Tensor elements, nonlinear optics

Symmetry is one of the most important issues in the field of second-order nonlinear optics. As an example, we will briefly demonstrate a method to determine the number of independent tensor components of a centrosymmetric medium. One of the symmetry elements present in such a system is a center of inversion that transforms the coordinates xyz as ... 

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... 

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)... 

Nonlinear optical techniques (SHG, SFG) Adsorption kinetics, interfacial coverage, reactioii kinetics, phase transitions, orientational order (average tilt angle), surface chirality. Intensity of the signal reflects the combined effect of interfacial coverage and orientational order. Tilt angles only obtainable if all non-zero elements of the hyperpolarizability tensor can be determined. 

Since second-order nonlinear optical materials are anisotropic, their optical properties are described by tensors as discussed previously in Sect. 2.1.2. For a nonlinear optical process, the -th order nonlinear polarization is due to n interacting electric held vectors and is described by an (n -I-1) rank tensor composed of 3"+ tensor elements. In nonlinear optics, several fields with different frequencies l can be present simultaneously so that the electric field and the polarization are represented by... 

When the optical frequencies involved in the nonlinear interaction are far from any resonance, the frequency dependence (or dispersion) of the optical response of the material can be ignored. In this case, the tensor elements are unchanged by the permutation of all Cartesian indices without changing the frequency arguments ... 

From the two descriptions of the Pockels effect in the frameworks of electro-op-tics and nonlinear optics, one can show that the electro-optic tensor elements and the second-order nonlinear susceptibility elements are related by... 

To understand and optimize the electro-optic properties of polymers by the use of molecular engineering, it is of primary importance to be able to relate their macroscopic properties to the individual molecular properties. Such a task is the subject of intensive research. However, simple descriptions based on the oriented gas model exist [ 20,21 ] and have proven to be in many cases a good approximation for the description of poled electro-optic polymers [22]. The oriented gas model provides a simple way to relate the macroscopic nonlinear optical properties such as the second-order susceptibility tensor elements expressed in the orthogonal laboratory frame X,Y,Z, and the microscopic hyperpolarizability tensor elements that are given in the orthogonal molecular frame x,y,z (see Fig. 9). 

Here, Pq is the permanent polarization, and and denote the second- and third-order nonlinear optical three-dimensional susceptibility tensors. The indices attached to the x tensors refer to the tensor elements, and the indices associated with the E values refer to the components of the electric field strength, here expressed in the laboratory frame. 

When a high-intensity laser beam impinges on material, its electromagnetic field induces electrical polarization that gives rise to a variety of nonlinear optical properties, because in this case the higher terms in Eq. (3-4) are not negligible. The determination of the coefficients and that serve to characterize the nonlinear properties is complicated by the fact that they are composed of many elements. With a being equal to two and three, and are composed of 3 = 27 and 81 elements, respectively. Fortunately, these tensors possess symmetry properties that can be invoked to reduce the number of independent elements, for instance, when the optical frequencies involved in the nonhnear interaction are far away from resonance (absorption) [15]. 

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,... 

The symmetry group of ZnO is P63mc (Hermann-Mauguin notation) and (Schoenflies notation). In optical nonlinearity nomenclature, hexagonal ZnO has class 6 mm symmetry. This being the case, many of the susceptibility tensor elements vanish. The only nonvanishing components are dn, di4 = dn, dn, dn = dn, and dn . 

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... 

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]... 

Optical Kerr Effect. Another important method used to characterize polymers is the optical Kerr effect (OKE). The optical Kerr effect differs from the quadratic electrooptic effect in that the birefringence effects are induced solely by an optical field (37). In this measurement, an intense linearly polarized pump pulse induces birefringence in the nonlinear sample through an intensity-dependent refractive index change. The sample is placed between crossed polarizers and a weak, typically tunable, continuous wave (cw) probe laser (usually at a different wavelength and polarized at 45° to the pump pulse) overlaps the pumped region. The increased transmission of the probe beam when the pump pulse arrives is proportional to (Xeff), a combination of elements of the tensor. Many... 

Dipolar chromophores must be assembled into a noncentrosymmetric lattice to translate molecular optical nonlinearity to maaoscopic electro-optic (EO) aaivity, which is one example of second-order NLO properties. The EO coefficient in units of pmV , rs3, is the principal element of the linear Pockel s EO effect tensor and denotes the magnitude of refractive index shift (Ai ) obtained for an applied low-frequency electric field. This... 

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