*4.1. Introduction*

In this section, we begin our careful examination of the mode of interrelation between the physics- and topology-based types of nonlocality introduced and discussed above.<sup>5</sup> Let the nonlocality domain of the electromagnetic medium, the region *D* ⊂ R<sup>3</sup> in (11), be bounded. Corresponding to (1), a similar operator equation in the frequency domain

representing the most general form of a nonlocal electromagnetic medium can be posited, namely

$$\mathbf{R}(\mathbf{r};\omega) = \mathcal{L}\_{\omega} \{ \mathbf{F}(\mathbf{r};\omega) \},\tag{20}$$

where the nonlocal medium linear operator is itself frequency dependent. For simplicity, and as stated before, whenever it is understood from the context that the material response operator is formulated in the frequency domain, all dependencies on *ω* appearing in its formal expression will be removed.

We are going to propose a change in the mathematical framework inside which electromagnetic nonlocality is usually defined. This will be done in two stages:


The key conceptual idea behind the entire theory presented here is that of the *topological microdomains* associated with the field theory of nonlocal continua, e.g., the electromagnetism of continuous media, which we first develop thematically in the next Section 4.2 before moving subsequently to the more rigorous and exact topological formulation of Section 5.

#### *4.2. The Concept of Topological Microdomains in Nonlocal Continuum Field Theories*

In conventional frequency domain local electromagnetism, the boundary-value problem of multiple domains is formulated as a set of coupled partial differential equations or integro-differential equations interwoven with each other via the appropriate intermaterial interface boundary conditions dictating how fields change while crossing the various spatial regions inside which the equations hold [28,40,41]. This has been traditionally achieved by taking up the electromagnetic response function **<sup>K</sup>**(**<sup>r</sup>**,**r** ; *ω*) as an essential key ingredient of the problem description, which traditionally has been exploited in two stages: First, the constitutive relations would enter into the governing equations in each separate solution domain. Second, the constitutive relations themselves are used in order to construct the proper electromagnetic boundary conditions prescribing the continuity/discontinuity behavior of the sought field solutions as they move across the various interfaces separating domains with different material properties.

Unfortunately, it has been well known for a long time that it is not possible to formulate a universal electromagnetic boundary condition for nonlocal media, especially for the case of spatial dispersion. This will be discussed later with more details in Section 8, but also see the discussion around additional boundary conditions (ABC) in Appendix A.1. For now, we concentrate on gaining a deeper understanding of the generic structure of spatial nonlocality in continuum field theories.

Consider the microdomain structure depicted in Figure 2. A key starting observation is how nonlocality forces us to associate with every spacetime point (**r**, *t*), or frequency space point (**r**; *<sup>ω</sup>*), a *topological neighborhood* of **r**, say *V***<sup>r</sup>**, such that **r** ∈ *V***<sup>r</sup>**. For now, let us assume that the spatial material domain *D* **r** is just an *open set* in the technical sense of the topology of the Euclidean space R<sup>3</sup> inherited from the standard Euclidean metric [59]. By restricting *D* to be open, we avoid the notorious problem of dealing with boundaries or interfaces between such (possibly overlapping) open sets. That is, the topological closure

of *D*, denoted by cl(*D*), is excluded from the domain of nonlocality. Let *D* be the maximal such topological neighbored for the problem under consideration.<sup>6</sup> We now associate with each point **r** a "smaller" open set *V***r** where the following holds:

$$\left| \forall \mathbf{r} \in D, \,\, \exists \, V\_{\mathbf{r}} \subset D \text{ such that } \mathbf{r} \in V\_{\mathbf{r}} \text{ and } \,\, V\_{\mathbf{r}} \text{ is open.} \right| \tag{21}$$

Note that the assumed openness of *D* makes the above construction technically possible. We will call the proposition (21) the *principle of nonlocal microdomain generation*. It formally captures the main content of the structure of nonlocality at the microscopic level. In Section 7, a practical example taken from nonlocal semiconductor metamaterials will be investigated in depth in order to illustrate the applicability of (21).

**Figure 2.** The microtopological structure of nonlocal metamaterial systems includes more than just the three-dimensional spatial domains D*<sup>n</sup>*, *n* = 1, 2, ... It is best captured by classes V(<sup>D</sup>*n*) composed of various open sets *V***r** ⊂ D*n* based at each point **r** ∈ D*<sup>n</sup>*. On every such subset a vector field is defined, representing the external field excitation field. The collection of all vector fields on a given set *V***r** gives rise to a linear topological function space F(*<sup>V</sup>***r**). The topologies consisting of the base spaces D*<sup>n</sup>*, the nonlocal microdomains *V***<sup>r</sup>**, and the function spaces F(*<sup>V</sup>***r**), collectively give rise to a total "macroscopic" topological structure (superspace) that is considerably more complex than the base spaces D*<sup>n</sup>*.

Now, instead of considering fields like **<sup>R</sup>**(**r**) and **<sup>F</sup>**(**r**) defined on the entire maximal domain of nonlocality *D* (which can grow "very large") we propose to reformulate the problem of nonlocal continua as a *topologically local*<sup>7</sup> structure by exploiting the fact that the physics of field–matter interactions gives the field response at location **r** due to independent excitation fields essentially confined within a "smaller domain" around **r**, namely the open set *<sup>V</sup>***<sup>r</sup>**.<sup>8</sup>

Furthermore, if the response at another *different* point **r** = **r** is needed, then a *new*, generally different, "small" open set *V***r** will be required. That is, in general we allow that

$$V\_{\mathbf{r}} \neq V\_{\mathbf{r'}} \tag{22}$$

even though it is expected that typically there should be some overlap between these two small local domains of electromagnetic nonlocality in the sense that

$$V\_{\mathbf{r}} \cap V\_{\mathbf{r'}} \neq \bigotimes \tag{23}$$

especially if the nonlocality radius |**r** − **r**| is small.

The following fundamental collection of "smaller" sets, where a metric scale characterizing "smallness" is not implied, written as

$$\{V\_{\mathbf{r}}, \mathbf{r} \in D\},\tag{24}$$

will be dubbed *nonlocal microdomains*, or just *microdomains* in short. A possible precise definition is given next.

**Definition 1** (**Nonlocal microdomains: the physics-based scenario**)**.** Consider a material domain *D* with the associated nonlocal response function **<sup>K</sup>**(**r** ,**<sup>r</sup>**). We define the (physicsbased) nonlocal microdomain *V***r** ⊂ *D*, labeled by **r** ∈ *D*, as the interior of the compact<sup>9</sup> support of **<sup>K</sup>**(**r** ,**<sup>r</sup>**). The support itself is defined by the standard formula

$$\text{supp}\,\mathbb{K}(\mathbf{r}',\mathbf{r}) := \text{cl}\_D\{\mathbf{r}' \in D\_\prime \, \left\|\, \mathbb{K}(\mathbf{r}',\mathbf{r})\right\| \neq 0\},\tag{25}$$

where · is a suitable tensor norm, for example the matrix norm.<sup>10</sup> The topological closure operator cl*D* is here taken with respect to the total material space *D* where the latter is viewed as a topological space on its own.

**Remark 5** (**Microdomain topology**)**.** By Definition 1 above, the nonlocal microdomain *V***r** is always open. It can be shown that the collection of open sets {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ *D*} induces a topology on the total space occupied by the nonlocal material (the details are omitted since they are lengthy though straightforward.) In what follows, this topology will be referred to by the term *microdomain topology*. The set of physics-based nonlocality microdomains (microdomains for short), as constructed in Definition 1, explicate the fine microtopological structure of nonlocal electromagnetic domains at a spatial scale different from that of the (topologically "larger") material domain *D* itself and are fundamental for the theory developed in this paper.

**Remark 6** (**Discrete topology in local continua**)**.** In local media, the microdomains topology reduces to the trivial discrete topology

$$\{\{\mathbf{r}\}, \mathbf{r} \in D\} \tag{26}$$

since the external field interacts only with the point **r** at which it is applied and hence

$$V\_{\mathbf{r}} = \{\mathbf{r}\}\tag{27}$$

holds as the "smallest" possible topological microdomain in that rather special case. Therefore, the microdomain topology is interesting only for the case of physics-based nonlocality, e.g., the scenario of EM microdomains discussed in more details in the examples and applications below. In particular, from the point of view of this article, *local* metamaterials are not topologically interesting.

### *4.3. Construction of Excitation Field Function Spaces on the Topological Microdomains of Nonlocal Media*

After enriching the MTM domain *D* with the finer topology of nonlocality microdomains *V***<sup>r</sup>**,**<sup>r</sup>** ∈ *D*, we wish to equip this total medium with additional mathematical structure based on the physics of field–matter interaction. Consider the set of all sufficiently differentiable vector fields **<sup>F</sup>**(**r**) defined on *V***<sup>r</sup>**, **r** ∈ *D*. This set possesses an obvious complex *vector space structure*: for any two complex numbers *a*1, *a*2 ∈ C, the sum

$$a\_1\mathbf{F}\_1(\mathbf{r}) + a\_2\mathbf{F}\_2(\mathbf{r})$$

is defined on *V***r** whenever **<sup>F</sup>**1(**r**) and **<sup>F</sup>**2(**r**) are, while the null field plays the role of the origin. In what follows, we will denote such function spaces by F(*<sup>V</sup>***r**) or just F if it is understood from the context on which material spatial domains the fields are defined.

**Remark 7** (**The excitation field function space and Sobolev spaces**)**.** It is possible to equip F(*<sup>V</sup>***r**) with a suitable topology in order to measure how "near" to each other are any two fields defined on *V***<sup>r</sup>**, e.g., see [59,62,63]. Therefore, in this manner F(*<sup>V</sup>***r**) acquires the structure of a *topological vector space* [59]. In particular, it can be made a *Sobolev space*, where the latter is not only a Banach space (normed space), but also a Hilbert space (inner product space) [64–66].

The detailed construction of a Sobolev space on a given microdomain is not needed for what follows in this paper, but can be found in the literature, including the references quoted in this Remark 7.

#### *4.4. The Global Topological Structure of Nonlocal Electromagnetic Material Domains: First Look*

In light of the analysis above, each microdomain *V***r** induces an infinite-dimensional linear function space (Sobolev space) F(*<sup>V</sup>***r**) indexed by the position **r** ∈ *D*, with the corresponding topology being essentially determined by the geometry of *V***<sup>r</sup>**. On the other hand, this latter geometry is obtained from the *physics* of field–matter interaction in nonlocal media. Consequently, the physical content of nonlocal materials is encoded at the level of the topological microstructure encapsulated by the following formal scheme:

$$D \ni \mathbf{r} \xrightarrow[\text{Physics-based Nonlocality}]{\text{Physical Data}} V\_{\mathbf{r}} \xrightarrow[\text{Sobolev Space}]{\text{Mathematical Data}} \mathcal{F}(V\_{\mathbf{r}}) \tag{28}$$

Let us first identify the main relevant collections of subsets needed in order to understand the formal set-theoretic structure of the problem. We begin by

$$\mathcal{V}(D) := \{ V\_{\mathbf{r}} \subset D \mid \mathbf{r} \in D, V\_{\mathbf{r}} \text{ is open} \}, \tag{29}$$

as the class of physics-based nonlocal microdomains (Definition 1). On the other hand, it is also possible to introduce the useful construction

$$\mathcal{G}[\mathcal{V}(D)] := \{ \mathcal{F}(V\_{\mathbf{r}}) \, | \, \mathbf{r} \in D, \mathcal{F}(V\_{\mathbf{r}}) \text{ is a Sobolev function space} \}, \tag{30}$$

as a convenient class into which we collect all the function spaces of excitation fields on each nonlocality microdomain *V***r** as spanned by the position index **r** ∈ *D* (see Remark 7 for the construction of each such function space.) It follows then that (28) can be neatly captured by the ordered triplet

$$D \times \mathcal{V}(D) \times \mathcal{G}[\mathcal{V}(D)]. \tag{31}$$

We wish now to unpack this compact structure in a careful, step-by-step manner, proceeding as follows:


 3. We further emphasize that the various sets *V***r** ∈ V(*D*) constitute an open cover of *D*, that is, we have

$$D = \bigcup\_{\mathbf{r} \in D} V\_{\mathbf{r}}.\tag{32}$$

In this way, the model can accommodate excitation fields **<sup>F</sup>**(**r**) applied at every point in **r** ∈ *D*.


