Multiplexing image detector method for digital sun sensors with arc-second class accuracy and large FOV Minsong Wei,1,2,3 Fei Xing,1,2,3,* Zheng You,1,2,3 and Geng Wang1,2,3 1 Department of Precision Instrument, Tsinghua University, Beijing 100084, China State Key Laboratory of Precision Measurement Technology and Instruments, Tsinghua University, Beijing 100084, China 3 Collaborative Innovation Center for Micro/Nano Fabrication, Device and System, Tsinghua University, Beijing 100084, China * [email protected]
2
Abstract: To improve the accuracy of digital sun sensors (DSS) to the level of arc-second while maintaining a large field of view (FOV), a multiplexing image detector method was proposed. Based on a single multiplexing detector, a dedicated mask with different groups of encoding apertures was utilized to divide the whole FOV into several sub-FOVs, every of which would cover the whole detector. In this paper, we present a novel method to analyze and optimize the diffraction effect and the parameters of the aperture patterns in the dedicated mask, including the aperture size, focal length, FOV, as well as the clearance between adjacent apertures. Based on the simulation, a dedicated mask with 13 × 13 various groups of apertures was designed and fabricated; furthermore a prototype of DSS with a single multiplexing detector and 13 × 13 sub-FOVs was built and test. The results indicated that the DSS prototype could reach the accuracy of 5 arc-second (3σ) within a 105° × 105° FOV. Using this method, the sun sensor still keeps the original features of low power consumption, small size and high dynamic range when it realizes both high accuracy and large FOV. ©2014 Optical Society of America OCIS codes: (120.6085) Space instrumentation; (120.4640) Optical instruments; (040.1880) Detection.
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#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23094
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1. Introduction Sun sensors are one of the most crucial components in the Attitude Determination System (ADS) of satellites [1]. Compared with star trackers, another component in ADS with high accuracy [2,3], digital sun sensors (DSS) are advantageous because of its small size, light weight, low power consumption, and the capability of functioning under high dynamic conditions [4,5]. In recent years, several DSS have been developed to fulfill different application requirements [6–11]. A mini-DSS with an APS + chip which was implemented by Delft University of Technology and TNO can achieve an accuracy of 0.03° in the whole 102° × 102° FOV, with 69 × 52 × 14mm3 in size and 72g in weight [9]. Another DSS developed by TNO has the accuracy of 0.024° (2σ) within 120° × 120° FOV, with a power consumption of 1 W, 130 × 120 × 45mm3 in size and 310g in weight [11]. SS441, a commercial product of Sinclair Interplanetary could realize an accuracy of 0.11° (2σ) in ± 70° FOV with 34 × 32 × 21mm3 in size and 34g in weight [7]. Currently, the rapid development of advanced earth-observing satellites and long-range laser links for satellite communications has led to the requirement for more and more stringent attitude measurement accuracy (arc-second class) [12–14]. However, the accuracy of state-of-the-art DSS is approximately 0.01° ~0.1° as mentioned above, and therefore, the improvement on DSS accuracy is of great help to broaden the application of DSS with the superiority of miniaturization, low power consumption and high dynamic property. Based on the principle of the digital sun sensor, the trade-off between large FOV and high resolution has always hindered the development of the digital sun sensor with a large FOV and high accuracy. Several different methods have been proposed to improve the performance of the sun sensors. For instance, additional fish-eye lens, which is mounted just above the mask plane, was introduced to expand the FOV to hemispherical dimensions [11]. However, the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23095
accompanying problems are both the increment of optical system complexity for the design of the fish-eye lens and the decrease in the accuracy within the whole FOV from 0.02° to 1°. Another method of multiple diode detectors was employed to increase the FOV of a fine analogue sun sensor with the accuracy of 0.1° and the FOV of 120° [15,16]. Such approach cannot be applied to a DSS since the multiple digital image detectors would significantly increase the system size as well as power consumption. A DSS with the accuracy of 3.6 arcsecond was developed by Japan Aerospace Exploration Agency, however, its FOV was seriously decreased to 1° × 1°, far less than the application requirement [17]. To accomplish the goal of both large FOV and high accuracy, the design of multiple pinholes in a dome-like structure was also proposed [18,19]. Nevertheless, it would bring forth the increment of difficulty of manufacture, assemble as well as alignment for the fabrication and assemble of the dome and the multiple pinholes in it. Moreover, none of the above proposals could achieve our requirement of both arc-second class accuracy and large FOV. In our previous work, we introduced the basic principle of multiplexing image detector and Electrical Rolling Shutter (ERS) to realize a DSS with high precision and update rate [20]. A planar mask with dedicated encoding apertures was introduced to divide the whole FOV of the digital sun sensor with the name XDSS into several sub-FOVs without no additional optical lenses or other structures needed. In this work, we modelled the multiplexing image detector method and deduced the geometric condition for continuous measurement of the incident sun angle within the whole FOV. Since the increment on the focal length between the mask plane and the image detector plane was utilized to improve the resolution of XDSS, the ratio between the aperture size and focal length will decrease, resulting in significant diffraction effect. Thus, we analyzed the diffraction spot based on Fresnel-Kirchhoff formula and optimized the pattern parameters for high accuracy achievement in this article. Based on a single multiplexing detector and a dedicated mask, a prototype of XDSS with 13 × 13 sub-FOVs was built and test. With the multiplexing image detector method, the XDSS prototype can achieve the 5 arc-second accuracy within the 105° × 105° FOV. 2. Multiplexing image detector approach for high resolution and large FOV 2.1 Principle of the DSS The measurement of incident sun angles is primarily implemented through the mask and the image detector (CCD or CMOS) of a DSS, and Fig. 1 reveals the geometric relation between mask plane and the image detector plane.
z θ
Mask
α
O
X
β
xt l
Sun spot
Incident sunray
Y
h
y o
x
yt
Image detector
Fig. 1. Schematic of DSS with planar image detector.
By analyzing the location of the sun spot on image detector plane, the sun angles (α, β) as well as the incident angle (θ) can be calculated as:
xt h
α = arc tan
,
(1)
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23096
yt , h
β = arc tan
(2)
)
(
l (3) tan 2 α + tan 2 β . Where, xt and yt denote the distance between the center of current sun spot and its origin center along x axis and y axis, respectively, l denotes the linear distance between the center of current sun spot and its origin center, and h denotes the height from the mask plane to the image detector plane. Take the incident sun angle for example, the equation of resolution dθ can be presented as:
θ = arc tan = arc tan h
cos 2 (θ ) (4) d (l ). h Where, d(l) denotes the distance resolution of the image detector. To achieve the goal of arc-second level accuracy, the resolution of the DSS should satisfy the arc-second level. Based on Eq. (4), improvement on the distance resolution of the image detector would improve the resolution of the DSS. Thus, to further improve the distance resolution of the image detector from pixel to sub-pixel, centroiding algorithm [10,21] is employed to determine the center of the sun spot achieved on the image plane. Once a sun spot is detected on the image plane, a region with m × n pixels will be extracted as the region of interest (ROI) of the sun spot, and then the coordinate of the sun spot center can be calculated through Eq. (5) and Eq. (6). dθ =
m
xc =
n
∑∑ [ P(i, j ) × i]
=i 1 =j 1 m n
∑∑ P(i, j )
(5)
,
=i 1 =j 1 m
yc =
n
∑∑ [ P(i, j ) × j ]
=i 1 =j 1 m n
∑∑ P(i, j )
(6)
.
