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如图1所示,半无限一维光子晶体结构模型由左手材料L和右手材料R沿Z轴方向周期排列而成。其中L层为左手材料,其物理厚度为
${d_{\rm L}}$ ,介电常数和磁导率分别为${\varepsilon _{\rm L}}$ 与${\mu _{\rm L}}$ ,R层为右手材料,其物理厚度为${d_{\rm R}}$ ,介电常数和磁导率分别为${\varepsilon _{\rm R}}$ 与${\mu _{\rm R}}$ ,C层为L介质表面覆盖介质薄层,其厚度为${d_{\rm C}}$ ,最左侧A区是入射介质,为无限大的空气层,其介电常数和磁导率分别为${\varepsilon _0}$ 与${\mu _0}$ 。 -
为最大限度的减小不规则排列、厚度标量等因素对光子晶体带隙的影响,可通过参数直接匹配的方法使光子晶体的平均折射率
$\overline n = (2{n_{\rm L}}{d_{\rm L}} + {n_{\rm R}}{d_{\rm R}})/\varLambda = 0$ ,式中$\varLambda = 2{d_{\rm L}} + {d_{\rm R}}$ 是光子晶体的晶格常数。因为光子晶体介质界面处折射率高低转换以及局域电场的变化由平均折射率$\overline n $ 是否等于零条件决定,当光子晶体满足$\overline n = 0$ 时出现不同于布拉格散射所形成的带隙,称零平均折射率带隙[2,5]。故在光子晶体的平均折射率$\overline n = 0$ 前提下匹配左、右手材料的参数${\varepsilon _{\rm L}}$ 、${\varepsilon _{\rm R}}$ 、${\mu _{\rm L}}$ 、${\mu _{\rm R}}$ 、${d_{\rm L}}$ 和${d_{\rm R}}$ ,对于左手材料${n_{\rm L}} = - \sqrt {{\varepsilon _{\rm L}}{\mu _{\rm {\rm L}}}}$ ,右手材料${n_{\rm R}} = \sqrt {{\varepsilon _{\rm R}}{\mu _{\rm R}}}$ 。以图1模型结构为例,当电磁波沿$z$ 正向传播时,若TE偏振的磁场量平行于$x$ 轴,则TM偏振的电场矢量平行于$y$ 轴。根据Bloch-Floquet定理可得:$$\cos ({K_{\rm B}}\varLambda ) = \dfrac{1}{2}Tr({M_1}{M_2}) = {\varsigma ^{\alpha \gamma }} - \eta _ + ^{\alpha \gamma }{\xi ^{\alpha \gamma }}$$ (1) 式中:M为介质层内电场的传输矩阵;
${K_{\rm B}}$ 为布洛赫波矢;${\varsigma ^{\alpha \gamma }} = \cos ({k_{\alpha z}}{d_\alpha })\cos ({k_{\gamma z}}{d_\gamma })$ ,${\xi ^{\alpha \gamma }} = \sin ({k_{\alpha z}}{d_\alpha })\sin ({k_{\gamma z}}{d_\gamma })$ ;$\eta _ \pm ^{\alpha \gamma } = ({\eta _\alpha }/{\eta _\gamma } \pm {\eta _\gamma }/{\eta _\alpha })$ ,$(\alpha ,\gamma = {\rm L},{\rm R})$ 。当光子晶体的表面被其他介质层所替代或被截断时,光子晶体的原有周期结构被打破,表面布洛赫波矢[10-13,15]为${{K}_{\rm B}} = iq + m{\text{π}} /\varLambda$ 。则公式(1)写成:$${( - 1)^n}\cosh (q\varLambda ) = {\varsigma ^{{\rm L}{\rm R}}} - \eta _ + ^{{\rm L}{\rm R}}{\xi ^{{\rm L}{\rm R}}}$$ (2) 将半无限结构光子晶体与表面替代介质联合起来并利用布洛赫定理进行数学处理[10-13,15],可得:
$$ \begin{split} &({\varsigma ^{{\rm L}{\rm R}}} - \eta _ + ^{{\rm L}{\rm R}}{\xi ^{{\rm L}{\rm R}}}) -\\ &\dfrac{{({\varPsi ^{{\rm R}{\rm L}}} + \eta _ + ^{{\rm L}{\rm R}}{\varPsi ^{{\rm L}{\rm R}}})({\varPsi ^{{\rm R}s}} + \eta _ + ^{s{\rm R}}{\varPsi ^{s{\rm R}}}) - \eta _ - ^{{\rm L}{\rm R}}\eta _ - ^{s{\rm R}}{\xi ^{{\rm L}s}}}}{{({\varsigma ^{s{\rm R}}} - \eta _ + ^{s{\rm R}}{\xi ^{s{\rm R}}})}} - 1 = 0 \end{split} $$ (3) $$\begin{split} &\tanh (q\varLambda ) =\\ &\frac{{({\varPsi ^{{\rm R}{\rm L}}} + \eta _ + ^{{\rm L}{\rm R}}{\varPsi ^{{\rm L}{\rm R}}})({\varPsi ^{{\rm R}s}} + \eta _ + ^{s{\rm R}}{\varPsi ^{s{\rm R}}}) - \eta _ - ^{{\rm L}{\rm R}}\eta _ - ^{s{\rm R}}{\xi ^{{\rm L}s}}}}{{({\varsigma ^{{\rm L}{\rm R}}} - \eta _ + ^{{\rm L}{\rm R}}{\xi ^{{\rm L}{\rm R}}})({\varsigma ^{s{\rm R}}} - \eta _ + ^{s{\rm R}}{\xi ^{s{\rm R}}})}} - 1 = 0 \end{split} $$ (4) 公式(3)是光子晶体表面波随频率
