• Photonics Research
  • Vol. 5, Issue 2, 57 (2017)
Zhen-Zhen Liu1, Qiang Zhang1, Yuntian Chen2、3, and Jun-Jun Xiao1、*
Author Affiliations
  • 1College of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China
  • 2School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China
  • 3Wuhan National Laboratory of Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China
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    DOI: 10.1364/PRJ.5.000057 Cite this Article Set citation alerts
    Zhen-Zhen Liu, Qiang Zhang, Yuntian Chen, Jun-Jun Xiao. General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss[J]. Photonics Research, 2017, 5(2): 57 Copy Citation Text show less

    Abstract

    The exceptional point (EP) is one of the typical properties of parity–time-symmetric systems, arising from modes coupling with identical resonant frequencies or propagation constants in optics. Here we show that in addition to two different modes coupling, a nonuniform distribution of gain and loss leads to an offset from the original propagation constants, including both real and imaginary parts, resulting in the absence of EP. These behaviors are examined by the general coupled-mode theory from the first principle of the Maxwell equations, which yields results that are more accurate than those from the classical coupled-mode theory. Numerical verification via the finite element method is provided. In the end, we present an approach to achieve lossless propagation in a geometrically symmetric waveguide array.
    H=[β1+jγ1κ12κ21β1+Δβ+jγ2],(1)

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    β±=β1+Δβ+j(γ1+γ2)2±[Δβ+i(γ2γ1)]24+κ12κ21.(2)

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    jddz(aAaB)=(βA+jγAκAκBβB+jγB)(aAaB),(3)

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    κA(B)=ωϵ02SA(B)(ϵcoϵb)uA(B)*·uB(A)dxdy,(4)

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    γA(B)=ωϵ02SA(B)ϵA(B),IuA(B)*·uA(B)dxdy.(5)

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    t×em+jβmz×em+=jωμhm+,(6)

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    t×hm+jβmz×hm+=jωϵ0ϵrem+,(7)

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    E=(nanen)exp[j(ωt+βz)],(8)

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    H=(nanhn)exp[j(ωt+βz)].(9)

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    t×(nanen)+jβz×(nanen)=jωμ(nanhn),(10)

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    t×(nanhn)+jβz×(nanhn)=jωϵ0(ϵr+Δϵ)(nanen).(11)

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    nan[j(ββm)pmn+bmn+jkmn]=0.(12)

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    bmn={·(hm+×en)·(hn×em+)}dxdy,(13)

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    pmn={z·(hm+×en)z·(hn×em+)}dxdy,(14)

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    kmn=ωϵ0Δϵen·em+dxdy.(15)

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    nan[(ββn)pmn+kmn]=0.(16)

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    H·a=H21·H1·a=βa,(17)

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    am[(ββm)pmm+kmm]=0.(18)

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    [β1p11k11β2p12k12β1p21k21β2p22k22][a1a2]=β[p11p12p21p22][a1a2].(19)

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    Zhen-Zhen Liu, Qiang Zhang, Yuntian Chen, Jun-Jun Xiao. General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss[J]. Photonics Research, 2017, 5(2): 57
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