{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T20:01:46Z","timestamp":1790107306638,"version":"4.0.1"},"reference-count":29,"publisher":"Optica Publishing Group","issue":"28","license":[{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Appl. Opt."],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"<jats:p>We report a framework for complete pupil function retrieval of optical systems (phase\n\t\t\t\t\tand amplitude) based on the analysis of a single intensity pattern\n\t\t\t\t\tarising from point-source imaging. The approach implements\n\t\t\t\t\tunconstrained fits over sets of linearly independent, real-valued,\n\t\t\t\t\tphase and amplitude diffraction modal functions (DMF), defined in the\n\t\t\t\t\timage space of the system. DMFs are derived from explicit expressions\n\t\t\t\t\tinvolving the optical parameters of the system, and from the base(s)\n\t\t\t\t\tone chooses to represent the amplitude and\/or phase pupil maps.\n\t\t\t\t\tNaturally, such bases can be the Zernike basis, although this is not a\n\t\t\t\t\tlimitation. Beyond the capability to efficiently retrieve complete\n\t\t\t\t\tpupil functions from single inputs, the approach features other assets\n\t\t\t\t\tby construction, among which: i) it can readily recurse to\n\t\t\t\t\tadvanced\/rigorous diffraction integrals; ii) it can cope with\n\t\t\t\t\tincomplete datasets, and iii) it allows the evaluation of metrics at\n\t\t\t\t\tany step of the retrieval process, which can be used to assess its\n\t\t\t\t\tphysical consistency. This paper gives both theoretical foundations\n\t\t\t\t\tand practical implementation guidelines for the method. Its accuracy\n\t\t\t\t\tand robustness to noise are thoroughly investigated in a series of\n\t\t\t\t\tnumerical examples, as preparatory work for experimental\n\t\t\t\t\tapplications.<\/jats:p>","DOI":"10.1364\/ao.593268","type":"journal-article","created":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T15:00:07Z","timestamp":1779980407000},"page":"9627","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Analytical framework for complete pupil function retrieval from single diffraction patterns"],"prefix":"10.1364","volume":"65","author":[{"given":"Pierre-Antoine","family":"Frugier","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris Cit\u00e9, CEA, CNRS, AIM"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,9,21]]},"reference":[{"key":"ao-65-28-9627-R1","doi-asserted-by":"publisher","first-page":"2758","DOI":"10.1364\/AO.21.002758","type":"journal-article","volume":"21","author":"Fienup","year":"1982","journal-title":"Appl.\n\t\t\t\t\tOpt."},{"key":"ao-65-28-9627-R2","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1109\/MSP.2014.2352673","type":"journal-article","volume":"32","author":"Shechtman","year":"2015","journal-title":"IEEE Signal Process.\n\t\t\t\t\tMag."},{"key":"ao-65-28-9627-R3","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1137\/S003614450139075","type":"journal-article","volume":"44","author":"Luke","year":"2002","journal-title":"SIAM Rev."},{"key":"ao-65-28-9627-R4","first-page":"237","type":"journal-article","volume":"35","author":"Gerchberg","year":"1972","journal-title":"Optik"},{"key":"ao-65-28-9627-R5","doi-asserted-by":"publisher","first-page":"1747","DOI":"10.1364\/AO.32.001747","type":"journal-article","volume":"32","author":"Fienup","year":"1993","journal-title":"Appl.\n\t\t\t\t\tOpt."},{"key":"ao-65-28-9627-R6","doi-asserted-by":"publisher","first-page":"1737","DOI":"10.1364\/AO.32.001737","type":"journal-article","volume":"32","author":"Fienup","year":"1993","journal-title":"Appl. Opt."},{"key":"ao-65-28-9627-R7","doi-asserted-by":"publisher","first-page":"410","DOI":"10.7498\/aps.30.410","type":"journal-article","volume":"30","author":"Yang","year":"1981","journal-title":"Acta Phys. Sin."},{"key":"ao-65-28-9627-R8","doi-asserted-by":"publisher","first-page":"1160","DOI":"10.1364\/JOSAA.32.001160","type":"journal-article","volume":"32","author":"Antonello","year":"2015","journal-title":"J. Opt.\n\t\t\t\t\tSoc. Am. A"},{"key":"ao-65-28-9627-R9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S1076-5670(05)41001-0","type":"journal-article","volume":"141","author":"Mugnier","year":"2006","journal-title":"Adv. Imaging Electron.\n\t\t\t\t\tPhys."},{"key":"ao-65-28-9627-R10","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1117\/12.7972989","type":"journal-article","volume":"21","author":"Gonsalves","year":"1982","journal-title":"Opt.\n\t\t\t\t\tEng."},{"key":"ao-65-28-9627-R11","first-page":"189","type":"journal-article","volume":"11","author":"Thimons","year":"2018","journal-title":"SIAM"},{"key":"ao-65-28-9627-R12","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1364\/AO.33.000209","type":"journal-article","volume":"33","author":"Yang","year":"1994","journal-title":"Appl. Opt."