Identification and correction of Sagnac frequency variations: an implementation for the GINGERINO data analysis

Ring laser gyroscopes are top sensitivity inertial sensors used in the measurement of angular rotation rates. It is well known that the response of such remarkable instruments can in principle access the very low frequency band, but the occurrence of nonlinear effects in the laser dynamics imposes severe limitations in terms of sensitivity and stability. We report here general relationships aimed at evaluating corrections able to effectively account for nonlinear laser dynamics. The so-derived corrections are applied to analyse thirty days of continuous operation of the large area ring laser gyroscope GINGERINO leading to duly reconstruct the Sagnac frequency $\omega_S$. The analysis shows that, on the average, the evaluated corrections affect the measurement of the Earth rotation rate $\Omega_E$ at the level of 1 part in $1.5\times10^{3}$. Among the identified corrections, the null shift term $\omega_{NS}$ is the dominant one. It turns out proportional to the optical losses $\mu$ of the ring cavity, which are changing in time at the level of $10\%$ within the considered period of thirty days. The time behaviour is reconstructed based on available signals (interferogram and mono-beam intensities), and the Allan deviation of the estimated $\Omega_E$ shows a remarkable long term stability, leading to a sensitivity better than $10^{-10}$rad/s with more than $10$s of integration time, and approaching $(8.5\pm 0.5)\times 10^{-12}$rad/s with $4.5\times10^{5}$s of integration time.


Introduction
Ring laser gyroscopes (RLGs) are inertial sensors based on the Sagnac effect [1,2,3]. They are largely used for inertial navigation, and applications in geodesy, geophysics and even for General Relativity tests are foreseen [4]. Since 2011 we are studying the feasibility of the test of Lense-Thirring dragging of the rotating Earth at the level of 1% with an array of large frame RLGs [5,6,7]. For that purpose it is necessary to push the relative accuracy of the Earth rotation rate Ω E measurement in the range from 1 part in 10 9 up to 1 part in 10 12 .
RLG consists of a laser with a cavity comprising of three or four mirrors, rigidly attached to a frame; large frame RLGs are utilised to measure the Earth rotation rate, being attached to the Earth crust. Because of the Sagnac effect, the two counter-propagating cavity modes have slightly different frequency, and the beat note of the two beams is proportional to the angular rotation rate of the ring cavity. Large frame RLGs are the most sensitive instruments for inertial angular rotation measurements. The Sagnac frequency of a RLG is in fact proportional to the component of the angular velocity − → Ω felt by the instrument along the normal to the cavity plane: where A is the area of the ring cavity, L is its perimeter, λ the wavelength of the light, and θ is the angle between the area versor of the ring and − → Ω . For RLGs lying horizontally (area versor vertical) θ is the co-latitude angle, while for RLGs aligned at the maximum Sagnac signal θ = 0. Eq. 1 defines the scale factor S, which is a function of the geometry and of λ, quantities than can be measured with a very high accuracy.
Further to sensitivity, other key points of such instruments rely on their broad bandwidth, which can span from kHz down to DC, and their very large dynamical range. In fact the same device can record microseismic events and high magnitude nearby earthquakes [8], being the signal based on the measurement of the beat note between the two counter-propagating beams. It has been proven that large size RLGs, equipped with state of the art mirrors, can reach the relative precision of 3 parts in 10 9 with one day of integration time, in the measurement of Ω E [1]. If shot noise limited, sensitivity level scales with the square of the size of the ring cavity [1]. However, other limitations can affect the measurement. It is well known that the nonlinear laser dynamics plays a role in determining the RLG signal. We have recently developed a model to reconstruct the Sagnac frequency ω S starting from the measured beat note ω m and using the mono-beam signals 1 . The Sagnac angular frequency can be expressed as the linear sum of six contributions. The first one, ω S0 , is dominant, it takes into account the so called back-scatter noise, as already discussed in [9].
In this paper we complete the analysis, providing the necessary information to evaluate the other terms. This analysis has been applied to thirty days of continuous operation of GINGERINO and the results are discussed. The so called null shift, ω N S , is the dominant contribution after ω S0 . The main originality of this paper is the direct reconstruction of this term for RLG signals. It demonstrates that, differently from what assumed in previous approaches, the null shift affects not only accuracy, but also sensitivity of the apparatus, since it is linked to the optical losses µ, which depend on time.

