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仿真中将石墨烯建模为有厚度的3-D材料,利用介电常数εg来表征其电特性。εg可由石墨烯的表面电导率σg导出,二者的关系为εg=1+iσg/(ωε0dg)[10],其中, ω为入射光角频率,ε0为空气的绝对介电常数,dg为单层石墨烯的厚度。电导率σg由带内部分和带外部分组成。在太赫兹波段, μc$ \gg $kBT0条件下,σg可表示为[11]:
$ {\sigma _{\rm{g}}} = \frac{{{e_0}^2{\mu _{\rm{c}}}}}{{p{{\rlap{--} h}^2}}}\frac{{\rm{i}}}{{w + {\rm{i}}{\tau ^{ - 1}}}} $
(1) 式中,μc为石墨烯化学势,受外加电压和化学掺杂控制,kB为玻尔兹曼常数,T0为绝对温度,e0为单位电荷,$\rlap{--} h$为约化普朗克常量,p为周期,w为宽度,τ为弛豫时间,与石墨烯质量有关。
为了获得结构的吸收率,将采用严格耦合波分析(rigorous coupled-wave analysis,RCWA)法对结构进行仿真。RCWA法广泛用于求解周期结构电磁衍射问题,通过矩阵分析方法可求得电磁波的传输率T和反射率R以及吸收率A=1-T-R。具体求解步骤如下[12]:首先对光栅进行适当分层,然后对每一层内的电磁场进行傅里叶级数展开,导出光栅上层和其下层的电磁场表达式;其次,利用傅里叶级数对光栅的介电常数做展开,根据麦克斯韦方程组推导出耦合波方程;接下来,利用电磁边界条件解出每一层本征模式场的振幅、传播系数等物理参量,并确定光栅衍射效率;最后,通过分别累加前向衍射效率和后向衍射效率来获得传输率和反射率。
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图 1中给出了嵌有石墨烯条带的缺陷光子晶体结构。F-P谐振腔顶部和底部反射镜分别由多层介质(SiO2/Si)m和(Si/SiO2)n构成,其中m和n代表周期数;太赫兹范围内SiO2和Si的介电常数ε1和ε2分别为3.9和11.9,每个周期内二者的厚度d1和d2分别设为λ0/($4\sqrt {{\varepsilon _1}} $)和λ0/($4\sqrt {{\varepsilon _2}} $),其中, λ0为参考波长;腔内填充介质为SiO2,腔长dc设为2d1;石墨烯条带置于谐振腔正中间,占空比η=w/p;电磁波以TM波的形式入射到结构上,设入射角为θ。取参考波长λ0=60μm,则底部反射镜有效反射带宽为4.12THz~5.88THz[13]。为了实现双模完美吸收,相关参量设置如下:μc=0.6eV,τ=0.5ps,m=4, n=10, p=0.8λ0, η=0.6, θ=0°。
结构的光响应频谱如图 2所示。由于底部多层介质的光子禁带效应,系统传输率在整个目标频段内为0,以至A=1-R,提高了系统的重吸收能力。在5.1537THz和5.1970THz处,吸收率达到了0.9882和0.9825,实现了双模完美吸收,两模式依次命名为模式1和模式2。当结构阻抗等于自由空间阻抗时,结构反射率为0,系统能够获得全吸收。结构的等效阻抗能够表示为[14]:
$ Z = \sqrt {\left[ {{{\left( {1 + {S_{11}}} \right)}^2} - {S_{21}}^2} \right]/\left[ {{{\left( {1 - {S_{11}}} \right)}^2} - {S_{21}}^2} \right]} $
(2) 式中,S11和S21分别代表端口1的反射系数和端口1到端口2的传递系数。图 3中给出了阻抗Z随频率的变化情况。这里Re(Z)和Im(Z)分别代表Z的实部和虚部。显然,在模式1和模式2处,Re(Z)≈1和Im(Z)≈0成立,结构获得了良好的阻抗匹配。
为了说明吸收模式的起源,图 4中给出了光子晶体缺陷腔内部模式1和模式2对应的归一化磁场H分布。可见,两模式下腔内均有纵向磁场局域,其表现出F-P谐振的特点,而在石墨烯的位置,磁场表现出横向局域,其反映出SPP共振的特点。所以,两完美吸收模式是在F-P谐振和SPP共振耦合作用下产生的。
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通过电调谐石墨烯的化学势能够显著改变石墨烯器件的性能。图 5中给出了系统吸收谱随μc的变化情况。当μc较低时,系统仅有一个弱的F-P谐振吸收模,此时石墨烯表现出弱的金属性。随着μc增加,石墨烯的金属性不断增强,以至其SPP共振效应开始凸显。此外,F-P谐振模式向高频方向移动,这种现象可用微扰理论来解释。腔内谐振波长偏移满足以下关系[15]:
$ \Delta \lambda /\lambda = \Delta \varepsilon \cdot \Delta V/V $
