  The initial optical design for the multi-component zoom optical system with
fixed distance between focal points is challenging but rewarding. Here, we
propose a systematic approach to tackle this problem via matrix optics for the
four-component zoom system. The main aim of this paper is to obtain the
optimization starting point of four-component zoom system and determine the
zoom trajectory. In the design process, the properties of the system matrix of
zoom system are analyzed by paraxial ray tracing. Zoom conditions are
implemented by restricting the specific matrix elements, and the zoom
trajectory is determined by solving the zoom equations iteratively. The
efficiency of the proposed method is demonstrated through three numerical
examples under two different structures, including a design example with an 8X
zoom capability. The results show that the zoom capability has a significant
improvement. We believe that the proposed method is a practical and powerful
tool for the paraxial design of complex zoom optical systems.
