Nonlinear dielectric media

We consider a nonlinear dielectric medium in one dimension. The relation between the polarization P and the electric field E is given by ... 

A linear dielectric medium is characterized by a linear relation between the polarization density and the electric field, P = e0xE, where eo is the permittivity of free space and x is the electric susceptibility of the medium. A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between P and E, as illustrated in Figure 4.19. 

Before turning our attention to the free energy functionals, we recall a few fundamental concepts that will be used throughout in the following. We start from the general expression for the electrostatic energy of a charge density p0 in a nonlinear dielectric medium [22] ... 

Nonlinear optical (NLO) properties of organic polymers can be viewed as dielectric phenomena, and their origins can conveniently be explained by considering a planar wave propagation through a nonlinear dielectric medium [1-4]. In a dielectric medium the polarization P induced by the incident field E can be written as a power series of the field strength as follows ... 

When a strong static electric field is applied across a medium, its dielectric and optical properties become anisotropic. When a low frequency analyzing electric field is used to probe the anisotropy, it is called the nonlinear dielectric effect (NLDE) or dielectric saturation (17). It is the low frequency analogue of the Kerr effect. The interactions which cause the NLDE are similar to those of EFLS. For a single flexible polar molecule, the external field will influence the molecule in two ways firstly, it will interact with the total dipole moment and orient it, secondly, it will perturb the equilibrium conformation of the molecule to favor the conformations with the larger dipole moment. Thus, the orientation by the field will cause a decrease while the polarization of the molecule will cause an... 

Directly following the development of the optical laser, in 1961 Frankel et al. [10] reported the first observation of optical harmonics. In these experiments, the output from a pulsed ruby laser at 6943 A was passed through crystalline quartz and the second harmonic light at 3472 A was recorded on a spectrographic plate. Interest in surface SHG arose largely from the publication of Bloembergen and Pershan [11] which laid the theoretical foundation for this field. In this publication, Maxwell s equations for a nonlinear dielectric were solved given the boundary conditions of a plane interface between a linear and nonlinear medium. Implications of the nonlinear boundary theory for experimental systems and devices was noted. Ex-... 

The properties of a dielectric medium through which an electromagnetic (optical) wave propagates are completely described by the relation between the polarization density vector P(r, t) and the electric-field vector E r, t). It was suggested that P(r, t) could be regarded as the output of a system whose input was E(r, t). The mathematical relation between the vector functions P(r, t) and E(r, t) defines the system and is governed by the characteristics of the medium. The medium is said to be nonlinear if this relation is nonlinear. 

Figure 4.19 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium...

The propagation of light in a nonlinear medium is governed by the wave equation, which was derived from Maxwell s equations for an arbitrary homogeneous dielectric medium,... 

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],... 

A dielectric medium is said to be linear if the vector held P r, t) is linearly related to the vector held E r, t). This approximation is always used in the held of linear optics but fails in the case of nonlinear optics as will be discussed in more details in Sect. 3. A medium is said to be nondispersive if its response is instantaneous, meaning that the polarization at time t depends only on the electric held at that same time t and not on prior values of E. In most dielectrics the response time is very short (femtosecond or picosecond response times), but the fact that it is nonzero has huge consequences as will be discussed later. A medium is said to be homogeneous if the response of the material to an electric held is independent of r. A medium is said to be isotropic if the relation between E and P is independent of the direction of the held vector E. In the simplest case, when the medium satishes all these conditions, the vectors P and E at any position and at any time are parallel and proportional and related to each other by... 

During 1980-1981 the possibility of the existence of nonlinear surface polaritons of various types was predicted in the literature (18), (19)-(20). In particular, Tomlinson (19) and Maradudin (21) derived s-polarized surface polaritons at a plane interface between two dielectrics, one of which has an isotropic and linear (e1) dielectric constant whereas the dielectric constant of the other is that of a nonlinear uniaxial medium... 

When two laser beams (Eu nonlinear optical medium, the nonlinear dielectric polarization, PNL, exhibits the following forced oscillations,... 

The relation between the polarization density P of a dielectric medium and the electric field E is linear when E is small, but becomes nonlinear as E acquires values comparable with interatomic electric fields (10 to 10 V/cm). Under these conditions the relation between P and E can be expanded in a Taylor s series... 

Nonlinear Optical Effects with Lasers We noted in Section 6B-7 that when an electromagnetic Wave is transmitted through a dielectric medium, the electromagnetic field of the radiation causes momentary distortion, or polarization, of the valence electrons of the molecules that make up the medium. For tirdinary radiation the extent of polarization P is directly proportional to the magnitude of the electric field Fl of the radiation. Thus, we may write ... 

Figure 12.20. Schematic depiction of total internal reflection -- transmission switching by a dielectric (medium 1) cladded nonlinear material (medium 2). Two optional thin films of index, on both sides of the nonlinear material, are used for maximizing the transmission of the device in the transmission state. Also shown are the optical electric fields for the reflected, the transmitted, and the incident light.

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

A final point worth mentioning is the effect of local fields on the optical nonlinearities of strongly QC nanostructures. These arise from embedding QD s in a medium of different dielectric constant (2). One requires to know how the field intensity inside the particle varies at saturation in excitonic absorption. This has been approached theoretically by defining a local field factor f such that Em = f Eout (2). The factor f depends on the shape of the QD and the dielectric constant of the QD e = + E2 relative to that of the surrounding medium. Here... 

Figure 3.1 shows a simplified picture of an interface. It consists of a multilayer geometry where the surface layer of thickness d lies between two centrosymmetric media (1 and 2) which have two different linear dielectric constants e, and e2, respectively. When a monochromatic plane wave at frequency co is incident from medium 1, it induces a nonlinear source polarization in the surface layer and in the bulk of medium 2. This source polarization then radiates, and harmonic waves at 2 to emanate from the boundary in both the reflected and transmitted directions. In this model, medium 1 is assumed to be linear. 

In much of the above analysis, the relative magnitude of the surface and bulk contribution to the nonlinear response has not been addressed in any detail. As noted in Section 3.1, in addition to the surface dipole terms of Eq. (3.9), there are also nonlocal electric-quadrupole-type nonlinearities arising from the bulk medium. The effective polarization is made of a combination of surface nonlinear polarization, PNS (2co) (Eq. (3.9)), and bulk nonlinear polarization (Eq. (3.8)) which contains bulk terms y and . The bulk term y is isotropic with respect to crystal rotation. Since it appears in linear combination with surface terms (e.g. Eq. (3.5)), its separate determination is not possible under most circumstances [83, 129, 130, 131]. It mimics a surface contribution but its magnitude depends only upon the dielectric properties of the bulk phases. For a nonlinear medium with a high index of refraction, this contribution is expected to be small since the ratio of the surface contribution to that from y is always larger than se2(2co)/y. The magnitude of the contribution from depends upon the orientation of the crystal and can be measured separately under conditions where the anisotropic contribution of vanishes. 

---