Bound states in the continuum through environmental design

TL;DR

This paper introduces a novel environmental engineering approach to realize bound states in the continuum (BICs) in photonic systems, enabling control over radiation channels and the creation of complex BIC configurations.

Contribution

The study presents a new paradigm for BIC realization by environmental design, demonstrating BICs in photonic crystal slabs within engineered environments, including lines, points, and their intersections.

Findings

BICs can be engineered through environmental control of radiation channels.

Photonic crystal slabs exhibit both isolated and line BICs in different Brillouin zone regions.

Line intersections of BICs and leaky resonances can produce exceptional points and bulk Fermi arcs.

Abstract

We propose a new paradigm for realizing bound states in the continuum (BICs) by engineering the environment of a system to control the number of available radiation channels. Using this method, we demonstrate that a photonic crystal slab embedded in a photonic crystal environment can exhibit both isolated points and lines of BICs in different regions of its Brillouin zone. Finally, we demonstrate that the intersection between a line of BICs and line of leaky resonance can yield exceptional points connected by a bulk Fermi arc. The ability to design the environment of a system opens up a broad range of experimental possibilities for realizing BICs in three-dimensional geometries, such as in 3D-printed structures and the planar grain boundaries of self-assembled systems.

Equations

$${\partial }_{t}a=-(i{\omega }_0+\gamma )a+K^T{s}_{\text{in}},$$

$$s_{\text{out}}=C{s}_{\text{in}}+Da.$$

$$C{D}^{\ast }=-D.$$

$$d_0^{\ast }(r+t)=-d_0,$$

$$\hat{H}={\omega }_D-i\gamma +\left(v_{gy}\delta k_y-i\gamma \right){\hat{\sigma }}_z+v_{gx}\delta k_x{\hat{\sigma }}_x,$$

$${\omega }_{\pm }={\omega }_D-i\gamma \pm \sqrt{v_{gx}^2\delta k_x^2+v_{gy}^2\delta k_y^2-{\gamma }^2-2i\gamma v_{gy}\delta k_y}.$$

Full text

Bound states in the continuum (BICs), which are radiation-less states in an open system whose frequency resides within the band of radiative channels, have recently attracted a great deal of interest for their applications in producing vector beams from surface emitting lasers Meier et al. (1998); Imada et al. (1999); Noda et al. (2001); Miyai et al. (2006); Matsubara et al. (2008); Iwahashi et al. (2011); Kitamura et al. (2012); Hirose et al. (2014) and enhancing the resolution of certain classes of sensors Yanik et al. (2011); Zhen et al. (2013); Romano et al. (2018). Originally proposed in 1929 in a quantum mechanical context von Neumann and Wigner (1929), BICs have now been found in a broad range of physical systems, such as photonic crystal slabs Paddon and Young (2000); Pacradouni et al. (2000); Ochiai and Sakoda (2001); Fan and Joannopoulos (2002); Hsu et al. (2013a, b); Yang et al. (2014); Zhen et al. (2014); Zhou et al. (2016); Gao et al. (2016); Kodigala et al. (2017); Zhang et al. (2018a); Minkov et al. (2018), waveguide arrays Plotnik et al. (2011); Weimann et al. (2013); Corrielli et al. (2013), strongly coupled plasmonic-photonic systems Azzam et al. (2018), metasurfaces Koshelev et al. (2018) acoustics Parker (1966, 1967); Cumpsty and Whitehead (1971); Koch (1983); Parker and Stoneman (1989); Evans et al. (1994), and water waves Ursell (1951); Jones (1953); Callan et al. (1991); Retzler (2001); Cobelli et al. (2009, 2011). Additionally, lines of BICs were recently found in composite birefringent structures Gomis-Bresco et al. (2017); Mukherjee et al. (2018). In principle, BICs can be classified into three main categories Hsu et al. (2016): those which are engineered using an inverse construction method, those which are protected by symmetry or separability, and those which can be found ‘accidentally’ through tuning a system’s parameters. In practice however, systems supporting BICs from the first category are difficult to experimentally realize due to the high degree of fine-tuning required. Thus, much of the current excitement surrounding BICs has focused on systems which feature symmetry-protected and accidental BICs; moreover, these BICs have been shown to possess topological protection that guarantees their existence under perturbations to the system Zhen et al. (2014); Bulgakov and Maksimov (2017); Xiao et al. (2017); Zhang et al. (2018b); Doeleman et al. (2018); Takeichi and Murakami (2018).

