邊界態在界面上發生穿透行為時, 佔很重要的影響。而邊界態是耗散的形式, 但一般用平
面波展開法算出光子晶體的能帶, 因為我們取的是週期性邊界條件(k 是實數), 則所得到的
波是propagation of waves。故我們無法去模擬出耗散波的形式。因此在本論文中, 我們將
平面波展開法做一些變形, 讓我們可以給定頻率, 反推回所滿足的k 點, 而我們可以得到有
實數和複數的k, 分別對應到的是propagation modes 跟evanescent modes。此方法我們
稱擴展平面波法。
接著我們去討論當在真空中的單頻波, 去入射半無限大的光子晶體的情形。其中透射波和
反射波我們是用擴展平面波法所算出的propagation modes 和evanescent modes, 再利用
界面上的連續條件, 去線性組合出來。再利用Poynting’s vector 的性質(∇·S = 0), 去
算出穿透率和反射率。借由所算出的穿透率和反射率, 我們去驗証看是否能有效地模擬出邊
界態。
本論文中, 我用擴展平面波法分別去算正方晶格和三角晶格的kx , 然後去計算出正方晶
格上的穿透率和反射率。

The boundary states are very important when the external plane waves are emitted
to the crystal , and the form of The boundary states are evanescent . However, people
use the periodic boundary conditions(k is real ) and the plane-wave expansion method
to calculate the band of the photonic crystal , they can’t simulate the behavior of
evanescent modes . In this thesis , we modified the eigenvalue equation used by the
plane-wave expansion method(PWE) to the equation in which the eigenvalue is kx , and
the inputs are ω~k,n and ky . Besides the real kx and complex kx represent propagation
and evanescent modes respectively . This way we called it the extended plane-wave
expansion method(EPWE) .
Then we discuss the situation that an external plane wave in the vacuum emitted to
a semi-infinite photonic crystal . In this situation , based on the continuous conditions
at the incident surface , we can use the propagation and evanescent modes to obtain
the transmissive and reflective waves ; and we used the property of the Poynting’s
vector (∇·S=0) to calculate the transmittance and reflectance . According to
the calculation of the transmittance and reflectance , we judge whether the evanescent
modes can simulate the boundary states effectively .
In this thesis , we employ the extended plane-wave expansion method to calculate
the kx for the square and triangular lattice , and we calculate the transmittance and
reflectance for the square lattice .

1 .Kazuaki Sakoda,” Numerical analysis of the interference patterns in the optical
transmission spectra of a square photonic lattice,”Vol. 14, No. 8/August 1997/J. Opt.
Soc. Am. B

2 .Kazuaki Sakoda,”Transmittance and Bragg reflectivity of two-dimensional photonic
lattices”,PHYSICAL REVIEW B 51.12

3 .Kazuaki Sakoda,”Optical transmittance of a two-dimensional triangular photonic
lattice”,PHYSICAL REVIEW B 51.4672

4 .Kazuaki Sakoda and Hitomi Shiroma,”Numerical method for localized defect
modes in photonic lattices”,PHYSICAL REVIEW B 56 .4830

5 .K.Sakoda ,”Numerical analysis of the interference patterns in the optical transmission
spectra of a square photonic lattice”,J.Opt.Soc.Am.B 14 , 1961

6 .Young-Chung Hsue and Tzong-Jer Yang,”Applying a modified plane-wave expansion
method to the calculations of transmittivity and reflectivity of a semi-infinite
photonic crystal” ,PHYSICAL REVIEW E 70, 016706

7 .Kazuaki Sakoda,”Optical Properties of Photonic Crystals”/Springer

8 . 溫熙森,”光子/聲子晶體理論與技術”/科學出版社