Theoretical introduction to cross-polarized wave generation

Theoretical introduction to cross-polarized wave generation

This paragraph introduces theoretically the process of cross-polarized wave generation. An historical introduction is presented in section 4.3.XPW generation is a four-wave mixing process, governed by the anisotropy of the real part of the crystal third-order nonlinearity tensor χ(3), where an intense linearly polarized incident wave generates a new linearly polarized wave in the orthogonal direction. This process has im- portant applications as a nonlinear filter to increase the temporal contrast of ultra-short laser pulses and to broaden their spectrum (section 4.2). In the first section I derive the system of differential equations governing XPW generation and analyze the contribution from different terms. Due to the fact that this nonlinear process is generated in a crystal lattice, these terms depend on the angles of propagation of the waves with respect to the crystal axis. This analysis continues in the second section with the deriva- tion of the system in Hamiltonian form and the analysis of the trajectories in phase space. All these results are derived for monochromatic plane waves. This analysis is then extended for ultra-short pulses. One of the most interesting properties of XPW generation is that the process is automatically phase-matched and is therefore adapted for applications involving ultra-short pulses. For this kind of pulses temporal compression is not trivial and it is interesting to un- derstand the influence of quality of the recompression on the efficiency of the process and on the generated spectrum. This will be presented in the third section for pulses as short as 10 fs. To describe correctly the process for sub-10 fs pulses the propagation in the crystal needs to be taken into account. In this case a system of nonlinear partial differential equations needs to be solved. This is done using a numerical method called the split-step Fourier method and it is the subject of the fourth section.

Cross-polarized wave generation

The above expressions (3.10) do not convey new physical information. They simply mean that rotating β by 90◦, the B-wave would become A, and the A-wave would become B. They could Investigation of Eq. (3.11b) shows that γ2 (β, ϕ, ϑ) reaches its global maxima (with equal absolute values of γ2) when ϕ = 2nπ/4 and ϑ = 2 (m + 1) π/4, or when ϕ = 2 (n + 1) π/4 and ϑ = 2mπ/4 where m, n = 0, 1, . . .. This means that, to reach the maximum of |γ2| for specific values of β-angle, the light propagation direction k should lie in the (x, y), (x, z), orInvestigation of Eq. (3.11b) shows that γ2 (β, ϕ, ϑ) reaches its global maxima (with equal absolute values of γ2) when ϕ = 2nπ/4 and ϑ = 2 (m + 1) π/4, or when ϕ = 2 (n + 1) π/4 and ϑ = 2mπ/4 where m, n = 0, 1, . . .. This means that, to reach the maximum of |γ2| for specific values of β-angle, the light propagation direction k should lie in the (x, y), (x, z), or 1], also known as holographic-cut orientations. For the eight equivalent to [111] propagation directions γ2 = γ4 = 0 for any angle β and the orthogonal generation term is always zero regardless of the polarization state or orientation, and the medium shows isotropic behavior.

In the non-depleted regime, the efficiency of XPW generation is proportional to the square of γ2 ([8, 9]). The dependence of this coefficient on angle β is plotted on Fig. 3.2 for three orientations: the most efficient one [101] (holographic-cut), [001] (z-cut), and for the polarization preserving orientation [111] (Fig. 3.3). In calculations γ0 = 1 and σ = −1.2 (typical values for the BaF2 anisotropy [6]) were used. All the firsts experiments of XPW generation were done using z-cut crystals. This was an academic choice because in this case the k-axis is along the z or equivalent axis. This theoretical analysis demonstrates that the maximum absolute value of γ2 for [101]- cut is 12% greater than the maximum for z-cut and this corresponds to an almost 26% increase in the XPW efficiency. This advantage does not depend on the particular choice of the values of γ0 and σ since γ2 depends linearly on them for arbitrary crystal orientation i.e γ2 ∝ γ0σ.