1. Introduction

Symmetry is common in the natural world. For example, the roads of the incident and reflected light or sound are always symmetric. However, such a common characteristic cannot hold all the time in quantum physics. The chiral edge modes observed in the quantum Hall effect exhibit nonreciprocal propagation properties in opposite directions [1]. Similarly, the one-way electromagnetic (EM) modes are permitted to propagate unidirectionally, and they have been investigated and observed in many physical systems such as the magneto-optical (MO) heterostructure [2–8]. The one-way EM modes in the MO-based structures refer to as unidirectional (one-way) surface magnetoplasmons (SMPs) that are sustained at the surfaces of the MO materials. In 2009, the first experimental realization of unidirectional SMPs was achieved in a photonic crystal (PhC) consisting of yttrium-iron-garnet (YIG) rods [9]. Other groups have also explored many interesting phenomena, such as slow light around the exceptional points [10], strong handedness-switchings [11], and broadband field enhancement [12], in the nonreciprocal systems. Recently, the discussion about breaking the time-bandwidth limit by using the MO heterostructure has received much attention. Tsakmakidis et al. claimed that in the MO heterostructure, the time-bandwidth (T-B) limit, which was believed to be a fundamental limit in both engineering and theory, can be overcome due to the broken Lorentz reciprocity [2,13]. Mann et al. challenged such a claim with the coupled-mode theory and argued the causality between the nonreciprocity and breaking the T-B limit [14,15]. Shortly after that, Buddhiraju [16] and Gangaraj [17] et al. proposed that the one-way window of the MO heterostructure may be closed due to the nonlocal effect which is characterized by spatial dispersion. We note here that, in this work we will only consider the local cases because of two reasons: (1) the $k = 0$ modes should not be significantly affected by the nonlocal effect [6,18,19], and (2) the experimental verification of the one-way propagation in the microwave regime guarantees the correctness of the theoretical analysis of the local microwave one-way system [4,9], while the terahertz INZ mode in the nonlocal condition is a novel research topic that we will study in our future work. Besides, our group reported several MO-based one-way nonreciprocal waveguides where many interesting phenomena such as slow wave [20], truly trapping of robust unidirectional rainbow [21,22], and bidirectionally slowed down rainbow trapping and releasing [23] were illustrated. It is inspiring to use the simple powerful MO heterostructure to build novel devices with new functionalities.

Metamaterials (MMs) with fantastic electromagnetic characteristics have been an attractive topic in the past decades. Many different MMs such as negative-index MMs [24,25], hyperbolic metamaterials [26,27], and epsilon-negative MMs [28] were proposed by several research groups. Moreover, the index-near-zero (INZ) MMs, including the epsilon-near-zero (ENZ) MMs with vanishing permittivity [29–32], the $\mu$-near-zero MMs with vanishing permeability [33], double-near-zero (DNZ) MMs with both permittivity and permeability approaching zero [34–36], and the INZ MMs proposed in the PhCs [37,38], is one of the research focus. There are three typical ways to build the ENZ MMs: (1) utilizing the plasma materials such as indium-tin-oxide (ITO) where the ENZ region is always near the plasma frequency [39,40], and (2) building the rectangle waveguide filled with magneto-optical materials while the ENZ region near the cutoff frequency [41], and (3) fabricating layered metal-dielectric configurations, and according to the effective medium theories (EMTs), the real part of the relative permittivity may vanish at discrete frequencies [42]. We note that the group velocities ($v_\mathrm {g}$) in most of the ENZ MMs tend to zero since the imaginary part of the permittivity is negligible as well, leading to forbidden energy transmission [43]. The DNZ MMs were proposed to solve such a problem [34,44,45]. However, both the ENZ and DNZ materials suffer from the narrow operating band and complex manufacturing processes [46]. On the other hand, the INZ materials have wide applications in energy harvesting based on supercoupling effect [39,47], wavefront shaping [34], terahertz lens [32], acoustic MMs or metasurface [48–51], quantum optics [52], switch of total transport, and total reflection [35,53]. We emphasize that even though the experimental realization of the INZ modes has been proposed in all-dielectric systems at Dirac point frequencies [54], it is still of significant interest to dig novel tunable INZ materials with nonzero $v_\mathrm {g}$.

