The star sensor is a high-precision space attitude optical sensor that has been used extensively in the aerospace field. Before the star sensor is actually in orbit, it is necessary to complete star sensor system error calibration in a ground environment [1–3]. However, the variability of weather and climate will inevitably cause inconvenience for the development of star sensor equipment and technology research, and the target imaging simulator is particularly important as the equipment for ground testing of star sensors. Therefore, it is necessary to replace real stars by developing target imaging simulators for experiments and tests [4–6].

Star simulators are divided into calibration star simulators and function test star simulators according to different working methods. In a calibration-type star simulator, a star point plate is placed on the focal plane of a collimating objective lens. After the light source is irradiated, the image will be imaged to infinity to simulate a star. Calibration-type star simulators are also called static star simulators [7–9]. This type of star simulator has high requirements for the parallelism of the outgoing beam and the stability of the star point position. It is used for star sensor calibration and performance testing. The function detection star simulator uses a thin-film-transistor liquid crystal display (TFT-LCD), digital micromirror device (DMD) Liquid crystal on silicon (LCoS) and other display devices to generate real-time star maps for the functional testing of star point detection and star map recognition of star sensors [10–12].

Dai et al. [13] used a DMD as a display device to design a dynamic star simulator with a band range of 600–700 nm, an exit pupil distance of 60 mm, an exit pupil diameter of 7 mm, and a field of view of 28.6°. The focal length of the projection system designed by it was 32.64 mm. Meng et al. [14] used the dynamic star simulation of LCoS stitching to improve and optimize the traditional stitching method and correction method. The exit pupil distance of the optical system is 35 mm, the exit pupil diameter is 20 mm, and the field of view angle is 12°. The focal length of the projection system is 73.07 mm. The exit pupil distances of the dynamic star simulators introduced above are relatively short. Zhao et al. [15] designed a large exit pupil distance, large field of view, and wide-band LCD scene simulation system. The exit pupil distance is 700 mm, the exit pupil diameter is 40 mm, the field of view is 7°, and the working band is 400 nm–1,000 nm. the entire optical system is designed with 13 lenses, the focal length of the optical system is 170 mm, and it also adopts the secondary imaging method,but the optical system has a large number of lenses, with an F/# of 4.25, the optical system is relatively easier to design. Zhang et al. [16] used LCoS to design a dynamic star simulator with a working wavelength of 450–1,000 nm and a long exit pupil distance of 1,250 mm. It has the advantages of a wide band and super long exit pupil distance. The angle is 4°, the focal length of the projection optical system is 300.7 mm, and the total length is 365.1 mm. Because of its large focal length, it can be designed with seven lenses after one imaging.

Starting from the target imaging simulator with a long exit pupil distance, this paper focuses on the design of a projection optical system based on TFT-LCD. Combined with specific design parameters, the design steps and design results are given, and the optical system is simulated and analyzed by software. Irradiance uniformity and magnitude error that meet the requirements of the system are obtained, which verifies the feasibility of the system.

The main function of the target imaging simulator is to test star sensor star pattern recognition and star tracking capabilities. The simulator is composed of a light source, illumination optical system, TFT-LCD, and projection optical system. The projection optical system is composed of primary imaging and secondary imaging systems. An overall schematic diagram of the system is shown in Fig. 1.

Figure 1. Overall schematic diagram of the system.

In the target imaging simulator, the light emitted by the light source is uniformly illuminated by the lighting system and evenly irradiated on the display device TFT-LCD. The digital target and star background images generated by the computer image generation subsystem are converted into two-dimensional light by the TFT-LCD. The modulated visible light image passes through the projection optical system and exits in parallel, and is coupled to the optical entrance pupil of the star sensor for detection, capture, identification and tracking experiments.

3.1. Main Parameters of the Optical System

In the visible light system, a liquid crystal light valve with a pixel size of 7 μm × 7 μm and a number of pixels of 1,920 × 1,080 is used as a display device, and its 1,024 × 1,024 part is used as an effective light-emitting unit, and the radius of its circumscribed circle is used as the half-field image height of the system.

The image height of the half-field of view is obtained as shown in Eq. (1):

In order to meet the requirement that the field of view in all directions of the optical system is greater than or equal to 4.5°, the maximum field of view of the system is set to 6.4°, which is _√2 times the minimum field of view, which can make full use of the 1,024 × 1,024 pixels of the liquid crystal. The focal length of the system is shown in Eq. (2):

The relevant indicators of the optical system are determined according to the indicators required by the simulator, as shown in Table 1.

TABLE 1. Main parameters of optical system.

