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class="c-article-identifiers" data-test="article-identifier"> <li class="c-article-identifiers__item" data-test="article-category">Article</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item">Published: <time datetime="2024-11-12">12 November 2024</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">Expanding momentum bandgaps in photonic time crystals through resonances</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" 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Wang<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-5997-1130"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-5997-1130</a></span><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a></sup><sup class="u-js-hide"> <a href="#na1">na1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-P_-Garg-Aff3" data-author-popup="auth-P_-Garg-Aff3" data-author-search="Garg, P." data-corresp-id="c2">P. Garg<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-7262-3085"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-7262-3085</a></span><sup class="u-js-hide"><a href="#Aff3">3</a></sup><sup class="u-js-hide"> <a href="#na1">na1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-M__S_-Mirmoosa-Aff4" data-author-popup="auth-M__S_-Mirmoosa-Aff4" data-author-search="Mirmoosa, M. S.">M. S. Mirmoosa</a><sup class="u-js-hide"><a href="#Aff4">4</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-A__G_-Lamprianidis-Aff3" data-author-popup="auth-A__G_-Lamprianidis-Aff3" data-author-search="Lamprianidis, A. G.">A. G. Lamprianidis</a><sup class="u-js-hide"><a href="#Aff3">3</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-C_-Rockstuhl-Aff2-Aff3" data-author-popup="auth-C_-Rockstuhl-Aff2-Aff3" data-author-search="Rockstuhl, C.">C. Rockstuhl</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-5868-0526"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-5868-0526</a></span><sup class="u-js-hide"><a href="#Aff2">2</a>,<a href="#Aff3">3</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 6 authors for this article" title="Show all 6 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-V__S_-Asadchy-Aff5" data-author-popup="auth-V__S_-Asadchy-Aff5" data-author-search="Asadchy, V. S.">V. S. Asadchy</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-9840-4737"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-9840-4737</a></span><sup class="u-js-hide"><a href="#Aff5">5</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/nphoton" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Nature Photonics</i></a> (<span data-test="article-publication-year">2024</span>)<a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">7931 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">178 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/s41566-024-01563-3/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/condensed-matter-physics" data-track="click" data-track-action="view subject" data-track-label="link">Condensed-matter physics</a></li><li class="c-article-subject-list__subject"><a href="/subjects/metamaterials" data-track="click" data-track-action="view subject" data-track-label="link">Metamaterials</a></li><li class="c-article-subject-list__subject"><a href="/subjects/photonic-crystals" data-track="click" data-track-action="view subject" data-track-label="link">Photonic crystals</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>The realization of photonic time crystals is a major opportunity but also comes with considerable challenges. The most pressing one, potentially, is the requirement for a substantial modulation strength in the material properties to create a noticeable momentum bandgap. Reaching that noticeable bandgap in optics is highly demanding with current, and possibly also future materials platforms because their modulation strength is small by tendency. Here we demonstrate that by introducing temporal variations in a resonant material, the momentum bandgap can be drastically expanded with modulation strengths in reach with known low-loss materials and realistic laser pump powers. The resonance can emerge from an intrinsic material resonance or a suitably spatially structured material supporting a structural resonance. Our concept is validated for resonant bulk media and optical metasurfaces and paves the way towards the first experimental realizations of photonic time crystals.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41567-023-02023-5/MediaObjects/41567_2023_2023_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41567-023-02023-5?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41567-023-02023-5">Photonic metamaterial analogue of a continuous time crystal </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">20 April 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-024-46642-9/MediaObjects/41467_2024_46642_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-024-46642-9?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s41467-024-46642-9">Ultrafast all-optical second harmonic wavefront shaping </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">20 March 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41377-019-0225-z/MediaObjects/41377_2019_225_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41377-019-0225-z?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41377-019-0225-z">Nonreciprocal metasurface with space–time phase modulation </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">18 December 2019</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732399418, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Main"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Main</h2><div class="c-article-section__content" id="Sec1-content"><p>Within the domain of time-varying systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Galiffi, E. et al. Photonics of time-varying media. Adv. Photon. 4, 014002–014002 (2022)." href="/articles/s41566-024-01563-3#ref-CR1" id="ref-link-section-d139125714e455">1</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Ptitcyn, G., Mirmoosa, M. S., Sotoodehfar, A. & Tretyakov, S. A. A tutorial on the basics of time-varying electromagnetic systems and circuits: historic overview and basic concepts of time-modulation. IEEE Trans. Antennas Propag. 65, 10–20 (2023)." href="/articles/s41566-024-01563-3#ref-CR2" id="ref-link-section-d139125714e458">2</a></sup>, an intriguing concept that has recently garnered interest is that of photonic time crystals (PTCs)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zurita-Sánchez, J. R., Halevi, P. & Cervantes-González, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t). Phys. Rev. A 79, 053821 (2009)." href="#ref-CR3" id="ref-link-section-d139125714e462">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lustig, E., Sharabi, Y. & Segev, M. Topological aspects of photonic time crystals. Optica 5, 1390–1395 (2018)." href="#ref-CR4" id="ref-link-section-d139125714e462_1">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Asgari, M. M. et al. Photonic time crystals: theory and applications. Preprint at https://arxiv.org/abs/2404.04899 (2024)." href="/articles/s41566-024-01563-3#ref-CR5" id="ref-link-section-d139125714e465">5</a></sup>. Photonic time crystals are artificial materials whose electromagnetic properties are uniform in space but periodically modulated in time. This temporal periodicity generates a momentum bandgap in which light exponentially grows over time. Such behaviour results in exotic light–matter interaction, including amplification of spontaneous emission of an excited atom<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Lyubarov, M. et al. Amplified emission and lasing in photonic time crystals. Science 377, 425–428 (2022)." href="/articles/s41566-024-01563-3#ref-CR6" id="ref-link-section-d139125714e469">6</a></sup>, subluminal Cherenkov radiation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Dikopoltsev, A. et al. Light emission by free electrons in photonic time-crystals. Proc. Natl Acad. Sci. USA 119, e2119705119 (2022)." href="/articles/s41566-024-01563-3#ref-CR7" id="ref-link-section-d139125714e473">7</a></sup>, superluminal momentum-gap solitons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Pan, Y., Cohen, M.-I. & Segev, M. Superluminal k-gap solitons in nonlinear photonic time crystals. Phys. Rev. Lett. 130, 233801 (2023)." href="/articles/s41566-024-01563-3#ref-CR8" id="ref-link-section-d139125714e477">8</a></sup> and others.</p><p>Light amplification in the momentum bandgap of PTCs is a parametric process, distinct from traditional methods in nonlinear optics<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Asgari, M. M. et al. Photonic time crystals: theory and applications. Preprint at https://arxiv.org/abs/2404.04899 (2024)." href="/articles/s41566-024-01563-3#ref-CR5" id="ref-link-section-d139125714e484">5</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Khurgin, J. B. Photonic time crystals and parametric amplification: similarity and distinction. ACS Photon. 11, 2150–2159 (2024)." href="/articles/s41566-024-01563-3#ref-CR9" id="ref-link-section-d139125714e487">9</a></sup>. In particular, although backwards optical parametric amplification under transverse pumping shares similarities in geometry and medium eigenmodes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Ding, Y. J., Lee, S. J. & Khurgin, J. B. Transversely pumped counterpropagating optical parametric oscillation and amplification. Phys. Rev. Lett. 75, 429–432 (1995)." href="/articles/s41566-024-01563-3#ref-CR10" id="ref-link-section-d139125714e491">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Lanco, L. et al. Semiconductor waveguide source of counterpropagating twin photons. Phys. Rev. Lett. 97, 173901 (2006)." href="/articles/s41566-024-01563-3#ref-CR11" id="ref-link-section-d139125714e494">11</a></sup>, it is limited by an inherent finite growth of light energy due to the oscillatory nature of light propagation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Khurgin, J. B. Photonic time crystals and parametric amplification: similarity and distinction. ACS Photon. 