To this end, in what follows, we propose the tunable INZ modes with vanishing wavenumber ($k=0$) in the simple MO heterostructures. The explored INZ modes are similar to the Dirac point modes illustrated in the PhCs since the effective refractive indices of both the INZ and the Dirac point modes are approaching zero. Moreover, based on our theoretical analysis, the working frequencies of such MO INZ modes can be continually adjusted by modifying the thickness parameters of the structure or the external magnetic field. Therefore, compared to the previously reported INZ modes, our proposed adjustable INZ modes in this work are more favorable in many fields, such as the realization of strong nonlinearity, cloaking, wireless communication, and wavefront shaping. As a proof of concept, we further design several microwave and terahertz perfect optical buffers where the EM wave can be slowed down with zero phase shift.

2. Slowed down unidirectional SMPs

In our previous works [20,21,23,55], cutting off the MO material has been proved to be an efficient way to engineer the dispersion of the EM modes (e.g., SMPs) in the nonreciprocal systems. In this paper, we will show how such a technique can be used to achieve the tunable INZ modes. We first study the propagation properties in a metal-dielectric-YIG-metal structure shown in Fig. 1(a). Since the metals can be treated as perfect electric conductors at microwave frequencies, such a configuration is named the ’EDYE’. We note that the dielectric, in this subsection, is set to be air ($\varepsilon _r=1$). An external magnetic field ($H_0$) was applied to the YIG layer in the $-z$ direction, and the permeability of YIG, in this case, has the form [56]

 

Fig. 1. (a) The schematic of the EDYE model. (b) The dispersion diagrams of SMPs in the metal-dielectric-YIG-metal (EDYE) structure as $D=D_1=(0.1\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$ (red line) and $D=D_2=(0.01\lambda _\mathrm {m},0.01\lambda _\mathrm {m})$ (dashed black line). The yellow shaded areas indicate the complete one-way propagation (COWP) band while the gray and cyan shaded regions respectively demonstrate the compressed bulk zones and band gaps in the $D_1$ case. The blue dashed lines are light lines. The other parameters are $\varepsilon _r=1$, $\omega _0=\omega _\mathrm {m}$, $\varepsilon _\mathrm {m}=15$ and $\omega _\mathrm {m}=2\pi \times 5\times 10^9$ rad/s.

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In Fig.1 (b), two different thickness parameters were considered to show the dispersion properties in the EDSE configurations. The red line indicates the dispersion curve of the SMPs as $D=D_1=(0.1\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$ ($\lambda _\mathrm {m}=c/f_\mathrm {m}\approx 60$ mm, c is the speed of light in vacuum), and $\omega _0=\omega _\mathrm {m}$ ($H_0\approx 1785.7$G). It is clear that there is a COWP band (the yellow shaded area) which is limited by two AFs, i.e. $\omega _\mathrm {AF}^-$ and $\omega _\mathrm {AF}^+$. The gray-colored areas represent the bulk zones of YIG, which are much narrower than the ones in the infinite YIG cases [57], and a new band gap (the cyan zone) emerges below the compressed bulk zone 1. As a contrast, the dashed black line demonstrates the dispersion diagram in a ultra-thin EDYE structure with $D=D_2=(0.01\lambda _\mathrm {m},0.01\lambda _\mathrm {m})$. In this case, the bandwidth of the COWP band is analogous to the previous one in the $D_1$ case since the AFs remain the same. We note that the bulk zones 1 and 2 are both narrowed down (not shown in Fig. 1(b)) due to the ultra-thin vertical width [20], and the broadband gap region, which should have potential uses in the design of the optical isolator, is observed as well. The dispersion curve of the SMPs (black line) is much more horizontal than the red one, implying smaller group velocities and slow waves.

