Off-axis point spread function reconstruction for single conjugate adaptive optics.

Modern giant segmented mirror telescopes (GSMTs) such as the Extremely Large Telescope, which is currently under construction, depend heavily on adaptive optics (AO) systems to correct for atmospheric distortions. However, a residual blur always remains in the astronomical images corrected by single conjugate AO (SCAO) systems due to fitting and bandwidth errors, which can mathematically be described by a convolution of the true image with a point spread function (PSF). Due to the nature of the turbulent atmosphere and its correction, the PSF is spatially varying, which is known as an anisoplanatic effect. The PSF serves, e.g., as a quality measure for science images and therefore needs to be known as accurately as possible. In this paper, we present an algorithm for PSF reconstruction from pupil-plane data in directions apart from the guide star direction in an SCAO system. Our algorithm is adapted to the needs of GSMTs focused on estimating the contribution of the anisoplanatic and generalized fitting error to the PSF. Results obtained in an end-to-end simulation tool show a qualitatively good reconstruction of the PSF compared to the PSF calculated directly from the simulated incoming wavefront as well as stable performance with respect to imprecise knowledge of atmospheric parameters.

Robust and bias-free localization of individual fixed dipole emitters achieving the Cramér Rao bound for applications in cryo-single molecule localization microscopy.

Single molecule localization microscopy (SMLM) has the potential to resolve structural details of biological samples at the nanometer length scale. Compared to room temperature experiments, SMLM performed under cryogenic temperature achieves higher photon yields and, hence, higher localization precision. However, to fully exploit the resolution it is crucial to account for the anisotropic emission characteristics of fluorescence dipole emitters with fixed orientation. In case of slight residual defocus, localization estimates may well be biased by tens of nanometers. We show here that astigmatic imaging in combination with information about the dipole orientation allows to extract the position of the dipole emitters without localization bias and down to a precision of 1 nm, thereby reaching the corresponding Cramér Rao bound. The approach is showcased with simulated data for various dipole orientations, and parameter settings realistic for real life experiments.

Retinal adaptive optics imaging with a pyramid wavefront sensor.

The pyramid wavefront sensor (P-WFS) has replaced the Shack-Hartmann (SH-) WFS as the sensor of choice for high-performance adaptive optics (AO) systems in astronomy. Many advantages of the P-WFS, such as its adjustable pupil sampling and superior sensitivity, are potentially of great benefit for AO-supported imaging in ophthalmology as well. However, so far no high quality ophthalmic AO imaging was achieved using this novel sensor. Usually, a P-WFS requires modulation and high precision optics that lead to high complexity and costs of the sensor. These factors limit the competitiveness of the P-WFS with respect to other WFS devices for AO correction in visual science. Here, we present a cost-effective realization of AO correction with a non-modulated P-WFS based on standard components and apply this technique to human retinal in vivo imaging using optical coherence tomography (OCT). P-WFS based high quality AO imaging was successfully performed in 5 healthy subjects and smallest retinal cells such as central foveal cone photoreceptors are visualized. The robustness and versatility of the sensor is demonstrated in the model eye under various conditions and in vivo by high-resolution imaging of other structures in the retina using standard and extended fields of view. As a quality benchmark, the performance of conventional SH-WFS based AO was used and successfully met. This work may trigger a paradigm shift with respect to the wavefront sensor of choice for AO in ophthalmic imaging.

Phase-contrast THz-CT for non-destructive testing.

A new approach for image reconstruction in THz computed tomography (THz-CT) is presented. Based on a geometrical optics model containing the THz signal amplitude and phase, a novel algorithm for extracting an average phase from the measured THz signals is derived. Applying the algorithm results in a phase-contrast sinogram, which is further used for image reconstruction. For experimental validation, a fast THz time-domain spectrometer (THz-TDS) in transmission geometry is employed, enabling CT measurements within several minutes. Quantitative evaluation of reconstructed 3D printed plastic profiles reveals the potential of our approach in non-destructive testing of plastic profiles.

Nonlinear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber-Kaczmarz iterations.

Accurate and robust wavefront reconstruction methods for pyramid wavefront sensors are in high demand, as these sensors are planned to be part of many instruments currently under development for ground-based telescopes. The pyramid sensor relates the incoming wavefront and its measurements in a nonlinear way. Nevertheless, almost all existing reconstruction algorithms are based on a linearization of the model. The assumption of a linear pyramid sensor response is justifiable in closed-loop adaptive optics (AO) when the measured phase information is small, but, depending on the system, may not be feasible due to unpreventable errors such as non-common path aberrations. In order to solve the nonlinear inverse problem of wavefront reconstruction from pyramid sensor data, we introduce two new methods based on the nonlinear Landweber and Landweber-Kaczmarz iterations. Using these algorithms, we experience high-quality wavefront estimation, especially for the non-modulated sensor, by still keeping the numerical effort feasible for large-scale AO systems.

Convolution- and Fourier-transform-based reconstructors for pyramid wavefront sensor.

