Nonlinear molecules permutational symmetry

From the form of the polarization it is clear that in order to observe any nonlinear optical effect, the input beams must not be copropagating. Furthermore, nonlinear optical effects through the tensor y eee requires two different input frequencies (otherwise, the tensor components would vanish because of permutation symmetry in the last two indices, i.e., ytfl eee = Xijy ) For example, sum-frequency generation in isotropic solutions of chiral molecules through the tensor y1 1 1 has been experimentally observed, and the technique has been proposed as a new tool to study chiral molecules in solution.59,61 From an NLO applications point of view, however, this effect is probably not very useful because recent results suggest that the response is actually very low.62... 

If some of the involved frequencies are degenerate the above expressions for 2D-molecules can be further simplified due to the permutations symmetry of the nonlinearities. The two-photon absorption is frequently characterized with a coefficient <5 defined as the loss of photons through the two-photon process upon propagation along the z-direction... 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating the corresponding chiral surfaces [68, 69 and 70]. From the point of view of performing surface and interface spectroscopy with nonlinear optics, we must first examine the nonlinear response of the bulk liquid. Clearly, a chiral liquid lacks inversion symmetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the permutation symmetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

The number of independent nonzero tensor elements depends on the nonlinear optical process and on the symmetry of the molecule, see for instance Bogaard and Orr (1975). For example, is symmetric with respect to the permutation of the second and third indices (see O Eq. 11,100) and this can be used to simplify the equations for the parallel and perpendicular components. For nonzero frequencies, the number of independent tensor components to be computed decreases when Kleinman s symmetry (Kleinman 1972) is assumed - that is, we assume that we can permute the indices of the incoming light without changing the corresponding frequencies,... 

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