不繞射的彎曲光束近因為具有不繞射特性、在自由空間中無外力加速以及自我恢復的良好特性，所以這幾年來一直被廣泛討論。而Airy加速光線就是最早被討論的不繞射的彎曲光束，它的概念是來自於1978年時，Berry M.V.在量子力學上提出Airy波包在空間上是一個不繞射的解並且是一個拋物線軌跡。而在光學上，Airy加速光線是來自於Helmholtz方程式在近軸近似的解，並且其光束無法克服較大的彎曲軌跡並且很快就會在彎曲處消逝掉。於是非近軸近的不繞射彎曲光束就顯得很重要，像是圓形加速光線以及Mathieu加速光線，這兩者分別來自於Helmholtz方程式在圓柱座標下的解以及在橢圓座標下的解。於是在本論文將利用FDTD(Finite Difference Time Domain)模擬這三種不繞射的加速彎曲光束：Airy加速光線、圓形加速光線以及Mathieu加速光線並且與理論的結果做比較。並且為了更了解這三種光束的特性，我們還分析了Airy加速光線、圓形加速光線以及Mathieu加速光線三者的波印庭向量各分量的貢獻。

若要產生Airy波包，傳統上我們是讓非秒雷射透過一個cubic相位片再透過透鏡，然而這個方法並不是適用於微小的積體光路裡，所以我們透過特殊的金屬狹縫排列使得通過去的TM波產生Airy加速光線。

The non-diffracting accelerating beams exhibit special wave propagation properties such as beam curving or accelerating without external forces and self-healing after obstacles. Airy accelerating beam was first proposed for electron wave function in quantum mechanics in 1979 and then demonstrate in optical waves in 2007. The optical Airy accelerating beam is a solution of the paraxial wave equation which is similar to the Schrodinger equation and exhibits a parabolic trajectory. However, the Airy accelerating breaks down at large angle bending due to its paraxial approximation. Several non-paraxial accelerating beams were proposed in 2012. For example, the circular accelerating beam and the Mathieu accelerating beam are the non-paraxial solution of the Helmholtz equation in the polar coordinates and the elliptical coordinates respectively. In this thesis, three types of non-diffracting accelerating beams: the Airy accelerating beam, circular accelerating beam and the Mathieu accelerating beam were analyzed by Finite-Difference Time-Domain method.

The difference between the non-paraxial accelerating beams and the paraxial Airy accelerating beam were compared and discussed. Specifically, the electric field and magnetic field contributions and the Poynting vectors were shown for both the TE and TM modes. The self-healing properties of these three type accelerating beams though the small obstacles were demonstrated.

To realize the generation of the Airy accelerating beam, metallic slit array were designed to lunch the Airy accelerating with TM waves. These non-diffracting accelerating beams might have potential applications in optical tweezers for moving nano-particles in curved trajectories and manipulating bio cells in tissues.

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