### 7.9 Laguerre–Gauss modes

Laguerre–Gauss modes are another complete set of functions, which solve the paraxial wave equation.
They are defined in cylindrical coordinates and can have advantages over Hermite–Gauss modes in the
presence of cylindrical symmetry. More recently, Laguerre–Gauss modes are being investigated in a
different context: using a pure higher-order Laguerre–Gauss mode instead of the fundamental
Gaussian beam can significantly reduce the impact of mirror thermal noise on the sensitivity
of gravitational wave detectors [54, 12]. Laguerre–Gauss modes are commonly given as [50]
with , and as the cylindrical coordinates around the optical
axis. The letter is the radial mode index, the azimuthal mode
index
and are the associated Laguerre polynomials:
All other parameters are defined as above for the Hermite–Gauss modes.
The dependence of the Laguerre modes on as given in Equation (146) results in a spiraling phase
front, while the intensity pattern will always show unbroken concentric rings; see Figure 45.
These modes are also called helical Laguerre–Gauss modes because of the their special phase
structure.

The reader might be more familiar with a slightly different type of Laguerre modes (compare Figure 46
and Figure 47) that features dark radial lines as well as dark concentric rings. Mathematically, these
can be described simply by replacing the phase factor in Equation (146) by a sine
or cosine function. For example, an alternative set of Laguerre–Gauss modes is given by [55]

This type of mode has a spherical phase front, just as the Hermite–Gauss modes. We will refer to this set as
sinusoidal Laguerre–Gauss modes throughout this document.
For the purposes of simulation it can be sometimes useful to decompose Laguerre–Gauss modes
into Hermite–Gauss modes. The mathematical conversion for helical modes is given as [7, 1]

with real coefficients
if . This relates to the common definition of Laguerre modes as as follows:
and .
The coefficients can be computed numerically by using Jacobi polynomials. Jacobi polynomials
can be written in various forms:
or
which leads to