Understanding the dispersive properties of photonic crystals is a fundamental and well-studied problem. However, the introduction of singular permittivities and damping complicates the otherwise straightforward theory. In this paper, we study photonic crystals with a Drude–Lorentz model for the permittivity, motivated by halide perovskites. We demonstrate how the introduction of singularities and damping affects the spectral band structure and show how to interpret the notion of a “band gap” in this setting. We study a one-dimensional model for which we present explicit solutions.

## I. INTRODUCTION

Photonic crystals present a variety of interesting and useful wave transmission properties and have become ubiquitous in wave physics. Even very simple photonic crystals, such as those composed of periodically alternating layers of non-dispersive materials, can display exotic dispersive properties. As a result, they are able to support band gaps: ranges of frequencies that are unable to propagate through the material.^{25} These band gaps are the fundamental building blocks of the many different wave guides and wave control devices that have been conceived. Notable examples include flat lenses,^{20} invisibility cloaks,^{17} rainbow trapping filters^{27} and topological waveguides.^{11}

When working at certain electromagnetic frequencies (which often includes the visible spectrum), it is important to take into account the oscillatory behaviour of the free electrons in a metal. This behaviour leads to resonances at characteristic frequencies and gives metals a highly dispersive character (even before the introduction of macroscopic structure, as in a photonic crystal). Several different models exist to describe this behaviour. Most models are variants of the Lorentz oscillator model, whereby electrons are modelled as damped harmonic oscillators due to electrostatic attractions with nuclei.^{15} A popular special case of this is the Drude model, in which case the restoring force is neglected (to reflect the fact that most electrons in metals are not bound to any specific nucleus, so lack a natural frequency of oscillation). Many other variants of these models exist, for instance by adding or removing damping from the various models, cf. Ref. 21 or Ref. 14, and by taking linear combinations of the different models, as in Ref. 22.

A key feature that unites dispersive permittivity models is the existence of singularities in the permittivity. The position of these poles in the complex plane, which correspond to resonances, are one of the crucial properties that determines how a metal interacts with an electromagnetic wave. In conventional Lorentz models the poles appear in the lower complex plane.^{15} The imaginary part of the singular frequency is determined by the magnitude of the damping, and the singularities accordingly fall on the real line if the damping is set to zero. In the Drude model, the removal of the restorative force causes the singularities to fall at the origin and on the negative imaginary axis.

A particularly important example of dispersive materials, that are central to the motivation for this study, are halide perovskites. They have excellent optical and electronic properties and are cheap and easy to manufacture at scale.^{16} As a result, they are being used in many applications, including optical sensors,^{10} solar cells^{24} and light-emitting diodes.^{28} The dielectric permittivity of halide perovskites has been shown to depend heavily on excitonic transitions, leading to a permittivity that has symmetric poles in the lower complex plane.^{16}

There is a range of methods that can be used to capture the spectra of photonic crystals. For one-dimensional systems, explicit solutions typically exist and transfer matrices are particularly convenient. These were used for Drude materials in Ref. 23 and for undamped Lorentz materials in Ref. 14, for example. In multiple dimensions, studies often resort to numerical simulation (for instance with finite elements). A valuable approximation strategy is a multi-scale asymptotic method known as high-frequency homogenisation,^{9} which can be extended to approximate the dispersion curves in dispersive media.^{26}

In this work, we will study photonic crystals composed of metals with permittivity inspired by that of halide perovskites, in the sense that it has symmetric poles in the lower complex plane. After setting out the Floquet–Bloch formulation of the periodic problem in Sec. II, we will study the one-dimensional periodic Helmholtz problem in Subsection II D. In Sec. III, we retrieve the dispersion relation which characterizes the halide perovskite system and in Sec. IV, we show how its properties depend on the characteristics of the dispersive permittivity (namely, being real or being complex and having poles either on or below the real axis).

## II. PROBLEM SETTING

### A. Initial Helmholtz formulation

*D*

_{1},

*D*

_{2}, …,

*D*

_{N}which together occupy a bounded domain $\Omega \u2282Rd$, for

*d*∈ {1, 2, 3}. The collection of particles Ω will be the repeating unit of the periodic photonic crystal. We suppose that permittivity of the particles is given by a Drude–Lorentz-type model, given by

*ɛ*

_{0}denotes the background dielectric constant and

*α*,

*β*,

*γ*are positive constants.

