Laskennallisen tekniikan koulutusohjelma Kandidaatinty¨o
Tomi Krokberg
Mathematical methods in adaptive optics
Supervisor: Assoc. Prof. Tapio Helin
Lappeenranta-Lahti University of Technology School of Engineering Science
Degree Programme in Computational Engineering Tomi Krokberg
Mathematical methods in adaptive optics
Bachelor’s Thesis 2020
27 pages, 10 figures
Supervisor: Assoc. Prof. Tapio Helin
Keywords: adaptive optics; mathematics; inverse problems; statistics; OOMAO;
In this thesis, we made a literature review of adaptive optics in astronomy and to the mathe- matical methods behind it. Adaptive optics is used to correct wavefront perturbations caused by the atmosphere when imaging astronomical objects. We also simulated an adaptive optics system using an OOMAO environment in Matlab.
We start by defining the mathematical preliminaries: random fields, power spectrums and inverse problems. Random fields and power spectrums are used when modeling the atmo- spheric turbulence. Reconstructing the wavefront shape from measurements is an inverse problem.
After that, we will introduce the basic adaptive optics system and its components: refer- ence star, atmosphere, wavefront sensor and a deformable mirror. We showcase airflow in the atmosphere to be turbulent with Reynolds number and introduce statistical models for turbulence using Kolmogorov and von Karman spectrums. We also showcase the working principle of the Shack-Hartmann wavefront sensor and how it can be used to reconstruct the shape of the arriving wavefront.
In OOMAO simulations we take a look at how reference star magnitude affects our imaging quality and also how integrator controller gain behaves. We noticed imaging quality to be stable until certain magnitude, after which it started to decrease slowly as magnitude contin- ued to increase. There was a second critical point in magnitude, after which imaging quality decreased fast and stabilized. Integrator gain was noticed to decrease steadily between these two points.
Lappeenrannan-Lahden teknillinen yliopisto School of Engineering Science
Laskennallisen tekniikan koulutusohjelma Tomi Krokberg
Matemaattiset menetelm¨at adaptiivisessa optiikassa
Kandidaatinty¨o 2020
27 sivua, 10 kuvaa
Ohjaaja: apulaisprofessori Tapio Helin
Avainsanat: adaptiivinen optiikka; matematiikka; inversio-ongelmat; tilastotiede; OOMAO
Kandidaatinty¨oss¨a tehtiin kirjallisuuskatsaus adaptiiviseen optiikkaan t¨ahtikuvauksessa sek¨a matemaattisiin menetelmiin sen taustalla. Lis¨aksi adaptiivisen optiikan toimintaa simuloitiin Matlabille tehdyss¨a OOMAO ymp¨arist¨oss¨a. Adaptiivista optiikkaa k¨aytet¨a¨an korjaamaan ilmakeh¨an aiheuttamia v¨a¨aristymi¨a t¨ahdist¨a saapuville aaltorintamille.
Ty¨on alussa esitell¨a¨an aiheeseen oleellisesti liittyv¨at matemaattiset aiheet: satunnaiskent¨at, tehospektrit ja inversio-ongelmat. Satunnaiskent¨at ja tehospektrit liittyv¨at ilmakeh¨an
mallintamiseen, kun taas v¨a¨aristyneen aaltorintaman rekonstruktointi on invesio-ongelma.
T¨am¨an j¨alkeen esitell¨a¨an adaptiivisen optiikan systeemi ja sen komponentit: t¨ahti, ilmakeh¨a, aaltorintamasensori ja mukautuva peili. Ilmakeh¨a todistetaan k¨ayt¨ann¨oss¨a aina turbulent- tiseksi Reynoldsin luvun avulla. Turbulenssirintamille esitell¨a¨an Kolmogorovin ja von Kar- manin spektrimallit. Aaltorintamasensorin kohdalla esitell¨a¨an Shack-Hartmann sensori, sek¨a miten sen mittauksista voidaan johtaa aaltorintaman muoto.
OOMAO simulaatiossa esitell¨a¨an miten t¨ahden magnitudi vaikuttaa kuvanlaatuun, sek¨a integraattori-ohjaimen vahvistuksen k¨ayttytymist¨a. Kuvanlaadun havaittiin pysyv¨an vakiona tiettyyn magnitudiin asti, josta se rupesi laskemaan hitaasti magnitudin kasvaessa. Magni- tudin kasvaessa tarpeeksi havaittiin toinen piste, jonka j¨alkeen kuvanlaatu laski nopeasti ja stabiloitui. Integraattori-ohjaimen vahvistuksen havaittiin laskevan tasaisesti n¨aiden kahden pisteen v¨aliss¨a.
