Space-time superoscillations

Domain: neuroscience
Published: 2026-02-02
Editorial outcome: major_revision

Abstract
Superoscillation (SO) refers to the phenomenon in which a wave field locally oscillates at a rate exceeding its highest spatial or temporal Fourier component. SO has enabled light to be focused into arbitrarily small hotspots, forming the basis of superresolution imaging and metrology far beyond the Abbe-Rayleigh diffraction limit. Here we show that spatial and temporal superoscillations can occur simultaneously at the same point in space-time, a phenomenon we term space-time superoscillation (STSO). We demonstrate STSOs in a band-limited version of supertoroidal light pulses, a recently introduced family of space–time nonseparable finite-energy solutions of Maxwell’s equations. Our results reveal a new regime of extreme spatio-temporal field structuring, with implications for ultrafast metrology, light–matter interactions, and deep-subwavelength control of electromagnetic waves.

Introduction
Superoscillations (SOs) have emerged as a powerful concept in wave physics, enabling local oscillation rates that exceed the highest frequency present in the global spectrum1,2. This counterintuitive phenomenon—originally developed for waves in one, two, or three dimensions—has inspired applications in optical superresolution imaging, precision metrology, and structured light engineering1–5. The concept has recently been extended to ultrafast space-time nonseparable structured light fields8–10. Among them are supertoroidal pulses (STPs), a class of solutions to Maxwell’s equations of rich topological structure11. A basic member of this family is the “Flying Doughnut” pulse—a wave packet with high energy confinement and non-trivial topology12–15. Finite-energy solutions require α ≥ 1; larger values of α yield increasingly rapid local field variations and stronger confinement16. STPs can be TE (transverse electric) or TM (transverse magnetic). Here we focus on the TE case without loss of generality. The full field expressions are provided as11:

Eθ(α) = −2αi f0 (q1 + iτ)α (q2 − iσ)α / [r2 + (q1 + iτ)(q2 − iσ)]α+1 (1)

Hr(α) = 2αi f0 (q1 + iτ)α (q2 − iσ)α / [r2 + (q1 + iτ)(q2 − iσ)]α+1 (2)

Hz(α) = −4αf0 (q1 + iτ)α (q2 − iσ)α / [r2 + (q1 + iτ)(q2 − iσ)]α+1 (3)

Here we show that the band-limited version of STPs include regions where both the local spatial wave vector and local frequency of oscillation in time exceed the respective global spectral bounds. We term this phenomenon space–time superoscillation (STSO).

Results
Band-limited supertoroidal pulses

An STP is characterized by two length parameters, q1 and q2, related to the central wavelength of the wave packet and the Rayleigh range, respectively, and a real dimensionless parameter α, which controls the pulse topology11. In the cylindrical coordinate system, the instantaneous electric field can be written as Eθ(r,z,t) = A(r,z,t)eiφ(r,z,t), where A and φ correspond to the amplitude and phase.

Although STPs are localized finite-energy solutions to Maxwell’s equations, they are not band-limited. We therefore introduce a band-limited version of STPs, i.e. the STPs with their time domain and space domain spectra truncated at f < fm and k < km, respectively. The entire spectrum of the STP is confined on the surface of light cone, i.e the conic surface with unit slope of its generatrix in the coordinate (kr, kz, 2πf/c), where the spatial wave vectors are related to the spatial frequencies by kr = 2πfr/c and kz = 2πfz/c. In the following, we will consider band-limited pulses:

E~c(r,z,t) = ∫−1 1 ∫−1 1 ∫−1 1 E~(fr, fz, f)B(f)exp[i2π(frr + fzz + ft)] J0(2πrfr) fr dfr dfz df (4)

where E~(fr, fz, f), the spectrum of a standard STP, is truncated by function B(f) cut-off frequencies, fm and km = 2πfm/c, and J0(2πrfr) is the 0-th order Bessel function of the first kind (see details in Supplementary Material S2). From now on, we will be analyzing the properties of this band-limited version of STP.

Coexistence of spatial and temporal superoscillation

We illustrate the presence of STSO, i.e. a point in space-time where spatial and temporal SOs exist simultaneously, where both spatial (along the radial direction) and temporal phase gradients, ∂φ/∂r and ∂φ/∂t, exceed km and fm correspondingly, by considering a TE-mode band-limited version of an STP with α = 50, q2 = 50q1, q1 = 1 (see Fig. 1a for the electric field distribution Eθ(r,z,t) in space-time, and Fig. 1b for the field isosurface structure). The spatial (radial) variation of the electric field at (z=0, t=ts) is presented in Fig. 1c, where we observe that a segment in the off-axis region (r=rs) oscillates considerably faster than the harmonic oscillation of the maximal radial frequency km (dashed red line). Similarly, in Fig. 1d, we show the temporal variation of the electric field at (z=0, r=rs) which around t=ts exhibits oscillations faster than frequency fm (dashed red line). Therefore, the field at the focal plane (z=0) exhibits STSO.

