## Abstract

Ultrawideband unidirectional surface magnetoplasmons (USMPs) are theoretically achieved in both planar and coaxial configurations without an external magnetic field. It is shown that the two proposed structures, which are formed by ferrite layers with opposite remanences, can support USMP without low-frequency cutoff, thus the USMP bandwidth is increased at least twice compared to (microwave) USMPs previously reported. For the coaxial structure, the dispersion of USMP is closely dependent on the thickness of the ferrite layers, and based on this, USMP can be slowed down significantly by reducing the ferrite-layer thickness. Moreover, by using a tapered coaxial structure, subwavelength focusing can be realized with a giant enhancement of the magnetic field.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Unidirectional electromagnetic (EM) modes are such modes that are allowed to travel in only one direction, and because of the absence of back-propagating mode in the system, they are immune to backscattering at imperfections or bends. These modes have important potential applications in classical and quantum information processing, including isolation [1,2], circulation [3,4], and resonance without the time-bandwidth constraint [5]. The unidirectional EM mode was first proposed from analogue of quantum Hall edge states in photonic crystal (PhC) [6–8], then it was experimentally verified by using magnetic-optical (MO) PhC in the microwave regime [9]. Later, another type of unidirectional EM mode was also proposed in the form of surface magnetoplasmon (SMP) [10,11]. Since then, unidirectional EM modes have received increasing attention [12–18]. Owing to simple configuration and large bandwidth, unidirectional surface magnetoplasmon (USMP) seems to be more attractive compared to unidirectional PhC edge modes.

Both types of unidirectional EM mode rely on MO material and external magnetic field, and the latter causes difficulties in their practical applications. To overcome this problem, topologically protected states, which are capable to suppress backscattering in the systems without external magnetic field, were proposed and studied, but they often involve complicated field polarizations [19–21]. In the microwave domain, however, there is a simple way to solve the above problem. Ferrite materials, which are commonly used as MO materials in microwave domain, can preserve magnetization after removing external magnetic field. Hence, ferrite materials with remanent magnetization possess gyromagnetic feature in the absence of external magnetic field [22]. Recently, it was reported that USMPs can be sustained by an interface between ferrite material with remanence and dielectric. The dispersion characteristic of SMPs in the proposed system with remanence is similar to that in the system with external magnetic field applied, e.g., the USMP behaviour and SMP resonance, the latter leads to a low-frequency cutoff for USMPs [23].

However, different from SMPs, the dispersion characteristic of bulk modes in ferrite material is significantly modified after removing the external magnetic field [23]. It is known that USMP can exhibit complete unidirectional propagation (CUP) (that is immune to backscattering) only when it is located within the bandgap of the ferrite material. For ferrite material under external magnetic field, there exists one bandgap between two separate bulk-mode zones [24,25]. In the ferrite material with remanence, the lower bulk-mode zone vanishes, then the bandgap extends down to zero and its bandwidth is increased almost by twice [23]. So it seems possible that in a system constructed with ferrite material with remanence, USMP maybe possess ultra-wide CUP band without low-frequency cutoff. But such phenomenon is not reported yet. In this paper, we propose and investigate unidirectional waveguides formed by ferrite layers with opposite remanences, and both planar and coaxial configurations are considered. We will show that compared to the previous USMP in [23], the proposed USMPs possess an ultra-wide CUP band without low-frequency cutoff, whose bandwidth is increased at least by twice. Moreover, by utilizing the dependence of the USMP dispersion on the ferrite-layer thickness, we will show that subwavelength focusing with extremely enhanced magnetic field can be realized by using a tapered coaxial structure.

