臺灣博碩士論文加值系統

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。

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.

目錄
中文摘要 ……………………….…………………………………………………… i
英文摘要…………………………..………………………………………………… ii
誌 謝..…………………………….………………………………………………… iv
目 錄..……………………………..…………………………………………………… v
圖目錄..…………………………….………………………………………………… vii
第一章 緒言...…………………………………………………………………………1
1.1. 前言...……………………………………………………………………1
1.2. 動機...……………………………………………………………………1
1.3. 論文架構...………………………………………………………………2
第二章 繞射理論...……………………………………………………………………4
2.1. Kirchhoff及Rayleigh-Sommerfeld 傳播原論…………………………4
2.2. Fresnel diffraction傳播理論……..……………………………………..8
2.3. Fraunhoffer diffraction傳播理論...……………………………………10
2.4. Angular spectrum傳播理論…...………………………………………10
第三章 論文回顧...…………………………………………………………………..14
3.1. 利用角頻譜及Fraunhoffer之混合作法來求傾斜場..……………......15
3.2. 利用Rayleigh- Sommerfeld diffraction 求傾斜場…………….……...16
3.3. 利用角頻譜來求傾斜場 ……………………………………………...19
第四章 以角頻譜分析光場傳輸到傾斜面原理…………….……………………20
4.1. 以角頻譜分析光傳播到傾斜面的原理……………………………20
4.1.1. 旋轉矩陣……………………………………………………20
4.1.2. 衰勢波及反向波 …………………………………………22
4.1.3. Jacobin因子 …………………………………………….23
4.2. 內差法…………………………………………………………………25
4.3. 頻率平移法……………………………………………………………28
4.4. 角頻譜的問題…………………………………………………………29
4.4.1. 取樣點數跟取樣點距………………………………………29
4.4.2. 輸入場跟繞射場大小………………………………………31
4.4.3. 因旋轉矩陣產生的中心偏移頻率…………………………31
第五章 模擬之分析與比較…………………………………………………………34
5.1. 角頻譜利用內插法求的傾斜光場……………………………………34
5.2. 角頻譜利用頻率平移法求的傾斜光場………………………………39
5.2.1. 二維的傾斜場………………………………………………39
5.2.2. 三維的傾斜場………………………………………………41
5.3. 比較內插法及頻率平移法求得之繞射場……………………………43
5.4. 模擬分析和結果討論及後續之研究…………………………………47
5.4.1. 傾斜繞射場範圍之極限...………………………….………47
5.4.2. 以角頻譜分析遠場繞射之問題……………………………48
5.4.3. 點距及開口大小對繞射場之影響…………………………49
第六章 結論…………………………………………………………………………51
參考文獻………………………………………………………………………………52
附錄
A:「使用角頻譜模擬自由空間光場在遠場及近場的繞射現象」
B: 「探討以角頻譜方法分析繞射光場傳播在傾斜面上的內插問題」
發表於OPT2006台灣光電科技研討會壁報及口頭論文


圖目錄

圖2-1 Kirchhoff 平面孔徑的繞射光學架構………………………………………...4
圖2-2 Kirchhoff平面孔徑繞射模型…………………………………………….……6
圖2-3 Rayleigh-Sommerfeld透過一平面孔徑之光學架構……………………….…8
圖2-4繞射光學架構 ……………………………………………………………....…9
圖2-5波向量示意圖…………………………………………………………………11
圖2-6角頻譜示意圖…………………………………………………………………12
圖2-7角譜傳播示意圖………………………………………………………………13
圖 3-1 Ganci的光學系統………………………………….………………………….15
圖 3-2 Leseber實驗之光學架構示圖……………………………………………....17
圖4-1反向波示意圖…………………………………………………………………23
圖 4-2傾斜造成Jacobin的產生……………………………………………………..24
圖 4-3光學架構圖及計算運作流程……………………………………….……… ..27
圖 4-4內插法示意圖………………………………………………………………... 27
圖4-5頻率平移法示意圖……………………………………………………………29
圖 4-6有無中心頻率fx`(fx,fy)之比較……………………………………….……….32
圖 4-7有無中心頻率fz`(fx,fy)之比較……………………………………….……….34
圖 5-1以內插法計算傾斜0°及10°平面之光場強度分布模擬………………….…35
圖5-2以內插法計算傾斜20°及30°平面之光場強度分布模擬…………..………36
圖 5-3以內插法計算傾斜40°及50°平面之光場強度分布模擬..……….........……37
圖 5-4以內插法計算傾斜60°及70°平面之光場強度分布模擬……….......………38
圖 5-5方孔跟圓孔的繞射場型之比較…………………………………………… ...39
圖5-6利用頻率平移法求得之二維傾斜場………………………………………....40
圖5-7利用頻率平移法求得之二維傾斜場之整合圖…………………….………...41
圖 5-8未考慮中心頻率及jacobin的傾斜平面繞射場…………………….……… .42
圖 5-9考慮中心頻率及jaconbin的傾斜平面光場………………………………….43
圖 5-10在z=0.1mm時比較內插法跟頻率平移法求得之繞射場………………….45
圖 5-11在z=1mm時比較內插法跟頻率平移法求得之繞射場……………………46
圖5-12觀測平面之傾斜角的極限…………………………………………..………47
圖5-13用角頻譜在不同參數條件下求得遠場………………………………….…48
圖5-14制孔徑大小及取樣點距來觀察繞射場的變化……………………………..50

參考文獻
[1]J. W. Goodman, Introduction to Fourier Optics,2nd Ed., McGraw-Hill, New York, 1996.
[2]Eugene Hecht, Optics,4nd Ed,Addison Wesley, 2002.
[3]EAMON LALOR ,“Conditions for the Validity of the Angular Spectrum of Plane Waves”, JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, VOL 58, NUM 9,pp.1235-1237,1968.
[4]S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, pp. 158-160 ,1981.
[5]K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Optica Acta 30, pp. 673-679,1983.
[6]H. J. Rabal, N. Bolognini, and E. E. Sicre, “Diffraction by a tilted aperture: Coherent and partially coherent cases,” Optica Acta 32, pp. 1309-0311 ,1985.
[7]D. Leseberg and C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, pp. 3020-3024,1988.
[8]T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, pp. 556-558 ,1992.
[9]T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, pp. 299-305 ,1993.
[10]N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A15, pp. 857-867,1998.
[11]Alfredo Dubra and Jose A. Ferrari “Diffracted field by an arbitrary aperture”Am.J.Phys.Vol.67,No.1,pp87-92,1999
[12]N. Delen and B. Hooker, “Verification and comparison of a fast Fourier transform-based full diffraction method for tilted and offset planes“, APPLIED OPTICS, Vol. 40, No. 21,PP.3525-3531,2001
[13]K. Matsushima, H. Schimmel, and F. Wyrowski, ”Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A20, pp. 1755-1762 ,2003.
[14]Gokhan Bora ESMER and Levent ONURAL” Simulation of scalar optical diffraction between arbitrarily oriented planes” IEEE. 0-7803-8379-6/04,2004.
[15]Lingfeng Yu, Yingfei An, and Lilong Cai “Numerical reconstruction of digital hologramswith variable viewing angles”Optics Express, Vol. 10, Issue 22, pp. 1250-1257,2002.
[16]S. De Nicola, A. Finizio, and G. Pierattini P. Ferraro, and D. Alfieri “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes” Optics Express, Vol. 13, Issue 24, pp. 9935-9940,2005.
[17]A.Katrich “Angular spectrum representation in nonparaxial propagation” IEEE,1-4244-0490-8/06,pp.605-607,2006.