### 7.10 Tracing a Gaussian beam through an optical system

Whenever Gauss modes are used to analyse an optical system, the Gaussian beam parameters (or
equivalent waist sizes and locations) must be defined for each location at which field amplitudes are to be
computed (or at which coupling equations are to be defined). In our experience the quality of a computation
or simulation and the correctness of the results depend critically on the choice of these beam parameters.
One might argue that the choice of a basis should not alter the result. This is correct, but there is a
practical limitation: the number of modes having non-negligible power might become very large if the beam
parameters are not optimised, so that in practice a good set of beam parameters is usually
required.
In general, the Gaussian beam parameter of a mode is changed at every optical surface in a well-defined
way (see Section 7.11). Thus, a possible method of finding reasonable beam parameters for every
location in the interferometer is to first set only some specific beam parameters at selected
locations and then to derive the remaining beam parameters from these initial ones: usually it is
sensible to assume that the beam at the laser source can be properly described by the (hopefully
known) beam parameter of the laser’s output mode. In addition, in most stable cavities the light
fields should be described by using the respective cavity eigenmodes. Then, the remaining beam
parameters can be computed by tracing the beam through the optical system. ‘Trace’ in this
context means that a beam starting at a location with an already-known beam parameter is
propagated mathematically through the optical system. At every optical element along the
path the beam parameter is transformed according to the ABCD matrix of the element (see
below).