11, 2150–2159 (2024)." href="/articles/s41566-024-01563-3#ref-CR9" id="ref-link-section-d139125714e498">9</a></sup>. Nevertheless, each process has an optimal regime for practical use. Backward optical parametric amplification is beneficial for moderate values of parametric gain (coupling coefficient) due to the positive feedback in space. Conversely, PTCs excel when the parametric gain is sufficiently large while pump depletion is still minor<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Khurgin, J. B. Photonic time crystals and parametric amplification: similarity and distinction. ACS Photon. 11, 2150–2159 (2024)." href="/articles/s41566-024-01563-3#ref-CR9" id="ref-link-section-d139125714e502">9</a></sup>. Moreover, amplification in PTCs has advantages compared with difference frequency generation processes: it is phase-independent, eigenmodes grow temporally rather than spatially, and phase-matching is easier to achieve (refer to section 4.3 of ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Asgari, M. M. et al. Photonic time crystals: theory and applications. Preprint at https://arxiv.org/abs/2404.04899 (2024)." href="/articles/s41566-024-01563-3#ref-CR5" id="ref-link-section-d139125714e506">5</a></sup>).</p><p>Although PTCs were experimentally confirmed at microwave frequencies<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Reyes-Ayona, J. R. & Halevi, P Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals. Appl. Phys. Lett. 107, 074101 (2015)." href="#ref-CR12" id="ref-link-section-d139125714e513">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Park, J. et al. Revealing non-hermitian band structure of photonic floquet media. Sci. Adv. 8, eabo6220 (2022)." href="#ref-CR13" id="ref-link-section-d139125714e513_1">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e516">14</a></sup>, designing them at optical frequencies remains a prime challenge. The material temporal modulation must be extremely fast—typically twice the oscillation frequency of the light that probes the response—and the relative change of the refractive index of the crystal <i>Δ</i><i>n</i>/<i>n</i> must be near unity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Zurita-Sánchez, J. R., Halevi, P. & Cervantes-González, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t). Phys. Rev. A 79, 053821 (2009)." href="/articles/s41566-024-01563-3#ref-CR3" id="ref-link-section-d139125714e528">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Lustig, E., Sharabi, Y. & Segev, M. Topological aspects of photonic time crystals. Optica 5, 1390–1395 (2018)." href="/articles/s41566-024-01563-3#ref-CR4" id="ref-link-section-d139125714e531">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Hayran, Z., Khurgin, J. B. & Monticone, F. ℏω versus ℏk: Dispersion and energy constraints on time-varying photonic materials and time crystals. Opt. Mater. Express 12, 3904–3917 (2022)." href="/articles/s41566-024-01563-3#ref-CR15" id="ref-link-section-d139125714e534">15</a></sup>. This can be achieved only with all-optical modulation of the refractive index<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Williamson, I. A. D. et al. Integrated nonreciprocal photonic devices with dynamic modulation. Proc. IEEE 108, 1759–1784 (2020)." href="/articles/s41566-024-01563-3#ref-CR16" id="ref-link-section-d139125714e538">16</a></sup> (for example, via third-order optical nonlinearities). In particular, the nonlinear Kerr effect under the assumption of an undepleted pump is described by linear coupled-mode equations with a time-varying material refractive index. However, nonlinear effects in low-loss materials are very weak<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Borchers, B., Brée, C., Birkholz, S., Demircan, A. & Steinmeyer, G. ünter Saturation of the all-optical Kerr effect in solids. Opt. Lett. 37, 1541–1543 (2012)." href="/articles/s41566-024-01563-3#ref-CR17" id="ref-link-section-d139125714e543">17</a></sup>, yielding a relative change in the refractive index to reach a maximum of less than 1%. Transparent conductive oxides that exhibit relative changes in the refractive index of the order of 100% were recently suggested as alternative material candidates to synthesize PTCs<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Alam, M. Z., De Leon, I. & Boyd, R. W. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science 352, 795–797 (2016)." href="#ref-CR18" id="ref-link-section-d139125714e547">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bohn, J., Luk, T. S., Horsley, S. & Hendry, E. Spatiotemporal refraction of light in an epsilon-near-zero indium tin oxide layer: frequency shifting effects arising from interfaces. Optica 8, 1532–1537 (2021)." href="#ref-CR19" id="ref-link-section-d139125714e547_1">19</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zhou, Y. et al. Broadband frequency translation through time refraction in an epsilon-near-zero material. Nat. Commun. 11, 2180 (2020)." href="#ref-CR20" id="ref-link-section-d139125714e547_2">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caspani, L. et al. Enhanced nonlinear refractive index in ε-near-zero materials. Phys. Rev. Lett. 116, 233901 (2016)." href="#ref-CR21" id="ref-link-section-d139125714e547_3">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Tirole, R. et al. Double-slit time diffraction at optical frequencies. Nat. Phys. 19, 999–1002 (2023)." href="/articles/s41566-024-01563-3#ref-CR22" id="ref-link-section-d139125714e550">22</a></sup>. However, they demand<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Hayran, Z., Khurgin, J. B. & Monticone, F. ℏω versus ℏk: Dispersion and energy constraints on time-varying photonic materials and time crystals. Opt. Mater. Express 12, 3904–3917 (2022)." href="/articles/s41566-024-01563-3#ref-CR15" id="ref-link-section-d139125714e554">15</a></sup> extremely high pumping power densities of up to tens of TW cm<sup>–</sup><sup>3</sup>. When combined with substantial material dissipation, this could lead to rapid thermal damage to the material. Therefore, only time refraction has been experimentally realized in transparent conducting oxides so far<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Lustig, E. et al. Time-refraction optics with single cycle modulation. Nanophotonics https://doi.org/10.1515/nanoph-2023-0126 (2023)." href="/articles/s41566-024-01563-3#ref-CR23" id="ref-link-section-d139125714e561">23</a></sup>.</p><p>We introduce a distinct approach to designing PTCs to avoid the aforementioned materials-related obstacles. We capitalize on artificial composites that support high-quality resonances rather than seeking new materials with improved nonlinear characteristics. Interestingly, a similar generic idea was recently suggested in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Khurgin, J. B. Energy and power requirements for alteration of the refractive index. Laser Photon. Rev. 18, 2300836 (2024)." href="/articles/s41566-024-01563-3#ref-CR24" id="ref-link-section-d139125714e568">24</a></sup>; however, no practical design or implementation was demonstrated. The approach based on resonant artificial composites allows us to create PTCs with pronounced momentum bandgaps with a substantially reduced required modulation strength (<i>Δ</i><i>n</i>/<i>n</i> ratio)—in reach with known low-loss materials and realistic laser pump powers. This provides the first material platform to realize a PTC at optical frequencies. We validate our concept of a resonant PTC for bulk materials and optical metasurfaces, assuming realistic material losses.</p></div></div></section><section data-title="Results"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Results</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3">Photonic time crystals made of intrinsically resonant media</h3><p>In the following, we initially demonstrate that temporally modulating the resonance frequency of a bulk material with Lorentzian dispersion substantially increases the momentum bandgap size compared with modulating the corresponding plasma frequency.</p><p>For a Lorentzian material, if the plasma frequency changes in time, the temporal permittivity in the frequency domain is written as <span class="mathjax-tex">$\epsilon (\omega ,t)=1+{\omega }_{{\rm{p}}}^{2}(t)/({\omega }_{{\rm{r0}}}^{2}-{\omega }^{2}+j\gamma \omega )$</span> (refer to section 4.3 of ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Mirmoosa, M. S., Koutserimpas, T. T., Ptitcyn, G. A., Tretyakov, S. A. & Fleury, R. Dipole polarizability of time-varying particles. New J. Phys. 24, 063004 (2022)." href="/articles/s41566-024-01563-3#ref-CR25" id="ref-link-section-d139125714e724">25</a></sup>). Here, <i>ω</i><sub>p</sub>(<i>t</i>), <i>ω</i><sub>r0</sub> and <i>γ</i> represent the plasma frequency, resonance frequency and damping factor, respectively. Considering a time-harmonic modulation of the plasma frequency with magnitude <i>m</i> and modulation frequency <i>ω</i><sub>m</sub>, that is, <span class="mathjax-tex">${\omega }_{{\rm{p}}}^{2}(t)={\omega }_{{\rm{p0}}}^{2}[1+m\cos ({\omega }_{{\rm{m}}}t)]$</span>, the above expression for the permittivity reduces to <span class="mathjax-tex">$\epsilon (\omega ,t)=1+\chi (\omega )[1+m\cos ({\omega }_{{\rm{m}}}t)]$</span>, where <i>χ</i>(<i>ω</i>) denotes the stationary susceptibility of the material.