### 7.2 Transverse electromagnetic modes

In general, any solution of the paraxial wave equation, Equation (113), can be employed to
represent the transverse properties of a scalar electric field representing a beam-like electro-magnetic wave.
Especially useful in this respect are special families or sets of functions that are solutions of the paraxial
wave equation. When such a set of functions is complete and countable, it’s called a set of transverse
electromagnetic modes (TEM). For instance, the set of Hermite–Gauss modes are exact solutions of the
paraxial wave equation. These modes are represented by an infinite, countable and complete set of
functions. The term complete means they can be understood as a base system of the function space
defined by all solutions of the paraxial wave equation. In other words, we can describe any
solution of the paraxial wave equation by a linear superposition of Hermite–Gauss modes:
which in turn allows us to describe any laser beam using a sum of these modes:
The Hermite–Gauss modes as given in this document (see Section 7.5) are orthonormal so that
This means that, in the function space defined by the paraxial wave equation, the Hermite–Gauss functions
can be understood as a complete set of unit-length basis vectors. This fact can be utilised for the
computation of coupling factors. Furthermore, the power of a beam, as given by Equation (108), being
detected on a single-element photodetector (provided that the area of the detector is large with respect to
the beam) can be computed as
or for a beam with several frequency components (compare with Equation (76)) as