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Robust topological one-way edge states in radius-fluctuated photonic Chern topological insulators

Jianfeng Chen, Qiumeng Qin, Chaoqun Peng, and Zhi-Yuan Li

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Abstract

Recent developments in topological photonics have shown that the introduction of disorders can yield the innovative and striking transport phenomena. Here, we theoretically investigate topological one-way edge states in radius-fluctuated photonic Chern topological insulators (PCTIs), which are composed of two-dimensional gyromagnetic photonic crystals with cylinder site fixed but with cylinder radius fluctuated. We use a fluctuation index to characterize the degree of radius fluctuation, employ two empirical parameters to inspect the evolution of topological one-way edge states, and verify the stability of topological one-way edge states by calculating massive samples with various random numbers. We find that as the radius-fluctuation strength increases, there arises a competition between topological one-way edge state, Anderson localization state and trivial bulk state. We reveal that the Anderson localization state appears far more easily in the radius-fluctuation PCTI with even a weak strength compared with the position-perturbed PCTI with a strong randomness. We also demonstrate that the topological one-way edge states are protected against a strong fluctuation much larger than the fabrication errors in practical experiments. Our results show that the PCTIs consisting of gyromagnetic photonic crystals have a high-tolerance for the material and sample fabrication errors, and this would provide a deeper understanding of fundamental topology physics.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Topological insulators have attracted increasing attention and triggered numerous rapidly developing research fields, because of their intriguing aspects in both fundamental physics and potential applications [1–3]. A key feature of topological insulators is the existence of topologically protected bandgap and edge states or surface states that are highly robust against defects, disorders and even obstacles on the transport path. One striking example of topological insulators is the one-way transport quantum Hall system (or Chern topological insulator), in which the external magnetic field breaks the time-reversal symmetry and makes the emergence of nonzero topological band invariants (or Chern numbers) [4]. The counterparts in classical wave systems have also been reported in both theory and experiment [5–16], particularly in the realm of photonics [5–14].

To date, photonic Chern topological insulators (PCTIs) have been found to be useful not only in demonstrating some fascinating topological effects that are difficult to reach in condensed matter physics, such as topological one-way fibers [17,18], topological Anderson localizations [19,20], topological antichiral one-way edge states [21,22], but also in providing potential applications in photonic integrated circuit and topological quantum computing [23–31]. So far, plenty of PCTIs have focused on periodic and quasiperiodic systems, such as photonic crystals [32–35] and quasicrystals [36–38]. One typical path towards the realization of PCTIs is to immerse a two-dimensional (2D) gyromagnetic photonic crystal (GPC) of a square lattice in an external direct current magnetic field [7,8], as shown in Fig. 1(a).

 figure: Fig. 1.

Fig. 1. Sketch maps of photonic Chern topological insulator. (a) Photonic Chern topological insulator of a square lattice. (b) Position-perturbed photonic Chern topological insulator. (c) Radius-fluctuated photonic Chern topological insulator.

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However, nonuniformities unavoidably occur in the fabrication of materials, samples and devices of 2D GPCs. Among them are material imperfections of crystal “atom” (gyromagnetic cylinders), deviation of “atom” position, fluctuation of “atom” radius, configuration nonuniformity of “atom,” nonuniform magnetization of “atom,” and so on. It is then interesting to in examine the influence of these disorders or fluctuations on the transport properties of electromagnetic (EM) waves in GPCs. Mansha et al. pointed that amorphous gyromagnetic photonic lattices also support topological one-way edge states [39], and subsequently, Zhou et al. demonstrated that topological one-way edge states can persist into the amorphous regime prior to the glass-to-liquid transition [40]. Besides, Yang et al. proposed that topological one-way edge states can exist in position-disordered GPCs although gyromagnetic cylinders strongly deviate from their periodic lattice sites [41]. On the other hand, some studies revealed that counterintuitively, adding disorders can turn a photonic topological trivial insulator into a nontrivial insulator called the photonic topological Anderson insulator [19,20]. These theoretical and experimental results have drawn enormous interests in the role of disorder in GPCs [42–45].

Nevertheless, the vast majority of disordered GPCs have focused on the position-perturbed GPCs [36–38], as illustrated in Fig. 1(b). Yet, previous works in dielectric photonic crystals have shown that usually photonic bandgaps are far more sensitive to disorders with a radius randomness than with a position randomness [46,47]. So, it is interesting to have a look at the influence of radius-fluctuation randomness upon topological one-way edge states. Besides, in addition to the machining errors of gyromagnetic cylinders, other types of perturbations, such as the material imperfections of gyromagnetic cylinders, configuration nonuniformity of gyromagnetic cylinders and nonuniform magnetizations of gyromagnetic cylinders, might also exist and they can still be approximately described as being equivalent to the radius fluctuation. Then it is also of technological means to see the influence of radius-fluctuation randomness on practical topological photonic devices.

