Abstract

The strong stray light has huge interference on the detection of weak and small optical signals, and is difficult to suppress. In this paper, a miniaturized baffle with angled vanes was proposed and a rapid optimization model of strong light elimination was built, which has better suppression of the stray lights than the conventional vanes and can optimize the positions of the vanes efficiently and accurately. Furthermore, the light energy distribution model was built based on the light projection at a specific angle, and the light propagation models of the vanes and sidewalls were built based on the Lambert scattering, both of which act as the bias of a calculation method of stray light. Moreover, the Monte-Carlo method was employed to realize the Point Source Transmittance (PST) simulation, and the simulation result indicated that it was consistent with the calculation result based on our models, and the PST could be improved by 2-3 times at the small incident angles for the baffle designed by the new method. Meanwhile, the simulation result was verified by laboratory tests, and the new model with derived analytical expressions which can reduce the simulation time significantly.

© 2017 Optical Society of America

1. Introduction

The strong stray light has huge harmfulness to the optical imaging system and is difficult to be suppressed and eliminated, especially for the detection of weak and small optical signals, such as the particles in the field of biology [1–3], the stars in the astronomy [4,5], and the machine vision in the navigation [6,7]. The star tracker, as an optical attitude sensor based on star detection, has been widely used for the vision navigation of aerospace crafts nowadays [8–11]. Since the light from stars is extremely weak, which sometimes is less than1-billionth of the stray light from the sun, therefore, the strong light from the sun, the earth, the moon or the aerospace craft body will undermine the star image quality, reduce the Signal to Noise Ratio (SNR), or even lead to the star tracker failure directly [12–14].

In order to decrease the interference, a baffle with multi-vanes is traditionally employed to minimum the strong light. In the aspects of the baffle design, a series of methods have been proposed and applied in the past decades, such as increasing the number of stages [15], optimizing the inclination and chamfering of vanes [16,17], cutting part structure of the baffle for special orbits [18,19], and programing the baffle and vanes design [20]. All these methods are based on the geometry principle, which requires that the light with the incident angle greater than the exclusive angle cannot enter into the lens directly or with scattering once [21]. However, when the light leaves from the vanes and sidewalls of the baffle after being absorbed by the black material, the absorptivity of which is always between 95%~97% [22, 23], it is not suitable for the star tracker with one-stage baffle, and lots of scattered light will access the lens after scattering twice out of control based on the traditional method. On one hand, the black material with extremely high absorptivity can improve the elimination performance of the stray light, such as the carbon nanotubes with the absorptivity of 99.95% [24], however, it is not firm enough for application on the baffle. On the other hand, the suppression performance of the stray light can also be improved by optimizing the baffle structure, in order to achieve accurate optimization result of the baffle, plenty of light rays are needed for the traditional ray tracing simulation; however, this method is time-consuming and not accurate with less light ray. Therefore, it is necessary to propose a new method to optimize the baffle structure quantitatively and rapidly.

In this paper, we designed a miniaturized baffle with angled vanes and proposed a mathematical optimization method for the baffle design based on lambert scattering. Through this method, the optimal angles of different vanes can be designed efficiently, which can minimize the PST of the baffle, and the PST is the ratio of stray light illuminance on the exit of the baffle to source illuminance at the entrance of the baffle, thus, an optimal baffle is achieved based on the requirement of the stray light suppression, which is the infinite wide parallel light and the spectrum range is visual light in this wok. Further simulation and laboratory experiment indicated that the calculation results with the new method were consistent with the simulation results based on Monte- Carlo ray-tracing method and the laboratory test results, and the PST could be improved by 2-3 times at the small incident angles, which are the most challenging part for the vast stray light in the baffle, compared with the traditional vertical vanes method. This new method can achieve the optimal stray light suppression performance of the baffle with a less time-consuming.