*α*describes the strength of the interactions,

*β*determines the natural resonant frequency and

*γ*is the damping factor. This is motivated by the measured permittivity of halide perovskites, as reported in Ref. 16. We choose to use this expression as a canonical model for dispersive materials whose permittivities have singularities in the complex frequency space. Notice that (2.1) is singular at two complex values of

*ω*. These are given by

*α*,

*β*and

*γ*we can force these singularities to lie in the lower half of the complex plane (

*γ*> 0), on the real line (

*γ*= 0) or to vanish completely (

*β*=

*γ*= 0). We will make use of this property when trying to interpret the dispersion diagrams we obtain in the following analysis. We suppose that the particles are surrounded by a non-dispersive medium with permittivity

*ɛ*

_{0}. We assume that the particles are non-magnetic, meaning the magnetic permeability

*μ*

_{0}is constant on all of $Rd$.

*ω*. This is a reasonable model for the scattering of transverse magnetic polarised light (see e.g., Ref. 18, Remark 2.1 for a discussion). The wavenumber in the background $Rd\\Omega \u0304$ is given by

*k*

_{0}≔

*ωɛ*

_{0}

*μ*

_{0}and we will use

*k*to denote the wavenumber within Ω. Let us note here that, from now on, we will suppress the dependence of

*k*

_{0}and

*k*on

*ω*for brevity. We, then, consider the system of equations

*u*

_{in}is the incident wave, assumed to satisfy $(\Delta +k02)uin=0$, and the appropriate outgoing radiation condition depends on the dimension of the problem and of the periodic lattice.

### B. Periodic formulation

*N*particles is repeated in a periodic lattice Λ. We suppose that the lattice has dimension

*d*

_{l}, in the sense that there are lattice vectors $l1,\u2026,ldl\u2208Rd$ which generate Λ according to

*N*=6 particles with lattice dimension

*d*

_{l}=1 is shown in Fig. 1. The dual lattice of Λ, denoted by Λ*, is generated by the vectors $\alpha 1,\u2026,\alpha dl$ satisfying

*α*

_{i}·

*l*

_{j}= 2

*πδ*

_{ij}for

*i*,

*j*= 1, …,

*d*

_{l}. Finally, the

*Brillouin zone*

*Y** is defined by

**0**is the zero vector in $Rd\u2212dl$. The Brillouin zone

*Y** is the space that the reduced unit cell of reciprocal space.

### C. Floquet–Bloch theory

In order to study the problem (2.7), we will make use of Floquet–Bloch theory.^{13} Let us first give certain definitions which will help with the analysis of the problem.

*A function* $f(x)\u2208L2(Rd)$ *is said to be* *κ**-quasiperiodic, with quasiperiodicity* *κ* ∈ *Y***, if* *e*^{−iκ·x}*f*(*x*) *is* Λ*-periodic.*

*(Floquet transform). Let*$f\u2208L2(Rd)$

*. The Floquet transform of*

*f*

*is defined as*

*κ*-quasiperiodic in

*x*and periodic in

*κ*. The Floquet transform is an invertible map $F:L2(Rd)\u2192L2(Y\xd7Y*)$, with inverse given by

*g*(

*x*,

*κ*) is extended quasiperiodically for

*x*outside of the unit cell

*Y*.

*κ*-quasiperiodic radiation condition is subtle to define in general, particularly since it depends on the dimensions of both the lattice and the physical space. The condition is required so that the solution (which, of course, is

*κ*-quasiperiodic in the

*d*

_{l}directions in which the lattice is periodic) contains only outgoing radiation in the

*d*−

*d*

_{l}directions perpendicular to the axes of periodicity. See, for example Refs. 4–7, for more details. Clearly, as we will see in Sec. II D, this condition is not needed for the one-dimensional periodic transmission problems that will be the centrepiece of this work.

