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Dynamics of a ring-laser gyroscope with backscattering
Abstract
We analyze the standard third-order model describing a laser gyroscope with backscattering in the two limiting cases of dissipative and conservative coupling. In both cases we discuss the steady-state and periodic solutions, determine their stability, and construct the locking diagram. In particular, in the dissipative coupling case, an exact periodic solution is found analytically. It is also proved that the concepts developed in the passive-cavity problem are useful tools to understand the dynamics of the laser gyroscope.
... Backscattering has been studied across the electromagnetic spectrum, from microwave to optical frequencies. It is of practical importance for applications in ring lasers [8,9] and gyroscopes [10,11], and it is fundamentally connected to topological systems [12]. In particular, backscatter immunity is a defining characteristic of topologically protected edge channels, even in the presence of disorder [13][14][15][16]. ...
Optical cavities have found widespread use in interfacing to quantum emitters. Concerns about backreflection and resulting loss, however, have largely prevented the placement of optics such as lenses or modulators within high-finesse cavities. In this work, we demonstrate a million-fold suppression of backreflections from lenses within a twisted optical cavity. We achieve this by quantitatively exploring backscatter in Fabry–Perot resonators, separating the effect into three physical sectors: polarization, mode envelope, and transverse mode profile. We describe the impact of each of these sectors and demonstrate how to minimize backreflections within each. This culminates in measured effective reflectivities below the part-per-billion level for the fundamental mode. Additionally, we show that beams carrying orbital angular momentum experience up to ${10^4}$ 10 4 times additional suppression, limited only by the density of states of other cavity modes. The understanding and techniques described in this work could expand the utility of optical resonators in topics ranging from quantum optics and cavity quantum electrodynamics to ring resonators and laser gyroscopes.
... Backscattering has been studied across the electromagnetic spectrum, from microwave to optical frequencies. It is of practical importance for applications in ring lasers [8,9] and gyroscopes [10,11], and is fundamentally connected to topological systems [12]. In particular, backscatter immunity is a defining characteristic of topologically protected edge channels, even in the presence of disorder [13][14][15][16]. ...
Optical cavities have found widespread use in interfacing to quantum emitters. Concerns about backreflection and resulting loss, however, have largely prevented the placement of optics such as lenses or modulators within high-finesse cavities. In this work, we demonstrate a million-fold suppression of backreflections from lenses within a twisted optical cavity. We achieve this by quantitatively exploring backscatter in Fabry-P\'erot resonators, separating the effect into three physical sectors: polarization, mode envelope and spatial mode profile. We describe the impact of each of these sectors, and demonstrate how to minimize backreflections within each. This culminates in measured effective reflectivities below the part-per-billion level for the fundamental mode. Additionally, we show that beams carrying orbital angular momentum experience up to $10^{4}$ times additional suppression, limited only by the density of states of other cavity modes. Applying these ideas to laser gyroscopes could strongly suppress lock-in, thereby improving sensitivity at low rotation rates.
... The precision of such inertial navigation is strongly dependent on the number of inertial sensors, among which is the gyroscope. The records for precision and bias stability of industrially available gyroscopes are traditionally held by ring-laser gyroscopes and fiber-optic gyroscopes, the ideas of which were developed by the end of the 20th century [4][5][6][7][8]. Based on the Sagnac effect, their precision is proportional to the surface enclosed by the optical light path [9,10]. ...
A rotation sensor is one of the key elements of inertial navigation systems and compliments most cellphone sensor sets used for various applications. Currently, inexpensive and efficient solutions are mechanoelectronic devices, which nevertheless lack long-term stability. Realization of rotation sensors based on spins of fundamental particles may become a drift-free alternative to such devices. Here, we carry out a proof-of-concept experiment, demonstrating rotation measurements on a rotating setup utilizing nuclear spins of an ensemble of NV centers as a sensing element with no stationary reference. The measurement is verified by a commercially available MEMS gyroscope.
A model of the interference of fields from backscattering point sources in a ring optical resonator is presented. Examples of calculations of complex coupling coefficients in ring resonators of laser gyroscopes are given. Comparison of the calculation results with the results of model experiments demonstrates good agreement and makes it possible to determine dissipative and conservative backscattering components of individual resonator mirrors. The possibilities of reducing the dissipative and conservative backscattering components in the process of ring resonator alignment are discussed.
