the matrix above is the round-trip matrix, the q at that point is , or . Note that it is necessary that to have a real spot size (Stability Criterion). The width is generally less than 1 mm for small lasers, but it increases approximately with . For calculation of beam positions for misaligned mirrors, see Polarization The polarization of rings exhibits particular features: Planar rings are either s-polarized, i.e. perpendicular to the ring plane, or p-polarized, in the plane; non-planar rings are circularly polarized. The Jones calculus is used to calculate polarization. Here, the column matrix signifies the electric field components in-plane and off-plane. To study further the transition from planar rings to non-planar rings, reflected amplitudes rp and rs as well as phase shifts upon mirror reflection χp and χs are introduced in an extended mirror matrix . Also, if the reference planes change, one needs to refer the E-vector after reflection to the new planes with the rotation matrix . Analysis of a skew-square ring by the Jones calculus yields the polarization in a ring. (A skew-square ring is a plane square ring where one mirror is lifted out of the plane of the other mirrors by a (dihedral) angle θ and tilted accordingly.) Following the Jones’ vector around the closed circuit, one gets (Note that the polarization at the end of the loop has to equal the polarization at the start). For small loss differences and small phase shift differences , the solution for is , where . If the dihedral angle θ is large enough, i.e. if , the solution of this equation is simply , i.e. a definitely non-planar beam is (left-handed or right-handed) circularly (not elliptically) polarized. On the other hand, if (a planar ring), the formula above results in p or s reflection (linear polarization). A planar ring, however, is invariably s-polarized because the losses of the multilayer mirrors used are always less in s-polarized beams (at the so-called “Brewster angle”, the reflected p-component even vanishes). There are at least two interesting applications: 1. The Raytheon ring laser. The fourth mirror is elevated by a certain amount over the plane of the other three. The Raytheon ring laser works with four circular polarizations, where now the difference of the differences represents twice the Sagnac effect. This configuration is in principle insensitive to drift. The scheme of detection is also more immune to