</p><p>When solving for the eigenmodes to Maxwell’s equations in this time-varying medium, the temporal modulation splits the degenerate eigenmodes at <i>ω</i> = <i>ω</i><sub>m</sub>/2, resulting in a momentum bandgap (see derivations in Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">1</a>). To illustrate this phenomenon, we plot the dispersion relation within the frequency range [0, <i>ω</i><sub>m</sub>] in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1a</a>, assuming <i>m</i> = 0.2. Inside the momentum bandgap, the eigenfrequency has two complex conjugate solutions corresponding to exponentially decaying and growing modes in time.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Band structures of time-varying Lorentzian material."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1: Band structures of time-varying Lorentzian material.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="550"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p><b>a</b>, Band structure of a time-modulated bulk material when its plasma frequency is modulated harmonically. <b>b</b>, Dispersion relations of a stationary (unmodulated) bulk material when its plasma frequency is <i>ω</i><sub>p1</sub> (red line) and <i>ω</i><sub>p2</sub> (green line). <b>c</b>, Band structure of a time-modulated bulk material when its resonance frequency is modulated harmonically. <b>d</b>, Dispersion relations of a stationary (not modulated) bulk material when its resonance frequency is <i>ω</i><sub>r1</sub> (red line) and <i>ω</i><sub>r2</sub> (green line). In all of the panels, the wavenumber is normalized by <span class="mathjax-tex">${k}_{{\rm{r0}}}={\omega }_{{\rm{r0}}}\sqrt{{\epsilon }_{0}{\mu }_{0}}$</span>, and the material parameters are chosen as <i>m</i> = 0.2, <i>ω</i><sub>m</sub>/2 = 0.95<i>ω</i><sub>r0</sub>, <i>γ</i> = 0 and <i>ω</i><sub>p0</sub> = 3.5<i>ω</i><sub>r0</sub>. A time convention of e<sup><i>j</i><i>ω</i><i>t</i></sup> is considered, with <i>j</i> being the imaginary unit.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>To analytically determine the bandgap size, the weak-modulation approximation is used<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Asadchy, V. et al. Parametric Mie resonances and directional amplification in time-modulated scatterers. Phys. Rev. Appl. 18, 054065 (2022)." href="/articles/s41566-024-01563-3#ref-CR26" id="ref-link-section-d139125714e1162">26</a></sup> to calculate the bandgap edges (<i>k</i><sub>−</sub> and <i>k</i><sub>+</sub> indicated in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1a</a>). However, this powerful tool provides a closed-form solution for the edges and highlights a crucial observation: the bandgap edges <i>k</i><sub>−</sub> and <i>k</i><sub>+</sub> precisely correspond to the eigenwavenumbers <i>k</i><sub>1</sub> and <i>k</i><sub>2</sub> of two non-modulated materials with scaled susceptibilities <i>χ</i>(<i>ω</i>)(1 <span class="stix">∓</span> <i>m</i>/2) evaluated at the frequency <i>ω</i> = <i>ω</i><sub>m</sub>/2 (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">1</a>). This correspondence is general and applies to all PTC topologies with <i>m</i> <span class="stix">≪</span> 1 explored in this work, including the low-loss materials. Accordingly, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1b</a> depicts the dispersion relations of the two stationary materials with those scaled susceptibilities. As <i>χ</i> is linearly proportional to <span class="mathjax-tex">${\omega }_{{\rm{p}}}^{2}$</span>, the two materials can be alternatively described with scaled plasma frequencies <span class="mathjax-tex">${\omega }_{{\rm{p1}}}^{2}={\omega }_{{\rm{p0}}}^{2}(1-m/2)$</span> and <span class="mathjax-tex">${\omega }_{{\rm{p2}}}^{2}={\omega }_{{\rm{p0}}}^{2}(1+m/2)$</span>. Thus, by plotting the dispersion relation of the material with no modulation, one can determine the momentum bandgap size for a given modulation. We emphasize this property in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1a,b</a> with the vertically shaded region. Importantly, this correspondence provides a simple visual insight into how a given material dispersion affects the bandgap size and position. It is evident from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1a,b</a> that the temporal modulation of the plasma frequency results in a relatively narrow bandgap due to the tiny variation of the dispersion curves of a stationary material with fixed resonance frequency (<i>k</i><sub>1</sub> ≈ <i>k</i><sub>2</sub>).</p><p>By contrast, the situation drastically differs when the resonance frequency <i>ω</i><sub>r</sub> varies in time. To show this, we assume that the intrinsic resonance frequency changes harmonically so that <span class="mathjax-tex">${\omega }_{{\rm{r}}}^{2}(t)$</span><span class="mathjax-tex">$={\omega }_{{\rm{r0}}}^{2}[1+m\cos ({\omega }_{{\rm{m}}}t)]$</span>. Following the above-described correspondence between the stationary and modulated material scenarios, we plot in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1d</a> the dispersion relation of a stationary bulk material for two values of the resonance frequency: <span class="mathjax-tex">${\omega }_{{\rm{r1}}}^{2}={\omega }_{{\rm{r0}}}^{2}(1+m/2)$</span> and <span class="mathjax-tex">${\omega }_{{\rm{r2}}}^{2}={\omega }_{{\rm{r0}}}^{2}(1-m/2)$</span>. We observe that for the same modulation strength of <i>m</i> = 0.2, the two dispersion curves now differ substantially at <i>ω</i> = <i>ω</i><sub>m</sub>/2. For <i>ω</i><sub>r2</sub> < <i>ω</i><sub>m</sub>/2 < <i>ω</i><sub>r1</sub>, the horizontal dashed line at <i>ω</i> = <i>ω</i><sub>m</sub>/2 intersects with only one of the dispersion curves at <i>k</i> = <i>k</i><sub>1</sub> (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1d</a>), indicating a semi-infinite momentum bandgap extending from <i>k</i> = <i>k</i><sub>1</sub> to <i>k</i> = +<i>∞</i>, as shown by the grey-shaded region. The prediction from the dispersion curves of the stationary materials agrees excellently with the rigorously computed band structure (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">2</a>) shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1c</a>. Moreover, the imaginary part of the dispersion as a function of the momentum (orange dotted line) sensitively depends on the modulation frequencies, which might allow us to engineer the amplification rates.</p><p>In this conceptual analysis, we disregard possible effects of the non-local response of the material at very high <i>k</i>-values<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Buddhiraju, S. et al. Absence of unidirectionally propagating surface plasmon–polaritons at nonreciprocal metal–dielectric interfaces. Nat. Commun. 11, 674 (2020)." href="/articles/s41566-024-01563-3#ref-CR27" id="ref-link-section-d139125714e1795">27</a></sup>, which would make the bandgap in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1</a> finite but still very large. As per our suggestion in the last section, non-local effects are fully considered. Furthermore, we also neglected losses in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1</a>. Possible losses cause a positive offset in the imaginary parts of eigenfrequencies, reducing the temporal amplification rates (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">3</a>). However, the bandgap size remains and the amplification rate is still much higher compared with modulating the plasma frequency in the lossless case.</p><p>The temporal modulation of the intrinsic resonance frequency of bulk material is achieved, for example, through strong dynamic electric biasing<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Serra, J. C. & Silveirinha, M. G. Homogenization of dispersive space-time crystals: anomalous dispersion and negative stored energy. Phys. Rev. B 108, 035119 (2023)." href="/articles/s41566-024-01563-3#ref-CR28" id="ref-link-section-d139125714e1811">28</a></sup> but, in practice, could be very challenging. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig2">2a</a> illustrates this scenario in which an external electric field modulates the effective spring constants <i>κ</i> of each nucleus-electron oscillator. However, instead of modulating the intrinsic resonance frequency of the natural atoms of a material, we propose a metamaterial concept where the resonances of spatially structured meta-atoms are modulated from which a metamaterial is assembled, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig2">2b</a>. Probably, the most practical scenario is when the properties of the material from which the meta-atoms are made are modulated, which can be achieved with current optical modulation techniques<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Alam, M. Z., De Leon, I. & Boyd, R. W. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science 352, 795–797 (2016)." href="#ref-CR18" id="ref-link-section-d139125714e1824">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bohn, J., Luk, T. S., Horsley, S. & Hendry, E. Spatiotemporal refraction of light in an epsilon-near-zero indium tin oxide layer: frequency shifting effects arising from interfaces. Optica 8, 1532–1537 (2021)." href="#ref-CR19" id="ref-link-section-d139125714e1824_1">19</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zhou, Y. et al. Broadband frequency translation through time refraction in an epsilon-near-zero material. Nat. Commun. 