Nonlinear optics, linear polarizability

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

The proportionality constants a and (> are the linear polarizability and the second-order polarizability (or first hyperpolarizability), and x(1) and x<2) are the first- and second-order susceptibility. The quadratic terms (> and x<2) are related by x(2) = (V/(P) and are responsible for second-order nonlinear optical (NLO) effects such as frequency doubling (or second-harmonic generation), frequency mixing, and the electro-optic effect (or Pockels effect). These effects are schematically illustrated in Figure 9.3. In the remainder of this chapter, we will primarily focus on the process of second-harmonic generation (SHG). 

If a highly polarizable group is introduced into a receptor molecule, substrate binding should cause substantial perturbations, so that the recognition event would be converted into a non-linear optical signal. Such recognition-dependent nonlinear optical probes may be derived for instance from polyenes such as those shown in Figure 20, from inclusion complexes [8.94a] or from donor-acceptor calixarenes [8.94b]. 

In this introductory chapter the concepts of linear and nonlinear polarization are discussed. Both classical and quantum mechanical descriptions of polarizability based on potential surfaces and the "sum over states" formalism are outlined. In addition, it is shown how nonlinear polarization of electrons gives rise to a variety of useful nonlinear optical effects. 

Thus, just as linear polarizabilities are frequency dependent, so are the nonlinear polarizabilities. Perhaps it is not surprising that most organic materials, with almost exclusively electronic nonlinear optical polarization, have similar efficiencies for SHG and the LEO effect. In contrast, inorganic materials, such as lithium niobate, in which there is a substantial vibrational component to the nonlinear polarization, are substantially more efficient for the LEO effect than for SHG. 

In the above equation a is the linear polarizability. The terms 3 and Y, called first and second hyperpolarizabilities, describe the2 nonlinear optical interactions and are microscopic analogues of x and x... 

Electro- and magnetooptical phenomena in colloids and suspensions are widely used for structure and kinetics analysis of those media as well as practical applications in optoelectronics [143,144]. The basic theoretical model used to study optical anisotropy of the disperse systems is the noninteracting Brownian particle ensemble. In the frame of this general approximation, several special cases according to the actual type of particle polarization response to the applied field may be distinguished (1) particles with permanent dipole moments, (2) linearly polarizable particles, (3) nonlinearly polarizable particles, and (4) particles with hysteretic dipole moment reorientation. 

Periodic oscillations in this dipole can act as a source term in the generation of new optical frequencies. Here a is the linear polarizability discussed in Exps. 29 and 35 on dipole moments and Raman spectra, while fi and x are the second- and third-order dielectric susceptibilities, respectively. The quantity fi is also called the hyperpolarizability and is the material property responsible for second-harmonic generation. Note that, since E cos cot, the S term can be expressed as -j(l + cos 2 wt). The next higher nonlinear term x is especially important in generating sum and difference frequencies when more than one laser frequency is incident on the sample. In the case of coherent anti-Stokes Raman scattering (CARS), X gives useful information about vibrational and rotational transitions in molecules. 

The extensive jt-delocalized system of metal-dithiolene complexes is also responsible for the nonlinear optical properties (NLO) which have been recently reviewed . The interaction of radiation with the matter induces an instantaneous displacement (polarization Pq = /X = aE, where a is the linear polarizability) of the electronic density away from the nucleus at small field (linear optics). At high fields (laser light) the polarizability of the molecule can be driven beyond the linear regime and a nonlinear polarization is induced (NLO) = aE + fiE" + y E + and for the bulk material... 

Studies of nonlinear optical phenomena in conjugated polymers date back to the earliest studies of the polydiacetylenes [219], The large oscillator strength associated with the p-p transition gives rise to a relatively large linear electronic polarizibility. The early work speculated that because of the implied delocalization of charge in the excited state, conjugated polymers would offer opportunities as NLO materials. 

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 ... 

It is clear that the route followed hitherto can be extended to an arbitrary order in the perturbation, and thus provide the description of general multi-photon interactions. However, since very high light intensities are needed in order to observe tlie nonlinear responses, the predominant interest in optics are focused at processes incorporated in the linear polarizability and the first- and second-order hyperpolarizability. The expression for the general-order nonresonant response... 

Our discussion has so far been concerned with the microscopic response of a molecule to an external electric field, and thus with an expansion of the molecular energy in orders of the response with respect to the external field, giving rise to the molecular (hyper)polarizabilities. Although experimental data for nonlinear optical properties of molecules in the gas phase do exist [55], the majority of experimental measurements are done in the liquid or solid states, as these states also are the ones that are of greatest interest with respect to developing materials with specifically tailored (non)linear optical properties. 

There are two major ways to view tlie vibrational contribution to molecular linear and nonlinear optical properties, i.e. to (hyper)polarizabilities. One of these is from the time-dependent sum-over-states (SOS) perturbation theory (PT) perspective. In the usual SOS-PT expressions [15], based on the adiabatic approximation, the intermediate vibronic states K, k> are of two types. Either the electronic wavefunction... 

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