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Abstract
We investigate the optical resonances in coupled meta-atoms with hybrid interaction pathways. One interaction pathway is the directly near-field coupling between the two meta-atoms. The other interaction pathway is via the continuum in a waveguide functioned as a common bus connecting them. We show that by properly introducing gain or loss into the meta-atoms, the hybrid optical system becomes parity-time ($\mathcal {PT}$) symmetric, in which the effective coupling rate can be customized by manipulating the length of the waveguide. At the exact phase of the customized $\mathcal {PT}$ symmetry, the coupled meta-atoms support discrete super-resonant modes that can be observed from the transmission spectra as extremely sharp peaks. At an exception point where the eigenmodes coalesce, albeit the transmission curve is flat, a high-Q factor of the localized field in the meta-atoms can be obtained. Similarities of the super-resonance with the bound states in the continuum (BICs) are discussed. This investigation promotes our understanding about the ways in realizing high-Q optical resonance especially by manipulating the distributions of loss and gain via the concepts of $\mathcal {PT}$ and BICs. Many attractive applications are expected.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past years, many researchers have made attempts to transfer the concepts and technologies from quantum physics to optics. Conventional quantum states on electrons face the complications arising from decoherence and many-body effects. On the contrary, optical waves can be easily controlled in the macroscopic scale by properly engineering optical waveguides and resonators. Among all the optical-quantum analogues investigated so far, bound states in the continuum (BICs) [1–6] are the most appealing one. A BIC is a perfectly confined state that coexists with a continuous spectrum of radiating waves. As an embedded non-leaky mode, it does not couple with the continuous spectrum. The high-$Q$ factor of a BIC can promise many attractive applications [1].
The intrinsic physical mechanisms of BICs are the overlap of multiple eigenstates and the constructive interference of the leaky radiation [1]. General examples of BICs are these ones protected by symmetry, separability, and through parameter tuning [1]. Among various formula proposed for the explanation of BICs, the temporal coupled-mode (TCM) theory [7,8] is very successful in the study of Fabry-Perot BICs (FP-BICs) [9–17]. In TCM, in order to describe the leaky nature of each constituent optical resonance, the diagonal elements of the Hamiltonian matrix are complex. An analysis on the existence of a real eigensolution of the non-Hermitian Hamiltonian can explain the interference effect in forming a FP-BIC.
On the other hand, in recent years we have also witnessed the advances of non-Hermitian optics especially the parity-time ($\mathcal {PT}$) symmetry and exceptional points (EPs) [18–22]. In $\mathcal {PT}$-symmetric optics, the interplay of gain and loss produces many unprecedented novel optical effects [18–22]. Since both BICs and $\mathcal {PT}$ symmetry are focused on the coupling among different optical resonances and on the manipulation of the loss characteristics of the coupled mode, it is then an interesting question that whether these two concepts can be combined together in order to reveal new physics and to demonstrate other novel optical effects [23–30].
In this article we pay attention to the optical resonance of coupled meta-atoms (or subwavelength resonators) with hybrid interaction pathways. One interaction pathway is the directly near-field evanescent coupling when the meta-atoms are placed closely together. The other interaction pathway is via the continuum in a waveguide functioned as a common bus connecting these two meta-atoms. The interaction strengths of both pathways can be controlled by proper structural design. Although similar configurations were shown to support FP-BICs [10–18] and phase-coupled plasmon-induced transparency [31] in the passive scenario, our attention is paid to the active case. We show that by properly introducing gain or loss into the meta-atoms, the TCM Hamiltonian of the hybrid system can transfer into a perfect $\mathcal {PT}$-symmetric one. The off-diagonal effective coupling rate is customizable by simply manipulating the length of the waveguide. Many features of the customized $\mathcal {PT}$ symmetry can be observed from the transmission spectra in the waveguide. The formation of discrete optical super-resonances at the exact $\mathcal {PT}$ phases and a BIC-like resonance at EPs are analyzed and discussed. This investigation promotes our understanding about the ways in realizing high-$Q$ resonances of optical waves by manipulating the distributions of loss and gain, and promises many attractive applications.