=i 1 =j 1
Where, xc and yc denote the coordinate of the sun spot center in two directions, respectively, P(i, j) denotes the gray value of the pixel in line i and column j within the ROI. FOV indicates the range of sun angles which can be measured by DSS, as shown in Fig. 2. The FOV in x direction and y direction can be determined by Eqs. (7) and (8), respectively. FOVx
z Mask
Y
O
X
h
FOVy
y
o
xs
ys
x
Image detector
Fig. 2. FOV formation of the DSS.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23097
x FOVx = 2 arc tan s , 2h
(7)
y (8) FOVy = 2 arc tan s . 2h Where xs and ys are the length of the active area of the image detector in two directions. 2.2 Operation Principle of the multiplexing image detector approach Based on the DSS principle, it is apparent that the resolution is proportional to h while FOV is inversely proportional to h. In this case, if a higher resolution could be achieved with a larger h in Eq. (4), the FOV would become smaller based on Eq. (7) or Eq. (8) and vice versa. Such constrain between resolution and FOV has fundamentally obstructed the achievement of both high accuracy and large FOV. Therefore, the multiplexing image detector method was introduced to conquer this obstacle, as shown in Fig. 3(b). By employing the multiplexing image detector method, only one image detector and one mask will be used in our design, which significantly reduces the system complexity and has none of the drawbacks discussed in the introduction section. The mask pattern was redesigned from one group, which is utilized in traditional DSSs, to several distinctive groups. The image pattern that is obtained by the image detector would vary directly with the incident sun angle and hence, the whole FOV would be enlarged. Generally, there is only one pattern group on the mask of traditional DSS, which was demonstrated in Fig. 3(a), and it might be one aperture, one slit or one array of apertures. Based on Eq. (7) or (8), the FOV depends on just two parameters, height and the size of active area of the image detector. In contrast, there are several distinctive groups of pattern on our dedicated mask and each group can be employed to determine the incident sun angle individually. Therefore, one sub-FOV would be defined and determined by one group. For example, there are three groups of pattern on the mask, as shown in Fig. 3(b), and the whole FOV of DSS based on multiplexing detector is composed of three sub-FOVs: Sub-FOV1, Sub-FOV2, and Sub-FOV3. The equation to calculate the whole FOV can be presented as:
Where lpattern respectively.
ld +l pattern (9) FOV =2 arc tan . 2h and ld denote the size of the image detector and the pattern on the mask,
FOV
FOV Sub-FOV3
Sub-FOV1
Mask #3 Pattern lpattern h
Mask Pattern
#1
(a)
#2 h
Image detector ld
Sub-FOV2
Image detector ld (b)
Fig. 3. FOV of the DSS (a) without multiplexing image detector method and (b) with multiplexing image detector method [20].
When a higher resolution could be gained by the increment of h, the FOV of the DSS without multiplexing image detector would decrease, given a determined image detector size; while the whole FOV of the DSS with multiplexing image detector could be compensated by
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23098
increasing the length of the pattern and therefore, based on Eq. (9), the constrain between large FOV and high resolution could be resolved. The innovative design will dramatically improve the resolution of the sun sensors without sacrificing the FOV and then both high accuracy and large FOV could be achieved in one DSS. 2.3 Modeling analysis of the multiplexing image detector approach To further analyze the relation between the mask pattern and the FOV, the multiplexing image detector approach has been modeled in Fig. 4. FOV0
FOV-1 FOV-n
lm
lm
dm
lm
FOV1 lm
lm
FOVn
Pattern lpattern
h
Image detector ld Fig. 4. Schematic of the mask pattern based on multiplexing image detector.
We take one axis for example to deduce the formula concerning the mask pattern and the FOV. In Fig. 4, h is the height from the mask plane to the image detector plane, lm is the length of every pattern, dm is the separation between each two adjacent patterns, lpattern is the central distance between two patterns which are at the edge of the whole pattern and the total number of groups is (2n + 1). Therefore, the pattern length lpattern can be calculated as: (10) l pattern =2n ⋅ d m . The sub-FOV corresponding to each group can be calculated through Eqs. (11)-(13):
l -l FOV0 : − arc tan d m 2h
2d +l -l FOV1 : arc tan m m d 2h
ld -lm , arc tan 2h ,
ld -lm +2d m , arc tan 2h
(11)
,
(12)
2nd m +lm -ld ld -lm +2nd m (13) FOVn : arc tan , arc tan . h 2 2h And the whole FOV composed of each sub-FOV (FOV0, FOV1, FOV-1 …… FOVn, FOVn) is determined by Eq. (14):
ld +l pattern -lm l -l +2nd m (14) FOV =2 arc tan d m =2 arc tan . 2h 2h To realize the continuous coverage of the incident sun angle within the whole FOV, the adjacent sub-FOV should be overlaid and it could be confined through geometrical limitation, as presented by Eq. (15): (15) d m + lm ≤ ld . In this case, it is guaranteed that there will be at least one group of pattern completely imaged on the detector at any incident angle within the whole FOV. Moreover, on the transition condition of two adjacent sub-FOVs, another new group of pattern will be obtained by the detector before the image of the former group moving out of the active area of the detector, and by such restriction, there will be no gap between sub-FOVs.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23099
The whole FOV of the DSS with multiplexing image detector method can be precisely calculated by Eq. (14); while, given the length of every pattern lm is far less than the length of the pattern lpattern and the length of the image detector ld, this equation could be simplified to Eq. (9). Furthermore, Eq. (15) provides the geometric restriction for the implementation of continuous measurement of the incident sun angle within the whole FOV and hence all incident sun angles within the whole FOV will be determined successfully once such geometric condition is met. Based on the model analysis, the details of the mask design will be presented in Section 4. 3. Optical simulation To achieve the optimized design of the DSS optical system, the diffraction on the image detector plane should be analyzed on the basis of Fresnel diffraction theory [11,22,23]. As shown in Fig. 5, since the incident sun rays can be simplified to be collimated and the aperture is square in shape, the complex attitude at point P on the image detector plane, E ( P) , can be determined as the superposition of every wavelet which origins from the point (x0, y0) within the aperture area, like point Q in Fig. 5. Based on Fresnel-Kirchhoff formula [24–26], the intensity distribution of diffraction image can be calculated through Eq. (16). ] 0 A e [0 (16) E ( P) = ∫∫ dσ . iλ aperture r Where λ is the wavelength; A is constant which indicates the electric field intensity of the incident sun ray; k is the wave vector; r is the distance between area element dσ on the mask plane and the image point P; a and b denote two direction cosines of the incident sun ray. ik x cos( a ) + y cos( b ) + r
Y
Y0
X
dσ
Q
X0
A
Z r
A* Mask
P
Image detector
Fig. 5. The diffraction model of DSS with a square aperture on the mask plane.
