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Abstract
We propose a flexibly designed photonic system based on ultrathin corrugated metallic “H-bar” waveguide that supports spoof surface plasmon polariton (SPP) at microwave frequencies. Five designs were presented, in order to demonstrate flexibility according to varying height, period, core width, rotation, and shifting on the “H-bar” unit of the waveguide. The propagation constant between two hybrid designs of period and height structure was then shown in order to study the coupling effect. Next, we constructed a coupled waveguide array that followed the Su-Schrieffer-Heeger (SSH) model. This model was constructed by a hybrid design with the identical propagation constant of each waveguide, except it had dimerized spacing. The propagation feature of topological zero mode was then observed as theoretically expected in the dimerized array. Our proposed spoof SPP waveguide array has great flexibility to be used as a powerful experiment platform, particularly in photonic simulation of the quantum or topological phenomena described by Schrödinger equation in condensed matters.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon polariton (SPP) [1–3] at optical frequencies is a localized surface wave arising from collective oscillation of electrons when coupled with electromagnetic field at the metal/dielectric interface. Owning to its remarkable capability of confining the electric field within subwavelength regions [4–8], the excitation of SPP has developed various fruitful applications in the field of plasmonics [9,10]. However, when the frequency is reduced to the terahertz and microwave bands that do not support the existence of SPP, the metal will behave like a perfect electric conductor (PEC) rather than a plasma with a negative dielectric constant. In 2004, Pendry et al. proposed an elegant metamaterial approach to predict that surface waves confined on corrugated metal surfaces are able to mimic the familiar optical dispersion characteristics of SPP, subsequently given rise to the concept of spoof SPP [11–13]. Later, more complicated structures supporting spoof SPP are booming out, for instance, corrugated metallic wires [14], dominoes [15], grooves or wedges plasmons [16]. Following the development of spoof SPP, many functional devices such as ring resonators [17], directional couplers [18], and power splitters [19] were engineered and reinvented. However, in these devices, the spoof plasmon waveguide was found to surrender to the fundamental power limitation due to its three-dimensional (3D) structure against the high-density integration. In order to compensate for design flaws, in 2013, Shen and Cui proposed conformal surface plasmons (CSPs) [20–22], referring to spoof SPP propagating at ultrathin corrugated metallic waveguides. The waveguide deposited on FR4 substrate can be folded [21], bent [21], or even twisted [21] to manage the CSP at a broad microwave frequency bandwidth. Afterwards, much more functional components had also been constructed based on CSP, such as broadband and high-efficiency converters [23], frequency splitters [24,25], directional couplers [26,27] and wavelength demultiplexers [28,29]. Surprisingly, we noted that few investigations had been reported yet on the array composed of coupled spoof SPP waveguides.
In the optical regime, waveguide arrays [30–32] and layered metamaterials [33,34] have attracted much attention not only owning to their controllability of diffraction management [35,36] but also acting as an equivalent photonic platform for exploring quantum physics phenomena [36,37], yielding to equivalence between the coupled mode theory (CMT) in the waveguide array and the Schrödinger’s equation in solid state physics. Hence, the discrete diffraction of light in coupled waveguides is strongly dependent on the propagation constant (on-site potential) and coupling coefficient (hopping), and then the array of field propagation is considered as the crystal (or lattice) on which electron wavefunction evolves [36]. By carefully designing the underlying periodic potentials, the propagation pattern on the array is capable of mimicking the evolution of relativistic quantum particles, such as described Dirac fermions [38,39]. Therefore, optical analogues of such important quantum phenomena as Klein tunneling [40,41], Zitterbewegung [40], the relativistic gauge field [40], and photonic topological insulators [36,37,40] can be realized in the framework of the generalized coupled waveguide arrays, without requiring special synthetic media with subwavelength controllability. Such optical simulations offer plenty of benefits, such as direct measurement of wavefunction dynamics and long coherence time of light beams. However, it is still difficult to control the propagation of optical waveguide modes on curved surfaces flexibly and conformably [21,22,37] for the advanced functionalities realization. Compared with traditional dielectric waveguides [7–9], flexible ultrathin corrugated metallic waveguides offer much more freedom to alter structural parameters to control the propagation constants and coupling coefficients. In particular, based on microwave near-field detection, ultrathin metallic corrugated waveguides can allow for a diversity of new designs and geometries of corrugation to explore new physics phenomena and advanced devices.
