# Free-Space QKD with Modulating Retroreflectors Based on the B92 Protocol
## 2 MRR-QKD With B92  
First, let us review the scheme proposed in RM2018. Four pairs of MQWs were used to realize polarization modulation, where each pair corresponded to one of the four BB84 states (0, 45, 90, and 135\({}^{\circ}\)). While the transmission loss of the MQW modulator is set to the lowest (highest), the MQW is in the \"on\" (off) state. The polarization modulation is realized by controlling the states of MQWs. For example, when the MRR terminal sends a horizontal state, the pair of MQWs corresponding to the horizontal state will be \"on\" and the others will be \"off\". More details are shown in Appendix C.  
The extinction ratio of the MQW between the \"on\" and \"off\" states is only approximately 7.5 dB, which is too small. Unwanted light will leak from the MQW, which are \"off\", resulting in extra QBER. To decrease the QBER, RM2018 uses a stack of two MQWs packaged together to achieve an extinction ratio \(r\) of 15 dB in every path. However, this will greatly increase the complexity and cost of the system, and the extinction ratio will remain unsatisfactory.  
Our MRR-QKD system, based on the superposition principle of two mutually perpendicular waves, uses only three MQW modulators in total and eliminates unexpected noise leakage. The configuration of the MRR-QKD system is shown in Figure 1.  
The system works as follows. First, the interrogator generates an interrogation beam, which is a classical light pulse with right-hand circular polarization the same with RM2018, and sends it to the MRR terminal through a free-space channel. After arriving at the MRR terminal, the interrogation beam passes through MQW3, which is used to modulate decoy states. Then, it is split into two beams with the same intensity by a PBS, the horizontal  
Figure 1: Schematic of the modulating retroreflector-quantum key distribution (MRR-QKD) system. LD: Laser Diode; WP: Wave Plate; BS: Beam Splitter; PBS: Polarizing Beam Splitter; SPD: Single Photon Detector; RR: Retro-reflector.  
state is transmitted, and the vertical state is reflected. These two beams will pass forward through an MQW and be reflected by a retroreflector, and then pass backward through the MQW again. Therefore, the intensity of these two beams are modulated by MQW1 (and MQW2) twice. Finally, they are combined by the PBS, and the output is a linearly polarized beam with polarization direction \(\theta\) (see Appendix A).  
\[|\psi_{out}\rangle=|\beta_{\theta}\rangle=|\cos\theta\beta_{H}\rangle|\sin \theta\beta_{V}\rangle \tag{1}\]  
where \(|\beta_{\theta}\rangle\) is the linearly polarized coherent state. \(\beta\) is the amplitude of the beam, and \(\theta=\arctan(\sqrt{M_{2}/M_{1}})\) is its polarization direction. \(M_{1}\) and \(M_{2}\) are the modulation efficiencies of MQW1 and MQW2, respectively.  
The B92 protocol is adopted in our system. Each retroreflected beam is in one of two linear polarizations with polarization directions of \(22.5^{\circ}\) and \(67.5^{\circ}\) by adjusting the ratio of the modulation efficiency of MQW1 and MQW2. This is slightly different from the general B92 protocol.  
The beam is attenuated and modulated by MQW3 to generate signal or decoy states. After the beam is transmitted back and received by the interrogator, it is still linearly polarized because of the stability of the polarization in free space [26]. However, its polarization direction is changed because of the relative orientation of the interrogator and the MRR terminal. A half-wave plate is used for polarization compensation. The incident beam will be converted to \(0^{\circ}\) (\(45^{\circ}\)) linear polarization. The following procedures, such as measurement and post-processing, are the same as those of the general B92 protocol [25; 27].