Xinyang Pan(潘昕陽), Haitao Li(李海濤), Weijie Dong(董為杰), Xiaoxi Zhou(周蕭溪),Gang Wang(王鋼),?, and Bo Hou(侯波)

1School of Physical Science and Technology&Collaborative Innovation Center of Suzhou Nano Science and Technology,Soochow University,Suzhou 215006,China

2School of Optical and Electronic Information,Suzhou City University,Suzhou 215104,China

3Wave Functional Metamaterial Research Facility,The Hong Kong University of Science and Technology(Guangzhou),Guangzhou 511400,China

Keywords: Dirac point,transition,moving,guided mode

1.Introduction

The nodal degeneracies, such as Dirac points (DPs) and Weyl points (WPs), in band structures in various lattice systems may provide a platform to conduct the synergy investigation between the different fields of physics.[1-12]For instance, the structural transition between type-I and type-II Dirac points is considered in condensed matter physics and is proposed to explore the black hole physics and astrophysics.[13-17]Therefore, the design of nodal degeneracies and the study of their transitions in momentum space are of fundamental interest.Through dispersion engineering,the two-dimensional (2D) DPs can be realized with photonic guided modes along the direction where continuous translational symmetry retains.[18-20]Straightforwardly, a two-band degeneracy can be designed between the transverse electric(TE) and transverse magnetic (TM) guided modes for a slab of dielectric medium, and a simple approach is to introduce the in-plane anisotropy to cause the band crossing between TE and TM branches with mirror symmetry protection.[21]Furthermore, because of the simultaneous positivity of the slopes of both guided modes, the resultant degeneracy is essentially of type-II cone in the continuous medium case, as illustrated in Fig.1(a).To evoke the type-I cone and the transition between both cones, Bloch periodicity is employed to reverse the slope of the band, where the dispersion of the guided modes is back-folded within the first Brillouin zone arising from spatial periodicity in Bloch lattices.[18,22]Notably, the graphene lattice is an excellent 2D platform to engineer the type-I Dirac cone.Even remarkably, double Dirac cone (type-I) can be realized with photonic periodical structures through aligning TM-polarized cone and TE-polarized cone at the same frequency.[23,24]However,how to obtain the type-I cone in the uniform medium lack of the discrete translational symmetry is a question.

Fig.1.Sketches of guided modes (upper row) and zoomed Dirac cones(lower row)around the degenerate point in a uniaxial crystal slab waveguide with in-plane optical axis for (a) static case and (b) moving case, respectively.The slab with the finite thickness along the x direction is illustrated by the insets.The moving direction is assumed along the optical axis, i.e.,horizontal direction, labeled as z axis in the insets.The dark red lines represent TE modes and the dark blue lines represent TM modes, and the dash line denotes the light line in air.

Here, we propose a method to realize the 2D type-I DPs in the uniform dielectric medium slab.This method dates to the 19th century when Fresnel predicted the dragging of light in moving media.[25]The Fresnel’s theory was verified by Fizeau’s celebrated experiment with flowing water,exposing the difference of the speed of light in the up-stream and down-stream direction.[26]In the study, we find that moving the dielectric slab can lift the otherwise overtilted Dirac cone along the counter-moving direction, because the dragging effect manifests very different intensity to TE and TM guided modes.Therefore, we may realize the transition of the Dirac cone from type-II to type-I in the continuous medium through tunning the moving speed of the slab,as shown in Fig.1(b).