For these reasons, in this paper we make a systematical theoretical investigation on the radius-fluctuated PCTIs composed of gyromagnetic cylinders immersed in an air background. The positions of gyromagnetic cylinders are fixed and arrayed in a square lattice, while their radii are fluctuated, as shown in Fig. 1(c). We use a fluctuation index (κ) related to fluctuation strength to characterize the degree of radius fluctuation, employ two well-defined empirical parameters (i.e., the quality of edge localization Cs and the one-way property of edge propagation Rs) to inspect the evolution of topological one-way edge states, and verify the stability of topological one-way edge states by calculating massive samples with various random numbers (ξ). We also demonstrate several distinguished transport behaviors of EM waves in PCTIs at different fluctuation strengths by exhibiting the electric field intensity distributions and energy flux (Poynting vector) distributions. Our results may provide a deeper understanding of disordered topological photonics, and will also be of significance in exploring and designing practical topological photonic devices.

2. Photonic Chern topological insulator of a square lattice

We first consider a type of 2D PCTI consisting of a square lattice GPC, which has been proposed and implemented [7,8]. The array of gyromagnetic cylinders is immersed in air, and a metallic cladding is used near the upper edge of GPC to form a transport waveguide, as shown in Fig. 2(a). The relative permittivity and permeability of the air are ε1 = 1 and μ1 = 1, respectively. The lattice constant is a = 3.87 cm, the radius of gyromagnetic cylinder is r = 0.11a, and the gyromagnetic cylinders are made from yttrium-ion-garnet (YIG). The relative permittivity and permeability of these YIG cylinders are ε2 = 15.26 and μ1 = 1 respectively in the absence of external magnetic field. When a direct current magnetic field is applied along the out-of-plane (+z) direction, there induces a strong gyromagnetic anisotropy in YIG cylinders so that their permeability becomes a tensor as follows

$$\hat{\mu }\textrm{ = }\left( {\begin{array}{ccc} {{\mu_r}}&{i{\mu_k}}&0\\ { - i{\mu_k}}&{{\mu_r}}&0\\ 0&0&1 \end{array}} \right), $$
where ${\mu _r} = 1 + \frac{{{\omega _m}{\omega _0}}}{{\omega _0^2 - {\omega ^2}}}$, ${\mu _k} = \frac{{\omega {\omega _m}}}{{\omega _0^2 - {\omega ^2}}}$, with ${\omega _0} = 2\pi \gamma {H_0}$ being the resonance frequency, γ=2.8 MHz/Oe being the gyromagnetic ratio, and ${\omega _0} = 2\pi \gamma {M_0}$ being the characteristic circular frequency, where M0 = 1780 Gauss is the saturation magnetization. The intensity of applied external magnetic field is H0 = 1800 Gauss, and the damping coefficient of YIG is ignored (α=0). All the simulations are performed by using the commercial software COMSOL MULTIPHYSICS with RF module in frequency domain, and only the E polarization state (where the electric field E is parallel to the z-axis direction) is considered.

 figure: Fig. 2.

Fig. 2. Photonic Chern topological insulator composed of a gyromagnetic photonic crystal of a square lattice. (a) Schematic diagram. (b) Projected band structure. (c) Straight-line transport. (d) 90°-turn transport. (e) Immune-obstacle transport. The yellow and gray rectangles indicate the perfect electric conductors and the scattering boundary conditions respectively in simulations. The yellow stars represent the point sources oscillated at a frequency of fs = 4.50 GHz, and the purple arrows point to the transport directions of energy fluxes.