List of symbols and abbreviations 5
1 INTRODUCTION 6
1.1 Background . . . 6 1.2 Research objective and the scope of the thesis . . . 7 1.3 Research methodology and arrangements . . . 7
2 MATHEMATICAL PRELIMINARIES 8
2.1 Probability distributions . . . 8 2.2 Power spectrum . . . 9 2.3 Inverse problems . . . 10
3 ADAPTIVE OPTICS 13
3.1 Atmospheric turbulence . . . 16 3.2 AO correction . . . 18
4 OOMAO SIMULATION 20
4.1 Enviroment . . . 20 4.2 Methods . . . 21 4.3 Results and discussion . . . 21
REFERENCES 26
Figures 28
OOMAO Matlab based simulation tool for adaptive optics ELT Extremely Large Telescope
GS Guide star
NGS Natural guide star LGS Laser guide star WFS Wavefront sensor AO Adaptive optics
SH Shack-Hartmann
OPL Optical path lenght
Cn2 Index of refraction structure constant r0 Fried parameter
Re Reynolds number
Sφ Strehl ratio
λ Wavelength
1 INTRODUCTION
This work showcases the mathematical foundation of adaptive optics (AO) used in telescope imaging. We will focus on systems that use one guide star and Shack-Hartmann sensor as the wavefront sensor. We will also investigate how the magnitude of the guide star affects the quality of the scientific image using a Matlab based AO simulation.
1.1 Background
When collecting images from astronomical (stellar) objects using astronomical telescopes, both light collecting area and angular resolution are important. Increasing the light collecting area (diameter) allows us to image dimmer objects and can also increase the resolution by reducing the amount of diffraction happening at the optics. However, after some point we get no more resolution benefit from increasing the diameter, as the effects of the atmosphere will be the limiting factor instead of diffraction. This threshold is called atmospheric seeing and as we have already reached it, the most common way to express the quality of telescopes is by using its collection diameter, as opposed to angular resolution [2].
The perturbation of wavefronts travelling through the atmosphere is caused by turbulent airflow of different temperatures. This is the same effect that causes the stars to flicker at night when looking at the sky. This also causes point-like sources like stars to look like blurry disks when imaged. So most often when we are imaging the sky from the ground, the limiting factor to the resolution will be distortion caused by the turbulence and not the diffraction caused by limited telescope optics.
To combat this issue, adaptive optics were developed. Adaptive optics uses a deformable mirror to physically correct the deformed wavefront. This will result in an image that is closer to a diffraction-limited image. To achieve this, we use a wavefront sensor and a guide star to estimate the distortion that happens in the atmosphere. Using this knowledge we can apply such transformation to the optics of the telescope (mirror), so that it straightens the incoming wavefront. The guide star can be any bright object close to the object of interest.
Usual options for guide stars include natural objects (NGS), like astronomical objects, and artificially created guide stars (LGS), which are usually made with lasers.
This technology is already widely used in modern telescopes and is also going to be used in the upcoming European Extremely Large Telescope (ELT). ELT will have a massive 37 m collection diameter, being the largest optical telescope in the world once finished. It
promises to capture 13 times more light than largest optical telescopes existing today and deliver sharper images than what Hubble space telescope can resolve [8].
1.2 Research objective and the scope of the thesis
In this thesis, we will discuss the basic mathematics and ideas behind adaptive optics. We will focus on systems which use one guide star. The most important parts will be under- standing what happens to the light when it penetrates the atmosphere, how we can model it and how we can measure it using a Shack-Hartmann wavefront sensor. We will also study how the magnitude of the guide star and closed-loop integrator gain contribute to imaging quality using simulations.
1.3 Research methodology and arrangements
Majority of this work is based on a literature review. In the final section ”OOMAO simu- lation”, we run some simulations on Matlab based Adaptive optics simulator called Object- Oriented Matlab Adaptive Optics, or OOMAO for short, and discuss the results.
2 MATHEMATICAL PRELIMINARIES
Before discussing the mathematical methods behind adaptive optics, let us introduce the main related mathematical concepts. Here we introduce the basic ideas regarding random fields and ill-posed inverse problems. Random fields are needed when we study the atmo- spheric turbulence and trying to resolve the form of the perturbed wavefront at the wavefront sensor can be considered an ill-posed inverse problem.
