## Abstract

Photonic crystal (PhC) consisting of a square lattice of ferrite rods with remanence in the air is theoretically investigated. It is shown that such a PhC can possess a large band gap with a nontrivial topological property. By interfacing the PhC with different cladding materials, which have (bulk-mode) band gaps with different topological properties, different types of one-way modes that are immune to backscattering are achieved in the absence of an external magnetic field. It is numerically demonstrated that by using such one-way modes, waves can be flexibly and effectively manipulated in PhC systems without almost any power loss.

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

## 1. Introduction

One-way 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 can be immune to backscattering. With such unusual properties, one-way EM modes have important potential applications in classical and quantum information processing. One-way EM modes were first proposed by Haldane and Raghu as analogues of quantum Hall edge states in photonic crystals (PhCs) [1–4]. It was predicted that one-way modes can be confined at the edges of certain two-dimensional (2D) PhCs made of magneto-optical (MO) materials, where the time-reverse (${T}$) symmetry is broken by applying a dc magnetic field. These edge modes possess group velocities pointing in only one direction, determined by the direction of the external magnetic field [5,6]. The existence of one-way (EM) modes was first experimentally verified by Wang and his colleagues using MO PhCs in the microwave regime [7], since then one-way modes have received increasing attention. Different type of one-way EM modes was also proposed in the form of surface magnetoplasmons (SMPs) [8,9]. Due to their simple configurations, one-way SMPs have been intensively researched recently [10–14]. More recently, however, it was reported that one-way SMPs may vanish or become leaky in the presence of material nonlocality [15,16]. Compared to one-way SMPs, one-way PhC modes seem more attractive due to the robust mechanism. In addition, the characteristics of one-way PhC modes, including operation frequencies, can be flexibly tailored by adjusting the geometric parameters of PhCs.

The exsitence of one-way PhC modes relies on photonic band gaps that possess nontrivial topological properties [5,17]. Such band gaps are created by breaking ${T}$ symmetry in PhCs, and this can be achieved by applying external magnetic field. But the use of external magnetic field causes difficulties for one-way modes in their practical applications. To overcome this problem, some topologically protected states, which are unidirectionally propagating in the systems without external magnetic field, were proposed, but they often involve complicated field polarizations [18–20]. In the microwave domain, however, there may be a simple way to solve the above problem. Ferrites, which are commonly used as MO materials in microwave domain, can preserve magnetization after removing external magnetic field [21]. So ferrites with remanence possess gyromagnetic feature in the absence of external magnetic field. It is interesting if one-way modes can be realized using PhCs made of ferrites with remanence, but no research is reported about it so for.

In this paper, we will investigate a PhC formed by a square array of ferrite rods with remanence in air. We will show that such PhC can have a large band gap with nontrivial topological property. Using this PhC, two types of interface waveguide are constructed by using different cladding materials, and the changes of gap Chern number across the interfaces in them are different. We will carefully investigate the guiding characteristics of the two types of waveguides. These waveguides also provide new examples to examine if Hatsugai’s relation between edge states and Chern numbers is applied to photonic systems [22]. Furthermore, we will demonstrate that in PhC systems one-way modes can be used to manipulate waves flexibly and variously almost without any power loss.

## 2. Photonic crystal with remanence

We consider a photonic crystal (PhC) consisting of a square lattice of ferrite rods with radius $b$ in air, as illustrated in the inset of Fig. 1(a), where the lattice constant is denoted by $a$. In this PhC, the ferrite rods with relative permittivity ${\varepsilon _m}$ possess remanent magnetization ($\boldsymbol{M}_r$) in the axial direction (i.e., the -z direction). Due to the gyromagnetic anisotropy induced by the remanence, the relative permeability of the ferrite rods is frequency-dependent and takes the tensor form [21]

We investigate the band structure of the proposed PhC for the TM polarization. For simplicity, we neglect the effects of material dispersion and loss, assuming a frequency-independent permeability tensor (with $\nu =0$), because only phenomenon in operating frequency is of our interest. We will show later that our result will not be substantially affected by these effects. According to the Bloch’s theorem, the nonzero components of EM fields in the PhC can be expanded in a series of plane waves

