**Appendix A**

### *Appendix A.1. Survey of the Literature on Nonlocal Metamaterials*

Appendix A.1.1. Introduction

We first provide a non-exhaustive and selective review of the development of nonlocal electromagnetic materials research. More information and proposals regarding engineering applications are given in Appendix A.3, where additional references can be found. The main propose behind this literature overview is to sugges<sup>t</sup> that the area of nonlocal metamaterials research might be approached as an approximately coherent field of investigation, i.e., more than just being merely a technical sub-discipline selected from within the sciences of metals, semiconductors, plasma, periodic structures, metasurfaces, etc. In fact, one of the main objectives of this paper is to demonstrate that a unified theoretical treatment of the entire subject is mathematically possible (the superspace formalism.) However, one needs to be convinced first of the presence of substantial past researches into this area. Hence, convincing readers not familiar with the topic about the long and very rich history of investigations into various nonlocal phenomena in material systems is one of the objectives of this Appendix.

### Appendix A.1.2. Historically Important Examples

Some of the physical phenomena that cannot be understood using local electromagnetic theory include spatial dispersion effects [83], extreme negative group velocity and negative refraction [52,94], new diffraction behavior in optical beams [95], superconductivity [96], natural optical activity [16,97,98], non-Planck equilibrium radiation formulas in nonlocal plasma [99]. Outside electromagnetism but within wave phenomena, there also exists processes that cannot be fully accounted for through simple local material models, for instance, we mention phase transitions, Casimir force effects [100], and streaming birefringence [9]. By large, *spatial dispersion* has attracted most of the attention of the various research communities working on nonlocal electromagnetic materials. Indeed, few booklength researches on spatial dispersion already exist in literature, most notably [14–16,83]. We provide additional remarks on the history of spatial dispersion in Appendix A.2.

### Appendix A.1.3. General Theories of Nonlocal Continua

The majority of the published research on nonlocal media and nonlocal electromagnetism tend to focus on applications and specialized materials (see the majority of the references quoted below). Few exceptions include investigations attempting to approach the subject at a more general level. For example, from the perspective of general thermodynamics, see [8,9]. A unified perspective inspired by condensed-matter physics, especially plasma physics, can be found in [47]. Within nanoscale electrodynamics, nonlocality was treated broadly as an essential feature of microscopic interactions at the nano- and mesoscopic scales [50,53]. Some of the topics reexamined within the framework of a general nonlocal field–matter interaction theory include the applicability of optical reciprocity theorems [101–104], energy/power balance [105], quantization [106–108], operator methods [87], extension of spatial dispersion to include inhomogeneous media [76], and alternative formulations of spatial dispersion in terms of the Jones calculus [109].

### Appendix A.1.4. Semiconductors, Metals, Plasma, Periodic Structures

The bulk of the available literature on nonlocality is concentrated in the very large area of general field–matter interactions. There already exists a well-attested body of research on nonlocality in metals based on various phenomenological approaches, e.g., see [110] for a general review. Nonlocality has also been extensively investigated in dielectric media, for example semiconductors [83,111]. A comprehensive recent review of nonlocality in crystal structures is provided in [112], which updates the classic books [16,55]. Moreover, numerous researches conducted within condensed-matter physics and material science implicitly or explicitly assume that nonlocality is essentially based on microscopic (hence quantum) processes, and develop an extensive body of work where the spatially dispersive dielectric tensor is deployed as the representative constitutive material relation [37,38,50,53,54]. On the other hand, one can also treat nonlocality without resort to spatial dispersion by modeling certain classes of material media as periodic structures [113], e.g., photonic nanocrystals [114], where the susceptibility tensor is derived from the symmetry of the overall structure [37,89,115] or from the lattice dynamics approach [92,116].

### Appendix A.1.5. Boundary Conditions in Nonlocal Metamaterials

For solving nonlocal problems, several methods have been proposed in order to deal with the notorious problem of the lack of exact universal nonlocal response models at the intermaterial interface between a nonlocal domain and other media. The so-called additional boundary condition (ABC) approach adjoins new boundary conditions to the standard Maxwell's equations in order to account for "additional waves" excited at the interface, which otherwise would not be explicable by the standard local theory alone [16]. However, it must be noted that without exception all ABC formulations are inherently model-specific since each boundary condition model presupposes a particular type of nonlocal media, or simply just postulates specific ABCs based on their ease of use in applications, e.g.,

see [55,56,80,117–119]. We note that such ABC formalisms are not inevitable since there exists several boundary-condition free formulations, e.g., see [50,53,89].<sup>29</sup>

### Appendix A.1.6. Computational Techniques

For performing full-wave field analysis in the presence of nonlocal materials, a number of discretization strategies have been proposed. For example, an FDTD-based method was suggested to deal with metallic spatially dispersive objects [120]. The formulation, discretization, and solution of surface integral equations for nonlocal plasmonic materials were also attempted in [121,122], where the reduction of the electromagnetic problem to a finitematrix form was achieved using the RWG basis functions. Moreover, specialized methods were proposed for various possible scenarios involving nonlocal field–matter interactions, such as nonlocal dielectric profile retrieval from measurable data [123], iterative solutions of nonlocal wave equations [124,125], applications of the derivative expansion method to nonlocal plasma analysis [126], application of Kramers–Kronig relation method [127], application of the Pade approximation to homogenization [128].

