Tensor components, nonlinear optics

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

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 ... 

Any symmetry operation is required to leave the sign and magnitude of physical properties unchanged and therefore y xxx = 0. The same line of reasoning can be used to show that all tensor components will vanish under inversion. Hence, second-order nonlinear optical properties are not allowed in centrosymmetric media using the electric dipole approximation. The presence of noncentrosymmetry is one of the most stringent requirements in... 

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... 

Assuming the tensor component of xf x parallel to the polymer chain direction is dominant for the nonlinear optical response, one can use the following equation ... 

Electric Field Induced Second-Harmonic Generation. An essential aspect of the development of materials for second-order nonlinear optics is the determination of the p tensor components. The technique that has been developed to accomplish this is called electric field induced second harmonic generation (EFISH) (13,14). 

For dipolar chromophores that are the subject of this chapter, only one component of the molecular hyperpolarizability tensor, Pzzz, is important. Thus, the summation in Eq. (8) disappears. Electric field poling induces Cv cylindrical polar symmetry. Assuming Kleinman [12] symmetry, only two independent components of the macroscopic second-order nonlinear optical susceptibility tensor... 

If the molecule or the solid has a center of inversion symmetry, then the permanent dipole moment m0 vanishes, as do the even-rank tensors j8 and 8 and the even-rank tensors y<2), y<4), and All matter, with or without a center of inversion symmetry, has nonzero values for the odd-rank molecular tensors a and y, and all odd-rank tensors ytensor components is vastly reduced. The components have values that depend seriously on the frequency of the electromagnetic radiation used to probe them. A practical application of nonlinear optics is frequency-doubling of the high-powered... 

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 ... 

Under the influence of an optical pump, the molecular angular distribution described by Equation 12.4 can be considerably modified. In turn, this results in modification of the X ijkl tensor components. Further, we discuss the influence of a polarized pump beam on third-order nonlinear phenomena such as third harmonic generation (THG) [(described by (-3a),ft>,w,a>) coefficient], electric field induced second harmonic generation (EFISH) [x / kl -2(0, (o, o), 0)] and degenerate four-wave mixing (DFWM) X kl ... 

Most linear optical phenomena such as refraction, absorption and Rayleigh scattering are described by the first term in Eq. (1) where is the linear susceptibility tensor. The higher order terms and susceptibilities are responsible for nonlinear optical effects. The second-order susceptibility tensor T underlies SFG, whereas and BioCARS arises within As we are concerned with optical effects of randomly oriented molecules in fluids, we need to consider unweighted orientational averages of the susceptibility tensors in Eq. (1). We will show that the symmetries of the corresponding isotropic components and correspond to time-even pseudoscalars the hallmark of chiral observables [2]. 

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. 

For the structure of tensor components of the nonlinear material coefficient see Table 7.3. Piezo-optical, elasto-optical and electro-elastic coefficients for a-quartz are listed in Table 7.4. 

With reference to the above expressions, it is important to specify that in the case of an oscillating electric field as that associated with an incident electromagnetic wave, which is a common experimental situation in nonlinear optics, the components of the tensors and also depends on the frequency of the... 

Other symmetry rules for the components of in the various crystal classes can be found in standard textbooks of nonlinear optics [3, 4], It is interesting to remark that has the same symmetfy properties of the piezoelectfic tensor [5]. 

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,... 

Table 8 shows experimental second-order nonlinear optical susceptibilities for different tensor components dll and various fundamental wavelengths. The quantities dn are defined as follows ... 

If an optical frequency co is close to one-photon electronic resonance with an intermediate state /, the terms including the denominator -co — 0 will be dominant and we obtain a good approximation by retaining only these (resonant nonlinear Raman spectroscopy). In the absence of such additional electronic resonances, we can neglect the damping coefficient for all denominators whose absolute value is far from zero and write them in terms of real Raman scattering tensor components for the transition from... 

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