{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T15:44:00Z","timestamp":1783007040810,"version":"3.54.5"},"reference-count":42,"publisher":"Wiley","issue":"9","license":[{"start":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:00:00Z","timestamp":1777939200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:00:00Z","timestamp":1777939200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S02297X\/1"],"award-info":[{"award-number":["EP\/S02297X\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Nanophotonics"],"published-print":{"date-parts":[[2026,5,13]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>We propose a new family of paraxial Gaussian modes, the Euler\u2010top Gaussian modes. Motivated by  angular momentum spin algebra, these modes are defined as eigenfunctions of a natural quadratic combination of spin\u2010like Fradkin\u2013Stokes parameters . Together with Hermite\u2013Gaussian, Laguerre\u2013Gaussian and Ince\u2013Gaussian modes, these new modes complete the natural (quadratic) families of Gaussian eigenmodes, up to  transformations. Euler\u2010top Gaussian modes with positive eigenvalues have a real Laguerre\u2013Gaussian appearance, and those with negative eigenvalues have a coupled Hermite\u2013Gaussian appearance with fourfold symmetry. Certain other modes quantise a separatrix state between the two regimes. Their structure is motivated by paths on the Gaussian ray\u2010orbital Poincar\u00e9 sphere, analogous to the polhode curves describing the angular momentum of an Euler top, that is, a classical rigid body with three different axes of inertia. Various properties of these new modes are considered, including their analogy to Lipkin\u2013Meshkov\u2013Glick modes from nuclear physics.<\/jats:p>","DOI":"10.1002\/nap2.70077","type":"journal-article","created":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T04:35:44Z","timestamp":1778042144000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Euler\u2010Top Gaussian Modes: Structured Beams From Quadratic Angular Momentum Dynamics"],"prefix":"10.1002","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1147-1804","authenticated-orcid":false,"given":"Mark R.","family":"Dennis","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy University of Birmingham  Birmingham UK"},{"name":"Centre for Doctoral Training in Topological Design University of Birmingham  Birmingham UK"},{"name":"H H Wills Physics Laboratory University of Bristol  Bristol UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kerr","family":"Maxwell","sequence":"additional","affiliation":[{"name":"School of Physics and Astronomy University of Birmingham  Birmingham UK"},{"name":"Centre for Doctoral Training in Topological Design University of Birmingham  Birmingham UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,5,5]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41566\u2010021\u201000780\u20104"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1088\/2040\u20108978\/19\/1\/013001"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.45.8185"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.45.1973"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1364\/JOSA.61.000446"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.48.656"},{"key":"e_1_2_8_8_1","doi-asserted-by":"publisher","DOI":"10.1364\/AOP.3.000161"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.522573"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.522574"},{"key":"e_1_2_8_11_1","volume-title":"Lasers","author":"Siegman A. E.","year":"1986"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1364\/OL.29.000144"},{"key":"e_1_2_8_13_1","doi-asserted-by":"publisher","DOI":"10.1088\/1367\u20102630\/ad17dc"},{"key":"e_1_2_8_14_1","volume-title":"A Treatise on the Analytical Dynamics of Particles and Rigid Bodies","author":"Whittaker E. T.","year":"1917"},{"key":"e_1_2_8_15_1","volume-title":"Classical Mechanics","author":"Goldstein H.","year":"1980"},{"key":"e_1_2_8_16_1","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2015.0441"},{"key":"e_1_2_8_17_1","doi-asserted-by":"publisher","DOI":"10.1364\/OPTICA.4.000476"},{"key":"e_1_2_8_18_1","doi-asserted-by":"publisher","DOI":"10.1088\/2515\u20107647\/ab011d"},{"key":"e_1_2_8_19_1","doi-asserted-by":"publisher","DOI":"10.1088\/2040\u20108986\/ab2c52"},{"key":"e_1_2_8_20_1","first-page":"229","volume-title":"Quantum Theory of Angular Momentum","author":"Schwinger J.","year":"1965"},{"key":"e_1_2_8_21_1","doi-asserted-by":"publisher","DOI":"10.1364\/OL.24.000430"},{"key":"e_1_2_8_22_1","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.16.002914"},{"key":"e_1_2_8_23_1","doi-asserted-by":"publisher","DOI":"10.1088\/1464\u20104258\/6\/5\/001"},{"key":"e_1_2_8_24_1","doi-asserted-by":"publisher","DOI":"10.1364\/OL.35.003535"},{"key":"e_1_2_8_25_1","doi-asserted-by":"publisher","DOI":"10.1088\/2040\u20108986\/ac3676"},{"key":"e_1_2_8_26_1","volume-title":"Geometric Optics on Phase Space","author":"Wolf K. B.","year":"2004"},{"key":"e_1_2_8_27_1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.107.L031502"},{"key":"e_1_2_8_28_1","first-page":"9","article-title":"Th\u00e9orie Nouvelle de la Rotation des Corps","volume":"16","author":"Poinsot L.","year":"1851","journal-title":"Journal de Mathematiques Pures et Appliquees"},{"key":"e_1_2_8_29_1","doi-asserted-by":"publisher","DOI":"10.1080\/09500348714551321"},{"key":"e_1_2_8_30_1","volume-title":"Geometric Quantization","author":"Woodhouse N. M. J.","year":"1998"},{"key":"e_1_2_8_31_1","unstructured":"K.Maxwell \u201cStructured Gaussians From Geometric Quantisation \u201d preprint (2025) https:\/\/arxiv.org\/abs\/2503.20775."},{"key":"e_1_2_8_32_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002\u20109947\u201096\u201001559\u20100"},{"key":"e_1_2_8_33_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1666449"},{"key":"e_1_2_8_34_1","doi-asserted-by":"publisher","DOI":"10.1080\/0020739920230607"},{"key":"e_1_2_8_35_1","unstructured":"H.Fong \u201cSquircular Calculations \u201d preprint (2016) https:\/\/arxiv.org\/abs\/1604.02174v1."},{"key":"e_1_2_8_36_1","volume-title":"Angular Momentum in Quantum Physics","author":"Biedenharn L. C.","year":"2010"},{"key":"e_1_2_8_37_1","volume-title":"Modern Quantum Mechanics","author":"Sakurai J. J.","year":"1993"},{"key":"e_1_2_8_38_1","doi-asserted-by":"publisher","DOI":"10.1016\/0029\u20105582(65)90862\u2010X"},{"key":"e_1_2_8_39_1","doi-asserted-by":"publisher","DOI":"10.1088\/0305\u20104470\/34\/15\/305"},{"key":"e_1_2_8_40_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41598\u2010018\u201020486\u2010y"},{"key":"e_1_2_8_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01036350"},{"key":"e_1_2_8_42_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0030\u20104018(00)00736\u20107"},{"key":"e_1_2_8_43_1","doi-asserted-by":"publisher","DOI":"10.1364\/OL.29.001870"}],"container-title":["Nanophotonics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nap2.70077","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/nap2.70077","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nap2.70077","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T13:05:59Z","timestamp":1778850359000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nap2.70077"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,5]]},"references-count":42,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2026,5,13]]}},"alternative-id":["10.1002\/nap2.70077"],"URL":"https:\/\/doi.org\/10.1002\/nap2.70077","archive":["Portico"],"relation":{},"ISSN":["2192-8614","2192-8614"],"issn-type":[{"value":"2192-8614","type":"print"},{"value":"2192-8614","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,5]]},"assertion":[{"value":"2025-12-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-03-12","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-05-05","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70077"}}