Issue #1/2023

Microoptical Gyros Based on Whispering Gallery Mode Resonators

**Yu. V. Filatov, A. S. Kukaev, V. Yu. Venediktov, A. A. Sevryugin, E. V. Shalymov**Microoptical Gyros Based on Whispering Gallery Mode Resonators

DOI: 10.22184/1993-7296.FRos.2023.17.1.26.44

Optical gyros, such as ring laser gyros and fiber optical gyros, have become a basis for strapdown inertial navigation systems due to a number of advantages (larger dynamic range of the measured velocities; high stability of scale factor, insensitivity to linear acceleration and G-stress; smaller readiness time, etc.). Despite success in its development, ring laser and fiber optic gyros are unsuitable for using in control systems of small portable devices because of its large size and weight. Now the actual task is miniaturization of optical gyros, or development and research of microoptical gyros.

Optical gyros, such as ring laser gyros and fiber optical gyros, have become a basis for strapdown inertial navigation systems due to a number of advantages (larger dynamic range of the measured velocities; high stability of scale factor, insensitivity to linear acceleration and G-stress; smaller readiness time, etc.). Despite success in its development, ring laser and fiber optic gyros are unsuitable for using in control systems of small portable devices because of its large size and weight. Now the actual task is miniaturization of optical gyros, or development and research of microoptical gyros.

Теги: angular velocity sensor microoptical gyros reciprocal frequency shift whispering gallery mode resonators взаимный сдвиг частоты датчик угловой скорости микрооптические гироскопы резонаторы галереи шепчущих мод

Microoptical Gyros Based on Whispering Gallery Mode Resonators

Yu. V. Filatov, A. S. Kukaev, V. Yu. Venediktov,

A. A. Sevryugin, E. V. Shalymov

Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

Optical gyros, such as ring laser gyros and fiber optical gyros, have become a basis for strapdown inertial navigation systems due to a number of advantages (larger dynamic range of the measured velocities; high stability of scale factor, insensitivity to linear acceleration and G-stress; smaller readiness time, etc.). Despite success in its development, ring laser and fiber optic gyros are unsuitable for using in control systems of small portable devices because of its large size and weight. Now the actual task is miniaturization of optical gyros, or development and research of microoptical gyros.

Keywords: Microoptical Gyros, Whispering Gallery Mode Resonators, angular velocity sensor, Reciprocal frequency shift

Received on: 31.10.2022

Accepted on: 21.11.2022

During the last decade main activities in the area of developing the microoptical gyro were concentrated on the scheme of device based on the use of passive ring single-mode cavities, which are usually produced with the use of planar integral optical technologies.

Whispering gallery modes resonators can be also used as the gyro sensitive element instead of planar single-mode resonator. This is due to their following properties: highest optical quality factors, small eigenmodes volume, compactness and relative ease of fabrication. In this work we consider effects arising in the whispering gallery modes resonator during its rotation and possible ways of their application as sensing element of microoptical gyro.

1. Introduction

Miniaturization is one of the crucial processes that determines modern technologies, devices and even the lifestyle. If we have a glance at the history of gyroscope evolution, we can see that mechanical gyros were invented prior to the optical ones and their miniaturization was performed earlier as well. MEMS gyros are already widespread and cover the huge part of the sensors market owing to its usage in UAVs, robotics, and even smartphones. However, advantages that made optical gyros to take its place are pushing forward the idea of designing a microoptical gyro (MOG).

The sensing element of most MOGs is a passive ring resonator (PRR). Its type determines the manufacturing technology of the entire instrument, the potential sensitivity, the minimum possible dimensions and many other characteristics of the gyroscope. Therefore, MOGs are conveniently classified based on the type of PRR used (see Fig. 1).

Usually waveguide PRRs are used as a sensitive element. Most often they are made of planar waveguides. They can also be made of fibre or photonic crystals. In this case, only single-mode waveguides are used. As in all optical gyroscopic systems, the use of multimode waveguides is impossible because of the mode dispersion. For the same reason, conventional open resonators cannot be used. Although it is possible to use one of the specific subspecies – confocal ring resonators. MOGs based on waveguide resonators are well described in [1] and on ring confocal resonators in [2]. The resonators of the whispering gallery modes stand apart. They have recently been considered as promising sensitive MOG elements. Let us consider them in more details.

The whispering gallery phenomenon has been known for several centuries due to various architectural monuments. For example, the “Echo Wall” that encircles the courtyard of the Temple of the Imperial Heavens in Beijing (Fig. 2). The wall is made of special bricks from Linqing City, Shandong Province. These bricks are well processed and have the same structure. If you break such a brick, you can see that there are practically no pores in it. Such bricks reflect the acoustic wave perfectly and absorb almost none of its energy. The phenomenon is that the sound (whisper) travels along the concave surface of the wall, and not along the shortest path.

Similar phenomena is found under the domes of some cathedrals. For example, at St. Paul’s Cathedral in London (Fig. 2), where this phenomenon was investigated and scientifically described by Lord Rayleigh in the 19th century. For its description, the scientist has introduced the term – Whispering Gallery Mode (WGM) [3]. Later, optical WGMs were also discovered and corresponding resonators were developed. Usually these are dielectric axially symmetric resonators with smooth edges that support WGM due to total internal reflection on the resonator surface (capillary, sphere, toroid, disk, ring, racetrack, and bottleneck resonators) [4]. Optical WGM resonators are characterized by ultra-high Q-factor (up to 109 and more), compact dimensions (from several centimeters to several microns), and a limited number of natural frequencies. In this regard, they are considered as promising sensitive elements for various compact devices and systems. In particular, such resonators can be used to measure angular velocity.

2. Measuring the angular velocity from the reciprocal frequency shift of the whispering gallery modes

Consider the cross-section of a WGM resonator (Fig. 3). Let us assume that radiation is introduced into and out of the resonator through an auxiliary waveguide connected to the resonator due to the effect of optical tunneling. As known, centrifugal forces caused by the rotation of material objects can lead to their mechanical deformation. This is also true for the resonators of the whispering gallery modes. Rotation of WGM resonators relative to the inertial space causes changes in the radius of the working section of the resonator equal to ΔR. The WGM frequencies are inversely proportional to the radius of the cross-section of the resonator, i. e. decrease with increasing radius [3]. Thus, under the action of centrifugal forces, a reciprocal (the same for opposite directions of the resonator bypass) shift WGM ΔfC arises (Figure 4).

The reciprocal frequency shift caused by the rotation of the spherical WGM resonator around the axis perpendicular to its cross-section and passing through its center (around the main axis of sensitivity) is described by the following equation [5]:

, (1)

where fm,CW и fm,CCW are the frequencies of the WGM waves bypassing the resonator clockwise and counterclockwise, respectively; fm is the WGM frequency of the stationary resonator; R0 is the radius of the working section of the stationary resonator; KC is the scale factor of the effect; ρ is the resonator density; ν is Poisson’s ratio; G is shear modulus.

It is clear that the scale factor KC is determined not only by the properties of the resonator material and its size but also depends on its shape. When using WGM resonators of a different shape (for example, toroidal), equation 1 changes somewhat, but the quadratic dependence of the shift ΔfC on the angular velocity Ω and the radius of the working section R0 is preserved.

