Super-resolution wavefront reconstruction in adaptive-optics with pyramid sensors
Carlos M. Correia, Charlotte Z. Bond, João Aveiro, François Leroux, Cédric Taïssir Heritier, Christophe Vérinaud
备注：
Paper has a major mathematical flaw that needs to be addressed (in the introduction). The results are still verified and stand good, but the underlying justification for why one can perform super-resolution with a pyramid is not accurate nor complete
Super-resolution (SR) refers to a combination of optical design and signal processing techniques jointly employed to obtain reconstructed wave-fronts at a higher-resolution from multiple low-resolution samples, overcoming the intrinsic limitations of the latter. After compelling examples have been provided on multi-Shack-Hartman (SH) wave-front sensor (WFS) adaptive optics systems performing atmospheric tomography with laser guide star probes, we broaden the SR concept to pyramid sensors (PyWFS) with a single sensor and a natural guide star. We revisit the analytic PyWFS diffraction model to claim two aspects: i) that we can reconstruct spatial frequencies beyond the natural Shannon-Nyquist frequency imposed by the detector pixel size and/or ii) that the PyWFS can be used to measure amplitude aberrations (at the origin of scintillation). SR offers the possibility to control a higher actuator density deformable mirror from seemingly fewer samples, the quantification of which is one of the goals of this paper. A super-resolved PyWFS is more resilient to mis-registration, lifts alignment requirements and improves performance (against aliasing and other spurious modes AO systems are poorly sensitive to) with only a factor up to 2 increased real-time computational burden.
We consider the mathematical background of the wavefront sensor type that is widely used in Adaptive Optics systems for astronomy, microscopy, and ophthalmology. The theoretical analysis of the pyramid sensor forward operators presented in this paper is aimed at a subsequent development of fast and stable algorithms for wavefront reconstruction from data of this sensor type. In our analysis we allow the sensor to be utilized in both the modulated and non-modulated fashion. We derive detailed mathematical models for the pyramid sensor and the physically simpler roof wavefront sensor as well as …
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 non-linear 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 AO when the measured phase information is small but may not be reasonable in reality due to unpreventable…
Wavefront sensing and control are important for enabling one of the key advantages of using large apertures, namely higher angular resolutions. Pyramid wavefront sensors are becoming commonplace in new instrument designs owing to their superior sensitivity. However, one remaining roadblock to their widespread use is the fabrication of the pyramidal optic. This complex optic is challenging to fabricate due to the pyramid tip, where four planes need to intersect in a single point. Thus far, only a handful of these have been produced due to the low yields and long lead times. To address this, we …
In this paper, we address the inverse problem of fast, stable, and high-quality wavefront reconstruction from pyramid wavefront sensor data for Adaptive Optics systems on Extremely Large Telescopes. For solving the indicated problem we apply well-known iterative mathematical algorithms, namely conjugate gradient, steepest descent, Landweber, Landweber-Kaczmarz and steepest descent-Kaczmarz iteration based on theoretical studies of the pyramid wavefront sensor. We compare the performance (in terms of correction quality and speed) of these algorithms in end-to-end numerical simulations of a clos…
Adaptive optics systems correct atmospheric turbulence in real time. Most adaptive optics systems used routinely correct in the near infrared, at wavelengths greater than 1 micron. MagAO- X is a new extreme adaptive optics (ExAO) instrument that will offer corrections at visible-to- near-IR wavelengths. MagAO-X will achieve Strehl ratios greater than 70% at H-alpha when running the 2040 actuator deformable mirror at 3.6 kHz. A visible pyramid wavefront sensor (PWFS) optimized for sensing at 600-1000 nm wavelengths will provide the high-order wavefront sensing on MagAO- X. We present the optica…
