Transverse Mode - Laser Modes

Laser Modes

In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with a Laguerre polynomial. The modes are denoted TEM_pl where p and l are integers labeling the radial and angular mode orders, respectively. The intensity at a point r,φ (in polar coordinates) from the centre of the mode is given by:

where ρ = 2r2/w2, and L_pl is the associated Laguerre polynomial of order p and index l. w is the spot size of the mode corresponding to the Gaussian beam radius.

With p=l=0, the TEM₀₀ mode is the lowest order, or fundamental transverse mode of the laser resonator and has the same form as a Gaussian beam. The pattern has a single lobe, and has a constant phase across the mode. Modes with increasing p show concentric rings of intensity, and modes with increasing l show angularly distributed lobes. In general there are 2l(p+1) spots in the mode pattern (except for l=0). The TEM_0i* mode, the so-called doughnut mode, is a special case consisting of a superposition of two TEM_0i modes (i=1,2,3), rotated 360°/4i with respect to one another.

The overall size of the mode is determined by the Gaussian beam radius w, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation. Higher order modes are relatively larger compared to the TEM₀₀ mode, and thus the fundamental Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity.

In many lasers, the symmetry of the optical resonator is restricted by polarizing elements such as Brewster's angle windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated TEM_mn with m and n being the horizontal and vertical orders of the pattern. The electric field pattern at a point (x,y,z) for a beam propagating along the z-axis is given by

where, and are the waist, spot size, radius of curvature, and Gouy phase shift as given for a Gaussian beam; is a normalization constant; and is the kth physicist's Hermite polynomial. The corresponding intensity pattern is

The TEM₀₀ mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing m and n show lobes appearing in the horizontal and vertical directions, with in general (m + 1)(n + 1) lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode.

The phase of each lobe of a TEM_mn is offset by π radians with respect to its horizontal or vertical neighbours. This is equivalent to the polarization of each lobe being flipped in direction.

The overall intensity profile of a laser's output may be made up from the superposition of any of the allowed transverse modes of the laser's cavity, though often it is desirable to operate only on the fundamental mode.