To further extend the application of spoof SPP into topological photonics, a photonic platform consisted of ultrathin corrugated coupled metallic waveguides is proposed. Referring to the dispersion relations, the propagation constants of spoof plasmon modes are numerically simulated on five types of designed waveguides, and also in good agreement with those of experimental near-field measurements at a fixed frequency 17 GHz. We have found that the dispersion of the “H-bar” unit is more sensitively affected by altering the height and period parameters. Next, in the design of coupling two different designed waveguides, the nearly total energy from one waveguide appears to be completely coupled to the other one with the same propagation constant, on the contrary only portion energy was partly transferred in the presence of a mismatch propagation constant. As an implementation, finally, we had constructed the Su-Schrieffer-Heeger (SSH) [42] model via these two types of designs and eventually observed the topological zero-mode on the same “H-bar” with two staggered spacings acted as the single and double bonds. The model also had been widely studied in topological photonics [36], which had brought many underlying concepts related to topological insulators, such as zero-modes (topologically-protected edge states), soliton (domain walls) and Zak phase (topological invariants). In supporting spoof SPP, the topology language of our photonic platform has both practical and fundamental significance. Practically, the introduction of topological protection into density integration can enhance the robustness of signal integrity with defects and disorders, with significant features desirable in next-generation circuits. Fundamentally, the spoof SPP waveguide array provides a macroscopic quantum simulation platform for studying the topological phases described by the Schrödinger’s equation in the condensed matters.
2. Dispersion relation on “H-bar” unit of ultrathin corrugated metallic waveguide
Here, we proposed five types of flexible designs based on “H-bar” units, that are, varying height structure (h, to change the height of “H-bar”, short named HS), period structure (p, the period of “H-bar”, PS), width structure (w, the core width of the through bar, WS), shifting structure (up-down shift the bar relating the through bar, SS), and rotating structure (to rotate the angle relating to the through bar, RS), as schematically shown in Fig. 1(a). The existed spoof SPP on the ultrathin corrugated metallic waveguide hold similar dispersion characteristics as SPP in optical frequency, whose dispersion curves significantly are deviated from the light line, as shown in Figs. 1(b) and 1(c). The dispersion curves are calculated by the finite-difference time-domain (FDTD) method. Especially, in these five types of designs, the dependence of the height (h, HS) and period (p, PS) structure of the “H-bar” strongly affects the dispersion (or propagation constant) of the eigenmode. The cut-off frequency of spoof SPP turns to be suppressed into lower-frequency from 24.81 GHz to 64.22 GHz as the height of “H-bar” decreases from h = 5.0 mm to h = 1.5 mm with fixed other parameters at w = 1.0 mm and p = 2.0 mm. This means that increasing the height h can enlarge the propagation constant (β) and thus lead to more confined electric fields of spoof SPP.
Fig. 1 (a) The five flexible designs structure, height (h, HS), period (p, PS), width (w, WS), shifting (SS) and rotating (RS). The dashed line structures are the different parameters of each design. (b) The dispersion relation of height design (h = 5.0 mm to h = 1.5 mm), the light line is denoted by a black line. (c) The dispersion relation of period design (p = 0.5 mm to p = 4.0 mm). The blue dashes line is the operation frequency 17 GHz. (d) and (e) The propagation constant (β, mm−1) distribution in different parameters of height and period designs.
In comparison with the HS design, the light line will shift to a counterclockwise direction with decreasing the period of “H-bar” from p = 4.0 mm to p = 0.5 mm and the other parameters fixed at w = 1.0 mm and h = 4.0 mm, and consequently, the cut-off frequency will increase from 35.27 GHz to 22.69 GHz, as shown in Fig. 1(c). The dispersion curves for the other three designs are shown in the extended data Fig. 7. The “H-bar” unit dispersion has a strong relation with the structure parameters, which indicates to the flexibly tunability for the propagation of spoof SPP within a broadband frequency range. For example, fixing the excited frequency at 17 GHz, the propagation constant (β) is from 0.57 mm−1 to 0.39 mm−1 (Δβ≈0.18 mm−1) with altering the height from h = 5.0 mm to 1.5 mm for the HS design, as shown in Fig. 1(d). And the propagation constant (β) varies from 0.47 mm−1 to 0.52 mm−1 (Δβ≈0.05 mm−1) with altering the period from p = 4.0 mm to p = 0.5 mm for the PS design, as shown in Fig. 1(e).