**Remark 8** (**Topology, physics, and multiple scales**)**.** It is interesting to observe how, within the framework proposed above, some sort of delicate constructive "division of labor" is seen to emerge into the picture, where a fruitful interaction between physics and mathematics generates the various required multiscale topological microstructures characteristic of nonlocality in continuum field theories. This is also the source of some potential difficulties hidden in the formal set-theoretic structure (31). Indeed, we will next try to smooth out the differences between the two main substructures V(*D*), which is principally controlled by physics, on one side, and G[V(*D*)], which is dominated by purely mathematical considerations. One way to achieve a resolution of this philosophical tension between the physical and mathematical is by developing the entire theory of the set-theoretic structure (31) in a form that can encode all of its main substructures within a single, rich enough "metastructure": the Banach vector bundle *superspace* (see Section 5 for the detailed construction).

As can be seen from Remark 8, there is indeed some strong motivation to search for alternative formulations of physical theory in complex and rich systems such as nonlocal material continua, where there exists multiple spatial topological scales. It will be seen that the superspace theory appears to provide some form of rare direct and transparent unity between physics and topology in this regard. In order to reach there, gradual, step-by-step changes in the conventional formulation of continuum field theory will be introduced. We now begin to look into such a reformulation, starting with a straightforward one.

### *4.5. A Reformulation of the Nonlocal Continuum Response Function*

It is now possible to provisionally construct the nonlocal continuum response function by working on the fundamental topological domain structure (31) instead of the global domain *D*, the later being the favored arena of conventional continuum field theory that we would like to ultimately move beyond. Again, for concrete expressions, the special case of electromagnetic theory will be presupposed but it should always be kept in mind that the mathematical structure of the theory is quite general and applies to all nonlocal continuum field theories governed by an abstract material response function model, such as the one discussed in Section 3.

We start by noting that the response field **<sup>R</sup>**(**r**) can be re-expressed by the map

$$\mathbf{R}: D \times \mathcal{V}(D) \to \mathbb{C}^3,\tag{33}$$

where the codomain is taken to be C<sup>3</sup> because the electric or magnetic response functions **D** or **B**, respectively, are complex vector fields in the frequency domain.<sup>11</sup> The value of the EM nonlocal response field due to excitation field **<sup>F</sup>**(**r**) applied at a microdomain *V***r** can be computed by means of

$$\mathbf{R}(\mathbf{r};\omega) = \int\_{V\_{\mathbf{r}}} \mathbf{d}^3 r' \,\mathbf{K}(\mathbf{r}, \mathbf{r}'; \omega) \cdot \mathbf{F}(\mathbf{r}'; \omega) . \tag{34}$$

Although (34) may appear at first sight to be only slightly different from (11), the underlying difference between the two formulas is significant. In essence, the construction of the EM response field **<sup>R</sup>**(**r**) via the map (33) amounts to *topological localization* of electromagnetic nonlocality, since in the latter case, the EM response function **<sup>K</sup>**(**<sup>r</sup>**,**r**; *ω*) is no longer allowed to extend globally onto "large and complicated material domains." Indeed, with the recipe (34) only the response to "small"–or more rigorously *topologically local*12—domains, namely the microdomains *V***<sup>r</sup>**, is admitted. On the other hand, in order to find the response field **<sup>R</sup>**(**r**) everywhere in *D*, one needs to use sophisticated topological techniques to extend the response from one point to another until it covers the entirety of *D*. This local-to-global extension application of differential topology is discussed in detail in Section 5.3 and again briefly in Appendix A.11.

In such a manner, it becomes possible to provide an alternative, more detailed explication of the behavior of the medium at topological interfaces (boundary conditions in

nonlocal metamaterials are treated–provisionally–in Section 8) and also explore the effect of the topology of the bulk medium itself on the allowable response functions and the production of non-trivial edge state, with obvious applications to emerging areas such as nonlocal metamaterials.<sup>13</sup>

### **5. The Fiber Bundle Superspace Formalism in the Field Theory of Generic Nonlocal Continua**

Here, an outline of the direct construction of a fiber (Banach) bundle over an entire (global) nonlocal generic material domain is given, where our purpose is to attach to every point **r** ∈ *Ui* a *fiber* F*i*, actually a vector space in our case. The contents of this section are the most technically advanced in this paper. Readers interested in applications may skim through Sections 5.1 and 5.2, skip Section 5.3, then move directly to Section 6 for a general summary of the fiber bundle algorithm. Concrete computational models are outlined in Section 7 using a practical nonlocal model, while additional remarks and discussions about current and future uses of the theory are provided in Appendixes A.3 and A.11. However, even readers not fully familiar with the differential manifold theory will benefit from reading the present technical section, because we strive to illustrate the physical intuition behind the various mathematical computations and steps therein.

#### *5.1. Preparatory Step: Promoting the Material Domain D to a Manifold* D

In order to investigate in depth the fundamental physico-mathematical constraints imposed on nonlocal continua, the domain *D*, which we have working with so far as the main total spatial space of the material, should be upgraded in complexity to the higher level of a *differential manifold*, the latter which posses a quite rich and sophisticated structure that allows performing calculus and geometrical reasoning simultaneously [26,57,62,63,68]. There are several reasons why this is highly desirable:


For all these reasons, it is desirable to strive to furnish the domain *D* with the most general and flexible mathematical apparatuses available to us, which, in this case, amounts to equipping the material/metamaterial spatial domain with a *smooth manifold structure*.

We quickly illustrate how this can be accomplished. If we denote by D a three-manifold (three-dimensional smooth manifold), then, since *D* ⊂ R3, there is a natural differential structure defined on *D*, inherited from the ambient three-dimensional Euclidean space itself. (Throughout this paper, such differential three-manifold structure will be presupposed as the de facto space for the *total*, i.e., largest, material space.)

Following the standard theory of smooth manifolds, let

$$(\mathcal{U}\_{i\prime} \phi\_i) \tag{35}$$

be a countable collection of charts (an atlas), labeled by

$$
\lambda \subset I \subset \mathcal{N},
\tag{36}
$$

where *I* is an index set. Together, the devices (35) and (36) can equip D ⊂ R<sup>3</sup> with a differential three-manifold structure. For simplicity, we will refer to the points of the manifold D by **r**, i.e., using the language of the global (ambient) Euclidean space R3. Symbolically, by adding a differential manifold structure, we effected the transformation

*i*

$$D \xrightarrow[\text{Introduction of \"differential\" amplitudes\" }]{\text{Inset of smooth manifold structure}} \mathcal{D} \tag{37}$$

This well-known construction [26,57,63] constitutes the differential atlas on D, which will be used in what follows.

#### *5.2. Attaching Fibers to Generic Points in the Nonlocal Material Manifold D*

Our current goal is to attach a *vector fiber* (a linear function space in this case) at every point **r** ∈ D, namely the function space F(*<sup>V</sup>***r**) introduced in Section 4.3. It turns out that accomplishing this requires finding suitable "compatibility laws" dictating how coordinates change when two intersecting charts *Ui* and *Uj* interact with each other, which is typical in such types of constructions [26]. In particular, we will need to later find the law of mutual transformation of vectors in the fibers <sup>F</sup>(*<sup>V</sup>φi*(**r**)) and <sup>F</sup>(*<sup>V</sup>φj*(**r**)). Here, the expression

$$\mathcal{F}\left(V\_{\phi\_i(\mathbf{r})}\right) \tag{38}$$

means the fiber space attached to the point whose coordinates are *φi*(**r**), i.e., the *function* space where all functions are expressed in terms of the language of the *i*th chart (*Ui*, *φi*(**r**)).

In this connection, the major technical problem facing us is a mathematical one induced by the physics of the situation. We first isolate and describe the main problem by the following brief technical resume:

*Since the differential structure associated with charts*

$$\{ (\mathcal{U}\_{i\prime}\phi\_i(\mathbf{r})), i \in I\_{\prime} \}$$

*can be fixed by essentially mathematical considerations alone, while the collection of microdomains*

$$\mathcal{V}(\mathcal{D}) = \{V\_{r\prime} \: r \in \mathcal{D}\}$$

*is solely determined by the physics of electromagnetic nonlocality (See Remark 3 and Section 4), there is no direct and simple way to determine and express the vector transformation*

$$\mathcal{F}(V\_{\phi\_{\vec{\imath}}(r)}) \longrightarrow \mathcal{F}(V\_{\phi\_{\vec{\jmath}}(r)})\_{\prime}$$

*because several different coordinate patches other than Ui and Uj, belonging to the differential three-manifold* D *atlas, might be involved in geometrically building up the microdomain V<sup>r</sup>.*

The above technical problem will be solved in Section 5.3 by using the technique of *partition of unity* borrowed from differential topology [26,57,62]. It will allow us to split up each full microdomain *V***r** into several suitable sub-microdomains (details below), which can be later joined up together in order to give back the original EM nonlocality microdomain *V***<sup>r</sup>**.

For now, we start by recalling that the microdomain structure represented by the collection V(D) = {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ D} is an *open cover* of the manifold D. Therefore, and since the

material domain manifold D possesses a *countable* topological base [59], it contains a locally finite open cover subordinated to V(D) [26,57].<sup>15</sup> This implies that an atlas (*Ui*, *φi*), *i* ∈ *I*, with diffeomorphisms

$$
\phi\_i \colon \mathcal{U}\_i \to \mathbb{R}^3,\tag{39}
$$

describing the differential structure of the manifold D exists such that the elements {*Ui*, *i* ∈ *I*} constitute the above mentioned locally finite subcover *subordinated* to the microdomains collection V(D). Moreover, the images *φi*(*Ui*) are *open balls* centered around 0 in R<sup>3</sup> with finite radius *a* > 0 (henceforth, such balls will be denoted by *Ba*) [26].

In this way, the physics-based open cover set V(D) provides a first step toward the construction of a complete *topological* description of the *physics-based* nonlocal microdomain structure. The reason is that the coordinate patches (*Ui*, *φi*), *i* ∈ *I*, are *subordinated* to the microdomains {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ D} [26].

It is also known that there exists a *partition of unity* associated with the D-atlas (*Ui*, *φi*), *i* ∈ *I*, constructed above summarized by the following lemma [26,57,62,63,68]:

**Lemma 1** (**Partition of Unity**)**.** There is a collection of functions

$$
\psi\_i: \mathcal{U}\_i \subset \mathcal{D} \to \mathbb{R} \tag{40}
$$

satisfying the following requirements:


$$\text{supp}\,\forall\_i \subset \mathcal{U}\_{i\prime} \tag{41}$$

holds. Recall that the support is defined as the (topological) closure of the set

$$\{\mathbf{r} \in \mathcal{D} \, | \, \psi\_i(\mathbf{r}) \neq 0\}. \tag{42}$$

See for example [30,68,69].


$$\sum\_{i \in I\_{\mathbf{r}}} \psi\_i(\mathbf{r}) = 1,\tag{43}$$

where the sum is always convergen<sup>t</sup> because the set *I***r** is finite.

**Remark 9.** It can be shown that the sets

$$(\phi\_i^{-1}(B\_{a/3}), i \in I) \tag{44}$$

where *Ba*/3 is a standard Euclidean ball centered at the origin with radius *a*/3, already cover D [57]. Moreover, the closure

$$\text{cl}\{\phi\_i^{-1}(B\_{a/3})\}\tag{45}$$

may be taken to constitute the support of *ψi*(**r**), while [26,57,70]

$$\mathbf{r} \notin \text{supp}\{\psi\_i(\mathbf{r})\} \implies \psi\_{\bar{i}}(\mathbf{r}) = 0. \tag{46}$$

The partition of unity functions *ψi* can be computationally constructed using standard methods, most prominently the bump functions, see [57,71] for details.

The motivation behind the deployment of the partition of unity technique and how it immediately arises in connection with our fundamental EM nonlocal structure should now be clear. We have found that the following three-step process is natural:

1. Initially, the *physics*-based collection of sets

$$\mathcal{V}(\mathcal{D}) = \{V\_{\nu} \mathbf{r} \in \mathcal{D}\}\_{\nu}$$

for example, the EM nonlocal microdomain structure based on each point **r** in the nonlocal metamaterial D, is obtained using a suitable physical microscopic theory or some other procedure.<sup>17</sup>

2. Introduce a differential atlas

$$(\mathcal{U}\_{i\prime}\phi\_i(\mathbf{r})), \ i \in I\_\star$$

on the smooth manifold D subordinated to V(D) and representing the nonlocal material domain under consideration.