Analysis
The present analysis starts from the model of the laser dynamics recently developed [9], based on the Aronowitz model of RLG [10,11,12], which describes the laser dynamics through several dimensionless parameters (Lamb parameters). The polarization of the laser plasma is described up to the third order expansion in power of the field [10,12]. In the case of large frame RLG, to avoid mode competition, two Neon isotopes, 20 N e and 22 N e, are utilised, and the laser is set close to threshold to guarantee single-mode operation. This particular choice allows for some simplifications: the Lamb parameters of cross-saturation θ 12 and θ 21 can be neglected, and we can assume the self-saturation terms are equal each other, i.e., β 1 = β 2 = β. The model provides two analytical expressions for the Sagnac angular frequency: a non-linear relationship (eq. 8 of ref. [9]), which connects all the laser parameters with each other, and an approximate decomposition of ω S as a linear sum of six terms, which is the collection of the first and second order expansion as a function of the two main terms: In the present analysis only the first order terms of the Sagnac angular frequency are considered: ω N S1 = ω N S , ω K1 = ω K . A preliminary study has shown that the second order terms can be neglected, since too small compared with the present sensitivity of our prototype. The first term ω S0 does not require knowledge of the laser dynamics, and has been already illustrated in the previous paper [9]. Here we will report the detailed expressions to evaluate ω N S and ω K .

Implementation of the Model
To explicitly evaluate ω N S and ω K , the mono-beam signals are exploited [11,12,13]. We make use of the DC components, P H1 and P H2, of the monobeam signals, their AC components at the beat frequency ω m /2π, I S1 and I S2 , and the relative phase between the two AC components. Dissipative processes (like diffusion or absorption from the mirrors) can produce non reciprocal losses between the two counter-propagating beams, so that two distinct loss parameters µ 1 = µ 2 must be used. Without loss of generality, it is possible to take µ 1 = µ and µ 2 = µ + δµ, where µ represents the reciprocal losses term and δµ the non reciprocal ones. The plasma dispersion function depends on several parameters: gas pressure, atomic characteristics of the two isotopes, line width of the excited atoms, resonance frequency of the two counter propagating modes, area of the beam profile at the discharge, and temperature of the plasma.
Assuming that β 1 = β 2 = β (a rough evaluation reported in [11] gives β 1 −β 2 10 −14 ), combining the information of the mono-beam DC signals P H1 and P H2 with the plasma dispersion function, the gains of the two beams Gain1 and Gain2 are evaluated. The non reciprocal loss term δµ is hence determined.
Using other relationships reported in the literature [13] it is possible to write ω N S and ω K as a function of the known quantities ω m , P H1, P H2, I S1 , I S2 , , and µ. µ can be evaluated off-line with 1% precision by measuring the ring down time of the cavity. The analytical calculation shows that µ is a multiplicative factor in ω N S , which accordingly can be evaluated by means of a best fit. The term ω N S depends on µ, δµ, PH1, PH2, I S1 and I S2 , and decreases linearly for increasing perimeter length L. It exhibits also a small modulation as a function of , according to: The best case is when δµ = 0, leading, with the parameters of GINGERINO, to ω N S 10µrad/s, which gives ω N S ∼ 2mrad assuming µ = 2 × 10 −4 . In such conditions, the amplitude of the modulation induced by the cos(2 ) term is in the nHz range. In this best case the null shift term is about 1 part in 10 6 of Ω E , but its fluctuations are no larger than 1 part in 10 11 . A similar evaluation on the presently available data of GINGERINO leads to δµ = 0.02µ, ω N S ∼ 1rad/s, with a modulation due to the cos(2 ) term of 10µrad/s. The evaluation of ω K requires some further assumptions. ω K oscillates at the Sagnac frequency, containing sin(ω S t) and cos(ω S t) terms. However, due to non-linear dynamics, the long time average of such oscillating terms can deviate from zero. Therefore, we can express ω K as the product of a small parameter M K with a given function of known parameters. The value of M K and its very slow variations can be found by a best fit procedure. In the appendix, a Mathematica notebook is reported showing the details of the symbolic calculations leading to ω N S and ω K functions, and the estimation of the required parameters. We note that, according to our model, ω K can be written as the ratio of two polynomials where the parameter µ enters in one term, preventing direct evaluation of such a parameter through a simple fit procedure. However, we have checked that, with typical µ values of 100 − 1ppm, variations in calculated results are not significant. In the following, we report the calculations for a general RLG dynamics, and their implementation for the specific case of GINGERINO.