To clarify the slow-wave phenomenon occurred in the ultra-thin EDYE structure, we studied two cases of the EDYE structure. In the first case, constant $d_2$ ($=0.15\lambda _\mathrm {m}$) and five different $d_1$ were considered. The corresponding zoomed-in dispersion diagrams around the COWP region are demonstrated in Fig. 2(a), in which we find that when reducing the thickness of the air layer ($d_1$), the dispersion branches with $k<0$ rise up distinctly while most of the dispersion branches in the $k>0$ region locate near $\omega =\omega _\mathrm {AF}^+$. In the second case, we considered the constant $d_1$ ($=0.1\lambda _\mathrm {m}$) and five different values of $d_2$, i.e. $0.15\lambda _\mathrm {m}$, $0.05\lambda _\mathrm {m}$, $0.03\lambda _\mathrm {m}$, $0.02\lambda _\mathrm {m}$ and $0.1\lambda _\mathrm {m}$. The corresponding dispersion diagram is shown in Fig. 2(b) where the dispersion branches with $k>0$ descend whereas the dispersion branches with $k<0$ are nearly the same. It is obvious that the group velocities of SMPs decrease when decreasing the value of $d_1$ (see Fig. 2(a)) or $d_2$ (see Fig. 2(b)).

 

Fig. 2. The relation between the thickness parameter $D=(d_1,d_2)$ and the dispersion curves of SMPs. (a) The dispersion diagrams for five different values of $d_1$ when $d_2=0.15\lambda _\mathrm {m}$. (b) The dispersion diagrams for five different values of $d_2$ when $d_1=0.1\lambda _\mathrm {m}$.

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More clear results are displayed in Fig. 3, in which the corresponding group velocities and wavenumbers of the SMPs are illustrated in the COWP band $1.5\omega _\mathrm {m}<\omega <2\omega _\mathrm {m}$. In Fig. 3(a), we set $d_2=0.15\lambda _\mathrm {m}$, and the red, the cyan, and the blue solid lines represent the values of $v_\mathrm {g}$ in the cases of $d_1=0.1\lambda _\mathrm {m}$, $d_1=0.03\lambda _\mathrm {m}$ and $d_1=0.01\lambda _\mathrm {m}$, respectively. As expected, the solid lines drop down when decreasing the value of $d_1$. Moreover, similar results are shown in Fig. 3(b) for constant $d_1$ cases. On the other hand, it is quite different between constant $d_2$ cases and constant $d_1$ cases in values of k of the unidirectional modes. As shown in Fig. 3, the dashed lines illustrate the changing of k of the SMPs. In the case of $d_1=0.1\lambda _\mathrm {m}$, the wavenumbers of SMPs in the COWP band are constantly greater than zero. On the contrary, for $d_2=0.15\lambda _\mathrm {m}$ case, the wavenumbers can be positive or negative depending on the values of $d_2$ and frequency. Besides, as demonstrated in Fig. 3(a), the EM modes with $k=0$ (marked by points) and $v_\mathrm {g}\neq 0$, i.e. the index-near-zero (INZ) modes, are found in the EDYE structures. Deriving from Figs. 2 and 3, it is believable that the INZ modes can be manipulated in the whole COWP band by engineering the thickness of air ($d_1$). In another word, the EDYE configuration, as discussed above, can support the tunable INZ modes by simply engineering the thickness parameter of the structures. Therefore, due to the slow-wave and tunable characteristics, these simple configurations are promising for designing optical functional devices, and one of its direct uses is designing the perfect optical buffer (POB) with near-zero phase shift. It is also worth noting that varying the external magnetic field, as presented in our previous works, should be another efficient way to engineer the INZ modes [20,22].

 

Fig. 3. The wavenumbers (k) and the group velocities ($v_\mathrm {g}$) of SMPs (a) in Fig. 2(a) and (b) in Fig. 2(b) as functions of $\omega$. The solid lines represent the values of $v_\mathrm {g}$ whereas the dotted lines show the corresponding values of k. The other parameters are the same as in Fig. 1.

