Ray-transfer matrices is one of the possibile methods to describe optical systems in the paraxial approximation. It is widely used for first-order layout and for the purpose of analyzing optical systems (Gerrard and Burch, 1975).
The form obtained like this is complicated and it is difficult to trace the ray without making use of a computer. Therefore, a linearized form of this law would be helpful for thinking about the optical system, and this is the motivation for starting with a paraxial layout.
It would be a precious tool for analyzing optical instruments if the approximated description would also allow for cascading subsystems to describe a compound system. The method of ray-transfer matrices provides this advantage and cascading of subsystems is performed by matrix multiplication.
Another aspect, which might be sometimes underestimated, is that paraxial descriptions, and especially the matrix method, provide a convenient shorthand notation to communicate and discuss ideas to other optical designers.
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