Here, we simplify the incident sun ray as collimated light; however, there will be an angular sun size of 0.53° given the actual sun size. If we take the angular sun size into account, the sun, which functions as planar source, can be considered as the set of unit point sources. The sun ray from any unit point source dS will be collimated with an additive incident angle ranged from 0° to 0.265°, given its location on the planar source and the Eq. (16) can be applied to each unit point source. Therefore, the final result of the sun spot is the superposition of the diffraction spots from all the unit point sources [26]. If we fix all the parameters, including r and the aperture area, the simulation results of the diffraction spot by
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23100
using both the simplified model without the angular sun size and the theoretical model with the angular sun size have been shown in Fig. 6. 25
25
1 0.9
Intensity
0
0
1
(a) (b)
0.8
0.8
0.7
0.6
central principal maximum width
0.6 0.5 0.4 0.3
0.4 0.2
0.2
0
0.1 -25 -25
0
25
(a)
-25 -25
0 0
25
(b)
-25 -20 -15
y axis -10
-5
0
5
10
15
20
25
x axis / pixel
x axis
(c)
Fig. 6. The diffraction spots at the incident sun angle of 0° with different models (a) simulation result with theoretical model (b) simulation result with simplified model (c) the intensity of the central line along x axis of both diffraction spots.
According to Figs. 6(a) and 6(b), the diffraction spot in the theoretical model will be larger than that in the simplified model, resulting from the superposition of the diffraction spots from all the unit point sources in the planar source, and Fig. 6(c) indicates that the contrast of the diffraction spot in the theoretical model decreases since its average intensity of the central line is higher than that in the simplified model. However, rather than the brightness change in the whole image plane, the width of the central principal maximum of both two diffraction spots is the same, demonstrating that the simplified model could be an effective alternative for diffraction analysis and the computation complexity could be reduced by using simplified model as well. Therefore, based on Eq. (16), numerical simulation can be conducted to analyze the image size with respect to the distance between mask plane and image detector plane, h. As shown in Fig. 7, when the incident sunray is perpendicular to the mask plane, the diffraction images of a 20 pixels × 20 pixels (1pixel = 5.3μm) aperture with respect to different h have been simulated and the light intensity has been normalized. The results indicated that the diffraction effect is not notable with a smaller h, as shown in Fig. 7(a), and with the increase of h, the diffraction effect is more and more striking and the size of image spot also increases, as shown in Figs. 7(b) and (c). 1 0.8 0.6 0.4 0.2 0
(a)
(b)
(c)
100um
Fig. 7. The diffraction spots with respect to different h (a) h = 10mm (b) h = 15mm (c) h = 20mm.
To achieve the goal of arc-second accuracy, h is calculated to be 17.2mm to guarantee the arc-second resolution. According to the analysis results, the diffraction effect is no longer negligible. Thus, the relation between the aperture size and the size of its diffraction spot has been studied. When the incident sunray is perpendicular to the mask plane, Fig. 8 shows the central width of diffraction spots that we will get on the image detector plane with different aperture sizes. At first, the maximum intensity of the diffraction spot will rise because more light will go through the aperture with the increase of the aperture size, and then the maximum intensity will decrease with further increase of the aperture size which resulted
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23101
from the divergent spot. Meanwhile, with the increase of the aperture size, the width curve of the diffraction spot goes sharp at first indicating that the contrast of the diffraction spot gets enhanced. However, once the maximum intensity of the diffraction spot is reached, the simulated results also indicated that there is a reduction in contrast of the diffraction spot with further increase of the aperture size. 3.5 18pixels×18pixels 22pixels×22pixels
3
26pixels×26pixels 30pixels×30pixels 32pixels×32pixels 34pixels×34pixels
2.5
Intensity
36pixels×36pixels 2
1.5
1
0.5 0 -30
-10
-20
0
10
20
30
Width of the diffraction spot / pixel
Fig. 8. Simulated results of the diffraction spot width with different aperture sizes.
The conditions of oblique incidence are taken into account as well. The numerical simulation has been conducted to analyze the diffraction effect with oblique incident sunray and the result is shown in Fig. 9. The incident sun ray is oblique along x axis with different angles and the aperture size is fixed. Based on the simulation results, the diffraction spot will stretch in the same direction (x axis) and if the incident sun angle gets larger, the diffraction spot will stretch longer along the incident direction. x axis 1 0.8 0.6 0.4 0.2 0
(a)
(b)
(c)
(d)
100um
Fig. 9. The diffraction spots at different incident sun angles (a) θ = 0° (b) θ = 20° (c) θ = 35° (d) θ = 50°.
To further analyze the diffraction spots with different aperture sizes on the conditions of oblique incidence, the central width of the diffraction spot along the x axis has been simulated at the incident angle θ = 50°, which is oblique along the same axis. The results with different aperture sizes are shown in Fig. 10. Compared with Fig. 8, on the conditions of oblique incidence, change of the central width of the diffraction spots along the oblique direction has the similar trend with that on the condition of perpendicular incidence. At first, a flat curve with respect to a smaller aperture size gradually turns to a steep curve with the increase of the aperture size, indicating the enhancement in the contrast of the diffraction spot within the certain range. Then, with further increase of the aperture size, the width curve goes lower in the middle with a higher shoulder at both sides, indicating the contrast of the diffraction spot decreases.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23102
2.5 25pixels×25pixels 30pixels×30pixels 35pixels×35pixels 40pixels×40pixels 45pixels×45pixels 50pixels×50pixels
2
55pixels×55pixels
Intensity
1.5
1
0.5
0 -75
-60
-45
-30
-15
0
15
30
45
60
75
Width of the diffraction spot / pixel
Fig. 10. The width of diffraction spots with different aperture sizes at θ = 50°.