11, 2180 (2020)." href="#ref-CR20" id="ref-link-section-d139125714e1824_2">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Caspani, L. et al. Enhanced nonlinear refractive index in ε-near-zero materials. Phys. Rev. Lett. 116, 233901 (2016)." href="/articles/s41566-024-01563-3#ref-CR21" id="ref-link-section-d139125714e1827">21</a></sup>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Conceptual realization method of time-varying medium with resonance frequency modulated."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2: Conceptual realization method of time-varying medium with resonance frequency modulated.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig2_HTML.png?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="314"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p><b>a</b>, Illustration of the temporal modulation of the intrinsic resonance frequency of a bulk material. The modulation can be described by a time-varying effective spring constant <i>κ</i>(<i>t</i>). <b>b</b>, Illustration of the temporal modulation of the resonance frequency of a metamaterial consisting of meta-atoms. The modulation can be achieved in various ways, including modulating the permittivity <i>ϵ</i><sub>r</sub> or permeability <i>μ</i><sub>r</sub> of the material from which the meta-atoms are made, the size of the meta-atoms <i>R</i>, or the metamaterial periodicity <i>d</i>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec4">Photonic time crystals based on resonant metasurfaces</h3><p>Next, we realize the PTC in a metasurface geometry to steer our consideration to a practical scenario. The metasurface is assumed to support surface waves bounded to the <i>x</i><i>y</i>-plane (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3a</a>). The meta-atoms are deeply sub-wavelength and all identical, resulting in a spatially homogeneous metasurface at <i>z</i> = 0. Similarly to bulk PTCs that exhibit momentum bandgaps for bulk propagating eigenmodes, the metasurface-based counterparts support momentum bandgaps for surface eigenmodes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e1897">14</a></sup>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Analysis of time-varying LC resonant metasurface."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3: Analysis of time-varying <i>LC</i> resonant metasurface.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig3_HTML.png?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="297"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p><b>a</b>, Generic illustration of a time-varying resonant <i>LC</i> metasurface. Each meta-atom is described by a time-modulated surface capacitance <i>C</i>(<i>t</i>) and a constant surface inductance <i>L</i>. The red arrow indicates a surface eigenmode whose amplitude is growing in time due to its wavenumber being inside the momentum bangdap of the metasurface. <b>b</b>, Upper: dispersion relations of a stationary metasurface for the two scaled values of the surface capacitance. Lower: band structure of the time-modulated resonant metasurface. Here, <span class="stix">ℜ</span> and <span class="stix">ℑ</span> represent real and imaginary parts of the complex frequency, respectively. The horizontal axis is normalized by <i>k</i><sub>r0</sub> = <i>ω</i><sub>r0</sub>/<i>c</i><sub>0</sub> where <i>c</i><sub>0</sub> is the speed of light in vacuum. The grey region emphasizes the bandgap in the lower panel and the eigenwavenumber variation at <i>ω</i> = <i>ω</i><sub>m</sub>/2 in the upper panel. <b>c</b>,<b>d</b>, Magnetic field snapshots for momenta <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 5<i>k</i><sub>r0</sub> and <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 10<i>k</i><sub>r0</sub> above the metasurface at the time moment when modulation switches on (<i>t</i> = 0) and after some time passed (<i>t</i> = 30<i>T</i><sub>m</sub>, where <i>T</i><sub>m</sub> = 2π/<i>ω</i><sub>m</sub>). The complete field evolution animation is available in Supplementary Video <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">1</a>. <b>e</b>, Time evolution of the magnetic field above the metasurface (<i>z</i> = 0) calculated with full-wave simulations and analytically from the band structure. The theoretical amplitude is calculated as <span class="mathjax-tex">${H}_{y}(t)={H}_{y0}\exp [\Im (\omega )t]$</span> where <i>H</i><sub><i>y</i>0</sub> = 1 A m<sup>–1</sup> is the initial field at <i>t</i> = 0 and <i><span class="stix">ℑ</span></i>(<i>ω</i>) = 0.025<i>ω</i><sub>m</sub>. <b>f</b>, Imaginary part of the eigenfrequency for metasurfaces with different quality factors. The real part of eigenfrequency is fixed as <i><span class="stix">ℜ</span></i>(<i>ω</i>) = <i>ω</i><sub>m</sub>/2. The black dashed line separates propagating waves (p.w.) and surface waves (s.w.). The quality factor of the stationary metasurface can be qualitatively described as a quality factor of an <i>R</i><i>L</i><i>C</i> circuit (where <span class="mathjax-tex">$R=\sqrt{{\mu }_{0}/{\varepsilon }_{0}}$</span> is the free-space characteristic impedance, which is connected in parallel to the metasurface equivalent circuit), that is, <span class="mathjax-tex">$Q={\omega }_{{\rm{r0}}}{C}_{0}\sqrt{{\mu }_{0}/{\varepsilon }_{0}}$</span> (refer to section 6.1 of ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Pozar, D. M. Microwave Engineering (John Wiley and Sons, 2011)." href="/articles/s41566-024-01563-3#ref-CR41" id="ref-link-section-d139125714e2284">41</a></sup>). The quality factor is tuned by varying the values of <i>C</i><sub>0</sub> and <i>L</i> while keeping <span class="mathjax-tex">${\omega }_{{\rm{r0}}}=1/\sqrt{L{C}_{0}}$</span> constant. Importantly, such a regime of the open bandgap for propagating modes is inherent to metasurfaces with high quality factors and cannot occur in non-resonant metasurfaces such as those in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e2355">14</a></sup>. <b>g</b>, The magnetic field evolution for a dipole source positioned 1.5 wavelengths above the time-varying <i>L</i><i>C</i> metasurface with <i>Q</i> = 122. Due to the infinite bandgap, all momenta <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> are amplified by the metasurface. In Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3c,d,g</a>, the excitation source is turned off after modulation starts; <i>m</i> = 0.2 in all of the panels.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>A resonant impenetrable metasurface can be described by effective reactive parameters: the surface capacitance <i>C</i> and surface inductance <i>L</i> connected in parallel (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3a</a>)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Tretyakov, S. Analytical Modeling in Applied Electromagnetics (Artech House, 2003)." href="/articles/s41566-024-01563-3#ref-CR29" id="ref-link-section-d139125714e2401">29</a></sup>. Such a generic <i>L</i><i>C</i>-model describes the interaction of light with a time-modulated metasurface independent of any specific geometry and operational frequency. At optical frequencies, as we show in the following section, such a resonant metasurface can be a two-dimensional array of spherical nanoparticles made of a material with a time-modulated dielectric constant. At microwave frequencies, the implementation can be a mushroom-type high-impedance surface with varactors embedded between adjacent patches<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e2411">14</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Sievenpiper, D., Zhang, L., Broas, RomuloF. J., Alexopolous, N. G. & Yablonovitch, E. High-impedance electromagnetic surfaces with a forbidden frequency band. IEEE Trans. Microw. Theory Tech. 47, 2059–2074 (1999)." href="/articles/s41566-024-01563-3#ref-CR30" id="ref-link-section-d139125714e2414">30</a></sup>.</p><p>In our model, the surface capacitance varies time-harmonically as <span class="mathjax-tex">$C(t)={C}_{0}[1+m\cos ({\omega }_{{\rm{m}}}t)]$</span>, whereas the surface inductance <i>L</i> is constant. Such a configuration effectively modulates the resonance frequency in time. Without loss of generality, we consider transverse-magnetic polarization for the surface waves. Similarly to the observations in the previous section, the momentum bandgap size of such a time-varying metasurface can be estimated by the dispersion curves of two stationary <i>LC</i> surfaces with surface capacitance <span class="mathjax-tex">${C}_{1}={C}_{0}(1+\frac{m}{2})$</span> and <span class="mathjax-tex">${C}_{2}={C}_{0}(1-\frac{m}{2})$</span> (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">4</a>). For verification, we plot the dispersion curves of the two stationary surfaces and the time-modulated metasurfaces for <i>m</i> = 0.2 in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3b</a>. When modulating at <i>ω</i><sub>m</sub> = 2<i>ω</i><sub>r0</sub>, the momentum bandgap has only one edge, and the size extends from <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> ≈ 4<i>k</i><sub>r0</sub> to infinity, in the absence of non-local effects. We stress that akin to the bulk scenario, the bandgap size remains the same if realistic material losses are included (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">4</a>).</p><p>Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3c,d</a> shows fields calculated from numerical simulation for modes with two distinct momenta, that is, <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 5<i>k</i><sub>r0</sub> and <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 10<i>k</i><sub>r0</sub> inside the momentum bandgap. Both modes are effectively amplified after modulating for 30<i>T</i><sub>m</sub>. The simulated field amplitude evolution matches very well with theoretical predictions, as illustrated in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3e</a>.