2.1 Probability distributions
A probability distribution is a mathematical function describing the likelihood of random events and their likelihoods of happening. Probability distributions can be discrete or con- tinuous. A dice throw is a classical example of a discrete probability distribution. All the events are as likely to happen and so we have an even likelihood of 16 for every six cases.
However, in our case, it is more useful to look at continuous distributions. These have an in- finite amount of cases that can happen and the probability is thought of as what is the chance of the result being in any measurable set. They can be often characterized by probability densities f with limited integrals, where the likelihood of eventx∈[a,b]is calculated as
P(x∈[a,b]) = Z b
a
f(x)dx. (1)
Here we denotexas the random variable,aandbas the limits ofx(wherea<b) and f(x)as our probability density function, whereR−∞∞ f(x)dx=1. One of the most common continuous distributions is called Gaussian distribution or normal distribution. It can be defined as
f(x) = 1 σ√
2πe−12(x−µσ )2 (2)
whereµ∈Ris the mean (expected value) andσ>0 standard deviation. This is an important distribution for multiple reasons. These include it being tractable analytically, symmetry and ability to approximate other distributions given mild conditions and a large number of samples [4]. Due to these properties, it is often used to model-independent random variables with unknown distributions.
The showcased one-dimensional distribution can be extended to multiple dimension. Now for random vectorXXX = [X1,X2, . . . ,Xn]T, we have the probability density function as:
f(XXXrrr) = 1
(2π)n/2|Σ|1/2e−(XXXrrr−µµµ)TΣ−1(XXXrrr−µµµ)/2 (3)
whereµµµis an n×1 vector representing the expected values of random vectorXXX, Σ is the n×n variance-covariance matrix of XXX and XXXrrr ∈Rn×1 [1]. We denote this n-dimensional normal density byNn(µµµ,Σ). Whenn>1, these are considered multivariate Gaussian.
Now that we know what a multivariate Gaussian distribution is, we can talk about Gaussian random fields. A Gaussian random field is a collection of random variables{ξ(sss):sss∈Rn} such that given any choice of distinct values[(sss1), . . . ,(sssn)], vectorξξξ = [(ξ(sss1), . . . ,ξ(sssn)]is a multivariate Gaussian. Here ξ(sss)is a random variable and sss is a spatial variable used to index it [5].
2.2 Power spectrum
Power spectrum gives us an alternative method to analyse signals. By using the Fourier transform, we can transform signals from time to frequency domain. It is now possible to examine from which frequency components the original signal is made of and how much they contribute to it. Letxxx,ωωω∈Rn and define the n-dimensional Fourier transform as the following pair
(F f)(ωωω) = fˆ(ωωω) = Z
Rn
e−2πixxx·ωωωf(xxx)dxxx (4) and
(F−1f)(xxx) = f(xxx) = Z
Rn
e2πixxx·ωωωfˆ(ωωω)dωωω (5) wherexxxis the spatial variable andωωω is the spatial frequency. We will use the wave number κ
κκ=2πωωω instead of spatial frequency. This will simplify the form a bit and help us later.
From here, we can define the power spectrum of f(xxx)as the squared modulus of its Fourier transform
Φf(ωωω)≡ |fˆ(ωωω)|2 (6) As the power contained in the given frequency must be the same no matter if we interval in terms ofωωωorκκκ, we must haveΦf(κκκ)dnκ=Φf(ωωω)dnω, so
Φf(κκκ)≡(2π)−n|fˆ(κκκ)|2. (7) For stochastic processes we can use the modified definition
Φf(κκκ)≡ lim
V→∞
1 V Z
V
f(ωωω)exp(−iκκκ×xxx)dnx
2
. (8)
This power spectrum entirely determines the statistical properties of normally distributed random processes [11].
2.3 Inverse problems
First, we have to define what is an inverse problem. According to Neto [14] a classical example in the mathematical community is: ”Can one hear the shape of a drum?”. So, instead of trying to figure out an effect (sound) that is caused by an action (hitting a drum), we are trying to solve a problem where we know the result and are interested in what caused it. This is a lot harder to solve than the corresponding direct problem, and in this case it turned out that you are not able to solve the shape of a drum based on the sound it makes.
There are two drums with different shapes making the same sound.
