Third Order Effects

The quadrupolar effects of order higher than two (7) are usually assumed to be negligible, especially at high magnetic fields. However, once the first- and second-order effects are removed, the measurement of third-order contributions becomes realistic. It can be easily shown that, similar to the first-order case, the CT and all symmetric MQ transitions (q = 0) are free of the third-order contribution, which thus can be safely ignored in DAS, DOR, and MQMAS experiments [161,162]. This is not the case for transitions between non-symmetric spin states, such as the STs. Indeed, numerical simulations of the third-order effect have explained the spectral features that have been observed in 27A1 STMAS spectra of andalusite mineral [161]. 

Where P is the polarisation and the others the linear (1) and non-linear, second (2) and third order (3) terms. Examples of important second order effects are frequency doubling and linear electro-optic effects (Pockles effect), third order effects are third-harmonic generation, four-wave mixing and the quadratic electro-optic effect (Ken-effect). 

Second, if each run is performed only once, the effect standard deviation can still be estimated because high-order effects should be zero. A non-zero estimate of a third-order effect, therefore, may be attributed to random error and used to estimate the standard deviation of all effects. If m high-order effects can be calculated, the standard deviation of the effect is estimated as... 

The structure/property relationships that govern third-order NLO polarization are not well understood. Like second-order effects, third-order effects seem to scale with the linear polarizability. As a result, most research to date has been on highly polarizable molecules and materials such as polyacetylene, polythiophene and various semiconductors. To optimize third- order NLO response, a quartic, anharmonic term must be introduced into the electronic potential of the material. However, an understanding of the relationship between chemical structure and quartic anharmonicity must also be developed. Tutorials by P. Prasad and D. Eaton discuss some of the issues relating to third-order NLO materials. 

Since semiconductor materials appear to be general candidates for third order effects, it is not surprising that photoconductive charge transfer complexes such as poly(vinyl carbazole)-trinitrofluorenone (PVK-TNF) would exhibit modest DFWM.(142) The observed nonlinearity at 602 nm, a wavelength absorbed by the CT complex, ranges from 0.2 - 2 x 10"11 esu, increasing as the molar fraction of TNF in the complex, PVK TNF, increased. Charge... 

The photorefractive effect is classified here as a special third order effect for several reasons. First, it is perhaps the least well understood, mechanistically. Second, it represents the area of greatest current... 

The fundamental component (aE) is linear in E and represents the linear optical properties discussed above. The second (jfiE-E) third ( yE-E-E) and subsequent harmonic terms are nonlinear in E and give rise to NTO effects. The / and values are referred to, respectively, as the first and second hyperpolarisabilities. The second harmonic term gives rise to second harmonic generation (SHG), the third results in frequency tripling effects, and so on. Importantly, since only the time-averaged asymmetrically induced polarisation leads to second-order NLO effects, the molecule and crystal must be non-centrosymmetric, otherwise the effects will cancel one another. Third-order effects, however, may be observed in both centrosymmetric and non-centrosymmetric materials. 

Similarly, we have effective second- and third-order effective polarizabilities... 

Before we examine how second- and third-order N LO effects are related to nonlinear polarization, we briefly examine an important symmetry restriction on second-order NLO properties. From Eq. (5), we can see that P(E) = P(0) + xmE + x E2 + x i)E3+... and P -E) = P(0) - x 1)E + xp)E2 - x Eh... we can also see from Fig. 11.1 that P(E) + P(-E) if%(2) + 0. In a centra symmetric material, P(E) is necessarily equal to P(-E) and, therefore, P(0), and other even-order terms must be zero. Therefore, for second-order effects to be observed in a molecule or material, the molecule or material must be non-centrosymmetric. However, no such requirement applies to odd-order processes, such as third-order effects [Fig. 11.1 shows P(E) = P(-E) for a material with only linear and cubic susceptibilities non-zero]. 

Just as second-order effects involve the interaction of two electric fields with the electrons of a material, third-order effects involve three electric fields, 1 2 and 3. In the special case where all these three fields have the same frequency and where y(21 is zero (which is necessarily the case in centrosymmetric materials -see Section 11.1.2), we see that... 

Electric Field Definition and Prefactors of Third-Order Effects. . . 128... 

In a strong electric field second-order and even third-order effects may be present ... 

NLO susceptibilities (second- and third-order effects, respectively). 

At time t = 0, a dc field is applied it produces an electric field-induced Pockels effect (EFIPE), which is solely due to a third-order effect Eq) in the case of the copolymer because the molecules are not oriented by the dc field alone at room temperature, but which also contains a part due to the rotation, in a polar manner, of free chromophores in the guest-host system (induced The value of x is measured from the modulations of ATR... 

Thus, in both cases, the molecular unit can be tailored to meet a specific requirement. A second crucial step in engineering a molecular structure for nonlinear applications is to optimize the crystal structure. For second-order effects, a noncentrosymmetrical geometry is essential. Anisotropic features, such as parallel conjugated chains, are also useful for third-order effects. An important factor in the optimization process is to shape the material for a specific device so as to enhance the nonlinear efficiency of a given structure. A thin-film geometry is normally preferred because nonlinear interactions, linear filtering, and transmission functions can be integrated into one precise monolithic structure. 

All isothermal calculations discussed here employ Lennard-Jones potential functions and, unless otherwise stated, simulate free-boundary conditions. The neglect of three-particle interactions for a similar (Barker-Fisher-Watts) isolated pair potential has been shown to produce effects that are quite small for Ar systems. For clusters of more than three particles, the third-order potential energy terms 3 increase as the number of three-particle interactions increases. In the limit of zero temperature, where the third-order effects are most prevalent, 3 of the 13-particle Ar cluster (although already 60% of its bulk value) is less than 4.5% of the cluster s total potential energy. For a five-particle Ar cluster, 3 is less than 3% of the total potential energy. 

Since the current density in the bulk measures the surface charges, the time-dependent current-density functional theory (CDFT) appears to be a way to investigate this problem. At least the results presented by de Boeij et al. [171] for the bulk susceptibility and by van Faassen et al. [172] for the polarizability of linear chains are encouraging, although this may not be the case for second-and third-order effects. 

In the case of non-degenerate frequencies, the nonlocal third-order effects may give rise to chiral pump-probe spectroscopies. The only observation of a coherent Raman optical activity process to date is also due to a third-order pseudoscalar. Spiegel and Schneider have observed Raman optical activity in coherent anti-Stokes Raman scattering in a liquid of (-l-)-trans-pinane and report chiral signals that are 10 of the conventional electric-dipolar CARS intensity [23],... 

This work provides a relatively comprehensive review of studies involving ruthenium coordination and organometallic complexes as nonlinear optical (NLO) compounds/materials, including both quadratic (second-order) and cubic (third-order) effects, as well as dipolar and octupolar chromophores. Such complexes can display very large molecular NLO responses, as characterised by hyperpolarizabilities, and bulk effects such as second harmonic generation have also been observed in some instances. The great diversity of ruthenium chemistry provides an unparalleled variety of chromophoric structures, and facile Ru" —> Ru" redox processes can allow reversible and very effective switching of both quadratic and cubic NLO effects... 

The third-order NLO effects are described by xi. i) susceptibility on macroscopic level and by second hyperpolarizability y tensor on microscopic one. In contrary to the second order NLO effects the third order effects are present in all molecules and in bulk materials. There exists also a similar relationship between die corresponding bulk susceptibilities and the molecular hyperpolarizabilities as in the... 

We now see why the third order effect 234+135 was found to be large in the extrusion spheronization experiment. The defining relation shows that these second-order interactions 234 and 135 are confounded with the block effect -6. 

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