2. Theory and simulation
2.1 Theory of the customized $\mathcal {PT}$ symmetry
Figure 1 shows the structure under investigation, which consists of two meta-atoms and a curved waveguide in coupling them together. The localized resonances in the two meta-atoms have the same angular frequency of $\omega _0$. Unlike former studies on passive FP-BICs [10–18], we further assume that appropriate gain or loss are applied to the meta-atoms. Consequently, the coupled system could be modeled in TCM as
Here $\omega ^{\pm }$ are the eigenfrequencies, and $\Psi =(E_1,E_2)^T$ is the complex fields in the two meta-atoms. The non-Hermitian Hamiltonian $H$ is
Fig. 1. (a) Schematic of the energy-level diagram under investigation and the two interaction pathways between the two meta-atoms. (b) Structure design in our COMSOL simulation. Width of the waveguide is $D = 125$ $\mu$m. The semiminor (semimajor) axis of each meta-atom is $r = D/2$ ($R = D$). The distance $d=20$ $\mu$m between the two meta-atoms determines $\kappa$, while the distance $s= 5$ $\mu$m between the meta-atoms and the waveguide determines $\gamma$. Refractive index of the meta-atoms and the waveguide are $n_r = 2$, and the surrounding media are assumed to be metal-like with $\epsilon _m=-300$. Characteristics of the hybrid system can be detected from the transmission spectra $T$ of the extended waveguide.
In the absence of the artificially introduced gain and loss, i.e. $g_1=g_2=0$, the system is passive and supports FP-BICs which could trap standing waves to achieve a high-$Q$ resonance. Generally, now the eigenfrequencies are $\omega ^{\pm }=\omega _0\pm \kappa -j\gamma (1\pm e^{j\phi })$. In other words, $Re\{\omega ^{\pm }\}=\omega _0\pm (\kappa +\gamma \sin \phi )$, and $Im\{\omega ^{\pm }\}=-\gamma (1\pm \cos \phi )$. When $\phi$ is an integer multiple of $\pi$, e.g. $\phi ^+_{\textrm {BIC}}=(2m-1)\pi$ ($\phi ^-_{\textrm {BIC}}=2m\pi$) where $m$ is an integer, a FP-BIC is obtained at $\omega _{\textrm {BIC}}^{+}$ ($\omega _{\textrm {BIC}}^{-}$) which is a real number. The real eigenfrequency implies that a totally destructive interference of the leaked fields from the two meta-atoms occurs, and the eigenmode is perfectly confined in the system. At the same time, the other eigenmode at $\omega _{\textrm {BIC}}^{-}$ ($\omega _{\textrm {BIC}}^{+}$) is a more lossy mode due to the constructive interference, and leaks at the twice rate as that of an individual meta-atom [1].
From the above short review we can see the study on BICs requires people to find the conditions in realizing real eigenfrequencies of the Hamiltonian $H$. With versatile combination of parameters, it is indeed feasible to achieve an accidental real solution of Eq. (3). However, here we are interested in a set of solutions that can be described analytically. Toward this target, we can firstly eliminate the background loss outside the square-root arithmetic of Eq. (3), i.e.
which gives $\omega ^\pm =\omega _0\pm \sqrt {\kappa _{\textrm {eff}}^2-(g_1-\gamma )^2}$. Now we can see that once $k_{\textrm {eff}}$ is a real number, which can be achieved by tuning the phase delay $\phi$ that is proportional to $L$, the coupled system would become to a perfect $\mathcal {PT}$ symmetric one. To be more explicit, two sets of customized $\mathcal {PT}$ symmetry can be achieved when $\phi$ is just at the middle of two adjacent FP-BICs $\phi ^\pm _{\textrm {BIC}}$. The first set of perfect $\mathcal {PT}$ symmetry ($\mathcal {PT}$1) requires where $m$ is an integer, and the eigenfrequencies are given byThe second set of perfect $\mathcal {PT}$-symmetry ($\mathcal {PT}$2) is achieved when
and the eigenfrequencies areThe waveguide lengths $L$ of adjacent $\mathcal {PT}$1 and $\mathcal {PT}$2 phases are different by half of the effective wavelength.