With its high sensitivity, the Pyramid wavefront sensor (PyWFS) is becoming an advantageous sensor for astronomical adaptive optics (AO) systems. However, this sensor exhibits significant non-linear behaviours leading to challenging AO control issues. In order to mitigate these effects, we propose to use, in addition to the classical pyramid sensor, a focal plane image combined with a convolutive description of the sensor to perform a fast tracking of the PyWFS non-linearities, the so-called optical gains (OG). We show that this additional focal plane imaging path only requires a small fractio…
The success of ground-based instruments for high contrast exoplanet imaging depends on the degree to which adaptive optics (AO) systems can mitigate atmospheric turbulence. While modern AO systems typically suffer from millisecond time lags between wavefront measurement and control, predictive wavefront control (pWFC) is a means of compensating for those time lags using previous wavefront measurements, thereby improving the raw contrast in the post-coronagraphic science focal plane. A method of predictive control based on Empirical Orthogonal Functions (EOF) has previously been proposed and de…
Electro-Multiplying CCDs offer a unique combination of speed, sub-electron noise and quantum efficiency. These features make them extremely attractive for astronomical adaptive optics. The SOUL project selected the Ocam2k from FLI as camera upgrade for the pyramid wavefront sensor of the LBT SCAO systems. Here we present results from the laboratory characterization of the 3 of the custom Ocam2k cameras for the SOUL project. The cameras showed very good noise ($0.4e^-$ and $0.4-0.7e^-$ for binned modes) and dark current values ($1.5e^-$). We measured the camera gain and identified the dependenc…
It is by now well known that pyramid based wavefront sensors, once in closed loop, have the capability to improve more and more the gain as the reference natural star image size is getting smaller on the pyramid pin. Especially in extreme adaptive optics applications, in order to correct the non-common path aberrations between the scientific and sensing channel, it is common use to inject a certain amount of offset wavefront deformation into the DM(s), departing at the same time the pyramid from the optimal working condition. In this paper we elaborate on the possibility to correct the low ord…
The Narrow Field InfraRed Adaptive Optics System (NFIRAOS) for the Thirty Meter Telescope (TMT) will use a natural guide star (NGS) Pyramid Wavefront Sensor (PWFS). A 32-mm diameter Fast Steering Mirror (FSM) is used to modulate the position of the NGS image around the tip of the pyramid. The mirror traces out a circular tip/tilt pattern at up to 800 Hz (the maximum operating frequency of NFIRAOS), with a diameter chosen to balance sensitivity and dynamic range. A circular dither pattern at 1/4 the modulation frequency is superimposed to facilitate optical gain measurements. The timing of this…
The next generation of Extremely Large Telescope (24 to 39m diameter) will suffer from the so-called "pupil fragmentation" problem. Due to their pupil shape complexity (segmentation, large spiders ...), some differential pistons may appear between some isolated part of the full pupil during the observations. Although classical AO system will be able to correct for turbulence effects, they will be blind to this specific telescope induced perturbations. Hence, such differential piston, a.k.a petal modes, will prevent to reach the diffraction limit of the telescope and ultimately will represent t…
Extremely Large Telescopes have overwhelmingly opted for the Pyramid wavefront sensor (PyWFS) over the more widely used Shack-Hartmann WaveFront Sensor (SHWFS) to perform their Single Conjugate Adaptive Optics (SCAO) mode. The PyWFS, a sensor based on Fourier filtering, has proven to be highly successful in many astronomy applications. However, it exhibits non-linearity behaviors that lead to a reduction of its sensitivity when working with non-zero residual wavefronts. This so-called Optical Gains (OG) effect, degrades the close loop performance of SCAO systems and prevents accurate correctio…
The MMTO Adaptive optics exoPlanet characterization System (MAPS) is an ongoing upgrade to the 6.5-meter MMT Observatory on Mount Hopkins in Arizona. MAPS includes an upgraded adaptive secondary mirror (ASM), upgrades to the ARIES spectrograph, and a new AO system containing both an optical and near-infrared (NIR; 0.9-1.8 um) pyramid wavefront sensor (PyWFS). The NIR PyWFS will utilize an IR-optimized double pyramid coupled with a SAPHIRA detector: a low-read noise electron Avalanche Photodiode (eAPD) array. This NIR PyWFS will improve MAPS's sky coverage by an order of magnitude by allowing r…
