## Abstract

The creation of artificial structures with very narrow spectral features in the terahertz range has been a long-standing goal, as they can enable many important applications. Unlike in the visible and infrared, where compact dielectric resonators can readily achieve a quality factor (Q) of 10^{6}, terahertz resonators with a Q of 10^{3} are considered heroic. Here, we describe a new approach to this challenging problem, inspired by the phenomenon of extraordinary optical transmission (EOT) in 1D structures. In the well-studied EOT problem, a complex spectrum of resonances can be observed in transmission through a mostly solid metal structure. However, these EOT resonances can hardly exhibit extremely high Q, even in a perfect structure with lossless components. In contrast, we show that the inverse structure, a periodic array of very thin metal plates separated by air gaps, can exhibit non-trivial bound states in the continuum (BICs) reflection resonances, with arbitrarily high Q, and with peak reflectivity approaching 100% even for a vanishingly small metal filling fraction. Our analytical predictions are supported by numerical simulations, and also agree well with our experimental measurements. This configuration offers a new approach to achieving ultra-narrow optical resonances in the terahertz range, as well as a new experimentally accessible configuration for studying BICs.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Bound states in the continuum (BICs) are non-radiative states whose frequencies lie inside the continuum spectrum of spatially extended states (radiating waves) [1,2]. This concept was originally proposed by von Neumann and Wigner in 1929, in the context of quantum mechanics [3], and has recently become a topic of interest in optics [1]. Normally, a state whose frequency is inside the continuum spectrum of radiating waves has a finite lifetime, because the state is coupled with the radiating waves, leading to a leaky state whose energy can be dissipated by the radiating waves. However, in some circumstances, a state can be decoupled from the radiating waves with the same frequency. This type of states are BICs whose lifetimes are infinitely large. Since structures with BICs have, theoretically, infinitely large quality factors (Q), they have promising applications in optics and photonics, including lasing [4–6], sensing [7,8] and filtering [9].

The scattering of electromagnetic waves by periodic metallic structures is a long-standing topic of interest [10–14]. In particular, a 1D array of slits in an opaque metal screen is of unique importance due to its geometrical simplicity. While bulk metals are highly reflective for a broad frequency range of electromagnetic waves, extraordinary optical transmission (EOT) resonances with nearly unity transmittance can be found on such a structure, either due to the excitation of cavity modes inside the slits or due to coupled surface modes on both sides of the metal screen [15–17]. There have been many theoretical and experimental studies of this general structure [18–26], but so far no investigation of resonant phenomena in its complementary structure, a periodic array of thin metal plates separated by large air gaps. Although the only geometrical difference between the two structures is that the filling fraction of metal approaches unity in the EOT structure but zero in the converse structure, we find that the scattering properties are quite dramatically different, due at least in part to the presence of BICs in the latter case. Since these metal plate arrays can be easily fabricated, such structures have already been used as artificial dielectric materials in the microwave and terahertz range [27–31]. However, an analysis of the resonant features is conspicuously lacking.

In this article, we show that this structure can exhibit narrow reflection resonances with nearly unity reflectance for a *p*-polarized incident wave, even if the metal filling fraction is very small. Hence, in analogy with EOT, we describe these as extraordinary optical reflection (EOR) resonances. We theoretically demonstrate that Qs of EOR resonances can diverge for normal incident angle and for particular plate separations (or plate lengths), indicating the existence of BICs, which has not been found in the EOT structures or other periodic metallodielectric structures with anomalously large reflections [32]. We compare our theoretical results to experimental results, using a structure designed to produce resonances in the terahertz range, and find excellent agreement, validating the theoretical approach.

## 2. Theoretical results

Figure 1 illustrates the structure of the thin-metal-plate array and the definition of the parameters used throughout this article: the length (*L*) and thickness (*a*) of the metal plates, and the separation (*d*) between neighboring plates. The array is illuminated by *p*-polarized plane waves with an incident angle of *θ*. We assume that the metals in the structure are perfect electric conductors (PECs), which is a reasonable approximation for the terahertz and microwave regimes [33]. Conventional modal expansion method [34] or the S-matrix method [35] can be used to solve this scattering problem. However, the solutions from these methods are very complicated, and are not useful for extracting analytical expressions. Thanks to the mirror-symmetry of the structure, we can decompose the original problem into a symmetric sub-problem and an anti-symmetric sub-problem [36], and solve each one by the modal expansion method [34]. In this way, our solution is readily approximated in a compact analytic expression which provides more physical insight.

