Joint-optimization of phase-diversity and adaptive optics

Fast & Furious: a potential wavefront reconstructor for extreme adaptive optics at ELTsVisa Korkiakoski and Christoph U. KellerLeiden Observatory

Niek DoelmanTNO Science and Industry

Raluca Marinica and Michel VerhaegenDelft Center for Systems and ControlI will talk about a new wavefront sensing technique that we have recently been developing at the Leiden Observatory.1OutlineScope: real-time WF sensing for XAO at ELTsWhy to investigate new wavefront sensing methods?Issues with roof sensorFast & FuriousA new focal plane wavefront sensing algorithmBrief introduction & current statusHere is the outline. At first, I try to give some reasons for why are looking for totally new ways to make the wavefront sensing at systems like EPICS.Then, I will briefly introduce the new algorithm we have developed. We call it the Fast & Furious, for obvious reasons. 2Issues with roof sensorRequirements for EPICS XAO WFS driven by the need for very high contrastA possible option is a roof-sensor

Detailed simulations: see (Korkiakoski et al 2010)So far, the best known choice for the EPICS final stage XAO wavefront sensor is the roof sensor. (See the illustration: the beam is split, and four pupil images are recorded at detectors.) The choice is driven by the need for a very high contrast, especially at low spatial frequencies, and the roof sensor is shown to perform well. We have done extensive and detailed simulations during the phase-A study, and verified this. Here is an example how the residual wavefront looks like during the AO loop. The simulations include, for instance, full end-to-end model of both the 1st and 2nd stage AO loops, different chromaticity effects, wind-caused jitter and telescope spiders and segment errors. The obtained contrast is close to the theoretical photon-noise limit.3Issues with roof sensorFulfills the requirements, but has some challenges:NonlinearityFast WF reconstruction still to be solvedShould we fix the nonlinearity by modulation?

Residual WF (6 /D mod)RS measurement (modulation 6 /D)NO!However, the roof sensor has two major issues: it is non-linear, and so far no fast WF reconstructors exist. The existing fast WF reconstruction methods have been designed for linear sensors, and in particular for slope sensors like SH-WFS. However, the roof sensor is really different from those. It is well known that the linearity of the roof sensor can be increased by modulation, i.e. blurring the signal by rotating the intermediate focus. This comes at the cost of the sensitivity. Here is example how the residual WF would look like when using slightly modulated roof sensor. Clearly, large low-order errors remain. They are not corrected, because the sensor doesnt see them. Here are shown what the sensor actually records. These errors would decrease the contrast by a factor of 5-10, which is not acceptable.4Issues with roof sensorThe performance does not scale as you expectSpiders & windshake at E-ELT + nonlinear sensor can cause unexpected results

Residual WF (6 /D mod)RS measurement (modulation 6 /D)And there is one more thing we learned from these simulations. The performance of the AO systems doesnt scale as you would expect. Simple analytic models predict only slight performance degradation when you go from zero to 6 l/D modulation. We are dealing with non-linear sensors, and the E-ELT with spiders and lots of wind-shake. E.g., see here the island modes caused by masked sub-apertures. So, if you study a system with 80x80 WF sampling, you cannot conclude that a same setting would do fine at a resolution of 200x200.5Issues with roof sensorTraditional WF reconstruction requires a command matrix having a size of 55000 x 30000Fast reconstruction requires approximationsSo far, no detailed performance comparisons have been shown

Zonal interaction matrix: 3% non-zero elementsCommand matrix:0% non-zero elementsSo, when using the roof sensor, we think it is necessary to model accurately its full diffraction properties. This results in a command matrix with a huge size. Somehow, the fast roof sensor reconstruction would need to grasp the full properties of this matrix-multiplication without making too much destructive assumptions. So far, we dont know how to do that.6Fast & FuriousWe have been studying an alternative wavefront sensing option: no WFS at allAO residual< 1 rad rmsDMFocal planeScience cameraFast & Furious algorithmControl & DM fittingCorrected WFImagesWF estimatesDM commandsIn the meantime, we have been studying alternative options to do the wavefront sensing: the fast and furious algorithm. The idea is that we get a pre-corrected beam from 1st stage AO, and it gets reflected from the XAO DM. The beam is not split, but imaged at the science camera. The recorded images are fed to the Fast & Furious algorithm. The algorithm computes a wavefront estimate. Then, we calculate the corresponding DM commands and apply dynamic control. We run the algorithm iteratively, in a closed loop, until convergence.7Fast & FuriousHow does F&F differ from other focal plane methods?Linear approximationCut down the complexityUse of pupil symmetries & phase diversityExtremely fast + smaller role for the diversityIterative WF correctionApplicable to systems with a deformable mirrorThe first question might be: how does the Fast & Furious differ from the other focal plane wavefront sensing methods? There are three main features. First, it uses a linear small-phase approximation to reduce the complexity, in a same way actually several other algorithms do. Second, to make it possible to retrieve the phase from focal-plane images, it is using, in addition to the phase diversity, the knowledge that pupil function is symmetric. This makes it possible to make the algorithm extremely fast, and the symmetry assumption also means we need to rely much less on the phase-diversity information to do the reconstruction. Thirdly, the algorithm has an iterative nature, which makes it well suited for applications with a deformable mirror.8Fast & FuriousAll the needed pieces have been around, we combine earlier resultsSmall-phase solution (Gonsalves 2002)Approximate complex amplitudes linear function of WFUse the symmetry properties of pupilMuch of the WF information can be obtained from a single PSF measurement, use diversity for the restIterative phase-diversityUse the adaptive optics DM change as a diversity (Gonsalves 2010)It should be noted here that all the pieces we use have been around for a while. We have managed to combine them, with small changes, to give excellent results.The main source for inspiration is the small-phase solution by Gonsalves. The paper is only 2 pages long, but most of the math used in Fast & Furious is there. It was that paper that first proposed using the linear approximation, phase diversity and pupil symmetries to do the phase retrieval.The second piece was also published by Gonsalves. The idea is the use the change in the AO system DM as a phase-diversity.In the Fast & Furious, we combine these two ideas.

