Multi-beam klystron (MBK) is a promising high power microwave device with the traits of high power, high efficiency, high frequency, etc. For the high power relativistic MBK, the multi-beam rotation around an axis may reduce the transmission efficiency obviously due to the effect of space electromagnetic field. In previous researches, the influence of mirror-image electromagnetic field is ignored, which can play a leading role in some cases. In this study, we present a method by taking into account the mirror-image effect to analyze the angular drift of multi-beam in the hollow cylindrical waveguide. The hollow cylindrical waveguide is a part of relativistic MBK such as input cavity and transition section, which is just behind the diode. In this method, the equation of the multi-beam angular drift is deduced and analyzed quantitatively. Based on the equation, the expression of the angular velocity about the multi-beam in the waveguide is derived, meanwhile the minimum equilibrium magnetic field, called Brillouin magnetic field, is obtained. To verify the effectiveness of the method, numerical simulations are carried out by the three-dimensional (3D) code and the results show good agreement with the theoretical results. The theoretical analysis and simulation results show that the mirror-image electromagnetic field can dominate the multi-beam angular motion in some conditions, especially when the number of the multi-beams and the distance between the conducting wall and the multi-beam are both small. In this case, the mirror-image electromagnetic field can be much higher than the self-induced electromagnetic field. Nevertheless, as the the number of the multi-beams or the distance between the conducting wall and the multi-beam increases, the mirror-image electromagnetic field decreases and approaches to zero rapidly and the self-induced electromagnetic field controls the angular movement. Interestingly, in general cases, it is found that the change rate of the angular speed is not related to the number of multi-beams, nor the radius of waveguide, nor the distance between the multi-beam, nor waveguide, etc, except for the accelerating voltage. In addition, we experimentally investigate the angular drift of the multi-beam at a voltage of about 670 kV, current of about 7 kA and length of waveguide about 100 mm. The experimental results show that the multi-beam distorts obviously, which changes the beam spot shape from circle to ellipse. To solve this problem, we simultaneously investigate the multi-beam emission and transmission in simulation experiment. The analogue results not only reveal that the distortion is mainly caused by the emission of the multi-cathode rods, but also provide a new phenomenon that the angular drift distance in the accelerating gap of the diode is twice as large as that in the cylindrical hollow waveguide due to the low beam speed along the axis and high electrostatic field in the accelerating region. It is also found that the distortion is more evident as the rod radius decreases. Furthermore, we propose an optimization design to improve the relativistic multi-beam system by inclining the multi-cathode rods, which is proved to be effective by simulation. This study could provide theoretical basis for studying the relativistic MBK.