A simple mathematical inverse problem that a lot of people have solved in their life is fit- ting a line or a curve to given points. If we take a very simple example of three points [(1,1),(2,2),(4,4)], it is quite obvious that the line connecting those points would simply be
f(x) =x.
Figure 1.Measurement points and a simple fitted line plotted
However, when we apply these kinds of problems in real life there are many sources of inaccuracy. This causes noise to the measurements and leads to uncertainty when we are
trying to solve the problem. Given that we would have measured that same line, but this time with some random noise, we could have been given a result of[(1,1.1),(2,3.3),(4,3.9)].
Figure 2. Noisy measurement points and two fitted lines plotted. Each line has their own error function which is minimized.
Now we can see that we can’t simply connect all the points with one single line. We would have to make some guess on how to minimize the error in the model and as a result, use some sort of a function to describe the error. Depending on what type of error function we use, the results would look slightly different. On Figure 2 the blue line is fitted using the least-squares method and the red with the minimum absolute error. As a result, we can see the two lines differ from each other. This is an example case of an ill-posed inverse problem.
These problems have more than one possible solution and as such are considered ill-posed.
We can also take this idea of noise a bit further. We can also consider how much does the error in the measurement affects the result of our evaluation. We can talk about well- conditioned and ill-conditioned problems/evaluations. In a well-conditioned problem, a small error in the measurement does not cause a big change to the results. If a small er- ror in the measurement causes big changes to the results we can talk about ill-conditioned
problems. A problem can inhibit both of these features. For example if we think of function f(x) = 1x and we are sampling f(x), small variations in measurements near zero cause big changes in x. On the other hand while sampling larger values, for example f(x)>1, the problem is well-conditioned (the opposite effect regarding noise is observed).
3 ADAPTIVE OPTICS
Here we introduce foundations of an AO system. It consists of a reference star, atmosphere, wavefront sensor, and a deformable mirror. The following section is mainly based on work by Hickson [11] unless otherwise stated.
A reference star is a bright object used to measure the perturbation caused by the atmosphere.
These can be divided into two categories, natural guide stars (NGS) and laser guide stars (LGS). NGS is the simpler option of these two as you don’t need any extra hardware to create them. Any sufficiently bright object within close proximity to the imaged subject should be proficient. The problem with this system is that the image quality decreases as a function of angular distance from the NGS and these guide stars can not be found everywhere on the night sky. To combat this issue LGS were created. By using a powerful laser, it is possible to create an artificial bright object anywhere in the sky to use as a guide star. One way to create a LGS is to use scattering from the molecules in the lower atmosphere to return light back to the telescope. The problem with this method is that it works better in denser air, meaning most of the returning light will be from the lower 15 km of the atmosphere. This means we are not capturing all the effects of the atmosphere and it can be observed as noise in the measurements. The second, more complex option, is to use a laser that is tuned to 589 nm, theD2resonance line of atomic sodium. It excites sodium atoms in the mesospheric sodium layer 90 km above the sea level. This way we can create the artificial star high up into the atmosphere, minimizing the noise caused by photons returning from the lower atmosphere.
Figure 3.Computer generated image of ELT and its LGS system in action. [9]
Next, the light from the reference star will travel through the atmosphere. On its way down, it will pass through turbulent air. This turbulence is caused on a larger scale by sunlight and the diurnal cycle. On a smaller scale it is caused by wind. This complex and unpredictable air movement gives rise to air pockets of different temperatures called eddies. Due to the
index of refraction of air being temperature sensitive, these pockets of air refract the light differently. This causes perturbation to the originally flat wavefront, as a result of optical path length (OPL) changing. OPL is the geometric length of the light and varies based on the refractive index of the medium passed through [12]. Now, as mentioned earlier, the turbulent air causes random air pockets with varying temperatures, and as such these air pockets have varying refractive indices. This means different OPLs occur at different parts of the wavefront. The result will be that the parts of the wavefront arrive at separate times (the wavefront is perturbed). As the formation of these air pockets can be considered random, the result is that the refractive index of the atmosphere is also random and so is the resulting phase of the wavefront [16].
Now that we have a bright enough reference star and its wavefronts are perturbated in the atmosphere, we can use a wavefront sensor to measure the perturbation that happens in the atmosphere. There are multiple types of wavefront sensors to choose from, including Shack- Hartmann sensor, curvature sensor and pyramid sensor. The most recent of these, pyramid sensor, uses irradiance patterns to measure the phase error. Curvature sensor on the other hand uses irradiance fluctuations to measure out wavefronts. Shack-Hartmann sensor, being the one we will be focusing on, uses a lenslet array and a detector plane to measure local slopes of the wavefront.
