Characterization of a Push-Pull Membrane Mirror for an
80
Master Erasmus Mundus in Photonics Engineering, Nanophotonics and Biophotonics Europhotonics MASTER THESIS WORK Characterization of a Push-Pull Membrane Mirror for an Astronomical Adaptive-Optics System María Barrera Verdejo Supervised by Dr. Alan Watson and Dr. Salvador Cuevas Cardona (Universidad Nacional Autónoma de México, UNAM) Dr. Santiago Royo Royo (Universidad Politécnica de Cataluña, UPC) Dr. Uli Lemmen (Karlsruhe Institute of Technology, KIT) Presented in Barcelona, on 10 th Sept 2012 Registered at
Characterization of a Push-Pull Membrane Mirror for an
Europhotonics
María Barrera Verdejo
Supervised by
Dr. Alan Watson and Dr. Salvador Cuevas Cardona (Universidad
Nacional Autónoma de México, UNAM)
Dr. Santiago Royo Royo
Dr. Uli Lemmen (Karlsruhe Institute of Technology, KIT)
Presented in Barcelona, on 10th Sept 2012
Registered at
Tú eres tu sonrisa, los lugares que tus ojos vieron y las personas
que por el camino fueron escuchadas por tus oídos. Las historias
que contar, y las que no se cuentan. Las carcajadas que regalaste y
aquellos por quien una lágrima fue derramada. La tierra que tus
pies pisaron y los pasos que hubo que retroceder. Las puertas que
en otros un día abriste. Eres las olas que tocaron tus pies y las
partidas de parchís que perdiste. Eres el tiempo que has esperado y
el que te queda por delante cargado de "aprender"... Gracias a
todas las personas que han hecho posible esta
maravillosa experiencia, especialmente a mi familia y animales
acuáticos como pescados y ranas. Gracias también a todo el
Instituto por su gran acogida, en particular a Alan, Salvador, Álex
y El Playa, por darnos muchos quehaceres y hacernos el día a día
más ameno. Otro sincero agradecimiento a Ramón Vilaseca, al cual
espero no molestar más ahora que todo esto va acabando. Y es
preciso no olvidar el importante soporte
proporcionado por la Fundación La Caixa, sin cuyo apoyo, todo
habría sido mucho más gris. Por último, un pequeño consejo: si no
les gusta el picante, no viajen a México.
Characterization of a Push-Pull Membrane Mirror for an Astronomical
Adaptive-Optics
System
Contents
1 Introduction 6 1.1 Motivation of the work . . . . . . . . . . . .
. . . . . . . . . . 6 1.2 Deformable membrane mirrors on adaptive
optics for Astronomy 8
2 Theoretical static mirror model 10 2.1 Push-pull mirror device .
. . . . . . . . . . . . . . . . . . . . . 10 2.2 Developement of
model . . . . . . . . . . . . . . . . . . . . . . 12
3 Experimental static mirror characterization 18 3.1 Comparison
between model and experimental results . . . . . 18
3.1.1 Method of measurement and Zygo error . . . . . . . . 18 3.1.2
Analysis of the mirror in rest position . . . . . . . . . . 21
3.1.3 Data comparison . . . . . . . . . . . . . . . . . . . . .
23
3.2 Membrane tension estimation . . . . . . . . . . . . . . . . . .
26 3.3 Measurements of maximum stroke . . . . . . . . . . . . . . .
. 26 3.4 Considerations of required stroke . . . . . . . . . . . .
. . . . 31 3.5 Repeatability and hysteresis . . . . . . . . . . . .
. . . . . . . 34
3.5.1 Electronics and mirror control . . . . . . . . . . . . . .
38
4 Dynamic characterization of the mirror 40 4.1 Description of the
system . . . . . . . . . . . . . . . . . . . . . 40 4.2 Bandwidth
measurements . . . . . . . . . . . . . . . . . . . . 43
5 Conclusions 48
B Appendix: Matlab code to study mirror rest position 65
C Appendix: Matlab code to compare mathematical model and real Zygo
data 68
D Appendix: Matlab code to analyse repeatability of the mir- ror
73
2
List of Figures
1 Difference between simpler PAN mirrors and Saturn one under
study. First has one single set of actuators and second is built
using two of them. . . . . . . . . . . . . . . . . . . . . . . . .
10
2 Electrodes distribution. Right image represents back actua- tors.
Left image, front ones. . . . . . . . . . . . . . . . . . . .
11
3 Graphical explanation of the 5 different cases under study. . .
14 4 Influence matrix. Each circle represents the influence of
each
actuator. By inverting the influence matrix, one can generate the
desidered wavefront. . . . . . . . . . . . . . . . . . . . . .
17
5 Average surface of the reference mirror, extracted from 10
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 20
6 Variance of the samples about the mean surface of the refer- ence
mirror, extracted from 10 samples. . . . . . . . . . . . . .
20
7 Average surface of the equilibrium position of Saturn mirror,
extracted from 10 samples of said position. . . . . . . . . . . .
22
8 RMS surface, referred to the average surface, of the equilib-
rium position of Saturn mirror, extracted from 10 samples of said
position. . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
9 Graph of the influence calculated with the model on the inter-
est area when actuator 9 is on. . . . . . . . . . . . . . . . . . .
24
10 Graph of the influence on the interest area when actuator 9 is
on, measured with zygo. . . . . . . . . . . . . . . . . . . . . .
24
11 Calculated difference between the model and data measured with
Zygo over the area of interest. . . . . . . . . . . . . . . .
25
12 Maximum peak-valley values measured on Zygo over the active area
(11 mm diameter) for (a) only 9th electrode on and (b) for the
electrodes under 9 on and the rest off. It makes an stroke of
little bit less than 4 µm. . . . . . . . . . . . . . . . . 29
13 Different scheme on the measurements of stroke. Adaptica
measures deformation achieved on the whole membrane while useful
stroke is only considered in the laboratory for the active area of
the device. . . . . . . . . . . . . . . . . . . . . . . . .
30
14 Repeatability results. On the left axis, in blue, standard de-
viation from the average is represented. On the right axis, a
normalization to percentage of said standard deviation is shown. On
x axis, all the electrodes are found. . . . . . . . . . 36
15 Hysteresis results. Each of the six graphs, for every kind of
electrode, are shown. Region out of interest is coloured in blue.
37
3
16 Transfer function: real delivered voltage to the mirror as a
function of the selected input voltage percentages. Red line
represents the theoretical value and blue one the obtained
measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
17 Oaxaca design based on NIR achromatic Mirrors for the 2.1m SPM
Telescope. . . . . . . . . . . . . . . . . . . . . . . . . . .
42
18 Setup used to measure the bandwidth of the Saturn mirror. . . 45
19 Examples of images on the MicroLens Array when a tilt
move-
ment in X (a) and Y (b) directions created on the mirror and when a
defocus (c) is produced by moving the 9th electrode. . 46
20 Normalized amplitude of the oscillations to calculate the band-
width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 47
4
1 Introduction
The word astronomy has its origin on the Greek terms astron ”star”
and -nomy, from nomos, ”law”. It literally means ”law of the
stars”. This sci- ence is devoted to the study of the celestial
objects (such as stars, planets, galaxies...), their understanding
and facts related to their behavior. One of the most wonderful
facts involving this field is that many sciences gather towards one
common target: the understanding of the universe. Astron- omy would
not find any sense without the involvement of chemistry, physics,
meteorology, physical cosmology and a long et cetera.
1.1 Motivation of the work
Astronomy is maybe one of the oldest sciences. It starts back
several thou- sands of years ago. And this is due to the important
fascination the firma- ment has called on civilizations. When one
looks up to the celestial vault in a clear night, cannot feel other
way but overwhelmed. The never-ending and unknown heaven, standing
there, dark, plenty of tiny whitish dots. And Babylonians, Greeks,
Egyptians, Mayas... also felt curious about what there was over
them. They dyed their ideas with religious tints and associated
their discoveries to gods or spirits.
Thanks to their studies of our sky, there is nowadays a huge and
transcen- dental legacy. And not only related to wisdom on stars
and planets, but also to buildings, instruments or tools. One
example of it is the very well known El Caracol observatory in
Chichen Itza (Mexico) was built in an attempt to try to get closer
to the god Venus and to its representation as a shape of a planet.
This Maya wonder was one of the observatories this culture built up
around 906 A.D. to watch Yucatan sky. Thanks to it, they could
learn inter- esting facts such as Venus cycle duration and point
out that five Venus cycles correspond to eight solar years. It is
guessed that they could also observe many amazing astronomical
events such as eclipses, equinoxes or solstices.
Not only Mayas were fascinated by Venus, and the sky in general.
Also Babylonians were interested on it already for long. This
civilization first realized the periodicity of the astronomical
phenomena and applied their strong mathematical knowledge to
predict events in the sky. They classified stars and constellations
and were even able to predict planets movement. Babylonians will
set up the astronomical knowledge that will stand for base to the
rest of cultures.
Following the Babylonians, Greek civilization achieved also
important discoveries such as the measurement of the Earth diameter
or the distance and sizes of the Moon and Sun.
6
But even if all ancient cultures spent many years staring at the
sky and gave big steps to the understanding of the firmament, their
observations were just eye-based. Until the first telescope was
fabricated, nobody could approach to closer see planets and stars.
And it did not happen until the very beginning of seventeenth
century. Who invented the telescope is still in controversy, but is
seems we have to thank Hans Lippershy his contribution to science
in 1609. This device for seeing things far away as if they were
nearby allowed Galileo Galilei, who was the first in using this
device for as- tronomical applications, to confirm and support
Nicolaus Copernico previous theories, developed on his De
revolutionibus, where the heliocentric system was proposed and
described.
Many discoveries came by the hand of the telescope invention and
thanks to outstanding scientists such as Kepler, Euler, Lagrange,
Newton, Fraun- hofer or Kirchhoff. This revolutionary tool that
allowed to count craters on the Moon suffered a big development
over the years.
Astronomical knowledge grew in an important way in the twentieth
cen- tury due to many other techniques to extract information from
our sky were also developed, like space telescopes. In this modern
astronomy, the discov- ery of the Milky Way and other galaxies, the
understanding of strange bodies (black holes, quazars, radio
galaxies, etc.), the appearance of Big Bang theory among others are
some of the most remarkable discoveries.
Other significant improvement in the nineties in the observation of
the sky was the introduction of adaptive optic techniques.