2. Design of a miniaturized baffle with angled vanes

In order to achieve extremely high stray light elimination, vanes are required in the baffle to increase the scatter times before the stray light entering the lens. There are mainly two kinds of vanes structures for the baffle- angled vanes and vertical vanes, and the angled vanes can miniaturize the baffle size and have better performance on stray light elimination than the vertical vanes. The number of the vanes is related with the length and width of the baffle, the principle of the minimum vanes is that the sidewall cannot be viewed from the lens directly, and the extra vanes have no effect on the stray light suppression, moreover, the distance between the vanes is decided by the minimum PST. In this paper, three angled vanes are employed in the baffle design. However, different from the geometrograph method [21], the angled vanes are determined by the boundary incident light line, the field of view (FOV) line and the sidewall of the baffle, as shown in Fig. 1, AA´ and BB´ denote the exit and the entrance of the baffle separately; α, β, and γ denote the angles of the vanes, and area a, b, and c denote the visible areas of the lens when the incident angle of the light greater than α.

 

Fig. 1 Cutaway view of the imaging system and the baffle with angled vanes.

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When the angles of the vanes are confirmed, the angle between the sidewall and the vane should be manufactured accurately at the specific position. Because when the angle is greater than design, the stray light elimination performance will get worse, and when the angle is smaller than design, part of the stray light will access into the lens after scattering once. Meanwhile, the edge of the vane should be sharp. The angles of the vanes are determined by the optimization method as follows.

3. Optimization method of the stray light elimination for the baffle

The structure of the baffle and the positions of vanes are determined by the geometry principle for the traditional method, which can only design the baffle roughly, and cannot optimize the elimination performance of the stray light. Given that different vane structures of the baffle have an effect on the elimination performance of the stray light, especially for the light with a small incident angle, the stray light elimination performance can be improved dramatically by adopting the optimized vanes, compared with the simple geometry method. Therefore, it is of great significance to optimize the baffle vanes, which can improve the resolution and the detection of the weak and small target for the optical imaging system.

In order to achieve the optimal results, Monte- Carlo ray-tracing simulation is always adopted; however, the tracing result varies with the amount of light ray and is not accurate. Meanwhile, because each simulation needs a new mechanical structure and simulation circumstance, this method is very inefficient. For the purpose of optimizing the vanes of the baffle efficiently, according to the practical propagation process of the light ray and the scattering characteristic on the surface of the baffle, the energy entering into the lens can be mainly divided into four parts, which are the energy from the first, second, and third cavity- E1, E2, E3, and the third- opposite cavity of the baffle- E4. Moreover, because approximately 95% energy entering into the lens comes from the two-time scattering light, therefore, we just considered the two-time scattering energy in this mathematic model. When the light enters into the baffle, at first, it will distribute on the vanes and sidewalls of different cavities, after the first absorption by the black material, it will scatter to the visible areas of different vanes, then, after the second absorption, the scatter light will access into the lens. Therefore, based on the above propagation process, E1, E2, E3 and E4 can be calculated with the equation below:

E1=Ein(1η)2s1es1rp1.
E2=Ein(1η)2(v2ev2r+s2es2r)p2.
E3=Ein(1η)2(v3ev3r+s3es3r)p3.
E4=Ein(1η)2(s1es1r_opp+s2es2r_opp+s3es3r_opp)p3.
where Ein denotes the energy entering into the baffle, η denotes the absorptivity of the surface material on the baffle, and p1, p2 and p3 denote the ratios of light entering into the lens from area a, b and c respectively. s1e, s2e, and s3e denote the ratios of energy irradiated on the sidewalls of the baffle; similarly, v2e and v3e denote the ratios of energy irradiated on the vanes of baffle. Meanwhile, s1r represents the energy ratio of the light irradiated on area a from sidewall 1, s2r and v2r represent the energy ratio of the light irradiated on area b from sidewall 2 and vane 2, and s3r and v3r represent the energy ratio of the light irradiated on area c from sidewall 3 and vane 3. Similarly, s1r_opp, s2r_opp and s3r_opp denote the energy ratio of the light irradiated on opposite area of c from sidewall 1, 2 and 3, respectively. For the reason that the energy irradiated on opposite areas of a and b is very little, it is not taken into account in this model. Based on the aforementioned analysis, the total energy entering into the lens can be expressed by Eq. (5):
Eout=E1+E2+E3+E4.
where Eout denotes the total energy entering into the lens with scattering twice. According to our analysis, the optimization method of the stray light elimination for the baffle mainly contains two parts- the energy distribution on different surfaces and the scattering ratios between different surfaces, which will be analyzed in 3.1 and 3.2. The design parameters of the baffle in this paper are listed in Table 1, based on which the length and entrance aperture of the baffle can be achieved. The exit aperture is corresponding to AA’ in Fig. 1, the half-FOV is angle between the FOV line and the center line, the earthlight exclusive angle indicates that the earthlight cannot enter into the lens directly when the incident angle greater than the earthlight exclusive angle, and the sunlight exclusive angle indicates that the sunlight cannot enter into the lens less than scattering twice when the incident angle greater than the sunlight exclusive angle.

