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Abstract
In this work, a theorem is proved stating that in various types of waveguides with mirror reflection symmetries, the electromagnetic duality correspondence between eigenmodes of complementary structures induces counterpropagating spin-polarized states. The mirror reflection symmetries may be preserved around one or more arbitrary planes. Pseudospin-polarized waveguides supporting one-way states manifest robustness. This is similar to topologically non-trivial direction-dependent states guided by photonic topological insulators. Nevertheless, a remarkable aspect of our structures is that they can be implemented in extremely broad bandwidth by simply using complementary structures. Based on our theory, the concept of the pseudospin polarized waveguide can be realized using dual impedance surfaces ranging from microwave to optical regime. Consequently, there is no need to employ bulk electromagnetic materials to suppress backscattering in waveguiding structures. This also includes pseudospin-polarized waveguides with perfect electric conductor-perfect magnetic conductor boundaries where the boundary conditions limit the bandwidth of waveguides. We design and develop various unidirectional systems and the spin-filtered feature in the microwave regime is further investigated.
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1. Introduction
The transformation of electromagnetic power and/or signals from one end to the other with the least amount of potential electromagnetic (EM) energy loss is accomplished by waveguiding structures, which are regarded as supporting components [1,2]. Open waveguides integrating complementary metasurfaces have been implemented as a result of recent advancements in the development of artificially patterned planar structures [3–5]. Metasurfaces are easily fabricated and can control the propagation of surface waves by varying the sizes of their unit cells. [6–9]. Due to their planar structure, metasurfaces can also be integrated with microwave and nanophotonic devices [10–12]. Recently, there has been an illustration of a new electromagnetic mode known as a line wave (LW) which occurs based on complementary impedance boundaries concept [3].
By imposing the inversion symmetries, a pseudospin-polarized line mode is created at the interface of two complementary metasurfaces. The LW demonstrates robust unidirectional propagation and broad operating bandwidth. Additionally, symmetry-protected transport of pseudospin states has been discovered and examined in a channel with an effective perfect magnetic conductor (PMC) and perfect electric conductor (PEC) boundaries [13]. Nevertheless, the operational bandwidth of a spin-polarized waveguide is constrained by the employment of dual EM boundary conditions in the form of PECs and PMCs. Time-reversal invariant photonic topological insulators exhibiting pseudospin-polarized one-way transport have been proposed as well [14–19]. The protected propagation of interface states in these systems arises from energy bands with nontrivial topological features. Alternatively, photonic systems with broken time-reversal symmetry support one-way edge states confined to the interface between two magneto-optic crystals. These nonreciprocal structures, which are the precise counterparts of quantum Hall effect exhibit robustness against structural deformations [20–24]. Robust transport implementations based on magneto-optic effects and photonic topological insulators suffer from restricted bandwidth and inevitable design complexities [25–27]. Here, we investigate whether pseudospin transport is possible for any arbitrary system whose boundary conditions are created by complementary metasurfaces.
In this paper, we prove that the protected transport characteristic within an ultra-wide frequency range can be achieved in a straightforward manner by imposing complementary impedance boundary conditions. Next, we design numerous broadband spin-filtered waveguides and investigate immune spin-dependent transport. In pseudospin-polarized waveguides with complementary impedance boundaries, spatial inversion symmetries lead to forming polarization-momentum locked spin-up forward and spin-down backward states, which are robust against spatial perturbations. In such systems, disturbances do not establish backward waves and they are strongly filtered owing to preserving time-reversal symmetries. Our findings are universal as they pave the way for generalizing the idea of spin-polarized protected transport into all types of classical waveguides. We examine our theory on some open and closed waveguides. The waveguides studied in this research are formed by complementary metasurfaces having isotropic sheet impedances. Our waveguides are suitable for applications in sensing, chiral quantum coupling, directional couplers, one-way system isolators, and reconfigurable devices. This work opens a wide avenue for designing and developing new waveguides that support one-way states.
2. Theorem of pseudospin states
There is an EM duality for the eigenstates on complementary metallic ultrathin structures inserted in a homogenous, isotropic, and source-free medium [28]. The explicit duality correspondence of EM fields on complementary formed thin metallic films warrants that the surface modes on dual-complementary structures share strictly the same dispersion. Indeed, this spatial correlation of EM fields is a connotation of Babinet’s principle [29].
