Curvature radius and width

The beam has a spot size w: ${\displaystyle \left|E\right|=\left|E_{0}\right|e^{-{\frac {r^{2}}{w^{2}}}}}$,

where ${\displaystyle E_{0}}$ 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

${\displaystyle {\frac {1}{q}}={\frac {1}{R}}-j{\frac {\lambda }{\pi w^{2}}}}$.

The ring design uses a matrix M₁ = ${\displaystyle \left({\begin{matrix}1&d\\0&1\\\end{matrix}}\right)}$ for a straight section and M₂ = ${\displaystyle \left({\begin{matrix}1&0\\-{\frac {1}{f}}&1\\\end{matrix}}\right)}$ 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:

${\displaystyle f=f_{x}={\frac {R_{M}}{2}}\cdot \cos \theta }$,

for oblique incidence at angle θ, perpendicular to the plane:

${\displaystyle f=f_{y}={\frac {R_{M}}{2}}\cdot {\frac {1}{\cos \theta }}}$,

resulting in astigmatic beams.

The matrices have

${\displaystyle \left|M_{1}\right|=\left|M_{2}\right|=1}$.

A typical design of a rectangular ring has the following form:

${\displaystyle \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{4}=\left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}=\left(M_{1}\cdot M_{2}\right)_{4}\cdot \left(M_{1}\cdot M_{2}\right)_{3}\cdot \left(M_{1}\cdot M_{2}\right)_{2}\cdot \left(M_{1}\cdot M_{2}\right)_{1}\cdot \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}}$

${\displaystyle \left({\begin{matrix}A&B\\C&D\\\end{matrix}}\right)\cdot \left({\begin{matrix}r\\r'\\\end{matrix}}\right)_{1}}$

(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.^[14]

The requirement that the ray is to be closed is therefore ${\displaystyle \left|{\begin{matrix}A&B\\C&D\\\end{matrix}}\right|=1}$.

Propagation of complex curvature

The complex curvatures q_in and q_out in a section of a beam circuit with the section matrix ${\displaystyle \left({\begin{matrix}A_{s}&B_{s}\\C_{s}&D_{s}\\\end{matrix}}\right)}$ is

${\displaystyle q_{out}={\frac {A_{s}q_{in}+B_{s}}{C_{s}q_{in}+D_{s}}}}$. In particular, if the matrix above is the round-trip matrix, the q at that point is

${\displaystyle q={\frac {Aq+B}{Cq+D}}}$,

or

${\displaystyle {\frac {1}{q}}={\frac {1}{R}}-j{\frac {\lambda }{\pi w^{2}}}={\frac {D-A}{2B}}-j{\frac {\sqrt {1-({\frac {A+D}{2}})^{2}}}{B}}}$.

Note that it is necessary that ${\displaystyle \left|{\frac {A+D}{2}}\right|\leq 1}$

to have a real spot size (Stability Criterion). The width is generally less than 1 mm for small lasers, but it increases approximately with ${\displaystyle {\sqrt {L}}}$ . For calculation of beam positions for misaligned mirrors, see ^[15]

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^[14] is used to calculate polarization. Here, the column matrix

${\displaystyle \left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)}$

signifies the electric field components in-plane and off-plane. To study further the transition from planar rings to non-planar rings,^[16] 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

${\displaystyle M_{refl}=\left({\begin{matrix}r_{p}e^{j\chi _{p}}&0\\0&-r_{s}e^{j\chi _{s}}\\\end{matrix}}\right)}$. Also, if the reference planes change, one needs to refer the E-vector after reflection to the new planes with the rotation matrix

${\displaystyle M_{rot}=\left({\begin{matrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{matrix}}\right)}$.

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

${\displaystyle \left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)=\left(M_{refl_{4}}\right)\left(M_{rot_{4}}\right)............\left(M_{refl_{1}}\right)\left(M_{rot_{1}}\right)\left({\begin{matrix}E_{p}\\E_{s}\\\end{matrix}}\right)}$ (Note that the polarization at the end of the loop has to equal the polarization at the start). For small loss differences ${\displaystyle \delta =\delta _{p}-\delta _{s}=(1-r_{p})-(1-r_{s})}$ and small phase shift differences ${\displaystyle \chi =\chi _{p}-\chi _{s}}$, the solution for ${\displaystyle E_{p}/E_{s}}$ is

${\displaystyle {\frac {E_{p}}{E_{s}}}=\pm j{\sqrt {1-(\gamma /\theta )^{2}}}+\gamma /\theta }$, where ${\displaystyle \gamma ={\frac {1}{\sqrt {2}}}(\delta -j\chi )}$. If the dihedral angle θ is large enough, i.e. if

${\displaystyle \gamma /\theta <<1}$, the solution of this equation is simply ${\displaystyle E_{p}/E_{s}=\pm j}$, i.e. a definitely non-planar beam is (left-handed or right-handed) circularly (not elliptically) polarized. On the other hand, if ${\displaystyle \gamma /\theta >>1}$ (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 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.

2. If the fourth mirror is suspended such that it can rotate around a horizontal axis, the appearance of ${\displaystyle E_{p}}$ 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.

Lock-in and pulling

These are new phenomena in rings. The lock-in frequency f_L, 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_t, the actual signal frequency f is

${\displaystyle f=f_{t}{\sqrt {1-({\frac {f_{L}}{f_{t}}})^{2}}}}$. 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.

When the dependence on perimeter L is taken into account for large rings, the relative difference between theoretical output frequency f_t and actual output frequency f is inversely proportional to the fourth power of L:

${\displaystyle {\frac {f_{t}-f}{f_{t}}}\cong {\frac {1}{2}}({\frac {f_{L}}{f_{t}}})^{2}\propto {\frac {1}{L^{4}}}}$.

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^[6] 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 frequency of about 20 Hz.