The solutions to (2.8) typically take the form of a countable collection of spectral bands, each of which depends continuously on the Bloch parameter *κ*. The goal of our analysis is identifying and explaining the gaps between the spectral band. At frequencies within these band gaps, waves do not propagate in the material and their amplitude decays exponentially. As a result, they are the starting point for building waveguides and other wave control devices.

For real-valued permittivities, it is straightforward to define band gaps as the intervals between the real-valued bands.

*(Band gap for real permittivities). A frequency* $\omega \u2208R$ *is said to be in a band gap of the periodic structure* $D$ *with real permittivities if it is such that (2.8) admits non-trivial solutions only for purely imaginary Bloch parameter* *κ*.

We are interested in materials for which the permittivity takes complex values, corresponding to the introduction of damping to the model. We elect to keep the frequency $\omega \u2208R$ as a real number but allow the Bloch parameter *κ* to take complex values. In which case, the imaginary part of *κ* describes the rate at which the waves amplitude decays as it propagates in space. This choice is made for mathematical convenience (as will become clear below) and can be understood as encoding the attention of a wave propagating through the system as a function of space rather than time. It should be noted that it is also quite common to do the opposite and force *κ* to be real while allowing *ω* to be complex valued. This was done, for example in Refs. 3 and 26, and corresponds to viewing the attenuation as a function of time rather than space. Different conventions are typically adopted by different communities. For the one-dimensional problems considered in Sec. II D, it is clear that both approaches are mathematically equivalent.

In the real-valued case, it is clear that *κ* belongs to the Brillouin zone *Y** (which has the topology of a torus, due to the periodicity in *κ*). When *κ* is complex valued, its real part still lives in *Y** but its imaginary part can take arbitrary values. Thus, *κ* lives in a subset of the complex plane that is isomorphic to $Y*\xd7R$. This can be thought of as a “generalised” Brillouin zone; this idea has been used to describe the spectral convergence of non-Hermitian systems in Ref. 2.

When there is damping in a system (characterized by a complex permittivity) and the frequency *ω* is chosen to be real, the Bloch parameter *κ* will typically have a non-zero imaginary part. As a result, it is less clear how to distinguish between spectral bands and band gaps. Nevertheless, we can understand a band gap as a region of frequency space in which the wave experiences attenuation beyond the material damping alone, due to the geometric structure. In which case, the imaginary part of *κ* will experience a local peak. With this in mind, we make the following definition of a band gap for a system with complex material parameters.

*(Band gap for complex permittivities). We define a band gap for complex permittivities to be frequencies* $\omega \u2208R$ *for which (2.8) admits a non-trivial solution with quasiperiodicity* $\kappa \u2208C$ *that is such that* $|I(\kappa )|$ *is in a neighbourhood of a local maximum.*

We will study the problem in the one-dimensional setting. In one dimension, the problem is easier to manipulate and we are able to retrieve explicit expressions. Hence, we can get a variety of results concerning the characteristics of the quasiperiodic system. In particular, our main goal is to obtain the dispersion relation, an expression which relates the quasiperiodicities $\kappa \u2208C$ with the frequencies $\omega \u2208R$, and study its properties.

### D. One dimension

## III. DISPERSION RELATION

### A. Analytic expression

*ρ*as

*Let*

*u*

*denote a solution to (2.9). Then,*

*u*

*is given by*

*where*$A,B\u2208C$

*are two constants.*

Using the boundary conditions (2.11), we can obtain the dispersion relation for the one-dimensional problem. This is a well-known result, that first appeared in a quantum-mechanical setting^{12} and has since been shown to describe a range of periodic classical wave systems also Refs. 1 and 19. We include a brief proof, for completeness.

