Nonlinear susceptibility tensors

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

The second-order nonlinear susceptibility tensor ( 3> 2, fOj) introduced earlier will, in general, consist of 27 distinct elements, each displaying its own dependence on the frequencies oip cci2 and = oi 012). There are, however, constraints associated with spatial and time-reversal symmetry that may reduce the complexity of for a given material [32, 33 and Ml- Flere we examine the role of spatial synnnetry. 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating die corresponding chiral surfaces [, and 70]. From the point of view of perfomiing surface and interface spectroscopy with nonlinear optics, we must first examhie the nonlinear response of tlie bulk liquid. Clearly, a chiral liquid lacks inversion synnnetry. 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 pemiutation synnnetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

We now consider this issue in a more rigorous fashion. The inference of molecular orientation can be explamed most readily from the following relation between the surface nonlinear susceptibility tensor and the molecular nonlinear polarizability... 

Notice that /pijn = nonlinear susceptibility tensor elements... 

There are different paths to achieving surface specificity. One can exploit optical susceptibilities and resonances that are nonzero only at the surface or only for the molecular species of interest adsorbed on the surface. Examples include the use of second-order nonlinear mixing processes such as second harmonic generation7-9 for which the nonlinear susceptibility tensor is nonzero only where inversion symmetry is broken. Spectroscopic techniques with very high selectivity for molecular resonances such as surface-enhanced infrared or Raman spectroscopy10-12 may also be used. 

By measuring the temperature dependence of second-harmonic generation (SHG) of the neodymium laser wavelength at 1.06 pm in sodium nitride for five different radiation geometries which correspond to the five nonvanishing components of the nonlinear susceptibility tensor, Vogt etal. 3 2) could determine these tensor components and the coherence length 1 = 4 2 re the... 

In the case of second-harmonic generation, the second-order nonlinear susceptibility tensor elements are symmetric in their last two Cartesian indices and are unchanged by the permutation of their second and third frequency arguments because they are identical. Thus, Eq. (28) can be rewritten in the simplified form... 

The nonlinear response of the interface may then be characterized in terms of a surface (or interface) nonlinear susceptibility tensor xp This quantity relates the applied electromagnetic fields to the induced... 

In order to obtain a useful material possessing a large second order nonlinear susceptibility tensor % 2) one needs to use molecules with a large microscopic second order nonlinear hyperpolarizability tensor B organised in such a way that the resulting system has no centre of symmetry and an optimized constructive additivity of the molecular hyperpolarizabilities. In addition, the ordered structure thus obtained must not loose its nonlinear optical properties with time. The nonlinear optical (NLO) active moieties which have been synthesized so far are derived from the donor-rc system-acceptor molecular concept (Figure 1). 

Let us first give a sketch of the underlying theory. We consider an interface formed by two bulk media with centrosymmetry. Because of the broken symmetry at the interface, the second-order nonlinearity of the interface layer becomes nonvanishing (under the electric-dipole approximation). We can use a surface nonlinear susceptibility tensor to... 

It is seen that for high peak power applications a low refractive index is required. In the crystal case, to avoid SHG (second-harmonic generation), a centro-symmetric structure has to be searched for, because the so-called Miller rule (Miller 1964) gives for the allowed elements of a nonlinear susceptibility tensor for second-harmonic generation. 

Symmetry point group Dielectric tensor Elastic tensor ors , Piezoelectric tensor dimn Elasto-optic tensor Pmn Electrooptic tensor rmk Nonlinear susceptibility tensors x > 1 X< > Table number... 

As with the evaluation of the linear susceptibility, one can determine the properties of the nonlinear susceptibility tensor from Eqs. 2 and 3. In regions of low dispersion the x ifk tensor exhibits special symmetry properties that are referred to as Kleinman... 

Parametric Processes. Parametric processes are those in which photon energy is conserved while the light propagates through the medium. Parametric phenomena of particular interest for devices due to the real part of the nonlinear susceptibility tensors include... 

Cholesterics belong to the Doo symmetry group which allows for one independent non-vanishing element of namely x zxy = Xly (The director is along the X axis). For non-absorbing materials Kleinman s rule holds also, according to which the elements of a nonlinear susceptibility tensor are invariant under any permutation of their indices.Taking into account this latter restriction one finds that xixy vanishes as well, so finally no SHG is allowed. 

We obtain now an expression for the components of the NLC nonlinear-susceptibility tensor due to a shift of the phase-transition temperature. In our case the third-order nonlinear polarizability can be written in the form... 

Starting from this relation we can formally introduce the cubic nonlinear susceptibility tensor whose components are given by... 

Now we turn to the second term in (6.6) with the nonlinear susceptibility tensor We assume that the incident wave contains only two frequencies (0 and 2 With 0) = co (02) we have the detailed description... 

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. 

The nonlinear susceptibility tensor can then be represented as a 3 X 6 matrix containing 18 elements. In the transparent region, i.e. outside of absorption bands, one may assume the validity of the Kleinman symmetry condition, vv hich states that the indices /, /, k may be freely permuted ... 

The nonlinear optical susceptibilities can be calculated, in principle, on the basis of the density matrix formalism. However, one can often draw some conclusions about the nonlinear optical output from symmetry considerations. The nonlinear susceptibility tensors reflect the structural symmetry of the crystal since they are determined by its electronic or vibrational states. 

Second-order nonlinear optical properties describe the coupling interaction between two electric fields (as described in Equation 3.129) and the crystal. For the ideal wurtzite ZnO, with the 6 mm symmetry, there are four nonvanishing second-order nonlinear susceptibility tensor elements, = Xm- and... 

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