The effect of singularities and damping on the spectra of photonic crystals

Alexopoulos, Konstantinos; Davies, Bryn

doi:10.1063/5.0164213

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 filters²⁷ 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 cells²⁴ 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

Let us consider

N \in N

particles D₁, D₂, …, D_N which together occupy a bounded domain

Ω \subset R^{d}

⁠, 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

ε (ω) = ε_{0} + \frac{α}{1 - β ω^{2} - i γ ω},

(2.1)

where ɛ₀ 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 is singular at two complex values of ω. These are given by

ω_{\pm}^{*} = \frac{1}{2 β} (- i γ \pm \sqrt{4 β - γ^{2}}) .

(2.2)

By varying the parameters 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 ɛ₀. We assume that the particles are non-magnetic, meaning the magnetic permeability μ₀ is constant on all of

R^{d}

⁠.

We consider the Helmholtz equation as a model for the propagation of time-harmonic waves with frequency 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

R^{d} \ \bar{Ω}

is given by k₀ ≔ ωɛ₀μ₀ 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 k₀ and k on ω for brevity. We, then, consider the system of equations

\{\begin{cases} Δ u + ω^{2} ε (ω) μ_{0} u = 0 & in Ω, \\ Δ u + k_{0}^{2} u = 0 & in R^{d} \ \bar{Ω}, \\ u |_{+} - u |_{-} = 0 & on \partial Ω, \\ \frac{\partial u}{\partial ν} |_{+} - \frac{\partial u}{\partial ν} |_{-} = 0 & on \partial Ω, \\ u (x) - u_{i n} (x) & satisfies the outgoing radiation condition as | x | \to \infty, \end{cases}

(2.3)

where u_in is the incident wave, assumed to satisfy

(Δ + k_{0}^{2}) u_{i n} = 0

⁠, and the appropriate outgoing radiation condition depends on the dimension of the problem and of the periodic lattice.

B. Periodic formulation

We will assume that the collection of 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

l_{1}, \dots, l_{d_{l}} \in R^{d}

which generate Λ according to

Λ ≔ {m_{1} l_{1} + \dots + m_{d_{l}} l_{d_{l}} | m_{i} \in Z} .

(2.4)

The fundamental domain of the lattice Λ is the set

Y \in R^{d}

given by

Y ≔ {c_{1} l_{1} + \dots + c_{d_{l}} l_{d_{l}} | 0 \leq c_{1}, \dots, c_{d_{l}} \leq 1} .

(2.5)

An example of a lattice of 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

α_{1}, \dots, α_{d_{l}}

satisfying α_i · l_j = 2πδ_ij for i, j = 1, …, d_l. Finally, the Brillouin zone Y* is defined by

Y^{*} ≔ (R^{d_{l}} \times {0}) / Λ^{*},

(2.6)

where 0 is the zero vector in

R^{d - d_{l}}

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

FIG. 1.

A periodic array of halide perovskite particles. Here, we have six particles D1, …, D6 repeated periodically in one dimension. Each of them has the halide perovskite permittivity ɛ(ω) defined by (2.1).

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A periodic array of halide perovskite particles. Here, we have six particles D₁, …, D₆ repeated periodically in one dimension. Each of them has the halide perovskite permittivity ɛ(ω) defined by (2.1).

The periodic structure, denoted by

D

⁠, is given by

D = ⋃_{i = 1}^{N} (⋃_{m \in Λ} D_{i} + m) .

Hence, the problem we wish to study is the following:

\{\begin{cases} Δ u + ω^{2} ε (ω) μ_{0} u = 0 & in D, \\ Δ u + k_{0}^{2} u = 0 & in R^{d} \ \bar{D}, \\ u |_{+} - u |_{-} = 0 & on \partial D, \\ \frac{\partial u}{\partial ν} |_{+} - \frac{\partial u}{\partial ν} |_{-} = 0 & on \partial D, \\ u (x_{l}, x_{0}) & satisfies the outgoing radiation condition as | x_{0} | \to \infty . \end{cases}

(2.7)

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.

Definition 2.1.

A function $f (x) \in L^{2} (R^{d})$ is said to be κ-quasiperiodic, with quasiperiodicity κ ∈ Y*, if e^−iκ·xf(x) is Λ-periodic.

Definition 2.2

(Floquet transform). Let

f \in L^{2} (R^{d})

. The Floquet transform of f is defined as

F [f] (x, κ) ≔ \sum_{m \in Λ} f (x - m) e^{- i κ \cdot x}, x \in R^{d}, κ \in Y^{*} .

We have that

F [f]

is κ-quasiperiodic in x and periodic in κ. The Floquet transform is an invertible map

F : L^{2} (R^{d}) \to L^{2} (Y \times Y^{*})

⁠, with inverse given by

F^{- 1} [g] (x) = \frac{1}{| Y_{l}^{*} |} \int_{Y^{*}} g (x, κ) d κ, x \in R^{d},

where g(x, κ) is extended quasiperiodically for x outside of the unit cell Y.

Let us define

u^{κ} (x) ≔ F [u] (x, κ)

⁠. Then, applying the Floquet transform to , we obtain the following system:

\{\begin{cases} Δ u^{κ} + ω^{2} ε (ω) μ_{0} u^{κ} = 0 & in D, \\ Δ u^{κ} + k_{0}^{2} u^{κ} = 0 & in R^{d} \ \bar{D}, \\ u^{κ} |_{+} - u^{κ} |_{-} = 0 & on \partial D, \\ \frac{\partial u^{κ}}{\partial ν} |_{+} - \frac{\partial u^{κ}}{\partial ν} |_{-} = 0 & on \partial D, \\ u^{κ} (x_{l}, x_{0}) & is κ - quasiperiodic in x_{l}, \\ u^{κ} (x_{l}, x_{0}) & satisfies the κ - quasiperiodic radiation condition as | x_{0} | \to \infty . \end{cases}

(2.8)

The appropriate 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.

