本論文從理論與實驗上去探討研究掠角X光背向繞射其光學特性及可能的應用。
本研究的理論計算基於X光動力繞射理論，使用了Stetsko與Chang於1997年提出的直角座標系統演算法，將基本波場方程轉換為特徵方程，以求解其特徵值和特徵向量。我們應用此演算法去處理Si (12 4 0)的24光掠角X光背向繞射。我們的計算解出了96個特徵模態與所對應的晶體內電場值，並進一步獲得晶體外在實驗上可觀察到的繞射角分布強度。
計算結果顯示，被激發的Si (12 4 0)背向繞射在正向入射與掠角入射兩種不同繞射幾何展現出非常不一樣的光學行為。
透過司乃耳定律(Snell’s law)，我們推導出在掠角X光背向繞射時，其入射角與相對應晶體的斜切角，滿足一個簡單的靜力關係。
實驗在台灣光子源09A光束線上進行。使用光束線的超高能量解析分光儀，我們成功地量測到掠角X光背向繞射，是已知世界上第一個數據。實驗顯示，掠角X光背向繞射所呈現的繞射強度分布、繞射中心點位置的偏移、以及多光繞射的抑制，皆與我們理論計算所預測的結果相符。這項發現意味著基於掠角X光背向繞射的X光光學元件將不需依賴深度蝕刻技術。此外，當觀察改變入射光能量時，掠角X光背向繞射的繞射圖樣也顯示出劇烈變化。這說明掠角X光背向繞射對能量極為敏感，並展現出其異常的角色散(angular dispersion)特性。
進一步透過X光動力繞射計算，我們發現當晶體斜切角接近90度的時候，將會產生最大的角色散率(angular dispersion rate)。後續的實驗中，我們證實了這項預測。在一片斜切角為89.926度Si (12 4 0) 所進行的掠角X光背向繞射，我們測量到了角色散率為 44.4 ± 9.3 μrad/meV，此一數值遠超過之前其他研究團隊的成果。我們比較了半動力計算的DuMond分析法，靜力繞射與全動力繞射理論所預測掠角X光背向繞射的角色散率(angular dispersion rate)的極限。
掠角X光背向繞射所展示的繞射現象為X光繞射物理開啟了新的視野。這種掠角X光背向繞射所呈現的異常角散率，亦揭示了製造先進同步輻射光學元件的新應用方向。我們期待掠角X光背向繞射在未來能夠應用於低維度材料的研究。

This dissertation investigates the novel diffraction phenomena of grazing-incidence X-ray back diffraction (GIXBD) through simulations based on dynamical X-ray diffraction theory, subsequently verified by experiments conducted at an ultrahigh-resolution synchrotron radiation beamline.
The theoretical framework adopting Stetsko and Chang’s Cartesian coordinate algorithm (1997) is to transform the fundamental equations into an eigenvalue problem. The simulation has taken into consideration the 24-beam Si (12 4 0) back diffraction at grazing incidence. The 96 eigenmodes and their corresponding electric fields inside the crystal were obtained, giving rise to the measurable diffraction intensity outside the crystal. The calculation results show that the excited Si (12 4 0) back diffraction displays very different optical behaviors in the two different diffraction geometries: normal-incidence and grazing-incidence. The incident angle and the refracted angle satisfy a simple kinematic relationship that can be directly derived from Snell's Law.
The experiments were carried out at the 09A beamline at Taiwan Photon Source. Employing an ultra-high resolution (∆E/E="2.8"×〖"10" 〗^(-"8" )) setup, we have successfully obtained the world’s first GIXBD pattern.
Experiments show that the diffraction intensity distribution of GIXBD, the shift of the diffraction center point position, and the suppression of multiple diffraction are all consistent with the results predicted by our theoretical calculations. This discovery suggests that X-ray optical devices based on GIXBD will not need to rely on deep etching technology. Moreover, the diffraction pattern of GIXBD manifests drastic changes when changing the incident energy. This implies that GIXBD is extremely sensitive to energy and exhibits its abnormal angular dispersion characteristics.
Through further X-ray dynamic diffraction calculations, we found that when the miscut angle θ_m of the crystal is close to 〖"90" 〗^°, the substantial yet finite angular dispersion rate (ADR) will be generated. We confirmed in subsequent experiments that for GIXBD on a piece of Si (12 4 0) with a miscut angle 〖"89.926" 〗^°, we measured an ADR of 44.4 ± 9.3 μrad/meV. This value far exceeded the results of the other previous research ever reported. We compared the limits of the ADR of GIXBD predicted by the DuMond analysis method of semi-dynamic calculation, kinematic approach and fully dynamical diffraction theory.
The novel diffraction phenomena of GIXBD bring about new horizon in X-ray physics. The substantial ADR generated by GIXBD lays the groundwork for the creation of high-performance X-ray optical devices. The prospective application of GIXBD in the realm of low-dimensional materials is highly anticipated.

