# DC Voltage Induces Quadratic Optical Nonlinearity in Ion-Exchanged Glasses at Room Temperature
## 3 Results and Discussion
### The EFISH vs. the IE Duration  
For the SON arising under the action of an applied static electric field (the EFISH effect), one can write [5]  
\[\chi^{(2)}_{eff}\sim\chi^{(3)}\cdot E_{DC}, \tag{1}\]  
where \(\chi^{(3)}\) and \(\chi^{(2)}_{eff}\) are the third-order optical susceptibility and the effective (electric-field-induced) second-order optical susceptibility, respectively; \(E_{DC}\) is the inner static electric field generated by the Maxwell-Wagner charge. Note that \(E_{DC}\) and, consequently, \(\chi^{(2)}_{eff}\) generally depend on the depth coordinate (\(x\)) since the distribution of \(E_{DC}\) is defined by the gradient distribution of slow K\({}^{+}\) ions formed in the course of IE. We assume that the electric field is concentrated in the region thinner than the SHG coherence length, \(l_{coh}=\frac{\lambda}{2\Delta n}\), where \(\lambda\) is the fundamental wavelength and \(\Delta n\) is the difference of the glass indices at the fundamental and SH wavelengths. Our estimation for the fundamental wavelength of 1064 nm is that \(l_{coh}\) is about 40 \(\upmu\)m for the soda-line glass used, which is far more than the depth of the potassium penetration in the glass under the conditions used [37]. Thus, we can consider the following relation for the intensity of the SH signal:  
\[I_{2\omega}\sim\left|\int\chi^{(3)}\cdot E_{DC}(x)dx\right|^{2}\sim\left|\chi ^{(3)}U_{K}\right|^{2}, \tag{2}\]  
where \(U_{K}\) is the voltage drop at the K\({}^{+}\)-enriched layer, which is directly proportional to the voltage applied, \(U\), if the layer is relatively thin. \(U_{K}\) directly depends on the resistance of this layer; therefore, \(U_{K}\propto d_{eff}\), where \(d_{eff}\) is the effective depth of the penetration of K\({}^{+}\) ions in the glass. Since IE is a diffusive process, the effective depth is proportional to the squared root of the IE duration, \(d_{eff}\sim\sqrt{t_{IE}}\). In summary, the SH signal \(I_{2\omega}\) should linearly depend on the duration of IE, \(t_{IE}\):  
\[I_{2\omega}\sim t_{IE}. \tag{3}\]  
We fabricated a series of specimens with the duration of IE varied from 15 min to 8 h and measured the MF of these specimens after the application of 1300 V DC voltage to them for 30 min. Note that the SON of each specimen before applying DC voltage was not different from the SON of the virgin glass slide, representing purely the surface nonlinearity. In Figure 2, we present the dependence of the maximal SH signal on the duration of IE. In Figure 2, one can observe the linear dependence (red dashed line) expected from the estimations given above. Plotted on the uniform scale, the MFs of the specimens are shown in subfigures in Figure 2.  
Note that longer durations or higher temperatures of IE may result in a stronger effect as they increase the depth of the penetration of K\({}^{+}\) ions in the glass, i.e., the thickness of the nonlinear region. However, when this depth approaches a noticeable fraction of \(l_{coh}\) (that is about 40 \(\upmu\)m in the studied glass), the strengthening of the effect would slow down due to the phase-mismatching of the SHG. Thus, in our case, the limiting depth of IE is about 40 \(\upmu\)m.  
Additionally, we measured the nonlinear signal from the specimen exchanged for 2 h as a function of applied voltage. We used transparent ITO electrodes, set the angle of the fundamental light incidence corresponding to the maximal SH signal and gradually increased the applied voltage from 200 V to 1400 V with a 200 V step. We kept the specimen under each voltage for \(\sim\)20 min to ensure it became fully charged and only then measured the SH signal. In Figure 3, we demonstrate the maximal SH signal as a function of the square of the applied voltage, \(U^{2}\). The dependence exhibits a linear trend with decent accuracy. This is in agreement with Equation (2) because \(U_{K}\) is in a linear proportion with the voltage \(U\) applied to the specimen.