Design of terahertz reconfigurable devices by locally controlling topological phases of square gyro-electric rod arrays

: In topological photonics, there is a class of designing approaches that usually tunes topological phase from trivial to non-trivial in a magneto-optical photonic crystal by applying an external magnetic field to break time reversal symmetry. Here we theoretically realize topological phase transition by rotating square gyro-electric rods with broken time reversal symmetry. By calculating band structures and Chern numbers, in a simple square-lattice photonic crystal, we demonstrate the topological phase transition at a specific orientation angle of the rods. Based on the dependence of topological phase on the orientation angle, we propose several terahertz devices including an isolator, circulator and splitter in a 50x50 reconfigurable rod array by locally controlling topological phases of the rods. These results may have potential applications in producing reconfigurable terahertz topological devices.


Introduction
Photonic topological insulator (PTI) in a photonic system is an analogue of electronic topological insulator in an electronic system [1][2][3].It can support topologically protected defect-immune edge state thus draws much attention.According to the process of development, there are several PTIs realized by different methods [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].They can be divided into two classes.One class breaks time reversal symmetry while the other keeps time reversal invariant.In the first class, the simplest way to break time reversal symmetry is to apply an external magnetic field.This kind of PTI was first proposed in photonic crystal by Raghu and Haldane with photonic quantum anomalous Hall (QAH) effect [4,5] and then experimentally demonstrated in magneto-optical photonic crystal composed of gyromagnetic materials [6].More researches on this class keep going in recent years [7][8][9][10][11][12].Because the time reversal symmetry of the gyro-magnetic (or gyro-electric) crystal is broken by magnetic field, the topological protected edge waves in this class are strictly nonreciprocal.In the second class with time reversal invariant, PTI can be realized by using different mechanisms.One approach is based on photonic quantum spin Hall (QSH) effect (relying on the symmetry of the electromagnetic field).This kind of PTI have been realized in many systems including bi-anisotropic metacrystals [13] and C6-symmetric all-dielectric photonic crystal [14,15], where two pseudospin states in the photonic system play the same roles of two spin states in the electronic system.Another approach relies on photonic quantum valley Hall (QVH) effect.Several groups have already realized this kind of PTI in all-dielectric photonic crystals [16,17] or bi-anisotropic metamaterials [18] or surface plasmon crystals [19], where valley symmetry is broken and the edge states are valley dependent.Because of invariant time reversal symmetry, the topological protected edge waves in the polarized state To realize magnetic field provide enou years, it is fo InSb) can be terahertz (TH resonance eff obtain THz on in terahertz ba Relying o this paper des rod array.Thi Under a relati transits from calculations o the rods with functionalities The robust edg agation.h broken time slas is necessar e in the optic onance effect i symmetry with us THz topolog ials are propo photonic crysta t soon.gical phases on ncluding isolato otonic crystal f we find the topo he orientation numbers.By n manipulate w onfigurable rod tor rod array arran onstant, L is the si enting the rod orie ed along z-axis.ork consists of ith air, as show s a.The + x ax ng at the beg he + x axis and only consider e BZ edges ar ecome gyro-ele ngth direction ge states exist e reversal sym ry for the gyro cal or microwa in semiconduc h a relatively s gical devices b osed [25].Sim al [26]. where ω c = eB/m* is the electron cyclotron resonance frequency (e is the charge of the electron and m* is the effective mass of the electron), ω is the angular frequency, γ is the collision frequency of the charge carriers characterizing the material loss, ε ∞ and ω p is the high frequency limit permittivity and the bulk plasma frequency of the semiconductor respectively.Here we use indium antimonide (InSb) semiconductor to achieve high ω c under a relatively small magnetic field at THz range.The material parameters of InSb at room temperature are given by ε ∞ = 15.68,ω p = 12.6 THz, m* = 0.014m 0 (m 0 is the free electron mass in vacuum) [28].At the working frequency of 3.5 THz in this work, the elements in the permittivity tensor are evaluated as ε 1 = 0.585, ε 2 = 0.190 under B = 0.8 T and ε 3 is fixed to be 0.670 no matter how B varies.For simplicity, the material loss is neglected in most of the simulations.
For this kind of semiconductor, only the H-polarized dispersion relation of the bulk modes depends on the cyclotron resonance frequency.To calculate the band structure of the photonic crystal, we solve the Maxwell's equations in the H-field form where μ(r) and ε(r) are the permeability and permittivity tensors respectively.ω is the eigenfrequency, which is solved by using the finite-element method in the commercial software COMSOL Multiphysics.The topological properties of the proposed lattice can be explored by calculating Chern number of the nth band (the band with the lowest energy is the 1st band) where the integral is over the whole BZ and the nth band Berry curvature is expressed as [29] ( ) A k denoting the Berry connection of the nth band energy u nk .In some cases, when two adjacent bands show mutual degeneracy points, it is better to calculated the composite Chern number of the two bands defined by [30] 1 Here (

