傳統的完美導體球及介電質球的散射問題均有其正解，介電質球的散射問題更是早在1908年就被解出，但是當介電質球含部分金屬蓋的情況下就無所謂正解，僅能用數值解表示。
一般使用模態匹配的方法來解含金屬蓋介電質球的散射問題時，一開始數學式子就不是完全精確的式子，而且用矩陣解數值解時，需要 大小的計算量，但是當我們使用積分方程式（TMIE）來解這個問題時，不但求出其嚴謹的積分方程式，而且數值解矩陣僅需要 大小的計算量。
含金屬蓋多層同心介電質球的主要應用元件就是龍波球（Luneberg lens），本論文將試著應用含金屬蓋多層同心介電質球的結果去分析龍波球的場型。

The problems of the scattering off the perfect conductor sphere and the dielectric sphere have the exact solutions. But there are no exact solutions for the scattering off a multi-layered dielectric sphere with a metallic cap like the Lunberg lens reflectors which is used as a strong omni-directional reflector found in many microwave applications.
Haruo Sakurai applied the modal expansion technique and point-matching method (PMM) to study the scattering of the Lunberg lens reflectors. The problem is eventually formulated as 2MN by 2MN simultaneous matrix equation with M regions each having 2N unknowns due to two set of coupled polarization vectors. Strictly speaking, the formulae of the mode matching method for the problem of the scattering of the dielectric sphere are not exact. Furthermore, the size of the simultaneous matrix equation is also unnecessarily too larger.
In this thesis, we employ an integral equation formulation in the Frequency-domain together with the modified impedance transformation technique for the spherically layers to study the scattering of the Lunberg lens reflectors. We show that the formulae of the integral equation are exact and using an equivalent matrix equation, that the entire problem can be reduced to a N by N matrix equation where N is the number of terms of the expansion of the unknown field in the opening.
To verify our formulation we compute the total field of the plane wave incident upon the multi-layered micro lenses and compared the results with those from the geometric optics. We get good agreement for the regions that both theories apply. Small discrepancy is also observed and is consistent with the theoretical prediction.

誌謝 I
中文摘要 III
英文摘要 IV
目錄 V
圖表目錄 VII
第一章 導論 1
1-1 背景 1
1-2 動機與目的 4
第二章 等效直角座標結構 5
2-1 等效阻抗的轉換 7
2-2 主要理論推導 8
2-3 數值模擬結果 16
2-4 結論 24
第三章 赫茲向量（Hertz Vector） 25
3-1 電赫茲向量 26
3-2 馬克思威爾方程式的二元性與對稱性及磁赫茲向量 29
3-3 赫茲向量在球型結構場的解 31
第四章 理論基礎 34
4-1 場型 36
4-2 利用 與 來產生 與 40
4-3 邊界條件 43
4-4 特殊函數 45
第五章 多層同心介電質球的阻抗轉換 50
5-1 阻抗轉換通式 52
5-2 MIE Solution 60
5-3 數值模擬結果 65
5-4 結論 71
第六章 含金屬蓋單層介電質球 72
6-1 金屬球（電牆）反射係數 74
6-2 殘餘反射係數與穿透係數 76
6-3 以邊界條件帶出橫向模態積分
方程式 81
6-4 數值解 84
第七章 數值模擬結果 90
7-1 等效直角座標結構 91
7-2 多層同心介電質球 100
7-3 含金屬蓋單層介電質球 105
第八章 結論與未來工作 113
參考文獻 116
中英對照表 117

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