Tables Icon

Table 1. The design parameters of the baffle

3.1 Energy distribution model among different cavities of the baffle

While the incident light varied from perpendicular to oblique relative to the baffle, the projection of the entrance aperture and vanes on the incident direction changes from the circle to the ellipse. According to the characteristic, the incident light energy on different cavities and surfaces can be calculated based on the projection areas, as shown in Fig. 2, and the light incident angle is 36°. Based on the ellipse equation, D/2 or Dv/2 denote the semi-major axis of the ellipse, and Dcosφ/2 or DVcosφ/2 denote the semi-minor axis of the ellipse, therefore, the projection equations of the entrance aperture and the vane top-edge can be derived as shown in Eqs. (6) and (7) respectively.

x2(D/2)2+y2(Dcosφ/2)2=1.
x2(DV/2)2+(y+MV)2(DVcosφ/2)2=1.
where D denotes the diameter of the entrance, φ denotes the light incident angle relative to the baffle, DV denotes the diameter of the vane top-edge, and MV denotes the ellipse center displacement of the vane top-edge on the Y axis. With the vanes in different positions, the projection areas of the cavities can be calculated based on the integral difference between Eqs. (6) and (7). Therefore, the energy ratios of the incident light can be derived when the vane in different positions with 36°incident light, as shown in Fig. 3, with the increase of vane’s angle, the energy ratio in the cavity increases synchronously. Similarly, based on the ellipse equation of the vane bottom, Eqs. (6) and (7), the incident light energy distribution on different sidewalls (s1e, s2e, s3e) and vanes (v2e, v3e) can also be derived, as shown in Fig. 4, with the angle increase of vane 2, the energy on sidewall 1 increases synchronously, and the energy on other surfaces are related with the angle between vane 2 and vane 3.

 

Fig. 2 The projection area in the light incident direction.

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Fig. 3 The energy ratios when the vane in different positions with 36° incident light.

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Fig. 4 The incident light energy distribution on different (a) sidewalls and (b) vanes.

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3.2 Light propagation model of the surfaces based on Lambert scattering

For the light entering into the baffle, it will scatter to the stereoscopic space after being absorbed by the surface material, which has the characteristic of Lambert scattering. However, it is difficult to quantify the stereoscopic distribution of the scatter light in the baffle. Since the baffle structure was distributed symmetrically around the center line, the stereoscopic scattering model can be simplified to the plane scattering model, and the luminous flux of the plane scattering within φ degree can be expressed as Eq. (8):

