1. Introduction

In recent years, dual liquid crystal polarization gratings (LCPGs) have shown good promise as beam steering devices for free-space optical communications, countermeasures, laser weapons, and fiber optic switches where beam alignment and target tracking are required [1], [2]. In 2009, Kim proposed the beam control concept of "dual liquid crystal polarization gratings", which consists of two independently rotating polarization gratings interferometrically inscribed by orthogonal circular polarization [3–5]. When the dual LCPGs are rotated, the outgoing beam diffracted by the two LCPGs is scanned within a range determined by the period of the LCPGs and their relative orientation [6].

At present, the steering angle of the beam of the traditional mechanical scanning device is usually linear with the rotation angle of each axis, which simplifies the rotation control structure of the rotating parts, so it is widely used in the photoelectric capture tracking system, but the universal frame type is large in size and slow in response frequency. Dual LCPGs with a thin plate containing polarization grating to replace the volume and weight of the Risley prism elements, and the use of their highly polarization-sensitive diffraction [7], [8]. The establishment of a device for tracking targets with dual LCPGs is an improvement from the dual prisms target tracking device, and the basic premise of research on rotating dual prisms and rotating dual LCPGs in the theory and application of target tracking is to resolve the mathematical relationship between the rotation angle of the two prisms or LCPGs and the position of the target, which can be divided into forward and reverse solutions. The two-step method was first proposed for optimizing the inverse solution process of the rotating Risley prisms, which reduces the operational clutter by simplifying the variables. The forward and reverse solutions for rotating Risley prisms beam pointing were then derived, and the two-step method was applied to find the beam pointing altitude and azimuth angles [9–14]. In the process of research on LCPGs [15], a large-angle non-mechanical beam deflection system based on polymer LCPGs is designed to achieve coarse deflection control at 1064 nm with a transmission rate of 66$\%$-70$\%$. In [16], a super binary coarse deflection design method based on polymer LCPGs is proposed, demonstrating that the transmission rate can be improved by optimizing the substrate and electrode materials. In [17], a polarized laser beam steering system based on multi-stage rotating PGs is proposed, and the beam steering performance of two and four rotating grating schemes is analyzed theoretically, concluding that the grating can provide a faster rotation than the prism.

The reverse solution in the above studies is not the reverse solution required in the tracking application, but only the mathematical reverse solution of the forward solution pointed by the outgoing beam. At this stage, the scanning and pointing problems based on dual LCPGs are forward solutions with simple algorithms and the emitted beam is always in the direction of the grating optical axis. Therefore, the forward beam deflection formula derived in previous papers is only applicable when the incident light is incident along the optical axis of the grating [6]. Other studies on LCPGs have focused on improving their beam steering performance and beam energy utilization, and no target tracking device based on dual LCPGs and the inverse solution to tracking formula has been established in the literature. When tracking a target using dual LCPGs, the beam diffracted by the dual LCPGs reaches the tracked target, and the beam reflected from the target then passes through the dual LCPGs, which is the case when the light with incident angle enters the LCPGs. Therefore, both forward and reverse solutions are required for target tracking applications, making the algorithm more complex. The reason why the dual LCPGs cannot be used for target tracking is that the relationship between the target position and rotation angle of dual LCPGs is non-linear and does not introduce the miss distance into the formula, but our study solves this problem and uses the formula as the basis for the design of tracking experiments to verify the usability of the proposed formula. It is also important to note that the small size and light weight of the dual LCPGs are important features that can be installed on other integrated optical mechanisms or on-board platforms, whether compared to the coarse tracking device of the gimbal frame or the Risley prism.

In this paper, we first propose the relationship between the miss distance, target position, and the rotation angle of the LCPGs, give the derivation process of the proposed formula and establish a target tracking system based on rotating dual LCPGs, and describe the closed-loop target tracking process based on the formula. We establish the optical structure of the dual LCPGs in ZEMAX with the period is 8.222$\mu$m to verify the availability of the formula, comparing the output altitude angle and azimuth angle of the formula and the ZEMAX simulation. We build a test optical platform and the dual LCPGs servo unit can be calculated the altitude angle and azimuth angle of the target position according to the proposed formula. We design and complete the tracking process described in the second chapter with the coarse tracking detector and the servo system forms a closed loop tracking. Statistical analysis of the remaining miss distances gives a coarse tracking accuracy that meets our proposed technical specification.

