# Laser Transverse Modes

The operating modes of a physical system are formed by physical laws and boundary conditions. Consider a guitar string that is plucked. The physical law is the restoring force that tends to bring the string to its initial, straight position. The boundary conditions are that on either end of the string, the amplitude of the vibration must be zero. The superimposed waveforms that form the mode move with a fixed velocity that is dependant on the properties of the string, and have wavelengths equal to 2L/n, where L is the length of the string, and n is a positive integer. These are the modes of the system.

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 (Erf) for this equation are a series of axially symmetric modes, which, for stable resonators, are closely approximated by Laguerre-Gaussian functions, denoted by TEMrf. 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.