## 2. Unidirectional planar waveguide

The planar waveguide proposed for ultrawideband USMP is schematically depicted in Fig. 1(a), which consists of two ferrite slabs of the same thickness ($d$) sandwiched between a pair of metal slab. The metal slabs are assumed to be perfect electric conductor (PEC), which is a good approximation in the microwave regime. Both ferrite slabs possess remanence, and the remanent magnetization vectors in them have the same amplitude ($M_r$) but opposite directions, i.e., $\textbf {M}_r^+=M_r \hat y$, $\textbf {M}_r^-=-M_r \hat y$. The relative permeabilities for the ferrite slabs take the form [23]:

Suppose that SMP propagates along the z direction in the proposed waveguide. The electric field of SMP is transversely polarized (TE), and its nonzero component ($E_y$) can be written as

We first analyze the dispersion relation (4) for low frequencies $\omega \ll \omega _r$. In this case, $\alpha \approx \sqrt {\epsilon _r} k_r$, where $k_r=\omega _r/c$ ($c$ is the light speed in vacuum), then Eq. (4) becomes

Evidently, SMP has no low-frequency cutoff as expected, and at low frequencies, it is only allowed to propagate forward with $k>0$. From the dispersion relation (4), we find there exists an asymptotic frequency of $f_{sp}=f_r$ (where $f_r=\omega _r/2\pi$) for SMP, at which $k\rightarrow +\infty$. This asymptotic frequency is related to the magnetostatic resonance at the interface between the ferrite and metal, and it doesn’t occur in the simple system consisting of semi-infinite ferrite materials with $\textbf {M}_r^+$ and $\textbf {M}_r^-$, in which SMP has the dispersion relation

which can be directly derived from Eq. (4) by letting $d\rightarrow +\infty$. The asymptotic frequency $f_r$ leads to a high-frequency cutoff for SMP in the waveguide.In the proposed waveguide, there also exist bulk modes that have a real transverse component of wave vector in the ferrite materials. However, different from the previous unidirectional waveguide [23], in which two ferrite materials (beside the dielectric layer) with opposite remanences are semi-infinite, the ferrite materials in the present waveguide are terminated by metal slab, and consequently bulk modes transform into discrete bound modes, due to the field confinement in the transverse direction. For convenience, we refer to these bound modes in the waveguide as bound bulk modes. Evidently, the dispersion of bound bulk modes is closely dependent on the thickness of the ferrite layers. So it is desired that by reducing the ferrite-layer thickness, the bulk-mode zone shifts up and then the ferrite-layer bandgap broadens. The dispersion relation for bound bulk mode can be derived from Eq. (4) by replacing $\alpha$ with $ip$, which yields

where $p=\sqrt {\epsilon _r\mu _{\mathrm {v}}k_{0}^{2}-k^{2}}=\sqrt {\epsilon _r(k_0^2-k_r^2)-k^2}$ being a real-valued number in the lossless case ($\nu =0$). Obviously, bound bulk modes require $k_0>k_r$, i.e., $f>f_r$, so they are located above the SMP band. Therefore, the whole dispersion curve for SMP is a CUP band, which is terminated by $f_{sp}$ from above, and the CUP bandwidth is equal to $f_r$. In contrast, the CUP band for the previous waveguide with remanence ranges from $f_r/2$ to $f_r$ and the bandwidth is only $f_r/2$. So the CUP bandwidth for the present waveguide is twice larger than previous one.To validate our analysis above, we numerically calculate the dispersion relations for SMP and bound bulk mode with Eqs. (4) and (7). In this paper, the parameters of the ferrite materials with remanence are taken as follows: $\epsilon _r=15$ and $f_r=3.587$ GHz [22,23]. The dispersion relation for SMP is shown in Fig. 1(b), where $d=10$ mm as an example. The shaded area in Fig. 1(b) represents the zone where the bound bulk modes are located, and the boundary corresponds to the dispersion curve for the lowest-order bulk mode. As expected, the whole dispersion curve for SMP is a CUP band with $k\ge 0$, as it lies below the bulk-mode zone. The dispersion feature of SMP in the present waveguide is quite different from that in the previous waveguide [23]. In the previous waveguide, which is a ferrite-dielectric-ferrite structure, there exist two SMP modes with odd and even symmetries. For the odd mode, the electric field vanishes at the axial line of the waveguide, so such mode cannot occur in the present waveguide, in which the electric field of SMP peaks at the interface of the two ferrite layers. For the even mode in the previous waveguide, the dispersion band has two branches with $k<0$ and $k>0$, and they propagate in the opposite directions. The dispersion branch with $k<0$ has an asymptotic frequency of $f_{sp}=f_r/2$, which results from the resonance of SMP at the interface between ferrite and dielectric, and the asymptotic frequency determines the lower limit of the CUP band. Obviously, by removing the dielectric layer in the present waveguide, this SMP resonance and even the dispersion branch with $k<0$ are suppressed, and consequently the lower-frequency cutoff of the CUP band vanishes.

