{"id":"https://openalex.org/W7221018155","doi":"https://doi.org/10.1364/oe.618209","title":"Lie algebra of transverse-magnetic waves in one-dimensional layered media","display_name":"Lie algebra of transverse-magnetic waves in one-dimensional layered media","publication_year":2026,"publication_date":"2026-10-08","ids":{"openalex":"https://openalex.org/W7221018155","doi":"https://doi.org/10.1364/oe.618209","pmid":"https://pubmed.ncbi.nlm.nih.gov/42972705"},"language":"en","primary_location":{"id":"doi:10.1364/oe.618209","doi":"https://doi.org/10.1364/oe.618209","is_oa":true,"landing_page_url":"https://doi.org/10.1364/oe.618209","pdf_url":null,"source":{"id":"https://openalex.org/S178776955","display_name":"Optics Express","issn_l":"1094-4087","issn":["1094-4087"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-1","ki-jl-1","medline","norway-2","oarl","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315679","host_organization_name":"Optica Publishing Group","host_organization_lineage":["https://openalex.org/P4310315679"],"host_organization_lineage_names":["Optica Publishing Group"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optics Express","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","datacite","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.1364/oe.618209","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110225514","display_name":"Yong Xu","orcid":"https://orcid.org/0000-0001-5849-032X","observed_orcids":["https://orcid.org/0000-0001-5849-032X"]},"institutions":[{"id":"https://openalex.org/I859038795","display_name":"Virginia Tech","ror":"https://ror.org/02smfhw86","country_code":"US","type":"education","lineage":["https://openalex.org/I859038795"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Yong Xu","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, USA"],"raw_orcid":"https://orcid.org/0000-0001-5849-032X","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, USA","institution_ids":["https://openalex.org/I859038795"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110225514"],"corresponding_institution_ids":["https://openalex.org/I859038795"],"apc_list":{"value":2270,"currency":"USD","value_usd":2270},"apc_paid":{"value":2270,"currency":"USD","value_usd":2270},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.55352498,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"34","issue":"22","first_page":"41006","last_page":"41006"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10245","display_name":"Metamaterials and Metasurfaces Applications","score":0.6704000234603882,"subfield":{"id":"https://openalex.org/subfields/2504","display_name":"Electronic, Optical and Magnetic Materials"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10245","display_name":"Metamaterials and Metasurfaces Applications","score":0.6704000234603882,"subfield":{"id":"https://openalex.org/subfields/2504","display_name":"Electronic, Optical and Magnetic Materials"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10666","display_name":"Photonic Crystals and Applications","score":0.25999999046325684,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10295","display_name":"Plasmonic and Surface Plasmon Research","score":0.02019999921321869,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/transverse-magnetic-waves","display_name":"transverse magnetic waves","score":0.972000002861023},{"id":"https://openalex.org/keywords/multilayer-structures","display_name":"multilayer structures","score":0.8510000109672546},{"id":"https://openalex.org/keywords/infinitesimal-generator","display_name":"infinitesimal generator","score":0.8500000238418579},{"id":"https://openalex.org/keywords/dielectric-materials","display_name":"dielectric materials","score":0.8640000224113464},{"id":"https://openalex.org/keywords/polarization-angle","display_name":"polarization angle","score":0.9929999709129333},{"id":"https://openalex.org/keywords/photonic-band-gap","display_name":"photonic band gap","score":0.9909999966621399},{"id":"https://openalex.org/keywords/inverse-design","display_name":"inverse design","score":0.9670000076293945},{"id":"https://openalex.org/keywords/optical-response","display_name":"optical response","score":0.5979999899864197}],"concepts":[{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.6355999708175659},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.48260000348091125},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.45089998841285706},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.43380001187324524},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.32679998874664307},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.32010000944137573},{"id":"https://openalex.org/C44886760","wikidata":"https://www.wikidata.org/wiki/Q1758425","display_name":"Wave propagation","level":2,"score":0.31150001287460327},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2957000136375427},{"id":"https://openalex.org/C154236941","wikidata":"https://www.wikidata.org/wiki/Q942347","display_name":"Physical