3. Finally, the same atlas is linked to a set of functions *ψi*(**r**) (partition of unity) that can be recruited as "topological bases" in order to expand any differentiable field excitation function into sum of individual sub-fields defined on open subsets of the material domain D (see Section 5.3).

The three-step process outlined above is summarized in Figure 3, illustrating how to progressively construct micro-coordinate systems allowing one to see through increasingly smaller spatial scales in the fundamental characterization of electromagnetic material nonlocality.

**Figure 3.** The three-step process of constructing micro-coordinate representations of material nonlocality starting from the nonlocal microdomain set and ending with the partition of unity on the material continuum's superspace.

The key idea to be developed next is that both the base manifold D and the nonlocal physics-based microdomains *V***r** are described *locally* (in the topological sense18) by the *same* collection of charts, namely (*Ui*, *φi*(**r**)), *i* ∈ *I*. This will permit us to construct a direct unified description of *both* the base manifold D *and* its fibers, i.e., the linear topological function spaces F(*<sup>V</sup>***r**), the latter being the model of the physical electromagnetic fields exciting the nonlocal material D.

The construction of a fiber bundle superspace for nonlocal electromagnetic materials will be completed in two steps:


We start with Step I, while we leave the more complicated Step II to Section 5.3.

Consider the (*Ui*, *φi*(**r**)), *i* ∈ *I*, as our atlas on the three-manifold D introduced in Section 5.1. At each point **r** ∈ *Ui*, we attach a linear topological space F(*Ui*) defined as the Sobolev space

$$\mathcal{W}^{p,2}(\mathcal{U}\_i),\ p \ge 1,\tag{47}$$

of functions on the open set *Ui*, i.e., we write

$$\forall i \in I, \quad \mathcal{F}(L\_i) := \{\psi\_i(\mathbf{r})\mathbf{F}(\mathbf{r}), \mathbf{r} \in lL\_i, \text{ is in the Sobolev space } \mathbb{W}^{p,2}(\mathcal{U}\_i)\},\tag{48}$$

where **<sup>F</sup>**(**r**) is a suitable *Cp*,<sup>2</sup> vector field.

**Remark 10** (**Sobolev Spaces**)**.** For the precise technical definition of the infinite-dimensional Sobolev function space *Wp*,<sup>2</sup>(*Ui*), see [64,65]. Appendix A.5 provides some additional information on the literature. Section 4.3 gives a simplified intuitive definition of the physics-based function space F*i*, in particular see Remark 8. The intricate details of the theory of such Sobolev function spaces will not be needed for our immediate purposes in what follows (compare with Remark 11).

Physically, the multiplication of the global excitation field **<sup>F</sup>**(**r**) by *ψi*(**r**) in constructions like (48) above and (50) below effectively "localizes" (in the topological sense) the field into a smaller compact subdomain, namely the support of the "topological localization basis function" *ψi*(**r**) itself. Moreover, because the *Cp*-functions *ψi*(**r**) have *compact* supports satisfying the inclusion restrictions

$$\text{supp}\{\psi\_i\} \subset \mathcal{U}\_{i\prime} \quad i \in I\_{\prime} \tag{49}$$

it follows that F(*Ui*) is effectively a *local* Sobolev space on *Ui* [66]. Alternatively, it is also possible to seek different constructions, such as the one captured by the following remark.

**Remark 11.** We may define a less complicated function space on *Ui* using the following construction:

$$\forall i \in I, \quad \mathcal{F}'(Ll\_i) := \{\psi\_l(\mathbf{r})\mathbf{F}(\mathbf{r}), \mathbf{r} \in Ll\_i, \text{is an element of a } \mathbb{C}^p \text{sup-norm function space}\}, \tag{50}$$

where the *Cp*-sup-norm is defined by

$$<\langle |\psi\_i(\mathbf{r})\mathbf{F}(\mathbf{r})|\rangle\rangle := \sup\_{\mathbf{r}\in\text{supp}} \{\psi\_i\} \left[\psi\_i(\mathbf{r})\mathbf{F}(\mathbf{r})\right].\tag{51}$$

In the case of F(*Ui*), one may further consider only *Cp*-vector excitation fields **<sup>F</sup>**(**r**). A choice of which linear function space to work with depends on the particular application under consideration. In what follows, we further simplify our notation by writing F*i* instead of F(*Ui*) whenever the partition of unity's differential atlas' coordinate patches *Ui* are used.

### *5.3. Direct Construction of Bundle Homomorphism as Generalization of Linear Operators in Electromagnetic Theory*

We now demonstrate how the material constitutive relations in conventional (local) continuum theory may be absorbed into a new structure, the *bundle homomorphism*, which is the most natural generalization of linear operators in local electromagnetism taking us into the enlarged stage of the generic nonlocal medium's superspace formalism. In the future, these bundle homomorphisms may be discretized using topological numerical methods, e.g., see [72]. In what follows, we focus on the rigorous exact construction using the technique of partition of unity, which allows computations going from local to global domains.<sup>19</sup>

5.3.1. The Basic Definition of the Nonlocal Material, (or Continuum, Metamaterial (MTM), etc.), Banach (Fiber) Bundle Superspace

The initial step in formally defining the proposed nonlocal MTM bundle superspace is the following disjoint union construction:

**Definition 2** (**Preliminary Definition of the Bundle Superspace**)**.** Let the material continuum's superspace be denoted by M, which is also called the *total bundle space*. We define this space as the disjoint union of all spaces F*i* of the form:

$$\mathcal{M} := \{ (\mathbf{r}, \mathcal{F}\_i) \, | \,\forall i \in I, \mathbf{r} \in \mathcal{U}\_i \subset \mathcal{D} \}. \tag{52}$$

Associated with M is a surjective map

$$
v: \mathcal{M} \to \mathcal{D},
\tag{53}$$

which "projects" the fiber onto its corresponding point in the base manifold D, i.e., *p*((**<sup>r</sup>**, F)) := **r**.

**Remark 12** (**Other constructions of bundle spaces**)**.** In mainstream literature, the fiber bundle concept is often approached in a manner slightly differently from that of Definition 2. Indeed, the *fiber* of M at **r** ∈ D is *defined* as the set *<sup>p</sup>*<sup>−</sup><sup>1</sup>(**r**), but provided the map *p* is already given as part of the bundle's initial data. However, in this paper, we *construct* the bundle data starting with the physics-based topological structure (31).

**Remark 13** (**Fiber Projections and Local Isomorphisms**)**.** The map *p* is called the *projection* of the vector bundle M onto its *base space* D. Moreover, from now on, we will also use the notation F**r** to denote the fiber *<sup>p</sup>*<sup>−</sup><sup>1</sup>(**r**). By construction, it should be clear that

$$\forall i \in I: \quad p^{-1}(\mathbf{r}) = \mathcal{F}\_i \iff \mathbf{r} \in \mathcal{U}\_i. \tag{54}$$

From the topological viewpoint, the material continuum superspace M manifests itself *locally* as a product space in the form

$$
\mathcal{U}\_i \times \mathcal{F}\_i. \tag{55}
$$

In other words, the map *p* should behave locally as a conventional projection operator; i.e., in a local domain *Ui*, the material's total bundle space M is isomorphic to *Ui* × F*i*, and *p*(*Ui* × F*i*) should be isomorphic to *Ui*. Symbolically, we have:

$$\mathcal{M} \cong\_{\text{locally}} \mathcal{U}\_{i} \times \mathcal{F}\_{i}, \quad p(\mathcal{U}\_{i} \times \mathcal{F}\_{i}) \cong\_{\text{locally}} \mathcal{U}\_{i}, \tag{56}$$

for all *i* ∈ *I*, and where ∼=locally means local topological (in this case also smooth) isomorphism.<sup>20</sup>

In order to complete the specification of the nonlocal material continuum superspace, we next construct the linear function space *Xi* defined by

$$\forall i \in I, \ X\_i := \left\{ \psi\_i \left[ \phi\_i^{-1} (\mathfrak{X}) \right] \mathbf{F} \left[ \phi\_i^{-1} (\mathfrak{X}) \right] \text{ is an element of a Sobolev space for all } \mathfrak{X} \in B\_d \right\}, \tag{57}$$

which is the Sobolev space of *Wp*,<sup>2</sup>(*Ba*) functions on the Euclidean 3-ball *Ba*. Here, each function is defined with respect to the *local* coordinates

$$\overline{\mathfrak{X}} := \phi^{-1}(\mathbf{r}), \ \mathbf{r} \in \mathcal{U}\_i. \tag{58}$$

In fact, it should be straightforward to deduce from the above that there exists maps

$$
\pi\_i \colon p^{-1}(\mathcal{U}\_i) \to \mathcal{U}\_i \times \mathcal{X}\_{i\prime} \tag{59}
$$

for all *i* ∈ *I*, that are *isomorphisms* (*diffeomorphism* in our case), where such diffeomorphism may be expressed by

$$\forall i \in I:\ p^{-1}(\mathcal{U}\_i) \cong \mathcal{U}\_i \times \mathcal{F}\_i. \tag{60}$$

We also add that the fact of (59) actually playing the role of such an isomorphism would naturally follow from the respective definitions of the spaces F*i* and *Xi*, as specified by (48) and (57), and from the proposition that each *φi* is a diffeomorphism from *Ui* into R<sup>3</sup> (or, equivalently, to the unit 3-ball *Ba* with radius *a* instead of R3.) Furthermore, note that by construction the diffeomorphism *τi* satisfies

$$\text{proj}\_1 \circ \tau\_{\bar{i}} = p\_{\prime} \tag{61}$$

where proj1 is the standard projection map defined by proj1(*<sup>x</sup>*, *y*) := *x*. Finally, if we restrict *τi* to *<sup>p</sup>*<sup>−</sup><sup>1</sup>(**r**), the resulting map

$$\left. \tau\_{i} \right|\_{p^{-1}(\mathbf{r})} : p^{-1}(\mathbf{r}) \to \{\mathbf{r}\} \times X\_{i} \tag{62}$$

is a (linear) topological vector space isomorphism from F**r** to *Xi*; namely, we have

$$\forall i \in I, \ \mathbf{r} \in \mathcal{U}\_i: \qquad \mathcal{F}\_{\mathbf{r}} \cong X\_i. \tag{63}$$

**Remark 14.** The charts (*Ui*, *τi*) are called *trivialization covering* of the vector bundle M. They provide a coordinate representation of local patches of the vector bundle. (The global topology of the bundle, however, is rarely trivial [62].) Since here all maps are *Cp* smooth, *τi* are also called *smooth* trivialization maps. The complete derivations of the diffeomorphism (60) and the topological vector space isomorphism (63) are straightforward, but lengthy, and the full proofs are omitted.

Consider now two patches *Ui* and *Uj* with *Ui* ∩ *Uj* = ∅. By restricting *τi* and *τj* to *Ui* ∩ *Uj*, two diffeomorphisms

$$\begin{aligned} \pi\_i &: p^{-1}(\mathcal{U}\_i \cap \mathcal{U}\_j) \to (\mathcal{U}\_i \cap \mathcal{U}\_j) \times \mathcal{X}\_{i\prime} \\\\ \pi\_j &: p^{-1}(\mathcal{U}\_i \cap \mathcal{U}\_j) \to (\mathcal{U}\_i \cap \mathcal{U}\_j) \times \mathcal{X}\_{j\prime} \end{aligned} \tag{64}$$

are obtained, which together imply in turn that

$$(\mathcal{U}\_i \cap \mathcal{U}\_j) \times X\_i \cong (\mathcal{U}\_i \cap \mathcal{U}\_j) \times X\_j. \tag{65}$$

or, equivalently, the following expected Banach space isomorphism:

$$X\_{\mathbf{i}} \cong X\_{\mathbf{j}\prime} \tag{66}$$

In particular, it can be shown that the composition map

$$
\pi\_{\dot{\jmath}} \circ \pi\_{i}^{-1} : (\mathcal{U}\_{i} \cap \mathcal{U}\_{\dot{\jmath}}) \times X\_{i} \to (\mathcal{U}\_{i} \cap \mathcal{U}\_{\dot{\jmath}}) \times X\_{\dot{\jmath}} \tag{67}
$$

possesses the simple form

$$
\pi\_{\rangle} \circ \pi\_{i}^{-1}(\mathbf{r}, \mathbf{F}) = (\mathbf{r}, \mathcal{g}(\mathbf{r})\mathbf{F}),
\tag{68}
$$

with the following formal structure:

$$\forall \mathbf{r} \in \mathcal{U}\_{i} \cap \mathcal{U}\_{j}, \mathbf{F} \in \mathcal{X}\_{i\prime} \quad \exists \mathbf{g} \in \mathcal{L}(X\_{i\prime}, X\_{j}), \tag{69}$$

where the abstract vector linear space

$$\mathcal{L}(X\_i, X\_j) \tag{70}$$

is defined as the space of all linear operators [26]

$$\mathcal{S}: X\_i \to X\_j \tag{71}$$

on Banach vector spaces. In particular, *g*(**r**) is a *Cp*-Banach space isomorphism. **Remark 15.** In the mathematical literature, the smooth maps *τj* ◦ *τ*<sup>−</sup><sup>1</sup> *i* are called the *vector bundle transition maps*. They are essential technical tools for computing global data by starting from local data then gluing them together. For example, they will be used in Sections 5.3.2 and 5.3.3 as part of the toolbox needed in the process of generalizing local information into global domains.