Definition 2.3

(Band gap for real permittivities). A frequency $ω \in R$ 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 $ω \in R$ 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 $Y^{*} \times R$ ⁠. 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.

Definition 2.4

(Band gap for complex permittivities). We define a band gap for complex permittivities to be frequencies $ω \in R$ for which (2.8) admits a non-trivial solution with quasiperiodicity $κ \in C$ that is such that $| I (κ) |$ 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 $κ \in C$ with the frequencies $ω \in R$ ⁠, and study its properties.

D. One dimension

Let us treat the Helmholtz problem in the one-dimensional case. We will work on the interval [−1, 1], with [−1, 0) denoting the background and [0, 1) the particle. A schematic depiction of this is given in Fig. 2. Hence, the problem reads as follows:

\frac{d}{d x} (\frac{1}{ε (x, ω)} \frac{d u}{d x}) + ω^{2} μ_{0} u (x) = 0,

(2.9)

on the domain [−1, 1], where

ε (x, ω) ≔ \{\begin{cases} ε_{0}, & x \in [- 1, 0) (background), \\ ε (ω), & x \in [0, 1] (particle), \end{cases}

(2.10)

with the boundary conditions

u (1) = e^{2 i κ} u (- 1) and \frac{d u}{d x} (1) = e^{2 i κ} \frac{d u}{d x} (- 1) .

(2.11)

FIG. 2.

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The one-dimensional setting. The periodically repeated cell is of length 2. Here the interval [−1, 0) is the background and the interval [0, 1) is the particle.

III. DISPERSION RELATION

A. Analytic expression

We will now retrieve an expression for the solution to . Let us define the quantities

σ_{0} ≔ ω \sqrt{ε_{0} μ_{0}}, and σ_{c} ≔ ω \sqrt{ε (ω) μ_{0}} .

(3.1)

For many of the results that follow, the crucial quantity will be the contrast between the material inside the particles and the background medium. With this in mind, we introduce the frequency-dependent contrast ρ as

ρ (ω) ≔ \frac{σ_{c}}{σ_{0}} = \sqrt{\frac{ε (ω)}{ε_{0}}} .

(3.2)

Then, the following expression holds for the solution to .

Lemma 3.1.

Let u denote a solution to (2.9). Then, u is given by

u (x) = \{\begin{cases} A ρ \sin (σ_{0} x) + B \cos (σ_{0} x), & x \in [- 1, 0) \\ A \sin (σ_{c} x) + B \cos (σ_{c} x), & x \in [0, 1], \end{cases}

(3.3)

where

A, B \in C

are two constants.

Proof.

We know that a solution to must be given by

u (x) = \{\begin{cases} A_{1} \sin (ω \sqrt{ε_{0} μ_{0}} x) + B_{1} \cos (ω \sqrt{ε_{0} μ_{0}} x), & x \in [- 1, 0), \\ A_{2} \sin (ω \sqrt{ε (ω) μ_{0}} x) + B_{2} \cos (ω \sqrt{ε (ω) μ_{0}} x), & x \in [0, 1], \end{cases}

(3.4)

where

A_{1}, A_{2}, B_{1}, B_{2} \in C

are constants to be defined. This, also, gives

\frac{d u}{d x} (x) = \{\begin{cases} A_{1} ω \sqrt{ε_{0} μ_{0}} \cos (ω \sqrt{ε_{0} μ_{0}} x) - B_{1} ω \sqrt{ε_{0} μ_{0}} \sin (ω \sqrt{ε_{0} μ_{0}} x), & x \in [- 1, 0), \\ A_{2} ω \sqrt{ε (ω) μ_{0}} \cos (ω \sqrt{ε (ω) μ_{0}} x) - B_{2} ω \sqrt{ε (ω) μ_{0}} \sin (ω \sqrt{ε (ω) μ_{0}} x), & x \in [0, 1] . \end{cases}

Now, from the boundary transmission conditions in , we require

\lim_{x \to 0^{-}} u (x) = \lim_{x \to 0^{+}} u (x) and \lim_{x \to 0^{-}} \frac{d u}{d x} (x) = \lim_{x \to 0^{+}} \frac{d u}{d x} (x) .

These conditions mean we must have that

B_{1} = B_{2}

and

A_{1} = \sqrt{ε (ω) / ε_{0}} A_{2} = ρ (ω) A

⁠, which gives the desired result.□

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 setting¹² 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.

Theorem 3.2

(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

κ \in C

satisfies the dispersion relation

\cos (2 κ) = \cos (σ_{0}) \cos (ρ σ_{0}) - \frac{1 + ρ^{2}}{2 ρ} \sin (σ_{0}) \sin (ρ σ_{0}) .

(3.5)

Proof.