Figure 2 shows the variation of the real part of the eigenfrequencies at the $\mathcal {PT}$1 and $\mathcal {PT}$2 phases when $\kappa =4\gamma$. We can see the eigenfrequencies are symmetric with respect to $g_1=\gamma$. When the $\mathcal {PT}$ phase is conserved (exact $\mathcal {PT}$), each scenario supports two real eigenfrequencies. When $g_1$ is sufficient different from $\gamma$, EPs can be formed, across which the eigenfrequencies become complex.
Fig. 2. Variation of the eigensolutions of $\mathcal {PT}$1 and $\mathcal {PT}$2 versus $g_1$ when $\kappa =4\gamma$.
Above analysis shows that with properly introduced gain or loss $g_{1,2}$, an effective $\mathcal {PT}$ symmetry can be customized in the coupled meta-atoms. The splitting of the eigenfrequencies is determined by the values of $\kappa$ and $\gamma$, the gain difference $g_1-g_2 = 2(g_1-\gamma )$, and the phase delay $\phi =k_{\textrm {eff}} L$. With properly structural design the coupled system can be held within the exact $\mathcal {PT}$ phase with two discrete real eigenfrequencies. From Eqs. (6) to (9) we can see a unique feature of the structure investigated here is that the length $L$ can manipulate the set of $\mathcal {PT}$ symmetry ($\mathcal {PT}$1 or $\mathcal {PT}$2) achieved in the system. In the next subsection we would provide an example on the phase transition from $\mathcal {PT}$1 to $\mathcal {PT}$2 by tuning $L$.
2.2 Super-resonance at the exact $\mathcal {PT}$ phase
To prove the customized $\mathcal {PT}$ symmetry can be really realized and to reveal features of the real eigenfrequencies at the exact $\mathcal {PT}$ phase, we perform full-field finite element optical simulations (COMSOL Multiphysics) on the designed structure shown in Fig. 1. Refractive index of the meta-atoms and the waveguide are $n_r = 2$, and width of the waveguide is $D = 125$ $\mu$m. Because TCM requires that the investigated resonance inside each meta-atom could couple into the two directions of the waveguide simultaneously and equivalently, the two meta-atoms are set to be elliptical with the semiminor axis ($r = D/2$) much smaller than the effective wavelength $\lambda /n_r$. The semimajor axis ($R = D$) is oriented normal to the waveguide. Such an orientation forces the excitation of a linear polarized resonant mode inside each meta-atom. Rings or wispergalley resonators with radius much larger than the wavelength cannot be utilized here because the clockwise/counterclockwise helical modes would couple only to an individual direction of the waveguide. Distance between the two meta-atoms is $d = 20$ $\mu$m, whereas the distance of each meta-atom to the adjacent waveguide is $s = 5$ $\mu$m. To avoid field leakage from the curved waveguide and the meta-atoms, the surrounding media are assumed to be metal-like with $\epsilon _m=-300$.
The gain and loss are presented by a nonzero imaginary component $n_i$ of the refractive index $n=n_r+jn_i$ in the two meta-atoms. Value of $n_r$ is kept at 2, whereas $n_i$ is much smaller than $n_r$. Consequently, it is a good approximation that $g$ is proportional to the perturbation $n_i$ here, i.e. $g =\beta n_i$. Best fitting shows that $\beta = 2\pi \times 161$ GHz within the frequency regime we are interested in. In the COMSOL simulation, we first introduce a proper $n_i$ to the first meta-atom. In order to satisfy Eq. (5), the value of $n_i$ in the second meta-atom is then carefully modified until the Eigenfrequency Solver in COMSOL gives two real solutions of the investigated structure.
