Susceptibility tensors

The appropriate (complex) susceptibility tensor for this generator is = co. + 01 + co ). 

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Consider an isotropic medium that consists of independent and identical microscopic cln-omophores (molecules) at number density N. At. sth order, each element of the macroscopic susceptibility tensor, given in laboratory Cartesian coordinates A, B, C, D, must carry s + 1 (laboratory) Cartesian indices (X, Y or Z) and... 

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. 

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

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... 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization Ps 2a)) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor co, m) is large enough to dwarf the contribution from higher... 

The electric quadrupole Q 2co) involves both the gradient of the electromagnetic incident electric field E u)) and the gradient of the electric quadrupole susceptibility tensor Xq 2o), CO, co). This problem is nonetheless solved by the mere addition of supplementary terms in the surface susceptibility tensor. As a result, the surface susceptibility tensor becomes an effective tensor instead of a purely surface specific one [27,38] ... 

Interestingly, the contributions from the gradient of the electromagnetic field across the interface, Tfg xx and Tfg zzz, which scale with the mismatch in the optical dielectric constants of the media forming the interface [37], only appear in the susceptibility tensor components nd xl zzz- Therefore, these contributions may be rejected with a... 

TABLE 1 Absolute Magnitude of the Nonvanishing Susceptibility Tensor Component at Several Air-Liquid and Liquid-Liquid Interfaces [40]. All units are in esu but Transformation into SI Units is Obtained Using the Relationship 1 esu = 3.72 x 10 m ... 

If the electric dipole contribution dominates in the total SH response, the macroscopic response can be related to the presence of optically nonlinear active compounds at the interface. In this case, the susceptibility tensor is the sum of the contribution of each single molecule, all of them coherently radiating. For a collection of compounds, it yields ... 

The surface susceptibility tensor of a chiral surface possesses different symmetry properties as compared to the surface susceptibility tensor of an isotropic surface. The main difference for a chiral surface arises from the axes OX and the O Y, the two axes in the plane of the surface, which are no longer indistinguishable. The nonvanishing elements of the susceptibility tensor are then [52] ... 

Another determination of the surface equilibrium entails the use of the coupling of the DC electric field present at charged interfaces with the electromagnetic field, as described in the theoretical section. Integration of the nonlinear polarization over the whole double layer leads to the following expression of the effective susceptibility tensor ... 

Here E and E are electric field amplitudes on the surface and in vacuo interrelated by the Fresnel formulae c0 is the velocity of light in vacuo x°jf (K,[Pg.57]

Quantitative Determination of Electric and Magnetic Second-Order Susceptibility Tensors of Chiral Surfaces... 

Nonvanishing Components of Second-Order Susceptibility Tensor for Second-Harmonic Generation in Electric-Dipole Approximation for Achiral and Chiral Isotropic (i.e. isotropic in the plane of the film) Films0... 

The coefficients /, g, and h are unique for each second-harmonic signal and depend on the three susceptibility tensors. We normalize the relative values of the tensor components to = 1- The task is then to determine the complex values of the other 14 tensor components (see Table 9.2). A sufficient number of 8 independent measurements is provided by the p- and s--polarized components of the reflected and transmitted second-harmonic signals for the two orientations of the sample shown in Figure 9.17. The change in sample orientation corresponds to a coordinate transformation that reverses the... 

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