This effect can be used to measure angular velocity. It is clear that the heavier and softer the resonator, the more noticeable the effect. When using resonators made of soft polymers, the described effect can prevail over other effects causing the frequency shift of WGM [6]. For example, consider a WGM spherical resonator made of polydimethylsiloxane with 1/60 part of a hardener (60:1 PDMS) with the following parameters: R0 = 0.5 mm; ρ = 960 kg / m3; ν = 0.49; G = 1 000 Pa. In this case, let us assume that light with a frequency fm of about 2 · 1014 Hz (corresponding to a wavelength in a vacuum of about 1.5 μm). Based on equation 1, the reciprocal frequency shift ΔfC will correspond to the graph in figure 5. Thus, if the frequency fm, is known (measured in advance), then by scanning the resonator by frequency, it is possible to determine the reciprocal shift ΔfC WGM and calculate the corresponding angular velocity.

3. Problems associated with measuring the angular velocity from the reciprocal frequency shift of the whispering gallery modes and their solution

3.1. Influence of Environmental Factors

There are several problems that complicate the measurement of the angular velocity from the reciprocal displacement of the WGM. For example, the influence of various external factors (temperature, pressure, etc.) on the radius of the working section of the resonator. This results in a parasitic reciprocal frequency shift that is difficult to distinguish from ΔfC. This problem can be solved by passing to the measurement of the angular velocity by the splitting of the frequencies of neighboring WGMs with different azimuthal indices [7]. It is known that when WGM resonators rotate, not only the radius of their working section changes but also their shape. For example, when a spherical WGM resonator rotates, it is deformed into an ellipsoid of revolution [6]. Usually, the modes of a spherical resonator with different azimuthal indices n, but the same polar m are degenerate in frequency [3]. When rotating, the deviation of the resonator shape from the spherical removes this degeneracy. For an ellipsoid with a small eccentricity, the following expression is correct [3]:

, (2)

where Δa is the deviation of the smallest semi-axis of the ellipsoid from the radius of the ball. Based on equation 2, the frequency difference between adjacent WGMs with different azimuthal indices (differing by the index n by one):

. (3)

The deviation of the smallest semi-axis of the ellipsoid from the sphere radius Δa is proportional to the change in the radius of the WGM resonator working section ΔR. The aspect ratio depends on the resonator material. For the WGM spherical resonator considered above (made of 60:1 PDMS with ν = 0.49) Δa = 2ΔR. Considering this, we transform equation 3:

. (4)

Thus, as a result of the action of centrifugal forces, the WGM frequencies with different values of the azimuthal indices are split. By registering the frequency difference between adjacent WGMs, the rotation speed of the resonator can be determined. A change in the environmental parameters leads to an isotropic change in the resonator radius and the appearance of a corresponding reciprocal frequency shift of the WGM, but the shape of the resonator is preserved. Therefore, when measuring the angular velocity from the splitting of adjacent modes of a spherical resonator with different azimuthal indices, it is not necessary to stabilize the environmental parameters. However, the described method of measuring the angular velocity also has a serious drawback. By comparing equations 1 and 4, it is easy to see that the splitting of WGM frequencies ΔfE caused by centrifugal forces is always less than the reciprocal frequency shift ΔfC, i. e. when measuring the angular velocity from the splitting of neighboring WGMs, the sensitivity is reduced by a factor of М:

. (5)

For the WGM resonator considered above, n ≈ m = 2 930 and М ≈ 244. Using equation 4, we can determine the WGM frequency splitting ΔfE observed at Ω = 50 rad/s. It will be around 22 MHz. Nevertheless, the width of the peak of a spherical WGM resonator can be several hundred kHz or less [3], which makes it possible to distinguish individual WGM spectral lines with different azimuthal indices in the resonator spectrum and to measure the angular velocity by ΔfE.

3.2. Cross Sensitivity

Another problem associated with measuring the angular velocity from the reciprocal displacement of the WGM is the presence of parasitic cross-sensitivity (side axes of sensitivity). Since the orientation of the rotation axis is generally unknown, the sensitivity to the component of the angular velocity perpendicular to the main axis of sensitivity can lead to uncertainty in the measurement results. Using computer models set by the finite element method, the cross-sensitivity of various types of WGM resonators was analyzed: spherical, toroidal, disk-shaped, and bottleneck [9–11].

It has been determined that the magnitude and the sign of this cross-sensitivity depend on the shape of the WGM resonator used. When the spherical WGM resonator rotates around the side axis perpendicular to the main sensitivity, its working section is deformed from a circle to an ellipse. In this case, the minor axis of the ellipse corresponds to the side axis. The higher the angular velocity, the greater the eccentricity of the ellipse. In this case, modeling showed that the path traveled by the waves in one round trip of the resonator decreases. The resonator frequency of the spherical WGM resonator is shifted in the opposite direction to the shear direction corresponding to rotation around the main axis of sensitivity [9]. With a similar rotation of toroidal or disc-shaped resonators, the working section is also deformed from a circle into an ellipse. In this case, in contrast to the spherical resonator, the path traveled by the waves increases in one round of the resonator. The frequency of the WGM of a torus-shaped and disk-shaped resonators is shifted in the direction coinciding with the shear direction observed when rotating around the main axis of sensitivity [10]. Bottleneck resonators are different from others. They are remarkable in that the sign of their cross-sensitivity can change when their geometrical parameters change [11]. Such WGM resonators represent a cylindrical stem with a thickening that plays the role of a resonator (Figure 6). In this case, both parts of the resonator can be made either from the same or from different materials. The geometry of bottleneck resonators can be described quite accurately using the parameters r, R0 и rst, indicated in figure 6 [12].

Moreover, at certain values of the geometric parameters (specific values depend on the materials used) bottleneck resonators, their cross-sensitivity becomes negligible compared to the sensitivity to rotation around the main axis. For example, this is observed in a bottleneck resonator made entirely of fused silica with r = 250 μm, R0 = 600 μm, and rst = 108 μm.

It is also possible to solve the problem associated with parasitic cross-sensitivity by using a triad of WGM resonators of any type, the working sections of which are mutually orthogonal to each other.

3.3. Measurement Uncertainty and Nonlinearity of the Output Characteristic

Other significant problems are the reciprocal nature of the frequency shift (see equation 1 and fig. 5), which does not allow to determine the sign of the angular velocity (the object rotates clockwise or counterclockwise), and the nonlinearity of the scale factor (the square-law dependence of the effect on the angular velocity), which leads to a decrease in sensitivity to low angular velocities (fig. 5). To solve these problems, a method was developed to reduce the nonlinearity of the scale factor, based on the application of the initial offset of the operating point of the output characteristic [13]. The prospects of using both constant and alternating initial displacements (bias) were considered. The use of a constant bias is justified when the measured angular velocity Ωx is much less than the bias Ωb. Such a bias can be realized, for example, by placing a WGM resonator on the motor axis. Suppose, when the engine is turned on, the resonator begins to rotate around the main axis of sensitivity in the positive direction (counterclockwise) with a certain constant angular velocity Ωb. As a result of such rotation, the WGM resonator is deformed, the radius of its working section increases and the frequency of its mode fm decreases by some value Δfb and becomes equal to fb. (fig. 7).