Furthermore, the sizes of diffraction spots with respect to different incident angles and various aperture sizes have been simulated and the intensity threshold for spot extraction is set to 0.4 after the normalization of the diffraction spot image. Figure 11 demonstrates the number of illuminated pixels which will be extracted as a sun spot and it can be utilized to evaluate the size of the diffraction spot. Within the range of a small aperture size, which is from 20pixels × 20pixels to 30pixels × 30pixels as shown in Fig. 11, the size of diffraction spot gets larger with the increase of the incident angle; while the size of diffraction spot gets smaller before it gets larger with the increase of the incident angle when the aperture size is larger than 35pixels × 35pixels. On the other hand, given a certain incident angle, the minimum diffraction size can be achieved with different aperture size. For example, when the incident sunray is perpendicular to the image detector plane (θ = 0°), the extracted sun spot will achieve its minimum with the aperture size of 30pixels × 30pixels, approximately; and the size of extracted sun spot with the aperture size of 40pixels × 40pixels is much larger than its minimum. While the extracted sun spot will achieve its minimum with the aperture size of 40pixels × 40pixels, approximately, at the incident angle of 50° and the size of extracted sun spot with the aperture size of 20pixels × 20pixels is much larger than that. 450 20pixels×20pixels
400
25pixels×25pixels 30pixels×30pixels
Illuminated pixels
350
35pixels×35pixels 37pixels×37pixels 40pixels×40pixels
300 250 200 150 100 50 0 0
5
10
15
20
25
30
35
40
45
50
Incident angle / °
Fig. 11. The illuminated pixels of the diffraction spot with respect to different incident angles as well as various aperture sizes.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23103
The analyses discussed above have demonstrated that the diffraction problem should be carefully considered and numerical simulations are quite necessary. The relevant results of the numerical simulation will be applied in the mask pattern design and optimization process. 4. Mask pattern design and optimization 4.1 Pattern design based on the multiplexing image detector method As discussed in Section 2.2, the pattern group on the mask should be distinguishable from each other and an elaborative mapping code is introduced to achieve the arrangement. In our design, every group of pattern is composed of three encoding apertures aligned in a line with distance information and these apertures are defined as “positioning marks” since each group could be employed individually to determine a certain sub-FOV. The layout of the pattern design on the mask is revealed in Fig. 12.
dmx dmy lmy
Encoding apertures
Fig. 12. The schematic of the mask design with detail view.
As shown in the detail view of one group of encoding apertures, L1 is the distance between the left and middle apertures and L2 is the distance between the right and middle apertures. Both L1 and L2 are in the unit of pixel. In our mask pattern design, the groups in the same line are coded with the same L1 and disparate L2 while the groups in the same column are coded with the same L2 and disparate L1, such as (Ns, Ns) for the central group of encoding apertures, (Ns, Ns + 1) for the group to the left of the central group and (Ns + 1, Ns) for the group just above the central group, where Ns is the clearance between two adjacent encoding apertures. Based on such coordinate-type mapping code, each group of encoding apertures will be determined by (L1, L2) exclusively. Thus, once a frame of image is obtained by the image detector, the distance information (L1, L2) will be calculated following the sun spot extraction, and then the extracted group of encoding apertures will be determined according to the mapping code. Eventually, the current sun angles can be calculated based on the determined encoding apertures. 4.2 Optimal design of the mask parameters Based on the numerical simulation and analysis above, the aperture size should be optimized to minimize the diffraction effect and achieve a concentrated sun spot. For this purpose, the size of diffraction spots with respect to various apertures was simulated and if the intensity threshold for spot extraction is set to 0.4 after the normalization of the spot image, the results were demonstrated in Fig. 13. Figure 13(a) reveals the diameter of the diffraction spot regarding to different aperture sizes with the incident sun ray perpendicular to the mask plane. The diameter of the diffraction spot will firstly decrease and then increase during the increment of the aperture size within the featured range; and the minimum diameter can be obtained at the range of aperture size of 26~32pixels, which is consistent with the analysis in Section 3. Figure 13(b) reveals the number of illuminated pixels which will be extracted as a sun spot regarding to different aperture sizes at the same condition. Akin to the phenomenon of Fig. 13(a), the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23104
number of illustrated pixels will firstly decrease and then increase during the increment of the aperture size within the same range; and the minimum number can be obtained at the aperture size of 32pixels. Therefore, the size of diffraction spot is around its minimum when the aperture size is 169.6um × 169.6um. Similarly, the size of the different apertures with respect to different sub-FOVs can be optimized in the same criterion. Diameter of the diffraction spot /×5.3um
Illustrated pixel number /number of pixels 450
25
400
20
350 15
300
10
250 200
5 0
Aperture size /×5.3um 35
150 15
20
25
100 35 15
30
20
25
(a)
30
(b)
Fig. 13. Aperture size optimization regarding (a) diameter of the diffraction spot and (b) illuminated pixel number of the diffraction spot.
To prevent the interference between the image spots of two adjacent apertures, we analyze the diffraction effect with the incident sun angle of 50° once the aperture size is optimized, as shown in Fig. 14. We assume that the threshold for sun spot extraction is 0.4 after the light intensity is normalized. Then, the part of two adjacent sun spots with the intensity larger than half threshold should not be overlaid and hence the clearance should be larger than 50pixels. To strictly eliminate the interference between two adjacent sun spots, the clearance value Ns is doubled to 100pixels. 75
1 0.9
50
y axis x axis
Intensity
y / ×5.3μm
0.8 25 0 -25
0.7 0.6 0.5
threshold
0.4 0.3 0.2
-50
0.1 -75 -75
-50
-25
0
25
x / ×5.3μm
50
75
0
-75
-50
-25
0
25
50
75
Position/ ×5.3μm
Fig. 14. Clearance optimization with incident sun angle of 50°.
Based on the design optimization, the pattern on the mask was composed of 13 × 13 groups of encoding apertures, as shown in Fig. 12, resulting in 13 × 13 sub-FOVs. The minimum of L1 and L2 is set to 0.53mm based on the clearance value. Thus, the distance between the adjacent groups is 3.392mm (dmx) × 3.816mm (dmy) and the pattern size of the encoding group is 1.272mm (lmx) × 0.424mm (lmy). According to the geometric restriction for continuous measurement of the incident sun angle, Eq. (15), examination calculation indicates that there is no gap between sub-FOVs within the whole FOV. 5. Experiments and results To verify the performance improvement of the XDSS based on multiplexing image detector method, the XDSS prototype, which is mainly composed of the mask fabricated by MEMS procedures, the housing as well as the electrical system, was tested. The XDSS prototype is 96mm × 80mm × 41.5mm in size and 182 g in weight with the power consumption of 500mW. The test system includes a sun simulator, which provides the collimated light with a
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23105
field angle of 32′ (approximately equals to the angular sun size), and a three-axis gimbals rotary table with the position accuracy of 0.0001°. The sun sensor prototype was fixed on the rotary table and arbitrary sun incident angles can be established with the rotation of the threeaxis gimbals rotary table, as shown in Fig. 15. In the performance experiment, the FOV and accuracy have been examined specifically. XDSS prototype Sun simulator
Rotary table
Fig. 15. Performance test of the XDSS prototype.
The control step of the FOV test was 0.5° along two axes and 5° off two axes. Test results suggest that the FOV of the XDSS prototype was 105° × 105° and there were no blind zone for sun angle measurement within the whole FOV. To achieve the arc-second accuracy, the intrinsic parameters of the XDSS prototype are precisely on-site measured with the optical tomography technology based on feedback of microchip Nd:YAG laser [27]. The system accuracy is analyzed based on the error at each tested point and the error is defined as the difference between the input incident angle determined by the rotary table and the output value provided by the XDSS prototype. The accuracy test results of both sun angles (α, β), as shown in Fig. 16, demonstrated that the accuracy of the XDSS prototype was 5 arc-second (3σ) and the arc-second level accuracy were achieved. Error/″
10
α β
Yerror Xerror
8 6 4 2 0 -2 -4 -6 -8
0
10
20 30 Incident angle/°
40
50
Fig. 16. Measurement error statistics of sun sensor performance test.
