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Xinξtics Deformable Mirror Amanda Beasley June 1, 2007 1 Introduction The Research School of Astronomy and Astrophysics (RSAA) was loaned a deformable mirror (DM) by Gemini in return for its characterisation. This DM was manufactured to test an actuator design to be incorporated into the Gemini Multi-Conjugate Adaptive Optics program. The characterisation work took place primarily in the Academic Labo- ratory in RSAA’s new Advanced Instrumentation and Technology Centre (AITC) on Mt Stromlo (Figure 1). The experiments were undertaken be- tween July 2006 and May 2007. 2 Equipment In this section the DM and the equipment required to characterise it will be described. 2.1 The DM The DM loaned to us by Gemini was manufactured by Xinξtics Inc.1 Its mir- ror face is 33mm in diameter, with 37 actuators arranged in a square pattern, 5mm apart. Figure 2 shows a close up view of the actuator arrangement. As can be seen from this ﬁgure, the DM was initially uncoated. About half way through the project the mirror was coated by the Physics Department at the Australian National University (ANU). The actuators of the DM are piezoelectric which means they mechani- cally deform when a voltage is applied. The actuators push on the surface of the mirror, which is made of ultra-low-expansion (ULE) glass. The voltage range of the actuators is -20V to +20V with a bias voltage of -40V. The actuators can move the surface of the mirror up to several microns. See Figure 3 for a simple diagram of how a DM works. 1 See http://www.xinetics.com/ 1 Figure 1: Academic Laboratory in the AITC on Mt Stromlo where the experiments took place. Figure 2: Close up view of the face of the DM when it was uncoated. The actuators can clearly be seen 5mm apart across the 33mm diameter mirror face. 2 Figure 3: Simple diagram of a DM. The actuators push on the surface of the mirror, deforming it. Picture from Cerro Tololo Inter-American Observatory (2001). The DM actuators are powered using a high voltage (HV) driver obtained from Starpoint Adaptive Optics.2 The HV driver is sent commands from the computer via two Digital-to-Analogue (D-A) cards, which are in turn sent commands by a user-manipulated LabView program. This program was written by Murray Dawson (Head of Electronics at Mt Stromlo), and the user-interface can be seen in Figure 4(a) with Figure 4(b) showing the actuator/channel numbering system. The electronics set-up on the bench can be seen in Figure 5. 2.2 The Shack-Hartmann Wavefront Sensor To examine the shape of the surface of the DM a wavefront sensor was used. This was a Shack-Hartmann (SH) type wavefront sensor , commonly used in adaptive optics (AO) systems. A diagram of the optical setup of the SH used in the experiments can be seen in Figure 6(a). The primary components of the SH are a lenslet array and a charge-coupled device (CCD) camera. Light is shone onto the surface under test and is reﬂected onto the camera through the lenslet array. Each lenslet (diameter ∼ 0.3mm in our instrument) makes a subimage (spot) on the camera. By comparing the positions of the spots from a ﬂat reference mirror and the test object, the local gradient of the wavefront can be found. As will be seen in Section 3 this information can then be used to reconstruct the wavefront and hence determine the shape of the surface of the mirror. The SH used in this experiment was an Optino supplied by Spot-Optics of Italy.3 It has a 70x70 lenslet array and came with two cameras. The 2 See http://www.starpointao.com/ 3 See http://www.spot-optics.com/ 3 (a) LabView interface (b) DM actuator conﬁguration Figure 4: (a) LabView interface: the lower panel allows the user to select the channel (actuator) and the voltage required. There is also a facility to load, save and clear the text ﬁle containing the voltage settings. (b) The numbering of the actuators/channels on the DM. 4 Figure 5: The electronics setup to drive the DM. There are 7 white ribbon cables plugged into the back of the DM (middle right) that each drive a row of actuators. The black box in the centre of the picture is the HV driver. The grey cables plug into D-A cards in the computer (middle left). SBIG4 camera is suited to working in low light as it has large (9µm) pixels, and the Pixelink5 camera can operate at high frame rates (14fps at full resolution, faster at lower resolution). Since the collimating lens is only 10mm in diameter the SH also has a beam expander. This consists of two lenses positioned so as to increase the beam size, in our case to around 40mm, which means we had around 23x23 lenslets to sample the DM. The beam expander lenses are plano-convex to minimise the introduction of spherical aberration. A photograph of the SH and the beam expander on the bench can be seen in Figure 6(b). The beam expander is about 40cm long and the Optino is about 20cm long. There are two motorised stages which allow the user to control the position of two lenses – one of the beam expander lenses, and the collimating lens in the Optino. Collimation of the light in the system is important to ensure the accuracy of the measurements of the test object. The software interface to the SH is called Sensoft. The ﬁrst step required by Sensoft is to take a reference image. This is an image of a nominally ﬂat mirror and would normally look like Figure 7(a), where the subimages made by the lenslets are evenly spaced and all have about the same intensity. The 4 See http://www.sbig.com/ 5 See http://www.pixelink.com/ 5 (a) Diagram of the SH wavefront sensor (b) The SH on the bench Figure 6: (a) Diagram of the optical setup of the SH used in the experiments. (b) Photo of the Spot-Optics SH on the bench. The gold box contains the beam expander, and the silver box is the Optino which contains the lenslet array and beamsplitter. The camera is the black box on the side of the Optino. 