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Abstract
The emergence of Dirac points (DPs) characterizes the topological phase transition and the gapless interface states in composite metal-dielectric metamaterials. In this work, we study a kind of compound plasmonic-dielectric periodic structure (PDPS) which sustains both plasmonic modes and multiple photonic modes. The structure has primitive cell consisting of four layers made from triple constituent components. Due to the generalized Su-Schrieffer-Heeger, DPs can emerge at the Brillouin zone center. More specifically, in weak plasmonic-photonic mode interaction regime, multiple DPs would emerge at the Brillouin zone center and edge due to the band folding, from the perspective of general effective medium. From the rigorous field analysis, the origin of these DPs is clearly demonstrated. These interleaved DPs behave as the intermediate transitions of the surface impedance for the PDPS and raise fully spanned topological interface states originated from 0 to 2nd-order photonic bands in the PDPS. The cases of combining our PDPS with either a plasmonic or dielectric homogenous medium are presented.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Dispersion singularities and the associated localized states in physical systems have attracted considerable interest and are crucial to understand a multitude of physical phenomena occurring on artificially structured surface [1–4]. Dirac points (DPs) [5–8] and Weyl points (WPs) [9–12], where conical intersections in dispersive energy surfaces are present, are two typical classes of degeneracy. Accompanied by the topological phase transition, DPs or WPs behave as intrinsic signatures to characterize the properties of the topological phases [4]. Due to their peculiar properties and associated intriguing phenomena, these singularities have been widely studied in condensed matter physics, photonics, and acoustics [13–22]. In terms of DP-associated light manipulation, optical periodic lattice has been the major platform in both fundamental and practical aspects [23–28]. The basic binary dielectric or metal-dielectric waveguide arrays have shown great capability to control light dynamics such as topological interface confinement [29], negative refraction [27], subwavelength solitons [30], etc.
Two typical schemes to determine the position of DPs have been illustrated: the real part of the average permittivity of the superlattices vanishes, where the Zak phase changes [31]; the opposite sign of the inter- and intra-coupling coefficients in Su-Schrieffer-Heeger (SSH) model can also predict the presence of degeneracy at Brillouin zone (BZ) center [32–34]. When the SSH model is degenerated to the situation with identical staggered hopping coefficients, a degeneracy point occurs at the edge of the BZ [22,29]. It can be understood as the folding effect of the irreducible band structure, which has been further employed in synthetic space to explore the topological phase transition at the BZ edge [22] and the high dimensional topological physics like WPs [29]. The efficiency of the spatial averaged permittivity requires the multilayers in the scale much smaller than the wavelength and the weak interaction between constituents in the waveguides. Whereas the SSH model [35] is usually applied to quite strong interactions in the two-modes coupling model. Both of them cannot interpret the presence of DPs at BZ edge in plasmonic waveguide arrays. Besides, the bands evolution arising from the weak interactions among multiple modes has not been widely investigated. In this paper, we use the effective medium theory (EMT) [36] to demonstrate the interleaved degeneracies at BZ center and edge inspired by the fold of band due to the enlarged superlattice. In the multilayered plasmonic-dielectric waveguides, the higher-order photonic modes are predominant in the relatively short wavelength range. From the rigorous field analysis, the origin of these DPs at BZ center and edge is illustrated. All of these can be identified from the bands structure and fields distributions of discrete model. Based on the intermediate DPs, the gapless interface states between the waveguide array and the connected homogeneous medium (free space and metal) are spanned in the full gap regions.
2. Analysis based on effective medium theory
Binary metal-dielectric metamaterials have been investigated in past decades, where the surface plasmon polaritons (SPPs) are localized along the interface between metal and dielectric. SPPs are transverse magnetic (TM) polarized electromagnetic waves. The SPP modes localized along both sides of metal are coupled via metal whose coupling coefficient is negative. As such, the negative refraction [27] and Zitterbewegung effect [28] have been observed and widely investigated. Apart from the plasmonic mode, the high-index photonic modes can also be involved to tune the band structure and the resultant beam propagation. Simultaneously, the width of the high-index dielectric layer can be tuned to support multiple photonic modes. As such, the multilayered superlattice composed of metal, high- and low-index dielectric is a feasible platform to investigate the mutual interactions between the high-index photonic and plasmonic modes.
