variations of the optical properties of engineered materials it is possible to realize advanced functionalities and new wave phenomena, such as time interfaces, time scattering and time crystals. In turn, suitable combinations of spatial and temporal structuring of a material have been enabling space-time- or four-dimensional metamaterials with enhanced degrees of freedom for wave control [574].

These seemingly disconnected research lines are likely to offer tremendous opportunities when combined, with interesting prospects for fundamental research breakthroughs and for applications across different scales. Indeed, nonlocalities broadly arise in natural and artificial materials, both in space and time, and are particularly relevant in the context of resonant wave phenomena, at the basis of metamaterials and metasurfaces. For instance, nonlocality engineering in time has been considered in the context of temporal metamaterials [575], and space-time nonlocality engineering in metasurfaces has been recently explored [576] to enable analog image processing in space and time. By engineering the spatial and temporal dispersion of a passive metasurface [Fig. 32(a)], it is possible to perform space-time differentiation of an incoming image stream, enabling neuromorphic event detection through an ultrathin, passive metasurface [576]. This initial proposal leaves open several questions: to what extent is it possible to engineer nonlocalities both in space and time, leading to superior control over spatial and frequency dispersion in metamaterials? What range of applications can benefit from this control? What are the fundamental limits in engineering space-time nonlocalities? What role do nonlocalities play in the emerging field of space-time metamaterials?

<figure><img src="image_37.png" /><figcaption>**Fig. 32.** (a) A space-time nonlocal metasurface performing derivatives both in space and time, with application as an analog-domain neuromorphic event detection device: transmission is only allowed for the edges of objects that are changing in time. Adapted with permission from Ref. [576] (Copyright © 2024 American Physical Society). (b) Experimental demonstration of a cascade of time-derivative metasurfaces, each of them independently transmits the time-derivative of the input wave. As a bi-layer, the measured output is the second-derivative in time of the incoming signal. Adapted with permission from Ref. [577] (Copyright © 2024 Nature Springer).</figcaption></figure>

## Challenges and opportunities

A first step towards the experimental demonstration of space-time nonlocal metasurfaces was recently shown in Ref. [577], in which passive metasurfaces performing the time-derivatives of the incoming signal were demonstrated by tailoring their frequency dispersion. By cascading two of such metasurfaces performing the first derivative of the incoming signal [Fig. 32(b)], a second-derivative device was also demonstrated, showing opportunities to realize more complex
operations by simply stacking multiple devices. While a metasurface showcasing engineered nonlocalities in both space and time has not been demonstrated, this initial experiment has highlighted a few challenges: the limited spatial extent over which a metasurface engages with the wave in the propagation direction makes the control over nonlocalities inherently difficult. In the devices demonstrated in Ref. [577], the event detection response is limited to temporal variations in the picosecond range, which may be of interest for ultrafast imaging, but limits other applications. For significantly slower temporal dynamics the demonstrated metasurfaces operate at much reduced efficiency, and better responses can be obtained only at the price of significantly increased resonance Q factors, which is challenging for free-space operation and limits the overall operational bandwidth. Another related challenge is reconfigurability: even though these nonlocalities control both spatial and temporal dispersion, the underlying devices are static in nature, which is appealing in terms of scalability and ease of implementation, but it implies a non-reconfigurable response, which limits the application space. Moreover, the demonstrated responses have so far been limited to linear operations, implying that the resulting application is limited to Fourier-based operations. Extensions towards nonlinear space-time operations may largely broaden the degree of applicability.

Engineered space-time nonlocality opens interesting opportunities not only in the field of
analog-based image processing, but also in other contexts. Metasurfaces with engineered
nonlocalities both in space and time can shape the coherence profile of their own thermal emission
and photoluminescence [522], converting oscillations of matter driven by incoherent processes
to efficient radiation with large temporal and spatial coherence, in the range of hundreds of
oscillation periods and free-space wavelengths. By combining these nonlocal phenomena with
local structuring in space of the underlying resonances, it is also possible to tailor the polarization
and wavefront profile at will [578], offering a new platform for compact, efficient and cheap
custom light generation. Yet, several challenges remain: taming incoherent processes is difficult,
making these metasurfaces sensitive to fabrication defects and disorder. The enhanced coherence
is fundamentally limited by these practical factors, as well as by the extent of the area over which
these devices can be reliably fabricated. Space-time nonlocality emerges as a very important
element in the accurate modeling of time and space-time metamaterials. Similar to how spatial
nonlocalities play an important role in capturing the response of metamaterials [579], temporal
nonlocalities cannot be neglected in the proper modeling of the wave scattering at temporal
inter interfaces [580,581]. As a result, the microscopic implementations of time- and space-time
metamaterials, and the associated nonlocalities, are expected to play a crucial role in determining
the overall wave response, and their engineering may be leveraged to enhance its control.
Theoretical progress in the understanding of these interactions in space-time metamaterials, and
experimental progress on the implementation of space-time nonlocality engineering will be
fundamental to enable this vision.

Future developments to address challenges

The growing interest in engineered nonlocalities is driving impactful research that holds
the promise to address the challenges mentioned in the previous section, and leverage the
many available opportunities. The possibility of reconfiguring spatial nonlocalities has been
recently demonstrated in proof-of-concept metasurface devices relying on phase-change materials
[582,583], which may be extended to reconfigurable space-time nonlocality. Electro-optical or
all-optical modulation may extend the degrees of freedom that can be reconfigured, opening
dramatically the design and application. Space. Similarly, the limitation of resonance quality-
factors of free-space metasurfaces can be addressed with tailored designs and improved fabrication
techniques. Recent efforts have enabled demonstrations with quality-factors comparable to
integrated photonic platforms [584], opening many opportunities in the context of space-time
nonlocality engineering. Similar to diffractive nonlocal metasurfaces [571], which combine
local wavefront engineering in the spatial domain with nonlocality engineering in momentum space, future developments may consider space-time local and nonlocal responses coming together in the metasurface platform, offering opportunities to tailor dynamically space-time dispersion of complex wavefronts, further expanding the degree of control over waves. Finally, enhanced nonlinearities in metasurfaces, relying on strong light-matter interactions, as well as on material engineering as in the case of multiple quantum wells [224], hold the promise to further broaden the opportunities and impact of space-time nonlocalities by extending it to the control of nonlinearly-generated signals through wave mixing, as well as to perform nonlinear image processing in space and time.

### Concluding remarks

The fields of nonlocal metasurfaces and of space-time metamaterials have been thriving in recent years, and the initial attempts of merging these research areas have been showcasing tremendous opportunities for the future of metamaterials and metasurfaces, providing enhanced wavefront control, signal processing and neuromorphic computing, and coherence control in space and time over thermally generated light or photoluminescence. Once mature, space-time nonlocality engineering holds the promise to be another exciting toolbox for photonic engineers and metamaterial scientists.

**Funding.** U.S. Department of Defense; Simons Foundation (SFIMPS-EWP-00008530-10).

## 26. Spectral and angular control of light with nonlocal metasurfaces

JUNG-HWAN SONG & MARK L. BRONGERSMA

### Overview

Many emerging imaging, sensing, communication, display, and non-linear optics applications require compact optical elements that can selectively manipulate light waves at very well-defined wavelengths and/or angles of incidence. In this Roadmap article we argue that nonlocal metasurfaces are ideally suited for this purpose. We further highlight that a careful spectral engineering of the optical materials loss in such optical elements opens a unique opportunity to fully-decouple the optical functions at different illumination wavelengths.

### Current status

The field of metasurface flat-optics has opened a myriad of new ways to control the flow of light beyond the capabilities of polished pieces of glass and geometrically-shaped metallic mirrors. Metasurface optical elements can be created by judiciously nanostructuring thin films of metal or semiconductor materials. They effectively harness light scattering and optical interference processes to achieve a very wide range of valuable optical functions [585,586]. Low-cost and large-area nanofabrication techniques as well as advanced computational tools are now propelling these optical elements into real, commercial applications [587] and highly-integrated device technologies [588,589]. Metasurfaces can broadly be divided into two distinct classes based on their operational principle. There are local metasurfaces for which the optical interaction between neighboring nanostructures or “meta-atoms” is weak and nonlocal metasurfaces in which the coupling is substantial and can extend over a long range. Recent work provides helpful insights into the ways that the meta-atom geometry can be engineered to transition between local and nonlocal behaviors [590]. Each type has their unique application spaces, advantages, and limitations. The design of local metasurfaces is conceptually very simple as one can think of each nanostructure imparting independent, local, spatially-varying amplitude, phase, and polarization changes on an optical wavefront. These optical manipulations can be controlled through judicious choices of the nanostructure sizes, shapes, orientations and spatial arrangements.
A growing intuition about the operation of local metasurfaces together with powerful rapid-design software tools based on topological optimization, inverse design and deep-learning principles has facilitated rapid progress in their development [591–593]. At the same time, it is worth pointing out that there is always some degree of optical coupling between meta-atoms present and this can result in notable performance limitations in terms of the achievable diffraction efficiency [594]. In contrast, nonlocal metasurfaces strategically take advantage of the optical coupling and collective responses of nanostructures to boost efficiency. Based on their distinct nature, local and nonlocal metasurfaces display notably different spectral and angular responses. For local metasurfaces these properties are intimately connected to those of the meta-atom building blocks they are constructed from. They can be either non-resonant, truncated waveguides [595] or employ optically-resonant plasmonic [596,597] or Mie [598] resonators with low optical quality factor (Q) resonances. As a result, the nanostructures deliver display light scattering responses that are broad in terms of their spectral and angular characteristics. As the nanostructures in a local metasurface more-or-less act independently, these traits are naturally transferred to the entire optical element. This is advantageous when the goal is to realize optical elements that can operate across a broad spectral range and need to deliver a high numerical aperture. However, this is detrimental in applications that require the selective manipulation of light waves at well-defined frequencies and/or angles incidence. This is an area where nonlocal metasurfaces can shine.

## Challenges and opportunities

Many modern day optics applications require extreme control over the spectral and angular properties of light. For example, in imaging and microscopy spectral control helps to mitigate aberrations and produce sharper images. Spectroscopy and sensing tools require spectrally-selective optical excitation and analysis of light to analyze materials properties, chemical composition, and molecular structure. In optical communication systems, spectral and angular control enables the precise management of light signals for efficient data transmission, multiplexing, and de-multiplexing. Local metasurfaces have displayed challenges in delivering the required spectral and angular control, but nonlocal metasurfaces can capitalize on the optical interaction between the building blocks to achieve such functions in an integrated form factor. To understand why nonlocal metasurfaces can provide superior control over the angular and spectral properties of light, we investigate how they are designed and operate at a conceptual level. In the design of most local metasurface structures, one arranges the meta-atoms to achieve a transmission function $t(x, y)$ that is dependent locally on the spatial coordinates x and y. For example, in a metalens the transmission function is designed to achieve a local space-variant phase shift that shapes the wavefront in the same fashion as a polished lens [599]. In contrast, a nonlocal metasurface is designed to deliver a transmission function $t(k_x, k_y)$ that is dependent on the wave-vectors $k_x$ and $k_y$ of the incident light [570,600] (see also Section 27 of this Roadmap). In other words, this type of metasurface will differently manipulate light waves with distinct incident angles. Figure 33 illustrates in a tutorial example how the spectral- and angle-dependent behaviors can be engineered. Figure 33(a) shows a very basic nonlocal metasurface comprised of a 240 nm thick single-mode silicon nitride ($Si_3N_4$) slab waveguide with a surface-relief grating etched into its top surface [601]. Light incident on this optical element can follow two distinct pathways. Most commonly, the incident light waves follow a direct transmission pathway through the patterned nitride layer. The spectral transmission behavior for this pathway mimics the transmission for a regular, low quality-factor ($Q \sim 2$) Fabry-Pérot resonator and only a gentle buildup of field can be seen in the optical simulation. However, at selected wavelength and incident-angle combinations, the grating can resonantly couple free-space light waves to quasi-guided modes. Such coupling is facilitated by the power of constructive interference, where all of the small grating teeth can work in concert to scatter and effectively redirect the incident light waves into the nitride waveguide. For the discussion that follows, it is important to note that this can lead to a notable energy
storage in the waveguide and a very substantial increase in the electric field in the waveguide. This field build up can be seen in the optical simulation for the second pathway. In the same way the grating can couple the light into the waveguide, it also causes the guided light to slowly leak back out of the structure. In the forward direction the leaked waves interfere with the directly transmitted light. On the grating coupling resonance, this interference is destructive and gives rise to a pronounced dip in the spectral transmission spectra [Fig. 33(b)]. The amplitude of the grating elements determines the coupling efficiency in/out of the waveguide and thus also the radiative quality factor ($Q_r$) for the resonances. The period controls the wavelength at which the light is coupled. In this example, a grating period of 390 nm places the resonance at 630 nm for normally-incident light with a transverse magnetic (TM) polarization (the magnetic-field is pointing along the grooves of the grating). The grating depth was chosen to be 90 nm to produce a resonance with a quality factor $Q = 60$.

<figure><img src="image_38.png" /><figcaption>**Fig. 33.** (a) A cross sectional schematic of a basic nonlocal metasurface [601]. Incident transverse magnetic (TM) polarized light can take a direct (1) and indirect, resonant pathway (2) through this optical element. The corresponding field distributions are shown underneath the schematic. (b) The simulated transmittance spectrum of the metasurface at normal incidence shows a pronounced dip that results from destructive interference between the direct and indirect pathways on resonance. (c) Simulated angular transmittance spectra of the metasurface on and near the resonant wavelength $λ_0$. The case without the surface relief grating on the nitride layer is shown for reference. Reproduced with permission from Ref. [601] (Copyright © 2022 Nature Springer).</figcaption></figure>

Figure 33(c) shows the simulated angle-dependent transmittance on resonance ($λ_0$ = 630 nm) and for illumination wavelengths 3 nm above and below it. It highlights that the sharp transmission feature also naturally comes with a narrow angular selectivity. The dispersion relation of the nitride waveguide mode governs for what pairs of wavelengths and angles light coupling can be achieved. For analog optical computing, image processing, and the compression of free-space in optical systems, it is critical to have nonlocal metasurfaces with well-defined dispersive behaviors that can be engineered through their geometric parameters. In other applications, it can be desirable to fully decouple the optical functions at different wavelengths and angles of incidence. Eye tracking is an illustrative application and we discuss it below. Figure 34(a) shows a photograph of a basic eye tracker prototype where we patterned a large-area high-Q, nonlocal metasurface on a pair of glasses [18]. A near-infrared (NIR) light-emitting diode emitting at 870 nm illuminates the eye and the metasurface redirects the scattered light from the eye toward a miniature camera that is attached to one of the arms. The metasurface should provide the person wearing the glasses with an unperturbed view of the outside world, while a high-speed camera can image and follow the eye's motion. This can in principle be accomplished with other compact optical elements (e.g. gratings, miniature prisms, local metasurfaces, and waveguides) placed directly in front of the eye, but these dispersive components inevitably produce unwanted rainbows [Fig. 34(b)]. This is very distracting when used in environments with bright optical sources, such as the sun. A high-Q, nonlocal metasurface can provide a solution to this problem by performing different, independent operations on light waves at different frequencies and
incident angles. In our example, all light waves in the visible (VIS) spectrum should be allowed to pass completely unperturbed, independent of the incident angle. However, in the NIR we aim to only redirect light across a narrow band of wavelengths and a small range of angles to the camera for imaging.

