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An active segmented cophasing imaging testbed with a high accurate optical adjustment and control in nanoscale is set up. A two-dimensional dispersed fringe analysis method is used to calculate the piston error between the adjacent segments. Segments alignment of tip-tilt is fulfilled by overlapping diffraction pattern centroids of the individual segment on the focal plane. The cophasing errors are corrected by sensitive micro-displacement actuators under a computer closed-loop control. A FISBA interferometer is introduced to evaluate that a final total residual surface error of about 0.08λ RMS and the residual piston error of 0.026λ RMS are achieved (λ = 633nm).
©2011 Optical Society of America
1. Introduction
The aperture of telescope primary mirror must be enlarged for a high resolution. This makes optical processing and testing, and transportation or launch very difficult. If the diameter of the primary mirror is over 10m, such as 30m caliber extremely large telescope (ELT), the manufacturing of 30m optical glass in one piece is impossible. A segmented and deployable primary mirror was adopted to avoid these problems. In 1991, the Keck telescope with a segmented primary mirror with approximately 10m caliber was set up, and a diffraction limited image at infrared was realized. This verifies successfully that adopting a segmented primary mirror in telescope for a high resolution is feasible. After that, many telescopes with segmented primary mirrors with diameters in the range from 8 to 10m have been constructed. Currently, some 30 to 50m caliber segmented primary telescopes are under design, such as the European Extremely Large Telescope (EELT) [1] in the European Southern Observatory and the Thirty Meter Telescope (TMT) [2]. The 42m caliber primary mirror of the EELT will be made of 984 segments. The 30m caliber primary mirror of the TMT will be made of 492 segments.
The relative position of the segments of primary mirror will be affected by external disturbances, such as vibrations of the structure, air disturbance, and gravitational load. Therefore, the segments of primary mirror must be cophased by an active control to achieve a spatial resolution comparable to that of a monolithic mirror. Many cophasing methods have been studied, such as broadband Shack–Hartman [3], narrowband Shack–Hartman [4], phase diversity [5,6], Zernike phase filtering sensor [7,8], pyramid sensor [9], curvature sensing [10], and dispersed fringe sensor [11]. These methods have different capture range and accuracy from each other.
But for the ground-based segmented primary telescope with a diameter over 10m, it is very difficult to realize diffraction-limited imaging because of the atmospheric turbulence although with an adaptive optical correction. Therefore, a space-based telescope has been developed. Also for a high resolution and taking into account launch, a deployable and segmented primary mirror was adopted. After launch and deployment, the segments must be cophased to achieve a spatial resolution comparable to that of a monolithic mirror also. The performance of this kind of space-based telescope may be compromised by some factors such as manufacturing and alignment errors, on orbit thermal deformations, segment deployment errors, launch induced misalignments, and spacecraft jitter [12]. Therefore, some test-beds, on the ground, have been designed and built up to research some key technologies and express in hardware technologies capable of mitigating these risk factors. The Developmental Cryogenic Active Telescope Testbed (DCATT) [13,14] was set up to validate the feasibility of various technologies. The Wavefront Sensing and Control Testbed (WCT) [15] and Active Phasing Experiment [16] were established to validate wavefront sensing (including tip-tilt and piston errors and high-order aberrations) and control algorithms. The Test bed PHASAR [17] was built up to demonstrate segmented cophasing imaging principle and validate the method of detecting cophasing error. The Low Cost Space Imager (LCSI) [18] was used to demonstrate imaging principle of a sparse aperture optical system. The “experimental system of the segmented mirror active optics” was set up at the Nanjin Astronomical Instruments Research Center of China [19] to validate real-time control algorithms. “A segmented mirror imaging system” at the Harbin Institute of Technology of China [20] was used to evaluate image quality for segmented mirror synthetic aperture optics.