Figure 1(a) schematically shows the compound plasmonic-dielectric periodic structure (PDPS) studied in this work: two low-index dielectric slabs (layer A and C, both silica with refractive index ${n_1} = {n_3} = 1.44$) are connected by high-index dielectric (layer D, silicon with ${n_4} = 3.48$) and plasmonic (layer B, silver with relative permittivity ${\varepsilon _2}$) slabs. The permittivity of silver is described by the well-known Drude model ${\varepsilon _2} = {\varepsilon _\infty } - {{\omega _p^2} / {\omega ({\omega + i\gamma } )}}$ with permittivity ${\varepsilon _\infty } = 3.7$, plasmon frequency ${\omega _p} = 1.38 \times {10^{16}}\textrm{ rad} \cdot {\textrm{s}^{ - 1}}$, and damping factor $\gamma = 2.723 \times {10^{13}}\textrm{ rad} \cdot {\textrm{s}^{ - 1}}$. In the long wavelength limit, the system can be described by an effective medium as illustrated in Fig. 1(b).
Fig. 1. (a) Schematic of the multilayered structure: each unit cell consists of silica-silver-silica-silicon layers. (b) The homogeneous medium with permittivity $\varepsilon _x^{\textrm{eff}}$ and $\varepsilon _y^{\textrm{eff}}$ based on EMT. (c) Band structure of (a) based on TMM. (d) Band structure of (b) for effective homogeneous medium. I, II and III are the three actual DPs.
Based on the transfer matrix method (TMM) and the Bloch theorem [37,38], Fig. 1(c) shows the obtained TM band structure of the PDPS. Here, ${k_x}$ is the Bloch wave number along x-axis (normal to the layer plane), ${k_y}$ the propagation constant along y-axis (parallel to the layer plane), and ${k_0} = 2\pi /\lambda $ is the wave number in free space with wavelength $\lambda $. Interestingly, it can be observed that DPs alternatively appear at the BZ center and edge. In the long wavelength limit, the PDPS can be regarded as a homogeneous effective medium with permittivity components $\varepsilon _x^{\textrm{eff}}$ and $\varepsilon _y^{\textrm{eff}}$
The homogenization of the PDPS by the EMT is approximate, and there still exist weak interactions among photonic and SPP modes, giving rise to the lift of the degenerate lines at the boundaries. As such, the band folding degenerate line becomes isolated DP, e.g., “I” and “II” in Fig. 1(c). In order to further demonstrate the formation of these degeneracy, we decompose the periodic array [see Fig. 1(a)] into the discrete systems including a simple dielectric-metal-dielectric (DMD) slab [labeled as “SiO2/Ag/SiO2” in Fig. 2(a)], a high refractive index waveguide [labeled as “SiO2/Si/SiO2” in Fig. 2(a)]. Their combination is shown in Fig. 2(b). By rigorous mode analysis [39], the silver slab with ${d_2} = 10$ nm basically holds two gap SPP modes with even and odd parities, whose dispersion is respectively plotted as violet and olive solid curves in Fig. 2(a). For index-guided photonic modes, the silicon waveguide with ${d_4} = 200$ nm supports multiple bands with different orders, respectively plotted as violet (even) and olive (odd) dashed curves in Fig. 2(a). These multiple photonic modes are responsible for the interleaved DPs. The main difference between these two types of modes is that the propagation constant of even SPP mode is less than that of odd SPP, while it is opposite for photonic modes [27]. This indicates the origin of anomalous coupling through metal. Note that two straight red solid lines in Fig. 2(a) are the light lines for homogenous silica and silicon, respectively.
