## Abstract

Large ring lasers have exceeded the performance of navigational gyroscopes by several orders of magnitude and have become useful tools for geodesy. In order to apply them to tests in fundamental physics, remaining systematic errors have to be significantly reduced. We derive a modified expression for the Sagnac frequency of a square ring laser gyro under Earth rotation. The modifications include corrections for dispersion (of both the gain medium and the mirrors), for the Goos–Hänchen effect in the mirrors, and for refractive index of the gas filling the cavity. The corrections were measured and calculated for the $16\text{\hspace{0.17em}}{\mathrm{m}}^{2}$ Grossring laser located at the Geodetic Observatory Wettzell. The optical frequency and the free spectral range of this laser were measured, allowing unique determination of the longitudinal mode number, and measurement of the dispersion. Ultimately we find that the absolute scale factor of the gyroscope can be estimated to an accuracy of approximately 1 part in ${10}^{8}$.

© 2017 Optical Society of America

## 1. INTRODUCTION

Ring laser gyroscopes (RLGs) have become established as highly sensitive instruments for measuring rotation relative to an inertial frame. Large RLGs now have the resolution and stability to enable measurement of subtle aspects of Earth rotation kinematics [1–3]. Furthermore, they represent the only measurement technique in space geodesy that provides a direct access to the instantaneous axis of rotation of the Earth, thus providing direct access to polar motion. With a significant reduction of systematic measurement errors and a moderate improvement in sensitivity they will also allow to enter the regime of sub-diurnal polar motion, which is out of reach for the other geodetic techniques, such as very long baseline interferometry and the global navigational satellite systems. Apart from these practical goals there is a growing interest in exploring the possibility of measuring absolute Earth rotation rates to allow detection of relativistic precessions of the Earth’s inertial frame relative to the celestial frame defined by International Earth Rotation Service observations of quasars coordinated by the International Earth Rotation and Reference System Service [4]. The relativistic precessions are 9–10 orders of magnitude smaller than the (approximately constant) Earth rotation rate [4]. Making such absolute measurements will require careful elimination of systematic errors, since the relativistic effects on the rotation rate of the Earth appear as a tiny offset from the slow variations of the rotational velocity of the Earth. In an earlier paper [5] we described how backscatter-induced rotation-rate errors are generated, and how they can be corrected. For Earth rotation rate measurements, backscatter errors are of magnitude typically 1–10 parts per million (ppm), and variable in time. The residual systematic errors described in the present paper are smaller by typically 2 orders of magnitude, and much more stable in time. They only become relevant when the backscatter corrections have been made accurately. From physical considerations we expect that backscatter does not interact with the processes described in this paper, and we expect simple additivity of the rotation rate offsets. The usual Sagnac equation [6] for a ring laser gyro is

where ${f}_{s}$ is the Sagnac frequency (Hz), and $A,\mathrm{\Omega},\lambda $, and $P$ are, respectively, the vector area enclosed by the beam, the vector rotation rate (rad/s) relative to an inertial frame, the optical wavelength, and the length of the perimeter. In the special case of a square ring laser cavity, $A$ and $P$ are nominally related by $A={P}^{2}/16$. Also $P=N\lambda $ where $N$ is the longitudinal mode number. If we consider only the case where the RLG is aligned with the rotation axis, $\mathbf{A}\xb7\mathrm{\Omega}$ may be replaced with $A\mathrm{\Omega}$. Substituting all these back into Eq. (1), we get This suggests that for such a square RLG the scale factor ($N/4$) should take only discrete values, because $N$ must be exactly a whole number. Therefore the scale factor could be determined to ppb accuracy or better if the value of $N$ could be established. However, Eqs. (1) and (2) are only approximations, as we shall show. Our goal is to replace them with*correct*expressions for a square RLG.

#### A. Grossring RLG (G)

Our method is illustrated using the single axis “G” RLG located at the Geodetic Observatory Wettzell (Germany) [1–3], at latitude 49.145°. This instrument was designed and built for the specific purpose of monitoring Earth rotation. It is nominally 4 m square, and is in a local horizontal plane. The four mirrors are spherical, with a radius of curvature of 4 m (nominal, measured as 4.05 m). The cavity is very precisely engineered using the low thermal expansion material Zerodur. The laser operates on the 633 nm He–Ne transition. Under Earth rotation, the Sagnac frequency for G, from Eq. (1), is approximately 348.5 Hz.