As seen in Fig. 1(b), the dispersion curve for SMP in the proposed waveguide is almost a linear function except for a very small interval near $f_{sp}$, and it can be well approximated by Eq. (6) (represented by dashed line), which is the dispersion relation for SMP at the single interface between semi-infinite ferrite materials with opposite remanences. Evidently, in a small interval around $f_r$, SMP sustained by the interface of the ferrite layers with opposite remanences strongly couples with the magnetostatic waves at two ferrite-metal interfaces, and as a result, the dispersion curve of SMP is split into two branches. The lower branch is terminated by the resonant frequency ($f_r$) of the magnetostatic wave, and the upper branch (with $f>f_r$) further couples with bound bulk mode, then making the bulk-mode zone asymmetric about $k=0$. To show this, the magnetic field amplitude for SMP at a frequency close to $f_r$ (marked by B point) is plotted in the inset of Fig. 1(b), where the field distribution at the central frequency of the CUP band (A point) is also included for comparison. As a dispersive medium, ferrite with remanence is inherently lossy. It is interesting to investigate the loss effect on the dispersion characteristic of SMP in the waveguide. Thus we numerically solve Eq. (4) for the case of $\nu =0.001$, and the results are plotted in Fig. 2. In the loss case, the propagation constant of SMP becomes a complex number. Figure 2(a) shows the phase constant [$\mathrm {Re}(k)$] of SMP versus frequency. The phenomenon of asymptotic frequency disappears in the presence of ferrite loss, and the maximal value of the phase constant is $5.44k_r$ in the interval around $f_r$. In this realistic structure, the dispersion curve of SMP is only locally distorted by its coupling with the magnetostatic waves (at the ferrite-metal interfaces), and the CUP band is still well preserved. Figure 2(b) shows the attenuation coefficient [$\mathrm {Im}(k)$] versus frequency. The attenuation coefficient is almost constant over the most part of the CUP band, and at the central frequency of the CUP band, it equals $0.002k_r$. But the attenuation coefficient becomes very large in the neighborhood of $f_r$, and its maximal value is up to $1.27k_r$.

The proposed two-dimensional (2D) waveguide can be easily mapped into a three-dimensional (3D) waveguide. In the 2D waveguide, the EM field of SMP extends uniformly along the $y$ direction, and the electric field just points in this direction. Suppose that the 2D waveguide is truncated in the $y$ direction by using a pair of PEC slabs, which is separated by a distance $w$, the truncated field distributions will be preserved in the constructed 3D system. For the frequency range of our interest, when $w$ is smaller than a certain value, this 3D waveguide can only support USMP with field uniform in the $y$ direction, and its dispersion relation is the same as that for the 2D waveguide. To verify the wideband CUP behaviour in the 3D waveguide, we simulate wave transmission in it by using finite element method (FEM). In the simulation, we take $d=10$ mm, $w=6$ mm, and $\nu =0.001$. A linear electric current is used to excite wave in the middle of the 3D waveguide. The simulated electric field amplitudes are displayed in Fig. 3(a) for $f=1.5$ GHz and in Fig. 3(c) for $f=2.5$ GHz, and unidirectional propagation is observed for both cases. For comparison, similar simulations are also made for the previous 2D waveguide in [23]. Figure 3(b) shows the electric field amplitudes for $f=1.5$ GHz, and bidirectional propagation is observed, i.e., the excited wave simultaneously propagates along the $+z$ and $-z$ directions. Figure 3(d) shows the electric field amplitudes for $f=2.5$ GHz, and in this case, the excited wave only propagates forward. The results from the simulations for the 3D waveguide agree well with our dispersion analysis for the related 2D waveguide.