Nowadays, it has al- ready allowed ground based telescopes to
produce images as sharp as those gotten from Hubble Space
Telescope. It was first proposed by Babcock in 1953 and, later but
independently, by Linnick (1957). But their theoretical development
were still too expensive to be built for an astronomical appli-
cation. Meanwhile, military applications were found and, by the end
of the seventies, adaptive optics systems were already spread on
defense purposes. It was a great tool to compensate the undesidered
effect of atmosphere on the focusing of a laser beam on remote
targets, to see in the dark with IR lights, to track enemy missile
plumes and et cetera. A very different aim from observing celestial
beauty. And all these techniques were tagged as classified.
It would be later in the beginning of the nineties when most
important knowledge on AO was declassified. And, far to be
forgotten, these techniques became so attractive and useful to
astronomers eyes, who are still using them nowadays.
According to the human need of going one (or couple of) steps
further in the clarity and resolution of imaging stars, one crashes
against the wall im- posed by atmosphere aberrations. Atmosphere
capricious behavior brought observers many headaches. Due to
changes on particles concentrations, tur-
7
bulence brought by gradients on temperature or different winds
speeds; the index of refraction has a strong dependence on the
position. And, in addi- tion, to make this problem even more
complex, this dependence is completely random.
So, the main idea is to try to correct those aberrations introduced
by atmosphere with some kind of feedback. A wavefront sensor, for
instance a ShackHartmann sensor, usually detects and analyses the
distorted wavefront. Once the aberration is known, its correction
is performed by means of other device, tipically a deformable
mirror, which is monitored by a computer. This cycle must be fast,
in the order of milliseconds, in order to achieve good real-time
images.
There are many kinds of deformable mirrors , such as bimorph
mirrors, liquid crystal mirrors or thermal mirrors, but deformable
membrane ones present several advantages. Their properties are low
cost, good optical power, achromaticity and good dynamic behavior
[2],[3]. Nevertheless they are still limited by their maximum
stroke.
Particularly, here in this work, the study and characterization of
one or these devices is going to be carried out: a push-pull
membrane deformable mirror. By means of front and rear actuators
the membrane is going to be deformed. Compared to classical
membrane mirrors, with only one set of actuators, this kind of
configuration presents many advantages, such as more flexibility,
higher accuracy or double stroke, since the membrane can be pushed
and pulled.
Its applications and contributions to astronomical observations
seem to be promising. Nowadays, a growing number of observatories
are incorporating to their facilities adaptive optic systems
following different configurations, but this push-pull mirror is a
relatively novel application in astronomy. So it can really be
worth to spend some time in the deep understanding of this kind of
device and its behaviour.
1.2 Deformable membrane mirrors on adaptive optics for
Astronomy
Adaptive Optics is an old technique which has been succesfully
applied to many fields of science: Medicine, industry, military
uses or Astronomy [10]. Some of its remarkable applications are
highly precise laser welding and cut- ting, ophthalmology laser
surgery, optical tweezers, atmosphere distortion correction and
even biological imaging.
There is a broad family of tools that can be used for an AO system.
In this work, the study of one of these devices is going to be
developed: the
8
deformable push-pull mirror from Adaptica: the Saturn mirror [7].
Its intro- duction to AO presents many advantages when compared
with other different devices such as liquid crystal modulators,
thermal mirrors or bimorph mir- rors. Even if their use for some
technological applications is still restricted by their maximum
deformation and spatial resolution, the improved features they
present are strong: good optical power, low cost, achromaticity, no
hysteresis and large dynamic range among others.
The aim of this thesis is to test these and other features to
investigate if the Saturn mirror is adequate for the astronomical
project OAXACA. This project is been developed by the
Instrumentation Department at Instituto de Astronoma of Universidad
Nacional Autonoma de Mexico (UNAM) [6]. Oaxaca is leaded, and also
the whole department, by Dr. Alan Watson. Dr. Salvador Cuevas and
Beatriz Sanchez are responsible of said project as well. Their goal
is to provide an adaptive optics mode of imaging to the largest
telescope in Mexico up to date: 2.1 m telescope in National
Astronomical Observatory San Pedro Martir (OAN/SPM).
In Oaxaca, the Saturn mirror will be used in three different steps.
First, in the laboratory, where it will be tested and studied.
Second will be an intermediate trial in the National Astronomical
Observatory of Tonanzintla, (OAN/T). In this observatory, placed in
Puebla, Mexico, there is a one meter telescope. Eventhough it is
not its final goal, for proximity to Mexico City and UNAM, the
system where Saturn is integrated will be tested there.
Once it is working under OAN/T conditions, which are worse, the
setup will be moved to its last step in San Pedro Martir, with the
target of getting a finest observation of the sky. It is predicted
that this system will be ready by summer 2013.
Thus, the contribution to Oaxaca project with this thesis is the
whole .characterizacion and test of one of the devices involved in
the AO system for said 2.1 m telescope in OAN/SPM. Following the
introduc- tion to the work written in this chapter, a theoretical
model to describe the mirror deformation will be developed in
chapter 2. The third chapter will be focused on the static
characterization of the mirror. Thereby, in section 3.1 a deep
comparison between said model and empirical measurements is carried
out. Features like optical aberrations of the mirror at rest
position, tension estimation over the mirror membrane,
repeatability and hysteresis are going to be analyzed.
Some dynamic measurements are also taken and studied in chapter 4.
The aim is to measure the working bandwidth of Saturn. Finally, the
last chapter will be devoted to sketch some conclusions.
9
2 Theoretical static mirror model
One of the first steps towards the good understanding of the AO
system is the characterization of the mirror. To know its
behaviour, its performance and characteristics is the only way of
making it perform as we desire. The main feature under study is
going to be the so-called Influence Matrix. A model on Matlab will
be developed for that aim. Then, results will be compared to real
measurements taken in the laboratory.
2.1 Push-pull mirror device
The mirror under study is a membrane deformable mirror aimed for
adaptive optics applications. It presents an important novelty with
respect to older mirrors: its capability of being pushed and
pulled.
So far, simpler PAN deformable mirrors were based on one membrane
and a single layer of actuators that, applying a voltage on them,
are able to induce a deformation on the membrane. Push-pull
mirrors, on the other hand, are fabricated with two layers of
actuators: one at the front and other at the rear, see Fig. 1. More
precisely, the mirror used in this work is Saturn from Adaptica
Srl. This new configuration will clearly allow an increase of the
maximum stroke at the membrane in a factor of two [1].
Figure 1: Difference between simpler PAN mirrors and Saturn one
under study. First has one single set of actuators and second is
built using two of them.
The Saturn mirror includes a 5 µm thick conducting and reflecting
mem- brane. Said membrane is set between the two actuators layers.
The central actuator which is placed on the top is transparent to
allow light pass towards the membrane.
10
At that point, may result interesting to present the structure of
the elec- trodes of Saturn mirror, figure 2. The right part of the
figure shows the distribution of the electrodes at the back of the
mirror, 32 in total, arranged in rings. Those actuators are
intended to be fabricated in such way that all of them have same
area. The left part of the image shows the configuration
corresponding to electrodes at the front of the membrane. They are
16 and are found arranged as a central big electrode and a ring of
smaller electrodes around it. Then, the total number of electrodes
Ne will be 48.
Figure 2: Electrodes distribution. Right image represents back
actuators. Left image, front ones.
The pressure induced by the jth electrode on this membrane is
propor- tional to the applied voltage as follows:
Pj = εo 2
)2
(1)
Where d is the separation between actuators and membrane (this
case, for Saturn, it equals 105 µm) and εo is the dielectric
constant in vacuum. That way the effect over the whole membrane can
be represented as a linear combination of the different influences
of each electrode separately. That is the reason why, in order to
correct aberrations in a proper maneer, the accuracy of the
calculation on those influence functions is crucial.
Small displacements M(r, θ) on the membrane can be modeled
obtaining the solution to Poisson’s equation in cylindrical
coordinates [2],[4]:
∇2M(~r) = P (r, θ)
T (2)
11
Where T is the tension on the membrane per unit length, which in
the following is going to be considered as a constant. Vf (~r) and
Vb(~r) are the volt- ages of the front and back electrodes. The
appropriate boundary condition for equation (3) is
M(r = rm) = 0 (4)
Where rm is the maximum radius of the membrane. With these
approach, the model of the mirror deformation can be now
started.
2.2 Developement of model
Solutions of Poisson equation, eq.(3), for the Saturn mirror will
allow to obtain the influence matrix and thus, the model for the
mirrror behavior. In order to come up to the solution, several
approximations have been taken into account [4].
First consideration is that, given an electrode position, its
associated pres- sure is only caused by said electrode. This can be
assumed in the case that the ratio of electrode width to d is much
larger than 1. Also small displace- ments have to be considered.
For them to be small it is required that the angle between the
tangent at the deformation point on the membrane surface and its
equilibrium position, is smaller than 3 degrees. Last approximation
to be assumed is that the tension over the membrane does not depend
on the position and is unaffected by surface deformations.
Then, the solution to eq.(3) is given by [4],[8]:
(5)
]] P (r, θ)
And can be written as a linear combination of solutions in the
form:
Mi = 1
AijPj (6)
Where Aij represents the proportionality coefficient derived from
the so- lution of the poisson equation and Mi is the displacement
produced over the
12
ith point due to the presence of the Ne, the total number of
electrodes. As said before, the displacement on the boundary of the
membrane is assumed to be zero.
Those coefficients should be understood as the surface displacement
caused by unit pressure from one single active electrode and can be
linearly combined with the rest of influence functions to give the
total surface displacement on the membrane, according to which
electrodes are active or not. To simplify notation, A matrix will
represent the elements of Aij, and will be named influence
matrix.
To completely define the actuators action, an influence matrix for
each of the electrodes has to be defined. According to eq. (5) and
(6), one can give the expression of the influence coefficients.
Thus, its general form comes given by:
(7)
] DSij
] Where the term DSij = sin n(θ2j−θi)−sin n(θ1j−θi) is used to
simplify
terms. The notation used in this and next equations can be
explained as follows (see also figure 3):
• ri represents the surface point, where the influence is being
calculated,
• r1j is understood as the first radial limit of the jth electrode
which is creating the action and r2j is the second radial limit of
said electrode,
• θ1j and θ2j are the lower and upper limit for the angle the jth
electrode define,
• θj is equal to the difference on the limit angles θ2j −
θ1j.
Performing integrations, one can get analytical expressions for the
co- efficients Aij. Nevertheless, it is necessary to distinguish
between different cases according to the values, limits and sizes
of each electrode and thus, an addaptation to the structure of
Saturn mirror is needed. If integration on eq. (7) is performed,
analytical expressions for Aij are obtained [4]. According to this,
five different cases of influence are discribed. Those cases come
defined below and are shown graphically in figure 3:
13
Figure 3: Graphical explanation of the 5 different cases under
study.