Currently the cavity is filled with a gas mix consisting of $0.100\text{\hspace{0.17em}}\mathrm{mb}\text{\hspace{0.17em}}{}^{20}\mathrm{Ne}$, $0.098\text{\hspace{0.17em}}\mathrm{mb}\text{\hspace{0.17em}}{}^{22}\mathrm{Ne}$, and $9.682\text{\hspace{0.17em}}\mathrm{mb}\text{\hspace{0.17em}}{}^{4}\mathrm{He}$ (with pressures measured at a temperature of 285.5 K, using a MKS Baratron 626AX11MCD pressure gauge; accuracy 0.25%). The refractivity of the He may be calculated as $0.31830\times {10}^{-6}$ using data and algorithms given in Stone and Stejskal [7], and that of the Ne as $0.01254\times {10}^{-6}$ using data from [8]. Assuming additivity of the refractivities at low pressures, these give a value for the refractive index $n=1+0.3308\times {10}^{-6}$. (In principle $n$ has dispersion, but it is very small: a 1% change in wavelength generates only a 0.001% change in refractivity and consequently dispersion in $n$ is ignored.)

RLGs such as G use mirrors (“supermirrors”) characterized by very low optical scattering and reflection losses. Typically one of these mirrors is constructed with multiple $\lambda /4$ dielectric layers of alternating high and low refractive index, on a highly polished glass substrate. There are well-established methods for calculating the properties of such mirrors. For the present work we need to know the optical phase shifts associated with reflections from the mirrors. Appendix A gives details of the calculations.

Gain is provided by exciting a plasma in a small portion (2 cm) of the path (16 m). The plasma is energized by capacitive coupling of a radio-frequency driving voltage. The cavity optical losses are parameterized by the ringdown time, measured as 0.80 ms. The optical loss per circuit of the cavity is then 66.7 ppm and therefore for steady-state output beam power this is also the single-pass gain through the plasma tube. The dispersion associated with the spectral gain is estimated in Appendix B.

## 2. OPTICAL FREQUENCIES OF A NON-ROTATING RLG

We first consider the optical frequencies at which a ring laser cavity may oscillate, i.e., the resonant frequencies of the active cavity, for the moment ignoring the Sagnac effect. We assume that the Sagnac splitting of the resonant frequencies is symmetrical, and that the mean of the clockwise and counterclockwise resonant frequencies for a rotating cavity is the same as the resonant frequency of the non-rotating cavity. Resonance requires the total phase variation around the cavity at any instant to be a whole-number multiple of $2\pi $ rad. The total phase has several contributing terms:

- (ii)
**Gouy phase:**the Gouy phase shift occurs in the propagation of laser beams and causes the phase velocity within a narrow laser beam to be slightly larger than for a plane wave of the same frequency [9]. For a square cavity of side length $L$ and identical spherical mirrors of radius of curvature $R$, the total Gouy phase around the cavity for the ${\mathrm{TEM}}_{00}$ mode is (This is derived from equation 19.21 in [9], making use of symmetry in a square cavity with spherical mirrors and 45° angles of incidence.) The Gouy phase associated with a laser cavity is independent of the wavelength (and therefore frequency) and also the refractive index of the medium in the cavity. - (iii)
**Mirror reflection phase:**for an ideal Bragg stack mirror, at the design wavelength (or frequency) the reflected amplitude has a phase of $\pi $ relative to the incident amplitude, and after four reflections round a square cavity there should be zero net reflection phase change. Non-ideal layer thicknesses may cause phase anomalies at the design frequency, and furthermore the phase varies nearly linearly with optical frequency at frequencies near the design value [10]. We define ${\varphi}_{M}$ as the mean*departure*from phase change $\pi $ in reflection for the four mirrors, and it may be expressed as with $\mathrm{\Delta}f$ the deviation in frequency from the central design value. - (iv)
**Phase changes in the laser gain medium:**the spectral gain of the laser gain medium is accompanied by a frequency-dependent phase change ${\varphi}_{K}$. This varies approximately linearly within the few hundred megahertz near the center of the laser gain curve [11], so ${\varphi}_{K}$ can be written as with $\mathrm{\Delta}f$ as the frequency deviation from the gain curve peak. ${\varphi}_{K}$ is proportional to the single-pass fractional gain, which at equilibrium is equal to the round-trip cavity loss, which is in turn inversely proportional to the ringdown time. In practice, for a He–Ne laser system with approximately equal quantities of ${}^{20}\mathrm{Ne}$ and ${}^{22}\mathrm{Ne}$, the gain curve is closely symmetrical about the frequency of maximum gain. The laser operating frequency is always close to this peak gain and ${\varphi}_{K}$ has a zero-crossing very close to peak gain, so for present purposes it is accurate enough to take the term ${\varphi}_{K0}$ as zero within 0.0001 rad.