From Lemma 3.1, we have that u is given by . Then, using , we have

\{\begin{cases} [\sin (σ_{c}) + e^{2 i κ} ρ \sin (σ_{0})] A + [\cos (σ_{c}) - e^{2 i κ} \cos (σ_{0})] B = 0, \\ [σ_{c} \cos (σ_{c}) - e^{2 i κ} ρ σ_{0} \cos (σ_{0})] A - [σ_{c} \sin (σ_{c}) + e^{2 i κ} σ_{0} \sin (σ_{0})] B = 0 . \end{cases}

(3.6)

We observe that for to have a non-zero solution, it should hold

[\sin (σ_{c}) + e^{2 i κ} ρ \sin (σ_{0})] \cdot [σ_{c} \sin (σ_{c}) + e^{2 i κ} σ_{0} \sin (σ_{0})]

(3.7)

+ [\cos (σ_{c}) - e^{2 i κ} \cos (σ_{0})] \cdot [σ_{c} \cos (σ_{c}) - e^{2 i κ} ρ σ_{0} \cos (σ_{0})] = 0,

(3.8)

which gives

\sqrt{ε (ω)} e^{4 i κ} + [\frac{ε_{0} + ε (ω)}{\sqrt{ε_{0}}} \sin (σ_{0}) \sin (σ_{c}) - 2 \sqrt{ε (ω)} \cos (σ_{0}) \cos (σ_{c})] e^{2 i κ} + \sqrt{ε (ω)} = 0 .

(3.9)

Making some algebraic rearrangements, we observe that

\begin{aligned} 2 \sqrt{\frac{ε (ω)}{ε_{0}}} [\cos (σ_{0}) \cos (σ_{c}) - \cos (2 κ)] - \frac{ε_{0} + ε (ω)}{ε_{0}} \sin (σ_{0}) \sin (σ_{c}) = 0 . \end{aligned}

(3.10)

Finally, making the substitutions σ_c = ρσ₀ and

\sqrt{ε (ω)} = ρ \sqrt{ε_{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 $R (κ) \in Y^{*} = [- π / 2, π / 2)$ ⁠.

FIG. 3.

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The dispersion relation of the halide perovskite photonic crystal. We model a material with permittivity given by (2.1) with α = 1, β = 1 and γ = 0.5. The frequency ω is chosen to be real and the Bloch parameter κ allowed to take complex values. The permittivity is singular at two points, which are in the lower complex plane and are symmetric about the imaginary axis, as indicated in the sketch on the right and (the real parts) by the crosses on the frequency axes of the plots.

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 $κ \in C$ ⁠, the frequencies $ω \in R$ 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.

Lemma 3.3

(Opposite quasiperiodicities). Let $κ \in C$ be a complex quasiperiodicity satisfying the dispersion relation (3.5) for a given frequency $ω \in R$ . Then, the opposite quasiperiodicity, i.e., −κ, satisfies the same dispersion relation.

Proof.

We just have to use that cos(·) is an even function. Then, if $κ \in C$ is such that (3.5) holds, from the fact that cos(−2κ) = cos(2κ), we get that $- κ \in C$ also satisfies (3.5). This concludes the proof.□

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

In the analysis that will follow, it will be useful to be able to describe the behaviour of the real and imaginary part of the quasiaperiodicity with respect to the permittivity. In particular, we will decompose both the quasiperiodicity κ and ρ into real and imaginary parts, and we will derive this dependence from the dispersion relation. Since

κ \in C

and

ρ \in C

⁠, let us define:

κ = κ_{1} + i κ_{2} and ρ = ρ_{1} + i ρ_{2},

(3.11)

with

κ_{1}, κ_{2}, ρ_{1}, ρ_{2} \in R

⁠. We will also define

L_{1}

and

L_{2}

⁠, which depend on

ω \in R

⁠, as follows:

\begin{aligned} L_{1} (ω) ≔ & \cos (σ_{0}) \cos (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) - \frac{\sin (σ_{0})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} [ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2}) \sin (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) \\ - ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2}) \cos (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})], \end{aligned}

(3.12)

and

\begin{aligned} L_{2} (ω) ≔ & \cos (σ_{0}) \sin (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2}) + \frac{\sin (σ_{0})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} [ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2}) \sin (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) \\ + ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2}) \cos (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})], \end{aligned}

(3.13)

where we note that ρ₁, ρ₂ and σ₀ all depend on the frequency ω, as specified in and . Then, we have the following result.

Proposition 3.4.

Let

κ \in C

, given by (3.11), satisfying the dispersion relation (3.5) for a given frequency

ω \in R

. Then, its real and imaginary parts are given by

R (κ) = \pm \frac{1}{2} \arccos (\frac{L_{1}}{\cosh (2 I (κ))}),

(3.14)

and

I (k) = \frac{1}{2} a r c s i n h (\pm \sqrt{\frac{1}{2} [L_{1}^{2} + L_{2}^{2} - 1 + \sqrt{{(1 - L_{1}^{2} - L_{2}^{2})}^{2} + 4 L_{2}^{2}}]}),

(3.15)

where

L_{1}

and

L_{2}

are given by (3.12) and (3.13), respectively. We, also, note that the choice of + or − should be the same in and .

Proof.

From , the dispersion relation becomes

\cos (2 κ_{1} + i 2 κ_{2}) = \cos (σ_{0}) \cos (σ_{0} ρ_{1} + i σ_{0} ρ_{2}) - \frac{1 + {(ρ_{1} + i ρ_{2})}^{2}}{2 (ρ_{1} + i ρ_{2})} \sin (σ_{0}) \sin (σ_{0} ρ_{1} + i σ_{0} ρ_{2}),

which is,

\begin{array}{l} \cos (2 κ_{1}) & \cosh (2 κ_{2}) - i \sin (2 κ_{1}) \sinh (2 κ_{2}) \\ = \cos (σ_{0}) [\cos (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) - i \sin (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})] \\ - \frac{1 + ρ_{1}^{2} + 2 i ρ_{1} ρ_{2} - ρ_{2}^{2}}{2 (ρ_{1}^{2} + ρ_{2}^{2})} (ρ_{1} - i ρ_{2}) \sin (σ_{0}) [\sin (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) + i \cos (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})] . \end{array}

Taking real and imaginary parts, we obtain, for the real part,

\begin{align} \cos (2 κ_{1}) \cosh (2 κ_{2}) & = \cos (σ_{0}) \cos (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) - \frac{\sin (σ_{0})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} [ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2}) \sin (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) + \\ - ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2}) \cos (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})], \end{align}

