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Abstract
In this article, we developed a generalized coupled-mode theory for mixing an isolated state with a continuum having an intrinsic energy gap, which dubbed as “the bound states in the gapped continuum” (BIGC). We investigated the mixture interaction by mimicking the Su-Schrieffer-Heeger model in an optical coupled waveguide array (WA), and presented a unified engineering mechanism for topologically-protected zero modes, Fano resonance, and Tamm surface states, even though those phenomena are diverse in topological insulators, atomic physics and semiconductors, respectively. By tuning the on-site potential and coupling strength of the isolated state, we found the unified operating characteristics for zero modes, Fano resonance, and Tamm states, with demonstrating their localization, transmission spectra, and distinct evolution dynamics explicitly. As an extension for triple-modes coupling, two special sandwich-like configurations are studied: the isolated-continuous-isolated and continuous-isolated-continuous configurations lead to adiabatic eliminations and domain walls, respectively, revealing possible applications and wide connections in many fields of physics and optics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coupled mode theory (CMT) is ubiquitous as a principle (or tool) for engineering in science and technology, and has been widely used across many disciplines for decades [1,2]. Implementations of CMT differ, but share the underlying assumption that a solution to a perturbed or weakly interacting system can be decomposed into a linear combination of its possible eigenmodes. For example, CMT has been widely utilized as a basic analytical formulation for different material systems such as optical waveguides [3,4], photonic crystals [5,6], semiconductors lasers and devices [7–9], phenomena such as Fano resonance [10–13], discrete solitons [14–16], Anderson localization [17–19], and also provides access to parity-time symmetry systems [20–23], and topological photonics [24–28].
Figure 1 depicts three typical setups in the CMT: the couplings between two isolated modes, two continuous modes and the hybrid continuous-isolated modes, respectively. In optics, the guide modes couple and transfer energy (or intensity) when the electric field amplitudes of two waveguides overlap (evanescent field), indicating the fundamental typical properties of mode-coupling. The coupling between adjacent waveguides leads to the superposition of two eigenmode, with emergence of, with emergence of symmetric and antisymmetric modes as shown in Fig. 1a. Extensively, Fig. 1b showed the gap generation when two continuous modes couple in semiconductor physics as well as topological insulators. The gap generation classifies the phase transition from insulators and metals, and the topology of those emerging gaps can further enable to classify the insulator phases into topologically trivial or non-trivial families. As a non-trivial exemplified, the gap generation is easily explained as well from the coupled-mode theory for two continuous bands crossing. Interestingly, Fig. 1c showed the Fano resonance emerging from the “bound states in the continuum” (BIC), the mixture of the localized and continuous states [29]. The physics of Fano resonance and its application have been widely investigated in a number of physical and optical systems [30]. The microscopic origin of the Fano resonance arises from the constructive and destructive interference of a narrow discrete resonance with a broad spectral line or continuum, resulting in asymmetric profile of the localized state which was Lorentzian line shape before interference which also related to the dynamics of CMT [11].
Fig. 1. The generalized coupled mode theory (CMT) for isolated-isolated states, continuous-continuous states, and mixed isolated-continuous states. (a) The symmetric and antisymmetric modes in waveguide systems, (b) gap generation in semiconductor physics, and (c) Fano resonance in scattering phenomena are unified in the CMT framework.
Among these areas, topologically-protected zero modes, Fano resonance, and Tamm surface states are repeatedly discussed for their distinct and attractive confinement properties features, and diverse applications in numerous independent branches of physics and optics. Zero modes are protected by sublattice symmetry in topological insulators, Fano resonance widely emerges from BIC, and Tamm states are well represented in surface physics and semiconductor physics [5,10,31–34]. Under the framework of CMT, surprisingly, these three localized mechanisms can be nicely demonstrated in a unified way via the interaction of isolated states and the ‘gapped’ continuum.
This work extends the coupling process of the isolated state with the gapped continuum, as compared with the conventional ‘gapless’ continuum (Fig. 1c). In contrast to the gapless continuum, the gapped continuum gives rise to robust protected modes, stabilizing the system against perturbations inherent in fabrication techniques. Owing to the inherent similarity between coupling in waveguides for optical fields and hopping in lattices for electrons, the optical waveguide array (WA) is chosen to mimic the Su-Schrieffer-Heeger (SSH) model [31] as the gapped continuum in this work [32,35,36]. By tuning the on-site potential of isolated states, and coupling strengths between isolate and the continuous states, the engineering from gap-protected zero modes, Fano resonance, and Tamm surface states is achieved. Then two applications based on zero modes, and Tamm state engineering, namely, the adiabatic elimination (AE) and domain walls, are demonstrated in the SSH dimerized WA model. We addressed that the CMT is not only applied to the coupling of the coupling of isolated-continuous (IC) states, but also to isolated-isolated (II) and continuous-continuous (CC) states, which have been intensively studied. A brief discussion of this extension is given at the end of this work.
