• Photonics Research
  • Vol. 6, Issue 5, A31 (2018)
D. J. Nodal Stevens1, Benjamín Jaramillo Ávila2, and B. M. Rodríguez-Lara1、3、*
Author Affiliations
  • 1Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, Mexico
  • 2CONACYT–Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico
  • 3Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico
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    DOI: 10.1364/PRJ.6.000A31 Cite this Article Set citation alerts
    D. J. Nodal Stevens, Benjamín Jaramillo Ávila, B. M. Rodríguez-Lara. Necklaces of PT-symmetric dimers[J]. Photonics Research, 2018, 6(5): A31 Copy Citation Text show less

    Abstract

    We study light propagation through cyclic arrays, composed by copies of a given PT-symmetric dimer, using a group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggests the use of these devices as output port replicators where the dynamics is controlled by the impinging light field. This is confirmed in good agreement with finite element propagation in an experimentally feasible necklace of passive PT-symmetric dimers constructed from lossy and lossless waveguides.
    izE0(z)=iγE0(z)+gdeiϕdE1(z)+gNeiϕNE2N1(z),izE1(z)=iγE1(z)+gdeiϕdE0(z)+gNeiϕNE2(z),izE2j(z)=iγE2j(z)+gdeiϕdE2j+1(z)+gNeiϕNE2j1(z),izE2j+1(z)=iγE2j+1(z)+gdeiϕdE2j(z)+gNeiϕNE2j+2(z),izE2N2(z)=iγE2N2(z)+gdeiϕdE2N1(z)+gNeiϕNE2N3(z),izE2N1(z)=iγE2N1(z)+gdeiϕdE2N2(z)+gNeiϕNE0(z),(1)

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    iz|E(z)=H^|E(z),(2)

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    |E(z)=(E0(z)E1(z)E2(z)E3(z)E2N2(z)E2N1(z))}(j=0)th  dimerk=0k=1}(j=1)th  dimerk=0k=1}(j=N1)thdimerk=0k=1.(3)

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    H^=(H^dH^0^20^H^+H^+H^dH^0^0^0^H^+H^d0^0^0^0^0^H^dH^H^0^0^H^+H^d),(4)

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    H^d=(iγgdeiϕdgdeiϕdiγ),H^+=(0gNeiϕN00),H^=(00gNeiϕN0),(5)

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    H^d=γiσ^z+gdeiϕdσ^++gdeiϕdσ^,H^±=gNe±iϕN,(6)

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    σ^z=(1001),σ^+=(0100),σ^=(0010).(7)

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    C^N=(0000110000010000000000010),C^N=(0100000100000000000110000).(8)

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    H^=1^NH^d+C^NH^++C^NH^,(9)

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    |E(z)=j=0N1k=01E2j+k(z)|jN|k2,(10)

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    F^N=1Nj,k=0N1ei2πNjk|jk|,(11)

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    Λ^N=F^NC^NF^N=j=0N1ei2πNj|jj|.(12)

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    |E(z)=(F^N1^2)|A(z),(13)

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    iz|A(z)=H^D|A(z)(14)

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    H^D=j=0N1|jj|H^j,(15)

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    H^j=iγσ^z+[Γjσ^++Γj*σ^],(16)

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    Γj=gdeiϕd+gNeiϕNei2πNj.(17)

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    λ±j=±|Γj|2γ2,=±gd2+gN2+2gdgNcos(ϕd+ϕN+2πNj)γ2,(18)

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    γ2g=cos2πNj,j=0,1,2,N2(N21),(19)

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    |E(z)=U^(z)|E(0),(20)

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    U^(z)=j=0N1F^N|jj|F^NeiH^jz,(21)

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    eiH^jz=1^2cosΩjz+izH^jsinc  Ωjz,(22)

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    Ωj=|Γj|2γ2,(23)

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    eiH^jz=1^2cosΩjz+iΩjH^jsinΩjz,ΩjR.(24)

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    eiH^jz=1^2+izH^j,Ωj=0.(25)

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    eiH^jz=1^2cosh|Ωj|z+i|Ωj|H^jsinh|Ωj|z,ΩjC.(26)

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    |E(0)=F^N|j|ψ(0),=1Na=0N1ei2πNaj|a|ψ(0);(27)

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    |E(z)=1Np=0N1ei2πNjp|peiH^jz|ψ(0),(28)

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    Pj,k(z)=|k|eiH^jz|ψ(0)|2k=01|k|eiH^jz|ψ(0)|2,(29)

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    limzPj,0(z)=γ|Ωj|2γ,limzPj,1(z)=γ+|Ωj|2γ,(30)

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    iz(E1(z)E2(z)E3(z)E4(z))=(βR+iβIgd0gNgdβRgN00gNβR+iβIgdgN0gdβR)(E1(z)E2(z)E3(z)E4(z)),(31)

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    iz(E1(z)E2(z)E3(z)E4(z))=(iγgd0gNgdiγgN00gNiγgdgN0gdiγ)(E1(z)E2(z)E3(z)E4(z)),(32)

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    Pj,k(z)=|Ey[dx(1)j+k,dy(1)j,z]|2|Ey[dx(1)j,dy(1)j,z]|2+|Ey[dx(1)j+1,dy(1)j,z]|2.(33)

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    D. J. Nodal Stevens, Benjamín Jaramillo Ávila, B. M. Rodríguez-Lara. Necklaces of PT-symmetric dimers[J]. Photonics Research, 2018, 6(5): A31
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