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In the last decade, increasing attention has been drawn to high-precision optical experiments, which push resolution and accuracy of the measured quantities beyond their current limits. This challenge requires to place optical elements (e.g., mirrors, lenses) and to steer light beams with subnanometer precision. Existing methods for beam direction computing in resonators, e.g., iterative ray tracing or generalized ray transfer matrices, are either computationally expensive or rely on overparameterized models of optical elements. By exploiting Fermat's principle, we develop a novel method to compute the steady-state beam configurations in resonant optical cavities formed by spherical mirrors, as a function of mirror positions and curvature radii. The proposed procedure is based on the geometric Newton method on matrix manifold, a tool with second-order convergence rate, that relies on a second-order model of the cavity optical length. As we avoid coordinates to parametrize the beam position on mirror surfaces, the computation of the second-order model does not involve the second derivatives of the parametrization. With the help of numerical tests, we show that the convergence properties of our procedure hold for non-planar polygonal cavities, and we assess the effectiveness of the geometric Newton method in determining their configurations with high degree of accuracy and negligible computational effort.

We present the design, implementation, and characterization of a heterodyne laser interferometer for nanometer displacement metrology. The purpose is to monitor the 3-D shape of a large optomechanical structure planned for future general relativity experiments. Reaching the target 10(-11)-m displacement uncertainty over 7-m distances and many days' integration periods is a challenging task. The solution here investigated consists of a nonpolarizing Mach-Zehnder layout, featuring an optical cancelable circuit and a holey folding mirror. The instrument working principle and the method for online phase reconstruction are presented, as well as the complete hardware configuration used. The several sources of noise are investigated mathematically and, whenever possible, verified experimentally. The displacement gauge was tested up to one day of continuous data acquisition, showing nanometer-level performance down to 100 mHz, while air index variations and mechanical instabilities are currently the main limiting factors at lower frequencies. This experiment has brought into light many technical issues that will constitute precious "lessons learned" for the future improvements of the system.

In recent years, efficient optimization algorithms for Nonlinear Model Predictive Control (NMPC) have been proposed, that significantly reduce the on-line computational time. In particular, direct multiple shooting and Sequential Quadratic Programming (SQP) are used to efficiently solve Nonlinear Programming (NLP) problems arising from continuous-time NMPC applications. One of the computationally demanding steps for on-line optimization is the computation of sensitivities of the nonlinear dynamics at every sampling instant, especially for systems of large dimensions, strong stiffness, and when using long prediction horizons. In this paper, within the algorithmic framework of the Real-Time Iteration (RTI) scheme based on multiple shooting, an inexact sensitivity updating scheme is proposed, that performs a partial update of the Jacobian of the constraints in the NLP. Such update is triggered by using a Curvature-like Measure of Nonlinearity (CMoN), so that only sensitivities exhibiting highly nonlinear behaviour are updated, thus adapting to system operating conditions and possibly reducing the computational burden. An advanced tuning strategy for the updating scheme is provided to automatically determine the number of sensitivities being updated, with a guaranteed bounded error on the Quadratic Programming (QP) solution. Numerical and control performance of the scheme is evaluated by means of two simulation examples performed on a dedicated implementation. Local convergence analysis is also presented and a tunable convergence rate is proven, when applied to the SQP method., version 2

A model based on Lamb's theory of gas lasers is applied to a He-Ne ring laser gyroscope in order to estimate and remove the laser dynamics contribution from the rotation measurements. The intensities of the counter-propagating laser beams exiting one cavity mirror are continuously observed together with a monitor of the laser population inversion. These observables, once properly calibrated with a dedicated procedure, allow us to estimate cold cavity and active medium parameters driving the main part of the nonlinearities of the system. The parameters identification and noise subtraction procedure has been verified by means of a Monte Carlo study of the system, and experimentally tested on the G-Pisa ring laser oriented with the normal to the ring plane almost parallel to the Earth rotation axis. In this configuration the Earth rotation-rate provides the maximum Sagnac effect while the contribution of the orientation error is reduced at minimum. After the subtraction of laser dynamics by a Kalman filter, the relative systematic errors of G-PISA reduce from 50 to 5 part in 10^3 and can be attributed to the residual uncertainties on geometrical scale factor and orientation of the ring., corrected typos