For longitudinal mode number $N$, the lasing condition for the cavity is

We express the path-length phase ${\varphi}_{P}=2\pi Pn{f}_{\text{opt}}/c$, where ${f}_{\text{opt}}$ is the optical frequency and $c$ is the vacuum speed of light. Then with some rearrangement the resonance condition for mode number $N$ becomes It is useful to define a*reduced optical frequency*${f}_{\text{opt}}^{*}$ as $cN/(nP)$, related to ${f}_{\text{opt}}$ by

## 3. EXPERIMENTAL DETERMINATION OF N FOR THE G RLG

The frequency spacing of longitudinal modes (or free spectral range) of the cavity, which we denote by $F$, is the optical frequency difference between modes with consecutive values of $N$. For compactness we combine the two frequency-dependent phase terms from the mirrors and gain medium, putting ${\varphi}_{D}=4{\varphi}_{M}+{\varphi}_{K}$. Then putting values $N+1$ and $N$ into Eq. (8) and taking the difference, we get

This is the beat frequency that may be measured by running the laser with different longitudinal mode numbers in the two circumferential directions (after correcting for the Sagnac frequency) [11], or running in multimode [12]. The mode spacing is*independent*of $N$ provided that we remain within the region of linear dispersion for ${\varphi}_{D}$. It is useful to define the

*restored mode spacing*${F}^{*}$ by i.e., starting with the observable mode spacing $F$ a correction is applied for the frequency pulling caused by dispersion. With the above definitions of ${f}_{\text{opt}}^{*}$ and ${F}^{*}$, the longitudinal mode number is then given simply by

#### A. Experimental Determination of $\mathsf{N}$ for the G Ring Laser

G has a nominal perimeter of 16 m, so we expect a mode number of approximately $25.3\times {10}^{6}$. In September 2015 its optical frequency was measured by comparison with a frequency comb system (Menlo FC1500-250-WG), using the same experimental setup as in [13]. Measurements were made over several hours, with a resolution of 0.1 MHz and repeatability of around 0.2 MHz, limited by the short-term drift in the laser. The comb was locked to a hydrogen maser frequency standard. The measured mean frequency was $473612720.56\pm 0.15\text{\hspace{0.17em}}\mathrm{MHz}$. Simultaneous with the optical frequency measurement, the mode spacing was measured by running the laser at a beam power where weak extra longitudinal modes were running. These were spaced six mode numbers either side of the principal mode. (Such apparently large frequency separations are typical of these large ring lasers run in this manner [12,14].) The extra modes, beating against the principal mode, generate a small modulation of the beam power at (in this case) 6 times the mode spacing. The modulation was measured by monitoring one beam with a fast photodetector, taking the output to a VHF radio receiver, and measuring the frequency by standard RF techniques. These measurements were also locked to the same hydrogen maser standard as used in the optical frequency determination. In this way the mode spacing $F$ was measured as 18.734 380 691 MHz. This measured value allows a preliminary improved calculation of the perimeter as $P=c/F$ giving a value 16.0023 m, from which the mean side length is 4.0006 m. Using the measured radius of curvature of the mirrors ($4.05\pm 0.04\text{\hspace{0.17em}}\mathrm{m}$), Eq. (4) gives total Gouy phase ${\varphi}_{G}=6.487\pm 0.045\text{\hspace{0.17em}}\mathrm{rad}$. For the reflection phase shifts of four mirrors, our best estimate of ${\varphi}_{D0}$ is $0.045\pm 0.012\text{\hspace{0.17em}}\mathrm{rad}$ (Appendix A). The reduced optical frequency is then calculated from Eq. (9) as