</p><p>The size and strength of the momentum bandgap improve as the quality factor of the metasurface increases. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3f</a> shows that metasurfaces with a higher <i>Q</i>-factor provide wider momentum bandgaps for surface waves with larger amplification rates, assuming the same modulation function. In comparison, the metasurface discussed in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3b–e</a> has a quality factor of <i>Q</i> = 2.44. Moreover, for sufficiently large <i>Q</i>-factors (<i>Q</i> ≥ 9.75), a second momentum bandgap opens inside the light cone, that is, for propagating waves. The size of the second bandgap grows with the quality factor of the metasurface because resonances with longer lifetimes suffer from smaller radiation losses and require weaker modulation to maintain the same amplification rate. When the quality factor takes sufficiently large values, the two bandgaps merge and the metasurface can amplify incident waves with all possible momenta <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3f</a>).</p><p>We place a dipole emitter above the metasurface to demonstrate this infinite momentum bandgap (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3g</a>). The dipole radiation includes a wide spectrum of momenta, as shown in the upper panel of the figure. Once the temporal modulation of the metasurface is on, waves with all different momenta are amplified and radiated in the specular and retro-directions with respect to the source; see the lower panel in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3g</a>. This leads to interesting possibilities such as amplified emission and lasing of light from a radiation source<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Lyubarov, M. et al. Amplified emission and lasing in photonic time crystals. Science 377, 425–428 (2022)." href="/articles/s41566-024-01563-3#ref-CR6" id="ref-link-section-d139125714e2763">6</a></sup>. In contrast to the idea suggested in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Lyubarov, M. et al. Amplified emission and lasing in photonic time crystals. Science 377, 425–428 (2022)." href="/articles/s41566-024-01563-3#ref-CR6" id="ref-link-section-d139125714e2767">6</a></sup>, due to the substantially enhanced bandgap, it is possible here to amplify emission with a large and, in principle, tunable spectrum of wavenumbers. This provides opportunities for beam shaping of the amplified signal and for creating perfect lenses<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966 (2000)." href="/articles/s41566-024-01563-3#ref-CR31" id="ref-link-section-d139125714e2771">31</a></sup>. Indeed, the evanescent wave content of the source radiation can be reconstructed effectively thanks to the amplification of the wide range of <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub>. In Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">5</a>, we demonstrate that evanescent and propagating wave components of the radiating dipole are amplified by the metasurface in reflection and transmission regimes.</p><h3 class="c-article__sub-heading" id="Sec5">Optical implementation</h3><p>To provide a feasible optical realization of the resonant PTC, we consider a penetrable metasurface surrounded by air and consisting of dielectric nanospheres that are made of a material with a time-varying permittivity (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4a</a>). Each nanosphere effectively behaves as an <i>L</i><i>C</i> resonator as it supports Mie resonances<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Rahimzadegan, A., Alaee, R., Rockstuhl, C. & Boyd, R. W. Minimalist Mie coefficient model. Opt. Express 28, 16511–16525 (2020)." href="/articles/s41566-024-01563-3#ref-CR32" id="ref-link-section-d139125714e2799">32</a></sup>. For simplicity, we initially ignore material dispersion. The permittivity associated with each nanosphere reads <span class="mathjax-tex">$\varepsilon (t)=1+{\chi }_{0}[1+m\cos ({\omega }_{{\rm{m}}}t)]$</span>. Varying the permittivity in time modulates the Mie resonance frequencies of the nanospheres (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig2">2b</a>). In the following, we rely on the T-matrix method to study the optical response from such a metasurface<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Garg, P. et al. Modeling four-dimensional metamaterials: a t-matrix approach to describe time-varying metasurfaces. Opt. Express 30, 45832–45847 (2022)." href="/articles/s41566-024-01563-3#ref-CR33" id="ref-link-section-d139125714e2903">33</a></sup> (see <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec7">Methods</a> and Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">6</a> for details).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Illustration of the momentum bandgap enhancement for surface waves."><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4: Illustration of the momentum bandgap enhancement for surface waves.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig4_HTML.png?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="382"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p><b>a</b>, A representative design of a PTC based on an optical time-varying resonant metasurface made of dielectric nanospheres. The nanospheres are arranged in an infinite square lattice with period <i>a</i>. The radius of each nanosphere is <i>R</i>. The purple arrow indicates a surface eigenmode whose amplitude grows in time due to its wavenumber being inside the momentum bandgap of the metasurface. <b>b</b>, Band structure of a time-invariant metasurface. The colour denotes the lowest singular value <span class="mathjax-tex">${S}_{\min }$</span> of the matrix in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ3">3</a>). The white dashed lines represent the light lines. <b>c</b>, The two lowest-order Mie coefficients of an isolated time-invariant nanosphere. The vertical scale is the same as in <b>b</b>. <b>d</b>, Band structure of a time-varying metasurface with modulation frequency <i>ω</i><sub>m1</sub> (non-resonant case). <b>e</b>, Magnified band structure in the green highlighted region of <b>d</b>. <b>f</b>, The corresponding imaginary part of the frequency for fixed <i><span class="stix">ℜ</span></i>(<i>ω</i>) = <i>ω</i><sub>m1</sub>/2. <b>g</b>, Band structure of a time-varying metasurface with modulation frequency <i>ω</i><sub>m2</sub> (resonant case). <b>h</b>, Magnified band structure in the green highlighted region of <b>g</b>. <b>i</b>, The corresponding imaginary part of the frequency for fixed <i><span class="stix">ℜ</span></i>(<i>ω</i>) = <i>ω</i><sub>m2</sub>/2. <b>j</b>, Imaginary part of the eigenfrequency <i><span class="stix">ℑ</span></i>(<i>ω</i>) at the bandgap centre and the relative amplification momentum bandwidth <span class="mathjax-tex">$\Delta {k}_{| | }^{{\rm{a}}}=2| ({k}_{| | }^{{\rm{a2}}}-{k}_{| | }^{{\rm{a1}}})/({k}_{| | }^{{\rm{a2}}}+{k}_{| | }^{{\rm{a1}}})|$</span> as a function of the damping factor <i>γ</i>. Here, <span class="mathjax-tex">${k}_{| | }^{{\rm{a1}}}$</span> and <span class="mathjax-tex">${k}_{| | }^{{\rm{a2}}}$</span> correspond to those values of <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> for a bandgap where <i><span class="stix">ℑ</span></i>(<i>ω</i>) = 0 (see Supplementary Fig. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">7a</a>). Note that the band structures shown in <b>b</b>, <b>d</b> and <b>g</b> accommodate the eigenmodes for both transverse magnetic and transverse electric polarizations; however, in <b>e</b>,<b>f</b>,<b>h</b>,<b>i</b> we optimize the metasurfaces for transverse-electric waves. Further note that the momentum bandgap in <b>g</b> is incomplete because bands exist at <i><span class="stix">ℜ</span></i>(<i>ω</i>) > <i>ω</i><sub>m2</sub>/2 within the bandgap. Nevertheless, in contrast to spatial photonic crystals with their energy bandgaps, the modes inside of an incomplete momentum bandgap are always dominant due to their amplifying nature<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Lustig, E., Sharabi, Y. & Segev, M. Topological aspects of photonic time crystals. Optica 5, 1390–1395 (2018)." href="/articles/s41566-024-01563-3#ref-CR4" id="ref-link-section-d139125714e3325">4</a></sup>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>First, using equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ3">3</a>) in the <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec7">Methods</a>, we calculate the band structure of a time-invariant metasurface, substituting <i>m</i> = 0 and <i>ω</i><sub>m</sub> = 0 (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>). One can see nearly flat bands near the <i>M</i> and <i>Γ</i> points of the spatial Brillouin zone due to the Mie resonances of the single nanosphere. For comparison, we plot the dipolar Mie coefficients of a time-invariant isolated nanosphere in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4c</a> (ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Rahimzadegan, A., Alaee, R., Rockstuhl, C. & Boyd, R. W. Minimalist Mie coefficient model. Opt. Express 28, 16511–16525 (2020)." href="/articles/s41566-024-01563-3#ref-CR32" id="ref-link-section-d139125714e3366">32</a></sup>); this figure shows magnetic and electric dipolar resonances, which explain the flat bands in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>.