In the present chapter we study a nonlinear response of an optical matrix formed by the K2HPO4 crystal doped with TiO2 nanoparticles. Such doped matrix is a nonlinear optical system that is characterized by the cubic non-linear optical response at picosecond laser pulses. Laser pulses release photoelectrons from nanoparticles, which emerge as free carriers on the nanoparticles' surface generating an electric field in local area of the K2HPO4 matrix, which results in the phase transition from the paraphase to the ferroelectric phase state. The appeared ferroelectric phase induces a large polarization around TiO2 nanoparticles, which in turn immediately produces a nonlinear optical response to the laser pulse of the inverse sign, such that the laser beam becomes more focused. The gigantic non-linear susceptibility ??(3) responsible for the phenomenon of focusing of the laser beam is calculated by using the pseudospin model for the description of ferroelectric crystals and the expressions for nonlinear-susceptibility tensor components computed by other researchers.
TopIntroduction
The third-order nonlinear optical effects (including nonlinear absorption and refraction) break the diffraction limit and form superresolution nanoscale spot (Wei, 2015). Especially important are the characteristics of the third-order effects. When a light beam with a frequency of ω is incident on the isotropic nonlinear medium, the nonlinear effect occurs, and the second-order nonlinear susceptibility 
can be neglected. The whole polarization is presented as
,where
and
the linear and third-order nonlinear polarization, respectively, and, correspondently, they are provided with the linear
and third-order nonlinear
susceptibility.
The single crystal potassium dihydrogen phosphate KH2PO4 is characterized by a unique set of properties, such as a wide range of optical transparency, nonlinear, electrooptical and piezoelectric effects. However, one of the main weaknesses of the crustal is its relatively low quadratic susceptibility. A possible way to increase the susceptibility and, subsequently, the efficiency of the three-wave processes is by altering its structure through a formation of nanocomposite medium (Grachev et al., 2012; Gayvoronsky et al., 2012, 2013). Nanoparticles incorporation into the KH2PO4 matrix was realized in order to design a novel lasing medium, which could result in the appearance of third-order nonlinear 
susceptibility. One of such nanoparticles is titanium dioxide TiO2 especially in the anatase phase.
A successful growth of high quality KH2PO4 (KDP) crystals with incorporated TiO2 anatase nanoparticles was demonstrated by Grachev et al. (2012). Those doped crystals of KH2PO4 were studied by using transmission and scanning electron microscopy, energy dispersive X-ray analysis, Fourier transformation infrared spectra, electron paramagnetic resonance spectra, and nonlinear optics. It was revealed that TiO2 nanoparticles are embedded in the KH2PO4 not chaotically, but as layers separated at a distance of about 15 μm.
As Grachev et al. (2012) and Gayvoronsky et al. (2012, 2013) shown, the incorporation of anatase nanoparticles into the KH2PO4 crystal changes the sign of the refractive nonlinear optical response relatively to that of the pure KH2PO4 crystal matrix. The phenomenon is associated with the overlapping of the energy states of intrinsic defects in the crystal matrix and the surface state of TiO2 nanoparticles.
TiO2 nanoparticles with an average diameter 2R=15 nm are uniformly distributed in plains of the KH2PO4 crystal. The density of TiO2 in the KH2PO4 crystal varies from 1016 to 1017 m-3. This allows one to determine an average distance between these nanoparticles equal to 15 μm in each plain of the KH2PO4.