We define *A _{s}* to be the complex scattered field amplitude when both sides of the array are illuminated by in-phase uniform plane waves with the same incident angle of

*θ*(the symmetric case), and

*A*to be the scattered field amplitude when both sides of the array are illuminated by out-of-phase plane waves (the anti-symmetric case). The complex reflectance

_{a}*r*and transmittance

*t*are related to

*A*and

_{s}*A*by $r=\left({A}_{s}+{A}_{a}\right)/2$ and $t=\left({A}_{s}-{A}_{a}\right)/2$. We use the modal expansion method to solve for

_{a}*A*and

_{s}*A*[34]. Due to the symmetry of each sub-problem, we only need to match the boundary conditions at one surface, rather than both surfaces as in the conventional methods [34,35], so the analysis is significantly simplified.

_{a}As a specific example, we consider an array of thin metal plates with $L=4$ mm and $a=30$ μm. We define ${k}_{//}=k\mathrm{sin}\theta $ to be the parallel wave number of the incident wave, where $k=2\pi f/c$ is the free-space wave number. Figure 2(a) shows the computed |*r*| as a function of both ${k}_{//}$ and frequency at a fixed plate separation of $d=1$ mm, and Fig. 2(b) shows the computed |*r*| as a function of both *d* and frequency at a fixed incident angle of $\theta ={10}^{\circ}$. These results are calculated by retaining the free-space modes from (−25^{th})-order to 25^{th}-order, and the waveguide modes from TM_{0} (TEM) to TM_{30}. In these figures, we clearly observe narrow EOR resonances. Although there are other resonances at higher frequencies (not shown in the figures), we concentrate on the regions where only the zeroth-order free-space mode is propagating (the regions below the green dashed lines in Fig. 2). In this region, the ( ± 1^{st})-order (and higher-order) free-space modes are evanescent surface waves. We observe some interesting properties of these resonances: they have asymmetric Fano line-shapes with peak reflectance as high as unity; surprisingly, their linewidths completely vanish for zero incident angle [see the green ellipse in Fig. 2(a)]; and, these linewidths approach infinitesimally small values for particular plate separations for oblique incidence [see the green ellipses in Fig. 2(b)]. The unusual behaviors on the linewidths are in marked contrast to well-known EOT resonances in the 1D arrays of narrow metallic slits, which always have non-vanishing linewidths, regardless of the geometry of the structure or the angle of illumination.

## 3. Physical insight

To clarify this surprising result, we derive an analytic expression which approximates the rigorous calculation. We assume that the plate thickness is much smaller than the plate separation, i.e. $a\ll d$. Under this assumption, the only possible propagating waveguide modes in the region of interest are the TEM and TM_{1} modes [37], since the cut-off frequency of the TM_{2} waveguide mode, $2\pi c/d$, is always slightly larger than the cut-off frequency of the first-order free-space mode at normal incidence, $2\pi c/\left(d+a\right)$. As a result, we retain only the TEM and TM_{1} terms in the modal expansion. For the free-space fields, both the propagating (0^{th}-order) mode and the evanescent [( ± 1^{st})-order and higher-order] modes play important roles, so we retain the three most significant terms in the modal expansion (0th- and ( ± 1^{st})-order). We also assume that the incident angle is small, i.e. ${k}_{//}\ll k$. With these approximations, it can be shown that the solution to each sub-problem is $A=\left(1+iP\right)/\left(1-iP\right)$ with the following definitions:

*x*) is equal to tan(

*x*) for the symmetric sub-problem and -cot(

*x*) for the anti-symmetric sub-problem. We have $\left|A\right|=1$, since

*P*is a real number in the region of interest. This can also be justified by conservation of energy, since by assumption there is no loss in the structure. In this case,

*A*

_{s}and

*A*

_{a}are vectors on the complex plane of unit length, and their phase difference completely determines |

*r*| and |

*t*|. The expression for the phase of

*A*

_{s}or

*A*

_{a}is $arg\left(A\right)=2\mathrm{arctan}\left(P\right)$, where

*P*is defined in Eq. (1). We note that the first term in

*P*is a slowly varying function, so the frequencies of the EOR resonances are largely determined by the second term of

*P*, which is $N/D$. At frequencies that are far from the zeros of

*D*, this second term is negligible, and the first term of

*P*provides a slowly-varying background phase of

*kL*for the symmetric sub-problem and $kL-\pi $ for the anti-symmetric sub-problem. Since there is a phase difference of

*π*between these two,

*A*

_{s}and

*A*

_{a}point in opposite directions, leading to $\left|r\right|=0$ and $\left|t\right|=1$. However, when the frequency approaches and passes a value such that $D=0$, the second term of

*P*diverges. As a result, one of the phasors

*A*

_{s}or

*A*

_{a}rapidly rotates by 2

*π*while the other does not, leading to a resonance in |

*r*|. The asymmetric Fano line-shapes are caused by the presence of other resonances, which slightly shift the starting positions of the rapid 2

*π*rotations.