Figure 3 shows the transmission spectra $T$ in a chosen $\mathcal {PT}$1 phase at different $g_1$ values. We can see two very sharp transmission peaks $T$ can be obtained. Quality of the resonance can be represented by the $Q$ factor defined by the maximum field $E_{\textrm {max}}$ in the meta-atoms with respect to the incidence $E_{\textrm {in}}$, i.e. $Q=|E_{\textrm {max}}/E_{\textrm {in}}|^2$. The $Q$ factor can reach the order of $10^8$ at the high-$T$ peaks, representing an extremely enhancement of the optical field in the meta-atoms. Such a kind of resonance can be termed super-resonance. From Fig. 3 we can also see when $g_1$ decreases from zero ($g_2$ increases from $2\gamma$), the two super-resonant peaks approach each other. This feature is in agreement with the curves shown in Fig. 2. With further decreased $g_1$ till $g_1 =-\kappa$, the two transmission peaks coalesce together in forming an EP, which is discussed in the next subsection.
Fig. 3. Transmission spectra $T$ at different values of $g_1$ in a structure design supporting a $\mathcal {PT}$1 phase. Insets are the distribution of field amplitude at the high-$T$ peaks. Note that $T$ is plotted by the logarithm function (log).
The two high-$T$ peaks of super-resonance bear great similarities with BICs and super-cavity modes [1–17]. They are all of extremely large $Q$ factors (even infinite) or ultra-narrow FWHM. Furthermore, they are all associated with real eigenfrequencies of the Hamiltonian [1]. For BICs and super-cavity modes, the real eigenfrequencies are related to the null leakage of field into the environment, which can be achieved by destructive interference, symmetric protection, or inherent symmetric mismatch with bulk modes [1]. For the super-resonance presented here, they are associated with the exact phase of the customized $\mathcal {PT}$ symmetry. To verify that the super-resonance is closely related to the balance condition of Eq. (5), in Fig. 4 we show the transmission spectra $T$ at different $g_1$ values when $g_2 = 2\gamma$. We can see the super-resonance can be observed only at $g_1 = 0$. A tiny perturbation from $g_1 = 0$ would violate Eq. (5) and the transmission $T$ would drop strongly even when the perturbation provides additional gain, i.e. $g_1 + g_2 > 2\gamma$. Figure 4 proves the importance of the subtle balance in the customized $\mathcal {PT}$ symmetry, that the eigenmode with a complex eigenfrequency is weakly localized in the coupled meta-atoms. It also implies that the super-resonance is not an ordinary lasing phenomenon because an increased gain would reduce, other than enhance, the transmission spectra $T$ of field. Such an effect is similar to the singular exceptional resonance discussed in gain-mediated metamaterials [32].
Fig. 4. Transmission spectra verse $g_1$ when $g_2 = 2\gamma$. When $g_1 = 0$ an exact $\mathcal {PT}$1 phase is achieved. Apart from $g_1 = 0$, the values of $\omega ^\pm$ are complex and the magnitude of $T$ reduces sharply.
Figure 3 only displays the scenario of a $\mathcal {PT}$1 phase at a proper waveguide length of $L_{\mathcal {PT}1}$. The case of a $\mathcal {PT}$2 phase can be found as well when the waveguide length $L$ is properly modified by $\delta L =\lambda _{\textrm {eff}}/2$. To see how these two sets of $\mathcal {PT}$ phases transit into each other, we study the situations of different $\delta L$ values so that the length $L$ increases slowly from $L_{\mathcal {PT}1}$ ($\delta L=0$ $\mu$m) to $L_{\mathcal {PT}2}$ ( $\delta L=224$ $\mu$m). Figure 5 shows the results that display the transition from $\mathcal {PT}$1 to $\mathcal {PT}$2. Note that the super-resonant peaks of $T$ at the two exact $\mathcal {PT}$ phases are reduced by 8 orders in order to get comparable magnitudes with other curves and to get much information about the variation of the positions of $T$.