2. Target tracking principle based on rotating dual liquid crystal polarization gratings

2.1 Model of a rotating dual LCPGs

Figure 1 is a schematic diagram of a rotating dual LCPGs beam system, where both the LCPG 1 and the LCPG 2 are written by orthogonal circular polarization interference. The LCPG 1 and LCPG 2 are placed parallel to the common optical axis, and a right-handed coordinate system is established with the opposite direction of the optical axis as the positive z-axis and the grating line as the positive x-axis as shown in the figure, the outgoing beam is uniquely determined by the polar coordinates. Figure 2 shows $\Phi$ is the altitude angle of the target position, $\Theta$ is the azimuth angle of the target position, $\Phi _{x}=\Phi \cos \Theta$, $\Phi _{y}=\Phi \sin \Theta$, the coordinate z-axis is its common rotation axis, that is, the system optical axis. LCPG 1 and LCPG 2 are rotated independently on the common optical axis, with counterclockwise rotation around the z-axis as the positive direction. ${\theta _{1}}$ is the rotation angle position of LCPG 1 around the z-axis, and ${\theta _{2}}$ is the rotation angle position of LCPG 2 around the z-axis. By varying ${\theta _{1}}$ and ${\theta _{2}}$, the outgoing beam can be adjusted in any direction within the conical observation field defined by the maximum altitude angle ${\Phi _{max}}$ that the system can achieve.

 

Fig. 1. Schematic of rotating dual LCPGs beam system

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Fig. 2. Representation of the polar coordinate system’s altitude angle and azimuth angle

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As shown in Fig. 3, when the target is in the observation field of the system, the light source illuminates the target, and the light reflected from the target is diffracted by LCPG 1 and the LCPG 2 in turn to form an image point on the detector; because the optical path is reversible, the light beam is diffracted by LCPG 2 and the LCPG 1 and then emitted to the target position. The controller outputs a voltage signal for motor 1 to rotate the LCPG 1 to the new angle position ${\theta _{11}}$ and for motor 2 to rotate the LCPG 2 to the new angle position ${\theta _{22}}$. $\Delta \theta _{1}=\theta _{1}-\theta _{11}$ is the rotation angle of the LCPG 1 and $\Delta \theta _{2}=\theta _{2}-\theta _{22}$ is the rotation angle of the LCPG 2 according to the rotation angle position ${\theta _{1}}$ of the LCPG 1, the rotation angle position ${\theta _{2}}$ of the LCPG 2, and the altitude angle ${\Phi _{0}}$ and the azimuth angle ${\Theta _{0}}$ of the image point on the detector. To control the rotation of the LCPGs to track the target position, it is necessary to determine the relationship miss distance, target position, and rotation angle of dual liquid crystal polarization gratings.

 

Fig. 3. Schematic of rotating dual LCPGs target tracking device

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2.2 Relationship between miss distance, target position, and rotation angle of the LCPGs

We represent the diffracted beam of the LCPG in directional cosine space. The outgoing beam diffracted by the LCPG has a conical shape and any of the outgoing beams can be represented in directional cosine space as:

So the directional cosine of the outgoing beam vector can be expressed as:

To facilitate the representation of the directional cosine in terms of the altitude angle $\Phi$ and azimuth angle $\Theta$ that we need, we put the expressed directional cosine into a spherical coordinate system, which facilitates the representation of the incident beam vector using the altitude angle $\Phi$ and azimuth angle $\Theta$.