When the described system (engine with a WGM resonator) rotates relative to inertial space with a certain desired velocity Ωx, the resonator mode will experience an additional frequency shift Δfx:

. (6)

If the sought rotation velocity Ωx coincides in sign with the angular velocity of the bias Ωb, then the total angular velocity of the WGM resonator increases. In this case, the frequencies of the mode fm,CW and fm,CCW become lower than fb (in figure 7 corresponds to fm,CW1) and the additional frequency shift Δfx will be positive. If the desired rotation speed Ωx has a sign opposite to the angular velocity of the bias Ωb, then the total angular velocity of the WGM resonator will decrease. In this case, the frequencies of the mode fm,CW и fm,CCW will become higher than fb and the additional frequency shift Δfx will be negative. Considering that fm – fm,CCW = KC(Ωb + Ωx)2 и Δfb = KC(Ωb)2, we can transform equation 6:

. (7)

Since the rotation speed of the bias Ω, and the frequency fb are known, then by scanning the resonator by frequency, it is possible to determine Δfx and calculate the corresponding angular rotation speed of the entire system Ωx. As can be seen from equation 7, for |Ωb| >> |Ωx| the shift Δfx is nonreciprocal (changes sign when the direction of rotation is changed).

As a result, the use of a constant bias makes it possible to judge not only about the magnitude of the angular velocity Ωx but also about its sign (the system rotates clockwise or counterclockwise). Under the condition |Ωb| >> |Ωx| the contribution of the second member in equation 7 is insignificant and the output characteristic becomes close to linear. For clarity, figure 8 shows the output characteristics for the above-considered WGM resonator, corresponding to different values of the constant bias Ωb.

Based on the graphs in figure 8 and equation 7, we can conclude that it is inadmissible to use a constant bias when determining the values of angular velocities |Ωx| ≥ 0.5|Ωb|, since in this case there is the measurement uncertainty Ωx. It is also easy to see that the higher the bias value Ωb, the greater the sensitivity to the measured angular velocity Ω¹ and the lower the nonlinearity of the output characteristic. Using equations 1 and 7, we can estimate how many times the sensitivity to angular velocity increases when using a constant bias:

. (8)

It should be noted that the use of constant bias can lead to the appearance of an additional measurement error due to the unevenness of Ωb. This places high demands on the motor used to create the constant bias and reduces interest in its practical implementation.

When using an alternating bias to measure the angular velocity from the reciprocal WGM frequency shift, the requirements for the characteristics of the drive are softer. Such a bias can be realized, for example, by fixing the WGM resonator on a piezoceramic rod or cylinder. When a periodic electrical signal is applied to electrodes placed in a certain way on the surface of piezoceramics [14, 15], torsional vibrations can be excited in the piezoceramic base and the WGM resonator fixed on it. In this case, the angular velocity of the alternating bias will be determined by the following equation:

, (9)

where Ωb0 is the amplitude of the alternating bias; νb0 are the oscillation frequency of the alternating bias; t is time. By the described method, using piezoceramic rods and cylinders, it is possible to excite torsional vibrations with a frequency from hundreds of Hz to units of GHz and amplitude of up to 1 000 rad / s and more [14].

Now let us analyze how the resonator frequencies of the spherical WGM resonator considered above change with an alternating bias. Let us set the following parameters of the alternating bias: amplitude Ωb0 = 40 rad/s, vibration frequency νb0 = 100 kHz. If a piezoceramic with a WGM resonator fixed on it is motionless relative to the inertial space, then, based on equations 1 and 9, the presence of an alternating bias Ωb leads to the appearance of a harmonic shift of the WGM frequency. Moreover, it is obvious that the frequency of this harmonic shift will be equal to the doubled frequency of the alternating bias (2νb0), and the amplitude is equal to KCΩb02.

Now, suppose that the described system (piezoceramics with a WGM resonator) began to rotate with a certain required speed Ωx relative to inertial space. In this case, for simplicity, we assume that the value of Ωx during the period of oscillation of the bias (τ = 1 / νb0) is close to a constant (changes in Ωx for τ are an order of magnitude less than the current value of Ωx). Then the total angular velocity of the resonator is the sum of a conditionally constant (slowly varying) value Ωx and a periodic alternating bias Ωb (see figure 9). As a result, the amplitude of the frequency shift of the whispering gallery mode changes. The extrema of the frequency shift of the whispering gallery mode become periodically equal to either Δf1, or Δf2 (Figure 9). Moreover, their difference depends on the value of the sought angular velocity:

. (10)

Thus, if, when using an alternating bias, we track the magnitude of the shift ΔfC and calculate the value of Δfx, then using equation 10 it is possible to determine the desired angular velocity Ωx. Also, using a variable bias, it is possible to measure the angular velocity in the case when the value of Ωx dynamically changes over the period τ. However, the changes in Ωx are averaged over the period of the bias oscillation τ.

Let us consider the advantages of using an alternating bias. For clarity, figure 10 shows the output characteristics corresponding to different values of the amplitude of the alternating bias Ωb0.

As seen from figure 10 and equation 10, the use of alternating bias allows one to judge not only about the magnitude of the angular velocity Ωx but also about its sign. In this case, the output characteristic is linear. It is also easy to see that the higher the amplitude of the alternating bias Ωb0, the greater the sensitivity to the measured angular velocity Ωx. Using equations 1 and 10, one can estimate how many times the sensitivity to angular velocity increases when using an alternating bias:

. (11)

From equation 11 it is obvious that the use of an alternating frequency bias makes it possible to increase the sensitivity to the angular velocity when the condition Ωb0 > 0,25Ωx is satisfied. The required range of operation of angle sensors rarely exceeds ±100 rad / s, while values of the amplitude of the alternating bias Ωb0 over 1 000 rad/s are achievable. Thus, the use of an alternating frequency bias makes it possible to increase the sensitivity in the entire operating range of the angular velocity sensors. It is also worth noting that this allows to solve other problems described above related to the measurement of the angular velocity from the reciprocal shift of the WGM. The influence of various external factors (temperature, pressure, etc.) and parasitic cross-sensitivity can lead to a reciprocal shift of the WGM frequencies Δf. The magnitude of this parasitic shift Δf is constant relative to the sought angular velocity Ωx and the angular velocity of the bias Ωb (i. e., the angular velocity when rotating around the main axis of sensitivity). Thus, when calculating the difference Δf1 – Δf2, the parasitic component Δf is excluded (see equation 10 and figure 10). Frequency shift Δf does not affect the calculated value Δfx, which is proportional to the sought angular velocity Ωx.

4. Measuring angular velocity using the Sagnac effect

Optical WGM resonators can also be used as sensing elements of resonator type micro-optical gyroscopes [16]. In this case, the nonreciprocal shift of the WGM frequencies, caused by the Sagnac effect, serves as a measure of the angular velocity. Consider the cross-section of a WGM resonator (Figure 3). As noted above, if the resonator is stationary relative to the inertial space, then the WGM frequencies for the opposite directions of its bypass (clockwise and counterclockwise) are equal. When the resonator rotates, due to the Sagnac effect, the WGM frequencies are shifted in opposite directions (Figure 11).