6. Conclusion In this paper, a novel multiplexing image detector approach has been discussed to improve the performance of the XDSS. By using a dedicated mask with several groups of encoding apertures to subdivide the whole FOV, the fundamental constrain between large FOV and high accuracy was resolved. The geometric condition for continuous measurement of the incident sun angle within the whole FOV was derived based on the modeling of the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23106
multiplexing image detector approach. Numerical simulation was also presented to accomplish the optimized design of the apertures and the mapping method was also introduced to achieve the uniqueness of each group of encoding apertures. The XDSS prototype with optimized mask was tested on the rotary table and relative results indicate that the FOV is 105° × 105° and accuracy is 5 arc-second (3σ) which is 10 times better than the state-of-art sun sensors. The XDSS based on multiplexing image detector method will be qualified for the application in high precise attitude determination system. Acknowledgments This work has been carried out in the State Key Laboratory of Precision Instrument Measurement, Tsinghua University under the financial support by the China National 863 Projects (No. 2012AA121503, No. 2012AA120603) and China NSF projects (No. 61377012, No. 60807004).
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23107
2
Abstract: To improve the accuracy of digital sun sensors (DSS) to the level of arc-second while maintaining a large field of view (FOV), a multiplexing image detector method was proposed. Based on a single multiplexing detector, a dedicated mask with different groups of encoding apertures was utilized to divide the whole FOV into several sub-FOVs, every of which would cover the whole detector. In this paper, we present a novel method to analyze and optimize the diffraction effect and the parameters of the aperture patterns in the dedicated mask, including the aperture size, focal length, FOV, as well as the clearance between adjacent apertures. Based on the simulation, a dedicated mask with 13 × 13 various groups of apertures was designed and fabricated; furthermore a prototype of DSS with a single multiplexing detector and 13 × 13 sub-FOVs was built and test. The results indicated that the DSS prototype could reach the accuracy of 5 arc-second (3σ) within a 105° × 105° FOV. Using this method, the sun sensor still keeps the original features of low power consumption, small size and high dynamic range when it realizes both high accuracy and large FOV. ©2014 Optical Society of America OCIS codes: (120.6085) Space instrumentation; (120.4640) Optical instruments; (040.1880) Detection.
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#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23094
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1. Introduction Sun sensors are one of the most crucial components in the Attitude Determination System (ADS) of satellites [1]. Compared with star trackers, another component in ADS with high accuracy [2,3], digital sun sensors (DSS) are advantageous because of its small size, light weight, low power consumption, and the capability of functioning under high dynamic conditions [4,5]. In recent years, several DSS have been developed to fulfill different application requirements [6–11]. A mini-DSS with an APS + chip which was implemented by Delft University of Technology and TNO can achieve an accuracy of 0.03° in the whole 102° × 102° FOV, with 69 × 52 × 14mm3 in size and 72g in weight [9]. Another DSS developed by TNO has the accuracy of 0.024° (2σ) within 120° × 120° FOV, with a power consumption of 1 W, 130 × 120 × 45mm3 in size and 310g in weight [11]. SS441, a commercial product of Sinclair Interplanetary could realize an accuracy of 0.11° (2σ) in ± 70° FOV with 34 × 32 × 21mm3 in size and 34g in weight [7]. Currently, the rapid development of advanced earth-observing satellites and long-range laser links for satellite communications has led to the requirement for more and more stringent attitude measurement accuracy (arc-second class) [12–14]. However, the accuracy of state-of-the-art DSS is approximately 0.01° ~0.1° as mentioned above, and therefore, the improvement on DSS accuracy is of great help to broaden the application of DSS with the superiority of miniaturization, low power consumption and high dynamic property. Based on the principle of the digital sun sensor, the trade-off between large FOV and high resolution has always hindered the development of the digital sun sensor with a large FOV and high accuracy. Several different methods have been proposed to improve the performance of the sun sensors. For instance, additional fish-eye lens, which is mounted just above the mask plane, was introduced to expand the FOV to hemispherical dimensions [11]. However, the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23095
accompanying problems are both the increment of optical system complexity for the design of the fish-eye lens and the decrease in the accuracy within the whole FOV from 0.02° to 1°. Another method of multiple diode detectors was employed to increase the FOV of a fine analogue sun sensor with the accuracy of 0.1° and the FOV of 120° [15,16]. Such approach cannot be applied to a DSS since the multiple digital image detectors would significantly increase the system size as well as power consumption. A DSS with the accuracy of 3.6 arcsecond was developed by Japan Aerospace Exploration Agency, however, its FOV was seriously decreased to 1° × 1°, far less than the application requirement [17]. To accomplish the goal of both large FOV and high accuracy, the design of multiple pinholes in a dome-like structure was also proposed [18,19]. Nevertheless, it would bring forth the increment of difficulty of manufacture, assemble as well as alignment for the fabrication and assemble of the dome and the multiple pinholes in it. Moreover, none of the above proposals could achieve our requirement of both arc-second class accuracy and large FOV. In our previous work, we introduced the basic principle of multiplexing image detector and Electrical Rolling Shutter (ERS) to realize a DSS with high precision and update rate [20]. A planar mask with dedicated encoding apertures was introduced to divide the whole FOV of the digital sun sensor with the name XDSS into several sub-FOVs without no additional optical lenses or other structures needed. In this work, we modelled the multiplexing image detector method and deduced the geometric condition for continuous measurement of the incident sun angle within the whole FOV. Since the increment on the focal length between the mask plane and the image detector plane was utilized to improve the resolution of XDSS, the ratio between the aperture size and focal length will decrease, resulting in significant diffraction effect. Thus, we analyzed the diffraction spot based on Fresnel-Kirchhoff formula and optimized the pattern parameters for high accuracy achievement in this article. Based on a single multiplexing detector and a dedicated mask, a prototype of XDSS with 13 × 13 sub-FOVs was built and test. With the multiplexing image detector method, the XDSS prototype can achieve the 5 arc-second accuracy within the 105° × 105° FOV. 2. Multiplexing image detector approach for high resolution and large FOV 2.1 Principle of the DSS The measurement of incident sun angles is primarily implemented through the mask and the image detector (CCD or CMOS) of a DSS, and Fig. 1 reveals the geometric relation between mask plane and the image detector plane.
z θ
Mask
α
O
X
β
xt l
Sun spot
Incident sunray
Y
h
y o
x
yt
Image detector
Fig. 1. Schematic of DSS with planar image detector.