6 reference mirror is then removed and the test object (in this case the DM) is put in place. The test object must be aligned with the reference image in order for the subimages in the reference and test image to be matched by the software, and this can be done with the help of Sensoft which locates the “centre of mass” of each image. Then an image of the test object is taken and Sensoft determines the shape of the surface of the test object by comparing the reference image with the test image (this process is described further in Section 3). An example of a test image of the DM is shown in Figure 7(b). There was an initial problem with using the SH to measure the shape of the DM because the DM was uncoated (see Figure 2). Because the uncoated DM only had around 4% reﬂectivity it was found that the internal reﬂections in the beamsplitter in the Optino were of approximately the same bright- ness, as can be seen in Figure 8. These “ghost” subimages caused problems because they prevented Sensoft from giving the user the correct information to align the test object with the reference image. The issue was overcome by using a “dark” frame. By default, Sensoft has dark frames stored for the individual cameras which are then subtracted from the ﬁnal image to help compensate for defects in the CCD chip. The solution employed to overcome the ghost spots was to replace the original dark frame with an image of the ghost spots only. This was obtained by simply covering the front of the beam expander and using Sensoft to take an image. The ghost subimages were then automatically subtracted from the image, allowing the alignment and rest of the data-taking to go ahead as normal. It was found in later experiments that using the non-standard dark frames caused the data-taking process to be somewhat unstable, and resulted in longer exposure times, but did not signiﬁcantly aﬀect the results. An example of the shape of the surface of the unpowered mirror can be seen in Figure 9(a). The mirror surface was also measured with an Intellium-Z100 interferometer.6 A result can be seen in Figure 10 which can be compared to Figure 9(a). The interferometer measured the same shape and surface characteristics as the Shack-Hartmann. Other information about the reference and test images and the measurement in general can be obtained from Sensoft. Examples include the signal-to-noise (S/N) ratio and the ellipticity of the spots. The S/N ratio for a measurement can be seen in Figure 9(b). Another feature is the ability to record measurements in a loop that repeats once per exposure time. The repeatability of Sensoft’s measurements were tested through experi- ments where the ﬂat reference mirror was measured against itself. A typical result can be seen in Figure 11. Our measurements approximately agreed with Spot-Optics acceptance test results of peak-to-valley (P-V) 15nm and root-mean-square error (RMS) 2nm. In another test, six measurements 6 See http://www.engsynthesis.com/products fizeau z100.php. 7 (a) Reference Image (b) Test Image Figure 7: (a) Typical reference image from Sensoft. (b) Test image of the DM from Sensoft where the central actuator has been driven by -5V. 8 (a) Percentage of total light at faces of beam splitter. (b) The ghost subimages are confused with the test subimages. Figure 8: (a) About 2% of the light source is reﬂected to the lenslet array in internal reﬂections in the beamsplitter. (b) The internal reﬂections from the beamsplitter were about the same brightness as the reﬂections from the DM, causing problems. The internal reﬂection is on the right and the DM reﬂection is on the left. 9 (a) Surface plot (b) S/N graph Figure 9: (a) A contour plot of the unpowered mirror – the peak-to-valley height here is 496nm and the RMS is 88nm (b) Signal to noise ratio plot. 10 Figure 10: The surface of the DM as measured by the interferometer. The test wavelength was 632nm so the P-V is about 390nm and the RMS is about 73nm This ﬁgure should be rotated 90 degrees clockwise before being compared to Figure 9(a). 11 Figure 11: Testing the reference mirror against itself shows the repeatability of the Sensoft measurements. The peak-to-valley here is 19nm and the RMS is 3.5nm. of the unpowered mirror were taken and the overall standard deviation of the mean of the reported RMS error was found to be 0.3nm. In the next section the method Sensoft uses to create these surface maps will be described, as well as our eﬀorts to reproduce its results. 