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3. Index-near-zero modes and perfect optical buffer

Zero phase shift (ZPS) is an important characteristic in many fields, for example, the near-field radio frequency identification (RFID) applications [58,59]. In this subsection, we will propose the two-dimensional (2D) and three-dimensional (3D) POBs in the microwave regime based on the EDYE configuration. The inset of Fig. 4(a) indicates the simulated magnetic field ($E_z$) in a thick-thin straight EDYE waveguide as $f=1.75f_\mathrm {m}$ and $\nu =10^{-3}\omega$ (the damping coefficient). The thickness parameters for the thick and thin parts are $D=D_1=(0.1\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$ and $D=D_3=(0.02\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$, respectively, and two parts have the same length of $L_1=L_2=60$ mm. We note here that the simulations were performed by utilizing the finite element method (FEM), and the lossy MO materials were considered in all simulations, with the aim of illustrating the loss effects on the nonreciprocity and the INZ characteristics. In the simulations, the EM mode was excited by a line current source at $x=20$ mm in the thick part, and it propagated along the air-YIG interface with the phase angle ($\Phi$) of $E_z$ changing in the $[-\pi,\pi ]$ region. Once the wave reaches and further transmits into the thin EDYE part, the curve (blue solid line in Fig. 4(a)) of $\Phi$ of $E_z$ becomes gentle since $|k|$ of the SMPs in the thick part is larger than the one in the thin part (see points in Fig. 2). It is quite inspiring to manipulate the phase-angle curve and to further make it nearly horizontal because the horizontal curve implies ZPS. In Fig. 4(b), $d_1$ is slightly enlarged, and we set $D=D_4=(0.0216\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$. In this case, $\Phi$ remains almost unchanged in the thin part except for the area near the rightmost end (PEC wall) of the waveguide. Since the EM wave can be slowed down with ZPS in the thin part, the structure demonstrated in the inset of Fig. 4(b) is suitable for the design of the POB. We note here that, in our previous work we have shown that the surface wave located in the light cone can efficiently couple with the fundamental modes of traditional dielectric waveguide [3]. Therefore, the above-mentioned INZ modes are promising in many fields, such as wireless communication and optical cloaking.

 

Fig. 4. The phase angle ($\Phi$) of $E_z$ along the air-YIG interface in the finite element method (FEM) simulations for (a) $D=D_3=(0.02\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$, and (b) $D=D_4=(0.0216\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$. The insets show the electric-field ($E_z$) distributions in the simulations where the structures are surrounded by PEC walls (the black areas). The other parameters are $f=1.75f_\mathrm {m}$, $L_1=L_2=60$ mm and $\nu =10^{-3}\omega$.