The next generation of giant ground and space telescopes will have the light-collecting power to detect and characterize potentially habitable terrestrial exoplanets using high-contrast imaging for the first time. This will only be achievable if the performance of Giant Segmented Mirror Telescopes (GSMTs) extreme adaptive optics (ExAO) systems are optimized to their full potential. A key component of an ExAO system is the wavefront sensor (WFS), which measures aberrations from atmospheric turbulence. A common choice in current and next-generation instruments is the pyramid wavefront sensor (PW…
MAPS, MMT Adaptive optics exoPlanet characterization System, is the upgrade of legacy 6.5m MMT adaptive optics system. It is an NSF MSIP-funded project that includes (i) refurbishing of the MMT Adaptive Secondary Mirror (ASM), (ii) new high sensitive and high spatial order visible and near-infrared pyramid wavefront sensors, and (iii) the upgrade of Arizona Infrared Imager and Echelle Spectrograph (ARIES) and MMT high Precision Imaging Polarimeter (MMTPol) science cameras. This paper will present the design and development of the visible pyramid wavefront sensor. This system consists of an acq…
Wavefront sensors encode phase information of an incoming wavefront into an intensity pattern that can be measured on a camera. Several kinds of wavefront sensors (WFS) are used in astronomical adaptive optics. Amongst them, Fourier-based wavefront sensors perform a filtering operation on the wavefront in the focal plane. The most well known example of a WFS of this kind is the Zernike wavefront sensor, and the pyramid wavefront sensor (PWFS) also belongs to this class. Based on this same principle, new WFSs can be proposed such as the n-faced pyramid (which ultimately becomes an axicone) or t…
The segmentation of the telescope pupil (by spiders & the segmented M4) create areas of phase isolated by the width of the spiders on the wavefront sensor (WFS), breaking the spatial continuity of the wavefront. The poor sensitivity of the Pyramid WFS (PWFS) to differential piston leads to badly seen and therefore uncontrollable differential pistons. In close loop operation, differential pistons between segments will settle around integer values of the average sensing wavelength. The differential pistons typically range from one to ten times the sensing wavelength and vary rapidly over time, l…
Most adaptive optics (AO) systems using pyramid wavefront sensors (PyWFS) to estimate the phase of the pupil field use mechanical modulation of the beam in order to increase the dynamic range in low-order modes so the PyWFS can usefully operate at low Strehl ratio. The tradeoff for this approach is reduced sensitivity, which, in turn, makes it difficult to attain a high Strehl ratio once the loop has been closed. We propose an algorithm that increases the dynamic range of the PyWFS without modulation. The proposed algorithm achieves this in two ways: 1) it allows the PyWFS to be treated with a…
The Provence Adaptive optics Pyramid Run System (PAPYRUS) is a pyramid-based Adaptive Optics (AO) system that will be installed at the Coude focus of the 1.52m telescope (T152) at the Observatoire de Haute Provence (OHP). The project is being developed by PhD students and Postdocs across France with support from staff members consolidating the existing expertise and hardware into an R&D testbed. This testbed allows us to run various pyramid wavefront sensing (WFS) control algorithms on-sky and experiment on new concepts for wavefront control with additional benefit from the high number of avai…
Adaptive optics systems are an essential technology for the modern astronomy for ground based telescopes. One of the most recent revolution in the field is the introduction of the pyramid wavefront sensor. The higher performance of this device is payed with increased complexity in the control. In this work we report about advances in the AO system control obtained with SOULat the Large Binocular Telescope. The first is an improved Tip/Tilt temporal control able to recover the nominal correction even in presence of high temporal frequency resonances. The second one is a modal gain optimization …
The wavefront sensors used today at the biggest World's telescopes have either a high dynamic range or a high sensitivity, and they are subject to a linear trade off between these two parameters. A new class of wavefront sensors, the Generalised Optical Differentiation Wavefront Sensors, has been devised, in a way not to undergo this linear trade off and to decouple the dynamic range from the sensitivity. This new class of WFSs is based on the light filtering in the focal plane from a dedicated amplitude filter, which is a hybrid between a linear filter, whose physical dimension is related to …
