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Abstract
Using two well-defined empirical parameters, we numerically investigate the details of the disorder-induced topological state transition (TST) in photonic Chern insulators composed of two-dimensional magnetic photonic crystals (MPCs). The TST undergoes a gradual process, accompanied with some interesting phenomena as the disorder of rod positions in MPCs increases gradually. This kind of TST is determined by the competition among the topologically protected edge state, disorder-induced wave localizations and bulk states in the system. More interestingly, the disorder-induced wave localizations almost have no influence on the one-way propagation of the original photonic topological states (PTSs), and the unidirectional nature of the PTSs at the edge area can survive even when the bulk states arise at stronger disorders. Our results provide detailed demonstrations for the deep understanding of fundamental physics underlying topology and disorder and are also of practical significance in device fabrication with PTSs.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Inspired by the intensive research on the topological materials in electronics [1–7], topological states in photonics have been a hot topic and have led to the prediction and realization of various photonic topological systems in recent ten years [8–28]. Just like the demonstration of the topological states in electronic systems, in periodic photonic structures the photonic topological states (PTSs) are also characterized by topological invariants such as Chern number and Z2 index defined in reciprocal space (wavevector space) [29,30]. According to the bulk-edge correspondence [8,9,31–33], topological bandgaps correspond to the topologically protected gapless edge (surface) states in two (three) dimensions. These topological states possess some astonishingly brand new properties, including one-way propagation and robustness against impurities or defects at edge (surface) area without backscattering and dissipation [10,14,15]. Due to their fascinating properties, the PTSs are thought to have potential applications in future photonic integrated circuit and topological quantum computing [34,35].
Although there have been plenty of theoretical and experimental studies on PTSs since they were first proposed [8], most of them were predicted and realized in periodic or quasiperiodic photonic structures [36–39]. However, in device fabrications it is impossible to fabricate perfect periodic or quasiperiodic photonic structures, and the production process will inevitably introduce disorders. In addition, theoretically studying the topological states in disordered photonic systems can deepen our understanding of the fundamental physics underlying topology and disorder [40–42]. In electronic systems, there have been a large number of studies on the topological states in disordered structures [43–46], and the calculation methods for topological invariants such as Bott index and Chern number in real space (coordinate space) have been developed as well [47–50]. Moreover, recently it was reported that the disorder could even induce the topological Anderson states in trivial systems [51–53]. However, it is quite surprising very few works have been done on topological states in disordered photonic systems up to now [54–56]. But the effect of disorder, especially the effect of continuously changed disorder on the PTSs, is of great significance in both theoretical research and device fabrication [57,58].
In this paper, we theoretically investigate the topological state transition (TST) induced by gradually increased disorder in photonic Chern insulators composed of two-dimensional (2D) magnetic photonic crystals (MPCs) [9,32]. Instead of calculating the topological invariants such as Bott index and Chern number, we employ two well-defined empirical parameters to perform our investigation [42,55]. Based on them, we clearly demonstrate the details of the TST as the position disorder of the rod elements in MPCs increases gradually.
2. Model and methods
2.1 Photonic Chern insulator systems
In our investigation, the selected photonic Chern insulator without disorder is a 2D square lattice MPC composed of magnetic rods immersed in air with external dc magnetic fields along rod axes (z direction). The lattice constant is a and rod radius is r as shown in the inset in Fig. 1(b). The relative permeability tensor of the magnetic rod takes the form of
Fig. 1. (a) Band structure of the 2D square lattice MPC. Blue (red) region denotes the topological bandgap. Chern numbers of the bands and bandgaps are marked. (b) Projected band diagram of the MPC. Topological edge state is denoted with blue (red) curve. Insert is the unit cell. Dashed (dotted) line denotes the frequency at a/λ=0.64 (0.87). (c) Sketch maps of the disordered MPC within a region of 6a×6a as η takes values of 0, 0.25, 0.5, 0.75 and 1, respectively, from left to right. Blue dots denote the rod elements in MPC.
The band structure of the MPC is shown in Fig. 1(a) and the Chern numbers of the bands and bandgaps are marked in the figure. In Fig. 1(b), we show the projected band diagram when the MPC is covered on the top by a metallic boundary along x direction [32]. Clearly, corresponding to the topological bandgap with Chern number Cgap=1 (-2) within the frequency range from a/λ=0.619 to 0.666 (from a/λ=0.848 to 0.895) in Fig. 1(a), there exists a single-mode (multi-mode) gapless edge state only with negative (positive) group velocity denoting the one-way propagation property in Fig. 1(b). Here λ is the wavelength in air background.