(3.16)

and, for the imaginary part,

\begin{align} \sin (2 κ_{1}) \sinh (2 κ_{2}) & = \cos (σ_{0}) \sin (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2}) + \frac{\sin (σ_{0})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} [ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2}) \sin (σ_{0} ρ_{1}) \cosh (σ_{0} ρ_{2}) \\ + ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2}) \cos (σ_{0} ρ_{1}) \sinh (σ_{0} ρ_{2})] . \end{align}

(3.17)

So, from and , we obtain the system

\{\begin{cases} \cos (2 κ_{1}) \cosh (2 κ_{2}) = L_{1}, \\ \sin (2 κ_{1}) \sinh (2 κ_{2}) = L_{2} . \end{cases}

(3.18)

From the first equation, we immediately see that

κ_{1} = \pm \frac{1}{2} \arccos (\frac{L_{1}}{\cosh (2 κ_{2})}),

then, substituting into the second equation gives

\sin [\arccos (\frac{L_{1}}{\cosh (2 κ_{2})})] \sinh (2 κ_{2}) = \pm L_{2} .

We know that for x ∈ [−1, 1], we have the identity

\sin [\arccos (x)] = \sqrt{1 - x^{2}}

⁠. Hence, from the above, we get

\sqrt{1 - \frac{L_{1}^{2}}{\cosh^{2} (2 κ_{2})}} \sinh (2 κ_{2}) = \pm L_{2} .

(3.19)

Similarly, using the fact that cosh²(x) − sinh²(x) = 1 for

x \in R

⁠, we find that

\sqrt{1 - \frac{L_{1}^{2}}{1 + \sinh^{2} (2 κ_{2})}} \sinh (2 κ_{2}) = \pm L_{2} .

(3.20)

Hence, we have

\sinh^{4} (2 κ_{2}) + (1 - L_{1}^{2} - L_{2}^{2}) \sinh^{2} (2 κ_{2}) - L_{2}^{2} = 0 .

Using the quadratic formula, this gives

\sinh^{2} (2 κ_{2}) = \frac{1}{2} [L_{1}^{2} + L_{2}^{2} - 1 + \sqrt{{(1 - L_{1}^{2} - L_{2}^{2})}^{2} + 4 L_{2}^{2}}],

and so, we get

κ_{2} = \frac{1}{2} arcsinh (\pm \sqrt{\frac{1}{2} [L_{1}^{2} + L_{2}^{2} - 1 + \sqrt{{(1 - L_{1}^{2} - L_{2}^{2})}^{2} + 4 L_{2}^{2}}]}) .

This gives the desired result.□

Remark.

Another way of viewing that the choice of + or − in (3.14) is the same as the one in (3.15) is from the fact that we have shown that if $κ \in C$ satisfies (3.5), then −κ does as well, but $\bar{κ}$ does not.

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.

Lemma 3.5.

Let the frequency-dependent contrast

ρ \in C

be given by (3.2). Then, it holds

\lim_{ω \to \infty} | ρ | = 1,

(3.21)

and

\lim_{ω \to \infty} | R (ρ) | = 1 and \lim_{ω \to \infty} | I (ρ) | = 0 .

(3.22)

Proof.

From , we have

ρ = \sqrt{1 + \frac{α}{ε_{0} (1 - β ω^{2} - i γ ω)}}

which gives directly

\lim_{ω \to \infty} | ρ | = 1

⁠. This can be rewritten as

ρ = \sqrt{1 + \frac{α}{ε_{0}} \frac{1 - β ω^{2}}{{(1 - β ω^{2})}^{2} + γ^{2} ω^{2}} + i \frac{α}{ε_{0}} \frac{γ ω}{{(1 - β ω^{2})}^{2} + γ^{2} ω^{2}}}

To ease the notation, let us write

a (ω) ≔ 1 + \frac{α}{ε_{0}} \frac{1 - β ω^{2}}{{(1 - β ω^{2})}^{2} + γ^{2} ω^{2}} and b (ω) ≔ \frac{α}{ε_{0}} \frac{γ ω}{{(1 - β ω^{2})}^{2} + γ^{2} ω^{2}} .

(3.23)

Then, we have

ρ = \pm (\sqrt{\frac{\sqrt{a {(ω)}^{2} + b {(ω)}^{2}} + a (ω)}{2}} + i \frac{b (ω)}{| b (ω) |} \sqrt{\frac{\sqrt{a {(ω)}^{2} + b {(ω)}^{2}} - a (ω)}{2}})

(3.24)

We observe, from , that, as ω → ∞,

a (ω) = 1 + O (\frac{1}{ω^{2}}) and b (ω) = O (\frac{1}{ω^{3}}),

(3.25)

which gives

\lim_{ω \to \infty} a (ω) = 1 and \lim_{ω \to \infty} b (ω) = 0 .

Also, since α, γ, ɛ₀ > 0, it holds that

\lim_{ω \to \infty} \frac{b (ω)}{| b (ω) |} = 1 .

Hence, combining these results, we get

\lim_{ω \to \infty} | ρ_{1} | = \lim_{ω \to \infty} |\sqrt{\frac{\sqrt{a {(ω)}^{2} + b {(ω)}^{2}} + a (ω)}{2}}| = 1

and

\lim_{ω \to \infty} | ρ_{2} | = \lim_{ω \to \infty} |\frac{b (ω)}{| b (ω) |} \sqrt{\frac{\sqrt{a {(ω)}^{2} + b {(ω)}^{2}} - a (ω)}{2}}| = 0 .