The selected playground and the first step of the analysis
Thirty days of continuous operation between June 16 and July 15 2018 have been selected for this test. After June 21, heavy operations took place in the underground laboratories where GINGERINO is placed for the building of another experiment. Disturbances induced in GINGERINO operation are well visible in the second half of the data stream. We included also these data to show that, even in non ideal conditions, very high sensitivity and stability can be obtained. The data are acquired with our DAQ system at 5kHz sampling rate [14]. The whole set of data has been analysed on a hourly base, and the terms ω m , ω N S (with µ = 1), ω ξ (with ξ = 1), and ω K (with M K = 1 and µ = 10 −4 ) evaluated and stored for offline analysis. In the calculation, average losses µ = 10 −4 are used to evaluate ω K , since it has been checked that the dependence on µ is negligible at the present sensitivity limit. The analysis, based on the Hilbert transform of interferograms and monobeam intensities, leads to the beat note frequency ω m /2π, the amplitudes of the mono beams at the beat frequency I S1 , I S2 , and the relative phase . In order to avoid spurious oscillations due to the boundaries between contiguous hours in the hourly-based analysis, for each hour 6 minutes of data are added at the beginning and at the end; these extra samples are removed after the analysis (overlap-save method). The analysed data are stored after decimation at 20 and 0.1Hz; the decimation is implemented via the standard Matlab function decimate, which applies 8th order Chebyshev Type I lowpass filter with cutoff frequency 0.8 × (F s/2), where F s is 20 or 0.1Hz.

The second step of the offline procedure
A second routine collects different days for longer analysis. The first operation is to identify the portions of data which are not at normal operation, typically less than 5 % of the data are removed. To this aim, we make use of the fringe contrast of the interferogram acquired at 20Hz rate. The choice of such a sampling rate minimizes the discarded sample. This operation associates the flag 1 to the good samples and the flag 0 to the bad ones; from this routine we get the mask to select data decimated at lower frequency.
In the present analysis, decimated at half an hour rate, 90% of data are selected. Both the intensity of the mono-beam signal (P H2), that is not used for the feedback control of the laser operation, and of the Discharge Monitor (DM , i.e. a photodiode looking at the laser discharge) change with time. This is a clear indication that µ changes with time; in particular, P H2 indicates the presence of non reciprocal losses δµ, while DM is proportional to the total excited atoms, and therefore to the total losses. DM and P H2 signals allow us to model the change of losses as a function of time through two form factors,

Reconstruction of the Sagnac frequency by a linear regression model
The vectors ω N S , ω N S × F F DM , ω N S × F F P H2 , ω ξ , and ω K are collected, and the linear regression method is utilised to determine the unknown parameters µ, M K and ξ, and evaluate ω S accordingly. Let us remind that ω S0 accounts for back scatter noise, and an additional term ω ξ has been implemented to account for inaccuracies on the measured quantities I S1,2 , P H 1,2 and [9]. In summary, in the procedure µ and M K are physical quantities which could be evaluated independently, while ξ compensates noises in the measured quantities. Fig. 1 shows data before (top panel) and after (bottom panel) the application of this procedure. The dispersion across the mean value is clearly suppressed using ω S , meaning that the new regressive analysis, which includes the terms ω N S and ω K , duly accounts for it. The Matlab function lm2 has been used to perform linear regression. Results indicate that the p-value, which tests the null hypothesis, is approximately zero for all vectors excluding  The first step of the analysis evaluates ω S0 , which accounts for back-scatter noise; ω ξ has been developed to further improve back-scatter cancellation, taking into account inaccuracies in the signals P H1, P H2, I S1,2 and .
the one related to ω K , whose p-value is 0.038. On the other hand, the relative error in the estimation of the M K term involved in the ω K expression is around 50%, whereas it amounts to 1 − 12% for all other vectors. 2 Fig. 2 shows the contribution of the laser systematics. A dominant role is played by ω N S , which produces a slowly variable level ranging between 180 − 200mHz, with standard deviation 2.3mHz. ω ξ gives a small correction with mean value −5.7mHz and standard deviation 2.3mHz. These two terms affect residuals, whereas ω K is smaller by more than a factor 10; it has a mean value compatible with zero and standard deviation 0.05mHz, see Fig. 3. Fig. 4 shows the evaluated losses. During the 30 day period considered, a variation of 12% is observed in µ, with a visible trend towards higher losses for increasing time.   ω m /2π, of the ω S0 /2π reconstructed in the first stage of our approach and that of the finally determined ω S /2π. It should be noticed that the distribution of ω m is highly non-Gaussian while that of both reconstructed frequencies becomes close to Gaussian, demonstrating that the nonlinear terms of dynamics are correctly accounted for. Moreover, ω S is shifted towards lower frequencies as a consequence of the null shift term ω N S . Fig. 6 reports the modified Allan deviation, expressed in angular rotation rate. We observe that, after the application of the procedure (red line), the variance is decreasing with the integration time up to more than two days. This suggests that with our procedure we are effectively correcting the long term laser dynamic effects. It is well known that RLGs are sensitive to Chandler Wobbler effect, which is typically below 100nrad. We have checked that the level reached in the long term stability is above the limit imposed by the Chandler Wobbler effect. The so far obtained long term instability is about 10 times larger than the scale factor changes induced by temperature variations considering 5 × 10 −6 / o C the thermal expansion coefficient of granite, the material used for the RLG frame, and 0.02 o C as typical temperature variation in the 30 day period. The obtained modified Allan deviation represents a fair improvement compared to the analysis in [14], carried out by Figure 6: Modified Allan deviation of the measured angular velocity Ω from the beat note (ω m ), ω S0 and evaluated ω S (blue, green and red lines, respectively), relative to the mean value, expressed in angular velocity [rad/s].
using available signals on a pure phenomenological basis, for instance the long time behaviour was cancelled using the DM signal.