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It is known that the existence of topological one-way edge states in a 2D periodic photonic crystal [see Fig. 2(a)] can be predicted by topological invariants (or Chern numbers), and the Chern number of band n is an integer defined by

$${C_n} = \frac{1}{{2\pi i}}\int\limits_{BZ} {{\textrm{d}^2}k\left( {\frac{{\partial A_y^{nn}}}{{\partial {k_x}}} - \frac{{\partial A_x^{nn}}}{{\partial {k_y}}}} \right)}, $$
where the k-space integral is performed over the first Brillouin zone and the Berry connection is given by
$${{\bf A}^{nn^{\prime}}}({\bf k} )\equiv i\left\langle {{{\bf E}_{n{\bf k}}}|{{\nabla_{\bf k}}} |{{\bf E}_{n^{\prime}{\bf k}}}} \right\rangle = i\int {{\textrm{d}^2}r\varepsilon ({\bf r} ){\bf E}_{n{\bf k}}^ \ast ({\bf r} )\cdot |{{\nabla_{\bf k}}{{\bf E}_{n^{\prime}{\bf k}}}({\bf r} )} |}, $$
where Enk is the periodic part of the electric-field Bloch function, the asterisk denotes complex conjugation, and ε(r) denotes the dielectric function. Our numerical calculation results show that the Chern numbers of the first, second and third bands are 0, –1 and +2 respectively, and the Chern numbers of the second and third bands have been added in the projected band structure, as illustrated in Fig. 2(b). As the gap Chern number Cgap=∑Cn can be calculated by summing the Chern numbers of all bands below the bandgap, the second bandgap (between the second and third bands) possesses a Cgap of –1 and thus is topologically nontrivial. On the other hand, for a line-defect waveguide, the number of topological one-way edge state is determined by ΔCgap of two composite GPC structures, and the sign of ΔCgap determines the transport direction of the topological one-way edge state. It can be seen that the waveguide channel between GPC (Cgap=–1) and metal plate (Cgap = 0) holds ΔCgap=+1, and thus it can support a topological one-way edge state propagating along the + x direction.

Figure 2(b) shows the projected band structure of the magnetized GPC, which is calculated by adopting a supercell consisting of eight YIG cylinders in one column. There appears a dispersion curve (blue) residing in the bandgap and connecting the second and third bands. The sign of its slope is always positive, so the GPC supports a topological one-way edge state propagating rightwards, in exact agreement with the theoretical prediction. In numerical studies of the transport behavior of topological one-way edge state, we use a configuration illustrated in Figs. 2(c)–2(e). We first set a perfect electric conductor (PEC, yellow rectangle) near the upper edge and the scattering boundary conditions (gray rectangles) near the left, right and lower edges, to form a straight-line waveguide at the upper edge. We then place a point source (yellow star) oscillating at a frequency of 4.50 GHz in the center of the upper edge. As seen in Fig. 2(c), the topological one-way edge state is excited and unidirectionally transports in the straight-line waveguide along the + x direction. Further on, we replace the right scattering boundary with a PEC to form a 90°-turn waveguide. As plotted in Fig. 2(d), the topological one-way edge state can smoothly bypass the sharp corner without any backscattering. Finally, we add a metallic obstacle on the right edge. One can see from Fig. 2(e) that the topological one-way edge state still can pass through the metallic obstacle by creating a new one-way waveguide channel between the metallic obstacle and GPC, showing the transport of topological one-way edge state immune from obstacle. Therefore, for a PCTI without disorder, there exists a topologically protected one-way edge state that is perfectly robust against sharp corner and metallic obstacle on its transport path.

3. Radius-fluctuated photonic Chern topological insulator

To embed disorders in photonic Chern topological systems, we introduce the random fluctuation to gyromagnetic cylinder radius according to rij = (κξ+1)r for the gyromagnetic cylinders located at ith row and jth line in the square lattice. r is the radius of unperturbed gyromagnetic cylinder, r = 0.11a, the parameter κ describes the radius-fluctuated strength ranging from 0 to 1, and the random number ξ has a uniform distribution between –1 and 1.

It is known that the Chern numbers are computed using Bloch band states, which in turn owe their existence to the translational symmetry of the lattice. Consequently, in the regime with significant randomness, the calculation of Chern number becomes formidably challenging, attributing to the absence of the lattice periodicity. Recently, some authors have proposed various methods to characterize the topological invariants in an aperiodic system [48–50]. Here we focus on the change of topologically protected one-way edge states when the phase transitions of PCTIs occur, so we employ two effective empirical parameters Cs and Rs that represent the quality of the edge localization and the one-way property of the edge propagation respectively to feasibly detect the topological phase transition as follows:

