## 5 Attenuation Method Experiment
### Phase II: Attenuation of Service Channels
#### 5.2.2 Raman Noise Estimation  
Raman noise is therefore the main limiting element of the whole scheme; for this reason, it is appropriate to calculate its approximate optical power. Based on the formulas from [27] given below, its values for the smallest (+0 dB) and largest (+20 dB) used service channel attenuations were calculated. * Raman noise power generated in the forward direction (\(\rightarrow\)): \[P_{\overline{SpR\overline{S}}}=P_{in}\cdot e^{-A}\cdot\rho(\lambda)\cdot L \cdot\Delta\lambda.\] (1)
* Raman noise power generated in the backward direction (\(\leftarrow\)): \[P_{\overline{SpR\overline{S}}}=P_{in}\cdot e^{-A}\cdot\rho(\lambda)\cdot \frac{sinh(L\alpha)}{\alpha}\cdot\Delta\lambda.\] (2)  
Both service channels contribute to Raman noise, which is generated in both forward and backward directions. Since the two channels emit in different directions, only forward Raman noise of CH30 and backward Raman noise of CH29 are incident on the QKD detector. The total malicious Raman noise power is then determined as the sum of these two outputs. The illustration can be found in Figure 4 below. \begin{table}
\begin{tabular}{c c c c} \hline
**SFP 29 and 30** & **Key Rate (kbps)** & **QBER (\%)** & **Visibility (\%)** \\ \hline Turned off & 2.07 & 3.04 & 98.60 \\ +20 dB & 2.04 & 3.22 & 98.40 \\ +10 dB & 1.59 & 3.41 & 97.70 \\ +7 dB & 1.26 & 4.10 & 97.20 \\ +5 dB & 0.65 & 4.45 & 97.40 \\ +4 dB & non-functional & 5.05 & 97.10 \\ +3 dB & non-functional & 5.27 & 96.70 \\ +2 dB & non-functional & 5.32 & 96.50 \\ +1 dB & non-functional & 6.56 & 96.00 \\ +0 dB & non-functional & 8.96 & 94.80 \\ \hline \end{tabular}
\end{table} 
Table 2: Quantum channel performance when combined with both service channels. The first column contains the attenuation added by the VOA to the already installed 20 dB. \begin{table}
\begin{tabular}{c c c c} \hline
**Connection** & **Key Rate (kbps)** & **QBER (\%)** & **Visibility (\%)** \\ \hline Direct & 2.11 & 3.33 & 98.70 \\
20 km & 1.21 & 4.05 & 97.40 \\ \hline \end{tabular}
\end{table} 
Table 3: Difference in quantum channel performance depending on the route length. Since Raman scattering is a nonlinear phenomenon, the calculation must be made in basic units (not decibels). Parameters of the following values are used for the calculation with Formulas (1) and (2):  
* **Output power--\(P_{out}\)**--optical power of the service channel (both have the same value) entering the first DWDM filter. This power must be converted to mW. It can be calculated as \(P_{out}=P_{in}\cdot e^{-A}=P_{in}\cdot e^{-A\cdot L}\), where \(A\) denotes the attenuation. \[P_{out|+0\text{ dB }}=-31.5\text{ dBm}=7.08\cdot 10^{-4}\text{ mW},\] \[P_{out|+20\text{ dB }}=-51.5\text{ dBm}=7.08\cdot 10^{-6}\text{ mW}.\]
* **Raman scattering coefficient--\(\rho(\lambda)\)**--determined from the Raman cross-section graph in [27] for channels two and three positions away from the quantum channel in the Stokes scattering region. \[\rho(\lambda)=1.85\cdot 10^{-9}\text{ }\frac{1}{\text{km}\cdot\text{nm}}.\]
* **Bandwidth--\(\Delta\lambda\)**--the region of the spectrum in which the Raman scattering power is measured. The DWDM channel has a width of 100 GHz, which must be converted to nanometers. \[\Delta\lambda=0.8\text{ nm}\approx\Delta\upsilon=100\text{ GHz}.\]
* **Route length--\(L\)**--the part of the route where Raman scattering and other nonlinear effects occur. \[L=20\text{ km}.\]
* **Attenuation coefficient--\(\alpha\)**--calculated based on the difference between input and output optical power, and then converted to \(\text{km}^{-1}\). \[\alpha=0.3\text{ }\frac{\text{dB}}{\text{km}}=6.91\cdot 10^{-2}\text{ km}^{-1}.\]  
Based on these values and formulas, both malicious components and their sum can then be calculated, thus expressing the total malicious Raman noise power that passes through the 100 GHz DWDM filter to the single-photon QKD detector. These results limit the range of SpRS optical power.