To embed disorders in the system, we introduce random displacements to the rod positions in MPC defined as $d_{x(y )}^i = \eta \xi _{x(y )}^i({a - 2r} )$ for the ith rod along x (y) direction, with η a parameter quantitatively describing the disorder strength from 0 to 1, and $\xi _{x(y )}^i$ a random number with uniform distribution between -0.5 and 0.5 [54]. A set of $\xi _{x(y )}^i$s denote a random seed ξx(y) for the disordered MPC. Figure 1(c) plots the sketch maps of MPC structures as disorder strength η increases from 0 to 1 with the same random seed.
As the disorder of the rod positions increases, the band structure becomes ill-defined and the calculation of topological invariant becomes formidably challenging in reciprocal space. In real space, topological invariants such as Bott index and Chern number have been developed to describe the topological properties of aperiodic systems [47–50]. Despite this, here in order to get detailed demonstrations, we employ the full-wave simulations based on the finite element method (COMSOL Multiphysics) to calculate two well-defined empirical parameters Cs and RAB to inspect the TST in our disordered system. Cs is used to characterize the edge confinement of the PTSs, and RAB is used to describe the one-way propagation.
2.2 Model setting and calculation of Cs and RAB
The calculation model of the MPC without disorder is shown in Fig. 2(c). It contains 40 (20) rods (denoted with blue dots) in x (y) direction, and is covered on the top by a metallic boundary (PEC, denoted with a black sold line) with a distance of a/2 from the centers of the outmost rods. On another three sides, the MPC is surrounded by perfect matched layer (PML, shown as the light green areas) to absorb the incident electromagnetic (EM) wave. The EM radiation is excited by a point source (denoted with a violet star) located at the middle point between two outmost rods near the PEC. Πs (denoted by the green area) and Π represent the edge region and the whole area of the MPC in the calculation of Cs defined in Eq. (2), respectively. Points A and B are the locations of the energy density measurements defined in Eq. (3).
Fig. 2. (a) Cs and (b) RAB with the increment of disorder strength η. Vertical lines denote the disorder values at η=0.5, 0.53, 0.56, 0.595, 0.605, 0.64, 0.67, 0.755, 0.805, 0.88 and 0.93, respectively, from left to right. (c) Schematic diagram of the model without disorder in our calculations and simulations. Blue dots denote the rod elements in MPC. Metallic boundary (PEC) is denoted with the black sold line. Perfect matched layers (PML) are shown as the light green areas. Point source is denoted with the violet star near the PEC.Πs (denoted by green area) and Π represent the edge region and the whole area of the MPC, respectively. Points A and B are the locations of the energy density measurements.
The first empirical parameter Cs describes the edge confinement of the EM wave defined as
with Ω(x,y) the time averaged EM energy density at a certain frequency [55]. Π is the whole area of the MPC, and Πs denoted by the green area in Fig. 2(c) is the edge region within a distance of 2a away from the PEC. Cs≃1 denotes the well-defined edge states, and the deviation of Cs from one implies the degradation of the edge confinement of the EM radiation. The second parameter RAB is the isolation ratio defined as3. Results and discussions
3.1 TST in disordered MPC examples
First, we fix the frequency at a/λ=0.64 [dashed line in Fig. 1(b)] which lies in the middle of the lower topological bandgap (with Cgap=1) within the frequency range from a/λ=0.619 to 0.666. At the beginning, as an example, we focus on a randomly selected disordered MPC example with a fixed random seed ξx(y).With the increment of the disorder strength η, the calculated Cs and RAB are shown in Figs. 2(a) and 2(b), respectively, quantitatively demonstrating the disorder-induced TST. Correspondingly, in Fig. 3, we show the spatial distributions of the normalized electrical field mode |Ez| at several discrete values of η, illustrating the TST qualitatively and visually.
Fig. 3. Distributions of the normalized electrical field mode |Ez| at several discrete disorders, corresponding to the vertical lines in Figs. 2(a) and 2(b). The point source is denoted by a yellow star in each panel (see Visualization 1).
