計算分析變化單位晶格角度下二維光子晶體的最大完全能隙

Calculation of maximal full band gaps for two-dimension photonic crystals with variable unit lattice angle

陳楠(Nan Chen)

指導教授 ： 趙聖德

國立臺灣大學/工學院/應用力學研究所/碩士(2011年)

https://doi.org/10.6342/NTU.2011.02835

摘要

在以往的研究當中，對於光子晶體內介電區域的研究已經廣泛的被研究，然而尚未有在研究晶格角度對最大完全帶隙(Full Band Gap)的關連。我們主要依照平面波展開法(Plane Wave Expansion)的理論來研究將空氣圓柱擺放在介電區域(ε)或是介電圓柱(ε)擺設在空氣區域的狀況，擺設的方式又分為兩種，其一者為將一根圓柱擺設在單位晶格(unit cell)原點位置上，另一者為將兩根圓柱擺設在單位晶格內距原點1/6以及5/6的對角線位置上。使其為變化圓柱幾何半徑參數(radius/lattice constant=r/a)與介電區域的材料參數(ε)並且改變單位晶格基底向量角度參數(angle=θ)的二維光子晶體。找出其在 TM(transverse magnetic)與TE(transverse electric)模態電磁波下的最大完全帶隙。當圓柱擺在原點時，角度參數(θ)為60度時即為三角晶格(triangular lattice)，角度為90度即為四角晶格(square lattice)，而當圓柱擺在對角線上時，角度參數(θ)為60度即為六角晶格或稱蜂窩晶格（Hexagonal lattice）。
　　我們將探討此角度參數(θ)對最大完全能隙以及重光子態影響的趨勢。研究的主要的計算上是使用布洛赫理論(Bloch Theorem)模擬電磁波其在光子晶體結構內交互作用之結果。首先利用威格納-賽茲晶格(Wigner-Seitz cell)來定義週期性結構在倒晶格空間所對應的布里淵區(Brillouin Zones)來簡化週期性結構，在變化角度參數(θ)的狀況下，布里淵區不同於一般三角或四角晶格有高對稱性，而是由四個梯型組成的一個低對稱性的幾何區域，所以我們需要較多的波向量(k-point)來計算布洛赫向量與頻率的色散關係(dispersion relation)。然而原始晶格與倒晶格在角度參數(θ)上顯示出互為補角的關係，所以可知角度參數(θ)只要掃描到90度即可知道到90~180度之情形。
　　我們主要發現最大完全能隙出現在將空氣柱擺放在介電區域的晶格圓點的時候，在TM band2~3 與TE band1~2之間，頻率由ω＝0.387(ωa/2πc)到0.529(ωa/2πc)，能隙寬度0.142(ωa/2πc)，其為相對能隙寬度為30.98%，其角度參數(θ)為60度，介電常數(ε)為19，幾何常數(r/a)為0.493。在尋找最大完全能隙的四個案子中，有三個最大的都在角度參數(θ)為60度的時候，顯示三角晶格確實較容易找到最大完全能隙。然而有一個例外為，當將介電柱放在空氣區域的晶格原點時，最大完全能隙出現在布里淵區為低對稱性，角度為50度、介電常數(ε)為11.8，幾何常數(r/a)為0.2的時候，這顯示並不是布里淵區接近圓形一定會有較大的最大完全能隙出現。而11.8為矽材料的介電常數(ε)，若是有光子晶體製程被限制在晶格原點-介電圓柱的時候，我們推薦用角度50度的光子晶體來做製程。
　　我們發現介電常數(ε)、幾何常數(r/a)變大，能量填充率上升，會讓單柱Mie氏共振產生紅移，使得能帶與能隙紅移。當角度參數(θ)變大，能量填充率下降，晶格週期性間隔差異變小，能隙呈現些微紅移現象。是由於角度參數(θ)變大使柱體之間的距離變小，此距離拉近時對單柱Mie氏共振產生些微紅移。而且三個參數都有完全能隙下緣頻率在出現最大完全能隙參數點前的紅移速率大於出現後的紅移速率的現象。此現象可以在將來用在混用多種參數時，經由比較完全能隙下緣頻率的斜率來快速的找到最大完全能隙的參數點。
　　角度參數(θ)、材料參數(ε)與幾何參數(r/a)對於重光子現象的分析中，對於任意偏振方向的電磁波都是在擁有最大完全能隙參數點時有最強的重光子現象。而角度參數(θ)對重光子現象影響相較於材料參數(ε)與幾何參數(r/a)亦不明顯，因為只改變角度參數(θ)時，晶體內高介電區的廣度(幾何參數r/a)與強度(材料參數ε)都沒改變，造成變化角度時單柱散射體Mie氏散射模型是固定的。因此改變角度參數(θ)不能直接影響重光子態，而是透過柱體間的距離影響單柱的Mie氏共振來間接影響重光子態。