#### Appendix A.1.7. Novel Systems and Devices with New Electromagnetic Behavior

The idea of exploiting nonlocality to design and develop a new generation of metamaterials (MTMs) exhibiting novel EM behavior has also received a revival in recent years [32,47,129], though the basic concept in itself is not completely new, going back to at least the 1980s and possibly earlier [16]. Recent examples of research focused in explicating nonlocal behavior to harness the associated new physics include spatial dispersion in photonic crystals [130], wire media [131–134], semiconductor nanoparticles [135–138], optically nonlinear liquids [139], hyperbolic metamaterials [140], layered dielectric-metal structures [141,142] and thin films [143], plasma-based metamaterials [144–146], quantum wells [147], soliton interactions with matter [148–153], superconducting films [154] and circuits [155], plasmonic devices and structures [156–158], nanocubes [159], cloaking [160], Chern metamaterials [161] and superconductors [162], dispersion managemen<sup>t</sup> profiles [52,163], biomedical applications in materials [164], nonlocal antennas [165,166], and nonlocal uniaxial metamaterials [167]. Due to the practical importance of this area of research, we provide additional information in Appendixes A.3 and A.11.

### Appendix A.1.8. Homogenization

Numerous homogenization theories for nonlocal MTMs, where averaging operations are considered over multiple spatial scales, have been reported in the literature, e.g., see [133,168–171]. We note that the subject of estimating the effective electric and magnetic properties of electromagnetic metamaterials, with or without nonlocality, is enormous and it is beyond the scope of this paper to even summarize the main papers in the field. Nevertheless, it is curious to note that until fairly recently, most publications have tended to focus on *non*-spatially-dispersive media; hence, local scenarios are still dominant in the area of advanced artificial material systems. This situation has began to change in the last few years, and nowadays an increasing number of reports appear to move from the old opinion that "spatial dispersion is a bug" to the more positive and fruitful perspective that nonlocality may provide pathways to novel physical behavior that can be exploited for various applications in metamaterial system design. However, we also note that progress in this second direction, where nonlocality is embraced rather than being treated with suspicion, has been generally slow.

### Appendix A.1.9. Topological Materials and Photonics

A particularly interesting direction of research in nonlocal media is the recent subject of *topological photonics*. The main idea was inspired by previous researches in Chern insulators and topological insulators [10], where the focus has been on electronic systems. There, it has already been observed that the nonlocal behavior of the fermion wave function may exhibit a rather interesting and nontrivial dependence on the entire configuration space of

the system, in that case the momentum space (the wave vector **k** space). In addition to the already established role played by nonlocality in superconductors, quantum Hall effects are among the most intriguing physically observable phenomena that turned out to depend fundamentally on purely topological aspects of the electron wave function [96]. The major themes exhibited by electrons undergoing topological transition states include topological robustness of the excited edge (surface) states moving along a two-dimensional interface under the influence of an external magnetic fields. More recently, it was proposed that the same phenomenon may apply to photons (electromagnetism) [172], where the key idea is to use photonic crystals to emulate the periodic potential function experienced by electrons in fermion systems. However, since photons are bosons, transplanting the main theme of topological insulators into photonics is not trivial and is currently generating a grea<sup>t</sup> attention, see for example the extensive review article [11], which provides a literature survey of the field. One of the most important applications of topological photonics is the presence of "edge states", which are topologically robust unidirectional surface waves excited on the interface between two metamaterials with topologically distinct invariants. Since edge states are immune to perturbations on the surface, they have been advocated for major new applications where topology and physics become deeply intertwined [173]. Topology can also be exploited to devise non-resonant metamaterials [174] and to investigate bifurcation transitions in media [175]. Another different but related exciting subject illustrating the synergy between topology, physics, and engineering is non-Hermitian dynamics, especially in light of recent work related to the origin of surface waves [176,177], which is now being considered as essentially non-trivial topological effect. In Appendix A.11, the subject of topological photonics is taken up again but from the viewpoint of applications.

#### *Appendix A.2. On the History of Spatial Dispersion in Crystal and Plasma Physics*

Historically, spatial dispersion had been under the radar since the 1950s, especially in connection with researches on the optical spectra of material domains [77,78,178]. However, the first systematic and thorough treatment of the subject appeared in 1960s, prominently in the first edition of Ginzburg's book on plasma physics, which was dedicated to electromagnetic wave propagation in plasma media. The second edition of the book, published in 1970, contained a considerably extended treatment of the various mathematical and physical aspects of the electromagnetism of spatially dispersive media [14]. Spatial dispersion in crystals had been also investigated by Ginzburg and his coworkers during roughly the same time [179–181]. The book [83] contains good summaries on spatial dispersion research up to the end of the 1980s. More recently, media obtained by homogenizing arrays of wires, already very popular because of their connection with traditional (temporal) metamaterials, are known to exhibit spatial dispersion effects, though many researchers ignore that effect to focus on temporal dispersion [182–184]. Other types of periodic or large finite arrays composed of unit cells like spheres and desks also exhibit spatial dispersion effects [185]. Nonlinear materials with observable nonlocality have also been investigated in the optical regime [186]. More recently, much of the resurgence of interest in spatial dispersion can be traced back to the observation that nonlocal phenomena cannot be ignored at the nanoscale level [187], especially in problems of low-dimensional structures, such as carbon nanotubes [89,91,92,188] and graphene [189,190]. The subject was also introduced at a pedagogical level for applications involving current flow in spatially dispersive conductive materials, such as plasma and nanowires [191].

#### *Appendix A.3. Some Further Engineering Applications of Nonlocal Metamaterials*

The purpose of this Appendix is to provide a sample of some other current and future possible applications of nonlocal metamaterials based on the author's own experience, which may serve as a supplementary text to be read in conjunction with the general survey of Appendix A.1.

