## 4 Quantum Integrated Photonic Devices
### Fabrication and Characterization of a Quantum Project
The separations and the coupling lengths chosen, after the photo-reduction process, are found in the interval \(s\in[3,6]\)um and \(l\in[0.5,2]\)mm, respectively. We must indicate that under the ion-exchange process, the lateral diffusion increases the width of the channel guides up to \(4\) um with respect to the designed mask width of \(3\) um. An optical characterization of the directional couplers was then made. We denote as \(c_{o1}\) and \(c_{o2}\), respectively, the input complex amplitudes at the two guides of the coupler and as \(c_{1}\) and \(c_{2}\) the corresponding output complex amplitudes, so that the output powers are \(P_{i}=|c_{i}|^{2}\), \(i=1,2\). In the tests, light was coupled to one of the branches of the coupler and the power at each output was measured. Experimental results are shown in Table 1.  
Figure 15: Sketch of a directional coupler belonging to the series of directional couplers fabricated. The main parameters are indicated: separation \(s\) between channel guides and length \(l\) of the guides from after and up to before the Minford curves (MC). Some important conclusions can be obtained from the results shown in Table 1: the coupler \((s,l)=(3.0,2.0)\) is very close to a \(X_{\pi/2}\) coupler, and the coupler \((s,l)=(6.0,1.5)\) is very close to a \(X_{\pi/4}\) coupler. Alternatively, one can use a numerical linear fitting to obtain the optimal values of \((s,l)\) for fabricating a particular coupler \(X_{\theta}\); thus, Figure 16 shows the linear fit \(\theta(l)=a\,l+b\) for the three separations \(s\) considered. The term \(\theta(0)\) is the coupling introduced by the Minford curves and it is different from zero because \(l\) is the distance between the curved sections. Note that the above fit allows us to define an effective length \(l_{\text{eff}}\) such that \(\theta(l_{\text{eff}})=\kappa l_{\text{eff}}\), where \(\kappa\) is the coupling coefficient (\(a\equiv\kappa\)), and thus we can write the following relationship  
\[\theta(l)=\kappa\,l+\theta(0) \tag{15}\]  
By deriving \(\kappa\) and \(\theta(0)\) from the linear fit of the experimental coupling results we obtain the effective length \(l_{\text{eff}}=l+\theta(0)/\kappa\). In short, we have found the empiric parameters for fabricating simultaneously \(X_{\pi/2}\) and \(X_{\pi/2}\) couplers and thus to fabricate the three IQP devices described in the above subsection. The phase shifter \(\pi/2\) could be implemented by changing in an adiabatic way the length or the width of a channel guide. The experimental results for the directional coupler \((s,l)=(6.0,1.5)\) (\(X_{\pi/4}\) coupler) as a quantum projector can now be shown. The test was made by using a He-Ne laser source, so that we made a semi-classical optical characterization with coherent states, which is sufficient to characterize the directional coupler; we must stress, however, that with single-photon states the same results would be obtained. We excited different states at the input of the directional coupler by using an optical grating [194]. Indeed, let us consider an external optical grating with a periodic structure along the \(y\) axis (see, for instance, Figure 15). When it is normally illuminated by a Gaussian beam propagating along \(z\) direction, and the diffracted light is collected by a lens, then the diffraction orders \(\pm 1\) can be coupled to the inputs \(c_{01}\) and \(c_{02}\) of the directional coupler shown in Figure 15. Next, by translating the optical grating along \(y\) axis, different relative phases \(\delta\) between the amplitudes are obtained; in other words, the following input vector of amplitudes is obtained at the directional coupler: \(\mathbf{v}_{o}^{t}=(c_{01},\,\epsilon^{i\delta}c_{02})^{t}\), where superscript \(t\) stands for transposed. It is worth writing the corresponding input quantum state, i.e., the bimode coherent state \(|c_{01}\,\,\epsilon^{i\delta}c_{02}\rangle\).