231:
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.
226:, a seminal paper by Rosenthal appeared in 1962, which proposed what was later called a ring laser. While the ring laser shares with regular (linear) 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 (1 meter Ă 1 meter), the
17:
805:
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 Ă 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.
3770:
avoided. The whole ring is filled with a HeNe mixture of suitable partial pressures (up to a few hundred Pascal), to achieve lasing and good suppression of multiple pairs of modes. (Typically, the HeNe lasing gas at 633 nm is used; attempts for an argon ring laser failed.) Further, the lasing is excited with radio frequency to easily adjust the amplitude to just below the appearance of the second pair of modes. The
Rayleigh scattering of the HeNe gas is, at this time, negligible.
1662:
3511:. This equation says that even slightly above lock-in, there is already a reduction in frequency (i.e. pulling) relative to the theoretical frequency. In the presence of several satellites, only the major signal is pulled. The other satellites have their proper, non-pulled, frequency separation from the major signal. This opens the way to classic precision side-band spectroscopy as is known in microwaves, except that the ring laser has side bands down to nHz.
1358:
2918:
3378:(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 âBrewster angleâ, the reflected p-component even vanishes). There are at least two interesting applications:
1657:{\displaystyle \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{4}=\left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}=\left(M_{1}\cdot M_{2}\right)_{4}\cdot \left(M_{1}\cdot M_{2}\right)_{3}\cdot \left(M_{1}\cdot M_{2}\right)_{2}\cdot \left(M_{1}\cdot M_{2}\right)_{1}\cdot \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}}
2657:
4037:
It is essential for a high quality ring to use mirrors of very high reflectivity. Metallic mirror surfaces are inadequate for laser work (household Al-covered mirror surfaces are 83% reflective, Ag is 95% reflective). However, multilayer dielectric mirrors with 20â30 alternate (low L and high H index
2651:
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 Jonesâ vector around the closed circuit,
230:
has decided the first ring laser was built under Sperry scientist, Chao Chen Wang, (see US Patent 3,382,758) based on the Sperry laboratory records. Wang showed that simply rotating it could generate a difference in the frequencies of the two beams (Sagnac). An industry focusing on smaller ring laser
3765:
of the ring is given (474 THz), it remains to increase the circulating energy in the ring W and decrease the power losses dW/dt as much as possible. W is obviously proportional to the length of the ring, but must be limited to avoid multimodes. The power losses dW/dt however can be vastly decreased.
630:
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
3651:
The quality factor Q of the cavity, as well as the time duration of the measurement, determines the achievable frequency resolution of a ring to a large extent. The quality factor depends largely on the reflection properties of the mirrors. For high quality rings, reflectivities larger than 99.999%
275:
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.
114:
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
3836:
This equation gives rise to formidable quality factors. For a 4 m x 4 m ring equipped with 1 ppm mirrors (R = 1-10) we would get, at 474 THz, Q = 4Ă10. This quality factor produces a passive resonance line of rms = 5 Hz, which is eight orders of magnitude smaller than the atomic linewidth of
3769:
The power loss can be minimized by increasing the reflectivity of the mirrors to as close to 1 as possible and by eliminating other, spurious, sources of power loss, for example the inaccuracy of mirror curvature. Any interfaces or apertures that would decrease the quality factor of the ring are
804:
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
4475:
have potential applications in all-optical computing. One primary application is as an optical memory device where the direction of propagation represents either 0 or 1. They can maintain the propagation of light in exclusively the clockwise or counterclockwise direction as long as they remain
234:
The resolution of large ring lasers has dramatically improved, as a result of a 1000-fold improvement in the quality factor (see Table 1). 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
3642:
This is a huge advantage of large rings over small ones. As an example, small navigational gyros have lock-in frequencies on the order of 1 kHz. The first large ring had a lock-in frequency of about 2 kHz, and the first ring that could measure the Earth's rotation rate had a lock-in
3861:
has a gain bandwidth of about 2.2 GHz). (Note that for example in regular pendulums the Q is of the order of 10 and in wristwatch-type quartzes it is of the order of 10.) The active ring further reduces the linewidth by several orders of magnitude, and increasing the measuring time may
3381:
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
2247:
705:
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 for white noise.
2335:
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
4464:
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" (NPROs) or MISERs. There are also ring
1757:
2646:
97:
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
2545:
3412:
is extremely sensitive to the mirror's rotation. In a reasonable arrangement, an angular sensitivity of Âą3 picoradian or 0.6 microarcsecond is estimated. With a mass suspended on the rotatable mirror, a simple gravitational wave detector can be constructed.
3637:
46:) 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
251:
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.
4244:. Hereby, all calculations are strictly carried out up to the first power in the ks, assuming the materials are weakly absorbing. The final result, after the stack is matched to the incoming medium (vacuum) and to the substrate (the substrate index is n
4077:
The goal is to increase the quality factor of the cavity until
Rayleigh scattering of the HeNe gas in the cavity or other unavoidable loss mechanisms set a limit. For simplicity we assume normal incidence. Introducing the complex index of refraction
4218:
4069:
The losses are composed of scattering S, absorption A, and transmission T, such that 1 - r = S + A + T. Scattering is not treated here, because it is largely dependent on details of surface and interface treatment, and not easily analyzed.
3969:
In order to maximize the signal/noise ratio of a ring inside a given circle of radius r with n mirrors, a planar ring is advantageous over an equivalent nonplanar ring. Furthermore, a regular polygon has a maximal A/Ln ratio, with A/Ln =
2913:{\displaystyle \left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)=\left(M_{refl_{4}}\right)\left(M_{rot_{4}}\right)............\left(M_{refl_{1}}\right)\left(M_{rot_{1}}\right)\left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)}
3202:
4073:
r, A, and T are amenable to analysis. The losses are analyzed with a matrix method that, given the success of surface treatment and reduction of absorption, shows how many layers have to be applied to reduce transmission accordingly.
3382:
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.
1921:
1127:
110:
at 1.5 Îźm) 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.
3959:
3961:(Ď is the photon lifetime.) Thus, Q = ĎĎ. This is an extremely simple equation to measure Q in large rings. The photon lifetime Ď is measured on an oscilloscope, as the times are of the order of microseconds to milliseconds.
1824:
486:
4066:Îť/4 layers achieve reflection losses (1 - r) of single parts per million, and an analysis shows that losses of parts per billion can be achieved, if materials technology is pushed as far as is done with fiber optics.
1000:
894:
1060:
1277:
2393:
2039:
4452:
Ring lasers are sometimes modified to allow only one direction of propagation by placing a device into the ring which leads to different losses for different propagation directions. For instance, this could be a
346:
4440:
In the past it was thought that only small ring lasers avoid multimode excitation. However, if signal bandwidth is sacrificed, there is no known limit to ring laser size, either theoretically or experimentally.
2106:
3014:
3425:, is the frequency at which the difference between the beam frequencies becomes so small that it collapses, synchronizing the two counterrotating beams. Generally, if the theoretical frequency difference is f
3509:
3255:
701:
1668:
1198:
3736:
4422:
4437:â 40 cm (16 in), where S/N becomes proportional to L. Therefore, the sensitivity of large rings increases quadratically with size, hence the quest for ever larger ring lasers for research.
4027:
2552:
1348:
791:
622:
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.
587:
267:
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.
2299:
264:
3. Stability: The assembly must be attached to or built within a substance that changes minimally in response to temperature fluctuations (e.g. Zerodur, or bedrock for extremely large rings).
3831:
2418:
3523:
3060:
3376:
3291:
2096:
1764:
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.
929:
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.
5211:
G. Koppelmann, âZur
Theorie der Wechselschichten aus schwachabsorbierenden Substanzen und ihre Verwendung als Interferometerspiegelâ, Ann. Phys. (Leipzig) 7, pp. 388-396 (1960).
3342:
407:
4295:. Thus it remains to decrease the extinction coefficients. N is then an adjustable parameter to minimize the overall losses (stacks with up to 50 pairs have been published).
4784:
Schreiber, K.U.; Velikoseltev, A.; Rothacher, M.; Kluegel, T.; Stedman, G.E.; Wiltshire, D.L. (2004). "Direct measurement of diurnal polar motion by ring laser gyroscopes".
