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Auto gain control of EMCCD in Shack–Hartmann wavefront sensor for adaptive optics
Optics Communications 380 (2016) 469–475
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Auto gain control of EMCCD in Shack–Hartmann wavefront sensor for adaptive optics Zhaoyi Zhu a,b, Dayu Li a, Lifa Hu a, QuanQuan Mu a, Zhaoliang Cao a, Yukun Wang a, Shaoxin Wang a, Li Xuan a,n a b
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China University of Chinese Academy of Sciences, Beijing 100039, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2015 Received in revised form 26 May 2016 Accepted 6 June 2016 Available online 29 June 2016
Electron multiplying charge-coupled-device (EMCCD) applied in Shack–Hartmann wavefront sensor (S–H WFS) makes the wavefront sensing more efficient for adaptive optics (AO). However when the brightness of the observed target changes in large ranges in a few minutes, a fixed electron multiplying (EM) gain may not be optimum. Thus an auto-gain-control (AGC) method based on the spots image of the S–H WFS is proposed. The designed control value is the average value of the maximum signals of all the light spots in a frame. It has been demonstrated in the experiments that the control value is sensitive to the change of the target brightness, and is stable in the presence of detecting noises and turbulence influence. The goal value for control is predetermined based on the linear relation of the signal with the EM gain and the number of photons collected in sub-apertures. The conditions of the self-protection of the EMCCD are also considered for the goal value. Simulations and experiments indicate that the proposed control method is efficient, and keeps the sensing in a high SNR which reaches the upper SNR limit when sensing with EMCCD. The self-protection of the EMCCD is avoided during the whole sensing process. & 2016 Elsevier B.V. All rights reserved.
Keywords: Shack–Hartmann Wavefront Sensor Electron multiplying CCD Auto-gain-control Adaptive optics
1. Introduction Adaptive optics (AO) is now widely used in astronomical ground-based telescopes. The usual astronomical objects are very dim, which brings challenges to the AO systems, especially to the wavefront sensing. Many kinds of sensing techniques for AO have been proposed such as Shack–Hartmann [1], curvature [2], pyramid [3]. The researches about the precision of the sensors demonstrate that the noise of the CCD detectors is the main constraint when observing dim and weak objects. A new kind of CCD detector named Electron Multiplying Charge-Coupled-Device (EMCCD) is proposed [4] and has achieved sub-electron readout noise via the electron multiplication technique. It has been used in wavefront sensors and greatly improves wavefront sensing efficiency [5–9]. Shack–Hartmann wavefront sensor (S–H WFS) has been widely used in AO systems for its high speed and high precision [1,10,11]. It is designed to measure the distorted wavefront through a subaperture method. The light propagating through the lenslet array produces a light spots array on the CCD detector at the focal plane. The displacements of centroids are used to calculate the slopes of n
Corresponding author. E-mail address: [email protected] (L. Xuan).
http://dx.doi.org/10.1016/j.optcom.2016.06.014 0030-4018/& 2016 Elsevier B.V. All rights reserved.
the sub-wavefronts, and the aberrated pupil wavefront is reconstructed from the slopes [12–14]. With the EMCCD used in S–H WFS, higher precision has been achieved. Nevertheless, how to determine a suitable Electron Multiplying (EM) gain in observations is essential, especially when observing the satellites through the AO system, their brightness will change in a large range in a few minutes due to their fast traveling or movements. In this case, a fixed EM gain won't be optimum for high precision sensing. An auto-gain-control (AGC) method based on the changing brightness of the targets may be an available way. However there're still no reports about the AGC method for the S–H WFS. The conventional gain control methods basing on histogram [15] or thresholds [16] are proposed for the normal imaging systems, which are not suitable for the S–H WFS. The main reason is that the purposes of gain control in normal image systems are to improve the image contrast and to enhance the details of the image. While the signals of the S–H WFS are used for the centroids computation, and by setting a high gain with the EMCCD, the small signals are amplified and the influence of the readout noise is decreased. Another thing is that the outputs of the S–H WFS are typically divided spots in which lots of pixels are of little signals, which is different from the normal imaging systems. However, the conventional gain control methods take all the pixels into account, comparing, sorting, and counting, which are not suitable for S–H WFS. The third, a selfprotection process is applied in the EMCCD. A high EM gain may
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easily cause the self-protection when the brightness of the target changes in large scales. To resolve the problem, we present a simple and effective AGC method for the S–H WFS based on the light spots array of the S–H WFS. It is suited to the S–H WFS for its reasonable EM gain calculation, low-time cost and it is efficient to avoid the EMCCD self-protection. A method to control the multiplication gain of the EMCCD for the S–H WFS is designed in this paper. In Section 2, the light spots array of the S–H WFS is analyzed. The control value based on the output spots image of the S–H WFS is designed and how to compute a proper gain is studied. The self-protection process is also discussed in this section. In Section 3, we conduct simulations of the S–H WFS sensing in which the control value is analyzed. Validation experiments are given in Section 4. Finally, conclusions are given in Section 5.