$${C_s} = {{\int_{{\Pi _\textrm{I}}} {\zeta ({x,y} )dxdy} } / {\int_{{\Pi _\textrm{I}} + {\Pi _{\textrm{II}}}} {\zeta ({x,y} )dxdy} }}, $$
$${R_s} = {{\int_{{\Omega _\textrm{I}}} {\zeta ({x,y} )dxdy} } / {\int_{{\Omega _\textrm{I}} + {\Omega _{\textrm{II}}}} {\zeta ({x,y} )dxdy} }}, $$
where $\zeta ({x,y} )$ is the time-averaged field intensity of electromagnetic wave at a certain frequency. As shown in Fig. 2(a), ПI+ПII is the whole area of the GPC and ПI is the area with a distant of D away from the upper boundary, and the parameter D is set as equal to the excitation wavelength λ in our computation. Besides, ΩI and ΩII indicate the right area (green dotted rectangle) and left area (yellow dotted rectangle) respectively with a distance of D away from the point source, as illustrated in Fig. 2(a). It should be noted that for a GPC of a square lattice, the effect of radius-fluctuation on each boundary is identical, so it is sufficient to only consider one edge and choose the area with a distance of D away from the upper boundary when we calculate the quality of edge localization Cs and the one-way property of edge propagation Rs. So a large value of Cs represents the good edge localization, while a large modulus of Rs illustrates the good one-way property of edge propagation (here we mainly focus on the right-propagating edge state). For a GPC without randomness depicted in Fig. 2(c), it hosts the topological one-way edge state whose electric fields are perfectly confined in the upper edge and dwelled in the area ΩI, so both Cs and Rs are equal to 1.

Figure 3 shows the color contour plots of Cs and Rs versus the κ, in the range of κ=(0.0∼0.3) and within the frequency range of f = 4.0∼4.8 GHz respectively, where the high Cs and high Rs regions are colored in blue and purple respectively. As κ increases from 0.0 to 0.3, the frequency window of high Cs regions and high Rs (i.e., Cs > 0.5, Rs > 0.8) both moves downwards and gradually closes to zero, indicating that the right-propagating topological one-way edge states gradually disappears (as illustrated by a pair of red dotted lines in Fig. 3). Notably, the area ПI with a distance of D away from the upper boundary is much narrower than the whole area ПI+ПII, thus, Cs > 0.5 is sufficient to be considered as the strong edge localization condition. We divide the fluctuation strength into seven levels as κ0 = 0.00, κ1 = 0.05, κ2 = 0.10, κ3 = 0.15, κ4 = 0.20, κ5 = 0.25 and κ6 = 0.30. For a perfect GPC without fluctuation (i.e., κ0 = 0.00), the frequency range of topological bandgap is 4.25∼4.58 GHz (Δf = 0.33 GHz) and Cs = Rs = 1. While for κ2 = 0.10, the topological bandgap is reduced to 4.23∼4.45 GHz (Δf = 0.22 GHz), but no apparent degradation is observed for the values of Cs and Rs (the average of Cs is about 0.83, and Rs≈0.94). It means that the radius fluctuation κ2 = 0.10 barely leads to 17.0% of the energy leakage to the bulk and 6.0% of the energy back-reflection to the left, so the robustly protected one-way edge states are still well confined at the waveguide channel and propagate rightwards nearly perfectly. For κ4 = 0.20, a number much larger than the fabrication errors in realistic experiments, the average values of Cs (Rs) significantly decrease to 0.50 (0.81) and the topological bandgap shrinks to 4.21∼4.32 GHz (Δf = 0.11 GHz). In this case, more energy fluxes are transmitted into the bulk (about 50.0%) and backscattered to the left (about 19.0%), but the topologically protected right-propagating edge states still exist. However, for a sufficiently strong fluctuation κ6 = 0.30, the topological bandgap is closed and Cs (Rs) is degraded to an extremely small value about 0.13 (0.60). The energy fluxes are no longer well localized on the waveguide channel and the topologically protected right-propagating edge states disappear completely.

 figure: Fig. 3.

Fig. 3. The color contour plots of Cs and Rs versus κ. (a) The quality of edge localization Cs. (b) The one-way property of edge propagation Rs. The high Cs and high Rs regions are colored in blue and purple respectively, and a pair of red dotted lines is used to illustrate the process of bandgap closing. The frequency in κ0 = 0.00, κ2 = 0.10, κ4 = 0.20 and κ6 = 0.30 (green points) is f0 = 4.40 GHz, f2 = 4.33 GHz, f4 = 4.26 GHz and f6 = 4.19 GHz, respectively.

