### 8.1 Coupling of Hermite–Gauss modes

Let us consider two different cavities with different sets of eigenmodes. The first set is characterised by
the beam parameter and the second by the parameter . A beam with all power in the fundamental
mode leaves the first cavity and is injected into the second. Here, two ‘misconfigurations’ are
possible:
- if the optical axes of the beam and the second cavity do not overlap perfectly, the setup is
called misaligned,
- if the beam size or shape at the second cavity does not match the beam shape and size of the
(resonant) fundamental eigenmode (), the beam is then not mode-matched
to the second cavity, i.e., there is a mode mismatch.

The above misconfigurations can be used in the context of simple beam segments. We consider the case
in which the beam parameter for the input light is specified. Ideally, the ABCD matrices then allow one to
trace a beam through the optical system by computing the proper beam parameter for each beam segment.
In this case, the basis system of Hermite–Gauss modes is transformed in the same way as the beam, so that
the modes are not coupled.

For example, an input beam described by the beam parameter is passed through several optical
components, and at each component the beam parameter is transformed according to the respective ABCD
matrix. Thus, the electric field in each beam segment is described by Hermite–Gauss modes based on
different beam parameters, but the relative power between the Hermite–Gauss modes with different mode
numbers remains constant, i.e., a beam in a mode is described as a pure mode throughout the
entire system.

In practice, it is usually impossible to compute proper beam parameter for each beam segment as
suggested above, especially when the beam passes a certain segment more than once. A simple case
that illustrates this point is reflection at a spherical mirror. Let the input beam be described
by . From Figure 49 we know that the proper beam parameter of the reflected beam is

with being the radius of curvature of the mirror. In general, we get and thus two different
‘proper’ beam parameters for the same beam segment. Only one special radius of curvature would result in
matched beam parameters ().