) n n
A ⊕ + k is 2x2 matrix Berry connection between the nth band energy u nk and (n + 1)th band energy u (n+1)k , and Tr stands for the trace of the matrix.This definition can also be extended to more than two bands with degenerate points.The band and composite Chern numbers are obtained using the method in Ref [31].Furthermore, the gap Chern number C g is defined as the sum of the Chern numbers of all the bands below a band gap [10].This gap Chern number is used to examine the existence of the topological protected edge states between two topologically distinct insulators.

Topologi
At the beginn explained in t the photonic c 2(c) under an be reduced to small band ga Actually, whe point, which reversal symm phase.In cont near ω = 0.43 this cyan gap time reversal numbers of th bands, we dir numbers, as d should be me breaking the exchange of B between lowest two bands and higher bands at the original degenerate point, leading to nonzero composite Chern number of the lowest two bands.On the other hand, the cyan gap already exists without a magnetic field thus is not opened by the broken time reversal symmetry.There is no exchange of Berry phase between the lowest two bands and higher bands, giving rise to zero composite Chern number of the lowest two bands.We can conclude that the photonic crystal with different rod orientation may show different topological phase.
To further explain the topological phase transition process of the lowest band gap with different rod orientation, we calculate the band structures of the cases with other degrees in the range of 0°<θ<45° under the same magnetic field.We find the special case with θ = 33.2°, the transition angle for the topological phase, where its band structure is shown in Fig. 2(b).As pointed out in the purple circle, there are degenerate points between the 2nd and 3rd bands at M and M' points.This degeneracy also occurs without the magnetic field applied.After thorough calculations, we can describe the process of topological phase transition.As indicated by the arrow direction in Fig. 2, if we start from θ = 45°, in the process of clockwise rotating the rods from θ = 45° to θ = 33.2°, the lowest band gap is topologically trivial and becomes narrower until completely closed around θ = 33.2°(reduces to a degenerate point at the transition angle).When we keep rotating the rods from θ = 33.2° to θ = 0°, the lowest band is opened up again and maintains during the process.This time the gap turns to be topologically non-trivial, showing the photonic QAH phase.