A rotation sensor is one of the key elements of inertial navigation systems and compliments most cell phone sensor sets used for various applications. Currently, inexpensive and efficient solutions are mechanoelectronic devices, which nevertheless lack long-term stability. Realization of rotation sensors based on spins of fundamental particles may become a drift-free alternative to such devices. Here, we carry out a proof-of-concept experiment, demonstrating rotation measurements on a rotating setup utilizing nuclear spins of an ensemble of nitrogen vacancy centers as a sensing element with no stationary reference. The measurement is verified by a commercially available microelectromechanical system gyroscope.
We report the results of measuring the complex coupling coefficients in ring He – Ne lasers with a radiation wavelength of 632.8 nm. It is found that in assembling ring resonators from mirrors of the same quality, there is no correlation between the dissipative and conservative backscattering components. The analysis of the measurement results shows that this behaviour is associated with the speckle structure of the backscattered field in the ring resonator. The presence of backscattering (BS) sources with a random phase on the mirrors’ surface leads to a spread in the values of the BS parameters, which is described by the Rayleigh distribution function. Ways to reduce the spread in the values of dissipative and conservative backscattering components in assembling the ring resonators are discussed.
We report the results of measurements of nonlinear scale factor corrections, resulting from the backscattering effect in a laser gyro with a rectangular bias. It is shown that taking conservative and dissipative backscattering components into account allows one to obtain analytical relationships that adequately describe the frequency response of a laser gyroscope over the entire working range of angular velocities. Using these relationships to correct the frequency response of a laser gyroscope makes it possible to reduce the value of nonlinear corrections to 1 – 2 ppm.
This study experimentally reveals the self-biasing phenomenon in prism laser gyros. When the gyro operates in a specific double longitudinal mode and four-frequency oscillation state, the gyro can detect the normal angular velocity component of the Earth’s rotation without dithering, the self-biasing status is induced. In this study, a laboratory platform of the self-biasing laser gyro is established. The two longitudinal modes oscillate on both sides of the gain curve. The intensity ratio of the strong and weak modes is about 1.4 to 1, and the coupling effects of these two modes with parallel linear polarization have been discussed using Lamb theory. Because the adverse effects of biasing technical are avoided, the self-biasing gyro has potential to challenge the strategic precision.
Les gyroscopes laser à gaz constituent une solution technique de haute performances dans les problématiques de navigation inertielle. Néanmoins, pour de très faibles vitesses de rotation, les petites imperfections des miroirs de la cavité optique font que les deux faisceaux contra-propageant sont verrouillés en phase. En conséquence, les mesures en quadrature de leur différence de phase ne permettent plus de remonter directement aux vitesses de rotation à l'intérieur d'une zone autour de zéro, dite zone aveugle statique, ou, si l'on utilise une procédure d'activation mécanique, dite zone aveugle dynamique. Ce travail montre qu'il est néanmoins possible, en utilisant des méthodes issues du filtrage et de l'estimation, de remonter aux vitesses de rotation mêmes si ces dernières sont en zone aveugle. Pour cela, on part d'une modélisation physique de la dynamique que l'on simplifie par des techniques de perturbations singulières pour en déduire une généralisation des équations de Lamb. Il s'agit de quatre équations différentielles non-linéaires qui décrivent la dynamique des intensités et des phases des deux faisceaux contra-propageant. Une étude qualitative par perturbations régulières, stabilité exponentielle des points d'équilibre et applications de Poincaré permet de caractériser les zones aveugles statiques et dynamiques en fonction des imperfections dues aux miroirs. Il est alors possible d'estimer en ligne avec un observateur asymptotique fondé sur les moindre carrés récursifs ces imperfections en rajoutant aux deux mesures en quadrature celles des deux intensités. La connaissance précise de ces imperfections permet alors de les compenser dans la dynamique de la phase relative, et ainsi d'estimer les rotations en zone aveugle. Des simulations numériques détaillées illustrent l'intérêt de ces observateurs pour augmenter la précision des gyroscopes à gaz.
The degeneracy of the clockwise- and counterclockwise-traveling waves in a ring laser can be eliminated by linearly coupling the two waves by means of backscattering, which leads to a frequency splitting of the normal modes. In a theoretical treatment of this effect we stress the use of a standing-wave representation. We present, in particular, numerical results for the case in which the linear coupling is not much larger than the nonlinear coupling associated with gain saturation. We report experimental evidence of frequency splitting by analyzing the optical spectrum of a bidirectional ring dye laser with backscattering. The experimental results are in good agreement with the theoretical predictions.