3 Surface Shape Reconstructors 3.1 Modal The data that Sensoft provides (other than graphs) is in the form of Zernike coeﬃcients, each of which are associated with a polynomial in a set called Zernike Polynomials. These polynomials, weighted by the coeﬃcients, are then added over a unit circle to reconstruct the surface shape of the object under test as follows M W (x, y) = ak Zk (x, y), (1) k=1 where ak are the coeﬃcients, Zk (x, y) are the Zernike M polynomials and W (x, y) is the wavefront to be reconstructed. A list of the Zernike 12 polynomials used by Sensoft follows, where r is the radial coordinate and t is the angular coordinate. The numbers in brackets after the polynomial refers to its azimuthal frequency and radial degree, following Noll’s convention (Noll 1976). √ 3 −1 + 2r2 (0, 2 √ 2 2 −2r + 3r3 Cos[t] (1, 3 √ 5 1 − 6r2 + 6r4 (0, 4 √ 2 6r Cos[2t] (2, 2 √ 2 2r3 Cos[3t] (3, 3 √ 4 10r Cos[4t] (4, 4 √ 2 3 3r − 12r3 + 10r5 Cos[t] (1, 5 √ 7 −1 + 12r2 − 30r4 + 20r6 (0, 6 √ 10 −3r2 + 4r4 Cos[2t] (2, 4 √ 5 2 3r Cos[5t] (5, 5 √ 6 14r Cos[6t] (6, 6 4r7 Cos[7t] (7, 7 √ 3 2r8 Cos[8t] (8, 8 √ 2 3 −4r3 + 5r5 Cos[3t] (3, 5 4 10r3 − 30r5 + 21r7 Cos[3t] (3, 7 √ 2 5 −20r3 + 105r5 − 168r7 + 84r9 Cos[3t] (3, 9 √ 2 6 35r3 − 280r5 + 756r7 − 840r9 + 330r11 Cos[3t] (3, 11 √ 2 7 −56r3 + 630r5 − 2520r7 + 4620r9 − 3960r11 + 1287r13 Cos[3t] (3, 13 √ 14 6r2 − 20r4 + 15r6 Cos[2t] (2, 6 √ 3 2 −10r2 + 60r4 − 105r6 + 56r8 Cos[2t] (2, 8 √ 22 15r2 − 140r4 + 420r6 − 504r8 + 210r10 Cos[2t] (2, 10 √ 26 −21r2 + 280r4 − 1260r6 + 2520r8 − 2310r10 + 792r12 Cos[2t] (2, 12 √ 30 28r2 − 504r4 + 3150r6 − 9240r8 + 13860r10 − 10296r12 + 3003r14 Cos[2t] (2, 14 4 −4r + 30r3 − 60r5 + 35r7 Cos[t] (1, 7 √ 2 5 5r − 60r3 + 210r5 − 280r7 + 126r9 Cos[t] (1, 9 √ 2 6 −6r + 105r3 − 560r5 + 1260r7 − 1260r9 + 462r11 Cos[t] (1, 11 √ 2 7 7r − 168r3 + 1260r5 − 4200r7 + 6930r9 − 5544r11 + 1716r13 Cos[t] (1, 13 3 1 − 20r2 + 90r4 − 140r6 + 70r8 (0, 8 √ 11 −1 + 30r2 − 210r4 + 560r6 − 630r8 + 252r10 (0, 10 √ 13 1 − 42r2 + 420r4 − 1680r6 + 3150r8 − 2772r10 + 924r12 (0, 12 √ 15 −1 + 56r2 − 756r4 + 4200r6 − 11550r8 + 16632r10 − 12012r12 + 3432r14 (0, 14 √ 14 −5r4 + 6r6 Cos[4t] (4, 6 √ 3 2 15r4 − 42r6 + 28r8 Cos[4t] (4, 8 √ 22 −35r4 + 168r6 − 252r8 + 120r10 Cos[4t] (4, 10 √ 26 70r4 − 504r6 + 1260r8 − 1320r10 + 495r12 Cos[4t] (4, 12 13 A representation of some of the individual polynomials can be seen in Figure 12. The ﬁrst few polynomials depict classical optical aberrations such as defocus, coma and astigmatism. In all measurements in Sensoft the polynomials representing tip and tilt are automatically subtracted from the data (and are not included in the list above). An example of the Zernike coeﬃcients generated by Sensoft for the unpowered mirror can be found in Table 3.1. As an experiment the Zernike coeﬃcients provided by Sensoft were plot- ted in Matlab using the code “zernikes.m” (attached). As can be seen from Figure 13, which shows the shape of the mirror surface when one actuator was “poked” and the shape of the unpowered DM, the reconstruction us- ing Zernike coeﬃcients does not perfectly match that of the Sensoft output. The reason the two pictures do not match is due to the limited number of Zernike polynomial coeﬃcients provided by Sensoft and used by Matlab to reconstruct the wavefront. The highest order Zernike polynomial used in the reconstruction can be seen in Figure 14. The (spatial) period of the varia- tions in the polynomial at the edges is larger than the size of the “bump” the actuator makes, meaning the bump cannot be accurately reproduced with the number of polynomials provided. This suggests that Sensoft does not itself use the Zernike coeﬃcients and modal reconstruction to produce its graphs. The alternative method of obtaining the surface shape is zonal reconstruction which is discussed in the following section. A further experiment was carried out to ﬁnd out how much ﬁtting suc- cessive radial orders of Zernikes to the unpowered mirror improved the ﬁt. The results are in Figure 15 and show that the RMS error in the ﬁt does not change beyond about the 8th radial order. This indicates that including higher orders in the ﬁt does not make a diﬀerence to the accuracy of the ﬁt. 3.2 Zonal Zonal reconstruction is an alternative method of reconstructing the surface shape. As with the derivation of the Zernike coeﬃcients, zonal reconstruc- tion uses the surface gradients to directly determine the height of the mirror at each lenslet via least-squares error minimisation. In order to get the gradients the subimage positions must be known. Sensoft calculates the centroids of the reference and test subimages and outputs them to a binary ﬁle which can be read in Matlab. The code used to reconstruct the wavefront from this information is “chop spots.m” (attached). It takes the centroids, creates a gradient map g and then ﬁnds the wavefront W via the expression W = A−1 g, (2) 14 (a) Defocus (b) Coma (c) 3rd Order Spherical (d) 3rd Order Astigmatism (e) Trefoil (f) Quadratic Astigmatism Figure 12: A representation of a few Zernike polynomials. 