Transverse Modes

Transverse Modes

The fundamental TEM00 mode is only one of many transverse modes that satisfies the condition that it be replicated each round-trip in the cavity. Figure 36.9 shows examples of the primary lower-order Hermite-Gaussian (rectangular) modes.

Note that the subscripts m and n in the mode designation TEMmn are correlated to the number of nodes in the x and y direc- tions. 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 iden- tical, but for higher-order modes, they differ significantly, as shown in figure 36.10.

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 and space quadrature. (See W.W. Rigrod, “Isolation of Axi-Symmetric Optical-Resonator Modes,” Applied Physics Letters, Vol. 2 (1 Feb. ‘63), pages 51–53.)

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. Nonethe- less, 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.