent (1 + 1)D nonlinear propagation equation for the fundamental EH 11 mode of the HCF, neglecting spatial and plasma dynamics. 41,49 This approximation accurately describes ultrashort pulse propagation in the low-intensity regime, where the peak power and the peak intensity of the pulse remain, respectively, below the critical power for self-focusing and the threshold intensity for gas ionization. 52 When complete (2 + 1)D simulations are performed, the cylindrically symmetric pulse at the HCF output T z ( , , ) is then propagated in vacuum and focused onto a low density gas target to drive HHG, as depicted in Figure 1.…”

“…The first fiber stage is theoretically modeled with a (2 + 1)D multimode nonlinear propagation equation for the pulse complex envelope scriptE ( ρ , T , z ) , which can be written as ∂ ∂ z scriptE ( ρ , T , z ) = ( L̂ + N̂ false[ scriptE ( ρ , T , z ) false] ) scriptE ( ρ , T , z ) where the operator L̂ accounts for the linear propagation effects of diffraction, complete chromatic dispersion and linear losses, and N̂ includes the nonlinear effects of self-phase modulation (SPM), self-steepening, and photoionization and plasma absorption. A detailed description of each term and their mathematical form can be found in refs and . Here the pulse propagation equation is expressed in a local time T = t – z / v g measured in a reference frame traveling with the pulse at the group velocity v g , and the spatial pulse envelope is assumed to have cylindrical symmetry and depend only upon the radial coordinate ρ = x 2 + y 2 .…”

“…In the second substep, the nonlinearity is assumed to act alone, and eq with L̂ = 0 is integrated in the time domain with a fourth-order Runge–Kutta algorithm. In addition, to first explore different self-compression scenarios and perform systematic parameter scans, we reduce the (2 + 1)­D computationally demanding model to a time-dependent (1 + 1)­D nonlinear propagation equation for the fundamental EH 11 mode of the HCF, neglecting spatial and plasma dynamics. , This approximation accurately describes ultrashort pulse propagation in the low-intensity regime, where the peak power and the peak intensity of the pulse remain, respectively, below the critical power for self-focusing and the threshold intensity for gas ionization …”