where SNR is defined as ∑p(x, y)/∑n(x, y), and n(x, y) mainly consists of the photon spot noise and the readout noise, xcp is the actual centroid, xcn is the error centroid calculated by the noises. When the SNR is high enough, xc is close to xcp. In the EMCCD, the EM gain effects on the photons and the photon spot noise, so the SNR can be written as:
2. The control value design based on the spots image of the S–H WFS
where G is the set EM gain, and P is the real signals. F is the multiplying coefficient [4,19], N2 is the pixels number in the subwindow. sr2 is the variance of the readout noise. From Formula (3,4), we know that a higher SNR is gotten by setting a higher EM gain which leads to a higher precision. However there exists an upper SNR limit when working with the EMCCD, which is equal to
2.1. Analysis of the light spots array In practice, the adaptive optics system is installed at the focus of the telescope. The light from the target travels through the atmosphere and is focused into the adaptive optical system by the telescope, then incidents into the S–H WFS. Due to the turbulence of the atmosphere, the wavefront of the incoming light is disturbed, making the focal spots dislocated from the reference positions. The energy is uneven in the pupil due to the scintillation influence, and different energy is contained in each focal spot. Finally the spots are collected by the discrete pixels of the EMCCD. Fig. 1 is a frame of spot image in the experiments as a typical output of the S–H WFS. The centroid of each focal spot is calculated by the sub-window signals of the EMCCD by [12]:
xc =
∑ x ⋅ I (x, y) ∑ I (x, y)
(1)
where I(x, y) is the intensity of the pixel at coordinates (x, y), and xc is the calculated centroid of the spot in the x direction. The signals can be seen as the real signals combined with the noises, and written as:
I (x, y) = p(x, y) + n(x, y)
(2)
where p (x, y) stands for the real signals, and n (x, y) for the noises. Then based on Formula (2), xc is expressed as:
Fig. 1. An array of light spots in the S–H WFS.
xc = =
∑ x ⋅ p(x, y) ∑ x ⋅ n(x, y) SNR 1 ⋅ + ⋅ ∑ p(x, y) ∑ n(x, y) 1 + SNR 1 + SNR SNR 1 ⋅ xcp + ⋅ xcn 1 + SNR 1 + SNR
SNR =
G⋅P (F ⋅ G ⋅ P )2 + N2 ⋅ σr2
(3)
P
= 2
F ⋅P+
N 2 ⋅ σr2 G2
(4)
P /F2 . In addition N2 sr2 in formula (4) is a critical value for P, when P is lower than the critical value, the EM gain is needed and if P is larger than the critical value, the SNR without EM gain will be higher than that with EM gain, the EM gain should be off. For astronomical observation, targets are always very dim so the EM gain is necessary. 2.2. Principle of the auto-gain-control method From the typical output spot image of Fig. 1, the divided spots embody most features of this kind of sensor. A new method to compute the control value based on the divided spots is proposed here: pick up the maximum signal of each spot, and there will be as much maximum signals as the sub-apertures illuminated, then calculate the average of the chosen maximum signals, make the average value as the control value:
Vcon = mean(max{ In × n}, max{ In × n}, ⋯max{ In × n}) 1
2
k
(5)
where Vcon is the designed control value. In n is the discrete intensity distribution collected by the n n pixels in a sub-window. max{} is for picking the maximums, and mean() is for computing the average value. k is the number of the lenslets illuminated. The average method would reduce the influence of the atmospheric turbulence and detecting noise in the gain control process. Comparing with the reference method based on dynamic ranges of the histogram [15] and the thresholds [16], the average of the maximum signals is more suitable for the S–H WFS. The reference methods take all pixels into account for comparisons, sorting and counting. They neglect the characters of the spot images that the signal of each pixel and the histogram distribution changes rapidly due to the influence of the atmosphere. Another is that sometimes the telescope may miss following the targets due to their fast movement. In this situation only parts of the lenslets are illuminated, losing some image spots from the normal situations. Under this condition as shown in Fig. 2, the pixel numbers counted in the threshold method or the histogram method as control value will change, resulting that the predicted control values become invalid, and a wrong gain may be set. In our method, the average value is calculated by the illuminated spots and changes a little, which is concluded from the comparisons of the control values in the situations of different numbers of lenslets illuminated in Fig. 2(b). In the image system, the brightness of the CCD pixel has been
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Fig. 2. (a) A frame of spots image that parts of lenslets are illuminated, (b) comparisons of the control values in the situations of different lenslets illuminated.
and Gcurrent is the EM gain of the current frame. Fig. 3 presents the designed AGC control diagram. When the EMCCD self-protection process is triggered, it will reset the EM gain to 1, which should be avoided during the AO working. Normally, to avoid EMCCD protection, the output signal should be lower than 80% or 90% of the full-well signal of the EM register. Here in this paper, an OCam2 EMCCD produced by e2v technologies is applied, and a stricter limit is adopted. When there are more than 100 pixels above half of the full-well signal for 3 consecutive frames, the self-protection will work and set the gain to 1. The EM register of the OCam2 EMCCD is 14 bit, and the half full-well signal is 214/2 ¼8192 ADU. From Fig. 1, the highest energy of a spot may be contained in the maximum pixel. As our control value is calculated by all the maximum signals. Through the goal control value analysis, the possibility to trigger the protection is analyzed meanwhile by counting the number of pixels with signal larger than 8192ADU.