2.Electromagnetic parameters of moving uniaxial crystal slabs

In the following,we will discuss the dispersion relations in an inertial frame S fixed to the laboratory where a uniaxial medium is moving.We attach an inertial frame S＇to the medium(choosing the optical axis as one of coordinate axes,e.g.,z＇axis)where the medium is observed being rest and has the uniaxial permittivity tensor diag■ε＇,ε＇,ε＇z■and the uniaxial permeability tensor diag■μ＇,μ＇,μ＇z■(the isotropic permittivity or permeability case can be obtained through makingε＇=ε＇zorμ＇=μ＇z).Consider the simple case in which the coordinate axes of S and S＇are parallel to each other,that is,orthonormal vectors ?x‖?x＇, ?y‖?y＇,and ?z‖?z＇.The medium,i.e.,the frame S＇,is moving with velocity ˉvin the frame S.The Lorentz transformation of the electromagnetic field can be written as[27]

wherecis the speed of light in vacuum and

From Eqs.(1)and(2)we can obtain the constitutive matrix of the moving medium in the frame S:

in which we have assumed that the medium moves along the optical axis or thezdirection, namely,βx=0,βy=0, andˉβ=β?z.And we have

The bulk modes of EM wave in moving media have been investigated intensely.[27]For the ordinary wave,E-field has zero component along the propagating direction ?k= ˉk/kand nonzero components being transverse to the propagating direction.The dispersion relation can be obtained as follows:

wherek‖(k⊥) represents the wave number parallel (orthogonal) to the moving direction of the medium and satisfiesk2‖+k2⊥=k2.For the extraordinary wave,E-field has nonzero component being both parallel to ?kand transverse to ?k.The dispersion relation is

The dispersion of the ordinary wave is plotted in Fig.2,which shows the Fresnel drag by the moving medium.When the medium is at rest(β=0),the dispersion curve is displayed as a circle, where the nonmagnetic conditionμ＇z=μ＇=μ0has been applied.In the velocity range where 1-n＇2β2＞0,the curve is stretch into an ellipse which is symmetric about thek‖axis, we call it non-relativistic zone.For the velocities where 1-n＇2β2＜0, the ellipse is transformed into a hyperbola.This is the relativistic case and the velocity region is called the Cerenkov zone.The velocity that divides the nonrelativistic zone and the Cerenkov zone isβ=1/n＇,which is the speed of light that propagates in the stationary medium.In addition, the reversed transition of the iso-frequency contour from hyperbola to ellipse may be realized when assuming the hyperbolic property for the medium in the frame S＇,[28-32]seeing the supplementary material(Section A).

Fig.2.(a) Bulk dispersion of the ordinary wave at different velocities with n＇ =4.(b) The iso-frequency contour in the bulk medium at ω =0.3 GHz with different velocities labeled,respectively,by different colors.The green line represents the light cone in air at ω =0.3 GHz.The Cerenkov zone is pointed out with an arrow.The results of the extraordinary wave are similar,except the uniaxial anisotropy indicated by ε＇z/=ε＇,and are not plotted here.

3.Guided modes in the moving uniaxial crystal slab

Equation (12) is the dispersion relation of EM wave in the bulk moving material,and in contrast the investigated slab has a finite thicknessd, as shown in Fig.3(a).To solve the guided modes in the moving slab, we impose the boundary condition that requires both tangential electric field and magnetic field are continuous on the boundary.[27]In the first place,we consider the guided modes with the propagating direction along the optical axis,and the mirror symmetryMy:(x,y,z)→(x,-y,z) enables both TE and TM modes.Thus, we can obtain the characteristic equation for TE modes in the moving dielectric slab waveguide in the frame S:

in whichμzr=μz/μ0andkxsatisfies Eq.(12a)in the region I,seeing Fig.3(a).Likewise,we have the characteristic equation for TM mode:

in whichεzr=εz/ε0andkxsatisfies Eq.(12b) in the region I.In the above,αis the evanescent wavenumber in regions II and III,satisfyingk2z-α2=ω2μ0ε0.

Fig.3.(a) The schematic picture of the uniaxial crystal slab waveguide,where (x,y,z) denotes the reference frame S, (x＇,y＇,z＇) denotes the reference frame S＇,and(n＇o,n＇o,n＇e)is the dielectric principal axes for the uniaxial anisotropy.The red arrow represents the propagation direction of electromagnetic wave.The purple arrow shows the velocity of the slab relative to the frame S.The angle θ denotes the in-plane rotation of the optical axis and the velocity direction.The panel in the right shows the three layers structure in our system.(b)and(c)Dispersion diagram of the lowest band of TE and TM mode when θ =0,where the black,orange,and blue lines correspond to the velocities n＇β =0,n＇β =0.1,and n＇β =0.2,respectively,and green line represents the light line.

