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In many situations, it is necessary to match a laser beam precisely to another beam or a mode in order to obtain some kind of efficient coupling. Examples are:
- A beam from a laser has to be coupled into an optical fiber.
- A laser beam must be matched to a passive optical resonator which should act, e.g., as a spatial and/or spectral filter (→ mode cleaner cavities).
- For injection locking, the mode of a master laser has to be matched to that of the slave laser.
The necessary matching of modes means not only creating a good spatial overlap of the intensity profiles, but also matching the phase profiles. If the complex amplitude profiles of two beams are well matched in a certain plane, they will remain well matched during further propagation. Mode matching can be achieved by using suitable relay optics (typically some combination of curved mirrors or lenses), provided that the beam quality of the initial beam is close to diffraction-limited.
Mathematically, the quality of mode matching can be quantified with an overlap integral:
where E1 and E2 are the complex electric fields in a plane, referring e.g. to a laser beam and the field of a resonator or waveguide mode, and the integration spans the whole beam cross-section. That quantity is preserved during propagation in free space.
Note that an overlap integral as given above is useful, e.g., for calculating mode amplitudes. For many situations, such as coupling of optical power into a mode, the square of such an overlap integral is relevant.
If the beam from a frequency-tunable single-frequency laser hits a symmetric Fabry–Pérot interferometer and the laser frequency is tuned over the whole free spectral range of the resonator, the transmitted light can be used to analyze the degree of mode matching. For perfect matching to a cavity mode (typically the fundamental Gaussian mode), complete transmission of the resonator can be observed when the resonance condition is met, whereas other resonances (corresponding to other resonator modes) can not be excited.
See also: modes, cavities, laser beams, diffraction-limited beams
and other articles in the categories general optics, optical resonators
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