Third-order polarization propagator

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. 

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. 

X and x are the second and third-order nonlinear optical susceptibilities. Two important manifestations of optical nonlinearities are harmonic generation and refractive fndex modulation by electric and optical fields. Their origin can conveniently be explained by considering a plane wave propagation through the nonlinear medium. The polarization is then given by... 

The third-order and fifth-order polarizations propagating in the 2 2 - k direction are then... 

The TD approach is the natural candidate for the interpretation of data coming from pump-probe spectroscopies [13]. If both pump and probe fields are weak, the third-order time-dependent polarization can be computed propagating wave-packets with field-free Hamiltonians [14]. A nonperturbative approach can also be followed, numerically solving the TDSE with the external fields included in the Hamiltonian [15, 16]. Chapter 9 of this book presents the equation-of-motion phase-matching approach (EOM-PMA), which can be considered a mixed perturbative-nonperturbative method. 

Evaiuating aii the terms in the partitioned form of the poiarization propagator, Eq. (10.15), through third order one obtains the third-order polarization propagator approximation (TOPPA). The expressions for aii matrix eiements have been derived but oniy parts have been impiemented (Geertsen et at, 1991a). 

This keeps essentially the structure of the SOPPA equations but replaces in all matrix elements the first-order MP doubles correlation coefficients, Eq. (9.67), and the second-order MP singles correlation coefficient, Eq. (9.71), by coupled cluster singles and doubles amplitudes. In the earlier coupled cluster singles and doubles polarization propagator approximation (CCSDPPA) (Geertsen et al., 1991a), a precursor to SOPPA(CCSD), this was done only partially and in particular not in the second-order correction to the density matrix Very recently, a third method (Kjaer... 

The first two terms are electric quadrupole in character while the last term is magnetic dipolar. Under excitation by a single plane wave, the first term vanishes. In a homogeneous medium the second term vanishes by Gauss Law. The third term describes the induced polarization which is along the propagation direction. It can only radiate at the discontinuity of the surface. The full expression for the second-order nonlinear polarization in an isotropic medium is then written as the sum of the surface and bulk polarizations [78] ... 

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