In this paper, we present two novel algorithms for wavefront reconstruction from pyramid-type wavefront sensor data. An overview of the current state-of-the-art in the application of pyramid-type wavefront sensors shows that the novel algorithms can be applied in various scientific fields such as astronomy, ophthalmology, and microscopy. Assuming a computationally very challenging setting corresponding to the extreme adaptive optics (XAO) on the European Extremely Large Telescope, we present the results of the performed end-to-end simulations and compare the achieved AO correction quality (in terms of the long-exposure Strehl ratio) to other methods, such as matrix-vector multiplication and preprocessed cumulative reconstructor with domain decomposition. Also, we provide a comparison in terms of applicability and computational complexity and closed-loop performance of our novel algorithms to other methods existing for this type of sensor.

Efficient reconstruction method for ground layer adaptive optics with mixed natural and laser guide stars.

The imaging quality of modern ground-based telescopes such as the planned European Extremely Large Telescope is affected by atmospheric turbulence. In consequence, they heavily depend on stable and high-performance adaptive optics (AO) systems. Using measurements of incoming light from guide stars, an AO system compensates for the effects of turbulence by adjusting so-called deformable mirror(s) (DMs) in real time. In this paper, we introduce a novel reconstruction method for ground layer adaptive optics. In the literature, a common approach to this problem is to use Bayesian inference in order to model the specific noise structure appearing due to spot elongation. This approach leads to large coupled systems with high computational effort. Recently, fast solvers of linear order, i.e., with computational complexity O(n), where n is the number of DM actuators, have emerged. However, the quality of such methods typically degrades in low flux conditions. Our key contribution is to achieve the high quality of the standard Bayesian approach while at the same time maintaining the linear order speed of the recent solvers. Our method is based on performing a separate preprocessing step before applying the cumulative reconstructor (CuReD). The efficiency and performance of the new reconstructor are demonstrated using the OCTOPUS, the official end-to-end simulation environment of the ESO for extremely large telescopes. For more specific simulations we also use the MOST toolbox.

Finite element-wavelet hybrid algorithm for atmospheric tomography.

Reconstruction of the refractive index fluctuations in the atmosphere, or atmospheric tomography, is an underlying problem of many next generation adaptive optics (AO) systems, such as the multiconjugate adaptive optics or multiobject adaptive optics (MOAO). The dimension of the problem for the extremely large telescopes, such as the European Extremely Large Telescope (E-ELT), suggests the use of iterative schemes as an alternative to the matrix-vector multiply (MVM) methods. Recently, an algorithm based on the wavelet representation of the turbulence has been introduced in [Inverse Probl.29, 085003 (2013)] by the authors to solve the atmospheric tomography using the conjugate gradient iteration. The authors also developed an efficient frequency-dependent preconditioner for the wavelet method in a later work. In this paper we study the computational aspects of the wavelet algorithm. We introduce three new techniques, the dual domain discretization strategy, a scale-dependent preconditioner, and a ground layer multiscale method, to derive a method that is globally O(n), parallelizable, and compact with respect to memory. We present the computational cost estimates and compare the theoretical numerical performance of the resulting finite element-wavelet hybrid algorithm with the MVM. The quality of the method is evaluated in terms of an MOAO simulation for the E-ELT on the European Southern Observatory (ESO) end-to-end simulation system OCTOPUS. The method is compared to the ESO version of the Fractal Iterative Method [Proc. SPIE7736, 77360X (2010)] in terms of quality.

Kaczmarz algorithm for multiconjugated adaptive optics with laser guide stars.

We recently introduced the Kaczmarz algorithm for solving the atmospheric tomography problem in multiconjugate adaptive optics (MCAO). This iterative method solves the problem significantly faster than the standard matrix vector multiplication. We present the algorithm as well as an extension, which includes the effects of laser guide stars, such as the cone effect, tip/tilt indetermination, and spot elongation. We show that we can successfully cope with these effects and that the algorithm is suited for an MCAO system for the future generation of extremely large telescopes.

Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor.

We present a fast method for the wavefront reconstruction from pyramid wavefront sensor (P-WFS) measurements. The method is based on an analytical relation between pyramid and Shack-Hartmann sensor (SH-WFS) data. The algorithm consists of two steps--a transformation of the P-WFS data to SH data, followed by the application of cumulative reconstructor with domain decomposition, a wavefront reconstructor from SH-WFS measurements. The closed loop simulations confirm that our method provides the same quality as the standard matrix vector multiplication method. A complexity analysis as well as speed tests confirm that the method is very fast. Thus, the method can be used on extremely large telescopes, e.g., for eXtreme adaptive optics systems.

Numerical methods for the design of gradient-index optical coatings.

We formulate the problem of designing gradient-index optical coatings as the task of solving a system of operator equations. We use iterative numerical procedures known from the theory of inverse problems to solve it with respect to the coating refractive index profile and thickness. The mathematical derivations necessary for the application of the procedures are presented, and different numerical methods (Landweber, Newton, and Gauss-Newton methods, Tikhonov minimization with surrogate functionals) are implemented. Procedures for the transformation of the gradient coating designs into quasi-gradient ones (i.e., multilayer stacks of homogeneous layers with different refractive indices) are also developed. The design algorithms work with physically available coating materials that could be produced with the modern coating technologies.