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Next, we construct radius-fluctuated PCTIs with κ2 = 0.10, κ4 = 0.20 and κ6 = 0.30 based on the configuration displayed in Figs. 2(c)–2(e), and set a point source at the upper edge to observe the transport behaviors of EM waves. The excitation frequency in the case of κ0 = 0.00, κ2 = 0.10, κ4 = 0.20 and κ6 = 0.30 (green points in Fig. 3) is f0 = 4.40 GHz, f2 = 4.33 GHz, f4 = 4.26 GHz and f6 = 4.19 GHz, respectively. For κ2 = 0.10, as shown in Figs. 4(a1)-(a3), the electric fields are well concentrated on the waveguide, and the energy fluxes move unidirectionally even when there exists a 90° sharp corner and metallic obstacle on the path. Notably, there appears an Anderson localization state (localization A) induced by the radius fluctuation, where a part of electric fields is tightly trapped in a small region near the edge, in sharp contrast to the electric field distributions in the nonperturbed GPC [see Figs. 2(c)–2(e)]. In this case, the topological bandgap is still robust and the trivial bulk states are suppressed, so there holds a competition between topological one-way edge states and Anderson localization states.

 figure: Fig. 4.

Fig. 4. Electric field distribution in the radius-fluctuated photonic Chern topological insulators with different fluctuated strength κ. (a1)-(a3) κ2 = 0.10. (b1)-(b3) κ4 = 0.20. (c1)-(c3) κ6 = 0.30.

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As the fluctuation increases to κ4 = 0.20, as seen in Figs. 4(b1)-(b3), the topologically protected one-way edge states still can be excited, but their amplitudes are drastically reduced and more electric fields are localized at some regions such as localizations A and B. Interestingly, when the energy fluxes pass through the straight-line, 90°-turn, or metallic obstacle waveguides, the electric field distributions of localizations A and B are almost identical. However, when κ6 = 0.30, as illustrated in Figs. 4(c1)-(c3), the topological one-way edge states completely disappear, but there still exist Anderson localization states A and B induced by the strong radius-fluctuated randomness. It should be noted that, in this case, the electric field distributions of Anderson localizations are very sensitive to the waveguide structures, for instance, the electric field distributions of Figs. 4(c1)-(c3) are very distinguished from each other. This is because the topological protection breaks down and the competition now becomes the one between Anderson localization states and trivial bulk states. These simulation results show that as the fluctuation strength gradually increases, there appears a competition among the topological one-way edge states, Anderson localization states and trivial bulk states. On the one hand, when the fluctuation is weak, the topological bandgap is robust and the trivial bulk states are suppressed, then the competition is mainly between the topological one-way edge states and Anderson localization states. On the other hand, once the fluctuation is sufficiently strong, the topological bandgap is closed and the topological one-way edge states disappear, so that the competition is dominantly between the Anderson localization states and trivial bulk states.

4. Further discussions

To more clearly reveal the phase transition of PCTI and evolution of one-way edge states under various degrees of radius-fluctuation, we further investigate the transport behaviors of EM waves by plotting the energy flux (Poynting vector) distributions. Figures 5(a), 5(b), 5(c), and 5(d) illustrate the partial energy fluxes distributions corresponding to Figs. 2(c), 4(a1), 4(b1), and 4(c1), respectively. One can see that, for a PCTI with no radius-fluctuation (κ0 = 0.00), as shown in Fig. 5(a), the energy fluxes are strongly localized on the upper edge and rotate counterclockwise with a windmill shape around the gyromagnetic cylinders, forming a pure right-propagating edge state. Next, for the PCTI with a weak radius-fluctuated strength (i.e., κ2 = 0.10), whereas the majority of energy propagates rightwards along the upper edge, some energy is concentrated on the area of Localization A via the constructive interference of scattering fields, leading to an Anderson localization state with a counterclockwise energy flux circulation, as seen in Fig. 5(b). Then, as the fluctuation strength increases to κ4 = 0.20, as illustrated in Fig. 5(c), more and more energy fluxes disperse into the bulk and are focused on the wider region (Localization A) to form an Anderson localization state with a larger counterclockwise energy flux circulation, and some energy even flows to the regions 1 and 2. These transport characteristics mean that in this case, the one-way edge states, fluctuation-induced Anderson localization states and trivial bulk states coexist and will couple with each other. We proceed to discuss the transport phenomena of EM waves in the radius-fluctuated PCTI with a strong strength of κ6 = 0.30. As seen in Fig. 5(d), the vast majority of energy diffuses into the bulk of PCTI (e.g., Localization A and region 1) and even transmits to the left of the source (region 2). As a result, there are no longer the obvious one-way transport phenomena.

 figure: Fig. 5.

