## 2 Theory, Requirements and Synthesis
### Closed-Loop Control (CLC)
#### 2.3.2 The Process  
To set the ground for designing a CLC system, we must first understand the system to be tuned/controlled, i.e., the process. Figure 3 breaks down the closed-loop topology and shows the building blocks of the process. The process begins with the voltage-controlled current source \(i(u_{\mathrm{in}})\) that drives the TEC elements. Its time constant should be significantly greater than that of the TEC element, which is expected to be the slowest element in the chain. The TEC element converts the electrical current \(i_{\mathrm{in}}\) (not to be confused with the laser-supply current) into heat and temperature \(\phi_{\mathrm{TEC}}(i_{\mathrm{in}})\) change, resulting in the carrier  
Figure 3: The process with its constituent parts: A voltage-controlled current source (VCCS), a DFB, a Mach–Zehnder modulator (MZM), an MRR and a PIN-FET receiver. Figure 2: Block diagram showing a RoF link (turquoise polygon), a feedback topology (pink polygon), the process (beige polygon) under closed-loop control and the add/drop MRR enabling the sideband suppression and thereby limiting the power fading. wavelength \(\lambda_{\rm c}(\vartheta_{\rm TEC})\) change at the laser output. The corresponding 2-port model (shown in Figure 4a) with \(i_{\rm in}\) at the input and the optical power spectrum concentrated around \(\lambda_{\rm c}(i_{\rm in})\) at the output turns out to be a low-pass filter, featuring a 3 dB bandwidth (\(B\)) in the millihertz range and thus a relatively large time constant \(\tau\) in the range of a few seconds. For the laser used in our experiments, these two parameters were measured at \(B=f_{\rm C}\approx 17\) mHz and \(\tau_{\rm DFB}=1/2\pi f_{\rm C}\approx 9.4\) s, where \(f_{\rm C}\) represents the cutoff frequency. Additionally, we measured the slope (\(\Delta\lambda_{\rm pp}/\Delta f\approx-4.6\) dB/octave) to enable the extrapolation of the attenuation for higher modulation frequencies \(f_{\triangle}\). It must be noted that extrapolation can only be applied as a rough estimation, as the system is not linear. The mathematical approximation of the laser's output power spectrum is as follows:  
\[P_{\rm DFB}(\lambda,\lambda_{\rm c})=P_{\rm DFB,\partial}\,\delta[\lambda- \lambda_{\rm c}]\]  
where \(P_{\rm DFB,0}\) is the output optical power of the laser, \(\lambda\) is an independent variable, and \(\lambda_{\rm c}\) is a variable parameter, subject to tuning. This carrier is subsequently modulated by the RF signal at the Mach-Zehnder modulator (MZM) (Figure 4b), which in the first approximation introduces a 3-color (higher-order harmonics are neglected, assuming the MZM is biased at the quadrature point) signal to the MRR's input. All three resulting wavelength components are subject to temperature fluctuations \(\lambda_{\rm c}(\vartheta_{\rm TEC}),\ \lambda_{\rm USB}=\lambda_{\rm c}-\Delta \lambda,\ \lambda_{\rm LSB}=\lambda_{\rm c}+\Delta\lambda\), where \(\Delta\lambda\approx\frac{\lambda_{\rm c}^{2}}{\xi_{0}}f_{\rm RF}\), where \(c_{0}\) is  
Figure 4: Building blocks of the process under closed-loop control: (**a**) A heat-pump-controlled DFB laser and the corresponding low-pass behavior, (**b**) a MZM, its modulation characteristic and the spectral profile of the signal at its output, (**c**) a MRR employed in the input-drop direction, its optical transfer function and the slope detection principle, and (**d**) PIN-FET as a near DC power meter. the speed of light in vacuum and \(f_{\text{RF}}\) is the RF modulation frequency. The corresponding power spectrum is:  
\[P_{\text{MZM}}(\lambda)\propto\delta(\lambda-\lambda_{0})+[1-CSR][\delta(\lambda -\lambda_{\text{LSB}})+\delta(\lambda-\lambda_{\text{USB}})],\]  
where \(CSR\) stands for the carrier-to-sideband ratio (CSR).