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考虑将石墨烯插入介质1和介质2之间,结构模型如图 1所示。其中εj(j=1, 2)表示介质的介电常数,θ为入射角,Dr为GH位移。在接下来的分析中,假定介质1为空气,其介电常数ε1=1,介质2为半无限的六方氮化硼,其介电常数表示为:
$ {\varepsilon _{2, u}} = {\varepsilon _{\infty , u}}\left[ {1 + \frac{{{{\left( {{\omega _{{\rm{L}}0, u}}} \right)}^2} - {{\left( {{\omega _{{\rm{TO}}, u}}} \right)}^2}}}{{{{\left( {{\omega _{{\rm{TO}}, u}}} \right)}^2} - {\omega ^2} - {\rm{i}}\omega {\gamma _u}}}} \right] $
(1) 式中,ω为入射光的角频率,u=⊥或//,分别表示垂直于光轴所在平面和平行于光轴所在的平面,ε∞和γ分别代表高频介电常数和阻尼常数,ωLO和ωTO分别表示横向和纵向光学声子的共振频率。对于垂直于光轴所在的平面有:ε∞, ⊥=4.87,ωLO, ⊥=1610cm-1,ωTO, ⊥=1370cm-1,γ⊥=5cm-1;对于平行于光轴所在的平面有:ε∞, //=2.95,ωLO, //=830cm-1,ωTO, //=780cm-1,γ//=4cm-1。
ARTMANN利用稳态相位法对GH位移的物理机制进行了阐述,其所对应的位移量的表达式如下[4]:
$ {D_{\rm{r}}} = - \frac{\lambda }{{2{\rm{ \mathsf{ π} }}}}\frac{{\partial {\varphi _{\rm{r}}}}}{{\partial \theta }} $
(2) 式中,φr表示反射相位,可以从反射系数r=|r|eiφr计算出来。经过计算,给出了吸收材料反射光束的古斯-汉欣位移[18]:
$ {D_{\rm{r}}} = - \frac{\lambda }{{2{\rm{ \mathsf{ π} }}}}\frac{1}{{|r(\theta ){|^2}}}\left\{ {{\mathop{\rm Re}\nolimits} [r(\theta )]\frac{{{\mathop{\rm dIm}\nolimits} [r(\theta )]}}{{{\rm{d}}\theta }} - } \right.\left. {{\mathop{\rm Im}\nolimits} [r(\theta )]\frac{{{\mathop{\rm dRe}\nolimits} [r(\theta )]}}{{{\rm{d}}\theta }}} \right\} $
(3) 式中,Re[r(θ)]表示反射系数的实部,Im[r(θ)]代表反射系数的虚部。
结构的反射系数可以用传输矩阵方法进行计算。通过单层石墨烯的电磁场可由如下的传输矩阵进行连接:
$ \mathit{\boldsymbol{M}} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} {1 + \eta + \zeta }&{1 - \eta - s\zeta }\\ {1 - \eta + s\zeta }&{1 + \eta - \zeta } \end{array}} \right] $
(4) 对于TE波:
$ \left\{ {\begin{array}{*{20}{l}} {s = - 1}\\ {\eta = \frac{{{k_{2z}}}}{{{k_{1z}}}}}\\ {\zeta = \frac{{\sigma {\mu _0}\omega }}{{{k_{1z}}}}} \end{array}} \right. $
(5) 对于TM波:
$ \left\{ {\begin{array}{*{20}{l}} {s = 1}\\ {\eta = \frac{{{\varepsilon _1}{k_{2z}}}}{{{\varepsilon _{2, \bot }}{k_{1z}}}}}\\ {\zeta = \frac{{\sigma {k_{2z}}}}{{{\varepsilon _0}{\varepsilon _{2, \bot }}\omega }}} \end{array}} \right. $
(6) 则结构的反射系数可以表示为:
$ r=M_{21} / M_{11} $
(7) 式中,Mij为矩阵M的矩阵元,ε0和μ0分别代表真空的介电常数和磁导率。对于TE波,波矢量在z方向上的分量表示为${k_{1z}} = \sqrt {{\varepsilon _1}k_0^2 - k_x^2} $,${k_{2z}} = \sqrt {{\varepsilon _ \bot }k_0^2 - k_x^2} $;对于TM波,波矢量在z方向上的分量表示为${k_{1z}}\sqrt {{\varepsilon _1}k_0^2 - k_x^2} $,${k_{2z}} = \sqrt {{\varepsilon _ \bot }k_0^2 - \left( {{\varepsilon _ \bot }/{\varepsilon _{//}}} \right)k_x^2} $。
σ表示为石墨烯的电导率,一般由带内跃迁和带间跃迁两部分组成,其表达式如下:
$ \sigma = \frac{{{e^2}{E_{\rm{f}}}}}{{{\rm{ \mathsf{ π} }}{\hbar ^2}}}\frac{{\rm{i}}}{{\omega + {\rm{i}}{\tau ^{ - 1}}}} + \frac{{{e^2}}}{{4\hbar }}\left[ {\theta \left( {\hbar \omega - 2{E_{\rm{f}}}} \right) + \frac{{\rm{i}}}{{\rm{ \mathsf{ π} }}}\lg \left| {\frac{{\hbar \omega - 2{E_{\rm{f}}}}}{{\hbar \omega + 2{E_{\rm{f}}}}}} \right|} \right] $
(8) 式中, Ef为石墨烯的化学势,τ为弛豫时间,e为元电荷的电荷量,$\hbar $为约化普朗克常量。
六方氮化硼作为单轴各向异性的非铁磁材料,在TM模式下,其布儒斯特角对应的表达式为[19]:
$ {\theta _{{\rm{TM}}}} = {\sin ^{ - 1}}\sqrt {\frac{{1 - {\varepsilon _1}/{\mathop{\rm Re}\nolimits} \left( {{\varepsilon _{2, \bot }}} \right)}}{{1 - \varepsilon _1^2/{\mathop{\rm Re}\nolimits} \left( {{\varepsilon _{2, \bot }}{\varepsilon _{2, //}}} \right)}}} $
(9)
石墨烯-六方氮化硼结构的古斯-汉欣位移
Goos-Hänchen shift in graphene-hexagonal boron nitride structure
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摘要: 为了研究由石墨烯覆盖半无限六方氮化硼结构中的古斯-汉欣位移性质,采用传输矩阵方法分析了结构参量对反射光古斯-汉欣位移的影响。结果表明, 通过合理调节石墨烯的化学势或层数,均可实现古斯-汉欣位移由正到负的一个转变;通过选取合适的参量,可实现较大的古斯-汉欣位移,其最大值约为波长的450倍。此研究结果对设计光开关、光学传感器件具有重要意义。Abstract: In order to study the properties of the Goos-Hnchen shift in semi-infinite hexagonal boron nitride covered by graphene, the influence of structural parameters on the Goos-Hnchen shift was analysised by using the transfer matrix method. The results show that: By reasonably adjusting the chemical potential or layer number of graphene, the transformation of Goos-Hnchen shift from positive to negative can be realized. By selecting the appropriate parameters, large Goos-Hnchen shifts can be realized and the maximum value is about 450 times of the wavelength. It is of great significance for the design of optical switches, optical couplers and other applications.
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Key words:
- physical optics /
- Goos-Hänchen shift /
- transfer matrix /
- graphene /
- hexagonal boron nitride
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