Fig. 5. Transmission spectra $T$ when $L$ increases. Here $g_1 = 0$ and $g_2 = 2\gamma$. The lower and upper scenarios are within the exact $\mathcal {PT}$ phases, and their amplitudes of $T$ are artificially decreased by 8 orders.
From Fig. 5 we can see at the exact $\mathcal {PT}$1 and $\mathcal {PT}$2 phases ($\delta L = 0$ $\mu$m and 224 $\mu$m, respectively), super-resonances with extremely high-$T$ peaks are obtained. The two sets of super-resonant peaks are at 306.82/307.76 GHz and 306.25/308.32 GHz, respectively. The best fitting with Eqs. (7) and (9) gives $\omega _0 = 2\pi \times 307.29$ GHz, $\gamma = 2\pi \times 0.26$ GHz and $\kappa = 2\pi \times 0.80$ GHz. Between these two exact $\mathcal {PT}$ phases the transmission is much less effectively. The amplitude of $T$ decreases sharply and FWHM increases strongly because now the effective coupling rate $\kappa _{\textrm {eff}}$ is complex and the perfect $\mathcal {PT}$ symmetry is broken. At the condition of $\phi = 2m\pi$ for FP-BIC ($\delta L = 112$ $\mu$m), the curve of $T$ is very flat. Figure 5 proves that the system we proposed here indeed provides versatile transmission features, which can be customized by manipulating not only the complex rate $\kappa$ of the directly coupling pathway but also $\gamma \exp (j\phi )$ of the indirectly pathway via the continuum in the waveguide.
2.3 BIC-like super-resonance at EP
Being an effective $\mathcal {PT}$-symmetric system where the off-diagonal element $\kappa _{\textrm {eff}}$ can be customized especially by changing the length $L$ of the waveguide, the coupled meta-atoms also support EPs. The positions of EPs, where both the eigenfrequencies and the eigenstates coalesce, can be found by requiring a null of the square-root arithmetic of Eqs. (7) and (9). For the $\mathcal {PT}$1 phase, the existence of EPs requires
While for the $\mathcal {PT}$2 phase,
In consistent with Fig. 2, the two solutions of Eqs. (10) and (11) are symmetric with respect to $g_1 =\gamma$.
From the COMSOL simulation we confirm the emergence of EPs. As shown in Fig. 6, when $g_1=-\kappa$ and $g_2=2\gamma +\kappa$, the two super-resonant peaks of $\omega ^\pm _{\mathcal {PT}1}$ shown in Fig. 3 almost overlap with each other around $\omega _0$. Their amplitudes reduce strongly. In the curve of $T$ we could only observe a shallow saddle at $\omega _0$ surrounded by two broaden peaks. Shape of the saddle is very sensitive to the dielectric constants utilized in the COMSOL simulation, a phenomenon in agreement with the high sensitivity of EPs versus perturbation [19,20]. Although the curve of $T$ is very flat, the $Q$ factor can still reach a huge value. We detect the maximum field $E_{\textrm {max}}$ in the two meta-atoms, and find an astonish phenomena that the position of the peak of the $Q$ factor, which can reach $6\times 10^4$, coincides with the saddle in $T$ at $\omega _0$, see Fig. 6(b).
Fig. 6. Spectra of (a) $T$ and (b) $Q$ factor at EP. (c) and (d) are the distributions of field amplitude when incident from opposite ports.
Above demonstrated features of high-$Q$ resonance with a flat $T$ curve are very similar to the well-known FP-BICs [1]. We confirm that when $g_1=g_2=0$, the FP-BICs in the investigated structure are characterized by a wide-band flat spectra ($T = 1$) with a single peak of high-$Q$ factor. Results in Fig. 6 are similar to these of FP-BICs, that the shallow saddle in $T$ cannot be well distinguished from the spectra, but the field is in fact strongly bound and enhanced inside. It implies that at this degeneracy point of EP, the resonant mode interacts with the radiation channels of the environment very weakly and it is protected inside the meta-atoms. Consequently, such a mode at EP can be termed a BIC-like super-resonance.
