• Photonics Research
  • Vol. 11, Issue 6, 936 (2023)
Jie Yang1,2, Xuezhi Zheng1,4,*, Jiafu Wang2,5,*, Anxue Zhang2..., Tie Jun Cui3 and Guy A. E. Vandenbosch1|Show fewer author(s)
Author Affiliations
  • 1WaveCoRE Research Group, KU Leuven, Leuven B-3001, Belgium
  • 2Xi’an Jiaotong University, Xi’an 710049, China
  • 3State Key Laboratory of Millimeter Wave, Southeast University, Nanjing 210096, China
  • 4e-mail: xuezhi.zheng@esat.kuleuven.be
  • 5e-mail: wangjiafu1981@126.com
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    DOI: 10.1364/PRJ.485625 Cite this Article Set citation alerts
    Jie Yang, Xuezhi Zheng, Jiafu Wang, Anxue Zhang, Tie Jun Cui, Guy A. E. Vandenbosch, "Polarization singularities in planar electromagnetic resonators with rotation and mirror symmetries," Photonics Res. 11, 936 (2023) Copy Citation Text show less

    Abstract

    In this work, we apply the group representation theory to systematically study polarization singularities in the in-plane components of the electric fields supported by a planar electromagnetic (EM) resonator with generic rotation and reflection symmetries. We reveal the intrinsic connections between the symmetries and the topological features, i.e., the spatial configuration of the in-plane fields and the associated polarization singularities. The connections are substantiated by a simple relation that links the topological charges of the singularities and the symmetries of the resonator. To verify, a microwave planar resonator with the D8 group symmetries is designed and numerically simulated, which demonstrates the theoretical findings well. Our discussions can be applied to generic EM resonators working in a wide EM spectrum, such as circular antenna arrays, microring resonators, and photonic quasi-crystals, and provide a unique symmetry perspective on many effects in singular optics and topological photonics.
    lz={j+Mq,j<M/2Mj+Mq,j>M/2,

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    E=|EL|eilLφL^+|ER|eilRφeiβR^.

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    I=(lRlL)/2.

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    lL=(j+1+qLM),lR=(j1+qRM),

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    I=1+(qLqR)M2.

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    Δφ=2N1+1M2jπ,ψ=2N2πjπ2N1+1M2j,

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    PiΓ=d(Γ)NRΓii*(R)·PR.(A1)

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    PRf(r)=f(R1·r),PRf(r)=R·f(R1·r).(A2)

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    PiΓf(r)=fiΓ(r),PiΓf(r)=fiΓ(r).(A3)

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    PRfiΓ(r)=Γii(R)fiΓ(r),PRfiΓ(r)=Γii(R)fiΓ(r).(A4)

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    L^=12(1+i),R^=12(1i).(B1)

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    fj(r)=fjL(r)L^+fjR(r)R^.(B2)

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    PRfj(r)=R·fj(R1·r)=R·[fjL(R1·r)L^+fjR(R1·r)R^].(B3)

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    R·L^=L^eiθ0,R·R^=R^eiθ0,(B4)

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    PRfj(r)=fjL(ρ,θθ0)eiθ0L^+fjR(ρ,θθ0)eiθ0R^.(B5)

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    PRfj(r)=eijθ0[fjL(ρ,θ)L^+fjR(ρ,θ)R^].(B6)

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    fjL(ρ,θθ0)=ei(j+1)θ0fjL(ρ,θ),(B7)

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    fjR(ρ,θθ0)=ei(j1)θ0fjR(ρ,θ).(B8)

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    fjL(ρ,θθ0)=ei(j+1+qLM)θ0fjL(ρ,θ),(B9)

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    fjR(ρ,θθ0)=ei(j1+qRM)θ0fjR(ρ,θ).(B10)

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    fjL(ρ,θ)=ujL(ρ,θ)ei(j+1+qLM)θ.(B11)

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    fjL(ρ,θθ0)=ujL(ρ,θθ0)ei(j+1+qLM)(θθ0)=ei(j+1+qLM)θ0·ujL(ρ,θ)ei(j+1+qLM)θ=ei(j+1+qLM)θ0·fjL(ρ,θ).(B12)

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    fjR(ρ,θ)=ujR(ρ,θ)ei(j1+qRM)θ.(B13)

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    lL=(j+1+qLM),lR=(j1+qRM).(B14)

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    I=12(lRlL)=1+(qLqR)M2.(B15)

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    fj(r)=ujL(ρ,θ)ei(j+1+qLM)θL^+ujR(ρ,θ)ei(j1+qRM)θR^.(B16)

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    f0A1(r)=f0(r)+Psf0(r),f0A2(r)=f0(r)Psf0(r).(B17)

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    s=s1=(cosθ0sinθ0sinθ0cosθ0).(B18)

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    s1·r=(cosθ0sinθ0sinθ0cosθ0)·(ρcosθρsinθ)=(ρcos(θ0θ)ρsin(θ0θ)),(B19)

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    s·L^=(cosθ0sinθ0sinθ0cosθ0)·12(1+i)=12(1i)eiθ0=R^eiθ0,(B20)

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    s·R^=(cosθ0sinθ0sinθ0cosθ0)·12(1i)=12(1i)eiθ0=L^eiθ0.(B21)

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    Psf0(r)=u0L(ρ,θ0θ)ei(qLM+1)(θ0θ)R^eiθ0+u0R(ρ,θ0θ)ei(qRM1)(θ0θ)L^eiθ0=u0L(ρ,θ)e+i(qLM+1)θR^+u0R(ρ,θ)ei(qRM1)θL^.(B22)

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    f0A1(r)=[u0L(ρ,θ)ei(qLM+1)θ+u0R(ρ,θ)ei(qRM1)θ]L^+[u0R(ρ,θ)ei(qRM1)θ+u0L(ρ,θ)e+i(qLM+1)θ]R^,(B23)

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    f0A2(r)=[u0L(ρ,θ)ei(qLM+1)θu0R(ρ,θ)ei(qRM1)θ]L^+[u0R(ρ,θ)ei(qRM1)θu0L(ρ,θ)e+i(qLM+1)θ]R^.(B24)

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    f0A1(r)=[u0L(ρ,θ)+u0R(ρ,θ)]ei(qM+1)θL^+[u0R(ρ,θ)+u0L(ρ,θ)]e+i(qM+1)θR^,(B25)

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    f0A2(r)=[u0L(ρ,θ)u0R(ρ,θ)]ei(qM+1)θL^+[u0R(ρ,θ)u0L(ρ,θ)]e+i(qM+1)θR^.(B26)

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    lL=(1+qM),lR=+(1+qM).(B27)

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    f4B1(r)=f4(r)+Psf4(r),f4B2(r)=f4(r)Psf4(r).(B28)

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    lL=(M2+1+qM),lR=(M21+qM).(B29)

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    S0=Ix+Iy=|Ex|2+|Ey|2,(C1)

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    S1=IxIy=|Ex|2|Ey|2,(C2)

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    S2=I45°+I45°=2Re(Ex*·Ey),(C3)

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    S3=ILCPIRCP=2Im(Ex*·Ey).(C4)

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    S12=S1+iS2=|S12|eiϕ12.(C5)

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    σ12=12πϕ12·ds.(C6)

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    ICorIV=σ122.(C7)

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    Jie Yang, Xuezhi Zheng, Jiafu Wang, Anxue Zhang, Tie Jun Cui, Guy A. E. Vandenbosch, "Polarization singularities in planar electromagnetic resonators with rotation and mirror symmetries," Photonics Res. 11, 936 (2023)
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