<figure><img src="image_39.png" /><figcaption>**Fig. 34.** (a) Optical image of an eyetracking prototype showing the highly transparent nonlocal metasurface in front of the eye [602]. (b) Photograph taken through a glass surface with a 200 nm thick Si₃N₄ anti-reflection coating patterned with a 30 nm thick Cr grating on top showing multiple strong rainbows. (c) Photograph taken through a nonlocal metasurface with a 3 nm thick pSi grating showing a strongly suppressed rainbow and a high, color balanced transmission. (d) Simulated zeroth-order transmittance spectra of the Cr grating and optimized GMR metasurface under normally-incident planewave illumination. The dashed blue curve shows the situation where the materials absorption in the Si is artificially set to zero. The polarization direction is in parallel to the grooves of the grating (TE). (e) Simulated diffraction efficiency into the +1 diffracted-order transmission for the 30 nm thick Cr grating (black solid curve) and the nonlocal metasurface under normally-incident TE polarized planewave incidence (red solid curve). Both structures reach ~10% efficiency at the target wavelength of 870 nm. The dashed blue curve shows the situation where the materials absorption in the Si is artificially set to zero. The horizontal dashed line indicates a 0.1% upper limit for visible light diffraction to avoid perceptible rainbows. The purple shaded region shows the absorption of pSi. Reproduced with permission from Ref. [602] (Copyright © 2021 Nature Springer).</figcaption></figure>

To see the benefits of our metasurface design, we first consider the properties of a conventional grating comprised of 30 nm thick chromium (Cr) strips whose dimensions were optimized to redirect 10% of a normally-incident NIR beam through the first diffracted order. Figure 34(b) shows the low transmission of this element and the intense rainbows that it creates. These aspects are shown quantitatively as the black curves in Fig. 34(d,e). The rainbow is formed because of the high efficiency for light redirection into the first diffracted order. Figure 34(c) shows the same scene as viewed through a high-Q, nonlocal metasurface that has 3 nm thick poly-crystalline silicon (pSi) strips placed on a Si₃N₄ waveguide. This nonlocal metasurface shows a very high transmissivity at normal incidence across the visible range (89% on average). This can be expected as the grating is at least 2 orders of magnitude thinner than the absorption depth in pSi. Moreover, with proper design the Si₃N₄ waveguide and silica cover can also serve as a high-performance, double-layer antireflection coating. The transmission spectrum for the nonlocal metasurface shows a sharp spectral feature near 870 nm that can be attributed to the excitation of the guided resonance. At this resonance, a second indirect pathway for the light through the structure. On resonance, about 40% of the light is taken out of the incident beam and redirected into four first-order diffracted beams (+1 and -1 in forward and backward directions). Based on the high diffraction efficiency seen at 870 nm, one would naturally expect to see high diffraction-efficiencies in the visible range as well. However, the guide mode resonances in the VIS (at 608 nm, 496 nm, and 470 nm) display a very low diffraction-efficiency (<0.07%), explaining the virtual absence of rainbows as shown by the solid red curve in Fig. 34(e). The difference in behavior in the VIS and NIR lies in the spectral absorption properties of the 3-nm-thin pSi metasurface elements. The significant suppression of diffractions in the VIS takes
place when the absorption coefficient of *p*Si is above 1 cm⁻¹ [purple shaded region in Fig. 34(e)]. This can be seen by analyzing the dashed curves in Fig. 34(d,e) that show the transmission and diffraction efficiency for the case that the materials absorption in *p*Si is artificially set to zero. They highlight how the materials absorption in the metasurface elements can control the flow of light through the device on resonance; In the presence/absence of materials absorption, the diffracted orders can selectively be turned off/on without notably impacting the response off-resonance. One key insight is that all of the light that is redirected by the nonlocal metasurface has to flow through the waveguide first (i.e. follows the resonant channel). Because of the high *Q* of the resonance, the light-matter interaction in the waveguide is very strong. As such, even very small amounts of materials absorption in the waveguide can dramatically impact the flow of light through the structure. When the absorption efficiency of the metasurface elements (*p*Si strips) exceeds their scattering efficiency, the guided mode is damped and all of the light will necessarily follow the direct path only. As *p*Si strongly absorbs light in the visible, the diffraction channels and rainbows are effectively suppressed. When the absorption is weak (for *p*Si in the infrared), the light will couple to the guided mode and be decoupled from the waveguide to the camera before it is absorbed. At a high level, this example shows that diffractive optical functions can be deactivated by introducing a little bit of materials absorption.

## Future developments and concluding remarks

The use of materials absorption in nonlocal metasurfaces brings the valuable opportunity to decouple optical functions at different wavelengths and angles. In the eye tracking example, this was utilized to achieve different behaviors in the visible (transparency) and infrared regime (imaging of the eye). By further engineering the spectral absorption behaviors, one can imagine achieving highly-selective transmission/reflection of light waves at certain frequencies and/or incident angles. Through the introduction of materials with dynamically tunable absorption features, it may also be possible to modulate such transmission speeds that are ultimately only limited by the quality factor of the nonlocal metasurface. Such optical elements will have a range of applications in optical imaging, computation, communication, and sensing.

**Funding.** Ministry of Education - Singapore (FY2024).

## 27. Nonlocal metalenses

URIEL LEVY

<figure><img src="image_40.png" /></figure>

### Overview

Metasurfaces are artificially engineered, subwavelength-thickness and subwavelength patterned optical surfaces that can control the amplitude, phase, and polarization of light. They can also enhance light-matter interactions at the nanoscale [585,599,603–607]. Such metasurfaces can be classified into local or nonlocal metasurfaces, depending on the interaction of light with the nanoscale meta-atoms of the metasurface. If the metasurface is based on the interaction of light with an individual meta-atom and long-range interactions does not play a role, the metasurface is coined “local metasurface”. If on the other hand, the effect depends not only on the local phase control arising from the independent response of each meta-atom but also on the nonlocal (collective) resonance excitation originating from the interactions between neighboring meta-atoms, the metasurface is coined “nonlocal metasurface”. The latter offers opportunities related to high *Q* resonance [584,608], wavelength selectivity and tunability [609–612]. One of the most dominant types of metasurfaces is the metalens, sometimes defined also as a metasurface lens. The metalens is a flat optical device that performs similar operations as conventional lenses. However, its phase is controlled by the interaction of light with the meta-atom, and
the diffraction angle is also dictated by the local periodicity of the metalens, which is typically chirped away of its center, to facilities larger diffraction angles that are needed to focus all incident parallel light rays to the same focal spot [613,614]. In essence, the metalens is an advanced version of the diffractive lens, and it offers several advantages over diffractive lens, the such as polarization control, continuous phase control using a binary structure, CMOS compatibility, and reduced thickness to name a few [615]. Being based on diffraction, metalenses are inherently dispersive, featuring strong chromatic aberrations [616]. This is perhaps one of the main, if not the major drawback of metalenses, limiting their applicability to narrow band use cases. However, for broadband illumination, it would be ideal if the metalens can support large bandwidth, or alternatively provide high transmission only at the wavelength which it is designed for, while not affecting other wavelengths to avoid blurred image. To do so, the metalens needs to be highly spectral selective, which is typically achieved using a high *Q* resonance mechanism. This is where nonlocal metalenses come into play.

## Current status

In recent years, several nonlocal metalenses have been demonstrated. Malek et al. [617] demonstrated a nonlocal metalenses based on nonlocal dielectric metasurfaces that offer both spatial and spectral control of light. The demonstrated metalens was focusing light exclusively over a narrowband resonance while leaving off-resonant frequencies unaffected. Meta-atoms were designed based on the concept of quasi bound state in the continuum (q-BIC), encoded with a spatially varying geometric phase. They achieved effective transmission (polarization conversion efficiency) of 8%, lower than the theoretical limit of 25%, and a *Q* factor of 86. It is mentioned that conversion efficiency can be further increased by better design, better fabrication and higher out of plane symmetry breaking. Yao et al. [618] also demonstrated a nonlocal high-*Q* Huygens' metalens operating under circular polarized illumination and explored its application in wavelength-selective imaging. To do so, an integrated-resonant unit (IRU) was designed to achieve a balance between *Q*-factor, efficiency, and the robustness of meta-atom's orientation. By introducing structural asymmetry in the configuration, they could excite the leaky q-BIC mode, resulting in a high *Q*-factor. Each IRU consisted of a crescent nanopillar made of amorphous silicon placed on the silica substrate. Three different designs were fabricated, with *Q* factor as high as ~100 and very high polarization conversion efficiency of ~70%. The high efficiency, effectively exceeding the theoretical limit of 25%, is attributed to varying the structural parameters to modulate the resonant wavelengths of magnetic dipole resonance and the q-BIC mode. Basically, subwavelength-thick local metalenses perform well only over a limited input angular range, whereas nonlocal metalenses can achieve high efficiency and diffraction-limited characteristics over a much wider field of view (FOV). Yet, such nonlocal metalenses are typically thicker, as they need a minimal thickness to provide the nonlocality. Since a large FOV requires an angle-dependent response (i.e., angular dispersion), a spatially localized incident wave must spread as it propagates through the metalens under space-angle Fourier transform. Thus, increasing the angular diversity can be facilitated by increasing the degree of nonlocality. Based on this, Li and Hsu [619] discussed the thickness bound for nonlocal wide field of view (FOV) metalenses. They were able to show that there is an intrinsic trade-off between achieving a desired broad-angle response and reducing the thickness of the metalens. This thickness bound originates from the Fourier transform duality between space and angle. When applied to wide-FOV lenses, their approach predicted the minimal thickness as a function of the FOV, metalens diameter, and numerical aperture. Their formalism can provide guidance for the design of nonlocal metasurfaces in general, and nonlocal metalenses in particular, with the understanding that mitigating high FOV by a metalens of negligible thickness might be an extremely challenging task. In another paper [620], the same authors introduced transmission efficiency limit for nonlocal metalenses. Using the same concept of describing the device as a
transmission matrix that relates the input to the output, they were able to develop a term that links the efficiency to various metalens parameters. Specifically, it was shown that if the input and the output apertures are equal, the efficiency drops with the increase of the numerical aperture (NA) following the relation $\eta \propto \sqrt{1 - NA^2}$, where $\eta$ is the efficiency. However, if the input aperture is set to an optimal value, an increased transmission efficiency bound is obtained significantly surpassing this relation.

## Challenges and opportunities

One of the major opportunities of nonlocal metalenses is the ability to control the amplitude, phase and polarization response as a function of angle (see also Section 26 of this Roadmap). This may provide an opportunity to address the challenge of reducing aberration in large FOV applications, typically encountered in modern imaging. For example, the main camera of a smartphone can now support FOV as high as ~800, and the ultra-wide camera supports an even higher FOV. This imposes a great challenge in the design process of the metalens. Clearly, a slightly thickener metalens which is designed to take advantage of the nonlocal phase response may cope better with the goal of supporting a higher FOV. Here, focusing on narrow band illumination may become helpful, because supporting simultaneously large bandwidth, large FOV and large Fresnel number (for high light collection efficiency) is perhaps beyond the capability of a wavelength or subwavelength-thick metalens. Another challenge of the nonlocal metalens is related to nanofabrication. While simulations show that high *Q* nonlocal metalenses are feasible, it is extremely difficult to achieve such high *Q* values in practice. Any small deviation from the design parameters due to fabrication imperfection may result in a significant decrease in the *Q* factor. This is particularly relevant for nonlocal metalens, relying on a collective response of the meta-atoms. In such a case, a small perturbation in one or a few meta-atoms will affect the response of the entire device. Considering the need for large apertures to support high light collection efficiency and high *Q* factor in collective modes, the material from which metalens is made is crucial. While for the near infrared (beyond ~1 micron wavelength) amorphous silicon is an excellent material choice, this is not the case for shorter wavelength, where the absorption of amorphous silicon is a limiting factor. Materials such as titanium dioxide are not CMOS compatible and are very difficult to handle over a large area. A promising direction for the implementation of nonlocal metasurfaces and particularly nonlocal metalenses operating in the visible range is to use silicon rich nitride (SRN). SRN films can be easily applied on almost any substrate using a relatively simple process, e.g. using plasma enhanced chemical vapor deposition (PECVD). The refractive index of the SRN film can be tuned by controlling the atomic ratio between silicon and nitride. We have recently implemented metasurfaces based on SRN for digital holography and structural colors applications, and a similar material platform can be applied for nonlocal metalenses [621,622]. As mentioned before, one of the major opportunities of nonlocal metalenses is the capability to achieve high *Q* factor. On the one hand, this requires a delicate design that combines the local phase response with the collective response of many meta-atoms. On the other hand, as discussed before, the interference between local modes and nonlocal modes is helpful in overcoming transmission limitations. The high *Q* factor implies that the design is sensitive to fabrication imperfection and environmental effects such as temperature drifts. Yet, it allows large tunability using e.g. the Pockels or the Kerr effect, the plasma dispersion effect in semiconductors, the thermo optic effect. Tunability can also be achieved by structural modifications (e.g. MEMS based device [623,624], and the use of external medium such as liquid [625] and even quantum vapor [626]. Real time tunability of nonlocal metalenses is probably one of the major research topics in the field.
## Future developments to address challenges

The previous sections have been dealing with the status of nonlocal metalenses and highlighted opportunities related to nonlocality. Yet, a single nonlocal metalens, while holding a great promise, will probably be unable to cope with the grand challenge of simultaneously providing diffraction limited imaging with ultra-high resolution over the entire visible range, with large Fresnel number and large FOV. One possible way to overcome this limitation is to consider a hybrid refractive-metasurface design, where most of the optical power will be delivered by the refractive optical system, whereas the metasurface/metalens will be used for improving the imaging quality by providing some sort of aberration correction. The metasurface/metalens can also be used to simplify the requirements from the refractive optical system. Another possible avenue is to use a multi-layer nonlocal metalens, providing additional degrees of freedom and a slightly larger thickness to improve the quality of the metalens. In our opinion, an even more promising direction is to use the hybrid concept of combining the nonlocal metalens with computational imaging approach. This has been recently demonstrated for a local flat lens [627]. As shown in Fig. 35, the flat lens produced a highly chromatic aberrated image. Yet, by applying a deep-learning-based approach, and a proper training procedure it was possible to overcome chromatic aberrations of flat lens-based imaging systems operating outdoors under normal ambient illumination conditions. This approach, combined with advanced nonlocal metalens design, may be used to achieve high quality imaging under different scenarios, ranging from ambient light conditions to multi-color LED illumination.

