- 學位論文
週期極化壓電超晶格平板的電磁輻射
The Electromagnetic Radiation of Periodically Polarized Piezoelectric Superlattice Plate
摘要
本論文探討波傳方向調變的厚度方向週期極化鈮酸鋰平板的極子波傳行為，週期極化的壓電材料會使得機械波與電磁波發生強烈的耦合，其統御方程必須兼顧牛頓第二運動定律與馬克士威方程式。本文採用平面波展開法求解週期結構的特性，首先推導廣義一維壓電超晶格的二維極子波傳的通解，再針對波傳方向調變厚度方向週期極化的鈮酸鋰材料係數求取其波傳通解，並計入超晶格平板與真空介面的邊界條件以獲得特解。對於有對稱邊界的平板邊界條件，板內的波傳解可解耦成對板厚度方向對稱與反對稱兩種模態，通解的疊加係數必須能夠滿足邊界條件才符合該壓電超晶格平板的頻散關係。本文首度使用正負調變材料係數特有的奇偶函數場型分解的特性計算極子波傳，利用壓電材料係數傅立葉級數偶數項都為零的特性可以將板內的波傳通解分解成偶數項位移耦合奇數項電場EUOE以及奇數項位移耦合偶數項電場OUEE的兩組模態，邊界條件也可分解成相對應的兩組，平板內的波傳可以解耦成對稱與反對稱配合EUOE與OUEE總共可分為四種波傳。由計算得到頻散曲線與對應的場型以探討機電耦合的現象與真空中電磁波傳的行為。由頻散曲線可以得知，奇偶函數場型的分解分開了布里淵區邊界的翻折現象，但其布里淵區邊界兩邊對應的場形仍符合週期結構的波傳行為，再計算能量分佈可以得知在機電強烈耦合的模態，其真空中電磁輻射有很長的衰減距離，電磁波能量確實可以傳遞至遠處。
並列摘要
This thesis studies the behavior of polaritons in a periodically polarized Lithium Niobate plate. The polarization of the plate is in the thickness direction. Periodically polarized piezoelectric structures enable strong coupling of acoustic waves and electromagnetic waves. The governing equations of such coupled waves have to take care of both Newton’s second law and Maxwell’s equations simultaneously. Plane wave expansion method is adopted to study the characteristics of this periodically polarized structure. Two-dimensional plane waves in the general piezoelectric one-dimensional superlattice were solved first. Then the general wave solutions of a Lithium Niobate plate that is modulated periodically in the wave propagation direction were found. The particular solutions have to satisfy the boundary conditions on the plate-vacuum interface. For a plate has symmetry with respect to the middle plane, the solutions can be decoupled into symmetric and antisymmetric modes. The symmetric or antisymmetric particular solution that composed of general solutions must satisfy the boundary conditions to fulfill the dispersion relation. A new concept of even and odd function decomposition that results from the positive-negative periodically varied material coefficients was proposed. The vanishing of even term Fourier coefficients of the periodically varied piezoelectric constant leads to a decomposition of even term displacement couples odd term electric field and odd term displacement couples even term electric field, the EUOE and OUEE modes. Both the general solutions and boundary conditions can be decomposed into EUOE and OUEE modes. Waves in the plate can be grouped into four kinds of modes, the combinations between one of symmetric and antisymmetirc, and one of EUOE and OUEE. Dispersion curves were then plotted, and the field pattern and energy distribution were computed to investigate the acoustic-electromagnetic coupling phenomenon and the electromagnetic waves in vacuum. The even and odd function decomposition separates the folding of the dispersive curves at the Brillouin zone boundary, but the waves still possess the characteristics of periodic structures. Electromagnetic waves in vacuum possess very long attenuation distances for strong coupling cases, that is, the signal can be detected far away from the plate.