1. Introduction
The AESOP[1] computer algebra ray tracing package has been updated to include geometric tracing of a paraxial lens. This document describes ray direction and optical path changes upon traversal of the ideal optical element known as a paraxial lens. This is simply an infinitely thin, aberration-free lens in the paraxial approximation.
2. Ray Position and Direction
Figure 1 illustrates
the geometry. The incident and refracted
beam directions are and
. The horizontal position vectors in the lens
plane and in the focal plane are
and
,
where
is a rotation composition defined by Figure 2. The horizontal position vectors are
illustrated in the lens plane in Figure 3. We also define
.
In the paraxial approximation, we assume
that ψ is small such that all input rays converge to
a point on the focal plane defined by the intersection of the chief ray with
the focal plane. An equivalent statement
is that is independent of position
in the lens plane. This approximation will require an adjustment
to the optical path, which will be derived in the next section.
From Figure 1 and Figure 2, we may write
where
For a skew ray in the paraxial approximation, the refracted ray direction vector can be decomposed as
(3)
(This is
seen most easily from Figure 1.) Let us now scale the distance vectors and
by the focal length f,
Then we have the result
(5)
where and
are given by eqs. (2).
3. Optical Path
For a
perfect, paraxial lens, the optical path will be independent of the position in the lens plane. Geometrically, the optical path of an axial
ray from the lens to its focus is, assuming index
in the medium,
The geometric path for a skew ray is
(7)
Thus, the optical path correction is
(8)
In tracing a ray through a paraxial
lens, must be added to the calculated geometric path
in order to produce a converging spherical wave front.
Normalizing again according to eq. (4), we have
A Taylor expansion yields
Also, we have
Using eqs. (11)
and (12),
eqs. (9)
and (10)
may be expressed in terms of only the incident beam direction and the position
in the lens plane, yielding the optical path correction . This is precisely what is needed in
geometrical ray tracing, where ray position, direction, and optical path are
calculated sequentially for each optical element in a system.
A special case exists for which it is not necessary to calculate for a skew ray either the intersection position or the optical path correction. This occurs if the surface following the paraxial lens is planar and parallel to the paraxial lens plane. Then we know that a) the intersection point will, for all skew rays, be identical to that of the chief ray, given by eqs. (1) and (2); and b) the optical path will be given simply by that of the chief ray, eq. (6). The circumstances of this special case, however, are rather uncommon for most systems, where the emphasis is on the effects of optical element misalignments. Thus, in general, the full skew ray calculations must be employed.