Φp=LdAθ=0θ=φsinθdθ=IN(1cosφ).
where Φp denotes the luminous flux of the plane scattering within φ degree for a unit area- dA, L denotes the brightness, which is a constant for the Lambert scattering. According to the baffle structure as shown in Fig. 1, the light propagation model from the surfaces of the vanes to the visible area a, b, or c can be extracted, as shown in Fig. 5(a), where s is the visible area of the vane. Based on Eq. (8) and this model, the luminous flux of the plane scattering for length- l without and with taking s into account can be derived in Eqs. (9) and (10) respectively.
{Φ1=A=0A=lΦpdAcosφ=A+mcosφ1[(A+mcosφ1)2+(msinφ1)2]12.
{Φ2=A=0A=lΦpdAcosφ=A+mcosφ1+scosβ[(A+mcosφ1+scosβ)2+(msinφ1ssinβ)2]12.
where Φ1 and Φ2 denote the luminous flux of the plane scattering for length- l without considering s and with considering s, φ1 denotes the angle between the invisible line- m and the vane. Therefore, the scattering ratio between the surface of the vane and the visible area –vxr can be derived in Eq. (11):
vxr=Φ1Φ2Φ=(Φ1Φ2)/A=0A=l2INdA.
where Φ is the luminous flux of the plane scattering for length- l in all directions, x is the mark of the vane with x = 2 or 3 in this structure.

 

Fig. 5 The light propagation plane model between the visible area and (a) the vane, (b) the sidewall.

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Similarly, the light propagation model from the surfaces of the sidewalls to the visible area a, b, and c can also be extracted, as shown in Fig. 5(b). Based on Eq. (8) and this model, the luminous flux of the plane scattering for length- J without considering s and with considering s in Fig. 5(b) can be derived in Eqs. (12) and (13) respectively.

{Φ3=A=JsA=JΦpdAcosφ=A+l1cosα[(A+l1cosα)2+(l1sinα)2]12.
{Φ4=A=JsA=JΦpdAcosφ=A+l1cosα+scosα[(A+l1cosα+scosα)2+(l1sinα+ssinα)2]12.
where Φ3 and Φ4 denote the luminous flux of the plane scattering for length- J without considering s and with considering s, Js denotes the start point of the sidewall irradiated by the light, α denotes the angle between the sidewall and the vane. Therefore, the scattering ratio between the surface of the sidewall and the visible area –sxr can be derived in Eq. (14):
sxr=Φ4Φ3Φ=Φ4Φ3A=JsA=J2INdA.
where x is the mark of the sidewall with x = 1,2 or 3 in this structure. According to Eqs. (8)-(14), the first scattering ratios when the light enters into the baffle – vxr and sxr can be achieved, similarly, the second scattering ratios after the light enters into the baffle – px and sxr_opp can also be achieved in the same way. Moreover, the incident light energy entering into the lens after scattering twice - Eout can be calculated, and based on this result, the structures of the vanes in the baffle can be optimized.

4. Simulation and calculation of the energy entering into the lens

With the angle increase of the incident light, the incident light energy entering into the baffle decreases following the cosine curve principle rapidly. Therefore, when optimizing the structure of vanes in the baffle, the PST performance in small incident angles should be considered, with the incident angle of 36° in this paper. Moreover, the baffle design will change along with the variety of the field of view, and while the sunlight exclusive angle increases, the angles of the vanes will increase at the same time. In order to achieve the optimal structure of the vanes, the positon traversal simulation was carried out, and the minimal angle pitch between the second vane and third vane was 11°. The simulation results of the baffle PST curves were shown in Fig. 6.

 

Fig. 6 The PST curves of the baffle with 36° incident angle.

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In Fig. 6, the different colors represented the positions/ angles of the second vane, and the result indicated that the position of the vane had great influence to the elimination performance of the stray light. The incident light energy entering into the lens was simulated by Monte- carlo ray-tracing method in TracePro, which is a ray tracing software. The illuminance at the entrance is 1000lux, and the baffle coatings comply with ABg scattering model, where the absorptance is 95%, g = 0, and b = 1. The simulation results of the energy entering into the lens were shown in Fig. 7.

 

Fig. 7 The simulation results of the energy entering into the lens with 36° incident angle.

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To achieve the simulation result, the specific mechanical structure and simulation circumstance in every data point should be built separately, meaning there are 10 × 14 mechanical structures and simulation circumstances to be designed and built for this case, typically one week work. Therefore, it is very inefficient to optimize the baffle design with the traditional simulation method. However, employing the optimization method proposed in this paper, the result can be achieved just a few seconds without any mechanical structure and simulation setting. The two-time scattering light energy entering into the lens at 36° is shown in Fig. 8(a), and the two-time scattering light energy from different cavities are shown in Fig. 8(b), where X axis denotes the angle of the second vane, Y axis denotes the angle between the second vane and the third vane, and Z axis denotes the two-time scattering light energy entering into the lens.