We prove the existence of STSO of the band-limited STP in the (r, t) domain at focus (z=0) for a fixed cut-off frequency km = 2c/q1. In the SO region, the local temporal frequency or spatial wave vector should exceed the corresponding maximum value (fm or km)1,17. Figure 2a1–a4 shows log10 |Re[Eθ(r,z=0,t)]|, unwrapped phase of φ(r,z=0,t), phase gradient, energy density and Poynting vector distributions, respectively, of the fundamental toroidal pulse (α=1). In Fig. 2a3, there is only a small region with temporal local wave vector ∂φ/∂t exceeding fm, but no spatial SO and thus no STSO. In Fig. 2a4, we also highlight the region of energy backflow in accordance with prior works11,18. Higher values of α lead to spatial and temporal superoscillations, which result in multi-cycle structure and spatio-temporal focusing. Figure 2b1–b4 shows the corresponding characteristics of a band-limited STP at α=50. Figure 2b3 shows the presence of spatial SOs, where the local wavevector exceeds km, and temporal SOs, where the local temporal frequency exceeds fm. Importantly, spatial and temporal SO regions overlap, resulting in STSOs. We can also observe that STSOs appear at low amplitude regions and are accompanied by areas of energy backflow, see inset to Fig. 2b4.

Behaviors of supertoroidal pulses of different orders

Figure 3a,b show temporal profiles of ∂φ/∂t at specific radius r=10q1 and spatial profiles of ∂φ/∂r at specific time t=2q1/c, respectively, for a series of band-limited STPs with different values of parameter α. The spatial and temporal oscillations become stronger and faster with increasing α. Figure 3c shows the ratio between the energy within the rapid oscillations region, ESTSO, and the total energy of the pulse, ET, as a function of parameter α. For α > 38, the energy of STSOs monotonically increases with increasing α, see Fig. 3c. In the SO region, the local temporal frequency or spatial wave vector should exceed the corresponding maximum value (fm or km). reaching values on the order of 10−3 of the total pulse energy—within reach of experimental detection19–21.

Spectral signatures outside of the light cone

A hallmark of SO is that local spectra contain frequency components that exceed the global band limit of the entire field1. Similarly, we demonstrate that the spectra of STSO segments of band-limited STPs contain components out of the light cone, i.e. they oscillate faster than what would be expected based on the pulse bandwidth. Figure 4 shows the entire spectra (a1–a4) and local spectra (b1–b4) of band-limited STPs of different order, α=1, 10, 50, and 100. The local spectra are retrieved by the Fourier transform of the local space–time segment Eθ(r~rs, z=0, t~ts) into the (kr, f) domain. The spectra of full pulses are confined on the surface of the light cone, whereas local spectra present spectral components outside the light cone. The presence of the off-light-cone components directly reveals the presence of super-oscillations. Whereas for the fundamental toroidal pulse (α=1), spectra are fully contained within the light cone, in the case of the STPs (α>1) the off-cone components become stronger with increasing value of α. Therefore, the field locally oscillates faster than permitted by their global spectra.