Let us assume that the reciprocal shift caused by centrifugal forces is negligible compared to the nonreciprocal shift of the WGM frequencies caused by the Sagnac effect (the resonator is made of a stiff material). Then the magnitude of the frequency shift is determined by the following equation [17]:

, (12)

where fm,CW и fm,CCW are the WGM frequencies of waves bypassing the resonator clockwise and counterclockwise, respectively; fm is the WGM frequency of the stationary resonator; R is the radius of the cross-section of the resonator; C is the speed of light in vacuum and Ω is the angular speed.

Thus, by scanning the resonator in frequency, it is possible to determine the nonreciprocal frequency shift ΔfS and calculate the angular velocity proportional to it. As can be seen from equation 12, the Sagnac effect is linear with respect to the angular velocity and allows to determine the direction of rotation. The nonreciprocity of the effect makes it possible to exclude the mutual shifts of the WGM frequencies when calculating the difference in resonator frequencies ΔfS. Including those caused by changes in the parameters of the external environment (temperature, pressure, etc.), as well as caused by centrifugal forces ΔfC.

However, centrifugal forces also cause a change in the radius of the working section of the WGM resonator (see equation 1), which is included in the scale factor of the Sagnac effect (see equation 12). As a result, it is possible to rewrite the expression that determines the nonreciprocal shift of the WGM frequencies, taking into account the influence of centrifugal forces:

. (13)

Thus, the scale factor of the Sagnac effect becomes nonlinear. If we do not take into account the dependence of the radius on the speed of rotation (due to the action of centrifugal forces), then an additional systematic error appears. When measuring the angular velocity using the Sagnac effect, stiff (fused silica or calcium fluoride) WGM resonators are usually used. In this case, the magnitude of the systematic error caused by the action of the centrifugal force is small and becomes significant only at high angular velocities (more than 103 rad / s). If necessary, you can eliminate this systematic error. For this, starting from expression 1, the correction to the radius of the working section of the WGM resonator is calculated: ΔR = (fm – fm, CW) R0/fm. Also, this technique allows you to exclude a systematic error caused by a change in the resonator radius due to a change in environmental parameters.

The limiting sensitivity of MOGs resonator is proportional to the Q-factor of the resonator used in them. The Q-factor of the best waveguide PRRs, usually considered as sensitive elements of MOGs, reaches 106–107 [16], while in WGM resonators it reaches above 109 [18]. However, due to the specific shape of some types of WGM resonators (spherical, bottle-shaped, etc.), they are difficult to integrate with other MOG elements. Other, more technologically advanced WGM resonators (toroidal, disc-shaped), are usually characterized by a lower Q-factor or pronounced nonlinear properties.

Let us compare the magnitudes of the WGM shifts caused by the Sagnac effect and the influence of centrifugal forces. We restrict ourselves to considering shifts only in spherical WGM resonators. Using equations 1 and 12, we can obtain equation describing the magnitude of the angular velocity at which these WGM shifts are equal:

. (14)

The graphs in figure 12 illustrate the dependence of the angular velocity Ωe on the resonator radius R0. At the same time, spherical WGM resonators made of fused silica, or calcium fluoride, a soft optical polymer (60 : 1 PDMS) are considered. Under the curves shown in figure 12 is dominated by the Sagnac effect, and above the influence of centrifugal forces. The softer and heavier the material and the larger the WGM resonator, the lower the angular velocity, the shifts ΔfC and ΔfS become equal.

In connection with the above, the idea of simultaneous use of two effects to measure the angular velocity seems to be very attractive. It can be implemented in different ways. First, when reaching a certain value of the angular velocity (when reaching Ωe), you can simply switch between the effects used for measurement. Those make measurements using the Sagnac effect at low angular velocities (less than Ωe). At high angular velocities (more than Ωe), measure the angular velocity from the frequency shift caused by centrifugal forces. It is also possible to realize the simultaneous measurement of angular velocity using both effects, using the fact that one of the effects is reciprocal and the other is nonreciprocal. To determine the WGM frequency shift caused by the Sagnac effect, based on the difference of the WGM frequencies bypassing the resonator in opposite directions: ΔfS = fm,CW – fm,CCW. Simultaneously, calculate the shift of WGM frequencies caused by centrifugal forces using the sum of the same WGM frequencies: ΔfC = 0,5[2fm – (fm,CW + fm,CCW)]. The result is two devices operating simultaneously and using different physical effects but on the same resonator. This should make it possible to use the advantages of both effects and to increase the accuracy of determining the angular velocity by complexing the measurement results.

5. Conclusion

The paper considered a method for measuring the angular velocity from the reciprocal frequency shift of the WGM caused by the influence of centrifugal forces. When using resonators made of soft polymers, this effect can prevail over other effects causing the WGM frequency shift. Nevertheless, several problems complicate in practice the measurement of the angular velocity from the reciprocal displacement of the WGM.

One of the problems is the influence of various external factors (temperature, pressure, etc.) on the radius of the working section of the resonator. This leads to a parasitic reciprocal frequency shift, which is difficult to distinguish from the shift ΔfC by which the angular velocity is determined. It was demonstrated that this problem can be solved by switching to measuring the angular velocity by splitting the neighboring WGM frequencies with different azimuthal indices. In this case, the sensitivity to the angular velocity decreases.

Another problem is the presence of parasitic cross-sensitivity (side axes of sensitivity). Since the orientation of the axis of rotation is generally unknown when measuring the angular velocity, the sensitivity to the component of the angular velocity of the perpendicular main axis of sensitivity can lead to uncertainty in the measurement results. It has been determined that the magnitude and sign of this cross-sensitivity depend on the type of shape of the WGM resonator used. Moreover, at certain values of the geometric parameters (specific values depend on the materials used) of bottleneck resonators, their cross-sensitivity becomes negligible compared to the sensitivity to rotation around the main axis. For example, this is observed for a bottleneck resonator made entirely of fused silica with r = 250 μm, R0 = 600 μm and rst = 108 μm. Also, this problem can be solved by using a triad of WGM resonators of any type, the working sections of which are mutually orthogonal to each other.

In addition, significant problems are the reciprocal nature of the frequency shift, which does not allow judging about the sign of the angular velocity, and the nonlinearity of the scale factor, which leads to a decrease in sensitivity to low angular velocities. To solve these problems, it has been proposed to use the starting offset of the operating point of the output characteristic. The prospects of using both constant and alternating initial displacements (bias) were considered. It is more promising to use an alternating bias. In this case, the output characteristic becomes linear. In this case, the sensitivity in the entire operating range of the angular velocity sensors increases by a factor of four equal to the quadruple ratio of the angular velocity of the bias to the measured angular velocity. It also allows you to solve other problems described above related to the measurement of the angular velocity from the WGM reciprocal displacement.

The work considered the Sagnac effect in resonators of whispering gallery modes and the features of its application for measuring the angular velocity of such resonators. In particular, it was demonstrated that due to the influence of centrifugal forces, the scale factor of the Sagnac effect becomes nonlinear. This can lead to an additional systematic measurement error. However, since when measuring the angular velocity by the Sagnac effect, hard WGM resonators are usually used, the magnitude of this systematic error is small and becomes significant only at high angular velocities (more than 103 rad/s). A method for compensating this error was described, which also makes it possible to eliminate the systematic error caused by a change in the resonator radius due to a change in the environmental parameters.