When η=0, there is no disorder in the system. Cs ≃ 1 and RAB≃-14 denote the well-confined edge state and its perfect one-way propagation to the right (from the point source to point B), as shown in Fig. 3(a). This is the case of the perfect PTS. As the disorder of the rod positions increases from η=0 to 0.5, it can be seen that there is no apparent decrement in the value of Cs and RAB keeps a large modulus larger than 12, which implies the robustly protected edge confinement of the one-way electromagnetic (EM) wave. Accordingly, Fig. 3(b) shows the spatial distribution of |Ez| at η=0.5, corresponding to the disorder at vertical green solid lines in Figs. 2(a) and 2(b). Although there are some deformations in the field mode, the one-way propagation of the edge wave is well protected, so is the PTS.
As the disorder increases from η=0.5 to 0.625, Cs shows an overall decreasing tendency, and it also has some particularly small values shown as the deep dives marked with d1 and d2 in Fig. 2(a). In this disorder range, the disorder of rod positions produces wave penetration into the MPC, leading to the overall decrement of Cs, and multi-scattering of the disordered rods may introduce wave localizations near the edge or even in the bulk of the system [59–61], inducing the sudden decreases of Cs. Although RAB gradually increases to about -5 in this disorder range, it still maintains a high quality isolation property of the edge waves. In Figs. 3(c)–3(f), we illustrate the field mode distributions at η=0.53, 0.56, 0.595 and 0.605, respectively, corresponding to the disorders at red dashed lines from left to right in Figs. 2(a) and 2(b). Clearly, wave localizations produce some hot spots near the edge [such as H3 in Fig. 3(f)] or even in the bulk [such as H1 in Fig. 3(c) and H2 in Fig. 3(f)] of the system [55]. But nonetheless, the one-way propagation of the PTS is well preserved.
As η continually increases from 0.625 to 0.83, Cs continually decreases to about 0.25 and the modulus of RAB reduces further to about 3. In this disorder region, the randomness of rod positions may destroy the topological bandgap of the MPC, leading to the propagation of the EM waves into the lattice bulk. In Figs. 3(g)–3(j), we illustrate the field mode distributions at η=0.64, 0.67, 0.755 and 0.805, respectively, corresponding to the disorders at black dotted lines from left to right in Figs. 2(a) and 2(b). Obviously, bulk states arise and begin to compete with the wave localizations and the topological edge state, and as the disorder enlarges, the EM radiation propagates into the bulk more easily. Even so, the unidirectional propagation of the edge waves can still be recognized.
When η is larger than 0.83, both Cs and the modulus of RAB take very small values compared with those in weak disorders, indicating that the EM radiations are ill confined at the edge area and their one-way propagations are also difficult to identify. Figures 3(k) and 3(l) show the field modes at η=0.88 and 0.93, respectively, corresponding to the disorders at blue dotted dashed lines in Figs. 2(a) and 2(b). It can be seen that the bulk states, together with the localization states, dominate the EM radiation, and the unidirectional propagation of the EM wave almost disappears at the edge area. The PTS is destroyed in this disorder region.
So far, for the fixed frequency at a/λ=0.64, by taking a randomly selected random seed ξx(y) as an example, we clearly depict the details of the TST in photonic Chern insulator as the disorder of the rod positions in MPCs gradually increases. The competition among the topological edge state, disorder-induced wave localizations and the bulk states determines the transition [59–61]. Clearly, the TST undergoes a gradual process as disorder increases continuously.The coexistence of the topological edge state and the disorder-induced wave localizations may be useful for the design of topological random lasers [62–65]. The unidirectional propagation of the EM wave at the edge area can survive even when the bulk states arise at stronger disorders, which relaxes the manufacturing requirement of the one-way waveguide in device fabrication with PTSs. To give a clearer demonstration of the TST in Figs. 2(a) and 2(b), as in Fig. 3, we simulate the propagation of the EM radiation when the disorder is gradually increased by steps of η=0.05 (see Visualization 1). Sometimes, we concern the transmittance properties of the edge states for practical applications. So, we also calculated the values of ΩA and ΩB in Eq. (3) to describe the transmission of the edge states from the source to left and right (see Supplement 1 for details). The results are in good agreement with those in Figs. 2 and 3.
Further, instead of using the fixed frequency, we perform the investigations for the disorder-perturbed MPC at continuous frequencies from a/λ=0.6 to 0.69 including the topological bandgap with Cgap=1 in Fig. 1. The random seed is set to the same as in the above research. Figure 4 shows the TST results, with the black dotted (red dashed) lines denoting the edges (center) of the bandgap of the MPC without disorder. At η=0, within the frequency range of the topological bandgap from a/λ=0.619 to 0.666, the PTSs are well confined at the edge area with Cs ≃ 1 [Fig. 4(a)] and propagate unidirectionally with RAB smaller than -12 [Fig. 4(b)]. As the disorder increases, the PTSs with frequencies near the bandgap edges are bleached. The PTSs located at the center of the bandgap have the best robustness, immune to the stronger disorders. When the disorder dominates the system, the topological edge states are annihilated at last. From Fig. 4, we can also see the competition of the topological protection, disorder-induced localizations and bulk states during the TST within the frequency range of the topological bandgap as the disorder increases gradually.
