• Photonics Research
  • Vol. 10, Issue 4, 1071 (2022)
Pingzhun Ma1, Junda Zhu2, Ying Zhong3, and Haitao Liu1、*
Author Affiliations
  • 1Institute of Modern Optics, College of Electronic Information and Optical Engineering, Nankai University, Tianjin Key Laboratory of Micro-scale Optical Information Science and Technology, Tianjin 300350, China
  • 2College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China
  • 3State Key Laboratory of Precision Measurement Technology and Instruments, Tianjin University, Tianjin 300072, China
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    DOI: 10.1364/PRJ.449154 Cite this Article Set citation alerts
    Pingzhun Ma, Junda Zhu, Ying Zhong, Haitao Liu. Theories of indirect chiral coupling and proposal of Fabry–Perot resonance as a flexible chiral-coupling interface[J]. Photonics Research, 2022, 10(4): 1071 Copy Citation Text show less

    Abstract

    The chiral coupling of an emitter to waveguide mode, i.e., the propagation direction of the excited waveguide mode is locked to the transverse spin (T-spin) of a circularly polarized emitter, has exhibited unprecedented applications in nanophotonics and quantum information processing. This chiral coupling can be largely enhanced in terms of unidirectivity, efficiency, and spontaneous emission rate by introducing resonant modes as coupling interfaces. However, this indirect chiral coupling still undergoes limitations in flexibility and miniaturization, and the underlying physical mechanisms are to be clarified. Here, we present an intuitive and rigorous approach for analyzing the direct/indirect chiral coupling, and thereout, derive some general relations between the chiral-coupling directionality and the T-spin of the field or emitter. Based on the theories, we propose an indirect chiral-coupling system on the platform of surface plasmon polariton (SPP), with a nanocavity supporting Fabry–Perot (FP) resonance of dual SPP modes serving as a novel coupling interface. The FP resonance provides flexible design freedoms which can modulate the chirality of the T-spin (and the resultant chiral-coupling directionality) to flip or disappear. A unidirectivity up to 99.9% along with a high coupling efficiency and enhancement of spontaneous emission rate is achieved. Two first-principles-based SPP models for the reciprocal and original problems are built up to verify the decisive role of the FP resonance in achieving the chiral coupling. The proposed theories and novel chiral-coupling interface will be beneficial to the design of more compact and flexible chiral-coupling systems for diverse applications.
    β+,m=E,mcoupling(r0)·pψ,m|ψ+,m,β,m=E+,mcoupling(r0)·pψ,m|ψ+,m,

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    ψ1|ψ2=z=z1(E2×H1E1×H2)·zdxdy,

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    β+,m=βm+iβm||=Ex,,mcoupling(r0)+iEz,,mcoupling(r0)ψ,m|ψ+,m,(3a)

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    β,m=βmiβm||=Ex,+,mcoupling(r0)+iEz,+,mcoupling(r0)ψ,m|ψ+,m,(3b)

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    Ex,,mcoupling(r0)=Ex,+,mcoupling(r0),Ez,,mcoupling(r0)=Ez,+,mcoupling(r0).

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    Ex,,mcoupling(r0)=iEz,,mcoupling(r0)Ex,+,mcoupling(r0)=iEz,+,mcoupling(r0),

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    Ex,+,mcoupling(r0)=iEz,+,mcoupling(r0)Ex,,mcoupling(r0)=iEz,,mcoupling(r0),

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    g=S3(r0).

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    g=2Im(pxpz*)|px|2+|pz|2.