4104:
620:
3102:
2325:
5290:
Tartaglia, Angelo; Di
Virgilio, Angela; Belfi, Jacopo; Beverini, Nicolò; Ruggiero, Matteo Luca (15 February 2017). "Testing general relativity by means of ring lasers".
514:
378:
4094:
is the extinction coefficient) of the high-index material h ), and a corresponding complex index for the low-index material l , the stack is described by two matrices:
5199:
F. AbĂŠlès, âInvestigations on the propagation of sinusoidal electromagnetic waves in stratified media: application to thin filmsâ, Ann. de
Physique 5, 596-640 (1950).
3763:
3410:
924:
3109:
631:
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:
409:
is the rotation rate vector, Îť is the vacuum wavelength, L is the perimeter. (For complicated geometries like nonplanar rings or figure-8 rings, the definitions
1845:
4916:
Chiao, R.Y.; Moulthrop, A.A.; Levinson, M.T. (1984). "A Josephson gyroscope using superfluids". In Jacobs, S.F.; Physics of Quantum Electronics (Group) (eds.).
119:. Since typically the achievable quality factors are low, such rings cannot be used for research where quality factors above 10 are sought and are achievable.
1069:
4283:), where the first term is the AbÊlès limit, the second term the Koppelmann limit. The first term can be made as small as desirable by increasing the stack,
2397:
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
3766:
The ensuing decreased signal output power is not critical, as modern silicon detectors have low noise, and for very low signals photomultipliers are used.
3873:
3652:(R = 1-10 ppm) are indispensable. At this time, the main limitation of mirrors is the extinction coefficient of the evaporated high-index material TiO
796:
To decrease the line width further, long measurement times are necessary. A measurement time of 243 days reduced the Ď to 50 nHz in the Grossring.
1770:
414:
2303:
to have a real spot size (Stability Criterion). The width is generally less than 1 mm for small lasers, but it increases approximately with
937:
816:
1012:
1208:
81:. The response to a rotation is a frequency difference between the two beams, which is proportional to the rotation rate of the ring laser (
4925:
4580:
2341:
2242:{\displaystyle {\frac {1}{q}}={\frac {1}{R}}-j{\frac {\lambda }{\pi w^{2}}}={\frac {D-A}{2B}}-j{\frac {\sqrt {1-({\frac {A+D}{2}})^{2}}}{B}}}
1928:
276:
The standard matrix methods for the beam characteristics â curvature and width â are given, as well as the Jones calculus for polarization.
293:
2923:
932:
The phase is spherical with radius of curvature R. It is customary to combine radius of curvature and spot size into a complex curvature
5123:
Bilger, H.R.; Wells, P.V.; Stedman, G.E. (1994). "Origins of fundamental limits for reflection losses at multilayer dielectic mirrors".
3514:
When the dependence on perimeter L is taken into account for large rings, the relative difference between theoretical output frequency f
709:
For low-Q rings, an empirical relation for 1/f noise has been ascertained, with the one-sided frequency power spectral density given by
4900:
3434:
3344:, i.e. a definitely non-planar beam is (left-handed or right-handed) circularly (not elliptically) polarized. On the other hand, if
1752:{\displaystyle \left({\begin{matrix}A&B\\C&D\\\end{matrix}}\right)\cdot \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}}
85:). The difference can easily be measured. Generally however, any non-reciprocity in the propagation between the two beams leads to a
3207:
2547:. Also, if the reference planes change, one needs to refer the E-vector after reflection to the new planes with the rotation matrix
636:
1138:
3670:
2641:{\displaystyle M_{rot}=\left({\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{matrix}}\right)}
258:
1. Size: Larger ring lasers can measure lower frequencies. The sensitivity of large rings increases quadratically with size.
4308:
3664:
It is quite important for large rings to increase the quality factor Q, because it appears as 1/Q in the expression for noise.
3973:
1291:
712:
4654:
Rosenthal, A.H. (1962). "Regenerative Circulatory Multiple-Beam Interferometry for the Study of Light-Propagation Effects".
2920:(Note that the polarization at the end of the loop has to equal the polarization at the start). For small loss differences
525:
240:
2540:{\displaystyle M_{refl}=\left({\begin{matrix}r_{p}e^{j\chi _{p}}&0\\0&-r_{s}e^{j\chi _{s}}\\\end{matrix}}\right)}
2255:
5187:
P. Rouard, âĂtudes des propriĂŠtĂŠs optiques des lames metalliques très mincesâ, Ann. Phys. (Paris) 7, pp. 291-384 (1937).
102:
as wave guides, obviating the use of mirrors. However, even rings using the most advanced fiber working in its optimal
3778:
5042:
Bilger, H.R.; Stedman, G.E.; Wells, P.V. (1990). "Geometrical dependence of polarization in near-planar ring lasers".
4732:
239:, New Zealand in 1992 was sensitive enough to measure the Earth's rotation, and a 4 m Ă 4 m ring built in
3632:{\displaystyle {\frac {f_{t}-f}{f_{t}}}\cong {\frac {1}{2}}({\frac {f_{L}}{f_{t}}})^{2}\propto {\frac {1}{L^{4}}}}
3773:
For mirrors of proper curvature (spherical shape is acceptable) and equal reflectances r, the quality factor is
4757:
Stedman, G.E.; Bilger, H.R. (1992). "Ringlaser, an ultrahigh-resolution detector of optical nonreciprocities".
4502:
4472:
3019:
1761:(for the equivalent rays where r = distance of equivalent ray from the axis, râ = the slope against the axis).
3347:
3262:
3656:. The size and shape of the cavity as well as the presence of interfaces also influence the quality factor.
4492:
2046:
59:
16:
4989:
J. T. Verdeyen, âLaser Electronicsâ, Third Edition, Prentice Hall Series in Solid State Electronics, 1981.
4507:
3385:
2. If the fourth mirror is suspended such that it can rotate around a horizontal axis, the appearance of
51:
4444:
One of the major advantages of large rings is a quartic reduction of lock-in and pulling in large rings.
235:
dramatic increase in measurement time (see section on Line Width). A 1 m Ă 1 m ring built in
55:
4213:{\displaystyle \left({\begin{matrix}1&j/(n_{r}-jk_{r})\\(n_{r}-jk_{r})&1\\\end{matrix}}\right)}
3296:
383:
5349:
5309:
5264:
5255:
Bilger, H.R.; Stedman, G.E.; Li, Z.; Schreiber, U.; Schneider, M. (1995). "Ring Lasers for Geodesy".
5245:
R. R. Simpson and R. Hill, âRing laser geometry and sizeâ, Roy. Aeron. Soc. London, UK, 25 Feb. 1987.
5230:
5132:
5086:
5051:
5008:
4957:
4846:
4803:
4705:
4663:
4608:
4536:
4497:
4458:
74:
43:
32:
5077:
Hoeling, B.; Leuchs, G.; Ruder, H.; Schneider, M. (1992). "An argon ion ring laser as a gyroscope".
4733:"Ăffet tourbillonnaire optique. La circulation de L'Ăther lumineux dans un interfĂŠrographe tournant"
926:
is the peak field of the beam, E is the field distribution, and r is the distance off beam center.
47:
595:
5325:
5299:
5102:
4819:
4793:
4480:
3065:
2306:
4999:
Bilger, H.R.; Stedman, G.E. (1987). "Stability of planar ring lasers with mirror misalignment".
491:
354:
115:
such devices are known as "non-planar ring oscillators" (NPROs) or MISERs. There are also ring
5148:
5024:
4921:
4896:
4862:
4624:
4576:
284:
The following equations can be used to calculate the signal-to-noise ratio, S/N for rotation.
4892:
4884:
4599:
Duling III, I.N. (1991). "All-fiber ring soliton laser mode locked with a nonlinear mirror".
3197:{\displaystyle {\frac {E_{p}}{E_{s}}}=\pm j{\sqrt {1-(\gamma /\theta )^{2}}}+\gamma /\theta }
793:, with Aâ4. It is notoriously difficult to reduce line width in the presence of this noise.
