Engineering topological interface states in metal-wire waveguides for broadband terahertz signal processing

Abstract Innovative terahertz waveguides are in high demand to serve as a versatile platform for transporting and manipulating terahertz signals for the full deployment of future six-generation (6G) communication systems. Metal-wire waveguides have emerged as promising candidates, offering the crucial advantage of sustaining low-loss and low-dispersion propagation of broadband terahertz pulses. Recent advances have opened up new avenues for implementing signal-processing functionalities within metal-wire waveguides by directly engraving grooves along the wire surfaces. However, the challenge remains to design novel groove structures to unlock unprecedented signal-processing functionalities. In this study, we report a plasmonic signal processor by engineering topological interface states within a terahertz two-wire waveguide. We construct the interface by connecting two multiscale groove structures with distinct topological invariants, i.e., featuring a π-shift difference in the Zak phases. The existence of this topological interface within the waveguide is experimentally validated by investigating the transmission spectrum, revealing a prominent transmission peak in the center of the topological bandgap. Remarkably, we show that this resonance is highly robust against structural disorders, and its quality factor can be flexibly controlled. This unique feature not only facilitates essential functions such as band filtering and isolating but also promises to serve as a linear differential equation solver. Our approach paves the way for the development of new-generation all-optical analog signal processors tailored for future terahertz networks, featuring remarkable structural simplicity, ultrafast processing speeds, as well as highly reliable performance.

In the THz frequency regime, metals are generally considered perfect conductors, as the negligible penetration of the electromagnetic field leads to highly delocalized surface plasmon polaritons (SPPs) akin to grazing-incidence light fields.The propagation characteristics of THz SPPs along the metal-air interface can be described by their dispersion relation, i.e., a map between the angular frequency ω and the propagation constant β(ω) of the fundamental mode.When periodic structures are engraved on the metal, surface waves resembling the behavior of SPPs, so-called spoof SPPs [1], can be still sustained, and their dispersion relation can be tailored by varying the geometry of the periodic structures.For a metal-insulatormetal plasmonic waveguide structure [2] with periodic grooves engraved on both metallic surfaces (counter-facing towards each other), so-called a spoof-insulator-spoof (SIS) waveguide structure, the analytical expression of its dispersion relation is given by [3]: where 0 k c   denotes the propagation constant of plane waves in free space, c being the speed of light.The width w, depth d, and period p describe the geometry of the grooves, where g is the size of the air gap that separates the two flat metallic surfaces.When the duty cycle w/p is equal to 0 (or the depth d is 0), this condition indicates that there are no grooves etched on the metal, and thus, the dispersion relation has an exact linear dependence on the frequency (i.e., no dispersion).When the geometry of the grooves is in the subwavelength scale, the effective medium theory can be applied, thus the effective refractive index neff is expressed as [3]: Based on Eqs.S1 and S2, it is clear that the propagation characteristics of the spoof SPPs can be easily tuned by engineering the depth of the grooves d, the duty cycle w/p, as well as the gap size g between the two metal surfaces.In principle, the THz guidance in metal-wire waveguides also relies on the propagation of SPPs along the metal-air interface [4].In our study, we aim to tailor the propagation characteristics of the SPPs confined in-between two metal wires by corrugating the wire surfaces with periodic grooves, as shown in Fig. S1(a).Although such a structure can be also considered as a SIS waveguide, its dispersion relation cannot be directly expressed using Eq.S1.This is because Eq.S1 can only be used to describe the propagation characteristics of the spoof SPPs confined inbetween two plane conductors.Due to the non-planar surface of the metal wires, the depth of the grooves d and the air gap between the two wires g are not constant along the cutting direction (ydirection), in turn making it extremely difficult to derive an analytical expression of the dispersion relation.In order to achieve the most accurate results, finite-difference-time-domain (FDTD) simulations were performed.The dispersion relation shown in Fig. S2 describes the propagation characteristics of the spoof SPPs for a depth d = 40 µm, a width w = 35 µm, and a period p = 80 µm.The dispersion relation shows a cut-off frequency fc at ~1.2 THz, demonstrating that the spoof SPPs at such a frequency are stopped (propagation velocity equals 0) and the THz frequencies above fc cannot be guided within the waveguide.By adjusting the geometry of the grooves to the wavelength scale, the cut-off frequency fc can be accordingly shifted to the lower frequency range, in turn narrowing the operating bandwidth.Therefore, we could expect that, in principle, it is possible to modify the propagation velocity and the cut-off frequency of the propagating spoof SPPs, by tailoring the geometry of the grooves, where we change the depth d (or duty cycle w/p) while keeping the periodicity; however, with such a procedure, a Bragg resonance cannot be achieved within the operating THz bandwidth.
In order to overcome this issue and introduce a Bragg resonance without influencing the bandwidth, we introduce the concept of multiscale structures into the THz regime.A multiscale structure is achieved by superimposing a wavelength-scale periodic modulation Λ onto the subwavelengthscale periodic grooves.Such a multiscale structure can be interpreted as the combination of two sub-cells with different propagation constants βi, where the subscripts i=1,2 correspond to each individual sub-cell.Any geometrical difference (depending on d and w/p) between the sub-cells results in different neff, in turn leading to a periodic modulation at the wavelength scale, as shown in Fig. S1(b).The dispersion relation βm of the spoof SPPs propagating along such a multiscale structure can be obtained from the Bloch theorem [5]: In our design shown in Fig. S1(b), we have p1=80 µm, p2=60 µm, and T=2p1+2p2=280 µm.In particular, for the sub-cell 2, the duty cycle w/p equals 0 (or d=0).The solution of Eq.S3 exhibits a bandgap at the irreducible Brillouin zone boundary [3], where βm=π/T.In this condition, the upper limit of the bandgap edge occurs at fbandgap= βmc/2π=c/(2T), which also satisfies the Bragg condition.The simulated dispersion relation in Fig. S2 confirms the existence of a Bragg bandgap at 0.53 THz.By simply altering the period of the wavelength-scale modulation Λ, the location of such a Bragg bandgap can be tuned within a bandwidth as large as ~1 THz.Based on Eq.S3, it is clear that the concept of multiscale structures offers more degrees of freedom to tailor the spectral response of the entire structure and, as such, it is considered an effective tool for manipulating the properties of the spoof SPPs that propagate in metal-wire waveguides.
Practical considerations: For sample fabrications, we usually engrave the designed multiscale structures along one of the two wires in the TWWG, in order to avoid the alignment of the grooves on both sides.The misalignment of the grooves can lead to a phase shifts between the THz electric field propagating along the two wires, in turn introducing additional loss on the relatively higher frequency side [6].

Fig
Fig. S1 (a) Schematics of the TWWG with subwavelength-scale periodic grooves.The grooves are engraved on both wires and face each other.(b) Schematics of the TWWG with multiscale grooves.

Fig. S2 .
Fig. S2.Simulated dispersion relations for the plain TWWG (no grooves) and the TWWGs with subwavelength-scale and multiscale grooves.