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 mirror 73
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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 actuators. 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 reference 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 equilibrium position of Saturn mirror, extracted from 10 samples of said position. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9 Graph of the influence calculated with the model on the interest 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 deviation 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
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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 bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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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 science 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 firmament 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 interesting 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.
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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 astronomical 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 century 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 already 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 application. 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-
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bulence brought by gradients on temperature or different winds speeds; the index of refraction has a strong dependence on the position. And, in addition, 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
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deformable push-pull mirror from Adaptica: the Saturn mirror [7]. Its introduction to AO presents many advantages when compared with other different devices such as liquid crystal modulators, thermal mirrors or bimorph mirrors. 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 introduction 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.
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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 membrane. 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.
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At that point, may result interesting to present the structure of the electrodes 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 proportional 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)
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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 voltages 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 pressure 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 displacements 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 solution of the poisson equation and Mi is the displacement produced over the
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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 coefficients 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:
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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 =
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• 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)
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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 membrane can be gotten. By inverting the influence matrix [10], one can generate the desidered wavefront to correct a given aberration.
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Figure 4: Influence matrix. Each circle represents the influence of each actuator. By inverting the influence matrix, one can generate the desidered wavefront.
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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 results
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 theoretically 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
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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.
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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.
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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 measurements 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 deformation 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.
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Figure 7: Average surface of the equilibrium position of Saturn mirror, extracted 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.
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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
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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.
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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.
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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 estimation 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) deformation 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 maximum stroke and their understanding are different. For Adaptica the maximum 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 measurements 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 appropriate 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 measures 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 correction 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 maximum 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 introduced 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 aberrations 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 requirements 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 hysteresis. 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 understood 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 working 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 deviation 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 differences on repeatability values are coming from or why the maximum deformation 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 function of the selected input voltage percentages. Red line represents the theoretical 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 amplitude. 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 bandwidth.
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 different 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 electrodes. 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 developed. 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 measurements 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 important 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
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
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 ) ) ) ) ) ) ;
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
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 ) )
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 ) )
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 ) ) ) ) ) ) ;
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
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
62
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

characterization of a push-pull membrane mirror for an

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