Fig. 3. The control diagram for the designed AGC.
derived to be a linear relationship of the incident light intensity and the EM gain as B ¼k I G naming the exposure function, in which B is the brightness of the pixels, k is the coefficient of the linearity, I is the collected energy of the pixels and G is the set EM gain [18]. The proposed control value should follow the linear relation which will be demonstrated in the simulation and experiment. In addition in our control method, a goal control value is used and predetermined to conquer the demands based on the observing condition in the control process. The control value is an average of pixel brightness, so is the goal control value. From B ¼k I G, the relations will be:
Vcon = k ⋅ Itarget ⋅ G
(6)
Vgoal = k ⋅ Itarget ⋅ Ggoal
(7)
where Vcon is the calculated control value with the collected energy Itarget and the set gain G. Vgoal is the predetermined goal value and Ggoal is the corresponding set gain. From Eqs. (6) and (7), the Ggoal can be determined as:
Ggoal = Vgoal/Vcon ⋅ G
(8)
Typically in the EMCCD system, the control value is calculated from the output image and the gain is set to multiply the next frame of image, so the EM gain will be calculated by:
Gnext = Vgoal/Vcurrent ⋅ Gcurrent
(9)
where Gnext is the calculated EM gain and then is set to the EMCCD
3. Simulations 3.1. S–H WFS simulation A simulation of the sensing process of the S–H WFS combined with the AGC is made in this paper [11]. Table 1 is the parameters of the S–H WFS used in the experiments. The simulations of the atmospheric affection are based on an optical device named turbulence simulator [20]. Distorted wavefronts in the sub-apertures are from the phase plate data of the turbulence simulator. Different photon numbers collected in the sub-apertures are set to simulate the influence of the scintillation, and the photon numbers appeared in this paper present the mean photon numbers collected in all sub-apertures in a frame. The normalized intensity distribution function of each focal spot is computed from the dissected sample wavefronts by the Discrete Fourier Transform theory. The pixel signals of each spot are gotten by multiplying the normalized intensity distribution function with the photons number collected in the sub-aperture, and the photon spot noise submitting the Poisson distribution is added then. The gotten pixel signal will be multiplied by an auto-controlled EM gain. A second time of Poisson function effects on the signals to simulate the multiplying influence [19]. A Gaussian readout noise with zero mean and a variance of sr2 ¼36 ADU2 is then added to the pixel signals. Usually the pupil of the telescope is circular or ring, so the lenslets at the edge of the lenslet array will be unlighted and a frame of spot image is shown in Fig. 1.
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Table 1 Parameters of the S–H WFS components, lenslet array and CCD220a (airy spot size follows the formula Dairy ¼2 1.22 Dlen λ/f with λ ¼0.55 um). Parameters
Value
lenslet array lenslet diameter lenslet focal length CCD pixels CCD pixel size pixels for each lenslet airy spot size pixels for an airy spot Readout noise
20 20 (square) 288 um 19.35 mm (calibrated) 120 120 48 um 288 um/48 um ¼ 6 6 90 um 90 um/48 um ¼ 1.88 1.88 6ADU/pixel
a CCD220 developed by e2v technologies (UK) 2 2binning in use.
Fig. 4. Comparisons of the average value with the maximum under unchanging brightness, RMS of the average is 114ADU, and RMS of the maximum is 510ADU.
3.2. Characteristics of the control value Based on the simulations, we have gotten frames of spots image, and the proposed average control value and the maximum value of each frame are calculated. Fig. 4 compares the average control value with the maximum in frames of spots images. Results demonstrate that the proposed control value stays stable in the presence of detecting noise and turbulence. Due to the turbulence, the maximum value varies rapidly in large range from frame to frame with a root mean square (RMS) value of 510 ADU of 500 frames, but the average value changes a little with a RMS value of 114 ADU that the influence of turbulence and detecting noise can be ignored in the view of control systems. Moreover, the control values are compared under different strengths of turbulence. The size of the sub-aperture on the aperture of the telescope is designed to be 10 cm. Different r0 as 10 cm, 9 cm, 8 cm, 7 cm present different strengths of turbulence, and the scintillation constants are 0.0348, 0.0415, 0.0505, 0.0631 respectively when considering a single layer of turbulence at distance L ¼10 km. Fig. 4 is the errorbar of the control values under different strengths of turbulence. With different photons collected in a sub-aperture, the control values under different r0 are almost the same. Meanwhile the linear relation of the control value with the photons number and the EM gain is also proved in Fig. 5. As mentioned above, a predetermined goal control value is needed. It is first determined based on the S–H WFS simulations and then verified in the experiments. The proposed AGC method is aiming to increase the sensing precision of the S–H WFS, and must avoid the self-protection of the EMCCD. For the EMCCD we experiment with, the self-protection process is described before, and
Fig. 5. Errorbar of the control values under different strengths of turbulence for different photons collected in a sub-aperture.
