Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities

Abstract

We present a detailed study of soliton compression of ultra-short pulses based on phase-mismatched second-harmonic generation (i.e., the cascaded quadratic nonlinearity) in bulk quadratic nonlinear media. The single-cycle propagation equations in the temporal domain including higher-order nonlinear terms are presented. The balance between the quadratic (SHG) and the cubic (Kerr) nonlinearity plays a crucial role: we define an effective soliton number -related to the difference between the SHG and the Kerr soliton numbers -and show that it has to be larger than unity for successful pulse compression to take place. This requires that the phase mismatch be below a critical level, which is high in a material where the quadratic nonlinearity dominates over the cubic Kerr nonlinearity. Through extensive numerical simulations we find dimensionless scaling laws, expressed through the effective soliton number, which control the behaviour of the compressed pulses. These laws hold in the stationary regime, in which group-velocity mismatch effects are small, and they are similar to the ones observed for fiber soliton compressors. The numerical simulations indicate that clean compressed pulses below two optical cycles can be achieved in a β-barium borate crystal at appropriate wavelengths, even for picosecond input pulses.

Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression

“…Two different nonlocal response functions appear naturally, one with a localized amplitude -representing the stationary regime -and one with a purely oscillatory amplitude -representing the nonstationary regime. In the presence of GVM they are asymmetric and thus give rise to a Raman effect on the compressed pulse.In the theoretical analysis we may neglect diffraction, higher-order dispersion, cubic Raman terms, and selfsteepening to get the SHG propagation equations for the FW (ω 1 ) and SH (ω 2 = 2ω 1 ) fields E 1,2 (z, t) [3, 9]:where η = 2/3 for Type I SHG [3]. κ j = ω 1 d eff /cn j , d eff is the effective quadratic nonlinearity, ρ j = ω j n Kerr,j /c, and n Kerr,j = 3Re(χ (3) )/8n j is the cubic (Kerr) nonlinear refractive index.…”

“…The scaling conveniently gives the SHG soliton number [3,4], N In the cascading limit ∆β ≫ 1 the nonlocal approach takes U 2 (ξ, τ ) = φ 2 (τ ) exp(−i∆βξ), keeping its time dependence but neglecting the dependence on ξ of φ 2 . To do this the coherence length L coh = π/|∆k| must be the shortest characteristic length scale in the system, which is true in all cascaded compression experiments.…”

Few-cycle solitons and supercontinuum generation with cascaded quadratic nonlinearities in unpoled lithium niobate ridge waveguides

“…However, such a self-defocusing nonlinearity should work with the normal GVD to excite solitons, so the window of operation is actually from 1100 nm to the GVD transition position at 1900 nm. Within such a "compression window" [17] the phase-mismatch parameter is actually below the critical value Δk c referring the balance between the cascaded and the Kerr nonlinearity, see Fig. 3(c).…”

“…Figure 4(d) shows the electric field amplitude of the compressed pulse, with clean and single-cycle profile. The quality factor (defined in [17]) is around 0.4. Actually, Fig.…”

Cascaded Soliton Compression of Energetic Femtosecond Pulses at 1030 nm

“…(1) shows that the effective SPM can be expressed by an effective nonlinear refractive index n I eff = n I casc + n I Kerr . For the case ∆k > 0 and n I eff < 0 we can introduce an effective soliton order N Kerr , which characterizes the cascaded soliton behavior [8]. We have previously studied cascaded soliton pulse compression at λ 1 ≃ 1.03 − 1.06 µm: in Ref.…”