1. Introduction
Risley prisms are a widely used beam-steering mechanisms, with their compact structure, high precision, low moment of inertia, and low cost [
1,
2], they are often used as beam-pointing devices and beam-scanning devices in laser communication [
3], large-FOV imaging [
4], photoelectric tracking [
5], lidar [
6], laser scanning [
7], and other fields. Risley prisms consist of a pair of optical wedges that rotate around a common axis independently, which enables high-resolution arbitrary beam-pointing over a wide range.
The features and advantages of Risley prisms make them suitable as a coarse or fine pointing mechanism for laser communication acquisition, tracking, and pointing (ATP) systems, especially for inter-satellite and airborne laser communication.
Figure 1 shows the optical layout of the laser communication system based on the Risley prism’s coarse tracking mechanism. The accuracy of the beam-steering mechanism of the ATP system is one of the key factors affecting the system’s performance. The pointing accuracy of the Risley prism is mainly determined by two factors: the accuracy of the solution method and the impact of the error on the solution. The former refers to the precision of solving the mathematical relationship between the rotation angle of prisms
and the spatial orientation of the outgoing beam
. The definition of the prism rotation angle
and the deflection angle and azimuth of the outgoing beam
are shown in
Figure 2. The trapezoid formed by the dotted line in the figure is the principal section of the prism, and the red arrows represent the light rays.
The solution consists of two aspects: solving the direction of the emitted light according to the prism’s rotation angles, called the forward solution, and solving the prism’s rotation angles with a known direction of the emitted beam, called the inverse solution. Yang used the vector form of Snell’s law to perform nonparaxial ray tracing and gave an exact forward analytical solution for the first time [
8]. For the inverse solution, the two-step method [
9], table look-up method, third-order approximate expression solution [
10], damped least-squares iterative solution [
11], and forward iterative optimization solution [
12] have been proposed.
The systematic errors of Risley prisms also impact the accuracy of the forward and inverse solution. The systematic error refers to the deviations between the actual and theoretical parameters of the Risley prisms system, which usually has a greater impact than the solution errors. For the impact of systematic errors on the performance and forward solution accuracy, a lot of work has been carried out. Horng investigated the optical distortions of scan patterns caused by the component errors and alignment errors of prisms [
13]. Zhou et al. analyzed the influence of component errors, prism orientation errors, and assembly errors on pointing accuracy. The allowances of the error sources for a given pointing accuracy were evaluated [
14].
There are relatively few studies focusing on compensating for the impacts of system errors. Bravo-Medina et al. proposed an error correction method based on the paraxial approximation model [
15]. Under the paraxial approximation model, an error vector was added to compensate for the system error. The method model is simple, easy to implement, and considers the influence of the thickness and spacing of the prism on the outgoing beam. However, the solutions obtained by the paraxial approximation method will deviate from the exact solution, especially when the deflection angle is large [
16]. So, it is not suitable for large-angle deflection applications.
Li et al. proposed an error parameter identification method [
17]. This method is based on an analytical model of achromatic Risley prisms. The genetic algorithm was used to fit the actual parameters of the physical model, which can achieve higher accuracy than the approximate model. However, not all errors are included in the error fitting of the article. The Risley prism used in the article has a small pointing range (±3°), so errors that have little effect on pointing accuracy are ignored. In a system with a large pointing range, the beam-pointing deviation caused by these errors cannot be ignored.
However, the above studies all focus on the impact of errors on the forward solution; there are few studies on the impact of errors on the inverse solution. Considering the accuracy and the real-time and data volume requirements of laser communication, the two-step method is the most suitable inverse solution method. Since the solution of the two-step method is based on the ideal motion trajectory of the spot, the offset of the actual motion trajectory caused by the systemic error of the prism will cause the pointing error of the inverse solution. This paper investigates the inverse solution error of Risley prisms induced by systemic error and proposes a method based on pointing-field transformation for correction.
The outline of this paper follows. A more accurate error model is established in
Section 2. In
Section 3, the causes of the inverse solution error are explained, and simulations are performed to analyze the impact of the inverse solution error on pointing accuracy. Furthermore, the correction algorithm is proposed. In
Section 4, the feasibility of the correction algorithm is verified by simulation and experiment. Conclusions are drawn in
Section 5.
2. Error Model of Risley Prisms and Forward Solution Error Correction Method
2.1. Forward Solution and Inverse Solution of Risley Prisms
The precise forward and inverse solution of Risley prisms is derived by nonparaxial ray tracing through the vector form of Snell’s law shown in Equation (1) [
18].
where
and
are the refractive indices of the medium on both sides of the refractive surface,
is the unit normal vector of the refraction interface,
is the unit vector of the incident light, and
is the vector of the exit light. The notations of the prism surface normal vector and the ray direction vector are shown in
Figure 3. The refractive indices of prism 1 and prism 2 are denoted by
and
, respectively. The direction of the incident light is defined as the reverse direction of the
Z-axis, so its vector is
.