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To further explore the POB theory based on the EDYE configuration, the frequencies ($\omega$) of the INZ modes are plotted as a function of $d_1$ for three different $d_2$, and the results are shown in Fig. 5(a). As one can see, the black star is on the red line which indicates the case of $d_2=0.15\lambda _\mathrm {m}$, and the corresponding frequency $\omega =1.75\omega _\mathrm {m}$ while $d_1\approx 0.0216\lambda _\mathrm {m}$. Thus, our calculation fits well with the simulation results illustrated in Fig. 4(b). The green and blue lines indicate the cases of smaller $d_2$, i.e. $d_2=0.02\lambda _\mathrm {m}$ and $d_2=0.01\lambda _\mathrm {m}$. It is obvious that when decreasing $d_2$, the line drops down, which implies the decreasing $\omega$ of the INZ modes. One can derive the same conclusion from Fig. 2(b), in which the dispersion curve drops down when decreasing $d_2$. Notably, the INZ region can be efficiently varied in the COWP band by modifying the thickness parameter while the COWP band can be tuned by changing the external magnetic field [60]. The INZ modes/region mentioned here can be dynamically switched from ’on’ to ’off’ by controlling the external magnetic field. In this work, we just focus on the cases of the stable external magnetic field. Based on the above results, we further designed a complete POB consisting of three straight EDYE parts. As shown in the inset of Fig. 5(b), the values of $d_1$ in the three parts are $d_1=0.1\lambda _\mathrm {m}$, $d_1=0.039\lambda _\mathrm {m}$ and $d_1=0.1\lambda _\mathrm {m}$, respectively. In the simulation, we set the operating frequency ($f$) to be $1.6f_\mathrm {m}$ (red star in Fig. 5(a)). As shown in Fig. 5(b), the launched wave transmits through the second part and recovers in the third part, and the phase angle of the wave is nearly unchanged during the propagation process in the second part. Due to the coupling effect between the modes in the thick and thin parts, a slight phase shift is found at the thick-thin interfaces (see the upper inset of Fig. 5(b)). An efficient way to reduce the coupling effect is to add a tapered EDYE configuration between the thick and thin EDYE parts. Figure 5(c) shows the designed thick-tapered-thin (TTT) EDYE structure. In the simulations of the TTT structure (Fig. 5(c), lower panel), we find that the effective wavelength ($\lambda _\mathrm {ef}$) is gradually lengthened along the $+x$ direction in the tapered part, and $\lambda _\mathrm {ef}$ tends to be near-infinite in the ultra-thin part. More importantly, the phase shift induced by the propagation in the ultra-thin part can be less than $0.3\% \Phi _0$ ($\Phi _0$ is the incident phase angle) when $L_2=3L_1=180$ $\mathrm{\mu}$m.

The 2D POB theory based on the EDYE configuration can also be used to design realistic 3D POBs. The schematic of a designed 3D POB, which is also constructed by three EDYE parts, is shown in Fig. 6, and the direction of the external magnetic field is assumed to be $-z$. According to the analysis of Figs. 2 and 3, both the air and YIG layers of the thin part of the 3D POB are designed to be narrower than the thick parts to efficiently slow the wave, which is different from the above 2D POB. As an example, the thickness parameters in the three parts of the 3D POB are set respectively to be $D=D_6=(0.05\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$, $D=D_7=(0.01452\lambda _\mathrm {m},0.02\lambda _\mathrm {m})$ (marked by the green star in Fig. 5(a)) and $D=D_6$, and we further performed the FEM simulation in such a 3D EDYE structure. As shown in Fig. 7(a) and the inset of Fig. 7(b), similar to the 2D simulations presented in Figs. 4 and 5, the wave unidirectionally propagates from the first thick part and travels through the ultra-thin part and is trapped in the rightmost end of the last part. Figure 7(b) demonstrates the changing of the phase angle of the wave along $y=d_{10}/2$ ($d_{10}$, the air thickness in the thin part) in the XY plane ($z=\mathrm {w}/2$ where w is the lateral thickness of the structure). As expected, the clear ZPS is also observed in the thin part of such a 3D one-way waveguide. In a sense, our designed microwave POB can also work as a type of cloaking device since the INZ wave travels through the structure with ZPS and without reflection.

 

Fig. 5. (a) Theory of the tunable POB based on the EDYE configuration. Solid lines illustrate the numerical solutions of $k=0$ when $d_2=0.15\lambda _\mathrm {m}$ (red line), $d_2=0.02\lambda _\mathrm {m}$ (green line) and $d_2=0.01\lambda _\mathrm {m}$ (blue line), respectively. (b) The phase angle of $E_z$ along the air-YIG interface in a microwave POB for $f=1.6f_\mathrm {m}$. The thickness parameter in the compressing part is $D=D_5=(0.039\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$ (red star in (a)). The inset illustrates the distribution of simulated $E_z$. (c) The schematic and the FEM simulation of a tapered EDYE structure.

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Fig. 6. The schematic of a three-dimensional (3D) POB.