In addition, the work compares the values of the WGM shifts caused by the Sagnac effect and the influence of centrifugal forces and considers the idea of simultaneous use of these effects for measuring the angular velocity.

Acknowledgments

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. FSEE-2020-0005).

About the authors

Venediktov Vladimir Yurievich, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-0728-2050

Kukaev Alexander Sergeevich, Cand. of Technical Sciences, associate professor, Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-9525-8412

Sevryugin Alexander Alexeevich, Cand. of Physical and Mathematical Sciences, senior lecturer, Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

Filatov Yury Vladimirovich, Doctor of Technical Sciences, Professor, Head of Department Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-4388-8033

Shalymov Egor Vadimovich, Cand. of Technical Sciences, associate professor,Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-0731-6978

Yu. V. Filatov, A. S. Kukaev, V. Yu. Venediktov,

A. A. Sevryugin, E. V. Shalymov

Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

Optical gyros, such as ring laser gyros and fiber optical gyros, have become a basis for strapdown inertial navigation systems due to a number of advantages (larger dynamic range of the measured velocities; high stability of scale factor, insensitivity to linear acceleration and G-stress; smaller readiness time, etc.). Despite success in its development, ring laser and fiber optic gyros are unsuitable for using in control systems of small portable devices because of its large size and weight. Now the actual task is miniaturization of optical gyros, or development and research of microoptical gyros.

Keywords: Microoptical Gyros, Whispering Gallery Mode Resonators, angular velocity sensor, Reciprocal frequency shift

Received on: 31.10.2022

Accepted on: 21.11.2022

During the last decade main activities in the area of developing the microoptical gyro were concentrated on the scheme of device based on the use of passive ring single-mode cavities, which are usually produced with the use of planar integral optical technologies.

Whispering gallery modes resonators can be also used as the gyro sensitive element instead of planar single-mode resonator. This is due to their following properties: highest optical quality factors, small eigenmodes volume, compactness and relative ease of fabrication. In this work we consider effects arising in the whispering gallery modes resonator during its rotation and possible ways of their application as sensing element of microoptical gyro.

1. Introduction

Miniaturization is one of the crucial processes that determines modern technologies, devices and even the lifestyle. If we have a glance at the history of gyroscope evolution, we can see that mechanical gyros were invented prior to the optical ones and their miniaturization was performed earlier as well. MEMS gyros are already widespread and cover the huge part of the sensors market owing to its usage in UAVs, robotics, and even smartphones. However, advantages that made optical gyros to take its place are pushing forward the idea of designing a microoptical gyro (MOG).

The sensing element of most MOGs is a passive ring resonator (PRR). Its type determines the manufacturing technology of the entire instrument, the potential sensitivity, the minimum possible dimensions and many other characteristics of the gyroscope. Therefore, MOGs are conveniently classified based on the type of PRR used (see Fig. 1).

Usually waveguide PRRs are used as a sensitive element. Most often they are made of planar waveguides. They can also be made of fibre or photonic crystals. In this case, only single-mode waveguides are used. As in all optical gyroscopic systems, the use of multimode waveguides is impossible because of the mode dispersion. For the same reason, conventional open resonators cannot be used. Although it is possible to use one of the specific subspecies – confocal ring resonators. MOGs based on waveguide resonators are well described in [1] and on ring confocal resonators in [2]. The resonators of the whispering gallery modes stand apart. They have recently been considered as promising sensitive MOG elements. Let us consider them in more details.

The whispering gallery phenomenon has been known for several centuries due to various architectural monuments. For example, the “Echo Wall” that encircles the courtyard of the Temple of the Imperial Heavens in Beijing (Fig. 2). The wall is made of special bricks from Linqing City, Shandong Province. These bricks are well processed and have the same structure. If you break such a brick, you can see that there are practically no pores in it. Such bricks reflect the acoustic wave perfectly and absorb almost none of its energy. The phenomenon is that the sound (whisper) travels along the concave surface of the wall, and not along the shortest path.

Similar phenomena is found under the domes of some cathedrals. For example, at St. Paul’s Cathedral in London (Fig. 2), where this phenomenon was investigated and scientifically described by Lord Rayleigh in the 19th century. For its description, the scientist has introduced the term – Whispering Gallery Mode (WGM) [3]. Later, optical WGMs were also discovered and corresponding resonators were developed. Usually these are dielectric axially symmetric resonators with smooth edges that support WGM due to total internal reflection on the resonator surface (capillary, sphere, toroid, disk, ring, racetrack, and bottleneck resonators) [4]. Optical WGM resonators are characterized by ultra-high Q-factor (up to 109 and more), compact dimensions (from several centimeters to several microns), and a limited number of natural frequencies. In this regard, they are considered as promising sensitive elements for various compact devices and systems. In particular, such resonators can be used to measure angular velocity.

2. Measuring the angular velocity from the reciprocal frequency shift of the whispering gallery modes

Consider the cross-section of a WGM resonator (Fig. 3). Let us assume that radiation is introduced into and out of the resonator through an auxiliary waveguide connected to the resonator due to the effect of optical tunneling. As known, centrifugal forces caused by the rotation of material objects can lead to their mechanical deformation. This is also true for the resonators of the whispering gallery modes. Rotation of WGM resonators relative to the inertial space causes changes in the radius of the working section of the resonator equal to ΔR. The WGM frequencies are inversely proportional to the radius of the cross-section of the resonator, i. e. decrease with increasing radius [3]. Thus, under the action of centrifugal forces, a reciprocal (the same for opposite directions of the resonator bypass) shift WGM ΔfC arises (Figure 4).

The reciprocal frequency shift caused by the rotation of the spherical WGM resonator around the axis perpendicular to its cross-section and passing through its center (around the main axis of sensitivity) is described by the following equation [5]:

, (1)

where fm,CW и fm,CCW are the frequencies of the WGM waves bypassing the resonator clockwise and counterclockwise, respectively; fm is the WGM frequency of the stationary resonator; R0 is the radius of the working section of the stationary resonator; KC is the scale factor of the effect; ρ is the resonator density; ν is Poisson’s ratio; G is shear modulus.

It is clear that the scale factor KC is determined not only by the properties of the resonator material and its size but also depends on its shape. When using WGM resonators of a different shape (for example, toroidal), equation 1 changes somewhat, but the quadratic dependence of the shift ΔfC on the angular velocity Ω and the radius of the working section R0 is preserved.

This effect can be used to measure angular velocity. It is clear that the heavier and softer the resonator, the more noticeable the effect. When using resonators made of soft polymers, the described effect can prevail over other effects causing the frequency shift of WGM [6]. For example, consider a WGM spherical resonator made of polydimethylsiloxane with 1/60 part of a hardener (60:1 PDMS) with the following parameters: R0 = 0.5 mm; ρ = 960 kg / m3; ν = 0.49; G = 1 000 Pa. In this case, let us assume that light with a frequency fm of about 2 · 1014 Hz (corresponding to a wavelength in a vacuum of about 1.5 μm). Based on equation 1, the reciprocal frequency shift ΔfC will correspond to the graph in figure 5. Thus, if the frequency fm, is known (measured in advance), then by scanning the resonator by frequency, it is possible to determine the reciprocal shift ΔfC WGM and calculate the corresponding angular velocity.

