Soliton delay driven by cascading and Raman responses

Abstract

Summary form only given. Quadratic cascading response is evoked during the ultrafast and phase mismatched (cascading limit) second harmonic generation (SHG) process, which becomes more and more recognized alongside with typical nonlinear phenomena such as nonlinear phase change, pulse intrinsic self-steepening (SS) and material Raman effects. The mean value (local component) of this cascading response has been widely investigated and known as cascading quadratic nonlinearity (has a soliton number N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">casc</sub> ) which gives rise to a Kerr-like phase change and is tailored by the phase mismatch (Δk) between the fundamental wave (FW) and the second harmonic (SH) [1]. Moreover, the first order of the cascading response is revealed as an effective SS term [2,3], which adds to the intrinsic SS and induces shock front on pulses. Then, such SS effects will cause pulse delay when operating with material dispersions. Meanwhile, first order Raman response will also cause pulse delay by continuously red-shifting the pulse spectrum. Hence, there comes a pulse delay competition between the cascading and Raman responses.In this work, we analytically and numerically study the soliton pulse delay driven by first order cascading and Raman responses and demonstrate a potential delay balance by tuning the cascading delay time through Δk.Analytically, in the cascading limit, the coupled wave equations governing the FW and SH can be degenerated to the famous nonlinear Schrödinger like (NLS-like) equation governing an undepleted FW , in which the cascading and Raman responses are both included [4]. Then by leaving the first order responses and eliminating the higher order terms, the FW amplitude and phase (written as: UFW = A(ξ,τ)eiφ(ξ,τ)) equations are derived as [4]: (in dimensionless form and in dispersionless condition) ∂ 2 2 = N ffA2 - 2τRNέ biCA α + 2N Ncubic + 2τN / I AZ ∂τ c c- c c NA2 aAaφ (4N as 3N ubi + 2τ ατ' aξ aίj where N ff = N2casc - N2cubic scales the total self-defocusing nonlinearity necessary to hold the soliton propagation with normal dispersion, τc and τR are cascading and Raman delay time and τc ∝ GVM/Δk. It is noted that the pulse amplitude is strongly dependent on cascading terms (Ncasc and τc) while the phase is dominated by Raman effects (τc term), especially after the soliton formation (where ∂φ ∂τ = 0). Numerical results are shown by solving the above mentioned NLS-like equation. Fig. 1(a) shows that the cascading response gives rise to shock front in the dispersionless condition and causes slow pulses (Fig. 1(b)) with normal dispersion, tuned by Δk. Then, in Fig. 1(c), with strong material Raman effects over the cascading, fast pulses are driven as more red-shifted pulses would travel faster with normal dispersion. At last, introducing a stronger cascading delay time (with smaller Δk), fast pulses are tuned back to the zero delay position (Fig. 1(d)).