5317:
5272:
5140:
5094:
5059:
5016:
4965:
4854:
4811:
4766:
4713:
4671:
4616:
4544:
4054:
3839:
227:
78:
3741:
3388:
1916:{\displaystyle \left({\begin{matrix}A_{s}&B_{s}\\C_{s}&D_{s}\\\end{matrix}}\right)}
902:
4040:
1122:{\displaystyle \left({\begin{matrix}1&0\\-{\frac {1}{f}}&1\\\end{matrix}}\right)}
5313:
5268:
5136:
5090:
5055:
5012:
4961:
4850:
4807:
4709:
4667:
4612:
4540:
4837:
Sayeh, M.R.; Bilger, H.R. (1985). "Flicker noise in frequency fluctuations of lasers".
4644:
L. E. Nelson et al., âUltrashort-pulse fiber ring lasersâ, Appl. Phys. B 65, 277 (1997)
4454:
86:
2041:. In particular, if the matrix above is the round-trip matrix, the q at that point is
5343:
5063:
4770:
82:
5329:
5106:
4823:
4693:
3954:{\displaystyle W=W_{0}e^{-{\frac {\omega t}{Q}}}\equiv W_{0}e^{-{\frac {t}{\tau }}}}
236:
99:
3518:
and actual output frequency f is inversely proportional to the fourth power of L:
5321:
5169:
Macleod, H.A. (1992). "New techniques revolutionize thin-film optical coatings".
4466:
116:
4858:
4568:
4548:
128:
Table 1. ~10 improvement in the resolution of large rings from 1972 to 2004.
103:
39:
4970:
4945:
4220:
r = l,h, which are multiplied in pairs, according to the size of the stack: M
1819:{\displaystyle \left|{\begin{matrix}A&B\\C&D\\\end{matrix}}\right|=1}
481:{\displaystyle {\vec {A}}={\frac {1}{2}}\oint {{\vec {r}}\times }d{\vec {l}}}
4029:
which itself has a maximum at n = 4, hence a planar square ring is optimal.
995:{\displaystyle {\frac {1}{q}}={\frac {1}{R}}-j{\frac {\lambda }{\pi w^{2}}}}
889:{\displaystyle \left|E\right|=\left|E_{0}\right|e^{-{\frac {r^{2}}{w^{2}}}}}
70:
5152:
5028:
4866:
4675:
4628:
1055:{\displaystyle \left({\begin{matrix}1&d\\0&1\\\end{matrix}}\right)}
5144:
5020:
4920:. Vol. 487. SPIEâthe International Society for Optical Engineering.
4815:
4620:
1272:{\displaystyle f=f_{y}={\frac {R_{M}}{2}}\cdot {\frac {1}{\cos \theta }}}
4798:
35:
traveling in opposite directions ("counter-rotating") in a closed loop.
5098:
2388:{\displaystyle \left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)}
5276:
4717:
2034:{\displaystyle q_{out}={\frac {A_{s}q_{in}+B_{s}}{C_{s}q_{in}+D_{s}}}}
341:{\displaystyle {\frac {{\vec {\Omega }}\cdot {\vec {A}}}{\lambda L}}}
69:
gyroscope, the two counter-propagating waves are slightly shifted in
1129:
for a mirror of focus length f. The relation between mirror radius R
243:, Germany improved the precision of this measurement to six digits.
5304:
3009:{\displaystyle \delta =\delta _{p}-\delta _{s}=(1-r_{p})-(1-r_{s})}
1133:
and focus length f is for oblique incidence at angle θ, in plane:
223:
28:
15:
3862:
additionally decrease the linewidth by many orders of magnitude.
2327:. For calculation of beam positions for misaligned mirrors, see
4918:
Physics of optical ring gyros: 7â10 January 1984, Snowbird, Utah
4887:. In Arecchi, F.T.; Stitch, M.L.; Bass, M.; et al. (eds.).
255:
Critical variables in the construction of a ring laser include:
3504:{\displaystyle f=f_{t}{\sqrt {1-({\frac {f_{L}}{f_{t}}})^{2}}}}
1353:
A typical design of a rectangular ring has the following form:
3250:{\displaystyle \gamma ={\frac {1}{\sqrt {2}}}(\delta -j\chi )}
1203:
for oblique incidence at angle θ, perpendicular to the plane:
696:{\displaystyle \sigma ={\sqrt {\frac {hf_{0}^{3}}{2PQ^{2}T}}}}
5207:
5205:
4878:
4876:
1193:{\displaystyle f=f_{x}={\frac {R_{M}}{2}}\cdot \cos \theta }
4883:
Statz, H.; Dorschner, T.A.; Holz, M.; Smith, I.W. (1985).
3731:{\displaystyle Q=2\pi f_{0}{\frac {W}{-{\frac {dW}{dt}}}}}
1767:
The requirement that the ray is to be closed is therefore
4417:{\displaystyle {\text{ }}S/N\propto L^{3}^{\frac {1}{2}}}
3421:
These are new phenomena in rings. The lock-in frequency f
3870:
The integral of the definition equation for Q above is:
1842:
in a section of a beam circuit with the section matrix
4303:
The perimeter dependence of the Signal/Noise ratio is
4022:{\displaystyle {\frac {r}{2}}{\frac {\cos(\pi /n)}{n}}}
1343:{\displaystyle \left|M_{1}\right|=\left|M_{2}\right|=1}
786:{\displaystyle S_{1/f}={\frac {A}{Q^{4}}}(f_{0}^{2}/f)}
4694:"Rotation rate sensing with traveling-wave ring laser"
4113:
2875:
2666:
2580:
2449:
2350:
1854:
1779:
1717:
1677:
1622:
1409:
1368:
1078:
1021:
126:
4311:
4107:
3976:
3876:
3781:
3744:
3673:
3526:
3437:
3391:
3350:
3299:
3265:
3210:
3112:
3068:
3022:
2926:
2660:
2555:
2421:
2344:
2309:
2258:
2109:
2049:
1931:
1848:
1773:
1671:
1361:
1294:
1211:
1141:
1072:
1015:
940:
905:
819:
715:
639:
598:
582:{\displaystyle S_{\delta f}={\frac {hf^{3}}{PQ^{2}}}}
528:
494:
417:
386:
357:
296:
4985:
4983:
4981:
4594:
4592:
4939:
4937:
3257:. If the dihedral angle θ is large enough, i.e. if
2294:{\displaystyle \left|{\frac {A+D}{2}}\right|\leq 1}
5195:
5193:
4416:
4212:
4021:
3953:
3825:
3757:
3730:
3631:
3503:
3404:
3370:
3336:
3285:
3249:
3196:
3096:
3054:
3008:
2912:
2640:
2539:
2387:
2319:
2293:
2241:
2090:
2033:
1915:
1818:
1751:
1656:
1342:
1271:
1192:
1121:
1054:
994:
918:
888:
785:
695:
614:
581:
508:
480:
401:
372:
340:
5118:
5116:
3826:{\displaystyle Q={\frac {\pi L}{2\lambda (1-r)}}}
5241:
5239:
4640:
4638:
3837:the Ne line (a 1:1 mixture of the two isotopes
2405:as well as phase shifts upon mirror reflection Ď
5220:M. Born, Optik (Springer-Verlag, Berlin, 1933).
261:2. Mirrors: High reflectivity is important.
8:
5233:, 6th ed. (Pergamon, Oxford, 1981), Chap. 1.