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This work theoretically demonstrated a concept of space-time super-oscillation by introducing the super-toroidal pulse with non-separable nature in space and time domain. Through extends the concept of super-oscillation to the space-time domain, super-oscillation effect can be created at a special location in space and at a specific moment in time. This is a quite interesting work in science. However, I think it may not suitable for publication in this journal for the following reasons:\n1. The content of the manuscript is heavily inclined towards physics, which do not align with the scope of this journal. While the theoretical insights into spatial-time super-oscillation are valuable, they are more suited to a specialized physics journal.\n2. The manuscript presents a well-structured theoretical framework for spatial-time super-oscillation. However, it lacks experimental validation. Without experimental results, it is difficult to fully assess the validity and applicability of the proposed theories. I strongly suggest that the authors conduct relevant experiments to support their theoretical work. This would significantly enhance the credibility and value of their research.\n3. Another major concern is the unclear application prospects of the research. We can see from the simulation results that the spectra of STSO segments of STPs contains components out of the light cone, and the spatial and temporal SOs overlap indicates the presence of STSOs. However, the STSO only exist at a specific space and time, and the extremely low amplitude also surrounded by a high amplitude field. How to use this physical effect in the practical scenarios. For a paper to be published in NC, it is essential that the potential applications of the research are clearly discussed, which could help readers understand the significance and relevance of the work.\nIn conclusion, I hope that the authors will consider these comments and revise their manuscript accordingly for future submission to a more suitable journal.", "reviewer_2": "The authors presented the theoretical manner of constructing superoscillating field in space-time domain, by extending the model presented in ref. [11] in the main text, which, honestly, is an intriguing and promising subject, and probably is a key to precisely manipulate ultrafast pulse of light. This theory is introduced completely and lucidly, establishing the important criteria for identifying superoscillation, and also providing adequate details for who follows this work. However, the authors merely find the superoscillatory cases, rather than reveal the underlying primary phenomena which are different from the relevant object in other fields.\nI strongly suggest to generalize this kind of spatiotemporal structures to have higher complexity, and more controllable degrees of freedom, e.g. the superoscillating spatiotemporal arrays [Opt. Lett. 45, 2538-2541 (2020); Nat. Commun. 14, 3237 (2023)].\nThe additional discussions uncovering the promising prospect on light-matter interaction is another way to increase the contents of this manuscript. I really look forward to the novel effects induced by the coupling between superoscillating pulse and new material, e.g. controlling the dynamics of phonons via the presented light field [Nat. Mater. 20, 607-611 (2021)].\nConventionally, superoscillating main lobes just occupy little energy, affected by the side-lobes. Here, the analysis of power occupation mentioned that the energy of superoscillating region obtains the higher ratio versus total energy, as index α increases, but more explanations were not given out.\nAccording to my knowledge, the superoscillating signal can be designed by superimposing the signals with different parameters. Is it possible to construct the space-time superoscillation in this way (the units are replaced by the fields with different parameters α)? Can it improve the energy occupation of superoscillating lobes?\nIt is better to interpret the motivation of implementing the “Fourier transform” in Eq. (S21) of supplementary material, which is distinct to the conventional one, i.e. Hankel transform.\nThe important way to construct the bandlimited supertoroidal pulse should be introduced in the main text, rather than in supplementary material.\nThe group velocity, as an essential index, is not given out in the manuscript, which determines the propagation of supertoroidal pulse.\nThe shapes of superoscillating pulse (α=50) is only exhibited in Fig. 1b, but the higher order cases were not given out in the appropriate figures.\nFinally, this manuscript presents an attractive concept, but more primary effects induced by this supersocillation are not revealed here. It would be remarkable to add more contents to this manuscript."}, "author_response": "Response to Reviewers\nReviewer #1 (Comments to the Author):\nThis work theoretically demonstrated a concept of space-time super-oscillation by\nintroducing the super-toroidal pulse with non-separable nature in space and time domain.\nThrough extends the concept of super-oscillation to the space-time domain, super-oscillation\neffect can be created at a special location in space and at a specific moment in time. This is\na quite interesting work in science. However, I think it may not suitable for publication in this\njournal for the following reasons:\nResponse: We thank the Reviewer for the positive comments. In the following we address\ntheir concerns.\n1. The content of the manuscript is heavily inclined towards physics, which do not align with\nthe scope of this journal. While the theoretical insights into spatial-time super-oscillation are\nvaluable, they are more suited to a specialized physics journal.