Traditionally, the appearance of accidental BICs is understood in terms of modal interference Friedrich and Wintgen (1985); Hsu et al. (2016); Gao et al. (2016), with two or more resonances of the device destructively interfering in the system’s radiation channels and resulting in a bound mode spatially localized to the device. This interpretation emphasizes how tuning the device’s parameters changes the spatial profile of its resonances to realize this modal interference, while considering the available radiation channels in the surrounding environment as fixed. This is because most previously studied systems with accidental BICs consider devices embedded in free space, where the outgoing propagating channels cannot be readily altered. However, from this argument it is clear that the environment is also important in determining the presence or absence of BICs: the environment’s properties dictate the number and modal profiles of the available radiation channels, and thus strongly constrain when it is possible to achieve the necessary modal interference. Yet thus far, the role of the environment in creating BICs has remained relatively unexplored.

In this Letter, we show that the properties of the environment play an important role in whether, and where, BICs can exist, regardless of the specific geometry of the device embedded in this environment. This argument is presented using temporal coupled-mode theory (TCMT) Haus (1983); Suh et al. (2004); Alpeggiani et al. (2017) and as such is completely general and applicable to all systems which exhibit BICs. As an example of this theory, we will then show that by embedding a photonic crystal slab into a photonic crystal environment, both isolated BICs and lines of BICs can be found in the resonance bands of the photonic crystal slab depending on the number of available radiation channels. Moreover, perturbations to the environment can shift the locations of the system’s BICs even when the photonic crystal slab layer remains unchanged, demonstrating that the environment of a system is an equal partner to the embedded device in determining the existence and types of BICs found in the system. Finally, we show that when two resonance bands of the photonic crystal slab undergo a symmetry protected band crossing, it is possible for a line of BICs to pass from one band to the other through a bulk Fermi arc. Understanding the relationship between the device and surrounding environment in forming BICs is a necessary first step towards realizing BICs in three-dimensional geometries, such as grain boundaries in photonic crystals Deubel et al. (2004) or self-assembled structures Blanco et al. (2000).

To illuminate the role of the environment in determining the presence and properties of BICs in a system, we first consider a photonic crystal slab embedded in an environment, such that the entire system is periodic in $x$ and $y$. The dynamics of an isolated resonance at any choice of in-plane wavevector, $\mathbf{k}_{\parallel}=(k_{x},k_{y})$, of the photonic crystal slab can be described using temporal coupled-mode theory (TCMT) as

[TABLE]

where the resonance, with amplitude $a$, frequency $\omega_{0}$, and decay rate $\gamma$, couples to both the incoming channels of the environment, $s_{\textrm{in}}$, via the coupling constants $K$, and the outgoing channels of the environment, $s_{\textrm{out}}$, via

[TABLE]

Here, $C$ represents the direct transmission and reflection through the photonic crystal slab, and is an $N\times N$ complex matrix, where $N$ is the total number of radiative channels both above and below the photonic crystal slab at frequency $\omega_{0}$ and $\mathbf{k}_{\parallel}$, while $D$ is an $N\times 1$ complex column vector that denotes the outcoupling of the resonance to each available radiation channel. Finally, if the system possesses $180^{\circ}$ rotational symmetry about the $z$-axis ($C_{2}$), such that $-\mathbf{k}_{\parallel}$ is equivalent to $\mathbf{k}_{\parallel}$, by time-reversal symmetry $C$ and $D$ can be shown to be related as Suh et al. (2004)

[TABLE]

In the language of TCMT, a BIC occurs when $D=0$, i.e. all of the complex outcoupling coefficients of the resonance to all of the available radiation channels are simultaneously zero. For this to occur accidentally with finite probability, there must be at least as many degrees of freedom of the system as there are unknown parameters of $D$. Thus, naively one would expect to require $2N$ degrees of freedom to find BICs, as there initially appear to be $2N$ unknown parameters in $D$. However, while the properties of the resonances, and thus $D$, of the photonic crystal slab are dependent upon the specific patterning of the slab, the direct scattering processes contained in $C$ are agnostic to this patterning, and can instead be considered using a homogeneous dielectric slab Fan and Joannopoulos (2002); Fan et al. (2003). As such, as $C$ is essentially constant for perturbations to the photonic crystal slab, the requirement of Eq. (3) represents a set of additional constraints on $D$, halving its number of unknown parameters. Thus, in general if an entire system is $C_{2}$ symmetric, one only needs $N$ degrees of freedom to find accidental BICs.

A second symmetry commonly present in photonic crystal slab systems is mirror symmetry about the $z=0$ plane ($\sigma_{z}$). Although this symmetry is not required to find BICs, its presence further reduces the number of free parameters among the components of $D$ as the outcoupling coefficients $d_{m}$,$d_{n}$, forming pairs $d_{n}=\sigma d_{m}$ with $\sigma=\pm 1$ depending on whether the resonance is even or odd about $z=0$ Hsu et al. (2017). Here, $m,n$ correspond to the two radiation channels that are mirror-symmetric partners in the regions of the environment which are above and below the photonic crystal slab that the resonance of interest is coupled to. Thus, mirror symmetry both halves the number of unknown parameters in $D$ and also halves the number of independent constraints represented by Eq. (3).

