我們的理論解釋了球型共振腔的光致激發頻譜的物理機制並利用模擬結果來解釋實驗。而其中考慮到的物理機制包括 (1)在微米球之下考慮依孔隙率(Porosity)修正之等效介電常數、 (2) 波色爾效應(Purcell effect)如何影響光子態密度 、(3)受激輻射和自發輻射之速率方程、(4) 激子偏振子(exiton-polariton)效應，同時在最後我們也簡略地介紹金屬奈米球的非局域性效應(non-local effect)。
主要的理論架構為 : 在球座標系統下將格林函數用米基底(Mie Basis)做展開，然後解戴森方程式(Dyson equation)得到滿足球型邊界條件之下微米球系統的完整解，這個方法使我們避開了其他人常遇到的困境。由於本徵值為複數，因此無法以數值方法得到完整基底，並且在對格林函數使用本徵函數展開時，必須重新對本徵函數做歸一化，此歸一化對漏溢模態(leaky mode)有收斂性的疑慮，在考慮波色爾效應的情形之下，該歸一化常數被物理量體積所吸收，因此稱之為波色爾效應的等效體積。
而我們在得到球型共振腔完整解型態的格林函數後，只需將展開係數做頻率的極點展開(pole expansion)，這使得我們可以在共振態上考慮波色爾效應和受激輻射與自發輻射的修正，最終得到能適切解釋實驗之模擬結果。而在我們發表的第二篇期刊中，加入了激子偏振子對折射率之效應，使我們的模擬結果能夠進一步地應用在近紫外光波段。
最後我們亦探討了金屬奈米球的量子效應，在多體理論隨機相位近似(Random phase approximation)之下得到非局域性介電常數，並與先前久保(Kubo) 的理論比對，我們的理論能夠更適用於遠紅外波段，同時也證明了在五奈米以下的金屬球，其非局域性量子效應是不可忽略的。

We present theoretical studies of photoluminescence (PL) spectra of spherical microcavities. Our theory explains possible physical mechanisms for the lineshapes of PL spectra and the simulation results based on our model match the experimental data very well. The physical mechanisms considered include: (1) the dielectric function modified by porosity in microspheres via an effective-medium theory, (2) the modification of photon density due to the Purcell effect, (3) the relation between rate equations and spontaneous and stimulated emission, and (4) exciton-polariton effect. In the end, we also consider the non-local effect of metallic nano-particles and their coupling with microcavities.
The main concept of theory: to expand the free space Green’s function in spherical coordinate with the Mie-basis and solve Dyson’s equation which satisfies the spherical form of boundary conditions to get the full Green’s function of microsphere. This strategy prevents the difficulty encountered by other researchers encounter. Since the eigenvalues are complex numbers, it is difficult to get the complete basis numerically. On the other hand, when Green’s function is expanded by eigen basis, the normalization of eigen function is necessary. This normalization is also ill-defined in terms of convergence for leaky modes. With the Purcell effect taken into account, this normalization is incorporated into the formalism through the effective volume of microsphere.
After we obtain the full Green’s function of spherical cavity, we do the pole expansion and extract the expansion coefficients for the Mie basis. This expansion allows us to modify our resonance modes with the Purcell effect, the spontaneous emissions and stimulated emissions. Including all of these effects, our theoretical simulation fits the experimental data well. For our publication in MRS Advances [46], the exciton-polariton effect is also involved so that our simulation can better fit the near ultraviolet (UV) emissions of zinc-oxide microsphere.
In the end, we also discuss the quantum effect of metallic nano-particles. The random phase approximation (RPA) in many-body theory is applied to get the non-local dielectric function. The result is also compared with Kubo’s theoretical result. Our theory works better in the infrared regime and also proves that the non-local quantum effect is not negligible when the size of metallic nanoparticle is below 5 nm.

摘要 ii
Abstract iv
誌謝 vi
List of Figures ix
Chapter 1. Introduction 1
Chapter 2. Porosity and Effective medium theory 10
Chapter 3. Green’s Function and Leaky mode 13
Chapter 4. Mode density and Purcell Effect 20
Chapter 5. Spontaneous and Stimulated Emission 27
Chapter 6. Polariton Effect 40
Chapter 7. Nonlocal Dielectric Function of Metallic Nanoparticles 45
Chapter 8. Conclusion 54
Chapter 9. Perspective 60
References 61
Appendix 67

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