The uncertainty is principally from the optical frequency measurement, with nearly insignificant contribution from the terms of Eq. (9). Our best initial estimate for the factor $\partial {\varphi}_{D}/\partial f$ in Eq. (11) is the sum of the dispersion due to four mirrors ($-5.595\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$: see Appendix A) and to the gain medium with 66.7 ppm single-pass gain ($-2.575\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$: see Appendix B) giving a total $-8.170\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$. Using this value in Eq. (11) gives a restored mode spacing: The uncertainty comes principally from uncertainty in the preliminary estimate of the mirror dispersion (Appendix A). Substituting values for ${f}_{\text{opt}}^{*}$ and ${F}^{*}$ into Eq. (12), we estimate $N=25280397.23\pm 0.28$. The mode number $N$ is required to be an integer, and on probabilistic grounds the experimental result very strongly favors the nearest integer value 25 280 397. To proceed further, we round the experimental value of $N$ to this value, and take it as*exact*. Inserting into Eq. (12) we can then recalculate a value for ${F}^{*}$ with much lower uncertainty: From Eq. (11), this is equal to $c/nP$, so a more accurate value for $P$ follows: Finally the square-bracketed dispersion correction factor on the right-hand side of Eq. (11) recurs in later expressions so it is useful to denote it as $D$: We can evaluate $D$ accurately from $D={F}^{*}/F$, as $1+2.5266\times {10}^{-7}$.

## 4. RELATIONSHIP BETWEEN AREA AND PERIMETER

#### A. Lateral Beam Shifts in Reflection: The Goos–Hänchen (GH) Effect

For reflections at structured dielectric surfaces such as the supermirrors in G, there are lateral shifts of the reflected beams [15]. A rough physical explanation invokes the incident light penetrating into the surface to some extent, so the effective surface of reflection is below the geometric surface. The magnitude $\gamma $ of the shift, measured perpendicular to the beam direction, is given by

where $\theta $ is the angle of incidence and $\varphi $ is the plane-wave reflection phase [15]. (Note that $\varphi $ is polarization-dependent.) Also, on a curved mirror there should be a small associated change in the direction of the reflected beam, compared to specular reflection at the outer surface. We assume that this is by an (unknown) angle $\alpha $. For a square cavity with curved mirrors the GH effect causes displacements of the beam positions on the mirrors. Figure 1 shows a representative case with greatly exaggerated GH shifts. The dashed line represents the alignment of the beam in the*absence*of a GH shift. The resteered beam, represented by the red line, is propagating clockwise. The symbols $a$ and $b$ represent the displacements at arrival and departure, respectively: a positive sign indicates the resteered beam at that point is outside the dashed line. (In the particular case drawn, $a$ is positive and $b$ is negative.) Detailed analysis shows that the value of $\alpha $ controls the orientation of the truncated square defined by the red line in the figure. (The figure is drawn for the case $\alpha =0$, and it can be seen that for this case the counterpropagating beams travel very slightly different paths.) It is clear that $a+b=\gamma $ in every case, to first order in $\gamma $, irrespective of $\alpha $. The area enclosed by the beam of the ring laser is determined by the average outward displacement of the beams relative to the dashed line in the figure. This is just the average of the quantities $a$ and $b$, and equals $\gamma /2$.

Then the total area enclosed by the beam is

Because $L=P/4$ for a square cavity, in terms of the perimeter we get An important point here is that the phase length of each side of the RLG is not changed by the GH effect. This may be seen by considering the sub-reflections from the layers of the mirror. The layers are designed such that the reflections are phase-coherent. The center of the departing beam in Fig. 1 is at the point marked 2 on the left-hand mirror, but it has the same phase as if the center were at the point marked 1. The path from 1 to the point marked 3 on the right-hand mirror has the same length (to first order in $\gamma $) as the path for zero GH shift.#### B. Area Enclosed by the Beam of the G Ring Laser

Given the structure of the mirror dielectric layers, the mirror reflection phase derivative with angle can be evaluated numerically, and this is done for the G mirrors in Appendix A. The calculated GH shift $\gamma $ is 0.1759 μm. Then the area correction factor $1+8\gamma /P$ in Eq. (20) is $1+8.78\times {10}^{-8}$. In order for the area of G to be determined by Eq. (20), the beam must describe a sufficiently accurate square. The planarity errors of G are below 0.05 mm, and the cosines of the associated dihedral angles are within 2 parts in ${10}^{10}$ of unity. The mirrors are mounted on the ends of four arms contacted to the top of, and radiating from a 4 m diameter Zerodur disc. The angles of the arms are accurate to better than 10 arc sec. The largest geometric uncertainty is in the uniformity of the arm ends from the center, estimated at $\pm 0.15\text{\hspace{0.17em}}\mathrm{mm}$. The associated perturbations to both area and perimeter are first order in this error, but the first-order perturbations cancel in Eq. (20), leaving second-order perturbations of around 3 ppb (determined by Monte-Carlo modeling).