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Fig. 7. (a) Simulated electric-field distribution in a 3D POB where the thickness parameters of the thick and thin parts are respectively $D=D_6=(0.05\lambda _\mathrm {m},0.15\lambda _\mathrm {m})$ and $D=D_7=(0.01452\lambda _\mathrm {m},0.02\lambda _\mathrm {m})$. The inset illustrates the distribution of $E_z$ in XY plane ($z=\mathrm {w}/2$) and the inset of (b) shows the corresponding distribution of $E_z$ in YZ planes (Left: $x=7$ mm. Right: $x=17$ mm.). (b) The phase angle of $E_z$ along the interface of the XY plane ($z=\mathrm {w}/2$) and the XZ plane ($y=d_{10}/2$).

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4. INZ mode-based POB in the terahertz regime

Nowadays, communication band gradually transfers from microwave to terahertz regime due to the increase in demand for enormous information. It is really meaningful to extend the microwave POB theory to the terahertz regime. As we reported in previous works [20,21,23], the COWP band may narrow down or vanish in sub-wavelength one-way terahertz waveguides. Moreover, to our knowledge, the INZ modes have never been reported in the one-way terahertz waveguide sandwiched by metals. In this subsection, we propose two types of terahertz POB utilizing the perfect magnetic conductor (PMC) boundary. As shown in Fig. 8, the leftmost two diagrams indicate the two kinds of one-way terahertz waveguides, and the upper one is the PMC-dielectric-semiconductor-PEC (MDSE) structure, while the lower one is the PMC-dielectric-semiconductor-PMC (MDSM) structure. Here, we note that the dielectric layers applied in the MDSE and MDSM waveguides are air and silicon, respectively, for engineering appropriate COWP bands. We emphasize that the PEC-dielectric-semiconductor-PEC (EDSE) and PEC-dielectric-semiconductor-PMC (EDSM) structures cannot sustain the INZ modes, and thus they were not considered in designing the terahertz POB. The dispersion relation of TM-polarization SMPs in the MDSE and MDSM structures can be derived by solving Maxwell’s equations, and they have the following forms

 

Fig. 8. Realization of the terahertz POB by utilizing PMC walls. The dispersion diagrams of PMC-air-InSb-PEC (MDSE) structures as (a) $D=D_8=(d_1,d_2)=(0.1\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ and (b) $D=D_9=(0.1\lambda _\mathrm {p},0.01\lambda _\mathrm {p})$. (c) and (d) demonstrate the dispersion diagrams of PMC-air-InSb-PMC (MDSM) structures as $D=D_{8}$ and $D=D_{10}=(0.01\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$, respectively. Two leftmost diagrams show the schematics of the MDSE and MDSM configurations. The other parameters are $\varepsilon _\mathrm {d1}=1$, $\varepsilon _\mathrm {d2}=11.68$, $\varepsilon _\infty =15.6$ and $\omega _\mathrm {c}=0.6\omega _\mathrm {p}$.