\nResponse: We respectfully disagree with the assertion that our work is not aligned with the\nscope of Nature Communications. As clearly stated on the journal's website: \"Nature\nCommunications is an open access, multidisciplinary journal dedicated to publishing high-\nquality research in all areas of the biological, health, physical, chemical, Earth, social,\nmathematical, applied, and engineering sciences. Papers published by the journal aim to\nrepresent important advances of significance to specialists within each field.\"\nOur work on space-time superoscillations represents a significant theoretical\nadvancement in physics, with potential applications in various fields such as optics, quantum\nmechanics, and materials science. Our findings extend the concept of superoscillations to\nthe space-time domain and will be of substantial interest to specialists within this research\narea. As such, they align well with the journal's commitment to publishing impactful\nresearch.\n2. The manuscript presents a well-structured theoretical framework for spatial-time super-\noscillation. However, it lacks experimental validation. Without experimental results, it is\ndifficult to fully assess the validity and applicability of the proposed theories. I strongly\nsuggest that the authors conduct relevant experiments to support their theoretical work. This\nwould significantly enhance the credibility and value of their research.\nResponse: We agree that experimental validation of theoretical predictions is important.\nHowever, the purpose of the current manuscript is different: it seeks to draw attention to the\nunexpected phenomenon of space-time superoscillations in space-time nonseparable\npulses. We argue that it is timely to present the theory of space-time superoscillations now,\nwhich can stimulate interest not only in the Optics community, but also by researchers\nworking in acoustics, quantum mechanics, fluid dynamics.\nWe also note that space-time superoscillating pulses are exact solutions of Maxwell’s\nequations, one of the most successful theories of classical physics. Thus, experimental work\nwould target the practicality of generating space-time superoscillations, rather their validity.\n3. Another major concern is the unclear application prospects of the research. We can see\nfrom the simulation results that the spectra of STSO segments of STPs contains\ncomponents out of the light cone, and the spatial and temporal SOs overlap indicates the\npresence of STSOs. However, the STSO only exist at a specific space and time, and the\nextremely low amplitude also surrounded by a high amplitude field. How to use this physical\neffect in the practical scenarios. For a paper to be published in NC, it is essential that the\npotential applications of the research are clearly discussed, which could help readers\nunderstand the significance and relevance of the work.\nResponse: The applications of STSOs should be clear and highly anticipated. The spatial\nsuperoscillation and temporal superoscillation have recently created advanced applications\nindividually. For example, the spatial superoscillation has recently enabled ultra-small super-\nresolved nanometric metrology [Science364,771-775(2019)], microscopy and imaging [Nat.\nRev. Phys. 4, 16–32 (2022), Nature Mater 11, 432–435 (2012)], even picometric metrology\n[Nat. Mater. 22, 844–847 (2023)]. In the case of temporal superoscillations, advanced\nspectroscopies have been proposed [Phys. Rev. Lett. 131, 153803 (2023)]. Therefore, by\ncombining for the first time these two forms of superoscillations into space-time\nsuperoscillations, we anticipate advanced applications in metrology, imaging, sensing, and\nspectroscopy, at ultra-fine spatial and ultrafast temporal resolution.\nWe have added specific examples and references to our revised manuscript to\nillustrate these potential applications. Additionally, we have included a section on how\nspace-time superoscillations can be used to enhance light-matter interactions, which could\nlead to novel applications in quantum technologies and material science. See added content\nin the second-last paragraph in Discussion highlighted in the main text.\nIn conclusion, I hope that the authors will consider these comments and revise their\nmanuscript accordingly for future submission to a more suitable journal.\nResponse: We hope that we have convinced the reviewer that our work is suitable for\npublication in Nature Communications.\nReviewer #2 (Comments to the Author):\nThe authors presented the theoretical manner of constructing superoscillating field in space-\ntime domain, by extending the model presented in ref. [11] in the main text, which, honestly,\nis an intriguing and promising subject, and probably is a key to precisely manipulate ultrafast\npulse of light. This theory is introduced completely and lucidly, establishing the important\ncriteria for identifying superoscillation, and also providing adequate details for who follows\nthis work.\nResponse: We thank the referee for the very positive comments.\nHowever, the authors merely find the superoscillatory cases, rather than reveal the\nunderlying primary phenomena which are different from the relevant object in other fields.\nResponse: We thank the reviewer for raising this important issue. We would like to stress\nthat the space-time superoscillatory pulses presented in our manuscript have no analog in\nother fields; this is the first illustration of space-time superoscillations (STSOs). Further, our\nwork invites key questions about the underlying physics of electromagnetic STSOs. For\ninstance, what is the role of space-time nonseparability in the existence of STSOs? Is there\na canonical route to the construction of STSOs? We discuss these questions in the second\nparagraph in Discussion highlighted in the main text.