# DLI-Based DP-QPSK Reception Scheme for Short-Range Optical Communication
## 4 FSR Optimization  
Traditionally, DLI is configured with an exactly one-symbol-period differential delay on one arm and a 45\({}^{\circ}\) (\(-\)45\({}^{\circ}\)) phase shift on the other, which means the FSR of the DLI is equal to the transmission symbol rate. It has already been shown through simulations that decreasing the differential delay led to increased chromatic dispersion tolerances in DLI-based BPSK demodulation [28; 29; 30]. This study first qualitatively analyzed the effect of FSR adjustment on DP-QPSK demodulation performance through a theoretical derivation. The two cases of FSR increasing and decreasing are shown in Figure 4.  
\(T<1/R_{sym}\)  
When decreasing the differential delay of the DLI to less than one symbol period, part of a symbol interferes with itself. The signals before the photodetectors can be expressed as:  
\[E_{1u,2u,1v,2v}\ =\ \left\{ \begin{array}{l}\frac{A}{2}e^{j\varphi(t)}[e^{i \left/(\varphi_{n}\pm\frac{\pi}{4})}\pm e^{j\varphi_{n}}]\,\ in\ t_{1},\\ \frac{A}{2}e^{j\varphi(t)}[e^{i\left/(\varphi_{n+1}\pm\frac{\pi}{4})}\pm e^{j \varphi_{n}}]\,\ in\ t_{2}.\end{array} \right.\right. \tag{8}\]  
The converted photocurrent is (taking \(E_{1u}(t)\) as an example here):  
\[i_{PD}(t)=\frac{\sqrt{2}\alpha A^{2}}{4}(\sqrt{2}+\cos\Delta\varphi_{n}-\sin \Delta\varphi_{n})+\frac{\sqrt{2}\alpha A^{2}}{4}(1-\cos\Delta\varphi_{n}+ \sin\Delta\varphi_{n})t_{1}R_{sym}. \tag{9}\]  
After the balanced photodetectors, the two electrical signals can be written as (the DC component is ignored here):  
\[ \begin{array}{l}i_{u}(t)=\frac{\sqrt{2}}{2}\alpha A^{2}(\cos\Delta\varphi_{ n}-\sin\Delta\varphi_{n})t_{2}R_{sym},\\ i_{v}(t)=\frac{\sqrt{2}}{2}\alpha A^{2}(\cos\Delta\varphi_{n}+\sin\Delta \varphi_{n})t_{2}R_{sym}.\end{array}  \tag{10}\]  
\(\bullet\)\(T>1/R_{sym}\)  
The situation of increasing the delay to greater than one symbol period is shown in Figure 4. The signals before the photodetectors can be expressed as:  
\[E_{1u,2u,1v,2v}\ =\ \left\{ \begin{array}{l}\frac{A}{2}e^{j\varphi(t)}[e^{i \left/(\varphi_{n+2}\pm\frac{\pi}{4})}\pm e^{j\varphi_{n}}]\,\ in\ t_{1},\\ \frac{A}{2}e^{j\varphi(t)}[e^{i\left/(\varphi_{n+1}\pm\frac{\pi}{4})}\pm e^{j \varphi_{n}}]\,\ in\ t_{2}.\end{array} \right.\right. \tag{11}\]  
The converted photocurrent is (taking \(E_{1u}(t)\) as an example here):  
\[i_{PD}(t)=\frac{\sqrt{2}\alpha A^{2}}{4}(\sqrt{2}+\cos\Delta\varphi_{n}-\sin \Delta\varphi_{n})+\frac{\sqrt{2}\alpha A^{2}}{2}[\cos(\Delta\varphi_{n+1}+ \Delta\varphi_{n}+\frac{\pi}{4})-\cos(\Delta\varphi_{n}+\frac{\pi}{4})]t_{1} R_{sym}. \tag{12}\]  
After the balanced photodetectors, the two electrical signals can be written as (the DC component is ignored here):  
\[ \begin{array}{l}i_{u}(t)=\alpha A^{2}[\cos(\Delta\varphi_{n}+\frac{\pi}{4} )t_{2}+\cos(\Delta\varphi_{n+1}+\Delta\varphi_{n}+\frac{\pi}{4})t_{1}]R_{sym},\\ i_{v}(t)=\alpha A^{2}[\cos(\Delta\varphi_{n}-\frac{\pi}{4})t_{2}+\cos(\Delta \varphi_{n+1}+\Delta\varphi_{n}-\frac{\pi}{4})t_{1}]R_{sym}.\end{array}  \tag{13}\]  
Figure 4: Two cases of free spectral range (FSR) adjustment. \(R_{sym}\) is the symbol rate, \(t_{1}\) is the period during which a single symbol interferes with itself or the next second one, and \(t_{2}\) is the period during which a single symbol interferes with its adjacent symbol. It should be noted that the above formulas were derived without considering the noise. Since \(i_{u}(t)\) and \(i_{v}(t)\) obtained using Equations (7), (10), and (13) are subsequently compared to the set threshold for a 0-1 judgment, the impact of noise cannot be ignored. Figure 5 gives a schematic diagram of the 0-1 judgment and the noise was introduced. It can be seen from Figure 5 that the smaller \(\left|i_{u,v}(t)\right|\) was, the greater the probability of misjudging will be in the presence of the noise. Based on the formula derivations for the three situations, a qualitative analysis could be conducted.