本論文中,我們以純量繞射理論中的Fresnel(近場)繞射公式來推導和分析用於傅氏轉換型雷射投影系統的光學成像系統之公式。傅氏轉換式雷射投影系統是將所顯示的影像之反傅立葉轉換優化成為系統的輸入圖案,以凸透鏡做轉換,在背焦平面上顯示輸出影像,此一影像僅佔系統背焦平面上的第零階繞射範圍,成像面積受限於輸入元件之像素大小而非常有限,本論文中我們利用凹透鏡之組合來將傅氏轉換的影像放大,目的是使投影系統能夠符合實用之要求。 在理論分析方面,我們的結果可方便用於影像放大系統的放大率和成像位置的關係與設定上,由於公式可歸納出放大率和各段距離的通式,故可適用於多透鏡組合之影像放大系統,最終目的是在有限的系統條件下(固定之輸入/輸出距離,總透鏡數目甚至可用之透鏡焦距),計算(設計)出最大放大率的透鏡系統架構。最後,我們利用空間光調變器(Spatial light modulator)作為輸入元件以實現純相位式繞射光學元件,以產生傅氏轉換型影像,再分別用一片和兩片凹透鏡來放大影像,實驗結果和理論公式推衍一致。本論文之結果可於未來再推廣至成像品質之分析和理論建立。
In this thesis, we used the scalar diffraction theory for the analysis of the formula of Fourier-transform laser projection system. Inverse Fourier transform of image will be optimized to become the input pattern to the system. Then, we obtained the image of Fourier transform of pattern in the back focal plane of the convergent lens. In the back focal plane, the image was only obtained in the zero order, and hence the size of image is limited to the number of pixels of input element, the wavelength, and the effective focal length. For this purpose, we used a combination of divergent lenses to enlarge the image to fulfill the demand of the projection systems. In the theoretical analysis, our results were useful for determining the magnification, the imaging distance, and their relationship. Based on this relationship, we can generalize a general formula from them. According to the theoretical formula, we were able to design the optical imaging system with a maximum magnification by given a set of parameters such as the input/out distances, the number of lenses, and the focal lengths of the lenses. Finally, we implemented the system with a spatial light modulator as the input device, a convergence lens for the Fourier-transform imaging, and a combination of divergent lenses to magnify the image. The magnification and the imaging distance measured in the experiment coincided with the results of the theoretical formula, the result of this thesis will benefit further analysis of imaging quality and establish its theory.