Chapter 1 Rays and paraxial approximations - OED

6 CHAPTER 1. RAYS AND PARAXIAL APPROXIMATIONS
where a is a characteristic length of the changes in the medium. Then we have
d2r −2n0r∆
=
dz2 a2n0 [1 − ∆(r/a) 2
2∆
≈ −
] a2
r, (1.13)
where we have assumed that ∆ is small. The solution of the above equation is given by
r(z) = r0 cos(κz) + r′ 0
κ
sin(κz), (1.14)
where κ = √ 2∆/a, r0 is the radius of the ray at z = 0, and r ′ 0 is the slope of the ray at
z = 0. Further, by taking the derivative with respect to z in Eq. (1.14) we find
r ′ (z) = −r0κ sin(κz) + r ′ 0 cos(κz). (1.15)
Hence for a graded-index fiber of length ℓ, the ABCD matrix is given by
1
A B cos(κℓ) κ =
C D
sin(κℓ)
. (1.16)
−κ sin(κℓ) cos(κℓ)
1.2.3 Mirrors
In some optical systems, flat mirrors are applied simply to reduce length of the optical
system. Basically, a flat mirror does not alter the distances and angles between two inde-
pendent rays and therefore all rays are transformed equally. Therefore, a flat mirror only
changes the direction of the optical symmetry axis.
A spherically curved mirror is similar to a lens, except that the reflected ray returns rather
than passes through to the other side. For a fixed coordinate system, this would result in
opposite signs for the slopes of the rays in case of a curved mirror compared to those for
the slopes in the case of a lens.
Because of the use of these mirrors in cascaded systems, which will be described below, it
is more convenient to take the direction of the ray as the positive z coordinate, i.e., for the
flat and the curved mirror, positive z is toward the mirror for the incident rays and away
from the mirror for reflected rays.
For a flat mirror, this means that the ABCD matrix is equal to the identity matrix and
therefore this optical element can be left out in the ABCD matrix formalism, which means
that the free-space sections before and after the flat mirror become a single section of free
space, with a length equal to the sum of the lengths of the two separate sections.