For concreteness, the parameters of the stationary medium are chosen asd= 2 mm,ε＇= 16ε0= (n＇o)2,ε＇z=100ε0=(n＇e)2, andμ＇=μ＇z=μ0, so thatn＇=n＇o=4.It is has known that the nonmagnetic planar slabs have been extensively used as basic waveguides in microwave engineering and devices where a broad horizon of dielectric materials, e.g., high-kprinted circuit board (PCB) and ceramics, is available.[33,34]The dispersion curves of the lowest band of TE and TM modes for different velocity are depicted in Figs.3(b)and 3(c).We limit the normalized wavenumberkz/(π/d)from-0.4 to 0.4 and calculate the characteristic equation in nonrelativistic case.When the waveguide stays stationary in the frame S(β=0),the dispersion relation ofωkzconsists of two branches which are symmetric about theω-axis(black line in Figs.3(b) and 3(c)), in agreement with the parity symmetry in the stationary system.The two branches correspond to the waves traveling in opposite directions, respectively.Furthermore,the slope of TE and TM modes are both positive(negative)in+zdirection(-zdirection).However,the parity symmetry is broken as soon as the medium begin to move,which causes the+zand-zdirections inequivalent.Then,as we increase the velocity, the asymmetry of the dispersion is more evident.What is more,comparingβ=0.1(that is,n＇β=0.4)with the case ofβ=0.2(that is,n＇β=0.8),the sign of group velocity(dω/dk)of TM mode has a switch along the negativekz-direction and the switching point is atkz ≈-0.104(π/d),whereas the group velocity of TE mode keeps the sign unchanged in the range fromkz=0 tokz=-0.4(π/d).In terms of the classification of DPs,where the type-I DP comes from the crossing of two bands with opposite slope, the normalized velocityβof the medium should not be lower than 0.1 to obtain the crossing between TE and TM bands with opposite slope.To display an explicit result,we take the velocityβ=0 andβ=0.16 in the following calculation.

4.Type-I Dirac degeneracy in the moving uniaxial crystal slab

As shown in Figs.4(a) and 4(b), the analytical results are solved using Eqs.(13) and (14).Whenβ=0, the TE2(subscript labeling the TE band number)band crosses linearly with the TM3(subscript labeling the TM band number)band at point A＇at the frequency 23.7 GHz, see Fig.4(a), where their slope is of the same sign.In contrast, whenβ=0.16,the degeneracy labeled as point A is reduced to 18.36 GHz due to the moving effect, see Fig.4(b), where their slope is noticed of opposite sign.The analytical results can be verified by numerical calculations.Using the eigenmode solver of finite-element simulation software COMSOL and representing the moving effect with bi-anisotropic parameters,we calculate the dispersion of the stationary and moving uniaxial crystal slab for TE and TM modes atβ=0 andβ=0.16, and plot them in Figs.4(a) and 4(b).It is seen that good agreement between analytical and numerical results is achieved.In addition,either TE or TM mode can be further classified with even or odd parity,because of the mirror symmetryMx(the system having the mirror planex=d/2).The parity is alternative for either TM or TE modes.Namely,TMnmodes with even mode numbernand TEnmodes with odd mode numbernare antisymmetric, and TMnmodes with oddnand TEnmodes with evennare symmetric.The numerical calculation reveals that TE2and TM3have the same parity,and the insets in Figs.4(a)and 4(b)are theHzmap of TE2mode from COMSOL.