Fig. 2. (a) Dispersion for a silver slab embedded in silica (SiO2/Ag/SiO2, solid lines) and for silicon embedded in silica (SiO2/Si/SiO2, dashed lines). (b) Dispersion for double silver gap waveguides connected by silicon layer. (c) and (d) The real part of ${H_z}$ distribution of the eigenstates at DPs “I” and “II”, respectively. Geometrical parameters are as follows: thickness of silver ${d_2} = 10$ nm, silicon ${d_4} = 200$ nm, silica ${d_1} = {d_3} = {d_0} = 50$ nm and ${d_1} + {d_3} + {d_4} = 300$ nm.
Figure 2(b) shows the corresponding dispersion of a discrete system with two coupled silver waveguides via a silicon layer. This can be considered as two even (odd) hybrid SPP modes coupled with the photonic modes. Importantly, the lower odd SPP band and the first-order photonic band intersect in Fig. 2(a), which results in the three lowest bands (two SPP-dominant bands with even and odd parity and an even photonic-dominant band) in Fig. 2(b). While for the even SPP band, it mainly intersects with the fourth-order photonic band which results in the other group of two hybrid SPP bands with even and odd parity and an odd photonic-dominant band in Fig. 2(b) around $0.45{\omega _p}$. Simultaneously, the other photonic bands are approximately overlapped in Figs. 2(a) and 2(b), as such we can conclude that the thin silver layer has little effect on these higher-order photonic modes.
Figure 2(c) shows the profiles of the degenerate eigenstates at DP “I” $(\lambda ,{k_x},{k_y})$ = $(634.9nm,0,1.51{k_0})$. The eigenstate with odd (even) parity originates from the second-to-lowest (third-to-lowest) dashed band in Fig. 2(b). Significantly, these two modes closely resemble the first-order and second-order photonic modes in Fig. 2(a). This can be distinguished from the eigenstates distribution of the separated structure in Fig. 5 of Appendix A. Similarly, Fig. 2(d) shows the profile of the degenerate eigenstates at DP “II” $(\lambda ,{k_x},{k_y}) = (1031.14\textrm{nm},\pi /\Lambda ,1.67{k_0})$. At this point, the eigenstate with odd (even) parity originates from the lowest (second-to-lowest) dashed band in Fig. 2(b). Here, these two modes are closely analogous to the zero- and first-order photonic modes in Fig. 2(a) as shown in Fig. 6 of Appendix A. These neighboring photonic modes at the degeneracy can be considered as the result of band folding due to EMT [see Fig. 1(d)]. In these two cases, the high-index photonic modes are predominant, and the silver imposes negligible effect on the band hybridization.
Fig. 3. (a) The real part of ${H_z}$ distribution of the eigenstates at DPs “III”. (b) Dispersion at the BZ center (${k_x} = 0$, dashed curves) and edge (${k_x} = {\pi / \Lambda }$, solid curves) in Fig. 1(c). The yellow (magenta) region in (b) represents the negative (positive) of the surface impedance.
Fig. 4. (a) Schematic of the unit cell, semi-infinite PDPS covered by air/silver and finite periods embedded in air/silver. (b) Projection of bulk bands (gray region) and the trajectories of the interface states for air (red solid curves) and silver (blue dashed curves). (c) Eigenmodes for the 10 periods embedded in air at wavelength $\lambda = 800$ nm, and (d) the corresponding reflectance as a function of wavelength and ${k_y}$ excited by a left-placed prism with permittivity ${\varepsilon _4}$. Loss of silver is considered in (d). The parameter $d = 10$ nm is selected to excite the evanescent waves which propagate along the interface in the air region.
Fig. 5. The field distribution ${H_z}$ for (a) the silver slab, (b) silicon slab corresponding to Fig. 2(a). 2(c) and 2(d) ${H_z}$ distribution of the discrete system in Fig. 2(b). The curves in 5(a) and 5(b) are obtained by TMM, and the patterns are obtained by numerical simulation by FEM. The working wavelength is at that of the DP “I”, i.e., $\lambda = 634.9$ nm.