• Case I: the influence of the central actuator over the central
point where ri = 0 and r1j = 0
Aij = 1
2π θj
[ r22j 2
)]] (8)
• Case II: the influence of the outer actuators over the central
point where ri = 0 and r1j > 0
Aij = 1
2π θj
[ r22j 2
)]] (9)
• Case III: influence of the jth electrode on the area outside the
ring the electrodes define, r2j < ri
(10) Aij =
14
• Case IV: influence of the jth electrode on the area inside the
ring of electrodes where it is found r1j > ri
(11) Aij =
α = −r2i (lnr2j − lnr1j), for n = 2,
α = r2i
(( ri r2j
(12)
• Case V: influence inside the ring the electrodes define r1j <
ri < r2j
Aij = 1
α = −r2i (lnr2j − lnri), for n = 2,
α = r2i
(( ri r2j
(14)
15
One can apply those complicated formulae on a software like Matlab
to be analysed and plotted. Appendix A anexes the code to generate
those results in said programming language. Final graphical
representation of the influence matrix is shown in Figure 4. Each
circle represents the influence of a single active electrode, from
1 to 48 actuators. In the figure, the electrodes are ordered
starting from the rear layer. First is central one, second is 31st
in figure 2, crossing the first ring until 32nd. Next will be 20th
and growing in the second ring up to 28th actuator. And so on.
Then, following the same criteria, front electrodes are
depicted.
Combining their action almost any desidered deformation on the mem-
brane can be gotten. By inverting the influence matrix [10], one
can generate the desidered wavefront to correct a given
aberration.
16
Figure 4: Influence matrix. Each circle represents the influence of
each ac- tuator. By inverting the influence matrix, one can
generate the desidered wavefront.
17
3 Experimental static mirror characterization
In this Chapter experimental results will be presented to fully
describe the characteristics of interest of the Saturn mirror. Some
features like influences measurement, hysteresis, repeatability
among others will be discussed. All those are crucial to be
well-known for the application we are dealing with.
3.1 Comparison between model and experimental re- sults
As explained on Section 2.2, a model has been created to describe
the static behaviour of the mirror. The influences of the
electrodes are studied the- oretically and results are presented on
figure 4. Next step is to go to the laboratory, take some
measurements and try to compare the results with the theoretical
conclusions that were obtained for the values of Aij. This part, in
the beginning understood as easy by the writer, was not that
simple. Hereinafter the procedure followed to develop this task is
explained in detail.
3.1.1 Method of measurement and Zygo error
In the Optics Laboratory of the Astronomy Institute, they own a
Zygo In- terferometer. It has a Fizeau configuration working with
phase shift. This tool is really precise and reliable. Measurements
with Zygo are based on optical interferometry measuring
displacements, surface figures, and optical wavefronts. High
precision interferometers can be used on a broad number of
applications but, in this case, it will be used to study the
deformation on the mirror surface.
A profile of the membrane deformation is obtained with this device
on the computer with the help of a software provided by the
company. Playing with it, one can obtain almost straightforward
interesting results such as profiles of deformation over selected
line, peak-valley differences or complete and detailed shape of the
surface. Zygo CCD camera detects with a sampling of aproximately 47
µm
pixel the properties of the surface under study.
Typically, images are exported from the Zygo and analysed in
Matlab. There are several possibilities of data exportation, but
the most suitable format to get the data to be later analyzed with
Matlab is the file extension .xyz. This file has a header where
some parameters of the simulation are specified and a body, where
three columns are found. Said colums correspond to X, Y and Z
values of the 3D function that represents the deformation of the
mirror surface. X and Y are the corresponding pixels of the camera,
taking as origin the left upper corner of said camera. Z is the
value of the
18
detected height of the mirror surface on each pixel. This value is
given in micrometers. If no value is found NaN (Not a Number) is
shown.
So, in order to calculate the error introduced by the Zygo
instrument, one can take a reference flat mirror. This mirror is a
known surface provided with the interferometer to be taken as
reference. It is a 6 inchs diameter mirror corrected to λ/20, which
means that, at the working frequency of ZYGO, He-Ne 633 nm, it has
a maximum peak valley of around 31 nm.
Ten samples have been taken from said surface. Once analysed,
average surface and rms have been calculated. Those results are
shown in figure 5 and 6 respectively.
19
Figure 5: Average surface of the reference mirror, extracted from
10 samples.
Figure 6: Variance of the samples about the mean surface of the
reference mirror, extracted from 10 samples.
20
From this data one can calculate the average error on the samples.
The mean rms of the surface in figure 6 is 1.8095 10−6µm2, which
means an average error over the sample of 1.3nm. Refering it to the
lambda we are dealing with, the error is smaller than λ/600. One
can conclude that this is a very good result and that the error
introduced by the instrument is non-significant at all. Now on, it
will not be taken into account.
3.1.2 Analysis of the mirror in rest position
Going back to the comparison of the data and the model, once needed
mea- surements are performed, the data from the Zygo must be
extracted and, as explained before, be analyzed on Matlab. An
adaptation of the data format is required to perform said
comparison.
First of all, it is necessary to select the area of interest in
both Zygo and model. It corresponds to the active area of the
Saturn mirror: 11 mm diameter circle. After that, a normalization
of said data is required, because the values for Aij calculated on
the model are just proportional to the de- formation of Saturn, as
is shown in eq. (6). The real value corresponding to the
deformation of the surface is also proportional to the tension over
the membrane and this is only an estimation, not a well-known
number.
For the model, this step is very simple: just divide by the maximum
of the function, see figure 9, and the values corresponding to the
mirror surface will vary between 0 and 1. For the Zygo data, it
requires little bit more of effort.
First of all, it cannot be supposed that the equilibrium position
of the mirror is completely flat, basically because it is not. So,
the idea is to take several measurements of said equilibrium
position at different points in time. Then, when a considerable
number of them is obtained, it is necessary to perform the
calculation of the average value of said position. To get that, 10
samples of said surface have been studied and the average surface
and standard deviation of the measurements about the mean have been
extracted. Those results are shown on figures 7 and 8
respectively.
21
Figure 7: Average surface of the equilibrium position of Saturn
mirror, ex- tracted from 10 samples of said position.
Figure 8: RMS surface, referred to the average surface, of the
equilibrium position of Saturn mirror, extracted from 10 samples of
said position.
22
It is interesting to make the reader pay attention to the different
scales on Z axis. On the Saturn mirror specification sheets, the
rms deviation from initial plane on the active region is pointed
out as ≤ 50nm. The mean rms deformation measured on these set of
data is around 37nm, which is surprisingly better than
expected.
According to these values, it can be concluded that the mirror has
good properties at rest position, because its deformation is not so
remarkable, eventhought it cannot be considered completely flat as
said before.
Once the average surface describing the initial plane is
calculated, one can subtract this information to the measurements
taken from Zygo, in order to study separately the influence of the
electrodes from other deformations. But still a normalization to
values from 0 to 1 has to be performed to be able to carry out the
difference with the model.
The code implementing the analysis of the mirror at rest position
can be found in Appendix B.
3.1.3 Data comparison
At this point, with the data ready to be compared, one can come
back to the comparison of the model and Zygo data. Two matrices of
points are treated here. But two of them with different sets of
points: model is defined in cylindrical coordinates, as was shown
in Section 2.2, and on the other hand, Zygo provides the data as a
cartesian net. In addition, there are some not defined values (NaN)
on the points where there was no information enough for a height to
be defined. So, an interpolation is required to fit the points from
one source to the other.
After following all those steps, one can get to the results on
figures 9 and 10. The first of the images shows the appearence of
the model on the area of interest. On the other hand, second figure
shows the data obtained with Zygo after the above explained
normalization process. In order to perform the comparison here,
central electrode of the front set has been chosen: number 9 on
figure 2. The main reason to study this electrode is because it has
the largest area and therefore, it is the electrode with higher
influence. Analogous processes can be performed to compare the rest
of the actuators with similar results.
Last step to get to the comparison is to perform the difference
between the two sets of data in figures 9 and 10. To do that, and
since the exact position of the center on Zygo data is completely
unknown, an optimization of the area under comparison is needed.
The aim is minimize the rms value of the difference according to
the size and position of the mask applied to the net of measured
data around what can be considered the central point
23
Figure 9: Graph of the influence calculated with the model on the
interest area when actuator 9 is on.
Figure 10: Graph of the influence on the interest area when
actuator 9 is on, measured with zygo.
24
Figure 11: Calculated difference between the model and data
measured with Zygo over the area of interest.
in a first approach. When this optimization is carried out, result
on figure 11 is obtained.
In a first look to figures 9 and 10, one can have the feeling that
both figures are quite similar. But it is analysing the result in
figure 11 when one can conclude that both are almost equal. Note
the difference on the axis. Calculating the rms of the difference
between model and measurements, the result is 0.0576, which,
written referred to percentage, a rms value of 5.76% is achieved.
Those differences are mainly due to the (always present?) problems
with the borders, due to reflections and others.
This can be considered as a good result and, therefore can be
concluded that the mathematical model developed to describe the
influences of the electrodes is reliable and works according to
measuremens of deformations on the real Saturn surface.
The set of programs developed on Matlab to carry out this
comparison can be found on Appendix C.
25
3.2 Membrane tension estimation
This section is going to be devoted to obtain an estimation for the
value of the tension on the mirror membrane. As writen in equation
(6), the deformation on the surface is proportional to the tension
on said membrane. This T has been considered constant as explained
in section 2.1.
With the measurements obtained so far, one can easily make an esti-
mation for the value of said force. Combining formulae (1) and (6),
and considering only the 9th electrode on (then Ne = 1) the value
of T can be isolated as follows:
T = A
( V
d
)2
(15)
Where d = 105µm is the separation between actuators and membrane;
εo = 8.85 · 10−12F/m is the dielectric constant; V = 227V is the
applied voltage; M = 1.72 · 10−6m represents the peak-valley
deformation measured over a diameter of 11 mm and finally, A =
7.69·10−6m2 is the proportionality constant calculated with the
model for the 9th electrode over the same region. The value
obtained for the applied voltage of 227 volts will be justified in
Section 3.5.1.
According to what has been analyzed so far, the dominant error in
this estimation corresponds to the 5% introduced on the value of
the model. So, with all those numbers it can be said that:
T ' 92± 5N/m (16)
As a conclusion, the value of the membrane tension should be around
92N/m.