Fig. 6. Numbers of pixels with signal larger than 8000ADU compared at different strengths of turbulence.
from the AGC simulations we finally determine the goal control value as 6000ADU. Fig. 6 plots the numbers of pixels of signal larger than 8000ADU during the AGC. Different situations of different strengths of turbulence are analyzed. The statistical results with AGC for different r0 are almost the same. Due to the influence of atmospheric turbulence, the recorded numbers change rapidly, but there isn't a frame that the pixels number is larger than 100. The self-protection is avoided.
4. Experimental results and discussion 4.1. Experiments Setup The experimental optical platform is built up and a series of experiments are made to test the new AGC method. A standard light source supplied by a programmable DC power is used to simulate the brightness change of the targets. In the experiments, a series of sine form signals of different changing periods are used. Two turbulence simulators are set in the optical path to generate atmospheric influences into the experiments [20], in which the first one is used to simulate the scintillation and the second is to generate the turbulent wavefronts, and the portion of the plate that the light beam passes through should conjugate with the pupil of the S–H WFS. Different sizes of the lighted portion will bring different strengths of turbulence; Rotating the phase plate can bring changing turbulence into the experiment and the Greenwood frequency is set to 50 Hz. The optical sketch is shown in Fig. 7.
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Fig. 7. The experiment optical path for testing the AGC method, (a) the optical path, (b) the turbulence simulator seen from front, the dark part is the turbulence plate, phase data is used in simulation.
Fig. 8. Errorbar under different strengths of turbulence for different photons collected in a sub-aperture. In the experiments the diameter of the sub-aperture is calculated to be 0.75r0 and 1.25r0.
4.2. Results 4.2.1. Auto-gain-control test The AGC method is realized and tested in the experiments. Firstly, the proposed control value is measured at different brightness and different strengths of turbulence. Different sizes of the light beam passing through the turbulence simulator bring different strengths of turbulence in the experiments. Two situations are tested in our experiments in which the sub-aperture diameter is calculated as 1.25r0 or 0.75r0. Fig. 8 plots the control value at different numbers of photons collected in a sub-aperture at different strengths of turbulence. The result is similar to the simulation results. The control value changes a little in different strengths of turbulence and is seen in a linear relation with
Fig. 9. The control value and the maximum change without AGC, measured at G ¼1.
photons number collected in the sub-apertures. Secondly, we programed the DC power to output sine form current to supply the standard light source, and the intensity of the light source will change as sine form. We record the control value which changes with the brightness of the light source at EM gain as 1, as shown in Fig. 9. Fig. 10 reveals the results under AGC and the goal control value is set as 6000ADU as predetermined. Under AGC the mean control value is kept at 6000ADU which fits the set goal value well and the RMS results demonstrates the stability and efficiency of the control method. The numbers of the pixels of signal larger above 8000ADU in a frame are recorded during the experiments, which are shown in Fig. 10(b). During the AGC test, the number of pixels of signal larger than 8000 in a frame is less than 100 and it doesn't cause the self-protection of the EMCCD. Actually different periods
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Z. Zhu et al. / Optics Communications 380 (2016) 469–475
especially when only a few photons are collected. The SNR achieved under AGC is much larger than that without EM gain for astronomical dim target observation as the line of gain ¼1.
5. Conclusions
Fig. 10. Results with AGC, (a) comparisons of the control value and the maximum value, the mean of Vcon is 5988ADU and the RMS is 207ADU, (b) numbers of pixels with signal larger than 8000ADU, the numbers are smaller than 100 that doesn't cause self-protection.
An AGC method is proposed which is suitable for the S–H WFS and it is proved that the new method improves the sensing precision. Based on the specialties of the output spot images of the S– H WFS, a new control value as the average of the maximum signals of all light spots in a frame of image is designed. Comparisons between the average control value and the maximum in frames of spots images show the feasibility of the control value in the simulations. The AGC method is realized with the EMCCD and tested in experiments. The testing results show that the average control value is sensitive to the target brightness changing, and stays stable under turbulence and detecting noise. The linear relation of the control value with the EM gain and the intensity of targets makes it simple to predetermine the goal value, and combined with the self-protection conditions the goal control value is set as 6000ADU. In the AGC experiments, the upper SNR limit working with the EMCCD is achieved for different numbers of photons collected in sub-apertures and in the whole process the self-protection is avoided. Furthermore, for our control value design method, it's proved to be practical for any gain control cases using EMCCD. This can be realize by choosing feature values such as mean or maximum of different parts of an image as the control value, and optimizing the goal control value to achieve the expected results.
Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 61405194). It's also supported by State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences.