Parameter	Index
Full Field of View (°)	≥4.5 × 4.5
Exit Pupil Diameter (mm)	≥50
Exit Pupillary Distance (mm)	≥1,000
Wavelength (nm)	400–850
Focal Length (mm)	90.66
Dispersion Angle (mrad)	≤0.15
Magnitude Error (Mv)	<0.2
Exit Pupil Uniformity (%)	>90

3.2. Design Thinking

In order to ensure that the liquid crystal light valve has a high utilization rate of light energy when simulating star-point projection, the optical system of the target imaging simulator should meet the requirements of an object space telecentric system, that is, the entrance pupil (before system inversion) should be located at infinity, and the exit pupil should be located at the focal plane of the image square of the system. However, the exit pupil distance required by the system is as large as 1,000 mm, which makes it difficult to meet the telecentric requirements for a primary imaging system. Therefore, a secondary imaging system needs to be used in the design. The principle of secondary imaging is shown in Fig. 2. The projection optical system is divided into a front and a rear groups. After the whole system is inverted, the diaphragm is set at the front of the system, 1,000 mm away from the first optical lens, and the diaphragm is the entrance pupil. The primary imaging system images the image of the entrance pupil (the exit pupil) to the primary image plane through the front group, then it uses the secondary imaging to conjugate the finite pupil to infinity through the rear group to realize the image space telecentric system (the system object-space telecentric structure before inversion). Since the secondary imaging system realizes the function of image rotation, the focal length of the entire optical system is a negative value. However, it does not affect the imaging field of view and image height of the system.

Figure 2. Schematic diagram of secondary imaging.

The front and a rear groups are designed separately, and the principle of pupil connection is met. The initial structure of the front group is the rear diaphragm telephoto lens, and the initial structure of the rear group is a double Gauss lens. After the initial structure of the front and rear groups of the projection optical system is selected, it is necessary to reasonably allocate the focal lengths of the front and rear groups. The application of Newton’s formula to analyze the combination of two light groups is as follows:

Substitute Eq. (3) into Eq. (4) to get:

In the formula, x′_F is the distance from the image focus of the rear group to the image focus of the combined system, f₁ and f₂ are the object focal lengths of the front and rear groups respectively, f ′₁ and f ′₂ are the image focal lengths of the front and rear groups, respectively, while d are the relative positions of the front group and the rear group.

Considering the total length of the entire system and the aberration balance, the focal length of the front group is set at about 180 mm, and d = 30 mm, substituting into Eq. (5) and Eq. (6), the focal length of the rear group is 50.24 mm. After the design is completed, the two systems are combined and optimized to balance the aberrations of the two systems to meet the optical parameters and image quality requirements of the entire optical system.

In the actual optical system optimization process, the RMS radius size of the spot diagram is always larger than the pixel size of the display chip, and the RMS radius size of different fields of view varies greatly. In order to improve the image quality of the optical system and reduce the number of lenses, an even-order aspheric surface is introduced into the secondary imaging system. The position of the aspheric surface is on the front and rear surfaces of the last lens. It can effectively correct spherical aberration and off-axis aberration, and is conducive to the correction of astigmatism, distortion, and coma. On the other hand, due to the small diameter of the lens, it is convenient for the processing and detection of the aspheric surface. The expression of an even-order aspheric surface is shown in Eq. (7):

In Eq. (7), h² = y² + x², c is the curvature at the apex of the aspheric surface, k is the conic coefficient of the quadric surface, and a_i is the coefficient of the high-order aspheric surface.

3.3. Design Results

The entrance pupil diameter of the visible light system is 50 mm, and the exit pupil distance is 1,000 mm. A larger exit pupil distance also determines that the system will have a larger aperture, which will also lead to a larger aperture aberration in the system, which brings certain difficulties to the design. The optimized system structure is shown in Fig. 3. The primary imaging system consists of five lenses, the secondary imaging system consists of four lenses, and the whole system consists of nine lenses. The total length of the system is 1,451.6 mm. The incident angles of the principal rays of each field of view to the image plane are less than 0.01°, which meets the requirements of an object space telecentric system.

Figure 3. System structure diagram. (a) 2D diagram of optical system, (b) magnified optical system structure.

The radius of curvature of each surface of the optical system, as well as the spacing between the lenses and the materials of each lens, are shown in Table 2. The cone coefficients and higher-order aspheric coefficients of even-order aspheric surfaces are shown in Table 3.

TABLE 2. Lens geometry and materials.