Figure 4.A 2d representation of the lenslet array found on Shack-Hartmann sensors
As visualised on Figure 4, the lenslet array shifts the focus point of the light beam according to the slope of the wavefront. So given a perfectly flat wave, the lenslet array would form a uniform grid on the imaging plane/detector. By measuring the difference between the actual focus point and the center point behind the lenslet, we can figure out the local slope of the wavefront in front of that lenslet. Using an array of these lenslets, we can have an idea of the overall shape.
Finally, we have to use that information to somehow correct the perturbation. To achieve this effect, so-called deformable mirrors are used. Deformable mirrors can physically de- form to correct the perturbation of an incoming wavefront. This way, once the wavefront is reflected off the mirror, it should be straight. However, due to measurement errors and limited deformability of the mirrors, there will always be some distortion left. There are multiple ways to achieve this deformation in the mirror. Some of the current technologies include piezoelectric, MEMS, segmented and adaptive secondary mirrors. MEMS or micro electro-mechanical systems are made using lithographic technology, the same technology semiconductor industry uses, giving it the highest amount of actuation points. However, this technique limits the size of the mirrors and their maximum stroke (how much they can de- form). Piezoelectric mirrors are based on the effect that piezoelectric elements deform when voltage is applied to them. Using actuators made from piezoelectric materials and a thin aluminized glass face-sheet connected to them, it is possible to bend the sheet by applying a voltage to the actuators. This technology has produced mirrors with 103actuators and sizes up to 30 cm. When we need larger mirrors, the two last methods are used. In a segmented mirror, the mirror is made of many smaller mirrors with three degrees of freedom: tip, tilt and height. This removes the size constraint, but it is possible to build single segments large enough where the response is too slow to correct atmospheric turbulence effectively.
The other option is called adaptive secondary mirrors, which uses magnetic actuators (voice coils) to deform a thin sheet used as a mirror. This sheet is originally in a convex or concave shape. Mirrors up to 8m in diameter have been made with this technology.
Figure 5.Visual representation of a deformable mirror [18, Figure 2.9].
Now that we have introduced how an AO system works, let us also introduce a way to evaluate its performance. Most common of these is called a Strehl ratio. It depicts the ratio of central intensities of an ideal diffraction-limited telescope and one where the perturbation
is taken into account. First, we define the diffraction-limited point spread function (PSF):
p0(r) =|F(χΩ)(r)|2=πD2 4λ2
2J1(πDr/λ) πDr/λ
2
(9) where p0(r)is the intensity of light at the focal plane as a function of distance from origin, Dthe diameter of the telescope,λthe wavelength of the ideal point source andJ1the Bessel function of first order. Bold letters denote vectors e.g r∈ R2. But as telescopes inside atmosphere are practically not only limited by diffraction, but also by the perturbation caused by the atmosphere. Letφ be the residual wavefront left after DM corrections. Now we can denote the combined PSF by pφ, so:
pφ(r) =|F(eiφχΩ)(r)|2 (10) Now, if we compare the central intensities of the diffraction-limited PSF and the one where the perturbation is taken into account, we have the Strehl ratio:
Sφ= pφ(0)
p0(0) (11)
As we can never reach higher intensities than what diffraction limits us to, the Strehl ratio is always between one and zero. It makes it easy to think of it as a ratio of how close to an optimal image we are getting [15].
3.1 Atmospheric turbulence
Turbulence appears almost always at atmospheric airflow. To understand why, we need to look at the study of fluid motion. Given a viscous fluid with an average velocity,vavg, and a characteristic size ofl, by changing the average velocity we can observe two states of distinct motion. At slow average speeds the flow is laminar: smooth and regular. This changes after some critical value and the flow turns turbulent. Turbulent motion is no longer smooth and regular, but unstable and random. This critical number is defined by non-dimensional Reynolds number
Re=vavgl
kv . (12)
Herekv is the kinematic viscosity of the fluid. For air this kinematic viscosity iskv=1.5∗ 10−5m2/sand if we also assume characteristic size ofl=10mand a velocity ofvavg=1m/s we get a Reynolds number of Re=6.7∗105. This is an relatively high Reynolds number with quite moderate assumptions, so it is safe to assume the airflow in atmosphere to be almost always turbulent [16, p. 58].