To provide an explicit example of how these constraints can be used to find BICs, consider the photonic crystal slab embedded in a photonic crystal environment shown in Fig. 1a. The presence of the photonic crystal environment breaks the degeneracy between the two polarizations of light in homogeneous media, splitting the light line, $\omega=c|\mathbf{k}_{\parallel}|$, into separate frequency cutoff bands, $\omega_{n}(\mathbf{k}_{\parallel})$, below which the $n$th radiative channel does not exist Johnson et al. (1999). As such, for the TE-like resonance band of the photonic crystal slab shown in Fig. 1b, the central region of this band in the Brillouin zone (below the dashed purple line in Fig. 1c) can couple to two radiation channels on each side of the slab, while the exterior region of the resonance band can only couple to a single radiation channel on each side.

First consider the single-radiation-channel region above the dashed purple line in Fig. 1c. Here, there initially appear to be $4$ unknown parameters in $D=(d_{\textrm{above}},d_{\textrm{below}})^{T}$, corresponding to two complex outcoupling coefficients to the two radiation channels. However, as the system is mirror symmetric about $z=0$ and the resonance band’s states are even under this symmetry, $D=d_{0}(1,1)^{T}$, with $d_{0}$ being the remaining complex free parameter. Moreover, one can show that the constraint represented by Eq. (3) in this region can be written as,

[TABLE]

which amounts to a constraint on the phase of $d_{0}$, as $|r|^{2}+|t|^{2}=1$, where $r$ and $t$ are the direct transmission and reflection coefficients and $C=(r,t;t,r)$. Thus, in this region there is only a single unknown parameter in $D$, and as there are two degrees of freedom in the system, $k_{x}$ and $k_{y}$, lines of BICs can be found in the resonance band in the single-radiation-channel region along portions of the edge of the Brillouin zone, as well as near portions of the $\Gamma$–$M$ line, as shown in Fig. 1c.

This same analysis can also be performed in the two-radiation-channel region of the resonance band, closer to the center of the Brillouin zone. In this region, one finds that $D$ has two unknown parameters, and thus it is possible to find isolated accidental BICs in this region as there are two degrees of freedom, similar to accidental BICs found in previous works of photonic crystal slabs embedded in homogeneous media Hsu et al. (2013b); Zhen et al. (2014). For the system shown in Fig. 1a, one finds an accidental BIC near $k_{x}=0$, $k_{y}=0.2\pi/a$, and a symmetry protected BIC at $\Gamma$ where both radiative channels are even under $C_{2}$ but the resonance band is odd, as marked in Fig. 1c.

Thus far, we have shown that the environment plays a significant role in determining the distribution of BICs in a given system. To demonstrate that the environment is an equal partner to the resonant device in determining the presence of BICs, we increase the symmetry of the photonic crystal environment ($C_{4v}$ as opposed to $C_{2v}$) but preserve the same photonic crystal slab, as shown in Fig. 2a. Thus, although the spatial profiles of the resonances of the photonic crystal slab remain the same, where they achieve complete destructive interference in the radiation channels of the new environment has changed, as can be seen by comparing the distribution of BICs found in Fig. 2c, to the distribution seen in Fig. 1c. In the $C_{4v}$ symmetric environment, the isolated accidental BIC in the two-radiation-channel region has merged with the isolated BIC at $\Gamma$ and the line of accidental BICs near the $\Gamma$–$M$ line has shifted to lie exactly along $\Gamma$–$M$ and become protected by mirror symmetry about the $x=y$ line.

There is a curious feature of the lines of BICs along $X$–$M$ and $Y$–$M$ found in both Figs. 1c and 2c: the lines of BICs appear to abruptly terminate prior to reaching $M$. Although such a termination is not precluded by the coupled-mode analysis previously discussed, the line of BICs along $X$–$M$ and $Y$–$M$ are not accidental, but are instead protected by symmetry, as the resonance band is odd about the $x=0$ ($y=0$) plane along this portion of the $X$–$M$ ($Y$–$M$) high symmetry line, while the radiative channel is even about the same plane, as shown in Figs. 3c and 3e. Thus, as the symmetry of the system has not changed at these points along high symmetry lines, it is strange that the modal profile of the resonance band would suddenly change to allow for the state to couple to the radiative channel. However, the disappearance of the line of BICs from the resonance band coincides with the location of an intersection with a second TE-like resonance band of the photonic crystal slab, shown in Fig. 3a and 3b. Elsewhere in the Brillouin zone these two resonance bands couple and exhibit an avoided crossing, but along the high symmetry line $X$–$M$ ($Y$–$M$) these two bands have opposite mirror symmetry about the $x=0$ ($y=0$) plane of the system, as shown in Figs. 3c and 3d, and thus exhibit a band crossing.