## 5. SPLITTING OF THE RESONANT FREQUENCIES FOR A ROTATING CAVITY: SAGNAC EFFECT

We now finally include rotation effects. In deriving an expression for the Sagnac effect, the usual approach is to derive the *difference in propagation time* between beams traversing the same path segment in opposite directions, in a rotating frame [16]. For one edge of a rotating polygon (defined by vertices ${V}_{1}$, ${V}_{2}$) the rotation causes a transit time perturbation

*independent of the medium in the segment of the cavity*, provided that the medium co-moves with the cavity, and the beam propagation in the medium is isotropic. In particular, a gas medium with non-zero refractivity in the cavity satisfies these conditions, provided that in the frame of the cavity the gas does not circulate. The expression Eq. (21) extends by simple summation to a complete closed path, and $A$ becomes the total area enclosed by the beam. The time perturbations generate extra phase terms, which, in an extension of Eq. (8), will generate different optical frequencies ${f}_{\text{opt}1}$ and ${f}_{\text{opt}2}$ in the two directions. Explicitly, the phase terms are

## 6. DISCUSSION AND CONCLUSION

Equation (26) has the same form as Eq. (2) but four correction factors are included. These, together with their numerical values for the G laser, are listed in Table 1. They are all close to unity but nevertheless they combine to give a very significant change to the Sagnac frequency for geodesy applications.

The last listed factor is the only one that depends on $N$, and only very weakly. If on some occasion the laser operates on a mode number adjacent to its present value, the Sagnac frequency will be changed but the correction factor is almost unaltered. The overall result of the corrections is an expression for the Sagnac frequency:

The accuracy in scale factor of G is limited not by the uncertainties in these corrections, but by geometric shape uncertainty, as described earlier. This is around 4 ppb when tangential as well as radial mirror position uncertainties are included. For G, running under the conditions described in this paper, Fig. 2(a) shows the raw measured Sagnac frequency, plotted versus time between mJD 57421 and 57445 (3–27 February 2016). The graph shows much structure: this is from geophysical effects and at this stage uncorrected backscatter perturbations. Figure 2(b) shows the same dataset (with 4 h averaging per data point), after backscatter corrections [5], and corrections for the following geophysical effects: diurnal polar motions, Chandler and annual wobbles, instantaneous Earth orientation, and local tidally induced tilts. The length-of-day anomaly has not been corrected. After correction, the Sagnac frequency has a mean value of 348.517015 Hz with an uncertainty estimated as 5.1 μHz, mostly due to residual uncertainty in the largest correction, which is that for backscatter. Variations in the length of day contribute to the actual rotation rate of the Earth at the level corresponding to 5–8 μHz for the Sagnac frequency of G, very close to the current resolution. From published International Earth Rotation Service data, the mean length-of-day anomaly during the period of the above data was 1.617 ms. The long-term mean length of one stellar day is 86164.0989 s so during the period of the measurement the mean stellar day length was 86164.1005 s. The mean Earth rotation rate was then $2\pi /86164.1005=7.2921150\times {10}^{-5}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{s}$. Recalling that for G, the relevant rotation rate is not $\mathrm{\Omega}$, but $\mathrm{\Omega}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta $ where $\theta $ is the misalignment from the Earth rotation axis (= effective co-latitude), then inserting values into Eq. (22) gives $\mathrm{cos}\text{\hspace{0.17em}}\theta =0.75621784\pm 0.00000004$ from which $\theta =40.868104\pm 0.000003\text{\hspace{0.17em}}\mathrm{deg}$. This is consistent with what we know of the co-latitude of G from conventional surveying and leveling measurements, but here we have at least an order of magnitude smaller uncertainty. If a high-accuracy absolute measurement of Earth rotation were desired using a RLG, there would be benefits in building a RLG larger than G. The correction factors of 1, with the exception of the refractive index correction, become closer to unity with increased size, and their uncertainties reduce. Although backscatter has not been dealt with in this paper, the susceptibility of the RLG to backscatter decreases very strongly as the laser gets larger [11]. We estimate a square RLG with side length $>10\text{\hspace{0.17em}}\mathrm{m}$ might have sufficient resolution to detect the general-relativistic Lense–Thirring precession [4]. For dimensional stability, rather than the monolithic low-thermal-expansion approach used for G, a method based on interferometric monitoring and active control would probably be necessary [18,19]. However, for any RLG with large misalignment to the Earth rotation axis, the uncertainty in determining the co-latitude $\theta $ would be likely to be a dominant overall uncertainty. The only feasible solution to this difficulty that we are aware of would be to tilt the RLG to make $\theta $ very close to zero, and then $\mathrm{cos}\text{\hspace{0.17em}}\theta =1$ with errors reduced to second order in $\theta $. From an engineering point of view, this would be more conveniently done for a RLG at high north or south latitude. Indeed the possibility of a large RLG at the south pole has been suggested previously [20], and this location has the advantage of maximizing the Lense–Thirring effect. On the other hand, a consequence of this approach would be that much of the structure apparent in Fig. 2(b), which results from variation in $\theta $ due to polar motion, and which is of interest to the geodetic community, would no longer be visible or measurable. There are a few large RLGs that opt for an equilateral triangular rather than square shape [21]. For a given perimeter a triangle encloses a smaller area than a square, but in compensation it has generally simpler engineering constraints. It is automatically planar, and ensuring that three sides are equal in length to sufficient accuracy automatically determines the corner angles. The derivation of scale factor presented in this paper for a square RLG is slightly altered for a triangle, but the general approach remains the same. In particular, wherever the mode number $N$ occurs here, for a triangle it must be changed to $N+\frac{1}{2}$, because of the odd number of phase-inverting reflections in a round-trip of the beam.## APPENDIX A: MODELING OF MIRROR PHASE PROPERTIES