9Fast & Furious: principleAn example: consider a sinusoidal wavefront

Wavefront (rad)See the math at (Keller et al 2012), (Korkiakoski et al 2012)I dont go here to details of the math, it can be found in our most recent publications. Instead, I try give some ideas how it works.As an example, consider a sinusoidal wavefront. 10Fast & Furious: principleMeasured PSFAn example: consider a sinusoidal wavefront

Wavefront (rad)See the math at (Keller et al 2012), (Korkiakoski et al 2012)

It will produce a image like that.11Fast & Furious: principleMeasured PSFAn example: consider a sinusoidal wavefront

Wavefront (rad)See the math at (Keller et al 2012), (Korkiakoski et al 2012)

??It is really not necessary to consider the wavefront as a sum these kind of sinusoids, when you study the math behind the Fast & Furious. However, I hope that it will give some insight to see how we can reconstruct the wavefront from such an image.12

See the math at (Keller et al 2012), (Korkiakoski et al 2012)Fast & Furious: principleMeasured PSFThe main lobe and 1st side lobes are accurately modeled by a linear approximation

The main lobe of the PSF, and the 1st side lobes are accurately described by a modeling the complex amplitudes as a linear function of the phase. 13

Linear approximationHere is the linear approximation of the PSF. Linear in the sense that complex amplitudes are a linear function of phase.14Measured PSF

The lobes further have a negligible contribution to the PSFHere you can see again the PSF you would measure. Almost the same, the only difference you can see are the small secondary lobes. With a small phase, the 2nd and further side lobes have a negligible contribution to the PSF. Those small secondary lobes, in fact, are caused by the the 2nd order quadric approximation of complex amplitudes.15

See the math at (Keller et al 2012), (Korkiakoski et al 2012)Fast & Furious: principleMeasured PSFThe main lobe and 1st side lobes are accurately modeled by a linear approximation

Pay attention to the conservation of the energy!The lobes further have a negligible contribution to the PSFHere, we also need to pay a bit attention to the conservation of the energy, though. When you look at the formula, it is obvious that the normalization is not explicitly included there. This issue has not been discussed by prior publications, but it is addressed in the recent paper by Christoph Keller. The best way to do follows directly from the approximations of the 2nd order effects, if you continue the Taylor expansion of complex amplitudes to quadric terms. The modification is small, but we found it to be very helpful. What we do, is that we scale the PSF estimate to correct Strehl ratio by adjusting only the constant term.But, as you see, this equation, as such, is still not very useful for phase-retrieval. To get the inversion done really fast, we need other simplicity assumptions.16See the math at (Keller et al 2012), (Korkiakoski et al 2012)Fast & Furious: principleThe next idea: use symmetryMeasured PSF

The pupil function is symmetric. Can use make use of that?We follow the Gonsalves approach, and use the symmetry relations. The aperture function is symmetric. To be more precise, it is even. Very interestingly, this turns out to be an important factor in simplifying the phase retrieval.17See the math at (Keller et al 2012), (Korkiakoski et al 2012)

Fast & Furious: principleEven PSF componentOdd PSF componentMeasured PSF

The next idea: use symmetryThe Fourier transform of even functions are real. Any function can be presented as a sum of even and odd components. The FFTs of even components are real, the FFTs of odd components are imaginary. When you do the math, you will find its necessary to make the analysis considering the odd and even parts of various quantities separately.Regarding our example, consider that that PSF is broken into even odd parts. This is a simple pixel-wise operation and takes no time.18

Also the wavefront can be broken to odd and even parts:Even PSF componentOdd PSF componentFast & Furious: principle