Fig. 5. Energy flux (Poynting vector) distributions in the radius-fluctuated photonic Chern topological insulators with different fluctuated strength κ. (a) κ0 = 0.00. (b) κ2 = 0.10. (c) κ4 = 0.20. (d) κ6 = 0.30. The plots of (a)-(d) illustrate the partial energy fluxes distributions corresponding to Figs. 2(c), 4(a1), 4(b1), and 4(c1) respectively. The size and direction of the blue arrows indicate the intensity and transport direction of the energy fluxes, respectively.

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These detailed calculations have verified that the topological one-way edge states still exist and are hardly affected by the sharp corner and metallic obstacle on the transport path even if the radius-fluctuation exceeds 20.0% (i.e., κ=0.20). As a sharp contrast, the bandgap of a dielectric photonic crystal will be closed when the radius fluctuation is only about 6.5% [46,47]. Beyond the one-way transport property, these results also reveal that the topological one-way edge states in PCTI are strongly robust against the radius-fluctuation. Besides, these results also reveal that the Anderson localization state appears far more easily in the radius-fluctuation PCTI with even a weak strength (e.g., 10.0%) in comparison with the position-perturbed PCTI with a strong randomness, where the Anderson localization state occurs until the randomness strength is over 25.0% [41]. This is because the presence of regular lattice can make radius-fluctuated PCTI easier to produce the constructive interference of scattering fields in some certain regions to form the Anderson localization state. In this case, the radius-fluctuation PCTI is far more sensitive to disorder than the position-perturbed PCTI, whose topological bandgap will not vanish even when the position-randomness is more than 40.0% [41]. Moreover, from a technical means of view, the radius-fluctuation of 20.0% is already a very strong fluctuation in the realistic experiments when referring to practically available accuracy of precision manufacturing technologies. Thus, the design of practical topological photonic devices is extremely tolerant to the fabrication errors caused by the radius-fluctuation in practical devices.

We go on to verify the stability of topological one-way edge states at a fixed fluctuation strength κ but with various random seeds. We obtain the plots of parameters Cs and Rs versus the random numbers by calculating and analyzing 300 samples with the same fluctuation strength κ. For κ1 = 0.05, the high Cs and high Rs regions are distributed in the frequency range of 4.25∼4.52 GHz, and the values of Cs and Rs at different random numbers are almost identical, as shown in Figs. 6(a) and 6(d). This indicates that the existence of topological one-way edge states at κ1 = 0.05 is extremely stable. Besides, for κ3 = 0.15, the high Cs and high Rs are basically resided in the frequency window of 4.21∼4.39 GHz, but their values rapidly reduce to 0.67 and 0.88 respectively and are not stable at each random number, as can be found in Figs. 6(b) and 6(e). As κ increases to κ5 = 0.25, the fragility of topological one-way edge states with different random numbers also grows. Figures 6(c) and 6(f) show that it is hard to see a clear high Cs and high Rs stripe regions, although there still exists an extremely narrow topological bandgap (4.20∼4.25 GHz) with Cs≈0.31 and Rs≈0.72. Thence, as the degree of radius fluctuation increases, the topological bandgaps are closed gradually and the topological one-way edge states becomes more and more fragile. These phenomena also can be seen in the electric field distributions with different fluctuated strengths in Fig. 4. One can see that with the increasing of the radius fluctuation, the bandgap is closed gradually, more and more energy flux is coupled into the backscattering channel, and the edge localization quality of one-way edge state is declined, meaning that the transport of one-way edge state becomes fragile. Especially, for an even stronger fluctuation (e.g., κ6 = 0.30), the topological one-way edge state disappears completely and the topological protection sharply breaks down, so that the topological bandgap becomes extremely fragile.

 figure: Fig. 6.

Fig. 6. The plots of parameters Cs and Rs versus the random numbers at different fluctuated strengths. (a)-(c) The quality of edge localization Cs. (d)-(f) The one-way property of edge propagation Rs. (a),(d) κ1 = 0.05. (b),(e) κ3 = 0.15. (c),(f) κ5 = 0.25. The random numbers are R = 300, and the high Cs and high Rs regions are colored in blue and purple respectively. The frequency range between a pair of red dotted lines in κ1 = 0.05, κ3 = 0.15 and κ5 = 0.25 is 4.25∼4.52 GHz, 4.21∼4.39 GHz and 4.20∼4.25 GHz, respectively.