This concludes the proof.□

Lemma 3.6.

As ω → ∞, we have that

| L_{1} | \leq 1 and L_{2} \to 0,

(3.26)

where

L_{1} = L_{1} (ω)

and

L_{2} = L_{2} (ω)

were defined in (3.12) and (3.13).

Proof.

From Lemma 3.5, we have that, as ω → ∞,

| ρ_{1} | \to 1, | ρ_{2} | \to 0

and from , we have that

σ_{0} \to \infty .

So, it is essential to understand the behaviour of σ₀ρ₂ as ω → ∞. Using the same notations as in the Proof of Lemma 3.5, we have, without loss of generality on the ± of ,

ρ_{2} = \frac{b (ω)}{| b (ω) |} \sqrt{\frac{\sqrt{a {(ω)}^{2} + b {(ω)}^{2}} - a (ω)}{2}},

and hence, from , we have

σ_{0} ρ_{2} = \sqrt{μ_{0} ε_{0}} \frac{b (ω)}{| b (ω) |} \sqrt{\frac{\sqrt{ω^{4} (a {(ω)}^{2} + b {(ω)}^{2})} - ω^{2} a (ω)}{2}} .

From , we see that, as ω → ∞,

ω^{4} a {(ω)}^{2} = ω^{4} + O (1), ω^{2} a (ω) = ω^{2} + O (1) and ω^{4} b {(ω)}^{2} = O (\frac{1}{ω^{2}}) .

Hence, as ω → ∞,

\sqrt{\frac{\sqrt{ω^{4} (a {(ω)}^{2} + b {(ω)}^{2})} - ω^{2} a (ω)}{2}} \to 0,

which gives,

\lim_{ω \to \infty} σ_{0} ρ_{2} = 0,

and so

\lim_{ω \to \infty} | \cosh (σ_{0} ρ_{2}) | = 1 and \lim_{ω \to \infty} | \sinh (σ_{0} ρ_{2}) | = 0 .

(3.27)

Thus, gives

\begin{array}{l} \lim_{ω \to \infty} | L_{1} | & = \lim_{ω \to \infty} |\cos (σ_{0}) \cos (σ_{0} ρ_{1}) - \sin (σ_{0}) \sin (σ_{0} ρ_{1})| \\ = \lim_{ω \to \infty} |\cos (σ_{0} (1 + ρ_{1}))| \leq 1, \end{array}

which is the desired bound for

L_{1}

⁠. Similarly, from the triangle inequality applied on , we have

\begin{aligned} | L_{2} | \leq | \sinh (σ_{0} ρ_{2}) | + \frac{1}{2 (ρ_{1}^{2} + ρ_{2}^{2})} & [| ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2}) ‖ \cosh (σ_{0} ρ_{2}) | \\ + | ρ_{1} | (1 + ρ_{1}^{2} + ρ_{2}^{2}) | \sinh (σ_{0} ρ_{2}) |] . \end{aligned}

(3.28)

Using Lemma 3.5 and , we obtain

\lim_{ω \to \infty} L_{2} = 0 .

This concludes the proof.□

Using these results, we will describe the behaviour of the imaginary part κ₂ of the quasiperiodicity $κ \in C$ as the frequency tends to infinity, i.e., ω → ∞.

Proposition 3.7.

Let us consider a complex quasiperiodicity

κ \in C

satisfying the dispersion relation (3.5) with

α, β, γ \in R_{> 0}

. Then, it holds that

\lim_{ω \to \infty} I (κ) = 0 .

(3.29)

Proof.

Indeed, since

κ \in C

⁠, let us define

κ_{1} ≔ R (κ)

and

κ_{2} ≔ I (κ)

⁠. Then, from , we have that κ₂ is given by

κ_{2} = \frac{1}{2} arcsinh (\pm \sqrt{\frac{1}{2} [L_{1}^{2} + L_{2}^{2} - 1 + \sqrt{{(1 - L_{1}^{2} - L_{2}^{2})}^{2} + 4 L_{2}^{2}}]})

From Lemma 3.6, we have that, as ω → ∞,

L_{1}

remains bounded, whereas

L_{2} \to 0

⁠. Thus, the following holds

\lim_{ω \to \infty} [L_{1}^{2} + L_{2}^{2} - 1 + \sqrt{{(1 - L_{1}^{2} - L_{2}^{2})}^{2} + 4 L_{2}^{2}}] = [L_{1}^{2} - 1 + | 1 - L_{1}^{2} |] = 0,

since we have that

| L_{1} | \leq 1

also from Lemma 3.6. Then, from the continuity of the arcsinh(·) function, the desired result follows.□

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

The first case we will consider is the one of a real-valued, non-dispersive permittivity, constant with respect to the frequency ω. In our setting this translates into having with β = γ = 0 and α > 0, i.e.,

ε (ω) = ε_{0} + α \in R_{> 0} .

(4.1)

This setting has been studied quite extensively. We refer to Ref. 8, as a classical reference for studying the dispersive nature of waves in periodic systems. In Fig. 4, we provide an example for the dispersion curves when the permittivity is constant and real. It is worth noting the following result.

FIG. 4.

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The dispersion relation of a photonic crystal with frequency-independent material parameters. We model a material with permittivity given by (2.1) with α = 1, β = 0 and γ = 0. The frequency ω is chosen to be real and the Bloch parameter κ allowed to take complex values. The permittivity is never singular in this case.

Lemma 4.1.

Let ɛ(ω) be the real-valued, non-dispersive permittivity given by (4.1). Then, if $κ \in C$ is a quasiperiodicity satisfying (3.5) for a given frequency $ω \in R$ , then so does $\bar{κ} \in C$ ⁠.

Proof.