\nI strongly suggest to generalize this kind of spatiotemporal structures to have higher\ncomplexity, and more controllable degrees of freedom, e.g. the superoscillating\nspatiotemporal arrays [Opt. Lett. 45, 2538-2541 (2020); Nat. Commun. 14, 3237 (2023)].\nResponse: Thank you for your suggested references, they are indeed useful, we have cited\nthem in our updated version. The STSO spots demonstrated in our manuscript are in\nisolated form, which are possible to be extended into more complex structured form, such\nas STSO spot array and multi-lobe forms. For instance, we can linearly superimpose a pulse\nchain of many STSOs to generate SO spatiotemporal arrays, analogous to the previous\nmethods to produce multi-lobe SO arrays in the spatial domain [Opt. Lett. 45, 2538-2541\n(2020); Nat. Commun. 14, 3237 (2023)]. See the end of the second paragraph in Discussion.\nThe additional discussions uncovering the promising prospect on light-matter interaction is\nanother way to increase the contents of this manuscript. I really look forward to the novel\neffects induced by the coupling between superoscillating pulse and new material, e.g.\ncontrolling the dynamics of phonons via the presented light field [Nat. Mater. 20, 607-611 (2021)].\nResponse: Thank you for your valuable suggestion and suggested reference. Indeed, we\nbelieve the STSO in light pulses will unlock nontrivial light-matter interaction, we added the\ndiscussion and cited the suggested reference. See the end of the second-last paragraph in\nDiscussion.\nConventionally, superoscillating main lobes just occupy little energy, affected by the side-\nlobes. Here, the analysis of power occupation mentioned that the energy of superoscillating\nregion obtains the higher ratio versus total energy, as index α increases, but more\nexplanations were not given out.\nResponse: We thank the reviewer for their valuable suggestion. The physical meaning of\nparameter α is to ensure finite energy localization of the pulse (see original work by\nZiolkowski: “Localized transmission of electromagnetic energy”, Phys. Rev. A 39, 2005\n(1989)). In particular, the parameter a α is a real dimensionless number that must satisfy α\n≥ 1 to ensure finite energy solutions, while α < 1 results in pulses of infinite energy, such as\nplanar waves and cylindrical waves. With increasing α, the local oscillations become\nincreasing faster leading to stronger confinement. However, the total energy contained in\nthe superoscillatory region remains relatively small (~3% or lower) in accordance with\nconventional superoscillations. See the added explanation and reference in the first and\nnext-to-last paragraphs of Results highlighted in the main text.\nAccording to my knowledge, the superoscillating signal can be designed by superimposing\nthe signals with different parameters. Is it possible to construct the space-time\nsuperoscillation in this way (the units are replaced by the fields with different parameters α)?\nCan it improve the energy occupation of superoscillating lobes?\nResponse: Thank you for your valuable insights. Indeed, the space-time superoscillatory\npulses can be constructed by an appropriate superposition of plane waves as presented in\nFig. 4a of the manuscript and Figure S2 of the Supplementary Materials. It is also likely that\na different basis may be more convenient (especially a cylindrically symmetric one) allowing\nto maximize the energy of the superoscillatory lobes, a question that we reserve for future\nwork. This is now discussed in paragraph 3 of the Discussion section.\nIt is better to interpret the motivation of implementing the “Fourier transform” in Eq. (S21)\nof supplementary material, which is distinct to the conventional one, i.e. Hankel transform.\nResponse: We thank the reviewer for this question. Since the monochromatic components\nof the pulse lie on the light cone, the temporal Fourier transform is sufficient.\nThe important way to construct the bandlimited supertoroidal pulse should be introduced in\nthe main text, rather than in supplementary material.\nResponse: Thanks for the suggestion. We have modified the main text accordingly. See\nthe second paragraph of Results.\nThe group velocity, as an essential index, is not given out in the manuscript, which\ndetermines the propagation of supertoroidal pulse.\nResponse: Theoretically, the group velocity of the STPs of different orders is always the\nconstant of c, see detailed proof in Supplementary Materials S1 around Eq. (S20) We add\nexplanation in the end of first paragraph of Results.\nThe shapes of superoscillating pulse (α=50) is only exhibited in Fig. 1b, but the higher\norder cases were not given out in the appropriate figures.\nResponse: We now include higher order cases in Fig. S1 of the supplementary material.\nFinally, this manuscript presents an attractive concept, but more primary effects induced\nby this supersocillation are not revealed here. It would be remarkable to add more\ncontents to this manuscript.\nResponse: We thank the referee for the very positive comments."}, {"round": 2, "reviewer_comments": {"reviewer_1": "In the revised manuscript, the authors have added some new content to the Discussion section, as well as a section explaining the bandlimited spatiotemporal STP pulse. While certain reviewer questions have been addressed, the revisions do not introduce any essential improvements to the original version.