2. Modelling
The dimerized SSH model can be easily achieved by applying binary coupling strengths to an optical WA, namely, designing a binary distance between adjacent waveguide channels. The WA material of modelling are not limited, in which the corresponding fabrication material, fabrication methods and structural parameters are generic, such as silicon-on-insulator structures by deep-UV lithography [35], and borosilicate glass by femtosecond laser direct writing [37], but these factors will not influence our discussion in this work. Thus, the SSH model for an optical WA is constructed as an example to define a gaped continuum in the following discussion. The corresponding dimerized Hamiltonian is given by:
3. The confinement of zero modes and Tamm states
Figure 1a and c show the energy spectrum of our hybrid SSH model of zero modes and Tamm states, namely, ‘bound states in the gapped continuum’ (BIGC), by tuning the on-site potential as depicted in the insets. The simulation parameters are κ0=0.5, δκ=-0.2, and β0=1 (all energy parameters are scaled by β0, thus, dimensionless), 2N + 1 = 41, δβ=0, 0.06, and 1.1, respectively, and V0=0.3. Usually the SSH model has two distinct phases (δκ<0, and δκ>0), one is topologically non-trivial and the other is trivial. The trivial phase (B-phase, δκ>0) is taken as the reference, and eventually the non-trivial phase (A-phase, δκ<0) is engineered for applications [39]. In Fig. 2a, the isolated state localized in the gap (A-phase) (δβ=0<Δ), is protected by the continuum and is called the protected zero mode. For δβ=1 in Fig. 2c, the on-site potential lies outside of the SSH spectrum and generates the second kind of localized state, i.e., Tamm surface state. Recently, a topological Tamm state was presented by L. Wang etc. [40] Since the traditional Tamm states is the defect surface mode in bulk without topological properties, a nontrivial SSH model is achieved by layer structure in 1D plasmonic crystals, which plays the role of topologically protected gap. As long as a defect surface mode located in this gap, the Tamm state earns topological properties. In our configuration, however, the Tamm surface state remains non-topological defect surface mode although the lattice is designed as SSH model.
Fig. 2. Eigenmode spectra for the mixing between the isolated state (n = 0) and the gapped continuum (2N = 40). The isolated state lies (a) in the gap (<Δ, in the inset), (b) in the bulk (∼ω, in the inset), and (c) outside the bulk spectra (>ω, in the inset), respectively. The insets in (a) and (c) show the confinement configuration of the eigenstates for the protected zero mode and Tamm surface state in lattice space.
Figure 2b shows the general case for the coupling of IC with the on-site potential, in which δβ=0.6 and the isolated state is clearly located in the continuum. If the difference of on-site potential δβ approaches 0.85 and the coupling between isolated and continuous states is much weaker (V0=0.06), Fano resonance occurs as expected (discussed in the next section). From these eigenmode diagrams, the differences between the two localized modes are not obvious, thus the confinement configuration of eigenstates is presented for the protected zero mode and Tamm surface state in lattice space as the insets in Fig. 2a and c, respectively. The eigenstate of the zero mode is protected by sublattice symmetry, and thus occupies only the even (or odd, depending on the count order) lattice. Here, the light is launched into the 0th waveguide, so, an even waveguide number is occupied in Fig. 2a inset for the zero mode and decays rapidly into the bulk. On the contrary, the Tamm surface state decays exponentially into the bulk without the protection of such sublattice symmetry, as shown in Fig. 2c inset.
Figure 3 shows the near-field propagation and its output for the three cases when the corresponding propagation constant difference of the isolated state were set to δβ=0, 0.06, and 1.1, respectively. The incident field was launched from the 0th waveguide to excite the zero mode and Tamm state. The field propagation retains its localization mostly on the 0th waveguide but still a small proportion of the field was scattered into the continuum due to the initial eigenstate projection. Meanwhile, Fig. 3a–3d and 3c–3f, show the near-field dynamics and final output of the zero mode and Tamm state with similar confinement at the WA boundary. The difference between their confinements is consistent with that of their eigenstates in Fig. 2a and c insets. To realize the three effects, we have to map the relative parameters into the experimental configurations, such as the materials of waveguide modes or resonator modes, the working frequency (microwave or optical), the geometric structures and fabrication parameters. Since the confinement and scattering effects is comparably robust against the perturbations, many microwave [41]and optical [15,17,28] platforms can be abled to conduct the experiment.
Fig. 3. Evolution dynamics in the WA for zero mode excitation (a), the bound state in the continuum (b), and Tamm state excitation (c), demonstrating even waveguide number occupation, decay in the bulk and exponential decay, respectively. Their respective output distributions are shown in (d–f).
