### 1.4 Plane-wave analysis

The main optical systems of interferometric gravitational-wave detectors are designed such that all
system parameters are well known and stable over time. The stability is achieved through a mixture of
passive isolation systems and active feedback control. In particular, the light sources are some of the most
stable, low-noise continuous-wave laser systems so that electromagnetic fields can be assumed to be
essentially monochromatic. Additional frequency components can be modelled as small modulations (in
amplitude or phase). The laser beams are well collimated, propagate along a well-defined optical axis
and remain always very much smaller than the optical elements they interact with. Therefore,
these beams can be described as paraxial and the well-known paraxial approximations can be
applied.
It is useful to first derive a mathematical model based on monochromatic, scalar, plane waves. As it
turns out, a more detailed model including the polarisation and the shape of the laser beam as well as
multiple frequency components, can be derived as an extension to the plane-wave model. A plane
electromagnetic wave is typically described by its electric field component:

with as the (constant) field amplitude in V/m, the unit vector in the direction of
polarisation, such as, for example, for -polarised light, the angular oscillation frequency of
the wave, and the wave vector pointing the in the direction of propagation. The
absolute phase only becomes meaningful when the field is superposed with other light
fields.
In this document we will consider waves propagating along the optical axis given by the z-axis, so that
. For the moment we will ignore the polarisation and use scalar waves, which can be written as

Further, in this document we use complex notation, i.e.,
This has the advantage that the scalar amplitude and the phase can be given by one, now complex,
amplitude . We will use this notation with complex numbers throughout. For clarity we
will simply use the unprimed letters for the auxiliary field. In particular, we will use the letter and also
and to denote complex electric-field amplitudes. But remember that, for example, in
neither nor are physical quantities. Only the real part of exists and
deserves the name field amplitude.