關鍵字

光子晶體 ； 光子能隙 ； 晶格角度 ； Mie氏共振 ； 重光子態

並列摘要

We are mainly in accordance with the theory of plane wave expansion method(PWE) to study the air-cylinder or dielectric-cylinder in two different locations of unit periodic lattice. We have three parameter which include the radius/lattice constant (r/a); the material dielectric (ε) and the base vector angle of unit cell (θ) in the two-dimensional photonic crystals. We find the maximum completely band gap (Full Band Gap) and the Heavy Photon effects under TM (transverse magnetic) wave and TE (transverse electric) wave. One located the cylinder at origin of the unit cell. The other one located two cylinder were to be furnished away from the origin within 1/6 or 5/6 of the diagonal line of the unit cell.
　　When the cylinder at origin, the angle parameter(θ) of 60 degrees is the triangular lattice, 90 degree is the square lattice, and when the cylinder placed on the diagonal line, the angle parameter(θ) of 60 degree is the hexagonal lattice or called honey lattice. We will discuss the point of angle parameter on the trend of the largest band gap. Calculation using the Bloch Theorem to simulate the interaction results of the electromagnetic waves in photonic crystal structures. However, we use the Wigner-Seitz cell which simplify the periodic structure to define the Brillouin zone of the reciprocal lattice space. Under varying angle, the Brillouin zone is different from triangular or square lattice with high symmetry, but formed by four trapezoidal with low symmetry. Therefore, we need more of the Bloch vector to calculate the relationship of the dispersion relation.Our main findings were as follows:
(1)The largest full band gap 0.142 (ωa / (2πc)) appears when the triangular lattice of air columns at the original, angle parameter (θ = 60), ε = 19, r / a = 0.493, changing the angle parameters (θ) proved changes were not greater than the triangular lattice band gap generation.
(2)When dielectric column located at origin, the largest full band gap appears in the angle parameter (θ) is 50 degrees, ε =11.8,r/a = 0.2, the pattern is not the higher symmetry of the Brillouin zone ,the display higher symmetry Brillouin zone not necessarily have high band gap.The dielectric constant 11.8 is silicon,this finding may help photonic crystals processes.
(3)Angle parameter (θ) have relation with periodic boundary condition and fill factor. when the angle becomes larger, fill factor dropped, then the periodic boundary of unit cell becomes larger, band gap showed red shift. Because only change the angle parameters, high dielectric region (geometric parameters) and strength (material parameters) did not change inside the unit cell, resulting in changes of angle Mie scatter model is fixed . The red shift of band gap are mainly due to scatterer distance reduce.
(4)Angle parameter (θ), geometric parameters (r / a) and material parameters (ε) in the band gap red shift analysis, the lower edge of the full band gap frequency before the maximum point appears redshift velocity greater than after the maximum point.Using in mix parameters, this phenomenon can fast search to the largest full band gap.
(5)geometric parameters (r / a) and material parameters (ε) and the angle parameter(θ) analysis are in the parameter point that have maximum full band gap have the strongest heavy photon effects. But angle parameter (θ) for the heavy photon effects is not so obvious, mainly because of changes in the angle parameters (θ) can not directly affect the heavy photon state, but through the scatterer distance with the Mie resonance to indirect affect the localization phenomenon in high dielectric area .

並列關鍵字

Photonic crystals ； Photonic band gap ； Lattice angle ； Mie resonance ； Heavy photon

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