5164:
5162:
4573:Encyclopedia of Laser Physics and Technology
4562:
4560:
4558:
2413:are introduced in an extended mirror matrix
77:is observed, which is used to determine the
3293:, the solution of this equation is simply
4885:"The multioscillator ring laser gyroscope"
5303:
4969:
4797:
4687:
4685:
4479:In 2017 a proposal was published to test
4403:
4391:
4367:
4361:
4354:
4335:
4320:
4312:
4310:
4188:
4172:
4152:
4136:
4124:
4112:
4106:
4002:
3987:
3977:
3975:
3939:
3935:
3925:
3901:
3897:
3887:
3875:
3788:
3780:
3749:
3743:
3705:
3696:
3690:
3672:
3621:
3612:
3603:
3591:
3581:
3575:
3562:
3551:
3534:
3527:
3525:
3493:
3481:
3471:
3465:
3454:
3448:
3436:
3396:
3390:
3354:
3349:
3319:
3310:
3304:
3298:
3269:
3264:
3217:
3209:
3186:
3172:
3160:
3146:
3129:
3119:
3113:
3111:
3088:
3079:
3073:
3067:
3055:{\displaystyle \chi =\chi _{p}-\chi _{s}}
3046:
3033:
3021:
2997:
2972:
2950:
2937:
2925:
2896:
2882:
2874:
2858:
2847:
2827:
2813:
2757:
2746:
2726:
2712:
2687:
2673:
2665:
2659:
2579:
2560:
2554:
2521:
2513:
2503:
2474:
2466:
2456:
2448:
2426:
2420:
2371:
2357:
2349:
2343:
2310:
2308:
2263:
2257:
2227:
2205:
2193:
2164:
2152:
2139:
2123:
2110:
2108:
2056:
2048:
2022:
2006:
1996:
1984:
1968:
1958:
1951:
1936:
1930:
1899:
1887:
1873:
1861:
1853:
1847:
1778:
1772:
1743:
1716:
1676:
1670:
1648:
1621:
1607:
1596:
1583:
1564:
1553:
1540:
1521:
1510:
1497:
1478:
1467:
1454:
1435:
1408:
1394:
1367:
1360:
1324:
1303:
1293:
1251:
1237:
1231:
1222:
1210:
1167:
1161:
1152:
1140:
1096:
1077:
1071:
1020:
1014:
983:
970:
954:
941:
939:
910:
904:
876:
866:
860:
856:
842:
818:
772:
766:
761:
746:
737:
724:
720:
714:
680:
662:
657:
646:
638:
603:
597:
570:
555:
545:
533:
527:
498:
493:
467:
466:
448:
447:
446:
433:
419:
418:
416:
388:
387:
385:
359:
358:
356:
316:
315:
301:
300:
297:
295:
3371:{\displaystyle \gamma /\theta >>1}
3286:{\displaystyle \gamma /\theta <<1}
38:Ring lasers are used most frequently as
4891:. Vol. 4. North-Holland. pp.
4692:Macek, W.M.; Davis Jr., D.T.M. (1963).
4519:
4426:This equation defines large rings with
2091:{\displaystyle q={\frac {Aq+B}{Cq+D}}}
4944:Schawlow, A.L.; Townes, C.H. (1958).
4090:is the real index of refraction and k
7:
4527:Post, E.J. (1967). "Sagnac effect".
3429:, the actual signal frequency f is
3016:and small phase shift differences
222:Shortly after the discovery of the
5292:The European Physical Journal Plus
390:
303:
14:
3738:. Since the operating frequency
3337:{\displaystyle E_{p}/E_{s}=\pm j}
271:The laser beam: theoretical tools
4740:Journal de Physique et le Radium
2252:Note that it is necessary that
1830:Propagation of complex curvature
1005:The ring design uses a matrix M
402:{\displaystyle {\vec {\Omega }}}
3643:frequency of about 20 Hz.
1283:resulting in astigmatic beams.
4400:
4388:
4358:
4341:
4194:
4165:
4158:
4129:
4010:
3996:
3817:
3805:
3600:
3572:
3490:
3462:
3244:
3229:
3169:
3154:
3003:
2984:
2978:
2959:
2224:
2202:
780:
754:
472:
453:
424:
393:
364:
321:
306:
1:
4946:"Infrared and optical masers"
1062:for a straight section and M
813:The beam has a spot size w:
27:are composed of two beams of
5064:10.1016/0030-4018(90)90374-3
4771:10.1016/1051-2004(92)90031-S
615:{\displaystyle S_{\delta f}}
3097:{\displaystyle E_{p}/E_{s}}
2320:{\displaystyle {\sqrt {L}}}
519:The noise frequencies are
212:
209:
198:
195:
184:
181:
170:
167:
156:
153:
5366:
5322:10.1140/epjp/i2017-11372-5
4859:10.1103/PhysRevLett.55.700
809:Curvature radius and width
509:{\displaystyle \oint {dl}}
373:{\displaystyle {\vec {A}}}
4759:Digital Signal Processing
4549:10.1103/RevModPhys.39.475
4483:by means of ring lasers.
4473:Semiconductor ring lasers
4971:10.1103/PhysRev.112.1940
4503:Semiconductor ring laser
1834:The complex curvatures q
287:The signal frequency is
93:Engineering applications
5257:IEEE Trans Instrum Meas
4493:Optical ring resonators
4676:10.1364/JOSA.52.001143
4508:List of laser articles
4418:
4214:
4023:
3955:
3827:
3759:
3732:
3633:
3505:
3406:
3372:
3338:
3287:
3251:
3198:
3098:
3056:
3010:
2914:
2642:
2541:
2389:
2321:
2295:
2243:
2092:
2035:
1917:
1820:
1753:
1658:
1344:
1273:
1194:
1123:
1056:
996:
920:
890:
787:
697:
616:
583:
510:
482:
403:
374:
342:
60:quantum-electrodynamic
21:
5229:M. Born and E. Wolf,
4567:Paschotta, R (2008).
4419:
4215:
4024:
3956:
3828:
3760:
3758:{\displaystyle f_{0}}
3733:
3634:
3506:
3407:
3405:{\displaystyle E_{p}}
3373:
3339:
3288:
3252:
3199:
3099:
3057:
3011:
2915:
2643:
2542:
2390:
2322:
2296:
2244:
2093:
2036:
1918:
1821:
1754:
1659:
1345:
1274:
1195:
1124:
1057:
997:
921:
919:{\displaystyle E_{0}}
891:
788:
698:
617:
584:
511:
483:
404:
375:
343:
280:Signal-to-noise ratio
56:Lense-Thirring effect
19:
5231:Principles of Optics
5145:10.1364/AO.33.007390
5021:10.1364/AO.26.003710
4816:10.1029/2003JB002803
4621:10.1364/OL.16.000539
4498:Ring laser gyroscope
4309:
4105:
3974:
3874:
3779:
3742:
3671:
3524:
3435:
3389:
3348:
3297:
3263:
3208:
3110:
3066:
3062:, the solution for
3020:
2924:
2658:
2553:
2419:
2342:
2307:
2256:
2107:
2047:
1929:
1846:
1771:
1669:
1359:
1292:
1209:
1139:
1070:
1013:
938:
903:
817:
800:Beam characteristics
713:
637:
596:
526:
492:
415:
384:
380:is the area vector,
355:
294:
75:interference pattern
44:ring laser gyroscope
5314:2017EPJP..132...73T
5269:1995ITIM...44..468B
5137:1994ApOpt..33.7390B
5091:1992ApPhB..55...46H
5056:1990OptCo..80..133B
5013:1987ApOpt..26.3710B
4962:1958PhRv..112.1940S
4851:1985PhRvL..55..700S
4808:2004JGRB..109.6405S
4731:Sagnac, G. (1914).
4710:1963ApPhL...2...67M
4668:1962JOSA...52.1143R
4613:1991OptL...16..539D
4541:1967RvMP...39..475P
3667:Definition of Q:
3417:Lock-in and pulling
771:
667:
129:
48:gravitational waves
5099:10.1007/BF00348612
4481:general relativity
4414:
4210:
4204:
4019:
3951:
3823:
3755:
3728:
3629:
3501:
3402:
3368:
3334:
3283:
3247:
3194:
3094:
3052:
3006:
2910:
2904:
2695:
2638:
2632:
2537:
2531:
2385:
2379:
2317:
2291:
2239:
2088:
2031:
1913:
1907:
1816:
1804:
1749:
1737:
1702:
1654:
1642:
1429:
1388:
1340:
1286:The matrices have
1269:
1190:
1119:
1113:
1052:
1046:
992:
916:
886:
783:
757:
693:
653:
612:
579:
506:
478:
399:
370:
338:
127:
22:
5277:10.1109/19.377882
5171:Laser Focus World
4927:978-0-89252-522-5
4718:10.1063/1.1753778
4582:978-3-527-40828-3
4411:
4385:
4315:
4017:
3985:
3947:
3914:
3821:
3726:
3723:
3627:
3597:
3570:
3557:
3499:
3487:
3227:
3226:
3178:
3135:
2315:
2279:
2237:
2233:
2221:
2185:
2159:
2131:
2118:
2086:
2029:
1267:
1246:
1176:
1104:
990:
962:
949:
882:
752:
691:
690:
577:
516:are to be used.)