Consider two interfaced complementary metasurfaces extended uniformly in the $\pm \mathrm {y}$ directions, as shown in Fig. 1. Assuming that the complementary structures are suspended in free space and coincided with the $\mathrm {x}-\mathrm {y}$ plane. At frequencies in which metasurfaces unit cell is subwavelength a circular patch metasurface and its Babinet complement, i.e., a circular mesh metasurface, exhibit a dominant mode of transverse-electric (TE) and transverse-magnetic (TM) waves, respectively. Let us consider that the TM surface comprises $E_x, H_y$, and $E_z$ field components at any point $(x,-y, z)$. According to the EM duality, field components of the complementary surface $H_x, E_y$, and $H_z$ at any point $(x, y, z)$ can be written as
Fig. 1. Two interfaced dual metasurfaces supporting pseudospin states at their interface line. The complementary metasurfaces satisfy $\varepsilon _r(x,-y, z)=\mu _r(x,y, z)$ , and $\mu _r(x,-y, z)=\varepsilon _r(x,y, z)$ symmetries because the they constitute a pair of mirror images.
Equation (1) describes the relationship between EM fields in pseudospin up state. According to the definition of the pseudospin up state, $E_y$ field component at any point $(x, y, z)$ and Hy field component at any arbitrary point $(x, -y, z)$ are in phase, while $H_{x,z}$ field components and $E_{x,z}$ field components are out of phase.
Note that the EM duality or Maxwell’s equations more generally, can be written in different systems of units, such as SI (International System of Units) or CGS (Centimeter-Gram-Second) units. Even though the units used may differ, the underlying equations remain unchanged and describe the same physical phenomena. It’s worth noting that in this case, the equations have been written using SI units [30–33]. The differential form of Maxwell’s equations describing the propagation of TM surface wave at point $(x, -y, z)$ are given by,
Similarly, the differential form of Maxwell’s equations for the complementary surface wave at point $(x, y, z)$ is given by
Combining (4), (7) and (3), (6) together with (1), (2), results in following
Equation (9) describes the mirror reflection symmetries. By considering the above equations the governing equation for the pseudospin up state can be written as
Equations (11), and (12) represent the mutual relationship between EM field components of the pseudospin down state. According to the definition of the pseudospin down state, $H_{\mathrm {y}}$ field component at any point $(x,-y, z)$ and $E_y$ field component at any point $(x, y, z)$ are out of phase while the $H_{x,z}$ field components and the $E_{x,z}$ field components are in phase. The linear combination of two surface modes with orthogonal polarizations propagating with the same momentum $\beta _{T E}=\beta _{T M}=\beta$ yields the pseudospin down state whose governing equation can be written as
3. Ultra-wide-band one-dimensional waveguide with symmetric line mode
There have been established conditions for the existence of the LW and the yielded mode by characterizing the interfaced planes merely by one-layer complementary sheets [3]. However, the LW suffers from asymmetric spatial field confinement in which higher field concentration is clearly seen on one side of complementary surfaces [4]. In this section, we aim to overcome this inherent limitation by using two pairs of complementary metasurfaces and properly preserving the mirror symmetries. Meanwhile, our complementary structures satisfy two main requirements of metasurfaces i.e., subwavelength thickness of array and subwavelength separation of the scatterer. For dispersive and isotropic complementary impedance surfaces characterizing the boundaries of our pseudospin-polarized waveguides the corresponding surface impedances for TM and TE polarized waves can be written as [3]
Substitution of Eq. (14) in Eq. (15) yields
Two impedance surfaces with opposite/complementary electromagnetic responses whose $\zeta$ satisfies Eq. (16) are dual EM metasurfaces. This corresponds to complementary impedance boundaries supporting TM and TE surface waves with identical wavenumbers. Thus, they show equal response to EM fields and preserve the mirror reflection symmetries.
The theorem of pseudospin states can be applied to establish new waveguiding phenomena having spin-filtered characteristics. This permits manipulating EM energy in a scattering-free way. Here, a new waveguide is presented that supports a symmetric LW. The proposed pseudospin-filtered waveguide provides extremely wide working bandwidth and better performance along bent paths. The waveguide consists of a strip line sandwiched between four impedance surfaces with complementary EM response, as shown in Fig. 2(a). The constituent impedance boundaries satisfy $\varepsilon _{r}(x, y, z)=\mu _{r}(x, y,-z)$ and $\mu _{r}(x, y, z)=\varepsilon _{r}(x, y,-z)$ symmetries about the conducting strip line which has the width of $w<\lambda _{0} / 15$. Consequently, a system of robust spin states is formed whose pseudospin polarizations are exclusively defined by the direction of wavevector.
