## 3 Measurement Techniques of the Electro-Optic Coefficient  
### Nonlinear Ellipsometry  
Nonlinear ellipsometry (NLE) is based on a single wavelength reflection configuration proposed, independently, by [PERSON] [13] and [PERSON] [14] in 1990. Since its introduction, the Teng and Man technique (TMT) has been widely used for the determination of the electro-optic susceptibility of organic polymer films and it demonstrated to be sensitive to a number of contributions to the measured signals that can lead to the evaluation of several nonlinear parameters [1,40,41]. The TMT permits the measurement of both \(\widetilde{\chi}^{(2)}(-\omega,\omega,0)\) and \(\widetilde{\chi}^{(3)}(-\omega,\omega,0)\)[42] complex susceptibilities. Moreover, it is sensitive to space distributions of charges trapped in the polymer films [1, 16, 43]. In order to be applied, the TMT needs to operate on samples in which the polymer film is sandwiched between two electrodes. In the case of polymers for \(\chi^{(2)}\) applications, the two electrodes are used either to pole the film or to apply a modulating voltage on the polymer film itself in order to measure the \(\chi^{(2)}\) values. As the TMT is based on the measurement of the electric field induced change of the optical phase difference experienced by the s (perpendicular to the incidence plane) and p (parallel to the incidence plane) components of a laser which propagates through the polymer film, at least one (reflection operation) of the electrodes must be transparent to the probe light. Usually, a semiconductive transparent oxide thin layer, deposited on a glass substrate is used as transparent electrode. A polymer film is then deposited by spin coating it and then coating it with a metal electrode. Some researchers used polymer films sandwiched between two transparent electrodes and performed the experiments in a transmittance configuration [44, 45]; in this case the samples were produced by spin coating pairs of polymer films onto oxide coated glass slides and then sandwiching them face to face; elimination of the air gap was guaranteed by baking at glass transition temperature under mechanical pressure. The experimental configuration in reflection mode for TMT is usually composed by a cw laser that impinges on the sample at an incident angle \(\alpha\) after passing through a polarizer, which sets its polarization at 45\({}^{\circ}\) with respect to the incidence plane, and a phase compensating device that changes the relative phase between the s and p components of the laser beam by an amount \(\psi_{c}\). After reflection at the polymer-metal interface, the beam is analyzed by a polarizer that is crossed with respect to the input one. The power of the beam transmitted by the whole system is detected by a photodiode. Between the electrodes it is applied either a dc voltage \(V_{p}\) during the poling procedure, or a modulated voltage _V(t)_ during the measurement of the electro-optic coefficient, given by:  
\[V(t)=V_{m}\cos(\Omega t)\qquad\mbox{with}\;\;\Omega<<\omega \tag{25}\]  
The optical power at the output of the whole system is modulated, due to the response of the polymer film, both at \(\Omega\) and \(2\Omega\) through the linear \(\chi^{(2)}\) and quadratic \(\chi^{(3)}\) electro-optic response. The average output power can be written as:  
\[P_{dc}=\frac{1}{4}P_{0}\big{|}r_{s}\big{|}^{2}\big{[}1+\big{(}\tan\Phi\big{)} ^{2}-2\tan\Phi\cdot\cos\big{(}\Psi_{ps}+\Psi_{c}\big{)}\big{]} \tag{26}\]  
where \(P_{0}\) is the input beam power after the first polarizer, \(\psi_{ps}\) and \(\psi_{c}\) are the phase differences between \(s\) and \(p\) polarizations originated by the propagation of the light in the polymer and in the phase compensator, respectively. \(\Phi\) is defined by the ellipsometry relation  
\[\frac{r_{p}}{r_{s}}=e^{\gamma\psi_{ps}}\,\cdot\tan\Phi \tag{27}\]  
where and \(r_{p}\) and \(r_{s}\) are the Fresnel field reflection coefficients for the \(p\) and \(s\) polarization, respectively.