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Figure 8(a) shows the dispersion curves of SMPs in the MDSE structure as $D=D_8=(0.1\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ ($\lambda _\mathrm {p}=150$ $\mathrm{mu}$m) and $\omega _\mathrm {c}=0.6\omega _\mathrm {p}$. It is clear that there are three discrete COWP bands, and three corresponding AFs are $\omega _{\mathrm {AF}}^{(1)}=0.6\omega _\mathrm {p}$, $\omega _{\mathrm {AF}}^{(2)} \approx 0.71\omega _\mathrm {p}$ and $\omega _{\mathrm {AF}}^{(3)}\approx 1.31\omega _\mathrm {p}$. The bulk zones are significantly compressed in this case, making a remarkable total bandwidth of the COWP bands. Moreover, a terahertz INZ mode (marked by a red point) with $k=0$ is found in the COWP 2 region. When further decreasing the thickness of InSb, as shown in Fig. 8(b) where $D=D_9=(0.1\lambda _\mathrm {p},0.01\lambda _\mathrm {p})$, the bulk zones nearly disappear in the $[0,1.5\omega _\mathrm {p}]$ band. Furthermore, the SMPs sustained at the air-InSb and PEC-InSb interfaces strongly coupled with each other when $\omega \approx \omega _\mathrm {s}$ ($\omega _\mathrm {s}=\sqrt {\omega _\mathrm {c}^2+\omega _\mathrm {p}^2}$), leading to the connecting dispersion curve (the upper red line). Besides, compared to the INZ mode in Fig. 8(a), the one in Fig. 8(b) rise up, which implies the higher working frequency of the one-way INZ mode. We further plotted the dispersion diagrams of the MDSM structures for $D=D_{8}=(0.1\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ (Fig. 8(c)) and $D=D_{10}=(0.01\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ (Fig. 8(d)) as $\omega _\mathrm {c}=0.6\omega _\mathrm {p}$. Four AFs in this condition are $\omega _{\mathrm {AF}}^{(1)} \approx 0.51\omega _\mathrm {p}$, $\omega _{\mathrm {AF}}^{(2)} \approx 0.74\omega _\mathrm {p}$, $\omega _{\mathrm {AF}}^{(3)} \approx 1.11\omega _\mathrm {p}$ and $\omega _{\mathrm {AF}}^{(4)} \approx 1.34\omega _\mathrm {p}$, respectively. As shown in Figs. 8(c) and 8(d), there are two COWP bands in which the EM waves have opposite propagation directions. More interestingly, the INZ modes can be found in all COWP bands once we carefully design the thickness of silicon (Fig. 8(d)). Since the INZ modes in the MDSE and MDSM waveguides have been observed and the INZ mode has been proved to be capable of being manipulated by changing the thickness of the semiconductor or the dielectric, we believe the tunable terahertz POBs can be achieved based on such two configurations.

In order to further demonstrate the relation between the INZ modes and the thickness parameters, we plotted $\omega$ as a function of $d_2$ for the MDSE waveguide with $d_1=0.1\lambda _\mathrm {p}$, as depicted in Fig. 9(a). As expected, the unidirectional INZ modes are found in a broad band within the COWP band. Note that the upper limit of the COWP band in Fig. 9(a) represents the cut-off frequency ($\omega _\mathrm {cf}$) of the lower bulk zone (see Fig. 8(a)). For the MDSM configuration, two kinds of INZ modes, i.e. the higher and the lower INZ modes (see Figs. 8(c,d)), were studied. Consequently, the INZ modes are tunable in almost the entire COWP bands. Based on the above results, we designed three types of terahertz POBs which are named the type-1, type-2, and type-3 POBs, respectively. The schematics of these POBs are exhibited at the top of Fig. 10. The thickness parameters of the thin parts of three POBs are $D=D_{11}=(0.1\lambda _\mathrm {p},0.05596\lambda _\mathrm {p})$ (the vertical line in Fig. 9(a)), $D=D_{12}=(0.0561\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ (the vertical line in Fig. 9(b)) and $D=D_{13}=(0.02945\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ (the vertical line in Fig. 9(c)), respectively. The simulated magnetic field $H_z$ of three thick-thin-thick waveguides are shown in the insets of Fig. 10, and the working frequencies were $f=f_\mathrm {p}$, $f=0.6f_\mathrm {p}$ and $f=1.25f_\mathrm {p}$, respectively. One should note that in the last simulation (Fig. 10(c)), the external magnetic field was reversed (see the top diagram of Fig. 10(c)) to support the forward-propagating (+x) EM wave. The lossy InSb with $\nu =0.001$ were considered in the FEM simulations to study the loss effect on the one-way INZ modes. The propagation properties of the launched EM waves in the simulations are similar to those in the microwave simulations (e.g. Figure 5(b)) demonstrated in the last subsections. Moreover, the phase angles remain the same in the thin parts of the three structures, which are in agreement with the calculation results illustrated in Fig. 9. Therefore, we conclude that the loss in the terahertz POBs will not significantly affect the propagation properties of the EM modes such as the INZ modes, and a similar result has been demonstrated in the INZ MMs with cylindrical defects [35]. Finally, according to our simulations, the transmission efficiencies of the INZ modes in the microwave/terahertz POBs can be more than $85\%$. This high-performance characteristic is due to the ’topologically’ unidirectional propagation properties in our design POBs [13], whereas the supercoupling transmission suggested in other ENZ/INZ structures always require ultra-small electrical scale(s) [29,47].