## 3. Unidirectional coaxial waveguide

We now consider a ferrite layered structure in coaxial configuration, as schematically depicted in Fig. 4(a). In this axisymmetric system, two ferrite layers with thicknesses $d_+$ and $d_-$ are sandwiched between a thin metal wire and a metal shell. In the ferrite layers, the remanent magnetization points in the $\phi$ direction or the reverse, i.e., $\textbf {M}_r^+=M_r \hat \phi$, $\textbf {M}_r^-=-M_r \hat \phi$, as indicated in Fig. 4(b), which shows the cross section of the 3D waveguide. In the cylindrical coordinates, the (relative) permeability tensor for the ferrite layers with remanent magnetization vectors $\textbf {M}_r=\pm M_r \hat \phi$ can be expressed in the same form as Eq. (1), and the tensor elements $\mu$ and $\kappa$ have the same expressions as those in Eq. (1) as well, i.e., $\mu = 1+i\nu \omega _r/\omega (1+\nu ^2)$, $\kappa = \omega _r/\omega (1+\nu ^2)$, where ${\omega _r}={\mu _0} \gamma {M_r}$.

In the considered coaxial system, SMP has an axisymmetric and TE-polarized field. The nonzero component of the electric field can be expressed as

where $k$ is the propagation constant. The electric-field amplitude $e_{\phi }\left ( r \right )$ in the inner ferrite layer with the magnetization $\textbf {M}_r^+=M_r \hat {\phi }$ satisfies the following equationThe electric-field component in the outer ferrite layer with magnetization $\textbf {M}_r^-=-M_r \hat {\phi }$ can be similarly solved for, and the solution has the form

As $E_{\phi }$ vanishes at the boundary of the outermost metal layer, we have $B_2=-[M_{-a,b}(\beta R)/W_{-a,b}(\beta R)] B_1$, where $R\approx d_- + d_+$. The magnetic field of SMP has nonzero components $H_r$ and $H_z$, and they can be derived straightforwardly from $E_{\phi }$. The z component ($H_{z}$) of magnetic field is found to be

The boundary conditions require both field components $E_{\phi }$ and $H_z$ to be continuous at the ferrite interface $r \approx d_+$, from which we obtain the dispersion relation for SMP

We first analyze the dispersion relation for low frequencies $f<<f_r$, and assume that $k_r R \ll 1$. In this case, $\beta \approx 2\sqrt {\epsilon _r}k_r$, then the dispersion relation (16) is simplified to

Obviously, SMP has no low-frequency cutoff in the coaxial structure. Moreover, for low frequencies, the dispersion curve of SMP lies in the region for $k>0$ and has a positive slope, which implies that SMP only propagates forward. From the dispersion equation (16), we find there also exists SMP resonance in the coaxial structure, and the asymptotic frequency ($f_{sp}$) is still equal to $f_r$, at which $k \to +\infty$. Figure 4(c) shows the dispersion curve (solid line) for SMP calculated with Eq. (16), and as an example, the thicknesses of the ferrite layers are set to be $d_-=d_+=10$ mm. The whole dispersion curve for SMP lies in the region for $k>0$ and the slope is always positive. Interestingly, except for a small interval near $f_r$, the dispersion curve is almost linearly varying, and it can be approximately described by Eq. (6), which is represented by the dashed line in Fig. 4(c). To verify the dispersion relation (16) for SMP, by using the FEM, we numerically solve for SMP in the coaxial structure, and the obtained results are plotted as open circles in Fig. 4(c), which agree well with those from Eq. (16). In this coaxial system, there also exist bound bulk modes as in the planar system, and these modes propagate bidirectionally. The bound bulk modes are represented by the shaded zone in Fig. 4(c), and the boundary corresponds to the lowest-order mode. Figures 4(d) and 4(e) show the field patterns for SMP and the bound bulk mode. For SMP, the field is located around the interface between the ferrite layers, whereas it is mainly distributed in the outer ferrite layer for the bound bulk mode.