Fig. 2. (a) Schematic of a pseudospin-polarized one-dimensional waveguide in which the space inside the waveguide is filled with Teflon with $\varepsilon _{r}=2.1, \delta _{t}=0.001$, (b) dispersion characteristics of the TE and TM surfaces with geometrical parameters, (c) $\zeta$ values of the corresponding complementary metasurfaces, and (d) electric field distribution of the LW at different frequencies.
Note that in the proposed waveguide the strip line is on the $x-y$ plane. Nevertheless, this circumstance does not violate the reflection symmetries preserving condition. In general, the strip line can be replaced by coplanar strips, spoof SPP structures, two interfaced complementary metasurfaces, and so on. The geometry of the proposed structure is characterized by the width of the strip ($w$) and separation distance between two complementary metasurfaces ($h$). If the value of $h$ remains in the subwavelength limit, i.e., $h<<\lambda$, the EM duality requirement and hence the mirror reflection symmetries are satisfied adequately well [28]. Since both parameters have subwavelength values, the waveguide supports the LW decaying away in both transverse directions from interface lines.
Figure 2(b) depicts the unit cell design of one-layer complementary metasurfaces and the corresponding dispersion relation for TM and TE surface waves propagation. The dispersion curves of freestanding complementary structures are nearly the same and they overlap over a wide frequency range. The corresponding plot of $\zeta$ values versus frequency is presented in Fig. 2(c). It is obvious that $\zeta$ parameters have the ratio of 4 as it was predicted in Eq. (16). The geometrical parameters of metasurfaces unit cells are engineered in Bisharat et al work to reduce the effect of the asymmetric behavior of LWs [3].
Nevertheless, however small the inequality is, it forces higher concentration of EM waves towards the surface with higher $\zeta$ values. Particularly, in case of a one-layer waveguide with complementary impedance boundaries, this asymmetric behavior leads to increasing scattered bend losses. Since the supporting complementary surfaces of the one-layer waveguide do not have a bandgap, the LW can be scattered into the surrounding area. The idea is that exploiting a bandgap structure prevents LWs scattering. However, this limits the operating bandwidth of these waves [5]. To alleviate this, we propose the use of a second layer that is complementary to the first one. This results in the equality of $\zeta$ about the $x-z$ plane; hence, EM waves confine symmetrically about the interface line, as shown in Fig. 2(d). Besides, the presence of the strip line, in turn, leads to decreasing waveguide bending loss. This is an alternative approach to improve waveguide performance.
Our pseudospin-polarized systems provide wideband operating bandwidth over conventional waveguides as they are safe from the bandwidth restriction connected by band gaps in photonic crystals. If the supporting metasurfaces share the same dispersion over a wide frequency range, constructive interaction between reciprocal counterparts of EM fields yields a pair of hybrid spin states exhibiting ultra-wide EM response. These states are quite robust against time-reversal symmetry-conserved disorders [3,35]. To analyze the ultra-wideband behavior of the proposed structure, we introduce a zigzag bend operating as a waveguide with multiple bends of angles 90-degree. The schematic of the zigzag pathway is shown in Fig. 3(a). The robustness of pseudospin states against sharp bends is preserved by boundary-mirror symmetries provided that the scatterer does not bring about a spin-flip transition. Figure 3(b) shows the propagation of the pseudospin-polarized LW along the bent path proving the backscattering immune spin-dependent transmission. As shown in Fig. 3(c), a transmittance (|S21|) of about $99 \%$ is obtained in the frequency band of 1-38 GHz for the straight waveguide. On the other hand, for the bent waveguide, a transmittance of about $95 \%$ is achieved in the frequency range of 1-34 GHz. This demonstrates that the bandwidth of the waveguide is not completely immune to perturbations. Note that waveguide losses increase proportionally with frequency. Especially, this is the case for the bent waveguide. Such perturbations do not scatter pseudospin states in reverse direction or break spin degeneracy as these states are decoupled. The dispersion diagram of the LW is presented in Fig. 3(d). The line mode forms the pseudospin-up state with wavenumber $+k$ and its time-reversal pair, i.e., the pseudospin-down state, with wavenumber $-k$. Furthermore, the wave has a nonlinear dispersion.
Fig. 3. (a) The schematic of a sharp bent waveguide with multiple bends, (b ) propagation of the hybrid spin state along sharp bends, (c) the simulated transmittance, indicating low backscattering losses and extreme bandwidth of the waveguide, and (d) dispersion characteristic of the LW.