Substitution of the expressions (2)–(4), (7) and (8) into Eqs. (5) and (6) yields

The above equations can be expressed in matrix form

where the vectors $[{\mathrm U}]$ and $[V_p]$ ($p=x,y$) are defined as $[{\mathrm U}]_{\boldsymbol{G}}=u_{\boldsymbol{G}}$ and $[V_p]_{\boldsymbol{G}}=v_{p,\boldsymbol{G}}$. The matrices $K_x$, $K_y$, $A$, and $B$ are defined as follows: $(K_x)_{\boldsymbol{G}^\prime,\boldsymbol{G}}=(G_x+k_x)\delta _{\boldsymbol{G}^\prime,\boldsymbol{G}}$, $(K_y)_{\boldsymbol{G}^\prime,\boldsymbol{G}}=(G_y+k_y)\delta _{\boldsymbol{G}^\prime,\boldsymbol{G}}$, where $\delta _{\boldsymbol{G}^\prime,\boldsymbol{G}}=1$ for $\boldsymbol{G}^\prime = \boldsymbol{G}$ and $\delta _{\boldsymbol{G}^\prime,\boldsymbol{G}}=0$ for $\boldsymbol{G}^\prime \ne \boldsymbol{G}$; $A_{\boldsymbol{G}^\prime,\boldsymbol{G}}=\alpha _{\boldsymbol{G}-\boldsymbol{G}^\prime }$, and $B_{\boldsymbol{G}^\prime,\boldsymbol{G}}=\beta _{\boldsymbol{G}-\boldsymbol{G}^\prime }$. By eliminating $[V_x]$ and $[V_x]$ from Eqs. (12)–(14), we obtain withNote that ${\cal H}$ is a function of $\boldsymbol{k}$. With Eq. (15), we can numerically solve for Bloch modes as an eigenvalue problem for the PhC with remanence. This formulation of the plane-wave expansion method also applies to the case when the PhC is unmagnetized. In this case, $B$ vanishes and the matrix in Eq. (15) reduces to ${\cal H}=A^{-1}(K_x^2+K_y^2)$. In this paper, the parameters of ferrite rods are taken as follows: ${\varepsilon _m}=15$, ${\omega _r}=7.174\pi \times {10^9}$ rad/s [21,23], and $b=0.132a$. In the lossless case, the tensor $\bar {\bar {\mu }}_m$ at the frequency $3.95$ GHz has the values of $\mu =1$ and $\kappa =0.91$.

We start with the band structure for a relevant PhC with ${T}$ symmetry, in which the ferrite rods of radius $0.132a$ are unmagnetized, i.e., $\bar {\bar {\mu }}_m=I$. The results are displayed in Fig. 1(a). This band structure contains a large band gap between the first and second bands. Higher bands are degenerate with a neighbouring band at discrete $k$ points, e.g., the second and third bands have a $M$-point degeneracy. Here, we are interested in band gaps created by ${T}$ breaking. Figure 1(b) shows the band structure for the PhC with remanence. The $M$-point degeneracy in Fig. 1(a) is lifted by applying ferrite remanence, then a new band gap opens between the second and third bands. The band gap ranges from $0.586(2\pi c /a)$ to $0.687(2\pi c /a)$, and it has a relative bandwidth of $16\%$. This band gap only changes a little when the material dispersion is taken into account, and the corrected relative bandwidth is $15\%$, as shown in Fig. 1(d). In the PhC with remanence, the band gap between the first and second bands is preserved. To clarify the topological properties of band gaps, we calculate the Chern numbers of the Bloch bands in the PhC with remanence. Note that for the ${T}$-symmetric PhC the Chern number of every band is zero. The Chern number of the nth photonic band in the PhC is given by [5,24–26]

The existence of TP band gap in a PhC relies critically on ${T}$ breaking in it. For PhCs with gyromagnetic materials described by Eq. (1), the strength of ${T}$ breaking equals $\kappa$ (assuming $\nu =0$), and it is nearly unity for our example in Fig. 1(b). Very different from conventional PhC band gaps, a TP band gap possesses nonzero Chern numbers in its both upper and lower bands. One of the key properties of the Chern number is that it is always an integer. Thus, there must exist a critical point of $\kappa$, where the Chern number of a photonic band changes abruptly. The evolution of band topology with $\kappa$ is of interest. To clarify this, we calculate the Chern number ($C_2$) of the second band in the PhCs for various $\kappa$ values, and the results are displayed in Fig. 1(c). Note that if the band gap between the second and third bands occurs, the gap Chern number just equals $C_2$. The critical $\kappa$ point is observed, and it is at nearly $10^{-6}$, which is far smaller than unity. So the PhC we consider exhibits strong ${T}$ breaking due to the use of ferrite material with remanence.

