折射與反射式光學曲面重建與其應用

Abstract

幾何光學設計的問題是當代許多應用領域包括汽車車燈，微型投影機，液晶螢幕背光板還有太陽能集光板等的設計都會面臨到的問題。 對非成像幾何光學中自由曲面的設計問題，基於能量守恆的原理可以推導出著名的非線性偏微分Monge-Ampere 方程。 Monge-Ampere 方程也同時出現在 differential geometry, mass transportation and geostrophic fluid 等研究領域 在本計畫中我們將延續先前計劃中所完成的工作。 我們已經成功的運用 Feng and Neilan 所提出的 vanished moment method 與 BCIZ 的有限元素法來計算Monge-Ampere 方程的數值解並驗證期數值解的收斂性。本計劃中我們一方面將利用條件優化的 conjugate gradient 方法來加速計算線性系統的解。我們將以雙調合算子來作為優化的preconditioner。 並用多重網格法來計算迭代過程中雙調合算子的解。 另一方面我們將發展折射與反對型自由曲面重建的演算法。 這方面我們不採用如Oliker, Neubauer, Halstead, Patow 與 Pueyo 等最佳近似的方法。 我們採用微分幾何的局部座標的自然概念來做為我們的計算基礎。 在一維曲面的重建我們已有初步的成果。我們將嘗試推廣我們的演算法到二維曲面並發展相關的應用。

The geometric optics design problems have been studied in many applications from automobile head-light design, solar energy reflector, LED illumination, micro projector and LCD backlight, etc. The image optical design mainly concerns the image quality generated from the designed optical system and the non-image optic design mainly considers transportation of energy flow in the designed optical system. In this proposal, we are particularly interested in the free-from surface (FFS) design problems in non-image optics. From the energy conservation principle, the model of freeform surface problems leads to a well known nonlinear partial differential equation: The Monge-Ampere equation. The Monge-Ampere equation also arises in various research fields including differential geometry, mass transportation, geostrophic fluid. In this proposal, we shall continue our unfinished works from previous project. We have successfully solved the Monge-Ampere equation by using the vanished-moment method, proposed by Feng and Neilan in 2007, and BCIZ element. In this project, we shall use the biharmonic operator as a preconditioner in solving the discrete linear system using CG method. Furthermore, we shall use the multigrid method to solve the biharmonic equation for computation efficiency. On the other hand, we shall develop the reconstruction algorithm for both the reflective and refractive free-form surfaces. Instead of using the optimization approach employed by Oliker, Neubauer, Halstead, Patow and Pueyo, our algorithm is based on the local coordinate representation naturally arisen from differential geometry. We shall test our algorithm for the 1D free-form curve reconstruction and generalize our algorithm to 2D surface reconstruction. We also look for interdisciplinary collaboration and industrial applications.