In this paper an active cophasing and aligning testbed with segmented mirrors (ACAT) is presented and used to validate a novel piston error detecting method [21–23], which is of a large capture range and high accuracy. The tip-tilt error is detected with the method of centroid sensing which has been widely used in segmented telescope. In addition, with this testbed, the cophasing imaging has been realized by correcting the cophasing errors with several sensitive micro-displacement actuators under a computer closed-loop control in nanoscale. Finally, an FISBA interferometer is introduced to evaluate the cophasing performance of the ACAT. A final total residual surface error of about 0.08λ RMS and the residual piston error of 0.026λ RMS have been achieved (λ = 633nm). The method of detecting and correcting cophasing error validated by the ACAT will be used in a 6m caliber deployable space-based segmented telescope under condition of the limiting beacon star magnitude of 6.
2. Testbed description
A block diagram of the ACAT set up in this paper is shown as Fig. 1 . It is a modular system that consists of a source module (SM), beam splitting module (BSM), piston error detection module (PEDM), tip-tilt error detection module (TEDM), sensitive micro-displacement actuator module (SMAM), segmented mirror module (SMM), FISBA spherical interferometer, laser plane interferometer (LPI), and Computer Control System (CCS). Its optical layout is shown as Fig. 2 .
All the modules of the ACAT are installed on a bed with upper and lower layers. Four lower temperature coefficient rigidity pillars are used to connect the upper layer and the lower layer. Segmented mirror and actuators coupled with the segments are placed on the lower layer, and the other modules are placed on the upper layer. The ACAT is installed on a vibration-isolated air base.
The SM consists of a continuous spectrum halogen light source with fiber bundle output, a group of lenses, and a 10μm pinhole (as a point object). The pinhole is illuminated by the light from the fiber bundle connected with the light source. The wave from the pinhole is collimated to provide a point object at infinity. After the BSM, the plane wave is converged and then reaches the spherical segmented mirror with a spherical curvature radius of 1500mm and a caliber of about 330mm. The segmented mirror is divided into three segments. One of them (Seg. 1) is fixed as a reference one. The others (Seg. 2 and Seg. 3) can be adjusted respectively in three degrees of freedom (piston, tip, and tilt) using piezoelectric transducer (PZT) actuators coupled with the back of the each adjustable segment in a range of 15mm with a resolution of 0.3nm. The each adjustable segment is coupled with the three PZT actuators. Waves reflected from the segmented mirror are recollimated and then incident to the PEDM and the TEDM after the BSM again. Piston and tip-tilt error between Seg. 1 and each adjustable segment are detected, respectively. According to the cophasing error, correction signals are calculated and output to the driver of the actuators by computer. Thus the position of each adjustable segment is corrected relative to Seg. 1, respectively. After several iterations, the piston and tip-tilt error between the segments will go steadily to a minimum. After phasing, the moveable plane reflector will be driven by a precision motor and inserted into the optical path to introduce the FISBA interferometer, which can evaluate the performance of the piston and tip-tilt error detection-correction closed-loop. The process of cophasing error detecting and correcting can be monitored in real time by the LPI.
3. Cophasing error detection and correction
3.1 Piston error detection
PEDM is designed to detect the piston errors, and a two-dimension dispersed fringe analyzing method [21–23] based on the principle of Rayleigh Interferometer is used to calculate the piston error between the adjacent segments.
A dispersed Rayleigh interferometer, shown as the left scheme in Fig. 3 , is formed by inserting a dispersive element into a Rayleigh interferometer with a broadband source S. It is used to measure optical path difference (OPD) δ between two parallel beams. A wave from the S is collimated by lens L1. A phase plate is used to introduce δ between the two parallel beams. Aperture stop with two same rectangle holes divides the collimated wave into two beams. The line linking the two centers of the rectangle holes is baseline. After the dispersive element whose dispersive direction is perpendicular to the baseline of aperture, the deviation angles for different wavelengths are not the same. Therefore, they are focusing on different position along the x axis by wavelength on the focal plane of imaging lens L2. Thus, a dispersed Rayleigh interference pattern (DRIP) is formed. DRIPs corresponding to the δ = 0 and δ = 15μm are shown as the right image in Fig. 3.
