本論文中研究了在現在與未來升級後的台灣光子源中三個與聚頻磁鐵技術有關的主題，第一是相位可調式聚頻磁鐵(adjustable phase undulator)，有別於傳統聚頻磁鐵藉由調整磁列間隙達到改變共振光子能量的方式，相位可調式聚頻磁鐵則是改變磁列的縱向相對位置調整光子能量，然而此種操作模式在橢圓極化模式中會產生無可避免的磁場橫向梯度場(~100 T/m)，此梯度場不但有機會降低同步輻射光源的品質還有可能改變儲存環中電子團的的工作條件，解析模型被提出以估計橫向梯度場對光源與電子團品質的影響，數值方法用來驗證解析模型對光源品質的估計。
第二個主題是在雙極小值垂直貝塔函數(betatron function)磁格中的串聯式聚頻磁鐵，為了達到更好的聚頻磁鐵輻射與電子團的橫向相空間匹配，三個四極磁鐵被安裝在儲存環的直段中間壓低垂直貝塔函數，然而，將一台聚頻磁鐵分成兩台和插入其中的四極磁鐵都造成聚頻磁鐵輻射中額外的相位延遲與軌跡的改變，導致傳統估計聚頻磁鐵輻射品質的方法不適用於此種特殊光源，基於維格納分布函數(Wigner distribution function)的數值方法被用來計算此種光源的光源亮度，相關的主題如光源的橫向同調性與兩台聚頻磁鐵的校準也一併討論。
最後一個主題是1:3利薩如曲線（Lissajous curve）式聚頻磁鐵，在傳統聚頻磁鐵中加入額外的不同週期長度磁列改變電子軌跡，達到降低近軸區域的輻射功率分布以減輕下游光束線中光學元件的熱附載的效果，初步的概念設計說明這樣的磁場分布與大小以現有的光學元件實際上是可以達到的，推廣版的聚頻磁鐵輻射解析表達式被用來評估聚頻磁鐵輻射的品質與降低熱附載的效果並以聚頻磁鐵輻射數值計算軟體驗證。

Three special topics related to undulator technology had been studied for Taiwan photon source (TPS) and future TPS-upgraded. The first one is the adjustable phase undulator (APU) which varies the resonance photon energy by changing the relative longitudinal position instead of the gap change between the magnet arrays. In an elliptically polarized undulator with APU mode, there exist unavoidable transverse field gradient (TFG) at the order of 100 T/m which may degrade the performance of a synchrotron light source and seriously affect the working conditions of the electron beam. Analytical model and the corresponding numerical results are given to evaluate the effects of the TFG on both the synchrotron radiation performance and electron beam behavior.
The second one is the tandem undulator in the double minimum beta-y lattice of TPS. The double minimum beta-y lattice is a special lattice in which a triple quadrupole magnets are installed to suppress the vertical beta-function for better phase space matching of the undulator radiation. However, the quadrupole magnets result in additional phase delay of the two radiation pulses that generated by each segment of the undulator and the trajectory variation of the electron beam. Conventional method cannot estimate the performance of such tandem undulators in the double minimum beta-y lattice. Numerical method that based on the Wigner distribution function is used to evaluate the performance of the tandem undulator radiation. Related issues such as the transverse coherence and the alignment of two undulators are also discussed.
Last one, the 1:3 Lissajous curve undulator in which additional magnet arrays with different period length are used to reduce the power on the paraxial region of the optical components in the beamline. A conceptual designation is shown to prove that it is possible to achieve enough magnetic field strength to reduce the synchrotron power. An analytical model as well as the numerical results are studied to estimate the performance of the undulator radiation and the power reduction.

摘要……………………………………………………………………………………………i
Abstract………………………………………………………………………………………ii
誌謝…………………………………………………………………………………………iii
Contents………………………………………………………………………………………iv
List of figures………………………………………………………………………………vii
List of tables…………………………………………………………………………………xii
Chapter 1 Introduction ………………………………………………………………………1
Chapter 2 Basic properties of the undulator radiation ……………………………………10
2.1 Spatial power distribution…………………………………………………………10
2.2 Spectral distribution………………………………………………………………11
2.3 Polarization………………………………………………………………………12
2.4 Transverse Coherence……………………………………………………………14
2.5 Brilliance……………………………………………………………………………17
2.6 Brilliance and transverse coherence……………………………………………21
2.7 Phase space matching…………………………………………………………23
2.8 Numerical calculation of the undulator radiation………………………………23
2.9 Effects on the equilibrium of the electron beam…………………………………25
Chapter 3 Adjustable phase undulator ……………………………………………………26
3.1 Magnetic-field characteristics of an elliptical polarized undulator……………26
3.2 Effects of transverse field gradient on a storage ring……………………………33
3.3 Effects of the transverse field gradient on Synchrotron Radiation………………41
3.3.1 Simple estimation for the brilliance and photon flux for TPS and TPS-upgraded………………………………………………………………………………41
3.3.2 Discussion of the brilliance for a filament electron beam ………………………45
3.3.3 Discussion of the brilliance for a finite emittance beam………………………49
3.4 Conclusion…………………………………………………………………………55
Chapter 4 Tandem undulators in the double-minimum beta-y lattice in Taiwan Photon Source………………………………………………………………………………………56
4.1 Theoretical background of undulator radiation…………………………………56
4.1.1 Radiation field of a tandem undulator…………………………………………58
4.1.2 Phase shifter………………………………………………………………………59
4.1.3 Quadrupole magnet and energy spread effects…………………………………61
4.1.4 Electron bunch effect on the on-axis Wigner distribution function……………64
4.2 Numerical analysis results using Wigner distribution function…………………64
4.2.1 Undulator configuration in the collinear case…………………………………68
4.2.2 Overall degree of coherence of the photon flux constraint in the aperture……73
4.3 Electron deviation between the tandem undulator and focusing effect…………74
4.4 Conclusions…………………………………………………………………………78
Chapter 5 1:3 Lissajous curve undulator…………………………………………………80
5.1 Theoretical estimation of the polarization rate and flux density in linear polarized mode……………………………………………………………………………………80
5.2 Conceptual designation of a 1:3 Lissajous curve undulator……………………85
5.3 Conclusions…………………………………………………………………………89
Chapter 6 Summary and outlook…………………………………………………………91
Reference……………………………………………………………………………………93
Appendix 1 Phase relation and dynamic field integral of the left/right operation of elliptical mode ……………………………………………………………………………100
Appendix 2 Phase relation and dynamic field integral of the inclined linear mode ……102
Appendix 3 Estimate of coefficient A in equation (3.25) ……………………………… 105
Appendix 4 Review of the brilliance related issues……………………………………… 107
Appendix 4.1 Coherent photon flux in an aperture and its degree of coherence……… 107
Appendix 4.2 Kim’s definition of coherent photon flux and its relation with brilliance110
Appendix 4.2.1 Difference of the coherent photon flux F_coh and F_coh in equation (2.11) and (2.21) ………………………………………………………………………101
Appendix 4.2.2 Coherent photon flux is calculated by using symmetrically placed pinholes ………………………………………………………………………………112
Appendix 4.3 Focusing of synchrotron radiation beam ……………………………113
Appendix 4.4 Summary………………………………………………………………115
Appendix 5 Adjustable phase undulator mode of EPU48 in current TPS ……116
Appendix 6 Evaluation of the double minimum beta-y lattice in possible NSLS-II upgrades……………………………………………………………………………………117
Appendix 7 Publication List …………………………………………………………120

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