An alternative approach is proposed to discuss mode coupling in bidirectional ring lasers that is induced by backscattering. It is shown that various features can be simply discussed in terms of the mode structure of the corresponding passive ring cavity. The nature of the backscattering is found to play a crucial role in determining the normal-mode structure. For instance, we show theoretically that, for a rotating ring laser (gyro), the characteristics of frequency locking are already present in the passive-mode structure if the mode coupling has a dissipative nature, i.e., if the backscattering originates in localized losses. If, on the other hand, the backscattering has a conservative nature, i.e., originates in steps of the refractive index, a frequency splitting is found in the passive-mode structure, making so-called oscillatory instability possible. Experimental observations are reported to support this point of view. The recently reported pi -phase jumps in He-Ne ring lasers are shown to fit naturally into this scheme. These jumps can be described as transitions between the normal modes of the passive ring cavity.
It is shown that the basic features of frequency locking in ring laser gyros can be modelled by linear coupled oscillators. The two- and four-mode cases are treated.
This paper presents a review of both active and passive ring laser devices. The operating principles of the ring laser are developed and discussed, with special emphasis given to the problems associated with the achievement of greater sensitivity and stability. First-principle treatments of the nature of quantum noise in the ring laser gyro and various methods designed to avoid low-rotation-rate lock-in are presented. Descriptions of state-of-the-art devices and current and proposed applications (including a proposed test of metric theories of gravity using a passive cavity ring laser) are given.
Jumps of + or - pi have been observed in the phase difference between the two counterpropagating modes of a HeNe ring laser whenever the complex backscattering coefficients satisfy a specified condition. The experiments made use of two retroreflectors that allowed light to be reflected back into the ring laser cavity with controllable amplitudes and phases. The phase jumps are usually accompanied by oscillatory instabilities of the two mode intensities. The observed behavior is found to be consistent with computer solutions of the coupled equations of motion.
By making provisions for retroreflecting the two emerging counterrotating waves back into the laser cavity with variable amplitudes and phases, backscattering effects in an He-Ne ring laser have been studied. The intensity fluctuations and the intensity cross correlation between the two counterpropagating modes were measured as the pump parameter and the phase of the backscattering are varied. The prediction, based on third-order laser theory, that equal-time moments depend on backscattering only through the combination R1 + R2(asterisk) (where R1 and R2 are complex backscattering coefficients), is confirmed. Good agreement between experiment and third-order laser theory is obtained so long as the normalized pump parameter is below approximately 100.
We consider two simple models which describe degenerate four-wave mixing and the degenerate parametric oscillator, respectively. In both cases we point out the existence of a Berry-like phase induced by the variation of the time. On leave of absence from the Physics Department, Beijing Normal University, Beijing, PR China.
It is shown that in an inhomogeneously broadened ring laser without detuning symmetric backscattering induces damped harmonic oscillations in the auto- and cross-correlation functions of the two-mode light intensities. The exact value of the oscillation frequency is derived. When the complex backscattering coefficients, R1, R2, satisfy the condition R1 + R2(asterisk) = 0 (off-phase backcoupling), the relaxation time of the correlation functions above threshold is obtained. The oscillations of the correlation functions are shown to decay more and more slowly as pump parameter increases. The results are in good agreement with the experimental data.
The third order equations of motion for a rotating, bidirectional, inhomogeneously broadened ring laser at line center with backscattering and asymmetry are solved exactly when the additive noise terms are negligible. The resulting solution for the relative phase of the two counterpropagating modes may exhibit steady state or transient oscillations. For certain initial conditions and operating parameters the phase solution is unstable. This gives rise to deterministic phase jumps in both the transient and steady state behavior. A series of phase jumps occurs if the system repeatedly crosses its unstable boundary. The jumps are multiples of π radians in magnitude, except for the case in which the phase discontinuity is caused by slowly varying operating parameters. Conditions are derived under which the frequency difference between the two modes (i) exhibits the well-known frequency lock-in effect at low rotation rates, or (ii) remains sensitive to cavity rotation regardless of rotation rate (lock-in suppression). Difficulties in achieving the latter result in a practical device are discussed.
The lock‐in threshold of a single‐mode 0.633‐μ HeNe laser was measured. Experiments were performed with a 1:1 mixture of the <sup>20</sup>Ne and <sup>22</sup>Ne isotope. The lock‐in threshold was found to increase with operation of the laser with smaller excess unsaturated gain. Minimum lock‐in threshold was found to occur at a higher frequency than the frequency at which the output power was a maximum. A rotational mode competition between the oppositely directed traveling waves was found for rotation rates less than the lock‐in threshold. The beam traveling with and against the direction of rotation was found to decrease and increase, respectively, with increasing rotation rate. Analysis showed that the single‐mode‐ring, laser equations, with mode coupling due to backscattering, satisfactorily explained the experiments.