3.3 Measurements of maximum stroke
It is considered an important feature to be well studied from the
Saturn mirror the maximum stroke it can achieve. It is essential to
know the limit of the deformation because the higher this value,
the larger is the correction of atmospheric aberrations one can
perform. We must be careful to distinguish the active area of 11 mm
diameter, which will correspond to the pupil in the AO system, and
the full clear aperture of the mirror of 19 mm diameter, which
includes regions beyond the pupil.
Electrode 9 is going to produce the highest deformation when acting
alone, see figure 2, since it is the largest actuator. If it is
activated to the maximum voltage allowed by the electronics, it
should provide a very big
26
deformation on the mirror membrane. Maximum peak-valley (P-V)
defor- mation given by this electrode over the clear aperture is
about 2,5 µm. This number is smaller if one reduces to the active
area, the real area of interest, where one is only able to see a
maximum P-V deformation of around 1.7 µm, as shown in figure
12(a).
The result is similar when all the electrodes behind the area of
actuator number nine are on, that is, electrodes 24, 31, 25, 61,
49, 43, 32, 20, 29, 54, 50, 57, 53, 47, 8, 4 and 28. One could have
switched on also electrodes on the outer ring, that is: 22, 18, 27,
56, 52, 63, 59, 55, 51, 45, 41, 6, 2, 30 and 26. But this set of
actuators does not contribute to induce a higher P-V deformation
over the active area. Instead, they will produce mainly piston and
tilt in the active area.
Then, in the described case the maximum P-V stroke over the active
area reaches little bit less than 4 µm, see figure 12, from +1.725
µm to -2.043 µm . One can try different configurations of
electrodes on and off, but a higher value is never achieved.
Comparing to data provided by the manufacturers, one finds little
bit of disagreement. As said in previous section, Adaptica gives as
a value of maximum stroke something ”bigger than 10µm, typically a
value around 14µm”. And this is not what is seen in the laboratory
at all.
We contacted Adaptica and concluded that our understanding of maxi-
mum stroke and their understanding are different. For Adaptica the
max- imum stroke is the maximum achievable deformation over the
whole clear aperture, while our measurements are with respect to
only the active area (see figure 13 for a better understanding).
Furthermore, for their measure- ments they use all of the
electrodes on one side or the other, including those in the outer
ring which contribute mainly piston within the active area. Thus,
their maximum stroke includes a large piston component over the
active area. In our opinion, there is little justification for
including this piston component since optically it is irrelevant in
the intended application of the mirror.
Then, it seems there is not such a big controversy, but the useful
stroke for the application here is much smaller than
expected.
To check the numbers stil make sense, one can use the estimation of
the tension obtained before, to calculate the expected theoretical
maximum stroke. One can come back to the formulae and remind that
the deformation M(~r) of the membrane is given by the solution to
equation (3). The appro- priate boundary condition for said
equation is M(r = rm) = 0., as explained on section 2.1. According
to [5], for a single circular electrode of diameter re centered on
the membrane, the exact solution is
27
2] for r < re and ln (rm/r) for r ≥ re.
(17)
Mmax = ε0r
2 eV
1
2
] . (18)
For the Saturn mirror rm = 8.5mm and large central electrode has re
= 5.425mm. Thus, for this electrode
Mmax ≈ 11.097µm (19)
Which seems to make sense according to the value provided by the
man- ufacturer.
28
Figure 12: Maximum peak-valley deformations measured on Zygo over
the active area (11 mm diameter) for (a) only 9th electrode on and
(b) for the electrodes under 9 on and the rest off. It makes the
total P-V stroke a little bit less than 4 µm.
29
Figure 13: Different scheme on the measurements of stroke. Adaptica
mea- sures deformation achieved on the whole membrane while useful
stroke is only considered in the laboratory for the active area of
the device.
30
3.4 Considerations of required stroke
When Saturn mirror was brougth as a candidate to perform aberration
cor- rection in Oaxaca project, specifications sheet was studied.
At that moment, the team saw a stroke bigger than 10µm and realized
it was more than enough to correct almost any atmosphere.
But now, it turns out that what the team considers the real useful
max- imum stroke is much smaller. Then, this situation can be
worrying. It is possible that with such a small deformation
provided by the mirror over the active area, the system is not able
to totally correct the aberrations intro- duced by the atmosphere.
And, if this is the case, the whole project of the telescope would
be worthless.
So, at this point, it is necessary to think about the properties of
the atmosphere over the telescope and theoretically calculate the
needed stroke on the mirror. The main question is how much
deformation on the mirror is needed to correct an aberration of
around 1 arcsec.This calculation is not a difficult task. One can
draw on Zernike polynomials to describe aberra- tions on an optical
system. This interpretation brings many advantages, but among
others, the easy threatment of atmospheric statistics and the
degrees of correction for said aberrations. It is interesting to be
able to calculate how much wave-front distortion is associated to
each kind of aberration. And also, how much error remains after
correcting a given aberration.
According to [10] and [9], there is a mean square residual error
associated to each Jth Zernique polynomial, shown in table 1. This
table presents the Zernike-Kolmogorov residual errors J and can be
interpreted as the remaining error once aberrations from 1 to J − 1
have been corrected.
They depend on the parameters D and ro, which are the diameter of
the telescope and the Fried parameter, respectively. This ro can be
seen as a measurement of how good (optically speaking) is the
atmosphere. It indicates the radius of the telescope where one can
observe under diffraction limit and the rms wavefront aberration is
smaller than unity. It is also related to the working
wavelength.
In the system here is described, the telescope diameter corresponds
to D = 2.1m, the wavelength is λ = 0.8µm and the Fried parameter at
said λ required to correct 1 arcsec is ro = 0, 1633. According to
that, one can evaluate the variance of the error for all Zernikes,
and it is shown also in table 1, on the third column.
To isolate the error associated to a single J :
σ2 J = J−1 −J (20)
This variance can be associated to a shift on the phase of the
wavefront
31
Z1 , Piston 1 = 1.0299(D ro
)5/3 1 = 72.7228
)5/3 2 = 41.0959
)5/3 3 = 9.4619
)5/3 4 = 7.8379
)5/3 5 = 6.2138
)5/3 6 = 4.5756
)5/3 7 = 4.1449
)5/3 8 = 3.7071
Table 1: Zernike-Kolmogorov residual errors associated to first
eight Zernike polynomials and their corresponding aberrations.
General values and values calculated for this telescope are shown
in 2nd and 3rd column.
caused by a stroke SJ as follows:
σJ = 2πSJ λ
(21)
Isolating SJ , one can easily obtain the needed stroke to correct
said σJ . In general, the most important aberrations are those of
lower J , however J = 1 is irrelevant for the configuration of
closed-loop the mirror is going to be working in.
Simple calculations can be performed in order to know the
deformation linked to each of them. For example, to correct the
tilt (Z2 and Z3), a stroke of S ′2−3 = 6.0762µm is required. For
the defocus, S ′4 = 0.97356µm will be enough. Note that these
values marked with prime are six times bigger that the ones given
by the formulae for the stroke SJ , due to the necessity of
covering the whole spectrum and thus the need of correcting from
−3σ to 3σ. The rest of the values associated to S ′J for each
Zernike are shown in table 2.
With those numbers, the Saturn mirror is really poor to correct
tilt on the system, but on the other hand, it could perfectly
correct defocus. Fortu- nately, as explained before, Saturn is not
the only mirror that is going to be used in the setup. There is
another device, a tilt mirror, just to correct the tilt aberration.
Then, stroke on the mirror membrane should be devoted to correct
from J = 4 on.
It would be also possible to calculate the stroke given by the
mirror associated to aberrations like astigmatism or coma, in order
to know if it fits the required S ′J on table 2. All the tools to
measure it are already
32
Zenike pol. σJ SJ(µm) S ′J(µm)
Z1 , Piston 5.6238 0.7160 4.296 Z2 , Tilt X 5.6244 0.7161 4.297 Z3
, Tilt Y 1.2744 0.1623 0.974 Z4 , Defocus 1.2744 0.1623 0.974
Z5 , Astigmatism 1.2799 0.1630 0.978 Z6 , Astigmatism 0.6563 0.0836
0.501
Z7 , Coma 0.8876 0.6617 0.505
Table 2: Needed stroke associated to correct each Zernike
polynomial. Cal- culations presented for the first seven Js.
available: the model, the estimation of the membrane tension and
the setup. This would be the next step that stays open for future
works and that, at some point will be performed by the team but
scapes of the aims of this thesis.
To sum up, it can be concluded that the mirror meets the require-
ments to work in the telescope and correct the aberrations produced
by the atmosphere, from defocus to higher Zernike
polynomials.
33
3.5 Repeatability and hysteresis
Another parameters to study mirror reliability are repeatability
and hystere- sis. These two features are necessary to be good to
have a nice performance in the system. And both are strongly
related. It can be said that it is just two ways of studying
similar characteristics, because one cannot be under- stood without
the other. But both kinds of measurements are interesting to be
confirmed.
To understand a good repeatability on the measurents, the shape on
the mirror surface must be the same for a given applied voltage at
different points in time. Otherwise one would not be able to know
how much preassure should be delivered to the mirror to get a
desidered deformation. To have an idea of the repeatibility, one
electrode of each type is chosen: two central ones and one actuator
from each ring. That is: electrodes 1, 9, 24, 26, 28 and 32 in
figure 2. Results can be extrapolated to the rest of actuators and,
in general, to any configuration since the effect of the electrodes
on the mirror is linearly dependent.
So, first step is taking several measurements in different random
moments for the same applied voltage. In this case maximum V is
going to be set, around 227 V, as will be seen in section 3.5.1, to
check the largest influence on the membrane. Then, each set of data
is compared to the rest and its error referred to the average value
is analyzed. Once one knows each error associated to every sample,
mean error is calculated.
This calculation can be understood in two different ways: absolute
error or relative error (normalized to the maximum peak-valley).
First one will come expressed in µm and second one as a percentage.
These calculations can be performed for each electrode and are
represented in figure 14.
From the figure it can be seen that repeatability is good because
errors are small. On the left axis standard deviation in µm is
found. The maximum value for these differences on the set of
samples is 200 nm over the whole surface (19 mm diameter), out of a
maximum deformation of around 2.5 µm. It corresponds to a
percentage of less than 8%. This is the worst case for the 9th
actuator, which is the biggest and thus, the electrode with highest
influence. It is important also to remark that the most important
differences are found on the borders of the mirror, out of the area
of interest. In terms of lambdas, this error has a value on the
order of a fourth of lambda. It is not so bad although it could be
better. But it is not a bad value at all taking into account that
the mirror is going to be working in closed-loop. For the rest of
the electrodes, the deviation is much smaller, because they have
smaller influence on the membrane deformation.
