Analysis of Faraday Mirror in Auto-Compensating Quantum Key Distribution

The 'plug & play' quantum key distribution system is the most stable and the earliest commercial system in the quantum communication field. Jones matrix and Jones calculus are widely used in the analysis of this system and the improved version, which is called the auto-compensating quantum key distribution system. Unfortunately, existing analysis has two drawbacks: only the auto-compensating process is analyzed and existing systems do not fully consider laser phase affected by a Faraday mirror (FM). In this work, we present a detailed analysis of the output of light pulse transmitting in a plug & play quantum key distribution system that contains only an FM, by Jones calculus. A similar analysis is made to a home-made auto-compensating system which contains two FMs to compensate for environmental effects. More importantly, we show that theoretical and experimental results are different in the plug & play interferometric setup due to the fact that a conventional Jones matrix of FM neglected an additional phase π on alternative polarization direction. To resolve the above problem, we give a new Jones matrix of an FM according to the coordinate rotation. This new Jones matrix not only resolves the above contradiction in the plug & play interferometric setup, but also is suitable for the previous analyses about auto-compensating quantum key distribution.