No.	Radius	Thickness	Material
1	239.285	22.171	H-ZPK2A
2	Infinity	1.000	-
3	433.925	22.277	H-FK95N
4	−335.122	1.810	-
5	171.650	21.288	H-ZPL7
6	Infinity	10.541	-
7	−382.203	10.000	H-ZF7LA
8	208.149	163.455	-
9	33.036	25.081	H-ZF88
10	22.422	30.067	-
11	−13.773	20.689	H-LAK12
12	−27.679	28.589	-
13	40.010	22.401	H-ZLAF69
14	−53.019	3.863	-
15	−33.618	4.494	H-ZF88
16	−573.139	14.934	-
17	20.706	25.027	OKP-4
18	450.095	17.341	-

TABLE 3. Even-order aspheric coefficient.

No.	Conic	Fourth Order Term	Sixth Order Term	Eighth Order Term
No. 17 in Table 2	−0.985	5.071E-06	1.224E-08	−4.730E-12
No. 18 in Table 2	−4.989	2.303E-05	7.541E-08	−4.491E-10

3.4. Image Quality Evaluation

Figure 4 shows the modulation transfer function (MTF) curves of different fields of view. It can be seen from the figure that at the Nyquist frequency of 72 lp/mm, the MTF of each field of view is close to the diffraction limit, and the MTF of the whole field of view is greater than 0.8, which has good imaging quality.

Figure 4. Modulation transfer function (MTF) curve of the optical system at 72 lp/mm.

Figure 5 shows spot diagrams of different fields of view. The spot diagram of the system reflects that the image plane has good symmetry and a certain degree of dispersion, and the RMS radii of the spot diagrams of each field of view are all less than 7 μm, that is, the size of one pixel.

Figure 5. Point plot of the system.

Figure 6 is a relative illuminance diagram. The relative illuminance of the lens is greater than 99%, the curve has no obvious inflection point, and the image uniformity is good.

Figure 6. Relative illuminance map of the system.

Figrue 7 is a graph of field curvature and distortion, and the distortion of each field of view is less than 0.04%.

Figure 7. Field curvature and distortion diagram of the system.

The calculation data of RMS radius and diffusion angle of each field spot diagram are shown in Table 4. Among them, the diffusion angle data is calculated by dividing the RMS diameter of each field spot diagram by the system focal length (90.66 mm). It can be seen that the dispersion angle data of each field of view of the optical system is less than the index requirement of 0.15 mrad.

TABLE 4. The size and dispersion angle of the system’s various field of view point charts.

Field (°)	RMS Radius (μm)	Dispersion Angle (mrad)
0	0.703	0.016
0.96	0.725	0.016
1.6	0.965	0.011
2.26	1.295	0.029
2.77	1.050	0.023
3.2	1.714	0.038

3.5. Thermal Analysis of Optical Systems

Changes in ambient temperature will cause changes in various parameters of optical elements. When the thermal expansion coefficient of the material used in the system is small, the impact of temperature on the optical system is relatively small. The lens barrel and spacer ring can be made of aluminum alloy with a relatively stable thermal expansion coefficient to ensure the temperature stability of the entire system. The target simulator designed in this article usually works in a 20 ℃ environment, adding multiple structures at two different temperatures, 10 ℃ and 30 ℃. Observe the change of the modulation transfer function in the optical system. The modulation transfer function of the optical system at different temperatures is shown in Fig. 8. From the comparison of Figs. 8(a)–8(c), it can be seen that the optical system has good stability at temperatures from 10 ℃ to 30 ℃.

Figure 8. Optical system modulation transfer function (MTF) at different temperatures. (a) MTF of optical system at 20 ℃ temperature, (b) MTF of optical system at 10 ℃ temperature, (c) MTF of optical system at 30 ℃ temperature.

3.6. Tolerance Analysis

After the system design is completed, tolerance analysis is required to simulate the actual lens processing and imaging conditions after assembly and adjustment. The results of tolerance analysis determine whether the optical system can be put into mass production, which is an important link in the design of the optical system. The tolerance factor of the optical system is unavoidable, and it needs to be limited within a reasonable range as much as possible. If it changes within this range, the performance of the optical system will not change greatly, and it can still meet the requirements of the system.

The tolerance distribution in this paper is shown in Table 5.

TABLE 5. Tolerance settings.

Tolerance Type	Tolerance Value
Radius (Fringes)	±2
Thickness (mm)	±0.025
Surface Decenter (mm)	±0.008
Surface Tilt (′)	±0.8
S + A Irregularity (Fringes)	±0.5
Element Decenter (mm)	±0.008
Element Tilt (′)	±0.8

The Monte Carlo method is used to randomly analyze 200 groups of lenses, and the tolerance analysis results are shown in Table 6. The tolerance analysis results show that within a given tolerance range, the optical system has a 90% probability that the RMS value of the spot diagram is less than 4.97 μm, which meets the optical index requirements of the dispersion angle.