From the Fraunhofer diffraction equation, it can be found that the angle of deflection of the positive and negative primary light at vertical incidence satisfies the equation:

When an incident at an arbitrary angle of inclination (deflection (incidence), azimuth (incidence)), the outgoing beam of order m diffracted through a liquid crystal polarization grating can be expressed as:

When the birefringent phase delay of the liquid crystal layer is an odd multiple of $\pi$ and the incident light is completely modulated into left-handedly or right-handedly polarized light, the theoretical diffraction efficiency of the diffracted beam passing through the LCPGs is 100$\%$ for the positive or negative level, and the outgoing beam passing through LCPG 1 and then through LCPG 2 is of opposite levels. Combined with the coordinate system established in this paper, since the opposite direction of the optical axis is the positive direction of the z-axis and the grating line is the positive direction of the x-axis to establish a right-handed coordinate system, the directional cosine (K, L, M) of the outgoing beam passing through the dual LCPGs when the left-rotating circularly polarized light is incident vertically can be obtained as [6]:

The incident beam is diffracted by dual LCPGs and the outgoing light is directed towards the target. The light reflected from the target is then captured by the detector through the dual LCPGs. This process can be understood as the introduction of an incident angle into the incident beam, which can represent the miss distance, and after dual LCPGs diffraction, the altitude angle and azimuth angle of the outgoing to the target can be obtained.Introduce an incident vector as:

By introducing the incident vector into the directional cosine of the outgoing beam passing through the dual liquid crystal polarization gratings at vertical incidence, we then establish the relationship between the miss distance, the target position, and the rotation angle of the dual LCPGs as follows:

So the altitude angle and azimuth angle of the outgoing beam reaching the target can be expressed as:

2.3 Target tracking process based on rotating dual LCPGs

Due to the limited inherent field of view of the detector, it is first necessary to complete the target tracking process based on guidance data for a given target position, after the target is imaged in the detector’s field of view, in the following steps so that the target is always imaged in the center of the detector’s field of view.

Step 1: The altitude angle ${\Phi _{0}}$ and the azimuth angle ${\Theta _{0}}$ of the target point on the detector are measured by the detector; the rotation angle ${\theta _{1}}$ of the LCPG 1 is measured by the angle encoder 1 and the rotation angle ${\theta _{2}}$ of the LCPG 2 is measured by the angle encoder 2; the altitude angle ${\Phi }$ and the azimuth angle ${\Theta }$ of the target position are calculated from the previously obtained ${\Phi _{0}}$, ${\Theta _{0}}$ , ${\theta _{1}}$ and ${\theta _{2}}$ according to the derived Eqs. (8)–(10).

Step 2: Using the azimuth angle $\Theta$ and the altitude angle $\Phi$ of the target calculated in step 1 as the pointing position of the beam at vertical incidence, find the inverse solution of the beam at vertical incidence according to the two-step method and obtain the new angles $\theta _{1}^{\prime }$ and $\theta _{2}^{\prime }$ for the rotation of the LCPG 1 and LCPG 2. Find the difference between the rotation angles of the LCPG 1 and the LCPG 2 at the pointing positions ($\Phi$, $\Theta$), where $|\Delta \theta |=\left |\theta _{2}-\theta _{1}\right |$. When the beam is incident vertically, the forward solution formula pointed by the beam needs to be substituted into Eq. (6) , Eq. (9) , Eq. (10) to obtain the formula proposed by Zhou Yuan [6]:

From Eqs. (11) and (12), the difference between the rotation angle of LCPG 1 and LCPG 2 is solved :

The first step of the two-step method: using Eqs. (4)–(7) as a theoretical basis, keep the LCPG 1 unchanged and make the LCPG 2 turn counterclockwise or clockwise to the desired $|\Delta \theta |$, the above two cases will result in the corresponding two beam pointing azimuths $\Theta ^{\prime }$ and $\Theta ^{\prime \prime }$. The second step of the two-step method: After rotating the LCPG 2 counterclockwise or clockwise to the desired $|\Delta \theta |$, the angles that LCPG 1 and LCPG 2 need to rotate clockwise simultaneously are $\Theta -\Theta ^{\prime }$ and $\Theta -\Theta ^{\prime \prime }$, which gives two sets of solutions for the new angles that LCPG 1 and LCPG 2 need to rotate to ($\theta _{1}^{\prime }$, $\theta _{2}^{\prime }$), and the two sets of solutions are as follows:

Step 3: According to step 1, $\theta _{1}$ and $\theta _{2}$ are the current angular positions of LCPG 1 and LCPG 2; according to step 2, Eq. (14) and Eq. (15) are the desired angular positions of LCPG 1 and LCPG 2. The servo controller calculates the angles of LCPG 1 and LCPG 2 to be rotated respectively, and the calculation process is as follows:

Step 4: The servo controller controls motor 1 to drive the LCPG 1 to the angular position $\theta _{1}^{\prime }$, and motor 2 to drive the LCPG 2 to the angular position $\theta _{2}^{\prime }$. The control algorithm uses a PID algorithm to control the rotation of the LCPG 1 and the LCPG 2 so that the target is always imaged to the center of the detector’s field of view. The detector and the grating servo system form a closed-loop tracking system, thus tracking the target stably and continuously.

3. Simulation to verify the relationship between miss distance, target position, and rotation angle of the LCPGs

3.1 ZEMAX design for dual LCPGs optical structure

The incident beam wavelength is set to 1064 nm, the grating period is 8.222$\mu$m, and the LCPG 1 and the LCPG 2 are placed parallel to the common optical axis, as shown in the Fig. 4 with the target position looking in the reverse direction of the optical axis, the grating line is the positive direction of the x-axis, the target position is uniquely determined by the polar coordinates ($\Phi$, $\Theta$). The coordinate z-axis is the common rotation axis of the LCPG 1 and the LCPG 2, i.e. the system optical axis. The dual LCPGs can be freely rotated counterclockwise or clockwise within the range (0$^{\circ }$, 360$^{\circ }$), and multiple structures are set up in ZEMAX to rotate the dual LCPGs. The incident beam is directed towards the target along the z-axis. We introduce an angle of incidence into ZEMAX to represent the miss distance, which is diffracted by dual LCPGs that can be rotated at different angles to obtain the altitude angle and azimuth angle ($\Phi$ , $\Theta$) of the outgoing beam to the target.

 

Fig. 4. Dual LCPGs optical structure.

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3.2 Verification process

We propose a non-linear formula between the miss distance, the target position, and the rotation angle of the dual LCPGs. Set the input amount as the miss distance and the rotation angle of the dual LCPGs, use the target position as the output amount, and compare the output amount of the formula and the output amount of the ZEMAX simulation, to verify the correctness of our proposed formula and propose the dual gratings coarse tracking accuracy needs to be no more than 0.45mrad (RMS). In the simulation process, we need to control the variables, control the miss distance, rotate the dual LCPGs to different angles and control the rotation angle of the dual LCPGs constant, change the miss distance, compare the difference between the equation of the target position and the ZEMAX simulation of the target position of the altitude angle and azimuth angle.

In ZEMAX, set the incident altitude angle and azimuth angle, set the miss distance altitude angle at 0.1$^{\circ }$, change the azimuth angle from 0$^{\circ }$ in ten-degree increments to 360$^{\circ }$, rotate the LCPG 1 counterclockwise by 10$^{\circ }$, and rotate the LCPG 2 counterclockwise by 80$^{\circ }$; in the proposed equations, set $\Phi _{0}$= 0.1$^{\circ }$, $\Theta _{0}$ from 0$^{\circ }$ in ten-degree increments to 360$^{\circ }$, $\theta _{1}$=10$^{\circ }$, $\theta _{2}$=80$^{\circ }$. The first eight sets of data from the ZEMAX simulation tracing and substitution into the equation are shown as examples in Table 1. The altitude angle comparison chart is shown in Fig. 5 and the azimuth angle comparison is shown in Fig. 6.

As can be seen from the data obtained, the difference between the target position altitude and azimuth data obtained from our proposed formula and the target position altitude and azimuth data obtained from the ZEMAX simulation is very small, leaving a lot of margin for mechanical errors and other systematic errors for the coarse tracking accuracy we want to obtain. Although we cannot traverse all the numerical simulations, we have verified the usability of our proposed formula by varying the amount of miss distance and the rotation angle of the LCPGs.

