According to the random characteristics of surface distribution of continuous phase plate (CPP), the characteristics of such random surface are analyzed from a statistical point of view. Based on the continuous phase plate with Gaussian distribution, the ergodic theorem is introduced to describe the statistical characteristics of CPP, and the qualification that fulfills the ergodic theorem is proved. The CPP which fulfills the ergodic theorem is proved to have better beam smoothing properties in both spatial and frequency domain by numerical simulation. The results show that after smoothing, the focal spot of random distortion beam is well controlled. The average PSD value of distortion wavefront is decreased by 50% or more. (C) 2011 Elsevier GmbH. All rights reserved.