Vibration-insensitive optical path length control transducer for ring laser gyroscope

Abstract

A ring laser gyroscope is an optical sensor based on the Sagnac effect. It has many outstanding advantages, such as a short start-up time, a high precision, a wide dynamic range and a stable scale factor. It is insensitive to accelerations and is in wide use in aircraft attitude control, north finding applications including navigation and space missions. The path length transducer in a ring laser gyroscope is required to stabilize the optical path of the laser beams in the cavity. The path length control transducer may tilt in complex environment, which will cause deformation of resonant optical path in resonant cavity and seriously influence the performance of ring laser gyroscope. Firstly, for the first time, finite element analysis software ANSYS was used to simulate and analyze the small deformation of ring laser gyroscope’s grooved mirror in three directions under typical random vibrational spectra, their value were 2.08 $"$ , 2.10 $"$ and 1.30 $"$ respectively, and the deformation in the diaphragm of resonant cavity was obtained by combining with matrix optical theory quantitatively. Secondly, the influence rule of different radius of curvature and isostructural parameters on the tilt of path length control transducer was sought through quantitative analysis. Finally, combining with the change law of axial displacement in literature [5], an evaluation criterion for optimization was given and the optimized parameters were listed for path length control transducer. The tilt was reduced to 0.33 $"$ after optimization, which was about 21.4% of current structure. The axial displacement reached 3.89 um, which was 1.84 times of existing structure. The research in the paper could provide reference for the design of path length control transducer of ring laser gyroscope.

Keywords

Ring laser gyroscope path length control transducer vibration finite element analysis

1. Introduction

Ring laser gyroscope (RLG) with wide dynamic range and bandwidth, high accuracy and reliability has important applications in many fields such as flight control, attitude measurement and inertial navigation system. Usually, a RLG with super high accuracy is achieved by a highly stable ring resonator cavity and a set of reflective mirrors of excellent quality. RLG is usually operated in a vibrational environment in practical engineering. The structural support transmits vibrations to the cavity and vibrational perturbations evoke elastic deformations of the cavity, which in turn change the optical length of the cavity [1–3]. The path length variations will affect the rotation measurements due to alternations of the lock-in threshold parameters [4,5]. It is very important to maintain a substantially constant optical path length during operations since the intensity of laser beam is dependent on the path length.

In order to maintain a constant path length, a path length control transducer (PLC transducer) is always employed [6,7]. The PLC transducer is usually assembled by a reflective mirror and a piezoelectric actuator which is attached to the reflective mirror. According to the piezoelectric effect, a given voltage can be converted into mechanical stress on the piezoelectric material, and then converted to an axial displacement to maintain a constant path length.

However, under vibrational perturbations, these perturbations induce elastic deformation along the instantaneous direction of random force, and the effect is that the PLC transducer or the reflective mirrors are not only axially deformed but also tilted. The tilt of the PLC transducer greatly influences the position of beam path and lock-in threshold. Therefore, we try to pursue optimal strategies to minimize the vibration sensitivity of the PLC transducer.

In this paper, the influence of elastic deformation along the tilted direction is provided. We explore various shapes of the PLC transducer to search designs that reduce the sensitivity of the cavity length to environmental vibration perturbations. A quantitative analysis of the elastic deformation of the PLC transducer provides valuable guidance for the design of ring resonator cavity. To our knowledge, however, a comprehensive investigation of the elastic deformation of ring resonator cavity at the modern precision level is not available in the literature. Here finite-element analysis is adopted for the first time to perform a detailed numerical analysis of various PLC transducer configurations and to identify optimal designs with improved vibration resistance.

2. Description of the PLC transducer

RLG is composed of a glass cavity and a dither mechanism, as shown in Fig. 1. The glass cavity is formed by four bars which are connected in a structure of a common foursquare block. The glass cavity is made of ZERODUR glass ceramic which has a very low coefficient of expansion. Two anodes and a cathode are sealed to the glass cavity to energize the beams of light. At each transition point of two adjacent bars, a PLC transducer is provided to reflect the light beams from one bar to the other. The PLC transducer consists of a grooved mirror and a piezoelectric actuator which is attached to the mirror. The piezoelectric actuator includes a base plate membrane and two ring-shaped lead-zirconate-titanate (PZT) ceramics. The base plate membrane is preferably comprised of materials with low thermal coefficient such as, invar or super invar material. A cross-section view of the PLC transducer is shown in Fig. 2 [5].

Fig. 1.

The Structure diagram of RLG.

Fig. 2.

