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Abstract
We have studied the coupling effect of topological photonic states in a double-channel magneto-optical photonic crystal waveguide by introducing a two-stranded ordinary Al2O3 photonic crystal as the coupling layer. There exist both M1 (odd) and M2 (even) one-way modes simultaneously in the bandgap. Interestingly, M1 mode is always a fast-light mode with large group velocity (vg) and large group velocity dispersion (GVD) regardless what the radius (RA) of Al2O3 rods is. However, when RA is appropriate, M2 mode becomes a very slow-light mode exhibiting near-zero vg and zero GVD simultaneously. The physical reason of such slow-light is attributed to the strong coupling effect between the one-way edge modes in both sub-waveguides. Furthermore, the simulation results show that the robustness of both the fast- and slow-light modes are extremely strong against perfect electric conductor defect and the one-way transmittance is close to 100%. Besides, the PEC defect can cause significant phase delay. These results hold promise for many fields such as signal processing, optical modulation, and the design of various topological devices.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Reducing the speed of light to an extremely low level, the so-called slow-light, has attracted a great deal of attention in the past decades for its promising properties in enhancing light-matter interaction strength and controlling the flow of light. Slow light propagations have been achieved by a variety of method, including optical fibers [1,2], metallic structures [3,4], and dielectric photonic crystals (PCs) [5,6]. As one kind of method to realize slow-light, PC waveguides have attracted much attention recently because they have many merits such as abilities of being integrated on optical chip and operating at room temperature [7–10].
The idea of topological photonics comes from the exciting developments of solid materials and the discovery of a new phase of substances called topological insulators [11,12]. In topological photonic systems, the one-way modes have provided a powerful platform to realize a wealth of potential applications, including robust one-way waveguides [13–18], valley states [19–22], topological light trapping [23], and especially slow-light [24–26]. In recent years, magneto-optical photonic crystals (MOPCs) have become one of the most prominent ways to create one-way modes due to the exertion of the external magnetic field and consequent broken time-reversal symmetry. Some works have shown that one-way modes based on MOPC can be used to overcome the back-scattering problems in slow-light systems, owing to its robust one-way properties [24,27]. In 2013, Yang et al. demonstrated the one-way slow-light state in a line channel comprising MOPCs and metal cladding [27] with a large group velocity (vg) but unfortunately together with huge group velocity dispersion (GVD). In 2016, Fang et al. proposed a new mechanism to achieve slow-light by the coupling of two magnetic surface modes [28]. In 2019, our group studied the generation of slow-light state with zero GVD but without topological protection by the strong coupling effect of two counter-propagating one-way modes in a single-channel MOPC waveguide [26]. Later, a broadband group dispersionless topological slow-light state based on the coupling effect of two co-propagating one-way modes in a MOPC system was further reported [29]. Besides, in 2020, Yoshimi et al. proposed another efficient way to achieve slow-light based on topological valley PCs which supports topological kink modes with large group indices over 100 within the topological bandgap [30].
Up to now, most studies on the one-way slow light states were based on single-channel waveguides constructed by MOPC/air/MOPC or metal/MOPC structures. However, there were rare researches about double-channel one-way waveguides composed of two MOPCs with an ordinary PC structure inserted between them. In fact, double-channel MOPC waveguides not only have more structural parameters for design, but also possess abundant physical phenomena such as slow-light to be explored. It is expected that they will provide an important platform to study the coupling effect of one-way modes in the two sub-waveguides, and the consequent unique group-dispersionless slow-light mode.
In this work, we propose a double-channel MOPC waveguide to study the band structure characteristics and the unique slow-light properties resulted from the coupling effect of one-way modes between two sub-waveguides. It is found that there always exist both fast-light (M1) and slow-light (M2) one-way modes simultaneously. The physical original of such two modes is explained by different coupling strength between one-way modes in the two sub-waveguides. Especially, with appropriate parameter conditions, the M2 mode becomes a very slow-light mode exhibiting near-zero vg and zero GVD simultaneously, which are well verified by the simulation results. These researches have potential for many fields such as signal processing, optical modulation, and the design of various topological devices.
