Ring laser

Ring lasers are composed of two beams of light of the same polarization traveling in opposite directions in a closed loop.
Ring lasers are used most frequently as gyroscopes in moving vessels like cars, ships, planes, and missiles. The world's largest ring lasers can detect details of the Earth's rotation. Such large rings are also capable of extending scientific research in many new directions, including the detection of gravitational waves, Fresnel drag, Lense–Thirring effect, and quantum-electrodynamic effects.
In a rotating ring laser gyroscope, the two counter-propagating waves are slightly shifted in frequency and an interference pattern is observed, which is used to determine the rotational speed. The response to a rotation is a frequency difference between the two beams, which is proportional to the rotation rate of the ring laser. The difference can easily be measured.
Generally however, any non-reciprocity in the propagation between the two beams leads to a beat frequency.

Engineering applications

There is a continuous transition between ring lasers for engineering application and ring lasers for research. Rings for engineering have begun to incorporate a vast variety of materials as well as new technology. Historically, the first extension was the use of fiber optics as wave guides, obviating the use of mirrors. However, even rings using the most advanced fiber working in its optimal wavelength range have vastly higher losses than square rings with four high-quality mirrors. Therefore, fiber optic rings suffice only in high rotation rate applications. For example, fiber optic rings are now common in automobiles.
A ring can be constructed with other optically active materials that are able to conduct a beam with low losses. One type of ring laser design is a single crystal design, where light reflects around inside the laser crystal so as to circulate in a ring. This is the "monolithic crystal" design, and such devices are known as "non-planar ring oscillators" or MISERs. There are also ring fiber lasers. Since typically the achievable quality factors are low, such rings cannot be used for research where quality factors above 10¹² are sought and are achievable.

History

Shortly after the discovery of the laser, a seminal paper by Rosenthal appeared in 1962, which proposed what was later called a ring laser. While the ring laser shares with regular lasers features like extreme monochromaticity and high directivity, it differs in its inclusion of an area. With the ring laser, one could distinguish two beams in opposite directions. Rosenthal anticipated that the beam frequencies could be split by effects that affected the two beams in different ways. Although some may consider Macek et al. has built the first large ring laser, the US patent office has decided the first ring laser was built under Sperry scientist, Chao Chen Wang, based on the Sperry laboratory records. Wang showed that simply rotating it could generate a difference in the frequencies of the two beams. An industry focusing on smaller ring laser gyros emerged, with decimeter-sized ring lasers. Later it was found that any effect that affects the two beams in nonreciprocal fashion produces a frequency difference, as Rosenthal anticipated. Tools to analyze and construct rings were adapted from regular lasers, including methods to calculate the signal-to-noise ratio and to analyze beam characteristics. New phenomena unique to rings appeared, including lock-in, pulling, astigmatic beams, and special polarizations. Mirrors play a much greater role in ring lasers than in linear lasers, leading to the development of particularly high quality mirrors.
The resolution of large ring lasers has dramatically improved, as a result of a 1000-fold improvement in the quality factor. This improvement is largely a result of the removal of interfaces that the beams need to traverse as well as the improvements on technology which allowed a dramatic increase in measurement time. A 1 m × 1 m ring built in Christchurch, New Zealand in 1992 was sensitive enough to measure the Earth's rotation, and a 4 m × 4 m ring built in Wettzell, Germany improved the precision of this measurement to six digits.

Construction

In ring lasers, mirrors are used to focus and redirect the laser beams at the corners. While traveling between mirrors, the beams pass through gas-filled tubes. The beams are generally generated through local excitation of the gas by radio frequencies.
Critical variables in the construction of a ring laser include:

Size: Larger ring lasers can measure lower frequencies. The sensitivity of large rings increases quadratically with size.
Mirrors: High reflectivity is important.
Stability: The assembly must be attached to or built within a substance that changes minimally in response to temperature fluctuations.
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.
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
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
The noise frequencies are
where is the one-sided power spectral density of quantum noise, h is the Planck 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 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. 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 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 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 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 R_M 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:
.
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 q_in and q_out in a section of a beam circuit with
the section matrix is
In particular, if 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. 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 r_p and r_s as well as phase shifts upon mirror reflection χ_p and χ_s 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. Following the Jones’ vector around the closed circuit, one gets
. 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 circularly polarized. On the other hand, if , the formula above results in p or s reflection. A planar ring, however, is invariably s-polarized because the losses of the multilayer mirrors used are always less in s-polarized beams. There are at least two interesting applications:

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 stray light etc. Raytheon's use of a Faraday element to split internal frequencies introduces however optical 1/f noise and renders the device non-optimal as a gyro.
If the fourth mirror is suspended such that it can rotate around a horizontal axis, the appearance of

is extremely sensitive to the mirror's rotation. In a reasonable arrangement, an angular sensitivity of
±3 picoradian or 0.6 microarcsecond is estimated. With a mass suspended on the rotatable mirror, a simple gravitational wave detector can be constructed.