## 2 Theory, Requirements and Synthesis
### Closed-Loop Control (CLC)
#### 2.3.3 Closed-Loop Control Synthesis  
As the laser is chirped in wavelength, the photodiode measures a wavelength-dependent optical power, similar to that shown in Figure 5. The three maxima correspond to the three colors of the ODSB signal, as they pass by the MRR's resonance. Unfortunately, the prominence of the three maxima depends on several factors, such as an MRR's OTF and the modulation frequency (hence a shift of the sidebands from the carrier) and on a CSR. It might also be the case that the three maxima merge into one, which is unfavorable. For the CLC system to work, the maxima must be pronounced as the change in the derivative sign, which tells us whether we are on the rising \(\left(\frac{\text{d}}{\text{d}t}>0\right)\) or falling \(\left(\frac{\text{d}}{\text{d}t}<0\right)\) edge of a local maximum. The input voltage signal consists of a manually set bias voltage \(U_{\text{man}}\), a slow-changing yet dynamic bias voltage from the integrator \(U_{\text{int}}\) and a triangle wave voltage \(u_{\triangle}(t)\) of a certain frequency and amplitude. The latter is intended for  
Figure 5: Characteristic of the process, input triangle sensing signal, output triangle sensing signals and their derivatives at two complementary bias points. derivative sign. Given the desired near-DC behavior of \(U_{\text{int}}\), we will merge it together with \(U_{\text{man}}\) into a single DC term \(U\), which yields:  
\[u_{\text{in}}(t)=U+u_{\triangle}(t)\]  
The included DC component \(U\) sets the bias wavelength, which through the CLC system's operation should tend towards the local maximum. To aim for one of the extreme (left or right) local maxima rather than the global maximum (in the middle), as desired, we can either set the initial bias (wavelength) point to their vicinity or involve a more advanced system that initially performs a sweep over wider range of wavelengths to locate the desired local maximum. When any of the three spectral components approaches the MRR's resonance, the varying wavelength is converted to amplitude variations, which are then captured by the PIN-FET photodiode and eventually converted back to an electrical (voltage) signal of similar (triangular) shape. \[u_{\text{out}}(t)=\overset{\sim}{U}+\overset{\sim}{u}_{\triangle}(t)\]  
In principle, this is similar to the principle of slope detection from radio communications. Since the slope of the filter \(\mathrm{d}P/\mathrm{d}\lambda\) is roughly the opposite on each side of the MRR resonance, the resulting voltage signal reverses in phase as the maximum point is passed. This means that the derivatives of the input signal \(\frac{\mathrm{d}u_{\text{in}}}{\mathrm{d}t}\) and the output signal \(\frac{\mathrm{d}u_{\text{out}}}{\mathrm{d}t}\) match in sign when a color approaches from the left side, and mismatch when it approaches from the right side. Both are approximately square waves. Now, if the product \(\frac{\mathrm{d}u_{\text{in}}}{\mathrm{d}t}\cdot\frac{\mathrm{d}u_{\text{out}} }{\mathrm{d}t}\) is taken, we should ideally get either a positive or a negative DC voltage, which indicates whether \(U\), and hence \(\lambda_{c}\), should be increased or decreased. The DC component \(\widetilde{U}\) is actually a measurement of the average received optical power by the photodiode and is not relevant to our CLC system's operation, since it operates with respect to the derivative of the signal. Therefore, in the current implementation documented later on, \(\overset{\sim}{U}\) is filtered out by the DC block capacitor of the PIN-FET module. Figure 6 shows the block diagram of the proposed CLC system. Let us assume that the integrator's output is neutral right after the startup. At this point, the signal \(u_{\text{in}}\) comes solely from the triangle-wave generator. As this signal from the input \(u_{\text{in}}\) propagates through the process, we obtain either an in-phase or out-of-phase triangle wave signal at the output \(u_{\text{out}}\).