As seen in Fig. 4(c), the lower-frequency cutoff ($f_{b,cf}$) of the bound bulk mode with the lowest order is below the SMP asymptotic frequency ($f_r$) in the coaxial structure, which is quite different from the situation in the planar structure. So for the coaxial structure with $d_-=d_+=10$ mm, the CUP band for SMP ranges from $0$ to $f_{b,cf}$, and the CUP bandwidth is only $\Delta {f_{\mathrm {CUP}}}=f_{b,cf}$, which is smaller than the value ($f_r$) for the planar waveguide. However, in the coaxial structure, the cutoff ($f_{b,cf}$) of the bulk mode can be adjusted upward by reducing the ferrite layer thickness, as it is guided with radial resonance. To show this, we calculate the dispersion relations for the coaxial waveguides with various ferrite-layer thicknesses. Figure 5(a) shows the dispersion relations of SMP and bound bulk modes for $d_-=d_+=6$ mm. In this case, as desired, the cutoff $f_{b,cf}$ rises above $f_r$, while the asymptotic frequency for SMP is preserved to be $f_r$. Therefore, SMP has a CUP bandwidth of $\Delta {f_{\mathrm {CUP}}}=f_r$ for $d_-=d_+=6$ mm. For $d_+$ fixed at $6$ mm, it is found that $\Delta {f_{\mathrm {CUP}}}=f_r$ for any $d_-$ value below $7.8$ mm, above which it almost linearly decreases with $d_-$, as shown in Fig. 5(b). In contrast, for $d_-$ fixed at $6$ mm, $\Delta {f_{\mathrm {CUP}}}$ is equal to $f_r$ for any $d_+$ value, as shown in Fig. 5(b). Therefore, $\Delta {f_{\mathrm {CUP}}}$ is insensitive to $d_+$, and this is consistent with the fact that the field of the bound bulk mode is mainly distributed in the outer ferrite layer.

To verify the CUP behaviour of SMP in the coaxial waveguide, we simulate wave transmission in it with FEM. In the simulation, a ring current source is placed in the middle of the waveguide (near the ferrite interface at $r=d_+$) to excite wave, and the frequency is set at $f=2.5$ GHz. The length of the waveguide is taken to be $60$ mm, and the ferrite thicknesses are $d_-=d_+=6$ mm. The ferrite loss is taken into account, and we take $\nu =0.001$. Figure 6(a) shows the simulated electric field (i.e., $E_\phi$) amplitude. As expected, excited SMP only propagates forward with field mainly distributed around the ferrite interface (at $r=d_+$). To verify the robustness of the USMP, a hollow ring (filled with air) is further placed at the ferrite interface as an obstacle in the coaxial waveguide. The simulated electric field is plotted in Fig. 6(b). Clearly, the traveling wave goes around the obstacle without any power loss, as no backward-propagating wave emerges in the left side of the source. Figure 6(c) shows the distribution (solid line) of electric field amplitude along the line at $r=d_+$. The electric field amplitude only drops in the local region of the obstacle, then completely recovers behind it. For comparison, the corresponding distribution (dashed line) in the uniform waveguide [i.e., in the case of Fig. 6(a)] is also included in Fig. 6(c). The field amplitudes for the two cases are almost the same except in the local region around the obstacle. Therefore, we conclude that the proposed coaxial waveguide can support USMP that is immune to backscattering at imperfection.

