有限元素時域波束傳播法之發展與應用

Abstract

傳統上，波束傳播法假設只具有前向傳播的波，若遇到結構上會產生反射波的情形便不適用。在本研究中，我們考慮有限元素時域波束傳播法(FETD-BPM)以處理在波導結構上會產生反射的情形，應用於橫向電場模態及橫向磁場模態。另外，我們引進了一完美匹配層(PML)作為邊界條件來吸收超出數值空間之電磁波。我們的演算法分別採用了近軸近似或帕帝(Padé)近似來處理窄頻或寬頻的脈衝光波問題。 為了驗證此方法的正確性，我們分析了簡單的平面光波導及光柵這兩個結構。接著，我們將有限元素時域波束傳播法應用到幾種不同的波導結構，如彎曲型波導和環型和碟型共振器，數值計算顯示，帕帝(Padé)近似在計算短如只有數飛秒的脈衝波仍具有優異的表現。 此外，我們亦以此方法分析了數個曾於文獻中被討論的二維高折射率對比直角轉彎波導元件的TE模態傳播特性。所分析的元件結構包括簡單的直角轉彎波導、直角轉彎波導加入方形共振腔結構、直角轉彎波導於轉折處外側切45°截面、直角轉彎波導於轉折處加入四分之一圓結構。所研究的波導具0.2微米的寬度，其傳播層折射率為3.2而覆蓋層折射率為1.0，分析的波長範圍由1.48微米至1.62微米。藉著在波導的轉折處加入某些共振腔結構，其透射效能可以顯著地提升。研究結果亦顯示對於含有曲線形或斜線形介電接面的結構而言，如直角轉彎波導於轉折處外側切45°截面及直角轉彎波導於轉折處加入四分之一圓結構，此方法分析所得之數值結果與以有限差分時域法分析所得的結果略有不同。

Traditionally, the beam propagation method (BPM) assumes only the forward propagating waves exist. It is difficult to take into account backward reflecting waves in the BPM. In this research, a time-domain beam propagation method (BPM) based on the finite-element scheme is described for the analysis of non-forward-propagating pulses of both transverse-electric and transverse-magnetic modes in waveguiding structures containing arbitrarily shaped discontinuities. In order to avoid nonphysical reflections from the computational window edges, the perfectly matched layer boundary condition is employed. The present algorithms use two kinds of approximations, the paraxial approximation and the Pade approximation, which can treat the narrow-band and the wide-band optical pulses, respectively. After validating this method for a simple slab waveguide and an optical grating with modulated refractive indexes, various optical waveguide structures, such as the bent waveguides, and the microcavity ring and disk resonators are simulated. The calculated results show that the Pade approximation can successful simulate a short pulse with only several femtoseconds. Besides, the method is employed to study several two-dimensional high-index-contrast 90-degree waveguide bend structures previously reported in the literature regarding their TE mode propagation characteristics. The device structures considered include the simple 90-degree-bend waveguide, the 90-degree-bend waveguide modified by a square resonator, the 90-degree-bend waveguide with a 45-degree-cut at its outer corner, and the 90-degree-bend waveguide modified by a quarter disk. The waveguide structure is assumed to have the core width of 0.2μm, the core index of 3.2, and the cladding index of 1.0, and the wavelength range is from 1.48μm to 1.62μm. By adding some resonator structure at the waveguide bend, the transmittance performance can be significantly improved. We also show that there are some differences in the numerical results compared with those calculated by using the finite-difference time-domain method for the 90-degree-bend waveguide with a 45-degree-cut at its outer corner and the 90-degree-bend waveguide modified by a quarter disk which have the curved/oblique shaped dielectric interfaces.