We have now succeeded in directly constructing a specialized smooth Banach vector bundle (M, D, *τ*, *p*) consisting of the nonlocal material continuum's total fiber bundle space M, the material domain's base three-manifold D, a set of smooth trivialization charts *τi*, *i* ∈ *I*, and a projection map *p*. The base manifolds D itself is described by a differential atlas (*Ui*, *φi*), also associated with the partition of unity (*Ui*, *ψi*), *i* ∈ *I* as per our discussion in Section 5.2 above. This incredible increase in the complexity of the mathematical space of nonlocal continuum field theory; that is, the transition from spacetime (or space–frequency) as the configuration space to a a larger superspace, here the fiber bundle space (which might be time- or frequency-dependent), is a direct expression of the very significant complexity and richness of the *physics* of nonlocal field theory in general.

As will be seen in the next Section 5.3.2, it is possible to demonstrate ye<sup>t</sup> another remarkable departure from conventional theory where the concept of *linear operator*, as such, a fundamental structural object in the mathematical and computational physics of local continuum field theories [65], is found to be generalizable to the concept of *homomorphism*, which is essentially topological in nature.

### 5.3.2. The Nonlocal Material Continuum Fiber Bundle Homomorphism

At this point, we need to describe how the *evaluation* process of the response field (33) may be formulated within the new enlarged framework of the fibered superspace M. The most obvious method is to introduce a *new* vector bundle with the base space being the same base space D, but with the fibers now taken as the complex Hilbert space C3. This is a well-known vector bundle, which we denote by R, and dub the *range vector bundle*. Formally, the structure of this vector bundle is expressed by the ordered quadruple (R, D, *τ* , *p* ), where *τ* and *p* are the range bundle R's smooth trivialization and projection maps, respectively. On the other hand, the *source vector bundle* is taken as M.

As a preparation for introducing the concept of the nonlocal continuum homomorphism, let us recap and comment on the overall physical process of exciting a material nonlocal continuous domain D as follows:


We turn now to a precise definition for convenient maps between bundle superspaces. Formally, we may directly use the standard concept of homomorphism in fiber bundle theory, adapted to our purposes in the following manner [57,68]:

**Definition 3** (**Bundle homomorphism**)**.** A smooth bundle homomorphism over a common base space D shared between the two vector bundles M and R is defined by the (smooth) map:

$$
\mathcal{L}: \mathcal{M} \to \mathcal{R} \tag{72}
$$

satisfying *p* ◦ L = *p*. Moreover, the restriction of L to each fiber *p*<sup>−</sup><sup>1</sup>(**r**) induces a linear operator on the corresponding vector space of that fiber. In effect, the following diagram

is commutative.

**Remark 16.** Because the nonlocal material continuum's superspace M and its range fiber bundle R both share an identical base manifold D, the action of the homomorphism L as a bundle map is effectively reduced to how it interacts with each fiber *p*<sup>−</sup><sup>1</sup>(**r**) by acting on the latter as a standard vector space linear operator. Therefore, a large portion of the conventional linear algebra and computational methods extensively deployed in the mathematical and numerical apparatus of local continuum field theory, such as nonlinear functional analysis [65], Hilbert space methods [64], and the Finite Element Method [75], may be reused as "sub-algorithms" within the larger, more general formalism of nonlocal continuum field theory proposed in this paper.

Now, since the Banach space *Xi* is isomorphic to *<sup>p</sup>*<sup>−</sup><sup>1</sup>(**r**), we may assemble the homomorphism L by specifying its *local* expression in each topological subdomain *Ui* ⊂ D of the open cover {*Ui*, *i* ∈ *<sup>I</sup>*}. In particular, we define the local action using the source and range bundles' trivialization maps *τi* and *τi* by the intuitively obvious formula:

$$
\pi\_i' \circ \mathcal{L}\_{\omega} \circ \pi\_i^{-1} : \mathsf{U}\_i \times \mathsf{X}\_i \to \mathsf{U}\_i \times \mathsf{C}^3 \,. \tag{73}
$$

with

$$
\pi\_i' \circ \mathcal{L}\_{\omega'} \circ \pi\_i^{-1} := (\mathbf{r}\_\prime \mathcal{L}\_{i,\omega} \mathbf{F}), \quad \mathbf{F} \in X\_{i\prime} \tag{74}
$$

where

$$\mathcal{L}\_{i,\omega} \colon X\_i \to \mathbb{C}^3 \tag{75}$$

is the linear operator defined by

$$\mathcal{L}\_{i\omega}(\*) = \int\_{\mathcal{U}\_i} \mathbf{d}^3 r' \,\overline{\mathbf{K}}(\mathbf{r}, \mathbf{r}'; \omega) \cdot (\*),\tag{76}$$

in which '\*' stands for an element of the smooth Banach function space *Xi*.

Therefore, within the frequency domain formulation of this paper, the operator L will leave every point in the base space D unchanged while mapping each smooth function on *Ui* (component of the total electromagnetic excitation field, see below) into its complex vector value in C<sup>3</sup> at **r** ∈ *Ui*. Physically, L*i* models a (topologically) localized "piece" of the global electromagnetic material operator mapping excitation fields **<sup>F</sup>**(**r**) to response fields **<sup>R</sup>**(**r**), where the entire physics here is restricted to the physics-based nonlocal subdomain *Ui*. The global operator itself is assembled by gluing together these small pieces using the partition of unity technique, as we endeavor to show next.

5.3.3. Computing Global Data Starting from Local Data

The final step is tying up together the fundamental source Banach bundle superspace M, range bundle R and the nonlocal microdomain physics space (31). The essential ingredients of the *physics* of nonlocal field–matter interaction are encoded in the geometrical construction of the collection of microdomains V(D) = {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ *<sup>D</sup>*}, and the excitation fields **<sup>F</sup>**(**r**) defined on them, i.e., the sets V(D) *and* the (excitation) function spaces G(D) combined together in one space, the superspace M.

So far, the vector bundle homomorphism L introduced above (Definition 3) can handle excitation fields supported on the open sets *Ui*, *i* ∈ *I*. However, the latter sets are *mathematical* fundamental building blocks, or "set-theoretic atoms", deployed in order to formally construct the source vector bundle superspace M. The question that will be addressed presently is the following one:

*How can we extend the description of the nonlocal continuum's response operators starting from excitation fields defined locally to excitation fields applied on the entire physical cluster of nonlocal microdomains* {*<sup>V</sup><sup>r</sup>*, *r* ∈ D)*?*

As mentioned before, it is the partition of unity (*Ui*, *ψi*), *i* ∈ *I*, what will make this expansion of the topological formulation technically feasible.

To see this, let us consider an electromagnetic field **<sup>F</sup>**(**r**) interacting with a nonlocal medium extended over the manifold D. Our goal is to compute the response field **<sup>R</sup>**(**r**); that is, at point **r**. Let us recall what the fundamental idea of EM nonlocality is: in order to compute the nonlocal material continuum's response at one point **r**, one must know the excitation field in the *entirety* of an open set *V***<sup>r</sup>**. This set *V***r** is one of these nonlocal microdomains composing D as per (29). Moreover, such *V***r** is also a topological neighborhood of its continuous index point **r** ∈ D (cf. Section 4.2). However, in general this microdomain will change depending on the position **r**. The goal now is to find **<sup>R</sup>**(**r**) using the vector bundle map L defined by (72) starting from the data:

1. Region *V***r**;

2. Vector field **<sup>F</sup>**(**r**) acting on *V***<sup>r</sup>**.

To accomplish this, we exploit the properties of the partition of unity functions *ψi* (Lemma 1) for expanding the excitation field **<sup>F</sup>**(**r**) over *all* patches *Ui* covering *V***<sup>r</sup>**, resulting in

$$\mathbf{F}(\mathbf{r}';\omega) = \sum\_{i \in I\_{\mathbf{r}}} \psi\_i(\mathbf{r}') \mathbf{F}\_i(\mathbf{r}';\omega),\tag{77}$$

where (43) was used. The truncated function **F***i* is equal to **<sup>F</sup>**(**r**) only if **r** ∈ *Ui* and zero elsewhere, i.e., we have

$$\mathbf{F}\_{i}(\mathbf{r}';\omega) := \begin{cases} \mathbf{F}(\mathbf{r}';\omega), & \mathbf{r} \in \mathcal{U}\_{i}, \\ 0, & \mathbf{r} \notin \mathcal{U}\_{i}. \end{cases} \tag{78}$$

Recall that according to Lemma 1, the set *I***r** is defined as the collection of indices *i* ∈ *I* of all *Ui* having the point **r** in their common set intersection; by construction, this index set *I***r** is always finite.

The main idea behind our construction should now become clear: while each truncated sub-field **F***i* fails to be differentiable (it is not even continuous), the multiplication by *ψi*(**r**) fixes this problem. In fact, each function

$$
\psi\_i(\mathbf{r}')\mathbf{F}\_i(\mathbf{r}';\omega) \tag{79}
$$

is a *smooth* component of the total excitation field **F** with support *fully* contained *inside* the coordinate patch *Ui*; that is, we have

$$\text{supp}\{\mathbf{F}\_i\} \subset \mathcal{U}\_i. \tag{80}$$

Consequently, the vector bundle map constructed in (72) can be applied to each such component field. From (73)–(75) and (77), the following can be deduced:

$$\mathbf{R}(\mathbf{r};\omega) = \sum\_{i \in I\_{\mathbf{r}}} \mathcal{L}\_{i,\omega} [\psi\_i(\mathbf{r}') \mathbb{F}\_i(\mathbf{r}';\omega)].\tag{81}$$

Finally, using (76), we arrive at our main superspace map theorem:

**Theorem 1** (**global superspace bundle map**)**.** For the fiber bundle superspace M of the nonlocal continuum whose differential manifold representation is D and the nonlocal continuum response function (tensor) **K**, the response and excitation fields **R** and **F** can be related to other other via the global bundle (superspace) map:

$$\mathbf{R}(\mathbf{r};\omega) = \sum\_{i \in I\_{\mathbf{r}}} \int\_{\mathcal{U}\_{i}} \mathbf{d}^{3}r' \,\overline{\mathbf{K}}(\mathbf{r},\mathbf{r}';\omega) \cdot \psi\_{i}(\mathbf{r}')\mathbf{F}\_{i}(\mathbf{r}';\omega)\_{\ast} \,\Big|\,\tag{82}$$

where *ψ<sup>i</sup>*, *i* ∈ *I*, are the partition of unity basis functions subordinated to the D-atlas (*Ui*, *φi*), *i* ∈ *I*.

Physically, Theorem 1 states that the nonlocal continuum's source bundle (superspace) M, the range bundle R, and the nonlocal response superspace map L, together, supply the fundamental formal scaffold upon which the material domain's response to generic excitation field, when the latter field operates on arbitrary configurations of nonlocality microdomain, can be constructed. By aggregating all of those physics-based microdomains constituting the topological microstructure of nonlocal processes in material continua, the main field-theoretic structures of the medium may be couched, computed and reformulated in the richer language of this more general superspace framework belonging to the Banach fiber bundle M instead of the position space D of conventional spacetime extensively used in local field theories. At this stage of our formulation, the vector bundle formalism of nonlocality becomes essentially complete, where the connection between the purely mathematical fiber superspace and the physical microdomain structures is secured by Theorem 1, especially the Formula (82).