By analyzing the location of the sun spot on image detector plane, the sun angles (α, β) as well as the incident angle (θ) can be calculated as:
xt h
α = arc tan
,
(1)
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23096
yt , h
β = arc tan
(2)
)
(
l (3) tan 2 α + tan 2 β . Where, xt and yt denote the distance between the center of current sun spot and its origin center along x axis and y axis, respectively, l denotes the linear distance between the center of current sun spot and its origin center, and h denotes the height from the mask plane to the image detector plane. Take the incident sun angle for example, the equation of resolution dθ can be presented as:
θ = arc tan = arc tan h
cos 2 (θ ) (4) d (l ). h Where, d(l) denotes the distance resolution of the image detector. To achieve the goal of arc-second level accuracy, the resolution of the DSS should satisfy the arc-second level. Based on Eq. (4), improvement on the distance resolution of the image detector would improve the resolution of the DSS. Thus, to further improve the distance resolution of the image detector from pixel to sub-pixel, centroiding algorithm [10,21] is employed to determine the center of the sun spot achieved on the image plane. Once a sun spot is detected on the image plane, a region with m × n pixels will be extracted as the region of interest (ROI) of the sun spot, and then the coordinate of the sun spot center can be calculated through Eq. (5) and Eq. (6). dθ =
m
xc =
n
∑∑ [ P(i, j ) × i]
=i 1 =j 1 m n
∑∑ P(i, j )
(5)
,
=i 1 =j 1 m
yc =
n
∑∑ [ P(i, j ) × j ]
=i 1 =j 1 m n
∑∑ P(i, j )
(6)
.
=i 1 =j 1
Where, xc and yc denote the coordinate of the sun spot center in two directions, respectively, P(i, j) denotes the gray value of the pixel in line i and column j within the ROI. FOV indicates the range of sun angles which can be measured by DSS, as shown in Fig. 2. The FOV in x direction and y direction can be determined by Eqs. (7) and (8), respectively. FOVx
z Mask
Y
O
X
h
FOVy
y
o
xs
ys
x
Image detector
Fig. 2. FOV formation of the DSS.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23097
x FOVx = 2 arc tan s , 2h
(7)
y (8) FOVy = 2 arc tan s . 2h Where xs and ys are the length of the active area of the image detector in two directions. 2.2 Operation Principle of the multiplexing image detector approach Based on the DSS principle, it is apparent that the resolution is proportional to h while FOV is inversely proportional to h. In this case, if a higher resolution could be achieved with a larger h in Eq. (4), the FOV would become smaller based on Eq. (7) or Eq. (8) and vice versa. Such constrain between resolution and FOV has fundamentally obstructed the achievement of both high accuracy and large FOV. Therefore, the multiplexing image detector method was introduced to conquer this obstacle, as shown in Fig. 3(b). By employing the multiplexing image detector method, only one image detector and one mask will be used in our design, which significantly reduces the system complexity and has none of the drawbacks discussed in the introduction section. The mask pattern was redesigned from one group, which is utilized in traditional DSSs, to several distinctive groups. The image pattern that is obtained by the image detector would vary directly with the incident sun angle and hence, the whole FOV would be enlarged. Generally, there is only one pattern group on the mask of traditional DSS, which was demonstrated in Fig. 3(a), and it might be one aperture, one slit or one array of apertures. Based on Eq. (7) or (8), the FOV depends on just two parameters, height and the size of active area of the image detector. In contrast, there are several distinctive groups of pattern on our dedicated mask and each group can be employed to determine the incident sun angle individually. Therefore, one sub-FOV would be defined and determined by one group. For example, there are three groups of pattern on the mask, as shown in Fig. 3(b), and the whole FOV of DSS based on multiplexing detector is composed of three sub-FOVs: Sub-FOV1, Sub-FOV2, and Sub-FOV3. The equation to calculate the whole FOV can be presented as:
Where lpattern respectively.
ld +l pattern (9) FOV =2 arc tan . 2h and ld denote the size of the image detector and the pattern on the mask,
FOV
FOV Sub-FOV3
Sub-FOV1
Mask #3 Pattern lpattern h
Mask Pattern
#1
(a)
#2 h
Image detector ld
Sub-FOV2
Image detector ld (b)
Fig. 3. FOV of the DSS (a) without multiplexing image detector method and (b) with multiplexing image detector method [20].
When a higher resolution could be gained by the increment of h, the FOV of the DSS without multiplexing image detector would decrease, given a determined image detector size; while the whole FOV of the DSS with multiplexing image detector could be compensated by
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23098
increasing the length of the pattern and therefore, based on Eq. (9), the constrain between large FOV and high resolution could be resolved. The innovative design will dramatically improve the resolution of the sun sensors without sacrificing the FOV and then both high accuracy and large FOV could be achieved in one DSS. 2.3 Modeling analysis of the multiplexing image detector approach To further analyze the relation between the mask pattern and the FOV, the multiplexing image detector approach has been modeled in Fig. 4. FOV0
FOV-1 FOV-n
lm
lm
dm
lm
FOV1 lm
lm
FOVn
Pattern lpattern
h
Image detector ld Fig. 4. Schematic of the mask pattern based on multiplexing image detector.
We take one axis for example to deduce the formula concerning the mask pattern and the FOV. In Fig. 4, h is the height from the mask plane to the image detector plane, lm is the length of every pattern, dm is the separation between each two adjacent patterns, lpattern is the central distance between two patterns which are at the edge of the whole pattern and the total number of groups is (2n + 1). Therefore, the pattern length lpattern can be calculated as: (10) l pattern =2n ⋅ d m . The sub-FOV corresponding to each group can be calculated through Eqs. (11)-(13):
l -l FOV0 : − arc tan d m 2h
2d +l -l FOV1 : arc tan m m d 2h
ld -lm , arc tan 2h ,
ld -lm +2d m , arc tan 2h
(11)
,
(12)
2nd m +lm -ld ld -lm +2nd m (13) FOVn : arc tan , arc tan . h 2 2h And the whole FOV composed of each sub-FOV (FOV0, FOV1, FOV-1 …… FOVn, FOVn) is determined by Eq. (14):
ld +l pattern -lm l -l +2nd m (14) FOV =2 arc tan d m =2 arc tan . 2h 2h To realize the continuous coverage of the incident sun angle within the whole FOV, the adjacent sub-FOV should be overlaid and it could be confined through geometrical limitation, as presented by Eq. (15): (15) d m + lm ≤ ld . In this case, it is guaranteed that there will be at least one group of pattern completely imaged on the detector at any incident angle within the whole FOV. Moreover, on the transition condition of two adjacent sub-FOVs, another new group of pattern will be obtained by the detector before the image of the former group moving out of the active area of the detector, and by such restriction, there will be no gap between sub-FOVs.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23099
The whole FOV of the DSS with multiplexing image detector method can be precisely calculated by Eq. (14); while, given the length of every pattern lm is far less than the length of the pattern lpattern and the length of the image detector ld, this equation could be simplified to Eq. (9). Furthermore, Eq. (15) provides the geometric restriction for the implementation of continuous measurement of the incident sun angle within the whole FOV and hence all incident sun angles within the whole FOV will be determined successfully once such geometric condition is met. Based on the model analysis, the details of the mask design will be presented in Section 4. 3. Optical simulation To achieve the optimized design of the DSS optical system, the diffraction on the image detector plane should be analyzed on the basis of Fresnel diffraction theory [11,22,23]. As shown in Fig. 5, since the incident sun rays can be simplified to be collimated and the aperture is square in shape, the complex attitude at point P on the image detector plane, E ( P) , can be determined as the superposition of every wavelet which origins from the point (x0, y0) within the aperture area, like point Q in Fig. 5. Based on Fresnel-Kirchhoff formula [24–26], the intensity distribution of diffraction image can be calculated through Eq. (16). ] 0 A e [0 (16) E ( P) = ∫∫ dσ . iλ aperture r Where λ is the wavelength; A is constant which indicates the electric field intensity of the incident sun ray; k is the wave vector; r is the distance between area element dσ on the mask plane and the image point P; a and b denote two direction cosines of the incident sun ray. ik x cos( a ) + y cos( b ) + r
Y
Y0
X
dσ
Q
X0
A
Z r
A* Mask
P
Image detector
Fig. 5. The diffraction model of DSS with a square aperture on the mask plane.