</p><p>We next plot the band structures of the suggested time-varying metasurfaces. We start with a modulation frequency of <i>ω</i><sub>m1</sub> = 2π × 183 THz, as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>. As <span class="mathjax-tex">$\frac{{\omega }_{{\rm{m1}}}}{2}$</span> is away from the flat bands, this configuration corresponds to a modulated non-resonant metasurface. Moreover, the dispersion around <span class="mathjax-tex">$\omega =\frac{{\omega }_{{\rm{m1}}}}{2}$</span> is linear (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>), indicating that the metasurface is spatially homogenizable near and below <span class="mathjax-tex">$\omega =\frac{{\omega }_{{\rm{m1}}}}{2}$</span> (refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Venkitakrishnan, R. et al. Lower limits for the homogenization of periodic metamaterials made from electric dipolar scatterers. Phys. Rev. B 103, 195425 (2021)." href="/articles/s41566-024-01563-3#ref-CR34" id="ref-link-section-d139125714e3521">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Garg, P. et al. Two-step homogenization of spatiotemporal metasurfaces using an eigenmode-based approach. Opt. Mater. Express 14, 549 (2024)." href="/articles/s41566-024-01563-3#ref-CR35" id="ref-link-section-d139125714e3524">35</a></sup>). Therefore, this time-varying metasurface is comparable in its response to a conventional (space-uniform) PTC with a modulated plasma frequency since the dispersion relation only weakly changes close to the considered <i>Γ</i>-point (compare with Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig1">1b,d</a>). Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4d</a> shows the band structure of this non-resonant metasurface. We choose <i>m</i> = 0.01, which corresponds to a relative refractive index change of <i>Δ</i><i>n</i>/<i>n</i> ≈ <i>m</i><i>χ</i><sub>0</sub>/(1 + <i>χ</i><sub>0</sub>) = 1%—close to the maximum attainable value in silicon before thermal damage occurs<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Cowan, B. Optical damage threshold of silicon for ultrafast infrared pulses. AIP Conf. Proc. 877, 837–843 (2006)." href="/articles/s41566-024-01563-3#ref-CR36" id="ref-link-section-d139125714e3560">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Polyanskiy, M. N., Pogorelsky, I. V., Babzien, M., Vodopyanov, K. L. & Palmer, M. A. Nonlinear refraction and absorption properties of optical materials for high-peak-power long-wave-infrared lasers. Opt. Mater. Express 14, 696–714 (2024)." href="/articles/s41566-024-01563-3#ref-CR37" id="ref-link-section-d139125714e3563">37</a></sup>. We observe the folding of the band structure of the time-invariant case<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Zurita-Sánchez, J. R., Halevi, P. & Cervantes-González, J. C. Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ϵ(t). Phys. Rev. A 79, 053821 (2009)." href="/articles/s41566-024-01563-3#ref-CR3" id="ref-link-section-d139125714e3567">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic Crystals: Molding the Flow of Light (Second Edition) 2 edn (Princeton Univ. Press, 2008)." href="/articles/s41566-024-01563-3#ref-CR38" id="ref-link-section-d139125714e3570">38</a></sup>. At the band crossing in the <i>M</i> − <i>Γ</i> region, the momentum bandgap appears, as seen in the magnified region (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4e</a>). The calculated relative gap width is very narrow, <i>Δ</i><i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 0.0183%. We also plot the imaginary part of the frequency <i><span class="stix">ℑ</span></i>(<i>ω</i>) for a fixed <span class="mathjax-tex">$\Re (\omega )=\frac{{\omega }_{{\rm{m1}}}}{2}$</span> (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4f</a>). As expected, <i><span class="stix">ℑ</span></i>(<i>ω</i>) reaches the largest absolute value in the middle of the bandgap. As the bandgap lies below the light line (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>), only surface-wave incident excitations couple to the modes inside it.</p><p>In sharp contrast to the previous case, when half of the modulation frequency corresponds to the magnetic Mie resonance of the nanospheres (see the dotted horizontal line <i>ω</i><sub>m2</sub>/2 = 2π × 196 THz in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a>), the bandgap dramatically expands. Keeping the same modulation strength <i>m</i> = 0.01 for a fair comparison, the relative bandgap size of this resonant metasurface reaches <i>Δ</i><i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 6.59%, calculated from the band structure in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4g–i</a>. Remarkably, the resonance of the metasurface leads to the widening of the bandgap by a factor of about 350 compared with the non-resonant case (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">7</a> for more details on effect of the structural resonances). It should be noted that the correspondence between a time-varying system and two stationary systems with scaled parameters also holds for this metasurface (see Supplementary Fig. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">6</a>). Comparing Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4f,i</a> reveals that the structural resonance not only enhances the bandgap size, but the amplification rate also substantially increases: <i><span class="stix">ℑ</span></i>(<i>ω</i>) is an order of magnitude larger for the resonant case.</p><p>We also study the effects of material losses by considering the nanospheres made from a time-varying dispersive material (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">8</a>). Depending on the damping factor <i>γ</i> of such material, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4j</a> shows the relative amplification bandwidth <span class="mathjax-tex">$\Delta {k}_{| | }^{{\rm{a}}}$</span> (defined in the caption) and the imaginary part of the eigenfrequency <i><span class="stix">ℑ</span></i>(<i>ω</i>) at the bandgap centre. Regarding this time-varying metasurface, modulation frequency <i>ω</i><sub>m2</sub> is assumed. As <i>γ</i> increases, the amplification rate of the eigenmode inside the bandgap <i><span class="stix">ℑ</span></i>(<i>ω</i>) decreases in amplitude. A similar trend is observed for the relative momentum bandwidth where amplification occurs. Nevertheless, amplification remains substantial over a wide momentum bandwidth of <span class="mathjax-tex">$\Delta {k}_{| | }^{{\rm{a}}} > 6 \%$</span> if <i>γ</i>/<i>ω</i><sub>r</sub> < 0.05. Serendipitously, this material loss threshold is substantially above the realistic damping factor <i>γ</i>/<i>ω</i><sub>r</sub> ≈ 10<sup>−5</sup> that silicon has at near-infrared wavelengths where the metasurface is designed to operate<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Vetterl, O. et al. Intrinsic microcrystalline silicon: a new material for photovoltaics. Sol. Energy Mater. Sol. Cells 62, 97 (2000)." href="/articles/s41566-024-01563-3#ref-CR39" id="ref-link-section-d139125714e3847">39</a></sup>.</p><p>Although the bandgaps in the above-considered scenarios occurred below the light line, we next optimized the metasurface to sustain bandgaps for propagating waves (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5a</a>). From Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4b</a> for a stationary metasurface, we observe a flat band within the light cone at <i>ω</i> = <i>ω</i><sub>m3</sub>/2 = 2π × 214 THz around the <i>Γ</i>-point. As the modes in the flat band lie within the light cone, they correspond to guided resonance modes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Fan, S. & Joannopoulos, J. D. Analysis of guided resonances in photonic crystal slabs. Phys. Rev. B 65, 235112 (2002)." href="/articles/s41566-024-01563-3#ref-CR40" id="ref-link-section-d139125714e3871">40</a></sup>. We obtain the band structure shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5b</a> by modulating the metasurface at <i>ω</i><sub>m3</sub> (<i>m</i> = 0.06 in this case). Remarkably, despite the small modulation strength, the momentum bandgap is very large and spans over a wide range of incident angles (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5c,d</a>): up to 54° in the incidence plane parallel to <i>M</i> − <i>Γ</i> and 33° parallel to <i>Γ</i> − <i>X</i>. Due to the almost constant <i><span class="stix">ℑ</span></i>(<i>ω</i>) inside the bandgap (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5d</a>), the waves that couple to the eigenmodes in the bandgap get amplified at nearly the same rates irrespective of the value of <b>k</b><sub><b><span class="stix">∣</span><span class="stix">∣</span></b></sub>. The bandgap in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5b</a> does not span all possible <b>k</b><sub><span class="stix">∣</span><span class="stix">∣</span></sub> values because of the relatively low quality factors of the magnetic Mie resonances in the nanospheres, as predicted in the <i>LC</i> circuit analysis and the limited value of <i>m</i>. We foresee that by tuning <i>ω</i><sub>m</sub>/2 to the frequency of the higher-order Mie resonances of the nanospheres, the bandgap size for the optical metasurface can be further enhanced. Finally, Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig5">5e</a> shows the loss influence on the metasurface operation similarly to the previous example. Here, the influence of losses is even less pronounced due to the larger modulation strength considered in this example.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Illustration of a large momentum bandgap for propagating waves."><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 5: Illustration of a large momentum bandgap for propagating waves.