15 Table 1: An example of the coeﬃcients generated by Sensoft for the un- powered mirror. The ordering of the coeﬃcients is the same as the list in Section 3 Coeﬃcient (nm) Angle (degrees) Defocus -37.5 – Tilt 4592.2 -69.4 Coma 65.1 146.4 SA3 -118.7 – Ast3 20.5 -4.4 Tricoma 41 -33.6 QuadAst 96.1 -3.5 SA5 -17.6 – SA7 -8.2 – SA9 15.5 – SA11 -18.9 – SA13 10.4 – Coma5 41.1 -67.6 Coma7 19.4 143.3 Coma9 11.8 -32.9 Coma11 5.9 150.7 Coma13 6 -62.1 . Ast5 31.5 -1.7 Ast7 13.4 66.5 Ast9 5.7 -76 Ast11 10 8.2 Ast13 11.5 -88.1 Ast15 3 0 Tricoma5 67 -58.6 Tricoma7 18.3 1.7 Tricoma9 2 -42.8 Tricoma11 2.8 -23.7 Tricoma13 2.5 45.9 QuadAst5 19.7 -15.7 QuadAst7 17 31.5 QuadAst9 18.5 -22.9 QuadAst11 11.4 23.4 Foil5 25.3 11.4 Foil6 29.3 22.4 Foil7 51.5 -18.8 Foil8 23.7 -19 16 (a) Actuator 13: Matlab result (b) Actuator 13: Sensoft result (c) Unpowered mirror: Matlab result (d) Unpowered mirror: Sensoft result Figure 13: A comparison of the outputs from Matlab and Sensoft. The Matlab output was created from the Zernike coeﬃcients provided by Sensoft. (a) and (b) are the results when actuator 13 was “poked” by 1V. (c) and (d) are results for the unpowered mirror. 17 Figure 14: The highest order Zernike provided by Sensoft (4,12). Figure 15: Fitting successive radial orders of Zernikes to a zero-volts mirror and showing the RMS error in the ﬁt. The curve levels at around 8th order indicating that further orders to not improve the ﬁtting accuracy. 18 where A−1 is the pseudo-inverse of the matrix A. The gradient map g is constructed using Fried geometry (Fried 1977) where the gradient at the point in the centre of a square of four spots is the average of the gradients at those four spots. The matrix A which is used in the reconstruction is put together to reﬂect this, and takes the form −1 1 0 . . . . . . −1 1 0 . . . . . . 0 −1 1 0 . . . 0 −1 1 0 . . . . . . . . . A= −1 −1 0 . . . . . . 1 . (3) 1 0 ... ... 0 −1 −1 0 . . . 0 1 1 0 ... . . . . . . An example of the gradient map, resulting wavefront and the equivalent Sensoft output can be seen in Figure 16. From this ﬁgure it can be seen that Matlab and Sensoft agree well. The reconstruction in Matlab is not perfect however, and this is due to how the edges of the spot images are handled. To do the least-squares reconstruction the grid must be a square, which necessitates padding the circular grid of spots (see Figure 7). The format of the centroids from Sensoft allowed this to be done automatically since each centroid in x and y was given a coordinate on a grid e.g. (x, y, x-coord) = (1,8, 420.5). This did, however, require reading in the centroid ﬁle as both unsigned-32 bit numbers and double-precision numbers. In the case of the Matlab code, the padding gradients are set to zero, which is simple to implement but is known to cause errors in the reconstruction (Poyneer, Gavel & Brase 2002). With this method of gaining useful qualitative data from Sensoft we were in a position to characterise the DM. 4 Characterising the DM By characterise we mean measure the response of the mirror. 4.1 Inﬂuence Functions The inﬂuence functions are the responses of the mirror to a unit movement of each actuator. They allow us to predict the overall shape of the mirror when various voltages are applied to diﬀerent actuators. The use of the inﬂuence functions will be described more fully in Section 5.1. The inﬂuence functions for this mirror were measured in a systematic way by measuring the unpowered shape then poking each actuator by +1V. To obtain the inﬂuence functions the unpowered shape of the mirror was subtracted from the one-volt shape as seen in Figure 17 to produce the results seen in Figure 18 for each of the 37 actuators. 19 (a) Gradient Map (b) Matlab Result (c) Sensoft Result Figure 16: Gradient map generated from the centroids of the reference and test images, along with the resulting Matlab and Sensoft output. The scales are in nanometers. 20 (a) Unpowered (b) +1V (c) The inﬂuence function Figure 17: Method of obtaining the inﬂuence functions – one DM actuator is poked and the surface of the unpowered mirror (a) is subtracted from the “poked” surface (b) to show the shape of the inﬂuence function (c). Units on the colour bar are in nm. 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 22 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Figure 18: All the inﬂuence functions arranged in actuator order on the mirror – subtract one from the caption number to compare the positions to that of Figure 4(b). The scale on the Matlab reconstruction was determined both geomet- rically and by comparing the Sensoft graphs of the central actuators with the peak height on the Matlab results. Geometrically 1 unit in Matlab is equivalent to 117nm and this can be seen as follows, referring to Figure 19. Each pixel on the camera is 9µm in diameter and the focal length of each lenslet is 41mm, thus one pixel subtends an angle of 0.0125◦ at its lenslet. Since the beam expander works at 4.7x magniﬁcation, the angle the pixel subtends at the DM is 0.0125/4.7 = 2.68×10−3 ◦ . This angle is the same as that made by a bump in the mirror of height h at the pixel separation (5mm), but because the measurement is made by reﬂecting oﬀ the mirror, the result must be halved, thus h can be expressed in nanometers as h = sin(2.67 × 10−3 ) × 5 ÷ 2 = 117nm. (4) Using the comparison method 1 unit in Matlab was found to be equiva- lent to 110nm. The Matlab code was thus modiﬁed to reﬂect this. The data from the inﬂuence functions gives important information on how well the mirror can be controlled and this will be seen in Section 5.1. To obtain further information from the inﬂuence functions 2D Gaussian ﬁts were made using Mathematica. The data for the heights and widths of the Gaussian ﬁts can be seen in Figure 20. It was found that the cen- tral actuators were ﬁtted by a Gaussian reasonably well as can be seen in Figure 21 with the residual variation being small. 