## 4. Microwave slowing and subwavelenth focusing

It is interesting to further investigate if the dispersion of SMP can be tailored by changing the ferrite-layer thickness in the coaxial waveguide. For simplicity, we set the thicknesses of the two ferrite layers to be the same, i.e., $d_+ = d_- =d$, and various values of $d$ are analyzed. Figure 7(a) shows the SMP dispersion curves for different $d$ values. Obviously, for small $d$ values, the dispersion curve remarkably deviates from the linear function Eq. (6) [dashed line in Fig. 4(c)]. So actually, the dispersion of SMP is closely dependent on the ferrite-layer thickness. As seen in Fig. 7(a), for a given frequency $f<f_r$, the propagation constant $k$ increases as $d$ reduces, thus the group velocity ($\mathrm {v}_g=d\omega /dk$) decreases. Figure 7(b) shows the dependence of $\mathrm {v}_g$ on $d$ for $f=f_r/2$, which is the central frequency of the CUP band. $\mathrm {v}_g$ decreases with decreasing $d$, and when $d<5$ mm, the variation is almost linear. We should indicate that all the results mentioned above are correct even in the loss case, as the loss effect is negligible on the SMP dispersion for the main part of the CUP band. Hence, for frequencies within the CUP band, waves can be slowed down without any reflection loss in a tapered coaxial structure, in which the ferrite-layer thickness ($d$) is reduced (even rapidly) along the propagation direction.

In a tapered coaxial waveguide, when wave is gradually slowed down, the energy is constantly concentrated as well. Based on the energy concentration of USMP, it is possible to realize subwavelength focusing with high efficiency. To confirm this idea, we perform the simulation of wave transmission within and outside a tapered coaxial structure, in which the ferrite-layer thickness $d$ is linearly reduced from $6$ mm to $0.1$ mm over a length of $30$ mm. In the simulation, wave was input into the tapered waveguide from the large port with $d=6$ mm, then was output into air from the small port with $d=0.1$ mm. The ferrite loss is taken as $\nu =0.001$. The operating frequency is set at $f=2.5$ GHz. Note that the length of the tapered waveguide is only quarter of the vacuum wavelength for the wave frequency. The simulated magnetic and electric field amplitudes are plotted in Figs. 8(a) and 8(b), respectively. Obviously, when the wave travels forward in the tapered structure, the field energy is constantly concentrated without any reflection loss. Interestingly, the energy concentration is mainly reflected in the enhancement of magnetic field, and this is more clearly illustrated in Fig. 8(c), which shows the $H$ and $E$ field distributions along the ferrite interface. We should point out that such magnetic field enhancement never occurs in conventional tapered waveguides. As seen in Fig. 8(a), the output wave forms a focal spot in air with extremely enhanced magnetic field. The transverse size ($\delta _t$) of the spot is only $0.03$ mm, which is far smaller than the (vacuum) wavelength ($\lambda _0=120$ mm). The output efficiency, which is the ratio of output power to input power, is found to be $\eta =87\%$. Evidently, the output efficiency is relevant to the ferrite loss. Figure 8(d) shows the dependence of $\eta$ on $\nu$. When $\nu$ grows from $10^{-4}$ to $10^{-2}$, $\eta$ decreases from $99.1\%$ to $22.6\%$. The dependence of $\delta _t$ on $\nu$ is also shown in Fig. 8(d). For $\nu =0.01$, we find that $\delta _t=0.004$ mm, which is about $10^{-5}\lambda _0$. Therefore, by using a tapered USMP coaxial structure, superfocusing can be realized with extremely enhanced magnetic field at microwave frequencies.

## 5. Conclusion

In summary, we have carefully studied SMPs in a planar and coaxial waveguides formed by ferrite layers with opposite remanences. It has been shown that both waveguides can support USMP at microwave frequencies in the absence of external magnetic field. Different from microwave USMPs previously reported, USMPs in the present waveguides have no low-frequency cutoff, so the CUP bandwidth is at least twice larger than the previous ones. Our numerical simulations have shown that USMPs in the present waveguides are immune to backscattering from imperfections. For the unidirectional coaxial waveguide, it has been shown that the dispersion of USMP is closely dependent on the ferrite-layer thickness. So it is available for USMP to be slowed down in a tapered coaxial waveguide, in which the ferrite-layer thickness is reduced along the propagation direction. Moreover, by using a tapered coaxial waveguide, we have numerically demonstrated that superfocusing can be realized with extremely enhanced magnetic field at microwave frequencies. The proposed unidirectional waveguides may open a new avenue for realizing various broadband microwave devices.

## Funding

National Natural Science Foundation of China (62075197).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## Data availability

No data were generated or analyzed in the presented research.

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