fluctuations (e.g. Zerodur, or bedrock for extremely large rings). 4. Gas: HeNe generates beams with the most desirable features for large ring lasers. For gyros, in principle any material that can be used to generate monochromatic light beams is applicable. The laser beam: theoretical tools For a ring as a measuring tool, Signal/Noise ratio and line widths are all-important. The signal of the ring as a rotation detector is used, whereas the all-pervasive white, quantum noise is the fundamental noise of the ring. Rings with a low quality factor generate additional low frequency noise. The standard matrix methods for the beam characteristics — curvature and width — are given, as well as the Jones calculus for polarization. Signal-to-noise ratio The following equations can be used to calculate the signal-to-noise ratio, S/N for rotation. The signal frequency is S = Δfs = 4, where is the area vector, is the rotation rate vector, λ is the vacuum wavelength, L is the perimeter. (For complicated geometries like nonplanar rings or figure-8 rings, the definitions and L = are to be used.) The noise frequencies are N = , where is the one-sided power spectral density of quantum noise, h is Planck's constant, f is the laser frequency, P includes all power losses of the laser beams, and Q is the quality factor of the ring. Line width Ring Lasers serve as frequency measuring devices. As such, single Fourier components, or lines in frequency space are of major importance in ring outputs. Their widths are determined by the prevailing noise spectra. The major noise contribution is typically white quantum noise If this noise is the only one present, the rms-line width sigma is obtained by corrupting the signal (represented by a δ function) with this noise in the interval 0-T. The result is: P should be maximized but kept below the level that generates additional modes. Q can largely be increased by avoiding losses (e.g. improving the quality of the mirrors). T is only limited by the stability of the device. T reduces the line width by the classic T−1/2 for white noise. For low-Q rings, an empirical relation for 1/f noise has been ascertained, with the one-sided frequency power spectral density given by , with A≃4. It is notoriously difficult to reduce line width in the presence of this noise. To decrease the line width further, long measurement times are necessary. A