 

Fig. 5. Target altitude angle of miss distance $\Phi _{0}$= 0.1$^{\circ }$, $\Theta _{0}$ $\ \in$ (0$^{\circ }$, 360$^{\circ }$).

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Fig. 6. Target azimuth angle for miss distance $\Phi _{0}$= 0.1$^{\circ }$, $\Theta _{0}$ $\ \in$ (0$^{\circ }$, 360$^{\circ }$).

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Table 1. Data table for miss distance $\Phi _{0}= 0.1, \Theta _{0} \in (0, 360)$

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Keeping the miss distance azimuth angle and azimuth angle set in ZEMAX unchanged, LCPG 1 is rotated 370$^{\circ }$ counterclockwise and LCPG 2 is rotated 80$^{\circ }$ counterclockwise. In the proposed equations, $\Phi _{0}$=0.1$^{\circ }$ and $\Theta _{0}$ are set from 0$^{\circ }$ to 360$^{\circ }$ in ten degree increments, $\theta _{1}$ =370$^{\circ }$, $\theta _{2}$=80$^{\circ }$. A set of data tables can also be obtained from the ZEMAX simulation tracing and substitution formulas, which are the same as those in Table 1. Keeping the miss distance azimuth angle and azimuth angle set in ZEMAX unchanged, LCPG 1 is rotated 350$^{\circ }$ clockwise and LCPG 2 is rotated 80$^{\circ }$ counterclockwise. In the proposed formulas, and are set from 0$^{\circ }$ in decimal increments to 360$^{\circ }$, $\theta _{1}$=-350$^{\circ }$ and $\theta _{2}$=80$^{\circ }$. A set of data tables can also be obtained from the ZEMAX simulation traces and substitution formulas, and the comparison shows that the data obtained is the same as in Table 1. The simulated data plots are also the same as Fig. 5 and Fig. 6. We have verified the same principle for LCPG 1 and LCPG 2 respectively by verifying multiple sets of data obtained, when the rotation angle of the LCPGs counterclockwise or clockwise rotation of 360$^{\circ }$ plus or minus integer multiples, as well as the rotation angle difference of 360$^{\circ }$ plus or minus integer multiples, the two cases, belong to the rotation of the loop and rotation reverse. Both cases yield the same target position data in the equation and ZEMAX simulation tracing. This phenomenon indicates that the rotation angle of the grating can be reduced in engineering tests, which can ensure the stability and continuity of the target tracking and prevent the loss of the target and is of guidance for engineering applications.

3.3 Comparative validation

To demonstrate the advantages of our proposed formula compared to the vertically incident formula [6], explain why we introduce the miss distance into the target position formula for the beam incident vertically to the rotating dual LCPGs. We apply each of the two formulas to target tracking, following the target tracking process in Chapter 2 for simulation. We set the LCPG 1 to rotate 10 degrees and the LCPG 2 to rotate 80 degrees in the simulation of the vertical incidence formula. In our proposed formula, the dual LCPGs rotate at the same degree, the azimuth angle of the miss distance azimuth is set to 80 degrees, and the altitude angle of the miss distance is set to 0.1 degrees in steps of 0.1 degrees until the altitude angle is 0.8 degrees. The target positions for each set of data from the two formulas are compared, the differences are listed in the Table 2 and the necessity of the existence of the new formula is explained.

Table 2. Difference between the target positions obtained from the old and new formulas

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Altitude angle comparison is shown in Fig. 7(a) and the azimuth angle comparison is shown in Fig. 7(b). We can find that a very small value of the miss distance results in an error of 11.7775mrad in the altitude angle of the target position and 50.0456mrad in the azimuth angle of the target position. We set the ratio of the grating period to the beam wavelength to 7.7274 and the rotation angle of the dual LCPGs to be the same in the proposed formula and the formula for vertical incidence presented in previous papers. If the previous formula is applied to the target tracking, it is assumed that the emitted beam is always along the optical axis of the grating and the miss distance is always zero, but in reality, the miss distance returned from the target has a numerical value. In the verification comparison, we set a smaller miss distance and the beam is deflected through the dual LCPGs and propagated onto the target. By substituting the miss distance and the rotation angle into the new formula, the actual position of the target can be found. Evaluate the difference between this position and the target position obtained by the original formula, and the resulting difference is the amount of error caused by the previous formula.