2. Double-channel MOPC waveguides design and band structure
Figure 1(a) shows the considered double-channel square MOPC waveguide formed by introducing a two-stranded square Al2O3 PC as the coupling layer. Both the MOPC and Al2O3 PC have the same lattice constant, i.e. aY=aA=a=3.87 cm. The upper and lower parts are MOPCs composed of YIG rods with radius RY=0.11a, and they are applied with external dc magnetic fields (i.e., +H and -H) in counter z-directions. After introducing the two-stranded Al2O3 PC with a radius of RA, two sub-waveguides with widths of wd1=wd2=1.52a are created. The relative permittivities of YIG and Al2O3 rods are ε1=15 and ε2=8.9, respectively. When applying a dc magnetic field H0 in the out-of-plane (+z) direction, gyromagnetic anisotropy will be induced strongly and the relative permeability of YIG rods becomes a tensor as follows [14, 31],
Fig. 1. (a) The supercell of the double-channel MOPC waveguide. (b-d) are the band structures for TM polarization with different radii of Al2O3 PC (RA=0.30a, 0.32a, 0.34a). There are three waveguide modes (blue, red, green). The blue and red lines present two typical defect modes supported in the double-channel waveguides. The purple dashed line in Fig. 1(c) denotes a typical frequency ωs=0.5572(2πc/a) which intersects with the blue and red defect curves at M1 and M2 points. The eigenfield distributions (|E| and Ez) corresponding to M1 and M2 are shown in the dashed frame. (M1a, M1b) and (M2a, M2b) denote the enginfield distributions in two sub-waveguides respectively.
In order to explore the characteristics of the double-channel structure, we will analyze the band structures under different RA by using finite element method (FEM). Only the TM polarization (i.e. the electric field parallel to the YIG rods) is considered in this work. We keep H0=1543 Gs, and the calculated band structures for RA=0.30a, 0.32a, and 0.34a are shown in Figs. 1(b)–1(d), respectively. Obviously, they all support several waveguide modes and we focus on the left blue one and the right red one, as shown in Figs. 1(b)–1(d). As RA increases from 0.30a to 0.34a, the left blue band shifts down from 0.572 to 0.552(2πc/a) and its curve shape hardly changes; while the right red band not only fluctuates over a wider frequency range, but also has a sensitive change of curve shape.
As for RA = 0.30a shown in Fig. 1(b), for the right red band curve, its left part is very smooth, but its right part climbs and eventually descends with increasing kx. What is more, the frequency range of this waveguide mode is too high to locate in the bandgap. That is, under the condition of RA=0.30a, the double-channel waveguide does not support slow light in the bandgap. As for RA=0.34a shown in Fig. 1(d), although the red band curve goes across the bandgap, it does not possess a mild stage but becomes even more tortuous, which implies that the condition of RA=0.34a is not suitable for slow light either.
What happens to the right red band curve if RA=0.32a? From the band structure of RA=0.32a shown in Fig. 1(c), one can find that the frequency range of the red band curve is within the bandgap. More importantly, the red band curve flattens out and continues to decline with increasing kx, meaning that its slope is always negative. As a result, one-way slow-light can be realized for RA=0.32a. To deeply explore the characteristics of such waveguide mode, we choose a typical frequency ωs=0.5572(2πc/a) (i.e. 4.319 GHz) in the bandgap which intersects with the left blue and right red bands at M1 and M2 points, respectively. Their eigenfield distributions (i.e., |E| and Ez) are shown in the dashed frame in Fig. 1. The mechanism of the formation of M1 mode (i.e., the fast-light) and M2 mode (i.e., the slow-light) can be understood from the interaction picture by analyzing the eigenfield distributions of sub-waveguides in Fig. 1. As shown in Fig. 1, for M1 mode (i.e., the fast-light mode), it can be regarded as the superposition of M1a and M1b modes (denoted by the two blue frames) whose eigenfields mainly locate inside the two sub-waveguides. Their eigenfield profiles almost do not overlap mutually in the Al2O3 PC coupling layer, meaning that M1a and M1b modes hardly couple with each other during the transporting process, so that the propagation velocity of M1 mode is almost unchanged and keeps fast. However, for M2 mode (i.e., the slow-light mode) which can be regarded as the superposition of M2a and M2b modes (denoted by the two red frames) of the sub-waveguides, the eigenfields of M2a and M2b modes mainly locate at both sides of each sub-waveguide, causing a strong electric field distribution in the Al2O3 PC coupling layer. This means that there exists strong energy exchange in the Al2O3 PC coupling layer during the transporting process. Therefore, the strong coupling of M2a and M2b modes results in very low group velocity of M2 mode and induces the flat dispersion accordingly.