### **6. Interlude: The Nonlocal Continuum Fiber Bundle Superspace Algorithm—Summary and Transition to Applications**

We review and summarize here the salient features of the fiber bundle superspace construction, carefully developed above, by explicitly outlining the algorithm implicit in the various detailed derivations of the previous sections. Our main objective in this short transitional section is to highlight again the fact, already discussed above, which is that our superspace formalism is based on estimating the *physics*-based nonlocality microdomain set V(D) = {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ D} associated with the nonlocal continuum D. These data can be obtained only through physical theory and/or measurement. However, once available, the construction of the fibered space proceeds in a computationally well-determined manner. We first summarize the algorithm then provide few additional preparatory remarks before moving to the more detailed and concrete computational examples of Section 7.

In Figure 4, we show two distinct points **r**1,**r**2 ∈ D and their associated microdomains *<sup>V</sup>***<sup>r</sup>**1 and *V***<sup>r</sup>**2 , respectively. From the locally finite subcover {*Ui*}*<sup>i</sup>*∈*<sup>I</sup>* subordinated to V(D) = {*<sup>V</sup>***<sup>r</sup>**,**<sup>r</sup>** ∈ D} we highlight two sets

$$\mathcal{U}\_i \subseteq \mathcal{V}\_{\mathbf{r}\_1}, \quad \mathcal{U}\_j \subseteq \mathcal{V}\_{\mathbf{r}\_{2'}} \tag{83}$$

where in general it is allowed that

$$V\_{\mathbf{r}\_1} \cap V\_{\mathbf{r}\_2} \neq \bigotimes\_{\prime} \quad \mathcal{U}\_i \cap \mathcal{U}\_j \not\equiv \bigotimes\_{\prime} \tag{84}$$

as could be inferred from a glance at the Figure itself. For the partition of unity (*Ui*, *ψi*)*<sup>i</sup>*∈*I*, which is subordinated to the open cover {*Ui*}*<sup>i</sup>*∈*I*, we also highlight the two compact sets

$$S\_i \coloneqq \text{supp}\{\psi\_i(\mathbf{r})\}, \quad S\_i \coloneqq \text{supp}\{\psi\_i(\mathbf{r})\},\tag{85}$$

forming the support of the corresponding partition of unity functions.

**Figure 4.** An example illustrating the various topological microstructures involved in modeling a generic nonlocal material. The microdomains *V***<sup>r</sup>**1 , *V***<sup>r</sup>**2 ∈ V(D) are open sets and belong to the nonlocal microstructure of the MTM D. The open sets *Ui* and *Uj* are the corresponding coordinate sets and partition of unity functions {*ψi*}*<sup>i</sup>*∈*I*'s domains subordinated to *V***<sup>r</sup>**1 and *V***<sup>r</sup>**2 , respectively. The compact sets *Si* and *Sj* are defined by *Si* := supp{*ψi*(**r**)} and *Si* := supp{*ψi*(**r**)}.

The nonlocal material continuum's superspace algorithm itself is summarized in Algorithm 1. Once the microdomain dataset V(D) is given, the construction proceeds automatically using the partition of unity basis functions (*Ui*, *ψi*)*<sup>i</sup>*∈*I*. The latter may be computed directly in terms of the standard bump functions, see [57,68,71], and also Remark 9.

Because of the fundamental importance of the physics-based nonlocality microdomain structure V(D), Section 7 will be entirely devoted to the explication of a quantitative practical example illustrating the origin of these microdomains in the concrete setting of a real-life advanced material system, including how the microdomain topology itself may be estimated in practice. In the subsequent sections Section 8 and Appendixes A.3 and A.11, we also explore the usefulness of the superspace homomorphism construction developed in Section 5 for reformulating boundary-value problems in the nonlocal continuum field theories of mathematical physics, besides also providing some hints and additional remarks on other current and future applications.

**Algorithm 1** The nonlocal continuum fiber bundle algorithm.


$$\mathcal{M} := \{ (\mathbf{r}, X\_i) | \forall i \in I, \mathbf{r} \in \mathcal{U}\_l \subset \mathcal{D} \}. \tag{86}$$

6. Construct the projection map *p* : M→D through the operation (**r**, *Xi*) → **r**.

7. Use (68) to transform vector from one fiber (function) space to another.

### **7. Applications to Advanced Materials: Nonlocal Inhomogeneous Semiconductors**

*7.1. Introduction*

A concrete example involving spatially-dispersive isotropic media is considered in this Section, where the intention is to provide an outline of how the intricate fiber bundle type

topological fine structure (the topology of microdomains attached to each point explored above as developed in detail in Section 5 and summarized in Section 6) may be estimated in actual practice. The contents of the example given below are rather detailed, and that is for two main reasons. First, in spite of the fact that nonlocal metamaterials are not proposed here for the first time, the author's experience indicates that there is still a general lack of appreciation of the subject in the large community, where most research on "metamaterials" concentrate on *temporally*-dispersive media. Because of that, we provide a very detailed example, including reintroducing some of the well-known physics of semiconductors (in some of the appendices) in order to make the presentation complete and self sufficient. Second, the detailed example to be found below is itself novel. The estimation of the microdomain nonlocal structure in inhomogeneous semiconductors seems to be achieved here for the first time. Therefore, it is a topic that could be treated not merely as an example illustrating the more general and abstract superspace theory developed in the earlier sections, but possibly as a stand-alone contribution to semiconductor materials and their physics. However, the main intention behind the inclusion of this highly-technical physical example continues to be the illustration of the fundamental superspace formalism. More detailed examinations of nonlocality in semiconductor metamaterials belong to a more specialized literature than the current article, whose main topic is the mathematical physics of nonlocal continuum field theories.

#### *7.2. A Topological Coarse-Grained Model for Inhomogeneous Nonlocal Material Domains*

A review of the homogeneous medium model of spatial dispersion is provided in Appendix A.6. Below, we describe a method that can help transitioning from the generic form (A3), valid for homogeneous nonlocal domains, to the *inhomogeneous* medium situation developed throughout this paper where nonlocality cannot be captured by a simple global dependence of the dielectric function on **k**. However, instead of working with the full nonlocal function **<sup>K</sup>**(**<sup>r</sup>**,**r** ), an alternative simplified model is proposed which we entitle *the topological coarse-grained model.* The idea is as follows. Consider a global material domain D, which is an open three-manifold, say an open subset of R<sup>3</sup> that may be either simply connected or disconnected.<sup>21</sup> The material is nonlocal and inhomogeneous. At each point **r** ∈ D, a microdomain, i.e., and open set *V***r** ⊂ D, is assigned. The medium is *locally isotropic and homogeneous* in the sense that within each microdomain we can describe the response to an external field excitation **E** by means of a relation similar to (34), namely:

$$\mathbf{D}(\mathbf{r};\omega) = \varepsilon\_0 \int\_{V\_\mathbf{r}} \mathbf{d}^3 r' \,\overline{\mathbf{K}}(\mathbf{r} - \mathbf{r}';\omega) \cdot \mathbf{E}(\mathbf{r}';\omega) . \tag{87}$$

That is, the only difference between (87) and (34) is that, in the former, we use the correct form of *homogeneous* nonlocality **<sup>K</sup>**(**r** − **r** ; *ω*) instead of **<sup>K</sup>**(**<sup>r</sup>**,**r** ; *<sup>ω</sup>*). Moreover, we have put the proper response and excitation fields **<sup>D</sup>**(**r**) and **<sup>E</sup>**(**r**) and inserted the free space permittivity *ε*0.

Fundamentally speaking, each material microdomain is now described by a *spatiallydispersive* model of the form (87). The "topological atoms" of nonlocality, namely the sets *V***<sup>r</sup>**, spanned by the continuous index **r** ∈ D, are each a spatially-dispersive "medium" on its own. As will be seen later in this section, the idea of the locally–spatially-dispersive nonlocal semiconductor system is to build an *inhomogeneous* metamaterial that goes *beyond* spatial dispersion by assembling a more general form of nonlocality using the spatiallydispersive material "atoms" *V***<sup>r</sup>**. In such systems, the engineered metamaterial is only *locally* spatial dispersive. On the other hand, at a larger spatial scale it does not follow the standard spatial dispersion law, but rather appears to belong to a more complicated class of nonlocal continua which, we believe, are best mathematically described using the fiber bundle superspace formalism of Section 5.

It may be seen then that as a topological coarse-grained process, the original inhomogeneous nonlocal medium, ultimately described by the material tensor **<sup>K</sup>**(**<sup>r</sup>**,**r** ; *<sup>ω</sup>*), is sub-decomposable into "small topological cells", the microdomains *V***<sup>r</sup>**, **r** ∈ D, such that

each "topological cell" or "atom" would in itself behave like a homogeneous nonlocal isotropic subdomain, hence may be described by (87), where the material tensor in that case takes the (topologically) locally correct form (A3). This can be considered a *quasi-local* model (also sometimes called *locally spatially dispersive*), where the global domain, electromagnetically speaking, is nonlocal, while, on the other hand, seen at the scale of a small region (cells) it would more or less behave like a typical electromagnetic local medium, see, for example, the discussion of some special cases of complex nonlocal crystals in [76].

**Remark 17.** We remind the reader again about the subtle difference between *mathematical* nonlocality and *physics*-based nonlocality, a distinction at the conceptual level that will become quite visible throughout this section. The term *local* is used in this paper in two senses. The first sense is the physical one in which *local* is set against *physical nonlocality*, which includes spatial dispersion (EM local/nonlocal.) On the other hand, in topology, a *local* property is that which holds in a small open neighborhood of a given point, in our case the topological microdomain *V***<sup>r</sup>**. The distinction between the two technical senses of the same term should always be clear from the context. In the few cases when there is a risk of confusion, we say *topologically local* to emphasize the second meaning above from *EM local.* (see also Remark 3 and Section 3.3).

Our key objective now is to first develop a simple estimation of the "size" of the nonlocal microdomains *V***<sup>r</sup>**. To do so, some metric methods must be introduced. An attractive approach would be to approximate the topology of the nonlocal metamaterial system using arrays of various *spheres*, and then use this array in order to obtain the topological content of the microdomain structure described in Section 4.

Let us illustrate the main ideas with a simple example first. Consider a point **r**1, which provides a label for one of the micro cells, we may deploy for creating a coarse-grained model for the inhomogeneous medium. To be more specific, let us construct the topological open ball defined by

$$B(\mathbf{r}, a\_{\mathbf{r}}) := \left\{ \mathbf{r} \in \mathbb{R}^3 \, \middle| \, d(\mathbf{r}, \mathbf{r}') < a\_{\mathbf{r}} \right\},\tag{88}$$

where *a***r** ∈ R<sup>+</sup> is a number quantifying the smallness of this "nonlocality ball" centered at **r**, while *d* is the distance metric. The number *a***r** will be determined later based on the actual physics of the problem.

Next, the fine-grained topological microdomain structure can be constructed by aggregating all these balls in order to produce a coarse-grained of the overall inhomogeneous nonlocal material domain D. The choice of the *shape* of the microdomain *V***r** as a sphere *<sup>B</sup>*(**<sup>r</sup>**, *<sup>a</sup>***r**) defined by (88) is justified by our earlier assumption that the material is (topologically) *locally* isotropic. However, note that *globally* electromagnetic processes need *not* behave as they do in isotropic domains.

In Figure 5, a diagrammatic depiction of the two local and global processes is provided where we illustrate:


As can be seen from the diagram, in the topological approach, there exists an open set (microdomain) *V***r** attached to each point **r** ∈ D such that nearby microdomains may *overlap* with each other; i.e., in such case the set

$$\mathbf{r} \cup\_{\mathbf{r}\_1, \mathbf{r}\_2 \in \mathcal{D}} \left[ V\_{\mathbf{r}\_1} \cap V\_{\mathbf{r}\_2} \right] \tag{89}$$

is not necessarily empty. On the other hand, the conventional approach to coarse-grained, depicted in Figure 5 (right), involves subdomains like *V* **<sup>r</sup>**1 and *V* **r**2 that are *non-overlapping*, leading to a grid-like structures or "tile covering up" of the material domain D where in general no holes are left.

