ADAPTIVE OPTICS
Adaptive Optics is a general technology for correcting images of objects seen through inhomogeneous medium. Originally developed by astronomers and the military to study objects outside the earth’s atmosphere, the technology now has applications in many different fields, from biological imaging to thermonuclear fusion.
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TABLE OF CONTENTS

How do you Build an AO System?

When my group started building AO systems in the early 1990s, there were very few components available to build an AO system and we had to build the deformable mirror, wavefront sensor and even the high speed computer ourselves. On the plus side, it did give us a lot of experience and this also meant that we were able to make many different types of deformable mirrors with different geometries, including some new designs. The original system ( called “ChAOS”) now resides in the Adler Planetarium in Chicago as a historical exhibit. It still looks good.

Making the Deformable Mirror

Our deformable mirrors were built using tubes of piezoelectric material about 1 inch long. These tubes change their length by a few microns when a voltage is applied across the inner and outer surfaces of the tube. A glass ball is used to interface the PZT tube with the faceplate. The actuators  were assembled in a baseplate and glued together so that the tops of the  glass spheres were all in a plane. A thin flat faceplate 1 mm was held in a vacuum chuck and glued onto the actuators, producing a deformable mirror than was flat to about ½ micron and could be moved over a range of 4 microns. Making these deformable mirrors is well within the capabilities of a skilled amateur and full details of how these mirrors are made is given in Mike Smutko's paper. A modest deformable mirror can be built for a few hundred dollars. Even lower cost mirrors can be made from bimorph plates.  Newer devices, using MEMS technology, can be bought ready-made for a few thousand dollars from ThorLabs.  A photograph of a collection of our DMs is shown below. These have 91 or 201 actuators. The best DMs that we make use a quasi-hex geometry, which reduces waffle pattern noise.                      
                                                                                                                     
Driving all these deformable mirrors is probably harder than making the mirror. Each actuator for either piezo, bimorph or MEMS device requires a few hundred volts to operate, and amplifiers operating at these voltage are both expensive and power hungry. However,the piezo actuators are built from a ceramic material and are basically capacitors. This enabled us to use a multiplexed drive system in which a single high voltage amplifier drove up to 16 actuators. It took 5 microseconds to charge each piezo so that the shape of the mirror could be updated in 100 microseconds.
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Building the Wavefront Sensor

choosing the Detector

  In this section I will assume that you have built a Shack Hartmann wavefront sensor that allows you to image the DM actuator pattern accurately onto the lenslet array and that you have a two dimensional detector to image the spots. After some processing of the CCD data you will end up with a series of measurements of the average slope of the wavefront across each subaperture. This is called the slope vector, although, in almost all calculations, what we actually need is the phase difference across the subaperture, which is the slope times the subaperture size, because most ways of reconstructing the wavefront from this data rely on a model relating these phase difference measurements to the phase of the wavefront at the edges of the subaperture.

The first step is to define the phase differences in terms of the phase points. The figure left shows one such arrangement, called the Fried geometry, often used with the Shack-Hartmann sensor. In this model, the first x phase difference is given by  0.5(2+5) - 0.5(1+4) where the integers correspond to actuator positions in the diagram. Different AO systems may have different geometries, but we can always define all the slopes measured by the wavefront sensor in terms of the actuator positions as a matrix equation:
                             
                                 
Where s is a vector containing a list of all the phase differences,  phi are the phases and A is a very sparse matrix called the geometry matrix. There exists another  inverse matrix, usually called A+, which is a non sparse matrix relating the phases to the phase differences:  
                    
This matrix gives us a recipe for calculating the phases from the phase differences - we take all the phase differences that we measure and weight each slope with a number that depends on the phase position j and the slope number i:
                                                         
There has been considerable discussion, often heated, over what the “best” coefficients are, but these differences are often only important when the source is faint and we are trying to optimize the signal/noise of the system. We should note that most Gradient and curvature wavefront sensors do not sense all possible modes. For instance, the Fried geometry, discussed above, cannot measure the so-called "Waffle mode".  The Waffle mode occurs when the actuators are moved in a checkerboard pattern( all white squares change position while the black squares remain stationary). This pattern can build up over time, it is reduced by using different actuator geometries or filtering the control signals so as suppress this mode. We should also note that the  geometry model is only an approximation to the  wavefront. Even if there the actuator positions reflect the true positions of the wavefront at these points, there are high frequency components in the wavefront that cause system errors. This error ( usually called "aliasing" ) can be reduced by proper design but introduces an additional term into the error budget.
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We need the work out an inverse to the geometry matrix.  Starting with the original matrix equation relating slopes to phase points, we note that A is not square ( there are about twice as many slopes as phase points) so that we cannot invert A directly. If we multiply both sides by A transpose we get a square matrix A transpose A and, following the steps shown left, we can at least write down an equation of the phase points in terms of the slopes. This reconstruction matrix is usually called the A+ matrix.