Here, we simplify the incident sun ray as collimated light; however, there will be an angular sun size of 0.53° given the actual sun size. If we take the angular sun size into account, the sun, which functions as planar source, can be considered as the set of unit point sources. The sun ray from any unit point source dS will be collimated with an additive incident angle ranged from 0° to 0.265°, given its location on the planar source and the Eq. (16) can be applied to each unit point source. Therefore, the final result of the sun spot is the superposition of the diffraction spots from all the unit point sources [26]. If we fix all the parameters, including r and the aperture area, the simulation results of the diffraction spot by
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23100
using both the simplified model without the angular sun size and the theoretical model with the angular sun size have been shown in Fig. 6. 25
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Fig. 6. The diffraction spots at the incident sun angle of 0° with different models (a) simulation result with theoretical model (b) simulation result with simplified model (c) the intensity of the central line along x axis of both diffraction spots.
According to Figs. 6(a) and 6(b), the diffraction spot in the theoretical model will be larger than that in the simplified model, resulting from the superposition of the diffraction spots from all the unit point sources in the planar source, and Fig. 6(c) indicates that the contrast of the diffraction spot in the theoretical model decreases since its average intensity of the central line is higher than that in the simplified model. However, rather than the brightness change in the whole image plane, the width of the central principal maximum of both two diffraction spots is the same, demonstrating that the simplified model could be an effective alternative for diffraction analysis and the computation complexity could be reduced by using simplified model as well. Therefore, based on Eq. (16), numerical simulation can be conducted to analyze the image size with respect to the distance between mask plane and image detector plane, h. As shown in Fig. 7, when the incident sunray is perpendicular to the mask plane, the diffraction images of a 20 pixels × 20 pixels (1pixel = 5.3μm) aperture with respect to different h have been simulated and the light intensity has been normalized. The results indicated that the diffraction effect is not notable with a smaller h, as shown in Fig. 7(a), and with the increase of h, the diffraction effect is more and more striking and the size of image spot also increases, as shown in Figs. 7(b) and (c). 1 0.8 0.6 0.4 0.2 0
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Fig. 7. The diffraction spots with respect to different h (a) h = 10mm (b) h = 15mm (c) h = 20mm.
To achieve the goal of arc-second accuracy, h is calculated to be 17.2mm to guarantee the arc-second resolution. According to the analysis results, the diffraction effect is no longer negligible. Thus, the relation between the aperture size and the size of its diffraction spot has been studied. When the incident sunray is perpendicular to the mask plane, Fig. 8 shows the central width of diffraction spots that we will get on the image detector plane with different aperture sizes. At first, the maximum intensity of the diffraction spot will rise because more light will go through the aperture with the increase of the aperture size, and then the maximum intensity will decrease with further increase of the aperture size which resulted
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23101
from the divergent spot. Meanwhile, with the increase of the aperture size, the width curve of the diffraction spot goes sharp at first indicating that the contrast of the diffraction spot gets enhanced. However, once the maximum intensity of the diffraction spot is reached, the simulated results also indicated that there is a reduction in contrast of the diffraction spot with further increase of the aperture size. 3.5 18pixels×18pixels 22pixels×22pixels
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Fig. 8. Simulated results of the diffraction spot width with different aperture sizes.
The conditions of oblique incidence are taken into account as well. The numerical simulation has been conducted to analyze the diffraction effect with oblique incident sunray and the result is shown in Fig. 9. The incident sun ray is oblique along x axis with different angles and the aperture size is fixed. Based on the simulation results, the diffraction spot will stretch in the same direction (x axis) and if the incident sun angle gets larger, the diffraction spot will stretch longer along the incident direction. x axis 1 0.8 0.6 0.4 0.2 0
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Fig. 9. The diffraction spots at different incident sun angles (a) θ = 0° (b) θ = 20° (c) θ = 35° (d) θ = 50°.
To further analyze the diffraction spots with different aperture sizes on the conditions of oblique incidence, the central width of the diffraction spot along the x axis has been simulated at the incident angle θ = 50°, which is oblique along the same axis. The results with different aperture sizes are shown in Fig. 10. Compared with Fig. 8, on the conditions of oblique incidence, change of the central width of the diffraction spots along the oblique direction has the similar trend with that on the condition of perpendicular incidence. At first, a flat curve with respect to a smaller aperture size gradually turns to a steep curve with the increase of the aperture size, indicating the enhancement in the contrast of the diffraction spot within the certain range. Then, with further increase of the aperture size, the width curve goes lower in the middle with a higher shoulder at both sides, indicating the contrast of the diffraction spot decreases.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23102
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Fig. 10. The width of diffraction spots with different aperture sizes at θ = 50°.
Furthermore, the sizes of diffraction spots with respect to different incident angles and various aperture sizes have been simulated and the intensity threshold for spot extraction is set to 0.4 after the normalization of the diffraction spot image. Figure 11 demonstrates the number of illuminated pixels which will be extracted as a sun spot and it can be utilized to evaluate the size of the diffraction spot. Within the range of a small aperture size, which is from 20pixels × 20pixels to 30pixels × 30pixels as shown in Fig. 11, the size of diffraction spot gets larger with the increase of the incident angle; while the size of diffraction spot gets smaller before it gets larger with the increase of the incident angle when the aperture size is larger than 35pixels × 35pixels. On the other hand, given a certain incident angle, the minimum diffraction size can be achieved with different aperture size. For example, when the incident sunray is perpendicular to the image detector plane (θ = 0°), the extracted sun spot will achieve its minimum with the aperture size of 30pixels × 30pixels, approximately; and the size of extracted sun spot with the aperture size of 40pixels × 40pixels is much larger than its minimum. While the extracted sun spot will achieve its minimum with the aperture size of 40pixels × 40pixels, approximately, at the incident angle of 50° and the size of extracted sun spot with the aperture size of 20pixels × 20pixels is much larger than that. 450 20pixels×20pixels
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Fig. 11. The illuminated pixels of the diffraction spot with respect to different incident angles as well as various aperture sizes.
