Linear and Nonlinear Polarization

Unfortunately, however, there are several alternative definitions of ft and y (and higher hyperpolarizabilities) and many studies do not explicitly state which convention is being followed. Therefore, one should be careful in comparing values from different studies. These are discussed in detail, along with conversion factors, in Ref. [1], For example, the (n - l)th-order hyperpolarizability is commonly equated to (1/n ) X (dn i/dEn), rather than simply to the differential. 

The nonlinear bulk polarization density is given by an expression analogous to Eq. (4)  

Before we examine how second- and third-order N LO effects are related to nonlinear polarization, we briefly examine an important symmetry restriction on second-order NLO properties. From Eq. (5), we can see that P(E) = P(0) + xmE + x E2 + x i)E3+... and P -E) = P(0) - x 1)E + xp)E2 - x Eh... we can also see from Fig. 11.1 that P(E) + P(-E) if%(2) + 0. In a centra symmetric material, P(E) is necessarily equal to P(-E) and, therefore, P(0), and other even-order terms must be zero. Therefore, for second-order effects to be observed in a molecule or material, the molecule or material must be non-centrosymmetric. However, no such requirement applies to odd-order processes, such as third-order effects [Fig. 11.1 shows P(E) = P(-E) for a material with only linear and cubic susceptibilities non-zero]. 

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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. 

A current-voltage technique was developed by Rajeshwar iS9 to study oil shales. When an electric field is applied to a solid substance, the current flowing through it is time-dependent. Two types of polarization mechanisms have been used to explain this time dependency linear polarization and nonlinear polarization. A convenient method of distinguishing between these is to examine the In versus In V plots (where is current density and V is the voltage). Linear polarization will result in curves with a slope of -I-1, whereas nonlinear polarization will give linear In Jx versus In J plots of slope 1. The oil shales reveal a complex behavior involving both linear and nonlinear polarization effects. 

In Eqs. 1 and 2. the indices i, j. k. and I refer to the coordinate system of the bulk material and molecule, respectively. Illustrated in Fig. 1 are the linear and nonlinear polarizations with respect to electric field. The Fourier decomposition of this nonlinear polarization comprising components of zero frequency, the fundamental frequency, the second-harmonic frequency, the third-harmonic frequency, etc., is shown in Fig. 2. The effects up to the second order, which are easily observed experimentally, are called the optical rectification. P(0) linear electro-optic effect P((u) second-harmonic generation P(2a>), and third-harmonic generation P(3co). 

The introduction of the local field factor in nonlinear optics is more complex due to the interplay of linear and nonlinear polarization. The reader is referred to two textbooks [16,17] for a rigorous derivation. The correct second-harmonic polarization... 

The theoretical studies we review here have been stimulated by the seminal work [120-123] of, and performed in collaboration with, Marder and coworkers [124-126]. Their main thrust has been to offer a unified picture of linear and nonlinear polarization in 7r-conju-gated organic chromophores [124], which paves the way for the design of a wide range of novel materials. In the... 

A most important result is obtained by observing in Fig. 1.11 that the linear polarizability a, the first hyperpolarizability j8, and the second hyperpolarizability y are seen to be derivatives, with respect to the structural parameter BOA (influenced by the external field), of their next lower order polarization (for a) or polarizability (for )8 and y). These derivative relations therefore provide a unified description of the linear and nonlinear polarization and of the dependence of the polarizability and hyperpolarizabilities on the structure in linear polymethine dyes [124]. A consequence to be stressed is that it is not possible, for a given compound, to find conditions that would simultaneously optimize the p and y responses When p reaches a peak, y tends to a minimum. 

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. 

Polarization which can be induced in nonconducting materials by means of an externally appHed electric field is one of the most important parameters in the theory of insulators, which are called dielectrics when their polarizabiUty is under consideration (1). Experimental investigations have shown that these materials can be divided into linear and nonlinear dielectrics in accordance with their behavior in a realizable range of the electric field. The electric polarization PI of linear dielectrics depends linearly on the electric field E, whereas that of nonlinear dielectrics is a nonlinear function of the electric field (2). The polarization values which can be measured in linear (normal) dielectrics upon appHcation of experimentally attainable electric fields are usually small. However, a certain group of nonlinear dielectrics exhibit polarization values which are several orders of magnitude larger than those observed in normal dielectrics (3). Consequentiy, a number of useful physical properties related to the polarization of the materials, such as elastic, thermal, optical, electromechanical, etc, are observed in these groups of nonlinear dielectrics (4). 

The asymmetrical D-ji-A dyes, often referred to as push-pull polyenes, are an additional class of cyanine-like molecules of interest. Due to their dipolar nature, the linear and nonlinear optical properties of this series of dyes can be strongly influenced by solvent polarity [84]. The structures of a series of such dyes (G19,... 

Comparison of Linearly and Circularly Polarized Probes of Nonlinear Optical Activity of Chiral Surfaces... 

In terms of beam delivery, the DLW method is based on optical microscopy, confocal microscopy [4,6,13] and laser tweezers [14] (for reviews on laser tweezers see [ 15,16]). These techniques allow for a high spatial 3D resolution of a tightly focused laser beam with optical exposure of micrometric-sized volumes via linear and nonlinear absorption. In addition, mechanical and thermal forces can be exerted upon objects as small as 10 nm molecular dipolar alignment can be controlled by polarization of light in volumes of with submicrometric cross-sections. This circumstance widens the field of applications for laser nano- and microfabrication in liquid and solid materials [17-22]. 

Here, 33, 333, 3333, and 33333 correspond to linear and nonlinear dielectric constants and are tensors of rank 2nd, 3rd, 4th and 5th, respectively. Even-ranked tensors including linear dielectric constant 33 do not change with polarization inversion, whereas the sign of the odd-ranked tensors reverses. Therefore, information regarding polarization can be elucidated by measuring odd-ranked nonlinear dielectric constants such as 333 and 33333. 

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

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 ... 

The macroscopic optical responses of a medium are given by its linear and nonlinear susceptibilities, which are the expansion coefficients of the material polarization, P, in terms of the Maxwell fields, 1 3]. For a dielectric or ferroelectric medium under the influence of an applied electric field, the defining equation reads... 

From the point of view of a computational chemist, one of the most appreciated strengths of the polarization propagator approach is that, although being generally applicable to many fields in physics, it also delivers efficient, computationally tractable formulas for specific applications. Today we see implementations of the theory for virtually all standard electronic structure methods in quantum chemistry, and the implementations include both linear and nonlinear response functions. The double-bracket notation is the most commonly used one in the literature, and, in analogy with Eq. (5), the response functions are defined by the expansion... 

For a long time the finite oligomer approach was the only method available for determining linear and nonlinear polarizabilities of infinite stereoregular polymers. Recently, however, the problem of carrying out electronic band structure (or crystal orbital) calculations in the presence of static or frequency-dependent electric fields has been solved [115, 116]. A related discretized Berry phase treatment of static electric field polarization has also been developed for 3D solid state systems... 

As an introduction to Section 36.5, let us note that solitons, polarons and bipolarons are excitations of major importance which are inherently nonlinear and thus will favor nonlinear responses in the presence of external fields. Also, it must be realized that the quasi one-dimensionality of the polymer chains allows them to easily undergo structural distortions that result in a significant lowering of the first electronic excitation. Thus, in a perturbation scheme, the linear and nonlinear polarizabilities must be significantly enhanced, since they are inversely proportional to the energy of the electronic excitation. Since 1983, these facts have drawn our attention to the field of polarization of polymeric materials, which is described in detail in Section 36.5. 

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