Edge states
Robust unidirectional edge states are the most attractive feature which exist at the interface between two insulators with different topological phases.We calculate the projected band diagrams along the edge direction of the photonic crystal with broken time reversal symmetry.In our setting, the photonic crystal consists of an array of the supercell along the x direction.Each supercell (shown in Fig. 3(c)) has 20 unit cells with the same structural size as in Fig. 2 along the y direction.The upper and lower edges are formed by ending the upper and lower boundaries of the supercell with metal plates (set as perfect electric conductor) respectively.The Floquet periodic boundary condition (characterized by the Bloch wavevector k x ) is employed in the x direction.The band diagram for the case of θ = 0° is shown in Fig. 3(a) and the target band gap region is enlarged in Fig. 3(b) for more clear presentation.The lowest white gap region near ω = 0.435 (2πc/a) agrees with the green gap in Fig. 2(a).Inside the gap, the orange and red lines respectively present the unique edge state on the upper and lower edge.Each edge band locates partially in + k x region and partially in -k x region of the band gap, thus the upper and lower edge mode have a crossing in the gap while each edge band connects the bands above and below the gap.The number of edge modes in the band gap is determined by the difference in the gap Chern number of two adjacent insulators [10].Here in the gap there is one edge mode at a single edge that is in good agreement with its gap Chern number C g = + 1. ).If L nger take L such as not locate inside lower topologically trivial gap at the same time.When working at these frequencies, a large part of the energy will leak away from the lower sub array.
Based on the above we are now able to control the waveguiding direction by configuring every rotatable rod.So a reconfigurable square array with multipurpose property is proposed for THz applications.This array consists of 50x50 rotatable square rods with the same parameters as in Fig. 4. We fix four ports unchanged as the uniform interfaces for wave input or output as shown in four red frames in Fig. 5(a), (c) and (e).With suitable configuration by rotating every single rod inside the array, this structure can be used for several occasions.Firstly, by rotating left sub array to θ = 45° and right sub array to θ = 0° as shown in Fig. 5(a), we can get a 2-port THz isolator to let waves go straight unidirectionally.This isolator allows waves propagating only from Port 1 to Port 2 along the edge of two sub arrays as shown the field distribution at 3.5 THz in Fig. 5(b), while the backward propagating is forbidden.Secondly, by rotating 4 sub arrays near each corner of the whole array to θ = 45° and all the remaining rods to θ = 0° as shown in Fig. 5(c), we can obtain a 4-port THz circulator with rotational symmetry to let waves go circularly.When waves input from Port 1, most of the energies propagate along the artificial edge in counterclockwise direction and output from Port 2 but few parts arrive Port 3 and 4, as shown the field distribution at 3.5THz in Fig. 5(d).Because of the circulation property, THz wave can also input from Port 2 or 3 or 4 and will get similar circulated transmission.At last, by rotating the sub arrays in upper left corner as well as lower right corner to θ = 45° and all the remaining rods to θ = 0°, we can achieve a THz wave splitter to slit waves at the center junction as shown in Fig. 5(e).The waves inputting from Port 1 can split into two edge wave beams at the junction then output from Port 2 and 3 as shown in Fig. 5(f).Here the splitting tunnel is two-row wide at the junction along the y direction, leading to the transmittance of 42.2% at Port 2. The width of the tunnel at the junction will affect the splitting ratio.Obviously, the transmittance at Port 2 becomes lower with narrower junction.For example, if the tunnel is zero-row wide, the transmittance at Port 2 is 16.3%.As mentioned above, the orientations θ in different sub arrays are not necessarily to be 0° or 45°.There are more choices for rod orientations as long as we can make two adjacent sub arrays have different topological phase.In addition, more edge channels can be created by rotating every single rod to make the devices with more ports or guide waves against disorders such as misalignments of the rods along desired path inside the array.

Conclusion
We propose a new concept of designing terahertz reconfigurable devices by controlling local topological phases of square-rod photonic crystals.These devices are realized in a 50x50 rod array where the rods are arranged in a square lattice.Magneto-optical semiconductor InSb showing enough gyro-electric response under a relatively small magnetic field is used as the material of the rods.Our calculation results reveal the dependence of the topological phase on the rod orientation.In such a simple photonic crystal, the topological phase transits from trivial to non-trivial at a critical rod angle near 33 degree.The topological non-trivial phase keeps unchanged as long as the orientation of the rods is smaller than the critical angle.The ability of controlling robust wave propagation in the 50x50 domain by changing local topological phases allows us to design terahertz devices with different functionalities.The high isolation ratio of the isolator and circulator is ensured by the non-reciprocal propagations.The splitting ratio of the splitter can also be tuned by adjusting the splitting junction.InSb used in our design is a commercially available and CMOS compatible material, and it can be easily structured, e.g., with a dicing saw [28].To break the time reversal symmetry, we rely on a relatively small magnetic field lower than 1T, which makes our design feasible for the experimental realization.The concept of locally controlling topological phases paves a new way to realize terahertz reconfigurable devices with new functionality.
Fig.1air.In rotate also s2.TheoreticThe photonic rods arranged square rod is axis of the anticlockwise Because of th The Brillouin the inset in F magnetic fiel tensor can be Fig. 2 transit T. Th coveri gap C 2nd a 45°; T For the ca green gap is + sum of the Ch green gap is t 45°, the comp the lowest fo topologically is opened fro Fig. 3 ended the ga gap.A the su confinTo demon band for exam electric fields edges respect velocity), the direction.It is due to the sm because there3.3ReconfigNext, we nu propagation a first presentat constant a = 3 trivial band g region and m stands on the between the p frequency f = distribution in direction on illustrates the defect-immun