龍伯球透鏡之波傳散射與聚焦特性研究

Abstract

龍伯球透鏡(Luneberg lens)為一種介電係數與半徑成一固定比例的球形透鏡，大部分是由數層不同介電係數介電質組合而成，而其主要特性能夠把入射波聚焦到球表面一點，假使在透鏡表面處鍍上一圓錐角金屬層即為龍伯球透鏡反射器(Luneberg lens reflector)，能把入射波依平面波的型態反射回原入射端，通常用於天線、雷達、衛星通訊、軍事等方面。 而本論文主要在研究Luneberg lens reflector的波傳散射和聚焦的特性， 首先利用球諧向量函數Mie theory在球座標下表示整個電磁場，將平面波用球諧函數方式展開以利於分析，因為在邊界上是由金屬和介電質兩種材質所組成，無法依相同的邊界條件求得，所以採用PMM數值方法去求透鏡混合邊界外場的值，其方式主要是取有限個節點滿足其所在位置邊界條件的方程式去求解，而透鏡內依照每層的邊界條件向內求出各層的電場，最後可得到平面波入射到透鏡經過散射反射之後的電磁場。 在討論方面我們採用不同層數的透鏡去逼近理想值曲線，因為非理想Luneberg lens的關係所以會產生差異性以及精確度上的不同，於是我們要在不同層數與聚焦效果兩者之間取得平衡，還有在不同的波長下對於透鏡聚焦的效果的影響，透鏡的散射圖騰在鍍上金屬之後的差異性，以及整個波傳場圖在不同的波長下聚焦效果是否明顯，聚焦點是否確實處於球表面處和反射之後是否為完整的平面波型態，最後畫出球表面上的電場向量以及poynting vector以了解能量的流向是否有漸漸往球表面一點匯集的效果。

Luneberg lens is a spherical lens which its permittivity varies with normalized radial coordinates. It is fabricated by using different homogeneous dielectric centrically layered media. The lens can focus a parallel beam of rays from any direction exactly at a point on the surface. If it is made by adding a reflecting surface “metallic cap”, it is called Luneberg lens reflector. It can reflect incident wave into plane wave in incoming direction. It is usually applied in antenna, radar reflector, satellite-based, and mobile-communucation. The main points of this thesis are about wave propagation, scattering and focusing effect of Luneberg lens. First we use spherical harmonics, the Mie theory, to express the electric and magnetic field in spherical polar coordinate system. We expand a plane wave in vector spherical harmonics. This problem can not be solved by the same boundary condition, because there are two different materials, dielectric and metallic, on the surface. So we use point-matching method (PMM) to solve this mixed boundary value problem. The method consists of satisfying the boundary conditions in the contact plane only at a finite point. Then we use the boundary condition of each layer, we can find the coefficient of each layer. Finally we can find the all field of the incident wave into a Lungberg lens. In the discussion, we use different layers of lens to approach the ideal Lungberg lens. We take the balance in different layers and the focusing effect. What is the influence of the different wave length and what is the difference between Lungberg lens and Lungberg lens reflector in scattering patterns. We also provide a picture of wave propagation through Lungberg lens to see if the points are exactly on the surface, the incoming wave reflected by the metallic cap is a perfect plane wave or not. Finally, we provide the poynting vector on the special surface. The purpose is to see if the energy focuses to a point on the surface.