(3) 式中, λ为谐振波长,Δλ为波长的偏移量,Δε为腔的扰动引起的介电常数的变化,ΔV为扰动体积,V为腔的体积。μc增加导致Δε减小,所以F-P谐振模式蓝移。当μc>0.5eV时,系统在F-P谐振和SPP共振的双重作用下获得了两个完美吸收模式; 当μc=0.7eV,两模式完全耦合,二者峰峰距很小,其可用于探测毗邻的入射光。
石墨烯作为结构中唯一的吸波材料其尺寸直接影响着系统的光吸收性能。图 6中分别给出了石墨烯条带周期p和占空比η的变化对系统吸收率的影响。在图 6a中,随着p的增加,SPP共振模幅度和位置均随之改变,以至完美吸收模式数目不断变化。当p为0.7λ0或0.8λ0时,系统保持双模吸收,且两模式处于耦合状态。在图 6b中,随着η的增加,F-P谐振模式蓝移,而SPP共振模位置保持稳定。当η=0.7时,两模式完全耦合。可见,调节石墨烯条带周期和占空比可以直接控制模式间的耦合程度,改变模式峰峰距,这有利于对不同频差的双路光信号进行探测。对比完整石墨烯薄层加载的缺陷光子晶体吸波体[6],石墨烯条带加载方式能够提高系统调控的2个自由度。
图 7中给出了入射光的角度对系统吸收谱的影响。当入射光少许偏离垂直入射,模式数目增加。由于F-P谐振模式不会分裂,所以模式数目增加来源于SPP共振模式分离[16]。垂直入射时,±m级次SPP共振模式简并;斜入射时,简并性被破坏,模式开始分离,分别向高频和低频方向移动。随着入射角度增大,SPP共振模式衰减愈发严重。当θ=4°时,系统将仅仅保留高吸收的F-P谐振模式。
基于F-P谐振与SPP共振的石墨烯双模吸波体设计
Design of graphene double-mode absrober based on F-P resonance and SPP resonance
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摘要: 为了实现石墨烯的双模吸收,将石墨烯条带嵌入到缺陷光子晶体,采用严格耦合波法进行了仿真研究,并进行了关键参量的影响分析。发现由于Fabry-Pérot谐振和表面等离子激元共振的共同作用,系统在5.1537THz和5.1970THz处获得了两个完美吸收模式,此时结构阻抗等于自由空间阻抗。结果表明,电调谐石墨烯化学势不仅能够改变模式数目,还能够改变模式的耦合程度;当化学势为0.7eV时,两吸收模完全耦合;调节石墨烯条带的几何尺寸也能控制模式的耦合程度;当入射光偏离垂直入射时,模式数目直接受偏角大小的影响。该研究为太赫兹器件吸收谱的重构提供了思路。
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关键词:
- 光学器件 /
- 石墨烯吸波体 /
- Fabry-Pérot谐振 /
- 表面等离子激元共振 /
- 模式耦合
Abstract: To realize the double-mode absorption, the graphene ribbon was embedded into the defective photonic crystal. Based on the rigorous coupled wave method, it is found that the system achieves two perfect absorption modes at 5.1537THz and 5.1970THz, due to the Fabry-Pérot resonance and surface plasmon polariton resonance. At the same time, the impedance of the whole structure was equal to that of the free space. Analysis of key parameters indicate that both modes number and modes-coupling extent can be changed by tuning the chemical potential of graphene electrically. When the chemical potential is 0.7eV, two absorption modes are coupled wholly. Also, the coupling extent can be controlled by adjusting the geometry of graphene ribbon. When the incident light deviates from the normal incidence, the modes number is directly affected by the deflection. This study is helpful to reconstruct the absorption spectrum of terahertz devices. -
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