近簡併共振腔皮秒克爾鎖模雷射之非線性動力學

Abstract

我們研究皮秒克爾鎖模雷射在簡併共振腔架構附近之非線性動態行為。由實驗中我們觀察到無外部調變與回饋控制情況下自啟動克爾 鎖模雷射其自啟動行為。從產生雷射後至穩定鎖模期間，雷射輸出功 率呈現一短暫且連續不規則輸出（free-running spiking），並且每次持續的時間皆不相同。利用Grassberger-Procaccia 分析法對其暫態部分做相關維度（correlation dimension）分析，我們得到一非整數值亦即表示此一暫態行為呈現渾沌特性。利用自相關函數對其做進一步分析亦顯示其具有渾沌特徵。我們將此暫態行為等分成數段並計算各段之相關維度，我們發現相關維度由較高的非整數值逐漸減少直到於穩定鎖模區域時相關維度為零。 在皮秒自克爾鎖模鈦藍寶石雷射中我們在簡併共振腔附近觀察到脈衝串列振幅調變。隨泵浦功率的增加脈衝串振幅調變的包絡逐漸由單群分裂成兩群或三群且調變深度亦隨之加深，最後當泵浦功率增加至更高功率，脈衝串振幅調變變成雜亂無序。由於皮秒雷射之脈衝強度遠小於飛秒雷射，故此調變應該不是由高階光固子或色散所造成的。再者脈衝串調變可操作範圍僅15 微米，雷射共振腔的損耗應該沒有太大變化，因此調變行為亦非遲緩震盪（relaxation oscillation）所造成。由於產生調變行為的操作區域略為偏離簡併共振腔架構，故可能因雷射晶體中空間不均勻增益所產生的兩組未與縱模相互簡併之高階橫模相互競爭所造成調變行為。 藉由薄片近似（thin slab approximation）我們可以描述一脈衝於共振腔傳播時，共振腔架構對其橫模模態分佈影響。我們使用柯林繞射積分（Collin integral）和考慮自聚焦效應之速率方程式，以數值模擬研究皮秒鎖模雷射於簡併共振腔附近的動態行為。在不考慮自聚焦效應於速率方程式時，我們發現雷射輸出功率隨腔長的變化呈現一功率凹陷的分佈。然而於速率方程式中加入自聚焦效應時，脈衝串的包絡呈現各種狀態，其中包括連續輸出或週期、週期-2 以及不規則狀態。值得注意的是數值模擬結果類似於自啟動克爾鎖模雷射，於自啟動時雷射輸出行為發生於不穩定與連續波輸出操作區域之間。這些現象均與我們在實驗中所觀察所得是一致的。此外，由於起始電場之振幅與相位初始值來自於自發輻射，於不同次模擬結果，此短暫且連續不規則輸出持續時間不完全相同且呈現一指數函數遞減分布趨勢。其回歸映像（return map）呈現出在起始時具有奇異吸子的渾沌態，隨時間演化轉變成準週期態最後收斂於穩定態。理論模擬結果顯示自啟動鎖模從雷射於渾沌態轉為穩定鎖模態之自適行為應由自聚焦效應所造成。

We experimentally observed the transient state from the laser starting to reaching a stable mode-locking (ML) state in a self-starting Kerr-lens mode-locked Ti:sapphire laser without external modulation and feedback control. By Grassberger-Procaccia analysis for the transient state, the correlation dimension of the transient state is a non-integer which implies it is a chaotic state. The chaotic characteristic can be further confirmed by observing the revivals of the autocorrelation function for long delay time. Pulse-train modulation was observed in this laser with pump-power dependence when it was operated around the degenerate cavity configuration. By increasing the optical pumping power, the envelope of periodic amplitude modulation splits into two or three clusters with enhanced modulation depth, and the amplitude modulation eventually becomes disordered at higher pump power. The amplitude modulation may be supported by exciting two sets of non-degenerate longitudinally mode-locked supermodes due to spatially inhomogeneous gain modulation in the Ti:sapphire crystal. We also numerically studied suppressing chaos to reaching completely mode-locking in this self-starting Kerr-lens mode-locked (KLM) laser. By thin slab approximation, we can describe transverse effect of a pulse propagates in a resonator. Based on Fox-Li’s approach, we used the Collins integral and rate equations with and without the self-focusing effect, we found without the self-focusing effect typical laser output and the feature of a power dip agrees with the observation of experiment for all calculated cavity configurations around the degeneracy at various pump powers. However, by adding the self-focusing effect, the time evolution of the pulse-train envelope presents various states including continuous wave or periodic state and instability such as period, period-2, and irregular states. The simulated self-starting KLM output, which possesses transient irregularity before reaching a constant amplitude output, occurs between the instability and continuous wave regions. The different runs of the simulated self-starting from the spontaneous emission reveal the buildup time of mode-locking not only is sensitive to the initial condition but also presents the distribution with exponential decay. Its return map presents chaotic state with a strange attractor in the initial stage. It transits to the quasi-periodic state and finally converges to a fixed point with time evolution. The theoretical simulation reveals that the self-focusing effect is responsible for the self-adaptation.