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Xintics Deformable Mirror


									                      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

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.

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

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

   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

                            (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.

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
    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

                            (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

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
      See fizeau z100.php.

                            (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.

               (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.

                              (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.

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).

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
                         W (x, y) =         ak Zk (x, y),                 (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

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

    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)

        (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.

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

      (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.

     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.

    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.

                             (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.
                               (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.

           1              2              3              4              5              6              7              8

           9              10             11             12             13             14             15             16

          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
                                 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

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.

                          (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.
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
    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.

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

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

                        (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.

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.

                           (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.

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

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

      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

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

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.

                          (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
                               2           d
                             σ = aF                   ,                   (8)
    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.

              (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.

                              (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.

    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

Cerro Tololo Inter-American Observatory (2001), ‘Adaptive Optics Tutor-
    ial at CTIO’. Availabe from∼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.

                            (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


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