## 3. Robust one-way modes in PhCs

One-way EM mode may emerge when the proposed PhC with a TP band gap is interfaced with a cladding material, which has a bulk-mode band gap with different topological property. This can be achieved by using a metal slab or a same PhC but with opposite remanence, and here two types of interface waveguides are considered. The type-I waveguide is shown in Fig. 2(a), where the cladding material is a metal slab, separated by a distance of $(h-b)$ from the outermost row of ferrite rods. This waveguide may support single one-way mode, as the metal slab is topologically trivial (with zero gap Chern number). The type-II waveguide is shown in Fig. 2(b), which is formed by two PhCs with opposite remanences, separated by a distance of $2(h-b)$ from each other. This waveguide would support two one-way modes, if Hatsugai’s relation between edge states and Chern numbers is applied to the photonic system.

By using the finite element method (FEM), we numerically solved for modes in the type-I waveguide through solving Maxwell’s equations in the super cell of the system. The super cell is terminated by scattering boundary condition from below, and the periodic boundary condition (characterized by the Bloch wavevector $k_x$) is applied in the x direction. In the calculation, the metal slab is assumed to be perfect electrical conductor (PEC), which is a good approximation for the microwave regime. Our numerical calculations show that there exists only a single mode in the type-I waveguide when $h\le a$, as shown in Fig. 2(c), where the dispersion relations for various $h$ values are displayed. The mode (represented by solid line) always possesses a positive group velocity, implying that it is one-way mode propagating forward only. The dispersion curve of one-way mode is affected by $h$, but it is always almost linear around the midgap frequency, which means a low group dispersion there. The influence of material dispersion on one-way mode is also analyzed, and it is very weak. For instance, the dispersion relation (circles) of one-way mode with material dispersion almost overlaps with the one (solid line) without material dispersion for $h=0.6a$ [see Fig. 2(c)]. In the type-I waveguide, high-order modes may occur when $h>a$. These modes are allowed to propagate forward and backward, as shown in Fig. 2(c), where the dispersion bands for the waveguide with $h=1.2a$ are also included, and the dotted line corresponds to the high-order mode. Evidently, high-order mode is a regular mode, which is guided based on the total internal reflection at the boundaries of the PhC and the PEC.

The modes in the type-II waveguide can be numerically solved for similarly to that described above. Our numerical calculations show that in the type-II waveguide there really exist two one-way modes with positive group velocities. One mode (even mode) has electric field distribution symmetric about the waveguide axis (i.e., the central line of the system along the x direction), and the other (odd mode) has electric field distribution antisymmetric about the waveguide axis. For the odd mode, the electric field vanishes at the waveguide axis, and if a PEC plate is inserted there, the field pattern below it would not change. So the odd mode is physically identical to the one-way mode in the type-I waveguide with the same $h$ value. Figure 2(d) shows the dispersion bands for the even mode for different $h$ values. For band completeness, the wavevector ($k_x$) zone is shifted to the interval $[0,2\pi /a]$. As seen in Fig. 2(d), the even mode always has very low dispersion around the midgap frequency, like the one-way odd mode [or the one-way mode in the type-I waveguide, see Fig. 2(c)]. As in the type-I waveguide, one-way modes in the type-II waveguide are weakly affected by the material dispersion, as shown in Fig. 2(d), where the dispersion relation of the one-way even mode corrected with the material dispersion for $h=0.6a$ is plotted as circles for comparison. In the type-II waveguide, high-order even mode begins to appear in the TP band gap when $h > 0.6a$. The higher-order mode is a regular mode, allowed to propagate forward and backward.

