LED(發光二極體, light-emitting diode)光源具有能量轉換效率高、壽命長與體積小等優點，在未來幾年將成為人類大量應用的光源，例如應用在室內與室外的照明，還有液晶顯示器的背光模組。為了能夠增加LED光源的應用方式，本篇論文利用能量守恆與幾何光學的原理，找出描述自由曲面透鏡(Free-form lens)表面的微分方程式，利用數值方法計算出能夠使光線呈現角度均勻或面均勻分布的透鏡曲面，再將這些透鏡用光學模擬軟體使用光線追跡的方法來討論透鏡效果。而設計這些透鏡的主要目的是希望能夠增加光線的使用效率，並且減少裝置的成本與耗能，達到環保節能的效果。

Light Emitting Diode (LED) possesses several advantages, having high energy-transferring efficiency, long life and small physical size. As a result, Light Emitting Diodes (LED) should become a general light source in human life in a few years. For example, it can be used to indoor lighting and Liquid crystal display (LCD) backlight module. In this thesis, we design several free-form lenses which can produce uniform angular intensity distribution or uniform illuminance. In this thesis, we find an ordinary differential equation (ODE) describing the of free-form lens surface by using the concept of energy conservation and geometrical optics. We then calculate the free-form lens surface by solving the ODE. The free-form lens properties are verified by ray-tracing simulation package, LightTools. The purposes of these free-form lenses are increasing light-using efficiency, decreasing the light source cost and energy-saving.

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