對給定初階像差目標值之光學薄透鏡組設計方法與應用

Abstract

在光學設計的步驟中，薄透鏡組之高斯參數與初階像差設定，佔有極 重要之 部分。因此，求解各類薄透鏡組結構的方法，即成為實務設計時 重要之工具，且 為光學設計者研究之題材。本文提出四種由二或三個鏡 片所構成之薄透鏡組設計 方法。每一類型之鏡組，均將其光焦度與像差 聯立方程式，合併化簡成一元多次 方程式，從而求解出所有符合規格之 薄透鏡結構。 鏡組型式及其初階像差目標值共分為以下四類： 一、雙分離透鏡（球差、彗差、色 二、三膠合透鏡（球差、中心彗差、色差） 三、接觸式雙單及單雙透鏡（球差、中心彗差、色差、二次色差） 四、分離式雙單及單雙透鏡（球差、彗差、色差、二次色差） 傳統上，可用阻尼最小二乘法之優化方式求取鏡組初始結構。然而，優化 方 式有幾項缺點。首先，必須給定一個接近實際解之起始值，否則不一 定能收斂至 實際解。第二、對於不同的解需要不同的起始值。第三、無 法確定有多少組解。 第四、優化過程運算量大，較為耗時。相對於優化 求解法，本文所提出之演算法 具有多項優點。第一、不需要給予任何起 始猜測值。第二、可確定出最多可能解 之數目，並將所有的解予以求出 。第三、運算量小，求解速度快。 對於每種型式之鏡 組，均以含有詳細數值之應用範例，說明求解薄透鏡之過 程。最後並舉 例設計二個完整的之光學系統。第一個例子，設計一個與文獻中類 似之 消色差望遠鏡物鏡，並由兩者像差曲線一致之特性，說明薄透鏡設計之有 效 性。第二個例子示範由兩個鏡組所構成之鏡頭設計。Momiyama設計了 一種用於35 mm相機之複消色差攝遠鏡頭，該專利設計優異，惟因使用了 大型瑩石鏡片，價格 過於高昂而未曾商業化。我們應用所導出之演算法 ，設計了相同光學規格之鏡頭 ，其中並未使用高價玻璃材質，且達到了 更佳之複消色差品質。 Thin-lens design methods of various lens types with given Gaussain parameters and primary aberrations are important in optical system design, and have been the subject of research by optical designers. In this thesis, the thin-lens design methods of four lens types which consist of two or three elements are proposed. The four lens types and their aberration targets are as follows: 1) air spaced doublet (spherical aberration, coma and chromatic aberration) 2) cemented triplet (spherical aberration, central coma and chromatic aberration) 3) contacted singlet-doublet and doublet-singlet lenses (spherical aberration, central coma, chromatic aberration and secondary spectrum) 4) airspaced singlet- doublet and doublet-singlet lenses (spherical aberration, coma, chromatic aberration and secondary spectrum ) For each lens type, the formulas for the lens power and aberrations are combined into a polynomial equation in one variable. Hence, the number of solutions is obtained, and all the solutions can be solved effectively by algebraic or numerical methods. Simple thin-lens design examples are given to demonstrate the calculating processes. Two more examples are given. In the first example, a chromatic telescope objective is designed to show the usefulness of thin-lens design. In the second example, we design a new telephoto lens consisting of two components. The lens doesn't use expensive glasses and reaches better apochromatic performance than the patented structure of Momiyama. As compared with the damped least square optimization method, the proposed methods take some advantages:(1) they avoid the need and difficulty of making initial guesses, (2) the number of solutions are known and all the solutions can be found, (3) they are more quickly and effectively.