Mode filling factor of a laser with end-pumped configuration

The mode filling factor is an important parameter especially in designing an end-pumped laser system. Although the end-pumped structure has been used for several decades, an overlap coefficient is often confused by many laser technicians. So far, there have usually been several conventional expressions using different formulae. A researcher often puzzles by some problems such as “which equation is a suitable one” and “how accurate the equation is.” Thus, it is necessary to construct a precise theory to systematically analyze the mode filling factor and investigate the difference among several traditional repressions. In this paper, an analytical model is first developed to study the mode filling factor for a pump beam and a laser beam with the modes in any order numbers. Then, several cases are discussed in which a pump beam and a laser beam are with a fundamental mode and a high-order mode as well as a hybrid mode, respectively. It has been demonstrated that the nonignorable deviation in the mode filling factor will occur if the conventional expression for both the pump and laser beams with fundamental modes is used to describe the case in which the pump and laser beams are with high-order transverse-modes. Similar analyses have been rare so far to the best of our knowledge. The theoretical conclusion in this paper is thought to be valuable in the design of an end-pumped laser system.