Nonlinear polarizability tensor

Let us first consider the dispersion of the nonlinear polarizability tensor of third order. The relation of this tensor to the triple-time correlation function is determined by the following expression (50) (here and in the following h = 1) ... 

A relation analogous to (6.86) also obtains for the nonlinear polarizability tensor Xijtm- In this case, for uj > 0, uj > 0 and uj" > 0 as well,... 

Usually the dependence of the nonlinear polarizability tensors on the wavevec-tors k, k jk", etc. is neglected. In this approximation in crystals with an inversion center the third-rank tensor (u>, uiu>"), which changes sign by inversion, vanishes and nonlinear processes involving three photons are absent. Explicit expressions for the nonlinear polarizability tensors, similar to the tensor e u , k), can be obtained within a microscopic theory. Above we presented various methods for calculation of the tensor k). The important point was that to obtain this tensor it was necessary to establish the crystal polarizability in the linear approximation with respect to the electric field. The nonlinear polarizability tensors can be obtained in a similar way, but now the crystal polarizability must be determined by taking into account higher order terms with respect to the electric field (see (16)). 

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

A surface is illuminated with a high-intensity laser and photons are generated at the second-harmonic frequency through a nonlinear optical process. For many materials, only the surface region has the appropriate symmetry to produce an SHG signal. The nonlinear polarizability tensor depends on the nature and geometry of adsorbed atoms and molecules. 

Using the same straightforward thongh cumbersome algebra, the third-order density matrix element p 2(0 and the corresponding nonlinear polarizability tensor component jyich can be obtained. It suffices to note that p (t) will involve a triple product of d . E and oscillate as where co, co, and co" are the frequency compo-... 

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... 

A simple calculation for urea by Spackman is instructive. Urea crystallizes in an acentric space group (it is a well-known nonlinear optical material), in which the symmetry axes of the molecules coincide with the two-fold axes of the space group. All molecules are lined up parallel to the tetragonal c axis. If the electric field is given by E, and the principal element of the diagonalized molecular polarizability tensor along the c axis by oc , the induced moment along the polar c axis is... 

In the above experiments in which an electrode surface is examined, the electrode potential is fixed as the electrode surface is rotated and the SH anisotropy recorded. In a useful variation of this procedure, one can fix the angle of rotation and measure the SH response as a function of potential. Since the absolute orientation of the crystal is known, information can be derived about the various tensor elements describing the nonlinear polarizability from these crystalline surfaces. By fixing the angle and applying a transient potential pulse, one can perform time resolved measurements and watch the evolution of various adsorption and deposition processes along specific crystal axes. 

On the assumption of total symmetry of the tensor of third-order nonlinear polarizability c(— co coi, cog, cog), its non-zero and independent elements are the same as those of Table 12. Direct theoretical calculations of c = c(0 0,0,0) have been performed for the atoms of inert gases and some simple molecules. Values of the tensor elements = c(— cu cu, 0,0) have been determined for numerous molecules from static Kerr effect studies and values of c = c(— cd ot>,coi — col) from measurements of optical birefringence induced by laser li t. Measurements of second-harmonic generation by gases in the presence of a static electric field yield the tensor elements c " = c( — 2co co, to, 0), which can also be obtained from second-harmonic scattering in centro-symmetric liquids. The elements of the tensor c = c(— 3co co, co, co)... 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

Since p and E are vector quantities, a, P, 7, etc., are tensors. For example, the electric field vector in the first term will have three components in the molecular coordinate system. Each electric field component can contribute to polarization along each of the three directions in the molecular coordinate system. This triple contribution of electric field components leads to a total of nine elements to the second rank polarizability tensor. Similarly, there are 27 components to the P tensor and 81 components to 7. Molecular symmetry generally reduces these tensors to only a few independent elements. Unless the molecular coordinate system lacks an inversion center, the form of the odd-rank tensors such as P will lead to zero induced polarization in this representation of optical nonlinearities. For molecules such as benzene and polymers such as poly[bis(p-toluenesulfonate)diacetylene]... 

It is most important to note that in many cases of harmonic emission, a more completely index-symmetric form of the polarizability tensor is implicated. Consider once again the prototypical example of optical nonlinearity afforded by harmonic generation. When any harmonic is generated from a plane-polarized beam, in an isotropic medium, it produces photons with the same polarization vector as the incident light. In such a case the radiation tensor pyk becomes fully index-symmetric, and arguments similar to those given above show that only the fully index-symmetric part of the hyperpolarizability tensor, 3p(—2m co, co), can be involved. This does not mean that the tensor itself is inherently fully index-symmetric, but it does mean that experiments of the kind described cannot determine the extent of any index antisymmetry. 

In the preceding section it was shown that the formation of bound states of phonons leads to the appearance of a new type of resonance of the dielectric tensor ij(co). It is clear, of course (23), that the nonlinear polarizabilities should have analogous resonances, and this also concerns, besides biphonons, other types of bound states of quasiparticles, such as biexcitons, electron-exciton complexes, etc. 

Note that for H(uj + uj ) = Eb it is necessary to take the damping of the biphonon into account in the expression for A(w + u/,0). In this case, along with the real part of the tensor Xije u>,uj), an imaginary part is also present. This corresponds, as is well known, to the occurrence of two-photon absorption that is accompanied, in the given case, by the excitation of a biphonon. As applied to excitons this question has been discussed by Hanamura (51) within the framework of a somewhat different approach. We refer to it here (see also Fly-tzanis (52)) because both the generation of a second harmonic and two-photon absorption are processes that are completely described by the nonlinear polarizability of the crystal found above with bound states taken into account. Actually, these processes can be investigated by a single method. 

Consider now one of the examples which shows how a tensor of nonlinear polarizability Xije is formed in a superlattice. Suppose this tensor, the same as in cubic crystals of the group 7, /, takes the form... 

The vector of a permanent dipole moment Pe and polarizability tensor a y describe the linear (in field) electric and optical properties. The nonlinear properties are described by tensors of higher ranks (this depends on the number of fields included). For instance, the efficiency of mixing two optical waves of frequencies coi and CO2 is determined by polarization Py (co3) = E coi) Ey(co2) where E/(coi) and Ey((X)2)j are amplitudes of two interacting fields. Here is a third rank tensor of the electric hyperpolarizability. 

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... 

O Equation 11.84 defines the components of the dynamic electric dipole polarizability tensor, cCf,v -coa-, o)i), those of the first electric dipole hyperpolarizabUity,j8 v(/ (-Wa wi, W2), and those of the second electric dipole hyperpolarizabUity yf,v ((-coa coi, C02, (03). The combination of frequencies and the presence or absence of static fields characterize various nonlinear processes. The calculations can be done for any combination of frequencies satisfying O Eq. 11.85, but only selected examples of the most commonly studied optical phenomena will be discussed below (see, e.g., Willets et al. 1992 and Bogaard and Orr 1975 for more details). 

---