*(Dispersion relation). Let*

*u*

*denote the solution to (2.9) along with the boundary conditions (2.11). Then, for*

*u*

*to be non-trivial, the quasiperiodicities*$\kappa \u2208C$

*satisfies the dispersion relation*

*u*is given by (3.3). Then, using (2.11), we have

*σ*

_{c}=

*ρσ*

_{0}and $\epsilon (\omega )=\rho \epsilon 0$, we obtain the desired result.□

The dispersion relation (3.5) can be used to plot the dispersion curves. For a given frequency *ω*, *ρ*(*ω*) can be calculated to yield the right hand side of (3.5), which can subsequently be solved to find *κ*. This is shown in Fig. 3. Since *ɛ*(*ω*) is complex valued, *κ* will generally take complex values. We plot only the absolute values of both the real and imaginary parts; as we will see below, this is sufficient to characterise the full dispersion relation. Notice also that $R(\kappa )\u2208Y*=[\u2212\pi /2,\pi /2)$.

### B. Properties of the dispersion relation

The dispersion relation (3.5) describes the behaviour of the periodic system and reveals the relationship between the quasiperiodicities $\kappa \u2208C$, the frequencies $\omega \u2208R$ and the permittivity *ɛ*(*ω*) of the material. We can use it to derive some simple results about the dispersion curves. The first thing to understand is the symmetries of the dispersion curves.

*(Opposite quasiperiodicities). Let* $\kappa \u2208C$ *be a complex quasiperiodicity satisfying the dispersion relation (3.5) for a given frequency* $\omega \u2208R$*. Then, the opposite quasiperiodicity, i.e.,* −*κ**, satisfies the same dispersion relation.*

It is with Lemma 3.3 in mind that we are able to plot only the absolute values of the imaginary parts in Fig. 3 and the subsequent figures.

#### 1. Real and imaginary parts

*κ*and

*ρ*into real and imaginary parts, and we will derive this dependence from the dispersion relation. Since $\kappa \u2208C$ and $\rho \u2208C$, let us define:

*ρ*

_{1},

*ρ*

_{2}and

*σ*

_{0}all depend on the frequency

*ω*, as specified in (3.1) and (3.2). Then, we have the following result.

*Let*$\kappa \u2208C$

*, given by (3.11), satisfying the dispersion relation (3.5) for a given frequency*$\omega \u2208R$

*. Then, its real and imaginary parts are given by*

*and*

*where*$L1$

*and*$L2$

*are given by (3.12)*

*and (3.13), respectively. We, also, note that the choice of*+

*or*−

*should be the same in*(3.14) and (3.15)

*.*

*x*∈ [−1, 1], we have the identity $sin[arccos(x)]=1\u2212x2$. Hence, from the above, we get

^{2}(

*x*) − sinh

^{2}(

*x*) = 1 for $x\u2208R$, we find that

#### 2. Imaginary part decay

From (3.15), we obtain a result on the decay of the imaginary part of the quasiperiodicity *κ* as *ω* → *∞*. We will first state some preliminary results, before proving the main theorem.

*Let the frequency-dependent contrast*$\rho \u2208C$

*be given by (3.2). Then, it holds*

*and*

*ω*→

*∞*,

*α*,

*γ*,

*ɛ*

_{0}> 0, it holds that

*ω*→

*∞*,

*σ*

_{0}

*ρ*

_{2}as

*ω*→

*∞*. Using the same notations as in the Proof of Lemma 3.5, we have, without loss of generality on the ± of (3.24),

*ω*→

*∞*,

*ω*→

*∞*,

Using these results, we will describe the behaviour of the imaginary part *κ*_{2} of the quasiperiodicity $\kappa \u2208C$ as the frequency tends to infinity, i.e., *ω* → *∞*.

*Let us consider a complex quasiperiodicity*$\kappa \u2208C$

*satisfying the dispersion relation (3.5) with*$\alpha ,\beta ,\gamma \u2208R>0$

*. Then, it holds that*

*κ*

_{2}is given by

*ω*→

*∞*, $L1$ remains bounded, whereas $L2\u21920$. Thus, the following holds

The decay predicted by Proposition 3.7 is shown in Fig. 3. Due to the damping in the model, the imaginary part has discernible peaks at the first few gaps, but then decays steadily to zero at higher frequencies.

## IV. THE EFFECT OF SINGULARITIES AND DAMPING

As mentioned before, the dispersion relation of the halide perovskite particles leads to dispersion curves which are not trivial to understand in terms of the traditional viewpoint of band gaps. In order to understand the behaviour, we will examine each distint feature of the halide perovskite permittivity, to understand the effect it has on the spectrum of the photonic crystal.

