Third-order susceptibility phase matching

Of course, the frequencies and wave vectors fulfil the phase-matching conditions. The third-order susceptibility Xijw is a fourth-rank tensor having a priori 81 elements. In an isotropic material, there remain 21 non-vanishing elements, among which only three are independent [69]. The simplest case consists in a unique incident plane wave, linearly polarized. Indeed, the third-order polarization vector is then parallel to the electric field and reduces to the sum of two propagating terms, one oscillating at the wave circular frequency co, and another at the circular frequency 3(o. The amplitudes of these two contributions write, respectively. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

The global energy conservation condition, Eq. (4.11), is explicitly demonstrated for resonant processes up to third order (5 3), particularly for resonant passive processes, at exact resonance, where population change can be achieved at a nonquadrature level by the fields. The phase matching condition is assumed, Ak = 0. As before, the material resonance at the one-photon level is taken into account by the complex wave vector, k, = k -I-ik", whose imaginary part is absorbed into the amplitude, ,( ) = sxp[-k" r]. The electrical susceptibility is expressed in terms of the scalar Cartesian component, as given in Eq. (2.17). 

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