<figure><img src="image_41.png" /><figcaption>**Fig. 35.** Top row – 5 aberrated test-set images. Middle row – The corresponding ground truth images. Bottom row – The reconstructed images. Reprinted (adapted) with permission from Ref. [627] (Copyright © 2023 American Chemical Society).</figcaption></figure>

## Concluding remarks

The use of nonlocal metalenses in imaging systems may provide an enhanced tool for the realization of high quality imaging systems based on flat optics technology. The capability to control amplitude, phase and polarization as a function of incident angle and wavelength of illumination may pave the way for advanced imaging functionalities. Nonlocal metalenses may also provide multi-functionality, such as imaging and spectral filtering, imaging and FOV manipulations, and polarimetric imaging to name a few. While this chapter was focused on the visible and the near infrared spectral regime, nonlocal metalenses can also be used for imaging
applications at longer wavelengths, ranging from the mid infrared, through the thermal domain, and all the way to the terahertz. Finally, it is important to understand that while a single nonlocal metalens can be very useful for many use cases, it cannot cope with the grand challenge of performing diffraction limited imaging over large bandwidth, large FOV and high Fresnel number simultaneously. For such a task, applying hybrid approaches e.g. refractive lens and nonlocal metalens, or nonlocal metalens with additional computation imaging effort is a key for mitigating the stringent requirements of modern imaging systems.

**Funding.** Israel Innovation Authority; The Metamaterials and Metasurfaces consortium program.

## 28. Nonlocal metasurfaces for space compression

OLIVIA Y. LONG ID, CHENG GUO ID & SHANHUI FAN ID

### Current status

Miniaturization has been a common trend in the development of optical technologies, driven by the goal of improved efficiency, compactness, and portability of devices. With the rapid rise of artificial intelligence, there is an accompanying need for fast processing of large data inputs in a scalable way, further fueling the endeavor for smaller components [628]. The bulk of conventional optical setups consists of free-space, including space filled with uniform dielectric materials [629]. In recent decades, metasurfaces with subwavelength scatterers have attracted increasing attention for their ability to manipulate the phase, amplitude, and polarization of light, paralleling advances in nanofabrication techniques [630]. However, most research efforts have been toward designing more compact optical components such as lenses, bandpass filters, quarter-wave plates, and polarization converters [631]. Here, we focus on a complementary effort: the compression of free-space.

The propagation of light in free space over a distance d in the paraxial regime is described by a quadratic phase transfer function [629]:

$$H(k_{\parallel}) = e^{-ik_0 d} e^{i \frac{\lambda_0 d}{4\pi}} k_{\parallel}^2 \quad (66)$$

where $k_0$, $\lambda_0$, and $k_{\parallel}$ are the free-space wavevector, free-space wavelength, and transverse component of the wavevector, respectively. To compress free space, one aims to construct a device that achieves the same phase transfer function of Eq. (66), but with a physical thickness that is much smaller than the distance d. To achieve this aim, the dependence on the transverse wavevector component $k_{\parallel}$ naturally suggests the use of nonlocal metasurfaces, which manipulate light in the wavevector space. To effectively compress space, the designed metasurface, with a thickness $d_m$, should impart a phase factor $e^{i\phi k_{\parallel}^2}$, with the equivalent free-space propagation distance $d = 4\pi\phi/\lambda_0 > d_m$. The performance of the metasurface can thus be quantified by a compression ratio $R = d/d_m$. Moreover, the amplitude of each $k_{\parallel}$-component of the incident light should ideally be preserved over a wide frequency bandwidth and range of incident angles. Figure 36(a) schematically shows the free-space propagation distance d (top inset) that is replaced by transmission through a nonlocal metasurface (bottom inset). The compression ratio is determined by the quadratic dependence of the transmission phase on $k_{\parallel}$. A comparison of three ideal transmission phase profiles with different compression ratios is shown in Fig. 36(b), and the ideal transmission amplitude as a function of $k_{\parallel}$ is depicted in Fig. 36(c), where the magnitude remains unity for all angles of incidence.

Pioneering theoretical proposals for space compression harnessed guided mode resonances of photonic crystal slabs [632] and multilayer structures [633], while the first experimental demonstrations employed a uniaxial crystal and low-index homogeneous media [633] to achieve the desired transfer function [Eq. (66)]. To date, other nonlocal metasurface designs for space
<figure><img src="image_42.png" /><figcaption>**Fig. 36.** (a) Schematic of nonlocal metasurface for space compression. Top inset shows the equivalent free-space propagation through a distance *d* mimicked by transmission through the nonlocal metasurface with thickness *d*<*d*<sub>3</sub> (bottom inset). (b) Ideal phase dependence of metasurface transmission coefficient on the transverse wavevector *k*<sub>‖</sub> for designs with different compression ratios. Here, the effective free-space propagation distances of the curves are related by *d*<sub>1</sub> (*d*<sub>2</sub> (*d*<sub>3</sub>) due to the different quadratic curvatures. (c) Ideal magnitude of transmission coefficient, allowing unity transmission of light at all angles of incidence.</figcaption></figure>

compression (also referred to as “spaceplates”) include Fabry-Pérot resonators [634,635], 4f lens-based approaches [636], and Huygens’ metasurfaces [637], as well as photonic crystal and multilayer designs with additional functionalities and optimizations [638–640]. A visual summary of many of these design approaches is provided in Fig. 37.

Single guided-mode resonance designs have achieved compression ratios ranging from 5 to 144 with numerical apertures (NA) of 0.32 and 0.01, respectively, often with polarization dependencies and limited bandwidths of operation [632,634,638]. Multiple coupled resonances have been shown to yield wider operating bandwidths [634], and optimized multilayers have achieved larger compression ratios of 340 at the expense of a relatively small NA (0.017) [640]. Thus, specific platforms may be more favorable depending on the specific use case. Ongoing research strives to achieve the ideal structure with simultaneously high compression ratio, large NA, wide frequency bandwidth, and polarization independence. In this article, we focus on the approach of nonlocal metasurfaces for space compression.

## Challenges and opportunities

In general designs for space compression, one would like to have strong quadratic dependence of the phase $\phi(k_\parallel)$ on the transverse wavevector $k_\parallel$ to achieve higher space compression ratios. In the nonlocal metasurface approach, one uses resonances to achieve such quadratic dependence. However, since the strongest dependencies on $k_\parallel$ occur near resonances, such designs are limited by a narrow frequency bandwidth of operation. Thus, one challenge in this approach to space compression is the inherent tradeoff between the frequency bandwidth and the transverse wavevector range of operation (numerical aperture). Proposed approaches to achieve wider bandwidths to date include using multiple resonances [634].
(a)

Multilayers

<figure><img src="image_43.png" /></figure>

(b)

Photonic crystals

<figure><img src="image_44.png" /></figure>

(c)

Uniaxial crystals

<figure><img src="image_45.png" /></figure>

(d)

Fabry-Pérot

<figure><img src="image_46.png" /><figcaption>**Fig. 37.** Nonlocal metasurface designs for space compression. (a) Optimized multilayer designs. Top panel is reproduced with permission from Ref. [640] (Copyright © 2022 Optica Publishing Group) while bottom panel is reproduced with permission from Ref. [639] (Copyright © 2024 American Chemical Society). (b) Photonic crystal slab designs. Left panel is reproduced with permission from Ref. [632] (Copyright © 2020 Optica Publishing Group), while right panel is reproduced with permission from Ref. [638] (Copyright © 2022 American Physical Society). (c) Uniaxial crystal spaceplate. Reproduced with permission from Ref. [633] (Copyright © 2021 Nature Springer). (d) Coupled multi-resonator design [634] and single Fabry-Pérot cavity design from Ref. [635]. Top panel is reprinted (adapted) with permission from Ref. [634] (Copyright © 2021 American Chemical Society), while lower panel is reproduced with permission from Ref. [635] (Copyright © 2022 American Institute of Physics).</figcaption></figure>

Polarization-independent operation would also be advantageous for integration into existing optical systems. One drawback of many recent designs, including both designs using nonlocal metasurfaces and designs based on multilayer films, is that the optical response of the device differs for TM and TE polarized light. Recent work has proposed polarization-independent space compression using a photonic crystal slab structure that features bands with similar curvatures, resulting in similar resonance responses for each polarization [638]. It is of interest to achieve polarization independence as well as a large compression ratio over a sufficiently large numerical aperture.

Since nonlocal metasurfaces for space compression have been mostly applied to imaging applications thus far and are often used in conjunction with lenses, it is desirable to achieve designs with tunable compression ratios for the purpose of reconfigurable focal distances. Thus, another opportunity for future development is in improving the programmability of such nonlocal nanophotonic structures. Efforts in the area of active nonlocal metasurfaces have included electro-optic, mechanical, and thermo-optic tuning [641]. Challenges in such approaches include operation efficiencies, speed of reconfigurability, and maintaining a pure phase response throughout the tuning process.

There is ample opportunity to explore integration of space compression with other functionalities, with the goal of expanding the domain of applications beyond conventional imaging. For example, the narrow bandwidth and angular ranges of current designs could be harnessed to achieve simultaneous polarization conversion or diffraction suppression at other operating wavelengths, which can be useful in augmented reality and eye-tracking technologies [602,642].
Nonlocal spaceplates can also be co-designed with existing lenses and local metasurfaces for functionalities such as aberration correction, as demonstrated in Ref. [639].

Finally, in the design of nonlocal metasurface for space compression, a theoretical challenge is to understand the tradeoff between various performance metrics, such as frequency bandwidth, numerical aperture, polarization independence, and compression ratio. For multilayer and single-resonance structures, the tradeoffs between frequency bandwidth and numerical aperture, and between frequency bandwidth and compression ratio, have been examined in Ref. [643]. There have also been works on the fundamental constraints of the thickness of optical devices in general [644]. It would be of interest to apply and further develop the formalisms in these papers to understand the design tradeoff in nonlocal metasurfaces.

## Future developments to address challenges

One of the challenges in achieving widespread adoption is designing nonlocal metasurfaces with the target performance metrics such as numerical aperture, bandwidth, and efficiency. Thus, future development in the understanding of the spaceplate design space would help accelerate the integration of such devices in optical systems. Exploration of the design space of space-compressing nonlocal structures has thus far included gradient-based optimization [639] and exploiting topological properties in the structural parameter space [645]. With the rise of artificial intelligence, we anticipate further optimization of existing designs using machine learning techniques such as those based on neural networks.

The development of new techniques for fast programmability of the compression ratio would help increase the range of applications for space-compressing nonlocal metasurfaces. Technological advances in areas such as electrically and optically tunable materials would contribute to greater reconfigurability in both nonlocal and local metasurface designs [646]. The simulation of metasurfaces designed with such materials would also be an interesting theoretical opportunity to explore.

Given that nonlocal metasurfaces are a more recent development compared to local metasurfaces, future integration of the two could help address the challenge of increasing multifunctionality. By harnessing the momentum-space control of nonlocal metasurfaces and the local wavefront control of local metasurfaces, new emergent functionalities will enable next-generation optical systems. For instance, a recent work demonstrates the use of spaceplates in correcting both chromatic and spherical aberrations from refractive and local metasurface lenses [639]. Building upon this work, the development of adaptive, tunable devices for automatic aberration correction represents an exciting frontier. Such multifunctional capabilities further expand the potential applications of nonlocal metasurfaces. Furthermore, the co-design of nonlocal metasurfaces with a computational backend may help overcome some of the limitations of the metasurface alone, as demonstrated recently for local metasurfaces [647].

Beyond single-resonance-based designs, multiple coupled resonances offer promising opportunities for improved performance. The key challenge lies in optimizing the collective arrangement of resonance poles to achieve both unity transmission amplitude and the desired phase response across a broader bandwidth. This complex optimization problem is particularly well-suited for machine learning and AI methods, which can efficiently explore the vast design space.

While our discussion has focused on compression of free space, similar principles could be extended to guided waves. An intriguing question emerges: can we compress the propagation length in waveguide arrays or in multimode waveguides? This concept could significantly reduce the footprint of photonic links in integrated photonic circuits, opening new possibilities for compact optical interconnects.

Looking beyond electromagnetic waves, these space compression principles could be extended to other wave varieties, including electron, spin, and acoustic waves. Particularly promising is the application to electron optics, where compressed free-space propagation could enhance electron
microscopy capabilities. Similarly, the manipulation of spin and acoustic waves through space
compression could enable novel functionalities in materials science and device engineering.

Concluding remarks

With further advancements in key metrics such as numerical aperture, bandwidth, and compression ratios, the integration of nonlocal metasurfaces for space compression into everyday optical devices such as cameras could become a reality in the near future. Moreover, the functionality of nonlocal space compression, in combination with those of local metasurfaces, highlights the rapidly advancing techniques in the precise manipulation of light at an increasingly smaller scale. The demonstration of space compression points to the potential for many more possibilities in the area of nonlocal metasurfaces, which may expand beyond imaging.

**Funding.** Stanford University (Graduate Fellowship); Air Force Office of Scientific Research (MURI project, FA9550-21-1-0312).

29. Electro-optic spatiotemporal nonlocal metasurfaces

SERGEY I. BOZHEVOLNYI

Overview

Spatiotemporal light control with dynamic optical metasurfaces has been high on the research agenda, promising attractive solutions and new avenues for modern highly integrated optics and nanophotonics. Due to ultrafast responses of electro-optic materials, various electro-optic metasurface configurations have been scrutinized with the aim of realizing both efficient and ultrafast dynamic metasurfaces. Given fundamentally nm-thin metasurfaces and very weak electro-optic responses, advances towards this ambitious goal have been facing enormous challenges without any clear way through at hand. Here, several promising electro-optic metasurface configurations are presented that exploit nonlocal interactions between incident radiation and grating-excited waveguide modes propagating inside nm-thin electro-optic films. Challenges and opportunities for spatiotemporal light control with electro-optic nonlocal metasurfaces are discussed, outlining also future developments towards tackling the most serious challenges and exploiting the most exciting opportunities.