In both approaches, it should be noted, we find that each type of the two subdomains, whether *V***r** or *V***r** , was already assumed to be *homogeneous*. The disadvantage of the conventional approach is that any *abrupt* change in the electromagnetic properties of the material, experienced when transitioning between two neighboring subdomains through their interface region, often requires imposing a suitable "boundary condition" at this geometric interface in order to obtain an accurate computational assessment of the physics. On the other hand, this problem does *not* exist in the topological approach, illustrated in Figure 5 (left), because the microdomains are allowed to overlap, where common regions between overlapping microdomains are treated correctly using the partition of unity basis functions as described in Section 5.3.

**Figure 5.** Topological coarse-grained model for an inhomogeneous nonlocal material domain D (**left**) in comparison with a conventional coarse-grained process (**right**). The topological microdomains constitute an open cover of the domain in the sense that D = 4**r**∈D *V***<sup>r</sup>**, which is the obvious generalization of (32). Note how the topological approach allows overlapping microdomains, e.g., between microdomains *V***<sup>r</sup>**2 and *V***<sup>r</sup>**3 . The technique of the partition of unity will take care of electromagnetic data "repeated" in such regions of overlap by assigning proper weights that always sum to unity at each point in **r** ∈ D.

#### *7.3. Resonant Nonlocal Semiconductor Domains and the Nonlocal Exciton-Polariton Model*

A concrete application of the topological coarse-grained algorithm proposed in Section 7.2 is now in order. The specific nonlocal metamaterial is a semiconductor with dielectric function exhibiting a single strong resonant exciton transition at the frequency *ω*= *ω*e. We first examine in detail the nonlocal exciton–polariton model to be used below. For a review on the physics of exciton–polariton interactions in solids, see Appendix A.7.

A *polariton* is simply a "photon living inside a dielectric medium". The quantum of an electromagnetic wave inside a dielectric domain is often called polariton instead of photons (sometimes polaritons are called "dressed photons"). An *exciton–polariton* is a polariton coupled with a mechanical exciton, e.g., an electron-hole pair. The latter should be distinguished from other types of polaritons such as *phonon-polaritons* defined as polaritons coupled with *phonons*, the quantum of lattice vibrations [37].

It is well known from quantum theory that near resonance, the dielectric function of such semiconductor materials may be approximated by the formula [16,77–80]:

$$
\varepsilon(\mathbf{k}, \omega) = \varepsilon\_0 + \frac{\chi}{k^2 - \gamma^2(\omega)},
\tag{90}
$$

where

$$\chi = 4\pi \frac{a m\_{\text{e}}^{\star} \omega \iota\_{\text{e}}}{\hbar}, \quad \gamma^2(\omega) = \frac{m\_{\text{e}}^{\star}}{\hbar \omega\_{\text{e}}} \left(\omega^2 - \omega\_{\text{e}}^2 + \text{i}\omega \Gamma\right). \tag{91}$$

Here, *h*¯ is the reduced Planck constant, while *α* serves as the oscillator strength.<sup>22</sup> The effective mass of the exciton is denoted by *<sup>m</sup>*e .23. On the other hand, the *exciton lifetime τ*e is defined by

$$
\pi\_{\mathfrak{e}} := \frac{2\pi}{\Gamma},
\tag{92}
$$

hence, Γ can be thought of as the *exciton decay* or *relaxation rate*. We emphasize that any dependence of Γ and the oscillator strength *α* on *k* is ignored in the excitonic model (90).

In Appendix A.7, the physical origin of nonlocality in the semiconductor is revisited, where it is traced to the quantum mechanical energy–momentum relations of exciton– polaritons. In order to actually see significant nonlocal physics taking place in the excitonic material system described by (90), the following sufficient condition may be imposed:

$$
\Gamma \ll \frac{\hbar k^2}{2m\_e^\*}.\tag{93}
$$

It can be shown that under such Γ-bound, the kinetic energy term in (A12) can induce significant nonlocal effects in (90). One way to realize nonlocal (spatially dispersive) semiconducting metamaterials is to operate with intrinsic semiconductors satisfying (93) by keeping the temperature low and the material pure (undoped) [83].

The model described by (90) and (91) can be viewed as a natural generalization of the local Lorentz model widely utilized to model temporal dispersion in solids and plasma [77,79]. It represents the simplest nonlocal resonant model with a single strong resonance at a characteristic frequency, here *ω* = *ω*e. All other off-resonance excitonic transitions are gathered into the background dielectric constant *ε*0 for simplicity. For frequencies well below *ω ω*e, the exciton–polariton behaves essentially like a photon propagating in a medium with background permittivity *ε*0. For *ω* ! *ω*e, we again recover photons but usually with a background described by *ε*<sup>∞</sup>, the high-frequency limit of permittivity. In general, the difference between the static and high-frequency permittivities is quite small in the sense that

$$|\mathfrak{e}\_{0} - \mathfrak{e}\_{\infty}| \stackrel{\Leftrightarrow}{\llcorner} \mathfrak{e}\_{0} \text{.} \tag{94}$$

Hence, for simplicity, in this example the two permittivities are treated as identical (*<sup>ε</sup>*0 *<sup>ε</sup>*∞) since we are interested in the EM response around a single excitonic resonance while in fact the oscillator strength *α* in (90) is small. One consequence of this assumption is that the splitting between longitudinal and transverse modes can be neglected. Indeed, since the longitudinal and transverse frequencies *ω*L and *ω*T are related to each other via the relation [16]

$$\frac{\omega\_{\rm L}^2}{\omega\_{\rm T}^2} = \frac{\varepsilon\_0}{\varepsilon\_{\infty}},\tag{95}$$

then the assumption (94) is equivalent to neglecting the longitudinal–transverse splitting

$$
\omega\_{\rm L,T} := |\omega\_{\rm L} - \omega\_{\rm T}| \tag{96}
$$

in the sense that

$$
\omega\_{\rm L^\*} \llcorner \llcorner \omega\_{\rm T} \tag{97}
$$

A consequence of this is the near equality of the longitudinal and transverse frequencies, which allows us to considerably simplify the mathematical treatment.<sup>24</sup> In addition, assuming that the oscillator strength *α* in (90) is nearly the same for both the longitudinal and transverse part of the response function, then it follows that we need only work with a single *scalar* response function, namely the form (90) itself instead of the more general tensorial Formula (A3).<sup>25</sup>

Nonlocal effects associated with the model (90) emerge from the quantum mechanical nature of exciton–polariton interactions and the need to enforce conservation of energy/momentum as discussed in Appendix A.7, leading to the strong dependence on *k* observed in (90). There is ye<sup>t</sup> another physical explanation of nonlocality. Within the regime of the large exciton mass limit

$$m\_{\text{ef}}^{\*} \to \infty,\tag{98}$$

the kinetic energy term in (A12) drops out and the excitonic dielectric function (90) becomes local. This is why spatial dispersion is sometimes referred to as the "finite-mass model", with some suggestions that the origin of nonlocality in this case is the inertial effects of the exciton [79].<sup>26</sup> In what follows, we assume that the effective mass of the exciton is always finite and positive:

$$0 < m\_{\mathfrak{e}}^{\star} < \infty. \tag{99}$$

However, it should be noted that since excitons are *collective excitations* of solids [84,85], they may have negative mass [86]. While this will not be pursued here, the negativity of the excitonic mass may be exploited in order to further design and control the EM behavior of nonlocal MTMs constructed using excitonic semiconductors.

In order to gain a deeper insight into the various resonance structures of the exciton– polariton response function (90), we rewrite it in the equivalent form

$$\varepsilon(\mathbf{k},\omega) = \varepsilon\_0 + \frac{\chi/k\_\mathrm{e}^2}{k^2/k\_\mathrm{e}^2 + 1 - \omega^2/\omega\_\mathrm{e}^2 - \mathrm{i}\omega\Gamma/\omega\_\mathrm{e}^2},\tag{100}$$

where

$$k\_{\mathfrak{e}} := \frac{2\,\pi}{\lambda\_{\mathfrak{e}}} = \sqrt{\frac{m\_{\mathfrak{e}}^{\star}\omega\_{\mathfrak{e}}}{\hbar}}\tag{101}$$

is called the *exciton wave number*. The wavelength *λ*e is a fundamental resonance spatial scale, which we will refer to as the *exciton wavelength* and is given by

$$
\lambda\_{\rm e} = \frac{1}{2\pi} \sqrt{\frac{\hbar}{m\_{\rm e}^{\star}\omega\_{\rm e}}}.\tag{102}
$$

For example, with *h*¯ *ω*e = 2.5 eV and *<sup>m</sup>*e = 0.9*<sup>m</sup>*el, where *<sup>m</sup>*el is the electron mass, the exciton wavelength *λ*e is around 0.0293 nm, which is the same order of magnitude of interatomic spacing. The excitation field wavelength *λ* is at least one order of magnitude larger. Later we will show typical values for the topological microdomain radius *a***r**.

There are several fundamental spatial and temporal scales involved in the process of describing the generic nonlocal metamaterial domain D. The excitation field **<sup>E</sup>**(**r**) itself introduces its own temporal excitation period

$$T := \frac{2\pi}{\omega},$$

in addition to a purely spatial scale (wavelength) measured by the formula

$$
\lambda := \frac{2\pi}{k}.\tag{104}
$$

On the other hand, the excitonic transition as such is associated with the fundamental (temporal) transition period

$$T\_{\mathfrak{e}} := \frac{2\pi}{\omega\_{\mathfrak{e}}},\tag{105}$$

while a fundamental spatial scale

$$
\lambda\_{\mathfrak{e}} := \frac{2\pi}{k\_{\mathfrak{e}}} \tag{106}
$$

can be unambiguously linked to the exciton at the same time. Table 1 gives a summary of all these parameters with their meaning explicitly stated. Moreover, it will be demonstrated later that the radius of a topological microdomain *V***<sup>r</sup>**, which is based at a generic position **r** ∈ D, can be given by a special Formula (120). Nonlocality arises from the delicate interplay between all these different spatial and temporal scales. In what follows, we will emphasize their relative roles in determining the rich nonlocal microstructure of the material domain, while introducing quantitative calculations.


**Table 1.** A summary of the various spatial and temporal scales involved in understanding and designing generic nonlocal metamaterials with exciton–polariton resonance-type of nonlocality.

Armed with this typology of spatial and temporal scales, we are now better positioned to understand the resonance structure associated with the exciton–polariton nonlocal dielectric function (100). Figure 6 illustrates two cases of resonance where the value of the dielectric function is examined with respect to variations in the excitation field wave number *k* (or equivalently the wavelength *λ*). In order to focus on nonlocality, we only plot the nonlocal part of the total response, which is found here to be proportional to the dielectric residue

$$
\varepsilon(\mathbf{k}, \omega) - \varepsilon\_0. \tag{107}
$$

As we may infer from Figure 6, a strong resonance takes place when the ratio

$$\frac{k}{k\_{\text{e}}} = \frac{\lambda\_{\text{e}}}{\lambda} \tag{108}$$

becomes comparable in magnitude to the quantities remaining in the denominator of (100). That is, the spatial resonance condition is

$$\frac{k^2}{k\_\text{e}^2} + 1 - \frac{\omega^2}{\omega\_\text{e}^2} \sim \frac{\omega \Gamma}{\omega\_\text{e}^2}.\tag{109}$$

However, the condition (109) holds only if the imaginary part of the denominator of (100), i.e., the quantity *ω*Γ/*ω*<sup>2</sup> e , is relatively small. Otherwise, since *k* and *k*e are real, the ratio *k*/*k*e can never lead to strong resonance when the relaxation rate Γ is sufficiently large. Another way to say the same thing is the following: strong *spatial* resonances, whose main origin is nonlocality, can take place either when dissipation is small, or when the exciton lifetime is long enough. The latter scenario of long exciton lifetime is characterized by the condition 

$$
\frac{\omega \Gamma}{\omega\_{\text{e}}^2} \ll \left| 1 - \frac{\omega^2}{\omega\_{\text{e}}^2} \right|. \tag{110}
$$

In such case, it is evident that the appropriate spatial and temporal sufficient conditions needed to secure nonlocal resonance are mutually related by the simple relation

$$\frac{k^2}{k\_\text{e}^2} \approx \frac{\omega^2}{\omega\_\text{e}^2} - 1.\tag{111}$$

From this, it can be inferred that nonlocal resonances generally occur only for *<sup>ω</sup>*/*<sup>ω</sup>*e > 1. In Figure 6 (left), we can see that for the above-resonance condition of *<sup>ω</sup>*/*<sup>ω</sup>*e = 1.5, the nonlocal domain possesses a spatial resonance at roughly *λ* ≈ *λ*e. On the other hand, if we operate the material at larger frequency *<sup>ω</sup>*/*<sup>ω</sup>*e = 2.5, i.e., well above the exciton transition frequency, then spatial resonances may occur only at values of the excitation field wavelength *λ* that are considerably smaller than the exciton wavelength *λ*e.

