## Abstract

It is well known that defects, such as holes, inside an infinite photonic crystal can sustain localized resonant modes whose frequencies fall within a forbidden band. Here we prove that finite, defect-free photonic crystals behave as mirrorless resonant cavities for frequencies within but near the edges of an allowed band, regardless of the shape of their outer boundary. The resonant modes are extended, surface-avoiding (nearly-Dirichlet) states that may lie inside or outside the light cone. Independent of the dimensionality, quality factors and finesses are on the order of, respectively, ${\left(L/\lambda \right)}^{3}$ and $L/\lambda $, where *λ* is the vacuum wavelength and *L* >> *λ* is a typical size of the crystal. Similar topological modes exist in conventional Fabry-Pérot resonators, and in plasmonic media at frequencies just above those at which the refractive index vanishes.

© 2014 Optical Society of America

## 1. Introduction

Resonant cavities (RCs) are devices used to confine light or other wave disturbances. They are broadly characterized by a set of quality factors *Q*, defined as the ratio between the frequency *ω* and the width $\delta \omega $ of a particular mode, and finesses $F=\Delta \omega /\delta \omega $, where Δ*ω* is the separation between adjacent modes. The most common implementation of a RC involves a region defined by a mirrored surface. Mirrorless RCs are also well known. Examples include natural substances which rely on, e. g., total internal reflection to give confinement close to the boundary (as for whispering gallery modes [1]) and systems sustaining modes that lie outside the light cone such as, e. g., surface plasmons in metals. Here, we show that, in a narrow range of allowed frequencies, photonic crystals (PCs) [2,3] behave as natural, mirror-free RCs regardless of their shape (this is unlike PC cavities resulting from defects, such as holes, which operate at frequencies inside forbidden gaps [4]). Ignoring all but radiative losses, and for arbitrary dimensions, the corresponding states are surface-avoiding [5,6], extended modes with *Q* values on the order of ${\left(L/\lambda \right)}^{3}$ and $F~L/\lambda $, where *λ* is the wavelength in vacuum and $L>>\lambda $ is a typical length of the PC. Analogous results apply to phononic crystals as well as to plasmonic materials, that is, substances for which the real part of the permittivity is negative in a certain range, just above the plasmon frequency. In addition, our approach provides clear bounds to the range of applicability of the envelope function approximation [7–9] (especially with regard to reported observations of long-lived modes in finite-size PCs [10]) and a simple explanation as to why the lowest-order modes of a conventional Fabry-Pérot cavity show low intensities at the edges and, thus, little diffraction losses [11].

The high-*Q* modes discussed here should not be confused with the much-studied resonant guided modes supported by periodically patterned dielectric slabs [12], which have been used in, e. g., optical filters [13] and surface-emitting lasers [14]. These guided modes occur in two-dimensional PC slabs of a thickness comparable to *λ*, which contain an infinite number of in-plane periods. For a lossless medium, their quality factors are thus dominated by out-of-plane radiative losses. In contrast, here we consider PC structures of arbitrary dimension, with a large but finite number of periods, and analyze the radiation from the edge of the periodic region.

## 2. Proof: two-dimensional photonic crystals

For simplicity, we restrict the theoretical discussion to *E* modes in the two-dimensional case bar general arguments discussed later that apply as well to the vector wave equation in three dimensions. We assume that the PC is surrounded by vacuum and occupies a simply-connected region $\Xi $ in the (*x*, *y*) plane, of characteristic dimensions $L>>\lambda $, that has no holes and is bounded by the closed curve ${C}_{\Xi}$; see Fig. 1. The equation for the *z* component of the electric field, Ψ, is

*ε*is the permittivity, which is a periodic function of the position vector

**r =**(

*x*,

*y*) inside $\Xi $. The boundary conditions (BCs) require that the field and its normal derivative, ${\partial}_{\perp}\Psi $, be continuous at ${C}_{\Xi}$. Outside the PC, we choose solutions of the formwhich lead to complex eigenvalues $k=\left\{{\kappa}_{1},{\kappa}_{2},\mathrm{...}\right\}$. Here,

