在環形漸變折射率共振腔中模態存在的樣貌與二維各向同性諧振子波函數的樣貌非常相似，因此本文將針對二維各向同性諧振子的Schrödinger方程式與漸變折射率共振腔中的電磁波方程式進行比較，並依此定義出折射率的型式，而後再藉由方程式係數的對應關係計算出預期之模態頻率，藉此來瞭解諧振子量子化能階與光波模態對應的關聯性。
本論文討論二維各向同性諧振子的徑向量子數與角動量量子數與模態和品質因子的關係，並比較以層狀結構或者環形光子晶體所構成的等效折射率共振腔與原來的漸變折射率共振腔之結果的差異。模擬使用COMSOL 3.5a套裝軟體。當以諧振子模型計算出參考頻率後，即利用COMSOL模擬軟體對附近的頻率進行掃描計算以確定模態所在之正確頻率位置。本文中同時對E-Polarization及H-Polarization兩種偏振光進行分析，我們發現此兩種偏振光的模態由於方程式的差異對於諧振子模型的預測值有不同的偏離趨勢。

The mode patterns of electromagnetic fields in a circular gradient index resonator are similar to that of the eigen-wavefunctions of the 2-D isotropic harmonic oscillator system. In this thesis, we define the gradient index of the circular resonator according to the parabolic potential energy in the 2-D isotropic harmonic oscillator system and we compare the mode patterns with the corresponding eigensolutions of this reference Schrödinger equation. We derived the one-to-one correspondence between these two systems and extracted the reference mode frequencies from the energy spectrum of the oscillator system. We also replace the gradient index cavities by layered structures or circular photonic crystals whose local effective indexes are the same as the original gradient index cavities and compare their results. In the case of the 2-D isotropic harmonic oscillator, there are two quantum numbers to characterize the eigensolution, and we use COMSOL 3.5a to simulate mode patterns in a finite sized resonator under the influence of varying these two quantum numbers and changing the polarizations. Based on our numerical results, the relationships between the circular gradient index resonator system and 2-D isotropic harmonic oscillator system are revealed. We find that for TE and TM modes the frequency deviations from the reference frequencies of the modes have opposite tendencies because they satisfy different wave equations.

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