References
Fig. 11. Comparisons of SNR in sub-aperture with or without AGC, black dashed line for the limit SNR.
of sine form current are designed in the experiments, Fig. 9 only shows the fastest changing one as 9.6 s per period. When the brightness changes slower a better control result will be achieved and for the observation of the satellites the bandwidth of the AGC is enough. 4.2.2. Detecting SNR analysis Further analysis is done to verify the SNR improvement under AGC with different numbers of photons in a sub-aperture. The results are shown in Fig. 11 that the SNR reaches the upper limit for different numbers of photons collected in a sub-aperture. Without AGC, the EM gain should be set lower to avoid the selfprotection as shown in Fig. 11. For example, when the gain is set to 100, if more than 1800 photons are collected in a sub-aperture, the EMCCD will be protected. Meanwhile, the SNR achieved is lower
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Auto gain control of EMCCD in Shack–Hartmann wavefront sensor for adaptive optics Zhaoyi Zhu a,b, Dayu Li a, Lifa Hu a, QuanQuan Mu a, Zhaoliang Cao a, Yukun Wang a, Shaoxin Wang a, Li Xuan a,n a b
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China University of Chinese Academy of Sciences, Beijing 100039, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2015 Received in revised form 26 May 2016 Accepted 6 June 2016 Available online 29 June 2016
Electron multiplying charge-coupled-device (EMCCD) applied in Shack–Hartmann wavefront sensor (S–H WFS) makes the wavefront sensing more efficient for adaptive optics (AO). However when the brightness of the observed target changes in large ranges in a few minutes, a fixed electron multiplying (EM) gain may not be optimum. Thus an auto-gain-control (AGC) method based on the spots image of the S–H WFS is proposed. The designed control value is the average value of the maximum signals of all the light spots in a frame. It has been demonstrated in the experiments that the control value is sensitive to the change of the target brightness, and is stable in the presence of detecting noises and turbulence influence. The goal value for control is predetermined based on the linear relation of the signal with the EM gain and the number of photons collected in sub-apertures. The conditions of the self-protection of the EMCCD are also considered for the goal value. Simulations and experiments indicate that the proposed control method is efficient, and keeps the sensing in a high SNR which reaches the upper SNR limit when sensing with EMCCD. The self-protection of the EMCCD is avoided during the whole sensing process. & 2016 Elsevier B.V. All rights reserved.
Keywords: Shack–Hartmann Wavefront Sensor Electron multiplying CCD Auto-gain-control Adaptive optics
1. Introduction Adaptive optics (AO) is now widely used in astronomical ground-based telescopes. The usual astronomical objects are very dim, which brings challenges to the AO systems, especially to the wavefront sensing. Many kinds of sensing techniques for AO have been proposed such as Shack–Hartmann [1], curvature [2], pyramid [3]. The researches about the precision of the sensors demonstrate that the noise of the CCD detectors is the main constraint when observing dim and weak objects. A new kind of CCD detector named Electron Multiplying Charge-Coupled-Device (EMCCD) is proposed [4] and has achieved sub-electron readout noise via the electron multiplication technique. It has been used in wavefront sensors and greatly improves wavefront sensing efficiency [5–9]. Shack–Hartmann wavefront sensor (S–H WFS) has been widely used in AO systems for its high speed and high precision [1,10,11]. It is designed to measure the distorted wavefront through a subaperture method. The light propagating through the lenslet array produces a light spots array on the CCD detector at the focal plane. The displacements of centroids are used to calculate the slopes of n
Corresponding author. E-mail address: [email protected] (L. Xuan).
http://dx.doi.org/10.1016/j.optcom.2016.06.014 0030-4018/& 2016 Elsevier B.V. All rights reserved.
the sub-wavefronts, and the aberrated pupil wavefront is reconstructed from the slopes [12–14]. With the EMCCD used in S–H WFS, higher precision has been achieved. Nevertheless, how to determine a suitable Electron Multiplying (EM) gain in observations is essential, especially when observing the satellites through the AO system, their brightness will change in a large range in a few minutes due to their fast traveling or movements. In this case, a fixed EM gain won't be optimum for high precision sensing. An auto-gain-control (AGC) method based on the changing brightness of the targets may be an available way. However there're still no reports about the AGC method for the S–H WFS. The conventional gain control methods basing on histogram [15] or thresholds [16] are proposed for the normal imaging systems, which are not suitable for the S–H WFS. The main reason is that the purposes of gain control in normal image systems are to improve the image contrast and to enhance the details of the image. While the signals of the S–H WFS are used for the centroids computation, and by setting a high gain with the EMCCD, the small signals are amplified and the influence of the readout noise is decreased. Another thing is that the outputs of the S–H WFS are typically divided spots in which lots of pixels are of little signals, which is different from the normal imaging systems. However, the conventional gain control methods take all the pixels into account, comparing, sorting, and counting, which are not suitable for S–H WFS. The third, a selfprotection process is applied in the EMCCD. A high EM gain may
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easily cause the self-protection when the brightness of the target changes in large scales. To resolve the problem, we present a simple and effective AGC method for the S–H WFS based on the light spots array of the S–H WFS. It is suited to the S–H WFS for its reasonable EM gain calculation, low-time cost and it is efficient to avoid the EMCCD self-protection. A method to control the multiplication gain of the EMCCD for the S–H WFS is designed in this paper. In Section 2, the light spots array of the S–H WFS is analyzed. The control value based on the output spots image of the S–H WFS is designed and how to compute a proper gain is studied. The self-protection process is also discussed in this section. In Section 3, we conduct simulations of the S–H WFS sensing in which the control value is analyzed. Validation experiments are given in Section 4. Finally, conclusions are given in Section 5.