In order to exhibit the complete dispersion structure around the degenerate points A＇and A,we need calculate the band diagramω(ky,kz).In the calculation,we first rotate the in-plane dielectric principal axes around thexdirection by an angleθ, as shown in Fig.3(a).The non-diagonalized permittivity (permeability) tensor) after rotation can be written in the frame S as

in which

And the magnetoelectric coupling matrixξθafter rotation has the form in the frame S:

Theθ-dependence of EM parameters is derived in the supplementary material (Section B).Then, we assume that the wave still propagate along thezdirection and express the electric fields and magnetic fields in different regions.Because the mirror symmetryMyis broken due to the rotation, the guided modes are no longer pure TE or TM mode but are their combination which we call hybrid mode.The characteristic equation for hybrid mode is solved by matching boundary conditions, which gives us the dispersionωθ(kz).The calculated results are plotted in the right panels of Figs.4(a) and 4(b), and threeθ-cut dispersions for theβ=0.16 case are shown in Figs.5(a)-5(c).When the medium is in motion,we can see from Figs.5(a)-5(c) that the degeneracy of point A is lifted uponθbeing nonzero.So is the static case, too.This is because theMysymmetry does not exist anymore and meanwhile two degenerate bands have the same parity underMx.[21]

In the static case, TE2and TM3linearly cross at the degenerate point A＇and form an over-tilted Dirac cone, which gives rise to a type-II DP in momentum space,as shown in the right panel of Fig.4(a).In the moving case ofβ=0.16,both modes cross with positive and negative group velocities and display a type-I cone or DP, as illustrated in the right panel of Fig.4(b).Furthermore, the tilting term in the type-I DP Hamiltonian can be entirely neglected,seeing the supplementary Material(Section C),which demonstrates an upright conical band structure near the degenerate point.Therefore,by employing the motion of medium we can“drag”the Dirac point from type-II to type-I.

Fig.5.The dispersion ωθ(kz)of hybrid modes in the moving slab when(a)θ =2°,(b)θ =15°,and(c)θ =30°.

5.Discussion

It is also noted in Fig.4 that a lower-frequency degeneracy is caused by the crossing between the fundamental TE and TM modes (TE1and TM1).The calculation reveals that the degeneracy transforms topologically from type-II to type-I upon moving,see Figs.4(a)and 4(b),and furthermore the degeneracy is not lifted in threeθ-cut plots though its frequency and the momentum are shifting,see Figs.5(a)-5(c).In fact,it is a line degeneracy protected by the symmetryMx(TE1and TM1having the opposite parity underMx).The line degeneracy is the type-II Dirac line in the static case,[21]and its tilting will be regulated in aθ-dependent way by the velocity in the moving case.

On the other hand,the spatial inversion symmetry in our system is broken due to the moving,andω(kz)is not symmetrical along +kzand-kzdirections, and our study is mainly focused on the-kzdirection.As an accompanying effect,the type-II degeneracies in the+kzdirection become even more titled when the slab changes from the static state to the moving state, which is the consequence of the spatial inversion symmetry broken by the moving.

Last, we make a discussion from an experimental point of view.Although achieving a speed of 0.16cis challenging,a moving medium can be effectively simulated by using spacetime-modulated methodology.[35]Therefore, by changing the electromagnetic parameters of a static and anisotropic medium in both temporal and spatial domains,the similar effects to the band structure as those in our study would be anticipated both theoretically and experimentally.

6.Conclusion

In conclusion, the physics of Fizeau’s flowing water experiment has been applied to a photonic confined structure which is a continuous dielectric slab waveguide and where the uniaxial anisotropy is involved.The moving effect not only brings about non-reciprocity to the band structure in the upstream and down-stream directions,but also leads to the topological transition of local degenerate points within the band structure.We calculated an explicit case in which the type-II DP can be turned into type-I DP when the slab waveguide is moving.Our results provide a new approach to regulate the topology of degeneracy for 2D photonic bands in the continuous translational symmetry situation.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant No.12074279),the Major Program of Natural Science Research of Jiangsu Higher Education Institutions(Grant No.18KJA140003),and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.