Matlab code used on the analysis of the repeatability can be found
on
34
Appendix D. To analyze hysteresis, same electrodes than for
repeatability are chosen.
The idea now is to start from zero voltage and increase it until
the maximum value. Once at this point, decrease it until zero once
again and check that the mirror deformates the same for rising and
falling voltages. It is handy to compare the peak-valley value on
these measurements, instead of the whole surface. The results for
the chosen electrodes are shown in figure 15.
In figure 15 the measurements taken for smaller voltages are not
relevant results, specially for values between 0 and 20 %. This is
due to the oscilations on the initial position of the membrane. It
is not a constant value, as was pointed out in section 3.1.2, and
small voltages do not induce a very big change on the surface. But
this fact is not so worrying because those small percentages are
not going to be useful for this application on the telescope. So
let’s focus on the analysis of higher voltages, which is the
interesting region to be studied.
Then, from results in figure 15 can be seen also that the
peak-valley deformation is not exactly linearly proportional to the
increase of the voltage, as could be expected from equation 1.
Instead, this behaviour it is only found for higher voltage
percentages, more precisely from around 60% on, where hysteresis is
almost total.
In addition it is necessary to say that electrodes placed on the
borders are more noisy and the value for the maximum deformation
has to be chosen very carefully.
Total absence of hysteresis have been found, but if a bad behaviour
would have been found, it would not be really worrying because the
system is work- ing in closed-loop. Then, if the mirror would not
adopt the desidered position, it can be corrected in the next
iteration.
In conclusion, from this analysis one can say that the mirror
perfectly suits the requirements for a telescope in a closed-loop
configuration.
35
Figure 14: Repeatability results. On the left axis, in blue,
standard devia- tion from the average is represented. On the right
axis, a normalization to percentage of said standard deviation is
shown. On x axis, all the electrodes are found.
36
Figure 15: Hysteresis results. Each of the six graphs, for every
kind of electrode, are shown. Region out of interest is coloured in
blue.
37
3.5.1 Electronics and mirror control
Arrived to this point, it seems interesting to wonder where those
little dif- ferences on repeatability values are coming from or why
the maximum de- formation is not completely linear with the applied
voltage, as would have been expected. One possible explanation is
that electronic control on the device is not completely lineal. To
check that possibility, one can study the transfer function of the
system to obtain the output voltage of the source (or equivalently,
delivered voltage to the mirror), as a function of the selected
input voltage. This kind of study will also allow the team to
control in a more exact way how the mirror is been deformed, since
this deformation is theoretically expected to be proportional to
the square of the applied voltage, see eq. 3.
In this setup, the input voltage is controled on a simple user
interface on the computer as a percentage from 0 to 100%, the
maximum voltage provided by the source. Theoretically, output
values are supose to oscilate linearly from 0 volts to 250 volts.
But it is not exactly what is happening.
When one takes many different measurements on the provided voltage
to one of the channels of the mirror, one checks that the
repeatability of this delivered voltage is really high. Only
sometimes there are small differences on the milivolts scale, which
is a neglectible value over the total 250 V we are dealing
with.
But, on the other hand, the response is not as exact as could be
expected in the beginning. There is a little difference on the
slope of the transfer function from the theoretical value to the
real one. It can be seen on figure 16. In addition, for small
values, there is a shift of the output to start growing. Another
difference is the maximum output value: instead of been the
expected 250 V, one can only get to 227 V.
Nevertheless, it does not come as a big problem: small values are
not going to be used for this application on the telescope. And, on
the other hand, there is not such important difference on the final
stroke for 30 V less on the total applied voltage. The important
aspect is that the response is linear over the region of interest
and to understand how our system is behaving and the limit it is
able to reach.
38
Figure 16: Transfer function: real delivered voltage to the mirror
as a func- tion of the selected input voltage percentages. Red line
represents the theo- retical value and blue one the obtained
measurements.
39
4 Dynamic characterization of the mirror
One step further to the understanding of the mirror behaviour is to
perform some measurements in dynamic regime. As already explained,
the mirror is going to be working in a close-loop configuration,
that is, a feedback is going to be provided during its operation.
Therefore, it is interesting to check and conclude if the device is
fast enough to follow this feedback at the needed frequency to
correct sky aberrations. That is the reason why bandwidth
characterization is going to be performed. Nonetheless, when some
bibliography related to this topic is searched for, one notices
that there is almost nothing.
4.1 Description of the system
First of all, a general view of the complete system is going to be
presented for a better understanding. The schemes showed in this
chapter are designed by the Optics team of Oaxaca and the writer
did not have a contribution on said design stage.
The complete optical design is depicted in the layout of figure 17.
Light comes from the telescope and is focused on the telescope
focal plane (TFP).
It is interesting to list all the components and their purpose in
the setup. Table 3 shows it in detail. The lenses and mounts are
purchased in Edmund Optics. Once all the pieces are in the
laboratory, the assembly of the scheme can be started. It is not
easy to align everything and this task can last for long. To help
to this alignment process, two alignment telescopes are used. One
has to be really careful in the mounting of some lenses. They are
biconvex and they have two different faces that cannot be
inverted.
L1 collimates the light from the telescope and forms a pupil image
on the Tilt mirror (TM). L2 and L3 transfer the pupil image to the
deformable mirror, which will presumably correct the rest of
aberrations. Last branch in the scheme stands for reducing the
effective focal length, which is too large, and collimate the light
that is getting to the beam splitter. Half of the beam will go on
straight to the Science camera, where images will be analyzed; and
the other half will be deviated to the right. Here, a pupil image
will be formed on the Microlenses array by L7 and its image will be
taken by the WaveFront Sensor Detector. L8 and L9 form an optical
relay to transfer the focal plane of the micro lenses array onto
the waveform sensor detector. The focal plane FS1 coincides with
the focal plane in the camera.
On the left side of the beam splitter there is a small branch that
stands for calibration. There is a light source and lenses L10 and
L11, that will be used to find the correct position of the
microlens array and the wavefront
40
sensor detector.
Element Component
TFP Telescope Focal Plane L1, L2, L3, L4 and L6 NT47380
TM Tilt mirror DM Deformable mirror L5 NT45803
PM1 and PM2 Pupil masks L7 and L8 NT49358
L9 NT49354 L10 and L11 NT49362
PH2 Pin hole BS2 Beam Splitter FS1 Field stop
MLA MicroLens Array CCD Science camera WFS WaveForm Sensor
Detector
Table 3: List of components in the setup
41
Figure 17: Oaxaca design based on NIR achromatic Mirrors for the
2.1m SPM Telescope.
42
4.2 Bandwidth measurements
For the measurements taken in this section, a simplified scheme is
used. In figure 17 only second and third branches are included.
There is one more change. A source of light simulating the incoming
light from the telescope is placed instead of the pinhole in the
focal plane between L2 and L3. A picture of the real setup is shown
in figure 18.
In the WFS detector, images of the mirror surface are been taken.
One can control the devices through a computer in the laboratory.
Alan Watson created two different routines for this purpose. One of
them is programmed to control the CCD camera and the other is made
to control the mirror.
With first one, exposure time is set. Second monitors the sort of
signal (square, sinus, tilt aberration) is sent to the mirror, its
frequency and ampli- tude. The duration of the signal and
electrodes that are activated are also parameters of this function
controlling the mirror.
If a varying signal is sent to the mirror the change in the
received light over a given time can be monitored. If the signal
changes its amplitude, it will imply thus an oscillation on the
position of the reflected light by the surface of the mirror.
An indirect way of measuring the bandwidth of the device is
precisely based on this idea: the faster the oscillation of the
signal (higher frequency), the smaller is going to be perceived the
amplitude of the movement (or remain constant, if the mirror is
able to follow said signal).
Then, one can interpret the bandwidth as the point when, increasing
the frequency of the signal, the amplitude of the oscillations
decays to one half (or 3 dB, if talking about decibels). According
to that, several measurements with different waveforms and
frequency are taken.
The setup in the laboratory is going to allow the study of signals
up to around 600 Hz. The electronics, the processor of the computer
and other devices features are setting this upper limit on the
measurements.
Nevertheless, it is a reasonably good frequency range to study. For
the application here is going to be dealed with, the required
bandwidth is not so high. It is expected that approximately 500
cycles per second are going to be enough for the system installed
in OAN/Tonanzintla. For the final application of the mirror in
OAN/San Pedro Martir, which has a larger telescope diameter, a
behaviour around 250 Hz is going to give good results.
In figure 20 the results corresponding to those measurements are
shown. Four different deformations are studied: moving 9th
electrode alone and creating a defocus, moving only the 24th
electrode and tilt on X and Y axes. Examples of the images are
shown in figure 19.
Since all of them produce a different strength an thus deformation,
it
43
seems interesting to express the results as normalized value. The
procedure is as follows: a sweep in frequency is performed in
steps
of 100 Hz, except for the first value that is 10 Hz. In each of the
points, the camera remains open for 10 seconds during each
exposition. The values for the normalized amplitude of the
oscillation are depicted in figure 20.
The first surprise can be perceived is that the setup in the
laboratory is not able to reach the cut off frequency at 3dB. The
value of the amplitude, lowered as expected, is not reduced to its
half in the range the experiment allows to study. It can be seen
also that the decreasing of the amplitude of the oscillations is
quite small and almost not even noticeable.
Thus, it can be easily concluded that it has not been able to
measure the 3 dB bandwidth, but it is for sure above 600 Hz. So,
Saturn perfectly fits the requirements for both telescopes.
44
Figure 18: Setup used to measure the bandwidth of the Saturn
mirror.
45
(c) X-Tilt
Figure 19: Examples of images on the MicroLens Array when a tilt
movement in X (a) and Y (b) directions created on the mirror and
when a defocus (c) is produced by moving the 9th electrode.
46
Figure 20: Normalized amplitude of the oscillations to calculate
the band- width.
47
5 Conclusions
Adaptive Optics is, in the most general way, a widespread branch of
optics used mainly to correct aberrations in light. It has been
utilized in many dif- ferent fiels such as industry, ophtalmology,
medicine, military actions, surgery and astronomy. Its application
in this last science is as old as the early six- ties.
Nevertheless, it has been recently in the nineties when it reached
a big improvement.
Many different devices can be used in AO for beam correction means,
such as liquid crystal devices, acousto-optic tools, piezoelectric
mirrors or deformable membrane mirrors. This last option is not
such a new device, but it can be improved to a better version: the
push-pull deformable membrane mirror. It is based on a membrane
which can be deformed by means of an applied voltage, and has the
novel capability of been pushed and pulled using two sets of
actuators, in front and rear of said membrane.
