Chapter 1 Rays and paraxial approximations - OED

4 CHAPTER 1. RAYS AND PARAXIAL APPROXIMATIONS
n1 > n2, there is a critical angle of incidence beyond which transmission from medium 1
to medium 2 is impossible. In that case, only a reflected ray is present.
1.2 Rays in systems with radial symmetry
To understand the optical properties of several optical elements, such as lenses and optical
fibers, we will examine the effects of changes in the refractive index with respect to the ray
direction.
We consider a point on the ray, which is given by r = xux + yuy + zuz in an arbitrary
coordinate system. Further, let s denote the unit vector in the ray direction and let s be
the distance along the ray. In Section 1.3 of the syllabus on the theoretical background it
is shown that the following differential equation holds.
d
n
ds
dr
= ∇n. (1.5)
ds
The rays in many optical instruments and laser resonators or guiding systems have only
small slopes with respect to an axis of symmetry, as in the example of the quadratic index
fiber of the preceding section. In such cases, the rays are called paraxial rays and the
optics deriving from this condition is called first-order optics. We will assume that the
symmetry axis of the system coincides with the z-axis. Further, the distance of the ray
with respect to the symmetry axis can then be expressed in terms of the cylinder coordinate
r = x 2 + y 2 . In some cases, r will become negative, which should be understood as a
change in the cylinder coordinate φ by π radians.For this class of problems, it is convenient
to describe the effect of various optical components by ray matrices that relate output
radius (rout) and slope (r ′ out = drout
dz ) of a ray to the input radius (rin) and slope (r ′ in = drin
dz ).
For optical elements with flat or spherical interfaces, and for elements with constant or
(approximately) quadratic refractive index variation, the pertaining relation is (almost)
linear. For these cases, the linear relation is denoted as
rout
=
A
C
B
D
r ′ out
rin
r ′ in
. (1.6)
The advantage of the ABCD matrix formulation is mainly in its ease of handling combi-
nations of elements and we illustrate this with a few examples.