{"id":"https://openalex.org/W2970692298","doi":"https://doi.org/10.3390/axioms8030101","title":"Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes","display_name":"Mathematical and Numerical Modeling of On-Threshold Modes of 2-D Microcavity Lasers with Piercing Holes","publication_year":2019,"publication_date":"2019-09-01","ids":{"openalex":"https://openalex.org/W2970692298","doi":"https://doi.org/10.3390/axioms8030101","mag":"2970692298"},"language":"en","primary_location":{"id":"doi:10.3390/axioms8030101","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms8030101","pdf_url":"https://www.mdpi.com/2075-1680/8/3/101/pdf?version=1568286500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/8/3/101/pdf?version=1568286500","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003336167","display_name":"Alexander O. Spiridonov","orcid":"https://orcid.org/0000-0003-4043-7648"},"institutions":[{"id":"https://openalex.org/I21203515","display_name":"Kazan Federal University","ror":"https://ror.org/05256ym39","country_code":"RU","type":"education","lineage":["https://openalex.org/I21203515"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"Alexander O. Spiridonov","raw_affiliation_strings":["Laboratory of Computational Technologies and Computer Modeling, Kazan Federal University, 18 Kremlevskaya st., 420008 Kazan, Russia"],"raw_orcid":"https://orcid.org/0000-0003-4043-7648","affiliations":[{"raw_affiliation_string":"Laboratory of Computational Technologies and Computer Modeling, Kazan Federal University, 18 Kremlevskaya st., 420008 Kazan, Russia","institution_ids":["https://openalex.org/I21203515"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018340199","display_name":"Evgenii M. Karchevskii","orcid":"https://orcid.org/0000-0001-6931-425X"},"institutions":[{"id":"https://openalex.org/I21203515","display_name":"Kazan Federal University","ror":"https://ror.org/05256ym39","country_code":"RU","type":"education","lineage":["https://openalex.org/I21203515"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Evgenii M. Karchevskii","raw_affiliation_strings":["Department of Applied Mathematics, Kazan Federal University, 18 Kremlevskaya st., 420008 Kazan, Russia"],"raw_orcid":"https://orcid.org/0000-0001-6931-425X","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, Kazan Federal University, 18 Kremlevskaya st., 420008 Kazan, Russia","institution_ids":["https://openalex.org/I21203515"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034402440","display_name":"Alexander I. Nosich","orcid":"https://orcid.org/0000-0002-3446-8168"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alexander I. Nosich","raw_affiliation_strings":["Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, vul. Proskury 12, 61085 Kharkiv, Ukraine"],"raw_orcid":"https://orcid.org/0000-0002-3446-8168","affiliations":[{"raw_affiliation_string":"Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, vul. Proskury 12, 61085 Kharkiv, Ukraine","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5003336167"],"corresponding_institution_ids":["https://openalex.org/I21203515"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.9144,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.73139813,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"8","issue":"3","first_page":"101","last_page":"101"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10299","display_name":"Photonic and Optical Devices","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10299","display_name":"Photonic and Optical Devices","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11429","display_name":"Semiconductor Lasers and Optical Devices","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10666","display_name":"Photonic Crystals and Applications","score":0.9993000030517578,"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/lasing-threshold","display_name":"Lasing threshold","score":0.8518631458282471},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6468896865844727},{"id":"https://openalex.org/keywords/laser","display_name":"Laser","score":0.5774897933006287},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5650151371955872},{"id":"https://openalex.org/keywords/spontaneous-emission","display_name":"Spontaneous emission","score":0.5180918574333191},{"id":"https://openalex.org/keywords/directivity","display_name":"Directivity","score":0.44637688994407654},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.42757293581962585},{"id":"https://openalex.org/keywords/equilateral-triangle","display_name":"Equilateral triangle","score":0.4272312521934509},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.4185446500778198},{"id":"https://openalex.org/keywords/radiative-transfer","display_name":"Radiative transfer","score":0.4116935133934021},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.3963761627674103},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.33260291814804077},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.29661786556243896},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.23870006203651428},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17871510982513428},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.149783194065094},{"id":"https://openalex.org/keywords/telecommunications","display_name":"Telecommunications","score":0.11401408910751343}],"concepts":[{"id":"https://openalex.org/C40637687","wikidata":"https://www.wikidata.org/wiki/Q676237","display_name":"Lasing threshold","level":3,"score":0.8518631458282471},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6468896865844727},{"id":"https://openalex.org/C520434653","wikidata":"https://www.wikidata.org/wiki/Q38867","display_name":"Laser","level":2,"score":0.5774897933006287},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5650151371955872},{"id":"https://openalex.org/C111154242","wikidata":"https://www.wikidata.org/wiki/Q923545","display_name":"Spontaneous emission","level":3,"score":0.5180918574333191},{"id":"https://openalex.org/C189039984","wikidata":"https://www.wikidata.org/wiki/Q8225412","display_name":"Directivity","level":3,"score":0.44637688994407654},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.42757293581962585},{"id":"https://openalex.org/C146786541","wikidata":"https://www.wikidata.org/wiki/Q157002","display_name":"Equilateral