Fig. 4. (a) Cs and (b) RAB with the increment of η within the frequency range from a/λ=0.6 to 0.69. Black dotted lines and red dashed line denote the edges (a/λ=0.619, 0.666) and center (a/λ=0.64) of the bandgap of the MPC without disorder, respectively.
To see the influence of different random seeds ξx(y) on the TST in our system, we further perform the same calculations and simulations as in Figs. 2–4 for another two disordered examples with different random seeds ξx(y). Although there are some differences in Cs and RAB such as the dive positions in Cs in different examples, the same TST features can be obtained (see Supplement 1 for details).
3.2 Statistical analyses and size scaling computations
In order to get the statistical results, we focus on the PTS at the frequency of a/λ=0.64 as an example. The results are obtained by calculating 100 samples with different random seeds. The averaged Cs and RAB are shown with blue solid lines in Figs. 5(a) and 5(b), respectively, with the cyan shaded regions denoting their standard deviations. For clarity, the fluctuations of Cs and RAB, correspondingly represented with δCs and δRAB, are also shown with blue solid lines in Figs. 5(c) and 5(d), respectively. Clearly, before the disorder increases to the moderate value about η=0.4, Cs decreases very little compared to its initial value without disorder and RAB keeps its modulus larger than 12. Meanwhile, δCs keeps almost zero and δRAB takes very small values with δRAB/RAB<0.05. These results denote that the PTS is well protected in this weak disorder region.
Fig. 5. Statistical results and system size scaling comparisons. (a) Cs and (c) its fluctuation δCs, (b) RAB and (d) its fluctuation δRAB. Annotations of 40(20), 34(17) and 46(23) in legends denote the rod numbers in x(y) direction in three different size systems, respectively. Cyan shaded regions (error bars) in (a) and (b) denote the standard deviations in 40(20) [34(17) and (46)23] systems. The results are obtained from 100 samples with different ξx(y) at the frequency of a/λ=0.64.
Then, with the increment of disorder to about η=0.8, Cs decreases to about 0.25 and RAB increases to about -3, indicating the gradually weakened edge confinement and degraded unidirectional propagation of the original PTS. This disorder region corresponds to the gradual process of TST from the original PTS to trivial state. As for the fluctuations, both δCs and δRAB at first increase fast from η=0.4 to about 0.6, and then decrease slow from η=0.6 to 0.8. The larger fluctuations indicate the larger differences among disorder sample realizations. By rechecking the results in Fig. 2 and Fig. S1 (see Supplement 1 for details) with different random seeds, we find that the large fluctuations δCs and δRAB come from the formations of disorder-induced wave localizations. Different random seeds may induce edge or bulk hot spots at different disorder values as η gradually increases from η=0.4, leading to the fast increasing fluctuations in Cs and RAB. As η increases further from η=0.6, bulk states arise and weaken the effect of localized hot spots on the Cs and RAB, leading to the slow decreases of δCs and δRAB.
Next, as the disorder increases further from η=0.8, both Cs and the modulus of RAB take a slowly decreasing tendency until finally they reach their respective limits before η=1.0. This is due to the fact that in this disorder region the enhanced bulk states will diffuse more EM waves into the bulk and at last when η is large enough the bulk states will dominate the system. Correspondingly, in this disorder region, both δCs and δRAB also decrease slowly and arrive at their respective limits at last.
