Note that the subscripts n and m in the Eigenmode
TEM_{ nm} are correlated
to the number of nodes in the x
and y directions. In each case, adjacent lobes of the
mode are 180กใ out of phase.
The
propagation equation can also be written in cylindrical form
in terms of radius (r) and angle (f). The eigenmodes
(E_{rf}) for this equation are a series of axially
symmetric modes, which, for stable resonators, are closely
approximated by LaguerreGaussian functions, denoted by TEM_{rf}. For the lowest order mode, TEM_{00},
the HermiteGaussian and LaguerreGaussian functions are identical,
but for higher order modes, they differ significantly, as
shown in the figure below.

The
mode, TEM_{01}*, also known as the "bagel" or "doughnut"
mode, is considered to be a superposition of the HermiteGaussian
TEM_{10} and TEM_{01} modes, locked in phase
quadrature.
In
realworld lasers, the HermiteGaussian modes predominate
since strain, slight misalignment, or contamination on the
optics tends to drive the system toward rectangular coordinates.
Nonetheless, the LaguerreGaussian TEM_{10} "target"
or "bullseye" mode is clearly observed in wellaligned gasion
and helium neon lasers with the appropriate limiting apertures.