4.2 Hysteresis Because the actuators are piezoelectric they have hysteresis. This is the eﬀect by which setting a certain voltage will result in a given amount of movement of an actuator depending both on the current voltage on the actuator and which direction it moves. Naturally this is a problem if precise control of the mirror is required. To obtain an idea of how well the DM would perform in a realistic AO experiment, the typical size of the movement required by any given actuator was calculated. To do this, the Strehl ratio, which is a measure how well the AO system is working, was used. The Strehl ratio is the ratio of the peak intensity of an aberrated image to that of a perfect image and hence is less than one (Hardy 1998). A typical image corrected by AO is expected to have a Strehl ratio of about 0.5. This can be used to estimate the phase variance over the wavefront that produced the image, via the expression 2) S ≈ e−(σ (5) where S is the Strehl ratio and σ 2 is the phase variance. A sine wave having a period equal to the actuator spacing (5mm) and a phase variance derived from a Strehl ratio equal to 0.5 was found to have a peak-to-valley 23 Figure 19: The angle a camera pixel subtends at the DM gives the calibration between pixel scale and height of the mirror. Figure 20: The heights and widths of the Gaussian ﬁts to the inﬂuence function data. The red spots denote the edge actuators. 24 (a) Inﬂuence function (b) Gaussian ﬁt (c) Residual Figure 21: The residual from the Gaussian ﬁt to central actuator. The residual is small relative to the peak height. 25 Figure 22: The hysteresis curve of the DM, over the range of voltages ex- pected in an AO system. The arrows show the direction of the voltage change. The amount of hysteresis is 7%. height of 237nm, which in term of volts applied to the mirror, is around 2.6V. The hysteresis of the mirror was therefore measured over the range ± 2.6V. The results are presented in Figure 22. The stroke range over volt- age range is 686nm, and the width of the hysteresis curve at 0V is 46nm. Thus the hysteresis of the DM over a range of voltages typical of an adaptive optics system is approximately 7%. 5 Flattening the DM One of the ﬁrst tests of the ability to control the mirror is to ﬂatten it. We used two diﬀerent methods and also discovered the limits to how ﬂat the mirror would go. The code written to ﬂatten the mirror was designed to be as automatic as possible. There were limitations to this, however, since data needed to be passed between Sensoft, Matlab and LabView. The amount of user input needed was minimised to 3 steps – one to take the data in Sensoft, two, to input the ﬁle name of the image taken by Sensoft into Matlab, three, to load the voltages into LabView. To run this as a true closed loop system major modiﬁcations would be required though. 26 5.1 Using Inﬂuence Functions The ﬁrst method that was used to ﬂatten the mirror makes use of the infor- mation in the inﬂuence functions and the code to implement it can be found in “chop spots I.m” (attached). The way it works is to measure the current shape of the wavefront, ﬁnd the reverse of this shape then determine the volts that are required to produce this shape. Thus when these volts are applied to the mirror it will “cancel” out the current shape and make the mirror ﬂat. The voltages are calculated in the following way, where V (x) are the voltages to be applied to an actuator x, I is the inﬂuence matrix and −W is the reverse of the current wavefront V (x) = I −1 (−W ), (6) where I −1 is the pseudo-inverse of I. Because of the hysteresis mentioned in Section 4.2 the initial application of the ﬂattening-volts will not make the mirror as ﬂat as theoretically possible. Several iterations are required to obtain maximum ﬂatness. The results of ﬂattening the mirror can be seen in Figure 23 and his- tograms of the wavefront heights at the ﬁrst and last iteration can be seen in Figure 24. There are still lumps and bumps in the ﬂat mirror but they cannot be physically removed. The reason for this can be explained by examining the inﬂuence functions. To do this the individual inﬂuence func- tions were summed to create one image. Then the Fourier transform of each row of this image was taken, the magnitude was summed and plotted. The same was done for a single image of the simulated “ﬂat” mirror – a picture of the “mirror” that had been ﬂattened in software (with lenslet and actu- ator positions superimposed) is in Figure 25. The Fourier transform graph produced can be seen in Figure 26. This graph shows that the cross over point of the lines is at around 3 lenslets meaning that the actuators cannot physically aﬀect any feature on the DM that is smaller than 3 lenslets across. The disadvantage of ﬂattening the mirror using this method is that it depends on knowledge of the inﬂuence functions to a certain accuracy. 