Pupil Geometry Research Articles

Optimal pupil basis set for telescope-coronagraph design and perturbation analysis based on the method of moments.

This paper concerns optimization and analysis of telescope-coronagraph systems for direct imaging of exoplanets. In this paper, the coronagraph system, with arbitrary pupil geometry, is theoretically considered, and the governing equation for the pupil design is derived. The method of moments is applied to solve the generalized energy-concentration eigenvalue problem to obtain the optimal pupil apodization and complete sets of orthonormal basis functions for arbitrary pupil geometries. The method yields eigenvalues indicating the fraction of starlight energy encircled in the area of the focal-plane mask (FPM), where starlight can be occulted and/or nulled. In other words, a higher eigenvalue implies less leakage/spillover of light outside of the FPM region and into the planet-discovery zone. Thus, a higher eigenvalue supports better starlight suppression for a given type of coronagraph. This methodology is useful for semi-quantitatively ranking different modes of perturbation with respect to energy spillage in the focal plane independent of coronagraph design details. A model-order-reduction-based sensitivity analysis is conducted to investigate the coupling between different pupil modes induced by aberrations. A pupil mode recovery scheme is presented to offer a theoretically rigorous and computationally efficient approach to reconstruct the optimal pupil mode under an arbitrary phase perturbation. The reconstruction coefficients and recovery-effectiveness factors are derived theoretically and demonstrated numerically. Several numerical examples, including the LUVOIR A and B pupils, are provided to validate and demonstrate the applicability of the proposed methods. The reported methodology enables model-order reduction based on degree of focal-plane energy concentration and reconstruction of optimal pupil apodization vis-á-vis phase aberrations using a precomputed basis set. These features should enhance computational efficiency for coronagraph design and sensitivity analysis.

Confusion in differential piston measurement with the pyramid wavefront sensor

Context. The pyramid was proven to be a highly sensitive and versatile wave-front sensor (WFS) and has been selected to be installed on the single conjugate adaptive optics (AO) systems of the extremely large telescope (ELT). The pupil of the ELT is fragmented by the secondary support spider arms, which are larger than the spatial coherence length of the atmospheric turbulence. This causes a rupture of the incoming wavefront continuity, which means that we need to be able to measure the differential pistons across the spider arms to achieve full wavefront reconstruction. Aims. We investigate the reaction of the modulated pyramid WFS to discontinuous aberrations in presence of phase residuals after AO compensation for a range of expected observing conditions at the location of the ELT. We then explore some parameters of the sensor in order to improve its sensitivity to the wavefront discontinuities, including optical gain compensation, specific sensor modulation paths, and the number of faces of the pyramidal prism. Methods. We derived sensitivity loss and modal cross-talk strength coefficients around static post-AO residual phases using the COMPASS end-to-end AO simulation software. Moreover, extensive closed-loop AO simulations let us derive two wavefront error criteria that are appropriate for a fragmented pupil geometry. We used these to assess the wavefront reconstruction performance. Results. We show that on the ELT, the pyramid experiences a drastic loss in sensitivity and also non-linear modal cross-talks. Added to the limited capture range, this makes it poorly suited for phase discontinuity measurements at visible wavelengths. The strategies we studied to increase the sensitivity and reduce the modal cross-talk provide an improvement of the reconstruction for low D/r0(λWFS) values, that is, for a K-band pyramid. In presence of a large residual wavefront variance, however, a similar sensor in visible light fails to provide the necessary trade-off to measure both the continuous modes and the wavefront discontinuities. Conclusions. The ELT instruments, designed with only visible-band pyramid WFSs, will not be able to perform a direct measurement of the wavefront discontinuities. They will have to rely on Kolmogorov statistics to restore the continuity of the atmospheric corrugated wavefront. If any other source of discontinuities arises on the ELT, instruments will need an additional, dedicated WFS.