As the detail in Ref. [23], the intensity distribution of DRIP on the focal plane of L2 can be written as
where l is the length of the rectangle hole, w is the width of the rectangle hole, f is the focal length of the L2, S(λ) is the spectral intensity, x(λ) is the dispersive character of the dispersive element, η0 is the distance between the two centers of the two rectangle holes, [λmin, λmax] is the spectrum range of the source, and δ is OPD between the two parallel beams.δ consists of two parts and can be expressed as
where δd (λ) is the decimal fraction part of OPD for each wavelength and δi (λ) is the integer part of OPD for each wavelength.Each column of the DRIP is a quasi-monochromatic Rayleigh interference pattern whose intensity distribution (the intensity distribution of the DRIP along the y axis) can be expressed as
where γ is the visibility, γ∈[0,1], and it is related to the dispersive power and OPD.By simulating Eq. (3) under a certain wavelength, the relation between δd (λ) and main peak position ym (λ) is obtained, it can be expressed as
where T(λ) is spacial cycle of the interference pattern along the y axis, that is the distance between the main peak and the first side-lobe shown as (b) in Fig. 4 .
Fig. 4 Illustration of two-dimension dispersed fringe analyzing. (a) Intensity distribution of the DRIP. (b) Intensity distribution along the y axis.
A two-dimensional dispersed fringe analyzing method was proposed to calculate the piston error between the adjacent segments. Its principle illustration is shown as Fig. 4. As the details in Refs. [21,22], using this method, OPD δ can be obtained by three steps. At the first, ym(λ1), ym(λ2), T(λ1), and T(λ2) are determined by analyzing the intensity distribution of the DRIP along the y axis. And according to Eq. (4), δd (λ1) and δd (λ2) can be gained. Secondly, fringe-order difference between λ1 and λ2 should be counted along the x axis. And a coarse OPD δc can be acquired according to Eq. (5) shown as
where M is the fringe-order difference between λ1 and λ2 (M = n-m).The third step is to obtain an accurate δ. In Eq. (5), the measurement error of δd (λ) is magnified, so it is necessary to determine the fringe order K(λ) accurately for each wavelength with Eq. (6) shown as
where function round means to choose the nearest integer.Then, a fine test result δf (λ) can be obtained,
Finally, the more accurate result δ can be given by averaging δf (λi)(i = 1,2,…,N),In the ACAT, a PEDM is designed based on the principle described above. It consists of an aperture stop, an Amici prism, an imaging lens, and a complementary metal oxide semiconductor (CMOS). The aperture stop, shown as Fig. 5 , is placed on the exit-pupil plane on which the image size of the segmented mirror is about 25mm in diameter. A diagram of the optical path for detecting piston error between the adjacent segments is shown as Fig. 6 .
Waves reflected from the segmented mirror are recollimated and then incident to the PEDM. The aperture stop must be installed exactly to ensure that the waves reflected only by two adjacent segments can be sampled by the PEDM. Then the two-dimensional dispersed fringe analyzing method is used to calculate piston error between the two adjacent segments according to DRIP formed by the sampled waves. Rotating the PEDM 120° with a precision motor around the B axis in turn, the piston errors between Seg. 1 and Seg. 2, between Seg. 1 and Seg. 3, and also between Seg. 2 and Seg. 3 can be acquired respectively. States of the aperture stop during the rotating PEDM are shown as Fig. 7 .
Fig. 7 States of the aperture stop during the rotating PEDM. (a) Initial state. (b) Rotating 120°. (c) Rotating 120° further.
Measurement accuracy of the two-dimension dispersed fringe analyzing method is mainly affected by tip-tilt error between the segments, aberration of the wavefront sampled by the aperture stop, capture range, and SNR. The smaller capture range is the higher measurement accuracy will be obtained. Sensitivity to the segment vibration along the piston is obtained by experiments; it is better than 2nm.
