### 7.1 The paraxial wave equation

Mathematically, Gaussian modes are solutions to the paraxial wave equation – a specific wave equation
for electromagnetic fields. All electromagnetic waves are solutions to the general wave equation, which in
vacuum can be given as:
But laser light fields are special types of electromagnetic waves. For example, they are characterised by low
diffraction. Hence, a laser beam will have a characteristic length describing the ‘width’ (the dimension
of the field transverse to the main propagation axis), and a characteristic length defining
some local length along the propagation over which the beam characteristics do not vary much.
By definition, for what we call a beam is typically small and large in comparison, so
that can be considered small. In fact, the paraxial wave equation (and its solutions)
can be derived as the first-order terms of a series expansion of Equation (109) into orders of
[37].
A simpler approach to the paraxial-wave equation goes as follows: A particular beam shape shall be
described by a function so that we can write the electric field as

Substituting this into the standard wave equation yields a differential equation for :
Now we put the fact that should be slowly varying with in mathematical terms. The
variation of with should be small compared to its variation with or . Also the second
partial derivative in should be small. This can be expressed as
With this approximation, Equation (111) can be simplified to the paraxial wave equation,
Any field that solves this equation represents a paraxial beam shape when used in the form given in
Equation (110).