475:
456:
441:
427:
396:
367:
336:
324:
309:
220:
219:
5357:
5334:
5333:
5307:
5287:
5281:
5280:
5252:
5246:
5243:
5234:
5227:
5221:
5218:
5212:
5209:
5200:
5197:
5188:
5185:
5179:
5178:
5166:
5157:
5156:
5120:
5111:
5110:
5074:
5068:
5067:
5039:
5033:
5032:
4996:
4990:
4987:
4976:
4975:
4973:
4941:
4932:
4931:
4913:
4907:
4906:
4880:
4871:
4870:
4834:
4828:
4827:
4801:
4781:
4775:
4774:
4754:
4748:
4747:
4737:
4728:
4722:
4721:
4698:Appl. Phys. Lett
4689:
4680:
4679:
4651:
4645:
4642:
4633:
4632:
4596:
4587:
4586:
4564:
4553:
4552:
4524:
4457:combined with a
4423:
4421:
4420:
4415:
4413:
4412:
4404:
4398:
4397:
4396:
4395:
4386:
4381:
4380:
4362:
4340:
4339:
4324:
4316:
4313:
4294:
4219:
4217:
4216:
4211:
4209:
4205:
4193:
4192:
4177:
4176:
4157:
4156:
4141:
4140:
4128:
4065:
4063:
4062:
4051:
4049:
4048:
4028:
4026:
4025:
4020:
4018:
4013:
4006:
3988:
3986:
3978:
3960:
3958:
3957:
3952:
3950:
3949:
3948:
3940:
3930:
3929:
3917:
3916:
3915:
3910:
3902:
3892:
3891:
3860:
3858:
3857:
3849:
3846:
3845:
3832:
3830:
3829:
3824:
3822:
3820:
3797:
3789:
3764:
3762:
3761:
3756:
3754:
3753:
3737:
3735:
3734:
3729:
3727:
3725:
3724:
3722:
3714:
3706:
3697:
3695:
3694:
3660:Quality factor Q
3638:
3636:
3635:
3630:
3628:
3626:
3625:
3613:
3608:
3607:
3598:
3596:
3595:
3586:
3585:
3576:
3571:
3563:
3558:
3556:
3555:
3546:
3539:
3538:
3528:
3510:
3508:
3507:
3502:
3500:
3498:
3497:
3488:
3486:
3485:
3476:
3475:
3466:
3455:
3453:
3452:
3411:
3409:
3408:
3403:
3401:
3400:
3377:
3375:
3374:
3369:
3358:
3343:
3341:
3340:
3335:
3324:
3323:
3314:
3309:
3308:
3292:
3290:
3289:
3284:
3273:
3256:
3254:
3253:
3248:
3228:
3222:
3218:
3203:
3201:
3200:
3195:
3190:
3179:
3177:
3176:
3164:
3147:
3136:
3134:
3133:
3124:
3123:
3114:
3103:
3101:
3100:
3095:
3093:
3092:
3083:
3078:
3077:
3061:
3059:
3058:
3053:
3051:
3050:
3038:
3037:
3015:
3013:
3012:
3007:
3002:
3001:
2977:
2976:
2955:
2954:
2942:
2941:
2919:
2917:
2916:
2911:
2909:
2905:
2901:
2900:
2887:
2886:
2869:
2865:
2864:
2863:
2862:
2838:
2834:
2833:
2832:
2831:
2768:
2764:
2763:
2762:
2761:
2737:
2733:
2732:
2731:
2730:
2700:
2696:
2692:
2691:
2678:
2677:
2647:
2645:
2644:
2639:
2637:
2633:
2571:
2570:
2546:
2544:
2543:
2538:
2536:
2532:
2528:
2527:
2526:
2525:
2508:
2507:
2481:
2480:
2479:
2478:
2461:
2460:
2440:
2439:
2394:
2392:
2391:
2386:
2384:
2380:
2376:
2375:
2362:
2361:
2326:
2324:
2323:
2318:
2316:
2311:
2300:
2298:
2297:
2292:
2284:
2280:
2275:
2264:
2248:
2246:
2245:
2240:
2238:
2232:
2231:
2222:
2217:
2206:
2195:
2194:
2186:
2184:
2176:
2165:
2160:
2158:
2157:
2156:
2140:
2132:
2124:
2119:
2111:
2097:
2095:
2094:
2089:
2087:
2085:
2071:
2057:
2040:
2038:
2037:
2032:
2030:
2028:
2027:
2026:
2014:
2013:
2001:
2000:
1990:
1989:
1988:
1976:
1975:
1963:
1962:
1952:
1947:
1946:
1922:
1920:
1919:
1914:
1912:
1908:
1904:
1903:
1892:
1891:
1878:
1877:
1866:
1865:
1825:
1823:
1822:
1817:
1809:
1805:
1758:
1756:
1755:
1750:
1748:
1747:
1742:
1738:
1734:
1707:
1703:
1663:
1661:
1660:
1655:
1653:
1652:
1647:
1643:
1639:
1612:
1611:
1606:
1602:
1601:
1600:
1588:
1587:
1569:
1568:
1563:
1559:
1558:
1557:
1545:
1544:
1526:
1525:
1520:
1516:
1515:
1514:
1502:
1501:
1483:
1482:
1477:
1473:
1472:
1471:
1459:
1458:
1440:
1439:
1434:
1430:
1426:
1399:
1398:
1393:
1389:
1385:
1349:
1347:
1346:
1341:
1333:
1329:
1328:
1312:
1308:
1307:
1278:
1276:
1275:
1270:
1268:
1266:
1252:
1247:
1242:
1241:
1232:
1227:
1226:
1199:
1197:
1196:
1191:
1177:
1172:
1171:
1162:
1157:
1156:
1128:
1126:
1125:
1120:
1118:
1114:
1105:
1097:
1061:
1059:
1058:
1053:
1051:
1047:
1001:
999:
998:
993:
991:
989:
988:
987:
971:
963:
955:
950:
942:
925:
923:
922:
917:
915:
914:
895:
893:
892:
887:
885:
884:
883:
881:
880:
871:
870:
861:
851:
847:
846:
830:
792:
790:
789:
784:
776:
770:
765:
753:
751:
750:
738:
733:
732:
728:
702:
700:
699:
694:
692:
689:
685:
684:
668:
666:
661:
648:
647:
621:
619:
618:
613:
611:
610:
588:
586:
585:
580:
578:
576:
575:
574:
561:
560:
559:
546:
541:
540:
515:
513:
512:
507:
505:
487:
485:
484:
479:
477:
476:
468:
462:
458:
457:
449:
442:
434:
429:
428:
420:
408:
406:
405:
400:
398:
397:
389:
379:
377:
376:
371:
369:
368:
360:
347:
345:
344:
339:
337:
335:
327:
326:
325:
317:
311:
310:
302:
298:
228:US patent office
130:
79:rotational speed
5365:
5364:
5360:
5359:
5358:
5356:
5355:
5354:
5340:
5339:
5338:
5337:
5289:
5288:
5284:
5254:
5253:
5249:
5244:
5237:
5228:
5224:
5219:
5215:
5210:
5203:
5198:
5191:
5186:
5182:
5168:
5167:
5160:
5122:
5121:
5114:
5076:
5075:
5071:
5041:
5040:
5036:
4998:
4997:
4993:
4988:
4979:
4943:
4942:
4935:
4928:
4915:
4914:
4910:
4903:
4882:
4881:
4874:
4839:Phys. Rev. Lett
4836:
4835:
4831:
4799:physics/0406156
4786:J. Geophys. Res
4783:
4782:
4778:
4756:
4755:
4751:
4735:
4730:
4729:
4725:
4691:
4690:
4683:
4656:J. Opt. Soc. Am
4653:
4652:
4648:
4643:
4636:
4598:
4597:
4590:
4583:
4566:
4565:
4556:
4526:
4525:
4521:
4516:
4489:
4455:Faraday rotator
4450:
4448:Practical rings
4435:
4399:
4387:
4363:
4350:
4331:
4307:
4306:
4301:
4292:
4288:
4284:
4282:
4278:
4274:
4270:
4266:
4262:
4258:
4254:
4247:
4243:
4239:
4236:..............M
4235:
4231:
4227:
4223:
4203:
4202:
4197:
4184:
4168:
4162:
4161:
4148:
4132:
4119:
4108:
4103:
