Hermite–Gaussian Modes | <<< | >>> | 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 in optical phase, and possibly also a change in 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 distributions are 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 the x and y direction, respectively. The quantities w and R evolve in the 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 the x direction, and (2m + 1) in the 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 directions. 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
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