Nonlinear second-order process

Both second-order and third-order materials have technological applications because of their ability to convert low-frequency light to high-frequency light. However, the efficiency with which they are able to accomplish this feat decreases dramatically from second order to third order. Even the second-order process is small compared to the first-order process. From the design point, one faces a dilemma avoid the symmetry constraint and live with the low efficiency of third-order materials or adhere to the symmetry constraints and reap the benefit of better conversion. In Chapter 12, more about synthetic strategies as they relate to producing nonlinear optical materials will be covered. 

Fig. 16.9 (a) Concept of higher photon confinement at the tip apex by a nonlinear optical process. Efficiency of a first-order process such as spontaneous Raman or fluorescence directly reflects the field distribution at the tip apex while the efficiency of a second-order process such as SHG or SFG and a third-order process such as CARS shows further confinement due to the nonlinear response of the material, (b) Energy diagram of CARS process... 

At pH 6-9, the transient Mn tthaH(02") complex disappears by a second-order process that varies nonlinearly with [Mn TTHA] and almost linearly with [H" ]. The equilibrium constant, Kg, is also calculated from a study of [Mn TTHA] and pH dependence on the observed rate of disappearance of Mn tthaH(02 ) , using equation II. This kinetic equation is easily derived assuming a mechanism involving equilibria 6 and 9 and reaction 10. Equilibrium 6 is necessary to the overall mechanism, as the observed spontaneous disappearance of O2" is pH dependent (22). 

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. 

Sum-fretiuencv generation (SI Ci) is a nonlinear optical technique basoil on the interaction of two plu tons at a surface. The result of the wave-mixing interaction is the production of a single photon whose frequency is the sum of the incident frequencies. If the two incident photons are of Ihe same frequency, the technique is called second-harmonic generation because the exiting photon has a frequency Iwice iha of the incident photons. Because this is a weak second-order process, intense lasers must be used. 

The nonlinear polarization for the second-order processes can be written as... 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows a schematic of this effect at the molecular level where second-order effects in noncentrosymmetric molecules combine to yield a third-order response that may be difficult to separate from a pure third-order process. This form of cascading is complicated by the near-field relationships that appear in the interaction between molecules, but analysis of cascaded phenomena is of interest, because it provides a way to explore local fields and the correlations between orientations of dipoles in a centros5nnmetric material (52). 

The multiple regression analysis was performed for constructing the RSM-based burr size mathematical models. In the present investigation, the second order burr height and burr thickness mathematical models have been developed in terms of four process parameters, namely, cutting speed (v) feed rate (/), drill diameter (d) and point angle (. The nonlinear second order response surface equation is given by ... 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows... 

Fig. 2. Schematic showing a cascaded optical nonlinearity in which the second harmonic generated at one molecule combines with the fundamental by sum frequency generation at a nearby molecule to result in a third harmonic generated through a cascade of second-order processes.

The n term in the series represents an n order interaction with the light field. The polarization, here defined as Tr[/6/t], is the measurable quantity in an optical experiment. Each term in the series has a specific meaning. The first term, usually referred to as the susceptibility, gives rise to absorption. The second term, which averages to zero in an isotropic medium, gives rise to second-order processes the third term is the relevant term for virtually all nonlinear optical experiments, also called experiments for the name of the susceptibility matrix. Higher order terms are rarely considered in view of the complicated experimental configuration needed, and the equally involved mathematical analysis needed to separate it from lower order processes. Some experiments such as Stark fluorescence, reported for a few tautomeric compounds, can be classified as measurements. 

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]... 

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