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/5" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig5_HTML.png?as=webp"><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_Fig5_HTML.png" alt="figure 5" loading="lazy" width="685" height="190"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p><b>a</b>, Optical time-varying metasurface designed to exhibit a momentum bandgap for propagating waves. The purple arrow indicates a propagating eigenmode whose amplitude is growing in time. The geometry of the metasurface is the same as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig4">4a</a>. <b>b</b>, Band structure of a time-varying metasurface with modulation frequency <i>ω</i><sub>m3</sub>. <b>c</b>, Magnified band structure in the green highlighted region in <b>b</b>. <b>d</b>, The corresponding imaginary part of the frequency for fixed <i><span class="stix">ℜ</span></i>(<i>ω</i>) = <i>ω</i><sub>m3</sub>/2. <b>e</b>, Imaginary part of the eigenfrequency <i><span class="stix">ℑ</span></i>(<i>ω</i>) at the centre of the bandgap and the relative amplification bandwidth <span class="mathjax-tex">$\Delta {k}_{| | }^{{\rm{a}}}$</span> as a function of the damping factor <i>γ</i> for the bandgap shown in <b>c</b> and <b>d</b>. Note that the band structure shown in <b>b</b> accommodates the eigenmodes for both transverse-magnetic and -electric polarizations; however, in <b>c</b> and <b>d</b> we optimize the metasurface for transverse-electric waves.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41566-024-01563-3/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Discussion"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6">Discussion</h2><div class="c-article-section__content" id="Sec6-content"><p>Our findings underscore that by exploiting structural resonances in PTC metasurfaces, we can achieve great enhancement of the momentum bandgap size (350-times wider compared with the same metasurface operating far away from the structural resonances) with the modulation strength as small as 1%. In principle, a stronger resonance has the potential to decrease the required modulation strength further. A distinctive feature of our approach is that the achieved momentum bandgap can cover the entire <i>k</i>-space, encompassing both free-space propagating modes and surface modes. This fundamentally provides new physics compared with the implementations based on bulk media<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Lustig, E., Sharabi, Y. & Segev, M. Topological aspects of photonic time crystals. Optica 5, 1390–1395 (2018)." href="/articles/s41566-024-01563-3#ref-CR4" id="ref-link-section-d139125714e4073">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Hayran, Z., Khurgin, J. B. & Monticone, F. ℏω versus ℏk: Dispersion and energy constraints on time-varying photonic materials and time crystals. Opt. Mater. Express 12, 3904–3917 (2022)." href="/articles/s41566-024-01563-3#ref-CR15" id="ref-link-section-d139125714e4076">15</a></sup>, which only support propagating modes and non-resonant metasurfaces<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e4080">14</a></sup> that operate exclusively with surface modes. From this perspective, our approach offers novel opportunities for designing more complex photonic time and space-time crystals, and for amplifying the spontaneous emission of light from emitters near the structure. Indeed, our geometry is practical for the latter application since the emitter does not have to be immersed inside a solid material. Moreover, the proposed PTCs can be used for designing a perfect lens, a long-standing goal in optics, since the information of an object carried by evanescent modes can be effectively amplified, resulting in a highly resolved image of the object. Although our PTCs operate in the infrared spectrum, they can be implemented for the visible spectrum with other materials. In addition, the shape of the meta-atoms is not restricted to spherical, and they can be deployed over a substrate. We anticipate and encourage experimental endeavours to facilitate the proposed approach.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec7-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec7">Methods</h2><div class="c-article-section__content" id="Sec7-content"><h3 class="c-article__sub-heading" id="Sec8">Calculation of eigenmodes of a metasurface with time-varying nanospheres</h3><p>Eigenmodes of a scattering structure are self-standing modes that exist without incident excitation. We use the T-matrix method to evaluate the eigenmodes of the time-varying metasurface (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">6</a> for details). Combining Supplementary equations (<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">28</a>) and (<a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">29</a>), we write</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{\bf{A}}}^{{\rm{sca}}}={\left(\hat{{\bf{U}}}-{\hat{{\bf{T}}}}^{{\rm{(s)}}}(\omega )\sum _{{{\bf{R}}}^{{\prime} }\ne {\bf{0}}}{\hat{{\bf{C}}}}^{(3)}(-{{\bf{R}}}^{{\prime} }){{\rm{e}}}^{-j{{\bf{k}}}_{| | }{{\bf{R}}}^{{\prime} }}\right)}^{-1}{\hat{{\bf{T}}}}^{{\rm{(s)}}}(\omega ) {{\bf{A}}}^{{\rm{inc}}}\,.$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>Here, <b>A</b><sup>inc</sup> and <b>A</b><sup>sca</sup> refer to the incident and scattered field coefficients, respectively, defined on the basis of vector spherical waves. <span class="mathjax-tex">$\hat{\mathbf{T}}^{\mathrm {(s)}}$</span> is the T-matrix of an isolated sphere, <span class="mathjax-tex">$\hat{\mathbf{C}}^{(3)}$</span> is a coupling matrix, <b>Û</b> is the identity matrix and <b>R</b><i>'</i> is a lattice vector. Next, we use <b>A</b><sup>inc</sup> = 0 in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ1">1</a>). We can thus rewrite equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ1">1</a>) as</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\left(\hat{{\bf{U}}}-{\hat{{\bf{T}}}}^{{\rm{(s)}}}(\omega )\sum _{{{\bf{R}}}^{{\prime} }\ne {\bf{0}}}{\hat{{\bf{C}}}}^{(3)}(-{{\bf{R}}}^{{\prime} }){{\rm{e}}}^{-j{{\bf{k}}}_{| | }{{\bf{R}}}^{{\prime} }}\right){{\bf{A}}}^{{\rm{sca}}}=0\,.$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>Finally, for equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ2">2</a>) to have a non-trivial solution that is, <b>A</b><sup>sca</sup> ≠ 0, we arrive at the condition</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\left\vert \hat{{\bf{U}}}-{\hat{{\bf{T}}}}^{{\rm{(s)}}}(\omega )\sum _{{{\bf{R}}}^{{\prime} }\ne {\bf{0}}}{\hat{{\bf{C}}}}^{(3)}(-{{\bf{R}}}^{{\prime} }){{\rm{e}}}^{-j{{\bf{k}}}_{| | }{{\bf{R}}}^{{\prime} }}\right\vert =0\,.$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>Here, the values of <i>ω</i> and <b>k</b><sub><b><span class="stix">∣</span><span class="stix">∣</span></b></sub> for which equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ3">3</a>) is satisfied correspond to the location of the eigenmodes of the time-varying metasurface. Note that instead of calculating the determinant <i>D</i>, we minimize the lowest singular value <span class="mathjax-tex">${S}_{\min }$</span> of the matrix in equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41566-024-01563-3#Equ3">3</a>) to identify the eigenmodes of the system. The metric of the <span class="mathjax-tex">${S}_{\min }$</span> is advantageous over that of the determinant (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">9</a> for more details). Moreover, for plotting the band structures, we limit ourselves only to the dominant dipolar moments in the vector spherical expansion and to the three dominant frequency harmonics (see Supplementary Section <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41566-024-01563-3#MOESM1">6</a>). This assumption does not modify the response of the metasurface inside the momentum bandgap but only allows the elimination of side-bands in the band structure.</p><p>Further, note that for the calculations involving the metasurface made from nanospheres, the radius of the nanospheres <i>R</i> is fixed at 210.6 nm, and the lattice period <i>a</i> is set to 3<i>R</i>. Moreover, the material susceptibility for the calculations involving dispersionless nanospheres is chosen to be <i>χ</i><sub>0</sub> = 10.68. Such a value of <i>χ</i><sub>0</sub> corresponds to silicon, which is nearly lossless and dispersionless in the considered infrared frequency regime<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Vetterl, O. et al. Intrinsic microcrystalline silicon: a new material for photovoltaics. Sol. Energy Mater. Sol. Cells 62, 97 (2000)." href="/articles/s41566-024-01563-3#ref-CR39" id="ref-link-section-d139125714e5205">39</a></sup>.</p><h3 class="c-article__sub-heading" id="Sec9">Numerical simulations for a time-varying <i>LC</i> metasurface</h3><p>The simulation results presented in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3c–e,g</a> are generated using COMSOL Multiphysics. The metasurface is modelled as an <i>LC</i> parallel circuit with surface current density defined as</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{\bf{J}}}_{{\rm{s}}}={{\bf{J}}}_{{\rm{L}}}+\frac{d[C(t){\bf{E}}(t)]}{dt},$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>where <b>E</b>(<i>t</i>) is the tangential electric field on the surface. Here, <b>J</b><sub>L</sub> is related to <b>E</b>(<i>t</i>) by <span class="mathjax-tex">${\bf{E}}(t)=L\frac{d{{\bf{J}}}_{{\rm{L}}}}{dt}$</span>. This relationship is established via the boundary ordinary differential equation and differential algebraic equation module in COMSOL transient simulation.