In Fig. 5, we also give out the statistical results of TST of another two different size systems. The smaller system contains 34 (17) rods and the larger one contains 46 (23) rods in x (y) direction. Correspondingly, for the smaller (larger) system the points A and B in Fig. 2(c) are set to 14a (20a) away from the point source, which still located at the middle point of the outmost rods near the PEC. The averaged Cs and RAB from 100 samples, together with their standard deviations denoted by error bars are shown in Figs. 5(a) and 5(b), respectively, with black dashed lines for the smaller system and red dotted lines for the larger one. The corresponding fluctuations δCs and δRAB are shown in Figs. 5(c) and 5(d), respectively. For clarity, we take only 20 uniformly distributed disorders from η=0 to 1 in these two sizes, whereas we take 200 disorder values in the system studied above. Comparing the results in these different size systems, we can see that except for the difference in Cs in disorder range η>0.6, the curves of Cs and RAB, together with their fluctuations δCs and δRAB, show nearly identical evolution characteristics respectively within the entire range of disorder, denoting the same TST behaviors in each of the three systems. The difference in Cs can be explained as follows. In Fig. 2(c), large models enlarge the whole area Π of the system in y direction, while the edge area Πs is still defined within 2a near the PEC. Thus, larger model leads to smaller ratio of Πs to Π. As the bulk states propagate in the system in disorder range η>0.6, the smaller ratio Πs/Π results in smaller Cs correspondingly. Taking into account this trivial difference, we think the system used above is big enough for our TST studies.
3.3 TST within higher frequency range
Above researches concentrate on the PTSs within the frequency range from a/λ=0.619 to 0.666. For the PTSs within the higher frequency range from a/λ=0.848 to 0.895 (corresponding to the bandgap with Cgap=-2) shown in Fig. 1, using the method above, we also inspect their TST transitions induced by the gradually increased disorder of rod positions (see Supplement 1 for details). Comparing these two kinds of TST, we find that they share similar characteristics, with only a difference that the PTSs with lower frequencies are more stable than those with higher frequencies. This is because that in our systems the topological states are formed from the multi-scattering of EM waves, and lower frequency waves (with longer wavelength) are less susceptible from the disorder of rod positions in MPCs.
4. Conclusion
We demonstrate the details of the disorder-induced TST in photonic Chern insulators. Although the parameters we used are not well-defined topological order parameters such as Chern number and Bott index in topological systems, our numerical calculations and simulations clearly demonstrate the features of the TST as disorder increases gradually, and our results provide a detailed and visualized description of the TST in photonic topological systems.
The TST in our system is a gradual process. At first, the PTSs are immune to the weak disorders. As the disorder increases to moderate values, disorder-induced wave localizations may form hybrid states composed of one-way edge mode and localized hot spots near the edge or even in the bulk of the system. These hybrid states come from the competition between topological protection and wave localizations [59–61]. Most interestingly, these localized hot spots almost have no influence on the one-way propagation of the PTSs. This phenomenon may be used to design topological random lasers [62–65]. Then, with the further increase of the disorder, bulk states arise and come into the competition with the topological edge state and localized hot spots, leading to the decrement of the one-way property and destruction of the edge confinement of the original PTSs. Despite this, the unidirectional nature of the PTSs at edge area can still be recognized in this disorder range, which relaxes the manufacturing requirements in device fabrication with topological states. These two stages of competition exist successively in the range of moderate disorders, until at last the bulk states dominate the system and the PTSs are destroyed when the disorder grows large enough. Although we only consider the disorder of rod positions, we expect that our results are also suitable for other types of disorders, such as radius fluctuations of rod elements, which commonly occur in the fabrication of nanophotonic devices.
Our results differ significantly from those of a recent work by Liu et al. [55]. In their studies, the authors investigated the disorder-induced TST in photonic chiral hyperbolic metamaterial as the permittivity fluctuation increases gradually. They only focused on the competition between the topological protection and Anderson localization at the sharp transition stage. After the sharp state transition point, the topological states are destroyed. However, in our MPC systems, the topological photonic states are disturbed and destroyed gradually with no obvious sudden state transition as the position disorder of rod elements in MPC increases continuously. Our studies also differ from those of the work by Orazbayev et al. [66]. In their researches, the authors studied the robustness of subwavelength edge modes in metamaterial spin-Hall (SP) systems and vally-Hall (VH) insulators, two time-reversal symmetry preserving systems being under hot research in topological photonics in recent years [17,67]. Althouht our researches are performed in systems with broken time-reversal symmetry, we expect that our methods and simulations are also operable for photonic SP and VH systems, and even for topological photonic systems including nonlinear effects [68]. Our results provide further understandings for the fundamental physics underlying topology and disorder, and are also of significance in practice of topological device fabrications.
Funding
National Key Research and Development Program of China (2017YFA0303800); Natural Science Foundation of Shandong Province (ZR2018MA036, ZR2018MA039); National Natural Science Foundation of China (11604132, 61421001, 61574071); Ministry of Education of the People's Republic of China (JZW17SL01); Liaocheng University (318051705).
Acknowledgments
H.Zhang thanks the Introduction and Cultivation Plan of Youth Innovation Talents for Universities of Shandong Province.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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