3. Problems associated with measuring the angular velocity from the reciprocal frequency shift of the whispering gallery modes and their solution

3.1. Influence of Environmental Factors

There are several problems that complicate the measurement of the angular velocity from the reciprocal displacement of the WGM. For example, the influence of various external factors (temperature, pressure, etc.) on the radius of the working section of the resonator. This results in a parasitic reciprocal frequency shift that is difficult to distinguish from ΔfC. This problem can be solved by passing to the measurement of the angular velocity by the splitting of the frequencies of neighboring WGMs with different azimuthal indices [7]. It is known that when WGM resonators rotate, not only the radius of their working section changes but also their shape. For example, when a spherical WGM resonator rotates, it is deformed into an ellipsoid of revolution [6]. Usually, the modes of a spherical resonator with different azimuthal indices n, but the same polar m are degenerate in frequency [3]. When rotating, the deviation of the resonator shape from the spherical removes this degeneracy. For an ellipsoid with a small eccentricity, the following expression is correct [3]:

, (2)

where Δa is the deviation of the smallest semi-axis of the ellipsoid from the radius of the ball. Based on equation 2, the frequency difference between adjacent WGMs with different azimuthal indices (differing by the index n by one):

. (3)

The deviation of the smallest semi-axis of the ellipsoid from the sphere radius Δa is proportional to the change in the radius of the WGM resonator working section ΔR. The aspect ratio depends on the resonator material. For the WGM spherical resonator considered above (made of 60:1 PDMS with ν = 0.49) Δa = 2ΔR. Considering this, we transform equation 3:

. (4)

Thus, as a result of the action of centrifugal forces, the WGM frequencies with different values of the azimuthal indices are split. By registering the frequency difference between adjacent WGMs, the rotation speed of the resonator can be determined. A change in the environmental parameters leads to an isotropic change in the resonator radius and the appearance of a corresponding reciprocal frequency shift of the WGM, but the shape of the resonator is preserved. Therefore, when measuring the angular velocity from the splitting of adjacent modes of a spherical resonator with different azimuthal indices, it is not necessary to stabilize the environmental parameters. However, the described method of measuring the angular velocity also has a serious drawback. By comparing equations 1 and 4, it is easy to see that the splitting of WGM frequencies ΔfE caused by centrifugal forces is always less than the reciprocal frequency shift ΔfC, i. e. when measuring the angular velocity from the splitting of neighboring WGMs, the sensitivity is reduced by a factor of М:

. (5)

For the WGM resonator considered above, n ≈ m = 2 930 and М ≈ 244. Using equation 4, we can determine the WGM frequency splitting ΔfE observed at Ω = 50 rad/s. It will be around 22 MHz. Nevertheless, the width of the peak of a spherical WGM resonator can be several hundred kHz or less [3], which makes it possible to distinguish individual WGM spectral lines with different azimuthal indices in the resonator spectrum and to measure the angular velocity by ΔfE.

3.2. Cross Sensitivity

Another problem associated with measuring the angular velocity from the reciprocal displacement of the WGM is the presence of parasitic cross-sensitivity (side axes of sensitivity). Since the orientation of the rotation axis is generally unknown, the sensitivity to the component of the angular velocity perpendicular to the main axis of sensitivity can lead to uncertainty in the measurement results. Using computer models set by the finite element method, the cross-sensitivity of various types of WGM resonators was analyzed: spherical, toroidal, disk-shaped, and bottleneck [9–11].

It has been determined that the magnitude and the sign of this cross-sensitivity depend on the shape of the WGM resonator used. When the spherical WGM resonator rotates around the side axis perpendicular to the main sensitivity, its working section is deformed from a circle to an ellipse. In this case, the minor axis of the ellipse corresponds to the side axis. The higher the angular velocity, the greater the eccentricity of the ellipse. In this case, modeling showed that the path traveled by the waves in one round trip of the resonator decreases. The resonator frequency of the spherical WGM resonator is shifted in the opposite direction to the shear direction corresponding to rotation around the main axis of sensitivity [9]. With a similar rotation of toroidal or disc-shaped resonators, the working section is also deformed from a circle into an ellipse. In this case, in contrast to the spherical resonator, the path traveled by the waves increases in one round of the resonator. The frequency of the WGM of a torus-shaped and disk-shaped resonators is shifted in the direction coinciding with the shear direction observed when rotating around the main axis of sensitivity [10]. Bottleneck resonators are different from others. They are remarkable in that the sign of their cross-sensitivity can change when their geometrical parameters change [11]. Such WGM resonators represent a cylindrical stem with a thickening that plays the role of a resonator (Figure 6). In this case, both parts of the resonator can be made either from the same or from different materials. The geometry of bottleneck resonators can be described quite accurately using the parameters r, R0 и rst, indicated in figure 6 [12].

Moreover, at certain values of the geometric parameters (specific values depend on the materials used) bottleneck resonators, their cross-sensitivity becomes negligible compared to the sensitivity to rotation around the main axis. For example, this is observed in a bottleneck resonator made entirely of fused silica with r = 250 μm, R0 = 600 μm, and rst = 108 μm.

It is also possible to solve the problem associated with parasitic cross-sensitivity by using a triad of WGM resonators of any type, the working sections of which are mutually orthogonal to each other.

3.3. Measurement Uncertainty and Nonlinearity of the Output Characteristic

Other significant problems are the reciprocal nature of the frequency shift (see equation 1 and fig. 5), which does not allow to determine the sign of the angular velocity (the object rotates clockwise or counterclockwise), and the nonlinearity of the scale factor (the square-law dependence of the effect on the angular velocity), which leads to a decrease in sensitivity to low angular velocities (fig. 5). To solve these problems, a method was developed to reduce the nonlinearity of the scale factor, based on the application of the initial offset of the operating point of the output characteristic [13]. The prospects of using both constant and alternating initial displacements (bias) were considered. The use of a constant bias is justified when the measured angular velocity Ωx is much less than the bias Ωb. Such a bias can be realized, for example, by placing a WGM resonator on the motor axis. Suppose, when the engine is turned on, the resonator begins to rotate around the main axis of sensitivity in the positive direction (counterclockwise) with a certain constant angular velocity Ωb. As a result of such rotation, the WGM resonator is deformed, the radius of its working section increases and the frequency of its mode fm decreases by some value Δfb and becomes equal to fb. (fig. 7).

When the described system (engine with a WGM resonator) rotates relative to inertial space with a certain desired velocity Ωx, the resonator mode will experience an additional frequency shift Δfx:

. (6)

If the sought rotation velocity Ωx coincides in sign with the angular velocity of the bias Ωb, then the total angular velocity of the WGM resonator increases. In this case, the frequencies of the mode fm,CW and fm,CCW become lower than fb (in figure 7 corresponds to fm,CW1) and the additional frequency shift Δfx will be positive. If the desired rotation speed Ωx has a sign opposite to the angular velocity of the bias Ωb, then the total angular velocity of the WGM resonator will decrease. In this case, the frequencies of the mode fm,CW и fm,CCW will become higher than fb and the additional frequency shift Δfx will be negative. Considering that fm – fm,CCW = KC(Ωb + Ωx)2 и Δfb = KC(Ωb)2, we can transform equation 6:

. (7)

Since the rotation speed of the bias Ω, and the frequency fb are known, then by scanning the resonator by frequency, it is possible to determine Δfx and calculate the corresponding angular rotation speed of the entire system Ωx. As can be seen from equation 7, for |Ωb| >> |Ωx| the shift Δfx is nonreciprocal (changes sign when the direction of rotation is changed).