4. Ultra-wideband pseudospin-polarized slotline waveguide
To examine our theory, we investigate possible realization of an open boundary pseudospin-polarized system that can be implemented by establishing mirror reflection symmetries. We only consider a few scenarios, even though various possibilities might be considered. In this section, a new slotline waveguide with multi-octave bandwidths has been demonstrated providing the possibility to suppress backward losses. Consider a classical slotline waveguide, bounded by two parallel complementary boundaries, preserving the spatial inversion symmetries of the form $\varepsilon _{r}(x, y, z)=\mu _{r}(x, y,-z)$ and $\mu _{r}(x, y, z)=\varepsilon _{r}(x, y,-z)$, as shown in Fig. 4(a). Assuming that the conducting ground planes are surrounded by air.
Fig. 4. (a) A schematic representation of the proposed slotline waveguide, where the structural parameters are $g=0.5\,\mathrm{mm}$, and $d=0.4\,\mathrm{mm}$, (b) dispersion curves of two subwavelength complementary sheets. The unit cell period $p$ and the geometric parameter of the unit cell $a$ are set as $p=1.5\,\mathrm{mm}$, and $a=0.6,1,1.3\,\mathrm{mm}$. (c) the normalized near-field distribution in complementary-unit-cells, and (d) transmission and reflection coefficients of the proposed waveguide obtained for $\mathrm {a}=0.6$.
Alternatively, they can be centered in a dielectric material that has about the same permittivity and permeability, i.e., $\varepsilon (\mathrm {r}) \approx \mu (\mathrm {r})$. This qualifies to preserve mirror symmetries at any given point. Additionally, such mediums can be used as a substrate or a superstrate in pseudospin-polarized waveguides with arbitrary configurations. Figure 4(b) presents the dispersion characteristics of complementary metasurfaces forming the boundaries of the pseudospin-polarized slotline waveguide. The coincidence of the dispersion curves over a wide frequency range is warranted by the EM duality. As a result, the mutual interaction between two surface modes of the same momentum gives rise to wideband pseudospin states. The reciprocal relationship between $\mathrm {x}$ components of $\mathrm {E}$ and $\mathrm {H}$ fields pertinent to supporting complementary metasurfaces is shown in Fig. 4(c). This relationship is a direct consequence of the EM duality resulting in the formation of pseudospin states. The simulated scattering parameters, reflection coefficient (S11), and isolation (S21), of the new slotline waveguide, are presented in Fig. 4(d). The waveguide has a mean value of isolation $-0.13\,\mathrm{dB}$ over a broad frequency range of 1-68 $\mathrm {GHz}$. The spin-filtered slotline waveguide features a wide bandwidth. Supporting momentum-spin-locked states is another crucial aspect of the proposed waveguide. This is also inherent to every fast-decaying wave with evanescent characteristics in a transverse plane to wave momentum [36]. This universal property which enforces the handedness of waves to be defined uniquely by the transport direction has been observed in surface plasmon polaritons as well [37].
To demonstrate the directionality of decoupled pseudospin states, we change the boundary conditions in the proposed structure, as shown in Fig. 5(a). Since the time-reversal invariant waveguide prevents the propagation of $\psi ^+$ along the defect region, the system is a spin-filtered route and the definition of pseudospin states is tied to the direction of momentum, as shown in Fig. 5(b). Alternatively, the intensity profiles for the incident and filtered E fields confirm that robust spin states are locked to the wavevector. The result agrees well with our predicted theory. Figure 5(c) shows the unidirectional excitation of pseudospin states using a chiral point source carrying spin angular momentum [38–40]. The chiral source placed at the center of the waveguide consists of two out-of-phase electric point sources with orthogonal configurations. The optional excitation of spin states demonstrates that one-way pseudospin states are strictly characterized by the direction of propagation. In the same manner, pseudospin states can be excited via a pair of dual point sources (i.e., electric, and magnetic Hertzian dipoles) with specific phase relations [41]. The dispersion relation of the proposed slotline waveguide and mode distribution of a supercell is presented in Fig. 5(d). The propagation constant of the slotline fundamental mode is the same as the TEM mode which has a linear dispersion relation.
Fig. 5. (a) A schematic of the deformed waveguide where the boundary conditions are reversed in the structure, (b) spin-locking of a TEM mode at a pseudospin polarized open waveguide with complementary impedance boundaries. The obtained incident and filtered field profiles manifest that the structure is a spin-filtered waveguide, (c) unidirectional excitation of the pseudospin states proving polarization-momentum locking feature of the waveguide, and (d) the dispersion diagram and the mode distribution of the slotline waveguide.
