{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T06:49:45Z","timestamp":1771051785915,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T00:00:00Z","timestamp":1751328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Graphene interfaces in layered dielectrics can support unique electromagnetic modes, but analyzing these modes requires robust computational techniques. This work presents a numerical method for computing TE-polarized eigenmodes in a planar stratified dielectric slab with an infinitesimally thin graphene sheet at its interface. The governing boundary-value problem is reformulated as coupled initial-value problems and solved via a customized shooting method, enabling accurate calculation of complex propagation constants and field profiles despite the discontinuity at the graphene layer. We demonstrate that the graphene significantly alters the modal spectrum, introducing complex leaky and surface waves with attenuation due to graphene\u2019s conductivity. Numerical results illustrate how the layers\u2019 inhomogeneity and the graphene\u2019s surface conductivity influence mode confinement and loss. These findings confirm the robustness of the proposed computational approach and provide insights relevant to the design and analysis of graphene-based waveguiding devices.<\/jats:p>","DOI":"10.3390\/computation13070157","type":"journal-article","created":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T06:56:54Z","timestamp":1751353014000},"page":"157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Numerical Modeling of Electromagnetic Modes in a Planar Stratified Medium with a Graphene Interface"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0168-0282","authenticated-orcid":false,"given":"Eugen","family":"Smolkin","sequence":"first","affiliation":[{"name":"Academy of Technology and Environment, University of G\u00e4vle, 801 76 G\u00e4vle, Sweden"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Guo, S., Chen, J., Zhang, Y., and Liu, J. (2021). Graphene-Based Films: Fabrication, Interfacial Modification, and Applications. Nanomaterials, 11.","DOI":"10.3390\/nano11102539"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2971","DOI":"10.1007\/s11434-012-5270-4","article-title":"Nonlinear optical properties of graphene-based materials","volume":"57","author":"Liu","year":"2012","journal-title":"Chin. Sci. Bull."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2587","DOI":"10.1109\/TMTT.2011.2163168","article-title":"Towards a unified approach to electromagnetic fields and quantum currents from Dirac spinors","volume":"59","author":"Rozzi","year":"2011","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Marconcini, P., and Macucci, M. (2022). Transport simulation of graphene devices with a generic potential in the presence of an orthogonal magnetic field. Nanomaterials, 12.","DOI":"10.3390\/nano12071087"},{"key":"ref_5","unstructured":"Heydari, M.B., and Samiei, M.H.V. (2018). Plasmonic Graphene Waveguides: A Literature Review. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"100834","DOI":"10.1016\/j.photonics.2020.100834","article-title":"Novel analytical model of anisotropic multi-layer cylindrical waveguides incorporating graphene layers","volume":"42","author":"Heydari","year":"2020","journal-title":"Photonics Nanostructures\u2014Fundam. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"245435","DOI":"10.1103\/PhysRevB.80.245435","article-title":"Plasmonics in graphene at infrared frequencies","volume":"80","author":"Jablan","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.photonics.2014.06.005","article-title":"Electromagnetic waves reflection, transmission and absorption by graphene\u2013magnetic semiconductor\u2013graphene sandwich-structure in magnetic field: Faraday geometry","volume":"12","author":"Kuzmin","year":"2014","journal-title":"Photonics Nanostructures\u2014Fundam. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1134\/S002136401305007X","article-title":"Tunable hybrid surface waves supported by a graphene layer","volume":"97","author":"Iorsh","year":"2013","journal-title":"JETP Lett."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jaroniek, M., Czechowski, L., Kaczmarek, \u0141., Warga, T., and Kubiak, T. (2019). A New Approach of Mathematical Analysis of Structure of Graphene as a Potential Material for Composites. Materials, 12.","DOI":"10.3390\/ma12233918"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"037401","DOI":"10.1103\/PhysRevLett.91.037401","article-title":"Nonlinear properties of left-handed metamaterials","volume":"91","author":"Zharov","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shestopalov, Y.V., Smirnov, Y.G., and Smolkin, E.Y. (2022). Optical Waveguide Theory: Mathematical Models, Spectral Theory and Numerical Analysis, Springer.","DOI":"10.1007\/978-981-19-0584-1"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/7\/157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:02:21Z","timestamp":1760032941000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/7\/157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,1]]},"references-count":12,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["computation13070157"],"URL":"https:\/\/doi.org\/10.3390\/computation13070157","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,1]]}}}