TABLE 6. Results of Monte Carlo tolerance analysis.

Percentage of Monte Carlo Samples (%)	RMS Radius (mm)
90	0.00497153
80	0.00430926
50	0.00323120
20	0.00202021
10	0.00175798

TracePro software is used to verify whether the simulated magnitude error of the target imaging simulator in the range of −1 to +6 Mv and the uniformity of the exit pupil surface irradiation meet the requirements. Import the optical system structure into the simulation software, set the exit surface of the TFT-LCD as a rectangular light source, and set the exit pupil surface as the receiving surface. The optical system modeling is shown in Fig. 9.

Figure 9. Optical system modeling.

The relationship between the radiant flux of the light emitted by the TFT-LCD exit surface and the irradiance passing through the optical system to the exit pupil surface is shown in Eq. (8):

In Eq. (8), E is the illuminance value of the exit surface of the optical system, D₂ is the diameter of the entrance pupil of the optical system, τ^k is the transmittance of the entire optical system, and (1 − α)d is the transparency of the system.

The irradiance of the light source corresponding to the magnitude of −1 to +6 Mv is calculated by Eq. (8), as shown in Table 7.

TABLE 7. Conversion of light source irradiance.

Magnitude (Mv)	Irradiance (W/m2)	Light Source Irradiance (W/m2)
−1	4.39E-03	1.20E-06
0	1.75E-03	4.77E-07
1	6.70E-04	1.88E-07
2	2.77E-04	7.56E-08
3	1.11E-04	3.01E-08
4	4.41E-05	1.20E-08
5	1.75E-05	4.77E-09
6	6.70E-06	1.88E-09

Figure 10 is the irradiance diagram for tracing 100,000 rays of magnitude −1 Mv. Nine points on the detection surface are selected to measure the irradiance.

Figure 10. −1 Mv magnitude irradiance map.

The average value of the measured irradiance is 9.54E-09 W/m², and the magnitude error of −1 Mv calculated according to Eq. (9) is 0.0072 Mv, and the irradiance uniformity of −1 Mv magnitude calculated according to Eq. (10) is 95.47%.

In Eq. (9), M₀ is the magnitude value of the zero-magnitude star, E₀ is the irradiance value corresponding to the zero-magnitude star, and E is the irradiance value of the M magnitude star.

In Eq. (10), ε is the irradiance uniformity, E_max is the maximum value of the irradiance of the detection surface, and E_min is the minimum value of the irradiance of the detection surface.

Ray tracing is performed on magnitudes 0 to +6 Mv in turn, and the obtained magnitude error and irradiance uniformity are shown in Table 8.

TABLE 8. Magnitude error and irradiation uniformity.

Magnitude (Mv)	Magnitude Error (Mv)	Irradiation Uniformity (%)
−1	−0.0072	95.47
0	0.0049	91.07
1	0.0081	90.78
2	−0.0039	91.64
3	0.029	91.63
4	−0.0074	90.14
5	−0.0019	90.53
6	−0.0137	92.44

It can be seen from Table 8 that the simulation errors of magnitudes from −1 to +6 Mv are all less than 0.2 Mv, and the uniformity of irradiation is greater than 90%, meeting the simulation requirements of the real magnitude of the target simulator.

According to the technical index requirements of the target imaging simulator, this paper designs a visible light projection optical system with a long exit pupil distance. The entire projection optical system is composed of eight spherical lenses and a plastic aspheric lens, which minimizes the number of lenses while correcting aberrations. The optimization results show that at the cutoff frequency of 72 lp/mm, the MTF of the optical system is greater than 0.8, the distortion of each field of view is less than 0.04%, and the performance of the optical system is excellent. The optical system was modeled using TracePro software, and the analysis and verification of the target imaging simulator simulated magnitude error within the range of −1 to +6 Mv was less than 0.2 Mv, and the irradiance uniformity of the exit pupil surface was greater than 90%, which meets the usage requirements of the target imaging simulator. It provides a theoretical basis for the further design and research of the target imaging simulator under the condition of long exit pupil distance.

The authors acknowledge financial support from the Science and Technology Development Plan of Jilin Province of China (20220201089GX).

Science and Technology Development Plan of Jilin Province of China (20220201089GX).

The authors declare no conflicts of interest.

The data underlying the results presented in this paper are not publicly available at the time of publication and can be obtained from the authors upon reasonable request.