Modeling for optical ray tracing and error analysis

One of the most popular mathematical tools in fields of robotics, mechanisms, and computer graphics is the 4×4 homogeneous transformation matrix. We extend the use of that matrix to the optical domain of:

(1) skew ray tracing to determine the paths of skew rays; and

(2) error analysis to investigate the various deviations of imagines due to imperfect placement of optical elements.

In order to trace a skew ray, the reflection and refraction laws of optics are formulated in the language of homogeneous transformation matrices. Then an error matrix to describe the position errors and orientation errors of optical elements is introduced in order to analyze their effects on rays' path. This ray tracing procedure can result in very powerful and fast optical design programs. The error analysis can provide the sensitivity of each error component of elements to a system's accuracy and is crucial to upgrade the precision of optical systems in design stage.