## 3 Measurement Techniques of the Electro-Optic Coefficient
### Second Harmonic Generation  
Since the EO coefficient \(r\) is related with the second order susceptibility term \(\chi^{(2)}\) (see Equation (14)) it is possible to evaluate it by measuring the efficiency of the second harmonic generation process in polymeric films. Second harmonic generation was the first all-optical phenomenon shown in an experiment in nonlinear optics [52]. The experiment consisted in focusing a ruby laser beam on a crystalline quartz plate and revealing the second harmonic with a prism spectrometer on a photographic plate. The conversion efficiency has greatly improved with respect to the first experiments and second harmonic generation is now ordinarily used in photonic devices and applications. Moreover, due to the fact that the \(\chi^{(2)}\) coefficient is present only in non-centrosymmetric materials, SHG was widely used in order to study symmetry properties of bulk materials, thin films [53, 54] or surfaces [55, 56] with high sensitivity in terms of linear measurements [57, 58]. Usually, the polymeric samples used in SHG measurements are film deposited by spin coating on glass substrates. In order to prevent influencing the measurement, the substrate was an amorphous glass slide (no \(\chi^{(2)}\) contribution) without electrodes on the polymeric film. Since, after deposition, the orientation of the nonlinear chromophores is isotropic, the material must be poled before measurement. In this case, the poling technique used is the corona poling,as it does not require any contact electrode. An alternative poling procedure is the in-plane poling or lateral poling [59] where the electric field is applied between two coplanar electrodes that are deposited on the polymeric surface beside the region where the SHG measurements are performed. In the case of using an electrode for contact poling, the analysis of the multilayer structure should be performed as in [60]. A standard technique used in order to measure the electro-optic coefficient by SHG from poled samples is the Maker fringes scheme [18, 19, 20] where second harmonic signal is detected as a function of the incidence angle \(\alpha\). The fundamental beam at the angular frequency \(\omega\) is generated by a pulsed laser that can be suitably polarized (e.g., in s or p polarization state); likewise an analyzer selects either \(s\) or \(p\) polarization for the signal generated at \(2\omega\). The power generated by the sample can be calculated by means of the theoretical expression for the SHG power [19] given by:  
\[\mathrm{W}_{2\omega}(\alpha)=\left(\frac{512\pi^{3}}{A}\right)\!\!\left(t_{ \omega}(\alpha)\right)^{4}\!\left(T_{2\omega}(\alpha)\right)\!\mathrm{W}_{ \omega}^{2}\frac{\left(\sin\!\left(\Psi(\alpha)\right)\right)^{2}}{\left[ \left(n_{\omega}(\alpha)\right)^{2}-\left(n_{2\omega}(\alpha)\right)^{2} \right]^{2}}\!\left(d_{\mathit{eff}}(\alpha)\right)^{2} \tag{30}\]  
where \(A\) is the area of the transverse section of the fundamental beam, \(T_{2\omega}\) is the intensity transmittance of the output film/air interface at \(2\omega\), \(t_{\omega}\) is the field transmission coefficient of the input air/glass interface at \(\omega\) (neglecting the glass/film interface due to the small refractive index contrast), \(W_{\omega}\) represents the input power of the fundamental beam, \(n_{\omega}(\alpha)\) and \(n_{2\omega}(\alpha)\) are the polarization dependent refractive indices at \(\omega\) and \(2\omega\) and \(d_{\mathit{eff}}\) is the effective nonlinearity. The phase factor \(\Psi(\alpha)\) is given by:  
\[\Psi=\left(\frac{\pi L}{2}\right)\!\  
The previous procedure can be used to find the susceptibility coefficient \(\chi_{jk}^{(2)}(-2\omega,\omega,\omega)\) via the Equation (32). Then, in order to recover the electro-optic coefficient a modified expression of Equation (17) can be used [61; 62]:  
\[r_{33}(-\omega,\omega,0)=-\frac{f^{0}}{f^{\omega}}\frac{2}{n_{\omega}^{4}}\, \chi_{jk}^{(2)}(-2\omega,\omega,\omega) \tag{35}\]  
where the field factors Equations (7) and (8) are present.