Indeed, since

ε (ω) \in R_{> 0}

⁠, then

ρ \in R_{> 0}

⁠. Let us take

κ \in C

satisfying . Then, we can write κ = κ₁ + iκ₂, with

κ_{1}, κ_{2} \in R

⁠. Now, since ρ > 0, we can write

\cos (σ_{0}) \cos (ρ σ_{0}) - \frac{1 + ρ^{2}}{2 ρ} \sin (σ_{0}) \sin (ρ σ_{0}) ≕ A > 0 .

Thus, gives us

\cos (2 κ_{1} + 2 i κ_{2}) = A,

which becomes the following system

\{\begin{cases} \cos (2 κ_{1}) \cdot \cosh (2 κ_{2}) = A, \\ \sin (2 κ_{1}) \cdot \sinh (2 κ_{2}) = 0 . \end{cases}

This implies that

κ_{2} = 0 or κ_{1} = \frac{m}{2} π, m \in Z .

If κ₂ = 0, then

κ = κ_{1} \in R

⁠. Thus,

κ = \bar{κ}

⁠, which gives the desired result. If

κ_{1} = \frac{m}{2} π

⁠, for

m \in Z

⁠, then κ satisfies if and only if

\cosh (2 κ_{2}) = \pm A .

(4.2)

Since cosh(·) is an even function, we have that −κ₂ satisfies for the same frequency ω. Hence,

\bar{κ} = κ_{1} - i κ_{2}

satisfies 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

Let us now study the case where the permittivity has a dispersive (and singular) character with respect to the frequency ω, but there is no damping, i.e., we consider α, β > 0 and γ = 0. This implies that

ε (ω) = ε_{0} + \frac{α}{1 - β ω^{2}} .

(4.3)

The interesting aspect in this setting is the existence of real poles for the permittivity. They are given by

ω_{\pm}^{*} = \pm \frac{1}{\sqrt{β}} .

In Fig. 5, we observe that near the pole of the permittivity there are infinitely many band-gaps. This was similarly observed recently by Ref. 26. Noting that a band gap occurs when the magnitude of the right-hand side of is greater than one. We define the function

f (ω) ≔ \cos (σ_{0} (ω)) \cos (ρ (ω) σ_{0} (ω)) - \frac{1 + ρ {(ω)}^{2}}{2 ρ (ω)} \sin (σ_{0} (ω)) \sin (ρ (ω) σ_{0} (ω)),

(4.4)

which is the right-hand side of . We will prove that this takes values greater than one on a countably infinite number of disjoint intervals within any neighbourhood of the singularity. To do so, we will introduce the following notation, which will be used in our analysis.

FIG. 5.

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The dispersion relation of a photonic crystal with frequency-dependent material parameters that have a singularity. We model a material with permittivity given by (2.1) with α = 1, β = 1 and γ = 0. The frequency ω is chosen to be real and the Bloch parameter κ allowed to take complex values. The permittivity is singular when ω = 1. The lower two plots are display the same dispersion curves, zoomed into the region around the singularity.

Notation.

Let $x, y \in R$ . 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.

Theorem 4.2.

Let ω* denote a pole of the permittivity ɛ(ω) given by (4.3), i.e., $ω^{*} \in \{\pm \frac{1}{\sqrt{β}}\}$ ⁠. Then, for δ > 0, the intervals [ω* − δ, ω*) and (ω*, ω* + δ] contain infinitely many disjoint sub-intervals, denoted by $I_{i}$ and $J_{i}$ , i = 1, 2, …, respectively, that are band gaps.

Proof.

We will first prove this result for the interval [ω* − δ, ω), for δ > 0. It suffices to show that there are infinitely many points

ω_{i}^{†} \in [ω^{*} - δ, ω^{*})

⁠, i = 1, 2, …, for δ > 0 for which

f (ω_{i}^{†}) > 1

or

f (ω_{i}^{†}) < - 1

⁠. Then, the continuity of f around these points gives us the existence of intervals of the form

I_{i} ≔ [ω_{i}^{†} - s, ω_{i}^{†} + s]

⁠, for i = 1, 2, …, for small s > 0, such that,

f (ω) > 1 or f (ω) < - 1, \forall ω \in I_{i}, i = 1, 2, \dots .

From and , this gives

\cos (2 κ) > 1 or \cos (2 κ) < - 1, \forall ω \in I_{i}, i = 1, 2, \dots,

This is equivalent to the

I_{i}

’s, i = 1, 2, … being band gaps, since κ becomes complex in these intervals, i.e.,

| I (κ) | \neq 0

⁠. Hence, since

| I (κ) |

is continuous with respect to ω, we get that it has a local maximum in each of the

I_{i}

’s, for i = 1, 2, ….

We observe that

\lim_{ω ↑ ω^{*}} ε (ω) = + \infty

⁠. Then, this implies that

\lim_{ω ↑ ω^{*}} ρ (ω) = + \infty

⁠, and so, we get

\lim_{ω ↑ ω^{*}} \frac{1 + ρ {(ω)}^{2}}{2 ρ (ω)} = + \infty .

Also, as ω ↑ ω*, we have that σ₀ is constant and so, without loss of generality, we can assume that sin(σ₀) > 0 [the same argument holds for taking sin(σ₀) < 0]. Hence, there exists δ₁ > 0 such that for all ω ∈ [ω* − δ₁, ω*), we have that

\frac{1 + ρ {(ω)}^{2}}{2 ρ (ω)} > \frac{1}{\sin (σ_{0} (ω))} \geq 1 .

(4.5)

Now, since

\lim_{ω ↑ ω^{*}} ρ (ω) = + \infty

⁠, we have that

\lim_{ω ↑ ω^{*}} ρ (ω) σ_{0} (ω) = + \infty .