The Cross-section view of the PLC transducer.

According to the piezoelectric effect, a given voltage can be converted into mechanical stress on the piezoelectric material, and then converted to an electric displacement. This movement of PLC transducer is called as axial movement which is the work pattern of stable cavity length of ring laser gyroscope. The vibration of external environment may result in the deflection of reflective mirror perpendicular to axial direction, which is called as tilt, as shown in Fig. 3.

Fig. 3.

Tilt of the PLC transducer.

3. Quantitative analysis of cavity deformation

In ideal status, there is only axial movement of PLC transducer along with the center pole of grooved mirror. At this moment, the internal optical path of gyroscope forms closed optical path through holes and the central axis of diaphragm, the mode volume in resonant cavity reaches maximum value with minimum loss, and the gyroscope is in optimal status. When ring laser gyroscope works in a complex vibration environment of wide spectrum, it will cause deformation of optical resonant cavity and change of resonant optical path, which will affect the performance of ring laser gyroscope. When the optical resonant cavity is affected by external vibration, it is cavity length control transducer that causes deformation of cavity. When PLC transducer tilts, it will cause great change of optical path, as shown in Fig. 4.

Fig. 4.

Light change caused by light path variations.

The deformation of PLC transducer caused by vibrations includes range shift and angular deflection. The augmented matrix of ring-shaped resonant cavity can be expressed as below [8,9] if introduce the augmented transformation matrix 5 × 5: $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}r_{ox}\\[5.0pt] r_{ox}^{\prime }\\[5.0pt] r_{oy}\\[5.0pt] r_{oy}^{\prime }\\[5.0pt] 1\end{array}\right)=\left(\begin{array}{@{}ccccc@{}}A_{x} & B_{x} & 0 & 0 & E_{x}\\[5.0pt] C_{x} & D_{x} & 0 & 0 & F_{x}\\[5.0pt] 0 & 0 & A_{y} & B_{y} & E_{y}\\[5.0pt] 0 & 0 & C_{y} & D_{y} & F_{y}\\[5.0pt] 0 & 0 & 0 & 0 & 1\end{array}\right)\left(\begin{array}{@{}c@{}}r_{ix}\\[5.0pt] r_{ix}^{\prime }\\[5.0pt] r_{iy}\\[5.0pt] r_{iy}^{\prime }\\[5.0pt] 1\end{array}\right). & & \displaystyle\end{eqnarray}$ (1)

In which, the coordinate system x and y is horizontal and vertical direction in the plane of grooved mirror, and z direction is determined according to right hand rule. r _ix, r _iy, r _ox and r _oy express the distance from incident ray and emergent ray to x and y axis, which are referred to as the eccentricity of optical axis; $r_{ix}^{\prime }$ , $r_{iy}^{\prime }$ , $r_{ox}^{\prime }$ and $r_{oy}^{\prime }$ represent the angles of incident ray and emergent ray deflecting from optical axis, which are called as the tilt angle of optical axis. A _x, B _x, C _x and D _x show transformation matrix elements in tangent plane, and A _y, B _y, C _y and D _y represent transformation matrix elements in sagittal plane. E _x and E _y express variables caused by translation of grooved mirror along with coordinate axis, while F _x and F _y represent the impact of angular deflection of grooved mirror on transformation matrix.

For 5 perturbation items may existing in spherical mirror (δ_ix, δ_iy, δ_iz, θ_ix and θ_iy), θ_ix and θ_iy represent deflection angles with δ_ix, δ_iy and δ_iz of grooved mirror’s translation (in which δ_ix and δ_iy are grooved mirror’s axial displacement and δ_iz of grooved mirror’s radial displacement), and the augmented transformation matrix of light can be expressed as: $\begin{eqnarray}\displaystyle M(M_{i})=\left(\begin{array}{@{}ccccc@{}}1 & 0 & 0 & 0 & 2{\delta}_{iz}\sin A\\ -2/(R\cos A) & 1 & 0 & 0 & -2{\delta}_{iz}\tan A/R+2({\theta}_{ix}+{\delta}_{ix}/R)\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & -2\cos A/R & 1 & 2({\theta}_{iy}+{\delta}_{iy}/R)\\ 0 & 0 & 0 & 0 & 1\end{array}\right). & & \displaystyle\end{eqnarray}$ (2)

As shown in Fig. 4, P ₁, P ₂, P ₃ and P ₄ are 4 reflective mirrors, in which P ₁ and P ₂ are spherical mirrors with P ₃ and P ₄ of planar mirrors. The curvature radius of spherical mirror is R, the incident angle of light is A _i, and the changes of 6 points are mainly analyzed.