5.2 Using Sivaramakrishnan & Oppenheimer Method The method of Sivaramakrishnan & Oppenheimer (S&O) as described in Sivaramakrishnan & Oppenheimer (1998) has a dual function of ﬂattening the mirror whilst also determining the gains of the actuators. It does this iteratively with the gains of the actuators being updated each iteration. The loop starts by measuring the current height of the mirror at each of the actuators. It then computes the settings to apply to the actuators to ﬂatten the mirror (taking into account the need for the average voltage settings to not vary throughout the procedure). The settings are applied 27 Figure 23: The ﬂattened mirror using the inﬂuence function method. The peak to valley is 196nm and the RMS wavefront error is 33nm. and the mirror is measured. If the height of the mirror at an actuator is far from where it should be, the gain for that actuator is updated and new settings are calculated. This procedure was implemented in the Matlab code “chop spots SO.m” and “frontendSO.m” (both attached). In this code only the central 21 actuators were corrected due to diﬃculties and uncertainties in locating the coordinates of the outer actuators. These outer actuators were slaved to their nearest central actuator. This method relies on knowing, to high accuracy, the location of the ac- tuators. To calibrate the positions each time the experiment was performed, a group of actuators were poked by -5V (such as in Figure 16) and the lo- cation of the peaks were compared to stored coordinates of the actuators, which were then corrected by eye if necessary. The results for this ﬂattening procedure can be seen in Figure 27 with the progression of the RMS at the actuator locations at each iteration and the ﬁnal image of the ﬂat mirror. The iterations stop when no gains are changed in an iteration, where the tolerance for this change is set at the beginning of the experiment. It can be seen from the ﬁgure that the actuators lie very close to the same contour, hence the program believes that the mirror is ﬂat. Interestingly by comparing this ﬁgure with that of Figure 25 it can be seen that the “lumps” are in the same place, strongly indicating that the mirror is at the limit of its ﬂatness. Running the S&O scheme does not guarantee that all actuators will have their gains calibrated since if the height of the mirror at an actuator starts oﬀ close to where the ﬁnal height of the mirror is, the gain update will not be triggered. Figure 28 shows the ﬁnal gains in one experiment, where all the 28 (a) Unpowered – ﬁrst iteration (b) 5th iteration Figure 24: Histograms of the wavefront heights before and after the mirror is ﬂattened. The heights of the mirror are much more concentrated around zero by the ﬁfth iteration. 29 Figure 25: A simulated ﬂat mirror with lenslets (small spots) and actuators (large squares) superimposed. Figure 26: Magnitude of the Fourier Transform versus lenslet number. Blue is from the sum of the inﬂuence functions and red is from the image of a ﬂattened mirror. The lines cross at 3 lenslets indicating the physical limit to ﬂattening this mirror. 30 (a) Final mirror surface (b) RMS progression Figure 27: The results of ﬂattening the mirror using the S&O scheme. (a) The ﬁnal mirror surface showing the location of the actuators being con- trolled (black squares). (b) The progression of the RMS at each iteration. The RMS at the actuators on the ﬁnal iteration is 17nm. 31 Figure 28: Final gains after running the S&O scheme. Initial gains were set to -1, so it can be seen that some gains did not change. initial gains were set to -1. It can be seen that not all gains were updated. In order that all actuator gains are updated the mirror should now be increased by a given height (so diﬀerent voltages are applied to each actuator depending on their gain) and the S&O scheme run again. The non-updated actuators will have poked above the relatively ﬂat rest of the mirror and so will have their gains updated. An modiﬁed version of the S&O scheme was also attempted (see the ﬁnal portion of “chop spots SO.m”). In this method a 60x60 pixel box around the actuator location was made and the weighted-mean of the heights of the mirror in that box was used as the height of the mirror at the actuator. The weights were in the form of a Gaussian which was given a width correspond- ing to that of the actuator’s inﬂuence function. It was found, however, that this did not improve the “ﬂatness” of the mirror, since the program just updated the gains instead of ﬂattening between the residual bumps as it was hoped it would. To test how well the S&O scheme had worked on the central 21 actuators a makeshift aperture was stuck on the beam expander to mask the outer actuators. The P-V of the DM over the central portion was found to be around 190nm with an RMS of around 33nm when the gains no longer changed. This is comparable to the results of the inventors of this scheme (Sivaramakrishnan & Oppenheimer 1998) who obtained a P-V of 171nm with an RMS of 19nm on a DM with 241 controllable actuators. An example of a Sensoft image of the central part of the ﬂattened DM can be seen in 32 Figure 29: The central portion of the DM ﬂattened by the S&O scheme. The P-V is 190nm and the RMS is 32.7nm. Figure 29. 