(4.6)

This implies that, for all K > 0, there exists δ₂ > 0 such that for all ω ∈ [ω* − δ₂, ω*) it holds that |ρ(ω)σ₀(ω)| > K.

Now, let δ ≔ max{δ₁, δ₂} and let

I_{δ}^{(-)} ≔ [ω^{*} - δ, ω^{*})

⁠. Then, implies that the exist two families of infinitely many points in

I_{δ}^{(-)}

⁠, denoted by

{ω_{i}^{(+)}}_{i = 1, \dots, \infty}

and

{ω_{i}^{(-)}}_{i = 1, \dots, \infty}

⁠, such that, for all i = 1, …, ∞, we have

\sin (ρ (ω_{i}^{(+)}) ω_{i}^{(+)}) = 1 and \sin (ρ (ω_{i}^{(-)}) ω_{i}^{(-)}) = - 1 .

(4.7)

We also note that this implies, for all i = 1, …, ∞, that

\cos (ρ (ω_{i}^{(+)}) ω_{i}^{(+)}) = \cos (ρ (ω_{i}^{(-)}) ω_{i}^{(-)}) = 0 .

(4.8)

Thus, for all i = 1, …, ∞, we have

f (ω_{i}^{(+)}) = - \frac{1 + ρ {(ω_{i}^{(+)})}^{2}}{2 ρ (ω_{i}^{(+)})} \sin (σ_{0} (ω_{i}^{(+)})) < - 1

(4.9)

and

f (ω_{i}^{(-)}) = \frac{1 + ρ {(ω_{i}^{(-)})}^{2}}{2 ρ (ω_{i}^{(-)})} \sin (σ_{0} (ω_{i}^{(-)})) > 1

(4.10)

In particular, without loss of generality, let us assume that

ω_{0}^{(+)}

is the smallest of the elements in both families

{ω_{i}^{(+)}}_{i = 1, \dots, \infty}

and

{ω_{i}^{(-)}}_{i = 1, \dots, \infty}

⁠. Then, the periodicity of sin(·) shows that the elements of these families respect the following ordering:

ω_{0}^{(+)} < ω_{0}^{(-)} < ω_{1}^{(+)} < ω_{1}^{(-)} < \dots .

(4.11)

Now, the continuity of f around these points allows us to take s > 0 such that

f (ω) < - 1, \forall ω \in [ω_{i}^{(+)} - s, ω_{i}^{(+)} + s], i = 1, 2, \dots,

f (ω) > 1, \forall ω \in [ω_{i}^{(-)} - s, ω_{i}^{(-)} + s], i = 1, 2, \dots,

and

[ω_{i}^{(+)} - s, ω_{i}^{(+)} + s] ⋂ [ω_{i}^{(-)} - s, ω_{i}^{(-)} + s] = \emptyset, i = 1, 2, \dots .

Finally, the infinity of elements in the families

{ω_{i}^{(+)}}_{i = 1, \dots, \infty}

and

{ω_{i}^{(-)}}_{i = 1, \dots, \infty}

gives us the desired result.

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 $I (κ)$ of the quasiperiodicity κ as the frequency $ω \in R$ approaches a permittivity pole. In fact, we see that close to a pole, $| I (κ) |$ 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.

Corollary 4.2.1.

Let $ω \in R$ and $κ \in C$ be the associated quasiperiodicity satisfying the dispersion relation (3.5) and let f(ω) be the function defined in (4.4). Let $ω^{*} \in R$ 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 $I (κ (ω_{p})) = p$ . That is, $| I (κ (ω)) |$ takes arbitrarily large values as ω ↑ ω*. The same result holds as ω ↓ ω*.

Proof.

From Theorem 4.2, we have that for all K > 0, there exists δ > 0, such that, for all ω ∈ [ω* − δ, ω),

\frac{1 + ρ {(ω)}^{2}}{2 ρ (ω)} > K .

We have that cos(σ₀(ω))cos(ρ(ω)σ₀(ω)) remains bounded close to ω* and because of the continuity of sin(·), we can take sin(σ₀(ω)) > 0 in [ω* − δ, ω). Then, it follows that, for all K > 0, in [ω* − δ, ω),

\frac{1 + ρ {(ω)}^{2}}{2 ρ (ω)} \sin (σ_{0} (ω)) > K .

But, from Theorem , we have the existence of infinitely many points

ω_{0}^{(+)} < ω_{0}^{(-)} < ω_{1}^{(+)} < ω_{1}^{(-)} < \dots

in [ω* − δ, ω), for which,

\sin (ρ (ω) σ_{0} (ω))

oscillates between 1 and −1 in each of the intervals of the form

[ω_{0}^{(+)}, ω_{0}^{(-)}]

⁠,

[ω_{0}^{(-)}, ω_{1}^{(+)}]

⁠, …, denoted by

I_{i}

⁠, i = 1, 2, …. This implies that, for all K > 0, for all p ∈ [−K, K], there exists

ω_{p} \in I_{i}

⁠, 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*.

Now, from and , we get that

κ = \frac{- i}{2} \ln (f (ω) \pm \sqrt{f {(ω)}^{2} - 1}),

(4.12)

where ln(·) denotes the complex logarithm. We see that

I (κ) = \frac{1}{2} R (\ln (f (ω) \pm \sqrt{f {(ω)}^{2} - 1})) .

Although, since we are using the complex logarithm, we have that

R (\ln (f (ω) \pm \sqrt{f {(ω)}^{2} - 1})) = \ln |f (ω) \pm \sqrt{f {(ω)}^{2} - 1}| .

