臺灣博碩士論文加值系統

摘要
2004年張石麟教授等人已成功地觀察到X光共振腔中的共振干涉現象.然而,此一現象須由X光動力繞射理論方能解釋. 因此,本論文冀求透過理論計算的方式來嘗試對此一現象提供物理上的解釋. 吾人採用1997年由Stetsko博士與張教授所發展的解X光動力繞射方程式的演算法來處理此一問題. 此一演算法的精神主要在於解一矩陣形式的固有值代數方程 (eigenvalue-eigenvector equation).其所解出的固有向量 (eigenvector) 則代表晶體內繞射光的電場向量, 藉由電磁學中的邊界條件 (即電磁場的連續性), 即可得出晶體外繞射光的電場向量.
當X光的能量達到14.4388 keV並垂直入射矽質共振腔的 (12 4 0) 原子面時,將產生背向繞射光, 並在腔內來回反射, 產生同調干涉進而形成共振干涉的同心圓. 由於晶體的對稱性,此時將有24光同時滿足布拉格定律而被激發並傳遞. 因此,此一共振干涉同心圓還伴隨著複繞射所產生的九條通過圓心的徑向條紋.吾人的計算結果與此一現象相符. 固有值方程式中的固有值其實部給出了色散表面, 而虛部則對應到線性吸收係數. 繞射光的相位則由固有向量的實部及虛部所決定. 此外, 激發模態的計算則同時需要利用到固有值及固有向量. 此四個物理量的計算及繪圖必須將固有值及固有向量做一排序的動作,方能得出正確的結果. 此一過程須手動完成, 因此非常曠日費時. 若只需計算繞射光的強度則不須要做排序的動作. 仔細觀察色散表面及線性吸收係數, 吾人發現在全反射區域線性吸收係數特別大, 這表示吸收越大反射也越大.而繞射光的相位在此區域也變化得非常明顯.
張教授所使用的共振腔為在同一片矽晶圓表面蝕刻出相距100微米的兩片互相平行且同樣大小及厚度的平行板.吾人想瞭解在不同厚度及間距的情況下對共振腔的反射率及穿透率的影響. 因此,吾人也針對各種厚度及間距計算其反射率及穿透率.
總之,吾人已成功發展出此一電腦程式來做X光動力繞射計算,而計算的結果也與實驗結果一致.而色散表面, 線性吸收係數, 繞射光的相位及激發模態的計算則讓我們對此一共振現象有更深入的瞭解.

Abstract

This dissertation aims at developing a theoretical approach to treat the 24-wave diffraction occurred in a Fabry-Perot type of resonant cavity of silicon by solving an eigenvalue-eigenvector equation, derived by Stetsko and Chang, deduced from the X-ray multiple-wave dynamical diffraction equation. We anticipate the calculations can offer a crystal clear physical picture for the experimental resonance pattern of the 24-wave diffraction in the X-ray Fabry-Perot cavity, and aid us design an appropriate resonant cavity to fulfill the experimental purposes.
The resonance pattern accompanied by nine coplanar diffractions occurred in an X-ray Fabry-Perot cavity of silicon at photon energy of 14.4388 keV at which 24 beams are simultaneously excited has been realized by employing the back diffraction of (12 4 0) by Chang et al.. The calculation is in a good agreement with the observed one. The interference fringes of the concentric rings accompanied by nine diffraction lines passing through the center of rings distinctly show up in the calculated pattern. Meanwhile, the dispersion surface and the linear absorption coefficient have been mapped out from the solved eigenvalues. The excitation of mode and the phases of the diffracted waves have also been calculated from the solved eigenvalues and eigenvectors and from boundary conditions. According to the calculated dispersion surface and linear absorption coefficient, we find that in the total reflection region, most of energy is reflected off the crystal, and the absorption reaches the maximum value, and the phase of the diffracted waves changes drastically. In the exact 24-wave region, the phase of the diffracted wave changes further by doubling the values.
The effect of the crystal thickness and the gap width on the reflectivity and transmissivity are also surveyed. The larger the reflectivity and the lower the transmissivity, the thicker is the crystal plate. The more the number of the resonance peaks, the wider is the gap width between the two plates. The time response curves of the cavity show a periodic structure of time due to the interference among the resonance peaks.
In brief, we have established a theoretical approach to treat the multiple-wave diffraction in the Fabry-Perot type of resonant cavity. The calculation is in a good agreement with the observed one. The procedures for the dynamical calculations are also offered in this dissertation.