 

Fig. 8 The calculation results of two-time scattering energy (a) entering into the lens and (b) from different cavities at 36°.

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Comparing Fig. 7 with Fig. 8(a), it is obvious that the energy of light entering into the lens increases gradually after an initial decrease with the increase of the second vane angle, and the calculation result is of the same order of magnitude as the simulation. Meanwhile, the angle of the third vane has little influence on the result, compared with the angle of the second vane. Given that the light scattering more than three times and part of the subordinate stray light are not considered in this method, and the number of simulation rays also has influence on the simulation result, the results at the same point are slightly different for the two methods. However, such difference has no effect on the baffle optimization. As shown in Fig. 8(a), the minimum value for the second vane is around 48°, which is in accord with the simulation result. As shown in Fig. 8(b), the light energy from cavity 1 increases with the angle increase of the second vane, and the light energy from cavity 2 is related with the angle between vane 2 and vane 3. Meanwhile, the light energy from cavity 3 and cavity 3- opposite (cavity 3′) decreases with the increase of the angle of third vane.

5. Experiments and discussions

Based on the calculation results and the analysis in section 4, the angle of the second vane and the third vane were determined as 48° and 62° respectively. Moreover, the Monte-Carlo ray-tracing method was employed to realize the PST simulation by TracePro for the baffles with angled vanes and vertical vanes, the sizes of which were similar and the structures of the baffles were different, the comparison result was shown Fig. 9. Compared with the baffle designed by the geometrograph method, it was obvious that the PST performance was improved by 2-3 times at the small incident angles for the baffle designed by the new method in this paper.

 

Fig. 9 PST simulation performance of different baffles.

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As shown in Fig. 9, the PST of the baffle with angled vanes is lower than 5 × 10−6. Given that the PST of the lens is about 10−3, the illuminance of the stray light on the image detector is about 5 × 10−6 lux. According to the parameter of the star tracker, the illuminance of 5.5 stellar magnitude on the image detector is about 5.126 × 10−5 lux, the illuminance of the stray light is less than 5.5 stellar magnitude. Therefore, the star tracker with angled vanes baffle is available for 5.5 stellar magnitude. For some traditional baffles with vertical vanes, they can be processed through the machining one- time without split; however, for the baffle with angled vanes, it need to be disassembled simply to complete the machining, and the cutaway view of the baffle and the installation in the star tracker are shown in Fig. 10.

 

Fig. 10 (a) The cutaway view and (b) the installation of the baffle in the star tracker.

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As shown in Fig. 10, the material of the baffle is aluminum, and the absorptivity of the black material on the baffle surface is 95%. Moreover, the length of the baffle is 55mm, the maximum diameter is 50mm, and the mass is just 42g with the connection structure. In order to achieve the practical PST performance of the baffle, the baffle was installed in the star tracker and irradiated by the solar simulator, and the illuminance is 1000lux, as shown in Fig. 11(a). According to the gray value of the circumstance and the average gray values of the images from the star tracker in different angles, the normalized test result of PST was achieved, and the simulation result was also normalized, as shown in Fig. 11(b). Compared with the experiment result, the normalized simulation result fluctuated around it. Moreover, the fluctuation of the simulation result was caused by the number of the light rays to some extent, and has no influence on the baffle design. Therefore, the test result was consistent with the simulation result.

 

Fig. 11 (a) The PST test of the baffle with angled vanes, (b) the normalized PST result.