*ρ*and

*ϕ*are the polar coordinates, ${A}_{p}$ are constants and ${H}_{p}^{(1)}$ is the Hankel function of the first kind. Let ${\phi}_{i}\left(r\right)$ be the eigenmode associated with ${\kappa}_{i}$. We wish to show that there is a regime in which these functions behave as surface-avoiding, long-lived modes, that is, modes for which the ratio between the amplitude at the boundary and at the interior of $\Xi $ (and, thus, between the imaginary and real components of the eigenvalues) is infinitesimally small. The procedure we employ is described briefly as follows. First, we consider solutions to Eq. (1) that satisfy two additional, virtual BCs: $\Psi \equiv 0$ (Dirichlet condition) and ${\partial}_{\perp}\mathrm{ln}\Psi =i\theta \left(s\right)$ (impedance condition), where $\theta \left(s\right)$ is some arbitrary complex function that depends on the coordinate

*s*along the curve. Explicit expressions of $\theta $ for the PC problem and for substances with $0<\epsilon <<1$ will be derived later. Let ${\chi}_{t}$ (

*t*= 1, 2, ..) and ${k}_{t}$, with ${\int}_{\Xi}{\left|{\chi}_{t}\right|}^{2}d\Xi}=1$ be the eigenfunctions and the (real) eigenvalues for the Dirichlet problem. Next, we assume that $\theta $ is sufficiently large so that the theory of perturbation of boundary conditions [15] can be used to express the solutions for the impedance BC as a series expansion in the set $\left\{{\chi}_{t},{k}_{t}\right\}$. Then, we rely on the uniqueness of the actual solutions to obtain the expansion that relates $\left\{{\phi}_{i},{\kappa}_{i}\right\}$ to the Dirichlet set and, finally, we find the conditions that ensure the validity of our assumption that $\theta $ is large. Details are given below.

Consider the Green’s function satisfying the inhomogeneous equation

*G*in the Dirichlet set $\{{\chi}_{t},{k}_{t}\}$,

*t*[15]

The next step is to show that the actual eigenstates and those satisfying the Dirichlet condition are also related through a series expansion. To that end, consider one of the eigenfunctions that satisfies the actual BCs, say $\tilde{\phi}$, and, in particular, the function it specifies on the boundary $\tilde{\theta}(s)=-i{\partial}_{\perp}\mathrm{ln}\tilde{\phi}$. In turn, $\tilde{\theta}$ defines a complete set of impedance modes ${\eta}_{j}$, of eigenvalues ${K}_{j}$, in terms of which we can expand $\tilde{\phi}$ inside $\Xi $ as $\tilde{\phi}={\displaystyle {\sum}_{j}{C}_{j}{\eta}_{j}}$; ${C}_{j}$ are constants. We then obtain

where $\tilde{\kappa}$ is the eigenvalue associated with $\tilde{\phi}$. Because the solutions to Eq. (1) are unique, it is clear that, ignoring degeneracies, one of the members of the set $\{{\eta}_{j},{K}_{j}\}$ must be identical to $\tilde{\phi}$ and $\tilde{\kappa}$.It remains to be shown that there is a frequency range in which the PC satisfies the conditions leading to resonant, surface-avoiding modes, $\left|{G}_{tm}\right|<<\left|{k}_{t}{}^{2}-{k}_{m}{}^{2}\right|$ and $\left|{G}_{tt}\right|<<\text{\hspace{0.17em}}\text{\hspace{0.17em}}{k}_{t}^{2}$, which are required for the perturbation series to converge. We now show that these conditions are met and, moreover, that an explicit expression for $\theta (s)$ can be obtained in the immediate vicinity of a band minimum or maximum, inside the allowed regions. For definitiveness, we assume the band extreme to be at the center of the Brillouin zone, and the band to be parabolic, i. e., $\omega \approx {\omega}_{\text{G}}+\alpha {q}^{2}$ (${\omega}_{\text{G}}$ is the frequency of the band minimum or maximum and α is a constant). Inside the PC, the most general state is a superposition of Bloch-Floquet states of the form

where ${u}_{q}^{}$ is a periodic function. For*q*small compared to the size of the Brillouin zone, the states are approximately of the formwhere ${E}_{q}={\displaystyle {\sum}_{\left|q\right|=q}{A}_{q}^{}{e}^{iq.r}}$ is a slowly-varying envelope function, which vanishes at ${C}_{\Xi}$ for Dirichlet modes, and ${q}^{2}=(ck-{\omega}_{\text{G}}^{})/\alpha $;

*c*is the speed of light. Inserting the above expression in Eq. (8), we get

*k*, the Green’s functions are $~O({k}^{-1}{L}^{-3})$ (note that $\int |{u}_{0}{E}_{t}^{\text{D}}{|}^{2}d\Xi}=1$). Using that $\left|{k}_{m}^{2}-{k}_{t}^{2}\right|~O({L}^{-2})$ for allowed states in the vicinity of the gap, it follows from Eq. (7) that the lowest-order correction to the eigenfunctions is $O(\lambda /L)<<1$. Also from Eq. (7), and to first order in ${\theta}^{-1}$, the expression for the eigenfrequencies ${\Omega}_{t}=c{K}_{t}$ is