 

Fig. 7. Plot of (a) altitude angle and (b) azimuth angle of the target for two formulas.

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Because the target position is wrong, there will also be a difference in the angle of rotation of the grating that makes the miss distance zero, following the steps of the tracking process described in Chapter 2, which are solved backward with the wrong target position. By rotating the grating according to the resulting rotation angle, the outgoing beam will point away from the true target position, resulting in a loss of the target. In fact, when tracking a target using dual LCPGs, the beam diffracted by the dual LCPGs reaches the tracked target, and the beam reflected from the target then passes through the dual LCPGs, which is the case when the light with incident angle enters the LCPGs. Therefore, both forward and reverse solutions are required for target tracking applications, making the algorithm more complex. The previous formula for vertical beam incidence can only be used for beam pointing and does not apply it to target tracking applications. Therefore, it is necessary to introduce the miss distance into the vertical incidence formula to obtain the relationship between the miss distance, the target position, and the rotation angle of the liquid crystal polarization grating if the target is to be tracked stably.

4. Track experimental verification

4.1 Experimental steps

The coarse tracking accuracy of the dual LCPGs is mainly affected by dynamic hysteresis error and the angular resolution of the image elements. In the absence of subdivision, the image tracking algorithm error is considered at 1 pixel. The system input attitude perturbation characteristics use a sinusoidal signal with an amplitude of 5$^{\circ }$ and a frequency of 0.2hz to simulate the platform’s attitude perturbation, and a vibration angle signal calculated from the actually collected acceleration signal is used as the vibration perturbation. The above errors approximately follow a Gaussian distribution, and with a margin for other mechanical errors, we set the technical specification for the coarse tracking error under the action of the sinusoidal signal body perturbation to no more than 0.45mrad.

The coarse tracking servo unit is an important component of the dual LCPGs servo control system, which is mainly responsible for high probability, fast capture, and high stability and accuracy tracking. Therefore, it is necessary to test the system’s coarse tracking accuracy index and evaluate whether it can meet the technical indicators of no more than 0.45mrad proposed by us.

As shown in Fig. 8, the test optical platform is built. A diagram of the experimental setup is shown in Fig. 9. The detector frame rate is 100 frames$/$s, the focal length is 960 mm, the angular resolution is 15.625 $\mu rad$, the cell size is 15 $\mu m$, and the number of cells is 640*512. The experimental setup includes a laser, a parallel light tube, a six-degree-of-freedom pendulum, an off-axis optical antenna, and a coarse tracking detector. The experimental setup also has a fine-tracking subunit including a fine-tracking detector and a fine-tracking oscillator, which is used for subsequent experiments and is not used here for verification. First of all, the optical axis of the dual LCPGs servo control system is aligned with the optical axis of the parallel light tube. The laser emits a 1064nm laser, which is then used as the target to be tracked. The laser is emitted through the parallel light tube and is narrowed by the off-axis optical antenna, and the dual LCPGs servo unit rotates the two gratings by a specific angle according to the altitude angle and azimuth angle information given by the external guidance data so that the dual LCPGs points to the target so that the target enters its coarse tracking field of view, and the dual LCPGs is rotated to adjust the light spot received by the coarse tracking unit to the center of the field of view. The six-degree-of-freedom pendulum is then rotated in a counterclockwise direction by 9.5$^{\circ }$ and the dual LCPGs is also rotated in this state to bring the target into the coarse tracking field of view. The six-degree-of-freedom pendulum is then powered up and set to move at 5$^{\circ }$, 0.2hz sinusoidal command in this state.

 

Fig. 8. Experimental set-up diagram.

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Fig. 9. Experimental scenario diagram.

