The third-order susceptibility

Although two-photon transitions were the first multiphoton transitions to be considered, there is no reason to limit studies of atomic response to just two optical waves, once the possibility of nonlinear coupling between the atoms and the radiation field has been recognised. Indeed, one of the most important processes involves the generation of a higher frequency lo by the superposition of three waves oq, ll 2 and 0J3, a process referred to as four-wave mixing. 

This process involves the hyperpolarisability, or third-order susceptibility in terms of which the polarisation produced is expressed as 

The nonlinear susceptibility is evaluated using third-order perturbation theory, and resonant enhancement is readily demonstrated to occur. Four-wave mixing is a useful experimental technique to extend the energy range available to tunable dye lasers [468]. It is also of interest that processes involving excitation by three photons allow transitions between even and odd parity states to be excited, as do single-photon transitions. 

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Three Wave Mixing. A number of resonances are possible in the third-order susceptibility, These resonances may be due to... 

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. 

The reflectivity of the doped gel was found to be 31% of that from the CS2 reference. Assuming that the third-order susceptibility of carbon disulfide is 1.7 x 10 ... 

The third order optical susceptibility was measured for a series of transition metal tetrakis(cumylphenoxy)phthalocyanines at 1.064 pm. Metal substitution caused a dramatic variation in the third order susceptibility. The largest s were found in the Co, Ni, and Pt complexes. Metal substitution introduces low lying electronic states which can enhance the susceptibility in these phthalocyanines. A strategy for enhancing the figure of merit, x(3)/a> of centrosymmetric nonlinear optical materials is suggested. 

This paper is a more extensive survey of the influence of the metal on the hyperpolarizability of a series of the transition metal tetrakis(cumylphenoxy)-phthalocyanines (MPcCP4). The compounds chosen were those most closely related to PtPcCP4, the compound which showed the largest hyperpolarizibility in the previous study. Specifically, phthalocyanines substituted with the last four members of the first row transition metal series (Co, Ni, Cu, and Zn) and also with the Ni, Pd, Pt triad were prepared and studied. The near IR spectra of these tetrakis(cumylphenoxy)-phthalocyanines are briefly discussed. Speculation on how metal substitution can influence the third order susceptibility of a near centrosymmetric structure, like that of the phthalocyanines, is presented. 

As expected, the third order susceptibilities vary significantly with polymer orientation. It seems unlikely however that this feature alone will ever increase the values by more than an order of magnitude and further significant improvements will probably require more highly polarizable substituents, the introduction of... 

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

The linear susceptibility yy1 1 is related to optical refraction and absorption. The most common effects due to second-order susceptibility x(2) are frequency doubling x (-2co co, co) and the EO (Pockels) effect x(2)(- 0, co). The third-order susceptibility y 3) is responsible for such phenomena as frequency tripling and the Kerr effect. 

As one of the alternative methods for treating ultrafast pump-probe experiments [50], the third-order susceptibility method is often used. A main feature of this method is that it can treat the case of the overlapping region between the pump and the probe lasers, but one has to carry out the perturbation calculations to the third-order approximations. 

This expression introduces the third order susceptibility of the medium, a quantity not easy to be accurately determined for the small portions of solvent in which the nonlinearity effect is sizeable. In addition we remark that with the favourable exception of atomic ions which have a spherical symmetry, the solvent layer in question has an irregular shape (not directly amenable to the molecular shape because the chemical groups responsible for nonlinearities are not regularly placed on the molecular surface). For this reason the whole tensorial expression of xCi> with a position dependent formulation, should be used. 

In conclusion the contribution to the dielectric response given by the third order susceptibility has different sources with opposite signs. Molecular simulations on ions in solution show that both dielectric saturation and electrostriction effects are presumably present and that for ions with a high charge density electric saturation predominates. This suggestion is in agreement with the general consensus that dielectric saturation is the first element to consider in the description of nonlinearities. 

Complications arise because of the cross-terms in the square of the third order susceptibility ... 

These processes are directly proportional to the imaginary part of the third-order susceptibility y of the self- and cross-phase modulation process. 

In consequence, any dominating two-photon resonance has to be avoided to maintain a small phase of the third-order susceptibility yf3). 

Special care has to be taken if the signal frequency is identical with one of the input frequencies as e.g. in DFWM and Z-scan. If the condition co cOj in the third-order susceptibility is fulfilled, the multiplication of the... 

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 external electric field is assumed to be parallel to the x-axis. In the case of an isotropic solution only the element Axl4x( 3co,co,co,co) of the third-order susceptibility creates a polarization at 3 , which is parallel to the incident electric field Ea ... 

To determine the third-order susceptibility y a step wise procedure is employed (Fig. 4). A measurement of a fused silica plate calibrates the THG setup. The measurement of the same fused silica plate in air atmosphere leads to the contribution of the air, which is a constant background reducing the signal in-... 

THG measurements are performed for different concentrations. From the third-order susceptibilities y of the solution series the second-order hyperpolarizabilities y of the molecules are determined by analyzing the data points with... 

The third-order susceptibility of the solvent is the intersection of the concentration series with the axis at zero concentration (Fig. 5). The hyperpolarizability of the sample molecule is proportional to the slope of the concentration series. Generally, the molecular hyperpolarizability yis retrieved with an experimental error of about ten percent. 

In the degenerate four wave mixing (DFWM) experiment the third-order susceptibility 3)(-tt>,tt>,-CL>,CL>) with degenerate frequencies can be determined [22]. This nonlinear susceptibility is directly proportional to the nonlinear refractive index n2, which is used to describe optically induced refractive index changes. An advantage of this technique is the possibility to record the temporal shape of the third-order nonlinear optical signal. 

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