Appendix A.3.1. Communications Systems and Information Transmission

Nonlocal metamaterials offer a very wide range of potential applications in wireless communications and optical fibers. The basic idea is to introduce specially engineered nonlocal domains either as part of the communication channel (e.g., optical fibers, plasmonic circuits, microwave transmission lines) [189], or as a control structure integrated with existing antennas [129,192]. Spatial dispersion was also used as a method to engineer wave propagation characteristics in material domains, e.g., see [193] for applications to high-efficiency modulation of free-space EM waves. A general linear partial equation explicating how spatial and temporal dispersion can be jointly exploited to produce zero distortion (e.g., constant negative group velocity) was derived and solved in [52]. The main idea originated from the fact that one of the main sources of distortion in communication systems is that due to *non*-constant group velocity **<sup>v</sup>***g* := ∇**k***<sup>ω</sup>* [194,195]. Since **<sup>v</sup>***g* is a strong function of the dependence of the material response tensor **<sup>K</sup>**(**<sup>k</sup>**, *ω*) on both **k** and *ω*, *dispersion management equations* can be derived for several applications. For example, it was proved in [52] that in simple isotropic spatially dispersive media with high-symmetry, one may obtain exact solutions where the group velocity is constant at an entire frequency band. This happens because while strong temporal dispersion is present (which alone causes strong distortion), incorporating optimized spatially dispersive profiles leads to *complete compensation* (cancellation) of distortion, resulting in essentially a distortion-free communication channel. There are enormous potentials of research into this new exciting area. The reason is that most practical realizations of nonlocal metamaterials involve complex material response tensors, where the relevant mathematics of dispersion engineering is still underdeveloped (and in fact underappreciated by researchers), which implies that, to the best of our knowledge, relatively very little has been done in this emerging field so far.

### Appendix A.3.2. Electromagnetic Metamaterials

While this paper attempts to analyze and understand the general structure of nonlocality in generic field theories of continuous media, we have already mentioned above that artificial media, better known nowadays as metamaterials systems, could provide one of the most direct paths toward building new functional advanced materials and also providing models to further explore nonlocality both experimentally and numerically. As early as the 1960s, it was proposed that EM nonlocality can be exploited to produce materials with very unusual properties. For example, in [16], negative refraction materials were noted as one possible application of spatial dispersion where the path toward attaining this goes through controlling the direction of the group velocity vector. Since in nonlocal media, power does not flow along the Poynting vector [14], new (higher-order) effects were shown to be capable of generating arbitrary group velocity profiles by carefully controlling the spatial and temporal dispersion profiles. Overall, the ability of spatial dispersion to induce higher-order corrections to power flow is a unique advantage enjoyed by nonlocal metamaterials exhibiting weak or strong spatial dispersion in addition to normal dispersion. This extra spatial degrees of freedom provided by nonlocality was researched, reviewed and highlighted in many publications, including, for example, works such as [32,47,115,129,134,141,163,175,184,196].

### Appendix A.3.3. Near-Field Engineering, Nonlocal Antennas, and Energy Applications

Another interesting application of nonlocality in electromagnetic media is near-field engineering, a subject that has not ye<sup>t</sup> received the attention it deserves. It was observed in [129] that a source radiating in homogeneous, unbounded isotropic spatial dispersive medium may exhibit several unusual and interesting phenomena due to the emergence of extra poles in the radiation Green's function of such domains. Both longitudinal and transverse waves are possible (dispersion relations), and the dispersion engineering equations relevant to finding suitable modes capable of engineering desired radiation field patterns are relatively easy to set and solve. For example, by carefully controlling the modes of the radiated waves, it is possible to shape the near field profile, including total confinement of

the field around the antenna even when losses is very small, opening the door for applications like energy harvesting, storage, and retrieval in such media [197]. The direct use of especially-engineered nonlocal metamaterials, however, has been explored only for simple materials so far and mainly at the theoretical level [32]. However, the increasing importance of energy localization [198] at both the level of numerical methods [199] and the device level applications [200], sugges<sup>t</sup> the need to reconsider the role played by nonlocality in complex media.

On the other hand, away from the source region, the subject of *far*-field radiation by sources embedded into nonlocal media was investigated previously by some authors within the context of plasma domains [104]. Recently, it has been systematized into a general theory for nonlocal antennas with media possessing an arbitrary spatial dispersion profile [25,165,166,192]. However, no general theory exists for nonlocal media, which are inhomogeneous. The superspace formalism proposed in this paper may help stimulate research into this direction in order to overcome the limitations of the existing theory of nonlocal antenna systems.

### *Appendix A.4. On the Concept of Superspace*

The concept of superspace is not new, and has been proposed several times in both physics and mathematics. For a brief but general view on the definition of superspaces, see [201]. For example applications, various superspaces have been proposed as fundamental structures in quantum gravity [202,203], which are frequently infinite-dimensional. Superspace concepts are also now extensively researched in quantum field theory and the standard model of particle physics, e.g., see [204–206]. In general, dealing with topics such as supergravity, supersymmetry, superfields, superstrings, and noncommutative geometry often requires the use of one superspace formalism or another [204]. In mathematics and mathematical physics, where the concept itself originated, a notable recent example of the superspace concept includes *sheaves*, which are used in differential and algebraic topology and algebraic geometry and have numerous applications in physics [30,207,208].

In this paper, the superspace concept has very little to do with applications to supersymmetry or supergravity, such as the examples mentioned above (and many others we do not mention.) Instead, our use of the concept is more aligned with the mathematical practice of *extending* one space by *embedding* it into a larger superspace as in the schema:

$$\text{Space} \xrightarrow[\text{injection}]{\text{embedding}} \text{Superspace.}\tag{A1}$$

In other words, the embedded space is injected as a substructure into the (larger) embedding superspace. The key interest behind the formula superspace-as-embedding (A1), of course, goes beyond mere set-theoretic inclusion. We are not here trying simply to say that Space ⊂ Superspace, which would be devoid of mathematical substance. Instead, the main motivation behind the superspace construction (A1) is that the embedded Space becomes a *substructure* attached to or placed within the larger, embedding "container", which is here superspace.