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During the work of the coarse tracking unit, after capturing the target signal, the coarse tracking detector extracts the difference between the target image and the center point to obtain the miss distance information. According to the tracking process proposed in chapter 2, firstly the dual LCPGs servo unit can calculate the altitude angle and azimuth angle of the target position using the proposed formula according to the miss distance and the grating rotation angle at this time. Based on the calculated target position, the dual LCPG inverse solution formula can be used to obtain the new angular position that needs to be reached when the miss distance becomes zero. The grating servo units control the rotation of the dual LCPGs to carry out coarse tracking alignment of the target, coarse tracking detector, and servo system to form a closed-loop tracking, to ensure that the coarse tracking accuracy meets the proposed indicators, so that the target into the fine tracking field of view, to ensure that the fine-tracking subunit normal. At this point, the coarse tracking accuracy can be derived from the data obtained by statistical analysis based on the remaining miss distance. Rotate the six-degree-of-freedom pendulum counterclockwise and use the proposed formula to track the target, and continuously record 3000 sets of miss distance data after tracking. During the process of target tracking using the proposed formula when the six-degree-of-freedom pendulum rotates counterclockwise, 3000 sets of miss distance data after tracking are continuously recorded. Similarly, the six-degree-of-freedom pendulum clockwise movement according to the command of 5$^{\circ }$, 0.2hz, also continuously records three thousand sets of miss distance data after tracking.

4.2 Experimental results and discussions

The X-axis or Y-axis miss distance information cannot independently characterize the coarse tracking accuracy, in this paper, we convert the target position information and miss distance information into altitude and azimuth characterizations. As shown in Fig. 2, the X-axis miss distance is $\Phi _{\mathrm {x} 0}=\Phi _{0} \cos \Theta _{0}$ and the Y-axis miss distance is $\Phi _{y 0}=\Phi _{0} \sin \Theta _{0}$. Therefore, the remaining miss distance collected by us can be calculated to obtain the corresponding altitude angle and azimuth angle. Using the three thousand sets of data obtained by rotating the six-degree-of-freedom pendulum counterclockwise as an example, plotting the data yields Fig. 10. The altitude angle and azimuth angle converted by 3000 sets of data can all be shown in Fig. 10. In Fig. 10(a), it can be preliminarily determined that the altitude angle of the remaining miss distance is between 74$\mu$rad and 740$\mu$rad, and most of the altitude angle is near the smaller altitude angle, so it can be preliminarily judged that the coarse tracking accuracy is in the two angle ranges. In Fig. 10(b), it can be judged that the azimuth angle of the remaining miss distance is between 180 degree and 360 degree, i.e. the converted miss distance is in the third and fourth quadrants. Figure 10 can comprehensively display the large amount of data collected, preliminarily determine whether the accuracy of the rough tracking meets the indicators, preliminarily explain the usability of the proposed formula in the tracking application, and guide us to continue to the next step of the precision calculation.

 

Fig. 10. Plot of (a) altitude angle and (b) azimuth angle data for remaining miss distance.

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The six thousand sets of data obtained were subjected to data processing to calculate whether the tracking accuracy of the coarse tracking met the proposed index and to analyze whether the data obtained from the tracking could verify the usability of the proposed formula. Due to a large amount of data, we only give ten sets of data for the counterclockwise rotation of the six-degree-of-freedom swing table as an example in this paper, and the data are listed in Table 3.

Calculate the three sets of data separately for $\left (X_{i}-\bar {X}\right )^{2}$ and $\left (Y_{i}-\bar {Y}\right )^{2}$, N=3000, using the following Eq. (18) for the accuracy of RMS and substituting the above data into the formula:

Table 3. Example of twenty sets of data

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The tracking accuracy of the coarse tracking is obtained by substituting the data into Eq. (18) obtained by rotating the six degrees of freedom pendulum clockwise, $\delta _{\text {c}}=434.8727\mu$rad. Similarly, the tracking accuracy of coarse tracking obtained by rotating the six degrees of freedom pendulum counterclockwise is $\delta _{\text {cc}}=408.8645\mu$rad. The coarse tracking accuracy is therefore within 450$\mu$rad for the six thousand sets of data collected in either the counterclockwise or clockwise rotation of the pendulum, which meets our proposed specifications and validates the usability of our proposed formula.

