Tensors third harmonic generation

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

Characterization of Molecular Hyperpolarizabilities Using Third Harmonic Generation. Third harmonic generation (THG) is the generation of light at frequency 3co by the nonlinear interaction of a material and a fundamental laser field at frequency co. The process involves the third-order susceptibility x 3K-3 , , ) where —3 represents an output photon at 3 and the three s stand for the three input photons at . Since x(3) is a fourth (even) rank tensor property it can be nonzero for all material symmetry classes including isotropic media. This is easy to see since the components of x(3) transform like products of four spatial coordinates, e.g. x4 or x2y2. There are 21 components that are even under an inversion operation and thus can be nonzero in an isotropic medium. Since some of the terms are interrelated there are only four independent terms for the isotropic case. 

Under the influence of an optical pump, the molecular angular distribution described by Equation 12.4 can be considerably modified. In turn, this results in modification of the X ijkl tensor components. Further, we discuss the influence of a polarized pump beam on third-order nonlinear phenomena such as third harmonic generation (THG) [(described by (-3a),ft>,w,a>) coefficient], electric field induced second harmonic generation (EFISH) [x / kl -2(0, (o, o), 0)] and degenerate four-wave mixing (DFWM) X kl ... 

The tensors and 7 constitute the molecular origin of the second-and third-order nonlinear optical phenomena such as electro-optic Pock-els effect (EOPE), optical rectification (OR), third harmonic generation (THG), electric field induced second harmonic generation (EFI-SHG), intensity dependent refractive index (IDRI), optical Kerr effect (OKE), electric field induced optical rectification (EFI-OR). To save space we do not indicate the full expressions for and 7 related to the different second and third order processes but we introduce the notations —(Ajy,ui,cj2) and 7(—a , o i,W2,W3), where the frequency relations to be used for the various non-linear optical processes which can be obtained in the case of both static and oscillating monochromatic fields are reported in Table 1.7. 

Third-order nonlinear optical effects, such as third-harmonic generation or other kinds of four-wave mixing phenomena, occur in all media, irrespective of their symmetry. This follows from the parity-even property of the corresponding tensors. Consequently, third-harmonic generation can be observed both in... 

Finally, fhe application of higher order nonlinear optical processes such as third-harmonic generation (Berkovic 1995 Tsang 1995) or fourth-harmonic generation (Lee et al. 1997) could provide even more detailed interface information. For example, in the case of fourth-harmonic generation the induced polarization is given by Pj 4u>) = j k,l,m,nx]tL Ek u>)Ei co)Em u>)E io), i.e., the hyperpolarizability is a tensor of rank 5. Hence if is possible fo resolve up to five-fold surface symmetries. However, the absolute values are... 

In the case of third harmonic generation, the frequencies of the input beams are the same (coi = CO2 = CO3) and the tripled beam has a frequency CO4 = 3coi. Therefore, for the THG process, Xyu should be invariant when the relative positions ofjkl indices change. Consequently, the number of independent elements of the susceptibility tensor for the THG process I is reduced to just four %xxxx, Xzzzz, Xxxzz, and Xzzxx-Furthermore, when Kleinman symmetry conditions are adopted for experiments performed far from the resonances, one would have x xzz = Xzzxx> resulting in only three independent elements [206]. 

The odd order susceptibilities are nonzero in all materials. However, owing to the fact that x is a third rank tensor, the second order susceptibility is nonzero only in noncentrosym-metric materials, that is, materials possessing no center of symmetry. The focus of this paper is on second order processes, and the relationships between the bulk susceptibility, second harmonic generation, and the linear electro-optic effect. For second harmonic generation, Xijl is symmetric in ij, leading to the relationship between the second harmonic coefficient dijk and the bulk second order susceptibility x 2)[i2l... 

Since xl" is third-rank tensor, it possesses non-zero components only in the case of molecules which have no centre of symmetry the precetUng process of second harmonic generation will not interest us any further here. It will be remembered, howevor, that a static dectric Rdd acts in a manner to lower the symmetry of bodies and can thus endow an isotropic body, naturally possessing a centre of symmetry, with the symmetry of an axially symmetric body without a centre of symmetry. In this way, a naturally isotropic body without centre of symmetry, during the time it is acted on by a static electric field E, has the atnlity to cause a doubling of the oscillation frequency of an incident laser beam, since now we have in equation (29) ... 

On the assumption of total symmetry of the tensor of third-order nonlinear polarizability c(— co coi, cog, cog), its non-zero and independent elements are the same as those of Table 12. Direct theoretical calculations of c = c(0 0,0,0) have been performed for the atoms of inert gases and some simple molecules. Values of the tensor elements = c(— cu cu, 0,0) have been determined for numerous molecules from static Kerr effect studies and values of c = c(— cd ot>,coi — col) from measurements of optical birefringence induced by laser li t. Measurements of second-harmonic generation by gases in the presence of a static electric field yield the tensor elements c " = c( — 2co co, to, 0), which can also be obtained from second-harmonic scattering in centro-symmetric liquids. The elements of the tensor c = c(— 3co co, co, co)... 

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... 

Most of the studies on push-pull phthalocyanines have been carried out by means of Electric Field Induced Second Harmonic (EFISH) Generation experiments in solution. Since only one experimental condition is favorable for EFISH, namely, parallel polarizations for all optical and static fields, these experiments lead to only one observable the vector component along the charge transfer axis assuming that the dipole moment vector and the vector part of the third-rank tensor along tlie molecular z-axis are collinear). 

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