\nThe authors claim that spatial and temporal superoscillations can be simultaneously observed at a specific point in space and time using spatiotemporally structured supertoroidal pulses, based on theoretical analysis. However, no experimental evidence is provided to support this claim. As shown in Figure 1, the spatiotemporal superoscillations (STSOs) occur only at a particular off-axis position and moment, and with very low amplitude. Under such conditions, how can this effect be experimentally validated?\nFurthermore, the authors claimed that the bandwidth and pulse duration can be compressed by factors of 5 and 45, respectively, while applying STSO effect to a Ti:sapphire femtosecond laser. Yet, no detailed explanation is given regarding how this can be achieved, and no experimental data are presented. The feasibility and potential applications of this effect cannot be convincingly demonstrated based solely on calculated values. Similar concerns apply to the other proposed applications in metrology, imaging, and spectroscopy.\nIn summary, while the concept of spatiotemporal superoscillation is scientifically intriguing, a carefully designed experimental demonstration is essential to validate the proposed STSO effect. I strongly encourage the authors to devote effort on experimental verification. Without it, the manuscript may not meet the publication standards of a high-impact journal such as Nature Communication.", "reviewer_2": "In the revised manuscript, the authors make more discussions on the potential applications (required by reviewer #1), and the more theoretical analysis, which has addressed my previous comments. I have to say again that this topic is interesting and promising. However, the revisions still cannot resolve some of my concerns. In order to demonstrate the importance of space-time superoscillation, I expect that the authors were able to unravel the potential phenomena with the use of presented theory, e.g. the intriguing dynamics considering both space and time domains rather than only introducing the mathematical framework. In my opinion, the new physical concept is closely associate with relevant essential effects, which, however, the authors did not particularly mention in the revision. Honestly, the used mathematical techniques are not surprising, which were mentioned in the previous papers. As a result, more contents of revealing particular physical phenomena are necessary. Otherwise, as suggested by the reviewer 1, experimental demonstrations on the space-time concept would be helpful. Without these, I agree with reviewer 1 that the manuscript may be suitable to a specialized journal, rather than the Nat. Commun.", "reviewer_3": "The manuscript first proposed and systematically demonstrated the phenomenon of \"space-time superoscillation\", a groundbreaking concept in my opinion, which extends the concept of Michael Berry’s superoscillation from the pure space or time domain to the spacetime joint domain, connecting to the recently emerging topic of spatiotemporal optics and physics. This work has significant theoretical innovation and potential application values. With very solid theories, the STSO phenomenon is clearly demonstrated and verified through multiple dimensions such as local wave vector/frequency, energy distribution, energy flow, and spectral analysis. The experimental plan and the numerical simulation part are reasonably designed and fully demonstrated. The article has a clear structure and smooth language, professionally written, undoubtedly meeting the high-level standards of Nature Communications. I recommend publication without delay. Some suggested minor revisions are as following:\n1. The statement that \"the group velocity of the STPs of different orders is always the constant of *c*\" is a critical point. While the proof is correctly relegated to the supplement, its fundamental importance warrants a slightly more substantive note in the main text. Please consider adding a short justification, such as: \"This is expected as STPs are vacuum solutions to Maxwell's equations and thus propagate without dispersion.\"\n2. To enhance its depth, it would be valuable to briefly acknowledge the signal-to-noise challenge inherent in detecting STSOs, given their occurrence in low-field regions. A sentence could be added, for instance: \"We note that the practical observation of STSOs, like other superoscillatory features, will require techniques sensitive to weak signals amidst higher-amplitude backgrounds, a challenge successfully overcome in prior spatial superoscillation measurements XXX.\"\n3. In prior works, topological skyrmion textures were demonstrated in such STP pulses, I believe the authors may also need to discuss more on how the STSO associated with skyrmion textures can hatch more meaningful applications, for instance, deep-subwavelength sensing, metrology, and light-matter interaction.\n4. \"Abbey-Reileigh\" should be the more common \"Abbe-Rayleigh\".\n5. Unit writing: For example, \"6 fs long pulses\" is suggested to be changed to \"6-fs-long pulses\" or \"pulses with a duration of 6 fs\". It is suggested that \"~80 nm\" be changed to \"≈80 nm\" or \"about 80 nm\".\n6. \"In particular, the parameter a α...\" The beginning of this sentence is a bit awkward. It is suggested to modify it to \"The parameter α is a real dimensionless number that...\" .\n7. “Our work expands the concept of SO to include spatiotemporal effects, illustrated by the arbitrarily spatially and temporally confined focal spots in STPs. \"this sentence is a bit long and can be split or reassembled for clarity.\n8. The conclusion part uses two \"Finally\" in a row. It is suggested that the last \"Finally\" be changed to \"Furthermore\" or \"Beyond electromagnetism\"."