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Finally, as a typical example, we proceed to calculate the GPC under a nonuniform magnetization, which is a very common case in the actual experiments. As shown in Fig. 7(a), the radii of all gyromagnetic cylinder are r = 0.11a and the gyromagnetic cylinders of different colors represent that the gyromagnetic cylinders are immersed in different strengths of external magnetic field along the + z direction. We define the nonuniform magnetization to gyromagnetic cylinder according to Hij = gξ+H0 for the gyromagnetic cylinders located at ith row and jth line in the square lattice. H0 = 1800 Gauss is the external magnetic field intensity of uniformly magnetized gyromagnetic cylinder, the parameter g describes the nonuniform magnetization ranging from 0 to 1200 Gauss, and the random number ξ has a uniform distribution between –1 and 1. Figures 7(b) and 7(c) indicate the color contour plots of Cs and Rs versus the g, in the range of g = (0∼1200 Gauss) and within the frequency range of f = 4.0∼4.8 GHz respectively, where the high Cs and high Rs regions are colored in blue and purple respectively. One can see that as the nonuniform magnetization continuously increases from g = 0 to g = 1200 Gauss, the bandgap supporting the topological one-way edge state is closed gradually. These calculated results clearly show that although the GPCs are immersed in a strong nonuniform magnetic field, they can still support the existence of one-way edge states. These also indicate that in addition to the randomness of radius fluctuation, the realization of one-way edge states also has a very strong tolerance to the nonuniform magnetization.

 figure: Fig. 7.

Fig. 7. Schematic diagram of a nonuniformly magnetized gyromagnetic photonic crystal and the color contour plots of Cs and Rs versus nonuniform magnetization g. (a) Schematic diagram. The gyromagnetic cylinders of different colors represent that the gyromagnetic cylinders are immersed in different strengths of external magnetic field along the + z direction. (b) The quality of edge localization Cs. (c) The one-way property of edge propagation Rs. The high Cs and high Rs regions are colored in blue and purple respectively, and a pair of red dotted lines is used to illustrate the process of bandgap closing.

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It also should be noted that in practical experiments, the height of the gyromagnetic cylinder also will affect the robustness and even existence of one-way edge states. Generally, the crystal of gyromagnetic cylinders is placed in an air-loaded planar waveguide sandwiched between two parallel metallic layers that are used to completely forbid the electromagnetic waves from leaking off in the z direction. Because the one-way edge states we discussed here only exist under the TE polarization (where the electric field is parallel to the z-axis direction), so in experiments we must choose the appropriate height of gyromagnetic cylinders to ensure that the polarization is identical to the pure TE states. If the height is too low, the one-way edge state may not be activated, and if the height is too high, both TE polarization and TM polarization (where the magnetic field is parallel to the z-axis direction) modes will be excited and the coupling between them will make the transmission of one-way edge states extremely fragile. Thus, in the process of designing the experimental sample, we need to use the equation of minimum cut-off frequency of the waveguide cavity to determine the proper height of gyromagnetic cylinder ensuring that the one-way edge states can be excited effectively. On the other hand, when the height of gyromagnetic cylinder has been selected appropriately, with the increasing of height disturbance, the one-way edge states also will become fragile. The reason is because the gap between the top/bottom of gyromagnetic cylinder and the metallic layers will become the transport channel for the localized resonance and backscattering of electromagnetic waves.

5. Conclusion

In summary, we have investigated and demonstrated the topological one-way edge states in a 2D PCTI made from a square lattice GPC. We have found that as the radius-fluctuated strength increases, there emerges a competition between topologically protected one-way edge state, fluctuation-induced Anderson localization state and trivial bulk state. We have found that the Anderson localization state appears far more easily in the radius-fluctuation PCTI with even a weak strength compared with the position-perturbed PCTI with a strong randomness. We also have observed that at κ=0.20 (a very strong fluctuation in realistic experiments), the topological one-way edge states still exist and are hardly affected by the sharp corner and metallic obstacle on the transport path. However, as the radius-fluctuated strength continuously increases to κ=0.30, the topological one-way edge state disappears completely and the topological protection sharply breaks down, so that the topological bandgap becomes extremely fragile. Notably, although we only consider the randomness of radius fluctuation, these results are also suitable for other types of perturbations and randomness, such as the nonuniform magnetization and material defects. Our simulations may provide a deeper understanding for the disordered topological photonics, and will also be of significance in the design, fabrication and application of topological photonic samples and devices. Although our work has focused on GPCs, similar ideas can be generalized to other photonic systems, and more broadly to other bosonic platforms, such as acoustics, electrics, mechanics and more.

Funding

National Natural Science Foundation of China (11974119); Science and Technology Planning Project of Guangdong Province (2020B010190001); Guangdong Province Introduction of Innovative R&D Team (2016ZT06C594); National Key Research and Development Program of China (2018YFA 0306200).