Current status

Optical metasurfaces formed by surface arrays of nanoscale resonant elements constitute one of
the most vibrant and explosively developing fields in modern optics and photonics, primarily
due to their ability to effectuate complete control over the transmitted/reflected fields enabling
thereby diverse optical functionalities, including those not realizable with conventional bulky
optics [648]. Most metasurfaces are however static in nature with their optical responses being
determined at the design stage and set in the process of fabrication. Recent years have therefore
seen increasing efforts to realize dynamically controlled (tunable) optical metasurfaces, rapidly
progressing towards faster operation and realization of spatiotemporal light control, which is
immensely attractive for many fascinating phenomena waiting to be realized and exploited
in highly integrated optics and nanophotonics [649]. Among various types of dynamically
controlled metasurfaces, electrically tunable optical metasurfaces have shown great promise due
to their fast response time, low power consumption, and compatibility with existing electronic
control systems, offering unique possibilities for dynamic tunability of light-matter interactions
via electrical modulation. It should be noted that to exercise truly spatiotemporal light control,
the electrical operation bandwidth should exceed, preferably by orders of magnitude, the optical
bandwidth of the radiation to be controlled, requiring thereby ultrafast modulation [649]. The
linear electro-optic (Pockels) effect, which is found in several crystalline (ordered) media without center of symmetry, ensures inherently fast electrical control of material birefringence with femto- or even attosecond-short response times. The main challenge when utilizing the electro-optic materials in optical metasurfaces is associated with inherently small refractive index variations accessible, a circumstance that, in combination with fundamentally thin nature of metasurfaces, results in enormous difficulties in their practical realization.

Electro-optic metasurface configurations, whether based on electro-optic polymers [650] or lithium niobate (LN) films [610,651], are designed to have electro-optic thin films sandwiched between bottom and top electrodes that can modify the film refractive index and thereby control the metasurface response [610,650,651]. Since the film thickness is typically subwavelength and the electro-optically induced index changes are very small ($<10^{-3}$), one should exploit resonant configurations that would increase the effective interaction length either by multiple reflections across the metasurface layer (i.e., electro-optic thin film) in Fabry-Pérot resonators [651] or by the nonlocal interaction [125] of incident radiation with waveguide modes propagating along the metasurface layer [610,650]. The latter approach is somewhat alike to that based on the nonlocal interaction with surface lattice resonance (SLR) modes, typically enhanced by simultaneous excitation of localized nanoparticle Mie [650] or plasmon [651] resonances. The excitation of quasi-bound states in the continuum (qBIC) has also been exploited [650]. Considering the experimental characteristics demonstrated for these configurations [610,650,651], while those based on electro-optic polymers exhibited superior efficiency and bandwidth [650], LN-based electro-optic metasurfaces [610,651] offer potentially similar bandwidth along with the excellent environmental (temperature) stability, a feature that is in a stark contrast with electro-optic polymers exhibiting typically a glass temperature below 100°C.

## Challenges and opportunities

The prior research into electro-optic metasurfaces has convincingly demonstrated that the best performance is expected from the configurations involving the nonlocal interaction of incident radiation with waveguide modes propagating along the electro-optic layer. The design optimization goal is then to identify the physical mechanism along with the modes involved that would result in ultra-narrow transmission/reflection minima/maxima, since those would most efficiently be influenced even with very weak, electro-optically induced, refractive index changes. The guided mode excitation by gratings is governed by phase-matching that can result in very narrow transmission/reflection spectral features, often referred to as guided-mode resonances (GMRs), for very wide gratings and nearly parallel incident light [652]. The latter requirement is not compatible with designing compact and robust practical devices and components. One can circumvent this challenge by making the excited waveguide modes to bounce back and forth in a distributed Bragg resonator (DBR), thus effectively extending the interaction between guided modes and grating couplers. Fortunately, this possibility seems being naturally present for the waveguide mode excitation by normal light incidence [653].

Let us consider a generic LN-based configuration of electro-optic metasurfaces illuminated at normal light incidence (Fig. 38). The first-order phase matching between the incident light and counter-propagating waveguide modes requires that the grating period, $Λ$, matches the waveguide mode wavelength, $λ_w = λ/N_{eff}$, where $N_{eff}$ is the effective mode index and $λ$ is the wavelength in vacuum. Once this condition is satisfied, the waveguide modes experience the Bragg reflection in the second-order diffraction by the grating [653]. Thus, the resonant excitation of waveguide modes under normal light incidence is intrinsically linked to the DBR formation. The latter is associated with the occurrence of the photonic band gap reflected in the corresponding BIC and qBIC dispersion branches. The qBIC resonance is accessible at normal incidence, resulting typically in ultrahigh-Q resonances [608]. Note that the excitation of qBIC resonances in electro-optic metasurfaces, that cannot be avoided in the grating-assisted
coupling to waveguide modes at normal incidence, has already been exploited both explicitly [650] and implicitly [610]. The former (based on the electro-optic polymer) exhibited high modulation efficiency (60%) and broad bandwidth (3 GHz) when driven by ±100 V [650]. The latter (utilizing environmentally stable LN films) did not yet reach the same level of performance, but several avenues for the improvement are clearly in sight [610]. One of the matters to be considered is that strong gratings enhance not only the Bragg reflection, improving thereby the DBR performance, but also the out-coupling of waveguide modes, thus increasing the energy dissipation in the system. The challenge of optimally balancing the metasurface lateral extension and DBR loss to radiation out-coupling by the grating and absorption by electrodes is yet to be carefully considered from the viewpoint of reaching the utmost (efficient and fast) performance of electro-optic metasurfaces.

<figure><img src="image_47.png" /><figcaption>**Fig. 38.** Schematic of an LN-based electro-optic metasurface operating under normal light incidence that, under the phase-matching condition, results in the excitation of counter-propagating waveguide modes. These modes, upon interaction with the electrode grating structure, couple out of the waveguide and Bragg reflect into each other.</figcaption></figure>

## Future developments to address challenges

The general design direction can be formulated as realizing ultrahigh-Q resonant transmission/reflection with the relevant electromagnetic fields efficiently overlapping with the electro-optic layer and controlling (electrostatic) fields (assuming that the electrostatic field is oriented to make use of strong components of the electro-optic susceptibility tensor [608]). This requires minimizing the radiation losses, both by absorption and out-coupling. The former is unavoidable because of the presence of control electrodes (Fig. 38), but can be decreased by exploiting transparent conductive materials, e.g., indium tin oxide (ITO) [651], and/or increasing the layer thickness. It should be noted though that electrical characteristics of transparent oxides are significantly worse than noble metals, and their use might jeopardize the speed of operation, i.e., the modulation bandwidth. Increasing the dielectric layer thickness would also decrease the electrode capacitance and thereby increase the modulation bandwidth, but the driving voltage would proportionally increase as well, resulting in the overall increase of the electrical energy consumed by the device. Decreasing the radiation loss requires reaching a delicate balance between the strength of coupling between incident free propagating fields and counter-propagating waveguide modes and that of Bragg reflection of waveguide modes. Both scattering channels originate in the diffraction of waveguide modes by the grating, corresponding to the first- and second-order diffraction, respectively. For the configuration shown in Fig. 38, the design freedom is limited, since enhancing the scattering strength of individual grating elements (e.g., by making the metal ridges thicker and wider) would increase the corresponding diffraction efficiencies in a similar way. In that respect, recently introduced asymmetric (dielectric) gratings, in which a grating supercell contains several nonidentical grating elements, promise new avenues for reaching ultrahigh-Q coupling (when considering only radiative losses) by decreasing the grating asymmetry [608,653] and thereby boosting up the performance of electro-optic metasurfaces. Thus, for example, one can realize vanishingly weak in- and out-coupling diffraction channels, while maintaining very strong Bragg reflection and thereby forming a high-Q DBR for waveguide modes [653]. It should be emphasized that very strong Bragg reflection implies the possibility of
drastically reducing the grating lateral extension by effectively folding the waveguide propagation region, while maintaining the same overall resonance quality. This folding has immediate consequences for the electrode capacitance, which is proportional to the electrode system area, allowing thereby to drastically increase the operation bandwidth. Concomitantly, the area of individually addressed metasurface elements making up electro-optic spatiotemporal metasurfaces for dynamic molding of electromagnetic wavefronts could also be significantly reduced without loss of efficiency, increasing thus the spatial resolution available for spatiotemporal wave shaping [654]. Perfecting the operation of electro-optic metasurfaces both as a single array and as an ensemble of individually addressed metasurface arrays requires different optimization strategies and targets different application domains. Thus, achieving complete light extinction with the electro-optic metasurface, and thereby 100% modulation efficiency, would allow one to realize dynamic phase contrast imaging using the same approach as that demonstrated recently with static non-local angle-selective metasurfaces [601]. Dynamic functionality would in turn open other application directions, such as adjustable contouring of images features by gradually turning on (when applying the electrical control signal) the phase contrast. Combining active temporal and spatial field modulation with individually addressed arrays would enable unique optical functionalities, such as frequency mixing, harmonic beam steering and molding, as well as non-magnetic nonreciprocity among other things [654].

### Concluding remarks

The considered configuration of electro-optic nonlocal metasurfaces (Fig. 38) is deceitfully simple, leaving one wondering about its potential. Indeed, DBRs and associated bandgap dispersion branches are well known since the 1980s, when being extensively investigated and exploited for mode selection in distributed feedback semiconductor lasers [655]. However, the realization of waveguide grating couplers with metal (lossy) electrodes [610], addition of asymmetry into grating configurations [608,653], making beneficial use of nonlocality in electromagnetic interactions [523,610] and yet unexplored finite-size effects add up to resulting in the configuration that is surprisingly rich in physical phenomena and potential applications as discussed above. It should be noted that the presented here considerations are all concerned the case of normal light incidence. Deviations from normal incidence would result in the occurrence of two transmission/reflection minima/maxima (instead of one at normal incidence) corresponding to the coupling at two dispersion branches [653]. This would enable certain tunability of the operation wavelength by controlling the incident angle close to normal incidence. This tunability, although limited to the DBR bandgap, is a very interesting (and yet unexplored) feature of the considered electro-optic nonlocal metasurfaces that should broad and enhance their application potential. Overall, I believe that many exciting and unexpected developments in this area are just around the corner, waiting to be discovered and explored.

**Funding.** HORIZON EUROPE European Innovation Council (Pathfinder project (OPTIPATH, project No. 101185769)).

## 30. Cascaded diffractive nonlocal metasurfaces for highly multifunctional meta-optics

**ADAM OVERVIG**

### Current status

Metasurface optics leveraging nonlocal scattering effects (e.g., interelement coupling) are emerging as a new frontier for compact optical systems with high information density. Taking advantage symmetry-breaking design principles and the frequency- and angle-selective responses of quasi-bound states in the continuum, highly multifunctional meta-optics may be achieved
by cascading several metasurfaces [571,617]. While local metasurface optics often aims for broadband and angle-insensitive operation, such an approach encodes comparatively negligible information in the electromagnetic wave compared to having distinct encodings at each frequency and angle. Nonlocal metasurfaces therefore offer a platform for vastly increasing our command of light within compact form factors. Most notably, the selectivity and mutual transparency of several resonant metasurface layers simplifies the design of a cascaded meta-optic; in contrast, cascaded local meta-optics require redesigning all layers upon the introduction of each new layer. Hence, cascaded nonlocal meta-optics offers a greatly simplified design space and approach for highly multifunctional devices. Nonlocal metasurface optics may be distinguished into two categories [523]: periodic devices that manipulate many plane waves (operating solely in momentum space), and aperiodic devices that generalize plane wave selectivity to spatial selectivity (operating in both momentum and real space). The latter are referred to as “diffractive nonlocal metasurfaces”, a generalized category of device encompassing both periodic nonlocal metasurface or local metasurfaces [571]. Here, we emphasize the potential of a cascaded set of these devices to encode distinct functionalities across many discrete angles and/or frequencies. Their nonlocality is engineered through high quality-factor (Q-factor) guided resonances, called quasi-bound states in the continuum, whose dispersion yields a sharp dependence on incident momentum (encompassing a combination of both angular and frequency selectivity). Meanwhile, the local scattering of these delocalized modes is controlled similarly to local metasurfaces: by leveraging the selection rules governing the leakage of these states to free space, tailored geometric perturbations applied pointwise to each unit cell customizes the response beyond plane waves, i.e., to any spatial profile of Q-factor, phase, and polarization state. Since the resulting devices would be periodic but for the applied perturbation, only resonant states interact strongly with the patterning; non-resonant states only weakly interact. This mechanism results in high wavelength and frequency selectivity, or “confidentiality”, of the response: outside of resonances, the metasurfaces behave as unpatterned thin films. By simultaneously engineering the perturbation and the non-resonant (broadband) response, these devices can operate with resonant efficiency approaching unity for a reflection peak within a transmission window (i.e., a stopband filter in which the rejected band is reflected and spatially shaped) [617] or a transmission peak within a reflection window (i.e., a passband filter with broadband rejection and a spatially shaped passed band) [656]. With polarization manipulation, a converted state may pass in transmission while the broadband response is also transmissive, but the maximal efficiency is 25% [571,617].

A closely related class of metasurfaces employs the resonant phase of localized high Q-factor resonators to pattern diffracted light [657,658]. In these devices, the goal is frequency selectivity without angular selectivity, conferring high numerical aperture performance. While there inevitably exists some degree of nonlocality due to the long lifetime of the modes, flat dispersion can yield a moderate to strong degree of localization. This behavior is compatible with multi-frequency operation, but not multi-angle operation. In contrast, certain metasurface approaches have achieved modest angular-dependent wavefront transformation [659,660]. When rationally designed, these devices are primarily treated as if their unit cell response has angular dependence, but without explicit introduction of large interelement coupling [659]; when computationally designed [660], interelement coupling associated with resonances often arises during optimization. This suggests that rational control of interelement coupling is a crucial tool for highly multifunctional optics of this kind.