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6. Conclusions

In this paper, a miniaturized baffle with angled vanes and a rapid optimization method of strong stray light elimination based on Lambert scattering were proposed to suppress the interference to imaging. In order to improve the suppression performance of the stray light, three angled vanes were employed in the baffle, which had better suppression of stray light than the conventional vanes, and the angled vanes were determined by the boundary incident light line. Moreover, the light energy distribution on different surfaces of the baffle were determined by the light projection area at the specific angle, and the light propagation models of the vanes and sidewalls for the angled vanes baffle were built based on the lambert scattering. The two-time scattering light energy entering into the lens was calculated according to this two models efficiently, which can reduce the simulation time significantly. In addition, the Monte-Carlo ray-tracing method was employed to realize the PST simulation, and the PST test result was achieved based on the normalized gray value of the image. The laboratory test result indicated that it was consistent with the calculation and simulation result, and the PST performance could be improved by 2-3 times at the small incident angles for the baffle designed by the new method in this paper. Compared with the traditional method, this new method can optimize the baffle and provide an efficient method for baffle design quantitatively.

Funding

National Key Research and Development Program of China (2016YFB0501201); National Natural Science Foundation of China (NSFC) (51522505, 61377012, 61505094, 61605099).

Acknowledgments

We gratefully acknowledge the support of the State Key Laboratory of Precision Measurement Technology and Instruments.

References and links

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12. E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.

13. G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

14. G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

15. A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

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21. S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

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24. E. Theocharous, C. J. Chunnilall, R. Mole, D. Gibbs, N. Fox, N. Shang, G. Howlett, B. Jensen, R. Taylor, J. R. Reveles, O. B. Harris, and N. Ahmed, “The partial space qualification of a vertically aligned carbon nanotube coating on aluminium substrates for EO applications,” Opt. Express 22(6), 7290–7307 (2014). [PubMed]  

References

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  • |

  1. S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
    [PubMed]
  2. S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).
  3. J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
    [PubMed]
  4. P. R. Lawson and J. A. Dooley, “Technology Plan for the Terrestrial Planet Finder Interferometer,” NASA Sti/recon Technical Report N 6 (2005).
  5. N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).
  6. X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.
  7. Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
    [PubMed]
  8. C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).
  9. C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).
  10. T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
    [PubMed]
  11. M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).
  12. E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.
  13. G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).
  14. G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).
  15. A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).
  16. R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).
  17. R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).
  18. S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).
  19. V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).
  20. J. Haghshenas, “A theoretical method for vanes profile design in star sensor baffle,” in SPIE Asia Pacific Remote Sensing (2014), p. 92641R.
  21. S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).
  22. R. D. Seals and M. B. Mcintosh, “Advanced baffles: knife-edged diffuse-absorptive and dual reflective baffles,” in San Diego (1993).
  23. S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).
  24. E. Theocharous, C. J. Chunnilall, R. Mole, D. Gibbs, N. Fox, N. Shang, G. Howlett, B. Jensen, R. Taylor, J. R. Reveles, O. B. Harris, and N. Ahmed, “The partial space qualification of a vertically aligned carbon nanotube coating on aluminium substrates for EO applications,” Opt. Express 22(6), 7290–7307 (2014).
    [PubMed]

2017 (2)

2016 (1)

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

2014 (1)

2013 (1)

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

2012 (2)

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

2007 (1)

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

2005 (1)

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

2004 (1)

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

2003 (2)

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

2002 (1)

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

1996 (1)

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

1995 (1)

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

1989 (1)

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

1977 (2)

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

Ahmed, N.

Averbuch, Y.

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Bao, J.

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Breault, R. P.

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

Buffington, A.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Cady, E.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Chen, X.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Chunnilall, C. J.

Devasenathipathy, S.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Fidler, S.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Fox, N.

Gibbs, D.

Harris, O. B.

He, L.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Hick, P. P.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Hollinger, A.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Howlett, G.

Isbrucker, V.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Jackson, B. V.

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

Jensen, B.

Kasdin, N. J.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Kendrick, S.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

Kovacs, R.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

Kundu, K.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Laurin, D.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Liebe, C. C.

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

Liu, Z.

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Ma, H.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Macintosh, B.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Marchen, L.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Martin, S.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Meier, S. R.

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

Meinhart, C. D.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Mole, R.

Natan, M. J.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Penn, S. G.

