康托集光柵碎形繞射和空間的Weierstrass碎形繞射

Abstract

我們探討康托集和其光柵繞射圖形之解析解和數值模擬，採用兩種方法: 座標法和遞迴法，導出康托集光柵碎形繞射公式，公式中隱含該碎形如何自我複製和如何彼此分離某個距離形成雙狹縫。此外，類似的方法也可以被使用在推廣的康托集光柵碎形繞射。接下來，我們仔細分析Weierstrass-Mandelbrot碎形函數，考慮確定性、隨機性的相位。對於確定性的相位而言，此函數的趨勢可以藉由帕松累加公式獲得；對於另一種情況隨機性的相位而言，此函數的統計性質被仔細分析，並且其增加量可證明為平穩過程。最後，使用修改過空間的Weierstrass碎形加上平面波作為入射波，經由刀刃和單狹縫形成繞射。我們分析繞射圖案的解析解和數值模擬，並發現改變碎形維度和相位所造成的效果。為了保證模擬計算的精準，我們也提出了一個嚴謹的方法估計剩餘誤差。

Cantor set and its grating diffraction pattern are studied both analytically and numerically. Two approaches, coordinate approach as well as recursive approach, are provided to derive Cantor set grating diffraction formula which preserves how the fractal is replicated and separated by a distance to form a double slit. In addition, these methods are applied to generalized Cantor set grating diffraction. Next, Weierstrass-Mandelbrot fractal function is analysed in detail both with deterministic phase and with stochastic phase. For the case with deterministic phase, the trend of this function can be derived by employing Poisson summation formula. For the other case with stochastic phase, its statistical properties are investigated and its increment is shown to be stationary. Finally, a modified spatial Weierstrass fractal plus a plane wave is used as the incident wave for knife edge and single slit diffraction. Their diffraction patterns are studied both analytically and numerically. We also find the effects of varying its fractal dimension and phase. A rigorous proof to estimate the remainder for computational purpose is also developed.