And this new configuration will allow many advantages with respect
to older mirrors. For instance, the maximum deformation over the
membrane can be double than the deformation on a mirror with a
single layer of elec- trodes. It also presents a very good dynamic
response, achromaticity, no hysteresis and they are relatively
cheap devices.
But what is really innovative is its application on astronomy.
These push- pull mirrors have been used for other means, however,
watching the stars is a fascinating new posibility they are
sighting.
The features push-pull mirror presents are reasonably good and it
seems it can perform properly on sky observation. In addition, its
price is quite attractive, because very expensive devices have been
used so far. Concretely, Saturn mirror from Adaptica Srl. has been
analysed in this thesis.
This mirror is predicted to be used in the National Astronomical
Obser- vatory of San Pedro Martir, in a 2.1 m telescope, the
largest in Mexico. As an intermediate trial stage, it will be
installed in the National Astronomical Observatory of Tonanzintla,
in a 1 m telescope, with harder atmospherical requirements.
Then, after studying deeply its characteristics, it turned out that
the mirror satisfies all the requirements and it can perform in a
fine way for astronomical purposes. Previous to get to this
conclusion, several steps have been followed.
First of all, a theoretical model of the mirror behaviour has been
de- veloped. It describes the deformation over the membrane after
appliying a voltage on the actuators. For a clearer understanding,
and given the fact that the electrodes action is linearly
dependent, it was a clever idea to study independently their
influences. And the so-called influence matrix could be
48
defined. This behaviour could be found by solving the Poisson
equation with appropiate boundary conditions.
A second step, once the model is created, was the comparison of
said theoretical numbers to real values measured in the laboratory.
These mea- surements of the deformed mirror surface were taken with
the help of an interesting tool: a Zygo interferometer. It can give
results with an error on the order of several nanometers. After
lots of effort, it was demonstrated that the model sucessfully
fitted the real deformation on the membrane.
Some intermediate operations had to be performed to end up in this
im- portant conclusion. For instance: it was interesting to measure
a calibration surface to evaluate the error the interferometer was
introducing. It was also important to analyze the resting position
of the mirror, in order to calibrate the measuremnts and to use it
as a reference.
Another physical parameter that was good to calibrate was tension
on the membrane. It was considered a constant value and could be
easily estimated with the help of some basic equations.
Next remarkable characteristic of Saturn was its maximum stroke.
That is, the maximum peak-valley value the membrane can achieve by
applying a voltage. The stroke comes associated to a phase shift
the mirror is able to correct. In other words: the worse the sky
conditions, the higher the needed stroke. So, it was necessary to
make the calculations according to the atmosphere above the
telescopes. Thus, we made numbers and got the necessary deformation
for the atmospheric conditions in San Pedro Martir and Tonanzintla.
On the other hand, the maximum deformation achieved by Saturn was
measured in the laboratory. It could be concluded that the mirror
was adequate for the conditions of the observation. Never- theless,
an important controversia was found between the maximum stroke
measured with Zygo in the laboratory and the one given by
Adaptica.
Next stage was to check repeatability and hysteresis of the mirror.
Sev- eral measurements were taken for each kind of electrode with
same applied voltage. It turned out that repeatability was quite
good. There were only some differences found on the borders due to
undesidered reflections which are in principle unavoidable in this
kind of systems. On the other hand, to analyse hyteresis, maximum
stroke on the membrane was measured in each sort of actuator for
increasing and decreasing voltage values. The result was that
hysteresis was pretty good in almost all range of voltages except
for small V that produce tiny deformations. The explanation could
be found in the resting position of the mirror, which was not
completely flat and kind of random. Anyways, it was not such
important problem because these small voltages were not going to be
used in this application.
Last feature to study was the Saturn working bandwidth. It was
necessary
49
to check if the mirror was able to follow the changes leaded by a
given varying signal. If the signal sent to the mirror had a very
high frequency, that is, it was too fast, it may not be able to
move its membrane that quickly. But the results show that,
according to the requirements of the atmosphere over the
telescopes, it was more than valid and was going to behave
good.
To sum up, after analyzing the problems the atmosphere presented in
the two telescopes and thus the requirements it implied, it could
be concluded that the mirror had good features to work in Oaxaca
project. Due to its good characteristics: good dynamic behaviour,
good hysteresis properties and enough stroke; it was a good
candidate to successfully correct aberrations working together with
the tilt mirror in the setup.
50
ences matrix
1 function A front = i n f l u e n c e s f r o n t ( ) 2
3 r=linspace ( 0 , 8 . 5 , 2 1 0 ) ; %r a d i u s mm 4
theta=linspace (0 ,2∗pi , length ( r ) ) ; %ang le 5 n act =16; 6
a=max( r ) ; 7
8 A front=zeros ( n act , length ( r ) , length ( theta ) ) ; %i n
f l u e n c e matrix
9
10 % D e f i n i t i o n o f r e g i o n s 11 r a d i i =[5.425 7
.5 8 . 5 ] ; 12 ang l e s =[2∗pi 2∗pi / 1 5 ] ; 13
14 %CASE I 15 %I n f l u e n c e o f c e n t r a l e l e c t r o d
e on c e n t r a l p o i n t 16 for k=1: length ( theta ) 17 A
front (1 , 1 , k ) =1/(2∗pi )∗ ang l e s (1 ) ∗ r a d i i ( 1 ) ˆ
2∗ ( 1 / 2
log ( r a d i i ( 1 ) /a ) ) /2 ; 18 end 19
20 %CASE I I 21 %Rest o f e l e c t r o d e s i n f l u e n c i n g
on c e n t r a l p o i n t 22
23 for k=1: length ( theta ) 24 for j =1: n act 25 i f (1< j
)&&(j <17) 26 A front ( j , 1 , k ) =1/(2∗pi )∗ ang l e
s (2 ) ∗( r a d i i ( 2 )
ˆ2/2∗ ( 1/2 log ( r a d i i ( 2 ) /a ) ) r a d i i ( 1 ) ˆ2/2∗ (
1/2 log ( r a d i i ( 1 ) /a ) ) ) ;
27 end 28 end 29 end 30
31 %CASE I I I 32 %I n f l u e n c e on areas o u t s i d e the c i
r c l e e l e c t r o d e s
d e f i n e
52
33
36
37 for j =1: n act %j counts number o f ac tua tors , i counts p o
i n t under s tudy
38 for i =1: length ( r ) 39 i f j==1 %i n f l u e n c e o f c e n
t r a l reg ion on 40 i f ( r a d i i ( 1 )<r ( i ) ) &&
( r ( i )<r a d i i ( 3 ) ) %
f i r s t , second , t h i r d r i n g and out 41 for k=1: length (
theta ) 42 A front ( j , i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (1
)
∗( log ( a/ r ( i ) ) ) ∗( r a d i i ( 1 ) ˆ2) /2) sum( r ( i ) ˆ2
. / ( n . ˆ 2 . ∗ (
n+2) ) . ∗ ( ( r ( i ) . ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n)
) ) . ∗ ( ( r a d i i ( 1 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ ( sin (n∗(
ang l e s (1 ) theta ( k ) ) ) sin (n ∗ ( theta ( k ) ) ) ) ) )
;
43 end 44 end 45 e l s e i f (1< j )&&(j <17) %i n f
l u e n c e o f the f i r s t
r i n g over 2nd and 3 rd r i n g 46 i f ( r a d i i ( 2 )<r ( i
) ) && ( r ( i )<r a d i i ( 3 ) ) %
second and t h i r d r i n g 47 for k=1: length ( theta ) 48 A
front ( j , i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (2 )
∗( log ( a/ r ( i ) ) ) ∗( r a d i i ( 2 ) ˆ 2 r a d i i ( 1 ) ˆ2)
/2) sum( r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i ) . ˆ
( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ ( ( r a d i i ( 2 )
/ r ( i ) ) . ˆ ( n+2) ( r a d i i ( 1 ) / r ( i ) ) . ˆ ( n+2) ) .