It is clear that both types of waveguides can exhibit complete one-way propagation (COWP) when high-order modes are avoided. In this case, one-way mode would possess the unusual property of the complete suppression of backscattering. For the type-I waveguide with $h\le a$, the COWP band covers the whole TP band gap. This is also the situation for the type-II waveguide with $h\le 0.6a$. To verify the COWP properties of both waveguides, we performed the simulations of wave transmission in them with the FEM. In all following simulations, the $h$ parameter is set to be $0.6a$ for both types of waveguides, and the ferrite loss of $\nu =10^{-3}\omega _c$ is taken into account, where $\omega _c=0.636(2\pi c /a)$ being the midgap frequency. Figure 3(a) shows the real parts of simulated electric-field amplitudes in the type-I waveguide. In the simulations, a linear electric current source is used to excite wave at the midgap frequency. As seen in Fig. 3(a), the excited wave only propagates forward. The one-way propagation is robust against defect, as shown in Fig. 3(b), where a PEC slab of width $6a$ and thickness $0.2a$ is inserted into the waveguide. In this case, the wave completely circumvents the defect and continues to travel forward.

The same phenomenon is also observed in the type-II waveguide, as shown in Fig. 4. In Figs. 4(a)-(c), a linear electric current source is placed at the waveguide axis, so only one-way even mode is excited. Though the defect cannot block propagating wave, it can destroy the spatial symmetry of the fields when the defect is located asymmetrically about the waveguide axis, as shown in Fig. 4(c). In this case, the one-way even mode is partly converted into the one-way odd mode by the defect, and the mixture of two modes causes the field pattern to be asymmetric behind the defect. In Figs. 4(d)-(f), a pair of linear electric currents with $180^\circ$ phase difference between them are symmetrically placed on the two sides of the waveguide axis, thus only one-way odd mode is excited. The observed phenomena are similar to those in Figs. 4(a)-(c). Finally, we point out that the two proposed 2D waveguides can be easily mapped into equivalent three-dimensional (3D) waveguides. In the 2D waveguides, the EM fields of a mode extend uniformly along the z direction, and the electric field just points in this direction. Suppose that the 2D waveguides are truncated in the z direction by using a pair of PEC slabs, which are separated by a distance $w$. In the TP band gap, when $w$ is smaller than a certain value, the 3D waveguides can only support modes with fields uniform in the z direction, so the modal properties are physically identical to those in the 2D waveguides.

## 4. Wave manipulations without any power loss

The one-way modes can be immune to backscattering, and this offers flexible and diverse manners to manipulate waves in PhC systems. For example, by using a simple PEC slab, we can completely convert an even mode (with symmetric electric field) to an odd mode (with antisymmetric electric field) and vice versa, as illustrated in Figs. 5(a) and 5(b). In conventional waveguides, the mode conversion is often achieved by using a grating, which produces a weak coupling between two modes under the phase-matching condition, and a coupling length of at least several wavelengths is needed. In contrast, the mode conversion in Fig. 5 is rapidly finished over a length smaller than the wavelength, and the mechanism is completely different from the conventional one. As seen in Fig. 5(a), when the even mode, which is excited by the source and travels forward only, meets the PEC slab, it is equally split into two waves, which correspond to the one-way mode of type-I waveguide. As the PEC slab is properly placed and asymmetrical about the (type-II) waveguide axis, the two waves acquire a phase difference of $180^{\circ }$ between them after they go around the PEC slab from the upper and lower ends, respectively. So the odd mode is produced by the antisymmetric interference field behind the PEC slab. In Fig. 5(b), the odd mode excited by the source is split into two waves of $180^{\circ }$ phase difference by the same PEC slab, so the interference field behind the PEC slab is symmetric about the waveguide axis, thus the even mode is produced this time. Note that these mode conversions have no power loss if the materials in the PhC system are lossless. In the loss case of $\nu =10^{-3}\omega _c$, it is found that the power of the even (odd) mode converted from the odd (even) mode is only reduced by nearly $2.8\%$, due to the absorption loss in the ferrite rods around the PEC slab. Evidently, the influence of the material loss on the mode conversion is very weak.