## Challenges and opportunities

By leveraging several modes within each metasurface and/or several mutually transparent metasurfaces cascaded together [617], the future of highly multifunctional cascaded meta-optics looks highly promising. Here, we suggest a picture for how to consider the potential for these devices as multi-frequency devices, multi-angle devices, or both; we note that there are many open questions on the implementation physicality of such systems and nature of the information
encoding in the following, and hope that this viewpoint will motivate efforts to clarify bounds and constraints on these systems. Figure 39(a) depicts a cascaded meta-optic with multi-frequency operation. At three frequencies, independent wavefront modulation is imparted at each device for light incident with a lateral momentum $k_{in}$ near 0 (near normal incidence). Three example dispersion profiles are shown in Fig. 39(b), showing three shifted copies of the same mode (easily achieved by altering the thickness, duty cycle, or period of the metasurface). Each mode may be locally patterned at will, imparting independent operations to each resonant frequency [571,617]. Meanwhile, the broadband response is simple transparency (black arrow), indicating the scalability of this approach to an arbitrary number of frequencies according to: the bandwidth of the mutual transparency window of the devices, the Q-factors of each device, the required angular range to encode the desired functionalities, and the acceptable tolerances of the system. Realization of this vision began in Ref. [617], which showed up to four functionalities. Beyond this, by using multi-layer nanofabrication, truly dense compound meta-optics with dozens of wavelengths controlled within a narrow band of operation are possible. Such highly multi-color meta-optics promise unprecedented capabilities for structured light, spectropolarimetry, and spatiotemporal pulse shaping. Advances could open opportunities for imaging, augmented reality, and highly customized light sources.

<figure><img src="image_48.png" /><figcaption>**Fig. 39.** Cascaded nonlocal meta-optics. (a) For light incident at a fixed incident angle, cascaded resonant metasurfaces may achieve multi-wavelength operation. (b) Example use of three dispersive metasurfaces (marked as red, green, and blue) for multi-wavelength operation (c) For light incident at a fixed frequency, cascaded nonlocal metasurface may achieve multi-momentum operation. (d) Example use of three dispersive metasurfaces for multi-angle operation. In (a) and (c), the black arrow indicates non-resonant waves, which simply pass through the system.</figcaption></figure>

On the other hand, Fig. 39(c) depicts a cascaded meta-optic with multi-angle operation at a fixed frequency $\omega_{in}$ [Fig. 39(d)]. Compared to Figs. 39(a,b), this approach effectively trades degrees of freedom from frequency to angle by using the intrinsic spatial dispersion of the underlying modes [i.e., the very same stack of devices could be used as in Figs. 39(a,b)]. This scheme breaks the numerical aperture of the meta-optic into a number $N_k$ of small ranges $\delta k$, each controlled by a distinct mode and/or metasurface layer. Moreover, it will be constrained by reciprocity: the output waves of each metasurface must be orthogonal to each other, or else the efficiency will necessarily suffer. Such an approach, if suitably generalized, could extend
the multifunctionality of multi-angle approaches such as Refs. [659,660], greatly increasing the customizability of a meta-optic at given frequency.

However, the bandwidth and information capacity of such approaches are limited by constraints such as dispersion and causality. In general, due to the dual nature of frequency and time, the number $N_\omega$ of small ranges $\delta\omega$ will vary according to the lifetime $\tau$ of the optical modes: $N_\omega \propto 1/\delta\omega \propto \tau$, reminiscent of an “uncertainty principle”. Here, $\tau = Q/\omega_r$ where Q is the Q-factor and $\omega_r$ is the resonant frequency. This is paralleled in momentum and space, namely: $N_k \propto 1/\delta k \propto \text{Re}\,\xi$, where $\xi$ is the (complex) nonlocality length, and $\xi_0 = \text{Re}\,\xi$ characterizes the in-plane distance a mode travels before radiatively decaying. Moreover, the allowable $N_k$ for a given bandwidth $\delta\omega$ depends on the nature of dispersion (e.g., parabolic or linear). Figure 40(a) depicts parabolic dispersion of a resonant frequency $\omega_r$ with band curvature b near a band-edge mode (top), and the linear dispersion with group velocity v (bottom). The distance $\xi_0$ increases with the square root of lifetime or linearly with lifetime, respectively. Hence, the “diffusive” transport of a band-edge mode results in increased localization (i.e., decreased $\xi_0$) compared to the “advective” transport of a linear mode with the same $\tau$. In other words, for multi-angle devices: to increase bandwidth (decrease Q) for the same $N_k$, a linear dispersion should be used. While for multi-frequency devices: to increase localization (or range of angles) for a high-Q structure, parabolic dispersion should be used.

<figure><img src="image_49.png" /><figcaption>**Fig. 40.** Spatio-temporal dispersion of nonlocal metasurfaces. (a) Parabolic dispersion (top) results in diffusive transport with nonlocality length, $\xi_0$, varying as the square root of the lifetime of the mode, $\tau$. Linear dispersion (bottom) results in transport with $\xi_0 \propto \tau$. (b) Potential information density that may be encoded using both frequency and angle, as a function of $\xi_0$ and $\tau$. The upper right region is superluminal, forming an upper bound on the angular selectivity for a given spectral selectivity. (c) Four representative responses in the four extremes of (b).</figcaption></figure>

While the scheme in Fig. 39 depicts either frequency or angle multifunctionality, the ultimate case is where both the frequency and angle ranges are independently addressable. In this case, the total number of independent functionalities grows as $N_\omega N_k \propto \xi_0 \tau$, representing the potential information density of a cascaded meta-optic [Fig. 40(b)]. Figure 40(c) shows four representative dispersion relations of the resonant efficiencies for four extreme cases in Fig. 40(b). Note that by causality, no mode can access the region of Fig. 40(b) such that the distance traveled exceeds the speed of light $c$; we require that $\xi_0 < c\tau$. A given device platform generally falls in a limited region of Fig. 40(b); quasi-bound states have proven highly popular because they can rationally
design the lifetime at will using symmetry-breaking concepts, extending their applicability in this context. A more generalized class of high *Q*-factor metasurface with both continuously tunable spectral and angular selectivity, yet locally addressable scattering phase and polarization, is highly desirable as a tool for both multi-frequency and multi-angle meta-optics. Such a platform does not exist to our knowledge.

Importantly, we note that the implementation of such an information-dense meta-optic cannot be achieved for arbitrarily thin systems: the required thickness will grow as the required independent modes increases. In other words, intuitively, information capacity grows with volume: both physical aperture size and thickness [644]. In this sense, cascaded diffractive nonlocal meta-optics amount to a rationally designed scheme for scaling the volume layer-by-layer according to the desired information capacity.

## Future developments to address challenges

To begin to realize this potential information capacity with high efficiency, many challenges must be overcome. For instance, in contrast to the limitations of Ref. [617], for high efficiency as well as use of the polarization channels, a transmission mode nonlocal meta-optic should be developed that resonantly alters light upon transmission within a broadband transparent response. Such a response has not been demonstrated, yet may be possible with unidirectional guided modes [661], motivating the incorporation of point-wise symmetry-breaking design principles with directional control of the leakage of these states. Multi-layer nonlocal metasurfaces have sufficient degrees of freedom for such a task, and should be a priority in future research. Moreover, while flat bands and linear bands are achievable within periodic flat optics, even within a single system via a rational design paradigm [662], such approaches have not been shown to be compatible with spatially aperiodic functions of phase and polarization. A “knob” to tune the spatial dispersion that does not conflict with the knobs that spatially control the phase and polarization is highly desirable to create flexible cascaded meta-optics over a wide range of angular selectivity responses.

While the primary message here has been to point to the ceiling of potential information capacity, full access to the full information space is not realistic in the near term; practical choices should be made to make tangible progress. We believe that the potential information capacity of current high-*Q* and nonlocal metasurfaces is often too high as a starting point, and as a result the implementation is not sufficiently flexible; the sensitivities to experimental and design errors may inhibit progress. That is, we argue that we should first seek individual nonlocal metasurfaces associated with less information capacity (i.e., smaller $\xi_0$ and $\tau$) for use in cascaded meta-optics, and then increase the capacity of each layer in the future as processes mature. For instance, while linear dispersion is optimal for extreme values of $N_k$ for a given bandwidth, the moderate localization of a band-edge mode affords tolerances to experimental and design errors that a linear dispersion does not. Hence, parabolic bands are more practical for low to moderate $N_k$. Unfortunately, if the band edge mode of a system is limited to normal incidence, only one such mode may be used for a given operating frequency. This motivates the incorporation of custom band-edge angles [663] with spatially aperiodic perturbations. Moreover, while high *Q*-factors are attractive for information density, practical experimental considerations often limit *Q*-factors to moderate values of less than $10^3$ or suffer drastically from low efficiency. Ideally, for exploration of proofs-of-principle and for robust, high-efficiency operation over moderate bandwidths, the radiative *Q*-factor would be lower than 100; it is well-known how to increase it later. That is, an arbitrarily high *Q*-factor may be designed using quasi-bound state concepts, yet the symmetry-breaking approach of quasi-bound states often has a practical lower bound in *Q*-factor. This can be simply understood as follows: larger symmetry breaking perturbations yield lower *Q*-factor, but a geometric perturbation has an upper bound before geometric features overlap each other. Future efforts lowering the *Q* in diffractive nonlocal metasurfaces will offer increased reliability and flexibility in the design space of Fig. 40(b) while maintaining the symmetry-based pointwise control of light afforded by the selection rules.
Concluding remarks

While local meta-optics have been ushering in unprecedented information density encoded at will into flat optical devices, the new frontier of cascaded diffractive nonlocal meta-optics reveals that the local approach only begins to scratch the surface of what is possible. In nonlocal metasurfaces, selectivity to incident frequency and/or angle is possible. In diffractive nonlocal metasurfaces, the spatial customization of amplitude, phase, and polarization may be retained in addition to this selectivity. Here, we have emphasized that in cascaded diffractive nonlocal meta-optics, the non-resonant transparency of each layer renders each selectively encoded functionality mutually transparent (orthogonal) to one another, establishing a scalable approach to highly multifunctional meta-optics. Such an approach amounts to volumetric metamaterials composed of many two-dimensional layers that are rationally designed to achieve unprecedented customizability of light. Success in these endeavors promises advances across optics, including spectrospatial structuring of light, optical computing, and dense holographic data storage. Beyond linear optics, the strong light-matter interactions of the customized high Q-factor responses promise new pathways for highly efficient and multifunctional reconfigurable, nonlinear, and quantum volumetric metamaterials that are designable and manufacturable with standard procedures.

**Funding.** Air Force Office of Scientific Research (FA9550-24-1-0068).

Part V—Nonlocality, topology and nonreciprocity, extreme geometries and fundamental limits

31. Role of nonlocality in topological metamaterials

FILIPA R. PRUDÊNCIO & MÁRIO G. SILVEIRINHA

Current status

Topological photonics is an exciting and rapidly advancing field that brings the mathematical elegance of topology into the realm of electromagnetic waves and photonic materials [664,665]. At its heart lies the principle of bulk-edge correspondence, a profound result that serves as the cornerstone of the field's theoretical framework and practical applications. This principle states that the topological invariants, which are properties of the bulk band structure, dictate the existence and robustness of edge states that propagate along the material boundary [666]. From a practical standpoint, the bulk-edge correspondence manifests as a remarkable conservation law: at the junction of different topological materials, all of which share a band gap, the number of edge states propagating toward the junction must precisely equal the number of edge states leaving it [Fig. 41(a)] [666]. This conservation law ensures that waves traveling along such boundaries follow well-defined pathways, dictated by the topological invariants of the materials. If this correspondence were violated—if, for instance, there were more states entering the junction than leaving—it would imply the presence of a singular behavior in conservative systems [666–668]. Such a scenario might correspond to the material acting as an electromagnetic sink, where light could enter but not escape [Fig. 41(b)]. The absence of such singularities is a fundamental feature of nature's physical laws and reflects the robustness of topological protection in these systems.

However, this robustness relies on subtle physical mechanisms, and it is here that nonlocality plays a critical role. Nonlocality refers to a material's response that depends not only on the electromagnetic field at a given point but also on its spatial variations. In plasmonics, a well-known example of nonlocality is found in the drift-diffusion (hydrodynamic) model, where electron-electron repulsive interactions effectively establish a minimum localization length, thereby preventing extreme wave localization. Thus, diffusion effects inherently counteract forms of field localization that might otherwise compromise the integrity of the bulk-edge correspondence. In the context of topological metamaterials, nonlocality becomes more than a secondary effect: it is an essential feature. Nonlocal effects act as the key mechanism that
<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /><figcaption>**Fig. 41.** (a) Junction between different bulk materials that share a common band-gap. In a topological system, the number of incoming edge channels and the number of outgoing edge channels is exactly the same. (b) A system with an ill-defined topology is not bound by the constraint in panel (a). The ill-defined topology can be used to abruptly halt the wave propagation at a topological singularity with a massive field enhancement. Adapted with permission from Ref. [668] (Copyright © 2022 SPIE).</figcaption></figure>

ensures singularities, such as electromagnetic sinks, are naturally avoided. By mediating how the response to fast varying fields tailors the material’s photonic structure, nonlocality enforces the constraints required for bulk-edge correspondence to hold universally.

## Challenges and opportunities

The Chern theorem applies strictly to closed manifolds, which initially suggests that electromagnetic continua—systems without intrinsic periodicity—are unsuitable for nontrivial photonic topologies. For example, early studies revealed a fundamental challenge: conventional local models of materials, such as magnetically biased plasmas or ferrites, fail to produce a well-defined topological classification. This deficiency is attributed to the noncompact nature of the wavevector space in electromagnetic continua, which is identified with the Euclidean plane [669]. However, recent advances reveal that nonlocality can relax these constraints, serving as a foundational approach to resolve such issues [512,669,670]. Specifically, by introducing a dependence of the material response on the wavevector **k**, nonlocal models can effectively suppress nonreciprocal responses at large **k** [669]. This suppression eliminates the problematic contributions from high-**k** modes, ensuring that the band gaps in electromagnetic continua have well-defined topological invariants.

Interestingly, the importance of nonlocal effects extends beyond electromagnetic continua to periodic systems such as photonic crystals [671]. In these systems, the wavevector space is isomorphic to a torus and thus naturally satisfies the conditions of the Chern theorem. Nonetheless, the topology of photonic crystals can also become ill-defined when the nonlocal response of the materials is disregarded. This issue arises from the bosonic nature of electromagnetic waves, which gives photonic band structures properties that differ fundamentally from those of electronic band structures. Unlike electronic systems, which have a well-defined ground state (a finite number of bands below the Fermi level), photonic systems exhibit spectra that are unbounded from below. This property stems from the real symmetry of the electromagnetic field, which enforces a particle-hole symmetry in the spectrum: for every positive frequency mode, there is a corresponding negative frequency partner. Consequently, photonic systems always have an infinite number of bands below any given bandgap. This feature places the Chern theorem in a precarious position, as it traditionally relies on having a finite number of bands below the gap to compute the topological invariants.
This issue can be vividly illustrated with an example. Consider a system with an infinite number of isolated bands below a gap, each contributing alternately +1 or -1 to the Chern number [671,672]. The gap Chern number in this case is given by the non-convergent series $C_{\text{gap}} = 1 + (-1) + 1 + (-1) + \dots$, resulting in an ill-defined topology. Additionally, material dispersion can exacerbate the problem. For example, the combination of a flat band in a material and the band folding induced by periodicity can result in an accumulation of an infinite number of bands at certain resonance frequencies. This accumulation creates further challenges, as the corresponding contributions to the Chern number may again form a divergent series [671].

