本研究探討過去文獻上利用商用軟體 Lumerical 裡的套件—bandstructure tool 計算亞波長光柵波導折射率的方法，並將快速準絕熱動態(Fast Quasiadiabatic Dynamics, FAQUAD)運用在亞波長光柵波導分光器的設計上。研究發現用有效介質近似理論(Effective medium theory, EMT)計算電磁場和折射率，再利用時域有限差分法(Finite-Difference Time-Domain, FDTD)驗證 FAQUAD 可以縮短元件長度，使線性結構 20 μm 縮短到 9 μm，約莫是縮短了 55%，在 1.3μm~1.7 μm 最大 imbalance 為 1.2 dB。如果對頻寬要求不高，亦可以縮短至 5.76um，約莫是縮短了 71%，仍然有 50/50 的分光效果，在波長 1.3 μm~1.7 μm 最大 imbalance 為 1.77 dB。縱向寬度製程容忍度(-20 nm~20 nm)則在 0.23 dB 以下，光傳播方向的 fill factor 製程容忍度則在 1.53 dB 以下；在整體光學元件的 Excess loss 方面，非線性結構元件總長 39μm 的損失為 0.18 dB。
我們的研究驗證了利用 EMT 法等效光柵的電磁場、折射率，再做 FAQUAD 可以縮短 SWG adiabatic coupler 的長度。此外，EMT 法可以用來設計 FAQAUD 的結構，只是 MODE 模擬 EMT 波導元件和 FDTD 模擬亞波常光柵波導的結果差很多，因此不具參考性。

This thesis explores the calculation of effective refractive indices of sub-wavelength grating waveguides using methods in the literature, and applied the Fast Quasiadiabatic (FAQUAD) algorithm to sub-wavelength grating couplers. We find that using Effective Medium Theory (EMT) to calculate the electromagnetic field and refractive index; and then apply the FAQUAD can shorten the length of the component.
The linear coupling region is shortened from 20 μm to 9 μm, a 55% reduction.If the bandwidth requirement is relaxed, it can also be shortened to 5.76 μm which is about 71% shorter and still has a 50/50 splitting effect.
In the bandwidth part, the imbalance of the sub-wavelength grating waveguide after FAQUAD (1.3 μm~1.7 μm) is about 1.77 dB; the vertical width fabrication tolerance (-20 nm~20 nm) is below 0.23 dB; the fill factor fabrication tolerance in the light propagation direction is below 1.53 dB. In terms of the excess loss of the overall structure, the loss of the total length of the non-linear structural element of 39 μm is 0.18 dB.
Therefore, choosing the EMT method can reduce simulation time, and it is also much more convenient in calculation.

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