6 Phase Screen A phase screen is an optical component used in adaptive optics testing for simulating the turbulence of the atmosphere in the laboratory in a controlled and repeatable way. The phase screen used for this project was supplied by Lexitek, Inc.7 It is 10cm in diameter and 10mm thick. Embedded between two sheets of thick glass is a polymer that has had a pseudo-random pattern etched onto it. A photograph of the phase screen can be seen in Figure 30. The phase screen was supplied with a motorised stage which allows it to be rotated to any position, or sequence of positions repeatedly. The pseudo-random pattern in the phase screen simulates speciﬁcally Kolmogorov turbulence. This is a model of the turbulent motion of the atmosphere which has a power spectrum as follows Φ(κ) ∝ κ−5/3 (7) where κ = 2π/l is the wave-number and l is the size of an atmospheric eddy. The model is valid for a range of eddy sizes between the “outer-scale” (largest size) and “inner-scale” (smallest size). Another parameter associated with the turbulence in the atmosphere is known as Fried’s parameter, or r0 , and is a measure of the strength of 7 See http://www.lexitek.com/ 33 Figure 30: A photograph of the phase screen from Lexitek, Inc. the turbulence. r0 deﬁnes an aperture size over which the mean-square wavefront error is 1 rad2 (Hardy 1998) and so a small value of r0 corresponds to strong turbulence. r0 depends on wavelength to the 6/5th power. The phase screen was supplied with an r0 of 5mm which exactly matches the spacing of the actuators on the DM. This is because in a real AO system, the actuator spacing of the DM being used would be matched as near as possible to the average r0 for the telescope location. Because of this however, we needed to use a laser instead of a white light source, because of the r0 dependence on wavelength. The wavelength of operation was 632nm, corresponding to a red Helium-Neon laser. Setting up the phase screen as part of the experiment required a major rearrangement items on the bench. It was discovered that on changing the SH to work with laser light rather than white light, new challenges with Sensoft existed. It became a priority to arrange the components on the bench in such a way as to have the DM conjugate to the lenslet array. Conjugation in this way means that the image of the DM falls completely and exactly onto the lenslet array, but with the available arrangement of the SH lenses satisfying this condition was actually impossible. The image of the DM was brought as close to the lenslet array by physically moving the DM as close as possible to the SH. Without adding or changing lenses in the SH, this was the only option. To ﬁnd the conjugate point experimentally, laser light was reﬂected from the DM, then through a lens, projecting an image on the wall. An actuator of the DM was then poked, and a change in intensity of the image at the 34 Figure 31: A photograph of the image of the DM when the central actu- ator has been poked. The intensity variation on the image is clearly seen, indicating that the image is far from the conjugate point of the DM. location of the actuator was clearly seen (see Figure 31). A piece of card was then moved closer to the lens until the intensity change was no longer visible, which located the conjugate point. This measurement agreed with the distance obtained theoretically. A diagram of the ﬁnal setup of the experiment on the bench can be seen in Figure 32, and a photograph of the experiment can be seen in Figure 33. An initial experiment that was carried out was a simple look at the subimage picture formed through the phase screen. A comparison of images with and without the phase screen in place can be seen in Figure 34. From this ﬁgure it can be seen that the subimages are distorted and shifted when the phase screen is in place. This proved to be a challenge for Sensoft to handle as it often “found” the diﬀraction spikes of the main subimages as well as the subimages themselves. The next experiment was carried out to get an idea of the range of shapes of the wavefronts generated by the phase screen. As described in Section 2.2 a reference image is needed. In this case, the DM was actually used as the reference and so all measurements are taken relative to this. Figure 35 shows a set of 5 independent measurements taken with the phase screen rotated to diﬀerent angles, and the 6th image is when it is back at the same position as the ﬁrst measurement. It can be seen that there is a range of diﬀerently 35 Figure 32: A schematic of the experiment set up on the bench showing the position of the laser, DM and phase screen relative to the SH. Figure 33: A photograph of the bench set up. Top is the laser and collimat- ing lens. Far left is the DM, then moving to the right, the phase plate stage and SH can be seen. 36 (a) Without phase screen (b) With phase screen Figure 34: A comparison of the spot images with and without the phase screen in place. In (b) the subimages are both irregularly spaced and mis- shapen. 