Discussion and Conclusions
The Sagnac frequency ω S of the GINGERINO RLG has been evaluated taking into account the laser dynamics. The model leads to a linear sum of several terms, which can be determined with the available signals of beat frequency and mono-beams. The so-called back-scatter noise, which has been so far considered the most severe limitation of RLG, is accounted for by terms ω S0 and ω ξ . It turns out that the null shift term ω N S is the second dominant contribution. It depends linearly on cavity losses µ; this term definitely affects accuracy of the apparatus, since in the best case, if the losses are constant, it represents a non-negligible DC contribution. Furthermore, it is likely that losses are not constant with time, hence ω N S affects stability. If a long term stability of 1 part in 10 9 is required for the Ω E measurement, the requirement for the stability of µ with time, or equivalently for the relative accuracy in the evaluation of µ, is ω N S ω S × 10 9 . According to the model, it is straightforward to see that the null shift level is minimised when δµ 0. Changes of µ at the level of 10% are evident in GIN-GERINO. In order to follow those changes we have utilised the DC mono-beam signal of the uncontrolled beam, that is the signal which is not used in feedback loops controlling the laser operation (beam 2 in our case), and the discharge monitor DM , also called gain monitor. ω S is evaluated with a linear regression model, see Eq. 2. Remarkably, during the measurements considered in this paper, in particular after the twentieth day of the 30 day record, heavy activity was present for the construction of a new experiment inside the underground laboratory hosting the RLG. Despite that, and the fact that GINGERINO is not equipped with control of the geometrical scale factor (accordingly its wavelength is not fixed), in 5.4 days of integration time it has reached the relative stability of (1.7 ± 0.07) × 10 −7 [corresponding to a resolution in the angular velocity measurement of (8.5±0.5)×10 −12 rad/s]. The analysis poses the problem of the accuracy of angular velocity estimations; however, it is not straightforward to understand the accuracy of the final measurement, an issue to be addressed by future Monte Carlo studies. The analysis has also shown that losses are the main limitation of GINGERINO. The question is now which part of the apparatus is the main responsible for losses and for their behaviour as a function of time. The hypothesis is that the main contribution comes from the gain tube of the laser. In the standard high sensitivity RLG scheme, it acts as gain tube and as spatial filter. The laser discharge is in fact produced within a pyrex capillary, with inner diameter of 4 mm, which is a bit small compared to the waist of the beams. In the near future, a new capillary tube with slightly larger inner diameter will be installed in GINGERINO to limit losses. Furthermore, additional data, such as current of the discharge tube and laser wavelength, will be continuously recorded in order to better characterise the instrument operation in the long term. As far as the analysis is concerned, the effect of terms θ 1,2 will be included in the model. The data analysis will be extended in order to evaluate the null shift second order term ω N S2 and to better understand the influence of each parameter, in particular of the change of wavelength in the RLG operation, which for the present set of data matters since GINGERINO is free running and the wavelength can change with even small temperature variations.  Operation Parameters of GINGERINO Appendix_A.nb