We note that, for small ${k}_{//}$ and *a*, *D*_{0} [Eq. (5)] is a good approximation to *D*. The solid lines in Fig. 3(a) track the zeros of *D*_{0} as a function of *d* and frequency for $L=4$ mm. They approximate very well to the resonant frequencies rigorously calculated in Fig. 2(b) [square dots in Fig. 3(a)]. We also note that each zero of *D*_{0} corresponds to a pole of $\mathrm{f}\left({\beta}_{1}L/2\right)$, although slightly red-shifted. The poles of $\mathrm{f}\left({\beta}_{1}L/2\right)$ are ${\beta}_{1}L=\pi ,3\pi ,5\pi ,\mathrm{...}$ for the symmetric sub-problem and ${\beta}_{1}L=2\pi ,4\pi ,6\pi ,\mathrm{...}$ for the anti-symmetric sub-problem. These are exactly the symmetric and anti-symmetric standing-wave conditions for the TM_{1} waveguide mode, respectively. As a result, we conclude that these EOR resonances arise mainly due to the excitation of TM_{1} cavity modes. To confirm our conclusion, we perform finite-element method (FEM) simulations of a device illuminated by a Gaussian beam with an incident angle of 10°. While generally the structure is almost perfectly transparent, we observe reflection resonances near the frequencies predicted by our analytical method [Figs. 4(a) and 4(b)]. The first- and second-order TM_{1} cavity modes, corresponding to the first and second resonances, can be clearly seen in Figs. 4(c) and (d).

We now consider the linewidth of a resonance, which is related to the rate of the rapid 2*π* rotation of *A _{s}* or

*A*with frequency. This rate is determined by the value of $N/\left(\partial D/\partial \omega \right)$ near the resonant point. A smaller value of $N/\left(\partial D/\partial \omega \right)$ corresponds to a narrower linewidth. There are special cases where $N/\left(\partial D/\partial \omega \right)$, where the rapid 2

_{a}*π*rotation becomes a sudden (i.e., discontinuous) 2

*π*phase jump, which is equivalent to a zero phase change. As a result, the resonance disappears (i.e., has zero linewidth, or has infinitely large Q) if $N=0$ at the corresponding zero of

*D*. Another way to describe this situation is that whenever

*N*and

*D*have a common zero, the zero of

*D*is cancelled by the zero of

*N*, so the corresponding resonance disappears. Under the approximation that ${k}_{//}$ and

*a*are small, we find that

*N*vanishes for ${D}_{0}=0$ when either ${k}_{//}$ or ${N}_{0}$ vanish. We clearly observe this disappearing of EOR resonances for normal incidence (vanishing ${k}_{//}$) in Fig. 2(a) (green ellipse). For oblique incidence, i.e. ${k}_{//}\ne 0$, we see several examples of vanishing and reappearing resonances in Fig. 2(b) (green ellipses), due to ${N}_{0}=0$ at particular values of

*d*. By finding the common zeros of ${N}_{0}$ and ${D}_{0}$, we can find the approximate value of

*d*where this occurs, for each resonance. The dashed lines in Fig. 3(a) track the zeros of ${N}_{0}$, and Fig. 3(b) shows the Qs of the resonances rigorously calculated in Fig. 2(b). Whenever the solid and dashed lines with the same color in Fig. 3(a) intersect, ${N}_{0}$ and ${D}_{0}$with the same symmetry (red for the symmetric sub-problem; blue for the anti-symmetric sub-problem) have a common zero, and the cross-point gives an approximate condition for the Q of the corresponding resonance to diverge. We find a very good agreement between the approximation and the rigorous calculation [Fig. 3(b)] of the vanishing-linewidth conditions. The small discrepancies are due to the truncation of the mode expansions, as well as the neglecting of the higher-order terms of

*a*and ${k}_{//}$.

