System Matrix

For systems of multiple thick lenses, it is sometimes useful to represent the system by a system matrix. The matrix is built up by multiplying the refraction matrices and translation matrices. The positions of the principal planes, the front and back surface powers, and the equivalent focal length of Gullstrand's equation can be calculated from the system matrix.

This is the form of the system matrix used by Meyer-Arendt. An alternate form is used by Hecht.

Refraction and Translation Matrices

Refraction Matrix

The use of vergence to describe the refraction at an interface can be used as a starting point for developing a matrix to perform the refraction.

Translation Matrix

The geometry of translation across the thickness of a lens can be put in matrix form for rays satisfying the paraxial approximation.