37 shaped wavefronts produced, and that the ﬁrst and 6th images are the same. The maximum number of non-overlapping regions of 30mm diameter (DM diameter) that can be ﬁtted in the phase screen is ﬁve. With the Matlab code “chop spots I” slightly reconﬁgured for use in this setup, the ﬁrst experiment was to attempt to ﬂatten the wavefront through the phase plate. The inﬂuence function method was used (see Section 5.1) and run for six iterations (until the RMS error did not get smaller). The initial RMS was 147.0nm and after the 6th iteration it was 62.9nm. This is somewhat worse than the RMS wavefront error obtained when ﬂattening the mirror in Section 5.1 (RMS 33nm), and is to be expected. This is because the scale of some of the wavefront aberrations introduced by the phase screen are smaller than the DM can correct (see Section 5.1). Figure 36 shows the initial and ﬂattened shape of the wavefront. The wavefront ﬁtting error can be calculated using the expression (Hardy 1998) 5/3 2 d σ = aF , (8) r0 where aF is a constant (units rad2 ) dependent on the shape of the inﬂu- ence function, d is the size of a lenslet (subaperture) projected onto the DM, and r0 is Fried’s parameter. For the purposes of our system however, the accuracy of the correction is limited by the number of actuators, so d is the actuator spacing. In this case aF =0.24 for a Gaussian inﬂuence function, d=5mm and r0 =5mm, and so the RMS ﬁtting error is 49nm. The ﬁtting error agrees with the RMS error measured (63nm). The RMS ﬁtting error is a strong limit on the performance of the system and indicates the minimum wavefront RMS that can be expected, but in this experiment the limitation is not in the setup or operation of the wavefront sensor but in the DM and the shape of the inﬂuence functions. The second experiment that was carried out was an extension of the hys- teresis experiment seen in Section 4.2. In the experiment the wavefront was measured at ﬁve independent points of the phase plate around a complete revolution of the motorised stage. A 6th measurement was made back at the starting position. The voltages required to ﬂatten the wavefront at each of those points was calculated. Then, the voltages were applied to the DM in turn, with the phase screen in the corresponding position. The wavefronts were recorded again. The aim was to ﬁnd out how well the DM came back to its original position after going through a series of deformations typical of an AO system. The diﬀerence between the applied voltages at the ﬁrst and 6th position, and the shape of the corrected wavefront at those positions gives an indication of the controllability of the mirror in circumstances that are more realistic than those in Section 4.2. 38 (a) Image 1 (b) Image 2 (c) Image 3 (d) Image 4 (e) Image 5 (f) Image 6 is the same as Image 1 Figure 35: Six images taken at ﬁve independent rotations of the phase screen. The 6th image is the same as the ﬁrst image, indicating that the motorised stage has returned to its original position. 39 (a) Initial shape (b) Final shape Figure 36: The results of ﬂattening the wavefront through the phase screen. (a) The initial P-V is 755nm and the RMS is 147.0nm. (b) After six itera- tions the ﬁnal P-V is 475nm and the RMS is 62.9nm. 40 The changing position of the phase screen represents a moving atmosphere and the hysteresis of the actuators will prevent the shape of the DM being the same at the ﬁrst and last (identical) “atmospheric position”. The greater the diﬀerence in DM shape, the more signiﬁcant the hysteresis. Figure 37 shows the ﬁrst and 6th corrected wavefronts. It can be seen from this ﬁgure that the P-V and wavefront RMS error are very similar, but the shape is less convincingly the same. Analysing the wavefronts in Matlab indicates that the RMS of the diﬀerence between the two wavefronts is only around 20% less than the RMS of either wavefront. By contrast, the average of the diﬀerence in voltage settings between the ﬁrst and 6th position is a minimum of 60% less between any of the other positions, indicating that the voltage calculation is repeatable, but the hysteresis of the actuators prevented the application of similar voltages giving a similar DM surface shape. References Cerro Tololo Inter-American Observatory (2001), ‘Adaptive Optics Tutor- ial at CTIO’. Availabe from http://www.ctio.noao.edu/∼atokovin/ tutorial/part2/dm.html [Accessed 4 May 2007]. Fried, D. (1977), ‘Least-square ﬁtting a wavefront distortion estimate to an array of phase-diﬀerence measurements’, JOSA 67, 370–375. Hardy, J. W. (1998), Adaptive Optics for Astronomical Telescopes, Oxford University Press, pp. 206, 374. Noll, R. (1976), ‘Zernike polynomials and atmospheric turbulence’, JOSA 66, 207–211. Poyneer, L., Gavel, D. & Brase, J, M. (2002), ‘Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform’, JOSA A 19, 2100–2111. Sivaramakrishnan, A. & Oppenheimer, B. (1998), ‘Deformable mirror cali- bration for adaptive optics systems’, Proc SPIE 3353, 910–916. 41 (a) First wavefront (b) 6th wavefront Figure 37: How well the DM came back to its original position after going through a series of deformations typical of an AO system. (a) The P-V is 700nm and the RMS is 117.2nm. (b) The P-V is 710nm and the RMS is 109.7nm. 42