The most important thing to note here is that the latter superspace acquires a more coherent and fundamental status than the former. Eventually, Space becomes nothing but a mere "substructure" or "index space" of the more originary mother space that we originally called superspace. Strangely, with time, superspaces tend to become so familiar and basic to the degree one begins to call them regular spaces, while the original Space fades into oblivion. This last observation regarding the ontological primacy of the superspace concept over space can be best seen from the *converse* generative schema:

$$\text{Superspace} \xrightarrow[\text{projection}]{\text{dc-embedding}} \text{Space.}\tag{A2}$$

Here, we recover the original space through a *projection* operation by which a de-embedding of the superspace substructure, the interior placeholder occupied by space, is achieved by projecting the mother space, the superstructure, superspace, onto the substructure, space. It is really the purely formal *structural* relation dictating how sub- and super-structures are organized within a common unifying global schema what is at stake in such type of superspace theories, i.e., not just the simple set-theoretic inclusion of one space into another.

Both operations, the injection (A1) and projection (A2) are necessary to fully understand the idea of superspace in general. However, in practice, usually only one of them is emphasized on the expense of the other. It is rare to find in superspace theories that both projection and injection operations are allotted the same ontological status. For instance, in the fiber bundle approach adopted in the present paper, space is recovered (or generated) from the fundamental superspace through the projection map of the mother fiber bundle, which will send each fiber into its "representative point" in the base manifold. In this manner, regular space may be seen as if it was actually *generated* or "produced" by the more primordial superspace mother structure [18,209].

A specific example more related to the subject of nonlocal MTMs is the original superspace concept introduced earlier for the analysis of deformed crystal [210] and subsequently utilized for fundamental investigations of EM nonlocality in incommensurate (IC) superstructures in insulators [76]. Such modulated-structure materials possess spaces with dimensions greater than spacetime [211]. Nevertheless, for fairly concrete models one may exploit group theory to construct finite-dimensional (dimension > 4) approximations of them. The general theory of superspace formalisms in quasi-periodic crystals is presented in [212]. Other examples from condensed-matter physics where superspace methods where applied include mesoscopic superconductivity [213].

### *Appendix A.5. Guide to the Mathematical Background*

We provide a brief overview on how to read the mathematical portions of this paper and where to find detailed references that might be needed in order to expand some of the technical proof sketches provided in the main text. We emphasize that in this paper only the *elementary* definitions of


that are needed in order to understand the mathematical development. Here, we briefly go over the principal ideas behind each one of these four key mathematical topics listed above, providing also additional references for readers interested in learning more about the required background. The current Appendix is not intended as a complete review; some familiarity with all four elementary mathematical topics listed above is required for a complete understanding of the technical proofs and constructions found in Section 5.

### Appendix A.5.1. Topology on Smooth Manifolds

A *differential manifold* is a collection of fundamental "topological atoms" each composed of an open set *Ui* and a chart *φi*(*x*), which serves as a *coordinate system*, basically an invertible differentiable map to the Euclidean space R*<sup>n</sup>*. That is, locally, every manifold looks like a Euclidean space with dimension *n*. When the differentiable map is smooth, the differential manifold is called *smooth manifold.* The collection of open sets *Ui*, *i* ∈ *I*, where *I* is an index set, covers this *n*-dimensional manifold. Since some of these open sets are allowed to overlap, the crucial idea underlying the concept of the differential manifold is that over the common intersection region *Ui* ∩ *Uj*, there exists a smooth reversible coordinate transformation function mutually relating the two coordinates of the same abstract point when expressed in the two (generally different) languages belonging to the topological atoms *Ui* and *Uj*. *Note that the key concept of topology is how to propagate* *information from the local to the global levels*. In this sense, differential manifolds present elementary structure allowing us to rigorously conduct this process using the efficient apparatus of the differential calculus. Note that only the elementary definition of smooth manifolds is required in this paper, which can be found in virtually any book on differential or Riemannian geometry, e.g., see [26,30,57,62,65,70,204,214].

### Appendix A.5.2. Banach and Hilbert Spaces

A *Banach space* is a vector space equipped with a norm satisfying the standard properties that a generic norm should have (namely, being positive, being zero only for the null vector, scale linearity, and the triangle inequality [215].) Most importantly, Banach spaces are also required to be *topologically complete* in the sense that every Cauchy sequence converges to an element in the space itself. In this way, no "holes" are left in the space thus defined, hence one may deploy a Banach space in order to do analysis on operators as in solving differential equations or the analysis of numerical methods. A *Hilbert space* is a Banach space equipped with an inner product. An important fact to remember about Banach and Hilbert spaces is that when they are employed to model *function* spaces (as in this paper), they most often lead to *intrinsically-infinite* dimensional vector spaces [30].

### Appendix A.5.3. Banach and Hilbert Manifolds

A straightforward process of combining Banach or Hilbert spaces with differential manifolds leads to the concept of *Banach* or *Hilbert manifold*, which are prominent examples of *infinite-dimensional manifolds.* A Banach/Hilbert manifold is simply a differentiable/smooth manifold that is locally isomorphic to a Banach/Hilbert space instead of the regular *n*-dimensional Euclidean space R<sup>3</sup> invoked in the basic definition of an *n*-dimensional manifold. The isomorphism itself can be either differentiable or smooth, where a suitable derivative operator, such as the Fréchet derivative, may be defined on Banach/Hilbert spaces, leading to the resulting Banach/Hilbert manifold itself being either a differentiable or smooth infinite-dimensional manifold. A Banach/Hilbert manifold is then an intrinsically infinite-dimensional manifold. An elegant formulation of the theory of Banach manifolds can be found in Lang's text [26]. Applications of this theory in the general fields of analysis and geometry can be found in textbooks on global analysis, e.g., see [70,216]. In general, much of the theory of *n*-dimensional manifolds carry over unchanged into the case of infinite-dimensional manifolds. However, there exists some subtle technical differences, which are carefully highlighted in [214].