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Pompea, S. M.

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

Reveles, J. R.

Santiago, J. G.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Savransky, D.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Schechner, Y. Y.

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Shaklan, S.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Shang, N.

Stauder, J.

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Sun, T.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Takehara, K.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Taylor, R.

Theocharous, E.

Urtasun, R.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Vaglia, A. L.

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

van Breukelen, E.

E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.

Vanderbei, R. J.

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

Wang, G.

Wei, M.

Wereley, S. T.

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

Xing, F.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

You, Z.

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method of star tracker orientation for sun-synchronous orbit based on space light distribution,” Appl. Opt. 56, 4480–4490 (2017).

G. Wang, F. Xing, M. Wei, T. Sun, and Z. You, “Optimization method for star tracker orientation in the sun-pointing mode,” Chin. Opt. Lett. 15, 081201 (2017).

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

Zhang, Z.

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

Zhao, J.

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

Appl. Opt. (1)

Chin. Opt. Lett. (1)

Curr. Opin. Chem. Biol. (1)

S. G. Penn, L. He, and M. J. Natan, “Nanoparticles for bioanalysis,” Curr. Opin. Chem. Biol. 7(5), 609–615 (2003).
[PubMed]

Exp. Fluids (1)

S. Devasenathipathy, J. G. Santiago, S. T. Wereley, C. D. Meinhart, and K. Takehara, “Particle imaging techniques for microfabricated fluidic systems,” Exp. Fluids 34, 504–514 (2003).

IEEE Aerosp. Electron. Syst. Mag. (1)

C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10, 10–16 (1995).

IEEE Trans. Aerosp. Electron. Syst. (1)

C. C. Liebe, “Accuracy Performance of Star Trackers-A Tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38, 587–599 (2002).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Y. Schechner and Y. Averbuch, “Regularized image recovery in scattering media,” IEEE Trans. Pattern Anal. Mach. Intell. 29(9), 1655–1660 (2007).
[PubMed]

Opt. Express (1)

Proc. SPIE (8)

S. R. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526, 195–207 (2004).

N. J. Kasdin, S. Shaklan, E. Cady, S. Martin, L. Marchen, R. J. Vanderbei, B. Macintosh, and D. Savransky, “Technology demonstration of starshade manufacturing for NASA’s Exoplanet mission program,” Proc. SPIE 8442, 84420A (2012).

S. M. Pompea, “Star sensor baffle optimization: some helpful practical design rules,” Proc. SPIE 2684, 333–338 (1996).

A. Buffington, B. V. Jackson, and P. P. Hick, “Space performance of the multistage labyrinthine SMEI baffle,” Proc. SPIE 5901, 325–334 (2005).

R. P. Breault, “Problems and techniques in stray radiation suppression,” Proc. SPIE 107, 2–23 (1977).

R. P. Breault, “Vane structure design trade-off and performance analysis,” Proc. SPIE 967, 90–117 (1989).

S. Kendrick, R. Kovacs, and A. L. Vaglia, “Solar simulation for testing off-axis light attenuation of a star sensor assembly,” Proc. SPIE 107, 150–157 (1977).

V. Isbrucker, J. Stauder, D. Laurin, and A. Hollinger, “Stray light control for asteroid detection at low solar elongation for the NEOSSat micro-satellite telescope,” Proc. SPIE 8442, 256–257 (2012).

Sensors (Basel) (2)

T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors (Basel) 13(4), 4598–4623 (2013).
[PubMed]

J. Zhao and Z. You, “A Microflow Cytometer with a Rectangular Quasi-Flat-Top Laser Spot,” Sensors (Basel) 16(9), 1474 (2016).
[PubMed]

Other (6)

P. R. Lawson and J. A. Dooley, “Technology Plan for the Terrestrial Planet Finder Interferometer,” NASA Sti/recon Technical Report N 6 (2005).

M. Wei, J. Bao, F. Xing, Z. Liu, T. Sun, and Z. You, “System-on-a-Chip Based Nano Star Tracker and Its Real-Time Image Processing Approach,” in 30th AIAA/USU Small Satellite Conference(2016).