## 3. Examples

One-dimensional PCs provide the simplest example of RC behavior. Figure 2 shows the calculated transmission coefficient *T* for a periodic multilayer structure of total length *D* and period *d*. The frequency range shown is that of the second ‘optical’ band of the PC (in the first ‘acoustic’ band, *ω* → 0 for *q* → 0). The surface-avoiding, cavity modes occur near the edges of the allowed band, where the *q*-dependence of *ω* is quadratic. Figure 2(b) shows the intensity for the first three lowest-lying modes. Their envelopes are in a one-to-one correspondence with the intensity profiles of a mirrored RC (as well as with the quantum eigenfunctions of a particle in an infinitely deep potential well [17]), which can be ordered according to the number of zeros of the Dirichlet eigenfunctions. The results in Fig. 2(c) show that the distance in frequency between two arbitrary peaks and their width scale, respectively, like ${D}^{-2}$ and ${D}^{-3}$, as for the two-dimensional problem. Also note that the number of peaks for which *T* = 1 equals the number of cells [17].

The results of finite-element numerical simulations for a square lattice of rods, shown in Fig. 3(a), illustrate the topological properties of PC resonators. Figures 3(b) and 3(c) show, respectively, the calculated absorption for the cases where the PC outer boundary is a circle and a bow tie. The frequency range shown is in the vicinity of a forbidden gap which, according to calculations using the MIT Photonic-Bands software [18], is defined by band edges at the M- and the X-point of the Brillouin zone. The PC parameters chosen are such that this gap extends from $\omega ={\omega}_{\text{G}}\approx 0.28\times (2\pi c/a)$ (M-point) to $\omega \approx 0.42\times (2\pi c/a)$ (X-point); *a* is the lattice parameter. As for the one-dimensional PC, the surface-avoiding modes manifest themselves as the narrow peaks that occur just below and above the edges of the allowed bands. Contour plots of the field magnitude are shown for the two highest-lying modes. Consistent with our theoretical description, the corresponding envelope functions show, respectively, no nodes and a line of nodes, in close correspondence with the associated Dirichlet modes. The simulations reveal similar behavior at the X-point for both the circle and the bow tie. Other boundaries also tested give the same outcome.

Surface-avoiding modes occur also in homogeneous substances and, in particular, plasmonic media immediately above the plasmon frequency where $0<\mathrm{Re}(\epsilon )<<1$. With some modifications, the above analysis can be applied to such systems. Inasmuch as the PC problem is the optical counterpart to that ofa quantum particle in a periodic potential, the plasmonic problem is analogous to that of a particle in the piecewise constant potential $V={V}_{0}>0$ and $V=0$ inside and outside Ξ, respectively. The far-field, asymptotic expression of Eq. (2),

*R*are ${J}_{t}({n}_{\Xi}k\rho ){e}^{it\varphi}$ ($\rho <R$) and ${H}_{t}^{(1)}(k\rho ){e}^{it\varphi}$ ($\rho \underset{\u02dc}{>}R$) leading to $\left|\kappa \right|~{n}_{\Xi}R$; ${n}_{\Xi}$ is the

*ω*-dependent refractive index. Therefore, for $\epsilon =1$outside $\Xi $, $\left|G\right|$ is generally of order $\lambda /{L}^{3}$and $\left|{k}_{t}{}^{2}-{k}_{m}{}^{2}\right|~1/{({n}_{\Xi}L)}^{2}$. Thus, the first-order correction to the lowest-lying eigenfunctions is of order ${n}_{\Xi}^{2}(\lambda /L)<<1$, confirming our assertion that the expansion converges and, thus, that the eigenmodes are of the surface-avoiding type. To first-order, the new (complex) eigenfrequencies are

Consider now a plasmonic substance whose permittivity is given by

*R*made of such a substance. Close to Re(

*ε*) = 0, and for $\gamma <<{\omega}_{0}$, the dependence of the frequency on the wavevector is of the form $\omega \approx {\omega}_{\text{P}}+A{q}^{2}$ where