If the coupling to the radiative channels could be ignored such that the system were completely Hermitian, this accidental band crossing would occur at a Dirac point. Instead, the coupling of the resonance bands to the outgoing radiative channels results in this system being non-Hermitian. Moreover, the two resonance bands in question couple to the single available radiative channel at different rates, i.e. one resonance band possesses a line of symmetry-protected BICs while the other resonance does not. Due to this unequal radiative coupling, where the hypothetical non-radiating Hermitian system would possess a Dirac point connecting the two bands, these two resonance bands are instead joined by a bulk Fermi arc in the radiating non-Hermitian system Zhou et al. (2018). Bulk Fermi arcs occur generically in non-Hermitian systems and form when a Dirac point is split into two exceptional points Mailybaev et al. (2005) connected by a contour where the real part of the frequencies of the two resonance bands are equal, $\textrm{Re}[\omega_{+}]=\textrm{Re}[\omega_{-}]$. When two bands are joined at a bulk Fermi arc, they form two halves of a single Riemann surface.

In the vicinity of the bulk Fermi arc, the effective Hamiltonian for the systems considered here is

[TABLE]

which results in the spectrum of the resonance bands

[TABLE]

Here, $\delta k_{x}$ and $\delta k_{y}$ are the wavevector displacements from the underlying Dirac point at $\mathbf{k}_{\parallel,D}$ which has frequency $\omega_{D}$, $2\gamma$ is the radiative rate of the resonance band which couples to the single environmental channel, $v_{gx}$ and $v_{gy}$ are the group velocities describing the dispersion near the Dirac point, and $\hat{\sigma}_{x,z}$ are Pauli matrices. Equations (5) and (6) are written for the accidental band crossing along $X$–$M$, but letting $x\leftrightarrow y$ yields the correct set of equations for the accidental band crossing along $Y$–$M$. As can be seen, the spectrum given in Eq. (6) exhibits a pair of exceptional points at $(\delta k_{x},\delta k_{y})=(\pm\gamma/v_{gx},0)$, where $\omega_{+}=\omega_{-}$, and which are connected by a bulk Fermi arc along the contour $\delta k_{x}<|\gamma/v_{gx}|$ and $\delta k_{y}=0$.

With the knowledge that the two resonance bands are connected at a bulk Fermi arc, one can now understand the apparent abrupt termination of the lines of BICs in Figs. 1c and 2c. Along the $X$–$M$ high symmetry lines where $\delta k_{x}=0$, one resonance band remains a BIC with $\textrm{Im}[\omega]=0$, but the two bands switch when traveling through the bulk Fermi arc, as can be seen in Fig. 3a. This is because the two connected resonance bands form a single Riemann surface, and trajectories in wavevector space which pass through the middle of the bulk Fermi arc travel from the upper to lower band, or vice versa. We can confirm that near $M$ the symmetry of the upper and lower bands along the $X$–$M$ line has switched, such that the lower band exhibits the line of symmetry-protected BICs while the upper band does not, by viewing the modal profiles of the resonances on both sides of the bulk Fermi arc. As is shown in Figs. 3c-h, for $\delta k_{y}<0$, the odd $x$ symmetry mode is found on the upper resonance band, but for $\delta k_{y}>0$ this mode is found on the lower resonance band. Thus, the symmetry protected line of BICs does exist along the entire high symmetry line, but passes from the upper resonance band to the lower resonance band through a bulk Fermi arc.

In conclusion, we have demonstrated that the environment within which a device resides significantly constrains the nature of BICs in different parts of the device’s resonance spectrum. In particular, we have shown that photonic crystal slabs embedded in a 3D photonic crystal environment can not only exhibit BICs, but can achieve lines of BICs where the environment only possesses a single radiation channel. This capacity to realize such lines of BICs in photonic structures without birefringent materials may have applications in on-chip communication and beam steering of photonic crystal surface emitting lasers. Additionally, we have observed that such simple systems can exhibit highly non-trivial physics, such as a symmetry protected line of BICs switching from one band of resonances to another by traveling through a bulk Fermi arc. More broadly though, the ability to engineer the environment rather than the device to realize BICs in a system opens up a broad range of new experimental possibilities. First, given the advent of advanced 3D-printing techniques such as two-photon polymerization technology Deubel et al. (2004), we expect that structures such as the one described here can be straightforwardly fabricated. However, there are many photonic systems, such as planar grain boundaries in self-assembled structures Blanco et al. (2000), where controlling the specifics of the embedded device may be very difficult, but engineering the environment is trivial, that may yield an entirely different route to photonic BICs than has been previously studied.