where SNR is defined as ∑p(x, y)/∑n(x, y), and n(x, y) mainly consists of the photon spot noise and the readout noise, xcp is the actual centroid, xcn is the error centroid calculated by the noises. When the SNR is high enough, xc is close to xcp. In the EMCCD, the EM gain effects on the photons and the photon spot noise, so the SNR can be written as:
2. The control value design based on the spots image of the S–H WFS
where G is the set EM gain, and P is the real signals. F is the multiplying coefficient [4,19], N2 is the pixels number in the subwindow. sr2 is the variance of the readout noise. From Formula (3,4), we know that a higher SNR is gotten by setting a higher EM gain which leads to a higher precision. However there exists an upper SNR limit when working with the EMCCD, which is equal to
2.1. Analysis of the light spots array In practice, the adaptive optics system is installed at the focus of the telescope. The light from the target travels through the atmosphere and is focused into the adaptive optical system by the telescope, then incidents into the S–H WFS. Due to the turbulence of the atmosphere, the wavefront of the incoming light is disturbed, making the focal spots dislocated from the reference positions. The energy is uneven in the pupil due to the scintillation influence, and different energy is contained in each focal spot. Finally the spots are collected by the discrete pixels of the EMCCD. Fig. 1 is a frame of spot image in the experiments as a typical output of the S–H WFS. The centroid of each focal spot is calculated by the sub-window signals of the EMCCD by [12]:
xc =
∑ x ⋅ I (x, y) ∑ I (x, y)
(1)
where I(x, y) is the intensity of the pixel at coordinates (x, y), and xc is the calculated centroid of the spot in the x direction. The signals can be seen as the real signals combined with the noises, and written as:
I (x, y) = p(x, y) + n(x, y)
(2)
where p (x, y) stands for the real signals, and n (x, y) for the noises. Then based on Formula (2), xc is expressed as:
Fig. 1. An array of light spots in the S–H WFS.
xc = =
∑ x ⋅ p(x, y) ∑ x ⋅ n(x, y) SNR 1 ⋅ + ⋅ ∑ p(x, y) ∑ n(x, y) 1 + SNR 1 + SNR SNR 1 ⋅ xcp + ⋅ xcn 1 + SNR 1 + SNR
SNR =
G⋅P (F ⋅ G ⋅ P )2 + N2 ⋅ σr2
(3)
P
= 2
F ⋅P+
N 2 ⋅ σr2 G2
(4)
P /F2 . In addition N2 sr2 in formula (4) is a critical value for P, when P is lower than the critical value, the EM gain is needed and if P is larger than the critical value, the SNR without EM gain will be higher than that with EM gain, the EM gain should be off. For astronomical observation, targets are always very dim so the EM gain is necessary. 2.2. Principle of the auto-gain-control method From the typical output spot image of Fig. 1, the divided spots embody most features of this kind of sensor. A new method to compute the control value based on the divided spots is proposed here: pick up the maximum signal of each spot, and there will be as much maximum signals as the sub-apertures illuminated, then calculate the average of the chosen maximum signals, make the average value as the control value:
Vcon = mean(max{ In × n}, max{ In × n}, ⋯max{ In × n}) 1
2
k
(5)
where Vcon is the designed control value. In n is the discrete intensity distribution collected by the n n pixels in a sub-window. max{} is for picking the maximums, and mean() is for computing the average value. k is the number of the lenslets illuminated. The average method would reduce the influence of the atmospheric turbulence and detecting noise in the gain control process. Comparing with the reference method based on dynamic ranges of the histogram [15] and the thresholds [16], the average of the maximum signals is more suitable for the S–H WFS. The reference methods take all pixels into account for comparisons, sorting and counting. They neglect the characters of the spot images that the signal of each pixel and the histogram distribution changes rapidly due to the influence of the atmosphere. Another is that sometimes the telescope may miss following the targets due to their fast movement. In this situation only parts of the lenslets are illuminated, losing some image spots from the normal situations. Under this condition as shown in Fig. 2, the pixel numbers counted in the threshold method or the histogram method as control value will change, resulting that the predicted control values become invalid, and a wrong gain may be set. In our method, the average value is calculated by the illuminated spots and changes a little, which is concluded from the comparisons of the control values in the situations of different numbers of lenslets illuminated in Fig. 2(b). In the image system, the brightness of the CCD pixel has been
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Fig. 2. (a) A frame of spots image that parts of lenslets are illuminated, (b) comparisons of the control values in the situations of different lenslets illuminated.
and Gcurrent is the EM gain of the current frame. Fig. 3 presents the designed AGC control diagram. When the EMCCD self-protection process is triggered, it will reset the EM gain to 1, which should be avoided during the AO working. Normally, to avoid EMCCD protection, the output signal should be lower than 80% or 90% of the full-well signal of the EM register. Here in this paper, an OCam2 EMCCD produced by e2v technologies is applied, and a stricter limit is adopted. When there are more than 100 pixels above half of the full-well signal for 3 consecutive frames, the self-protection will work and set the gain to 1. The EM register of the OCam2 EMCCD is 14 bit, and the half full-well signal is 214/2 ¼8192 ADU. From Fig. 1, the highest energy of a spot may be contained in the maximum pixel. As our control value is calculated by all the maximum signals. Through the goal control value analysis, the possibility to trigger the protection is analyzed meanwhile by counting the number of pixels with signal larger than 8192ADU.