In order to study the change law of diaphragm, setting the diaphragm as starting point in the derivation process, the light transmits a circle around the whole resonant cavity and comes back to the diaphragm, and then we’ll know the change condition of diaphragm. The total transformation matrix is the product of transformation matrixes, and the computing method of total transformation matrix is as below: $\begin{eqnarray}\displaystyle M=T\_Lhalf\times M\_m1\times T\_L4\times M\_m4\times T\_L3\times M\_m3\times T\_L2\times M\_m2\times T\_Lhalf. & & \displaystyle\end{eqnarray}$ (3) In which, T _Lhalf is propagator matrix from diaphragm to spherical mirror P ₂, M _m1, M _m2, M _m3 and M _m4 represent the transformation matrix of grooved mirror in P ₁, P ₂, P ₃ and P ₄, and T _L1, T _L2, T _L3 and T _L4 express the propagator matrix of light transmitting in four sides of quadrilateral cavity. When the proper value of Matrix M is 1, the resonant optical path is self-consistent, i.e. the light will coincide again after transmitting one circle around resonant cavity. The light coincides after transmitting one circle around resonant cavity, and the self-consistency of optical path can be obtained as: $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}r_{x}\\[5.0pt] r_{x}^{\prime }\\[5.0pt] r_{y}\\[5.0pt] r_{y}^{\prime }\\[5.0pt] 1\end{array}\right)=M\left(\begin{array}{@{}c@{}}r_{x}\\[5.0pt] r_{x}^{\prime }\\[5.0pt] r_{y}\\[5.0pt] r_{y}^{\prime }\\[5.0pt] 1\end{array}\right) & & \displaystyle\end{eqnarray}$ (4) Substitute formula (1) into (4), and then $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cccc@{}}M_{11}^{-1} & M_{12} & M_{13} & M_{14}\\[5.0pt] M_{21} & M_{22}^{-1} & M_{23} & M_{24}\\[5.0pt] M_{31} & M_{32} & M_{33}-1 & M_{34}\\[5.0pt] M_{41} & M_{42} & M_{43} & M_{44}^{-1}\end{array}\right)\left(\begin{array}{@{}c@{}}r_{x}\\[5.0pt] r_{x}^{\prime }\\[5.0pt] r_{y}\\[5.0pt] r_{y}^{\prime }\end{array}\right)=-\left(\begin{array}{@{}c@{}}M_{15}\\[5.0pt] M_{25}\\[5.0pt] M_{35}\\[5.0pt] M_{45}\end{array}\right). & & \displaystyle\end{eqnarray}$ (5)

The r _x, $r_{x}^{\prime }$ , r _y and $r_{y}^{\prime }$ of solution to seek will be obtained through solving the matrix equation.

The PLC transducer is comprised of piezoelectric ceramic actuator and grooved mirror. The deformation of optical resonant cavity under impact of random vibration spectrum could be equivalent to applying random vibration load on piezoelectric ceramic actuator [10], and the equivalent acceleration speed is a = 40g _rms.

Setting the existing structure as an example, random vibration load is applied on piezoelectric ceramic actuator, the specific deformation amount of grooved mirror along with three directions of X, Y and Z can be obtained through simulation analysis by ANSYS, and the deformation condition of grooved mirror in Y direction after applying stress is shown in Fig. 5.

Fig. 5.

Strain distribution of the grooved mirror when stress is applied along Y axis.

The center (node: 48613) and edge (node: 42284) of grooved mirror are selected to be two feature points, and the deformation in three directions is shown in Table 1:

Table 1

Displacement of the feature points in grooved mirror

Node	x direction (m)	y direction (m)	z direction (m)	Sum (m)
42284	1.70e-09	5.76e-08	4.33e-08	7.21e-08
48613	1.47e-10	6.79e-09	4.42E-08	4.47e-08

It can be obtained as below through the relationship of triangle and proximate method of small angle $\begin{eqnarray}\displaystyle {\theta}_{y}={\displaystyle \frac{5.76\times 10^{-8}-6.79\times 10^{-9}}{0.005}}=1.02\times 10^{-5}∼\text{Radian}=2.10" . & & \displaystyle \nonumber\end{eqnarray}$

The small angular tilt of grooved mirror along with X, Y and Z directions is shown in Table 2 after applying equivalent random vibration of a = 40g _rms in the three directions:

Table 2

The small angular tilt of grooved mirror along with three directions

Direction	x direction (")	y direction (")	z direction (")
Magnitude	2.08	2.10	1.30

It can be obtained according to formula (5), $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}r_{x}=-{\displaystyle \frac{{\theta}_{x}R\cos A}{2}}\\[10.0pt] r_{x}^{\prime }={\displaystyle \frac{12{\theta}_{x}R\cos A}{3L-16R\cos A}}\\[10.0pt] r_{y}=-{\displaystyle \frac{{\theta}_{y}R}{2\cos A}}\\[10.0pt] r_{y}^{\prime }=-{\displaystyle \frac{12{\theta}_{y}R}{16R-3L\cos A}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (6)

Choose A = 45°, L = 0.4 m and R = 8 m, thus the distance of diaphragm deflecting from X axis is r _x = −28.5 μm with that from Y axis of r _y = −57.6 μm, and the angle of light in diaphragm deflecting from optical axis is $r_{x}^{\prime }=-1.58"$ and $r_{y}^{\prime }=-1.59"$ . The tilt of PLC transducer increases with the intensification of external random vibration, and the distances of light spot in diaphragm deflecting from central position also increase. If the tilt angle of PLC transducer reaches 5 $"$ , the distances of light spot in diaphragm deflecting from central position are r _x = −57 μm and r _y = −115.2 μm, which equal to 5.7% and 11.5% of the radius of diaphragm. If two PLC transducers tilt at the same time, the maximum distance of light spot in diaphragm deflecting from central position could reach 23%, which makes resonant light generate modulation and affects the performance of gyroscope. The calculated results are the same as literature [11]. Therefore, in order to improve the anti-vibration performance of optical resonant cavity, it shall try to reduce the tilt of PLC transducer in vibration environment in design.

4. Optimization design of structural parameters of PLC transducer

The influence of each parameter’s geometry is given below. The geometric parameters (shown in Fig. 2) of the piezoelectric actuator (thickness H₂ of the base plate membrane, thickness d and radius R of the PZT ceramics) and the grooved mirror (membrane thickness H₁, base radius r and membrane depth D), are examined. Setting Y direction as an example, analyses on the FEM are performed in the aforementioned geometric dimensions when the equivalent random vibration a = 40g _rms.

The membrane thickness H₁ of the grooved mirror is the most important structural parameter of PLC transducer, and its thickness directly influences axial displacement. Detailed study on the influence rule of structural parameters on axial displacement has been carried out in reference [5]. Therefore, the research on structural parameters in this paper is also focused on the membrane thickness H₁ of the grooved mirror and gives priority to the tilt condition of grooved mirror under random vibration.

The variation of thickness H₂ of the base plate membrane and tilt of grooved mirror under different thickness of grooved mirror (H₁ = 0.6 mm--1.4 mm) is given in Fig. 6. It can be seen from the figure that, the tilt of grooved mirror reduces greatly with the increase of the thickness H₁ of grooved mirror. Setting H₂ = 1.4 mm, the tilt angle for a membrane with the thickness of 0.6 mm is over 3.9 times of 1.4 mm. Similarly, the tilt of grooved mirror increases gradually with the increase of thickness H₂ of the base plate membrane, and it is more obvious when H₁ = 0.6 mm. The plots in Fig. 6 indicate that the influence exerted by the membrane thickness H₁ plays a dominant role in the tilt angle. According to literature [5], it could increase the displacement of PLC transducer if decrease membrane thickness H₁, but it could also increase the tilt of grooved mirror. Therefore, factors in two aspects shall be paid special attention when choose the core parameter of membrane thickness H₁.

Fig. 6.

Tilt angle of the grooved mirror versus thickness of base plate membrane for various membrane thicknesses.

The influence of the thickness d of PZT ceramics with the value of 0.2 mm to 0.6 mm on the tilt angle of the grooved mirror with the membrane thickness H₁ of 0.6 mm to 1.4 mm is shown in Fig. 7. Since the structural stiffness of PLC transducer is improved due to the increase of the thickness d of PZT ceramic, the tilt of grooved mirror reduces gradually with the increase of thickness d of PZT ceramic. According to literature [5], the thickness d of PZT ceramic has little impact on the displacement of PLC transducer. Therefore, when design the parameters of thickness d of PZT ceramic, it shall try to increase the thickness d of PZT ceramic in order to reduce the tilt of grooved mirror.

Fig. 7.

Tilt angle of the grooved mirror versus membrane thicknesses for various thickness of PZT ceramics.

