Simple Analytical Solutions for the Optical Properties of Multilayered Confocal Prolate Spheroids in the Quasistatic Approximation

Simple Analytical Solutions for the Optical Properties of Multilayered Confocal Prolate Spheroids in the Quasistatic Approximation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Simple Analytical Solutions for the Optical Properties of Multilayered Confocal Prolate Spheroids in the Quasistatic Approximation Mugahid Ali, Fumin Huang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8018783/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract We present simple analytical solutions to Laplace’s equation for multilayered confocal prolate spheroids within the quasi-static approximation, tailored to configurations where the polarisation of the incident field aligns with the spheroid’s major axis. The field potential in each region is expressed as a superposition of two analytic functions, with amplitude coefficients conveniently determined through a compact matrix formulation derived from the boundary conditions. A key advancement of this work is the derivation of an explicit expression for the amplitude coefficient K0 of the perturbed field at the surface of the multilayered spheroid, obtained from the ratio of two matrix determinants. This coefficient inherently incorporates all constituent material properties and geometrical parameters of the nanostructure. Building on this, we derive a simple and general expression for the effective polarisability α of the multilayered spheroidal nanoparticle, which serves as the foundation for calculating its optical characteristics, including scattering and absorption. In contrast to previous models, the proposed analytical expressions enable rapid and accurate evaluation of optical responses—such as scattering, absorption, field enhancement, and plasmonic resonance—across arbitrary multilayered spheroidal configurations. Numerical validations for two-, three-, and five-layer structures confirm the model’s accuracy and computational efficiency, underscoring its potential as a powerful analytical tool for the design and optimisation of complex plasmonic nanostructures.. Nanophotonics Plasmonics Quasistatic Approximation Stratified Spheroids Full Text Additional Declarations No competing interests reported. Supplementary Files ElectronicSupplementaryMaterial.pdf Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 16 Feb, 2026 Reviewers agreed at journal 05 Feb, 2026 Reviews received at journal 04 Feb, 2026 Reviewers agreed at journal 19 Jan, 2026 Reviews received at journal 09 Jan, 2026 Reviewers agreed at journal 06 Jan, 2026 Reviewers invited by journal 08 Dec, 2025 Editor invited by journal 24 Nov, 2025 Editor assigned by journal 18 Nov, 2025 Submission checks completed at journal 14 Nov, 2025 First submitted to journal 14 Nov, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8018783","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":557156463,"identity":"9ab7f1c5-8dbe-4049-ae46-75282115174c","order_by":0,"name":"Mugahid Ali","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoklEQVRIiWNgGAWjYDACdsYGhg8VQMYBIIM4wMzYwDjjjAFJWoCItw2khVh38Tczt27gnfdHnu8Ac+MDorRIHGZsuyG5zcBw5gHGZgPirAFpMdxmkGBwgLFNgigd8iAtiXPAWtp/EKXFAKTlYAPEFuLcZQjUcrPhmLHhzMOMzcQ5TO54+7Pbf2rk5PmOtz/8QJw1cMBMovpRMApGwSgYBXgAAPSqMlbFUHkGAAAAAElFTkSuQmCC","orcid":"","institution":"Trinity College Dublin","correspondingAuthor":true,"prefix":"","firstName":"Mugahid","middleName":"","lastName":"Ali","suffix":""},{"id":557156467,"identity":"54182503-47d8-4dcd-81e9-600489a9c3d5","order_by":1,"name":"Fumin Huang","email":"","orcid":"","institution":"Queen's University Belfast","correspondingAuthor":false,"prefix":"","firstName":"Fumin","middleName":"","lastName":"Huang","suffix":""}],"badges":[],"createdAt":"2025-11-03 11:53:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8018783/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8018783/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":98423252,"identity":"1122a499-afa9-49a4-9bdc-fe02e7e464d7","added_by":"auto","created_at":"2025-12-17 16:32:01","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7552140,"visible":true,"origin":"","legend":"","description":"","filename":"SIMPLE1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8018783/v1_covered_14ce52ab-975b-4ba7-9cae-4d4256034f08.pdf"},{"id":97942671,"identity":"0ab68803-09d3-4890-a70a-04dafa385cbe","added_by":"auto","created_at":"2025-12-11 04:32:03","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":3322704,"visible":true,"origin":"","legend":"","description":"","filename":"ElectronicSupplementaryMaterial.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8018783/v1/b8886b1d0620023e04544e22.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Simple Analytical Solutions for the Optical Properties of Multilayered Confocal Prolate Spheroids in the Quasistatic Approximation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"discover-nano","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"narl","sideBox":"Learn more about [Discover Nano](https://www.springer.com/journal/11671)","snPcode":"11671","submissionUrl":"https://submission.nature.com/new-submission/11671/3","title":"Discover Nano","twitterHandle":"@SpringerOpen","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Nanophotonics, Plasmonics, Quasistatic Approximation, Stratified Spheroids","lastPublishedDoi":"10.21203/rs.3.rs-8018783/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8018783/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present simple analytical solutions to Laplace’s equation for multilayered confocal prolate spheroids within the quasi-static approximation, tailored to configurations where the polarisation of the incident field aligns with the spheroid’s major axis. The field potential in each region is expressed as a superposition of two analytic functions, with amplitude coefficients conveniently determined through a compact matrix formulation derived from the boundary conditions. A key advancement of this work is the derivation of an explicit expression for the amplitude coefficient K0 of the perturbed field at the surface of the multilayered spheroid, obtained from the ratio of two matrix determinants. This coefficient inherently incorporates all constituent material properties and geometrical parameters of the nanostructure. Building on this, we derive a simple and general expression for the effective polarisability α of the multilayered spheroidal nanoparticle, which serves as the foundation for calculating its optical characteristics, including scattering and absorption. 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