• Photonics Research
  • Vol. 10, Issue 5, 1244 (2022)
Rui Zhou1, Hai Lin1,5,*, Yanjie Wu1, Zhifeng Li1..., Zihao Yu1, Y. Liu2,3,6,* and Dong-Hui Xu4|Show fewer author(s)
Author Affiliations
  • 1College of Physics Science and Technology, Central China Normal University, Wuhan 430079, China
  • 2School of Physics and Electronic Sciences, Hubei University, Wuhan 430062, China
  • 3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China
  • 4Department of Physics, Chongqing University, Chongqing 400044, China
  • 5e-mail: linhai@mail.ccnu.edu.cn
  • 6e-mail: yangjie@hubu.edu.cn
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    DOI: 10.1364/PRJ.452598 Cite this Article Set citation alerts
    Rui Zhou, Hai Lin, Yanjie Wu, Zhifeng Li, Zihao Yu, Y. Liu, Dong-Hui Xu, "Higher-order valley vortices enabled by synchronized rotation in a photonic crystal," Photonics Res. 10, 1244 (2022) Copy Citation Text show less

    Abstract

    Synchronized rotation of unit cells in a periodic structure provides a novel design perspective for manipulation of band topology. We then design a two-dimensional version of higher-order topological insulator (HOTI) by such rotation in a triangular photonic lattice with C3 symmetry. This HOTI supports the hallmark zero-dimensional corner states and, simultaneously, the one-dimensional edge states. We also find that our photonic corner states carry chiral orbital angular momenta locked by valleys, whose wave functions are featured by the phase vortex (singularity) positioned at the maximal Wyckoff points. Moreover, when excited by a fired source with various frequencies, the valley topological states of both one-dimensional edges and zero-dimensional corners emerge simultaneously. Extendable to higher or synthetic dimensions, our paper provides access to a chiral vortex platform for HOTI realizations in the terahertz photonic system.
    Pi=1(2π)2d2kTr[A^i],i=1,2,

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    A(k)iuk|k|uk=iunitcelld2rε(r)uk*[kuk],(A1)

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    Ω(k)k×A(k)=Ay(k)kxAx(k)ky.(A2)

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    Θh(k)Θ1=h(k).(B1)

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    h(k)Θ|ukn=Θh(k)|ukn=ϵn(k)Θ|ukn.(B2)

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    Θ|ukn=m|ukmVkmn,(B3)

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    Vkmn=ukm|Θ|ukn=ukm|ukn*(B4)

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    h(k)Θ|ukn=ϵn(k)Θ|ukn=ϵn(k)m|ukmVkmn.(B5)

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    h(k)Θ|ukn=h(k)m|ukmVkmn=mϵm(k)|ukmVkmn(B6)

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    Vkmn[ϵn(k)ϵm(k)]=0,(B7)

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    Vkmn=ukm|ukn*,(B8)

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    r^h(k)r^=h(Rk).(B9)

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    h(Rk)r^|ukn=r^h(k)|ukn=ϵn(k)r^|ukn.(B10)

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    r^|ukn=m|ukBkmn,(B11)

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    Bkmn=uRkm|r^|ukn(B12)

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    Bkmn[ϵn(k)ϵm(Rk)]=0(B13)

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    [Θ,r^]=0.(B14)

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    Θ(r^|ukl)=Θ(n|uRknBknl)=m,n|uRkmVRkmnBknl*.(B15)

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    r^(Θ|ukl)=r^(m|uknVknl)=m,n|uRkmBkmnVknl.(B16)

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    n(VRkmnBknl*BkmnVknl)=0(B17)

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    {rΠn}=TRS{rΠn*}.(B18)

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    ai·bj=2πδij.(C1)

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    P=eSBZTr[A(k)]d2k,(C2)

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    P=e01ds101ds2Tr[A(s1b1+s2b2)],(C3)

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    μie2π01ds101ds2Tr[A(s2b2+s1b1)]·bi,(C4)

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    P·bi=2πμi.(C5)

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    P·bi=(p1a1·bi+p2a2·bi)=2πpi.(C6)

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    p1=μ1mode,p2=μ2mode.(C7)

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    P=piaipiTnijaj.(C8)

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    P=piai(pi+ni)ai,(C9)

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    pjTnji=pi+ni.(C10)

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    p1=n2n13,p2=2n2+n13,  for  C3  symmetry.(C11)

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    p1=p2mod  e,forC3  symmetry.(C12)

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    T(dB)=10log10PoutPin=10log10poutdSpindS,(G1)

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    Rui Zhou, Hai Lin, Yanjie Wu, Zhifeng Li, Zihao Yu, Y. Liu, Dong-Hui Xu, "Higher-order valley vortices enabled by synchronized rotation in a photonic crystal," Photonics Res. 10, 1244 (2022)
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