### Appendix A.5.4. Sobolev Spaces

The most economic approach to constructing *Sobolev spaces* is to define them as Hilbert spaces consisting of (Lebesgue) square integrable functions that posses "generalized derivative", a concept in itself technical but straightforward. For the basic definition of Sobolev spaces and their applications to partial differential equations in mathematical physics and finite-element method in engineering, we recommend [65]. The subject of Banach manifolds is less commonly treated in the literature on Sobolev spaces than finite-dimensional manifolds. For a very readable account on the functional analytic background to the use of Sobolev spaces, see [65], while [64] provides information on the applications of Sobolev spaces in the analysis of linear partial differential equations. The generalization of the theory of Sobolev spaces into the wider setting of functions defined on differential manifolds is tackled in [66] (with applications to nonlinear functional analysis).

### Appendix A.5.5. Vector Bundles

The quite general structure known as *fiber bundles*, of which *vector bundles* are famous special cases, are now standard topics in both mathematics (topology, geometry, differential equations), theoretical physics (quantum field theory, cosmology, quantum gravity), and applied physics (condensed-matter physics, many-body problems). On the major importance of vector and fiber bundles within the overall area of modern fundamental

physics, see [30,204,208]. In quantum field theory, gauge field theories use vector bundles as essential ingredients in the standard model of particle physics [18,30]. The increasing importance of methods based on quantum field theory in applications to condensed-matter physics has contributed into making knowledge of fiber bundle techniques useful and more widespread in physical and engineering research than originally anticipated; e.g., see the area of the Berry phase and the associated gauge connection [11,96]. The key idea behind the vector bundle is to attach an entire vector space to every point on a base manifold. To be more specific, consider a differential manifold D serving as the base manifold. Each copy of the vector space that is attached to a point in this base space will be called the *fiber* at that point. The standard *tangent space* of a smooth manifold is the most obvious example of such vector bundles. However, more complicated structures than finite-dimensional tangent spaces can also be captured by a suitable vector bundle concept. In this paper, we have shown that physics-based nonlocality in material continua can be modeled, very naturally in the mathematical sense, by considering the Banach space of all excitation fields acting on the microdomains indexed by a point in the material configuration space (base space). The fiber bundle superspace formalism may then be seen as a highly efficient and economic apparatus available for encoding, storing, and processing a large amount of topological and geometrical data pertinent to the problems of nonlocality in physics and engineering since fiber bundles lend themselves easily to complex calculations. Readable technical descriptions of vector bundles can be found in [30,57,63,70].

Appendix A.5.6. Additional Remarks on the Use of Sobolev Spaces in the Fiber Bundle Superspace Formalism

In Section 4, we introduced Sobolev space over the open domain *D* instead of simply operating with the more generic Banach space. The reason behind our decision to invoke the more specialized (and technical) structure of a Sobolev space was mainly to actually simplify the technical development and in anticipation of future work on the superspace formalism. Indeed, in this paper, the fiber bundle M is referred to just as Banach bundle, not Sobolev bundle for the reason that all our essential results and insights apply to the more general concept of Banach space, which contains Sobolev spaces as a special case. In fact, Sobolev spaces are easier to work with in problems involving integro-differential equations such as nonlocal continuum field theory. Nevertheless, we only used the elementary definition of Sobolev space itself in Section 5, not its advanced properties. In particular, none of the other technical properties of Sobolev spaces are needed in the paper. Nevertheless, since in the future the material bundle space M is expected to be employed in order to construct solutions of Maxwell's equations in new form (i.e., in superspace instead of conventional spacetime), Sobolev spaces are projected to play the most important role since they have proved very efficient in analysis and the theory of partial differential equations [64].

### Appendix A.5.7. Partition of Unity Techniques

In analysis and differential topology, the title *the partition of unity lemma* refers to a somehow rather technical tool used by topologists and analysts in order to help propagate information from the local to the global setting. They were found to be quite handy and easy to apply. The main theorem (Lemma 1) permits us to move from one topological "atom" to another by "gluing" them together using smooth standard domain-division functions. The technique was stated and used only toward the end of Section 5 in order to justify expansions, such as (77) and can be skipped in first reading of the paper. Partition of unity is usually taught in all topology and some geometry textbooks, e.g., see [26,57,63,70].

### *Appendix A.6. The General Electromagnetic Model of Nonlocal (Spatially-Dispersive) Isotropic Domains*

One of the simplest—yet still demanding and interesting—nonlocal media is the special case of isotropic, homogeneous, spatially-dispersive, but optically inactive domains [14]. In this case, very general principles force the generic expression of the material response tensor to acquire the concrete form [13,15,16]:

$$\mathbf{K}(\mathbf{k},\omega) = \mathbf{K}^{\mathrm{T}}(k,\omega)(\mathbf{I} - \mathbf{k}\mathbf{\hat{k}}) + \mathbf{K}^{\mathrm{L}}(k,\omega)\mathbf{k}\mathbf{\hat{k}},\tag{A3}$$

where

$$k := |\mathbf{k}|, \; \hat{k} := \mathbf{k}/k,\tag{A4}$$

and **k** is the wave vector (spatial-frequency) of the field. The first term in the RHS of (A3) represents the *transverse* part of the response function, while the second term is clearly the *longitudinal* component, with behavior captured by the generic functions *<sup>K</sup>*<sup>T</sup>(*k*, *ω*) and *<sup>K</sup>*<sup>L</sup>(*k*, *<sup>ω</sup>*), respectively.<sup>30</sup> The tensorial forms involving the dyads ˆ*<sup>k</sup>*<sup>ˆ</sup>*k*, however, are imposed by the formal requirement of the need to satisfy the Onsager symmetry relations in the absence of external magnetic fields [16]

Using a proper microscopic theory, ultimately quantum theory, it is possible in general to derive fundamental expressions for the transverse and longitudinal components of the response functions in (A3) [14–16,37,38,50]. These forms are often obtained in the following way:


A concrete example is given in Section 7.3 to illustrate the use of such physics-based dielectric functions for the case of exciton–polariton-based semiconductor materials.

