Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

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 their various approximations. Additionally, we calculate adjoint operators which build preliminaries for the application of several iterative mathematical approaches for solving inverse problems such as gradient based algorithms, Landweber iteration or Kaczmarz methods.

[1]  Roberto Ragazzoni,et al.  Extended source pyramid wave-front sensor for the human eye. , 2002, Optics express.

[2]  Serge Meimon,et al.  Adaptive optics systems for HARMONI: a visible and near-infrared integral field spectrograph for the E-ELT , 2010, Astronomical Telescopes + Instrumentation.

[3]  Glen Herriot,et al.  Testing the pyramid truth wavefront sensor for NFIRAOS in the lab , 2016, Astronomical Telescopes + Instrumentation.

[4]  Ronny Ramlau,et al.  Wavefront reconstruction from non-modulated pyramid wavefront sensor data using a singular value type expansion , 2018 .

[5]  H. Brezis,et al.  Composition in fractional Sobolev spaces , 2001 .

[6]  Brent L Ellerbroek,et al.  Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[7]  R. Ragazzoni,et al.  Sensitivity of a pyramidic Wave Front sensor in closed loop Adaptive Optics , 1999 .

[8]  Ronny Ramlau,et al.  Nonlinear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber-Kaczmarz iterations. , 2018, Applied optics.

[9]  Pascal Jagourel,et al.  ATLAS: the E-ELT laser tomographic adaptive optics system , 2010, Astronomical Telescopes + Instrumentation.

[10]  R. Fefferman.  Singular integrals on product domains , 1981 .

[11]  Thierry Fusco,et al.  Development of a Pyramid Wave-front Sensor , 2013 .

[12]  Richard A Frazin,et al.  Efficient, nonlinear phase estimation with the nonmodulated pyramid wavefront sensor. , 2018, Journal of the Optical Society of America. A, Optics, image science, and vision.

[13]  Stefan Hippler,et al.  PYRAMIR: Exploring the On-Sky Performance of the World’s First Near-Infrared Pyramid Wavefront Sensor , 2010 .

[14]  Simon Thibault,et al.  INO Pyramidal Wavefront Sensor Demonstrator: first closed-loop on-sky operation at Mont-Mégantic Telescope , 2015 .

[15]  Jun Ho Lee,et al.  Pupil plane wavefront sensing with a static pyramidal prism: Simulation and preliminary evaluation , 2000 .

[16]  Elizabeth Daly,et al.  Ophthalmic wavefront measurements using a versatile pyramid sensor , 2010 .

[17]  Ronny Ramlau,et al.  Convolution- and Fourier-transform-based reconstructors for pyramid wavefront sensor. , 2017, Applied optics.

[18]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[19]  L. Busoni,et al.  The pyramid wavefront sensor for the high order testbench (HOT) , 2008, Astronomical Telescopes + Instrumentation.

[20]  C. Vérinaud,et al.  Adaptive optics for high‐contrast imaging: pyramid sensor versus spatially filtered Shack–Hartmann sensor , 2005 .

[21]  Ronny Ramlau,et al.  Preprocessed cumulative reconstructor with domain decomposition: a fast wavefront reconstruction method for pyramid wavefront sensor. , 2013, Applied optics.

[22]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[23]  Armando Riccardi,et al.  Joint MICADO-MAORY SCAO mode: specifications, prototyping, simulations and preliminary design , 2016, Astronomical Telescopes + Instrumentation.

[24]  Cristina Alvarez Diez.  A 3-sided Pyramid Wavefront Sensor Controlled by a Neural Network for Adaptive Optics to reach diffraction-limited Imaging of the Retina , 2006 .

[25]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[26]  G. Burton.  Sobolev Spaces , 2013 .

[27]  Francois Roddier,et al.  Adaptive Optics in Astronomy: Imaging through the atmosphere , 2004 .

[28]  Alexander Kreinin,et al.  Regularization Algorithms for , 2001 .

[29]  Marcos A. van Dam,et al.  Design of a truth sensor for the GMT laser tomography adaptive optics system , 2012, Other Conferences.

[30]  Paul L. Butzer,et al.  Fourier analysis and approximation , 1971 .

[31]  B. Welsh,et al.  Imaging Through Turbulence , 1996 .

[32]  Ronny Ramlau,et al.  Wavefront reconstruction for ELT-sized telescopes with pyramid wavefront sensors , 2018, Astronomical Telescopes + Instrumentation.

[33]  Laurent Jolissaint,et al.  Nonlinear estimation with a pyramid wavefront sensor , 2018, Astronomical Telescopes + Instrumentation.

[34]  Ronny Ramlau,et al.  Fast algorithm for wavefront reconstruction in XAO/SCAO with pyramid wavefront sensor , 2014, Astronomical Telescopes and Instrumentation.

[35]  Roberto Ragazzoni,et al.  On-sky test of the pyramid wavefront sensor , 2003, SPIE Astronomical Telescopes + Instrumentation.

[36]  Jean-Pierre Véran,et al.  Pyramid versus Shack-Hartmann: Trade Study Results for the NFIRAOS NGS WFS - eScholarship , 2015 .

[37]  J. Hardy,et al.  Adaptive Optics for Astronomical Telescopes , 1998 .

[38]  Roberto Ragazzoni,et al.  Adaptive Optics for Extremely Large Telescopes III A STUDY OF PYRAMID WFS BEHAVIOUR UNDER IMPERFECT ILLUMINATION , 2013 .

[39]  Roberto Ragazzoni,et al.  Expected gain in the pyramid wavefront sensor with limited Strehl ratio , 2016 .