Acknowledgments

The authors are grateful for the financial support from the National Natural Science Foundation of China (11974119), Science and Technology Project of Guangdong (2020B010190001), Guangdong Innovative and Entrepreneurial Research Team Program (2016ZT06C594), National Key R&D Program of China (2018YFA 0306200).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Sketch maps of photonic Chern topological insulator. (a) Photonic Chern topological insulator of a square lattice. (b) Position-perturbed photonic Chern topological insulator. (c) Radius-fluctuated photonic Chern topological insulator.

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Fig. 2.
Fig. 2. Photonic Chern topological insulator composed of a gyromagnetic photonic crystal of a square lattice. (a) Schematic diagram. (b) Projected band structure. (c) Straight-line transport. (d) 90°-turn transport. (e) Immune-obstacle transport. The yellow and gray rectangles indicate the perfect electric conductors and the scattering boundary conditions respectively in simulations. The yellow stars represent the point sources oscillated at a frequency of fs = 4.50 GHz, and the purple arrows point to the transport directions of energy fluxes.

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Fig. 3.
Fig. 3. The color contour plots of Cs and Rs versus κ. (a) The quality of edge localization Cs. (b) The one-way property of edge propagation Rs. The high Cs and high Rs regions are colored in blue and purple respectively, and a pair of red dotted lines is used to illustrate the process of bandgap closing. The frequency in κ0 = 0.00, κ2 = 0.10, κ4 = 0.20 and κ6 = 0.30 (green points) is f0 = 4.40 GHz, f2 = 4.33 GHz, f4 = 4.26 GHz and f6 = 4.19 GHz, respectively.

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Fig. 4.
Fig. 4. Electric field distribution in the radius-fluctuated photonic Chern topological insulators with different fluctuated strength κ. (a1)-(a3) κ2 = 0.10. (b1)-(b3) κ4 = 0.20. (c1)-(c3) κ6 = 0.30.

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Fig. 5.
Fig. 5. Energy flux (Poynting vector) distributions in the radius-fluctuated photonic Chern topological insulators with different fluctuated strength κ. (a) κ0 = 0.00. (b) κ2 = 0.10. (c) κ4 = 0.20. (d) κ6 = 0.30. The plots of (a)-(d) illustrate the partial energy fluxes distributions corresponding to Figs. 2(c), 4(a1), 4(b1), and 4(c1) respectively. The size and direction of the blue arrows indicate the intensity and transport direction of the energy fluxes, respectively.

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Fig. 6.
Fig. 6. The plots of parameters Cs and Rs versus the random numbers at different fluctuated strengths. (a)-(c) The quality of edge localization Cs. (d)-(f) The one-way property of edge propagation Rs. (a),(d) κ1 = 0.05. (b),(e) κ3 = 0.15. (c),(f) κ5 = 0.25. The random numbers are R = 300, and the high Cs and high Rs regions are colored in blue and purple respectively. The frequency range between a pair of red dotted lines in κ1 = 0.05, κ3 = 0.15 and κ5 = 0.25 is 4.25∼4.52 GHz, 4.21∼4.39 GHz and 4.20∼4.25 GHz, respectively.

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Fig. 7.
Fig. 7. Schematic diagram of a nonuniformly magnetized gyromagnetic photonic crystal and the color contour plots of Cs and Rs versus nonuniform magnetization g. (a) Schematic diagram. The gyromagnetic cylinders of different colors represent that the gyromagnetic cylinders are immersed in different strengths of external magnetic field along the + z direction. (b) The quality of edge localization Cs. (c) The one-way property of edge propagation Rs. The high Cs and high Rs regions are colored in blue and purple respectively, and a pair of red dotted lines is used to illustrate the process of bandgap closing.

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Equations (5)

Equations on this page are rendered with MathJax. Learn more.

μ ^  =  ( μ r i μ k 0 − i μ k μ r 0 0 0 1 ) ,
C n = 1 2 π i ∫ B Z d 2 k ( ∂ A y n n ∂ k x − ∂ A x n n ∂ k y ) ,
A n n ′ ( k ) ≡ i ⟨ E n k | ∇ k | E n ′ k ⟩ = i ∫ d 2 r ε ( r ) E n k ∗ ( r ) ⋅ | ∇ k E n ′ k ( r ) | ,
C s = ∫ Π I ζ ( x , y ) d x d y / ∫ Π I + Π II ζ ( x , y ) d x d y ,
R s = ∫ Ω I ζ ( x , y ) d x d y / ∫ Ω I + Ω II ζ ( x , y ) d x d y ,
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