5. Psedospin-polarized closed waveguide
Open waveguides are characterized by imperfectly reflecting or open boundaries while closed waveguides are described as being entirely enclosed with highly reflecting walls. In ordinarily closed waveguides EM waves are confined inside boundaries considered as either perfectly electric walls with zero impedances or perfectly magnetic walls with zero admittances [42]. On the contrary, a pseudospin-polarized closed waveguide can be created by properly preserving the mirror reflection symmetries.
We can use two pairs of perfectly reflecting electric and magnetic boundaries to form a pseudospin-polarized closed waveguide and establish the EM duality between walls [13]. Nonetheless, PMCs are generally realized with Sievenpiper high-impedance surfaces which have a limited bandwidth [43]. Chen et al proposed implementation of an effective PMC boundary using periodic PEC structures that have a non-resonant nature [13]. This method introduces additional mirror symmetries to realize a PMC boundary. However, it is very complicated to implement a closed pseudospin-polarized waveguide using periodic PEC boundary conditions. We propose the use of non-resonant complementary impedance boundaries which satisfy Eq. (16). This leads to overcoming bandwidth limitations associated with employing PEC and PMC boundaries in the previously studied pseudospin-polarized closed waveguide.
Figure 6(a) shows a schematic of the proposed spin-polarized closed waveguide. The surface impedances or admittances of walls, relating $\mathrm {E}$ and $\mathrm {H}$ field components that are tangential to the wall are both frequency and spatially dispersive. In general, the impedance of walls can be anisotropic, which is described by a tensor rather than scalar surface impedance. We assume that the waveguide is homogeneously filled with an isotropic material with $\varepsilon =1.4, \mu =1$. Therefore, the nearly matched $\varepsilon -\mu$ condition provides a good approximation of a time-reversal invariant system. Regarding the coordinate system in Fig. 6(a), the proposed closed waveguide is mirror-symmetric about the $x-y$ plane. As a result, the waveguide supports decoupled pseudospin states. A simulation plot of the scattering parameters, $S 11$ and $S 21$, is shown in Fig. 6(b); The fundamental mode of the proposed waveguide is wideband in the frequency range $7.5-11.8 \mathrm {GHz}$ with a $S 21$ value of about $-0.5\,\mathrm{dB}$. The vector eigenfields distributions of pseudospin states over the cross-section of the waveguide are presented in Fig. 6(c). There are two plane-polarized waves (i.e., spin-up backward and spin-down forward states) that are essentially determined by the direction of propagation. Consequently, the system forms one-way states which are associated with the EM duality. Figure 6(d) plots the dispersion relation of fundamental pseudospin states. The red and blue lines correspond to pseudospin-up and pseudospin-down states dispersion, respectively. As expected, the pseudospin TEM mode is the mode of propagation as the $\mathrm {E}$ and $\mathrm {H}$ fields are perpendicular to each other and to the direction of propagation.
Fig. 6. (a) A schematic of the proposed closed waveguide where the unit cell period is $2\,\mathrm{mm}$, and the gap spacing between adjacent patches is $0.9\,\mathrm{mm}$, (b) scattering parameters of the corresponding waveguide, (c) eigenvector fields distribution of two decoupled spin states over a cross section of the closed waveguide, and (d) dispersion diagram for the closed waveguide with complementary boundaries.
Preserving the EM duality about an arbitrary mirror plane provides a closed waveguide with a backscattering-immune transportation feature. Here, various types of perturbations are considered to show that pseudospin modes are directly characterized by the direction of momentum. For simplicity, we consider non-dispersive surface impedance walls. This means that the value of impedances is independent of tangential wavenumber of the incident wave and frequency. Consider the case where a hybrid polarization $\psi ^{-}$ propagates along consecutive bends, as shown in Fig. 7(a). To satisfy the boundary continuity, there is no way to excite the $\psi ^{+}$ polarization. Since the scatterer does not flip the pseudospin polarization, protection in the system takes effect and the mirror reflection symmetries preserve the propagating without remarkable reflections. To show the spin-filtered characteristic, a T-junction structure is designed, as shown in Fig. 7(b). The excited pseudospin up state at port 1 couples into port 4 while the reversal of boundary conditions along ports 2 and 3 prevents the coupling of $\psi ^{+}$ into $\psi ^{-}$. It is possible to form a $-3\,\mathrm{dB}$ coupler by changing the boundary conditions of the system. As shown in Fig. 7(c), EM waves split into ports 2 and 3 while scattering into port 4 is impossible. Consequently, the pseudospin states are merely defined according to the direction of propagation and the closed waveguide with complementary impedance walls demonstrates a spin-filtered route. As shown in Fig. 7(d), scattering parameters $S 21$ and $S 31$ of about −3dB and $S 41$ below $-30\,\mathrm{dB}$ are obtained over a wide frequency range. Hence, the propagation is completely filtered along port 4. This is also a demonstration that the structure is a spin-filtered waveguide. These results were obtained by a driven mode analysis of $\operatorname {HFSS}$ for $Z_{T E}=-j 2 \eta _{0}$ and $Z_{T M}=j \eta _{0} / 8$ with a side length of the waveguide of $6\,\mathrm{mm}$.