</p><p>At <i>t</i> = 0, a stationary <i>LC</i> surface is excited from the left by a transverse-magnetic-polarized surface wave with momentum <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 5<i>k</i><sub>r0</sub> (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3c</a>) and <i>k</i><sub><span class="stix">∣</span><span class="stix">∣</span></sub> = 10<i>k</i><sub>r0</sub> (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3d</a>). For <i>t</i> > 0, harmonic modulation of the effective capacitance is initiated, opening the momentum bandgap and resulting in an exponential growth of the surface mode and higher-order frequency harmonics, some of which propagate inside the light cone. This growth is observed in the magnetic-field snapshot at <i>t</i> = 30<i>T</i><sub>m</sub> (<i>T</i><sub>m</sub> = 2π/<i>ω</i><sub>m</sub>) depicted in the lower panel of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41566-024-01563-3#Fig3">3c</a>.</p><p>Temporal modulation keeps the momentum of eigenwaves unchanged but generates backward and forward harmonics with equal amplitudes, creating a standing wave along the horizontal direction<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Wang, X. et al. Metasurface-based realization of photonic time crystals. Sci. Adv. 9, eadg7541 (2023)." href="/articles/s41566-024-01563-3#ref-CR14" id="ref-link-section-d139125714e5480">14</a></sup>.</p></div></div></section> </div> <div> <section data-title="Data availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>The authors declare that the data supporting the findings of this study are available within the paper, in the <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/s41566-024-01563-3#Sec11">Supplementary Information</a>, and are available from the corresponding authors upon request.</p> </div></div></section><section data-title="Code availability"><div class="c-article-section" id="code-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="code-availability">Code availability</h2><div class="c-article-section__content" id="code-availability-content"> <p>The codes are available from the corresponding authors on reasonable request.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Galiffi, E. et al. 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M. <i>Microwave Engineering</i> (John Wiley and Sons, 2011).</p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/s41566-024-01563-3?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>X.W. and C.R. acknowledge support by the Helmholtz Association via the Helmholtz program ‘Materials Systems Engineering’ (MSE). P.G., A.G.L. and C.R. are part of the Max Planck School of Photonics, supported by the Bundesministerium für Bildung und Forschung, the Max Planck Society, and the Fraunhofer Society. P.G. acknowledges support from the Karlsruhe School of Optics and Photonics (KSOP). P.G. and C.R. acknowledge support by the German Research Foundation within the SFB 1173 (project ID no. 258734477). V.A. acknowledges the Research Council of Finland (project no. 356797), Finnish Foundation for Technology Promotion, and Research Council of Finland Flagship Programme, Photonics Research and Innovation (PREIN), decision number 346529, Aalto University. X.W. acknowledges the Fundamental Research Funds for the Central Universities, China (project no. 3072024WD2603) and the Scientific Research Foundation, Harbin Engineering University, China (project no. 0165400209002). We would like to thank S. Tretyakov for the fruitful discussions on the application of the designed PTCs for constructing a perfect lens.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><span class="c-article-author-information__subtitle u-visually-hidden" id="author-notes">Author notes</span><ol class="c-article-author-information__list"><li class="c-article-author-information__item" id="na1"><p>These authors contributed equally: X. Wang, P. Garg.</p></li></ol><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Qingdao Innovation and Development Base, Harbin Engineering University, Qingdao, China</p><p class="c-article-author-affiliation__authors-list">X. Wang</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Institute of Nanotechnology, Karlsruhe Institute of Technology, Karlsruhe, Germany</p><p class="c-article-author-affiliation__authors-list">X. Wang & C. Rockstuhl</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, Karlsruhe, Germany</p><p class="c-article-author-affiliation__authors-list">P. Garg, A. G. Lamprianidis & C. Rockstuhl</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Department of Physics and Mathematics, University of Eastern Finland, Joensuu, Finland</p><p class="c-article-author-affiliation__authors-list">M. S. Mirmoosa</p></li><li id="Aff5"><p class="c-article-author-affiliation__address">Department of Electronics and Nanoengineering, Aalto University, Espoo, Finland</p><p class="c-article-author-affiliation__authors-list">V. S. Asadchy</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-X_-Wang-Aff1-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">X. 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Mirmoosa</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=M.%20S.%20Mirmoosa" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=M.%20S.%20Mirmoosa" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22M.%20S.%20Mirmoosa%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-A__G_-Lamprianidis-Aff3"><span class="c-article-authors-search__title u-h3 js-search-name">A. G. Lamprianidis</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=A.%20G.%20Lamprianidis" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=A.%20G.%20Lamprianidis" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22A.%20G.%20Lamprianidis%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-C_-Rockstuhl-Aff2-Aff3"><span class="c-article-authors-search__title u-h3 js-search-name">C. Rockstuhl</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=C.%20Rockstuhl" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=C.%20Rockstuhl" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22C.%20Rockstuhl%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-V__S_-Asadchy-Aff5"><span class="c-article-authors-search__title u-h3 js-search-name">V. S. Asadchy</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=V.%20S.%20Asadchy" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=V.%20S.%20Asadchy" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22V.%20S.%20Asadchy%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>X.W. proposed the initial idea of the time-varying resonant structure for enhancing the momentum bandgap, and V.A. expanded this concept to optical metasurfaces to address the challenges in creating optical photonic time crystals. X.W., P.G., M.S.M., A.G.L. and C.R. contributed to the development of the theory of time-varying Lorentz media, whereas P.G., A.G.L. and C.R. worked on the theory of time-varying optical metasurfaces. P.G. performed the simulations of optical metasurfaces. X.W. and M.S.M. developed the theory and simulation of time-varying LC metasurfaces. X.W., P.G., M.S.M. and V.A. performed data analysis. All authors actively participated in discussions. X.W., P.G., M.S.M. and V.A. prepared the manuscript, with all authors contributing to its review and editing. V.A. and C.R. provided supervision, with C.R. also responsible for acquiring funding.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding authors</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:xuchen.wang@hrbeu.edu.cn">X. Wang</a> or <a id="corresp-c2" href="mailto:puneet.garg@kit.edu">P. Garg</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar2">Competing interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Peer review"><div class="c-article-section" id="peer-review-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="peer-review">Peer review</h2><div class="c-article-section__content" id="peer-review-content"> <h3 class="c-article__sub-heading" id="FPar1">Peer review information</h3> <p><i>Nature Photonics</i> thanks the anonymous reviewers for their contribution to the peer review of this work.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p><b>Publisher’s note</b> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section data-title="Supplementary information"><div class="c-article-section" id="Sec11-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec11">Supplementary information</h2><div class="c-article-section__content" id="Sec11-content"><div data-test="supplementary-info"><div id="figshareContainer" class="c-article-figshare-container" data-test="figshare-container"></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM1"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary information" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_MOESM1_ESM.pdf" data-supp-info-image="">Supplementary Information</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>Supplementary Sections 1–9.</p></div></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM2"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary video 1" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_MOESM2_ESM.mp4" data-supp-info-image="">Supplementary Video 1</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>The complete field evolution animation for Fig. 3c.</p></div></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM3"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary video 2" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41566-024-01563-3/MediaObjects/41566_2024_1563_MOESM3_ESM.mp4" data-supp-info-image="">Supplementary Video 2</a></h3><div class="c-article-supplementary__description" data-component="thumbnail-container"><p>The complete field evolution animation for Fig. 3d.</p></div></div></div></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. 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