Contents

1 Introduction 1

2 Multiple-beam X-ray diffraction…………………………………………………4
2-1 Introduction……………………………………………………………………..5
2-1-1 X-ray diffraction…………………………………………………………..5
2-1-2 N-beam diffraction………………………………………………………..5
2-1-3 Renninger scan……………………………………………………………6
2-2 Indexing…………………………………………………………………………7
2-2-1 Multiple-beam diffraction pattern…………………………………………8
2-2-2 Theoretical calculation of multiple-beam diffraction pattern……………..9
2-2-2-1 Construction of the orientation matrix…………………………...9
2-2-2-2 Determination of the coordinates of reciprocal lattice points…..11
2-2-2-3 Sorting…………………………………………………………..12
2-2-2-4 Plot of a theoretical multiple-beam diffraction pattern………….14
2-2-2-5 Outline of the indexing procedure……………………………….16
2-3 Intensity profile of three-beam diffraction……………………………………..17
2-4 Structure factor and electric susceptibility……………………………………..21
2-5 Extracting the triplet phase……………………………………………………..23
2-6 Application……………………………………………………………………...24

3 Dynamical theory of X-ray diffraction…………………………………………..27
3-1 Dynamical theory of X-ray diffraction…………………………………………28
3-1-1 Fundamental equation of wavefield………………………………………28
3-1-2 Eigenvalue-Eigenvector equation………………………………………...31
3-1-3 Boundary conditions for a single crystal plate…………………………...34
3-2 Dispersion surface………………………………………………………………40
3-2-1 Mapping the dispersion surface…………………………………………..40
3-2-2 Excitation of tie point: phase matching and boundary condition………...41
3-2-3 Poynting Vector…………………………………………………………..42
3-2-4 Bragg gap…………………………………………………………………43
3-3 Excitation of Mode and Mode of wave propagation……………………………46
3-4 Absorption and Linear absorption coefficient…………………………………..49
3-5 Example………………………………………………………………………...51
3-5-1 Reflectivity……..…………………………………….…………………..56
3-5-2 Intensity of wavefield…………………………..……….………………..57
3-5-3 Dispersion surface and linear absorption coefficients……………………59
3-5-4 Excitation of mode……………………………………………………….61
3-5-5 Phase of diffracted beam…………………………………………………62
3-6 Procedures for dynamical calculation………………………………………….64

4 X-ray Fabry-Perot cavity………………………………………………………..68
4-1 Laser cavity.........................................................................................................69
4-1-1 Fabry-Perot etalon (dielectric parallel plate).............................................70
4-1-2 Stability of laser cavity…………………………………………………..73
4-2 X-ray Fabry-Perot cavity....................................................................................74
4-2-1 Reflection and transmission ampliudes for X-ray Fabry-Perot cavity......74
4-2-3 Effect of the crystal thickness...................................................................77
4-2-4 Effect of the gap width..............................................................................82
4-2-5 Time response of the X-ray Fabry-Perot cavity........................................84
4-2-6 Angular dependence……………………………………………………..87
4-3 Intensity of standing wave……………………………………………………..98
4-3-1 Intensity inside the crystal plate………………………………………….98
4-3-2 Intensity inside the cavity……………………………………………….100
4-4 Dispersion surface and linear absorption coefficient………………………….104
4-5 Phase of the diffracted wave………………………………………………......107
4-6 Comparison of X-ray and optical Fabry-Perot cavity…………………………108
4-6-1 Mechanism of reflection………………………………………………...108
4-6-2 Condition of resonance………………………………………………….109
4-6-3 Stability of cavity………………………………………………………..110
4-7 Coherence...........................................................................................................112
4-7-1 Temporal coherence..................................................................................113
4-7-2 Spatial coherence......................................................................................117

5 X-ray 24-beam diffraction....................................................................................118
5-1 Conditions for dynamical calculations………………………………………...119
5-1-1 Structure of X-ray Fabry-Perot cavity…………………………………..119
5-1-2 Distribution of 24 reciprocal lattice points………………………...........120
5-1-3 Electric susceptibility……………………………………………………122
5-2 Intensity distributions………………………………………………………….123
5-2-1 Nine coplanar diffractions………………………………………………123
5-2-2 Asymmetric profile of the nine coplanar diffraction……………………145
5-3 Dispersion surface and linear absorption coefficient………………………….147
5-4 Excitation of mode…………………………………………………………….150
5-5 Phase of the diffracted waves………………………………………………….152

6 Conclusion………………………………………………………………………..154

Reference…………………………………………………………………………..156

Appendix……………………………………………………………………

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