It is also important to understand the statistics of these temperature inhomogeneities, so a model for turbulence was created in 1941 by Kolmogorov and Obukhov [3]. They supposed that the turbulence is generated on a larger scale L0, from where it progresses to smaller vortices as the larger ones transfer their energy to the smaller ones. At some point, when the energy cascade reaches small enough inner scale l0, viscous forces will dissipate the energy into heat. The difference between the inner and outer scale is called inertial range.
Velocities in this model are considered isotropic (i.e. no preferred direction) and there is a well-developed relationship between energy and scale (stationary). As such, turbulence can be modelled as a stationary, isotropic Gaussian random field. When considering the turbulent energy in Fourier domain, the energy at wave numberκ∼2π/l(wherelis the scale) is:
E(κ)∝κ−5/3, (13)
which is Kolmogorov’s law. We can also write it in terms of a three-dimensional spectral densityE(−→κ). From here, we denote three dimensional quantities with an arrow and two dimensional by bold face. This is related to the one-dimensional case as such:
E(κ)dκ=E(−→κ)d3κ=4(−→κ)κ2dκ (14) Thus,
E(−→κ)∝κ−11/3. (15) As turbulence mixes air, turbulence cells of different temperatures called eddies are formed.
These eddies do not affect the formation or behaviour of turbulence, but instead are sim- ply carried by it. As a result, the power spectrum of temperature fluctuations follows Kol- mogorov’s law
φT(−→κ)∝κ−11/3. (16) Also, as the index of refraction is mainly defined by its temperature, the power spectrum of the index of refraction is given by Kolmogorov’s law as well:
φn(−→κ) =cκ−11/3, (17) where c is a proportionality constant. Now, we also want to consider the statistical structure of the refraction index in a spatial domain. With Wiener-Khinchine theorem we have
Cn(r) =4π Z ∞
0
sin(κr)
κr Φn(κ)κ2dκ. (18) This integral diverges, but we can use the structure function instead:
Dn(r) =8π Z ∞
0
1−sin(κr) κr
Φn(κ)κ2dκ
=8πc Z ∞
0
1−sin(κr) κr
κ−5/3dκ
=4πΓ(−5/3)cr2/3
(19)
The coefficient in equation (19) is often denoted asCn2, the index of refraction structure constant. So,
Dn(r) =Cn2r2/3 (20)
whereCn2is a function of height. Now by combining (17) and (19) we have Φn(~κ) = 1
4πΓ(−5/3)·Cn2· |κ|−11/3 '0.033·Cn2· |κ|−11/3
(21)
as the Kolmogorov spectrum [15]. However, there was some disagreement on what theL0 should be and measurements done later supported it being variable [2]. As such, von Karman proposed an update to the model with an variableL0[13]. If we add the outer scale proposed by von Karman, we have the von Karman spectrum as
Φn(~κ)'0.33·Cn2(κ2+κ20)−11/6, (22) whereκ0=2π/L0. As the power spectrum can be used to fully describe normally distributed random processes, it is possible to generate the Gaussian random fields to describe the spa- tially variable index of refraction with them. When simulating these atmospheric conditions, Taylors frozen flow hypothesis is often used. It suggests that the changes in the structure of the turbulence are relatively slow, so that it is enough to move the turbulence pattern with the speed of the wind to get a good approximation.
As a conclusion let us define Fried parameter as r0=0.185 λ2
R∞
0 Cn2(h)dh
!3/5
. (23)
Fried parameter is one way to define the amount of turbulence in the atmosphere. It depends on the wavelengthλand as such the wavelength should also be also stated when talking about it (although if not stated, it is often assumed to be 500nm). Fried parameter is extremely useful to us, as it can be used to define at what aperture size the telescope is being seeing limited vs diffraction limited. Apertures larger than the Fried parameter are limited by seeing at the chosen wavelength, whereas apertures smaller are essentially diffraction limited [7].
3.2 AO correction
As already mentioned, Shack-Hartmann wavefront sensor measures the slope of the wave- front at multiple points. More precisely, it measures the spatial gradient of the wavefront.
Each of these points where the gradient is measured is called a subaperture. These consists of
a lenslet and a detector plane. The detector planes are often CCD sensors, with a minimum of 2x2 but usually 4x4 or more pixels. To calculate the wavefront slope~son the subaperture
sss=k|xxxs|
f1 (24)
is used, wherek=2π/λ,xxxs is the location of the spot focused on the detector and f1is the focal length of the subaperture lens. Here we denote two dimensional vectors by bold face.