As is well-known, an infinite sum of integers converges only if all but a finite number of terms are zero. Without a mechanism to suppress contributions from high-$k$ modes or to regulate the effects of band folding and dispersion, the topology of photonic systems cannot be reliably defined. Here again, nonlocality offers an elegant solution. By introducing a wavevector cutoff that suppresses nonreciprocal responses at large $k$, nonlocal effects provide the regularization needed to resolve these ambiguities. This ensures that the band gaps in realistic dispersive photonic crystals have well-defined topological invariants, allowing them to serve as robust platforms for topological photonics [671].

## Future developments to address challenges

Despite the important role of nonlocal effects in ensuring well-defined topologies in photonic systems, several challenges remain to be addressed. One critical issue is the limited knowledge of the material response at high wavevectors $k$, as most physical models are developed for small $k$, corresponding to long-wavelength physics. The behavior of materials at high $k$, which governs short-wavelength phenomena, is often poorly characterized, leading to uncertainties in the topological classification of materials.

It has been shown that the topology of a material critically depends on its high-$k$ response and that different types of nonlocality can lead to entirely different topological phases [512,671]. This concept can be geometrically visualized using an analogy with the theory of surfaces. Imagine a torus with a vanishing inner radius, which forms a cusp at its center, as illustrated in Fig. 42(aii). Mathematically, the cusp introduces a nondifferentiable point, rendering the topology ill-defined. This issue is resolved by an infinitesimal deformation: either making the inner radius finite, creating a torus with genus one, or severing the cusp, yielding a sphere with genus zero.

This analogy underscores that different perturbations can regularize an ill-defined topology, yielding different topological invariants. In photonic systems, starting with an ill-defined topology (e.g., a local material model), various types of nonlocality can similarly create distinct topological phases (Fig. 42). For example, mechanisms like diffusion, odd viscosity, or full wavevector cutoffs have all been proposed to regularize the topology of materials [512,669,673]. However, this raises a critical issue: the exact nature of the nonlocal response—and therefore the resulting topology—often depends on assumptions about the high-$k$ behavior, which is not always experimentally or theoretically well-constrained.

Importantly, similar to the topological properties, the number of unidirectional edge state channels also depends on the specific form of nonlocality controlling the material response [674–677]. The number of unidirectional edge states consistently aligns with the predictions of the bulk-edge correspondence [676,677]. While this may seem problematic, such ambiguity often affects only very short-wavelength waves. Fortunately, dissipation tends to suppress contributions from these short wavelengths, ensuring that in practical scenarios the physical response remains consistent across different nonlocal models. Nevertheless, a deeper understanding of the mechanisms that determine the nonlocal responses at high-$k$ in weakly dissipative systems remains an open problem.

Another significant challenge lies in extending the bulk-edge correspondence to non-Hermitian systems, such as those with loss and gain. Recent studies have demonstrated that in active systems,
<figure><img src="image_52.png" /><figcaption>**Fig. 42.** Top row: Photonic band structures of dispersive photonic crystals formed by a hexagonal array of air rods embedded in a magnetized electric plasma [geometry of the unit cell is sketched in the inset of panel bi]. The green insets give the numerically calculated gap Chern numbers. The dispersive host material is characterized by (a) local model, (b) hydrodynamic model, and (c) a full cutoff model. Bottom row: Geometrical illustration of the concepts of ill-defined topology and topology regularization. Similar to the torus with vanishing inner radius, the topology of the photonic crystal depends on the regularization procedure. Adapted with permission from Ref. [671] (Copyright © 2022 American Physical Society).</figcaption></figure>

the bulk-edge correspondence can fail due to the non-Hermitian skin effect. In such systems, the bulk spectrum can change dramatically when switching from periodic to open boundary conditions, due to wave localization along specific boundaries. An open question is whether it is possible to identify a subclass of non-Hermitian systems where bulk-edge correspondence can be preserved. For instance, could the correspondence be constrained to dissipative systems (with only loss) or stable systems (with the spectrum in the lower-half complex frequency plane)? Understanding these conditions could open new avenues for topological photonics, particularly in active and non-equilibrium platforms. Nonlocality, with its ability to mediate high-k responses, may play a crucial role in addressing these challenges, potentially enabling robust bulk-edge correspondence even in non-Hermitian contexts.

## Concluding remarks

Understanding and incorporating nonlocality into the design and analysis of photonic topological metamaterials is not merely an academic exercise but a practical necessity. It provides the physical mechanism that underpins the robustness of topological edge states, ensuring that light retains its predictable and protected pathways. Nonlocality, in this sense, serves as the bridge between the abstract mathematical elegance of topology and the physical reality of photonic materials, safeguarding the interplay of edge states and topological invariants that makes this field so vibrant.

**Funding.** Institution of Engineering and Technology; Simons Foundation (SFIMPS-EWP-00008530-10); FCT/MECI through national funds and when applicable co-funded EU funds under UID/50008; Instituto de Telecomunicações.
## 32. Nonlocality in nonreciprocal plasmonics

### S. ALI HASSANI GANGARAJ & CHRISTOS ARGYROPOULOS ID

**Current status**

Surface-plasmon polaritons (SPPs) can be excited along metal-dielectric interfaces giving rise to the emerging field of plasmonics. While most plasmonic systems are reciprocal, unidirectional (nonreciprocal) SPPs can be realized by biasing metallic media with a quantity that is odd under time reversal symmetry, typically a static magnetic field. For instance, one-way SPPs can be induced along the surface of magnetized highly doped semiconductors (acting as metals). Interestingly, the realization of nonreciprocal SPPs may lead to extreme wave physics phenomena which, however, cannot be characterized correctly unless nonlocality (a.k.a., spatial dispersion) is included in the analysis [675]. One method to achieve nonreciprocal SPP propagation is by introducing asymmetry in the conventional SPP flat dispersion band at the surface plasmon resonance. This is done by interfacing a magnetized plasmonic material with a transparent dielectric, leading to asymmetrical SPP dispersion bands. However, recently [675], it was argued that the presence of a flat asymptote band leads to a thermodynamic paradox, i.e., infinite photonic states, $k \in (k_{min}, \infty)$ within the unidirectional SPP frequency range $\omega \in (\omega_{SP}^{-}, \omega_{SP}^{+})$, resulting in infinite energy at finite temperatures. This issue arises from the locality assumption implying that the material stays polarized even near asymptotes where SPP modes have diverging wavenumbers. Interestingly, the inclusion of nonlocality corrects this unphysical behavior since it bends the dispersion upwards at large wavenumbers and close the nonreciprocal window, as depicted in Fig. 43(a), leading to asymmetric SPP propagation but in all directions [674]. Another type of unidirectional SPP is realized by interfacing a magnetized plasma with an opaque material. Here, one-way propagation stems from asymmetric cutoff at $k = 0$ for counter-propagating modes in the plasmonic bandgap, rather than asymmetric asymptotes described before, as shown in Fig. 43(b). This type of nonreciprocal SPPs remain unaffected by the strong nonlocality in biased semiconductors, such as indium antimonide (InSb), mainly due to the topological properties of the resulted magnetized plasma [674]. While topological protection suggests robustness against nonlocality, there are instances where nonlocality violates bulk-edge correspondence in topological photonics. One violation occurs when a magnetic wall on a magnetized semiconductor “short-circuits” the tangential SPP magnetic field [Fig. 43(c)], preventing SPP existence despite a non-zero gap Chern number difference. However, it was proven that SPP theoretically exists only at diverging wavenumbers [667]. Incorporating nonlocality keeps the SPP wavenumber large but finite, formally restoring bulk-edge correspondence. Nonetheless, SPPs with large wavenumbers are hard to excite and suffer attenuation due to Landau damping, even when lossless materials are considered [667]. The second violation of bulk-edge correspondence occurs when SPP dispersion does not span the entire bandgap due to an opaque material with a plasma frequency near the upper bandgap edge, causing an asymptote within the bandgap [Fig. 43(d)]. While the lower bandgap supports a topological state, no edge mode exists in the upper portion, violating the principle. Including nonlocality in InSb response bends the SPP band asymptote upward, leading to SPP leakage into the bulk of InSb [667]. This nonlocality-induced radiation loss quickly attenuates SPP power. Thus, while nonlocality restores bulk-edge correspondence, it is violated practically as SPPs are immediately attenuated, even in lossless materials. Hence, nonlocality's role should be treated cautiously in nonreciprocal plasmonic systems. The results in Ref. [667] clarify that although nonlocality does not break the unidirectionality of topological SPPs, we still need to consider the material spatial dispersion in our analysis to prevent paradoxical observations such as bulk-edge correspondence violation. The above discussion highlights the role of nonlocality in nonreciprocal plasmonics and paves the way to future experimental observations. By leveraging nonreciprocal SPPs via realistic materials, one can develop highly efficient and more compact, directionally controlled novel light transport
<figure><img src="image_53.png" /><figcaption>**Fig. 43.** (a) Typical band diagrams for nonreciprocal bulk modes and SPPs formed inside or at the interface of biased plasmonic material (InSb) and transparent dielectric (Si) under both local and nonlocal InSb material properties [674]. Solid black and blue lines represent the local bulk and SPP dispersion, while dashed lines represent the nonlocal case, with nonlocality of InSb accounted for by a hydrodynamic approach [674]. The applied bias on InSb is $\omega_c/\omega_p = 0.2$, where $\omega_c$ is the cyclotron frequency proportional to the bias strength and $\omega_p$ is the plasma frequency. In the nonlocal case, bulk mode asymptotes bend upward, closing the bandgap [674]. The gray strip indicates the nonreciprocal SPP operation region where $\omega_{SP}$ shifts to $\omega_{SP}^{\pm} = \omega_{SP} \pm \omega_c/2$, opening an asymmetric frequency window with a bandwidth of $\omega_c$ [674]. The solid blue line represents the local nonreciprocal response with two asymptotes at $\omega_{SP}^{\pm}$; while the dashed red line represents the nonlocal response, where the asymptotes bend upward, leading to bidirectional but asymmetric SPP propagation. The field profiles for local and nonlocal cases are represented in the inset [674], where it is clearly depicted that in the local case SPPs are strictly one-way but including nonlocality leads to a weak backward propagation. (b) Similar results but for opaque material at the interface leading to topological SPPs within the bulk mode bandgap, where one-way propagation stems from the nontrivial topological properties of the biased plasmonic material. In this case, as is clear from the field profiles, the inclusion of nonlocality does not alter the nonreciprocal SPP propagation [674]. (c),(d) Typical dispersion diagrams for the two classes of bulk-edge correspondence principle violations [667]. Solid black and red curves indicate bulk and surface modes, respectively, and gray areas highlight the bulk-mode bandgap. Nonreciprocal SPPs exist only in (d) which only covers the bulk band gap partially in the local case. Reproduced with permission from Ref. [674] (Copyright © 2019 Optica Publishing Group).</figcaption></figure>

systems, which promise to be fundamental components of envisioned advanced classical and quantum photonic technologies. Hence, nonreciprocal plasmonics offers a versatile tool for fine-tuning the nanophotonic device performance with applications in different technological frontiers, such as communication and computing.

## Challenges and opportunities

The investigation of nonlocal effects in nonreciprocal plasmonics is still in its infancy. The discussion presented above indicates that accounting for material nonlocality gives rise to complex wave phenomena, playing a crucial role in the nonreciprocity of the induced surface modes. While these findings highlight the potential of nonreciprocity to control light propagation in unprecedented ways, they also underscore the growing need to practically realize this effect by
using new photonic systems. Currently, the predominant method for achieving such response remains heavily reliant on magneto-optical effects that require static magnetic field bias. The inherent weaknesses of magneto-optical effects, along with the sensitivity of one-way SPP propagation achieved by this method to nonlocality, raise major concerns about the practical applicability of nonreciprocal plasmonics. Drifting electrons resulting in current bias is also an odd quantity under time reversal symmetry, similar to magnetic field. Therefore, instead of magnetic bias, one may use currents in both bulk conducting media (metals, semiconductors, and plasmas) and emerging two-dimensional (2D) materials such as graphene. The origin of this nonreciprocal effect is rooted in the Doppler effect due to the electrons' movement along the medium, which produces a Doppler frequency shift ($\omega \rightarrow \omega - k \cdot v_d$) in the material optical response, where $k$ is the wavenumber and $v_d$ is the drifting velocity [520]. The wavenumber inclusion in the material response indicates a form of nonlocality that can lead to nonreciprocity. Although the inclusion of material's natural nonlocality challenges the nonreciprocal response of plasmonics, paradoxically, induced nonlocality via current bias provides opportunities to achieve this response. The nonlocality-induced nonreciprocity can be intuitively explained as follows: SPPs involve collective electron-photon oscillations. When these modes are dragged or opposed by the drifting electrons, SPPs "see" different media when propagating along or against the current flow, i.e., $\omega(k) \neq \omega(-k)$. Different platforms can be used to demonstrate nonlocality-induced nonreciprocity. Recently, nonreciprocal SPPs propagation along a gold nanowire carrying a small electric current with a drift velocity around $v_d/c_0 \approx 10^{-6}$ was demonstrated [520]. Another work studied n-type InSb which allowed for even larger drift velocities up to $v_d/c_0 \approx 10^{-3}$ [678]. The latter study was focused not only on nonreciprocal SPP propagation, but also on the excitation of slow light, steerable SPPs, and exceptional point-like mode transitions at dispersion inflection points. Such nonlocality-induced nonreciprocity can be utilized in thermal photonics, specifically to enable nonreciprocal near-field radiative heat transfer between two planar bodies. Graphene has also emerged as a promising material for nonlocality-induced nonreciprocal plasmonics. Recent experiments demonstrated nonreciprocity with electrons drifting at speeds around its Fermi velocity [679]. Interestingly, a relevant theoretical work [680] highlighted the potential of nonlocality-induced nonreciprocity in enhancing nonlinear light-matter interactions in graphene plasmonics. By utilizing drifting electrons induced by current on a voltage-biased graphene sheet placed on a periodically corrugated silicon (Si) grating, it is possible to create asymmetric SPP modes with field enhancement shown in Fig. 44(a). Blocking the path of nonreciprocal SPPs enables the generation of dramatically enhanced and localized broadband electric field hotspots near the termination, since the radiation cannot be reflected back, with the intensity of these hotspots controlled by the speed of the drifting electrons. The resulted energy accumulation can significantly enhance third-order nonlinear optical effects in graphene, with predicted conversion efficiencies of up to 0.3% around the plasmon resonance frequency [680].

