{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T15:41:44Z","timestamp":1785426104001,"version":"3.56.0"},"reference-count":24,"publisher":"Optica Publishing Group","issue":"11","license":[{"start":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T00:00:00Z","timestamp":1779840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/doi.org\/10.1364\/OA_License_v2#VOR-OA"},{"start":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T00:00:00Z","timestamp":1779840000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/opg.optica.org\/policies\/opg-tdm-policy.json"}],"funder":[{"name":"National Science and Technology Council","award":["112-2112-M-A49-022-MY3"],"award-info":[{"award-number":["112-2112-M-A49-022-MY3"]}],"id":[{"id":"https:\/\/ror.org\/02kv4zf79","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":["opg.optica.org"],"crossmark-restriction":false},"short-container-title":["Opt. Express"],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>The connection between the spherical harmonics and the stationary orbital states is theoretically established using the rotational transformation and the SU(2) algebra. Through the inverse quantum Fourier transform, the spherical harmonics can be decomposed into the superposition of the stationary orbital states. Using the partial sums of the decompositions, the gradual change of superpositions during the generation of spherical harmonics can be visualized. Furthermore, the expansion coefficients of hyperbolic sine functions are utilized to analytically derive the time-varying wave packet state corresponding to the inclined circular orbit. The orthogonality of Euler functions is further used to show that a periodic integration of the wave packet state can exactly generate the stationary orbital states. These derived formulas are expected to be used to explore the resonant modes of optical microsphere cavities.<\/jats:p>","DOI":"10.1364\/oe.601105","type":"journal-article","created":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T05:00:10Z","timestamp":1779426010000},"page":"20621","update-policy":"https:\/\/doi.org\/10.1364\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Connecting the spherical harmonics with the periodic orbits: application to optical microsphere cavities"],"prefix":"10.1364","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1435-5366","authenticated-orcid":true,"given":"Y. F.","family":"Chen","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00se2k293","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. X.","family":"Hu","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00se2k293","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Y. H.","family":"Fang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00se2k293","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. L.","family":"Hsieh","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00se2k293","id-type":"ROR","asserted-by":"publisher"}]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. C.","family":"Liang","sequence":"additional","affiliation":[{"name":"National Yang Ming Chiao Tung University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. C.","family":"Tung","sequence":"additional","affiliation":[{"name":"National Taipei University of Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"285","published-online":{"date-parts":[[2026,5,27]]},"reference":[{"key":"oe-34-11-20621-R1","doi-asserted-by":"publisher","first-page":"446","DOI":"10.1364\/JOSA.61.000446","type":"journal-article","volume":"61","author":"Feldmann","year":"1971","journal-title":"J. Opt. Soc. Am."},{"key":"oe-34-11-20621-R2","doi-asserted-by":"publisher","first-page":"1973","DOI":"10.1103\/PhysRevA.45.1973","type":"journal-article","volume":"45","author":"Danakas","year":"1992","journal-title":"Phys. Rev. A"},{"key":"oe-34-11-20621-R3","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-010-9711-6_6","article-title":"The relation between quantum mechanics, classical mechanics and optics","volume-title":"Quantum Mechanics","author":"Blokhintsev","year":"1964"},{"key":"oe-34-11-20621-R4","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1103\/PhysRev.88.625","type":"journal-article","volume":"88","author":"Moshinsky","year":"1952","journal-title":"Phys. Rev."},{"key":"oe-34-11-20621-R5","author":"Schr\u00f6dinger","year":"1928"},{"key":"oe-34-11-20621-R6","doi-asserted-by":"publisher","first-page":"2447","DOI":"10.1364\/JOSAB.6.002447","type":"journal-article","volume":"6","author":"Bu\u017eek","year":"1989","journal-title":"J. Opt. Soc. Am. B"},{"key":"oe-34-11-20621-R7","doi-asserted-by":"publisher","first-page":"3399","DOI":"10.1088\/0305-4470\/25\/11\/039","type":"journal-article","volume":"25","author":"De Bievre","year":"1992","journal-title":"J. Phys. A: Math. Gen."