4102:
4100:
4093:
4089:
4085:
4081:
4061:
4058:
4057:
4056:
4053:
4047:
4044:
4043:
4042:
4039:
4038:of refraction)
4035:
3989:
3972:
3971:
3967:
3931:
3921:
3903:
3893:
3883:
3872:
3871:
3868:
3856:
3854:
3853:
3852:
3851:
3844:
3842:
3841:
3840:
3838:
3798:
3790:
3777:
3776:
3745:
3740:
3739:
3715:
3707:
3701:
3686:
3669:
3668:
3662:
3655:
3649:
3617:
3599:
3587:
3577:
3547:
3530:
3529:
3522:
3521:
3517:
3489:
3477:
3467:
3444:
3433:
3432:
3428:
3424:
3419:
3392:
3387:
3386:
3346:
3345:
3315:
3300:
3295:
3294:
3261:
3260:
3206:
3205:
3168:
3125:
3115:
3108:
3107:
3084:
3069:
3064:
3063:
3042:
3029:
3018:
3017:
2993:
2968:
2946:
2933:
2922:
2921:
2903:
2902:
2892:
2889:
2888:
2878:
2870:
2854:
2843:
2839:
2823:
2809:
2805:
2753:
2742:
2738:
2722:
2708:
2704:
2694:
2693:
2683:
2680:
2679:
2669:
2661:
2656:
2655:
2631:
2630:
2619:
2604:
2603:
2592:
2575:
2556:
2551:
2550:
2530:
2529:
2517:
2509:
2499:
2494:
2488:
2487:
2482:
2470:
2462:
2452:
2444:
2422:
2417:
2416:
2412:
2408:
2404:
2400:
2378:
2377:
2367:
2364:
2363:
2353:
2345:
2340:
2339:
2333:
2305:
2304:
2265:
2259:
2254:
2253:
2223:
2207:
2177:
2166:
2148:
2144:
2105:
2104:
2072:
2058:
2045:
2044:
2018:
2002:
1992:
1991:
1980:
1964:
1954:
1953:
1932:
1927:
1926:
1906:
1905:
1895:
1893:
1883:
1880:
1879:
1869:
1867:
1857:
1849:
1844:
1843:
1841:
1837:
1832:
1803:
1802:
1797:
1791:
1790:
1785:
1774:
1769:
1768:
1736:
1735:
1727:
1724:
1723:
1712:
1711:
1701:
1700:
1695:
1689:
1688:
1683:
1672:
1667:
1666:
1641:
1640:
1632:
1629:
1628:
1617:
1616:
1592:
1579:
1578:
1574:
1573:
1549:
1536:
1535:
1531:
1530:
1506:
1493:
1492:
1488:
1487:
1463:
1450:
1449:
1445:
1444:
1428:
1427:
1419:
1416:
1415:
1404:
1403:
1387:
1386:
1378:
1375:
1374:
1363:
1362:
1357:
1356:
1320:
1316:
1299:
1295:
1290:
1289:
1282:
1256:
1233:
1218:
1207:
1206:
1163:
1148:
1137:
1136:
1132:
1112:
1111:
1106:
1090:
1089:
1084:
1073:
1068:
1067:
1065:
1045:
1044:
1039:
1033:
1032:
1027:
1016:
1011:
1010:
1008:
979:
975:
936:
935:
906:
901:
900:
872:
862:
852:
838:
834:
820:
815:
814:
811:
802:
742:
716:
711:
710:
676:
669:
649:
635:
634:
628:
599:
594:
593:
566:
562:
551:
547:
529:
524:
523:
490:
489:
413:
412:
382:
381:
353:
352:
328:
299:
292:
291:
282:
273:
249:
142:
137:
125:
109:
106:range (e.g. SiO
95:
12:
11:
5:
5363:
5361:
5353:
5352:
5342:
5341:
5336:
5335:
5282:
5263:(2): 468â470.
5247:
5235:
5222:
5213:
5201:
5189:
5180:
5158:
5131:(31): 7390â6.
5112:
5069:
5034:
5007:(17): 3710â6.
4991:
4977:
4933:
4926:
4908:
4902:978-0444869272
4901:
4889:Laser Handbook
4872:
4829:
4792:(B6): B06405.
4776:
4749:
4723:
4681:
4662:(10): 1143â7.
4646:
4634:
4607:(8): 539â541.
4588:
4581:
4554:
4535:(2): 475â493.
4529:Rev. Mod. Phys
4518:
4517:
4515:
4512:
4511:
4510:
4505:
4500:
4495:
4488:
4485:
4449:
4446:
4433:
4410:
4407:
4402:
4394:
4390:
4384:
4379:
4376:
4373:
4370:
4366:
4360:
4357:
4353:
4349:
4346:
4343:
4338:
4334:
4330:
4327:
4323:
4319:
4300:
4297:
4290:
4286:
4280:
4276:
4272:
4268:
4264:
4260:
4256:
4252:
4245:
4241:
4237:
4233:
4229:
4225:
4221:
4208:
4201:
4198:
4196:
4191:
4187:
4183:
4180:
4175:
4171:
4167:
4164:
4163:
4160:
4155:
4151:
4147:
4144:
4139:
4135:
4131:
4127:
4123:
4120:
4118:
4115:
4114:
4111:
4098:
4091:
4087:
4083:
4079:
4059:
4045:
4034:
4031:
4016:
4012:
4009:
4005:
4001:
3998:
3995:
3992:
3984:
3981:
3966:
3965:Shape of rings
3963:
3946:
3943:
3938:
3934:
3928:
3924:
3920:
3913:
3909:
3906:
3900:
3896:
3890:
3886:
3882:
3879:
3867:
3864:
3855:
3843:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3796:
3793:
3787:
3784:
3752:
3748:
3721:
3718:
3713:
3710:
3704:
3700:
3693:
3689:
3685:
3682:
3679:
3676:
3661:
3658:
3653:
3648:
3645:
3624:
3620:
3616:
3611:
3606:
3602:
3594:
3590:
3584:
3580:
3574:
3569:
3566:
3561:
3554:
3550:
3545:
3542:
3537:
3533:
3515:
3496:
3492:
3484:
3480:
3474:
3470:
3464:
3461:
3458:
3451:
3447:
3443:
3440:
3426:
3422:
3418:
3415:
3399:
3395:
3367:
3364:
3361:
3357:
3353:
3333:
3330:
3327:
3322:
3318:
3313:
3307:
3303:
3282:
3279:
3276:
3272:
3268:
3246:
3243:
3240:
3237:
3234:
3231:
3225:
3221:
3216:
3213:
3193:
3189:
3185:
3182:
3175:
3171:
3167:
3163:
3159:
3156:
3153:
3150:
3145:
3142:
3139:
3132:
3128:
3122:
3118:
3091:
3087:
3082:
3076:
3072:
3049:
3045:
3041:
3036:
3032:
3028:
3025:
3005:
3000:
2996:
2992:
2989:
2986:
2983:
2980:
2975:
2971:
2967:
2964:
2961:
2958:
2953:
2949:
2945:
2940:
2936:
2932:
2929:
2908:
2899:
2895:
2891:
2890:
2885:
2881:
2877:
2876:
2873:
2868:
2861:
2857:
2853:
2850:
2846:
2842:
2837:
2830:
2826:
2822:
2819:
2816:
2812:
2808:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2767:
2760:
2756:
2752:
2749:
2745:
2741:
2736:
2729:
2725:
2721:
2718:
2715:
2711:
2707:
2703:
2699:
2690:
2686:
2682:
2681:
2676:
2672:
2668:
2667:
2664:
2636:
2629:
2626:
2623:
2620:
2618:
2615:
2612:
2609:
2606:
2605:
2602:
2599:
2596:
2593:
2591:
2588:
2585:
2582:
2581:
2578:
2574:
2569:
2566:
2563:
2559:
2535:
2524:
2520:
2516:
2512:
2506:
2502:
2498:
2495:
2493:
2490:
2489:
2486:
2483:
2477:
2473:
2469:
2465:
2459:
2455:
2451:
2450:
2447:
2443:
2438:
2435:
2432:
2429:
2425:
2410:
2406:
2402:
2398:
2383:
2374:
2370:
2366:
2365:
2360:
2356:
2352:
2351:
2348:
2332:
2329:
2314:
2290:
2287:
