{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:42:44Z","timestamp":1785436964709,"version":"3.56.0"},"reference-count":23,"publisher":"Optica Publishing Group","issue":"6","license":[{"start":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T00:00:00Z","timestamp":1770940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR"},{"start":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T00:00:00Z","timestamp":1770940800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"funder":[{"DOI":"10.13039\/501100006087","name":"Direcci\u00f3n General de Asuntos del Personal Acad\u00e9mico","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006087","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100005739","name":"Universidad Nacional Aut\u00f3noma de M\u00e9xico","doi-asserted-by":"crossref","award":["IT104525"],"award-info":[{"award-number":["IT104525"]}],"id":[{"id":"10.13039\/501100005739","id-type":"DOI","asserted-by":"crossref"},{"id":"https:\/\/ror.org\/01tmp8f25","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Appl. Opt."],"published-print":{"date-parts":[[2026,2,20]]},"abstract":"<jats:p>We study the propagation of refracted wavefronts through chromatically separated doublets, considering an incident plane wavefront incoming to the optical system. Based on Huygens\u2019 principle and Hilbert integral, we define piecewise functions to fully represent the shape of wavefronts propagated through two conic lenses immersed in air, having a tiny distance of separation between them. Alternatively, by varying the design parameters such as curvature radii, axial thicknesses, and refractive indices, it is possible to reduce the size and modify the shape of the caustic surfaces for evaluating the circle of least confusion. Finally, we provide two different approaches to represent the shape of propagated wavefronts at the second order, which are related to the envelopes for all refracted rays.<\/jats:p>","DOI":"10.1364\/ao.588889","type":"journal-article","created":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T14:00:10Z","timestamp":1769176810000},"page":"1841","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Huygens\u2019 principle and Hilbert integral to propagate wavefronts through chromatic separated doublets"],"prefix":"10.1364","volume":"65","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-9686-3787","authenticated-orcid":true,"given":"Ismael","family":"Vel\u00e1zquez-G\u00f3mez","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01tmp8f25","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8205-3733","authenticated-orcid":true,"given":"Maximino","family":"Avenda\u00f1o-Alejo","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01tmp8f25","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0136-6701","authenticated-orcid":true,"given":"Mart\u00edn","family":"Jim\u00e9nez-Rodr\u00edguez","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01tmp8f25","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2364-2966","authenticated-orcid":true,"given":"Osvaldo","family":"Ponce-Hern\u00e1ndez","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01tmp8f25","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5856-179X","authenticated-orcid":true,"given":"Gabriel","family":"Castillo-Santiago","sequence":"additional","affiliation":[{"name":"Universidad del Istmo"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3305-676X","authenticated-orcid":true,"given":"Rafael","family":"Izazaga-P\u00e9rez","sequence":"additional","affiliation":[{"name":"Instituto Nacional de Astrof\u00edsica \u00d3ptica y Electr\u00f3nica"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4709-1830","authenticated-orcid":true,"given":"Oscar G.","family":"Rodr\u00edguez-Herrera","sequence":"additional","affiliation":[{"name":"Monash University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,2,13]]},"reference":[{"key":"ao-65-6-1841-R1","first-page":"176","volume-title":"Fundamentals of Optics","author":"Jenkins","year":"1976"},{"key":"ao-65-6-1841-R2","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/B978-0-08-051009-5.50007-X","volume-title":"Lens Design Fundamentals","author":"Kingslake","year":"1978"},{"key":"ao-65-6-1841-R3","first-page":"178","volume-title":"Applied Optics & Optical Design","volume":"1","author":"Conrady","year":"1957"},{"key":"ao-65-6-1841-R4","doi-asserted-by":"publisher","first-page":"5663","DOI":"10.1364\/AO.40.005663","type":"journal-article","volume":"40","author":"Rayces","year":"2001","journal-title":"Appl. Opt."