The sensitivity of the the pyramid wavefront sensor (PyWFS) has made it a popular choice for astronomical adaptive optics (AAO) systems, and it is at its most sensitive when it is used without modulation of the input beam. In non-modulated mode, the device is highly nonlinear. Hence, all PyWFS implementations on current AAO systems employ modulation to make the device more linear. The upcoming era of 30-m class telescopes and the demand for ultra-precise wavefront control stemming from science objectives that include direct imaging of exoplanets make using the PyWFS without modulation desirabl…
Extreme adaptive optics (AO) is crucial for enabling the contrasts needed for ground-based high contrast imaging instruments to detect exoplanets. Pushing exoplanet imaging detection sensitivities towards lower mass, closer separations, and older planets will require upgrading AO wavefront sensors (WFSs) to be more efficient. In particular, future WFS designs will aim to improve a WFS's measurement error (i.e., the wavefront level at which photon noise, detector noise, and/or sky background limits a WFS measurement) and linearity (i.e., the wavefront level, in the absence of photon noise, alia…
The Multiconjugate Adaptive optics Demonstrator (MAD) has successfully demonstrated on sky both Star Oriented (SO) and Layer Oriented (LO) multiconjugate adaptive optics techniques. While SO has been realized using 3 Shack-Hartmann wavefront sensors (WFS), we designed a multi-pyramid WFS for the LO. The MAD bench accommodates both WFSs and a selecting mirror allows choosing which sensor to use. In the LO approach up to 8 pyramids can be placed on as many reference stars and their light is co-added optically on two different CCDs conjugated at ground and to an high layer. In this paper we discu…
SAXO+, the second-stage adaptive optics for SPHERE: NCPA compensation and dark-hole loop with a pyramid wavefront sensor
LIRA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CY Cergy Paris Université, CNRS（巴黎天文台，PSL大学，索邦大学，巴黎西岱大学，塞吉-蓬图瓦兹大学，法国国家科学研究中心）；Université Lyon 1, ENS de Lyon, CNRS, CRAL（里昂第一大学，里昂高等师范学校，法国国家科学研究中心）；Départment d’astronomie de l’Université de Genève（日内瓦大学天文学系）；MPIA, Max-Planck-Institut für Astronomie（马克斯·普朗克天文学研究所）；INAF Osservatorio di Astrofisica e Scienza dello Spazio di Bologna（意大利国家天体物理研究所博洛尼亚空间天体物理与科学观测站）；Université Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry（巴黎萨克雷大学，光学研究生院，法国国家科学研究中心，夏尔·法布里实验室）；Université Grenoble Alpes, CNRS, IPAG（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，行星与地球物理学研究所）；Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange（蔚蓝海岸大学，蔚蓝海岸天文台，法国国家科学研究中心，拉格朗日实验室）；Aix-Marseille Univ., CNRS, CNES, LAM（艾克斯-马赛大学，法国国家科学研究中心，法国国家空间研究中心，天体物理实验室）
No need to modulate: On-sky results of a Neural Network enhanced pyramid wavefront sensor and prospects for the ELTs
NOVA, Netherlands Research School for Astronomy（NOVA，荷兰天文学研究学校）；Leiden Observatory, Leiden University（莱顿天文台，莱顿大学）；Steward Observatory, University of Arizona（斯图尔德天文台，亚利桑那大学）；Center for Computational Astrophysics, Flatiron Institute（计算天体物理学中心，西蒙斯平研究所）；Subaru Telescope, National Astronomical Observatory of Japan（昴星团望远镜，日本国立天文台）；Astrobiology Center, National Institutes of Natural Sciences（天体生物学中心，自然科学研究院）；Wyant College of Optical Sciences, The University of Arizona（怀扬光学科学学院，亚利桑那大学）；Northrop Grumman Corporation（诺斯罗普·格鲁曼公司）；Draper Laboratory（德拉珀实验室）；Starfire Optical Range, Kirtland Air Force Base（星光光学靶场，柯克兰空军基地）
For ExAO instruments for the Giant Segmented Mirror Telescopes (GSMTs), alternative architectures of WFS are under consideration because there is a tradeoff between detector size, speed, and noise that reduces the performance of GSMT-ExAO wavefront control. One option under consideration for a GSMT-ExAO wavefront sensor is a three-sided PWFS (3PWFS). The 3PWFS creates three copies of the telescope pupil for wavefront sensing, compared to the conventional four-sided PWFS (4PWFS) which uses four pupils. The 3PWFS uses fewer detector pixels than the 4PWFS and should therefore be less sensitive to…
The Giant Magellan Telescope (GMT) design consists of seven circular 8.4-m diameter mirror segments that are separated by large > 30 cm gaps, creating the possibility of fluctuations in optical path differences due to flexure, segment vibrations, wind buffeting, temperature effects, and atmospheric seeing. In order to utilize the full diffraction-limited aperture of the GMT for natural guide star adaptive optics (NGSAO) science, the seven mirror segments must be co-phased to well within a fraction of a wavelength. The current design of the GMT involves seven adaptive secondary mirrors, an off-…
