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  • Received: Jul. 23, 2019

    Accepted: Aug. 30, 2019

    Posted: Dec. 5, 2019

    Published Online: Dec. 3, 2019

    The Author Email:

    DOI: 10.3788/COL201917.122402

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    . Parametric resonances in nonlinear plasmonics [Invited][J]. Chinese Optics Letters, 2019, 17(12): 122402

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d2X(t)dt2+γdX(t)dt+ω02X(t)=F(t),(1)

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d2X(t)dt2+γdX(t)dt+{ω02+2ω0δω[F(t)]}X(t)=0.(2)

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2P1(r,t)t2+γP1(r,t)t=ε0ωpl2E1(r,t),(3)

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P2(r,t)=ε0(ε21)E2(r,t)+P2NL(r,t).(4)

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P1(t)=n,m{rnRn1[Pn,m(e)(t)Yn,m(e)(θ,ϕ)+Pn,m(o)(t)Yn,m(o)(θ,ϕ)]}.(5)

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d2Pn,m(e,o)(t)dt2+γdPn,m(e,o)(t)dt+ωn2Pn,m(e,o)(t)=ωn2Sn,m(e,o)(t)n.(6)

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ωn=nωpl2nε+(n+1)ε2.(7)

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Sn,m(e,o)(t)=r=RYn,m(e,o)(θ,ϕ)P2NL(R,θ,ϕ,t)·r^sin(θ)dθdϕ.(8)

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d2Pn,0(e)(t)dt2+γdPn,0(e)(t)dt+[ωn2α1EP(t)]Pn,0(e)(t)=α2[Pn,0(e)(t)]2.(9)

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α1=4πχzzzn3ωn4ωpl2Gn,0(e,e);α2=χzzznε0ωn6ωpl4Fn,0(e,e,e),Fn,0(e,e,e)=02π0π{cosθz[Rn+2rn+1Yn,0(e)(θ,ϕ)]R×z[Rn+2rn+1Yn,0(e)(θ,ϕ)]RYn,0(e)(θ,ϕ)}sinθdθdϕ,Gn,0(e,e)=02π0π{cosθz[rY1,0(e)(θ,ϕ)]R×z[Rn+2rn+1Yn,0(e)(θ,ϕ)]RYn,0(e)(θ,ϕ)}sinθdθdϕ.(10)

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Pn,m(e,o)(t)=p(t)cos[ωntθ(t)]eγ2t,p(t)=p0cosh(α1Ap2ωnt);θ(t)=arccot[exp(α1Ap2ωnt)],(11)

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Wabs(t)=nR3α1App0232ε0ωpl2[2ωn+γsinh(α1Apt2ωn)]eγtnR3α1App02γ64ε0ωpl2exp[(ApAPPR)α1t2ωn].(12)

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Pn,m(e,o)(t)α2Q122ωn2+Q1cos(ωnt+θ1)+α2Q126ωn2cos(2ωnt+2θ1),θ1=12arccos(APPRAp);Q1=ωnα26γωn5(ApAPPR)21.(13)

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W¯abs(t)=320nR3γε0ωpl2ωn3α22(ApAPPR)21.(14)

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σNL=3nR340ε0ε2ωn3ωpl2α12α22IPPRIp(1IPPRIp),Ip>IPPR.(15)

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