Centrosymmetric media

Optical second-harmonic generation (SHG) has recently emerged as a powerful surface probe [95, 96]. Second harmonic generation has long been used to produce frequency doublers from noncentrosymmetric crystals. As a surface probe, SHG can be caused by the break in symmetry at the interface between two centrosymmetric media. A high-powered pulsed laser is focused at an angle of incidence from 30 to 70° onto the sample at a power density of 10 to 10 W/cm. The harmonic is observed in reflection or transmission at twice the incident frequency with a photomultiplier tube. 

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. 

Unlike linear optical effects such as absorption, reflection, and scattering, second order non-linear optical effects are inherently specific for surfaces and interfaces. These effects, namely second harmonic generation (SHG) and sum frequency generation (SFG), are dipole-forbidden in the bulk of centrosymmetric media. In the investigation of isotropic phases such as liquids, gases, and amorphous solids, in particular, signals arise exclusively from the surface or interface region, where the symmetry is disrupted. Non-linear optics are applicable in-situ without the need for a vacuum, and the time response is rapid. 

Optical second harmonic generation (SHG), which is the conversion of two photons of frequency u to a single photon of frequency 2co, is known to be an inherently surface-sensitive technique, because it requires a noncentrosymmetrical medium. At the interface between two centrosymmetrical media, such as the interface between two liquids, only the molecules which participate in the asymmetry of the interface will contribute to the SHG [18]. SHG has been used as an in-situ probe of chemisorption, molecular orientation, and... 

Any symmetry operation is required to leave the sign and magnitude of physical properties unchanged and therefore y xxx = 0. The same line of reasoning can be used to show that all tensor components will vanish under inversion. Hence, second-order nonlinear optical properties are not allowed in centrosymmetric media using the electric dipole approximation. The presence of noncentrosymmetry is one of the most stringent requirements in... 

The strong point of SFG is that the process is forbidden in centrosymmetric media (i.e. media with an inversion center). Therefore it occurs only at interfaces, where the sum-frequency response forms within a region of typically one nanometer thickness. Hence, neither the bulk of the catalyst nor the molecules in the gas phase contribute to the SFG spectrum. 

Second harmonic generation (SHG) is one of the most intensively studied nonlinear optical effects that have ever been combined with near-held scanning optical microscopy (Shen et al. 2000 Zayats and Sandoghdar 2000 Zayats and Sandoghdar 2001 Takahashi and Zayats 2002). SHG, which is an even-order nonlinear process, is forbidden in centrosymmetric media under the dipole approximation (Shen 1984). Non-centrosymmetric molecules and lattices are allowed to exhibit SHG light. The second-order nonlinear polarization for SHG (T shg) is given in a scalar form by... 

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 similar work, Sipe, Moss and Van Driel [84] determined a functional form of the rotational anisotropy for cubic centrosymmetric media. Their derived expressions for the total reflected p- and s-polarized SH fields from perfectly terminated (111) and (100) crystals under p- and s-polarized excitation take the form... 

In centrosymmetric media the P-E function must have odd symmetry, so that the reversal of E results in the reversal of P without any other change. The second-order nonlinear coefficient d must then vanish, and the lowest order nonlinearity is of third order. 

Vibrational sum-frequency spectroscopy (VSFS) is a second-order non-linear optical technique that can directly measure the vibrational spectrum of molecules at an interface. Under the dipole approximation, this second-order non-linear optical technique is uniquely suited to the study of surfaces because it is forbidden in media possessing inversion symmetry. At the interface between two centrosymmetric media there is no inversion centre and sum-frequency generation is allowed. Thus the asynunetric nature of the interface allows a selectivity for interfacial properties at a molecular level that is not inherent in other, linear, surface vibrational spectroscopies such as infrared or Raman spectroscopy. VSFS is related to the more common but optically simpler second harmonic generation process in which both beams are of the same fixed frequency and is also surface-specific. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

Since the first nonlinear susceptibility is a third-rank tensor, it is only non-zero in non-centrosymmetric media. To break the centrosymmetry of the macroscopic media, poling techniques using optical and electric fields have been developed, or use was made of the inherent polar ordering in Langmuir-Blodgett films and crystals with non-centrosymmetric point groups. 

When the incoming field E consists of two frequencies, O) and dipoles non-linearly oscillating at frequencies (o) + (o ) and (o) - (o ) are Induced as well. In the sum-frequency generation (SFG) technique one focusses on the output at frequency + co and in the difference-frequency generation (DFG) technique on - a) ). As is the case with SHG, SFG and DFG are electric dipole-forbidden in centrosymmetric media. 

Optical second-harmonic generation involves the non-linear conversion of two photons of frequency v to a single photon of frequency 2v. This process requires a non-centrosymmetric medium, one example being the interface between two centrosymmetric media. Since only a few molecular layers are involved in the symmetry breaking at the interface, the SHG process is a highly interface-selective optical probe. 

In order to describe the second-order nonlinear response from the interface of two centrosymmetric media, the material system may be divided into three 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 stmcture 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. Formally, we can describe this sheet by a nonlinear dipole moment per unit area, which is related to a second-order bulk polarization by hy P - fr, y, z, f) = P, (.r. v, f Here z is the surface normal direction, and the X and j axes represent the in-plane coordinates (figure Bl. 5.5). 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosymmetric media. Input waves at frequencies (o j and a 2, with corresponding wavevectors Aj(co j) and are approaching the interface from medium 1. Nonlinear...

Other nonlinear optical spectroscopies have gained much prominence in recent years. Two techniques in particular have become quite popular among surface scientists, namely, second harmonic (SHG) [55] and sum-frequency (SFG) [56] generation. The reason why both SHG and SFG can probe interfaces selectively without being overwhelmed by the signal from the bulk is that they rely on second-order processes that are electric-dipole forbidden in centrosymmetric media by breaking the bulk symmetry, the surface places the molecular species in an environment where their second-order nonlinear susceptibility, the term responsible for the absorption of SHG and SFG signals, becomes non-zero. 

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