### *Appendix A.7. Origin of Electromagnetic Nonlocality in Excitonic Semiconductors*

Appendix A.7.1. Review of the Semiconductor Physics of Excitons

Very early in the history of condensed-matter physics, excitons were introduced by Frenkel [84,85], and further developed by other researchers, such as Wannier [217]. In the late 1950s, excitonic phenomena were transplanted into a central stage in the framework of light-matter interaction through the concept of *exciton–polariton* [178], which will be defined below. Pekar [178], Ginzburg [77], and others [79,119,218,219] affirmed the nonlocal approach to exciton–polariton materials by explicitly highlighting the strong impact of spatial dispersion near excitonic resonances. The subject of excitons is vast and multidisciplinary. For extensive treatments covering various applications in physics, chemistry, and technology, see [16,55,86,220–222].

In order to understand the particular nonlocal model to be presented in Section 7.3, let us first briefly explain the relevant physics of exciton–polariton interactions and why they can lead to strong nonlocal response. In a direct-band gap semiconductor the minimum of the conduction band is aligned along the maximum of the valance band, allowing electronic transitions from lower (unexcited) to excited bands upon interaction with external EM fields. For insulating semiconductors of the II-VI and III-V groups, exciton transitions occur in the visible or near-ultraviolet range of the electromagnetic spectrum. By engineering the material/metamaterial parameters, these transition frequencies can be shifted.

It should be noted that in contrast to metals and plasma, no free charged carriers are assumed to exist in the material. An electron exiting the valance band after the absorption

of an external photon will leave behind a *hole*, which acts as an independent quasiparticle that can travel throughout the material in the form of a collective excitation [84,85,223,224]. The *exciton* is defined as a coupled pair composed of the two bound states of the electron and hole. Here, both electrons and holes must be understood as "dressed" particles (quasiparticles) with effective mass and charge different from those of the bare (noninteracting) particle [225]. We may apply the Bohr model to the exciton (electron-hole pair) with simple modifications that can be summarized by the following procedure:

1. The electron mass must be replaced by the reduced exciton mass

$$m\_{\rm tr} := \frac{m\_{\rm el} m\_{\rm h}}{m\_{\rm el} + m\_{\rm h}},\tag{A5}$$

where *<sup>m</sup>*el and *m*h are the electron and hole masses, respectively.


From this, it follows that the exciton binding energy *E*b is given by

$$E\_b = \frac{m\_\text{r}e^4}{2\hbar^2\varepsilon\_0^2}.\tag{A6}$$

Therefore, the total energy needed to create an exciton state is given by

*h*¯ *ω*e = *E*g − *E*b, (A7)

where *E*g is the semiconductor band gap energy. In most applications, the binding energy *E*b is in the order of meV, while *E*g is usually few eV. That is, the energy needed to create an exciton is slightly less than the band gap energy and typically we have

$$E\_{\mathsf{B}} \ll \ll E\_{\mathsf{N}}.\tag{A8}$$

However, it is recommended to include binding energy in some applications for accurate calculations to help explaining the fine structure of measured excitonic transitions.

### Appendix A.7.2. A Simple Explanation of How Nonlocality Emerges in the Excitonic Semiconductor

They key to the origin of nonlocality is the scenario when the excitation photon has an energy ¯*hω* that is *greater* than the minimum exciton energy (A7). In the case where

$$
\hbar\omega \sim \hbar\omega\_{\oplus} \tag{A9}
$$

the excess energy will be transformed into *kinetic energy*. Due to the conservation of momentum, the wave vector of the exciton is equal to the photon wave vector **k** and, hence, the exciton kinetic energy *E*kinetic e is given by

$$E\_{\rm e}^{\rm kinetic} = \frac{\hbar \mathbf{k} \cdot \hbar \mathbf{k}}{2m\_{\rm e}},\tag{A10}$$

where

$$m\_{\rm e} := m\_{\rm el} + m\_{\rm h} \tag{A11}$$

is the translational mass of the exciton in the effective-mass approximation [217]. Consequently, the total exciton energy *E*e is given by [228]

$$E\_{\rm e}(\mathbf{k}) = \hbar\omega\_{\rm e}(\mathbf{k}) = \hbar\omega\_{\rm e} + \frac{\hbar^2 k^2}{2m\_{\rm e}}.\tag{A12}$$

Consequently, the exciton frequency *<sup>ω</sup>*e(**k**) acquires a novel dependence on **k**, which is mainly due to the kinetic energy term in expression (A12). It is *precisely* such dependence that eventually leads to the emergence of electromagnetic nonlocality in semiconductors around excitonic resonances when photons couple with excitons. In other words, away from the excitonic transition regime, the effective dielectric function of the semiconductor exhibits only the typical dependence on *ω* (normal or temporal dispersion.)

