Hermite-Gaussian Modes | previous | next | feedback |
Definition: propagation modes or resonator modes which are described with Hermite-Gaussian functions
When light propagates in free space or in a homogeneous optical medium, its intensity profile will generally change during propagation. For certain electric field amplitude distributions, however, which are called modes, this is not the case: the shape of the amplitude profile remains constant, even though there may be a re-scaling of the profile, an overall change of optical phase, and possibly also a change of the total optical power.
For each combination of an optical frequency, a beam axis, a focus position, and some beam radius of a Gaussian beam in the focus, there is a whole family of Hermite-Gaussian modes (TEMnm modes, Gauss-Hermite modes). These are approximate solutions of the wave equation, valid for weak focusing (→ paraxial approximation). Their electric field distribution is essentially given by the product of a Gaussian function and a Hermite polynomial, apart from the phase term:
where Hn(x) is the Hermite polynomial with the non-negative integer index n. The indices n and m determine the shape of the profile in x and y direction, respectively. The quantities w and R evolve in z direction as described in the article on Gaussian beams.
The intensity distribution of such a mode (Figure 1) has n nodes in the horizontal direction and m nodes in the vertical direction. For n = m = 0, a Gaussian beam is obtained. This mode is called the fundamental mode or axial mode, and it has the highest beam quality with an M2 factor of 1. Other Hermite-Gaussian modes with indices n and m have an M2 factor of (2n + 1) in x direction, and (2m + 1) in y direction.
Figure 1: Intensity profiles of the lowest-order Hermite-Gaussian modes, starting with TEM00 (lower left-hand side) and going up to TEM33 (upper right-hand side).
A further generalization of the equation above would allow for different mode sizes and focus positions (→ astigmatism) for the x and y direction. The direction of the electric field, not specified in the equation above, determines the polarization.
Another frequently used mode family is that of Laguerre-Gaussian modes.
See also: modes, higher-order modes, Gaussian beams
Categories: general optics, resonators