3. Near-field measurement and deriving of propagation constant of spoof SPP
For validations of the relation between the dispersion and parameters, we fabricated the plasmon waveguide with different heights h1 = 3.5 mm and h2 = 4.5 mm for HS designs, as shown in Fig. 2(a), in which the ultrathin copper strips (t = 0.018 mm) are printed on an ultra-thin and flexible dielectric film (F4BK) with a thickness of 0.2 mm. In order to clearly demonstrate the dispersion for the HS design, the near-field distribution of the Ez component, which has a strong relation with the height parameters, is simulated and measured at the frequency of 17 GHz, as illustrated in Fig. 2(b). The total sample length is 40 cm and the near-field probe spacing is 1.4 mm. From comparable simulations and measurements, we extract and plot the amplitude of the electric field along the propagation direction. The gradual conversion was observed from the excited mode to the spoof SPP eigenmode. In the first half of the fabricated waveguide, the envelope of the input field propagates explicitly with mismatch propagation constants. With the increase of the propagation length, however, the envelope of the simulation and measurement well-matches automatically in the second half. Explicitly, the envelope of the excited mode for HS designs h1 has been taken 26.15 cm to convert into spoof SPP, in addition, taken 33 cm for h2. The near-field platform provides a powerful tool to analysis the propagation of spoof SPP. We can easily derive the propagation constant from the field distribution by Fourier transformation, which results into β = 0.46 ± 0.0087 mm−1 for h1 = 3.5 mm and β = 0.52 ± 0.0096 mm−1 for h2 = 4.5 mm, as shown in the Figs. 1(d) and 1(e). The extended data Fig. 8 gives the Fourier transform results of the mode propagation constants in the experiments and simulations in detail, with a deviation of less than 1%.
Fig. 2 Illustration of spoof surface plasmon polaritons in the ultrathin corrugated metallic waveguide. (a) Top view of the sample, two inserts with different height, h1 = 3.5 mm and h2 = 4.5 mm. (b) Ez near-field test of similar propagation evolution for h1 and h2. (c) and (d) From the top to the bottom respectively are the simulation, experimental results and comparison of the electric field along the propagation direction for sample h1 and h2.
4. Effect of mismatch value caused by hybrid designs in coupled mode theory
Now we consider the coupling dynamics and their mismatch effects between two different designed spoof SPP waveguides. Propagation effects can be described by coupled mode theory (CMT). The microwave field was input from entrance of a waveguide, and then the intensity of the two coupled waveguides evolves as follows:
where γ = ± (κ2 + Δ2)1/2, z is the propagation direction, A(z) and B(z) are the amplitude of the microwave filed at the A and B waveguides respectively, κ is the coupling coefficient, Δ is the mismatch value of the A and B waveguides (2Δ≡βb-βa). Interestingly, when Δ≠0, the energy will not be completely transferred to the other waveguide. The minimum intensity that remains lying on the A site depends on the F factor, which defines as F = (κ/γ)2 = 1/[1 + (Δ/κ)2]. To clearly investigate the relation between the coupling effect and the mismatch of propagation constants, HS and PS were chosen to form a coupled waveguide with spacing gap = 2.0 mm. The parameters of the PS are fixed with p = 0.5 mm and h = 4.0 mm, at the same time, several height values of HS are selected to adjust the propagation constant with p = 2.0 mm, as shown in Fig. 3(a). Referring to the dispersion curve, the propagation constant for PS is β = 0.52 ± 0.0104 mm−1, and the mismatches respectively are Δβ1 = −0.052 mm−1, Δβ2 = −0.033 mm−1, Δβ3 = 0.003 mm−1, Δβ4 = 0.032 mm−1and Δβ5 = 0.053 mm−1 with altering from h = 3.5 mm to h = 5.0 mm for HS, as shown in Fig. 3(b).Fig. 3 (a) The coupled waveguides with period structure (PS) and height structure (HS) design, the gap is 2.0 mm and the coupled length is 40 cm. The inserts give detailed experimental sample parameters with match and mismatch propagation constants. (b) The relation between the propagation constant and the height h (from h = 5.0 mm to h = 1.5 mm) for HS and the period p (from p = 0.5 mm to p = 4.0 mm) for PS. The dashed line frame represents the selected parameter in our simulations and experimental tests.