Fig. 6. Similar to Fig. 5. The working wavelength is at that of the DP “II”, i.e., $\lambda = 1031.14$ nm.
There exists an additional DP “III” at $(\lambda ,{k_x},{k_y}) = (607.83\,{\textrm{nm}},0,3.17{k_0})$ in Fig. 1(d), which is not located at the EMT bands. This is due to the coexistence of strong inter- and intra-coupling coefficients between the plasmonic modes [33,34]. In this case, the existence of this degeneracy corresponds to the opposite sign of inter- and intra-coupling coefficients, which can be described by the SSH model (see Appendix B). From the eigenstate profile of DP “III” in Fig. 3(a), it has been traced that the odd and even eigenstates come from the lowest two bands (solid curves) in Fig. 2(b). These two hybrid bands arise from the coupling of the odd SPP mode of silver slab and the zero-order high-index photonic mode identified in Figs. 2(a), 2(b) and Fig. 7 of Appendix A.
Fig. 7. Similar to Fig. 5. The working wavelength is at that of the DP “III”, i.e., $\lambda = 607.83$ nm.
3. Multiple sets of interface states
To this end, the mechanism of the interleaved DPs has been clearly elucidated with the EMT and anomalous SSH model. Certainly, it is anticipated that the phenomena would be richer when the frequency goes higher where more photonic bands would be involved in Fig. 2(b).
Figure 3(b) shows the dispersion relations at ${k_x} = 0$ (the dashed curves) and ${k_x} = {\pi / \Lambda }$ (the solid curves). Note that the red curves indicate the eigenstates with even parity, and the black curves denote the eigenstates with odd parity. Apparently, the parity of the eigenstates exchanges by crossing these DPs. And the colored regions representing bulk bandgaps are connected by the interleaved DPs (I, II, and III). Due to the existence of topological phase transition, it is expected that topological interface states exist. Note that the topological properties of the common gaps and the determination of the interface states for semi-infinite multilayered structures can be characterized by the surface impedance, which is defined as the ratio ${E_y}/{H_z}$ at the interface [24,29,38]. In our PDPS, the interface is truncated at the center of the silicon layer. An interface state is guaranteed when the following condition is satisfied:
where ${Z_l}$ (${Z_r}$) is the surface impedance of the medium on the left-hand (right-hand) side of an interface. Here, silver and air serve as two typical paradigms as shown in Fig. 4(a). The surface impedance of the left homogenous medium is ${Z_l} = {{{k_x}} / {\omega {\varepsilon _l}}}$, with ${k_x} = \sqrt {k_0^2{\varepsilon _l} - k_y^2} $ and ${\varepsilon _l}$ its relative permittivity. Notice that ${Z_r}\textrm{ = }{{{E_y}} / {{H_z}}}$ is a purely imaginary number in the gap which can be directly derived by the TMM and the signs of the imaginary ${Z_r}$ in the gap are marked in Fig. 3(b): the yellow (magenta) region represents negative (positive). Obviously, the DPs give rise to a discontinuous jump of the sign of ${\mathop{\rm Im}\nolimits} ({Z_r})$. Based on Eq. (3) and $k_0^2{\varepsilon _l} - k_y^2 < 0$, the sign of ${\mathop{\rm Im}\nolimits} ({{Z_l}} )$ is opposite for plasmonic and dielectric medium. The dispersion of the interface state predicted by Eq. (3) is shown by red solid and blue dashed curves in Fig. 4(b) for the truncated PDPS connected with free space (air) and silver, respectively. Strikingly, DPs serve as an intermediate transition point and the dispersion of interface states starts or terminates at this degenerate point.To confirm the existence and the propagation of the interface states more visually and intuitively, we select a case at wavelength $\lambda = 800$ nm for demonstration. Figure 4(c) shows the corresponding distribution of the eigenstates. Black squares represent the modes in bulk band, and two red circles appearing at respective gap represent the coupling results of the interface states localized along the both truncations of the finite lattices. As the refractive index of the ambient medium is unity, the dispersion is above the light line ${k_y} = {k_0}$. Different interface states hold different penetration depth which is determined by the purely imaginary value of the Bloch wave number ${k_x}$ in the gap [40]. And the attenuation length in the connected homogeneous medium is governed by ${k_x} = \sqrt {k_0^2{\varepsilon _l} - k_y^2} $.