Finally, we add that when the nonlocal response is plotted as function of *ω* instead of *k*, resonance structures similar to Figure 6 are obtained under the condition (110) since in that case (111) approximately holds. In general, we would expect that for the best operation of the designed nonlocal MTM (maximal nonlocal response), the operating frequency should

be selected to be as close as possible to the exciton transition frequency, i.e., we would like to maintain the material design condition

$$\frac{\omega}{\omega\_{\text{e}}} \approx 1,\tag{112}$$

which is needed since, in general, the excitonic relaxation rate Γ is never exactly zero and, hence, the condition (110) seldom holds otherwise for all frequencies.

**Figure 6.** The nonlocal spatial resonance structure of the exciton–polariton dielectric response as a function of the excitation field wave number *k*. The normalized response function (*ε*(**<sup>k</sup>**, *ω*) − *<sup>ε</sup>*0)/(*χ*/*k*e)<sup>2</sup> is plotted, where the dashed line is the absolute value, the solid line represents the real part, while the dotted line is the imaginary part. For both figures, Γ/*<sup>ω</sup>*e = 0.01. (**Left**) *<sup>ω</sup>*/*<sup>ω</sup>*e = 1.5. (**Right**) *<sup>ω</sup>*/*<sup>ω</sup>*e = 2.5.

*7.4. Quantitative Estimation of the Electromagnetic Nonlocality Microdomain Structure in the Exciton-Polariton Dielectric Model*

In the spatial domain, the dielectric function can be obtained by computing the inverse Fourier transform

$$
\varepsilon(\mathbf{r} - \mathbf{r}'; \omega) = \mathcal{F}\_{\mathbf{k}}^{-1} \{ \varepsilon(\mathbf{k}, \omega) \}. \tag{113}
$$

where F −1 **k** is the converse of the forward Fourier transformation defined by (17). We will need the following inverse Fourier transform relation (proved in Appendix A.9):

$$\mathcal{F}\_{\mathbf{k}}^{-1}\left\{\frac{\chi}{k^2 - \gamma^2(\omega)}\right\} = \frac{\chi}{4\pi} \frac{e^{\gamma''(\omega)|\mathbf{r}-\mathbf{r}'|}e^{-\mathbf{i}\gamma'(\omega)|\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|},\tag{114}$$

where

$$\gamma' = -\sqrt{\frac{m\_{\text{e}}^{\star}}{2\hbar\omega\_{\text{e}}}}\sqrt{\left(\omega^2 - \omega\_{\text{e}}^2\right) + \sqrt{\left(\omega^2 - \omega\_{\text{e}}^2\right)^2 + \left(\omega\Gamma\right)^2}},\tag{115}$$

$$\gamma^{\prime\prime} = -\sqrt{\frac{m\_{\text{e}}^{\star}}{2\hbar\omega\_{\text{e}}}}\sqrt{-\left(\omega^{2} - \omega\_{\text{e}}^{2}\right) + \sqrt{\left(\omega^{2} - \omega\_{\text{e}}^{2}\right)^{2} + \left(\omega\Gamma\right)^{2}}}.\tag{116}$$

Hence, by substituting (100) into (113) and using (114), we arrive at

$$\varepsilon(\mathbf{r} - \mathbf{r}'; \omega) = \underbrace{\varepsilon\_0 \delta(\mathbf{r} - \mathbf{r}')}\_{\text{local response}} + \underbrace{\varepsilon\_{\text{NL}}(\mathbf{r} - \mathbf{r}'; \omega)}\_{\text{nonlocal response}}.\tag{117}$$

The first terms in the RHS of (117) provides the background local response of the medium. On the other hand, all nonlocal effects are relegated to the second term in the RHS of (117):

$$\varepsilon\_{\rm NL}(\mathbf{r} - \mathbf{r}'; \omega) := \frac{a m\_{\rm e}^{\star} \omega\_{\rm e}}{\hbar} \frac{e^{-i \gamma'(\omega) |\mathbf{r} - \mathbf{r}'|}}{|\mathbf{r} - \mathbf{r}'|} e^{\gamma''(\omega) |\mathbf{r} - \mathbf{r}'|} \,\tag{118}$$

which is nothing but the Green's function of the electromagnetic semiconductor material system under investigation.

The Green's function (118) is the most fundamental physical quantity needed for the construction of the microdomain structure D of the nonlocal medium. It has some similarity with the scalar free-space Green function for radiation fields, i.e., spherical waves of the form:

$$\frac{\exp\left(\mathbf{i}k|\mathbf{r}-\mathbf{r}'|\right)}{|\mathbf{r}-\mathbf{r}'|}.\tag{119}$$

However, there are notable differences:


Indeed, as can be seen from (90), dissipation is controlled by the exciton lifetime *τ*e, or, equivalently, the decay rate Γ. Dissipation decreases as the lifetime increases, i.e., when Γ is small. Figure 7 illustrates some examples where we plot both *γ* and *γ* as functions of frequency. The frequency-dependent behavior observable there strongly depends on Γ/*<sup>ω</sup>*e, i.e., the ratio between the relaxation frequency and the excitonic transition frequency. For ratios as small as Γ/*<sup>ω</sup>*e = 0.1, the intensity of attenuation per unit length *γ* is nearly constant for *ω* > *ω*e, while it assumes higher values for frequencies below the *ω*e as can be seen from Figure 7a. This is consistent with a "high-pass filtering behavior" typical for this type of resonance phenomena, where waves are often excited with frequencies slightly larger than the cutoff threshold at *ω*e. For the propagation constant *γ* at the same relaxation-to-exciton transition ratio Γ/*<sup>ω</sup>*e, Figure 7b shows that it becomes nearly straight line. Such behavior, when combined with nearly constant per-unit-length attenuation, represents negligible dispersion effects. On the other hand, when Γ/*<sup>ω</sup>*e increases, we begin to see strong dispersion effects, manifested by non-constant per-unit-length attenuation and nonlinear phase-delay relations.

In fact, the attenuation process described by the per-unit-length rate *γ* is not merely an expression of dissipation, but is also *the* signature of nonlocality in exciton–polariton semiconductor materials. The medium response weakens as the distance from the source increases, while the characteristic length scale of this nonlocality radius is found to be solely controlled by *<sup>γ</sup>*. Figure 8 illustrates the real part of the dielectric function Green's function (118). The ability of the excitonic semiconducting medium to respond to spatially distant sources is graphically illustrated by its dielectric profile's functional spread around the origin |**r** − **r**| = 0. The size of the nonlocal domain is then directly reflected by the rapidity of the decay of the Green's function (118) as one moves away from **r**, which is the origin here.

**Figure 7.** Frequency dependence of *γ* (**a**) and *γ* (**b**) for several values of the exciton decay rate Γ. Here, *<sup>m</sup>*e = 0.9*<sup>m</sup>*e, where *m*e is the electron mass. The exciton transition frequency is *ω*e = 3.7977 × 10<sup>15</sup> rad/s (¯*h<sup>ω</sup>*e = 2.5 eV).

**Figure 8.** (**a**) Comparison between the real parts of the long-range decay of the excitonic nonlocal domain Green function *<sup>ε</sup>*NL(**<sup>r</sup>** − **r**) with and without the full spatial dependence, including the exponential short-range decay factor exp(*γ*|**r** − **r**|) for *γ* = 1 nm<sup>−</sup><sup>1</sup> and *γ* = 2 nm<sup>−</sup>1. (**b**) Frequency dependence of *a***r**, the radius of the topological microdomain *<sup>B</sup>*(**<sup>r</sup>**, *<sup>a</sup>***r**) centered at some generic point **r** in the nonlocal excitonic material domain D for several values of the exciton lifetime Γ−1. Here, *<sup>m</sup>*e = 0.9*me*, where *m*e is the electron mass. The exciton transition frequency is *ω*e = 3.7977 × 10<sup>15</sup> rad/s (¯*h<sup>ω</sup>*e = 2.5 eV).

### *7.5. The Locally-Homogeneous Model of Nonlocal Semiconducting Domains*

Quasi-inhomogeneous, also known as smoothly-inhomogeneous or locally spatially dispersive nonlocal media, are some of the simplest possible prototypes of general (inhomogeneous) nonlocal materials where the spatial dispersion model *<sup>ε</sup>*(**k**), with a dependence on only one spatial spectral variable **k**, is found to be not adequate for the mathematical description of the physics of the nonlocal system [76,87]. In contrast, one would need the considerably more complex spectral functions of the form *<sup>ε</sup>*(**<sup>k</sup>**, **<sup>k</sup>**), which are threedimensional spatial Fourier transformations of generic nonlocal response functions like (3) or (10). In general, there has been quite few investigations aimed at going beyond spatial dispersion in homogeneous media. Examples include inhomogeneous plasma, such as those in controlled-fusion reactors [88], cold collisionless magnetoplasma [88], the electrodynamics of nanostructures [89–92], and incommensurately-modulated superstructures in insulators [76,93].

Here, we will analyze a simple inhomogeneous model of semiconductors experiencing exciton–polariton transitions as outlined above. The EM nonlocal model is *locallyhomogeneous* in the sense that around each point **r** ∈ D there exists a *topologically*-local neighborhood, namely the microdomain *V***<sup>r</sup>**, inside which the medium can be modeled as a homogeneous and spatially dispersive domain for all **r** ∈ *V***r** (i.e., the second mention of "locally" here means *topological* nonlocality, see Remarks 3 and 17). It should be noted though that for maximum generality, we allow for variations in the spatial dispersion model to take place from one microdomain *V***r** to another.

We now wish to estimate the size of each nonlocality microdomain with the help of the exponential law in (118). Let us first expand the homogeneous model treated in Section 7.3 to the inhomogeneous setting of the present discussion, where currently we need to allow that at each point **r** ∈ D, the parameters of the original exciton–polariton model (100) would all become generally functions of the position. That is, in this more general case, one should write *<sup>γ</sup>*(**r**), *<sup>γ</sup>*(**r**), *<sup>ω</sup>*e(**r**), *<sup>α</sup>*(**r**), *<sup>m</sup>*e (**r**), etc, where it is understood that the medium's microscopic composition may change from one position to another.

The main formula for computing the size (radius) of the topological microdomain balls *V***r** = *<sup>B</sup>*(**<sup>r</sup>**, *<sup>a</sup>***r**) can be easily given by the following expression:

$$n\_{\mathbf{r}} \simeq \frac{1}{|\gamma''(\mathbf{r})|}. \tag{120}$$

Roughly speaking, the radius given by (120) quantifies the spatial extension of that characteristic phenomenon of field localization entailed by the presence in the medium Green function (118) of exponential factors like exp(−| *γ*|*r* ). Using the formula (116), the relation (120) becomes:

$$\left| a\_{\mathbf{r}} = \sqrt{\frac{2\hbar \left( m\_{\mathbf{e}}^{\*} (\mathbf{r}) \omega\_{\mathbf{e}} (\mathbf{r}) \right)}{1 - \frac{\omega^{2}}{\omega\_{\mathbf{e}}^{2} (\mathbf{r})} + \sqrt{\left( \frac{\omega^{2}}{\omega\_{\mathbf{e}}^{2} (\mathbf{r})} - 1 \right)^{2} + \frac{\Gamma^{2} (\mathbf{r})}{\omega\_{\mathbf{e}}^{2} (\mathbf{r})} \frac{\omega^{2}}{\omega\_{\mathbf{e}}^{2} (\mathbf{r})}}}}}}} \cdot \tag{121}$$

This expression (121) is illustrated with some basic examples as given in Figure 8b for various values of the crucial parameter Γ/*<sup>ω</sup>*e. When this ratio between the relaxation rate and the exciton transition frequency is small, the size of the EM nonlocality domain will increase due to the weakening of the corresponding nonlocality-based attenuation (field localization or confinement) processes. Conversely, one may control the size of each EM nonlocality microdomain *V***r** by modifying the ratio <sup>Γ</sup>(**r**)/*<sup>ω</sup>*e(**r**) evaluated at that position. This may provide a path toward an experimental realization of generalized nonlocal MTMs with controlled microtopological structures. In order to give a view on the numerical values of this structure, Table 2 provides some relevant microdomain data computed by means of the expression (121).

