以角頻譜計算繞射場之研究

The study of diffractive propagation on a tilted plane or through a tilted device with angular spectrum

王祥宇(Shiang-Yu Wang)

指導教授 ： 徐巍峰

國立臺北科技大學/電資學院/光電工程系研究所/碩士(2006年)

https://doi.org/10.6841/NTUT.2006.00011

摘要

本論文探討以角頻譜作為光傳播方式時之強烈的物理意義和快速的數學運算及無近似的特性來求傾斜平面的繞射場，而利用角頻譜之純量繞射理論在任兩非平行面間傳播的這個議題上，已有許多研究及分析，一開始多採用Rayleigh-
Sommerfeld diffraction integral來解決傳播到傾斜面的問題，它無須像Fresnel、Fraunhoffer需要近似，在計算上也可以利用快速傅立葉轉換，但它需要複雜的數學代換，後來的研究大都採用角頻譜來處理，在論文中我便是用角頻譜來傳播光場，它在計算上不須複雜的代換及近似，但會因傾斜而產生空間頻率座標對應的問題:一、就不能以快速傅立葉轉換順利運算，我們提出內插法或頻率平移法來解決此問題，在論文中會把兩法求得的繞射場來作一比較;二、還要考慮Jacobian來修正傾斜造成的能量不守恆。除了傾斜的問題，角頻譜本身還有觀測範圍的限制，即輸入場跟繞射場大小會相同，如要觀測較遠之繞射場，就要加大計算面積，也需要去探討各參數間的問題，即孔徑大小、取樣點距間的關係。

關鍵字

繞射光學 ； 角頻譜 ； 傾斜平面之光場 ； 純量繞射

並列摘要

An angular spectrum, with special features of obvious physical meaning, fast calculation and unnecessary approximation, can be used to facilitate optical diffraction.
In order to get a picture about the properties of angular spectrum as a method of light propagation, Several studies have been conducted to apply the scalar diffraction theory to examine optical diffraction on tilted plane. Pioneers in this field adopted the Rayleigh-Sommerfeld diffraction integral to tackle with the problem, using fast Fourier transform that required complicated substitution. Subsequent researchers started to use angular spectrum of plane wave to achieve faster calculation by saving fast Fourier transform from the trouble of complex approximation and substitution,and our study also use this way. The new method, though more efficient, fails to offer a workable solution to the problem about the corresponding spatial frequency on the tilted plane. Since this problem, if not solved, at first we can obstruct the direct and smooth application of fast Fourier transform, this paper proposes to solve the problem by means of interpolation and shift in frequency domain. Moreover, the Jacobin factor is used to compensate the problem that total energy cannot be conserved after rotational transformation. In addition to the above problems related to the tilted plane, angular spectrum of plane waves has its own restriction, notably its narrow observation range in diffraction plane . In order to expand the observation range, the paper has examined and analyzed the relationships between relevant parameters such as the sampling number, the size of aperture. As focus has been placed on near field in previous studies, the paper has further discussed the diffractive field by using angular spectrum of plane wave in far field.

並列關鍵字

Diffractive optics, angular spectrum ； optical tilted plane ； scalar diffraction theory

參考文獻

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