How do the odd/even PSF parts relate to odd/even WF parts?In the same way, also the wavefront can be broken into odd and even parts. The plots show the radial cuts of the sinusoid used in this example. The blue plot is a sum of red and green ones.The question is: can we find useful relations between the broken unknown odd/even WF parts and the measured odd/even PSF parts.19Fast & Furious: principle

The odd WF part can be reconstructed by a single FFT using the odd PSF Odd PSF componentSee the math at (Keller et al 2012), (Korkiakoski et al 2012)

1 FFTThe answer is: yes, we can. When you look at the math, it turns out that just from a single odd PSF part, it is possible to accurately reconstruct the odd WF part. The accuracy depends mainly on how big are the WF aberrations and what is the spectral distribution. Low-order modes can be corrected, at least to some extent, with errors lower than 1 rad rms error.Here, the sinusoidal aberration, with an rms of 0.25 rad, is almost perfectly reconstructed.In a typical random Kolmogorov-style wavefront, the amounts of odd and even parts are about the same. Thus, a single FFT alone can reconstruct half of the wavefront.20Fast & Furious: principle

Even WF is more tricky: using even PSF, we get only absolute values of the even WF in Fourier space. For the sign, use phase diversity.

Even PSF,Previous New odd PSFNew even PSF

Applied DMcorrection2 FFTsWith even modes, the situation is slightly more tricky. When looking at the math, it turns out that only the magnitude of the even WF, in Fourier space, can be detected based on a single measurement. To determine the sign, we need to use phase-diversity. So, we need one previous PSF, the applied DM change, that acts as a phase diversity and the new PSF. The DM change is such that it best compensates the current WF error estimate. With the Fast & Furious, it takes 2 FFTs to reconstruct the even wavefront.21Fast & Furious: principle

The real FFTs can be combined. In total, we need 2 complex FFTs.The other can be replaced by a small-kernel convolution.

Even PSF,Previous New odd PSFNew even PSF

Applied DMcorrection

2 FFTsWhen you consider the whole iteration step, it can be seen that the purely real-valued FFTs can be combined. So, in total, we only need 2 complex FFTs per iteration step. And the other FFT, used to transfer diversity shape to Fourier space, can actually be replaced as a small-kernel convolution.

22Fast & Furious: computationConventional reconstruction: O(N2), depends on the actuator & subaperture count Fast & Furious: O(N log N), depends on valid detector areaCan be implemented at an EPICS scale with current technology

Obviously, the reason why I am presenting the algorithm here, is that it is computationally extremely efficient, if you have a lot of degrees of freedom to correct the wavefront.The conventional reconstruction scales as N squared, depending on actuator and subaperture count.Fast & Furious scales as N log N, since most of its computation are the FFTs. The FFTs are done on the images recorded at the detector, so their size depends in principle on the detector resolution, but in practice also on the desired WF sampling. These requirements are small enough, so that we could implement the algorithm at EPICS scale with current technology.23Fast & Furious: resultsThe algorithm has been experimentally verifiedLow order correction (DM 37 actuators)Static phase: correct only non-common path aberrationsSee (Korkiakoski et al 2012), (Keller et al 2012)

SR: 0.66SR: 0.36SR: 0.79SR: 0.80SR: 0.3The recorded PSF with F&FAt the Leiden Observatory, we have carried out experiments to validate the algorithm. We used a low-order system with 37 actuators, and we considered only a static case, where the algorithm compensated only the non-common path aberrations. We saw that the Strehl gets from 0.3 to 0.8 in just a few iterations.See our publications for more details.24Fast & Furious: resultsThe performance exactly matches the simulations

SR: 0.66SR: 0.36SR: 0.79SR: 0.80SR: 0.3

Required convergence time as a function of noiseThe experimental results correspond exactly what we would anticipate based on simulations. The algorithm is robust also at relatively high noise levels.25Fast & Furious: summaryChallengesCannot deal with WF errors > 1 rad rms Atmospheric evolution can cause problemsWorks only with monochromatic lightFuture developmentsDealing with coronagraphs / camera saturationExperimental validation with high-order modes (SLM tests at Leiden Observatory, HOT at ESO, FFREE at IPAG)However, the algorithm is still just a promising idea, a lot of work is needed before we can consider the implementation. We have three major challenges. The biggest is that the algorithm cannot deal with errors higher than 1 rad rms. Many other WFSs, like the roof sensor, saturate outside the dynamic range. F&F, however, fail totally. Therefore, it might be restricted only to longer wavelengths. The second major problem is that the atmospheric evolution can disturb the diversity techniques. This probably will mostly hurt the higher-order modes, so we might have to consider correcting only the low spatial frequencies. Finally, the algorithm works only with monochromatic light. It might be possible to compensate this by specialized optics, though.In the near-future, we plan to continue developments with issues involved in coronagraphs, and also to conduct more validating experiments at Leiden and with our collaborators.

26Thank you for your time!Issues with roof sensorAre there simple ways for fast WF reconstruction for non-modulated roof sensor?

Signal measured when an actuator pushed