},{"key":"ao-65-6-1841-R5","doi-asserted-by":"publisher","first-page":"1864","DOI":"10.1364\/AO.24.001864","type":"journal-article","volume":"24","author":"Robb","year":"1985","journal-title":"Appl. Opt."},{"key":"ao-65-6-1841-R6","doi-asserted-by":"publisher","first-page":"13592","DOI":"10.1364\/OE.20.013592","type":"journal-article","volume":"20","author":"Albuquerque","year":"2012","journal-title":"Opt. Express"},{"key":"ao-65-6-1841-R7","first-page":"66","volume-title":"Modern Lens Design, A Resource Manual","author":"Smith","year":"1992"},{"key":"ao-65-6-1841-R8","doi-asserted-by":"publisher","first-page":"122220J","DOI":"10.1117\/12.2633203","type":"journal-article","volume":"122220","author":"Vel\u00e1zquez-G\u00f3mez","year":"2022","journal-title":"Proc. SPIE"},{"key":"ao-65-6-1841-R9","doi-asserted-by":"publisher","first-page":"1406","DOI":"10.1364\/AO.480947","type":"journal-article","volume":"62","author":"L\u00f3pez-Bautista","year":"2023","journal-title":"Appl. Opt."},{"key":"ao-65-6-1841-R10","first-page":"149","volume-title":"Optics","author":"Hecht","year":"2002"},{"key":"ao-65-6-1841-R11","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1002\/3527608176.ch12","volume-title":"The Mathematics of Geometrical and Physical Optics, The K-Function and Its Ramifications","author":"Stavroudis","year":"2006"},{"key":"ao-65-6-1841-R12","first-page":"62","volume-title":"The Optics of Rays, Wavefronts and Caustics","author":"Stavroudis","year":"1972"},{"key":"ao-65-6-1841-R13","first-page":"11","volume-title":"Microwave and Optical Ray Geometry","author":"Cornbleet","year":"1984"},{"key":"ao-65-6-1841-R14","first-page":"45","volume-title":"Handbook of Lens Design","author":"Malacara","year":"2004"},{"key":"ao-65-6-1841-R15","first-page":"1","volume-title":"Introduction to Geometrical and Physical Optics","author":"Morgan","year":"1953"},{"key":"ao-65-6-1841-R16","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1017\/CBO9781139644181","volume-title":"Principles of Optics","author":"Born","year":"1999"},{"key":"ao-65-6-1841-R17","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1364\/AO.58.000939","type":"journal-article","volume":"58","author":"Avenda\u00f1o-Alejo","year":"2019","journal-title":"Appl. Opt."},{"key":"ao-65-6-1841-R18","doi-asserted-by":"publisher","first-page":"2370","DOI":"10.1364\/JOSAA.25.002370","type":"journal-article","volume":"25","author":"Shealy","year":"2008","journal-title":"J. Opt. Soc. Am. A"},{"key":"ao-65-6-1841-R19","doi-asserted-by":"publisher","first-page":"1312","DOI":"10.1364\/JOSAA.28.001312","type":"journal-article","volume":"28","author":"Hoffnagle","year":"2011","journal-title":"J. Opt. Soc. Am. A"},{"key":"ao-65-6-1841-R20","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1364\/AO.20.000897","type":"journal-article","volume":"20","author":"Burkhard","year":"1981","journal-title":"Appl. Opt."},{"key":"ao-65-6-1841-R21","doi-asserted-by":"publisher","first-page":"1576","DOI":"10.1364\/OPTCON.566241","type":"journal-article","volume":"4","author":"Jim\u00e9nez-Rodr\u00edguez","year":"2025","journal-title":"Opt. Continuum"},{"key":"ao-65-6-1841-R22","doi-asserted-by":"publisher","first-page":"3062","DOI":"10.1016\/j.optcom.2008.02.003","type":"journal-article","volume":"281","author":"Gitin","year":"2008","journal-title":"Opt. Commun."},{"key":"ao-65-6-1841-R23","first-page":"73","volume-title":"Aberrations of the Symmetrical Optical System","author":"Welford","year":"1974"}],"container-title":["Applied Optics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=ao-65-6-1841&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T14:22:12Z","timestamp":1778682132000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=ao-65-6-1841"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,13]]},"references-count":23,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/ao.588889","relation":{},"ISSN":["1559-128X","2155-3165"],"issn-type":[{"value":"1559-128X","type":"print"},{"value":"2155-3165","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,13]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/AO.588889","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":"29 December 2025","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"23 January 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"13 February 2026","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"\u00a9 2026 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.","name":"copyright","label":"Copyright"}]}}