Fig. 7. (a) Guiding spin down state along multiple bends showing back scattering immune transportation, (b) energy transmission in a T-junction network, (c) the T-junction as a splitter based on spin-filtering feature, and (d) scattering parameters of the corresponding splitter structure.
The associated momentum-spin locking property can be exploited to form versatile network devices. Incorporating impedance surfaces with dual capacitive-inductive nature makes it possible the formation of unconventional passive circuits with dynamic configurations as well [38,44]. In particular, this includes graphene-based metasurfaces showing great potential applications in terahertz and optical regimes [45–48]. As an illustration, Figs. 8(a), and 8(b) show an add-drop filter design that enables transforming EM waves along arbitrary paths simply by switching a capacitive/inductive impedance to the inductive/capacitive impedance. Figure 8(c) shows a spin-filtered-based directional coupler that is able to couple EM power into desired branches. The structure consists of two closed waveguides with complementary impedance walls, which are joined at the interface plane. A rectangular hole at the interface is created to make possible energy coupling. The coupler design is based on the spin-filtered feature. This allows for controlling EM waves through desired paths inside the closed system. The coupler may equally split power between two output ports. However, unlike traditional directional couplers, steering EM energy in one direction, our structure is capable of coupling energy in desired directions [47]. For instance, reversal of boundaries along ports 3 and 4 prevents the propagation of spin states through these paths. As a result, EM waves have no choice but to couple into port 2, as shown in Fig. 8(d).
Fig. 8. (a), and (b) guiding EM energy in an add-drop filter through dynamic paths, (c) a hybrid directional coupler, and (d) unconventional steering of EM energy in a directional coupler with complementry impedance walls.
6. Conclusion
In summary, we have shown that the ultra-wideband pseudospin-dependent transport can be achieved simply using 2D patterned and complementary sheets. Our results are applicable to all traditional waveguides. This brings about a framework to study waveguiding mechanisms providing the opportunity to further explore this new research area. We proposed a new configuration supporting an ultra-wideband LW having a better performance along bent path. In addition, we examined all possible scenarios to prove that our theory is reliable. The possibility of generalizing the pseudospin polarized waveguide concept to a slotline waveguide is further explored. In particular, our theory opens an avenue in the field of closed waveguides, thus allowing for designing wideband closed waveguiding structures. Additionally, we demonstrated that designing closed network devices, such as a magic-T structure is viable by altering the orientation of complementary metasurfaces. It is worth mentioning that the concept of pseudospin-polarized waveguide can be applied to acoustic systems. Acoustic metamaterials preserving the EM duality can be used in order to form pseudospin waveguides [49–56]. This concept could potentially be applied to other structures like water wave systems provided that the EM duality is established between boundaries of the system [57,58]. All in all, we used the theorem of pseudospin states to develop new wideband open and closed pseudospin-polarized waveguides. The findings change our point of view regarding classical waveguides. The complementary metasurface-based waveguides developed using our theory not only have wideband characteristics but also show robustness to certain structural perturbations.
7. Numerical analysis
Full-wave simulation was performed using ANSYS HFSS, which is a finite element method (FEM)-based commercial software, to obtain the scattering parameters. A 3-D computational domain surrounded by a radiation box of height $\lambda / 2$ in the vacuum region was considered to study source-excited structures. Furthermore, for dispersion relations, numerical simulations were performed for a supercell by the eigenmode solver of COMSOL Multiphysics, which is a finite-element-based commercial software as well. This time the resulting curves were calculated without there being substrate materials. The boundaries of the supercell along the propagation direction are applied with a periodic boundary while the other borders are chosen as radiation boundary conditions.
Acknowledgments
The theorem of pseudospin states is proved by the author of this work
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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