### 2.1 Mirrors and spaces: reflection, transmission and propagation

The core optical systems of current interferometric gravitational interferometers are composed of two
building blocks: a) resonant optical cavities, such as Fabry–Pérot resonators, and b) beam splitters, as in
a Michelson interferometer. In other words, the laser beam is either propagated through a vacuum system or
interacts with a partially-reflecting optical surface.
The term optical surface generally refers to a boundary between two media with possibly different
indices of refraction , for example, the boundary between air and glass or between two types
of glass. A real fused silica mirror in an interferometer features two surfaces, which interact
with a reflected or transmitted laser beam. However, in some cases, one of these surfaces has
been treated with an anti-reflection (AR) coating to minimise the effect on the transmitted
beam.

The terms mirror and beam splitter are sometimes used to describe a (theoretical) optical surface in a
model. We define real amplitude coefficients for reflection and transmission and , with ,
so that the field amplitudes can be written as

The phase shift upon transmission (here given by the factor ) refers to a phase convention
explained in Section 2.4.
The free propagation of a distance through a medium with index of refraction can be described
with the following set of equations:

In the following we use for simplicity.
Note that we use above relations to demonstrate various mathematical methods for the analysis of
optical systems. However, refined versions of the coupling equations for optical components, including those
for spaces and mirrors, are also required, see, for example, Section 2.6.