The fundamental TEM₀₀ mode is only one of many transverse modes that satisfy the round-trip propagation criteria. The figure below shows examples of the primary lower-order Hermite-Gaussian (rectangular) solutions to the propagation equation.

Low-order Hermite-gaussian resonator modes

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 180° 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 Laguerre-Gaussian functions, denoted by TEM_rf. For the lowest order mode, TEM₀₀, the Hermite-Gaussian and Laguerre-Gaussian functions are identical, but for higher order modes, they differ significantly, as shown in the figure below.

Low-order axisymetric resonator modes

The mode, TEM₀₁*, also known as the "bagel" or "doughnut" mode, is considered to be a superposition of the Hermite-Gaussian TEM₁₀ and TEM₀₁ modes, locked in phase quadrature.

In real-world lasers, the Hermite-Gaussian modes predominate since strain, slight misalignment, or contamination on the optics tends to drive the system toward rectangular coordinates. Nonetheless, the Laguerre-Gaussian TEM₁₀ "target" or "bulls-eye" mode is clearly observed in well-aligned gas-ion and helium neon lasers with the appropriate limiting apertures.