As a result, the use of a constant bias makes it possible to judge not only about the magnitude of the angular velocity Ωx but also about its sign (the system rotates clockwise or counterclockwise). Under the condition |Ωb| >> |Ωx| the contribution of the second member in equation 7 is insignificant and the output characteristic becomes close to linear. For clarity, figure 8 shows the output characteristics for the above-considered WGM resonator, corresponding to different values of the constant bias Ωb.

Based on the graphs in figure 8 and equation 7, we can conclude that it is inadmissible to use a constant bias when determining the values of angular velocities |Ωx| ≥ 0.5|Ωb|, since in this case there is the measurement uncertainty Ωx. It is also easy to see that the higher the bias value Ωb, the greater the sensitivity to the measured angular velocity Ω¹ and the lower the nonlinearity of the output characteristic. Using equations 1 and 7, we can estimate how many times the sensitivity to angular velocity increases when using a constant bias:

. (8)

It should be noted that the use of constant bias can lead to the appearance of an additional measurement error due to the unevenness of Ωb. This places high demands on the motor used to create the constant bias and reduces interest in its practical implementation.

When using an alternating bias to measure the angular velocity from the reciprocal WGM frequency shift, the requirements for the characteristics of the drive are softer. Such a bias can be realized, for example, by fixing the WGM resonator on a piezoceramic rod or cylinder. When a periodic electrical signal is applied to electrodes placed in a certain way on the surface of piezoceramics [14, 15], torsional vibrations can be excited in the piezoceramic base and the WGM resonator fixed on it. In this case, the angular velocity of the alternating bias will be determined by the following equation:

, (9)

where Ωb0 is the amplitude of the alternating bias; νb0 are the oscillation frequency of the alternating bias; t is time. By the described method, using piezoceramic rods and cylinders, it is possible to excite torsional vibrations with a frequency from hundreds of Hz to units of GHz and amplitude of up to 1 000 rad / s and more [14].

Now let us analyze how the resonator frequencies of the spherical WGM resonator considered above change with an alternating bias. Let us set the following parameters of the alternating bias: amplitude Ωb0 = 40 rad/s, vibration frequency νb0 = 100 kHz. If a piezoceramic with a WGM resonator fixed on it is motionless relative to the inertial space, then, based on equations 1 and 9, the presence of an alternating bias Ωb leads to the appearance of a harmonic shift of the WGM frequency. Moreover, it is obvious that the frequency of this harmonic shift will be equal to the doubled frequency of the alternating bias (2νb0), and the amplitude is equal to KCΩb02.

Now, suppose that the described system (piezoceramics with a WGM resonator) began to rotate with a certain required speed Ωx relative to inertial space. In this case, for simplicity, we assume that the value of Ωx during the period of oscillation of the bias (τ = 1 / νb0) is close to a constant (changes in Ωx for τ are an order of magnitude less than the current value of Ωx). Then the total angular velocity of the resonator is the sum of a conditionally constant (slowly varying) value Ωx and a periodic alternating bias Ωb (see figure 9). As a result, the amplitude of the frequency shift of the whispering gallery mode changes. The extrema of the frequency shift of the whispering gallery mode become periodically equal to either Δf1, or Δf2 (Figure 9). Moreover, their difference depends on the value of the sought angular velocity:

. (10)

Thus, if, when using an alternating bias, we track the magnitude of the shift ΔfC and calculate the value of Δfx, then using equation 10 it is possible to determine the desired angular velocity Ωx. Also, using a variable bias, it is possible to measure the angular velocity in the case when the value of Ωx dynamically changes over the period τ. However, the changes in Ωx are averaged over the period of the bias oscillation τ.

Let us consider the advantages of using an alternating bias. For clarity, figure 10 shows the output characteristics corresponding to different values of the amplitude of the alternating bias Ωb0.

As seen from figure 10 and equation 10, the use of alternating bias allows one to judge not only about the magnitude of the angular velocity Ωx but also about its sign. In this case, the output characteristic is linear. It is also easy to see that the higher the amplitude of the alternating bias Ωb0, the greater the sensitivity to the measured angular velocity Ωx. Using equations 1 and 10, one can estimate how many times the sensitivity to angular velocity increases when using an alternating bias:

. (11)

From equation 11 it is obvious that the use of an alternating frequency bias makes it possible to increase the sensitivity to the angular velocity when the condition Ωb0 > 0,25Ωx is satisfied. The required range of operation of angle sensors rarely exceeds ±100 rad / s, while values of the amplitude of the alternating bias Ωb0 over 1 000 rad/s are achievable. Thus, the use of an alternating frequency bias makes it possible to increase the sensitivity in the entire operating range of the angular velocity sensors. It is also worth noting that this allows to solve other problems described above related to the measurement of the angular velocity from the reciprocal shift of the WGM. The influence of various external factors (temperature, pressure, etc.) and parasitic cross-sensitivity can lead to a reciprocal shift of the WGM frequencies Δf. The magnitude of this parasitic shift Δf is constant relative to the sought angular velocity Ωx and the angular velocity of the bias Ωb (i. e., the angular velocity when rotating around the main axis of sensitivity). Thus, when calculating the difference Δf1 – Δf2, the parasitic component Δf is excluded (see equation 10 and figure 10). Frequency shift Δf does not affect the calculated value Δfx, which is proportional to the sought angular velocity Ωx.

4. Measuring angular velocity using the Sagnac effect

Optical WGM resonators can also be used as sensing elements of resonator type micro-optical gyroscopes [16]. In this case, the nonreciprocal shift of the WGM frequencies, caused by the Sagnac effect, serves as a measure of the angular velocity. Consider the cross-section of a WGM resonator (Figure 3). As noted above, if the resonator is stationary relative to the inertial space, then the WGM frequencies for the opposite directions of its bypass (clockwise and counterclockwise) are equal. When the resonator rotates, due to the Sagnac effect, the WGM frequencies are shifted in opposite directions (Figure 11).

Let us assume that the reciprocal shift caused by centrifugal forces is negligible compared to the nonreciprocal shift of the WGM frequencies caused by the Sagnac effect (the resonator is made of a stiff material). Then the magnitude of the frequency shift is determined by the following equation [17]:

, (12)

where fm,CW и fm,CCW are the WGM frequencies of waves bypassing the resonator clockwise and counterclockwise, respectively; fm is the WGM frequency of the stationary resonator; R is the radius of the cross-section of the resonator; C is the speed of light in vacuum and Ω is the angular speed.

Thus, by scanning the resonator in frequency, it is possible to determine the nonreciprocal frequency shift ΔfS and calculate the angular velocity proportional to it. As can be seen from equation 12, the Sagnac effect is linear with respect to the angular velocity and allows to determine the direction of rotation. The nonreciprocity of the effect makes it possible to exclude the mutual shifts of the WGM frequencies when calculating the difference in resonator frequencies ΔfS. Including those caused by changes in the parameters of the external environment (temperature, pressure, etc.), as well as caused by centrifugal forces ΔfC.

