Mode matching

An optical cavity, a single-mode fiber and a waveguide each accept light efficiently in only one spatial mode, and for most of them that mode is close to a Gaussian beam with a definite waist radius at a definite position. Mode matching is the job of transforming the incoming beam so that its waist has the same size and sits in the same place, with the same axis. The fraction of power that enters the target mode is the squared magnitude of the normalized overlap integral of the two fields; power left over either reflects, couples into higher-order modes or is lost, and the mode mismatch loss entry covers that loss for fiber and waveguide junctions.

Efficiency

For two aligned Gaussian beams with waist radii $w_1$ and $w_2$ whose waists are separated by a distance $\Delta z$ along the axis, the power coupling efficiency is

A 10% error in waist size alone gives about 99% and a 20% error about 97%, so the size tolerance is loose. For equal waists the efficiency reduces to $1/[1 + (\Delta z/2z_R)^2]$ in terms of the Rayleigh range $z_R$: an offset of one Rayleigh range leaves 80%, and one of a fifth of a Rayleigh range 99.0%. Lateral offset and tilt reduce it further, and for a small waist are usually the tighter tolerances.

Designing the optics

The target mode is known from the cavity geometry, found from its round-trip ABCD matrix, or from the fiber's mode field diameter. The input beam is measured, then one or two lenses are chosen and placed, again with ABCD propagation, so that the output waist matches; two lenses are needed in general, because size and position are two conditions, and adjusting their spacing gives a continuously tunable match. Mode-matching calculators and the Gaussian beam formulas do the arithmetic; the practical difficulty is that the result is only as good as the measurement of the input beam.

Measuring it on a cavity

For a Fabry-Perot or other resonator, the laser frequency or the cavity length is scanned and the transmitted power recorded. A well matched and aligned beam shows only the fundamental resonances; misalignment adds odd-order transverse modes, and mismatch of waist size or position adds even-order ones. The mode-matching efficiency is the fundamental mode's peak divided by the sum of all the peaks in one free spectral range. Values above 90% are routine with care, and higher values are reached for ultrastable cavities used with Pound-Drever-Hall locking, where unmatched light adds offsets to the error signal.

References: H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966); A. E. Siegman, Lasers (University Science Books, 1986); D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).