[40]  Remko Stuik,et al.  METIS: the mid-infrared E-ELT imager and spectrograph , 2014, Astronomical Telescopes and Instrumentation.

[41]  Colin Bradley,et al.  Calibration and testing with real turbulence of a pyramid sensor employing static modulation. , 2009, Optics express.

[42]  Fernando Vargas-Martin,et al.  Quantitative phase microscopy of transparent samples using a liquid crystal display , 2013, Journal of biomedical optics.

[43]  S. Esposito,et al.  Pyramid Wavefront Sensor behavior in partial correction Adaptive Optic systems , 2001 .

[44]  Richard G Lane,et al.  Phase retrieval from subdivision of the focal plane with a lenslet array. , 2004, Applied optics.

[45]  Petru Mironescu,et al.  Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces , 2001 .

[46]  Miska Le Louarn,et al.  Comparison between a model-based and a conventional pyramid sensor reconstructor. , 2007, Applied optics.

[47]  Armando Riccardi,et al.  Closed-loop performance of pyramid wavefront sensor , 2000, Defense, Security, and Sensing.

[48]  Jörg Neunhäuserer.  Funktionalanalysis , 2020, Mathematische Begriffe in Beispielen und Bildern.

[49]  S. Sobolev,et al.  Einige Anwendungen der Funktionalanalysis auf Gleichungen der mathematischen Physik , 1964 .

[50]  Ronny Ramlau,et al.  Two novel algorithms for wavefront reconstruction from pyramid sensor data: Convolution with Linearized Inverse Filter and Pyramid Fourier Transform Reconstructor , 2017 .

[51]  Christopher Dainty,et al.  Linearity of the pyramid wavefront sensor. , 2006, Optics express.

[52]  Kjetil Dohlen,et al.  Analysis and mitigation of pupil discontinuities on adaptive optics performance , 2018, Astronomical Telescopes + Instrumentation.

[53]  Miska Le Louarn,et al.  Applying sensitivity compensation for pyramid wavefront sensor in different conditions , 2008, Astronomical Telescopes + Instrumentation.

[54]  Ronny Ramlau,et al.  Advanced wavefront reconstruction methods for segmented Extremely Large Telescope pupils using pyramid sensors , 2018 .

[55]  Byron Engler,et al.  Effects of the telescope spider on extreme adaptive optics systems with pyramid wavefront sensors , 2018, Astronomical Telescopes + Instrumentation.

[56]  Michael Shao,et al.  Extreme adaptive optics for the Thirty Meter Telescope , 2006, SPIE Astronomical Telescopes + Instrumentation.

[57]  Stefan Hippler,et al.  PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor , 2006, SPIE Astronomical Telescopes + Instrumentation.

[58]  Gerard Rousset,et al.  A modal approach to optical gain compensation for the pyramid wavefront sensor , 2018, Astronomical Telescopes + Instrumentation.

[59]  Simone Esposito,et al.  Adaptive optics for ophthalmic applications using a pyramid wavefront sensor. , 2006, Optics express.

[60]  Thierry Fusco,et al.  Experimental study of an optimised Pyramid wave-front sensor for Extremely Large Telescopes , 2016, Astronomical Telescopes + Instrumentation.

[61]  Andreas Obereder,et al.  Dealing with spiders on ELTs: using a Pyramid WFS to overcome residual piston effects , 2018, Astronomical Telescopes + Instrumentation.

[62]  Jean-Franccois Sauvage,et al.  Sensing and control of segmented mirrors with a pyramid wavefront sensor in the presence of spiders , 2017 .

[63]  E. Vernet,et al.  A pyramid wavefront sensor with no dynamic modulation , 2002 .

[64]  Jean-Pierre Véran,et al.  Development of a pyramidal wavefront sensor test-bench at INO , 2013 .

[65]  R. Ragazzoni.  Pupil plane wavefront sensing with an oscillating prism , 1996 .

[66]  Miska Le Louarn,et al.  Improving the performance of a pyramid wavefront sensor with modal sensitivity compensation. , 2008, Applied optics.

[67]  Christophe Verinaud,et al.  On the nature of the measurements provided by a pyramid wave-front sensor , 2004 .

[68]  P. Hinz,et al.  High Contrast Imaging in the Visible: First Experimental Results at the Large Binocular Telescope , 2016, 1609.05147.

[69]  Christophe Verinaud,et al.  Extreme adaptive optics simulations for EPICS , 2010 .

[70]  L. Busoni,et al.  Large Binocular Telescope Adaptive Optics System: new achievements and perspectives in adaptive optics , 2011, Optical Engineering + Applications.

[71]  L. Carbonaro,et al.  Wavefront sensor design for the GMT natural guide star AO system , 2012, Other Conferences.

[72]  Armando Riccardi,et al.  Laboratory test of a pyramid wavefront sensor , 2000, Astronomical Telescopes and Instrumentation.

[73]  I. Iglesias.  Pyramid phase microscopy. , 2011, Optics letters.

[74]  Roberto Ragazzoni,et al.  PWFSs on GMCAO: a different approach to the non-linearity issue , 2016, Astronomical Telescopes + Instrumentation.

[75]  T. Fusco,et al.  The adaptive optics modes for HARMONI: from Classical to Laser Assisted Tomographic AO , 2016, Astronomical Telescopes + Instrumentation.

[76]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[77]  Armando Riccardi,et al.  Laboratory characterization and performance of the high-order adaptive optics system for the Large Binocular Telescope , 2010 .