Saturation of the nonlinear refractive index in atomic gases

Christian Köhler, Roland Guichard, Emmanuel Lorin, Szczepan Chelkowski, André D. Bandrauk, Luc Bergé, and Stefan Skupin

Phys. Rev. A 87, 043811 – Published 10 April 2013

Abstract

Motivated by the ongoing controversy over the origin of the nonlinear index saturation and subsequent intensity clamping in femtosecond filaments, we study the atomic nonlinear polarization induced by high-intensity ultrashort laser pulses in hydrogen by numerically solving the time-dependent Schrödinger equation. Special emphasis is given to the efficient modeling of the nonlinear polarization at a central laser frequency corresponding to a wavelength of 800 nm. Here, the recently proposed model of the higher-order Kerr effect (HOKE) and two versions of the standard model for femtosecond filamentation, including either a multiphoton or tunnel ionization rate, are compared. We find that around the clamping intensity the instantaneous HOKE model does not reproduce the temporal structure of the nonlinear response obtained from the quantum-mechanical results. In contrast, the noninstantaneous charge contributions included in the standard models ensure a reasonable quantitative agreement. Therefore, the physical origin for the observed saturation of the overall electron response is confirmed to mainly result from contributions of free or nearly-free electrons.

Received 18 January 2013

DOI:https://doi.org/10.1103/PhysRevA.87.043811

Authors & Affiliations

Christian Köhler¹, Roland Guichard², Emmanuel Lorin³, Szczepan Chelkowski⁴, André D. Bandrauk⁴, Luc Bergé¹, and Stefan Skupin^5,6

¹CEA-DAM, DIF, F-91297 Arpajon, France
²CNRS, UMR 7614, LCPMR, 75231 Paris Cedex 05, France
³Carleton University, Ottawa (Ontario) K1S 5B6, Canada
⁴Département de chimie, Université de Sherbrooke, Sherbrooke (Québec) J1K 2R1, Canada
⁵Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
⁶Friedrich Schiller University, Institute of Condensed Matter Theory and Optics, 07743 Jena, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 87, Iss. 4 — April 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Model parameters extracted from 1D TDSE simulations. (a) $χ^{(3)}$ , (b) $χ^{(5)}$ , (c) $χ^{(7)}$ for $τ = 10$ fs (blue circles), $τ = 20$ fs (red squares) and $τ = 100$ fs (green crosses) as a function of the peak intensity $I$ . (d) The MPI cross section $σ_{6}$ extracted from 1D TDSE simulations [same color coding as in panels (a)–(c)].Reuse & Permissions
Figure 2
1D simulation results. For (a) $τ = 20$ fs and (b) $τ = 100$ fs the nonlinear polarization at $ω_{0}$ is shown for TDSE calculations (thick solid black lines with circles) and compared to the standard models (dashed red lines $\equiv$ MPI rate, thin solid blue $\equiv$ tunnel rate), the HOKE_35 model (dotted green lines) containing a $χ^{(3)}$ and $χ^{(5)}$ contribution only and the fitted HOKE_FIT model (dash-dotted green lines). Panel (c) distinguishes bound (green filled circles), continuum (blue squares) and bound-continuum (red crosses) electronic contributions to the nonlinear polarization (black circles) for $τ = 20$ fs (see text for details). Additionally, the continuum contribution from the standard model with tunnel ionization rate is shown (dashed blue line). In panel (d) the free-electron density according to TDSE simulations (thick solid black line) is compared to the one obtained from the MPI rate (dashed red line) and tunnel rate (thin blue line) as used in the standard models.Reuse & Permissions
Figure 3
1D nonlinear polarization in time domain. (a) Results from TDSE and approximate models for $τ = 20$ fs at $I = 20$ TW/cm $^{2}$ , where free-electron contributions are negligible and the response is governed by the instantaneous contribution from bound electrons. (b)–(e) Temporal shapes, where delayed charge contributions become important: (b) TDSE and HOKE_FIT, (c) TDSE and standard model with MPI rate, and (d) TDSE and standard model with tunnel rate for $τ = 20$ fs and $I = 35$ TW/cm $^{2}$ . (e) TDSE and standard models with MPI rate and tunnel rate for $τ = 100$ fs and $I = 24$ TW/cm $^{2}$ . All temporal field shapes were filtered in the Fourier domain using a hyperbolic tangent mask around $0.8 ω_{0} < ω < 1.2 ω_{0}$ . Color coding is the same as in Fig. 2.Reuse & Permissions
Figure 4
Model parameters extracted from 3D TDSE simulations. (a) $χ^{(3)}$ , (b) $χ^{(5)}$ , (c) $χ^{(7)}$ for $τ = 20$ fs (black squares), $τ = 100$ fs (magenta crosses) as a function of the peak intensity $I$ . (d) The MPI cross section $σ_{6}$ extracted from 3D TDSE simulations [same color coding as in panel (a)].Reuse & Permissions
Figure 5
3D simulation results. $P_{NL} (ω_{0})$ for (a) $τ = 20$ fs and (b) $τ = 100$ fs. $P_{NL} (t)$ from 3D simulations and standard model with (c) MPI rate, (d) tunnel rate, and (e) HOKE_FIT model for $τ = 20$ fs, $I = 50$ TW/cm $^{2}$ . (f) 3D TDSE simulations and HOKE_FIT model for $τ = 100$ fs, $I = 40$ TW/cm $^{2}$ . The pulse intensities are chosen to show the delayed nature of charge contributions.Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information