GINGERino is a large frame laser gyroscope investigating the ground motion in the most inner part of the underground international laboratory of the Gran Sasso, in central Italy. It consists of a square ring laser with a $3.6$ m side. Several days of continuous measurements have been collected, with the apparatus running unattended. The power spectral density in the seismic bandwidth is at the level of $10^{-10} \rm{(rad/s)/\sqrt{Hz}}$. A maximum resolution of $30\,\rm{prad/s}$ is obtained with an integration time of few hundred seconds. The ring laser routinely detects seismic rotations induced by both regional earthquakes and teleseisms. A broadband seismic station is installed on the same structure of the gyroscope. First analysis of the correlation between the rotational and the translational signal are presented., Accepted for publication in Review in Scientific Instruments

In recent years several fast Nonlinear Model Predictive Control (NMPC) strategies have been proposed, aiming at reducing computational burden and widening the scope of NMPC techniques. A promising approach is the Real-Time Iteration (RTI) scheme, where Nonlinear Programming (NLP) problems are parametrized by multiple shooting and only one Sequential Quadratic Programming (SQP) iteration is performed at every sampling instant. A computationally expensive step of RTI is the calculation of sensitivity information of nonlinear dynamics, especially for problems with large system dimensions or long prediction horizons. In this paper, an inexact sensitivity updating scheme to be used in the RTI framework is proposed, that allows to reduce the number of sensitivities updates over the prediction horizon at each sampling instant. A Curvature-like Measure of Nonlinearity (CMoN) of dynamic systems is used as a metric to quantify the linearization reliability, and to trigger sensitivity update only if needed. Numerical simulation results show that the proposed approach can significantly reduce the on-line computational efforts for sensitivity computations without major impact on the control performance.

Widespread application of real-time, Nonlinear Model Predictive Control (NMPC) algorithms to systems of large scale or with fast dynamics is challenged by the high associated computational cost, in particular in presence of long prediction horizons. In this paper, a fast NMPC strategy to reduce the on-line computational cost is proposed. A Curvature-based Measure of Nonlinearity (CMoN) of the system is exploited to reduce the required number of sensitivity computations, which largely contribute to the overall computational cost. The proposed scheme is validated by a simulation study on the chain of masses motion control problem, a toy example that can be easily extended to an arbitrary dimension. Simulations have been run with long prediction horizons and large state dimensions. Results show that sensitivity computations are significantly reduced with respect to other sensitivity updating schemes, while preserving control performance.

GINGER (Gyroscopes IN General Relativity) is a project aiming at measuring the Lense-Thirring effect, at 1% level, with an experiment on earth. It is based on an array of ring-lasers, which are the most sensitive inertial sensors to measure the rotation rate of the Earth. The GINGER project is still under discussion; the experiment G-GranSasso is an R&D experiment financed by INFN Group II, it is studying the key points of GINGER and at the same time developing prototypes. In the following the signal coming out of a ring-laser and the present sensitivity are described.The prototypes GP2 and GINGERino and the preliminary results are reported. This project is inter-disciplinary since ring-lasers provide informations for the fast variation of the earth rotation rate, they are used for the rotational seismology and for top sensitivity angle metrology.

The sensitivity achieved by large ring-laser gyroscopes will make it possible to detect faint relativistic effects related to the rotation of the Earth's mass. This task requires a strict control of the ring cavity geometry (shape and orientation), which can be performed by a novel network of portable heterodyne interferometers, capable of measuring the absolute distance betweeen two retro-reflectors with a nominal accuracy better than 1nm. First steps have been taken towards the realization of this device and a starting prototype of distance gauge is under development and test.

Two ring laser gyroscopes are being developed by the INFN in view of fundamental Physics applications. The scope of this activity is to ‘fill the gap’ between the present state of the art technology and the requirements in sensitivity and accuracy needed for general relativity tests. The first prototype, called GP2 and located at INFN Pisa, is dedicated to the interferometric control of the ring laser cavity form factor. The second prototype, called GINGERino, is a larger ring laser located deep underground (INFN LNGS) and has the scope of characterizing the low frequency rotational noise of the site. We show the most recent results of the two experiments.