E. van Breukelen, “Facet nano, a modular star tracker concept for highly miniaturized spacecraft,” in Proc. of the 60th International Astronautical Congress (2009), pp. 12–16.

J. Haghshenas, “A theoretical method for vanes profile design in star sensor baffle,” in SPIE Asia Pacific Remote Sensing (2014), p. 92641R.

R. D. Seals and M. B. Mcintosh, “Advanced baffles: knife-edged diffuse-absorptive and dual reflective baffles,” in San Diego (1993).

X. Chen, K. Kundu, Z. Zhang, H. Ma, S. Fidler, and R. Urtasun, “Monocular 3d object detection for autonomous driving,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition(2016), pp. 2147–2156.

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Figures (11)

Fig. 1
Fig. 1 Cutaway view of the imaging system and the baffle with angled vanes.
Fig. 2
Fig. 2 The projection area in the light incident direction.
Fig. 3
Fig. 3 The energy ratios when the vane in different positions with 36° incident light.
Fig. 4
Fig. 4 The incident light energy distribution on different (a) sidewalls and (b) vanes.
Fig. 5
Fig. 5 The light propagation plane model between the visible area and (a) the vane, (b) the sidewall.
Fig. 6
Fig. 6 The PST curves of the baffle with 36° incident angle.
Fig. 7
Fig. 7 The simulation results of the energy entering into the lens with 36° incident angle.
Fig. 8
Fig. 8 The calculation results of two-time scattering energy (a) entering into the lens and (b) from different cavities at 36°.
Fig. 9
Fig. 9 PST simulation performance of different baffles.
Fig. 10
Fig. 10 (a) The cutaway view and (b) the installation of the baffle in the star tracker.
Fig. 11
Fig. 11 (a) The PST test of the baffle with angled vanes, (b) the normalized PST result.

Tables (1)

Tables Icon

Table 1 The design parameters of the baffle

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E 1 = E i n ( 1 η ) 2 s 1 e s 1 r p 1 .
E 2 = E i n ( 1 η ) 2 ( v 2 e v 2 r + s 2 e s 2 r ) p 2 .
E 3 = E i n ( 1 η ) 2 ( v 3 e v 3 r + s 3 e s 3 r ) p 3 .
E 4 = E i n ( 1 η ) 2 ( s 1 e s 1 r _ o p p + s 2 e s 2 r _ o p p + s 3 e s 3 r _ o p p ) p 3 .
E o u t = E 1 + E 2 + E 3 + E 4 .
x 2 ( D / 2 ) 2 + y 2 ( D cos φ / 2 ) 2 = 1.
x 2 ( D V / 2 ) 2 + ( y + M V ) 2 ( D V cos φ / 2 ) 2 = 1.
Φ p = L d A θ = 0 θ = φ sin θ d θ = I N ( 1 cos φ ) .
{ Φ 1 = A = 0 A = l Φ p d A cos φ = A + m cos φ 1 [ ( A + m cos φ 1 ) 2 + ( m sin φ 1 ) 2 ] 1 2 .
{ Φ 2 = A = 0 A = l Φ p d A cos φ = A + m cos φ 1 + s cos β [ ( A + m cos φ 1 + s cos β ) 2 + ( m sin φ 1 s sin β ) 2 ] 1 2 .
v x r = Φ 1 Φ 2 Φ = ( Φ 1 Φ 2 ) / A = 0 A = l 2 I N d A .
{ Φ 3 = A = J s A = J Φ p d A cos φ = A + l 1 cos α [ ( A + l 1 cos α ) 2 + ( l 1 sin α ) 2 ] 1 2 .
{ Φ 4 = A = J s A = J Φ p d A cos φ = A + l 1 cos α + s cos α [ ( A + l 1 cos α + s cos α ) 2 + ( l 1 sin α + s sin α ) 2 ] 1 2 .
s x r = Φ 4 Φ 3 Φ = Φ 4 Φ 3 A = J s A = J 2 I N d A .

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