威格納分佈函數(Wigner distribution function)能廣泛應用在物理、電機和其它科學領域。本論文將首先對威格納分佈函數的定義做基本介紹，然後再分別推導點光源、平面波、與橫向位置平方成正比的相位訊號、緩慢變化的相位訊號、高斯形式的訊號、薄透鏡、薄物體、和自由空間所對應到的威格納分佈函數。得到的結果將進一步利用光線圖說明其物理意義。我們接著考慮一個球面波被光柵繞射後再前進至觀察面的狀況。藉由完成幾個藕合的威格納分佈函數的運算，觀察面上的威格納分佈函數可被推導出。為了對結果有較簡單的了解，我們使用了有限傅利葉級數的方法將觀察面上的威格納分佈函數展開成許多相同、放大、但在橫向有不同位移的光柵威格納分佈函數疊加。將觀察面上的威格納分佈函數對所有的頻率積分就可得到光強度。最後我們將以己知振幅透射函數的振幅式和相位式光柵為例，並用Matlab軟体計算和畫出在滿足分數Talbot效應所對應的觀察面上所應看到的光強度分佈。這些模擬的結果將相互比較並討論它們的異同性。

Wigner distribution function can be used widely in physics, electrical engineering, and other scientific areas. In this thesis, Wigner distribution function will be defined in terms of both position and frequency. Then, the Wigner distribution function for a point source, plane wave, square phase signal, slowly varying phase signals, thin lens, thin objects, and free space propagation will be derived respectively. In addition, we will use ray diagrams to explain the relations between rays and Wigner distribution functions in each case. Furthermore, we will consider that a spherical wave is diffracted by a grating and observed at a distance behind the grating. The Wigner distribution function, at the observation plane, can be derived through calculating several coupled Wigner distribution functions. By the use of finite Fourier series, the result can further be simplified to a summation over several identical, magnified but laterally displaced Wigner distribution functions of grating. The intensity at the observation plane can be obtained by integrating the Wigner distribution function over all frequencies. Moreover, we will use an amplitude grating and a phase grating as examples to calculate and plot the intensity distribution at planes which satisfy the condition of fractional Talbot effect. The results will be compared and explained.

I.背景介紹                                       1
II.2.1 Wigner Distribution function              3
2.2 WDF描述的一些例子                            5
2.2.1 點光源                                     5
2.2.2 平面波                                     6
2.2.3 與x平方成正比的相位訊號                    7
2.2.4 緩慢變化的相位訊號                         8
2.2.5 高斯形式的光波訊號                         9
2.2.6 薄透鏡                                    11
2.2.7 進入和通過薄物體的WDF關係式               12
2.2.8 在自由空間行進前後之WDF關係式             12
III.3.1 球面波被光柵繞射後的WDF                 15
3.2 自成像平面的WDF和光強度                     18
3.3 某些特定非自成像平面的WDF                   20
3.4 某些特定非自成像平面的光強度                23
3.5 以振幅光柵與相位光柵為例子                  24
3.5.1 以透光比例為0.6的振幅光柵為例             24
3.5.2 以相位光柵為例                            26
IV.結論                                         28
V.附錄 Matlab程式原始碼                         30
VI.參考文獻                                     33

圖表目錄
圖一 複振幅為點光源時，它的WDF 所對應的光線圖 37
圖二 複振幅為平面波時，它的WDF 所對應的光線圖 38
圖三 複振幅為與x 平方成正比的相位訊號時，它的WDF所對應的光線圖 39
圖四 複振幅為緩慢變化的相位訊號時，它的WDF 所對應的光線圖 40
圖五 複振幅為高斯訊號時，它的WDF 所對應的光線圖 41
圖六 系統為薄凸透鏡時，它的WDF 所對應的光線圖 42
圖七 系統為薄凹透鏡時，它的WDF 所對應的光線圖 43
圖八 當系統為自由空間時，它的WDF 所對應的光線圖 44
圖九 球面波通過光柵的示意圖 45
圖十 通過光柵後的WDF 所對應的光線圖 46
圖十一 擴散光光源在z=(a+b)平面時的WDF 所對應的光線圖 47
圖十二 透光比例為0.6 的振幅光柵示意圖 48
圖十三 相位光柵的示意圖 49
圖十四 使用振幅光柵，滿足α=p/q‧2d2/λ(q=2且p=1)時as乘上位移的光柵透射函數後的實部和虛部 50
圖十五 使用振幅光柵，滿足α=p/q‧2d2/λ(q=2且p=1)時的觀察面光強度分佈示意圖 51
圖十六 使用振幅光柵，滿足α=p/3‧2d2/λ(q=2且p=1)時，
不同的p 值在觀察面上的光強度分佈圖 52
圖十七 使用振幅光柵，滿足α=p/4‧2d2/λ(q=2且p=1)時，
不同的p 值在觀察面上的光強度分佈圖 53
圖十八 使用振幅光柵，滿足α=p/5‧2d2/λ(q=2且p=1)時，
不同的p 值在觀察面上的光強度分佈圖 54
圖十九 使用相位光柵，滿足α=p/q‧2d2/λ(q=2且p=1)時觀察面上光複振幅的相角圖 55
圖二十 使用相位光柵，滿足α=p/3‧2d2/λ時，觀察面上的光強度分佈圖 56
圖二十一 使用相位光柵，滿足α=1/q‧2d2/λ(q=2且p=1)時觀察面上的光強度分佈圖 57

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