Ultra-sensitive ring laser gyroscopes are regarded as potential detectors of the general relativistic frame-dragging effect due to the rotation of the Earth. Our project for this goal is called GINGER (gyroscopes in general relativity), and consists of a ground-based triaxial array of ring lasers aimed at measuring the rotation rate of the Earth with an accuracy of . Such an ambitious goal is now within reach, as large-area ring lasers are very close to the required sensitivity and stability. However, demanding constraints on the geometrical stability of the optical path of the laser inside the ring cavity are required. Thus, we have begun a detailed study of the geometry of an optical cavity in order to find a control strategy for its geometry that could meet the specifications of the GINGER project. As the cavity perimeter has a stationary point for the square configuration, we identify a set of transformations on the mirror positions that allows us to adjust the laser beam steering to the shape of a square. We show that the geometrical stability of a square cavity strongly increases by implementing a suitable system to measure the mirror distances, and that the geometry stabilization can be achieved by measuring the absolute lengths of the two diagonals and the perimeter of the ring.

He-Ne ring lasers are, actually, the most sensitive devices for inertial rotation measurements. Depending on their linear dimensions, they find applications in different fields: from inertial navigation to structural engineering, from metrology to geophysics and fundamental physics. Here we report the recent progresses toward the development of GINGER- Gyroscopes IN GEneral Relativity, a triaxial ultra-sensitive ring laser gyroscope whose primary target is a ground measurement the Lense-Thirring effect, a small shift of the Earth rotation rate foreseen by Einstein's General Relativity. This target implies a measurement of the Earth rotation rate with a sensitivity better than one part over 10 10 and so requires a high level of stability and accuracy of the laser array and of the gyroscope stability. The main part of the research activity presented in this paper involves, then, a strict control of the systematic errors related to the fluctuation of the optical cavity geometry and of the laser parameters.

After reviewing the importance of light as a probe for testing the structure of space-time, we describe the GINGER project. GINGER will be a three-dimensional array of large size ring-lasers able to measure the de Sitter and Lense-Thirring effects. The instrument will be located at the underground laboratory of GranSasso, in Italy. We describe the preliminary actions and measurements already under way and present the full road map to GINGER. The intermediate apparatuses GP2 and GINGERino are described. GINGER is expected to be fully operating in few years., 11 pages, in Comptes Rendus Physique, Special Issue "The Sagnac effect: 100 years later"

GINGER is a proposed tridimensional array of laser gyroscopes with the aim of measuring the Lense-Thirring effect, predicted by the General Relativity theory, in a terrestrial laboratory environment. We discuss the required accuracy, the methods to achieve it, and the preliminary experimental work in this direction., Submitted to International Journal Laser Physics, 9 pages, 5 figure

The sensitivity achieved by large laser gyroscopes opens the perspective of observing in a ground laboratories very thin relativistic effect related to the Earth rotating mass (gravitomagnetic effect or Lense-Thirring effect). The required accuracy asks for a strict control of the ring cavity geometry. Here we present a control procedure that can be applied in order to solve this task.

We present the results of the experimental investigation of ring lasers dynamics obtained by operating the G-Pisa gyroscope with the laser plane oriented almost perpendicularly to the local North direction. In this configuration, the Earth's rotation contribution to the Sagnac frequency is less sensitive to orientation errors, and more precise observations of the laser dynamical parameters can be achieved. Cavity losses and scattering parameters are monitored by observing the single beams intensity modulation at the Sagnac frequency while the active medium parameters are identified by a dedicated spectroscopic analysis of the plasma dispersion function. A summary of the performances of both the identification and the frequency denoising algorithm.