In particular, we will begin with the simplest case when the permittivity is real and constant with respect to the frequency *ω*. Then, we will introduce a dispersive behaviour to the permittivity by adding singularities at non-zero frequencies. We will initially suppose that these poles lie on the real axis and will study the behaviour close to these regions. Finally, we will study the effect of introducing a complex permittivity, corresponding to damping. Taken together, these results will allow us to explain the spectra the halide perovskite photonic crystal.

### A. Constant permittivity

*ω*. In our setting this translates into having (2.1) with

*β*=

*γ*= 0 and

*α*> 0, i.e.,

*κ*=

*κ*

_{1}+

*iκ*

_{2}, with $\kappa 1,\kappa 2\u2208R$. Now, since

*ρ*> 0, we can write

*κ*

_{2}= 0, then $\kappa =\kappa 1\u2208R$. Thus, $\kappa =\kappa \u0304$, which gives the desired result. If $\kappa 1=m2\pi $, for $m\u2208Z$, then

*κ*satisfies (3.5) if and only if

*κ*

_{2}satisfies (4.2) for the same frequency

*ω*. Hence, $\kappa \u0304=\kappa 1\u2212i\kappa 2$ satisfies (3.5) and this concludes the proof.□

Crucially, the dispersion curves shown in Fig. 4 consist of a countable sequence of disjoint bands in which *κ* is real valued. Between each band there is a band gap, defined in the sense of Definition 2.3, in which *κ* is purely imaginary, corresponding to the decay of the wave. The occurence of *κ* being either purely real or purely imaginary is the mechanism behind Lemma 4.1. As we will see below, when we add singularities or damping to the model, the band gap structure is less straightforward to interpret.

### B. Singular permittivity with no damping

*ω*, but there is no damping, i.e., we consider

*α*,

*β*> 0 and

*γ*= 0. This implies that

*Let* $x,y\u2208R$*. Then, we use* *x* ↓ *y* *when* *x* → *y* *and* *x* > *y**. Similarly, we use* *x* ↑ *y* *when* *x* → *y* *and* *x* < *y*.

Then, close to a permittivity pole, the following holds.

*Let* *ω** *denote a pole of the permittivity* *ɛ*(*ω*) *given by (4.3), i.e.,* $\omega *\u2208\xb11\beta $. *Then, for* *δ* > 0*, the intervals* [*ω** − *δ*, *ω**) *and* (*ω**, *ω** + *δ*] *contain infinitely many disjoint sub-intervals, denoted by* $Ii$ *and* $Ji$*,* *i* = 1, 2, …*, respectively, that are band gaps.*

*ω** −

*δ*,

*ω*), for

*δ*> 0. It suffices to show that there are infinitely many points $\omega i\u2020\u2208[\omega *\u2212\delta ,\omega *)$,

*i*= 1, 2, …, for

*δ*> 0 for which $f(\omega i\u2020)>1$ or $f(\omega i\u2020)<\u22121$. Then, the continuity of

*f*around these points gives us the existence of intervals of the form $Ii\u2254[\omega i\u2020\u2212s,\omega i\u2020+s]$, for

*i*= 1, 2, …, for small

*s*> 0, such that,

*i*= 1, 2, … being band gaps, since

*κ*becomes complex in these intervals, i.e., $|I(\kappa )|\u22600$. Hence, since $|I(\kappa )|$ is continuous with respect to

*ω*, we get that it has a local maximum in each of the $Ii$’s, for

*i*= 1, 2, ….

*ω*↑

*ω**, we have that

*σ*

_{0}is constant and so, without loss of generality, we can assume that sin(

*σ*

_{0}) > 0 [the same argument holds for taking sin(

*σ*

_{0}) < 0]. Hence, there exists

*δ*

_{1}> 0 such that for all

*ω*∈ [

*ω** −

*δ*

_{1},

*ω**), we have that

*K*> 0, there exists

*δ*

_{2}> 0 such that for all

*ω*∈ [

*ω** −

*δ*

_{2},

*ω**) it holds that |

*ρ*(

*ω*)

*σ*

_{0}(

*ω*)| >

*K*.