*A*is a constant and ${\omega}_{\text{P}}$ is the plasma or longitudinal-optical phonon frequency. Thus, ${v}_{\text{G}},{n}_{\Xi}\propto {L}^{-1}$ and, as for PCs, we get from Eq. (15) that $\delta \omega /\omega ~{\lambda}^{3}/{L}^{3}$. The results in Fig. 4 support these arguments. Curves were calculated using the exact expressions for Mie scattering [20]. The narrow peaks that occur just above ${\omega}_{\text{P}}$reflect the extended modes that set the sphere as a RC. As expected, the two lowest eigenmodes depicted in Fig. 4(b) show vanishing intensity at the surface of the sphere. As for PCs, the results at various radii shown in Fig. 4(c) indicate that the distance between peaks scales like ${R}^{-2}$ and that the widths of the peaks decrease as

*R*

^{−3}with increasing radius. The latter behavior persists until one reaches the point where the radiative width becomes smaller than the non-radiative one after which the peak intensity diminishes strongly. The existence of cavity-like, surface-avoiding modes can be established using simple, back-of-the-envelope arguments. The radiative lifetime of any given mode can be estimated aswhere $R$ is the reflectivity. Consider first plasmonic substances for $\omega \underset{\u02dc}{>}{\omega}_{\text{P}}$. At normal incidence, $R={({n}_{\Xi}-1)}^{2}/{({n}_{\Xi}+1)}^{2}\approx 1-4{n}_{\Xi}$. Recalling that, in the vicinity of a zero in the permittivity the dispersion is $\omega \approx {\omega}_{\text{P}}+A{q}^{2}$, for a mode of wavevector $q\sim {L}^{-1}$ we get ${v}_{\text{G}}=2Aq\propto {L}^{-1}$ and ${n}_{\Xi}\propto {L}^{-1}$. Thus, ${\tau}_{\text{R}}\propto {L}^{3}$ and, in addition, the frequency separation between modes in any dimension is $\propto {L}^{-2}$. Hence, the occurrence of resonances with $Q\propto {\left(L/\lambda \right)}^{3}$ and $F\propto L/\lambda $ is a consequence of the quadratic dispersion which gives both high reflectivity and small group velocity. It is apparent that these results apply also to PCs near a gap. Moreover, noting that a Fabry-Pérot resonator is basically a finite waveguide operating just above cutoff, these reasons account for the fact that, irrespective of the form of the mirrors, the field intensity at the edges is significantly lower than at the center [11].

Surface avoiding modes hold promise for applications requiring large cavity volumes, that is, large energy storage capacity. We note that it is unlikely that the features discussed here will be observed in natural substances since plasmonic materials are too lossy. The situation is much better with PCs where values as large as *Q* ~36000 have been reported [21] and metamaterials involving spoof plasmons [22] with *Q* ~700 [23].

In conclusion, we have shown that bulk photonic crystals and other systems sustain extended, surface-avoiding (nearly-Dirichlet) modes with large quality factors and finesses that scale like a power of the size of the system. The arguments presented here apply as well to acoustic and quantum mechanical waves. We note that surface-avoiding acoustic modes have already been observed experimentally in semiconductor superlattices [6].

## Appendix

In the following, we prove that $\nabla \mathrm{ln}{E}_{q}\approx ik{e}_{\rho}$ for allowed modes near a gap. First, we write Eq. (11) as ${\Psi}_{\lambda}+{\Psi}_{L}$ with ${\Psi}_{\lambda}(r)=\left({u}_{0}(r)-\u3008{u}_{0}\u3009\right){E}_{q}(r)$ and ${\Psi}_{L}(r)=\u3008{u}_{0}\u3009{E}_{q}(r)$; $\u3008{u}_{0}\u3009$ denotes the average over a unit cell. Because the length scale of the rapidly-varying function ${u}_{0}$ is *λ* and that of the envelope function is $L>>\lambda $, one can express Eq. (2) as the sum of two contributions involving terms with $p~O(L/\lambda )$ and $p~O(1)$, associated with ${\Psi}_{\lambda}$ and ${\Psi}_{L}$, respectively. As for plasmonic media, the asymptotic form, Eq. (14), is an excellent approximation to ${\Psi}_{L}$ (terms with *p* >> 1 are exponentially small) and, thus, $\nabla \mathrm{ln}{\Psi}_{L}=\nabla \mathrm{ln}{E}_{q}\approx ik{e}_{\rho}$ since $\left|{\partial}_{\phi}\Psi \right|/\left|{\partial}_{\rho}{\Psi}_{L}\right|<<L$.

## Acknowledgments

Work supported by AFOSR under Grant No. FA9550-09-1-0636 and by the MRSEC Program of the NSF under Grant No. DMR-1120923. RM acknowledges support of the Simons Foundation.

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