Using one-way modes, one can well solve the problem of transmission efficiency for power splitters. For conventional power splitters, the transmission efficiency closely depends on the matching between the modes in the junction region and modes in the input and output channels. The junction part of a splitter with high efficiency is often intricate and fine, and the useful bandwidth is very small. The contradiction between high efficiency and broad band in splitters may be resolved by using one-way modes. For example, using one-way modes, we can easily construct a broadband equal-power splitter with high efficiency. To show this, we consider a T-branch PhC splitter [see Fig. 6(b)] and simulate wave transmission in it. In this splitter, the input channel is the type-II one-way waveguide, and two output channels are formed by the interfaces of a PhC with remanence and a regular PhC. The regular PhC consists of a square lattice of alumina rods (with radius $0.082a$ and relative permittivity $\epsilon _r=10$) in air, and it is tilted $45^{\circ }$ to match the lattices of the PhCs with remanence. Both output channels support a single one-way mode, and the modes in them are physically identical but propagate in the opposite directions. Figure 6(a) shows the dispersion relation of the one-way mode in the upper output channel. Figure 6(b) shows the simulated electric field amplitudes for the midgap frequency. The wave power in the input channel is equally split and completely transmitted into the two output channels. The transmission efficiency is almost $100\%$ for $\nu =10^{-3}\omega _c$. Obviously, such equal-power splitting can be attained throughout the whole TP band gap, which has a relative bandwidth of $16\%$.

For conventional T-branch PhC splitters, power splitting ratio may be tunable by modifying the junction region. This also works for the above splitter constructed with one-way modes. But for the present splitter there are more flexible ways to effectively tune the splitting ratio. For example, we can tune the splitting ratio by using a PEC slab inserted into the input waveguide, as shown in Fig. 7. When the center of the PEC slab is not located at the waveguide axis, i.e., $y_c \ne 0$, the incident even mode is partly converted into the odd mode (almost without any power loss), thus the mixture of even and odd modes in the input channel causes an asymmetric field pattern in the junction region. Consequently, the modal amplitudes generated in the two output channels are not equal. We simulated wave transmission in the splitters for various $y_c$ values, while the operation frequency was kept at $\omega _c$. For each case, the transmission coefficient $\eta _1$ ($\eta _2$) was calculated, which is the ratio of the power in the upper (lower) output channel to that in the input channel, and we find that $\eta _1+\eta _2 \approx 1$ for $\nu =10^{-3}\omega _c$. Figure 7(a) shows simulated electric field amplitudes for $y_c=0.2a$. Obviously, the modal amplitude in the upper output channel is smaller than one in the lower output channel, and the splitting ratio ($\eta _1/\eta _2$) is found to be $1:2$. Figure 7(b) shows the simulated results for $y_c=0.1a$. In this case, the incident even mode is completely converted into the odd mode by the PEC slab, so the equal-power splitting is observed. Figure 7(c) shows the results for $y_c=0.24a$. This case the incident power is totally transmitted into the lower output channel. The transmission coefficients as a function of $y_c$ are plotted in Fig. 7(d), where the three cases of Figs. 7(a)–7(c) are marked by A, B, and C, respectively. Interestingly, using this simple strategy, the splitting ratio can be tuned from zero to infinity.

## 5. Conclusion

In summary, we have shown that a photonic crystal made of ferrite with remanence can possess a large band gap with nontrivial topological property. Using such a photonic crystal, two types of one-way waveguides can be constructed without external magnetic field, which support a single and two one-way modes, respectively. Our simulations have shown that the one-way modes in the both waveguides can be immune to backscattering. Based on the unusual properties of one-way modes, waves can be manipulated flexibly and effectively in photonic crystal systems. We have numerically demonstrated that mode conversion can be rapidly achieved almost without any power loss, and equal-power T-branch splitter with nearly $100\%$ efficiency is capable to have a broad band. Moreover, new manner for tuning the splitting ratio of a T-branch splitter has been demonstrated. The proposed one-way modes may open a new avenue for realizing various compact microwave devices with high efficiencies.

## Funding

National Natural Science Foundation of China (62075197, 62101496).

## 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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