Compared with existing techniques, such as broadband Shack–Hartman [3], narrowband Shack–Hartman [4], phase diversity [5,6], Zernike phase filtering sensor [7,8], pyramid sensor [9], curvature sensing [10], and dispersed fringe sensor [11], larger capture range and higher accuracy are outstanding advantages of the two-dimensional dispersed fringe analysis. The detailed comparable results are shown in Table 1 .
Table 1. Comparison of Cophasing Error Detection Methods
3.2 Tip-tilt error detection
The TEDM is designed to detect the tip-tilt errors between the adjacent segments. As shown in Fig. 8 , the TEDM consists of an aperture stop wheel, lens 3 (imaging lens), and a CMOS.
Fig. 8 Layout of TEDM and the structure of its aperture stop. (a) Layout of TEDM. (b) Structure of the aperture stop wheel.
Waves reflected from the segments are recollimated and then incident to the TEDM. Images of the pinhole formed by three segments are sampled by the CMOS. If the tip-tilt error is zero the images are overlapped. Otherwise, the images are separated from each other.
The aperture stop wheel, shown as Fig. 8(b), is placed on the exit-pupil plane of the ACAT. Four aperture stops A, B, C, and D on the wheel will be run into the optical path of the TEDM in turn by rotating the aperture stop wheel. Thus, the three images of the pinhole can be sampled, and then the images centroid positions are calculated respectively. Take the centroid position of the image formed by Seg. 1 as a reference, the tip-tilt errors between Seg. 1 and Seg. 2, and also between Seg. 1 and Seg. 3, can be obtained by calculating the centroid position differences.
With this method, the maximum error of tip-tilt detecting is 0.1 pixels. Shown as Fig. 2, in principle, the sensitivity to segment vibration along tip and tilt can be calculated as
where is the pixel size of the CMOS of the TEDM, ; is the focal length of the lens 2; is the focal length of the lens 3; and is the curvature radius of the segmented spherical mirror.Three images formed by the segments can be observed at the same time when the aperture stop D is turned into the optical path of the TEDM, and the tip-tilt state of the segmented mirror can be known. States of the aperture stop on the exit-pupil plane during rotating the wheel are shown as Fig. 9 .
Fig. 9 States of the aperture stop on the exit-pupil plane during rotating the wheel. (a) Aperture stop A is run into and on the exit-pupil plane. (b) Aperture stop B is run into and on the exit-pupil plane. (c) Aperture stop C is run into and on the exit-pupil plane. (d) Aperture stop D is run into and on the exit-pupil plane.
When this method is used in the 6m caliber deployable space-based segmented telescope mentioned in Section 1, a tip-tilt error detecting precision can be obtained under condition of the limiting beacon star magnitude of 6.
3.3 Cophasing error correction
Taking Seg. 1 as reference, the piston and tip-tilt errors between Seg. 1 and Seg. 2, between Seg. 1 and Seg. 3, and also between Seg. 2 and Seg. 3 can be corrected with a computer closed-loop control according to the errors detected by using the methods described above.
Due to a bigger tip-tilt error effects on the measurement accuracy of piston error, the tip-tilt error detecting and correcting should be carried out before the piston error detecting and correcting. The detection-correction flowchart is shown as Fig. 10 .
From Fig. 10 we can see that the cophasing error between Seg. 1 and Seg. 2 is corrected at first. The aperture stop wheel is rotated to make the aperture stop A in the optical path of the TEDM shown as Fig. 9(a). The image of the pinhole formed by Seg. 1 is sampled, and its centroid position is calculated as a reference. The aperture stop wheel is rotated further to make the aperture stop B in the optical path shown as Fig. 9(b). The centroid position of the pinhole image formed by Seg. 2 is calculated also. Then tip-tilt error between Seg. 1 and Seg. 2 can be acquired by comparing the centroid positions difference between these two images. According to the tip-tilt error, correction signals are calculated and output by a computer to drive the micro-displacement actuators coupled with Seg. 2, and the tip-tilt error is corrected under computer closed-loop control until the error is lower than an expected value.