Calculations are performed using Equation (1) in each refractive surface of the Risley prism, and the direction vector of the outgoing beam can be obtained:
where
K,
L,
M are the final calculation results; they are functions of the prism rotation angles
and
, and they are also the components of the outgoing light vector on the
X,
Y, and
Z axis.
The deflection angle and azimuth of the outgoing beam can be calculated through the direction vector:
The inverse solution of Risley prisms is solved by the two-step method. Since the angle between the two prisms determines the deflection angle of the outgoing beam. The first step is to calculate the included angle between two prisms , which corresponds to the deflection angle of the desired beam-pointing position. The current prism angle is known, assuming that one prism remains stationary and the other one rotates , calculating the azimuth angle of the beam at this time. This step makes the deflection angle of the outgoing beam reach the target value.
When two prisms rotate 360° simultaneously with the same speed and direction at a certain included angle, the deflection angle of the outgoing beam remains unchanged, and the azimuth angle changes continuously. Hence, the trajectory of the light on the OXY surface is a circle centered on the intersection of the Z-axis and the OXY surface. We use the abbreviation “TLS” to specifically refer to this trajectory. Therefore, the second step is to rotate the two prisms by
simultaneously in the same direction. According to the two-step method process, the final rotation angles required of prisms can be obtained:
or
Since there are two ways of rotation in the first step, two sets of solutions can be obtained.
2.2. Error Model of Risley Prisms
The error model of the Risley prisms is obtained by introducing systemic errors into the mathematical model of the Risley prisms. The systemic errors of Risley prisms can be divided into component errors and assembly errors [
14]. Component errors refer to the deviation of the prism wedge angle and the refractive index from the design value, which may change with temperature or wavelength. Assembly errors include prism rotation axis tilt, prism tilt, incident light tilt, and prism rotation angle error. Their description and notation are shown in
Figure 4.
Take prism 1 as an example. Ideally, the prism rotation axis coincides with the axis, is the actual rotation axis which deviates from the Z axis, and the error between the two is represented by . is the angle between the axis and plane OZX, which represents the vertical error of the axis. is the angle between the projection of the axis on the plane OZX and the axis, and represents the horizontal error of the axis. The coordinate system denotes the coordinate system of the tilting prism; the tilt error of the prism in the vertical and horizontal directions are denoted by , respectively.
In the same way, the tilt errors of the incident light in the vertical and horizontal directions are represented by , respectively.
In
Figure 2, the configuration when the rotation angle of the prisms is
is defined as the zero position of the Risley prisms. The analytical model of the Risley prisms is derived when the prisms are in the zero position; therefore, the rotation angle of the prisms is calculated from the zero position. In practice, the prisms cannot be precisely mounted at the zero position, as the prism rotation angle error refers to the angle between the actual initial mounting position of the prism and the zero position. The symbols
and
are used to denote the rotation angle errors of prism 1 and prism 2, respectively.
According to the errors defined in
Figure 4, the incident light vector
, the normal vectors of the prisms’ four surfaces
, and the prism rotation axis vectors
can be obtained. Let
; the transformation matrix of any vector rotating around
can be obtained by using Rodrigues’ rotation formula:
where
E is the unit matrix, and
is the rotation angle of prism 1.
When prism 1 rotates around , the normal vectors of its two surfaces are and . In the same way, the normal vectors of the two surfaces of prism 2 can be obtained: and , where the transformation matrix of any vector rotating around .
Substituting the above vectors into Formula (1) in turn for calculation, the emitted light vector can be derived.
2.3. Error Parameter Identification
On the basis of the error model, the exact or equivalent values of errors can be obtained by parameter identification [
17]. As shown in Equation (8), the error model describes the relationship between prism rotation angles
and errors and the deflection angle and azimuth of the outgoing beam
.
The essence of the parameter identification method is to fit the exact or equivalent value of the error through the experimental data.
Among the several systemic errors in the error model, the incident light tilts and prism rotation angle errors are constants, while the bearing rotation axis wobble is a systematic error that changes periodically with the rotation of the bearing. Therefore, we set the prism rotation axis tilt as a function of the prism rotation angles and . After the prism is mounted on the bearing, the angle between the prism and its rotation axis is a fixed value, so we use the rotation axis tilt to represent the prism tilt . With sufficient data and a suitable optimization algorithm, the exact or equivalent value of the errors shown in Equation (8) can be fitted by the least-squares method.