自感應透明現象在非線性共振光能隙結構裡之研究

Abstract

這篇論文針對自感應透明現象在均勻摻有二階共振原子的非線性光能隙結構裡作了理論性的研究。研究發現描述光脈衝在這種共振光能隙結構的Maxwell-Bloch 方程組可以化簡為等效的非線性耦合方程組。我們首先找到了這組非線性方程組的一個完整解析解，這個解描述了一串無損耗的光脈衝序列，解的波包項是由一帶有背景光常數的正弦函數所形成，而相位項則是會隨時間變化的調變項。依據Bragg 光固子理論，這組非線性方程組還可以進一步地化簡為非線性Schrödinger 方程式。這樣的模型描述了一個等價的物理機制：由摻雜的共振原子提供了光能隙結構額外的等效群速色散與等效的三階非線性項，因此Bragg 光固子可以因為整個介質群速色散與非線性項的平衡而無失真地在此光能隙結構傳播。由於整個模型已經將共振原子的效應考慮進介質裡，原始禁制能帶會被摻雜原子與光脈衝高光強度的非線性響應給平移。這導致即使一個自感應透明光固子的中心頻率落在原始的禁制能帶內，此光固子依然能穿透此光能隙結構，只要它滿足等效的Bragg光固子傳播機制。 研究發現這個現象的近似解並以實際的數值例子驗證之。最後，我們也發現了 Bragg 光固子可以存在於摻有均質共振原子且具有非同調幫浦光的光能隙結構裡。我們的研究將原本在一般共振介質裡的自感應透明理論擴展到均勻摻有二階共振原子的光能隙結構裡。

In this dissertation, we theoretically study the self-induced transparency (SIT) in a nonlinear photonic bandgap structure uniformly doped with resonant two-level atoms. We have found that the Maxwell-Bloch equations describing pulse propagation in such a uniformly doped photonic band gap structure can be reduced to effective nonlinear coupled-mode equations. An exact analytic pulse-train solution to these effective coupled-mode equations is obtained first. Such a distortionless pulse-train solution is given by sinusoidal functions with a DC background and a modulated phase. According to the theory of Bragg soliton, the effective nonlinear coupled-mode equations can be further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for Bragg soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original bandgap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if an SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg soliton propagation. An approximate soliton solution describing such coexistence has been found. Numerical examples of the SIT pulse train and SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. Finally, we show that the Bragg soliton also can coexist with a stead-state pulse in an amplifying nonlinear photonic bandgap structure uniformly doped with homogeneously broadening two-level atoms. Our studies extend the SIT theory for the solitary waves propagating in a nonlinear resonant photonic bandgap structure.