  In this paper, we propose a systematic approach to design a four-component
zoom system with fixed spacing between focal points based on matrix optics.
Since the more complex model the higher degree of freedom it has, the task of
determining the zoom trajectory is meaningful and challenging. The elements of
the system matrix imply the working state of the optical system, and axial
displacement equation for the desired zoom system are derived by restricting
specific matrix elements. Properly selected trajectory of the particular
component described by means of a parametric function can make the model become
solvable explicitly. Then the paraxial design problem is transformed into the
optimization of these parameters with regard to the merit functions
encompassing the primary aberration terms, compactness, smoothness of the
trajectories. We adopt Particle Swarm Optimization (PSO) algorithm to globally
optimize the parameters to retrieve the optimum zoom trajectory in specific
design criteria. The proposed method is demonstrated through two numerical
examples under different configurations. The simulation results demonstrate
that our proposed method can be a practical and powerful tool for paraxial
design of complex multi-group zoom optical systems.