optics","level":2,"score":0.29490000009536743},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.288100004196167},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2854999899864197},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2806999981403351},{"id":"https://openalex.org/C110277352","wikidata":"https://www.wikidata.org/wiki/Q467980","display_name":"Geometrical optics","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C42067758","wikidata":"https://www.wikidata.org/wiki/Q174102","display_name":"Refractive index","level":2,"score":0.2777000069618225},{"id":"https://openalex.org/C2988995629","wikidata":"https://www.wikidata.org/wiki/Q2915729","display_name":"Matrix algebra","level":3,"score":0.271699994802475},{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2615000009536743},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.25769999623298645},{"id":"https://openalex.org/C19819980","wikidata":"https://www.wikidata.org/wiki/Q947542","display_name":"Beam splitter","level":3,"score":0.2540000081062317},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.2526000142097473}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1364/oe.618209","doi":"https://doi.org/10.1364/oe.618209","is_oa":true,"landing_page_url":"https://doi.org/10.1364/oe.618209","pdf_url":null,"source":{"id":"https://openalex.org/S178776955","display_name":"Optics Express","issn_l":"1094-4087","issn":["1094-4087"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-1","ki-jl-1","medline","norway-2","oarl","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315679","host_organization_name":"Optica Publishing Group","host_organization_lineage":["https://openalex.org/P4310315679"],"host_organization_lineage_names":["Optica Publishing Group"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optics Express","raw_type":"journal-article"},{"id":"pmid:42972705","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/42972705","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optics express","raw_type":"Journal Article"},{"id":"doi:10.6084/m9.figshare.c.8755899","doi":"https://doi.org/10.6084/m9.figshare.c.8755899","is_oa":true,"landing_page_url":"https://doi.org/10.6084/m9.figshare.c.8755899","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Collection"},{"id":"doi:10.6084/m9.figshare.c.8755899.v1","doi":"https://doi.org/10.6084/m9.figshare.c.8755899.v1","is_oa":true,"landing_page_url":"https://doi.org/10.6084/m9.figshare.c.8755899.v1","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Collection"}],"best_oa_location":{"id":"doi:10.1364/oe.618209","doi":"https://doi.org/10.1364/oe.618209","is_oa":true,"landing_page_url":"https://doi.org/10.1364/oe.618209","pdf_url":null,"source":{"id":"https://openalex.org/S178776955","display_name":"Optics Express","issn_l":"1094-4087","issn":["1094-4087"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-b","jufo-1","ki-jl-1","medline","norway-2","oarl","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315679","host_organization_name":"Optica Publishing Group","host_organization_lineage":["https://openalex.org/P4310315679"],"host_organization_lineage_names":["Optica Publishing Group"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Optics Express","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1429299901","https://openalex.org/W2015760100","https://openalex.org/W2049161806","https://openalex.org/W2065469303","https://openalex.org/W2072537188","https://openalex.org/W2077423587","https://openalex.org/W2133483681","https://openalex.org/W2290995438","https://openalex.org/W2899580696","https://openalex.org/W3212803744","https://openalex.org/W4380791807","https://openalex.org/W4406867917","https://openalex.org/W4415150714","https://openalex.org/W4415164699"],"related_works":["https://openalex.org/W2023064924","https://openalex.org/W2099547810","https://openalex.org/W2952701101","https://openalex.org/W2345509676","https://openalex.org/W3140773821","https://openalex.org/W3081183381","https://openalex.org/W2090401562","https://openalex.org/W2077813793","https://openalex.org/W4410086270","https://openalex.org/W4241266723"],"abstract_inverted_index":{"We":[0,14,62],"present":[1],"an":[2,65],"algebraic":[3],"framework":[4],"for":[5,68,80],"analyzing":[6],"transverse-magnetic":[7],"(TM)":[8],"waves":[9],"in":[10],"one-dimensional":[11],"multilayer":[12],"structures.":[13],"show":[15],"that":[16],"their":[17],"behavior":[18],"is":[19],"governed":[20],"by":[21],"the":[22,41,59],"underlying":[23],"Lie":[24],"algebra":[25],"w":[26,45],"\u2245":[27],"sl":[28],"(2,":[29],"R":[30],")":[31],"and":[32,40,53,76],"identify":[33],"a":[34,50,72],"natural":[35],"correspondence":[36],"between":[37],"dielectric":[38],"media":[39],"infinitesimal":[42],"generators":[43],"of":[44,49],".":[46],"The":[47],"existence":[48],"Brewster":[51],"angle":[52],"photonic":[54],"bandgaps":[55],"emerges":[56],"naturally":[57],"from":[58,71],"Lie-algebraic":[60],"structure.":[61],"also":[63],"derive":[64],"analytical":[66],"formula":[67],"inverse":[69],"design":[70],"prescribed":[73],"optical":[74],"response":[75],"demonstrate":[77],"its":[78],"potential":[79],"invisibility":[81],"engineering.":[82]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-09T00:00:00"}
