Second-order nonlinear susceptibility origin

Induced Polarization and Origin of the Second Order Nonlinear Susceptibility... 

The subindices f, j, k indicate base directions (x,y,z) in Cartesian coordinates. Xfj i2co co,co) represents a second order nonlinear susceptibility tensor, a tensor or third rank, responsible for SHG process while Ej(a)) and Ej (co) are the laser field amplitudes in j and k directions. Pf 2coi) is the induced SHG nonlinear polarization in i direction at twice the fundamental frequency, which couples to and generates light field radiation Ei(2co) in twice the original angular frequency 2 CO. Summation over j and k is implied in Eq. (1) by the Einstein summation convention. Please notice that xfk(2(o cOy(o) vanishes for a medium with centro-sym-metry. For a non-centro-symmetry medium, not all the tensor elements exist. 

Lalama, S.J., Garito, A.F. Origin of the nonlinear second-order optical susceptibility of organic systems. Phys. Rev. A 20, 1179-1194 (1979)... 

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... 

In isotropic media the reflection of all vectors at the origin should not change the nonlinear susceptibility. This yields Xijk = —Xijk which could be only fulfilled by Xijk = 0. In all media with an inversion center the second-order susceptibility tensor vanishes This means, for instance, that optical frequency doubling in gases is not possible. 

Barnik et al. observed the same effect but deduced a much smaller nonlinear susceptibility from the measurements than Arakelyan et al. They provided a different interpretation as well in their opinion SHG was due to electric- quadrupole effects. (Quadrupole effects originate from the coupling between the polarization P and the spatial derivatives of the electric field.) As second-order quadrupole effects are not forbidden by the existence of an inversion centre the interpretation of Barnik et ai is compatible with the generally accepted symmetry properties of nematics. A detailed theoretical analysis of the problem was given by Ou-Yang et Their results indicate also that the quadrupole mechanism is responsible for SHG in MBBA rather than the existence of noncentrosymmetric blocks. 

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