### 7.5 Higher-order Hermite–Gauss modes

The complete set of Hermite–Gauss modes is given by an infinite discrete set of modes
with the indices n and m as mode numbers. The sum n+m is called the order of the mode. The term
higher-order modes usually refers to modes with an order . The general expression for
Hermite–Gauss modes can be given as [35]
with
and the Hermite polynomials of order n. The first Hermite polynomials, without normalisation, can
be written
Further orders can be computed recursively since
For both transverse directions we can also rewrite the above to
The latter form has the advantage of clearly showing the extra phase shift along the z-axis of
, called the Gouy phase; see Section 7.8.