2283:
2278:
2274:
2271:
2268:
2262:
2236:
2230:
2226:
2220:
2216:
2213:
2210:
2204:
2201:
2198:
2192:
2189:
2183:
2180:
2175:
2172:
2169:
2163:
2155:
2151:
2147:
2143:
2138:
2135:
2130:
2127:
2122:
2117:
2114:
2084:
2081:
2078:
2075:
2070:
2067:
2064:
2061:
2055:
2052:
2025:
2021:
2017:
2012:
2009:
2005:
1999:
1995:
1987:
1983:
1979:
1974:
1971:
1967:
1961:
1957:
1950:
1945:
1942:
1939:
1935:
1911:
1902:
1898:
1894:
1890:
1886:
1882:
1881:
1876:
1872:
1868:
1864:
1860:
1856:
1855:
1852:
1839:
1835:
1831:
1828:
1815:
1812:
1808:
1801:
1798:
1796:
1793:
1792:
1789:
1786:
1784:
1781:
1780:
1777:
1746:
1741:
1733:
1730:
1726:
1725:
1722:
1719:
1718:
1715:
1710:
1706:
1699:
1696:
1694:
1691:
1690:
1687:
1684:
1682:
1679:
1678:
1675:
1651:
1646:
1638:
1635:
1631:
1630:
1627:
1624:
1623:
1620:
1615:
1610:
1605:
1599:
1595:
1591:
1586:
1582:
1577:
1572:
1567:
1562:
1556:
1552:
1548:
1543:
1539:
1534:
1529:
1524:
1519:
1513:
1509:
1505:
1500:
1496:
1491:
1486:
1481:
1476:
1470:
1466:
1462:
1457:
1453:
1448:
1443:
1438:
1433:
1425:
1422:
1418:
1417:
1414:
1411:
1410:
1407:
1402:
1397:
1392:
1384:
1381:
1377:
1376:
1373:
1370:
1369:
1366:
1339:
1336:
1332:
1327:
1323:
1319:
1315:
1311:
1306:
1302:
1298:
1265:
1262:
1259:
1255:
1250:
1245:
1240:
1236:
1230:
1225:
1221:
1217:
1214:
1189:
1186:
1183:
1180:
1175:
1170:
1166:
1160:
1155:
1151:
1147:
1144:
1130:
1117:
1110:
1107:
1103:
1100:
1095:
1092:
1091:
1088:
1085:
1083:
1080:
1079:
1076:
1063:
1050:
1043:
1040:
1038:
1035:
1034:
1031:
1028:
1026:
1023:
1022:
1019:
1006:
986:
982:
978:
974:
969:
966:
961:
958:
953:
948:
945:
913:
909:
879:
875:
869:
865:
859:
855:
850:
845:
841:
837:
833:
829:
826:
823:
810:
807:
801:
798:
782:
779:
775:
769:
764:
760:
756:
749:
745:
741:
736:
731:
727:
723:
719:
688:
683:
679:
675:
672:
665:
660:
656:
652:
645:
642:
627:
624:
609:
606:
602:
573:
569:
565:
558:
554:
550:
544:
539:
536:
532:
504:
501:
497:
474:
471:
465:
461:
455:
452:
445:
440:
437:
432:
426:
423:
395:
392:
366:
363:
334:
331:
323:
320:
314:
308:
305:
281:
278:
272:
269:
248:
245:
218:
217:
214:
211:
208:
204:
203:
200:
197:
194:
190:
189:
186:
183:
180:
176:
175:
172:
169:
166:
162:
161:
158:
155:
152:
148:
147:
144:
139:
134:
124:
121:
107:
94:
91:
87:beat frequency
65:In a rotating
13:
10:
9:
6:
4:
3:
2:
5362:
5351:
5348:
5347:
5345:
5331:
5327:
5323:
5319:
5315:
5311:
5306:
5301:
5297:
5293:
5286:
5283:
5278:
5274:
5270:
5266:
5262:
5258:
5251:
5248:
5242:
5240:
5236:
5232:
5226:
5223:
5217:
5214:
5208:
5206:
5202:
5196:
5194:
5190:
5184:
5181:
5176:
5172:
5165:
5163:
5159:
5154:
5150:
5146:
5142:
5138:
5134:
5130:
5126:
5119:
5117:
5113:
5108:
5104:
5100:
5096:
5092:
5088:
5084:
5080:
5079:Appl. Phys. B
5073:
5070:
5065:
5061:
5057:
5053:
5049:
5045:
5038:
5035:
5030:
5026:
5022:
5018:
5014:
5010:
5006:
5002:
4995:
4992:
4986:
4984:
4982:
4978:
4972:
4967:
4963:
4959:
4956:(6): 1940â9.
4955:
4951:
4947:
4940:
4938:
4934:
4929:
4923:
4919:
4912:
4909:
4904:
4898:
4894:
4890:
4886:
4879:
4877:
4873:
4868:
4864:
4860:
4856:
4852:
4848:
4844:
4840:
4833:
4830:
4825:
4821:
4817:
4813:
4809:
4805:
4800:
4795:
4791:
4787:
4780:
4777:
4772:
4768:
4764:
4760:
4753:
4750:
4745:
4741:
4734:
4727:
4724:
4719:
4715:
4711:
4707:
4703:
4699:
4695:
4688:
4686:
4682:
4677:
4673:
4669:
4665:
4661:
4657:
4650:
4647:
4641:
4639:
4635:
4630:
4626:
4622:
4618:
4614:
4610:
4606:
4602:
4595:
4593:
4589:
4584:
4578:
4574:
4570:
4569:"Ring Lasers"
4563:
4561:
4559:
4555:
4550:
4546:
4542:
4538:
4534:
4530:
4523:
4520:
4513:
4509:
4506:
4504:
4501:
4499:
4496:
4494:
4491:
4490:
4486:
4484:
4482:
4477:
4474:
4470:
4468:
4462:
4460:
4456:
4447:
4445:
4442:
4438:
4436:
4429:
4424:
4408:
4405:
4392:
4382:
4377:
4374:
4371:
4368:
4364:
4355:
4351:
4347:
4344:
4336:
4332:
4328:
4325:
4321:
4317:
4304:
4298:
4296:
4249:
4206:
4199:
4189:
4185:
4181:
4178:
4173:
4169:
4153:
4149:
4145:
4142:
4137:
4133:
4125:
4121:
4116:
4109:
4095:
4075:
4071:
4067:
4064:
4050:
4032:
4030:
4014:
4007:
4003:
3999:
3993:
3990:
3982:
3979:
3964:
3962:
3944:
3941:
3936:
3932:
3926:
3922:
3918:
3911:
3907:
3904:
3898:
3894:
3888:
3884:
3880:
3877:
3865:
3863:
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2079:
2076:
2073:
2068:
2065:
2062:
2059:
2053:
2050:
2042:
2023:
2019:
2015:
2010:
2007:
2003:
1997:
1993:
1985:
1981:
1977:
1972:
1969:
1965:
1959:
1955:
1948:
1943:
1940:
1937:
1933:
1924:
1909:
1900:
1896:
1888:
1884:
1874:
1870:
1862:
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517:
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430:
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312:
288:
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279:
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246:
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187:
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140:
135:
132:
131:
122:
120:
118:
112:
105:
101:
92:
90:
88:
84:
83:Sagnac effect
80:
76:
72:
68:
63:
61:
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53:
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30:
26:
18:
5295:
5291:
5285:
5260:
5256:
5250:
5225:
5216:
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5177:(11): 116â9.