An undergraduate experiment to illustrate spatial transfer function concepts in Fourier optics

Fourier analysis is a key tool in physics and engineering in the broadest sense and is particularly used in the field of optics to design and study the properties of advanced imaging systems. Fourier optics as a distinct topic is usually taught in final-year undergraduate or first-year postgraduate studies, and a wide range of teaching approaches have been used to describe and explain its key concepts. Concerning practical work to accompany classroom lectures, the simplest laboratory experiments are based on studying how the properties of some diffracting aperture (pupil) are related to the corresponding Fraunhofer diffraction pattern, but this approach usually does not consider in detail the concepts of the optical transfer function and spatial frequencies. We here describe a simple experimental setup that fills this particular gap and that illustrates the spatial frequency-domain response of a simple optical system using incoherent light. The Modulation Transfer Function is directly measured for several different pupil geometries and shows very good agreement with theoretical calculations. This experiment clearly demonstrates spatial transfer function concepts in Fourier optics, complementing and extending other studies of Fourier transforms in physics that may consider similar ideas in a time and frequency signal-processing context.

Characterization of Single-mode Fiber Coupling at the Large Binocular Telescope

Optimizing on-sky single-mode fiber (SMF) injection is an essential part of developing precise Doppler spectrometers and new astrophotonics technologies. We installed and tested a prototype SMF-injection system at the Large Binocular Telescope in 2016 April. The fiber injection unit was built as part of the derisking process for a new instrument named iLocater that will use adaptive optics (AO) to feed a high resolution, near-infrared spectrograph. In this paper we report Y-band SMF coupling measurements for bright, M-type stars. We compare theoretical expectations for delivered Strehl ratio and SMF coupling to experimental results, and evaluate fundamental effects that limit injection efficiency. We find the pupil geometry of the telescope itself limits fiber coupling to a maximum efficiency of ρtel ≈ 0.78. Further analysis shows the individual impact of AO correction, tip-tilt residuals, and static (noncommon-path) aberrations contribute coupling coefficients of ρStrehl ≈ 0.33, , and ρncpa ≈ 0.8 respectively. Combined, these effects resulted in an average Y-band SMF efficiency of 0.18 for all observations. Finally, we investigate the impact of fiber coupling on radial velocity precision as a function of stellar apparent magnitude.

Pupil autoregulation impairment as an early marker of glaucomatous damage.

Glaucoma, a degenerative and progressive disease, leads to structural and functional changes in the optic nerve head and retinal ganglion cells (RGCs), while the vasculature of the iris stays intact. The aim of this study was to determine whether the coherence level associated with pupil geometry and peripheral arterial pulsation can be the basis for differentiating glaucoma and glaucoma-suspected patients from a control group. This is an investigator-initiated, single-center prospective cohort study. Patients with diagnosed glaucoma (glaucoma group - GG) or glaucoma suspects (glaucoma suspects group - GSG), as well as healthy participants (control group - CG), were prospectively enrolled in the study. Glaucoma-diagnosed patients and glaucoma suspects who converted to glaucoma within 5 years were included. All patients underwent a full ophthalmological examination that included measurements of the thicknesses of the retinal nerve fiber layer (RNFL) and the ganglion cell complex (GCC) along with other parameters. A custom-made pupilometer was synchronized with a pulsometer to simultaneously record an image of the pupil and the peripheral arterial pulsation signal. All readings were processed with a script developed by the researchers. The main indicator of an increased influence of the vascular structures of the iris on pupil geometry in the patients and CG were the coherence levels (levC) between parameters describing the pupillary shape and peripheral arterial pulsation. Differences in the median value of the levCpS, levCpε and levCpθ parameters between the GG and GSG compared to the CG were found (p < 0.001). During the follow-up period, a larger decrease was observed in RNFL thickness and GCC thickness in the GSG than in the GG (p < 0.05). Strong correlations were found between levCpS and RNFL and GCC loss among the GSG group (p < 0.001), while in the GG this parameter correlated with RNFL and GCC thickness (p < 0.001). This is the first attempt to relate changes in the neuronal signaling pathways in glaucoma to the vascular-dependent changes of pupil geometry. The findings presented herein suggest that this approach can be used to determine which glaucoma suspects have autonomic system impairment in the eye, increasing their probability of glaucoma conversion.