4. Fano resonance engineering between zero modes and Tamm states
When the isolated state lies in the bulk, the propagation distribution appears to be diffusive, without the typical resonant characteristics of diffraction management [3]. Figure 3b and 3e show the spreading field evolution and its output distributions. However, it is difficult to distinguish its dynamic properties from the bulk properties in the energy spectrum in Fig. 2b or field distribution in Fig. 3b. To demonstrate Fano resonance, the transmission spectrum is required and is calculated by the transfer matrix method [42], the results of which for different cases are given in Fig. 4. The simulation parameters are δβ=0.85 and V0 is set at 0, 0.06 (Fano resonance), 0.2, and 0.4, respectively. The spectrum shows that the isolated state is strongly affected by the continuum, especially leading to Fano resonance around the same energy level as the isolated state. Figure 4a shows the typical antisymmetric Fano linewidth in the transmission spectra at weak coupling V0=0.06 (red curve), this effect has been widely studied in WAs, atomic physics, and coupled quantum dots [11]. At V0=0 (black curve), without coupling to the isolated state, the transmission spectrum shows the bandwidth (one-branch) of the continuum ranging from 0.4 to 1.0, and the zero mode and Tamm state are outside this range, thus, have no contribution to the transmission spectrum.
Fig. 4. Fano resonance is antisymmetric in the transmission spectrum when the isolated state couples with the gapped continuum. (b) The relative energy level of isolated state is δβ=0.85 which lies in the range of bulk band from 0.4 to 1.0. The black curve for the uncoupled case (V0=0). (c) The interference origin of Fano resonance and (d) the evolution dynamics of Fano resonance when the 1st waveguide is excited at the input.
By increasing the coupling strength V0 (blue and green curves), the antisymmetric linewidth becomes broader, and ultimately disappears. The single transmission spectrum being split into two sections at V0=0.4. This indicates the hybridization of the isolated state cutting the continuum into two independent pieces, where one effective band spans from 0.4 to 0.85, and the other from 0.85 to 1.0. Since the two pieces of the transmission spectrum decrease to zero at the energy (δβ=0.85) of the isolated state but still touch, no gap emerges even though physically there are already two effective bands, which cause perturbations in the energy spectrum in Fig. 4b. Figure 4d shows the field evolution with initial input from the 1st waveguide (not from the 0th). The reason for this excitation is to create an interference loop of field scattering between the isolated state and continuum where one path passes through the isolated state and is then reflected into the continuum and the other passes directly into the continuum, as demonstrated in Fig. 4c. The interference thus leads to Fano resonance, whose transmission spectra is shown in Fig. 4a for different cases of coupling strengths.
5. Adiabatic elimination and domain walls for zero mode and Tamm state hybridization
Two sandwich-like structures are introduced to study the extensions of BIGC. The first is a ‘isolated-continuous-isolated’ (ICI) structure, which supports the adiabatic elimination technique [41,42], and the second is the ‘continuum-isolated-continuum’ (CIC) structure, which results in a domain wall [43–45]. Adiabatic elimination (AE) is a powerful decomposition technique in quantum optics and atomic physics, that allows one to eliminate certain degrees of freedom from the dynamics under investigation [41,42]. By tuning the mixed coupling between isolated and continuum states independently, effective coupling between the two isolated states is achieved by integrating out the irrelevant continuous bulk state. To apply the generalized CMT twice, the integration procedure requires approximation to the second-order. By applying this integration procedure, the irrelevant degrees of freedom of the continuum are removed, and the effective interaction between the two relevant localized modes is determined. Thus, the effective hybridization of states such as zero modes and Tamm states is eventually obtained in the standard procedure of adiabatic elimination, which yields to perturbation methods such as the Wolf-Schrieffer transformation and Green's function [42,46].
Figure 5b and 5d show the field evolution of AE in the ICI structure. The parameters are 2N + 2 = 8, V1=V8=0.1, δβ=0.1(1.1) in Fig. 5b (d). The field propagates along with 1st waveguide then couples into the inner waveguides and thus recovers from the 8th waveguide. The inner waveguides play the intermediate transition role, effectively coupling the two end states which hold no strong propagating fields and thus may be eliminated. Also, a similar AE pattern is found for the Tamm states as shown in Fig. 5c and 5d, in which the isolated propagation constant is tuned to exceed that of the bulk spectrum. In addition, the AE for zero modes can be explained by the finite size effect of the topological phase (A-phase) of SSH modelling. The two degenerate zero modes exist since the protection of sublattice symmetry. When the total waveguide number of the array reduces to 2N = 8, the zero modes overlap in the bulk and lead to hybridization and energy splitting. The effective coupling strength of hybridization is proportional to the decay factor e-γN, where the coefficient ${\gamma }$ relates to its gap [39]. The hybridization of Tamm states is also exhibited in sandwich structures, which shows the advantage of generalized CMT. Note that the Fano-type setup (when Δβ=0.85) is an especially subtle case related to coupling strengths in this framework. There is then no AE when the adiabatic condition is violated resonantly if the coupling is strong (the energy of the isolated state approaches the top of the upper band) and the isolated state can spread into the coupled continuum much more easily. However, as the coupling is decreasing, AE still appears sensitively since the isolated state is more like the general BIC case in the weakly coupling regime [47–49].