The band structure and the interface bands can be further corroborated by the reflectance spectrum (contour) of the finite system excited via a silicon prism [see the lowest panel in Fig. 4(a)]. To excite the propagation along $y$-axis, the separation d between the prism and the lattices is fixed at 10 nm. Here, the reflection dips represent the excited bands [41] and the loci of reflectance dip show excellent agreement with the dispersion curve of the 10 period structures. The results of the overlap of the interface states evanescently decaying into the bulk of PDPS give rise to two interface states (anti-symmetric and symmetric). This can be clearly observed in the Fig. 4(d). As these interface states are far away from the boundary of the bulk bands, the interface states are highly localized at the truncated interface. Due to the negligible imaginary part of propagation constants, the predicted interface states are preserved and can propagate along the interface with unnoticeable attenuation. Similarly, the results for using different covering medium to finite superlattices structure can also be obtained (see Fig. 9 of Appendix C).
Fig. 8. (a) Schematic of the SSH model: single confined mode of each waveguide and its evanescent coupling to the adjacent waveguides. (b) Two iso-frequency surface for dimensionless propagation constant $K({k_x},\eta )$.
Fig. 9. Similar to Fig. 4. The covered medium is substituted by silver. (c) and (d) correspond to the FEM results of the interface states at $\lambda = 1250$ nm in (b).
4. Conclusion
In conclusion, the origin and topological properties of interleaved DPs in plasmonic-dielectric multilayered structures at the BZ center and edge have been reported and investigated systematically in the context of the mutual interactions between photonic and plasmonic modes. Considering the role of high order photonic modes, the interleaved DPs are present at the BZ center and edge. Particularly, the presence of DP at the BZ center is due to the match of the anomalous and normal coupling coefficients and can be understood based on a general SSH model. As such, the full band surfaces are connected in the considered frequency range. Meanwhile, the surface impedances in the band-gap zones are examined and their signs are found exchanged across the interleaved DPs. The topological phase transition at these kink points and the interface states between the superlattices and covered homogeneous medium (such as silver and air) can be observed. These results may find potential applications in optical sensing and enhancement of nonlinear effects, as well as exploring new properties of topological state in very general 1D layered structure.
Appendix A: Fields distribution at the Dirac points
In order to elucidate the origin of the DPs, the considered plasmonic-dielectric superlattices can be separated into several discrete structures. They consist of a silver slab, a silicon slab, and their combination, as illustrated in Fig. 2.
Figure 5 shows the field distribution of the considered discrete structures at the wavelength $\lambda = 634.9$ nm of DP I. The field distributions are calculated by COMSOL Multiphysics using finite element method (FEM). These coincide with the results of TMM. In Fig. 5(a), two gap SPP modes are supported. One mode with ${k_y} = 1.444{k_0}$ is even, and the other with ${k_y} = 2.787{k_0}$ is odd. In Fig. 5(b), two photonic modes are supported. Zero-order mode with ${k_y} = 3.143{k_0}$ is even, first-order mode with ${k_y} = 2.013{k_0}$ is odd. The final results of these modes interactions in the combined structure are shown in Figs. 5(c) and 5(d). It is obvious that DP I is ascribed to the two modes in Fig. 5(c) based on the eigenmodes’ profiles in Fig. 2(c).
