# In-Band OSNR Measurement Method for All-Optical Regenerators in Optical Domain
## 2 Principle of the Measurement Method for All-Regenerator  
In what follows, we firstly describe the gain characteristics of signal and noise through the nonlinear power transfer function (PTF) and then put forward the optical-domain measurement method of OSNR for all-optical regenerators. Figure 1 illustrates the power spectrum variation of the signal and noise within the same bandwidth through a device with the nonlinear PTF by the VPIphotonics(tm) simulation software. The spectra of the input optical Gaussian pulse and amplified spontaneous emission (ASE) noise at the same average power are plotted in Figure 1a, corresponding to the power distribution histograms as shown in Figure 1b. The horizontal axis represents the probability of the instantaneous power occurrence. It is obvious that the power distributions of the optical pulse and ASE noise have a great difference. After those power components pass through a device with the nonlinear transfer as shown in Figure 1c, the resulting output power distributions have a distinct change as shown in Figure 1d, and then result in different gains of the signal and noise. From Figure 1b,d, the average power ratios of the pulse signal to ASE noise are respectively 1:1 and 1.64:1 before and after the nonlinear PTF. The increase in the power ratio means the improvement of OSNR. In the following, we take the all-optical four-wave mixing (FWM)-based regenerator as an example, to reveal the principle of the proposed in-band measurement method. Firstly, according to the PTF of the regenerator, we obtain the gain at the operating point, \(G_{0}=G_{0}(P_{\text{in}})\), in which \(P_{\text{in}}\) is the input power. For a degraded signal, the total powers at the input and output of the regenerator can respectively be expressed as \(P_{\text{i}}=P_{\text{si}}+P_{\text{ni}}\) and \(P_{\text{o}}=P_{\text{so}}+P_{\text{no}}\), where the s and n subscripts represent the signal component and ASE noise, and the i and o subscripts indicate the input and output of the regenerator, respectively. Note that \(P_{\text{so}}=G_{0}\cdot P_{\text{si}}\) at the operating point for the input and output signal. Thus, the average power of the output ASE noise can be expressed as  
\[P_{\text{no}}=P_{\text{o}}-P_{\text{so}}=GP_{\text{i}}-P_{\text{so}}=(G-G_{0}) P_{\text{si}}+GP_{\text{ni}} \tag{1}\]  
where \(G=P_{\text{o}}/P_{\text{i}}\) is the total gain for the input degraded signal. Then, the output \(OSNR\) is deduced from Equation (1) as follows:  
\[OSNR_{\text{out}}=\frac{G_{0}}{(G-G_{0})+G\cdot OSNR_{\text{in}}^{-1}} \tag{2}\]  
where \(OSNR_{\text{in}}=P_{\text{si}}/P_{\text{ni}}\) and \(OSNR_{\text{out}}=P_{\text{so}}/P_{\text{no}}\), corresponding to the input and output \(OSNR\) respectively. It should be noted that the gain \(G_{0}\) is dependent on the input signal power at the operating point. For this reason, Equation (2) is applied to the cases with the same gain for each regenerative level, that is, the operating points locate on a common linear gain curve [5]. Here, we introduce the parameter of the optical shaping figure (\(OSF\)) in the optical domain to evaluate the measure of improving \(OSNR\) as follows:  
\[OSF=OSNR_{\text{out}}/OSNR_{\text{in}} \tag{3}\]  
By substituting Equation (2) into Equation (3), the \(OSF\) parameter can then expressed as  
\[OSF=\frac{G_{0}}{G+(G-G_{0})\cdot OSNR_{\text{in}}}. \tag{4}\]  
From Equation (4), in the absence of noise, \(G=G_{0}\) and \(OSF\) = 0 dB, with no \(OSNR\) improvement. We can also deduce the condition for \(OSF\) > 0 dB as follows:  
\[\frac{OSNR_{\text{in}}}{1+OSNR_{\text{in}}}<\frac{G}{G_{0}}<1 \tag{5}\]  
Figure 1: The nonlinear transfer process of the signal pulse and amplified spontaneous emission (ASE) noise, (**a**) power spectra of Gaussian signal pulse and ASE noise, (**b**) histogram of instantaneous input power distribution, (**c**) the nonlinear power transfer function (PTF) and signal gain curves of the optical device, and (**d**) histogram of the signal and noise output from the nonlinear PTF.