Matrices in Paraxial Optics

A ray can be defined by its height and its direction


(the angle it makes with the optical
axis)

Translational Matrix

Simple tanslation of a ray in a homogeneous medium is take into consideration

Translation from point 0 to 1 with paraxial approximation :


αi = α0 and y1 = y0 + L tan α0 = y0 L α0

Rewrite the equation :

image

image

Refraction Matrix

A refraction of a ray at a spherical interface is take into consideration (paraxial approximation):

Ray coordinates before refraction (y,a)


Ray coordinates after refraction (y',a')

image

Paraxial form of Snell's Law (n theta = n' theta'): image

image

Reflection Matrix

Ray coordinates before refraction (y,a)


Ray coordinates after refraction (y',a')

image

Connect (y',a') to (y,a) by a ray transfer matrix for reflection by a concave mirror

Sign convention for angles :

(+) pointing upward

(-) pointing downward

Eliminate theta and theta' by using theta=theta' to get: image

image