Subsequently, the PEDM is rotated to ensure the waves reflected from Seg. 1 and Seg. 2 could be sampled shown as Fig. 7(a). The piston error between Seg. 1 and Seg. 2 can be obtained by using the two-dimension dispersed fringe analyzing method. According to the piston error, correction signals are calculated and output by a computer to drive the micro-displacement actuators coupled with Seg. 2, and the piston error is corrected under computer closed-loop control until the error is lower than an expected value.
Since the tip-tilt error correction and the piston error correction influence each other, these two correction processes should be repeated several times in turns until not only the tip-tilt error but also the piston error become lower than the expected value.
The above process will be carried out to correct the cophasing error between Seg.1 and seg.3, and also between Seg. 2 and Seg. 3 respectively.
It is necessary to check the cophasing error between Seg. 2 and Seg. 3 after phasing Seg. 1 and Seg. 3. Ordinarily, process of the cophasing error detection and correction shown as Fig. 10 should be repeated several times to make the segments in a better cophasing state.
4. Experiment and results
The ACAT is assembled and adjusted after finishing its each function module; the photo of the ACAT is shown as Fig. 11 . Because the cophasing detecting and correcting method aforementioned will be used in the space-based telescope, so influence of atmospheric turbulence on the detection can be ignored. Therefore, the ACAT is installed in a glass room shown as Fig. 11 in order to avoid the air disturbance affecting on experiments.
Fig. 11 The photo of ACAT. (1) SM, (2) TEDM, (3) PEDM, (4) FISBA, (5) plane reflector mirror, (6) moveable plane reflector, (7) BSM, (8) LPI, (9) SMM. (a) Layout of the ACAT modules on upper layer. (b) Layout of the ACAT.
According to the flowchart shown as Fig. 10, experiment of cophasing imaging has been carried out on the ACAT as below.
Initial state of the segments can be observed on the display screens of TEDM, PEDM and LPI, shown as Fig. 12 . The pinhole images formed by three segments are separated from each other in Fig. 12(a). It means that there are tip-tilt errors between the segments. DRIPs in Fig. 12(b) were formed by every two adjacent segments. There are many fringe orders in each DRIP, it means that there are piston errors between the adjacent segments. The interference fringes displayed in Fig. 12(c) was formed by the waves reflected from the three segments and reference wave from LPI respectively. Directions of three groups of fringe are different as well as their space. It means that not only tip-tilt error exists but also piston error exists between the adjacent segments.
Fig. 12 Initial cophasing state of the segments. (a) Tip-tilt state of segments. (b) Piston error between the adjacent segments. (c) Interference fringes formed by the waves from the three segments and the reference wave from LPI.
The co-phasing errors between Seg. 1 and Seg. 2 were detected and corrected firstly.
At first, the tip-tilt error between Seg. 1 and Seg. 2 was detected. Then the tip-tilt error correction signal was calculated according to the tip-tilt error and output by the computer to the driver of the micro-displacement actuators coupled with Seg. 2. After several iterations, the tip-tilt error was corrected and became lower than an expected value.
Next, the piston error was detected. The piston error correction signal was calculated according to the piston error and output by the computer to the driver of the micro-displacement actuators coupled with Seg. 2.
The process of the tip-tilt error detecting and correcting and the process of the piston error detecting and correcting were repeated several times until the tip-tilt error and piston error all became lower than an expected value. The results are shown in Fig. 13 . From Fig. 13(a), we can see that the pinhole images formed by Seg. 1 and Seg. 2 are overlapped. From Fig. 13(b), we can see that the residual piston error is 2nm.
