Second-order nonlinearities

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 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. 

The most significant symmetry property for the second-order nonlinear optics is inversion synnnetry. A material possessing inversion synnnetry (or centrosymmetry) is one that, for an appropriate origin, remains unchanged when all spatial coordinates are inverted via / —> - r. For such materials, the second-order nonlmear response vanishes. This fact is of sufficient importance that we shall explain its origm briefly. For a... 

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

The focus of the present chapter is the application of second-order nonlinear optics to probe surfaces and interfaces. In this section, we outline the phenomenological or macroscopic theory of SHG and SFG at the interface of centrosymmetric media. This situation corresponds, as discussed previously, to one in which the relevant nonlinear response is forbidden in the bulk media, but allowed at the interface. 

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... 

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 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... 

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... 

The applications of this simple measure of surface adsorbate coverage have been quite widespread and diverse. It has been possible, for example, to measure adsorption isothemis in many systems. From these measurements, one may obtain important infomiation such as the adsorption free energy, A G° = -RTln(K ) [21]. One can also monitor tire kinetics of adsorption and desorption to obtain rates. In conjunction with temperature-dependent data, one may frirther infer activation energies and pre-exponential factors [73, 74]. Knowledge of such kinetic parameters is useful for teclmological applications, such as semiconductor growth and synthesis of chemical compounds [75]. Second-order nonlinear optics may also play a role in the investigation of physical kinetics, such as the rates and mechanisms of transport processes across interfaces [76]. 

Heinz T F 1991 Second-order nonlinear optical effects at surfaces and interfaces Noniinear Surfaoe... 

Reider G A and Heinz T F 1995 Second-order nonlinear optical effects at surfaces and interfaces recent advances Photonio Probes of Surfaoes ed P Halevi (Amsterdam Elsevier) pp 413-78... 

Hicks J M, Petralli-Mallow T and Byers J D 1994 Consequences of chirality in second-order nonlinear spectroscopy at interfaces Faraday Disc. 99 341 -57... 

Fig. 1. Representative device configurations exploiting electrooptic second-order nonlinear optical materials are shown. Schematic representations are given for (a) a Mach-Zehnder interferometer, (b) a birefringent modulator, and (c) a directional coupler. In (b) the optical input to the birefringent modulator is polarized at 45 degrees and excites both transverse electric (TE) and transverse magnetic (TM) modes. The appHed voltage modulates the output polarization. Intensity modulation is achieved using polarizing components at the output.

Recently photorefractivity in photoconductive polymers has been demonstrated (92—94). The second-order nonlinearity is obtained by poling the polymer doped with a nonlinear chromophore. Such a polymer may or may not be a good photoconductor. Usually sensitizers have to be added to enhance the charge-generation efficiency. The sensitizer function of fuUerene in a photorefractive polymer has been demonstrated (93). 

Serious attempts to use LB films in commercial appHcations include the use of lead stearate as a diffraction grating for soft x-rays (64). Detailed discussion on appHcations of LB films are available (4,65). From the materials point of view, the abiHty to build noncentro symmetric films having a precise control on film thickness, suggests that one of the first appHcations of LB films may be in the area of second-order nonlinear optics. Whereas a waveguide based on LB films of fatty acid salts was reported in 1977, a waveguide based on polymeric LB films has not yet been commercialized. 

Equation (2) is identified as a second-order, nonlinear differential equation once, the curvature is expressed in terms of a shape function of the melt/crystal interface. The mean curvature for the Monge representation y = h(x,t) is... 

Acentricity greatly enhances the y-value (see 92 vs 91 and 90 or 101 vs 99 and 100, Fig. 8). Such a trend had been predicted for certain ranges of compounds by theory [137] however when the first hyperpolarizability, which determines second-order nonlinear optical properties, is maximized, y is predicted to be zero [138]. 

When combining Eqs. (A.l) and (A.4), we obtain a second-order nonlinear differential equation for /o(r) which is mathematically very difficult to solve. Therefore, in DH theory a simplified equation is used The exponential terms of Eq. (A.4) are expanded into series and only the first two terms of each series are retained [exp(y) 1 +y]. When we include the condition of electroneutrality and use the ionic strength we can write this equation as... 

Hyper-Raman spectroscopy is not a surface-specific technique while SFG vibrational spectroscopy can selectively probe surfaces and interfaces, although both methods are based on the second-order nonlinear process. The vibrational SFG is a combination process of IR absorption and Raman scattering and, hence, only accessible to IR/Raman-active modes, which appear only in non-centrosymmetric molecules. Conversely, the hyper-Raman process does not require such broken centrosymmetry. Energy diagrams for IR, Raman, hyper-Raman, and vibrational SFG processes are summarized in Figure 5.17. 

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] ... 

The SHG signal intensity, 1(2 ), arising at the liquid-liquid interface is known to be proportional to the square of the second-order nonlinear electric susceptibility, c , at the surface [18,28,29],... 

The determination of the number of the SHG active complex cations from the corresponding SHG intensity and thus the surface charge density, a°, is not possible because the values of the molecular second-order nonlinear electrical polarizability, a , and molecular orientation, T), of the SHG active complex cation and its distribution at the membrane surface are not known [see Eq. (3)]. Although the formation of an SHG active monolayer seems not to be the only possible explanation, we used the following method to estimate the surface charge density from the SHG results since the square root of the SHG intensity, is proportional to the number of SHG active cation com-... 

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