Susceptibility nonlinear

The optical response of materials to the interaction of the electric dipole of light in a stationary state is given by Eq. (5.8), where the dielectric polarization of light, P, is expanded as a function of the electric field of light, E. 

Relationship between polarizability and phase velocity and intensity of light Phase velocity of light wave in non-magnetic body 

Absorption of light by molecule Fermi s golden rule 

b transition rate, P i transition dipole moment, p ( E ) energy density of light per unit volume Einstein s A coefficient 

Frequency dependence of refractive index Kramers-Kroning equation 

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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. 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anliannonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies cOq and coq 2> respectively.

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. 

The second-order nonlinear susceptibility tensor ( 3> 2, fOj) introduced earlier will, in general, consist of 27 distinct elements, each displaying its own dependence on the frequencies oip cci2 and = oi 012). There are, however, constraints associated with spatial and time-reversal symmetry that may reduce the complexity of for a given material [32, 33 and Ml- Flere we examine the role of spatial synnnetry. 

Table Bl.5.1 Independent non-vanishing elements of the nonlinear susceptibility, for an interface in the Ay-plane for various syimnetry classes. When mirror planes are present, at least one of them is perpendicular to they-axis. For SFIG, elements related by the pennutation of the last two elements are omitted. For SFG, these elements are generally distinct any syimnetry constraints are indicated in parentheses. The temis enclosed in parentheses are antisymmetric elements present only for SFG. (After [71])...

The basic physical quantities that define the material for SHG or SFG processes are the nonlinear susceptibility elements consider how one may detemiine these quantities experimentally. For... 

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, ... 

Spatial synnnetry is one of the basic properties of a surface or interface. If the syimnetry of the surface is known a priori, then this knowledge may be used to simplify the fomi of the surface nonlinear susceptibility as discussed in section Bl,5,2,2. Conversely, in the absence of knowledge of the surface synnnetry, we may characterize the fonn of -iexperimentally and then make inferences about the synnnetry of the surface... 

All of the symmetry classes compatible with the long-range periodic arrangement of atoms comprising crystalline surfaces and interfaces have been enumerated in table Bl.5,1. For each of these syimnetries, we indicate the corresponding fonn of the surface nonlinear susceptibility With the exception of surfaces... 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating die corresponding chiral surfaces [, and 70]. From the point of view of perfomiing surface and interface spectroscopy with nonlinear optics, we must first examhie the nonlinear response of tlie bulk liquid. Clearly, a chiral liquid lacks inversion synnnetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the pemiutation synnnetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

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... 

For other purposes, obtaining a measure of the adsorbate surface density directly from the experiment is desirable. From this perspective, we introduce a simple model for the variation of the surface nonlinear susceptibility with adsorbate coverage. An approximation that has been found suitable for many systems is... 

We now consider this issue in a more rigorous fashion. The inference of molecular orientation can be explamed most readily from the following relation between the surface nonlinear susceptibility tensor and the molecular nonlinear polarizability... 

Notice that /pijn = nonlinear susceptibility tensor elements... 

On the other hand, the nonlinear optical properties of nanometer-sized materials are also known to be different from the bulk, and such properties are strongly dependent on size and shape [11]. In 1992, Wang and Herron reported that the third-order nonlinear susceptibility, of silicon nanocrystals increased with decreasing size [12]. In contrast to silicon nanocrystals, of CdS nanocrystals decreased with decreasing size [ 13 ]. These results stimulated the investigation of the nonlinear optical properties of other semiconductor QDs. For the CdTe QDs that we are concentrating on, there have been few studies of nonresonant third-order nonlinear parameters. 

For the application of QDs to three-dimensional biological imaging, a large two-photon absorption cross section is required to avoid cell damage by light irradiation. For application to optoelectronics, QDs should have a large nonlinear refractive index as well as fast response. Two-photon absorption and the optical Kerr effect of QDs are third-order nonlinear optical effects, which can be evaluated from the third-order nonlinear susceptibility, or the nonlinear refractive index, y, and the nonlinear absorption coefficient, p. Experimentally, third-order nonlinear optical parameters have been examined by four-wave mixing and Z-scan experiments. 

Wang, Y. and Herron, N. (1992) Size-dependent nonresonant third-order nonlinear susceptibilities of CdS clusters from 7 to 120 A. Int.J. Nonlinear Opt. Phys., 1, 683-698. 

For adsorbates on a metal surface, an SFG spectmm is a combination of resonant molecular transitions plus a nonresonant background from the metal. (There may also be a contribution from the water-CaF2 interface that can be factored out by following electrode potential effects see below.) The SFG signal intensities are proportional to the square of the second-order nonlinear susceptibility [Shen, 1984] ... 

These effects Q.2) are all driven by the same third-rank frequency dependent nonlinear susceptibility x2(-u>3 w,, u>2).d is sometimes preferred for second-harmonic generation (SHG). 

In a crystalline medium, the parametric gain (2) T2 is propor-tionnal to d2 Ip n-3 and the oscillation condition r2A2>aA where a is the signal residual absorption (dramatically increased by any crystalline defect), d the efficient phase-matched nonlinear susceptibility, n an average refractive index, Ip the pump intensity (limited by the optical damage threshold) and A the effective interaction length (also limited by any source of crystalline disorientation). 

The linear and nonlinear optical properties of one-dimensional conjugated polymers contain a wealth of information closely related to the structure and dynamics of the ir-electron distribution and to their interaction with the lattice distorsions. The existing values of the nonlinear susceptibilities indicate that these materials are strong candidates for nonlinear optical devices in different applications. However their time response may be limited by the diffusion time of intrinsic conjugation defects and the electron-phonon coupling. Since these defects arise from competition of resonant chemical structures the possible remedy is to control this competition without affecting the delocalization. The understanding of the polymerisation process is consequently essential. 

Third-Order Nonlinear Susceptibility in Multilayers of Polydiacetylene... 

The third order nonlinear susceptibility is an important optical property of materials because of its contributions to numerous nonlinear optical processes. (1)(2) With the growing interest in all-op ical signal processing it has been proposed (3) recently that x ( 1, 2 3) and especially the degenerate third order nonlinear susceptibility x 3 (-w, to, 00) [defined as x (< >)], be utilized through its contributions to the changes in dielectric constant e with optical field strength E ... 

In order to understand the nonlinear susceptibilities of the polydiacetylenes (or any material) one must have a detailed knowledge of the wavefunctions for many levels in the system. 

The experimental technique used to measure the third order nonlinear susceptibilities of polydiacetylenes was developed recently at our laboratory. (15) This technique utilizes the intensity-dependent dispersion relation of confined modes of PDA films and determines by measuring the change of coupling angles with... 

Chen, Y. J. Carter, G. M. "Measurement of Third Order Nonlinear Susceptibilities by Surface Plasmons," Appl. Phys. 

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