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    a+sym=α+sym+b+symusymrbsym,(9a)

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    b+sym=a+symusymrasym,(9b)

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    a+asym=α+asym+b+asymuasymrbasym,(9c)

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    b+asym=a+asymuasymraasym,(9d)

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    a+sym/asym=ψsym/asym|ψ+,totalψsym/asym|ψ+sym/asym,

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    a+=α+1u2rarb,(11a)

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    b+=α+ura1u2rarb,(11b)

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    a=α1u2rarb,(12a)

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    b=αura1u2rarb,(12b)

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    ψ+coupling(r)=a+symusymψRsym(r)+a+asymuasymψRasym(r),(13a)

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    ψcoupling(r)=asymusymψRsym(r)+aasymuasymψRasym(r),(13b)

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    β+=asymusymEx,Rsym(r0)+iaasymuasymEz,Rasym(r0)ψ|ψ+,(14a)

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    β=a+symusymEx,Rsym(r0)+ia+asymuasymEz,Rasym(r0)ψ|ψ+,(14b)

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    a+sym=asym=asym,a+asym=aasym=aasym,

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    2k0Re(nneffasym)L+arg(raasym)+arg(rbasym)=2nπ.

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    Ex,+coupling(r0)=Ex,coupling(r0)=Excoupling(r0)=asymusymEx,Rsym(r0),(17a)

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    Ez,+coupling(r0)=Ez,coupling(r0)=Ezcoupling(r0)=aasymuasymEz,Rasym(r0).(17b)

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    Γtotal=12Re[Ex,source(r0)+assymusymEx,Rsym(r0)]12Im[Ez,source(r0)+asasymuasymEz,Rasym(r0)].

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    δ(zz0)=12π+exp[ik(zz0)]dk,(A1)

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    j(r)=12π+pδ(xx0)δ(yy0)exp[ik(zz0)]dk.(A2)

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    Em(r,k)=Em(ρ,k)exp(ikz),(A3)

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    E(r,r0,k)=m=1Cm(r0,k)Em(ρ,k)exp(ikz),(A4)

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    Cm(r0,k)=p·Em(ρ0,k)exp(ikz0)2πi[ωωm(k)]Pm(k),(A5)

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    Pm(k)=+dxdy{Em(ρ,k)·[ωε(ω)]ω|ω=ωm·Em(ρ,k)Hm(ρ,k)·[ωμ(ω)]ω|ω=ωm·Hm(ρ,k)}.(A6)

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    E(r,r0)=+E(r,r0,k)dk.(A7)

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    E(r,r0)=m=1+f(k)dk,(A8)

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    f(k)=exp[ik(zz0)]2πi[ωωm(k)]Pm(k)p·Em(ρ0,k)Em(ρ,k).(A9)

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    +f(k)dk=+f(k)dk+C+f(k)dk=2πilimkkm(kkm)f(k)=exp[ikm(zz0)]vg,mPm(km)p·Em(ρ0,km)Em(ρ,km).(A10)

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    +f(k)dk=+f(k)dk+Cf(k)dk=2πilimkkm(k+km)f(k)=exp[ikm(zz0)]vg,mPm(km)p·Em(ρ0,km)Em(ρ,km),(A11)

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    E(r,r0)=m=1{Θ(zz0)A+,mEm(ρ,km)exp[ikm(zz0)]+Θ(z0z)A,mEm(ρ,km)exp[ikm(zz0)]},(A12)

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    A+,m=p·Em(ρ0,km)vg,mPm(km),A,m=p·Em(ρ0,km)vg,mPm(km).(A13)

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    vg,mPm(km)=ψ,m|ψ+,m,(A14)

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    E,mcoupling(r0)=Em(ρ0,km)exp(ikmz0),(A15a)

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    E+,mcoupling(r0)=Em(ρ0,km)exp(ikmz0).(A15b)

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    E(r,r0)=G(r,r0)·p,(A16)

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    G(r,r0)=m=11vg,mPm(km){Θ(zz0)Em(ρ,km)Em(ρ0,km)exp[ikm(zz0)]+Θ(z0z)Em(ρ,km)Em(ρ0,km)exp[ikm(zz0)]}.(A17)

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    Em(ρ,km)=Em*(ρ,km).(A18)

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    G(r,r0)=m=11vg,mPm(km){Θ(zz0)Em(ρ,km)Em*(ρ0,km)exp[ikm(zz0)]+Θ(z0z)Em*(ρ,km)Em(ρ0,km)exp[ikm(zz0)]},(A19)

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    β+,m=Ex,mcoupling(r0)iEz,mcoupling(r0)ψ,m|ψ+,m,(A20a)