 

Fig. 9. (a) The frequencies of the INZ modes in the MDSE waveguides as a function of $d_2$ as $d_1=0.1\lambda _\mathrm {p}$. The frequencies of (b) the lower INZ modes and (c) the upper INZ modes in the MDSM waveguides as functions of $d_1$ as $d_2=0.1\lambda _\mathrm {p}$. The other parameters are the same as in Fig. 8.

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Fig. 10. The phase angle of $H_z$ along the dielectric-InSb interface in (a) MDSE-based and (b,c) MDSM-based terahertz POBs. The thickness parameters of the thin parts of the POBs are respectively (a) $D=D_{11}=(0.1\lambda _\mathrm {p},0.05596\lambda _\mathrm {p})$, (b) $D=D_{12}=(0.0561\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$ and (c) $D=D_{13}=(0.02945\lambda _\mathrm {p},0.1\lambda _\mathrm {p})$. The operating frequencies in the simulations are $f=f_\mathrm {p}$, $f=0.6f_\mathrm {p}$ and $f=1.25f_\mathrm {p}$ in sequence. The InSb is considered to be lossy in the simulations with electron scattering frequency $\nu =0.001$.

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We emphasize here that the INZ mode in this work is related to the single unidirectional mode, implying the single-zero effective permeability in the microwave MO systems and the single-zero effective permittivity in the terahertz MO systems, which is different from the INZ mode in the DNZ media based on PhCs with Dirac-like cones that are 3-fold degeneracies at the Dirac points [61–63]. Moreover, the INZ modes in the DNZ systems always suffer from the half-wavelength scale and the opposite-sign group velocities at the Dirac point frequency, whereas our proposed MO INZ modes can work even in the deep-subwavelength system due to the robust unidirectionality, which may pave the route to ultra-compact optical communication. Furthermore, the MO INZ mode, because of the tunability, is with significant advantages compared to the one in the PhCs. On the other hand, one should also note that similar single-zero characteristics, e.g. near-zero effective permittivity and non-zero effective permeability, have been proposed in some works by using complicated techniques/devices such as photonic doping [64], metamaterial [65], and strong-coupling system [41]. Meanwhile, because of the reciprocity, there are at least two group velocities in the ENZ region of these systems. On the contrary, the MO INZ modes in this work are achieved relied on simple waveguide systems, and only a single group velocity is found at the specific INZ frequency.

5. Conclusion

We have studied the tunable index-near-zero (INZ) modes in simple MO heterostructures. As an example of the potential applications of the INZ modes, we have further proposed the low-loss microwave POBs and terahertz POBs. In the microwave regime, we have demonstrated that by carefully modifying the thickness ($d_1$) of the dielectric layer, the INZ modes can be tuned in a broad band (i.e. the COWP band). The group velocities $v_\mathrm {g}$ of the INZ modes were found to be nonzero, and $v_\mathrm {g}$ decreased when decreasing $d_1$, indicating the slowed wave in the thin MO structures. The slow-wave and tunable characteristics of the microwave INZ modes have been confirmed by the numerical simulations of the POBs.

In addition, we have extended our research to the terahertz regime. By using the PMC wall(s), three types of terahertz INZ mode-based POBs have been proposed based on the MDSE and the MDSM configurations. Moreover, the terahertz INZ mode, according to our analysis, is different from the microwave INZ mode. More specifically, the terahertz INZ mode can be engineered by changing the thickness of the semiconductor in the MDSE structure or by changing the thickness of the dielectric in the MDSM structure. We have also proposed a promising way to turn ’on’ or ’off’ the INZ region by easily varying the external magnetic field, making our MO materials-based INZ modes highly maneuverable in practice. Owing to the characteristics of one-way propagation, ZPS, and tunability, the microwave and terahertz INZ modes in the simple MO heterostructures have great potential for high-performance and directional wireless communication, cloaking, holographic imaging, and other optical functional devices in integrated optical communication besides POBs.