利用高斯光束在共振腔中的疊代映象研究固態雷射上與架構有關的動態行為

Abstract

我們由高斯光束在一般共振腔中傳播的動態特性架構出疊代映象，並利用留數定理研究映象的穩定性，在無耗損狀況下，映象的留數是共振腔G參數的乘積。當G參數的乘積等於1/2，1/4或3/4，□及 時滿足低階共振的條件，這些共振腔架構對非線性效應的反應很敏感；值得注意的是這些架構位在幾何穩定區中。再者，我們發現有耗損元件的共振腔他的動態行為類似有阻泥的震盪，他主要的行為可由無阻泥情況下的保守系統決定。 為驗證上述特性，我們利用端泵固態雷射進行實驗，在泵雷射的光斑比共振模光斑小的情況下，在上述架構基模的輸出功率會有凹陷的現象，而且會有雷射的閥值變低、光腰變小及遠場光束外型突變的現象。在凹陷中的一段特定位置中，還會產生比其他位置噪訊增為21倍的情形。 將映象分析方法應用到克爾效應鎖模雷射上，從物理系統趨向於低能量的概念出發，藉由比較鎖模態與連續態漢米爾頓量(Hamiltonian)的大小可定義相對穩定性；在這個準則下，我們獲得有利鎖模的範圍和實驗觀測到自啟動鎖模的範圍相符。若從雅可賓矩陣(Jacobian matrix)本徵值的大小代表系統受微擾後收斂速率的快慢出發，我們亦可獲得與實驗相符的結果。 進一步的在守恆系統下研究上述特定架構的非線性行為，經數值研究克爾效應鎖模雷射發現，在共焦及共心架構下會有多個基模的解存在，其中叉形分岔發生在較對稱性的架構，而在對稱性受到破壞後則產生鞍節分岔，從分岔的特性來講，我們建議在等臂且共焦附近的架構較可能在克爾效應鎖模雷射上觀測到雙穩態。此外，週期二、三、四依序發生在共振腔G參數為1/2，1/4或3/4， □及 附近，這些非線性行為都可由留數定理解釋；當系統的非線性效應繼續增加後，會有不規則輸出的行為發生。雖然實際雷射系統為耗損系統，但保守系統分析仍保留其物理特性，所以我們認為這些非線性行為可在克爾效應鎖模雷射上觀測到。

In a general optical resonator without gain or loss, Gaussian beam dynamics can be studied by using the residue of the conservative iterative map, which is a function of resonator G-parameters' product. We found, for the first time to our knowledge, there exist new singularity configurations within the geometrically stable regions if the system persists to nonlinear effect. These specific configurations correspond to G-parameters' product equals 1/2, 1/4 or 3/4, and □ as a result of low order resonances. Numerical simulation further show that the dynamics of a dissipative resonator with loss optical elements is equivalent to a damping oscillatory motion and can be determined by the corresponding conservative resonator without those loss elements. In an end-pumped solid-state laser, we experimentally observed a fundamental mode power dip when the cavity length was adjusted approaching these configurations under pumping spot size less than that of the cavity mode. We found the laser exhibits a relatively low threshold together with shrinking beam waist and unexpected far-field beam profile. Within the dip a specific region has irregular behavior of which noise fluctuation is about 21 times of that outside this region. We apply the map method for the unresolved self-starting issue of the Kerr-lens mode-locking (KLM) lasers. Since a physical system tends to stay at lower energy state, by minimizing Hamiltonian between KLM and cw operations we obtained the preferable Kerr-lens mode-locking regions agree with the experimental self-starting ones. Again, the same conclusion can be made by considering the system having dissipative elements, in which the modulus of eigenvalue of the Jacobian matrix represents the convergent rate of the system against a small perturbation and can be defined as the stability factor of the physical system. Further discussing the nonlinear behavior of KLM optical resonator in specific configurations without dissipative elements, we found due to residue equal to zero more than one spatial fundamental Gaussian modes may happen in the concentric and confocal resonators. The pitchfork bifurcation results in more symmetrical configurations and saddle-node bifurcation appears as the symmetry being broken. From the properties of bifurcation, we suggest that the equal-arm and near-confocal resonator is suitable for the emergence of bistability in KLM lasers. In addition, period doubling, tripling and quadrupling can occur at the configurations with cavity generalized G-parameters' product equals 1/2, 1/4 or 3/4, and ,□ respectively. The systems will result in irregular behavior if one further increases the nonlinear effect. Even though we just considered the conservative system in previous discussions, the predicted results agree with the experimental ones that are dissipative. Thus, the nonlinear behavior of KLM laser may be observed in experiments.