The coarse tracking devices that are widely used at this stage are large and heavy [18], [19]. If we want to mount them on other platforms, the coarse tracking devices should be as small and heavy as possible. In our research, we find a coarse tracking device like dual LCPGs, and after continuous exploration, we establish the relationship formula of miss distance, target position, and rotation angle of dual LCPGs. We simulate and verify that the difference between the proposed formula and the actual target position of tracking is extremely small. In the process of coarse target tracking, the dual LCPGs will perform beam pointing and when the beam reaches the target is reflected back and then imaged onto the detector by the dual LCPGs. After this step, the data of miss distance can be obtained, and the current grating rotation angle can also be obtained from the encoder. The current target position can be solved by substituting these two pieces of information into our proposed formula. We want to track the target stably and image the target always in the center of the detector. So we use the inverse solution process, assuming that the miss distance is zero, and the resulting rotation angle of the grating is the rotation position that the grating needs to reach this moment. This is how we build the system to continuously track a moving target coarsely.

The rotating dual liquid crystal polarization gratings device serves to control the beam to achieve coarse tracking of the target. The gimbal frame integrates detector devices and inertial devices on the multi-axis rotary frame and realizes the spatial movement of the beam by controlling the rotation of the multi-axis rotary frame. In contrast to gimbals, the rotating dual LCPGs are rotatably displaced rather than laterally displaced, and such components can be easily installed into the slot plate or other integrated optical mechanisms. More importantly, the dual LCPGs are much smaller and lighter than the gimbal frame. The Risley prisms have also been used for beam pointing control, and our dual LCPGs are an improvement on this. The dual LCPGs replace the bulky and heavy prismatic elements of the Risley prism with a thin plate containing the polarization grating and adopts their polarization-sensitive diffraction. This characteristic of small size and light weight makes the dual LCPGs occupy less space, which is an important feature of the coarse tracking device placed on the on-board platforms.

In summary, the rotating dual LCPGs are suitable for use as coarse tracking devices on on-board platforms. The reason why they cannot be used for target tracking is that the relationship between the beam pointing of the dual LCPGs, i.e.the position of the target, and the position of the rotation angle of the two LCPGs is non-linear, and the miss distance is not introduced into the relationship formula, and there is a lack of research in this area. Finding the mathematical relationship between the miss distance, the target position, and the rotation angle of dual LCPGs is the basis for target tracking applications. We propose the relationship between the three for the first time, design the tracking process based on the relationship, and realize the coarse tracking of the target using the rotating dual liquid crystal polarization grating for the first time. The coarse tracking accuracy of the system meets the proposed specification of no more than 0.45mrad. The results show that the proposed formula provides a good basis for stable target coarse tracking and is instructive for engineering applications.

5. Conclusion

In this paper, the relationship between miss distance, target position, and rotation angle of dual liquid crystal polarization gratings (LCPGs) is proposed for the first time, the derivation process of the proposed formula is given, and the incident vector is introduced into the direction cosine of the outgoing beam passing through the rotating dual LCPGs at vertical incidence. We establish a target-tracking system based on rotating dual LCPGs and describe the closed-loop target-tracking process based on the formula. In ZEMAX, we set up the optical structure of the dual LCPGs, and the difference between the altitude angle and azimuth angle of the target position derived from the proposed formula and the ZEMAX simulation is compared, thus verifying the correctness of our proposed equation. In the simulation validation, we also compare and analyze the proposed new formula with the formula for vertical incidence to illustrate the need to introduce the miss distance into the relationship. We build a test optical platform, the dual LCPGs servo unit according to the miss distance and the rotation angle of the grating at this time can calculate the altitude angle and azimuth angle of the target position according to the proposed formula. We design and complete the tracking process described in the second chapter with the coarse tracking detector and the servo system forms a closed loop tracking. Statistical analysis of the remaining miss distances gives a coarse tracking accuracy that meets our proposed technical specification of no more than 0.45mrad. The results show that the proposed formula provides a good basis for stable target coarse tracking and is instructive for engineering applications.