*δ*≔ max{

*δ*

_{1},

*δ*

_{2}} and let $I\delta (\u2212)\u2254[\omega *\u2212\delta ,\omega *)$. Then, (4.6) implies that the exist two families of infinitely many points in $I\delta (\u2212)$, denoted by ${\omega i(+)}i=1,\u2026,\u221e$ and ${\omega i(\u2212)}i=1,\u2026,\u221e$, such that, for all

*i*= 1, …,

*∞*, we have

*i*= 1, …,

*∞*, that

*i*= 1, …,

*∞*, we have

*f*around these points allows us to take

*s*> 0 such that

We note that for the neighborhood of the form *ω* ∈ (*ω**, *ω** + *δ*] for *δ* > 0, the proof remains the same with the slight change of taking the limits as *ω* ↓ *ω**.□

In addition to the occurrence of a countable number of band gaps close to the pole, in Fig. 5, we observe that there is an interesting behaviour of the imaginary part $I(\kappa )$ of the quasiperiodicity *κ* as the frequency $\omega \u2208R$ approaches a permittivity pole. In fact, we see that close to a pole, $|I(\kappa )|$ becomes arbitrarily big. This due to the resonance occurring here and is strongly related to the existence of infinitely many band gaps close to the pole. Actually, it is a corollary of Theorem 4.2.

*Let* $\omega \u2208R$ *and* $\kappa \u2208C$ *be the associated quasiperiodicity satisfying the dispersion relation (3.5) and let* *f*(*ω*) *be the function defined in (4.4). Let* $\omega *\u2208R$ *denote a pole of the undamped permittivity* *ɛ*(*ω*)*, given by (4.3). Then, for all* *K* > 0*, there exists* *δ* > 0*, such that for all* *p* ∈ [−*K*, *K*]*, there exists* *ω*_{p} ∈ [*ω** − *δ*, *ω**) *such that* $I(\kappa (\omega p))=p$*. That is,* $|I(\kappa (\omega ))|$ *takes arbitrarily large values as* *ω* ↑ *ω***. The same result holds as* *ω* ↓ *ω**.

*K*> 0, there exists

*δ*> 0, such that, for all

*ω*∈ [

*ω** −

*δ*,

*ω*),

*σ*

_{0}(

*ω*))cos(

*ρ*(

*ω*)

*σ*

_{0}(

*ω*)) remains bounded close to

*ω** and because of the continuity of sin(·), we can take sin(

*σ*

_{0}(

*ω*)) > 0 in [

*ω** −

*δ*,

*ω*). Then, it follows that, for all

*K*> 0, in [

*ω** −

*δ*,

*ω*),

*ω** −

*δ*,

*ω*), for which, $sin\rho (\omega )\sigma 0(\omega )$ oscillates between 1 and −1 in each of the intervals of the form $[\omega 0(+),\omega 0(\u2212)]$, $[\omega 0(\u2212),\omega 1(+)]$, …, denoted by $Ii$,

*i*= 1, 2, …. This implies that, for all

*K*> 0, for all

*p*∈ [−

*K*,

*K*], there exists $\omega p\u2208Ii$, for

*i*= 1, 2, …, such that

*f*(

*ω*

_{p}) =

*p*. Since this holds for all

*K*> 0, it translates to

*f*oscillating and taking all values between +

*∞*and −

*∞*in [

*ω** −

*δ*,

*ω**) as we get closer to

*ω**.

*ω** −

*δ*,

*ω**), the function

*f*(

*ω*) oscillates between +

*∞*and −

*∞*, we have that $f(\omega )\xb1f(\omega )2\u22121$ has the same behaviour in [

*ω** −

*δ*,

*ω**), but the oscillation takes place between 0 and +

*∞*. Finally, since ln(·) is an increasing function, we obtain the desired result. Let us note that the proof is the same when we consider

*ω*↓

*ω**, with the slight change that we consider neighborhoods of the form (

*ω**,

*ω** +

*δ*].□

### C. Complex permittivity

*β*=

*γ*= 0, i.e.,

*α*with successively larger imaginary parts, to show the effect of gradually increasing the damping. We see that the clear structure of successive bands and gaps is gradually blurred out, eventually to the point that the spectrum bears no clear relation to the original undamped spectrum.