Besides, it should be noted that the dispersion curve also depends on the distance between the two YIG PCs, that is, the number N of periods of Al2O3 PC. Figure 2 shows the evolution of dispersion curve depending on N. Obviously, as N increases from 0 to 3, the blue mode ascends firstly (from N=0 to 2) and then descends (from N=2 to 3), and its left part always has a fast group velocity. Differently, the red mode is more sensitive to N as compared with the blue mode, and it becomes one-way and very flat to exhibit very low group velocity when N=2. What’s more, as N=2 and 3, there appears an additional trivial two-way waveguide mode (i.e., the green mode) caused by the increase of the number of periods of Al2O3 PC. It is expected that more trivial two-way waveguide modes will emerge as N becomes much larger.
Fig. 2. The band structures for TM polarization with different N: (a) N=0; (b) N=1; (c) N=2; (d) N=3. The blue, red, and green lines present different modes supported in the double-channel waveguides. The purple dashed line in (c) denotes a typical frequency ωs=0.5572(2πc/a) which intersects with the two defect curves at M1 and M2 points, respectively.
3. Group velocity and group velocity dispersion characteristics of the double-channel waveguides
Next, we further study propagation characteristics of both M1 and M2 modes. We mainly focus on two important parameters vg (vg =dω/dkx) and GVD [GVD = d(1/vg)/dω2] which are often used to characterize the slow-light states. Figure 3 presents the vg and GVD curves calculated by FEM. As for M1 mode shown in Fig. 3(a1), vg is always negative and much less than zero. When kx increases from -0.5 to -0.3(2π/a), vg first descends from -0.243c to -0.35c and then ascends quickly to a maximum of -0.225c, indicating that M1 mode is a one-way fast-light mode. Figure 3(a2) shows that the corresponding GVD of M1 mode varies from -3 to 2.5, which would cause severe distortion of signal pulse. Although GVD=0 appears at kx1=-0.439 and kx2=-0.339(2π/a) (points 1 and 2), their group velocities are as high as vg1=-0.350c and vg2=-0.234c, respectively. These results imply that M1 mode is not suitable for pulse transmission.
Fig. 3. (a1), (b1) vg of M1 and M2 modes for RA=0.32a, respectively. (a2), (b2) GVD of M1 and M2 modes for RA=0.32a, respectively. The green dashed line presents a parallel line with GVD=0. The insets in Figs. 3(b1-b2) are partial enlargements of the purple dashed frames.
However, the case of M2 mode is quite distinctive from that of M1 mode. In Fig. 3(b1), vg of M2 mode is always negative and extremely close to zero, meaning that M2 mode supports one-way slow-light transmission with near-zero vg. Moreover, the vg curve has a very smooth part with fluctuation less than -0.007c [see the enlargement inset in Fig. 3(b1)]. Accordingly, the GVD curve in Fig. 3(b2) also possesses a very smooth part with small fluctuation around GVD=0 [see the enlargement inset in Fig. 3(b2)]. Obviously, M2 mode supports slow-light transmission with both near-zero vg and zero GVD simultaneously. Furthermore, within the smooth part of GVD curve, there exist three intersections of GVD=0 (points 3, 4, and 5), i.e., kx3=0.089, kx4=0.203, and kx5=0.291(2π/a). Their corresponding group velocities are vg3=-1.64×10−4c, vg4=-6.74×10−3c, and vg5=-2.96×10−3c, respectively. In other words, the double-channel waveguides structure designed here supports zero GVD slow-light transmission.