Fig. 3. The control diagram for the designed AGC.
derived to be a linear relationship of the incident light intensity and the EM gain as B ¼k I G naming the exposure function, in which B is the brightness of the pixels, k is the coefficient of the linearity, I is the collected energy of the pixels and G is the set EM gain [18]. The proposed control value should follow the linear relation which will be demonstrated in the simulation and experiment. In addition in our control method, a goal control value is used and predetermined to conquer the demands based on the observing condition in the control process. The control value is an average of pixel brightness, so is the goal control value. From B ¼k I G, the relations will be:
Vcon = k ⋅ Itarget ⋅ G
(6)
Vgoal = k ⋅ Itarget ⋅ Ggoal
(7)
where Vcon is the calculated control value with the collected energy Itarget and the set gain G. Vgoal is the predetermined goal value and Ggoal is the corresponding set gain. From Eqs. (6) and (7), the Ggoal can be determined as:
Ggoal = Vgoal/Vcon ⋅ G
(8)
Typically in the EMCCD system, the control value is calculated from the output image and the gain is set to multiply the next frame of image, so the EM gain will be calculated by:
Gnext = Vgoal/Vcurrent ⋅ Gcurrent
(9)
where Gnext is the calculated EM gain and then is set to the EMCCD
3. Simulations 3.1. S–H WFS simulation A simulation of the sensing process of the S–H WFS combined with the AGC is made in this paper [11]. Table 1 is the parameters of the S–H WFS used in the experiments. The simulations of the atmospheric affection are based on an optical device named turbulence simulator [20]. Distorted wavefronts in the sub-apertures are from the phase plate data of the turbulence simulator. Different photon numbers collected in the sub-apertures are set to simulate the influence of the scintillation, and the photon numbers appeared in this paper present the mean photon numbers collected in all sub-apertures in a frame. The normalized intensity distribution function of each focal spot is computed from the dissected sample wavefronts by the Discrete Fourier Transform theory. The pixel signals of each spot are gotten by multiplying the normalized intensity distribution function with the photons number collected in the sub-aperture, and the photon spot noise submitting the Poisson distribution is added then. The gotten pixel signal will be multiplied by an auto-controlled EM gain. A second time of Poisson function effects on the signals to simulate the multiplying influence [19]. A Gaussian readout noise with zero mean and a variance of sr2 ¼36 ADU2 is then added to the pixel signals. Usually the pupil of the telescope is circular or ring, so the lenslets at the edge of the lenslet array will be unlighted and a frame of spot image is shown in Fig. 1.
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Table 1 Parameters of the S–H WFS components, lenslet array and CCD220a (airy spot size follows the formula Dairy ¼2 1.22 Dlen λ/f with λ ¼0.55 um). Parameters
Value
lenslet array lenslet diameter lenslet focal length CCD pixels CCD pixel size pixels for each lenslet airy spot size pixels for an airy spot Readout noise
20 20 (square) 288 um 19.35 mm (calibrated) 120 120 48 um 288 um/48 um ¼ 6 6 90 um 90 um/48 um ¼ 1.88 1.88 6ADU/pixel
a CCD220 developed by e2v technologies (UK) 2 2binning in use.
Fig. 4. Comparisons of the average value with the maximum under unchanging brightness, RMS of the average is 114ADU, and RMS of the maximum is 510ADU.
3.2. Characteristics of the control value Based on the simulations, we have gotten frames of spots image, and the proposed average control value and the maximum value of each frame are calculated. Fig. 4 compares the average control value with the maximum in frames of spots images. Results demonstrate that the proposed control value stays stable in the presence of detecting noise and turbulence. Due to the turbulence, the maximum value varies rapidly in large range from frame to frame with a root mean square (RMS) value of 510 ADU of 500 frames, but the average value changes a little with a RMS value of 114 ADU that the influence of turbulence and detecting noise can be ignored in the view of control systems. Moreover, the control values are compared under different strengths of turbulence. The size of the sub-aperture on the aperture of the telescope is designed to be 10 cm. Different r0 as 10 cm, 9 cm, 8 cm, 7 cm present different strengths of turbulence, and the scintillation constants are 0.0348, 0.0415, 0.0505, 0.0631 respectively when considering a single layer of turbulence at distance L ¼10 km. Fig. 4 is the errorbar of the control values under different strengths of turbulence. With different photons collected in a sub-aperture, the control values under different r0 are almost the same. Meanwhile the linear relation of the control value with the photons number and the EM gain is also proved in Fig. 5. As mentioned above, a predetermined goal control value is needed. It is first determined based on the S–H WFS simulations and then verified in the experiments. The proposed AGC method is aiming to increase the sensing precision of the S–H WFS, and must avoid the self-protection of the EMCCD. For the EMCCD we experiment with, the self-protection process is described before, and
Fig. 5. Errorbar of the control values under different strengths of turbulence for different photons collected in a sub-aperture.