Similar to Fig. 5, Fig. 6 shows the field distribution of the discrete structures at the wavelength $\lambda = 1031.14$ nm of DP II. The gap SPP modes in silver slab and the photonic modes in silicon slab are also shown in Figs. 6(a) and 6(b), respectively. The symmetries of these modes are similar to those in Fig. 5. While their interactions give rise to the hybrid SPP modes in Figs. 6(c) and 6(d). Here, DP II locating at the BZ edge, originates from the accidental crossing between the mode with ${k_y} = 1.461{k_0}$ (similar to the first-order photonic mode) and the mode with ${k_y} = 1.827{k_0}$ (similar to the zero-order photonic mode which is influenced by the SPP mode).
Similar to Fig. 5, Fig. 7 shows the field distribution of the discrete structures at the wavelength $\lambda = 607.83$ nm of DP III. The gap SPP modes in silver slab and the photonic modes in silicon slab are also shown in Figs. 7(a) and 7(b), respectively. Differently, the photonic modes have been increased to three. Another second-order mode is added and it is even. Hybrid SPP modes of the combined discrete structure are shown in Figs. 7(c) and 7(d). Here, DP III originates from the mode with ${k_y} = 3.072{k_0}$ and the mode with ${k_y} = 3.218{k_0}$. These two involved states stem from the odd SPP mode in silver slab coupled with the zero-order photonic mode. Certainly, they can also be considered as the coupling between the SPP modes via the silver and silicon layer. This can be modeled as a general SSH model, which is discussed in Appendix B in detail.
Appendix B: Generalized discrete SSH model
As a matter of fact, a more general SSH model can be extended to elucidate the emergence of DP “III”, where the staggered hopping amplitudes have opposite sign. The frequency-dispersive plasmonic medium (silver) whose permittivity can be described by the Drude model ensures the opposite sign of the coupling coefficients by contrast to that of the dielectric layers. When the sum of the inter-coupling and intra-coupling coefficients vanishes, a DP arises at the BZ center.
As sketched in Fig. 8(a), assuming that each unit cell hosts two sites, one on sublattice ${H_{2n}}$ and the other on sublattice ${H_{2n + 1}}$. For simplicity, we suppose that each waveguide in the sublattices supports only one pure guided mode, as schematically illustrated by the blue and red curves in Fig. 8(a). Due to the spatial overlap of the mode profiles, the interaction between the nearest-neighbor sublattices can be described by the coupling via inter-coupling coefficient ${C_ - }$ and intra-coupling coefficient ${C_ + }$. Based on the coupled mode theory, a compact Hamiltonian for this SSH model can be written as [33]
The eigenvalue degeneracy is possible if and only if $|\eta |= 1$. Here, two specific cases emerge: (1) DPs arises at the BZ center when ${C_ - } + {C_ + } = 0$ (i.e., of the same amplitude but opposite sign); (2) DPs emerge at the BZ edge when ${C_ - } = {C_ + }$. One possible case for the presence of DP at the BZ edge can be simply realized by doubling the unit cell length of one single site [22]. The BZ folding effect can give rise to such band degeneracy [29].
Appendix C: Reflectance and interface for silver
In addition to the free space, another type of medium (silver) can also be applied to design the interface states localized at the interface. This has been theoretically analyzed and predicted in the main text. Here, the numerical verification for covered medium silver is addressed in Fig. 9. As the permittivity of silver is negative, the interface states are located at the opposite gap regions as shown in Fig. 9(b). The loci of the reflectance dip are in good agreement with the prediction in Fig. 4(b). To verify the existence of interface states, the fields of the modes in the gap at wavelength $\lambda = 1250$ nm are shown in Figs. 9(c) and 9(d). It is clear that these two modes are localized at the interface and propagating along the interface.
Funding
Shenzhen Municipal Science and Technology Plan (JCYJ20170811154119292, JCYJ20180306172003963).
Disclosures
The authors declare no conflicts of interest
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