1.Collaborative Innovation Center of Suzhou Nano Science and Technology, School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2.Key Lab of Thin Film Materials of Jiangsu Province, Suzhou 215006, China 3.School of Optoelectronic Science and Engineering, Soochow University, Suzhou 215006, China 4.Analysis and Testing Center, Soochow University, Suzhou 215123, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 11975163) and the Priority Academic Program Development of Jiangsu Province Higher Education Institutions (PAPD), China
Received Date:01 February 2021
Accepted Date:23 March 2021
Available Online:07 June 2021
Published Online:05 September 2021
Abstract:Photonic crystals are widely used in a class of narrow-band frequency selective filter due to their excellent ability to control electromagnetic waves, in which the working frequency depends on the structural parameters of the point defect resonant cavity of the photonic crystal, and the introduction of some dispersive media into the cavity makes the filter adjustable. In general, this kind of cavity-filter is very sensitive to the parameter disturbance of the cavity, and the quality factor of the filter can be reduced significantly by material loss. On the other hand, some studies have shown that there may be bound states at the interface between two different photonic crystals, and the bound state is often accompanied by narrow band and high transmittance, which implies that a narrow-band filter based on bound states is feasible. Importantly, filters based on bound states may be resistant to material loss to some degree. In this paper, a bound state related tunable narrow-band filter composed of a one-dimensional photonic crystal and a two-dimensional plasma photonic crystal is proposed, and the working frequency of the filter is located in the common band gap of the two photonic crystals. The COMSOL Multiphysics finite element simulation software is used to study the influences of geometric parameters of the one-dimensional photonic crystal and plasma parameters on the performance of the filter. It is found that the closer to each other the center frequencies and depths of the two different forbidden bands are, the greater the peak transmittance of the filter, in which the center frequency dominates, will be. On the other hand, the working frequency of the filter is directly proportional to plasma density and inversely proportional to collision frequency. The quality factor of the filter first increases and then decreases with the increase of plasma density, and decreases with the increase of collision frequency. The peak transmittance of the filter first increases and then decreases with the increase of plasma density, and decreases with the increase of plasma collision frequency. Finally, with the increase of collision frequency, both the peak transmittance and the quality factor decrease slightly, which indicates that the filter has a certain resistance to plasma loss. We believe that this work is helpful in investigating some new plasmonic photonic crystal filters. Keywords:plasma/ photonic crystal/ tunable filter/ bound state