Kernel formalism applied to Fourier-based wave-front sensing in presence of residual phases

In this paper, we describe Fourier-based wave-front sensors (WFSs) as linear integral operators, characterized by their kernel. In the first part, we derive the dependency of this quantity with respect to the WFS's optical parameters: pupil geometry, filtering mask, and tip/tilt modulation. In the second part, we focus the study on the special case of convolutional kernels. The assumptions required to be in such a regime are described. We then show that these convolutional kernels allow to drastically simplify the WFS model by summarizing its behavior in a concise and comprehensive quantity called the WFS impulse response. We explain in particular how it allows to compute the sensor's sensitivity with respect to spatial frequencies. Such an approach therefore provides a fast diagnostic tool to compare and optimize Fourier-based WFSs. In the third part, we develop the impact of the residual phases on the sensor's impulse response, and show that the convolutional model remains valid. Finally, a section dedicated to the pyramid WFS concludes this work and illustrates how the slope maps are easily handled by the convolutional model.

Modal-based phase retrieval using Gaussian radial basis functions.

In this paper, we propose the use of Gaussian radial basis functions (GRBFs) to model the generalized pupil function for phase retrieval. The selection of the GRBF hyper-parameters is analyzed to achieve an increased accuracy of approximation. The performance of the GRBF-based method is compared in a simulation study with another modal-based approach considering extended Nijboer-Zernike (ENZ) polynomials. The almost local character of the GRBFs makes them a much more flexible basis with respect to the pupil geometry. It has been shown that for aberrations containing higher spatial frequencies, the GRBFs outperform ENZ polynomials significantly, even on a circular pupil. Moreover, the flexibility has been demonstrated by considering the phase retrieval problem on an annular pupil.

Wide-band white light sparse-aperture Fizeau imaging interferometer testbed for a distributed small-satellites constellation.

Under the background of astronomical observation by a distributed small-satellites constellation, a Fizeau imaging interferometer testbed (FIIT) was designed and constructed at National Astronomical Observatories of the Chinese Academy of Sciences in the past four years that achieved wide-band white light (400-700nm) phased imaging at laboratory with a field of view (FOV) ∼2 arcmin. This study presents the design of FIIT, which is mainly composed of a light source module for simulating infinite objects, three sub-telescopes in a Golay-3 distribution (i.e., equilateral triangle), three tip/tilt correction modules and optical path delay modules, as well as three detecting telescopes and one beam-combining telescope for phased imaging. The three detecting telescopes could determine the piston and tip/tilt errors, which could be compensated for by optical path delay modules and fast steering mirrors in a close-loop mode. The performance of FIIT is also derived analytically and simulated by a ray-tracing model. The tolerance analysis of the pupil geometry and magnification matching are also performed for achieving a better FOV. The optical test of each module could efficiently validate them to achieve diffraction limit imaging. The preliminary experimental results for both point source and extend scene could sufficiently demonstrate the phased mode imaging of FIIT, thus validating the good prospects of this technique in the future.

Polynomial Apodizers for Centrally Obscured Vortex Coronagraphs

Abstract Several coronagraph designs have been proposed over the last two decades to directly image exoplanets. Among these designs, vector vortex coronagraphs provide theoretically perfect starlight cancellation along with small inner working angles when deployed on telescopes with unobstructed pupils. However, current and planned space missions and ground-based extremely large telescopes present complex pupil geometries, including large central obscurations caused by secondary mirrors, which prevent vortex coronagraphs from rejecting on-axis sources entirely. Recent solutions combining the vortex phase mask with a ring-apodized pupil have been proposed to circumvent this issue, but provide a limited throughput for vortex charges . We present pupil plane apodizations for charge 2, 4, and 6 vector vortex coronagraphs that compensate for pupil geometries with circularly symmetric central obstructions caused by on-axis secondary mirrors. These apodizations are derived analytically and allow vortex coronagraphs to retain theoretically perfect nulling in the presence of obstructed pupils. For a charge 4 vortex, we design polynomial apodization functions assuming a grayscale apodizing filter that represent a substantial gain in throughput over the ring-apodized vortex coronagraph design, while for a charge 6 vortex, we design polynomial apodized vortex coronagraphs that have total energy throughput for the entire range of central obscuration sizes studied. We propose methods for optimizing apodizations produced with either grayscale apodizing filters or shaped mirrors. We conclude by demonstrating how this design may be combined with apodizations numerically optimized for struts and primary mirror segment gaps to design terrestrial exoplanet imagers for complex pupils.