},{"key":"oe-34-11-20621-R8","doi-asserted-by":"publisher","first-page":"7751","DOI":"10.1088\/0305-4470\/36\/28\/305","type":"journal-article","volume":"36","author":"Chen","year":"2003","journal-title":"J. Phys. A: Math. Gen."},{"key":"oe-34-11-20621-R9","doi-asserted-by":"publisher","first-page":"032124","DOI":"10.1103\/PhysRevA.83.032124","type":"journal-article","volume":"83","author":"Chen","year":"2011","journal-title":"Phys. Rev. A"},{"key":"oe-34-11-20621-R10","doi-asserted-by":"publisher","first-page":"1600253","DOI":"10.1002\/andp.201600253","type":"journal-article","volume":"529","author":"Chen","year":"2017","journal-title":"Ann. Phys."},{"key":"oe-34-11-20621-R11","doi-asserted-by":"publisher","first-page":"30002","DOI":"10.1209\/0295-5075\/122\/30002","type":"journal-article","volume":"122","author":"Chen","year":"2018","journal-title":"EPL"},{"key":"oe-34-11-20621-R12","doi-asserted-by":"publisher","first-page":"22711","DOI":"10.1364\/OE.555913","type":"journal-article","volume":"33","author":"Wang","year":"2025","journal-title":"Opt. Express"},{"key":"oe-34-11-20621-R13","doi-asserted-by":"publisher","first-page":"073001","DOI":"10.1088\/2040-8986\/addf5e","type":"journal-article","volume":"27","author":"Omatsu","year":"2025","journal-title":"J. Opt."},{"key":"oe-34-11-20621-R14","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1364\/OPTICA.477076","type":"journal-article","volume":"10","author":"Jia","year":"2023","journal-title":"Optica"},{"key":"oe-34-11-20621-R15","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1364\/OPTCON.484492","type":"journal-article","volume":"2","author":"Hsieh","year":"2023","journal-title":"Opt. Continuum"},{"key":"oe-34-11-20621-R16","doi-asserted-by":"publisher","first-page":"5209","DOI":"10.1364\/OE.554614","type":"journal-article","volume":"33","author":"Chen","year":"2025","journal-title":"Opt. Express"},{"key":"oe-34-11-20621-R17","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.optcom.2018.06.017","type":"journal-article","volume":"427","author":"Wang","year":"2018","journal-title":"Opt. Commun."},{"key":"oe-34-11-20621-R18","doi-asserted-by":"publisher","first-page":"131233","DOI":"10.1016\/j.optcom.2024.131233","type":"journal-article","volume":"574","author":"Ma","year":"2025","journal-title":"Opt. Commun."},{"key":"oe-34-11-20621-R19","doi-asserted-by":"publisher","first-page":"1434","DOI":"10.1021\/acsphotonics.3c00041","type":"journal-article","volume":"10","author":"Machfuudzoh","year":"2023","journal-title":"ACS Photonics"},{"key":"oe-34-11-20621-R20","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/S0034-4877(15)60023-4","type":"journal-article","volume":"75","author":"Mojaveri","year":"2015","journal-title":"Rep. Math. Phys."},{"key":"oe-34-11-20621-R21","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1140\/epjp\/i2014-14164-5","type":"journal-article","volume":"129","author":"Calzada","year":"2014","journal-title":"Eur. Phys. J. Plus"},{"key":"oe-34-11-20621-R22","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1098\/rsif.2005.0031","type":"journal-article","volume":"2","author":"Galinski","year":"2005","journal-title":"J. R. Soc. Inter."},{"key":"oe-34-11-20621-R23","doi-asserted-by":"publisher","first-page":"1448","DOI":"10.1002\/mrm.20275","type":"journal-article","volume":"52","author":"Welch","year":"2004","journal-title":"Magn. Reson. Med."},{"key":"oe-34-11-20621-R24","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/j.mri.2009.07.010","type":"journal-article","volume":"28","author":"Costa","year":"2010","journal-title":"Magn. Reson. Imaging"}],"container-title":["Optics Express"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/opg.optica.org\/viewmedia.cfm?URI=oe-34-11-20621&seq=0","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T17:34:17Z","timestamp":1779903257000},"score":1,"resource":{"primary":{"URL":"https:\/\/opg.optica.org\/abstract.cfm?URI=oe-34-11-20621"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,27]]},"references-count":24,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2026]]},"published-print":{"date-parts":[[2026]]}},"URL":"https:\/\/doi.org\/10.1364\/oe.601105","relation":{},"ISSN":["1094-4087"],"issn-type":[{"value":"1094-4087","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,27]]},"assertion":[{"value":"Optica Publishing Group","name":"publisher","label":"This article is maintained by"},{"value":"https:\/\/doi.org\/10.1364\/OE.601105","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":"7 April 2026","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"20 May 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"27 May 2026","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}},{"value":"\u00a9 2026 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement","name":"copyright","label":"Copyright"}]}}