In the hybrid design, we can observe the variation of the energy coupling ratio with the propagation constant mismatch between the HS and PS. The parameters of the PS waveguide are p = 0.5 mm and h = 4.0 mm, and the HS waveguide are p = 2.0 mm and h = 5.0 mm, the energy excited at the PS site has been transferred 50% to the HS site with the mismatch Δβ = 0.053 mm−1. Most interestingly, almost the excited field 96% at the PS site h = 4.0 mm has transferred to the coupled HS site with a mismatch Δβ = 0.003 mm−1. These different coupling effects are shown in Figs. 4(c) and 4(d). Derived the intensity distribution on the PS and HS sites, the coupling length can be easily measured, which are respectively Lc = 3.89 cm for Δβ = 0.053 mm−1 and Lc = 4.88 cm for Δβ = 0.003 mm−1. We can derive the coupling coefficients κ1~0.0404 mm−1 and κ2~0.0328 mm−1 from the equation κLc = π/2. The derived coupling coefficients are shown in Fig. 4(a) with different mismatch propagation constants Δβ. Then, we figure the transfer ratio between the simulation and the F factor, which fit very well with each other, as shown in Fig. 4(b).
Fig. 4 (a) The coupling coefficient (κ) of different mismatch value. (b) the transferred efficiency of different mismatch value. (c) and (d) The field and intensity distribution of mismatch value Δβ = 0.053 mm−1 and Δβ = 0.003 mm−1 designed by the PS and HS, respectively.
5. Realization of topological zero-modes in coupled identical waveguides array and coupled hybrid waveguides array
To demonstrate the design flexibility of a spoof SPP waveguide, for example, we have aligned the coupled waveguides repeatedly into an array to mimic the well-known lattice model of a one-dimensional (1D) topological insulator, namely, the Su-Schrieffer-Heeger (SSH) model, that is protected with sublattice symmetry (or chiral symmetry). For our design of coupled spoof SPP waveguide, one way simply to study the SSH model and its zero-mode evolution is to extend the coupled mode theory as discussed in previous section 4, by coupling more waveguides into the array with alternating strong coupling and weak coupling coefficient between two neighboring sites. On the other hand, the equivalence between two adjacent waveguides in the array, thus, acts as a preservation of the sub-lattice symmetry, resulting in the emergence of chiral-symmetry-protected zero-mode on the boundary of array.
Figure 5 shows the calculation and realization of a protected zero-mode in a dimerized spoof SPP waveguide array. The fabricated array samples consisted of 10 identical waveguides (Fig. 5(b)), with alternating of waveguide spacing 4.2 mm and 1.0 mm, whose energy spectrum (propagation constant of array) were calculated with corresponding weak coupling and strong couplings, respectively. The bulk state in the spectrum are well-gapped and the two degenerate zero-modes lie in the band gap, as shown in Fig. 5(a). Obviously, the zero-modes have a large mismatch compare with other bulk modes. According to the coupled mode theory discussed in section 4, a larger mismatch yields to a lesser energy exchange, thus zero-modes are mostly confined at the array boundary and hardly spreads. In addition, it can be found from the eigen-mode distributions of the zero-modes that most of the energy are propagated locally on the first waveguide, as shown in Fig. 5(b). In the language of topological insulators, the zero-modes are protected by the induced gap of bulk states in energy band. For simulations and experiments with the same design structure, we demonstrated the dynamical evaluation of one of the zero-modes (i.e., the red dot in Fig. 5(a)) by injecting a microwave field from the first waveguide at the boundary. The achieved zero-mode behaves strong robustness against disorder and defect. Both the results of the simulation and near-field measurements are nearly perfect matched, excepting the slow decay of intensity due to the fabrication error in the sample after propagating the length L = 40 cm. Also, due to the strong confinements of spoof SPP on the waveguide, we noted that excitation of the zero-mode on the array coupled with a few waveguide numbers N = 10 enables to escape from the suffering caused by the finite size effect, which implies highly-efficiency of sample fabrication to photonic simulations.
Fig. 5 (a) The eigen-value of the 10 waveguides array. (b) The eigen-mode distributions of these two zero-modes. Most of the energy localized at the boundary site. (c) The sample and the result both in CST simulation and near-field measurement where the first waveguide was excited by input the microwave field with frequency 17 GHz. The scale bar is 10 cm.