GINGER (Gyroscopes IN GEneral Relativity) is a proposal for measuring in a ground-based laboratory the Lense-Thirring effect, known also as inertial frame dragging, that is predicted by General Relativity, and is induced by the rotation of a massive source. GINGER will consist in an array of at least three square ring lasers, mutually orthogonal, with about 6-10 m side, and located in a deep underground site, possibly the INFN - National Laboratories of Gran Sasso. The tri-axial design will provide a complete estimation of the laboratory frame angular velocity, to be compared with the Earths rotation estimate provided by IERS with respect the fixed stars frame. Large-size ring lasers have already reached a very high sensitivity, allowing for relevant geodetic measurements. The accuracy required for Lense-Thirring effect measurement is higher than 10-14 rad/s and therefore Earth angular velocity must be measured within one part in 10-9. A 3.6 m side, square ring laser, called GINGERino, has been recently installed inside the Gran Sasso underground laboratories in order to qualify the site for a future installation of GINGER. We discuss the current status of the experimental work, and in particular of the GINGERino prototype.

He-Ne ring laser gyroscopes are, at present, the most precise devices for absolute angular velocity measurements. Limitations to their performance come from the non--linear dynamics of the laser. Following the Lamb semi-classical theory, we find a set of critical parameters affecting the time stability of the system. We propose a method for estimating the long term drift of the laser parameters and for filtering out the laser dynamics effects from the rotation measurement. The parameter estimation procedure, based on the perturbative solutions of the laser dynamics, allow us to apply Kalman Filter theory for the estimation of the angular velocity. Results of a comprehensive Monte Carlo simulation and results of a preliminary analysis on experimental data from the ring laser prototype G-Pisa are shown and discussed., 22 pages, 11 figures

We present a summary of the recent results obtained with the “G-Pisa” laser gyroscope prototype in the field of rotational metrology. The experimental apparatus consists in an He-Ne laser having a planar square cavity 1.35 m in side that can be operated both in the vertical and in the horizontal plane. For about one year, the ring laser has been utilized by the Virgo gravitational wave interferometer with the aim of estimating and monitoring the local rotational noise which is degrading the performances of its inertial suspensions. Results in the field of environmental monitoring for the improvement of the suspension control as well as the results in the field of rotational seismology are presented. In the last part of the paper we present some considerations about the possibility of performing a ground-based General Relativity test using an array of ring lasers.

We present the experimental test of a method for controlling the absolute length of the diagonals of square ring laser gyroscopes. The purpose is to actively stabilize the ring cavity geometry and to enhance the rotation sensor stability in order to reach the requirements for the detection of the relativistic Lense-Thirring effect with a ground-based array of optical gyroscopes. The test apparatus consists of two optical cavities 1.32 m in length, reproducing the features of the ring cavity diagonal resonators of large frame He-Ne ring laser gyroscopes. The proposed measurement technique is based on the use of a single diode laser, injection locked to a frequency stabilized He-Ne/Iodine frequency standard, and a single electro-optic modulator. The laser is modulated with a combination of three frequencies allowing to lock the two cavities to the same resonance frequency and, at the same time, to determine the cavity Free Spectral Range (FSR). We obtain a stable lock of the two cavities to the same optical frequency reference, providing a length stabilization at the level of 1 part in $10^{11}$, and the determination of the two FSRs with a relative precision of 0.2 ppm. This is equivalent to an error of 500 nm on the absolute length difference between the two cavities.

Active control of the geometrical parameters of large ring-lasers arrays is a challenging task, requiring the development of an accurate length monitoring system acting on the edges of these devices. The proposed solution is based on what was outlined by NASA JPL for its discontinued Space Interferometric Mission (SIM). It will be made up of a network of compact laser heterodyne interferometers, working together to keep the geometrical frame constantly well known (and fixed) to nanometer accuracy. We took the first steps towards the realization of this device by focusing on its fundamental element, i.e. the compact heterodyne laser interferometer. A starting simplified prototype has been conceived and is now under development. Its preliminary design will let us to evaluate parts behavior, alignment issues and links between single components and overall gauge performances. Concerned distance is investigated by means of a reference and a measurement beam, obtained by spatial splitting of the laser source. The phase shift between two heterodyne beatings will provide a reliable signal for precise monitoring of a linear distance.