積體電路為利用半導體製程做於矽晶圓上的微小電路，使得電子能夠於晶片內傳輸信號來進行運算，但是電子傳輸信號於速度上有其極限，而光子相較於電子傳輸速度更快，故利用光子作為信號傳輸及運算的積體光路由此而生。積體光路由波導管負責傳遞光訊號，而由於尺寸關係，當兩根波導管互相靠近時其模態能量會相互交換，造成定向耦合(directional coupling)現象，故在縮小結構尺寸的同時又能控制波導間的耦合效應有其必要性，若將金屬添加到波導管結構中，使波導以表面電漿模態(surface plasmon polaritons mode)進行傳遞，則可以帶來許多好處，像是尺寸相較於純介電質波導可以更小、擁有可加偏壓的電極、可傳遞熱量等優勢，而若將金屬改為以介電質、金屬層層交互相疊構成的雙曲超穎材料(hyperbolic metamaterial)，則前述優勢不但可以保存，還能在不改變波導尺寸的前提下，藉由更改雙曲超穎材料特性調整其耦合長度，使其有更廣泛的應用。除了利用雙曲超穎材料，一般的介電質波導也能透過光學的PT對稱(Parity-time symmetry，宇稱時間對稱)來控制波導的耦合效應，藉由分別賦予兩波導管等值的增益及損耗，便能控制波導管的耦合行為，及控制兩波導管接收端的強度比例，以作為邏輯運算的用途。
本論文主要分析兩平行並排波導管其模態的耦合行為，雙曲超穎材料的部分，先利用二維空間的有限差分時域法(two-dimensional Finite-Difference Time-Domain，2D FDTD)中的一個分支，也就是1.5D compact FDTD來解出以雙曲超穎材料作為纖衣之電漿波導在波長1550 nm的表面電漿模態(雙曲超穎材料為金屬、介電質交互層層週期性排列的材料，其中金屬於該材料中所占的比例為其重要參數)。接著利用2D FDTD來確認模態於單一波導管中傳遞的行為及計算平行並排兩波導管的耦合長度，隨後利用1.5D compact FDTD結合波導耦合理論解出耦合長度，其結果與利用2D FDTD所計算的結果一致，但運算速度更快，原本兩天以上的計算變成只需要十分鐘內即可完成，大幅降低了所需時間，之後我們比較不同金屬占比下模態差異性及耦合長度，結果表明，金屬占比越低，模態耦合時能量受到金屬的衰減較少，故越容易傳遞至相鄰的波導管，使得耦合長度越低，金屬占比越高則反之，我們也利用1.5D compact FDTD分析了存在於雙曲超穎材料中的模態bulk plasmon polaritons mode。另外，由於可利用compact FDTD結合波導耦合理論快速解出耦合長度，故我們也利用此方法模擬含有"高度"參數的三維結構，結果表明，波導結構的"高度"越大，模態更容易侷限於波導管中，與鄰近波導管的能量交換越小，導致耦合長度越長。
PT對稱系統的部分，我們一樣利用1.5D compact FDTD及2D FDTD來進行分析，透過賦予兩波導管相同大小的增益及損耗來達成光學中PT對稱的兩種狀態，分別為PT對稱態(PT-symmetric phase)及PT破壞態(PT-broken phase)，在FDTD與理論公式能夠互相驗證的情況下，藉由改變兩波導管增益及損耗的大小，便能調整波導管的耦合行為。

The Metal-Insulator-Metal (MIM) waveguide can transmit signal by using surface plasmon polaritons (SPPs) mode. By change the metal claddings into hyperbolic metamaterials (HMM) to form HIH waveguide, it can also allow SPPs mode for transmission. When two parallel HIH waveguides placed side by side, their mode energy would be exchanged due to directional coupling. By conventional Finite-Difference Time-Domain (FDTD) and compact FDTD method, the coupling length can be found and can be controlled by changing the composition ratio of hyperbolic metamaterials without changing the total size of the cladding layer.
In addition to use the hyperbolic metamaterials, we can also use the Parity-time (PT) symmetry coupled waveguides system to control the coupling effect. By giving same gain and loss value to each of the coupled waveguides, the output strength of two waveguides would be controlled, such as (1,0) or (0,1). By combining HMM and PT symmetry, one can envision a future compact photonics logical operators.

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