measurement time of 243 days reduced the σ to 50 nHz in the Grossring. Beam characteristics The beam in ring lasers is typically excited by High-Frequency excitation of a laser gas. Although it has been shown that ring lasers can be excited in all kinds of modes, including microwave-related modes, a typical ring laser mode has a Gaussian, closed shape, given proper adjustment of mirror position The analysis of beam properties (curvature radius, width, position of waists, polarization) is done with matrix methods, where the elements of the closed beam circuit, mirrors and distances in between, are given 2 × 2 matrices. The results are distinct for circuits with n mirrors. Typically, there are n waists. For stability, there has to be at least one curved mirror in the circuit. Out-of-plane rings have circular polarization. The choice of mirror radii and mirror separation is not arbitrary. Curvature radius and width The beam has a spot size w: , where is the peak field of the beam, E is the field distribution, and r is the distance off beam center. The mirror sizes have to be chosen large enough to ensure that only very small portions of the gaussian tails are to be cut off, such that the calculated Q (below) is maintained. The phase is spherical with radius of curvature R. It is customary to combine radius of curvature and spot size into a complex curvature . The ring design uses a matrix M1 = for a straight section and M2 = for a mirror of focus length f. The relation between mirror radius RM and focus length f is for oblique incidence at angle θ, in plane: , for oblique incidence at angle θ, perpendicular to the plane: , resulting in astigmatic beams. The matrices have . A typical design of a rectangular ring has the following form: (for the equivalent rays where r = distance of equivalent ray from the axis, r’ = the slope against the axis). Note that in order for the ray to close on itself, the input column matrix has to equal the output column. This round-trip matrix is actually called ABCD matrix in the literature. The requirement that the ray is to be closed is therefore . Propagation of complex curvature The complex curvatures qin and qout in a section of a beam circuit with the section matrix is . In particular, if