{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T17:08:25Z","timestamp":1785431305516,"version":"3.56.0"},"reference-count":35,"publisher":"Optica Publishing Group","issue":"7","license":[{"start":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T00:00:00Z","timestamp":1774569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR-OA"},{"start":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T00:00:00Z","timestamp":1774569600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2021YFA1400700"],"award-info":[{"award-number":["2021YFA1400700"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62105320"],"award-info":[{"award-number":["62105320"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"},{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"name":"CAS Project for Young Scientists in Basic Research","award":["YSBR-049"],"award-info":[{"award-number":["YSBR-049"]}]},{"DOI":"10.13039\/501100017668","name":"Key Research and Development Program of Anhui Province","doi-asserted-by":"crossref","award":["202304a05020013"],"award-info":[{"award-number":["202304a05020013"]}],"id":[{"id":"10.13039\/501100017668","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Opt. Express"],"published-print":{"date-parts":[[2026,4,6]]},"abstract":"<jats:p>\n                    We present a method for nanoscale displacement sensing based on robust spin\u2013orbit conversion in tightly focused beams carrying net spin angular momentum. This mechanism naturally defines a two-dimensional linear ruler in the vicinity of the zeros of the complex longitudinal field component\n                    <jats:italic>E<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>z<\/jats:italic>\n                    <\/jats:sub>\n                    , which are surrounded by a first-order topological phase vortex. In this region, the real and imaginary parts of\n                    <jats:italic>E<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>z<\/jats:italic>\n                    <\/jats:sub>\n                    vary linearly and orthogonally with lateral displacements along the x and y directions without any beam shaping or resonant nanostructures. Experiments using a single gold nanoparticle and polarization-resolved back-focal-plane imaging confirm displacement sensing with a resolution below 1 nm. We further show that this linear ruler remains functional under strong optical aberrations. Our approach, therefore, offers a universal, resonance-free, and experimentally simple route to displacement metrology in complex and noisy environments\n                  <\/jats:p>","DOI":"10.1364\/oe.585476","type":"journal-article","created":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T06:00:06Z","timestamp":1773640806000},"page":"12507","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Robust linear ruler near a longitudinal optical vortex for sub-nanometer displacement sensing of a single nanoparticle"],"prefix":"10.1364","volume":"34","author":[{"given":"Han","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-3137-9314","authenticated-orcid":true,"given":"Di","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zheng","family":"Xi","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,3,27]]},"reference":[{"key":"oe-34-7-12507-R1","doi-asserted-by":"publisher","first-page":"061102","DOI":"10.1103\/PhysRevLett.116.061102","type":"journal-article","volume":"116","author":"Abbott","year":"2016","journal-title":"Phys. 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