Chapter 1 Introduction......1
Chapter 2 X-ray diffraction......4
2.1 Bragg’s law......4
2.2 Multiple diffraction......5
2.3 Orientation matrix......6
2.4 Back diffraction and asymmetric diffraction......9
Chapter 3 Grazing-incident X-ray back diffraction......11
3.1 Fundamental equations of X-ray......11
3.2 Cartesian coordinate algorithm......11
3.3 Boundary conditions......16
3.4 Diffraction intensity and asymmetric factors......20
3.5 GIXBD for two-beam Si (12 4 0)......21
3.5.1 Intensity......21
3.5.2 Dispersion surface and linear absorption coefficient......23
3.5.3 Excitation of mode and penetration depth......25
3.5.4 Snell’s law for GIXBD......26
3.5.5 Dispersion surface and reflectivity maps of GIXBD.......29
3.5.6 Energy-dependent of GIXBD intensity distribution......31
3.6 GIXBD for 24-beam Si (12 4 0)......34
3.6.1 Intensity maps......35
3.6.2 Snell’s law for 24-beam GIXBD......37
3.6.3 Dispersion surface and linear absorption coefficient......38
Chapter 4 Angular dispersion at extremely asymmetric diffraction......41
4.1 DuMond diagram analysis......41
4.2 Dynamical treatment......43
4.3 Kinematic treatment......45
Chapter 5 Experiments......49
5.1 Experimental setup......49
5.1.1 Beamline layout......49
5.1.2 High-resolution monochromator......51
5.2 Sample fabrication......53
5.3 The experimental procedure and energy calibration......56
Chapter 6 Experimental results......58
6.1 Realization of Grazing incidence X-ray back diffraction......58
6.1.1 The experimental procedure for GIXBD on X-ray resonator......58
6.1.2 Intensity distribution of NIXBD and GIXBD from X-ray resonator......58
6.2 Energy dependence of GIXBD......61
6.2.1 Experimental procedure of calibrating the incident energy......61
6.2.2 Intensity distribution of GIXBD by varying energy......62
6.3 Angular dispersion rate......63
6.3.1 Significant angular dispersion rate from GIXBD......63
6.3.2 Modified DuMond diagram......65
Chapter 7 Conclusion and prospect......68
7.1 Novel diffraction phenomena of GIXBD......68
7.2 Future perspectives......69
7.2.1 GIXBD simulation......69
7.2.2 GIXBD for sub-μeV monochromator and analyzer......70
7.2.3 GIXBD for X-ray resonator......71
7.2.4 Surface-confined eigenmodes interacting with low dimensional materials......73
Appendix A The derivation of fundamental equations......75
Appendix B The X-ray resonators......78
B.1 Optical Fabry-Perot resonator......78
B.2 X-ray Fabry-Perot resonator......82
B.2.1 Longitudinal coherence length......82
B.2.2 Equations of XFPR......83
References......85

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