Nonlinear polarization

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

Kinetic polarization Nonlinear behavior of an electrochemical cell caused by the slowness of the reaction at the surface of one or both electrodes. 

CH3CHO polar nonlinear molecule) has to be estimated Giacalone s simple method is used. 

Pulse polarization nonlinearity may occur during a single pulse if it lasts long enough to cause irreversible electrolysis. Pulses < 1 ms are mostly transferred to the tissue by capacitive current paths not implying electrolysis (non-faradaic current). The same rules are valid for large AC current polarization the higher the frequency, the better the linearity. 

C2//507/(polar nonlinear molecule) From Perry and Chilton [1984],... 

The non-resonant part (3)nr j e l and includes the contribution of every species present. It will usually show a weak dispersion, being approximately constant for extended frequency ranges. In this case, the permutation of the frequency arguments alone does not alter much the numerical value of the susceptibility. This is the Kleinman s symmetry conjecture which, combined with the strict intrinsic permutation symmetry allows to freely permute the Cartesian subscripts in jj(3)NR This property, although not exact, is of great relevance for the polarization nonlinear Raman techniques. 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

As described at the end of section Al.6.1. in nonlinear spectroscopy a polarization is created in the material which depends in a nonlinear way on the strength of the electric field. As we shall now see, the microscopic description of this nonlinear polarization involves multiple interactions of the material with the electric field. The multiple interactions in principle contain infomiation on both the ground electronic state and excited electronic state dynamics, and for a molecule in the presence of solvent, infomiation on the molecule-solvent interactions. Excellent general introductions to nonlinear spectroscopy may be found in [35, 36 and 37]. Raman spectroscopy, described at the end of the previous section, is also a nonlinear spectroscopy, in the sense that it involves more than one interaction of light with the material, but it is a pathological example since the second interaction is tlirough spontaneous emission and therefore not proportional to a driving field... 

Lobaugh J and Voth G A 1994 A path integral study of electronic polarization and nonlinear coupling effects in condensed phase proton transfer reactions J. Chem. Phys. 100 3039... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

A RIKES experunent is essentially identical to that of CW CARS, except the probe laser need not be tunable. The probe beam is linearly polarized at 0° (—>), while the polarization of the tunable pump beam is controlled by a linear polarizer and a quarter waveplate. The pump and probe beams, whose frequency difference must match the Raman frequency, are overlapped in the sample (just as in CARS). The strong pump beam propagating tlirough a nonlinear medium induces an anisotropic change in the refractive mdices seen by tlie weaker probe wave, which alters the polarization of a probe beam [96]. The signal field is polarized orthogonally to the probe laser and any altered polarization may be detected as an increase in intensity transmitted tlirough a crossed polarizer. When the pump beam is Imearly polarized at 45° y), contributions... 

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

As we shall discuss later in a detailed fashion, the nonlinear polarization associated with the nonlinear susceptibility of a medium acts as a source tenn for radiation at the second hamionic (SH) frequency 2co. 

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. 

Here the collection of frequencies in the snnnnation may inclnde new frequencies in addition to those in siumnation of equation B1.5.21 for the applied field. The total polarization can be separated mto Imear, Pj, and nonlinear, parts ... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

An alternative scheme for extracting all tliree isotropic nonlinear susceptibilities can be fomuilated by examining equation B 1.5.39. By choosing an appropriate configuration and the orientation of the polarization of die SH radiation e 2a) such that the SHG signal vanishes, one obtains, assuming only surface contribution with real elements, ... 

A schematic diagram of the surface of a liquid of non-chiral (a) and chiral molecules (b) is shown in figure Bl.5.8. Case (a) corresponds to oom-synnnetry (isotropic with a mirror plane) and case (b) to oo-symmetry (isotropic). For the crj/ -synnnetry, the SH signal for the polarization configurations of s-m/s-out and p-m/s-out vanish. From table Bl.5.1. we find, however, that for the co-synnnetry, an extra independent nonlinear susceptibility element, is present for SHG. Because of this extra element, the SH signal for... 

The nonlinear response of an individual molecule depends on die orientation of the molecule with respect to the polarization of the applied and detected electric fields. The same situation prevails for an ensemble of molecules at an interface. It follows that we may gamer infonnation about molecular orientation at surfaces and interfaces by appropriate measurements of the polarization dependence of the nonlinear response, taken together with a model for the nonlinear response of the relevant molecule in a standard orientation. 

Takmg advantage of the synunetry changes induced by the presence of a surface. Many nonlinear teclmiques rely on the fact that the surface breaks the centrosynuuetrical nature of the bulk. The use of polarized light can also discriminate among dipole moments in different orientations. 

Figure B2.1.1 Femtosecond light source based on an amplified titanium-sapphire laser and an optical parametric amplifier. Symbols used P, Brewster dispersing prism X, titanium-sapphire crystal OC, output coupler B, acousto-optic pulse selector (Bragg cell) FR, Faraday rotator and polarizer assembly DG, diffraction grating BBO, p-barium borate nonlinear crystal.

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