The Southern Robotic Adaptive Optics (SRAO) instrument will bring the proven high-efficiency capabilities of Robo-AO to the Southern-Hemisphere, providing the unique capability to image with high-angular-resolution thousands of targets per year across the entire sky. Deployed on the modern 4.1m SOAR telescope located on Cerro Tololo, the NGS-AO system will use an innovative dual-knife-edge wavefront sensor, similar to a pyramid sensor, to enable guiding on targets down to V=16 with diffraction limited resolution in the NIR. The dual-knife-edge wavefront sensor can be up to two orders of magnit…
Contactless actuators and pyramid wavefront sensor, the SPLATT concept for space active optics: an overview of the project and the last laboratory results
The astronomical applications greatly benefit from the knowledge of the instrument PSF. We describe the PSF Reconstruction algorithm developed for the LBT LUCI instrument assisted by the SOUL SCAO module. The reconstruction procedure considers only synchronous wavefront sensor telemetry data and a few asynchronous calibrations. We do not compute the Optical Transfer Function and corresponding filters. We compute instead a temporal series of wavefront maps and for each of these the corresponding instantaneous PSF. We tested the algorithm both in laboratory arrangement and in the nighttime for d…
Commonly used wavefront sensors, the Shack Hartmann wavefront sensor and the pyramid wavefront sensor, for example, have large dynamic range or high sensitivity, trading one regime for the other. A new type of wavefront sensor is being developed and is currently undergoing testing at the University of Arizona's Center for Astronomical Adaptive Optics. This sensor builds on linear optical differentiation theory by using linear, spatially varying halfwave plates in an intermediate focal plane. These filters, along with the polarizing beam splitters, divide the beam into four pupil images, simila…
Fourier-based wavefront sensors, such as the Pyramid Wavefront Sensor (PWFS), are the current preference for high contrast imaging due to their high sensitivity. However, these wavefront sensors have intrinsic nonlinearities that constrain the range where conventional linear reconstruction methods can be used to accurately estimate the incoming wavefront aberrations. We propose to use Convolutional Neural Networks (CNNs) for the nonlinear reconstruction of the wavefront sensor measurements. It is demonstrated that a CNN can be used to accurately reconstruct the nonlinearities in both simulatio…
The future upgrade of Keck II telescope's adaptive optics system will include a pyramid wavefront sensor working in the near-infrared (J and H band). It will benefit from the recently developed avalanche photodiode arrays, specifically the SAPHIRA (Selex) array, which provides a low noise ($<$ 1 e- at high frame rates). The system will either work with a natural guide star (NGS) in a single conjugated adaptive optics system, or in a laser guide star (LGS) mode. In this case, the pyramid would be used as a low-order sensor only. We report on a study of the pyramid sensor's performance via end-t…
The ELT is a 39m large, ground-based optical and near- to mid-infrared telescope under construction in the Chilean Atacama desert. Operation is planned to start around the middle of the next decade. All first light instruments will come with wavefront sensing devices that allow control of the ELT's intrinsic M4 and M5 wavefront correction units, thus building an adaptive optics (AO) system. To take advantage of the ELT's optical performance, full diffraction-limited operation is required and only a high performance AO system can deliver this. Further technically challenging requirements for th…
We present the last version of the PyrAmid Simulator Software for Adaptive opTics Arcetri (PASSATA), an IDL and CUDA based object oriented software developed in the Adaptive Optics group of the Arcetri observatory for Monte-Carlo end-to-end adaptive optics simulations. The original aim of this software was to evaluate the performance of a single conjugate adaptive optics system for ground based telescope with a pyramid wavefront sensor. After some years of development, the current version of PASSATA is able to simulate several adaptive optics systems: single conjugate, multi conjugate and grou…