∗ ( sin (n∗ ( ( j 1 ) ∗ ang l e s (2 ) theta ( k ) ) ) sin (n∗ ( (
j 2 ) ∗ ang l e s (2 ) theta ( k ) ) ) ) ) ) ;
49 end 50 end 51 end 52 end 53 end 54
53
55
56
57 %CASE IV 58 %I n f l u e n c e on areas i n s i d e the c i r c
l e e l e c t r o d e s d e f i n e 59
60 alpha=zeros (1 , length (n) ) ; 61
62 for j =1: n act %j counts number o f ac tua tors , i counts p o
i n t under s tudy
63 for i =1: length ( r ) 64 i f (1< j )&&(j
<17)&&(r ( i )<r a d i i ( 1 ) ) %i n f l u e n c
e
o f f i r s r i n g 65 alpha (1 ) = r ( i ) ∗( r a d i i ( 2 ) r a
d i i ( 1 ) ) ; 66 alpha (2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 2 )
) log ( r a d i i
( 1 ) ) ) ; 67 for m=3: length (n) 68 alpha (m)=r ( i ) ˆ2∗ ( ( r (
i ) / r a d i i ( 2 ) ) ˆ(m 2 )
( r ( i ) / r a d i i ( 1 ) ) ˆ(m 2 ) ) /(m 2 ) ; 69 end 70 for
k=1: length ( theta ) 71 A front ( j , i , k ) =1/(2∗pi ) ∗( ang l
e s (2 ) ∗( r a d i i ( 2 ) ˆ2/2∗ ( 1/2
log ( r a d i i ( 2 ) /a ) ) ( r a d i i ( 1 ) ˆ2/2∗ ( 1/2 log ( r
a d i i ( 1 ) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /(
a ˆ2) ) . ˆ n . ∗ ( r a d i i ( 2 ) . ˆ ( n +2) r a d i i ( 1 ) . ˆ
( n+2) ) . / ( n+2)+alpha ) . ∗ ( sin (n∗ ( ( j 1 ) ∗ ang l e s (2
) theta ( k ) ) ) sin (n∗ ( ( j 2 ) ∗ ang l e s (2 ) theta ( k ) )
) ) ) ) ;
72 end 73 end 74 end 75 end 76
77
78 % CASE V 79 % I n f l u e n c e o f e l e c t r o d e over i t s
zone 80
81 alpha=zeros (1 , length (n) ) ; 82 alpha1=zeros (1 , length (n)
) ; 83
84
85 for j =1: n act 86 for i =1: length ( r )
54
87 i f ( j==1 && r ( i )<=r a d i i ( 1 ) ) % i n f l u
e n c e o f c e n t e r on c e n t e r
88 alpha (1 ) = r ( i ) ∗( r a d i i ( 1 ) r ( i ) ) ; 89 alpha (2
) = r ( i ) ˆ2∗( log ( r a d i i ( 1 ) ) log ( r ( i ) ) )
; 90 for m=3: length (n) 91 alpha (m)=r ( i ) ˆ2∗ ( ( r ( i ) / r a
d i i ( 1 ) ) ˆ(m 2 )
1 ) /(m 2 ) ; 92 end 93 for k=1: length ( theta ) 94 A front ( j ,
i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (1 ) ∗( log ( a/ r ( i ) ) ) ∗(
r ( i )
ˆ2 ) /2) sum( r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i )
. ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ (
sin (n∗( ang l e s (1 ) theta ( k ) ) ) sin (n∗( theta ( k ) ) ) )
) ) +1/(2∗pi ) ∗( ang l e s (1 ) ∗ ( ( r a d i i ( 1 ) ˆ2) / 2∗ ( 1
/ 2 log ( r a d i i ( 1 ) /a ) ) ( r ( i ) ˆ2/2∗ ( 1/2 log ( r ( i
) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2) ) . ˆ
( n) . ∗ ( r a d i i ( 1 ) . ˆ ( n+2) r ( i ) . ˆ ( n +2) ) . / (
n+2)+alpha ) . ∗ ( sin (n∗( ang l e s (1 ) theta ( k ) ) ) sin (n∗(
theta ( k ) ) ) ) ) ) ;
95 end 96
97 e l s e i f ((1< j )&&(j <17)&&(r a d i i
( 1 )<r ( i ) ) && ( r ( i )<=r a d i i ( 2 ) ) ) %i
n f l u e n c e o f f i r s r i n g on f i r s t r i n g
98 alpha1 (1 ) = r ( i ) ∗( r a d i i ( 2 ) r ( i ) ) ; 99 alpha1
(2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 2 ) ) log ( r ( i ) )
) ; 100 for m=3: length (n) 101 alpha1 (m)=r ( i ) ˆ2∗ ( ( r ( i )
/ r a d i i ( 2 ) ) ˆ(m
2 ) 1 ) /(m 2 ) ; 102 end 103 for k=1: length ( theta ) 104 A front
( j , i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (2 ) ∗(
log ( a/ r ( i ) ) ) ∗( r ( i ) ˆ 2 r a d i i ( 1 ) ˆ2) /2) sum( r
( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i )
. ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ ( 1 ( r a d i i
( 1 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ ( sin (n∗ ( ( j 1 ) ∗ ang l e s
(2 ) theta ( k ) ) ) sin (n∗ ( ( j 2 ) ∗ ang l e s (2 ) theta ( k )
) ) ) ) ) +1/(2∗pi
) ∗( ang l e s (2 ) ∗ ( ( r a d i i ( 2 ) ˆ2) / 2∗ ( 1 / 2
55
log ( r a d i i ( 2 ) /a ) ) ( r ( i ) ˆ2/2∗ ( 1/2 log ( r ( i ) /a
) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2) ) . ˆ ( n)
. ∗ ( r a d i i ( 2 ) . ˆ ( n+2) r ( i ) . ˆ ( n+2) ) . / (
n+2)+alpha1 ) . ∗ ( sin (n∗ ( ( j 1 ) ∗ ang l e s (2 ) theta ( k )
) ) sin (n∗ ( ( j 2 ) ∗ ang l e s (2 ) theta ( k ) ) ) ) ) )
;
105 end 106 end 107 end 108 end
56
1 function A = i n f l u e n c e s r e a r ( ) 2
3 r=linspace ( 0 , 8 . 5 , 2 1 0 ) ; %r a d i u s mm 4
theta=linspace (0 ,2∗pi , length ( r ) ) ; %ang le 5 n act =32; 6
a=max( r ) ; 7
8 A=zeros ( n act , length ( r ) , length ( theta ) ) ; % i n f l u
e n c e matrix
9
10 % D e f i n i t i o n o f r e g i o n s 11 r a d i i =[1.18 3
.42 5 .425 7 .5 9 . 5 ] ; 12 ang l e s =[2∗pi 2∗pi/6 2∗pi /10 2∗pi
/ 1 5 ] ; 13
14 %CASE I 15 %I n f l u e n c e o f c e n t r a l e l e c t r o d
e on c e n t r a l p o i n t 16 for k=1: length ( theta ) 17 A(1 ,1
, k ) =1/(2∗pi )∗ ang l e s (1 ) ∗ r a d i i ( 1 ) ˆ 2∗ ( 1 / 2 log
(
r a d i i ( 1 ) /a ) ) /2 ; 18 end 19
20 %CASE I I 21 %Rest o f e l e c t r o d e s i n f l u e n c i n g
on c e n t r a l p o i n t 22
23 for k=1: length ( theta ) 24 for j =1: n act 25 i f (1< j
)&&(j<8) 26 A( j , 1 , k ) =1/(2∗pi )∗ ang l e s (2 ) ∗(
r a d i i ( 2 ) ˆ2/2∗ ( 1/2 log (
r a d i i ( 2 ) /a ) ) r a d i i ( 1 ) ˆ2/2∗ ( 1/2 log ( r a d i i
( 1 ) /a ) ) ) ; 27 e l s e i f (7< j )&&(j <18) 28
A( j , 1 , k ) =1/(2∗pi )∗ ang l e s (3 ) ∗( r a d i i ( 3 ) ˆ2/2∗
( 1/2 log (
r a d i i ( 3 ) /a ) ) r a d i i ( 2 ) ˆ2/2∗ ( 1/2 log ( r a d i i
( 2 ) /a ) ) ) ; 29 e l s e i f (17< j )&&(j <33) 30
A( j , 1 , k ) =1/(2∗pi )∗ ang l e s (4 ) ∗( r a d i i ( 4 ) ˆ2/2∗
( 1/2 log (
r a d i i ( 4 ) /a ) ) r a d i i ( 3 ) ˆ2/2∗ ( 1/2 log ( r a d i i
( 3 ) /a ) ) ) ; 31 end 32 end 33 end 34
35 %CASE I I I 36 %I n f l u e n c e on areas o u t s i d e the c i
r c l e e l e c t r o d e s
57
38 n=1:100; 39
40
41 for j =1: n act %j counts number o f ac tua tors , i counts p o
i n t under s tudy
42 for i =1: length ( r ) 43 i f j==1 %i n f l u e n c e o f c e n
t r a l reg ion on 44 i f ( r a d i i ( 1 )<r ( i ) ) &&
( r ( i )<r a d i i ( 5 ) ) %
f i r s t , second , t h i r d r i n g and out 45 for k=1: length (
theta ) 46 A( j , i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (1 ) ∗( log
(
a/ r ( i ) ) ) ∗( r a d i i ( 1 ) ˆ2) /2) sum( r ( i ) ˆ2 . / ( n .
ˆ 2 . ∗ (
n+2) ) . ∗ ( ( r ( i ) . ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n)
) ) . ∗ ( ( r a d i i ( 1 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ ( sin (n∗(
ang l e s (1 ) theta ( k ) ) ) sin (n ∗ ( theta ( k ) ) ) ) ) )
;
47 end 48 end 49 e l s e i f (1< j )&&(j<8) %i n f l
u e n c e o f the f i r s t
r i n g over 2nd and 3 rd r i n g 50 i f ( r a d i i ( 2 )<r ( i
) ) && ( r ( i )<r a d i i ( 5 ) ) %
second and t h i r d r i n g 51 for k=1: length ( theta ) 52 A( j ,
i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (2 ) ∗( log (
a/ r ( i ) ) ) ∗( r a d i i ( 2 ) ˆ 2 r a d i i ( 1 ) ˆ2) /2) sum(
r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i ) . ˆ ( 2∗n) a
. ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ ( ( r a d i i ( 2 ) / r ( i
) ) . ˆ ( n +2) ( r a d i i ( 1 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ (
sin (n∗ ( ( j 1 ) ∗ ang l e s (2 ) theta ( k ) ) ) sin (n∗ ( ( j 2
) ∗ ang l e s (2 ) theta ( k ) ) ) ) ) ) ;
53 end 54 end 55 e l s e i f (7< j )&&(j <18) %i n f
l u e n c e o f second r i n g
over 56 i f ( r a d i i ( 3 )<r ( i ) ) && ( r ( i
)<r a d i i ( 5 ) ) %
58
t h i r d r i n g 57 for k=1: length ( theta ) 58 A( j , i , k )
=1/(2∗pi ) ∗ ( ( ang l e s (3 ) ∗( log (
a/ r ( i ) ) ) ∗( r a d i i ( 3 ) ˆ 2 r a d i i ( 2 ) ˆ2) /2) sum(
r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i ) . ˆ ( 2∗n) a
. ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ ( ( r a d i i ( 3 ) / r ( i
) ) . ˆ ( n +2) ( r a d i i ( 2 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ (
sin (n∗ ( ( j 7 ) ∗ ang l e s (3 ) theta ( k ) ) ) sin (n∗ ( ( j 8
) ∗ ang l e s (3 ) theta ( k ) ) ) ) ) ) ;
59 end 60 end 61 e l s e i f (17< j )&&(j <33) %i n f
l u e n c e o f t h i r d r i n g
over 62 i f ( r a d i i ( 4 )<r ( i ) ) && ( r ( i
)<r a d i i ( 5 ) ) %
outer 63 for k=1: length ( theta ) 64 A( j , i , k ) =1/(2∗pi ) ∗ (
( ang l e s (4 ) ∗( log (
a/ r ( i ) ) ) ∗( r a d i i ( 4 ) ˆ 2 r a d i i ( 3 ) ˆ2) /2) sum(
r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i ) . ˆ ( 2∗n) a
. ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ ( ( r a d i i ( 4 ) / r ( i
) ) . ˆ ( n +2) ( r a d i i ( 3 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ (
sin (n∗ ( ( j 1 7 ) ∗ ang l e s (4 ) theta ( k ) ) ) sin (n∗ ( ( j
1 8 ) ∗ ang l e s (4 ) theta ( k ) ) ) ) ) ) ;
65 end 66 end 67 end 68 end 69 end 70
71
72
73 %CASE IV 74 %I n f l u e n c e on areas i n s i d e the c i r c
l e e l e c t r o d e s d e f i n e 75
76 alpha=zeros (1 , length (n) ) ; 77 alpha2=zeros (1 , length (n)
) ; 78 alpha3=zeros (1 , length (n) ) ;
59
79
80 for j =1: n act %j counts number o f ac tua tors , i counts p o
i n t under s tudy
81 for i =1: length ( r ) 82 i f (1< j )&&(j<8) %i n
f l u e n c e o f f i r s r i n g 83 i f ( r ( i )<r a d i i ( 1
) ) %on the c e n t r a l zone 84 alpha (1 ) = r ( i ) ∗( r a d i i
( 2 ) r a d i i ( 1 ) ) ; 85 alpha (2 ) = r ( i ) ˆ2∗( log ( r a d
i i ( 2 ) ) log (
r a d i i ( 1 ) ) ) ; 86 for m=3: length (n) 87 alpha (m)=r ( i )
ˆ2∗ ( ( r ( i ) / r a d i i ( 2 ) ) ˆ(m 2 ) ( r ( i ) / r a d i i (
1 ) )
ˆ(m 2 ) ) /(m 2 ) ; 88 end 89 for k=1: length ( theta ) 90 A( j , i
, k ) =1/(2∗pi ) ∗( ang l e s (2 ) ∗( r a d i i ( 2 ) ˆ2/2∗ ( 1/2
log (
r a d i i ( 2 ) /a ) ) ( r a d i i ( 1 ) ˆ2/2∗ ( 1/2 log ( r a d i
i ( 1 ) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2)
) . ˆ n . ∗ ( r a d i i ( 2 ) . ˆ ( n+2)
r a d i i ( 1 ) . ˆ ( n+2) ) . / ( n+2)+alpha ) . ∗ ( sin (n∗ ( ( j
1 ) ∗ ang l e s (2 ) theta ( k ) ) ) sin (n∗ ( ( j 2 ) ∗ ang l e s
(2 ) theta ( k ) ) ) ) ) ) ;
91 end 92 end 93 e l s e i f (7< j )&&(j <18) %i n f
l u e n c e o f the second
r i n g 94 i f ( r ( i )<r a d i i ( 2 ) ) %on 1 s t and c e n t
e r 95 alpha2 (1 ) = r ( i ) ∗( r a d i i ( 3 ) r a d i i ( 2 ) ) ;
96 alpha2 (2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 3 ) ) log (
r a d i i ( 2 ) ) ) ; 97 for m=3: length (n) 98 alpha2 (m)=r ( i )
ˆ2∗ ( ( r ( i ) / r a d i i ( 3 ) ) ˆ(m 2 ) ( r ( i ) / r a d i i (
2 ) )
ˆ(m 2 ) ) /(m 2 ) ; 99 end 100 for k=1: length ( theta ) 101 A( j ,
i , k ) =1/(2∗pi ) ∗( ang l e s (3 ) ∗( r a d i i ( 3 ) ˆ2/2∗ ( 1/2
log (
r a d i i ( 3 ) /a ) ) ( r a d i i ( 2 ) ˆ2/2∗ ( 1/2 log ( r a d i
i ( 2 ) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2)
) . ˆ n . ∗ ( r a d i i ( 3 ) . ˆ ( n+2)
r a d i i ( 2 ) . ˆ ( n+2) ) . / ( n+2)+alpha2 ) . ∗ ( sin (n∗ ( (
j 7 ) ∗ ang l e s (3 ) theta ( k ) ) ) sin (n∗ ( ( j 8 ) ∗ ang l e
s (3 ) theta ( k ) ) ) ) ) ) ;
102 end 103 end 104 e l s e i f (17< j )&&(j <33) %i
n f l u e n c e o f t h i r d
60
r i n g on 105 i f ( r ( i )<r a d i i ( 3 ) ) %1 st , 2nd and c
e n t e r 106 alpha3 (1 ) = r ( i ) ∗( r a d i i ( 4 ) r a d i i (
3 ) ) ; 107 alpha3 (2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 4 ) ) log
(
r a d i i ( 3 ) ) ) ; 108 for m=3: length (n) 109 alpha3 (m)=r ( i
) ˆ2∗ ( ( r ( i ) / r a d i i ( 4 ) ) ˆ(m 2 ) ( r ( i ) / r a d i i
( 3 ) )
ˆ(m 2 ) ) /(m 2 ) ; 110 end 111 for k=1: length ( theta ) 112 A( j
, i , k ) =1/(2∗pi ) ∗( ang l e s (4 ) ∗( r a d i i ( 4 ) ˆ2/2∗ (
1/2 log (
r a d i i ( 4 ) /a ) ) ( r a d i i ( 3 ) ˆ2/2∗ ( 1/2 log ( r a d i
i ( 3 ) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2)
) . ˆ n . ∗ ( r a d i i ( 4 ) . ˆ ( n+2)
r a d i i ( 3 ) . ˆ ( n+2) ) . / ( n+2)+alpha3 ) . ∗ ( sin (n∗ ( (
j 1 7 ) ∗ ang l e s (4 ) theta ( k ) ) ) sin (n∗ ( ( j 1 8 ) ∗ ang
l e s (4 ) theta ( k ) ) ) ) ) ) ;
113 end 114 end 115 end 116 end 117 end 118
119
120 % CASE V 121 % I n f l u e n c e o f e l e c t r o d e over i t
s zone 122
123 alpha=zeros (1 , length (n) ) ; 124 alpha1=zeros (1 , length
(n) ) ; 125 alpha2=zeros (1 , length (n) ) ; 126 alpha3=zeros (1 ,
length (n) ) ; 127
128 for j =1: n act 129 for i =1: length ( r ) 130 i f ( j==1
&& r ( i )<=r a d i i ( 1 ) ) % i n f l u e n c e o
f
c e n t e r on c e n t e r 131 alpha (1 ) = r ( i ) ∗( r a d i i (
1 ) r ( i ) ) ; 132 alpha (2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 1 )
) log ( r ( i ) ) )
; 133 for m=3: length (n) 134 alpha (m)=r ( i ) ˆ2∗ ( ( r ( i ) / r
a d i i ( 1 ) ) ˆ(m 2 )
1 ) /(m 2 ) ;
61
135 end 136 for k=1: length ( theta ) 137 A( j , i , k ) =1/(2∗pi )
∗ ( ( ang l e s (1 ) ∗( log ( a/ r ( i ) ) ) ∗( r ( i ) ˆ2
) /2) sum( r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) ) . ∗ ( ( r ( i ) .
ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗ (
sin (n∗( ang l e s (1 ) theta ( k ) ) ) sin (n∗( theta ( k ) ) ) )
) ) +1/(2∗pi ) ∗( ang l e s (1 ) ∗ ( ( r a d i i ( 1 ) ˆ2) / 2∗ ( 1
/ 2 log ( r a d i i ( 1 ) /a ) ) ( r ( i ) ˆ2/2∗ ( 1/2 log ( r ( i
) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( ( r ( i ) /( a ˆ2) ) . ˆ
( n) . ∗ ( r a d i i ( 1 ) . ˆ ( n+2) r ( i ) . ˆ ( n +2) ) . / (
n+2)+alpha ) . ∗ ( sin (n∗( ang l e s (1 ) theta ( k ) ) ) sin (n∗(
theta ( k ) ) ) ) ) ) ;
138 end 139
140 e l s e i f ((1< j )&&(j<8)&&(r a d i i (
1 )<r ( i ) ) && ( r ( i ) <=r a d i i ( 2 ) ) ) %i n
f l u e n c e o f f i r s r i n g on f i r s t r i n g
141 alpha1 (1 ) = r ( i ) ∗( r a d i i ( 2 ) r ( i ) ) ; 142 alpha1
(2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 2 ) ) log ( r ( i ) )
) ; 143 for m=3: length (n) 144 alpha1 (m)=r ( i ) ˆ2∗ ( ( r ( i )
/ r a d i i ( 2 ) ) ˆ(m
2 ) 1 ) /(m 2 ) ; 145 end 146 for k=1: length ( theta ) 147 A( j ,
i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (2 ) ∗( log ( a/ r ( i ) ) ) ∗(
r ( i ) ˆ 2
r a d i i ( 1 ) ˆ2) /2) sum( r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) )
. ∗ ( ( r ( i ) . ˆ ( 2∗n) a . ˆ (2∗n) ) . / ( a . ˆ ( 2∗n) ) ) . ∗
( 1 ( r a d i i ( 1 ) / r ( i ) ) . ˆ ( n+2) ) . ∗ ( sin (n∗ ( ( j
1 ) ∗ ang l e s (2 ) theta ( k ) ) ) sin (n ∗ ( ( j 2 ) ∗ ang l e s
(2 ) theta ( k ) ) ) ) ) ) +1/(2∗pi ) ∗( ang l e s (2 ) ∗ ( ( r a d
i i ( 2 ) ˆ2) / 2∗ ( 1 / 2 log ( r a d i i ( 2 ) /a ) ) ( r ( i )
ˆ2/2∗ ( 1/2 log ( r ( i ) /a ) ) ) ) sum( 1 . / ( n . ˆ 2 ) . ∗ ( (
r ( i ) /( a ˆ2) ) . ˆ ( n) . ∗ ( r a d i i ( 2 ) . ˆ ( n+2) r ( i
) . ˆ ( n+2) ) . / ( n+2)+ alpha1 ) . ∗ ( sin (n∗ ( ( j 1 ) ∗ ang l
e s (2 ) theta ( k ) ) ) sin (n ∗ ( ( j 2 ) ∗ ang l e s (2 ) theta
( k ) ) ) ) ) ) ;
148 end 149
150 e l s e i f ((7< j )&&(j <18)&&(r a d i i
( 2 )<r ( i ) ) && ( r ( i )<=r a d i i ( 3 ) ) )
%second r i n g
151 alpha2 (1 ) = r ( i ) ∗( r a d i i ( 3 ) r ( i ) ) ; 152 alpha2
(2 ) = r ( i ) ˆ2∗( log ( r a d i i ( 3 ) ) log ( r ( i ) )
62
) ; 153 for m=3: length (n) 154 alpha2 (m)=r ( i ) ˆ2∗ ( ( r ( i )
/ r a d i i ( 3 ) ) ˆ(m
2 ) 1 ) /(m 2 ) ; 155 end 156 for k=1: length ( theta ) 157 A( j ,
i , k ) =1/(2∗pi ) ∗ ( ( ang l e s (3 ) ∗( log ( a/ r ( i ) ) ) ∗(
r ( i ) ˆ 2
r a d i i ( 2 ) ˆ2) /2) sum( r ( i ) ˆ2 . / ( n . ˆ 2 . ∗ ( n+2) )
. ∗ ( ( r ( i ) . &c