Efficient injection from large telescopes into single-mode fibres: Enabling the era of ultra-precision astronomy

Photonic technologies offer numerous advantages for astronomical instruments such as spectrographs and interferometers owing to their small footprints and diverse range of functionalities. Operating at the diffraction-limit, it is notoriously difficult to efficiently couple such devices directly with large telescopes. We demonstrate that with careful control of both the non-ideal pupil geometry of a telescope and residual wavefront errors, efficient coupling with single-mode devices can indeed be realised. A fibre injection was built within the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument. Light was coupled into a single-mode fibre operating in the near-IR (J-H bands) which was downstream of the extreme adaptive optics system and the pupil apodising optics. A coupling efficiency of 86% of the theoretical maximum limit was achieved at 1550 nm for a diffraction-limited beam in the laboratory, and was linearly correlated with Strehl ratio. The coupling efficiency was constant to within <30% in the range 1250-1600 nm. Preliminary on-sky data with a Strehl ratio of 60% in the H-band produced a coupling efficiency into a single-mode fibre of ~50%, consistent with expectations. The coupling was >40% for 84% of the time and >50% for 41% of the time. The laboratory results allow us to forecast that extreme adaptive optics levels of correction (Strehl ratio >90% in H-band) would allow coupling of >67% (of the order of coupling to multimode fibres currently). For Strehl ratios <20%, few-port photonic lanterns become a superior choice but the signal-to-noise must be considered. These results illustrate a clear path to efficient on-sky coupling into a single-mode fibre, which could be used to realise modal-noise-free radial velocity machines, very-long-baseline optical/near-IR interferometers and/or simply exploit photonic technologies in future instrument design.

Optical performance of piezoelectrically actuated MEMS tunable lenses with various pupil geometries

A square geometry for the diaphragm of microelectromechanical systems (MEMS) tunable lenses with circular pupils is often used. This square diaphragm introduces aberrations that would not be present for a circular diaphragm and the question arises of whether it could be advantageous to also consider other than circular geometries for the pupil. To investigate this question, we have extended a previously established modeling framework for piezoelectrically actuated MEMS tunable lenses to devices with general polygonal-shaped pupils. It models the static optoelectromechanical coupling for symmetric configurations based on laminated-plate theory, linear piezoelectricity, and ray tracing. The framework helps to find geometrical parameters that give a diffraction-limited tunable lens with a minimum F-number. The tunable lens’ optical performance and its focusing capability, alone and in combination with a fixed lens, were calculated in terms of object distances and actuation voltages. Using the modeling framework, we show that the modulation transfer function of the tunable lens and the fixed lens combination remains the same up to a ±10-deg field of view after voltage adjustment to refocus on near objects. In addition, we found that pupil masking of the tunable lens can provide a beneficial tradeoff between the lens dioptric power and its RMS wavefront error.

Active compensation of aperture discontinuities for WFIRST-AFTA: analytical and numerical comparison of propagation methods and preliminary results with a WFIRST-AFTA-like pupil