其中, $ {Z}_{\rm{L}} $和$ {Z}_{\rm{R}} $分别表示左侧和右侧光子晶体的表面阻抗. PCs的有效阻抗$ {Z}_{\rm{e}\rm{f}\rm{f}} $可以表示为${Z}_{\rm{e}\rm{f}\rm{f}} = $$ {\sqrt{{\mu }_{\rm{e}\rm{f}\rm{f}}}}/{\sqrt{{\varepsilon }_{\rm{e}\rm{f}\rm{f}}}}$. 在公共带隙内, 如果两种PCs的$\varepsilon_{\rm{e}\rm{f}\rm{f}}$异号, 就可能满足条件$ {Z}_{\rm{L}} $ + $ {Z}_{\rm{R}}=0 $. 文献[19, 20]则指出, 两种PCs的$\varepsilon_{\rm{e}\rm{f}\rm{f}}$异号就是指两种PCs带隙的拓扑性质不同. 因此, 束缚态依赖两个拓扑性质不同的光子带隙. 若要改变带隙的拓扑性质, 需要调节PCs参数让带隙先关闭再打开, 使得上下能带发生交换[17,20]. 对于放电管构成的PPCs, 可以调节等离子体密度使次序为$ n $（$ n $为大于0的正整数）的带隙关闭, 然后增加等离子体密度重新打开带隙[21]. 图2展示了调节过程, 其中, 2维 (2D) PPCs参数为${a}_{x} \!=\! 40\;\rm{m}\rm{m}$, ${a}_{y} \!=\! 12\;\rm{m}\rm{m}$, ${r}_{1} \!=\! 4\;\rm{m}\rm{m}$, ${r}_{2} \!=\! 5\;\rm{m}\rm{m}$, ${N}_{2} \!=\! 10$, ${\nu }_{\rm{e}} \!=\! 0$. 计算中取$ n=1 $. 本文利用有限元仿真软件COMSOL对模型进行数值计算. 图 2 随等离子体密度增加, 2D PPC的1阶带隙经历了从关闭到再打开的过程 Figure2. As the plasma density increases, the first-order band gap of 2D PPC undergoes a process from closing to reopening
从图2中可以看出, 随等离子体密度增加, 带隙经历了从关闭到再打开的过程. 这种现象与波阻抗有关[22]. 详细地说, 调节等离子体密度可以让石英放电管的有效阻抗和背景介质的阻抗相等, 从而关闭带隙. 进一步增大等离子体密度会破坏阻抗匹配条件, 从而打开带隙. 带隙从关闭到重新打开的过程就是上下能带交换的过程, 带隙的拓扑性质因此发生变化. 在图1所示模型中, 首先调节等离子体密度$ {n}_{\rm{e}} $使2D PPCs的1阶带隙先消失再重新打开, 再调节1D PCs参数以确保两者的1阶带隙重合, 从而获得束缚态. 图3展示了调节过程, 其中, 图3(a)和图3(b)分别表示1D PCs和两种2D PPCs的能带图, 图中均以黄色区域表示公共带隙, 两者的区别在于图3(b)中2D PPCs的带隙是重新打开的带隙. 图3(c)表示上述两种情况下的透射谱, 图3(d)表示束缚态在滤波器中的场分布. 在计算中, 1D PCs的参数为$ L=20\;\rm{m}\rm{m} $, $ d=0.45 L $, $ {N}_{1}=3 $. 2D PCs的参数为$ {a}_{x}=40\;\rm{m}\rm{m} $, $ {a}_{y}=12\;\rm{m}\rm{m} $, ${r}_{1}= $$ 4 \;\rm{m}\rm{m}$, $ {r}_{2}=5\;\rm{m}\rm{m} $, $ {N}_{2}=10 $, $ {\nu }_{\rm{e}}=0 $. 对于图3(a), $ {n}_{\rm{e}}=0 $. 对于图3(b), $ {n}_{\rm{e}}=6\times {10}^{11}{\rm{c}\rm{m}}^{-3} $. 图 3 (a) $ {n}_{\rm{e}}=0 $时两种光子晶体的能带图; (b) ${n}_{\rm{e}}=6\times {10}^{11}\;{\rm{c}\rm{m}}^{-3}$时两种光子晶体的能带图; (c) $ {n}_{\rm{e}}=0 $和${n}_{\rm{e}} =6\;\times $$ {10}^{11}\;{\rm{c}\rm{m}}^{-3}$时滤波器的透射谱; (d) 电场强度沿模型边界的线分布, 其中, 插图表示滤波器中的电场分布 Figure3. (a) Band structure of two different PCs with $ {n}_{\rm{e}}=0 $; (b) band structure of two different PCs with ${n}_{\rm{e}}=6\times {10}^{11}\;{\rm{c}\rm{m}}^{-3}$; (c) trans-mission spectrum of the filter when $ {n}_{\rm{e}}=0 $ and ${n}_{\rm{e}}=6\times {10}^{11}\;{\rm{c}\rm{m}}^{-3}$; (d) the intensity of electric field along the line of the model boundary, where the inset shows the electric field distribution in the filter.
首先研究1D PCs参数对滤波器性能的影响, 图4(a)和图4(b)分别表示滤波器透射谱随1D PCs周期数和氧化铝厚度的变化情况. 在计算中, 2D PPCs参数不变, 即 $ {a}_{x}=40\;\rm{m}\rm{m} $, $ {a}_{y}=12\;\rm{m}\rm{m} $, $ {N}_{2}=10 $, $ {n}_{\rm{e}}=6\times {10}^{11}\;{\rm{c}\rm{m}}^{-3} $, $ {\nu }_{\rm{e}}=0 $. 与此同时, 保持1D PCs的周期长度$ L=20\;\rm{m}\rm{m} $不变. 在图4(a)中, 氧化铝厚度$ d $为$0.45 L$, 在图4(b)中, 1D PCs周期数${N}_{1}=3.$ 图 4 (a)滤波器透射谱与$ {N}_{1} $的关系; (b) 滤波器透射谱与d的关系 Figure4. (a) The relationship between the transmission spectrum of the filter and $ {N}_{1} $; (b) the relationship between the transmission spectrum of the filter and d.