*Appendix A.8. An Alternative Intuitive Derivation of the Dielectric Model* (90) *and the Quantum Origin of Nonlocality in Excitonic Semiconductors*

The model (90) itself may be intuitively derived as follows. A generic oscillator model is the one having the following well-attested Lorentzian expression:

$$\frac{1}{\omega\_{\text{\textquotedblleft}}^2 - \omega^2 - \text{i}\Gamma\omega}. \tag{A13}$$

This Lorentzian form models a large number of physical processes in nature, from lattice vibrations to electronic transitions and numerous many others [13,15,37,229]. Substituting the wave vector-dependent *ω*e expression (A12) into the above Lorentzian form (A13), the dielectric function formula (90) can be immediately obtained when we keep only quadratic terms of **k**. For a more careful quantum mechanical derivation, see [16,54,86].

### *Appendix A.9. Computation of the Inverse Fourier Transform* (114)

We start from the standard Fourier transform pair

$$\mathcal{F}\_{\mathbf{k}}^{-1}\left\{\frac{\chi}{k^2 - \gamma^2(\omega)}\right\} = \frac{\chi}{4\pi} \frac{e^{-i\gamma(\omega)|\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|}, \quad \text{Im}\{\gamma(\omega)\} < 0,\tag{A14}$$

where the spatial Fourier transform is defined by (17). The condition

$$\operatorname{Im}\{\gamma(\omega)\} < 0\tag{A15}$$

is due to the physical requirement that fields do not grow exponentially in passive domains [13]. We also have written |**r** − **r** | instead of |**r**| in anticipation of the fact that the inverse Fourier transform will produce a Green function.

Our main task now is to make a proper choice of the correct sign when performing the square root operation *γ*2. Let us write

$$
\gamma^2 = \text{Re}\left\{\gamma^2\right\} + \text{i } \text{Im}\left\{\gamma^2\right\}. \tag{A16}
$$

From (91), we have

$$\operatorname{Re}\left\{\gamma^2\right\} = \frac{m\_\text{e}^\star}{\hbar\omega\_\text{e}} \left(\omega^2 - \omega\_\text{e}^2\right), \text{ Im}\left\{\gamma^2\right\} = \frac{m\_\text{e}^\star}{\hbar\omega\_\text{e}}\omega\Gamma. \tag{A17}$$

On the other hand, *γ* also can take the form

$$\gamma(\omega) = \gamma'(\omega) + \mathrm{i}\gamma''(\omega),\tag{A18}$$

where both *γ* and *γ* are real. The goal now is to derive expressions for *γ* and *γ* in terms of Re0 *γ*2 1 and Im0 *γ*2 1 with the correct sign since the square root is a many-one function.

To accomplish this, we use the following elementary theorem: let *x*, *y*, *a*, *b* ∈ R. Then the square root of *x* + i*y* is given by

$$
\sqrt{\mathbf{x} + \mathbf{i}y} = \pm (a + \mathbf{i}b),
\tag{A19}
$$

where the following expressions hold

$$a = \sqrt{\frac{x + \sqrt{x^2 + y^2}}{2}}, \quad b = \frac{y}{|y|} \sqrt{\frac{-x + \sqrt{x^2 + y^2}}{2}}.\tag{A20}$$

Substituting (A17) into (A20), the following is obtained

$$a = \sqrt{\frac{m\_{\text{e}}^{\star}}{2\hbar\omega\_{\text{e}}}}\sqrt{\left(\omega^2 - \omega\_{\text{e}}^2\right) + \sqrt{\left(\omega^2 - \omega\_{\text{e}}^2\right)^2 + \left(\omega\Gamma\right)^2}}\tag{A21}$$

$$b = \sqrt{\frac{m\_{\text{e}}^{\star}}{2\hbar\omega\_{\text{e}}}} \sqrt{-\left(\omega^{2} - \omega\_{\text{e}}^{2}\right) + \sqrt{\left(\omega^{2} - \omega\_{\text{e}}^{2}\right)^{2} + \left(\omega\Gamma\right)^{2}}}.\tag{A22}$$

Here, we used the calculation

$$y / |y| = \operatorname{Im} \left\{ \gamma^2 \right\} / |\operatorname{Im} \left\{ \gamma^2 \right\}| = \omega \Gamma / |\omega \Gamma| = 1,\tag{A23}$$

which follows from the fact that *ω*, Γ > 0.

It remains now to find the correct signs. From (A14), the condition

$$
\gamma^{\prime\prime} = \text{Im}\{\gamma(\omega)\} < 0 \tag{A24}
$$

must be satisfied. Therefore, we choose the *negative* sign in (A19). The final expressions become *γ* = <sup>−</sup>*a*, *γ* = <sup>−</sup>*b*, and after inserting *γ* and *γ* into (A14), the required relation (114) is obtained.

#### *Appendix A.10. On Extending Definition 1 to Noncompact Regions*

The localization of the physics-based nonlocality microdomains estimated using the formula (121) is based on approximating the exact mathematical definition of the topological microdomain (Definition 1) by response kernel functions possessing spatial decaying exponential behavior as in (114). It might be advisable then to provide a modification of Definition 1 taking into account the noncompact setting, which is the scenario more often encountered in physical applications.

**Definition A1** (**Nonlocal Microdomains: The Noncompact Case**)**.** Consider a material domain *D* with the associated nonlocal response function **<sup>K</sup>**(**r**,**<sup>r</sup>**). Assume further that the following bound holds:

$$\left\|\left\|\mathbf{K}(\mathbf{r}',\mathbf{r})\right\|\right\| \leq A \exp\left(-a|\mathbf{r}-\mathbf{r}'|\right),\tag{A25}$$

for all **r** ∈ *D*, where *a* and *A* are some positive numbers. (Here, · is as in Definition 1). The form (A25) is called the *exponentially-decaying nonlocal kernel response function*. We define the effective (physics-based) nonlocal microdomain *V***r** ⊂ *D*, labeled by **r** ∈ *D*, as the open ball

$$\{\mathbf{r}' \in D\_{\prime} \: \left| \left| \overline{\mathbf{K}(\mathbf{r}', \mathbf{r})} \right| \right| < A \exp(-1) \},\tag{A26}$$

where *A* is as defined in (A25). In other words, an effective onlocality microdomain of this type, such as the one in Definition 1, is still locally compact.