**Table 2.** Topological microdomain data at a generic position **r** ∈ D. The exciton transition frequency is *fe* = 23,862 THz (¯*h<sup>ω</sup>*e = 2.5 eV), while *m* e = 0.9 *m*e. For the left table, Γ/*<sup>ω</sup>*e = 2 × 10−5.


**Remark 18.** The approximation (120), strictly speaking, is not compatible with Definition 1 since the latter is based on assuming that the material response kernel possesses a compact support. However, for all practical purposes, a decaying exponential can be taken to approximate the behavior of a function with compact support. Nevertheless, in a more careful future treatment it is always possible to modify the exact Definition 1 in order to incorporate the decaying-exponential response kernel as another valid example of effective physics-based nonlocality mathematically realized by a topologically-localized function. An elementary discussion of some possible such modifications is given in Appendix A.10.

### **8. Application to Fundamental Theory: Electromagnetic Boundary Conditions in the Fiber Bundle Superspace Formalism**

Armed with the general superspace formalism of nonlocal continua (Section 5) and the detailed practical example illustrating the theory (Section 7), we now turn to a brief reexamination of a topic in fundamental theory: the role of boundary conditions in nonlocal continuum field theories. The well-known tension between nonlocal electromagnetism and intermaterial interfaces has been already mentioned several times above. Here, we provide some application of the fiber bundle theory of Section 5, aiming at elucidating the nature of this tension, and we sugges<sup>t</sup> some possible new formulation of the problem.

The natural starting point is Figure 2, where a zoomed-in topological picture based on the general structures explicated in Section 4 is given. The focus now is on the *interface* between two generic nonlocal domains D*n* and D*<sup>m</sup>*. In traditional local electromagnetism, the constitutive relation material tensor **K***n* is usually exploited to deduce conditions dictating how various electromagnetic field components behave as they cross the D*n*/D*m*

intermaterial interface. However, even if each response function **<sup>K</sup>***n*/*m*(**<sup>r</sup>**,**r** ) was to be treated as one belonging to a spatially dispersive domain, i.e., replacing it by **<sup>K</sup>***n*/*m*(**<sup>r</sup>** − **r** ), the presence of a boundary between two *distinct* material profiles completely destroys the *translational symmetry* of the structure on which the very rigorous derivation of the specific spatially-dispersive nonlocal response tensor **<sup>K</sup>***n*/*m*(**<sup>r</sup>** − **r** ) was originally based.

The breakdown of translational symmetry in inhomogeneous crystal configurations was very clearly identified and explained by Agranovich and Ginzburg [16], together with several proposals for a solution of such unusual electromagnetic problem. For example, because it is evident that close to the intermaterial interface the response tensor of each medium, when seen from its own side while approaching the boundary, must be reverted back to the most general nonlocal form, namely **<sup>K</sup>***n*/*m*(**<sup>r</sup>**,**r** ) instead of **<sup>K</sup>***n*/*m*(**<sup>r</sup>** − **r** ), it was then proposed that one may use the former, more general, functional form, but only within a "thin transitional layer" that includes the intermaterial interface, ye<sup>t</sup> while additionally extending, along some necessarily "ambiguous distance", into the depths of the two material domains D*n* and D*m* on both sides of the boundary. Outside this fuzzy region, a gradual transition, or a continuously changing profile (a tapered channel), is introduced to proceed from the most general forms **<sup>K</sup>***n*/*m*(**<sup>r</sup>**,**r** ), valid in the vicinity of the intermaterial interface, to the special spatially dispersive forms **<sup>K</sup>***n*/*m*(**<sup>r</sup>** − **r** ), which are more accurate the further one goes away from the material boundary, where the latter response tensor functions are considered characteristic of "bulk" homogeneous material domains [16].

Another proposal is to keep using everywhere spatial dispersion profiles of the form **<sup>K</sup>***n*(**<sup>r</sup>** − **r** ), but introduce specialized *additional boundary conditions* (ABCs) at the intermaterial interface based on each particular problem under consideration. Although this latter approach is both mathematically and physically inconsistent (due to the breakdown of symmetry caused by the presence of intermaterial interfaces), it nevertheless remains popular because—at least in outline—nonlocal electromagnetism is thereby held up in a form as close as possible to familiar local electromagnetic theory methods, especially numerical techniques, such as finite element method (FEM) [44], method of moment (MoM) [46], and finite difference time-domain method (FDTD) [45], i.e., established full-wave algorithms where it is quite straightforward to replace one boundary condition by another without essentially changing much of the code.<sup>27</sup>

Nevertheless, both approaches discussed above require considerable input from the *microscopic* theory, mainly to determine the tapering transition region in the case of the first, and the ABCs themselves in the second. That motivated the third approach, called, the ABC-free formalism, where the relevant microscopic theory was utilized right from the beginning in order to formulate and solve Maxwell's equations. For example, in [50,53], a global Hamiltonian of the matter-field system is constructed and Maxwell's equations are derived accordingly. In [38], the rim zone (field attached to matter) is investigated using different physical assumptions to understand the transition from nonlocal material domains to vacuum going through the entire complex near-field zone. In [89], the symmetry group of carbon nanotubes was exploited to construct a set of Maxwell's equations in nonlocal nanoscale problems without using a homogenized electromagnetic field-based boundary condition.

We believe that the main common conclusion from all these different formulations is that in nonlocal electromagnetism *it is not possible in general to formulate the electromagnetic problem at a fully phenomenological level.* In other words, microscopic theory appears to be in demand more often than in the case of systems involving only local materials. However, since all existing solutions use the traditional spatial manifold D as the main configuration space, the question now is whether the alternative formulation proposed in this paper, the extended fiber bundle superspace formalism, may provide some additional insights into the problem of why nonlocal continuum field theory cannot be formulated in general for inhomogeneous domains as in the local version of that theory.

We provide a provisional elucidation of the topological nature of field theory across intermaterial interfaces by noting that, in Figure 2, it is not only the behavior of the fields

**<sup>F</sup>**(**r**) in the two domains that is mostly relevant, but also the *entire* set of local topological microdomains *V***r** clustered on both sides of the interface inside the material domains. More specifically, we attach a grea<sup>t</sup> importance to *how these microtopological domains, together with the corresponding set of excitation fields that are applied on them, would behave as they move across the boundary*. In *general (set-theoretic)* topology, *boundaries* are defined fully in terms of the behavior of *open sets* [58,59].

We now build on this key set-theoretic topological concept in order to illustrate how the problem of nonlocal inhomogeneous continuum field theory may be reformulated through the superspace formalism developed in Section 5. First, Figure 9 provides a finer or more structured picture of the topological content of Figure 2 based on replacing the spaces D*m* and D*n* by the corresponding Banach bundle superspaces M*m* and M*<sup>n</sup>*, respectively. The thick horizontal curved lines represent the base spaces D*m* and D*<sup>n</sup>*, while the wavy vertical lines stands for the fiber spaces *Xm* and *Xn* attached at each point **r** ∈ D*n*/*m* in the corresponding base manifolds. The double discontinuous lines at the "junction" of the two base spaces D*m* and D*n* indicate the joining together of the two vector bundles M*m* and M*<sup>n</sup>*.

**Figure 9.** An abstract representation of the topological fiber bundle superspace structure behind Figure 2.

It should become clear now that since the two nonlocal material domains possess an *extra structure*, namely that of the individual copies of the fibers, each a linear vector Banach space attached to every point in the base space, we must also indicate how the various elements belonging to the Banach function spaces, i.e., the fields defined on the microdomains *V***r** in Figure 2,<sup>28</sup> would behave as they cross the boundary separating the two material domains D*m* and D*<sup>n</sup>*. One obvious way to do this is to introduce a *bundle homomorphism* between the two vector bundles M*m* and M*n* over the interface submanifold *∂*D*mn* separating D*m* and D*<sup>n</sup>*. This mathematical object is similar to the nonlocal response map L introduced by (72).

The motivation behind introducing this bundle homomorphism is to serve as a "boundary condition operator" acting on the fiber bundle *superspaces* M*m* and M*n* instead of the conventional spaces D*m* and D*n* always used in local continuum field theories. We will not go here into a detailed construction of such a new fiber bundle super-operator. Instead, we provide some additional remarks to illustrate the broad outline of the key idea behind our proposal. A more detailed investigation of the intermaterial interface homomorphism will be given somewhere else.

In continuum field theories, the formal expression of the traditional boundary condition applied to the two materials' D*n* and D*m* base spaces (spacetime, space–frequency, or space differential manifolds) will be summarized by the symbolic formula

$$\mathcal{D}\_{\mathfrak{m}} \rightleftharpoons \mathcal{D}\_{\mathfrak{n}} \tag{122}$$

in order to highlight that in such traditional formulation, it is the direct geometric relations between the individual material manifolds that usually holds the center stage. For example, the electromagnetism of continuous media, is usually spelled out in the more specific space-limit form:

$$\begin{aligned} \lim\_{\mathbf{r}\to\partial\mathcal{D}\_{\mathrm{mn}}} \left\{ \mathbf{F}\_{\mathrm{m}}(\mathbf{r}) - \mathbf{F}\_{\mathrm{n}}(\mathbf{r}) \right\} &= \Gamma\_{b\_{1}}[\mathbf{F}\_{\mathrm{m}}(\mathbf{r}), \mathbf{F}\_{\mathrm{n}}(\mathbf{r})], \\ \lim\_{\mathbf{r}\to\partial\mathcal{D}\_{\mathrm{mn}}} \left\{ \mathbf{R}\_{\mathrm{m}}(\mathbf{r}) - \mathbf{R}\_{\mathrm{n}}(\mathbf{r}) \right\} &= \Gamma\_{b\_{2}}[\mathbf{R}\_{\mathrm{m}}(\mathbf{r}), \mathbf{R}\_{\mathrm{n}}(\mathbf{r})], \end{aligned} \tag{123}$$

where *∂*D*mn* is the boundary between D*m* and D*<sup>n</sup>*. Here, <sup>Γ</sup>*b*1 and <sup>Γ</sup>*b*2 are "base space boundary functions", which are not universal, but whose detailed expressions depends on the concrete content of the field theory and the material system under consideration.

On the other hand, in the superspace formalism of nonlocal metamaterials and continua, it can be seen that the various elements belonging to each fiber space *Xn*/*m* attached at the point **r** ∈ D*n*/*m* of the base manifolds, i.e., the excitation field functions operating on the microdomains *V***<sup>r</sup>**, **r** ∈ D*n*/*<sup>m</sup>*, are to be mapped onto each other via an expression of the form:

$$X\_m \rightleftharpoons X\_n : \lim\_{\mathbf{r} \to \partial \mathcal{D}\_{mn}} (X\_m - X\_n) = \Gamma\_f [X\_m, X\_n]. \tag{124}$$

Here, Γ*f* is a new "fiber superspace boundary function". The full formulation of (124) is considerably more complex than the local field-theoretic case of (122) and (123) due to the fact that, additionally, the boundary condition quantity Γ*f* must be also proven compatible with the detailed corresponding fiber bundle structures of the materials involved. Consequently, for the field theoretic treatment of complex nonlocal continuum systems, the *global* topology of the metamaterial superspaces M*m* and M*n* will have to be assessed and utilized in the process of formulating a generalized "superspace boundary condition" of the form (124).

We summarize our main provisional view on the status of boundary conditions in the nonlocal field theory of inhomogeneous continua as follows:


However, despite the fact that the full mathematical formulation of the proposed fiber bundle boundary condition homomorphism (124) is beyond the scope of this paper, it is hopped that the initial insight provided in this section can at least clarify the subject and stimulate further researches into the fundamental theory of nonlocal continua and metamaterials. Additional possible applications are given in the Appendixes A.3 and A.11.