triangle","level":2,"score":0.4272312521934509},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.4185446500778198},{"id":"https://openalex.org/C74902906","wikidata":"https://www.wikidata.org/wiki/Q1190858","display_name":"Radiative transfer","level":2,"score":0.4116935133934021},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.3963761627674103},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33260291814804077},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.29661786556243896},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.23870006203651428},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17871510982513428},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.149783194065094},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.11401408910751343},{"id":"https://openalex.org/C21822782","wikidata":"https://www.wikidata.org/wiki/Q131214","display_name":"Antenna (radio)","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms8030101","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms8030101","pdf_url":"https://www.mdpi.com/2075-1680/8/3/101/pdf?version=1568286500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:b250b4ac76824af7ae1a7b329e1237e9","is_oa":true,"landing_page_url":"https://doaj.org/article/b250b4ac76824af7ae1a7b329e1237e9","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 8, Iss 3, p 101 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/8/3/101/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/axioms8030101","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms8030101","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms8030101","pdf_url":"https://www.mdpi.com/2075-1680/8/3/101/pdf?version=1568286500","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2970692298.pdf","grobid_xml":"https://content.openalex.org/works/W2970692298.grobid-xml"},"referenced_works_count":53,"referenced_works":["https://openalex.org/W169455514","https://openalex.org/W1148034687","https://openalex.org/W1150036060","https://openalex.org/W1517670975","https://openalex.org/W1523113578","https://openalex.org/W1674614866","https://openalex.org/W1737832913","https://openalex.org/W1908688112","https://openalex.org/W1968327152","https://openalex.org/W1976485542","https://openalex.org/W1979536342","https://openalex.org/W1981641894","https://openalex.org/W1995004795","https://openalex.org/W1995158137","https://openalex.org/W1999805625","https://openalex.org/W2003966383","https://openalex.org/W2007945079","https://openalex.org/W2007995322","https://openalex.org/W2014830259","https://openalex.org/W2015785236","https://openalex.org/W2050732801","https://openalex.org/W2056346692","https://openalex.org/W2069206540","https://openalex.org/W2070385980","https://openalex.org/W2073855934","https://openalex.org/W2086310962","https://openalex.org/W2087800750","https://openalex.org/W2092327885","https://openalex.org/W2107806447","https://openalex.org/W2116470633","https://openalex.org/W2137346879","https://openalex.org/W2142008542","https://openalex.org/W2151738053","https://openalex.org/W2319274180","https://openalex.org/W2491055108","https://openalex.org/W2491168606","https://openalex.org/W2500975155","https://openalex.org/W2503693487","https://openalex.org/W2561899043","https://openalex.org/W2625997139","https://openalex.org/W2628447719","https://openalex.org/W2790373456","https://openalex.org/W2887536216","https://openalex.org/W2896115574","https://openalex.org/W2911859729","https://openalex.org/W2913747892","https://openalex.org/W2913830140","https://openalex.org/W2913867926","https://openalex.org/W4249973082","https://openalex.org/W4292027041","https://openalex.org/W4292361795","https://openalex.org/W6677156936","https://openalex.org/W6758916771"],"related_works":["https://openalex.org/W3204326392","https://openalex.org/W2508087100","https://openalex.org/W3161792707","https://openalex.org/W2500000936","https://openalex.org/W2086778274","https://openalex.org/W1989339034","https://openalex.org/W1984515769","https://openalex.org/W117604103","https://openalex.org/W4317722026","https://openalex.org/W3129462641"],"abstract_inverted_index":{"This":[0,63],"study":[1],"considers":[2],"the":[3,9,13,18,58,67,70,73,82,91,97,102,123,141,146,159,162,166,169,173],"mathematical":[4],"analysis":[5],"framework":[6,89],"aimed":[7],"at":[8],"adequate":[10],"description":[11],"of":[12,15,20,57,60,66,72,113,143,148,161,168,175],"modes":[14],"lasers":[16,27],"on":[17],"threshold":[19,103,142],"non-attenuated":[21],"in":[22,39],"time":[23],"light":[24,149],"emission.":[25],"The":[26],"are":[28],"viewed":[29],"as":[30,127],"open":[31],"dielectric":[32],"resonators":[33],"equipped":[34],"with":[35,40,81,132,158],"active":[36,129],"regions,":[37],"filled":[38],"gain":[41,104],"material.":[42],"We":[43],"introduce":[44],"a":[45,133],"generalized":[46],"complex-frequency":[47],"eigenvalue":[48,94],"problem":[49,68],"for":[50,151],"such":[51],"cavities":[52],"and":[53,78,101,145],"prove":[54],"important":[55],"properties":[56],"spectrum":[59],"its":[61],"eigensolutions.":[62],"involves":[64],"reduction":[65],"to":[69,122],"set":[71],"Muller":[74],"boundary":[75],"integral":[76],"equations":[77],"their":[79],"discretization":[80],"Nystrom":[83],"technique.":[84],"Embedded":[85],"into":[86],"this":[87],"general":[88],"is":[90,138],"application-oriented":[92],"lasing":[93,144],"problem,":[95],"where":[96],"real":[98],"emission":[99,176],"frequencies":[100],"values":[105],"together":[106],"form":[107],"two-component":[108],"eigenvalues.":[109],"As":[110],"an":[111,128],"example":[112],"on-threshold":[114],"mode":[115],"study,":[116],"we":[117],"present":[118],"numerical":[119],"results":[120],"related":[121],"two-dimensional":[124],"laser":[125],"shaped":[126],"equilateral":[130],"triangle":[131],"round":[134],"piercing":[135,170],"hole.":[136],"It":[137],"demonstrated":[139],"that":[140],"directivity":[147],"emission,":[150],"each":[152],"mode,":[153],"can":[154],"be":[155],"efficiently":[156],"manipulated":[157],"aid":[160],"size":[163],"and,":[164],"especially,":[165],"placement":[167],"hole,":[171],"while":[172],"frequency":[174],"remains":[177],"largely":[178],"intact.":[179]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1}],"updated_date":"2026-07-30T17:31:21.811387","created_date":"2025-10-10T00:00:00"}