It is noticeable in Fig. 6 that, although the introduction of damping has distorted the curves (compared to those for the undamped system in Fig. 4), there are still intervals of frequency in which behaviour reminiscient of a band gap will be observed. That is, in Figs. 6(a)–6(c) there are regions where the imaginary part of the Bloch parameter has clear peaks. As a result, a wave propagating through the system will experience stronger attenuation at these frequencies. This is due to effects from the periodic structure of the material, on top of the damping in the materials parameters, so it makes sense to refer to these regions as band gaps, as per the Definition 2.4.

In Fig. 6(d), the material damping is sufficiently large that its effect dwarfs any attenuation due to band gap effects. As a result, there are only minimal fluctuations of $I(\kappa )$ in the corresponding regions and no local maxima. Hence, in the sense of Definition 2.4, there are no band gaps when the damping is too large. This aligns with what would be experienced in practice, in settings where the material damping is so large that any damping due to band-gap effects cannot be observed.

In many ways, the spectrum we obtain in this setting appears to be similar to the actual halide perovskite particles, as plotted in Fig. 3. Indeed, all of the results proved in Subsection III B hold, with the exception of the imaginary part decay. In fact, the converse is true, as made precise by the following result.

*κ*

_{2}, denoting the imaginary part of $\kappa \u2208C$, is given by (3.15), where $L1$ and $L2$ are given by (3.12) and (3.13), respectively. We have that

*σ*

_{0}is linear with respect to

*ω*and

*ρ*does not depend on

*ω*in this setting. This implies

*κ*

_{2}| → +

*∞*as

*ω*→ +

*∞*. Indeed, (3.12) gives that

*ω*→ +

*∞*,

*C*

_{1}and

*C*

_{2}are both bounded and

*C*

_{1}−

*C*

_{2}≠ 0. Then, from (4.15), we get that

*ω*→ +

*∞*, $C\u03031$ and $C\u03032$ are both bounded and $C\u03031\u2212C\u03032\u22600$. Then, from (4.15), we get that

### D. Discussion

The analysis in this section can be used to understand the dispersion diagram for the halide perovskite photonic crystal that was presented in Fig. 3. There are two crucial observations. First, we saw in Sec. IV B that the introduction of singularities in the permittivity led to the creation of countably infinitely many band gaps in a neigbourhood of the pole, when the pole falls on the real axis. However, this exotic behaviour is not seen in Fig. 3, due in part to the introduction of damping causing the poles to fall below the real axis. This effect can also be exaplined in terms of the results in Sec. IV C, where we saw that the introduction of damping to a simple non-dispersive model smoothed out the band gaps. The behaviour shown in Fig. 3 is a combination of these phenomena.

## V. CONCLUSION

We have used analytic methods to understand the dispersive nature of photonic crystals fabricated from metals with singular permittivities. In particular, we considered a Drude–Lorentz model inspired by halide perovskites that has poles in the lower complex plane. For a one-dimensional system, we characterised the effect that each feature of this model has on the dispersion relation. In particular, we showed that the introduction of singularities leads to the creation of countably many band gaps near the poles, whereas the introduction of damping smooths out the band gap structure.

## ACKNOWLEDGMENTS

The work of K.A. was supported by ETH Zürich under the Project No. ETH-34 20-2. The work of B.D. was funded by the Engineering and Physical Sciences Research Council through a fellowship with Grant No. EP/X027422/1.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Konstantinos Alexopoulos**: Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal). **Bryn Davies**: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Zenodo at http://doi.org/10.5281/zenodo.8055547. No other data were generated in this project.

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*Mathematical and Computational Methods in Photonics and Phononics*

*Mathematical Surveys and Monographs Vol. 235*

*Electromagnetic Theory of Gratings*

*Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices*

*Plasmonics: Fundamentals and Applications*

*Asymptotic Models of Fields in Dilute and Densely Packed Composites*

*Photonic Band Gap Materials*