}, "author_response": "Response to reviewers' comments\nReviewer #1 (Remarks to the Author):\nIn the revised manuscript, the authors have added some new content to the Discussion\nsection, as well as a section explaining the bandlimited spatiotemporal STP pulse. While\ncertain reviewer questions have been addressed, the revisions do not introduce any essential\nimprovements to the original version.\nResponse: We thank the reviewer for acknowledging our revision. We believe we have\naddressed all comments raised in the previous round of review. We also address below all\ncomments raised in the current round.\n1. The authors claim that spatial and temporal superoscillations can be simultaneously\nobserved at a specific point in space and time using spatiotemporally structured\nsupertoroidal pulses, based on theoretical analysis. However, no experimental evidence is\nprovided to support this claim. As shown in Figure 1, the spatiotemporal superoscillations\n(STSOs) occur only at a particular off-axis position and moment, and with very low\namplitude. Under such conditions, how can this effect be experimentally validated?\nResponse: We argue that the novelty of the work is sufficient to warrant publication in\nNature Communications and expect that it will stimulate experimental investigations of\nSTSOs. We have extended our discussion on the practical considerations of generating\nSTSOs in in the last paragraph of first subsection in Discussion.\nRegarding “very low amplitude” issue, we would like to note that the amplitude of the STSOs\nis of the order 10-3 of the peak amplitude in a non-optimized case (see Fig. 2b), while the total\nenergy contained in the STSO regions can exceed 0.3% (see Fig. 3c). This is significantly\nhigher than what is typically encountered in conventional superoscillations and lies within\ncurrent experimental capabilities. For instance, in superoscillatory microscopy [Sci. Rep. 3,\n1715 (2013)], the SO hotspot is about 0.1% energy of the total energy. We include this\ndiscussion in the first section of Discussion.\n2. Furthermore, the authors claimed that the bandwidth and pulse duration can be\ncompressed by factors of 5 and 45, respectively, while applying STSO effect to a Ti:sapphire\nfemtosecond laser. Yet, no detailed explanation is given regarding how this can be achieved,\nand no experimental data are presented. The feasibility and potential applications of this\neffect cannot be convincingly demonstrated based solely on calculated values. Similar\nconcerns apply to the other proposed applications in metrology, imaging, and spectroscopy.\nResponse: We thank the reviewer for raising this point. We now describe in detail the\ncalculation of spatial and temporal compression —— “For instance, in the optical domain,\ncommercially available Ti:Sa lasers can provide up to Δλ = 200 nm bandwidth (FWHM) central\nat λ = 800 nm. The corresponding limit to spatial and temporal focusing are λ /2 = 400 nm and\n0 0\n= K 0 2 = 4.64 fs (where K=0.441 for Gaussian profile pulse), respectively. Based on the results\nc\nshown in Figs. 3(a) and 3(b), for STP of α=100, at STSO spot, the temporal and spatial local\nwavevectors are 7 times and 5 times larger than the central wavevector, respectively. Here, we\nevaluate central wavevector as k =1 (unit: 1/q ) as the central wavevector, it actually should be\nc 1\nless than 1, see Fig. 4a. Therefore, we evaluate such STSO in STP allow temporal and spatial\nconfinements of ~Δτ/7=0.66 fs and ~λ /2/5 = 80 nm, respectively. If we apply the temporal SO\n0\nseparately (the central peak of Figs. 3(a) over 45 times than k =1), the temporal confinements can\nc\nalso be ~45 times shorter than Δτ (about 0.1 fs).” We replaced the corresponding statement into\nthe more accurate in Discussion.\nOur results indicate that very rapid local spatial and temporal field fluctuations are possible,\nwhich could find applications in metrology, imaging, and spectroscopy. We agree with the\nreviewer that the experimental verification is important and constitutes the next step forward.\nIn summary, while the concept of spatiotemporal superoscillation is scientifically intriguing,\na carefully designed experimental demonstration is essential to validate the proposed STSO\neffect. I strongly encourage the authors to devote effort on experimental verification. Without\nit, the manuscript may not meet the publication standards of a high-impact journal such as\nNature Communication.\nResponse: We refer the reviewer to our response to his previous point 1.\nReviewer #2 (Remarks to the Author):\nIn the revised manuscript, the authors make more discussions on the potential applications\n(required by reviewer #1), and the more theoretical analysis, which has addressed my\nprevious comments. I have to say again that this topic is interesting and promising.\nResponse: We thank the reviewer for their supportive comments.\nHowever, the revisions still cannot resolve some of my concerns. In order to demonstrate the\nimportance of space-time superoscillation, I expect that the authors were able to unravel the\npotential phenomena with the use of presented theory, e.g. the intriguing dynamics\nconsidering both space and time domains rather than only introducing the mathematical\nframework. In my opinion, the new physical concept is closely associate with relevant\nessential effects, which, however, the authors did not particularly mention in the revision.\nHonestly, the used mathematical techniques are not surprising, which were mentioned in the\nprevious papers. As a result, more contents of revealing particular physical phenomena are\nnecessary.