The influence of the radius R of PZT ceramics with the value of 6 mm to 12 mm on the tilt angle of the grooved mirror with the membrane thickness H₁ of 0.6 mm to 1.4 mm is shown in Fig. 8. The tilt angle of the grooved mirror decreases linearly as a function of decrease in the radius R of PZT ceramics for various membrane thicknesses. Figure 9 depicts the influence of membrane depth D on the tilt angle for various membrane thicknesses H₁. For a fixed membrane thickness H₁, membrane depth D varies from 0.5 mm to 6.5 mm. The tilt angle of the grooved mirror presents a non-linear relationship with the membrane depth for a given membrane thickness. We can see that there is a noticeable increase in the tilt angle of the grooved mirror as the membrane thickness decreases. For each membrane thickness H₁, a peak will occur in the maximum tilt angle of the grooved mirror, therefore, we shall avoid this point when select the parameter of membrane depth D. According to literature [5], membrane depth D has little influence on the displacement of PLC transducer, thus the smaller of membrane depth D in design, the better.

Fig. 8.

Tilt angle of the grooved mirror versus radius of PZT ceramics for various membrane thicknesses.

In Fig. 10, the tilt angle of the grooved mirror versus base radius r for various membrane thicknesses H₁ is plotted. The base radius r is defined as the radius of the coating base of the grooved mirror (shown in Fig. 2). Increasing the base radius from 2 mm to 5 mm can result in the decrease of the tilt angle. When base radius r = 5 mm, there is little difference of tilt angle resulted by the change of membrane thicknesses. Based on literature [5], the increase of base radius r will cause the decrease of axial displacement of grooved mirror; therefore, factors in two aspects shall be considered comprehensively during the design of base radius r.

Fig. 9.

Tilt angle of the grooved mirror versus membrane depth for various membrane thicknesses.

Fig. 10.

Tilt angle of the grooved mirror versus base radius for various membrane thicknesses.

According to the above analysis, combining with the influence rule of structural parameters on the displacement of PLC transducer in literature [5], we need to find the optimal value of each single variable, determine a new set of parameters, and combine them into an optimal structure. In practice, we hope that the larger the displacement of PLC transducer changes, the smaller the tilt angle changes. We can set the existing structure parameters as the reference value. An evaluation function as shown in formula (7) is introduced to search the optimal parameters. $\begin{eqnarray}\displaystyle Opt=0.5\times (d/D)+0.5\times (T/t). & & \displaystyle\end{eqnarray}$ (7)

In which, D and T are the displacement and tilt angle of the existing PLC transducer, d and tT are the displacement and tilt angle obtained by the variable. The coefficients are set to 0.5 because the weight of the displacement and the tilt angle are the same. The bigger the Opt is, the better the value is. The simulation results are listed in Table 3. It is shown that careful selection of the structural parameters allows the tilt angle of the grooved mirror to be decreased to 0.33”, which is 21.4% of current structure; meanwhile, the displacement of PLC transducer could reach 3.89 μm, which is 1.84 times of current structure. The optimized structure could not only reduce the tilt of grooved mirror, but also improve the displacement of PLC transducer. The PLC transducer of the optimal design will be more thermally stable if reduce the frequency of mode resets within the RLG, and improve the anti-vibration performance of ring-shaped resonant cavity at the same time.

Table 3

An optimal design and simulation result

Component	Piezoelectric actuator			Grooved mirror			Simulation result
Parameter	Base plate membrane thickness H2	PZT ceramics thickness d	PZT ceramics radius R	Membrane thickness H1	Membrane base radius r	Membrane depth D	Tilt angle (”)	Displacement (μm)
Existing structure	0.8 mm	0.4 mm	12 mm	0.8 mm	3 mm	3 mm	0.42	2.11
Optimal structure	0.75 mm	0.8 mm	11 mm	0.6 mm	3.5 mm	1.5 mm	0.33	3.89

5. Conclusions

In this paper, we quantitatively analyzed the tilt angle of the PLC transducer. The deformation of PLC transducer due to random vibration will cause deflection of resonant optical path of optical resonant cavity, which will directly affect the performance of ring laser gyroscope. Elastic deformations of PLC transducer with various shapes have been quantitatively analyzed in great detail, and the result of optimal design for PLC transducer taking axial displacement and tilt into account is given. These studies have made it possible to develop the PLC transducer of several designs for various applications and provide guidelines for optimization to improve RLG performance over a wide external vibration condition.

Footnotes

Acknowledgements

This work was supported by the National Science Foundation of China under grant 61503399 and navy advanced project 3020107010204.

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