chapter 1, An Introduction to Tools and Concepts

Excerpt

1.1 Matrix Method

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 reason why the paraxial approximation is often used in the first phase of a design or of an optical analysis becomes obvious if we have a look at the law of refraction in vectorial form as follows:

where is the vector of the ray incident on the interface with the normal . This interface separates two homogeneous media with indices of refraction n₁ and n₂. The refracted ray is described by the vector . For optical-layout purposes, we need an explicit expression of this ray in terms of the other quantities because we have to trace the ray through the optical system. Using vector algebra, Eq. (1.1) can be rewritten in the following way: 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.

©2007 Society of Photo-Optical Instrumentation Engineers