# Phase Alignment of an Array Optical Telescope System Using Balanced Detection
## 2 Theories and Design
### Theories of the Structural Innovation  
Mach Zehnder Interferometer (MZI) is a key device of DPSK demodulation. Figure 2 shows the schematic structure diagram of an MZI, which includes two cross-couplers and two optical fibers. As the inputs of MZI, the signal light and zero value light are equally split into two optical signals by the first cross-coupler. Then, the information flow of adjacent bits is obtained by setting a delay of one bit on the upper arm. Additionally, the phase information of the light is converted into the intensity information of the light with interference occurring at the output port of the second cross coupler. The interference output of MZI and the input of the balance detector are connected by optical fibers. Finally, the demodulation of the DPSK optical signal is completed by using the electrical signal. This study improved the MZI and combined it with the balance detector to establish the structure of DPSK demodulation and phase detection. The difference is that the structure reduced the use of the front-end cross-coupler. Therefore, this design of the demodulator combines digital signal demodulation and phase difference detection. The structure diagram is shown in Figure 3.  
Figure 1: The array optical communication telescope system. Figure 2: Schematic structure diagram of MZI. Differently from direct detection, heterodyne detection is to input a channel of signal light and a channel of eigen light, and autodyne detection refers to the coherent detection of two optical signals from the same laser source [19]. Compared with Figures 2 and 3, the innovative method used a phase shifter to replace the first coupler of MZI in the structure. In terms of input limit, the innovative structure directly inputs two channels of signal light, but MZI inputs one channel of the optical signal and one channel of the zero-value optical signal. As shown in Figure 3, two optical signals were input to the one-bit delayer to complete the detection process of autodyne detection. Two forms of optical signals were obtained at the input port of the balanced detector. One was the interferometric phase length signal and the other was the interferometric phase extinction signal. \(E_{111n}\) and \(E_{121n}\) are the input ports of the receiver structure and \(E_{111n}=E_{0},E_{121n}=E_{0}\), and the signals obtained by the phase adjustment of the quadrature phase shifter are \(E_{11 Out}\) and \(E_{12 Out}\), which are described as Equation (1). \[ \begin{pmatrix}E_{11 Out}\\ E_{12 Out}\end{pmatrix} = \begin{pmatrix}1&0\\ 0&-j\end{pmatrix}\begin{pmatrix}E_{111n}\\ E_{121n}\end{pmatrix} = \begin{pmatrix}E_{0}\\ -jE_{0}\end{pmatrix}  \tag{1}\]  
The duration of the delayer is strictly controlled to one bit period. The input optical signals of the coupler are obtained as \(E_{211n}=E_{0},E_{221n}=-jE_{0}\), when adjacent symbols have the same phase. For the optical coupler, the input ports are \(E_{211n}\) and \(E_{221n}\), and the output ports are \(E_{con}\) and \(E_{des}\). The transmission matrix for the cross is described in Equation (2) [20; 21]. \[ \begin{pmatrix}E_{1 Out}\\ E_{2 Out}\end{pmatrix} =\alpha \begin{pmatrix}\sqrt{1-c}&pj\sqrt{c}\\ pj\sqrt{c}&\sqrt{1-c}\end{pmatrix}\begin{pmatrix}E_{11n}\\ E_{21n}\end{pmatrix}  \tag{2}\]  
where \(p\) is the case of conjugate with the value of \(+1\) or \(-1\); \(\alpha\) is the additional loss with the ideal value of \(0\) on a logarithmic scale; and \(c\) is the coupling coefficient, and the value is \(0.5\) with the splitting rate of \(50\):\(50\). Equation (3) describes the output signals of the cross couple in this case. \[ \begin{pmatrix}E_{con}\\ E_{des}\end{pmatrix} = \begin{pmatrix}\frac{\sqrt{2}}{2}&j\frac{\sqrt{2}}{2}\\ j\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{pmatrix}\begin{pmatrix}E_{211n}\\ E_{221n}\end{pmatrix} = \begin{pmatrix}\sqrt{2}E_{0}\\ 0\end{pmatrix}  \tag{3}\]  
Similarly, the input optical signals of the coupler are obtained as \(E_{211n}=E_{0},E_{221n}=jE_{0}\), when the phase difference between adjacent symbols is \(\pi\).