Fig. 13 The cophasing error between Seg. 1 and Seg. 2 was corrected. (a) The tip-tilt error between Seg. 1 and Seg. 2 was corrected. (b) The piston error between Seg. 1 and Seg. 2 was corrected and the residual error is 2nm.
Subsequently, after the aperture stop wheel and PEDM had been rotated, the pinhole image formed by Seg. 3 and the DRIP formed by waves from Seg. 1 and Seg. 3 were sampled. Then the former process was repeatedly applied to detect and correct cophasing errors between Seg. 1 and Seg. 3. The results are shown in Fig. 14 . We can see that the pinhole images formed by three segments are overlapped, and the residual piston error between Seg. 1 and Seg. 3 is 2nm.
Fig. 14 The cophasing error between Seg. 1 and Seg. 3 was corrected. (a) The tip-tilt error between Seg. 1 and Seg. 3 was corrected. (b) The piston error between Seg. 1 and Seg. 3 was corrected, and the residual error is 2nm.
Finally, the cophasing error between Seg. 2 and Seg. 3 was checked further. If the cophasing error is bigger than λ/20 (the expected value, λ = 633nm), the processes, finished just, must be repeated. The result is shown as Fig. 15 . The piston error between Seg. 2 and Seg. 3 is 8nm, smaller than the expected value.
After phasing the segmented mirror, the image displayed on LPI screen is shown in Fig. 16 . From Fig. 16, we can see that the directions of three groups of fringe are identical, and the same for their space. It means that not only tip-tilt error but also piston error have been corrected and cophasing imaging has been realized.
For evaluating performance of the co-phasing error closed-loop detection and correction on the ACAT, the moveable plane reflector was inserted into the optical path shown as Fig. 2 to introduce the FISBA interferometer, which measured the whole aperture PV and RMS values after the closed-loop correction. The measurement result is shown in Fig. 17 , PV = 0.53λ, RMS = 0.07λ, the piston error between Seg. 1 and Seg. 2 is 0.03λ, the piston error between Seg. 1 and Seg. 3 is 0.06λ, and the piston error between Seg. 2 and Seg. 3 is 0.03λ; typically the tip-tilt error is . According to Fig. 17(a), we can see that every segment figure error and the optical system error contribute more to the PV and RMS values of the whole aperture, instead of the tip-tilt and piston errors.
Fig. 17 Result evaluated by FISBA interferometer after closed-loop correction. Residual local tip-tilt errors between the three segments are typically PV: , and the individual segment figure error is about PV = 0.24λ, RMS = 0.03λ. (a) Original interferogram of FISBA interferometer. (b) Surface deviation calculated by FISBA interferometer.
We repeated the process of the cophasing errors detection and correction, and evaluation with FISBA for 10 times, and achieved typical values of PV = 0.50λ, RMS = 0.08λ; accuracy of detection and correction of the piston error is 0.026λ RMS, and for the tip-tilt error.
5. Conclusion
In this paper, an ACAT with a high accurate optical adjustment and control in nanoscale is set up. The method of two-dimensional dispersed fringe analysis based on the Rayleigh interferometer is used to calculate the piston error between the adjacent segments. The tip-tilt errors between the adjacent segments are obtained by calculating the differences between the every two centroid positions of the images which were formed by three segments on the focal plane. The cophasing errors can be corrected by several sensitive micro-displacement actuators in three degrees of freedom in a range of 15mm with a resolution of 0.3nm according to the error signals. The process of the cophasing errors detection and correction can be carried out under the computer closed-loop control. After closed-loop detection and correction, the FISBA interferometer is introduced to evaluate that a final total residual surface error of about 0.08λ RMS, and the residual piston error of 0.026λ RMS have been achieved (λ = 633nm). Then the algorithms of piston and tip-tilt error analyzing are validated.
Acknowledgments
This work was supported by the National Basic Research Program of China (973 Program, Grant No. 2009CB72400601).
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