},{"key":"ao-65-28-9627-R13","volume-title":"Linear Systems, Fourier Transforms,\n\t\t\t\t\tand Optics","author":"Gaskill","year":"1978"},{"key":"ao-65-28-9627-R14","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1080\/713820605","type":"journal-article","volume":"28","author":"Hopkins","year":"1981","journal-title":"Opt. Acta Int. J.\n\t\t\t\t\tOpt."},{"key":"ao-65-28-9627-R16","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1364\/JOSAA.19.000849","type":"journal-article","volume":"19","author":"Janssen","year":"2002","journal-title":"J. Opt. Soc.\n\t\t\t\t\tAm. A"},{"key":"ao-65-28-9627-R17","doi-asserted-by":"publisher","first-page":"2216","DOI":"10.1364\/JOSAA.17.002216","type":"journal-article","volume":"17","author":"Mahajan","year":"2000","journal-title":"J. Opt. Soc. Am. A"},{"key":"ao-65-28-9627-R18","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1364\/OL.31.000501","type":"journal-article","volume":"31","author":"Dai","year":"2006","journal-title":"Opt.\n\t\t\t\t\tLett."},{"key":"ao-65-28-9627-R19","article-title":"Framework for\n\t\t\t\t\textracting rotation invariant features for image classification and an\n\t\t\t\t\tapplication using Haar wavelets\u201d","volume-title":"7th WSEAS International Conference on Artificial intelligence,\n\t\t\t\t\tKnowledge Engineering and Data Bases (AIKED\u201908)","author":"Akle","year":"2008"},{"key":"ao-65-28-9627-R20","doi-asserted-by":"publisher","first-page":"080101","DOI":"10.3788\/COL201311.080101","type":"journal-article","volume":"11","author":"Shakibaei","year":"2013","journal-title":"Chin. Optic. Lett."},{"key":"ao-65-28-9627-R21","doi-asserted-by":"crossref","DOI":"10.1002\/0470085002","volume-title":"Diffraction, Fourier Optics and\n\t\t\t\t\tImaging","author":"Ersoy","year":"2007"},{"key":"ao-65-28-9627-R22","volume-title":"Principle of Optics","author":"Born","year":"1980"},{"key":"ao-65-28-9627-R23","volume-title":"Fourier Optics","author":"Goodman","year":"2005"},{"key":"ao-65-28-9627-R25","volume-title":"Star Testing Astronomical\n\t\t\t\t\tTelescope","author":"Suiter","year":"2009"},{"key":"ao-65-28-9627-R26","doi-asserted-by":"crossref","DOI":"10.1002\/0471725153","volume-title":"Regression Diagnostics: Identifying\n\t\t\t\t\tInfluential Data and Sources of Collinearity","author":"Belsley","year":"1980"},{"key":"ao-65-28-9627-R27","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1080\/713818292","type":"journal-article","volume":"17","author":"Hopkins","year":"1970","journal-title":"Opt. Acta Int. J.\n\t\t\t\t\tOpt."},{"key":"ao-65-28-9627-R28","doi-asserted-by":"publisher","first-page":"96130H","DOI":"10.1117\/12.2188921","type":"journal-article","volume":"9613","author":"Chipman","year":"2015","journal-title":"Proc. SPIE"},{"key":"ao-65-28-9627-R30","doi-asserted-by":"publisher","first-page":"3003","DOI":"10.1364\/AO.17.003003","type":"journal-article","volume":"17","author":"Harvey","year":"1978","journal-title":"Appl.\n\t\t\t\t\tOpt."},{"key":"ao-65-28-9627-R31","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1364\/JOSA.66.000207","type":"journal-article","volume":"66","author":"Noll","year":"1976","journal-title":"J. Opt. Soc. Am."},{"key":"ao-65-28-9627-R33","article-title":"Zernike polynomials and\n\t\t\t\t\tbeyond","volume-title":"Zernike Lecture Notes\n\t\t\t\t\t(pdf)","author":"Mahajan","year":"2012"}],"container-title":["Applied Optics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=ao-65-28-9627&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T19:10:42Z","timestamp":1790104242000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=ao-65-28-9627"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,21]]},"references-count":29,"journal-issue":{"issue":"28","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/ao.593268","relation":{},"ISSN":["1559-128X","2155-3165"],"issn-type":[{"value":"1559-128X","type":"print"},{"value":"2155-3165","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,21]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/AO.593268","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"research-article","name":"content_type","label":"Article type"},{"value":"Screened by Similarity Check","name":"cross_check","label":"Similarity check"},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"12 February 2026","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"28 May 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"21 September 2026","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article\u2019s title, journal citation, and DOI.","name":"copyright","label":"Copyright"}]}}