3. Discussion
In the above sections we prove the feasibility in realizing a customized $\mathcal {PT}$ symmetry in coupled meta-atoms by manipulating the interaction pathway via the continuum in a curved waveguide. Generally speaking, at the exact $\mathcal {PT}$ phase two super-resonant peaks at discrete frequencies could be obtained. However, dispersions are not explicitly considered in the proposed TCM theory, which makes it impossible to simultaneously realize the exact conditions of the two super-resonances. To be more explicit, once the value of $\kappa _{\textrm {eff}}$ is real for one eigenmode, the value of $\kappa _{\textrm {eff}}$ for the other eigenmode is generally complex due to the dispersions in $kappa$, $\gamma$, and $k_{\textrm {eff}}$. A nonzero imaginary component of $\kappa _{\textrm {eff}}$ would break the exact $\mathcal {PT}$ phase, and introduce additional broadening effect to the transmission peak. Nevertheless, the observation of a single super-resonant peak in future experiments can still prove the feasibility of our proposal and can promise many attractive applications.
As for experimental verification of the proposed super-resonant effect, it can be made in metallic slabs with etched structures. At microwave region the structure can be easily etched in some metal slab by various methods. Albeit at microwave region most metals are perfect electronic conductor, adjacent subwavelength resonators or meta-atoms and the waveguide can be still coupled together via the diffracted evanescent field from the etched edge. Distribution of field in the structure can be detected by using near field scanning techniques. Note that in the higher frequency regions such as the infrared and visible ones, some assumptions made in this article cannot be applied. In particular, the imaginary component of the negative permittivity cannot be ignored due to the strong dispersion in metal and semiconductors, and the causality principle required by the Kramers-Kronig relations must be satisfied [33]. Nevertheless, with the-state-of-the-art techniques on the fabrications of various optical micro-resonators and waveguides [34–41], an experimentally realization of the proposed scheme in optical bands is still feasible.
In our study we show that super-resonance can be achieved in the customized $\mathcal {PT}$-symmetric systems. The field can be enhanced in the meta-atoms by up-to $10^7$ times. Furthermore, in the COMSOL simulation we can see FWHM in $T$ is not a good character in representing the field enhancement effect (especially at EP where the transmission spectra is flat but the field is still greatly enhanced inside, see Fig. 6). This work promotes the advances of the optical-quantum analogues about $\mathcal {PT}$ and BICs, and pushes forwardly our efforts in engineering high-$Q$ optical resonance. The field enhancement effect in the super-resonance has many potential applications such as in achieving dark mode lasers, efficient nonlinear optical conversions, and for sensing purpose.
4. Conclusion
In summary, in this article we investigate the optical resonance in coupled meta-atoms with hybrid interaction pathways. One interaction pathway is the directly near-field coupling, and the other one is via the continuum in a waveguide connecting the meta-atoms. We show that with properly introduced gain and loss, the hybrid system can be customized to be $\mathcal {PT}$ symmetric, in which the effective coupling rate is controllable by manipulating the length $L$ of the waveguide. At the exact phase of the customized $\mathcal {PT}$ symmetry, the coupled meta-atoms support extremely sharp super-resonant peaks. By tuning $L$ the super-resonances at the two sets of exact $\mathcal {PT}$ phases ($\mathcal {PT}$1 and $\mathcal {PT}$2) can transit into each other. At EPs, BIC-like phenomena can be obtained. Similarities of the super-resonant phenomena with these of BICs are discussed. This investigation promotes our understanding about the ways in realizing high-$Q$ optical resonance especially by manipulating the loss and gain in optical systems. Many attractive applications can be imaged.
Funding
National Natural Science Foundation of China (11874228, 11974259).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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