Alternatively, we design a dimerized waveguide array consisted of hybrid coupled waveguides as suggested in section 4. The sub-lattices A and B of SSH modelling follow two different structural design as suggested in Fig. 3, where the waveguides on sub-lattice A are designed with PS-shaped “H-bar” units and those waveguide on sub-lattice B with HS-shaped “H-bar” units. We tune the mismatching of the propagation constants on the hybrid waveguide into zero, that is, equivalent to the construction of the sub-lattice symmetry in the hybrid waveguide array. Consequently, the zero-mode propagating along the boundary of hybrid dimerized waveguide array can also be realized as shown in Fig. 6, in which the stimulation array setup is mapped into the background.
Fig. 6 The evolution of topological zero-mode field distribution in hybrid HS and PS-structured SSH configuration. (a) The fabricated sample following the SSH model. (b) The experimental result. The scale bar is 5 cm.
More flexible designs indicate more possibilities of quantum simulation and less limitations on fabrication and measurement. The two types of array realization of the SSH model in Figs. 5 and 6 show one advantages of the tunable design of spoof SPP waveguide. But more importantly, the more interesting quantum simulation of 1D lattice crystal modelling can benefit from the flexibility and efficiency of the “H-bar”-typed corrugated metallic waveguide. By controlling the structure shape of the “H-bar” unit on the two coupled waveguides, one is capable of manipulating the on-site potential (βi) and the next-neighboring coupling amplitude (κij). On the other hand, we introduce the slowly-varying modulation of the H-bar unit and make the propagation constant varying along the propagation direction, which remarkably results into the time-dependent quantum simulation on the waveguide array.
6. Conclusion
In conclusion, the conventional spoof SPP waveguide with corrugated metallic structures were studied to achieve a photonic simulator and device implementation with high design flexibility. Two typical waveguide designs based on the minimal corrugated “H-bar” unit among five designs have been systematically investigated through the energy dispersion, the coupled mode theory and topological photonic simulation. Both the simulation and the near-field measurement confirmed consistently that the propagation constant of spoof waveguide mode is strongly affected by the tunable “H-bar” structural parameters, especially by the aspects of its height and period structure. The effects of the mismatch of hybrid designs for the HS and PS waveguides are also discussed in details. Later, as a non-trivial example, we designed two types of spoof plasmon waveguide array and observed the topologically-protected zero-mode based on the SSH model. Thus, the flexibility of these spoof waveguides may be expatiated to verify that this type of waveguide array and its near-field measurements can offer us a novel testing platform for the study of photonic devices and even more intricate quantum simulations.
7. Extended data
The dispersion relation curves for the width design, shifting design, and rotating design are shown in Figs. 7(a)-(c). It is obvious that width design has a huge deviation on different parameters (w = 0.5 mm to w = 4.0 mm), compared with the shifting design (s = 0.25 mm to s = 1.5 mm) and rotating design (θ = 0○ to θ = 12.5○) get fewer changes.
Fig. 7 The dispersion relation for another three designs (a) width design (w = 0.5 mm to w = 4.0 mm, with step = 0.5 mm), (b) shifting design (s = 0.25 mm to s = 1.5 mm, with step = 0.25 mm), (c) rotating design (θ = 0○ to θ = 12.5○, with step = 2.5○), and the light line is denoted by a black line.
The propagation constants in experiment (red line) and simulation (blue line) are gotten by the Fourier transform with the field distribution data, as shown in Fig. 8. Experimental results have a good agreement with the simulation results which have a deviation of less than 1%.
Fig. 8 Data from experiments and simulations were extracted and Fourier transformed to determine propagation constants. (a) Structural parameters with h = 3.5 mm and p = 2.0 mm, experimental and simulation propagation constant difference Δβ = 0.008 mm−1. (b) Structural parameters with h = 4.5 mm and p = 2.0 mm, the difference Δβ = 0.007 mm−1. (c) Structural parameters with h = 4.0 mm and p = 0.5 mm, the difference Δβ = 0.013 mm−1. (d) Structural parameters with h = 4.0 mm and p = 3.0 mm, the difference Δβ = 0.006 mm−1.
Funding
National Natural Science Foundation of China (Grant Nos. 11874266, 11604208, 61705131 and 61372048); Shanghai Science and Technology Committee (Grant Nos. 16ZR1445600 and 16JC1403100); ChenGuang Program (17CG49); DIP (German-Israeli Project Cooperation); U.S.-Israel Binational Science Foundation; PBC Programmed of the Israel Council of Higher Education.
Acknowledgments
The authors would like to thank Prof. Yaron Silberberg for discussion help.
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