自1970年雷射被發明以來，被用在資料傳輸於光纖通訊系統的載波頻率，已從毫米增至微米等級,千倍乃至於萬倍的增加。
光纖通訊系統以雷射為光源，並以分波多工技術為基礎的單模光纖為傳輸媒介，其傳輸容量達到3.2Tb/s或更多在今日已被實現。 過去十年來積體電路的快速重成長，很多心力已經投入在光積體電路核心參數:光波導模態的計算。在本論文裡，我們以新穎數值方法來研究波導問題，並以兩種不同結構說明。首先是縱向結構不變之光波導多模干涉耦合器，在這主題中，我們提出不同於一般的數值方法來探討光波在一任意折射率分佈之縱向結構不變之光波導裡傳播情形。於我們方法中，將電場及折射率以有限項傅立葉級數展開，將這些被展開的級數代入波動方程式，可導出二階矩陣微分方程式，我們已証明這矩陣方程式可被解出且在波導任意縱向位置的波場亦可得到。以本數值方法解此波導結構，可免除使用商業軟體光束傳播法。本研究主題証明了所提方法和商業軟體 近似法有相同結果。本論文另外主題是以新穎數值方法求脊形波導模態折射率，在本方法裡，將脊形波導之截面積分成很多區域，在每一區域將電場及折射率以有限項傅立葉級數展開，將這些展開級數代入波動方程式，而導出了可解得二階矩陣微分方程式。符合邊界條件下，脊形波導每一區域之特徵模態方程式能被導出，進而數值解得模態折射率。本新穎數值方法在本論文裡以三種不同幾何大小脊形波導來驗證，數值結果証明了本數值方法非常高效率及高精確度，其誤差達到10-5 ~10-6 左右。另外我們對以矽長在絕緣材料(SOI)之脊形波導提出新設計方法，本方法中以較高折射率薄膜長在波導脊形區域來降低波場在平面區域之散射場，進而使其形成圓形模場分佈，如此改善了傳統波導元件的分光效能。例如在y形分光器及多模干涉耦合器，它價降低了分光損失，光纖之耦合損失，和元件尺寸大小。

The invention of laser in 1970 has steadily increased the carrier frequency from the millimeter wavelength to micrometer range. Optical fiber communication systems using laser diode as the a light source and single-mode fiber as transmission media based on wavelength-division multiplexed (WDM) technology with the transmission capacity of 3.2 Tb/s or more are available now. With the rapid growth of semiconductor manufacturing technology, considerable efforts have been directed to computing the modes of optical rib waveguides, which form the important parts of photonic integrated circuits.
In this dissertation, we study the solutions of waveguide problems by novel numerical methods on two waveguide structures. The first one is a multimode interference coupler with longitudinally invariant structure. In this structure, we propose a method to study wave propagation in longitudinally invariant waveguides with arbitrary index profile. According to our method, both the electric field and the refractive index profile are expanded into two Fourier cosine series. With these series substituted into the wave equation, a differential matrix equation can then be obtained. We show that such a matrix equation can be explicitly solved and an expression for the wave field at any longitudinal position along an optical waveguide can be obtained. The solution proposed in this method indicates that our approach yields the same results as those obtained by using the beam propagation method with approximation. The second novel numerical method is on solving rib-type waveguide problems. A new semi-analytic method for solving the modal indices of the optical rib-type waveguide problems is presented. In this method, the cross-section of a rib-type waveguide is divided into several regions. In each region, the refractive index profile and field distribution are expanded into Fourier cosine series, and then are substituted in the wave equation. A second-order differential matrix equation is then derived for each region, and a closed-form solution can be obtained. Given the boundary conditions, an eigenmode equation for the rib waveguide can be derived and solved numerically to give the modal indices. The method proposed here is used to deal with three rib waveguides in three different geometric dimensions and/or compositions, respectively. Computational results indicate that our method is quite efficient, in terms of CPU time and its accuracy. The relative error in computing the modal index with the method is about 10-5 ~10-6. In addition, a new design for beam splitting components employing silicon-on-insulator rib waveguide structures is presented. In this design, a high index thin film layer is deposited in the rib section to reduce the wave field dispersive tails in the slab section. Accordingly, it renders the mode field a confined spot. In terms of the excess loss, fiber coupling loss and compactness of these components, this structure improves the beam splitting performance as compared with some conventional waveguide components such as branches and multimode interference couplers (MMICs).

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