\nResponse: The reviewer raises the question of the necessary and sufficient physical\nconditions for STSOs to emerge. While we recognize the importance of this question, it lies\nbeyond the scope of the current work that aims to highlight the existence of STSOs in space-\ntime nonseparable light fields. A brief discussion is included in the last paragraph of\nDiscussion, first paragraph of Future perspectives, in the main text.\nOtherwise, as suggested by the reviewer 1, experimental demonstrations on the space-time\nconcept would be helpful. Without these, I agree with reviewer 1 that the manuscript may be\nsuitable to a specialized journal, rather than the Nat. Commun.\nResponse: We argue that the novelty of the work is sufficient to warrant publication in\nNature Communications and expect that it will stimulate experimental investigations of\nSTSOs. We also note here the scope of Nature Communications:“Papers published by the\njournal aim to represent important advances of significance to specialists within each field”.\nReviewer #2 (Remarks on code availability):\nNone.\nReviewer #3 (Remarks to the Author):\nThe manuscript first proposed and systematically demonstrated the phenomenon of \"space-\ntime superoscillation\", a groundbreaking concept in my opinion, which extends the concept\nof Michael Berry’s superoscillation from the pure space or time domain to the spacetime\njoint domain, connecting to the recently emerging topic of spatiotemporal optics and physics.\nThis work has significant theoretical innovation and potential application values. With very\nsolid theories, the STSO phenomenon is clearly demonstrated and verified through multiple\ndimensions such as local wave vector/frequency, energy distribution, energy flow, and\nspectral analysis. The experimental plan and the numerical simulation part are reasonably\ndesigned and fully demonstrated. The article has a clear structure and smooth language,\nprofessionally written, undoubtedly meeting the high-level standards of Nature\nCommunications. I recommend publication without delay.\nResponse: We thank the reviewer for their supportive comments.\nSome suggested minor revisions are as following:\n1. The statement that \"the group velocity of the STPs of different orders is always the\nconstant of *c*\" is a critical point. While the proof is correctly relegated to the supplement,\nits fundamental importance warrants a slightly more substantive note in the main text. Please\nconsider adding a short justification, such as: \"This is expected as STPs are vacuum\nsolutions to Maxwell's equations and thus propagate without dispersion.\"\nResponse: We thank the reviewer for the insightful comment. We revised the sentence into\n“In the condition of propagation in vacuum without dispersion, the group velocity on axis\n(r=0) of the STPs of different orders is always the constant of c, see Supplementary Materials\nS1, while the velocity of the centroid of the whole structured pulse is less than c.” See revised\nstatement in the paragraph after Eq.(3).\n2. To enhance its depth, it would be valuable to briefly acknowledge the signal-to-noise\nchallenge inherent in detecting STSOs, given their occurrence in low-field regions. A\nsentence could be added, for instance: \"We note that the practical observation of STSOs, like\nother superoscillatory features, will require techniques sensitive to weak signals amidst\nhigher-amplitude backgrounds, a challenge successfully overcome in prior spatial superoscillation\nmeasurements XXX.\"\nResponse: In response to the reviewer’s comment, we added the following statement in\npage XX, paragraph XX: \"Moreover, the practical observation of STSOs, like other\nsuperoscillatory features, will require techniques sensitive to weak signals amidst higher-\namplitude backgrounds, a challenge successfully overcome in prior spatial superoscillation\nmeasurements [6,7,Sci. Rep. 3, 1715 (2013)].\" See the end of third paragraph of first\nsubsection of Discusion.\n3. In prior works, topological skyrmion textures were demonstrated in such STP pulses, I\nbelieve the authors may also need to discuss more on how the STSO associated with\nskyrmion textures can hatch more meaningful applications, for instance, deep-subwavelength\nsensing, metrology, and light-matter interaction.\nResponse: We thank the reviewer for their valuable comments. Indeed, the STSO\nphenomena are linked to the skyrmionic field structure. For instance, STSOs emerge in\nregions of rapid field reversal resulting from a nested skyrmionic topological structure [11],\nwhich can also be extended with nondiffracting robust propagation [26]. Recently, optical\nskyrmions found their applications in subwavelength sensing, metrology, and light-matter\ninteraction [27-29], therefore, we perspective the skyrmions with SO and STSO has potential\nto further enhance the deep-subwavelength resolution in these applications. We have added\nthe discussion in the second last paragraph of the Discussion.\n4. \"Abbey-Reileigh\" should be the more common \"Abbe-Rayleigh\".\n5. Unit writing: For example, \"6 fs long pulses\" is suggested to be changed to \"6-fs-long\npulses\" or \"pulses with a duration of 6 fs\". It is suggested that \"~80 nm\" be changed to \"≈80\nnm\" or \"about 80 nm\".\n6. \"In particular, the parameter a α...\" The beginning of this sentence is a bit awkward. It is\nsuggested to modify it to \"The parameter α is a real dimensionless number that...\" .\n7. “Our work expands the concept of SO to include spatiotemporal effects, illustrated by the\narbitrarily spatially and temporally confined focal spots in STPs. \"this sentence is a bit long\nand can be split or reassembled for clarity.\n8. The conclusion part uses two \"Finally\" in a row. It is suggested that the last \"Finally\" be\nchanged to \"Furthermore\" or \"Beyond electromagnetism\".\nResponse: We have cleared up these minor issues."}]

Decision letter