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23103
The analyses discussed above have demonstrated that the diffraction problem should be carefully considered and numerical simulations are quite necessary. The relevant results of the numerical simulation will be applied in the mask pattern design and optimization process. 4. Mask pattern design and optimization 4.1 Pattern design based on the multiplexing image detector method As discussed in Section 2.2, the pattern group on the mask should be distinguishable from each other and an elaborative mapping code is introduced to achieve the arrangement. In our design, every group of pattern is composed of three encoding apertures aligned in a line with distance information and these apertures are defined as “positioning marks” since each group could be employed individually to determine a certain sub-FOV. The layout of the pattern design on the mask is revealed in Fig. 12.
dmx dmy lmy
Encoding apertures
Fig. 12. The schematic of the mask design with detail view.
As shown in the detail view of one group of encoding apertures, L1 is the distance between the left and middle apertures and L2 is the distance between the right and middle apertures. Both L1 and L2 are in the unit of pixel. In our mask pattern design, the groups in the same line are coded with the same L1 and disparate L2 while the groups in the same column are coded with the same L2 and disparate L1, such as (Ns, Ns) for the central group of encoding apertures, (Ns, Ns + 1) for the group to the left of the central group and (Ns + 1, Ns) for the group just above the central group, where Ns is the clearance between two adjacent encoding apertures. Based on such coordinate-type mapping code, each group of encoding apertures will be determined by (L1, L2) exclusively. Thus, once a frame of image is obtained by the image detector, the distance information (L1, L2) will be calculated following the sun spot extraction, and then the extracted group of encoding apertures will be determined according to the mapping code. Eventually, the current sun angles can be calculated based on the determined encoding apertures. 4.2 Optimal design of the mask parameters Based on the numerical simulation and analysis above, the aperture size should be optimized to minimize the diffraction effect and achieve a concentrated sun spot. For this purpose, the size of diffraction spots with respect to various apertures was simulated and if the intensity threshold for spot extraction is set to 0.4 after the normalization of the spot image, the results were demonstrated in Fig. 13. Figure 13(a) reveals the diameter of the diffraction spot regarding to different aperture sizes with the incident sun ray perpendicular to the mask plane. The diameter of the diffraction spot will firstly decrease and then increase during the increment of the aperture size within the featured range; and the minimum diameter can be obtained at the range of aperture size of 26~32pixels, which is consistent with the analysis in Section 3. Figure 13(b) reveals the number of illuminated pixels which will be extracted as a sun spot regarding to different aperture sizes at the same condition. Akin to the phenomenon of Fig. 13(a), the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23104
number of illustrated pixels will firstly decrease and then increase during the increment of the aperture size within the same range; and the minimum number can be obtained at the aperture size of 32pixels. Therefore, the size of diffraction spot is around its minimum when the aperture size is 169.6um × 169.6um. Similarly, the size of the different apertures with respect to different sub-FOVs can be optimized in the same criterion. Diameter of the diffraction spot /×5.3um
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Fig. 13. Aperture size optimization regarding (a) diameter of the diffraction spot and (b) illuminated pixel number of the diffraction spot.
To prevent the interference between the image spots of two adjacent apertures, we analyze the diffraction effect with the incident sun angle of 50° once the aperture size is optimized, as shown in Fig. 14. We assume that the threshold for sun spot extraction is 0.4 after the light intensity is normalized. Then, the part of two adjacent sun spots with the intensity larger than half threshold should not be overlaid and hence the clearance should be larger than 50pixels. To strictly eliminate the interference between two adjacent sun spots, the clearance value Ns is doubled to 100pixels. 75
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Fig. 14. Clearance optimization with incident sun angle of 50°.
Based on the design optimization, the pattern on the mask was composed of 13 × 13 groups of encoding apertures, as shown in Fig. 12, resulting in 13 × 13 sub-FOVs. The minimum of L1 and L2 is set to 0.53mm based on the clearance value. Thus, the distance between the adjacent groups is 3.392mm (dmx) × 3.816mm (dmy) and the pattern size of the encoding group is 1.272mm (lmx) × 0.424mm (lmy). According to the geometric restriction for continuous measurement of the incident sun angle, Eq. (15), examination calculation indicates that there is no gap between sub-FOVs within the whole FOV. 5. Experiments and results To verify the performance improvement of the XDSS based on multiplexing image detector method, the XDSS prototype, which is mainly composed of the mask fabricated by MEMS procedures, the housing as well as the electrical system, was tested. The XDSS prototype is 96mm × 80mm × 41.5mm in size and 182 g in weight with the power consumption of 500mW. The test system includes a sun simulator, which provides the collimated light with a
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23105
field angle of 32′ (approximately equals to the angular sun size), and a three-axis gimbals rotary table with the position accuracy of 0.0001°. The sun sensor prototype was fixed on the rotary table and arbitrary sun incident angles can be established with the rotation of the threeaxis gimbals rotary table, as shown in Fig. 15. In the performance experiment, the FOV and accuracy have been examined specifically. XDSS prototype Sun simulator
Rotary table
Fig. 15. Performance test of the XDSS prototype.
The control step of the FOV test was 0.5° along two axes and 5° off two axes. Test results suggest that the FOV of the XDSS prototype was 105° × 105° and there were no blind zone for sun angle measurement within the whole FOV. To achieve the arc-second accuracy, the intrinsic parameters of the XDSS prototype are precisely on-site measured with the optical tomography technology based on feedback of microchip Nd:YAG laser [27]. The system accuracy is analyzed based on the error at each tested point and the error is defined as the difference between the input incident angle determined by the rotary table and the output value provided by the XDSS prototype. The accuracy test results of both sun angles (α, β), as shown in Fig. 16, demonstrated that the accuracy of the XDSS prototype was 5 arc-second (3σ) and the arc-second level accuracy were achieved. Error/″
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Fig. 16. Measurement error statistics of sun sensor performance test.
6. Conclusion In this paper, a novel multiplexing image detector approach has been discussed to improve the performance of the XDSS. By using a dedicated mask with several groups of encoding apertures to subdivide the whole FOV, the fundamental constrain between large FOV and high accuracy was resolved. The geometric condition for continuous measurement of the incident sun angle within the whole FOV was derived based on the modeling of the
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23106
multiplexing image detector approach. Numerical simulation was also presented to accomplish the optimized design of the apertures and the mapping method was also introduced to achieve the uniqueness of each group of encoding apertures. The XDSS prototype with optimized mask was tested on the rotary table and relative results indicate that the FOV is 105° × 105° and accuracy is 5 arc-second (3σ) which is 10 times better than the state-of-art sun sensors. The XDSS based on multiplexing image detector method will be qualified for the application in high precise attitude determination system. Acknowledgments This work has been carried out in the State Key Laboratory of Precision Instrument Measurement, Tsinghua University under the financial support by the China National 863 Projects (No. 2012AA121503, No. 2012AA120603) and China NSF projects (No. 61377012, No. 60807004).
#214882 - $15.00 USD Received 26 Jun 2014; revised 21 Aug 2014; accepted 9 Sep 2014; published 15 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023094 | OPTICS EXPRESS 23107