To estimate the spot location~xs, a centroid is calculated from the spot on the detector plane.
We can show that this centroid calculation is equivalent to computing the average wave front phase gradient over the subaperture:
sss(j) = Z
Ws j(xxx)∇φ(xxx,t)dxxx, (25) wheresss(j)is the measured slope of jth subaperture,∇φ(xxx,t)is the spatial gradient of wave- frontφ(xxx,t)andWs j is the jth subaperture weighting function. It is normalized as:
Z
Ws j(xxx)dxxx=1. (26)
Now by adding a random vectorsssn(j)describing all the sources of noise to equation 25, we have the measured slope
sssm(j) = Z
Ws j(xxx)∇φ(xxx,t) +sssn(j)dxxx. (27)
The estimate of φ(xxx,t)will be computed from the derivative information using a phase re- construction algorithm [16]. One common approach is to use a simple least-squares fit from slopes data. Other more specialized methods, like CuReD [17], have also been made. As with all ill-posed inverse problems, there isn’t one perfect way to solve this fit. This wave- front estimation is then used to drive the control voltages of the deformable mirror.
Given good conditions, this approximation of the wavefront is better the more subapertures the SH-WFS has. We can also increase the accuracy of single slope measurement by using more pixels per subaperture. However, if the guide star that is used is too dim and there isn’t enough light per subaperture, electronic/photon noise can start to dominate the actual signal, leading to inaccurate measurements. This is why it is important to balance both factors.
4 OOMAO SIMULATION
4.1 Enviroment
OOMAO or Object-Oriented Matlab Adaptive Optics is a Matlab toolbox designed to sim- ulate AO systems [6]. As the name suggests, it’s based on object-oriented programming paradigm and as such the elements of the systems are represented as classes. The notable classes used in this simulation are
• source, representing the science object and the guide star
• atmosphere
• telescope
• deformableMirror
• shackHartmann, the currently implemented wfs
These classes are linked as follows:
Figure 6.Class diagram of OOMAO [6, Figure 1].
This system allows us to control almost all parameters in the simulation. We can simulate natural or laser guide star/science object with varying magnitude and wavelength, multiple layers of atmospheric turbulence (each with their own parameters), and different sizes of telescopes.
4.2 Methods
First we define basic parameters used in this simulation. In all simulations a telescope with an aperture of 1 m and a sampling time of 2 ms is used. The WFS used a SH-WFS, with a 10x10 lenslet array having 10x10 pixels in each subaperture. NGS had a wavelength of 550 nm and science object (object we are imaging) has 806 nm. We had one turbulence layer at 5000 m, with a Fried parameter of 15cm at wavelength of 550 nm. This means we were well limited by atmospheric seeing instead of diffraction (see equation (23)). The turbulence was modelled using Von Karman spectrum with outer scale of 30 m. Wind speed was 10 m/s at the turbulence layer. All tests were run in closed loop (meaning the WFS measures wavefront residual after the DM) with an integrator controller.
We start by looking at the effect of the guide stars magnitude to the imaging quality. To achieve this, we varied the magnitude of the guide star between runs and within each run we optimized the integrator gain to represent the best quality we can achieve. As a metric for the imaging quality, Strehl ratio was used.
When this was done, we used the results to choose three different magnitudes that represent different levels of performance in the system: optimal, fading, worst-case. Then we locked the guide star magnitude, varying only the integrator gain. After that we took a look on how it affected the Strehl ratio.
4.3 Results and discussion
Before examining the results let us introduce the important parameters, magnitude and inte- grator gain. Magnitude is a unitless and logarithmic measure used to describe the brightness of astronomical objects. It is important to note here that magnitude works backwards from what might be commonly expected. The higher the magnitude of an object is, the dimmer it is. To give some reference, a very bright star has a magnitude of 1, and 6 is the typical limit for naked eyes under very good conditions [10]. Integrator gain on the other hand describes the strength of the correction we are making. Given a gain of one, we will deform the mirror
exactly as measured. However, with noisy measurements this often leads to overshoots and as such suboptimal correction. Now, as in closed-loop we are measuring the residual wave- front after the mirror, it is useful to do partial corrections (0.5 gain leads to DM deformation of half the measurement). This means we will approach the optimal state after each correc- tion and the speed which we approach it at is determined by the amount of gain. As a result, the problem of lower gains arises. They might be too slow to respond to the always changing conditions in atmosphere and that is why integrator gain should always be optimized.
