Extremely Step-like Modulated Photonic Time Crystals: Simultaneous Momentum-Frequency Bandgaps and Optical Response for Normal and Oblique Incidence

Extremely Step-like Modulated Photonic Time Crystals: Simultaneous Momentum-Frequency Bandgaps and Optical Response for Normal and Oblique Incidence | 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 Article Extremely Step-like Modulated Photonic Time Crystals: Simultaneous Momentum-Frequency Bandgaps and Optical Response for Normal and Oblique Incidence Kishwar Ali, Francesco Ferranti, Fabrizio Frezza, Giulio Antonini This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9585049/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Periodic variation in the constitutive parameters of a propagating bulk medium over time, while being uniform in space, referred to as periodically time-modulated photonic crystals (PTMPCs), exhibits a new control modality for wave-matter interaction. Following this, a step-like (square profile) modulation of permittivity and permeability has been utilized to achieve momentum bandgaps (MBGs) in the photonic band structure (PBS), whereas the modulation of conductivity has been overlooked. In this work, in the beginning, underlying PTMPCs are extremely modulated with a square profile of permittivity ε(t), permeability µ(t), and conductivity σ (t)—opening momentum and frequency bandgaps simultaneously, which has not been developed to date. The dispersion relation yielding dispersion diagram (PBS), is established with the virtue of the Bloch-Floquet theorem and by exploiting the continuity of electric displacement D(t) and magnetic flux B(t) across simultaneous discontinuities of ε(t), µ(t), and σ (t)—leading to the Krönig-Penney methodology. Furthermore, the amplification of eigenmodes inside the MBGs is manipulated by conductivity modulation strength (MS) m σ as well as temporal duty cycle. In addition, the amplitude distribution of higher-order eigenmodes generated by temporal modulation, namely, a frequency comb, is tailored with m σ in a specific band. Moreover, the engineering of the electric field E(t) propagation over time (and its frequency domain spectrum) with multiple MS schemes is presented. Subsequently, the optical response for a spatially finite temporal slab is studied under normal and oblique (sand p-polarization) incidence of a plane wave. The frequency comb in reflection and transmission spectra against operating frequency and incidence angle is analyzed. At last, the total absorptance for sand p-polarization is also discussed. Additionally, the MS of the temporal variation ranges between −1 < m ε,µ,σ < 1, whereas earlier studies restricted it to 0 < m ε,µ,σ < 1. Physical sciences/Materials science Physical sciences/Optics and photonics Physical sciences/Physics Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 16 May, 2026 Reviewers agreed at journal 15 May, 2026 Reviewers invited by journal 13 May, 2026 Editor invited by journal 11 May, 2026 Editor assigned by journal 05 May, 2026 Submission checks completed at journal 05 May, 2026 First submitted to journal 01 May, 2026 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9585049","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":639993740,"identity":"4b1dddc5-a5e1-4466-b33e-21a05488cec8","order_by":0,"name":"Kishwar Ali","email":"data:image/png;base64,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","orcid":"","institution":"University of L'Aquila","correspondingAuthor":true,"prefix":"","firstName":"Kishwar","middleName":"","lastName":"Ali","suffix":""},{"id":639993741,"identity":"811d374c-f109-46cb-94d8-6062915bbfcf","order_by":1,"name":"Francesco Ferranti","email":"","orcid":"","institution":"Luleå University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Francesco","middleName":"","lastName":"Ferranti","suffix":""},{"id":639993742,"identity":"b458c9d5-cdd5-41f6-93ff-b2055a1fcf84","order_by":2,"name":"Fabrizio Frezza","email":"","orcid":"","institution":"Sapienza University of Rome","correspondingAuthor":false,"prefix":"","firstName":"Fabrizio","middleName":"","lastName":"Frezza","suffix":""},{"id":639993743,"identity":"f7745102-ada8-43c1-b560-bd78cf6a1ecc","order_by":3,"name":"Giulio Antonini","email":"","orcid":"","institution":"University of L'Aquila","correspondingAuthor":false,"prefix":"","firstName":"Giulio","middleName":"","lastName":"Antonini","suffix":""}],"badges":[],"createdAt":"2026-05-01 10:23:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9585049/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9585049/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109279484,"identity":"1938b27c-e7c8-41a8-af14-67a18dc727b0","added_by":"auto","created_at":"2026-05-14 