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# Mode Matching

Definition: the precise spatial matching of the electric field distributions of laser beams and resonator modes or waveguide modes

German: Modenanpassung

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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:

The necessary matching of modes means not only creating a good spatial overlap of the intensity profiles, but also matching the optical 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. The following formula, involving the square of such an overlap integral, calculates the coupling efficiency concerning optical powers:

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.

Overlap integral can also be used for calculating complex mode amplitudes.

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.

## Questions and Comments from Users

2021-02-15

How does one account for polarization in this equation?

Not at all. The calculation is based on scalar fields, i.e., neglecting polarization. If you want to calculate the overlap often arbitrary polarized field with a polarized mode, you first have to decompose the field into its polarization components.

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