5174:
5170:
5128:
5124:
5085:(1): 46â50.
5082:
5078:
5072:
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5047:
5043:
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4842:
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4789:
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4762:
4758:
4752:
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4704:(3): 67â68.
4701:
4697:
4659:
4655:
4649:
4604:
4600:
4572:
4532:
4528:
4522:
4478:
4471:
4467:fiber lasers
4463:
4451:
4443:
4439:
4431:
4427:
4425:
4305:
4302:
4250:
4096:
4076:
4072:
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4036:
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3663:
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3384:
3380:
3259:
3106:
2654:
2650:
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2415:
2396:
2338:
2334:
2331:Polarization
2302:
2251:
2103:
2100:
2043:
1925:
1833:
1766:
1763:
1760:
1665:
1355:
1352:
1288:
1285:
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1202:
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1004:
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928:
898:
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704:
633:
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591:
521:
518:
411:
350:
289:
286:
283:
274:
266:
263:
260:
257:
254:
250:
247:Construction
237:Christchurch
233:
221:
196:8.6 ÎźHz
117:fiber lasers
113:
100:fiber optics
96:
66:
64:
52:Fresnel drag
37:
33:polarization
31:of the same
24:
23:
5350:Laser types
5044:Opt. Commun
4299:Large rings
4251:1 - r = (4n
3866:Measurement
290:S = Îfs = 4
210:50 nHz
182:31 mHz
168:68 mHz
154:4.5 Hz
141:measurement
25:Ring lasers
5305:1612.09099
4746:: 177â195.
4514:References
4459:polarizing
4086:) (where n
3647:The cavity
626:Line width
216:Schreiber
213:243 d
104:wavelength
67:ring laser
40:gyroscopes
20:Ring Laser
5298:(2): 73.
5125:Appl. Opt
5001:Appl. Opt
4950:Phys. Rev
4601:Opt. Lett
4575:. Wiley.
4476:powered.
4461:element.
4430:>>
4356:−
4348:−
4329:∝
4179:−
4143:−
4000:π
3994:
3945:τ
3937:−
3919:≡
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3803:λ
3792:π
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3560:≅
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3329:±
3275:θ
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3242:χ
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3204:, where
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3166:θ
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3152:−
3141:±
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3040:−
3031:χ
3024:χ
2991:−
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2928:δ
2652:one gets
2628:θ
2625:
2617:θ
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2608:−
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2598:
2590:θ
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2286:≤
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1614:⋅
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1571:⋅
1547:⋅
1528:⋅
1504:⋅
1485:⋅
1461:⋅
1264:θ
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1249:⋅
1188:θ
1185:
1179:⋅
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965:−
858:−
641:σ
605:δ
535:δ
496:∮
488:and L =
473:→
460:×
454:→
444:∮
425:→
394:→
391:Ω
365:→
330:λ
322:→
313:⋅
307:→
304:Ω
171:16 s
157:10 s
138:linewidth
71:frequency
62:effects.
5344:Category
5330:54670056
5153:20941300
5107:59388387
5029:20490127
4867:10032424
4824:14552631
4629:19773991
4487:See also
4267:) + 2Ď(k
3363:>>
3278:<<
1732:′
1637:′
1424:′
1383:′
241:Wettzell
199:8 d
188:Stedman
185:8 h
160:Stowell
5310:Bibcode
5265:Bibcode
5133:Bibcode
5087:Bibcode
5052:Bibcode
5009:Bibcode
4958:Bibcode
4893:231â327
4847:Bibcode
4804:Bibcode
4706:Bibcode
4664:Bibcode
4609:Bibcode
4537:Bibcode
4248:), is:
4033:Mirrors
202:Bilger
174:Bilger
146:source
123:History
73:and an
5328:
5151:
5105:
5027:
4924:
4899:
4865:
4822:
4627:
4579:
4314:
4289:< n
899:where
592:where
351:where
58:, and
5326:S2CID
5300:arXiv
5103:S2CID
4820:S2CID
4794:arXiv
4742:. 5.
4736:(PDF)
2409:and Ď
2401:and r
1838:and q
224:laser
29:light
5149:PMID
5025:PMID
4922:ISBN
4897:ISBN
4863:PMID
4625:PMID
4577:ISBN
4434:crit
4285:N (n
4275:)/(n
4082:- jk
3850:and
522:N =
207:2004
193:1996
179:1994
165:1993
151:1972
143:time
133:year
5318:doi
5296:132
5273:doi
5141:doi
5095:doi
5060:doi
5017:doi
4966:doi
4954:112
4855:doi
4812:doi
4790:109
4767:doi
4714:doi
4672:doi
4617:doi
4545:doi
4279:- n
4271:+ k
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4101:=
4055:TiO
4041:SiO
3991:cos
3104:is
2622:cos
2611:sin
2595:sin
2584:cos
2101:or
1923:is
1840:out
1258:cos
1182:cos
136:rms
5346::
5324:.
5316:.
5308:.
5294:.
5271:.
5261:44
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5238:^
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4469:.
4263:/n
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2098:,
1836:in
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2137:j
2129:R
2126:1
2121:=
2116:q
2113:1
2083:D
2080:+
2077:q
2074:C
2069:B
2066:+
2063:q
2060:A
2054:=
2051:q
2024:s
2020:D
2016:+
2011:n
2008:i
2004:q
1998:s
1994:C
1986:s
1982:B
1978:+
1973:n
1970:i
1966:q
1960:s
1956:A
1949:=
1944:t
1941:u
1938:o
1934:q
1910:)
1901:s
1897:D
1889:s
1885:C
1875:s
1871:B
1863:s
1859:A
1851:(
1814:1
1811:=
1807:|
1800:D
1795:C
1788:B
1783:A
1776:|
1745:1
1740:)
1729:r
1721:r
1714:(
1705:)
1698:D
1693:C
1686:B
1681:A
1674:(
1650:1
1645:)
1634:r
1626:r
1619:(
1609:1
1604:)
1598:2
1594:M
1585:1
1581:M
1576:(
1566:2
1561:)
1555:2
1551:M
1542:1
1538:M
1533:(
1523:3
1518:)
1512:2
1508:M
1499:1
1495:M
1490:(
1480:4
1475:)
1469:2
1465:M
1456:1
1452:M
1447:(
1442:=
1437:1
1432:)
1421:r
1413:r
1406:(
1401:=
1396:4
1391:)
1380:r
1372:r
1365:(
1338:1
1335:=
1331:|
1326:2
1322:M
1318:|
1314:=
1310:|
1305:1
1301:M
1297:|
1254:1
1244:2
1239:M
1235:R
1229:=
1224:y
1220:f
1216:=
1213:f
1174:2
1169:M
1165:R
1159:=
1154:x
1150:f
1146:=
1143:f
1131:M
1116:)
1109:1
1102:f
1099:1
1087:0
1082:1
1075:(
1064:2
1049:)
1042:1
1037:0
1030:d
1025:1
1018:(
1007:1
985:2
981:w
968:j
960:R
957:1
952:=
947:q
944:1
912:0
908:E
878:2
874:w
868:2
864:r
854:e
849:|
844:0
840:E
836:|
832:=
828:|
825:E
822:|
781:)
778:f
774:/
768:2
763:0
759:f
755:(
748:4
744:Q
740:A
735:=
730:f
726:/
722:1
718:S
687:T
682:2
678:Q
674:P
671:2
664:3
659:0
655:f
651:h
644:=
608:f
601:S
572:2
568:Q
564:P
557:3
553:f
549:h
543:=
538:f
531:S
503:l
500:d
470:l
464:d
451:r
439:2
436:1
431:=
422:A
362:A
333:L
319:A
108:2
42:(
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