Second-order nonlinear susceptibilities

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. 

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] ... 

Nonlinear optics deals with physical systems described by Maxwell equations with an nonlinear polarization vector. One of the best known nonlinear optical processes is the second-harmonic generation (SHG) of light. In this section we consider a well-known set of equations describing generation of the second harmonic of light in a medium with second-order nonlinear susceptibility %(2 The classical approach of this section is extended to a quantum case in Section IV. 

Induced Polarization and Origin of the Second Order Nonlinear Susceptibility... 

Shoji, I., Kondo T. A. and Ito R., Second-order nonlinear susceptibilities of various dielectric and semiconductor materials. Opt.Quantum Electr. (2002) 34 797-833. 

Moreover, the isolation of self-assembled LiNbOj powders using this route has added credibility to the methodology they were otherwise prepared by templating colloidal crystals of polyelectrolyte-coated spheres. The interest in LiNb03 inverse opals stems from the fact that they have a constant refractive index, but a spatially periodic second-order nonlinear susceptibility. Such nonlinear periodic structures allow for efficient qnasi-phase-matched second-order harmonic generation, which conld find applications where simultaneous conversion of multiple wavelengths is reqnired. Thns, in this chapter we will focus our... 

The SHG intensity from interfaces is determined by the second-order nonlinear susceptibility and the Fresnel coefficients. The SHG spectra of the probe pulses change depending on the transient electronic population and the orientation of the chromophores through these physical quantities. Hohlfeld and coworkers have studied hot electron dynamics in thin metal films by this technique [21]. From the transient response of the SHG intensity, electronic temperature decay due to the electron-phonon coupling in the metal substrate is extracted. Eisenthal and coworkers have studied ultrafast excited state dynamics of dye molecules at liquid interfaces [22]. Particularly, the isomerization dynamics of an organic dye at the interfaces was found to become significantly slower than in the bulk. 

In the case of second-harmonic generation, the second-order nonlinear susceptibility tensor elements are symmetric in their last two Cartesian indices and are unchanged by the permutation of their second and third frequency arguments because they are identical. Thus, Eq. (28) can be rewritten in the simplified form... 

From the two descriptions of the Pockels effect in the frameworks of electro-op-tics and nonlinear optics, one can show that the electro-optic tensor elements and the second-order nonlinear susceptibility elements are related by... 

Tsirelson et al. [194] extended Levine s model to account for the different types of bonds in second-order nonlinear susceptibility calculations. Applications to lithium formate deuterate (LiCOOH D2O) and related crystals with and without water molecules showed that the water can play a significant role in X -. They also proposed an expression for X which gave qualitative agreement with experiment for five ionic crystals. 

Aside from these approaches, empirical relations have been proposed to relate the second- and third-order nonlinear susceptibilities to their linear analogs. Among them are Miller s rule [209] for the second-order nonlinear susceptibility of ionic crystals as well as the expressions of Wang [210] and of Boling, Glass, and Owyoung [211] for the third-order nonlinear susceptibility in the low-frequency limit. The latter have been shown, by comparison with experiment, to possess their own range of validity. 

In Second-Harmonic Generation (SHG) experiments, an input beam of frequency (jt) incident in the material generates an output beam of frequency 2w. The response is described by the second-order nonlinear susceptibility ... 

The Maker fringe method is one of the most used methods for determining the second-order nonlinear susceptibility x /t of a crystal [27]. It is a relative method and is only useful when the second-harmonic signal is compared with the signal from a crystal with known values. The experiment is designed as follows. 

Because a strong electric field is required to align the molecules, further restrictions are imposed on the molecules they should have a permanent dipole moment. For instance, EFISHG can not be applied to measure the second-order nonlinear susceptibilities of octopolar molecules, even though at the molecular level, then-molecular hyperpolarizability, is non-zero. Also, EFISHG can not be used with ionic molecules or with a polar solvent. 

The general microscopic expression for the nth-order susceptibility contains n + 1 dipole moment matrix elements, involving n intermediate states. For the linear susceptibility there is only one intermediate state, and if the latter is a hybrid one, the corresponding dipole matrix elements are determined mainly by the Frenkel component of the hybrid state. Thus, the linear susceptibility of the hybrid structure contains the factor (dp/ap)2, as is seen from eqn (13.77). For the second-order nonlinear susceptibility x one must have two intermediate states or three virtual transitions. One of them may be a hybrid one, and as long... 

Defining the second-order nonlinear susceptibility for SHG as the relation between the square of the relevant components of the driving fields and the nonlinear source polarization. 

The second-order nonlinear susceptibility describing a surface or interface, as indicated by the microscopic form of equation B 1.5.30, is resonantly enhanced whenever an input or output photon energy matches a transition energy in the material system. Thus, by scanning the frequency or frequencies involved in the surface nonlinear process, we may perform surface-specific spectroscopy. This method has been successfully applied to probe both electronic transitions and vibrational transitions at interfaces. 

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. 

---