We have, of course, still not shown that we can invert A, and, indeed if you try this the inversion will fail. To understand why, and what to do about it, we should look at a simple 3x3 matrix drawn in the last section. The first x slope is given by a Fried geometry as 0.5(2+5) - 0.5(1+4). If we add a constant to either (2 and 4) or (1 and 5) the resulting slope is unchanged, even though the shape of the wavefront is very different. This is just an artifact of the original equation but is the reason for the "Waffle mode"- you get the same slopes if you piston all the odd or all the even phase points. You can overcome this  mathematically for the inversion problem either by setting one odd and one even point to zero, or (better) by setting the sum of all the odd points,and the sum of all the even points to zero, the so-called minimum norm solution. The mode is still undetected and can build up with time, but we will discuss this issue in the section after next.

 The first thing to do is to define how your actuator positions produce the slopes you hope to measure.
If you use Mathematica you can define this matrix, called the geometry matrix, for an m x n array of points as follows, this adds in the extra two rows needed to provide the minimum norm solution.

         

We will try this for a 3x3 actuator matrix, setting m=3 and n=3.
The geometry matrix produces an ordered list of slopes from the phases at the actuator positions.
In this example, the geometry matrix, A,  looks as follows:

                                         

The first row is the first phase difference. The actuator positions run from 1 to 9 left to right along the top, so this matrix says that the first x phase difference is given as 0.5(2+3)-0.5(1+4), where the integer numbers are the actuator positions. There are 4 x slopes and then four y slopes. The last two rows at the bottom add up the phases of all the odd phases and all the even phases and represent the unobserved Waffle mode terms. The matrix A transpose A is given is Mathematica as

                                      

and the final A+ matrix is given by

In this A+ matrix each row represents a phase point which is a sum of the 8 slope values multiplied by a weighting function. The last two columns are the waffle mode pistons which are set to zero and not used. Although we have not shown it here, this is the least squares estimator of the phases from the phase differences. You must be careful about numbering of your actuator positions and phase differences, so that the geometry matrix you use to calculate the least squares solution corresponds to the actuator positions. Measured phase differences and phase difference measurements round the edges and corners should be treated slightly differently, but this example illustrates the general principles of reconstruction.

If you have an adequate number of pixels/subaperture, you can (in principle) simply poke up each actuator in turn and measure the resulting phase differences, thus measuring the geometry A matrix rather than calculating it.  The measured A matrix can then be used to calculate the reconstructor M matrix. This turns out to be trickier than it sounds and, if you are starting off in AO, it may be faster to put in the required opto-mechanical adjustments and simplify the maths.

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 Noise Gain

Once you have the A matrix, you can work out the noise gain of the system. We can show that the noise gain for the least squares A+ reconstructor is given by:

                                                 Noise gain = Tr[Inverse[Transpose[ae].ae]]/(m x n)

Adaptive Optics as a Feedback Loop

It is important to realize that we are actually building  a sampled feedback loop drawn, outlined below. The wavefront sensor attempts to measure the slopes of the difference between the atmospheric wavefront and deformable mirror, integrated over the subapertures and over a sample time period t. This slope vector is then multiplied by a matrix M to obtain an updated vector ( or map ) of the difference between the wavefront and the positions of the DM actuators. M can be one of a large number of different matrices from sparse iterative matrices to dense matrices based on the statistics of the wavefront.


               

The reconstructor matrix M provides error signals to the actuator controller and can also control the spatial smoothing of the deformable mirror. The most common reconstructor used in adaptive optics probably the least squares reconstructor . This minimizes the fitting error between the atmosphere and DM but assumes that there is no correlation between the phases of adjacent actuators. The lack of spatial smoothing has been a serious criticism of this reconstructor.  Atmospheric turbulence has the form of a fractal and has much more power at large spatial scales so that the idea that the atmospheric wavefront can have big swings between adjacent phase points is physically unrealistic. An enormous amount of time and effort has gone into working out the optimum M matrix assuming given atmospheric wavefront statistics. We should note however that what we actually measure is the difference between the atmosphere and the DM.  This naturally takes out the high power at large spatial scales (in fact that is how AO works) and the correct statistics are not those of the atmosphere  but those of , which are, very approximately, uncorrelated between actuator positions. The least square solution is therefore a better solution than would first appear.  Whatever M matrix is used, the new phase differences between atmospheric wavefront and deformable mirror are fed to the control loop of each actuator. This controller provides the required actuator position. We often use simple first order difference equation for this step.

                                       

where g is the loop gain. This says that we just add a weighted value of the new difference measurement to each of the original actuator positions to  obtain a new position for the next sample.

    The bandwidth of the system is controlled by the sample frequency and the loop gain of the system. For the least squares reconstructor, the servo bandwidth is given by    and the noise gain of the system by  , where   is proportional to the measurement noise of the wavefront sensor. We should note that  the system noise gain increases rapidly as g approaches 2 and the servo becomes unstable. For this reason the loop gain is rarely set above one.
 
The approach used in ChAOS was to have a set of precomputed M matrices, each one optimized for a range of different observing conditions (source brightness and atmospheric seeing) and to change this matrix and the servo gain factor independently, by trial and error, so as to obtain the best system performance.

There exist general trades between how much computing we need to do (M can be sparse) and the sample frequency. Under conditions of high turbulence, significant scintillation exists, including singularities in the wavefront and different approaches must be used.

Non-Common Path Errors