${v_s}$ 变化的函数,也是表面替代介质与光子晶体表面界面之间的表面波色散方程,式中$\varPsi $ 为矩阵函数,$q$ 为布洛赫波矢。由公式(3)、(4)可得${q_s}$ 值,${q_s}$ 的大小与对应表面波具有正相关系,则从${q_s}$ 理论上可得半无限一维光子晶体表面波的性质。再结合磁场与电场的关系、电磁场的边界条件等,可得:$${{\rm e}^{i{K_{\rm B}}\varLambda }} = \frac{{\sin {k_1}{d_{\rm L}} + \gamma \cosh {k_1}{d_{\rm L}}}}{{\gamma \cosh {k_2}{d_{\rm R}} - \dfrac{{{F_1}}}{{{F_2}}}\sinh {k_2}{d_s}}} \equiv B(\omega )$$ (5) $${E^{P{\rm C}}}(z) = {C_1}\sin ({k_1}{d_{\rm L}}) + \gamma \cos ({k_1}{d_{\rm L}})$$ (6) $${H^{P{\rm C}}}(z) = - \dfrac{1}{{\varepsilon \mu }}\dfrac{{\partial {E^{P{\rm C}}}}}{{\partial z}}$$ (7) $B(\omega )$ 为色散函数,式中$\gamma $ 、${C_1}$ 系数由边界连续条件确定。结合公式(2)和(3)可得:$${{\rm e}^{i{K_{\rm B}}a}} = {( - 1)^m}{{\rm e}^{ - \eta a}} = s(\omega ) = B(\omega ) \pm \sqrt {B{{(\omega )}^2} - 1} $$ (8) 式中:正、负号对应
$m$ 为偶数与奇数情况。于是光子晶体带隙中的通带区域和禁带区域即可通过公式(8)来判断,当${{\rm e}^{i{K_{\rm B}}a}}$ <1为电磁波可传播的通带区域,当${{\rm e}^{i{K_{\rm B}}a}}$ >1时为禁止电磁波传播的禁带区域。
Band gap and local electric field characteristics of surface waves in left-handed and right-handed materials of photonic crystal
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摘要: 为研究与设计新型光波导和光学传感器,通过参数匹配,并利用平面波展开的传输矩阵理论及布洛赫定理,研究左右手材料光子晶体的能带结构、表面波局域电场分布等,结果表明:零平均折射率左右材料光子晶体能带中存在的半封闭状和封闭状禁带结构,且通带中的能级曲线由高频向低频方向振荡衰减并简并。添加表面覆盖层介质后,部分半封闭状及封闭状禁带中出现正向波和反向波分立能级,分立能级随覆盖层厚度增大向波矢减小方向移动,半封闭状禁带中的分立能级在覆盖层厚度达到一定数值时出现分裂现象。禁带中正向表面波局域电场极大值均处于覆盖层与光子晶体表面交界处附近,并随覆盖层厚度增大或远离交界处而衰减,封闭状禁带对应的局域电场极大值对覆盖层厚度的响应灵敏度弱于半封闭状禁带。禁带中的反向表面波局域电场及其极大值均处于光子晶体内部,而且随覆盖层厚度增大而增强,封闭状禁带对反向表面波的局域限制作用、表面波与入射光的耦合作用、局域电场对覆盖层厚度的响应灵敏度等强于半封闭状禁带。Abstract: The energy band structure and the surface wave local electric field distribution of the left-handed and right-handed photonic crystals materials, were studied based on parameter matching by using the transmission matrix theory of plane wave and Bloch theorem in order to study and design novel optical waveguide and optical sensor. The results show that there are semi closed and closed band gap structures in the energy band of the left-handed and right-handed materials with zero mean refractive index, and the energy level curve in the transmission band attenuates from high frequency to low frequency with oscillation. After the surface coating medium is added, the discrete energy level of forward and reverse waves appear in the partially semi-closed and closed band gap of the photonic crystal, the discrete energy level moves to lower wave vector with the increase of the coating thickness, and the discrete energy level in the semi-closed band gap splits when the coating thickness is at a certain value. In the band gap, the maximum value of the local electric field of the forward surface wave and the highest light intensity are near the junction of the coating and the surface of the photonic crystal, and decay with the coating thickness increasing or away from the junction. The response sensitivity of the maximum value of the local electric field corresponding to the closed band gap to the coating thickness is weaker than that of the semi closed band gap. In the band gap, the local electric field, the maximum value of the local electric field of the reverse surface wave and the highest light intensity are in the the photonic crystal, and increase with the coating thickness. The local restriction of the closed band gap on the reverse surface wave, the coupling effect of the surface wave and the incident light, and the response sensitivity of the local electric field to the coating thickness are stronger than those of the semi-closed band gap.
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