The next generation of Giant Segmented Mirror Telescopes (GSMT) will have large gaps between the segments either caused by the shadow of the mechanical structure of the secondary mirror (E-ELT and TMT) or intrinsically by design (GMT). These gaps are large enough to fragment the aperture into independent segments that are separated by more than the typical Fried parameter. This creates piston and petals modes that are not well sensed by conventional wavefront sensors such as the Shack-Hartmann wavefront sensor or the pyramid wavefront sensor. We propose to use a new optical device, the Hologra…
In this paper we report on the laboratory experiment we settled in the Shanghai Astronomical Observatory (SHAO) to investigate the pyramid wavefront sensor (WFS) ability to measure the differential piston on a sparse aperture. The ultimate goal is to verify the ability of the pyramid WFS work in closed loop to perform the phasing of the primary mirrors of a sparse Fizeau imaging telescope. In the experiment we installed on the optical bench we performed various test checking the ability to flat the wave-front using a deformable mirror and to measure the signal of the differential piston on a t…
Laser guide stars (LGS) are used in many adaptive optics systems to extend sky coverage. The most common wavefront sensor used in combination with a LGS is a Shack-Hartmann wavefront sensor (SHWFS). The Shack-Hartmann has a major disadvantage for extended source wavefront sensing because it directly samples the image. In this proceeding we propose to use the generalized-Optical Differentation Wavefront Sensor (g-ODWFS) a wavefront sensor for wavefront sensing of LGS. The g-ODWFS uses only 4 pixels per sub-aperture, has little to no aliasing noise and therefore no spurious low-order errors and …
The Magellan extreme adaptive optics (MagAO-X) instrument is a new extreme adaptive optics (ExAO) system designed for operation in the visible to near-IR which will deliver high contrast-imaging capabilities. The main AO system will be driven by a pyramid wavefront sensor (PyWFS); however, to mitigate the impact of quasi-static and non-common path (NCP) aberrations, focal plane wavefront sensing (FPWFS) in the form of low-order wavefront sensing (LOWFS) and spatial linear dark field control (LDFC) will be employed behind a vector apodizing phase plate (vAPP) coronagraph using rejected starligh…
Adaptive optics (AO) is a technique allowing to drastically improve ground-based telescopes angular resolution. The wavefront sensor (WFS) is one of the key components of such systems, driving the fundamental performance limitations. In this paper, we focus on a specific class of WFS: the Fourier-filtering wavefront sensors (FFWFS). This class is known for its extremely high sensitivity. However, a clear and comprehensive noise propagation model for any kind of FFWFS is lacking. Considering read-out noise and photon noise, we derive a simple and comprehensive model allowing to understand how t…
Current designs for all three extremely large telescopes show the overwhelming adoption of the pyramid wavefront sensor (P-WFS) as the WFS of choice for adaptive optics (AO) systems sensing on natural guide stars (NGS) or extended objects. The key advantages of the P-WFS over the Shack-Hartmann are known and are mainly provided by the improved sensitivity (fainter NGS) and reduced sensitivity to spatial aliasing. However, robustness and tolerances of the P-WFS for the ELTs are not currently well understood. In this paper, we present simulation results for the single-conjugate AO mode of HARMON…
We propose a design of an adaptive optics (AO) system for the high-resolution fiber-fed echelle spectrograph installed at the Nasmyth focus of the 6-m BTA telescope at the Special Astrophysical Observatory (SAO) of the Russian Academy of Sciences (RAS). The system will be based on a pyramid wavefront sensor and benefit from the experience of the Laboratoire d'Astrophysique de Marseille team in the field of adaptive optics. The AO will operate in the visible domain of 430-680 nm, in an f\30 input beam and provide correction for the on-axis source only. The main challenges in this particular des…
The ingot wavefront sensor (I-WFS) has been proposed, for ELT-like apertures, as a possible pupil plane WFS, to cope with the geometrical characteristics of a laser guide star (LGS). Within the study and development of such a WFS, on-going in the framework of the MAORY project, the final purpose of the I-WFS simulation is to estimate its performance in terms of wavefront aberration measurement capability. The first step of this analysis is to translate incoming wavefronts into the three pupil images, produced by the optical system. The intrinsic geometrical characteristics of the ingot optical…