16:29:20","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1566403,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9585049/v1_covered_0d4f280d-744a-4638-8b34-0334a87b9e7d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Extremely Step-like Modulated Photonic Time Crystals: Simultaneous Momentum-Frequency Bandgaps and Optical Response for Normal and Oblique Incidence","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9585049/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9585049/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Periodic variation in the constitutive parameters of a propagating bulk medium over time, while being uniform in space, referred to as periodically time-modulated photonic crystals (PTMPCs), exhibits a new control modality for wave-matter interaction. Following this, a step-like (square profile) modulation of permittivity and permeability has been utilized to achieve momentum bandgaps (MBGs) in the photonic band structure (PBS), whereas the modulation of conductivity has been overlooked. In this work, in the beginning, underlying PTMPCs are extremely modulated with a square profile of permittivity ε(t), permeability µ(t), and conductivity σ (t)—opening momentum and frequency bandgaps simultaneously, which has not been developed to date. The dispersion relation yielding dispersion diagram (PBS), is established with the virtue of the Bloch-Floquet theorem and by exploiting the continuity of electric displacement D(t) and magnetic flux B(t) across simultaneous discontinuities of ε(t), µ(t), and σ (t)—leading to the Krönig-Penney methodology. Furthermore, the amplification of eigenmodes inside the MBGs is manipulated by conductivity modulation strength (MS) m σ as well as temporal duty cycle. In addition, the amplitude distribution of higher-order eigenmodes generated by temporal modulation, namely, a frequency comb, is tailored with m σ in a specific band. Moreover, the engineering of the electric field E(t) propagation over time (and its frequency domain spectrum) with multiple MS schemes is presented. Subsequently, the optical response for a spatially finite temporal slab is studied under normal and oblique (sand p-polarization) incidence of a plane wave. The frequency comb in reflection and transmission spectra against operating frequency and incidence angle is analyzed. At last, the total absorptance for sand p-polarization is also discussed. Additionally, the MS of the temporal variation ranges between −1 \u003c m ε,µ,σ \u003c 1, whereas earlier studies restricted it to 0 \u003c m ε,µ,σ \u003c 1.","manuscriptTitle":"Extremely Step-like Modulated Photonic Time Crystals: Simultaneous Momentum-Frequency Bandgaps and Optical Response for Normal and Oblique Incidence","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-14 16:29:11","doi":"10.21203/rs.3.rs-9585049/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"259017923217128647255104506659472834828","date":"2026-05-16T05:52:59+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"165477475499225214969963454408583408713","date":"2026-05-15T15:20:25+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-05-13T08:58:10+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-05-11T05:19:52+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-05T10:45:00+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-05-05T10:44:53+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-05-01T10:06:38+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d08ecf18-57d0-47d6-a8f9-23d77dacaccb","owner":[],"postedDate":"May 14th, 2026","published":true,"recentEditorialEvents":[{"type":"reviewerAgreed","content":"259017923217128647255104506659472834828","date":"2026-05-16T05:52:59+00:00","index":35,"fulltext":""},{"type":"reviewerAgreed","content":"165477475499225214969963454408583408713","date":"2026-05-15T15:20:25+00:00","index":33,"fulltext":""},{"type":"reviewersInvited","content":"7","date":"2026-05-13T08:58:10+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-05-11T05:19:52+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-05-05T10:45:00+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-05-05T10:44:53+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-05-01T10:06:38+00:00","index":"","fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":68115555,"name":"Physical sciences/Materials science"},{"id":68115556,"name":"Physical sciences/Optics and photonics"},{"id":68115557,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-05-14T16:29:11+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-14 16:29:11","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9585049","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9585049","identity":"rs-9585049","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}