4. Numerical simulations and discussions
Based on the above discussions, we have known that M1 mode is a one-way fast-light mode while M2 mode is a one-way slow-light mode with near-zero vg and zero GVD. We now proceed to study their propagation behaviors by setting two point sources (i.e., the white stars in Fig. 4) oscillating at ωs=0.5572(2πc/a) in the two sub-waveguides. Figure 4 are the simulation results for M1 and M2 modes carried out by FEM. In Fig. 4(a1) of M1 mode, the energy distribution in the Al2O3 PC coupling layer is very weak, which is consistent with the prediction in the eigenfield analysis in Section 2. Due to such weak coupling effect, the Ez field mainly concentrates in the two sub-waveguides and propagates one-way along the left direction with relatively large vg. From the enlargement view in Fig. 4(a2) of the yellow dashed frame, one can clearly see that the energy rotates around each YIG rod above wd1 and below wd2 waveguides to propagate along the left direction, due to the topological protection property. In order to characterize the coupling strength, Fig. 4(a3) plots the amplitude of electric field (|E|) along the center of the upper Al2O3 PC layer denoted by the purple dashed line in Fig. 4(a1). Figure 4(a3) indicates that the amplitude near the point sources is the strongest at around 6×104 V/m but attenuates to zero sharply on the right side, while the amplitude to the left of the point sources is always stable at about 1.75×104 V/m, which reduces by 70.83% from 6×104 V/m. These data prove that M1 mode supports fast-light propagation due to the weak coupling effect of two sub-waveguide one-way modes.
Fig. 4. (a1), (b1) Ez field distributions of RA=0.32a for M1 and M2 modes. (a2), (b2) Poynting vector distributions in the yellow dashed frame area in (a1) and (b1). The white stars represent the two point sources. (a3), (b3) Amplitude of electric field along the centers of the upper rods of Al2O3 PC layer as denoted in (a1) and (b1). x0=-15.5a is a minimum amplitude position.
How about the transmission characteristics of M2 mode created by the strong coupling effect? From the simulation results of M2 mode in Fig. 4(b1), the EM wave also propagates one-way along the left direction. However, different from that of M1 mode, part of Ez field is concentrated along the neighboring YIG rods upper and below the two sub-waveguides, while most of Ez field is concentrated in the middle Al2O3 PC coupling layer, as shown in Fig. 4(b2). Due to the strong coupling effect in the Al2O3 PC coupling layer, the energy propagates very slowly and three energy transmission paths (i.e., both spiral energy transmission paths along the lower edge of the above YIG crystal and the upper edge of the lower YIG crystal, as well as the energy transmission path inside the middle Al2O3 crystal) are formed to support the one-way slow-light transmission, as shown in Figs. 4(b1) and 4(b2). Figure 4(b3) presents the amplitude of electric field along the centers of the upper Al2O3 rods [the dashed purple line in Fig. 4(b1)]. It clearly shows that the amplitude near the point sources is strongest at 1.6×106 V/m but immediately decays to zero in the right far field, just like that of M1 mode. Significantly, on the left-side, the |E| just reduces by 18.75% and stays stable around 1.3×106 V/m. It is noted that the stable electric field for M2 mode (1.3×106 V/m) is two orders stronger than that of M1 mode (1.75×104 V/m). Therefore, the physical picture of the slow-light mode (M2 mode) can be understood as follows: Due to the strong energy exchange between the two sub-waveguides through the Al2O3 PC coupling layer, the M2 mode emerges as a result of the strong coupling between two one-way modes originated from the upper and lower isolated sub-waveguides, and the energy flow rotates around the YIG rods to propagate leftwards quite slowly.
5. Robustness of the coupling modes against perfect electric conductor
We further study the robustness of the one-way modes of the double-channel waveguide against perfect electric conductor (PEC) defect. A PEC defect with a length of 5a and width of 0.1a is inserted vertically in the center of the double-channel waveguide to block the channel. All of the simulation parameters are the same as those in Fig. 4, and the simulation results are shown in Figs. 5(a1)–5(a3) and Figs. 5(b1)–5(b3).
Fig. 5. (a1), (b1) Ez field distributions when a PEC (5a×0.1a) is vertically and symmetrically inserted in the double-channel waveguide. (a2), (b2) Poynting vector distributions in the yellow dashed frame area in (a1) and (b1). (a3), (b3) Amplitude of electric field along the centers of the upper rods of Al2O3 PC layer as denoted in (a1) and (b1). x1 =-11a is the new minimum amplitude position [corresponding to x0 in Fig. 4(b1)] caused by the PEC defect.