A resonance with diverging Q indicates the existence of BIC [1,2]. The BICs that appear at normal incidence originate from the symmetry incompatibility. For normal incidence, the TM_{1} cavity modes (discrete states), which lead to the resonances, have odd symmetry, while the free-space propagating waves (spatially extended states) have even symmetry. As a result, the TM_{1} cavity modes are decoupled from the free-space propagating waves, leading to BICs with infinitely large lifetime. However, the BICs that appear in the case of oblique incidence are caused by a different mechanism, the coupling of resonances. It has been shown that when two nonorthogonal resonances have identical frequencies, the combined resonance can have vanishing linewidth [38,39]. This phenomenon is known as the Friedrich–Wintgen BICs [2,38].

We note that each zero of ${N}_{0}$ corresponds to a pole of $\mathrm{f}\left(kL/2\right)$, thus corresponding to a standing-wave condition for the TEM waveguide mode. If there are resonances corresponding to the TEM cavity modes, these resonances will be orthogonal to the EOR resonances with the same symmetry [39], leading to the BIC phenomenon. However, there are no observable TEM cavity modes in our structure. To explain this, we need to examine the scattering properties of the structure as it evolves from a narrow-slit array (the EOT structure) to a thin-plate array (the EOR structure). For a narrow-slit array, the transmittance is usually low except for some narrow resonances with nearly unity transmittance, which are caused by the excitation of TEM cavity modes [15–17]. As the metal filling fraction of the structure decreases, the linewidth of the TEM resonances increase; eventually, in the case of a thin-plate array, these TEM resonances become so broad that they merge into an almost uniform background with nearly unity transmittance. As a result, the TEM cavity modes do correspond to resonances in our structure, although they are too broad to be distinctly observed. These hidden TEM resonances can interact with the EOR resonances (TM_{1} modes) with the same symmetry, leading to non-trivial BICs with vanishing linewidths.

## 4. Experiments

We perform experimental measurements for direct comparison with our predictions. We fabricate a device by mechanically assembling a set of titanium plates with precision spacers, similar to ref [31]. The assembled structure has $L=4$ mm, $a=100$ μm and $d=1$ mm, which is close to the theoretical Friedrich–Wintgen BIC condition for the 4^{th} resonance ($L=4$ mm, $a=100$ μm and $d=0.98$ mm) and is designed to produce resonances in the terahertz spectral range. We measure its power transmission spectra at various angles of incidence, using a THz time-domain spectrometer [40]. In our measurements, the incident wave is approximately a Gaussian beam, with a 1/*e*-amplitude beam diameter of 10 mm at the input facet of the EOR device [Fig. 5(a)]. Figures 5(b) and 5(c) show the experimental and predicted power transmittance |*t*|^{2} as functions of frequency and *θ*, where the predicted result is computed using the same truncated expansions as used to derive the plots in Fig. 2. Figure 5(d) shows the comparison between the experimental and theoretical power transmittance spectra at $\theta ={26}^{\circ}$. We find remarkable agreement between the experimental and theoretical results. The discrepancies in the depth and width of the resonances are primarily caused by the finite beam radius (limited by the size of the device), imperfections in the device structure, and the fact that titanium is not a perfect metal [37], as well as due to the limited spectral resolution of the spectrometer. Despite these experimental limits, we nevertheless find BIC-induced vanishing of resonances for $\theta ={0}^{\circ}$ [green ellipse in Fig. 5(b)] and for the 4th resonance [yellow ellipse in Fig. 5(b)]. We also observe a reflectivity $\left(1-{\left|t\right|}^{2}\right)$ of ~0.8 at the peak of the 2^{nd} resonance near 0.163 THz, despite the fact that 90% of the structure's surface area is empty space. In the spirit of analogy to the EOT effect, we note that this corresponds to an area-normalized reflectivity far greater than unity.

## 5. Conclusion

In conclusion, we have shown that a periodic array of thin metal plates exhibits narrow reflection resonances with nearly unity reflectance, which we describe as extraordinary optical reflection (EOR). A particularly interesting feature of the EOR structure is that it supports BICs for zero incident angle (arising from the symmetry of the states), and also for particular plate separations at non-zero angle (arising from coupling between non-orthogonal states of the same symmetry). In both cases, these BICs can produce resonances with infinitely large Q. This feature is distinct from the behavior of the analogous and well-studied EOT structure. Given the relative ease with which an EOR structure of this type can be fabricated, and the unique linewidth behavior described here, this represents a new and potentially valuable approach to the realization of narrow linewidths in artificial structures, and a new experimentally accessible system for studying BICs. Since, theoretically, the linewidths of the resonances can be infinitesimal, this structure could have important applications in infrared, terahertz, and microwave optics.

## Funding

National Science Foundation (NSF) (1609521).

## Acknowledgments

We acknowledge useful discussions with Yasith Amarasinghe.

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