The mirrors of the G laser are of the conventional Bragg stack design, with alternating layers of high and low refractive index materials, tantalum pentoxide, and silica, respectively, of nominal phase thickness $\lambda /4$ at 45° angle of incidence. The stack begins and ends with the high refractive index material and has 35 layers. In addition, there is a topmost 36th layer of silica of thickness $\lambda /2$.

The mirror manufacturer (Research Electro-Optics) has provided us with appropriate nominal values for the layer refractive indices, of 2.11 and 1.47. We characterized the dispersion performance using the well-established method [10] of a “transfer matrix” calculation that recursively calculates the complex (amplitude) reflection coefficient, starting from only the substrate, then this substrate looked at through the deepest layer, then the substrate and first layer looked at through the next layer, and so on. We assumed zero losses in the dielectric materials. We refer to this as the *nominal model*. The computed dispersion for the nominal model, for s-polarization, is $-1.3987\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$.

We acknowledge that the actual mirror design (refractive indices and layer thicknesses) is expected to differ slightly from the nominal model, and the computed dispersion will therefore be in error. However, the procedure detailed in Section 3 of the paper requires only that the computed dispersion be close enough to the true value to allow an unambiguous estimate of the G cavity mode number. The computed dispersion is required to be within approximately 10% of its true value for mirror designs close to the nominal. Using the transfer matrix calculations, in turn it can be shown that refractive index errors must be $<0.1$ or layer thickness errors $<6\%$. But if these errors were exceeded it would have such large effects on the mirror transmission that the errors could not go unnoticed. Therefore we consider the computed dispersions from the nominal model to be good enough initial values.

In addition to the model calculations we measured several properties of a set of mirrors from the same manufacturer and with the same specifications as the G mirrors. The measured properties were the phase differences between reflections of $p$ and $s$ polarizations, $({\varphi}_{p}-{\varphi}_{s})$, at 45° incidence angle, and also a slightly smaller angle $\approx 43\xb0$. (These phase differences are readily measured by using a linearly polarized incident beam, axially rotated so it is composed of equal in-phase $p$ and $s$ amplitudes, and testing the reflected beam for elliptical polarization. The extra measurement is necessary to establish the sign of the phase difference at 45°.)