The new frontier in the quest for the highest contrast levels in the focal plane of a coronagraph is now the correction of the large diffractive artifacts effects introduced at the science camera by apertures of increasing complexity. The coronagraph for the WFIRST/AFTA mission will be the first of such instruments in space with a two Deformable Mirrors wavefront control system. Regardless of the control algorithm for these multi Deformable Mirrors, they will have to rely on quick and accurate simulation of the propagation effects introduced by the out-of-pupil surface. In the first part of this paper, we present the analytical description of the different approximations to simulate these propagation effects. In Annex A, we prove analytically that, in the special case of surfaces inducing a converging beam, the Fresnel method yields high fidelity for simulations of these effects. We provide numerical simulations showing this effect. In the second part, we use these tools in the framework of the Active Compensation of Aperture Discontinuities technique (ACAD) applied to pupil geometries similar to WFIRST-AFTA. We present these simulations in the context of the optical layout of the High-contrast imager for Complex Aperture Telescopes, which will test ACAD on a optical bench. The results of this analysis show that using the ACAD method, an apodized pupil lyot coronagraph and the performance of our current deformable mirrors, we are able to obtain, in numerically simulations, a dark hole with an AFTA-like pupil. Our numerical simulation shows that we can obtain contrast better than $2.10^{-9}$ in monochromatic light and better than 3.e-8 with 10% bandwidth between 5 and 14 lambda/D.

The Control System of a Distributed Aperture Imaging Testbed

Star-9 is an experimental demonstration of distributed-aperture imaging built at the Lockheed Martin Advanced Technology Center. White light from a scene generator enters an array of nine telescopes and is combined at a focused image plane. Relative aberrations from each telescope are regulated by a control system using phase diversity and active relay mirrors. A Weiner filter is applied to the image, and the resulting angular resolution and image quality are nearly diffraction-limited with the diameter of the array. The control system takes estimates from the phase diversity algorithm and uses them to command mirrors to adjust tip, tilt, and piston in each telescope. Each active mirror has inductive position sensors for local position control. The mirrors are actuated by picomotors and inchworm motors. In addition to tip/tilt/piston errors, the control system is equipped to correct for pupil geometry and rotation errors, and to perform field of regard steering. In this paper, we present details on the control system's local sensing, control law synthesis, and precision actuation methodology. We also describe the incorporation of phase diversity and automated phasing algorithms with the control system and give performance results.

Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils

An efficient way of estimating orthonormal aberration coefficients on variable noncircular pupils is proposed. The method is based on the fact that all necessary pieces of information for constructing orthonormal polynomials (via the Gram-Schmidt process) can be numerically obtained during a routine least-squares fit of Zernike polynomials to wavefront data. This allows the method to use the usual Zernike polynomial fitting with an additional procedure that swiftly estimates the desired orthonormal aberration coefficients without having to use the functional forms of orthonormal polynomials. It is also shown that the method naturally accounts for the pixelation effect of pupil geometries, intrinsic to recording wavefront data on imaging sensors (e.g., CCDs), making the coefficient estimate optimal over a given pixelated pupil geometry. With these features, the method can be ideal for real-time wavefront analysis over dynamically changing pupils, such as in the Hobby-Eberly Telescope (HET), which is otherwise inefficient with analytic methods used in past studies.

Pupil remapping for high contrast astronomy: results from an optical testbed

The direct imaging and characterization of Earth-like planets is among the most sought-after prizes in contemporary astrophysics, however current optical instrumentation delivers insufficient dynamic range to overcome the vast contrast differential between the planet and its host star. New opportunities are offered by coherent single mode fibers, whose technological development has been motivated by the needs of the telecom industry in the near infrared. This paper presents a new vision for an instrument using coherent waveguides to remap the pupil geometry of the telescope. It would (i) inject the full pupil of the telescope into an array of single mode fibers, (ii) rearrange the pupil so fringes can be accurately measured, and (iii) permit image reconstruction so that atmospheric blurring can be totally removed. Here we present a laboratory experiment whose goal was to validate the theoretical concepts underpinning our proposed method. We successfully confirmed that we can retrieve the image of a simulated astrophysical object (in this case a binary star) though a pupil remapping instrument using single mode fibers.