## Future developments to address challenges

Plasmonic systems exhibit unique properties, such as subwavelength confinement and strong field enhancement, making them ideal solutions to realize nanophotonic technologies. One of their key features is the extreme enhancement of nonlinear light-matter interactions [681]. In this context, two general strategies are employed traditionally to generate strong electric field enhancement: i) localized resonances and ii) slow-light effects with adiabatic impedance matching. However, these methods suffer from sensitivity to material absorption losses, limited bandwidth, and large device footprint due to the very long adiabatically tapered structures.

An alternative strategy to address these technological challenges is the use of terminated nonreciprocal plasmonic waveguides to enhance weak optical nonlinear effects. Towards this end, strong electric field hot spots were demonstrated [682] when the unidirectional SPP propagation is blocked suitably in a nonreciprocal plasmonic waveguide made of Si and magnetically biased InSb
<figure><img src="image_54.png" /><figcaption>**Fig. 44.** (a) Illustration of a graphene sheet placed on a periodically corrugated Si substrate. Current bias is applied to graphene shown by the red arrows. The realized nonreciprocal structure is terminated at the two ends by opaque barriers. The line graph shows the distribution of the electric field enhancement along the graphene sheet at the absorption peak frequency. Two strong asymmetric hot spots exist at the two ends of the graphene waveguide. The inset shows the spatial distribution of the electric field enhancement. Reprinted (adapted) with permission from Ref. [680] (Copyright © 2023 American Chemical Society). (b) Illustration of the terminated one-way waveguide geometry consisting of a Si grating on top of a magnetic biased InSb. The structure is terminated at one end by an opaque material. The line graph shows the realized field enhancement as a function of frequency. Vertical dashed lines indicate the broadband nonreciprocal operation which is achieved even in the presence of nonlocality. Reproduced with permission from Ref. [682] (Copyright © 2020 American Physical Society).</figcaption></figure>

with results depicted in Fig. 44(b). The obtained large and broadband electric field enhancement is ideal to boost the efficiency of various optical nonlinear processes. Additionally, impedance matching is automatically guaranteed in this nonreciprocal waveguide configuration due to the absence of a backward mode, eliminating the need for long adiabatic structures, hence, substantially reducing the device footprint. While plasmonic effects are typically constrained by material loss, in the limit of negligible material loss, other effects—such as nonlocality-induced surface-to-bulk mode coupling—can become a primary limitation on field localization. However, recent studies have shown that electric field hot spots achieved based on this method can lead to substantial second [683] and third [682] harmonic generation enhancement. As previously outlined, the weak magnetic response of materials at optical frequencies limits the effectiveness of magnetic bias for this purpose. These technological challenges necessitate concerted efforts to advance the underlying platforms. Drifting electrons offer an alternative method to break time reversal symmetry. Achieving such nonlocality-induced nonreciprocity requires high current bias values, which, although difficult, are not experimentally unattainable, as was recently confirmed in Ref. [679] through the demonstration of Fizeau drag in graphene. Nonlocality-induced nonreciprocity holds great promise for classical nanophotonic technologies, offering enhancement in both linear and nonlinear light-matter interactions. Similarly, in the realm of quantum technologies, nonreciprocal nanophotonic media can provide benefits, such as enhancing interactions between two-level emitters (e.g., atoms, molecules, quantum dots). This improved interaction between two-level emitters can serve as a key mechanism for boosting inter-atomic energy transport efficiency [684] and entanglement [685]. Such advances could, in turn, enable the generation and mediation of entanglement in multipartite
systems, consisting a fundamental objective in quantum technology. Although much of the theory behind nonlocal nonreciprocal systems has been developed, offering a glimpse to various potential groundbreaking applications, experiments of the presented effects are still lagging. The experimental demonstration of nonreciprocity-induced field hotspots in any optical system, either drift current-biased or magneto-optical, is predicted to be an ongoing active research area in the broad fields of photonics and optics. Efficient operation is required for these nonreciprocal effects to become practical which has not been experimentally demonstrated yet.

### Concluding remarks

The above discussion provides a glimpse into how nonlocality and its effects on nonreciprocal plasmonics are crucial not only to advance classical photonic processes, such as enhancing linear and nonlinear light-matter interactions and one-way waveguiding, but also in quantum optics, where induced nonlocality may enable more efficient interactions in multipartite emitter systems. Nonlocal effects are essential for correcting the unphysical behavior arising in nonreciprocal plasmonic systems, particularly when one-way SPPs rely on asymmetric dispersion asymptotes that could otherwise lead to thermodynamic paradoxes. However, this resolution may come at the cost of SPP unidirectionality, unless the one-way behavior originates from the nontrivial topological properties of the underlying platform, such as in the case of magnetized plasma-like materials. Although nonlocality can challenge nonreciprocal plasmonics in extreme cases, it also offers opportunities to achieve one-way propagation through nonlocality-induced effects. Introducing nonlocality via drifting electrons presents a promising alternative for creating a one-way path for edge states, particularly in semiconductors (n-type InSb) or emerging 2D conductive materials such as graphene. Future work should focus on optimizing these nonlocal effects, exploring innovative natural or artificial materials with strong nonreciprocal properties, and incorporating these advances into practical photonic devices for both classical and quantum optical applications.

**Funding.** National Science Foundation (DMR-2224456).

## 33. Nonlocal effects in singular plasmonic geometries unveiled by transformation optics

PALOMA A. HUIDOBRO ID, EMANUELE GALIFFI ID, FAN YANG ID & JOHN B. PENDRY ID

### Current status

Transformation optics (TO) has played a pivotal role in advancing metamaterials research [686]. This theoretical framework exploits the form-invariance of Maxwell's equations under coordinate transformations to prescribe the spatial tailoring of electromagnetic constitutive parameters that have to be implemented in a metamaterial in order to obtain a desired optical effect. Being a very general theory framework, TO has also been used as a design tool for nanoplasmonic structures. By concentrating light within subwavelength volumes, plasmonic nanostructures act as nanoscale light harvesters. At optical frequencies, free electrons in plasmonic materials interact with incident electromagnetic fields forming surface plasmon polaritons that oscillate with much shorter wavelengths than incident radiation. These nanostructures concentrate light at nanometer length scales, with the strongest localization occurring in singular plasmonic structures, that feature sharp edges, nanometer-sized gaps, or touching points, towards which the surface plasmon polaritons carry electromagnetic energy. Examples include nanotips, nanocrescents and touching spheres, shown in Fig. 45(a), as well as gratings with sharp edges or almost touching points, shown in Fig. 45(b). As the surface plasmon travels towards, e.g., the edge of a nanotip, its
<figure><img src="image_55.png" /><figcaption>**Fig. 45.** Plasmonic singularities unveiled by TO. (a) Gallery of singular plasmonic structures that localize the EM fields on a single hot spot: tip, crescents, touching nanoparticles and nanoparticle-on-mirror structures. All these structures transform from the same infinite “mother” structure, a planar waveguide, and thus share the same continuous spectrum in the local limit. (b) Two plasmonic gratings with periodic singularities that localize the EM fields at a series of spots. (c) Nonlocal TO calculation of a plasmonic nanotip. Spatial dispersion effectively blunts the tip and limits field enhancement at singular points. Reprinted (adapted) with permission from Ref. [71] (Copyright © 2012 American Chemical Society). (d) The scattering cross section of two touching 10 nm nanowires shows a discrete set of resonance peaks due to nonlocality (solid lines: TO, dots: FEM simulations). Reproduced with permission from Ref. [48] (Copyright © 2012 American Physical Society). (e) The reflectivity spectrum of a silver surface with a periodic array of sharp grooves displays dips at a discrete set of modes when nonlocality is taken into account, as opposed to the continuous spectrum of a local description (grey line). Reproduced with permission from Ref. [689] (Copyright © 2019 American Physical Society).</figcaption></figure>

wavelength shrinks and the energy density is localized [see Fig. 45(c)]. In an ideal, lossless system, this generates singular hot spots where the intensity of electromagnetic fields diverges.

TO brings further insight into this process by revealing “hidden symmetries” by relating a structure with no apparent symmetry to another more symmetrical one. For example: a cylinder can be transformed into a plane which explains degeneracy of the plasmonic modes of a cylinder. According to this framework, all the singular structures shown in Fig. 45 can be transformed into an extended and nonsingular one: a planar waveguide for the geometries in panel a and the smooth grating in b, and a waveguide array for the grating with sharp singularities in c. Additionally, TO is not limited to the strict quasistatic limit, and radiative corrections can be incorporated in the theory to accurately deal with larger structures.

We can thus exploit this connection to predict the optical response of a structure of interest by studying the modes in a highly symmetric structure, which allows for an analytical treatment [687]. For instance, in a planar waveguide (or waveguide array), surface plasmons excited at the origin travel away towards infinity. Singular coordinate transformations map infinity into singular points in the frame of a localized or grating structure. Consequently, all singular structures
share a broadband optical spectrum associated with the energy build up at the singular points. Interestingly, in the case of gratings, this can be understood as the compacting of one of the dimensions of the non-singular structure into the singularities [688]. Huge concentrations of electromagnetic energy at nanoscale volumes can then be exploited for highly sensitive sensing or improved photovoltaic devices.

However, this immense energy concentration has limits. Fields cannot be confined beyond the electronic screening length, where nonlocal effects come into play and impose a limit to the achievable field enhancement with these light harvesting structures. The dynamics of the free carriers in plasmonic media, which mediate a response to incident electromagnetic fields over finite distances, explain the failure of the local response approximation of classical electrodynamics.

## Challenges and opportunities

The diverging field concentrations in singular (or near-singular) plasmonic nanostructures, as well as the optically small length scales involved in the nonlocal response, make most theoretical approaches challenging or even inapplicable. For instance, numerical methods are limited by the size of the finite elements used to spatially discretize a structure. These must be smaller than the length scales involved in the problem, which prevents from realizing a singular point. On the contrary, the analytical nature of TO makes it an ideal tool for dealing with these singular structures, by exploiting the fact that they can be mapped to extended, non-singular structures. This capability, together with the insight provided by TO, represents a unique opportunity to understand the key role played by nonlocal effects in singular plasmonic geometries.

Reference [48] showed that nonlocal effects can be incorporated into the TO framework by considering a hydrodynamic description of the free carriers. Within this model, electron-electron interactions result in a wave-vector dependent, spatially dispersive, dielectric function for longitudinal modes. This reflects the fact that incident electromagnetic fields can excite longitudinal plasmon oscillations that are not described in the common local response approximation. As a result, surface charges accumulate on a finite-length layer with a non-vanishing decay length [see purple area in the sketch of Fig. 45(d), representing two touching nanowires]. The existence of this layer smooths out any singularity and has critical implications on the field enhancement capabilities and the optical response of these structures.

An example of field localization calculated by means of TO is shown in Fig. 45(c) for the case of a plasmonic nanotip [71]. Under a local-response approximation (top panel), the wavelength of the surface plasmons shrinks as they travel towards the singularity, a point that is never reached as it corresponds to infinity in the other frame. In contrast, when spatial dispersion in the metal is included, the tip effectively blunts, quantizing the modes, and limiting the smallest achievable wavelength for the surface plasmons (bottom panel).

Figure 45(d) displays the optical response (absorption cross section) of two touching nanowires calculated by means of TO. In a local approximation (grey line), the singular nature of this structure results in a broad absorption peak due to its continuous spectrum. On the other hand, mode quantization due to nonlocality results in a series of discrete absorption peaks (blue, green and red lines for different values of the nonlocal decay length). In sharp contrast, for the case of a single nanowire (black lines), nonlocality only results in a blueshift (dashed line).

TO also presents opportunities to understand singularities in plasmonic gratings. Subwave-length arrays of sharp grooves etched into metallic surfaces, such as gold (Au) or silver (Ag), convert these materials from efficient reflectors into good broadband absorbers. TO reveals how the surface plasmons localized at the two-dimensional (2D) surface have a three-dimensional (3D) nature with an additional wave-vector inherited from the transformed structure compacted into the singularities [688]. The missing selection rule for this extra wave-vector produces a broadband absorption spectrum, unlike conventional gratings with discrete absorption lines.
Figure 45(e) shows the broadband spectral response of a local singular silver metasurface with a 10 nm period calculated analytically with TO: a small decrease in reflectivity is seen for a broad band. By contrast, when nonlocality is considered, the spectrum splits into a discrete series of peaks with very low reflectivity, thus high absorption [689].

## Future developments to address challenges

So far, most approaches to nonlocality using TO have been limited to the hard wall approximation and the hydrodynamic model. However, this fails to account for spill-out of the electronic density away from the metal. These nonclassical spill-out effects can be incorporated to the TO framework by combining Feibelman *d*-parameters [308]. This relies on introducing the electronic scale length through the modification of the classical boundary conditions with the Feibelman *d*-parameters (see Sec. 12 of this Roadmap). These are a set of mesoscopic complex surface-response functions which play a role analogous to the local bulk permittivity, but for interfaces between two materials [258].

The approach of Ref. [308] combines the analytical power of TO with the capability of this mesoscopic model to accurately incorporate nonlocality, spill-out, and surface-enabled losses. Figure 46(b) shows the absorption efficiency spectra of two almost touching nanowires. While the hard-wall hydrodynamic model predicts a blueshift of the first resonance peak, the nonclassical, mesoscopic calculation predicts a redshift and the appearance of the Bennet mode (see inset panel), showing how this approach successfully captures both electron spill-out and nonlocal effects.

<figure><img src="image_56.png" /><figcaption>**Fig. 46.** (a) Nonlocality leads to saturation of the density of states in a singular graphene metasurface. The optical response of a graphene conductivity grating with closely vanishing values at periodic points is a balance between a continuous spectrum due to the singularity and a discrete set of resonances due to nonlocality. Reproduced with permission from Ref. [690] (Copyright © 2020 De Gruyter). (b) Absorption efficiency spectrum of two almost touching nanowires under three TO approaches: classical and local, classical and nonlocal (hydrodynamic model), and nonclassical (mesoscopic approach). Reproduced with permission from Ref. [308] (Copyright © 2020 American Physical Society). (c) Sketch (left) and TEM image (right) of a Au–Au touching dimer utilized for photocatalysis. The average size of Au nanoparticles is 18 nm. Reprinted (adapted) with permission from Ref. [691] (Copyright © 2023 American Chemical Society).</figcaption></figure>

On the other hand, tunneling and size quantization are not contemplated within this model. Therefore, an outstanding challenge would be to incorporate these effects into the TO approach.
This would be relevant for optical nanostructures with the sharpest singularities, since these effects are important for feature sizes below about 1 nm [258].

