{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:44:24Z","timestamp":1785437064604,"version":"3.56.0"},"reference-count":19,"publisher":"Optica Publishing Group","issue":"14","license":[{"start":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:00:00Z","timestamp":1783900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR"},{"start":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:00:00Z","timestamp":1783900800000},"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":["Opt. Lett."],"published-print":{"date-parts":[[2026,7,15]]},"abstract":"<jats:p>\n                    Conventional diffractive deep neural networks (D\n                    <jats:sup>2<\/jats:sup>\n                    NNs) rely on discretized phase distributions, which often lead to fabrication challenges, significant speckle noise, and sensitivity to perturbations. To overcome these limitations, we propose a D\n                    <jats:sup>2<\/jats:sup>\n                    NN architecture based on smoothly freeform surfaces with C\n                    <jats:sup>2<\/jats:sup>\n                    continuity. Specifically, we employ multi-level cubic B\u2011splines to represent the phase distribution of each diffractive layer, which naturally yields a smooth phase profile while preserving high design freedom. An explicit curvature constraint is further imposed to regularize the surface curvature, which facilitates fabrication by, e.g., diamond turning. Results show that, compared with a discrete-phase D\n                    <jats:sup>2<\/jats:sup>\n                    NN, our method even achieves a slightly higher accuracy of 95.97% on MNIST, while being more robust against alignment errors.\n                  <\/jats:p>","DOI":"10.1364\/ol.605776","type":"journal-article","created":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T18:00:09Z","timestamp":1782324009000},"page":"4020","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Freeform diffractive deep neural networks with intrinsically smooth layers plus a curvature constraint"],"prefix":"10.1364","volume":"51","author":[{"given":"Haijian","family":"Wang","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01skt4w74","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Beijing Institute of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zexin","family":"Feng","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01skt4w74","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Beijing Institute of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Luo","sequence":"additional","affiliation":[{"name":"Tsinghua University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,7,13]]},"reference":[{"key":"ol-51-14-4020-R1","doi-asserted-by":"publisher","first-page":"eadu0904","DOI":"10.1126\/sciadv.adu0904","type":"journal-article","volume":"11","author":"Wang","year":"2025","journal-title":"Sci. Adv."},{"key":"ol-51-14-4020-R2","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1038\/s41377-024-01543-w","type":"journal-article","volume":"13","author":"Bai","year":"2024","journal-title":"Light Sci. Appl."},{"key":"ol-51-14-4020-R3","doi-asserted-by":"publisher","first-page":"eadn2205","DOI":"10.1126\/sciadv.adn2205","type":"journal-article","volume":"10","author":"Zhang","year":"2024","journal-title":"Sci. Adv."},{"key":"ol-51-14-4020-R4","doi-asserted-by":"publisher","first-page":"2667","DOI":"10.1364\/PRJ.474535","type":"journal-article","volume":"10","author":"Zheng","year":"2022","journal-title":"Photonics Res."},{"key":"ol-51-14-4020-R5","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1126\/science.aat8084","type":"journal-article","volume":"361","author":"Lin","year":"2018","journal-title":"Science"},{"key":"ol-51-14-4020-R6","doi-asserted-by":"publisher","first-page":"1483","DOI":"10.1016\/j.eng.2020.07.032","type":"journal-article","volume":"7","author":"Chen","year":"2021","journal-title":"Engineering"},{"key":"ol-51-14-4020-R7","first-page":"1","author":"Zhou","year":"2023"},{"key":"ol-51-14-4020-R8","doi-asserted-by":"publisher","first-page":"130267","DOI":"10.1016\/j.optcom.2024.130267","type":"journal-article","volume":"556","author":"Wu","year":"2024","journal-title":"Opt. Commun."},{"key":"ol-51-14-4020-R9","doi-asserted-by":"publisher","first-page":"14501","DOI":"10.1364\/OE.558279","type":"journal-article","volume":"33","author":"Wang","year":"2025","journal-title":"Opt. Express"},{"key":"ol-51-14-4020-R10","doi-asserted-by":"publisher","first-page":"19298","DOI":"10.1364\/OE.598903","type":"journal-article","volume":"34","author":"Wang","year":"2026","journal-title":"Opt. Express"},{"key":"ol-51-14-4020-R11","doi-asserted-by":"publisher","first-page":"9364","DOI":"10.1364\/AO.54.009364","type":"journal-article","volume":"54","author":"Feng","year":"2015","journal-title":"Appl. Opt."},{"key":"ol-51-14-4020-R12","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1364\/OPTICA.395177","type":"journal-article","volume":"7","author":"Schmidt","year":"2020","journal-title":"Optica"},{"key":"ol-51-14-4020-R13","doi-asserted-by":"publisher","first-page":"31875","DOI":"10.1364\/OE.439641","type":"journal-article","volume":"29","author":"Doskolovich","year":"2021","journal-title":"Opt. Express"},{"key":"ol-51-14-4020-R14","doi-asserted-by":"publisher","first-page":"41041","DOI":"10.1364\/OE.533298","type":"journal-article","volume":"32","author":"Liao","year":"2024","journal-title":"Opt. Express"},{"key":"ol-51-14-4020-R15","doi-asserted-by":"publisher","first-page":"046001","DOI":"10.1117\/1.AP.1.4.046001","type":"journal-article","volume":"1","author":"Li","year":"2019","journal-title":"Adv. Photonics"},{"key":"ol-51-14-4020-R16","doi-asserted-by":"publisher","first-page":"6398","DOI":"10.1364\/AO.567302","type":"journal-article","volume":"64","author":"Li","year":"2025","journal-title":"Appl. Opt."},{"key":"ol-51-14-4020-R17","author":"Piegl","year":"1996"},{"key":"ol-51-14-4020-R18","doi-asserted-by":"publisher","first-page":"2415","DOI":"10.1364\/JOSAA.29.002415","type":"journal-article","volume":"29","author":"Yu","year":"2012","journal-title":"J. Opt. Soc. Am. A"},{"key":"ol-51-14-4020-R19","doi-asserted-by":"publisher","first-page":"113724","DOI":"10.1016\/j.optlastec.2025.113724","type":"journal-article","volume":"192","author":"Cai","year":"2025","journal-title":"Opt. Laser Technol."}],"container-title":["Optics Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=ol-51-14-4020&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T14:42:46Z","timestamp":1783953766000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=ol-51-14-4020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,13]]},"references-count":19,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/ol.605776","relation":{},"ISSN":["0146-9592","1539-4794"],"issn-type":[{"value":"0146-9592","type":"print"},{"value":"1539-4794","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,13]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/OL.605776","name":"articlelink","label":"Crossref DOI link to publisher maintained version"},{"value":"letter","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":"14 May 2026","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"23 June 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"13 July 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"}]}}