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    β,m=Ex,mcoupling(r0)+iEz,mcoupling(r0)ψ,m|ψ+,m,(A20b)

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    Ex,mcoupling(r0)=Ex,,mcoupling(r0)=Ex,+,mcoupling(r0),(A21a)

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    Ez,mcoupling(r0)=Ez,,mcoupling(r0)=Ez,+,mcoupling(r0).(A21b)

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    |β+,m|2=|Ex,mcoupling|2+|Ez,mcoupling|22Im[Ex,mcoupling(Ez,mcoupling)]|ψ,m|ψ+,m|2,(A22a)

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    |β,m|2=|Ex,mcoupling|2+|Ez,mcoupling|2+2Im[Ex,mcoupling(Ez,mcoupling)]|ψ,m|ψ+,m|2,(A22b)

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    g=|β+,m|2|β,m|2|β+,m|2+|β,m|2=2Im[Ex,mcoupling(Ez,mcoupling)]|Ex,mcoupling|2+|Ez,mcoupling|2=S3,(A23)

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    β+,m=Ex,,mcouplingpx+Ez,,mcouplingpzψ,m|ψ+,m=Ex,,mcouplingpxiEx,,mcouplingpzψ,m|ψ+,m=pxipzψ,m|ψ+,m/Ex,+,mcoupling,(A24a)

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    β,m=Ex,+,mcouplingpx+Ez,+,mcouplingpzψ,m|ψ+,m=Ex,+,mcouplingpx+iEx,+,mcouplingpzψ,m|ψ+,m=px+ipzψ,m|ψ+,m/Ex,+,mcoupling,(A24b)

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    |β+,m|2=|px|2+|pz|22Im(pxpz*)|ψ,m|ψ+,m/Ex,+,mcoupling|2,(A25a)

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    |β,m|2=|px|2+|pz|2+2Im(pxpz*)|ψ,m|ψ+,m/Ex,+,mcoupling|2.(A25b)

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    g=|β+,m|2|β,m|2|β+,m|2+|β,m|2=2Im(pxpz*)|px|2+|pz|2=S3,source,(A26)

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    as=bsurb,(B1a)

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    bs=γ+asura,(B1b)

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    as=γurb1u2rarb,(B2a)

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    bs=γ1u2rarb.(B2b)

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    ψtotal=ψsource+assymusymψRsym+asasymuasymψRasym,(B3)

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    β+=bssymusymχ+sym+bsasymuasymχ+asym,(B4a)

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    β=bssymusymχsym+bsasymuasymχasym,(B4b)

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    χsym/asymψ|ψ+=α+sym/asymψsym/asym|ψ+sym/asym,(B5a)

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    χ+sym/asymψ|ψ+=αsym/asymψsym/asym|ψ+sym/asym,(B5b)

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    γsym/asymψsym/asym|ψ+sym/asym=ERsym/asym(r0)·pσ,(B6)

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    γsymψsym|ψ+sym=Ex,Rsym(r0),(B7a)

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    γasymψasym|ψ+asym=iEz,Rasym(r0).(B7b)

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    β+=bssymusymαsymEx,Rsym(r0)γsymψ|ψ++bsasymuasymαasymiEz,Rasym(r0)γasymψ|ψ+,(B8a)

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    β=bssymusymα+symEx,Rsym(r0)γsymψ|ψ++bsasymuasymα+asymiEz,Rasym(r0)γasymψ|ψ+.(B8b)

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    β+=(α1u2rarb)symusymEx,Rsym(r0)ψ|ψ++(α1u2rarb)asymiuasymEz,Rasym(r0)ψ|ψ+,(B9a)

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    β=(α+1u2rarb)symusymEx,Rsym(r0)ψ|ψ++(α+1u2rarb)asymiuasymEz,Rasym(r0)ψ|ψ+,(B9b)

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    Pingzhun Ma, Junda Zhu, Ying Zhong, Haitao Liu. Theories of indirect chiral coupling and proposal of Fabry–Perot resonance as a flexible chiral-coupling interface[J]. Photonics Research, 2022, 10(4): 1071
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