Speckle v Non-Redundant Masking

Unlike their radio counterparts, optical astronomers have paid relatively little attention to the role of the aperture function when designing their high resolution imaging experiments. This is despite the fact that it can be easily modified with the use of a suitable pupil plane mask. While the defining characteristics of the two most favoured pupil configurations have never been in doubt – negligible atmospheric noise in the case of a non-redundant configuration of small holes and high photon rates for a filled pupil – the relative merits of these two choices in terms of imaging performance in the presence of turbulence appear not to have been carefully investigated until very recently.Most existing comparisons of fully-filled aperture (FFA) and non-redundant mask (NRM) based imaging strategies appear to have ignored a number of fundamental and practical difficulties that are often encountered in practice. In the intermediate regime, between very high and very low light levels, that characterizes most astrophysical applications hybrid imaging schemes seem most profitable. These utilize partially-redundant pupil geometries that combine the advantages of redundancy at low light levels without incurring the penalties associated with fully redundant beam recombination. Such pupil geometries are also useful in reducing the level of systematic effects that often plague speckle imaging experiments.

Diffraction-limited imaging with partially redundant masks: II Optical imaging of faint sources

In a recent paper [ J. Opt. Soc. Am. A9, 203 ( 1992)] the benefits of pupil apodization were examined for the near-infrared imaging of bright sources. In the current paper we extend these considerations to optical speckle imaging, in which photon noise rather than detector readout noise is important. We demonstrate that a one-dimensional pupil geometry (i.e., a thin slit) has several advantages over an unapodized aperture when faint sources are being observed through atmospheric turbulence. The use of a slit aperture does not decrease the signal-to-noise ratios of the power-spectrum and bispectrum measurements, and in many cases it increases them, despite the large reduction in signal level. The disadvantage of this apodization is a reduction in Fourier-plane coverage, which must be compensated for by observations with the slit aligned at several position angles. The performance of many of the current generation of photon-counting imaging detectors deteriorates at the high counting rates that can be experienced even when one is observing sources that are approaching the limiting magnitude of the speckle imaging technique. Under such conditions, we recommend the use of an apodized pupil, in contrast to the current preference for employing a neutral-density filter to reduce the detector count rate.

Diffraction-limited imaging with partially redundant masks I Infrared imaging of bright objects

The utility of partially redundant pupil geometries has been studied in the context of near-infrared speckle imaging with ground-based telescopes. Using both numerical simulations and experimental data collected with a 4-m-class telescope, we find that the decrease in redundancy resulting from apodizing the telescope pupil results in an enhancement of the quality of reconstructed images at high light levels. This improvement in imaging fidelity is particularly valuable when short-term variations in the statistics of the atmosphere make the seeing calibration of speckle interferograms difficult. However, the use of an apodizing mask necessarily restricts the faintest source that can be imaged, leading to a loss in sensitivity of one to two magnitudes. For many of the brighter near-infrared astrophysical sources in the sky that have been the subject of previous speckle-imaging studies, the use of a partially redundant pupil is expected to enhance the fidelity of the imaging procedure considerably.

Interference of diffracted multiple wavefronts in the geometric shadow of the Fresnel region

A general diffraction analysis is presented to determine the irradiance distributions produced by the interference of diffracted wavefronts from multiple pupil configurations. Specifically, interference phenomena in the geometrical shadows of the Fresnel region close to the diffracting apertures are considered. The effects of multiple pupil geometry variations and preliminary impact of phase errors are considered. Finally, physical insights are offered for the phenomena of diffractive interference in the geometric shadow.

Design Of A Wide Field Of View Phased Array Telescope

The design of the Multipurpose Multiple Telescope Testbed (MMTT) is described. The MMTT, whose construction is now nearly complete, is believed to be the first wide field of view phased array imaging telescope. The design philosophy and alignment requirements are discussed and the telescope is described. The array consists of four 20 cm telescopes and operates with visible light. The coherently combined image has a 30 arcmin field of view. A key design goal is the maintenance of lateral pupil geometry to within 1µm of the ideal. Two mirrors behind each afocal telescope combine the four beams into one and are actuated to control both tilt and piston. Together, the two beam combiner mirrors control lateral pupil geometry. A control system senses the lateral pupil geometry error by continuously measuring piston phase changes over the telescope field of view. A discrete time varying Kalman filter then processes these measurements to estimate lateral pupil geometry errors.