**Remark A1.** The physical meaning of the ball (A26) is that it effectively approximate the spatial region where most of the "energy" of the response is concentrated, hence providing a physical means to estimate practical microdomain physical problems, since the Coulomb

interaction and other types of molecular forces are long range forces and, hence, cannot be described by an exact compact support such as the one originally introduced in the Definition 1.

**Remark A2.** It is straightforward to modify Definition A1 to include other forms of decay functions weaker than the exponential form. For example, we may replace the exponential in (A25) by a decaying function 1/*rn*, where *n* is a suitable positive integer. Obvious modifications of the ball (A26) can then be subsequently made to construct the nonlocality microdomain.

### *Appendix A.11. Possible Applications of the Superspace Formalism to Fundamental Methods in Metamaterials Research*

### Appendix A.11.1. Estimating Fundamental Limitations on Nonlocal Metamaterials

Fundamental continuum response maps, such as L (72), can be completely reformulated in a different setting, that of the space of *vector bundle sections* [57,62,70]. The latter topic, the theory of sections, is an extremely well-developed subject in mainstream differential topology. In fact, in some cases the electromagnetic response field function **<sup>R</sup>**(**r**) itself may be obtained by working directly with the source bundle superspace M. For example, under some conditions, this can be achieved by replacing each fiber *Xi* by *Xi* × C3. In this way, the entire nonlocal response problem reduces to understanding how vector bundle sections interact with the topology of the underlying base manifold D. There is a large literature in differential topology and geometry focused on this latter technical mathematical problem, especially how local information can be transported from one place to another in order to extend local structures into global ones [26,57,63].

The author believes that by starting from local data in a given nonlocal metamaterial domain, e.g., the global shape of the device, the distribution of topological holes, etc, one may then use existing techniques borrowed from differential topology, e.g., the theory of characteristic classes, to determine allowable EM response functions that are in principle permissible at the global level. Engineers are typically interested in acquiring in advance the knowledge of what the best (or worst) performance measures obtainable from specific topologies are. Hence, reformulating the electromagnetism of nonlocal metamaterials in terms of vector bundles could be of help in this respect since it opens a pathway, within metamaterials research, toward a synergy between general topology, physics, and engineering.

### Appendix A.11.2. Numerical Methods

Traditional full-wave numerical methods are sometimes deployed in order to deal with nonlocal EM materials, often using the additional boundary conditions framework, in spite of the latter's lack of complete generality.<sup>31</sup> At the heart of the traditional approach to numerical methods in local electromagnetism is the concept of operators between linear spaces. However, by reformulating the source space of field–matter interaction in terms of a Banach bundle, it should be possible to reformulate Maxwell's equations to act on this extended geometric superspace instead of the conventional spacetime framework. As an alternative to the concept of the linear operator of classical mathematical physics and numerical methods, we now have the much more general and richer concept of bundle homomorphism developed in Section 5. Some of the advantages anticipated from such reformulation include


interaction regime is explicitly encoded into the *geometry* of the new expanded solution superspace M itself. Characterizing this geometry is then possible through a suitable discrete approximation of the *interior* microtopological content of the superspace (fiber bundle) structure itself; i.e., not just at the "exterior" parts often found in the boundary conditions of classical local field continuum theories, but also "going inside" the problem space as such.

3. It is also possible that such numerical methods may emerge as more computationally efficient and broader in applicability than the conventional methods rooted in local electromagnetism. One reason for this is that the Banach vector bundle formulation introduced in this paper is quite natural and appears to reflect the underlying physics of nonlocal metamaterials in a direct manner. From our general experience in numerical methods, "natural operations" tend to translate into numerical methods with better convergence, sensitivity, and robustness.

As directly related to the three possible advantages of the superspace formalism discussed above, we also add that in recent years the subject of *computational topology* has gained momentum, where some researchers are now building new numerical methods by exploiting the topological structure of the problems under considerations, e.g., see [72,230]. The fiber bundle superspace formalism of this paper might provide a way to link research done in electromagnetic and non-electromagnetic nonlocal materials with such advances in the computational and applied mathematical sciences.

### Appendix A.11.3. Topological Photonics

One of the main applications of the proposed vector bundle formalism is that it opens the door for a new way to investigate the topological structure of materials. It has already been noted that the nonlocal EM response is essential in topological photonics, e.g., see [11,161] and also Appendix A.1. Indeed, since in topological photonics the wave function of bosons, usually the Bloch state, is examined over the entirety of momentum space (usually the Brillouin zone), then it is the dependence of the EM response on **k** what is at stake, which naturally brings in nonlocal issues. But now since by using our theory we can associate with every nonlocal material a concrete fiber bundle superspace reflecting the rich information about the multiscale topological microdomain structure and the global shape of the material plus the impact of the boundaries separating various material domains, it is natural to examine whether a topological classification of the corresponding fiber bundles may lead to a new way to characterize the topology of materials other than the Chern invariants used extensively in literature. The advantage of the superspace approach in this case is that the complicated topological and geometrical aspects of the boundaries and inhomogeneity in nonlocal media can be encoded very efficiently in the local structure of the material fiber bundle. Using standard techniques in differential topology [62], it should be possible to propagate this local information to the global domain (the entirety of the system), for example by computing suitable fiber bundle topological invariants like its homology groups [70]. Our approach is then a "dual" to the standard approach since we work on an enlarged configuration space (spacetime or space–frequency), while the mainstream approach operates in the momentum space of the wave function.