Figure 7.Strehl ratio and optimal gain as a function of the guide star magnitude.
First, we measured the Strehl ratio as a function of the guide star magnitude using optimized gain, as demonstrated in Figure 7. We can observe a steady decrease of Strehl ratio and integrator gain from magnitude 6 to 8. Still, integrator gain decreases much faster as the system combats the growing amount of noise in the measurements. This leads to the system being less responsive and as such the Strehl ratio slowly decreases as the system is more and more behind the actual wavefront. Somewhere around 8.5 of magnitude, the measurements become only noise. This is reflected in optimal gain dropping to zero (meaning DM is no longer used). After that, the Strehl ratio becomes mostly stable, as it’s only affected by the turbulence in the atmosphere. We can use this information to see how much benefit we
gained from the AO system. Without AO, the Strehl ratio was about 0.4. With AO, we had Strehl ratios of over 0.85 and even in what could be considered last usable point, it was still approximately 0.6. We clearly made some drastic improvements to the imaging quality.
Now we can define the three different levels of performance in the system. Here we chose magnitude of 6.5 as optimal, 8 as fading and 9 as the worst case.
Figure 8. Strehl ratio as a function of the integrator gain at optimal conditions. Here we are using a guide star with a magnitude of 6.5
Starting with the Strehl ratio as a function of gain at a guide star magnitude of 6.5 (see Figure 8), we observe Strehl ratio generally increasing as the gain increases. This would mean the WFS receives enough light to make accurate measurements and the higher gain allows it to respond faster to the changes in the atmosphere. The majority of performance gain happens from increasing the gain from 0 to 0.4. From 0.6 gain forward there is no benefit to increasing the gain. It could be quite possible that as the wind speed is quite modest, the system is fast enough to respond to the changes in atmosphere even at mediocre gains. Here we reached a maximum Strehl ratio of about 0.8.
Figure 9. Strehl ratio as a function of the integrator gain at fading conditions. Here we are using a guide star with a magnitude of 8.
Next we change the guide star magnitude to 8 (see Figure 9). Now, from zero to about 0.3 the Strehl ratio increases fast. This means that there is some accuracy to the measurements.
But as we keep on increasing the gain, it becomes obvious that there are large amounts of noise as well. This leads to the Strehl ratio slowly decreasing as we pass the 0.4 gain point.
We reach a peak Strehl ratio of about 0.7 with a gain of 0.2, and it decreases almost back to the same levels from where we started without adaptive optics (0.4) when the gain reaches 1.
Figure 10.Strehl ratio as a function of the integrator gain at worst case conditions. Here we are using a guide star with a magnitude of 9
Finally, we have the worst case performance with a magnitude of 9 (see Figure 10). Here we observe a different kind of behaviour compared to the two cases before. Any increase in gain just lowers the Strehl ratio and at a gain of just 0.1, we have reached a Strehl ratio of almost 0. At this point the wavefront measurements are just noise, meaning when we base the deformation of mirrors on them, we only add extra perturbation to the wavefront. At this point the AO system becomes disadvantageous.
As a conclusion, it is now obvious that the magnitude of the guide star will make a change to our imaging quality up to a certain point. However, magnitudes smaller than 6 presented no improvement or degradation of our performance. This might be due to the fact that we simulated a natural guide star instead of a laser created. It could be quite possible, that after some point brighter laser guide stars degrade the performance of the system, as there are now more photons returning from the lower atmosphere (as opposed to NGS, where all light comes from outside the atmosphere). This, however, was not simulated in this work and as such is just an idea. It could be a subject of interest in future work.
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1 Simple inverse fit . . . 10
2 Noisy inverse fit . . . 11
3 ELT and LGS visualisation . . . 13
4 Shack-Hartmann sensor lenslets . . . 14
5 Deformable mirror . . . 15
6 OOMAO class diagram . . . 20
7 Strehl ratio and optimal gain as a function of the guide star magnitude . . . 22 8 Strehl ratio as a function of integrator controller gain at optimal conditions . 23 9 Strehl ratio as a function of integrator controller gain at fading conditions . 24 10 Strehl ratio as a function of integrator controller gain at worst-case conditions 25