If the layer thicknesses were correct, then $({\varphi}_{p}-{\varphi}_{s})$ at 45° would be zero, independent of refractive index values. In practice a non-zero mean value of 0.0117 rad was measured. Computations with the transfer matrix model show that as the layer thicknesses are varied, ${\varphi}_{s}$ and $({\varphi}_{p}-{\varphi}_{s})$ track each other closely and ${\varphi}_{s}$ can be estimated as $0.96({\varphi}_{p}-{\varphi}_{s})$, with a maximum error of less than 25% in the worst case. Therefore we can estimate an approximate value for the mean phase shift ${\varphi}_{s}$ of the mirror reflections, of $0.011\pm 0.003\text{\hspace{0.17em}}\mathrm{rad}$ per mirror. This value is used in Section 3, in the calculation of the reduced optical frequency.

Finally, the reflection phase derivative with respect to the angle of incidence (angular dispersion), required to calculate the Goos–Hänchen shift, may also be evaluated from the nominal model: $d{\varphi}_{s}/d\theta =-1.655$, at 45°. Like the spectral dispersion, this value is also subject to the uncertainty of the nominal model. However a better estimate can be derived, using the following argument: after the assignment of an exact mode number for G (Section 3), the total dispersion (gain medium together with mirrors) can be recalculated with greatly reduced uncertainty, from Eq. (17) using the accurate value for D. The result is $-8.474\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$. Subtracting off the gain medium dispersion (Appendix B) leaves $-5.904\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$ for four mirrors, which is 5.5% larger than the spectral dispersion from the nominal model. From trials using the transfer matrix method, in the “neighborhood” of the nominal model there is a very high correlation between the spectral dispersion and the angular dispersion of a mirror. Therefore a better estimate of angular dispersion can be made by increasing the nominal model value by the same 5.5%, to get $d{\varphi}_{s}/d\theta =-1.746$. This is the value that we have used to calculate the GH shift.

## APPENDIX B: GAIN MEDIUM DISPERSION

We calculate the spectral gain (the “gain curve”) of the G laser using the method detailed in [5], including the same homogeneous and inhomogeneous broadenings. The plasma temperature is estimated as 355 K, from auxiliary thermometry experiments involving power dissipation in and heat flow from the plasma. For this temperature and the gas composition and pressures specified in Section 1, the homogeneous broadening is estimated as 840 MHz. (All broadenings are expressed as full width at half-maximum.) The inhomogeneous broadening is dominated by thermal Doppler broadening (1420 MHz and 1360 MHz for the ${}^{20}\mathrm{Ne}$ and ${}^{22}\mathrm{Ne}$ isotopes respectively), but with a significant contribution from the 890 MHz isotopic separation of the two Ne isotopes.

Our method for calculating gain curves follows the treatment described by Siegman [9]. It involves a full numerical integration of Siegman’s equations 30.4–30.6. The gain at a chosen frequency within the line profile is attributable to a weighted summation of Lorentzian profiles. Each of these has width given by the homogeneous broadening, and they are distributed in accordance with the inhomogeneous broadening profile. An unnormalized numerical value for the gain at the frequency of interest is obtained by summing the gain contributions of the nearby Lorentzians.

Then a closely similar numerical integration sums the reactances of the Lorentzians, giving an unnormalized numerical value for the total phase change. This second integration exploits the simple relationship between the gain and phase change for a Lorentzian, as detailed in section 2.4 of [9]. The ratio of the unnormalized gain and reactance values gives the *phase shift of the gain medium per unit fractional gain* at the optical frequency of interest.

The actual fractional gain $g$ at the laser operating frequency may be calculated from the cavity ringdown time $\tau $ as $g={(F\tau )}^{-1}$ where $F$ is the longitudinal mode spacing, as in Section 3 of the main text. We normalize the previously unnormalized gain from the first integration to match this gain at the operating frequency. This then results in the actual gain at any other chosen frequency. Multiplying this by the phase shift per unit gain then gives the phase shift of the gain medium at that frequency.

The effect of gain saturation is included in the calculation by multiplying the inhomogeneous gain distribution by a saturation factor as in equations 30.4–30.6 of Siegman [9]. From beam power measurements we estimate the intracavity beam power of G is around $20\pm 5\%$ of the saturation intensity of the gain medium, using saturation intensities (slightly extrapolated) from [22].

The final result for G is a dispersive phase shift of $2.575\times {10}^{-8}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{MHz}$ in the vicinity of the operating frequency. The uncertainty in this is estimated to be $\approx 5\%$.

## Funding

Deutsche Forschungsgemeinschaft (DFG) (SCHR 645/6-1); Royal Society of New Zealand (10-UOC-40).

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