Furthermore, there are plasmonic materials beyond metals, and TO can be extended to treat them [692]. For instance, graphene, the paradigmatic example of a two-dimensional material, hosts surface plasmons when doped. Interestingly, singular plasmonic gratings can be realized on graphene by having periodic conductivity modulations that approach zero at the grating minima, see Fig. 46(a). Under THz illumination, this monoatomic structure shows a broad absorption band. The stronger the suppression of conductivity at the grating valleys, the more broadband the absorption spectrum is [688]. Reference [690] showed how, under a semiclassical description of graphene’s conductivity (a local analogue model), these conductivity gratings could probe the nonlocal response of graphene by far field measurements only. However, quantum calculations have found that plasmon propagation is damped by strong modulations of the local Fermi energy [693], pointing to the need to implement fully quantum conductivity models within optical calculations, where TO can be of help.

Finally, if TO is to be employed for devising applications, further advances are needed to treat more complex structures. For instance, TO-inspired singular plasmonic nanospheres have been proposed for catalysis applications at metal-semiconductor interfaces [691]. Three-dimensional hybrid nanostructures with plasmonic singularities, see Fig. 46(c), were shown to provide broad and strong light harvesting at the active site of a semiconductor, facilitating a chemical reaction. However, these and other fully three-dimensional structures are challenging to model with TO, with only a few examples of nanosphere dimers having been developed in the literature [694].

### Concluding remarks

Combining the TO framework with nonlocal hydrodynamic or nonclassical mesoscopic models offers a powerful tool to unveil the effect of spatial dispersion, electron spill out or surface-enabled losses in nanoplasmonic structures. These effects are especially important in singular or near singular geometries where TO is particularly interesting, as it can treat structures that are commonly very challenging to simulate. Additionally, TO is able to uncover the geometrical basis of their optical response, by linking hotspots of field concentration in these systems with points at infinity in partner structures, to which surface plasmons travel. In conclusion, TO enables analytical calculations in complex geometries, in the presence of nonlocal and retarded effects. This makes TO a critical tool to understand nonlocal effects in plasmonics.

**Funding.** Spanish Ministry for Science, Innovation and Universities (grant no. RYC2021-031568-I and project no. PID2022-141036NA-I00 and CEX2018-000805-M financed by MCIN/AEI/10.13039/501100011033 and FSE+); National Natural Science Foundation of China (grant No. 12204328); Sichuan Provincial Science and Technology Support Program (grant No. 2024NSFSC1351); Fundamental Research Funds for the Central Universities; Simons Foundation through a Junior Fellowship of the Simons Society of Fellows grant No. 855344.

## 34. The concept of overlapping nonlocality and the ultimate thickness limits of optics

DAVID A. B. MILLER

### Current status

In many optical and wave systems, the output at a given point or pixel depends on the input at many other points or pixels, which we could regard as a definition of nonlocal behavior [523,571]. For example, in an imager, the light on one output image pixel depends on the light over the entire input surface of the lens [644]. However, the light on another output pixel also depends on the input light over the same entire input lens surface. So, these nonlocalities are “overlapping”.
Such overlapping nonlocalities [644] can be found in many other systems, including metasurfaces for image differentiation [695] or space compression [634]; there, too, multiple adjacent input pixels contribute to each output pixel (Fig. 47).

<figure><img src="image_57.png" /><figcaption>**Fig. 47.** Illustrations (after Ref. [644]) of overlapping nonlocality and corresponding number *C* of required channels that must flow through a “transverse aperture” inside the device. (a) Input and output surfaces for an *N*-pixel imager, with a transverse aperture defined by cutting the system from top to bottom through the middle, giving *C* = *N*/2. (b) Side view of a hypothetical image differentiator where each output pixel depends on the input field on four adjacent pixels, giving *C* = 4. (c) A side view of a hypothetical arrangement of beamsplitters and phase-delay plates separating two orthogonal input functions (as given by their vectors of amplitudes on the line of *Q* = 7 input pixels) to separate outputs, with *C* = 2. (d) Side view illustration that folding the optics of an imager can actually increase the required thickness. Now all *N* pixels or channels must flow through the transverse aperture, so *C* = *N*, hence increasing the required transverse aperture size.</figcaption></figure>

To understand overlapping nonlocality, we can imagine making a cut through both the input and output surfaces, which will define a “transverse aperture” in the optics (Fig. 47). The key idea here is that, for the device to do what we want, some number *C* – the overlapping nonlocality [644] – of independent channels must pass through this transverse aperture from side to side. Because we believe that physically we can only get so many propagating channels through some finite cross-section (e.g., the number of waveguide modes that could fit in that cross-section), this transverse aperture must be large enough to carry this number *C* of channels.

This idea of overlapping nonlocality addresses a key question that is particularly important for metastructures: can we do what we want with just a single layer of material or metamaterial? Or, more generally, just how much thickness will that structure need? This approach also gives bounds for the thickness of imagers, for example [644]. Note that, as in Fig. 47(d), any given channel could be collecting from a large number *Q* of input pixels, and we could regard *Q* as the (simple) “nonlocality”. (To distinguish from nonlocality as a physical distance, we could call *Q* the “nonlocality number” of the channel.) The overlapping nonlocality *C* is not the same as *Q*; *C* depends on the number of separate output channels that much be constructed from those same input pixels.

This overlapping nonlocality can be precisely specified mathematically. Some matrix *D* describes the mathematical mapping between input pixels (which feed matrix columns) and output pixels (which are fed by matrix rows) (Fig. 48). We can write that matrix so that all the columns for pixels on the “left” of the dividing surface are to the “left” of some vertical line
through the matrix, and similarly the rows for all output pixels to the “left” of the dividing surface are “above” some horizontal line through the matrix. This defines the “quadrant” sub-matrices $D_{LR}$ for “left-inputs-to-right-outputs” and $D_{RL}$ for “right-inputs-to-left-outputs”. We can view optical systems quite generally as mode converters [696,697], mapping one set of orthogonal input functions one-by-one to orthogonal output functions; these pair of functions – one each in the input and output spaces – form the mode-converter (or communication mode) basis sets for the system. If we know the desired matrix $D$ for the system, we can establish these pairs of functions by singular value decomposition of $D$, a standard matrix operation. Importantly, this process, through the resulting singular values, also gives the coupling strengths of these pairs. From this process, we can establish just how many such couplings are really needed, above some coupling strength we consider negligibly weak. (Quite generally [698], such couplings anyway fall off rapidly past some finite number in physical wave systems.) For our matrices $D_{LR}$ and $D_{RL}$, we can therefore deduce numbers of channels $C_{LR}$ and $C_{RL}$ needed in each case. Reciprocity [644] requires we add these numbers to get the required overlapping nonlocality $C = C_{LR} + C_{RL}$.

<figure><img src="image_58.png" /><figcaption>**Fig. 48.** Illustration of the matrix multiplication $\phi = D\psi$ that represents the operation of the device from inputs $\psi$ to outputs $\phi$. The choice of the transverse aperture position on the input surface divides the matrix into “left” and “right” parts, which also correspond to the input positions to the left and the right of the transverse aperture position. The choice of the transverse aperture position on the output surface divides the matrix into “upper” and “lower” parts, which correspond to the left and right parts of the output surface (relative to the transverse aperture position). These divisions generate matrix quadrants $D_{LR}$ and $D_{RL}$. The singular value decomposition of these quadrants effectively defines numbers $C_{LR}$ and $C_{RL}$ of channels that are required to pass through the transverse aperture in the operation of the device. The overlapping nonlocality is then $C = C_{LR} + C_{RL}$.</figcaption></figure>

The number $C$ arguably then can tell us a thickness of some optical element, based on a “maximum number of modes” conjecture that

> “the number of available channels cannot exceed the number of modes of a waveguide with a cross-section given by the transverse aperture and made from the largest refractive index used in the design”

This approach [644] has given suggested minimum thicknesses for differentiators [695], space plates [634], and imagers; optimized designs are thicker than these limits by no more than a factor of 3. It also gives limits for thickness of optical neural networks [699,700]; helps suggest better designs for optical processing networks [700]; is consistent with subsequent more specific limits for metalenses [701], metalens system design [702], and space plates in the microwave region [703]; and will impose thickness limits on metastructures for aberration correction [639]. Note the following about the overlapping nonlocality $C$.

1) $C$ is not related to the physical size of the function being performed; a small camera and a large camera with the same number of image pixels have the same overlapping nonlocality.
2) $C$ is an effective mathematical dimensionality of our desired input-output relation, and in that sense is independent of how we implement the device physically. In optics we may use propagating modes in dielectric structures to carry these transverse channels, but the same number of channels is required for any other physical implementation, such as plasmonics or even electrical wires in radio-frequency systems (e.g., in “reflective intelligent surfaces” with non-diagonal reflection coefficients [704]).

This concept of overlapping nonlocality is a quite general property of any optical or wave device intended to implement some relation between inputs and outputs. Understanding and using this concept can arguably help us design and operate all such systems.

### Challenges and opportunities

Though the overlapping nonlocality $C$ is well defined just from what we want the optical or wave system to do, there are at least two challenges in using this approach to deduce the necessary thickness of the required system.

1) We need to understand whether our approach to building the system can perform “dimensional interleaving” [644]. If we think of our system as having its input degrees of freedom in two dimensions – $x$ and $y$ in Fig. 47(a) – we need to know if we can move those degrees of freedom between the $x$ and $y$ dimensions. Now, a photonic integrated circuit with a two-dimensional array of grating couplers can readily couple their outputs to a one-dimensional array of waveguides, for example, hence “interleaving” the input information from $x$ and $y$ degrees of freedom into a line in the $y$ direction [644]. Many other optical systems do not, however, do this interleaving; conventional imagers and lens optics, free-space propagation, dielectric stacks, and photonic crystals all preserve the degrees of freedom in each of the $x$ and $y$ directions [644]. If there is no dimensional interleaving, the required thickness is determined by the number of degrees of freedom in the $x$ or $y$ directions (whichever is larger) and on a one-dimensional version of channel counting in a waveguide, which generally will lead to a larger thickness than if we can perform dimensional interleaving.

We can form a “dimensional interleaving” conjecture here that

> “dimensional interleaving is not possible without segmenting the field or, equivalently, it requires a discontinuous mapping between input and output fields.”

This conjecture remains to be proved or disproved. Note, for example, that a hypothetical array of grating couplers segments the input optical field, essentially discontinuously. Another device that could do dimensional interleaving, a photonic lantern [705], also segments the field as the fused fibers at one end separate into discrete fibers at the other. This author knows of no way of performing such dimensional interleaving without segmenting the field somewhere; of course, this author’s possible ignorance proves nothing.

2) Using the overlapping nonlocality $C$ to deduce minimum thickness of an optical system that we have not yet designed has required the “maximum number of modes” conjecture stated above. This conjecture also remains to be proved or disproved.

One possible approach to the “maximum number of modes” conjecture could be to use a multiple scattering theorem [706] that relates dielectric constant contrast in a volume to achievable optical functions; this has been used successfully for limits in one-dimensional structures [706,707], though it has not so far been extended to two or three dimensions.

A limitation of this approach so far is that it is only analyzed for one frequency. Since different frequencies are anyway orthogonal functions in that they can separately propagate through the
same channel, each containing different information (e.g., amplitudes), this approach has nothing to say about additional limits or opportunities [708] in multi-frequency systems, and further such additional limits remain open questions.

## Future developments to address challenges

Clear future directions for understanding and applying overlapping nonlocality are to prove or disprove the “maximum number of modes” and “dimensional interleaving” conjectures stated above. These conjectures are, of course, interesting for their direct relevance to overlapping nonlocality. Possibly, however, proving or disproving them might require moving beyond our typical ideas for understanding waves; as such, addressing them might expose new areas of opportunity. A final comment on future directions is that this overlapping nonlocality work emphasizes the importance of analyzing nonlocality also in real space, not just in the *k*-space “plane-wave basis” approach. Such plane waves are, of course, mathematically completely nonlocal because they have infinite lateral extent. This approach has been powerful and successful in many ways. It is, however, quite awkward mathematically in a *k*-space approach to introduce a surface that divides real space into two parts. Possibly as a result, overlapping nonlocality, though implicit in any actual design based on *k*-space approaches, may be far from obvious in that description. Such plane-wave approaches are also generally problematic as we move to small structures — e.g., from several wavelengths to sub-wavelength sizes — where they are arguably not a good basis and can cause us to miss physics in the problem. For small structures, physically there are really no (plane) evanescent waves, for example [698]. To model waves in and out of such structures, spherical waves [698,709] are a better fundamental basis. Unlike plane waves, such spherical waves are countable, which can help in counting channels, and by using “bounding spheres” they can implicitly include the finite size of real objects [698,709]. These results remind us of the importance of thinking also in real space when analyzing wave systems and generally of being prepared to move beyond *k*-space descriptions.

## Concluding remarks

The idea of overlapping nonlocality is ultimately a simple one that we can deduce directly from what we want our optical system to do. Using it to predict requirements and limits in optical designs of various kinds looks to be a potential productive and useful direction. At the same time, it is exposing new conjectures and ways of thinking about wave devices that may test and expand our understanding and use of waves in general.

**Funding.** Air Force Office of Scientific Research (FA9550-21-1-0312).

## Closing remarks and back matter

### 35. Conclusion and outlook

FRANCESCO MONTICONE ID & N. ASGER MORTENSEN ID

<figure><img src="image_59.png" /></figure>

Assembling this Roadmap has given us a unique opportunity to work with many of the leading experts in this field to explore the rapidly expanding landscape of nonlocal responses in natural and artificial materials. Through more than thirty sections, this Roadmap highlights an impressive array of exciting new developments and emerging directions. It is clear that the multifaceted concept of nonlocality is playing an increasingly important role in optical materials science and technology, from advancing our understanding of anomalous and extreme forms of light-matter interactions and revealing new insights into the fundamental limits of wave physics, to unlocking new opportunities for optical devices based on nonlocality engineering. In the spirit of this Roadmap, we believe that future progress in this field will not be “local” but will depend on an extended range of contributions from multiple areas. We hope that this Roadmap,
bringing together insights from different corners of this field, serves as a valuable guide to the
current landscape of nonlocality in photonic materials and metamaterials and will inspire further
innovation in expanding its frontier.

**Acknowledgments.** The lead authors are grateful for the community's strong support in organizing this Roadmap.
We also extend our thanks to the Editorial Board of the journal for inviting us to take on this project.

**Disclosures.** The authors declare no conflicts of interest.

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