从图4(a)可以看出, 约4.08 GHz附近存在一个透射峰. 随着周期数量$ {N}_{1} $增大, 透射峰的中心频率有微小变化. 此外, 峰值透射率先增大后减小, 当$ {N}_{1}=3 $时, 峰值透射率达到最大. 在图4(b)中, 随着氧化铝厚度$ d $增大, 透射峰的中心频率向低频移动, 峰值透射率先增大后减小. 图4表明存在一个最佳1D PCs参数. 研究发现禁带匹配度可以衡量1D PCs参数对滤波器的影响. 禁带匹配度由禁带深度（以透射率为单位）和禁带的中心频率进行表征. 定义中心频率匹配度${{{\varDelta}} }_{\rm{f}}$为${{{\varDelta}} }_{\rm{f}}=(1- |f_1- \bar f|/\bar f)\times 100{\text{%}}$, 禁带深度匹配度$ {{{\varDelta}} }_{\rm{s}} $为${{{\varDelta}} }_{\rm{s}}= (1- |s_1-\bar{s}|/\bar{s}) \times 100{\text{%}}$. 其中, $ {f}_{1} $和$ {s}_{1} $分别是1D PCs的禁带中心频率和禁带深度, $ {f}_{2} $和$ {s}_{2} $分别是2D PPCs的禁带中心频率和禁带深度, $\bar{f}=\left({{f}_{1}+{f}_{2}}\right)/{2}$, $\bar{s}= $$ \left({{s}_{1}+{s}_{2}}\right)/{2}$. 图5展示了禁带匹配度随1D PCs参数的变化情况, 其中, 图5(a)展示了禁带匹配度与$ {N}_{1} $的关系, 图5(b)展示了禁带匹配度与$ d $的关系. 可以看到, 禁带匹配度随1D PCs参数不断变化, 并且当禁带匹配度最大时, 峰值透射率最高. 例如在图5(a)中, 当$ {N}_{1} \!=\!3 $时, $ {{{\varDelta}} }_{\rm{f}}\!=\!98{\text{%}} $, $ {{{\varDelta}} }_{\rm{s}} \!=\! 77.6{\text{%}} $, 两者均为最大值, 而在图4(a)中, $ {N}_{1}=3 $可以获得最大的峰值透射率. 因此, 匹配度越大, 峰值透射率越大, 该规律在图5(b)中同样存在. 有趣的是, 研究发现峰值透射率更依赖$ {{{\varDelta}} }_{\rm{f}} $. 例如在图5(b)中, 当$ d/L $分别为0.35和0.55时, 前者的$ {{{\varDelta}} }_{\rm{s}} $较小但$ {{{\varDelta}} }_{\rm{f}} $更大. 而在图4(b)中, $ d/L \!=\! 0.35 $可以获得更大的峰值透射率. 图 5 (a)禁带匹配度与$ {N}_{1} $的关系; (b)禁带匹配度与$ d $的关系 Figure5. (a) The relationship between the band gap matching degree and $ {N}_{1} $; (b) the relationship between the band gap matching degree and $ d $.
23.2.等离子体密度的影响 -->
3.2.等离子体密度的影响
为了进一步研究滤波器的性能, 图6展示了等离子体密度${n}_{{\rm{e}}}$对滤波器性能的影响. 其中, 图6(a)表示滤波器透射谱与$ {n}_{\rm{e}} $的关系, 图中还以点线表示透射峰中心频率随$ {n}_{\rm{e}} $的变化情况. 图6(b)表示滤波器峰值透射率和品质因子Q与$ {n}_{\rm{e}} $的关系, 其中, 品质因子Q定义为[16]： 图 6 (a) 滤波器透射谱与$ {n}_{\rm{e}} $的关系, 点线表示滤波器的中心频率随$ {n}_{\rm{e}} $的变化; (b) 滤波器峰值透射率以及品质因子与$ {n}_{\rm{e}} $的关系 Figure6. (a) The relationship between the transmission spectrum of the filter and $ {n}_{\rm{e}} $, while the dotted line shows the evolution of the center frequency of the filter with $ {n}_{\rm{e}} $; (b) the relationship between the peak transmittance of the filter as well as the quality factor and $ {n}_{\rm{e}} $
最后研究等离子体碰撞频率$ {\nu }_{\rm{e}} $对滤波器性能的影响, 计算结果如图7所示. 其中, 图7(a)表示滤波器透射谱与$ {\nu }_{\rm{e}} $的关系, 图中还以点线表示透射峰中心频率随$ {\nu }_{\rm{e}} $的变化情况. 图7(b)表示滤波器峰值透射率和品质因子与$ {\nu }_{\rm{e}} $的关系. 在计算中, $ {n}_{\rm{e}}=6\times {10}^{11}\;{\rm{c}\rm{m}}^{-3} $, 其余参数与图6一致. 图 7 (a) 滤波器透射谱与$ {\nu }_{\rm{e}} $的关系, 点线表示滤波器的中心频率随$ {\nu }_{\rm{e}} $的变化; (b) 滤波器峰值透射率以及品质因子与$ {\nu }_{\rm{e}} $的关系 Figure7. (a) The relationship between the transmission spectrum of the filter and $ {\nu }_{\rm{e}} $, while the dotted line shows the evolution of the center frequency of the filter with $ {\nu }_{\rm{e}} $; (b) the relationship between the peak transmittance of the filter as well as the quality factor and $ {\nu }_{\rm{e}} $.
图 1 滤波器示意图, 下图展示了两种光子晶体的原胞
图 2 随等离子体密度增加, 2D PPC的1阶带隙经历了从关闭到再打开的过程
图 3 (a) 
图 4 (a)滤波器透射谱与
图 5 (a)禁带匹配度与
图 6 (a) 滤波器透射谱与
图 7 (a) 滤波器透射谱与