However, centrifugal forces also cause a change in the radius of the working section of the WGM resonator (see equation 1), which is included in the scale factor of the Sagnac effect (see equation 12). As a result, it is possible to rewrite the expression that determines the nonreciprocal shift of the WGM frequencies, taking into account the influence of centrifugal forces:

. (13)

Thus, the scale factor of the Sagnac effect becomes nonlinear. If we do not take into account the dependence of the radius on the speed of rotation (due to the action of centrifugal forces), then an additional systematic error appears. When measuring the angular velocity using the Sagnac effect, stiff (fused silica or calcium fluoride) WGM resonators are usually used. In this case, the magnitude of the systematic error caused by the action of the centrifugal force is small and becomes significant only at high angular velocities (more than 103 rad / s). If necessary, you can eliminate this systematic error. For this, starting from expression 1, the correction to the radius of the working section of the WGM resonator is calculated: ΔR = (fm – fm, CW) R0/fm. Also, this technique allows you to exclude a systematic error caused by a change in the resonator radius due to a change in environmental parameters.

The limiting sensitivity of MOGs resonator is proportional to the Q-factor of the resonator used in them. The Q-factor of the best waveguide PRRs, usually considered as sensitive elements of MOGs, reaches 106–107 [16], while in WGM resonators it reaches above 109 [18]. However, due to the specific shape of some types of WGM resonators (spherical, bottle-shaped, etc.), they are difficult to integrate with other MOG elements. Other, more technologically advanced WGM resonators (toroidal, disc-shaped), are usually characterized by a lower Q-factor or pronounced nonlinear properties.

Let us compare the magnitudes of the WGM shifts caused by the Sagnac effect and the influence of centrifugal forces. We restrict ourselves to considering shifts only in spherical WGM resonators. Using equations 1 and 12, we can obtain equation describing the magnitude of the angular velocity at which these WGM shifts are equal:

. (14)

The graphs in figure 12 illustrate the dependence of the angular velocity Ωe on the resonator radius R0. At the same time, spherical WGM resonators made of fused silica, or calcium fluoride, a soft optical polymer (60 : 1 PDMS) are considered. Under the curves shown in figure 12 is dominated by the Sagnac effect, and above the influence of centrifugal forces. The softer and heavier the material and the larger the WGM resonator, the lower the angular velocity, the shifts ΔfC and ΔfS become equal.

In connection with the above, the idea of simultaneous use of two effects to measure the angular velocity seems to be very attractive. It can be implemented in different ways. First, when reaching a certain value of the angular velocity (when reaching Ωe), you can simply switch between the effects used for measurement. Those make measurements using the Sagnac effect at low angular velocities (less than Ωe). At high angular velocities (more than Ωe), measure the angular velocity from the frequency shift caused by centrifugal forces. It is also possible to realize the simultaneous measurement of angular velocity using both effects, using the fact that one of the effects is reciprocal and the other is nonreciprocal. To determine the WGM frequency shift caused by the Sagnac effect, based on the difference of the WGM frequencies bypassing the resonator in opposite directions: ΔfS = fm,CW – fm,CCW. Simultaneously, calculate the shift of WGM frequencies caused by centrifugal forces using the sum of the same WGM frequencies: ΔfC = 0,5[2fm – (fm,CW + fm,CCW)]. The result is two devices operating simultaneously and using different physical effects but on the same resonator. This should make it possible to use the advantages of both effects and to increase the accuracy of determining the angular velocity by complexing the measurement results.

5. Conclusion

The paper considered a method for measuring the angular velocity from the reciprocal frequency shift of the WGM caused by the influence of centrifugal forces. When using resonators made of soft polymers, this effect can prevail over other effects causing the WGM frequency shift. Nevertheless, several problems complicate in practice the measurement of the angular velocity from the reciprocal displacement of the WGM.

One of the problems is the influence of various external factors (temperature, pressure, etc.) on the radius of the working section of the resonator. This leads to a parasitic reciprocal frequency shift, which is difficult to distinguish from the shift ΔfC by which the angular velocity is determined. It was demonstrated that this problem can be solved by switching to measuring the angular velocity by splitting the neighboring WGM frequencies with different azimuthal indices. In this case, the sensitivity to the angular velocity decreases.

Another problem is the presence of parasitic cross-sensitivity (side axes of sensitivity). Since the orientation of the axis of rotation is generally unknown when measuring the angular velocity, the sensitivity to the component of the angular velocity of the perpendicular main axis of sensitivity can lead to uncertainty in the measurement results. It has been determined that the magnitude and sign of this cross-sensitivity depend on the type of shape of the WGM resonator used. Moreover, at certain values of the geometric parameters (specific values depend on the materials used) of bottleneck resonators, their cross-sensitivity becomes negligible compared to the sensitivity to rotation around the main axis. For example, this is observed for a bottleneck resonator made entirely of fused silica with r = 250 μm, R0 = 600 μm and rst = 108 μm. Also, this problem can be solved by using a triad of WGM resonators of any type, the working sections of which are mutually orthogonal to each other.

In addition, significant problems are the reciprocal nature of the frequency shift, which does not allow judging about the sign of the angular velocity, and the nonlinearity of the scale factor, which leads to a decrease in sensitivity to low angular velocities. To solve these problems, it has been proposed to use the starting offset of the operating point of the output characteristic. The prospects of using both constant and alternating initial displacements (bias) were considered. It is more promising to use an alternating bias. In this case, the output characteristic becomes linear. In this case, the sensitivity in the entire operating range of the angular velocity sensors increases by a factor of four equal to the quadruple ratio of the angular velocity of the bias to the measured angular velocity. It also allows you to solve other problems described above related to the measurement of the angular velocity from the WGM reciprocal displacement.

The work considered the Sagnac effect in resonators of whispering gallery modes and the features of its application for measuring the angular velocity of such resonators. In particular, it was demonstrated that due to the influence of centrifugal forces, the scale factor of the Sagnac effect becomes nonlinear. This can lead to an additional systematic measurement error. However, since when measuring the angular velocity by the Sagnac effect, hard WGM resonators are usually used, the magnitude of this systematic error is small and becomes significant only at high angular velocities (more than 103 rad/s). A method for compensating this error was described, which also makes it possible to eliminate the systematic error caused by a change in the resonator radius due to a change in the environmental parameters.

In addition, the work compares the values of the WGM shifts caused by the Sagnac effect and the influence of centrifugal forces and considers the idea of simultaneous use of these effects for measuring the angular velocity.

Acknowledgments

The work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. FSEE-2020-0005).

About the authors

Venediktov Vladimir Yurievich, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-0728-2050

Kukaev Alexander Sergeevich, Cand. of Technical Sciences, associate professor, Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-9525-8412

Sevryugin Alexander Alexeevich, Cand. of Physical and Mathematical Sciences, senior lecturer, Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

Filatov Yury Vladimirovich, Doctor of Technical Sciences, Professor, Head of Department Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-4388-8033

Shalymov Egor Vadimovich, Cand. of Technical Sciences, associate professor,Laser Measurement and Navigation Systems Department, St. Petersburg Electrotechnical University „LETI“ St. Petersburg, Russia

ORCID: 0000-0002-0731-6978

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