Hence, since we have shown that in [ω* − δ, ω*), the function f(ω) oscillates between +∞ and −∞, we have that

|f (ω) \pm \sqrt{f {(ω)}^{2} - 1}|

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

We will now study the effect that introducing damping though allowing the permitivitty to be complex has on our one-dimensional system. Starting from the straightforward real-valued, non-dispersive model considered in Sec. IV A, we subsequently add damping. For this, we take

α \in C

and β = γ = 0, i.e.,

ε (ω) = ε_{0} + α \in C .

(4.13)

The dispersion curves for this setting are shown in Fig. 6. They are plotted for 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.

FIG. 6.

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The dispersion relation of a photonic crystal with frequency-independent complex-valued material parameters. We model a material with permittivity given by (2.1) with $α \in C$ ⁠, β = 0 and γ = 0. The frequency ω is chosen to be real and the Bloch parameter κ allowed to take complex values. The permittivity is never singular in this case.

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 $I (κ)$ 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.

Lemma 4.3.

Let

κ \in C

and

ω \in R

be a quasiperiodicity and a frequency, respectively, satisfying the dispersion relation (3.5) with complex-valued, non-dispersive permittivity given by (4.13). Then,

\lim_{ω \to + \infty} | I (κ) | = + \infty .

(4.14)

Proof.

Let us recall that κ₂, denoting the imaginary part of

κ \in C

⁠, is given by , where

L_{1}

and

L_{2}

are given by and , respectively. We have that

\lim_{ω \to \infty} ρ (ω) σ_{0} (ω) = + \infty .

(4.15)

since σ₀ is linear with respect to ω and ρ does not depend on ω in this setting. This implies

\lim_{ω \to \infty} \sinh (ρ (ω) σ_{0} (ω)) = + \infty and \lim_{ω \to \infty} \cosh (ρ (ω) σ_{0} (ω)) = + \infty .

We note that it is enough to show that

\lim_{ω \to \infty} | L_{1} | = \lim_{ω \to \infty} | L_{2} | = + \infty,

since applying this on gives that |κ₂| → +∞ as ω → +∞. Indeed, gives that

L_{1} = C_{1} \cosh (σ_{0} ρ_{2}) - C_{2} \sinh (σ_{0} ρ_{2}),

where

C_{1} ≔ \cos (σ_{0}) \cos (σ_{0} ρ_{1}) - \frac{ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} \sin (σ_{0}) \sin (σ_{0} ρ_{1})

and

C_{2} ≔ \frac{ρ_{2} (ρ_{1}^{2} - 1 + ρ_{2}^{2})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} \sin (σ_{0}) \cos (σ_{0} ρ_{1}) .

Using the exponential formulation of the hyperbolic trigonometric functions, we get

L_{1} = \frac{C_{1} - C_{2}}{2} e^{σ_{0} ρ_{2}} + \frac{C_{1} + C_{2}}{2} e^{- σ_{0} ρ_{2}} .

Now, we observe that, as ω → +∞, C₁ and C₂ are both bounded and C₁ − C₂ ≠ 0. Then, from , we get that

\lim_{ω \to \infty} | L_{1} | = + \infty .

Similarly, gives that

L_{2} = {\tilde{C}}_{1} \sinh (σ_{0} ρ_{2}) - {\tilde{C}}_{2} \cosh (σ_{0} ρ_{2}),

where

{\tilde{C}}_{1} ≔ \cos (σ_{0}) \sin (σ_{0} ρ_{1}) + \frac{ρ_{1} (1 + ρ_{1}^{2} + ρ_{2}^{2})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} \sin (σ_{0}) \cos (σ_{0} ρ_{1})

and

{\tilde{C}}_{2} ≔ \frac{ρ_{2} (ρ_{2}^{2} - 1 + ρ_{1}^{2})}{2 (ρ_{1}^{2} + ρ_{2}^{2})} \sin (σ_{0}) \sin (σ_{0} ρ_{1}) .

Then, we can write

L_{2} = \frac{{\tilde{C}}_{1} + {\tilde{C}}_{2}}{2} e^{σ_{0} ρ_{2}} + \frac{{\tilde{C}}_{2} - {\tilde{C}}_{1}}{2} e^{- σ_{0} ρ_{2}} .

As before, we observe that, as ω → +∞,

{\tilde{C}}_{1}

and

{\tilde{C}}_{2}

are both bounded and

{\tilde{C}}_{1} - {\tilde{C}}_{2} \neq 0

⁠. Then, from , we get that

\lim_{ω \to \infty} | L_{2} | = + \infty .

This concludes the proof.□

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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Published open access through an agreement with JISC Collections

The effect of singularities and damping on the spectra of photonic crystals

I. INTRODUCTION

II. PROBLEM SETTING

A. Initial Helmholtz formulation

B. Periodic formulation

C. Floquet–Bloch theory

D. One dimension

III. DISPERSION RELATION

A. Analytic expression

B. Properties of the dispersion relation

1. Real and imaginary parts

2. Imaginary part decay

IV. THE EFFECT OF SINGULARITIES AND DAMPING

A. Constant permittivity

B. Singular permittivity with no damping

C. Complex permittivity

D. Discussion

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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The effect of singularities and damping on the spectra of photonic crystals

I. INTRODUCTION

II. PROBLEM SETTING

A. Initial Helmholtz formulation

B. Periodic formulation

C. Floquet–Bloch theory

D. One dimension

III. DISPERSION RELATION

A. Analytic expression

B. Properties of the dispersion relation

1. Real and imaginary parts

2. Imaginary part decay

IV. THE EFFECT OF SINGULARITIES AND DAMPING

A. Constant permittivity

B. Singular permittivity with no damping

C. Complex permittivity

D. Discussion

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

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Related Content

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