Fig. 6. Numbers of pixels with signal larger than 8000ADU compared at different strengths of turbulence.
from the AGC simulations we finally determine the goal control value as 6000ADU. Fig. 6 plots the numbers of pixels of signal larger than 8000ADU during the AGC. Different situations of different strengths of turbulence are analyzed. The statistical results with AGC for different r0 are almost the same. Due to the influence of atmospheric turbulence, the recorded numbers change rapidly, but there isn't a frame that the pixels number is larger than 100. The self-protection is avoided.
4. Experimental results and discussion 4.1. Experiments Setup The experimental optical platform is built up and a series of experiments are made to test the new AGC method. A standard light source supplied by a programmable DC power is used to simulate the brightness change of the targets. In the experiments, a series of sine form signals of different changing periods are used. Two turbulence simulators are set in the optical path to generate atmospheric influences into the experiments [20], in which the first one is used to simulate the scintillation and the second is to generate the turbulent wavefronts, and the portion of the plate that the light beam passes through should conjugate with the pupil of the S–H WFS. Different sizes of the lighted portion will bring different strengths of turbulence; Rotating the phase plate can bring changing turbulence into the experiment and the Greenwood frequency is set to 50 Hz. The optical sketch is shown in Fig. 7.
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Fig. 7. The experiment optical path for testing the AGC method, (a) the optical path, (b) the turbulence simulator seen from front, the dark part is the turbulence plate, phase data is used in simulation.
Fig. 8. Errorbar under different strengths of turbulence for different photons collected in a sub-aperture. In the experiments the diameter of the sub-aperture is calculated to be 0.75r0 and 1.25r0.
4.2. Results 4.2.1. Auto-gain-control test The AGC method is realized and tested in the experiments. Firstly, the proposed control value is measured at different brightness and different strengths of turbulence. Different sizes of the light beam passing through the turbulence simulator bring different strengths of turbulence in the experiments. Two situations are tested in our experiments in which the sub-aperture diameter is calculated as 1.25r0 or 0.75r0. Fig. 8 plots the control value at different numbers of photons collected in a sub-aperture at different strengths of turbulence. The result is similar to the simulation results. The control value changes a little in different strengths of turbulence and is seen in a linear relation with
Fig. 9. The control value and the maximum change without AGC, measured at G ¼1.
photons number collected in the sub-apertures. Secondly, we programed the DC power to output sine form current to supply the standard light source, and the intensity of the light source will change as sine form. We record the control value which changes with the brightness of the light source at EM gain as 1, as shown in Fig. 9. Fig. 10 reveals the results under AGC and the goal control value is set as 6000ADU as predetermined. Under AGC the mean control value is kept at 6000ADU which fits the set goal value well and the RMS results demonstrates the stability and efficiency of the control method. The numbers of the pixels of signal larger above 8000ADU in a frame are recorded during the experiments, which are shown in Fig. 10(b). During the AGC test, the number of pixels of signal larger than 8000 in a frame is less than 100 and it doesn't cause the self-protection of the EMCCD. Actually different periods
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especially when only a few photons are collected. The SNR achieved under AGC is much larger than that without EM gain for astronomical dim target observation as the line of gain ¼1.
5. Conclusions
Fig. 10. Results with AGC, (a) comparisons of the control value and the maximum value, the mean of Vcon is 5988ADU and the RMS is 207ADU, (b) numbers of pixels with signal larger than 8000ADU, the numbers are smaller than 100 that doesn't cause self-protection.
An AGC method is proposed which is suitable for the S–H WFS and it is proved that the new method improves the sensing precision. Based on the specialties of the output spot images of the S– H WFS, a new control value as the average of the maximum signals of all light spots in a frame of image is designed. Comparisons between the average control value and the maximum in frames of spots images show the feasibility of the control value in the simulations. The AGC method is realized with the EMCCD and tested in experiments. The testing results show that the average control value is sensitive to the target brightness changing, and stays stable under turbulence and detecting noise. The linear relation of the control value with the EM gain and the intensity of targets makes it simple to predetermine the goal value, and combined with the self-protection conditions the goal control value is set as 6000ADU. In the AGC experiments, the upper SNR limit working with the EMCCD is achieved for different numbers of photons collected in sub-apertures and in the whole process the self-protection is avoided. Furthermore, for our control value design method, it's proved to be practical for any gain control cases using EMCCD. This can be realize by choosing feature values such as mean or maximum of different parts of an image as the control value, and optimizing the goal control value to achieve the expected results.
Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 61405194). It's also supported by State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences.
References
Fig. 11. Comparisons of SNR in sub-aperture with or without AGC, black dashed line for the limit SNR.
of sine form current are designed in the experiments, Fig. 9 only shows the fastest changing one as 9.6 s per period. When the brightness changes slower a better control result will be achieved and for the observation of the satellites the bandwidth of the AGC is enough. 4.2.2. Detecting SNR analysis Further analysis is done to verify the SNR improvement under AGC with different numbers of photons in a sub-aperture. The results are shown in Fig. 11 that the SNR reaches the upper limit for different numbers of photons collected in a sub-aperture. Without AGC, the EM gain should be set lower to avoid the selfprotection as shown in Fig. 11. For example, when the gain is set to 100, if more than 1800 photons are collected in a sub-aperture, the EMCCD will be protected. Meanwhile, the SNR achieved is lower
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