Third-order susceptibility, components

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. 

If monochromatic and linearly polarized input fields are considered, the third-order susceptibility can be expressed by its tensor components, Xyjj, =... 

In the third-harmonic generation, the third-order susceptibility leads to a nonlinear polarization component which oscillates at the third-harmonic frequency of the incident laser beam. This leads to a light wave at the third-harmonic frequency of the fundamental wave. As optical frequencies are involved and since the output frequency is different from the input frequency only the electronic nonlinearities can participate without any contributions from thermal or orientational effects. Because one needs fast nonlinearities for all-optical signal processing, the main interest is directed towards the fast electronic nonlinearities. Therefore and also due to its simplicity, third-harmonic generation is a very attractive method to characterize newly developed materials. 

The in (18)-(21) are the nth-order susceptibilities. They are tensors of rank n + with 3 " components. Thus, a second-order susceptibility, is a third-rank tensor with 27 components and the third-order susceptibility, a fourth-rank tensor with 81 components. The number of independent and significant elements is (fortunately) much lower (see p. 131). 

The quadratic effect of an externally applied field on the refractive index n is described by the third-order susceptibility (- ) w,0,0) (Kerr susceptibility). The two independent components Yilzz and x ixx can be interpreted in terms of molar polarizabilities. The results for 2 symmetric molecules with only one significant component of the second-order polarizability are expressed in (113) and (114),... 

We may expect all discrepancies found in second-order susceptibilities to become still worse in higher order, and this is certainly the case. The calculation of third-order susceptibilities is a straightforward extension of the calculation of second-order susceptibilities. Symmetry analysis of the zincblende structure shows that there arc two independent components those in which all indexes are the same (xi n i s an example) and those in which the indexes arc in pairs (x,i22 isan example). We obtain the susceptibility by obtaining the polarization per bond to third order in the field, in analogy with Eq. (5-10) ... 

In order to understand the origin of the high values of the third-order nonlinearity observed, we picked polybenzidine for more detailed study. The real and imaginary components of the third-order susceptibility x are measured separately. The real part Rex leads to nonlinear refraction, and the imaginary part Imx ) is responsible for nonlinear absorption. The nonlinear-optical measurements were made using dilute solutions of the polymer in DimethyLsulfoxide/Methanol in the ratio 4 1 (DMSO MeOH). 

In the case of THG, the third-order susceptibility corresponds to a nonlinear polarization component, which oscillates at the third harmonic frequency of the incident laser beam. Regarding the simphfied case of an isotropic solution, only the element ) third-order susceptibility tensor creates... 

D is the dielectric displacement, Co is the permittivity of free space, Xi >> the dielectric susceptibility of ith order, is the applied electric field), one gets the values of Ihble 2. It should be pointed out that the component should vanish in unpoled samples for thermodynamic reasons. Blending obviously reduces the third-order susceptibility. 

As is the case with molecular quantities, Fourier components of E and P are accompanied by frequency-dependent, complex susceptibilities % The macroscopic susceptibilities are used in the physical description of NLO effects, such effects typically being analyzed using wave equations in which the nonlinear polarization produced by a given type of interaction constitutes a source term. Quantities other than the susceptibilities are often used for describing specific NLO interactions. The most useful of these are the electro-optic coefficient r related to co ca,0), the nonlinear refractive index ti2, related to the real component of the degenerate third-order susceptibility Re(x —m,ai)), and the two-photon absorption coefficient jSg, related to the corresponding imaginary component Im(x —[Pg.66]

The situation is simpler for random eollections of moleeules as in, for example, liquids or glasses. As mentioned above, because isotropic media possess a statistical center of symmetry, the second-order susceptibility vanishes. For the third-order susceptibility, only two tensor components, xfni and xfui obtained as a result of orientational averaging. Xnn nan be related to the molecular hyperpolarizability as follows ... 

In Cartesian coordinates obviously there are altogether 3" elements in the third-order susceptibility X > ], a fourth-rank tensor, since i, j, k, /) each has three components 1, 2, 3. In an isotropic medium with inversion symmetry, however, it can be shown that there are only four different components, three of which are independent ... 

Detection sensitivity is one of the key issues in CARS microscopy. This is an especially acute problem in applications where chemical selectivity of CARS perfectly suits the tracking of small changes in cells related to specific protein and DNA distributions, external drug delivery/distribution, etc. There is, however, a component in CARS signal that is not associated with a particular vibration resonance and therefore does not carry chemically specific information. Unfortunately, in many cases, it can distort and even overwhelm the resonant signal of interest. In modeled approach, the CARS response originates from the third-order nonlinear susceptibility, which... 

The focal helds set up a polarization in the material. In the case of CARS, we are interested in the polarization resulting from the combined action of the pump (of frequency co ) and Stokes (of frequency coj beams, which induce motions in the electron clouds that oscillate at frequency 2co - co, the anti-Stokes frequency. The ability of the material to oscillate at the anti-Stokes frequency when the pump and Stokes helds are present is given by the third-order nonlinear susceptibility The strength of the polarization is furthermore determined by the amplitude of the pump (E ) and Stokes (E ) driving helds. In the tensorial notation, where and I denote the polarization components of the nonlinear susceptibility, the third-order polarization in the polarization direction i is given as... 

The components of the third-order nonlinear susceptibility relevant to the CARS process are conveniently subdivided into two terms vibrationally resonant (X, ) and vibrationally nonresonant (Xnr) components. The total response of the material depends on the sum of these two terms ... 

The imaginary components of the third-order nonlinear susceptibility of palladium and iridium complexes of C6o and C70 were determined using saturation spectroscopy. In all cases, ImQ ) values are smaller than those of uncomplexed C60 (1.78 X 10 16 m2 V-2) or C70 (7.55 X 10 17 m2 V-2), a result explained by decreased conjugation in the molecule and consequent decreased electron delocalization, although differing photodynamics were not excluded. 

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