• Photonics Research
  • Vol. 11, Issue 5, B125 (2023)
Qianru Yang1、3, Hao Hu1, Xiaofeng Li2, and Yu Luo1、*
Author Affiliations
  • 1School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
  • 2School of Optoelectronic Science and Engineering & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
  • 3e-mail: qianru002@ntu.edu.sg
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    DOI: 10.1364/PRJ.472233 Cite this Article Set citation alerts
    Qianru Yang, Hao Hu, Xiaofeng Li, Yu Luo. Cascaded parametric amplification based on spatiotemporal modulations[J]. Photonics Research, 2023, 11(5): B125 Copy Citation Text show less

    Abstract

    Active devices have drawn considerable attention owing to their powerful capabilities to manipulate electromagnetic waves. Fast and periodic modulation of material properties is one of the key obstacles to the practical implementation of active metamaterials and metasurfaces. In this study, to circumvent this limitation, we employ a cascaded phase-matching mechanism to amplify signals through spatiotemporal modulation of permittivity. Our results show that the energy of the amplified fundamental mode can be efficiently transferred to that of the high harmonic components if the spatiotemporal modulation travels at the same speed as the signals. This outstanding benefit enables a low-frequency pump to excite parametric amplification. The realization of cascaded parametric amplification is demonstrated by finite-difference time-domain (FDTD) simulations and analytical calculations based on the Bloch–Floquet theory. We find that the same lasing state can always be excited by an incidence at different harmonic frequencies. The spectral and temporal responses of the space-time modulated slab strongly depend on the modulation length, modulation strength, and modulation velocity. Furthermore, the cascaded parametric oscillators composed of a cavity formed by photonic crystals are presented. The lasing threshold is significantly reduced by the cavity resonance. Finally, the excitation of cascaded parametric amplification relying on the Si-waveguide platform is demonstrated. We believed that the proposed mechanism provides a promising opportunity for the practical implementation of intense amplification and coherent radiation based on active metamaterials.
    ε(x,t)=εs[1+2αcos(KxΩt)+2βcosΩt],

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    Ezx=μ0Hyt,(A1a)

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    Hyx=ε(x,t)Ezt.(A1b)

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    Ez(x,t)=m,n=+Em,neikmxiωm,nt,(A2a)

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    Hy(x,t)=m,n=+Hm,neikmxiωm,nt,(A2b)

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    {kmEm,n=c01ωm,nH˜m,nkmH˜m,n=εr,sc01ωm,n(Em,n+αEm1,n+αEm+1,n+βEm,n1+βEm,n+1),(A3)

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    Ez(x,t)=p,m,n=+apEm,n(p)eikm(p)(x+Lm/2)iωm,nt,(A4a)

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    Hz(x,t)=p,m,n=+apHm,n(p)eikm(p)(x+Lm/2)iωm,nt,(A4b)

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    Ez(x=±L2+0+,t;ωm,n)=Ez(x=±L2+0,t;ωm,n),(A5a)

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    Hy(x=±L2+0+,t;ωm,n)=Hy(x=±L2+0,t;ωm,n).(A5b)

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    Ezr(x,t)=m,n=+Em,nreiωm,n[(x+L/2)/vs+t],(A6a)

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    Hzr(x,t)=m,n=+ηsEm,nreiωm,n[(x+L/2)/vs+t],(A6b)

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    Ezt(x,t)=m,n=+Em,nteiωm,n[(xL/2)/vst],(A6c)

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    Hzt(x,t)=m,n=+ηsEm,nteiωm,n[(xL/2)/vst],(A6d)

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    [E(x=L/2+0;ωm,n)H(x=L/2+0;ωm,n)]=Ti[EiEr],(A7a)

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    [E(x;ωm,n)H(x;ωm,n)]=Tp(x)ap for x=±L2+0,(A7b)

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    [E(x=L/2+0+;ωm,n)H(x=L/2+0+;ωm,n)]=Tt[EtO],(A7c)

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    [EtO]=T[EiEr]=[T11T12T21T22][EiEr],(A8)

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    {Er=T221T21EiEt=T11Ei+T12Er.(A9)

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    [T12IT22O][ErEt]=M[ErEt]=O.(A10)

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    ε(x,t)=εs(1+2αcosΩt).(B1)

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    Ez(x,t)=eikxiωt(E1eiΩt+E0).(B2)

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    δκ2δω˜2(α/2)2,(B3a)

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    [E0E1]=[1δκδω˜α/2].(B3b)

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    Ez(x,t)={ei(1+Δ)Ω2(txvs1)+r0ei(1+Δ)Ω2(t+xvs1)+r1ei(1Δ)Ω2(t+xvs1),x0ei(1+Δ)Ω2t[Aei(1+γ)Ω2xvs1+Bei(1+γ)Ω2xvs1C*cei(1γ)Ω2xvs1D*cei(1γ)Ω2xvs1]+ei(1Δ)Ω2t[Acei(1+γ)Ω2xvs1+Bcei(1+γ)Ω2xvs1+C*ei(1γ)Ω2xvs1+D*ei(1γ)Ω2xvs1],x[0,L]t0ei(1+Δ)Ω2[t(xL)vs1]+t1ei(1Δ)Ω2[t(xL)vs1],xL.(B4)

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    r=r0+r1=itan(αKL/4).(B5a)

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    t=t0+t1=exp(iKL/2)cos(αKL/4).(B5b)

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    Qianru Yang, Hao Hu, Xiaofeng Li, Yu Luo. Cascaded parametric amplification based on spatiotemporal modulations[J]. Photonics Research, 2023, 11(5): B125
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