Fig. 5. Configuration of “isolated-continuum-isolated” (ICI) sandwich-like structure for two zero modes (a) and two Tamm states (c). The adiabatic elimination is achieved for zero modes (b) (δβ=0.1) and Tamm states (d) (δβ=1.1) when total waveguide number is 2N + 2 = 8, and coupling strengths are V1= V8=0.1.
The second sandwich-like structure is the “continuous-isolated-continuous” (CIC) configuration, that is, the domain wall (DW) configuration [43,45]. Figures 5a and c show two kinds of tunable DWs in the experimental realization of waveguide arrays. At δβ=0 where the isolated state lies in the center of the gap, the protection is enhanced by both the valence and conduction continua, and as a result gives rise to a topological soliton, for example, as proposed in Polyacetylene (1979) [31]. In the context of topological insulators (TIs), the “continuous-isolated” configuration is viewed as a topological phase, and creates an interface between the normal insulators. The phase continuity of the band requires the gap to undergo an “open-closed-reopen” process topologically when the phase passes through the interface [44]. The ‘close’ process of gap generation leads to topological soliton excitation which is provisionally called “bulk-edge correspondence” [44,45], and the topologically protected interference called domain walls due to the topological phase transition. The propagation pattern of the protected zero mode DW is shown in Fig. 5b, with the same parameters set in Fig. 5b. Similarly, Figs. 6c and 6d show the evolution of the Tamm state DW. It is noted that the protection by gap generation actually has no contribution to the gap’s confinement of the Tamm state DW. The Tamm state domain wall is a defect-like confinement which is sensitive to the specific spatial configuration of the CIC setup.
Fig. 6. Configuration of a “continuum-isolated-continuum” (CIC) domain wall structure for two zero modes (a) and two Tamm states (c). The solitary dynamics is achieved for (b) Zero modes (δβ=0) and (d) Tamm states (δβ=1.1) when the total waveguide number is 40 and the coupling strengths are V0 = 0.3 for both cases.
The CMT formulation also includes the direct coupling between two isolated states and two continuums, which are considered. In optics, the guided modes couple and transfer energy (or intensity) when the electric field amplitudes of two waveguides overlap (evanescent field), which indicates the fundamental typical properties of mode-coupling. As a result, the superposition of the evanescent field between adjacent waveguides eigenstate distribution leads to the emergence of symmetric and antisymmetric modes, whose dynamics relate to the chemical reaction process (bonding or anti-bonding). On the other hand, the eigenmode coupling process is not restricted to optics, but occurs in in condensed physics such as in the zero mode, Majorana fermions, and solitons [39]. For the coupling between CC, the gap generation would emerge and result in the fields of semiconductor physics as well as topological insulators. Gap generation classifies the phase transition from insulators and metals, and as well the topological invariants emerging from gaps can further enable classification of the insulator phases into topologically trivial or non-trivial families. More fundamentally, gap generation as the typical demonstration in CMT offers the physical origin of mass terms in quantum field theories [50]. Here, we have extended the CMT to a more general coupling scheme that can be applied to isolated-isolated states, continuous-continuous states, and hybrid isolated-continuous states. Finding the corresponding on-site potential and/or the coupling strength, makes it possible to unify the inherent physical mechanism of all three states by this modified CMT, as well as the various applications behind them.
6. Summary
A generalized CMT is developed for the mixture of isolated states and the gapped continuum. By tuning the on-site potential and the coupling strength of the isolated state, a simple and consistent demonstration of achieving protected zero modes, Fano resonance, and Tamm surface states is presented. Furthermore, adiabatic elimination and domain wall dynamics are realized in two distinct sandwich-like structures, especially for the realization of novel distinguished hybridization and domain walls for Tamm states. We anticipate that this unified CMT formulation represents a compelling approach to re-examine and explore a number of phenomena and applications in science and technology.
7. Scattering matrix method
The lattice Hamiltonian of our configuration is given by:
Funding
Zhenjiang Key Laboratory of Advanced Sensing Materials and Devices (SS2018001); Talent Fund of Jiangsu University (17JDG014); German-Israeli Project Cooperation (DIP) (7123560301); BSF-NSF (2014719); Israeli Centers for Research Excellence; Crown Photonics Center.
Acknowledgments
The authors acknowledge, and would like to thank Prof. Yuanping Chen for useful discussions.
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