Obviously, the PEC defect hardly affects the power transmittance. The distribution of Ez field almost does not change before and after the PEC defect as EM wave propagates one-way from the source to the left direction. Actually, as shown by the enlargements in Figs. 5(a2) and 5(b2), inserting a PEC defect creates an equivalent channel between the YIG rods and the PEC wall, so that the EM wave can propagate along this equivalent channel, bypass the PEC defect, and continue to move leftwards. Furthermore, on both sides of the PEC in Figs. 5(a3) and 5(b3), the |E| strength is almost the same as the cases without PEC defects, maintaining almost complete one-way power transmittance along the channel. This property is quite different from that in an ordinary PC waveguide consisting of isotropic dielectric materials. In an ordinary PC waveguide, a defect introduced in the waveguide would arouse backscattering wave to reduce the transmittance in the forward direction. The strong robustness of one-way modes against PEC defect provides good tolerance for unidirectional waveguide fabrication in practice. However, it should be noted that when the symmetry of the PEC defect about y=0 is broken, the transport robustness of slow-light mode will be affected because another mode will be excited.
Furthermore, we discuss the influence of the PEC defect on the phase of one-way modes. Since the EM wave has to get around the PEC defect to continuously propagate to the left direction, it will cause an obvious phase delay accordingly. This can be verified by comparing the patterns between the perfect waveguide (Fig. 4) and the waveguide with a PEC defect (Fig. 5). Without loss of generality, we take M2 mode as an example to show this property in details. In Fig. 4(b1), x0=-15.5a is a minimum amplitude position of electric field for the perfect waveguide without a PEC defect, and the Bloch wavelength is about λB=18a estimated by the difference between the maximum and minimum amplitude positions of electric fields (i.e. λB/2 = 9a) in Fig. 4. In Fig. 5(b1), due to the phase delay caused by the insertion of the PEC defect, this corresponding minimum amplitude position shifts to x1=-11a. This phase delay is about 4.5/18×360°=90°. Therefore, it can be inferred that the longer the PEC defect, the larger phase delay it will cause. This property provides an effect way to design topological phase delay devices as desired.
As shown in Fig. 6(c), the transmission spectrum in the bandgap without and with the defect almost coincide, and their transmittances are nearly 100%. This indicates that the slow-light propagation has strong energy transmission robustness, which is consistent with the results shown in Figs. 5(a3) and 5(b3). Besides, it should be emphasized that for the case of the waveguide with a sharp bend, although the fast-light mode and slow-light mode can be excited simultaneously when the electromagnetic wave bypasses the sharp bend, the slow-light M2 mode is still dominant, because the existence of one-way edge state at the both side of the waveguide can support the electromagnetic wave to turn over the sharp bend smoothly and further to excite the M2 mode.
Fig. 6. (a) Ez field distribution without a PEC. (b) Ez field distribution with a symmetrical PEC respecting to the x axis (y=0). (c) The transmission spectra without and with a PEC defect.
6. Conclusions
In conclusions, we have analyzed the coupling effect in a double-channel MOPC with a two-stranded ordinary Al2O3 PC inserted in the waveguide. By adjusting the radius of Al2O3 rods (RA), several one-way waveguide modes are achieved in the bandgap. At the typical frequency of ωs=0.5572(2πc/a), for the odd mode M1, because of weak coupling effect, a fast-light mode is created and the EM wave is well confined inside the two sub-waveguides. Differently, for the even mode M2, due to the strong interaction of two one-way edge modes of sub-waveguides, the EM wave mainly concentrates in the Al2O3 PC coupling layer, and one-way slow-light with zero GVD and near-zero vg is obtained with appropriate RA. Additionally, simulation analyses indicate that the one-way double-channel waveguide are strongly robust against PEC defects, maintaining almost 100% one-way transmittance. Besides, it is found that a PEC defect can cause significant phase delay. These results have great prospects for many fields such as signal processing, optical modulation, and the design of various topological devices.
Funding
National Key Research and Development Program of China (2018YFA 0306200); National Natural Science Foundation of China (12074127, 11504114, 11974119); Guangzhou Science and Technology Program key projects (201904010105); Guangdong Province Introduction of Innovative R&D Team (2016ZT06C594); Special Project for Research and Development in Key areas of Guangdong Province (2020B010190001); Fundamental Research Funds for the Central Universities (2019ZD50); Dongguan Introduction Program of Leading Innovative and Entrepreneurial Talents.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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