頻率解析光閘法(frequency-resolved optical gating, FROG)是一種廣泛用於超短脈衝雷射的量測方法。實驗上，使用一臂具延遲(delay)的干涉儀進行自我干涉，記錄延遲長度對應的頻譜構成的二維圖像，被稱為FROG trace。傳統上會使用迭代演算法(Iterative algorithm)，花費數秒以還原待測脈衝的資訊。在量測超短脈衝的超寬頻譜時，使用FROG會遭遇晶體的頻率響應問題，造成FROG trace在頻率軸上被裁切。以往這種被裁切的FROG trace只有極少數迭代演算法才能完成相位還原，並且有所限制。
本研究建立了一個基於卷積神經網路(Convolutional Neural Networks, CNN)的演算法，能夠以毫秒級的計算時間得到足夠可靠的脈衝資訊。亦能成功達成裁切FROG trace的相位還原，並得到與迭代演算法相比更好的統計結果。因此，此方法對於超寬光譜之脈衝的即時量測與監控提供了基礎。

Frequency-resolved optical gating (FROG) is a common mothed for retrieving the complex field of ultrashort pulse from experimentally taken spectrogram (i.e. FROG trace). Traditionally, a FROG trace is processed by some iterative algorithm(s), and takes several seconds to arrive at the solution. FROG trace would be spectrally truncated when the nonlinear crystal failed to provide sufficient phase-matching bandwidth particularly in supercontinuum pulse measurement. Only a few algorithms can work with truncated FROG trace, and is somehow limited.
In this work, we develop a convolution neural network based mothed to retrieve spectral phase from spectrally truncated FROG traces in only few milliseconds, while the pulse reconstruction remains acceptable. This mothed is promising for real-time characterizations of supercontinuum pulses.

摘要 I
ABSTRACT II
致謝 III
目錄 IV
圖目錄 V
一、緒論 1
二、基礎理論與模擬 3
2.1頻率解析光閘(FREQUENCY-RESOLVED OPTICAL GATING, FROG) 3
2.1.1數學模型與實驗架構 3
2.1.2相位重建－主成分投影演算法(principal component generalized projections algorithm, PCGPA) 4
2.1.3相位重建－重疊關聯成像(Ptychography) 7
2.1.4非完整取樣之成因與重建限制 8
2.2 脈衝庫建立 9
2.2.1 模擬脈衝生成 10
2.2.2 脈衝相位生成 11
2.2.3 複數脈衝生成 13
2.3脈衝庫之多樣性分析 14
2.3.1脈衝庫之複雜度分析 14
2.3.2脈衝庫之差異度分析 15
2.3.3脈衝庫之統計分布 16
2.3.4脈衝庫之時域差異度分析 17
2.4基於二倍頻之SHG-FROG圖樣生成 20
2.5脈衝相位還原正確性之評估 21
2.6窗口縮放(SCALING)之可行性討論 21
三、研究方法 22
3.1類神經網路簡談 22
3.1.1發展與應用 22
3.1.2類神經網路與迭代演算法之優劣勢分析 22





3.2類神經網路結構 23
3.2.1類神經元設計 23
3.2.2損失函數(loss function)設計與矩陣化 25
3.2.3神經網路結構選擇 28
3.2.3.1 DNN 深度神經網路結構 28
3.2.3.2 CNN 卷積神經網路結構 28
3.2.4反向傳播(Back propagation) 演算法 30
3.2.5神經權重與激勵函數 33
3.2.6最佳化器(Optimizer) 35
3.2.7訓練集(train)、驗證集(validation)與測試集(test) 37
3.2.8殘差神經網路(ResNet) 39
3.2.9類神經網路設計 41
3.3訓練軟硬體之取捨 43
3.3.1解析度與傅立葉轉換 43
3.3.2深度學習資料庫選擇 44
3.3.3使用中央處理器(CPU)及圖形處理器(GPU)進行計算的差異 45
四、研究成果 46
4.1原始脈衝的頻譜破碎程度對還原成功率之影響 46
4.2相位追回演算法對各頻譜裁切條件之相位還原比較 47
4.2.1 PCGPA演算法對各裁切條件之相位還原統計 48
4.2.2 Ptychography演算法對各裁切條件之相位還原統計 49
4.2.3類神經網路演算法對各裁切條件之相位還原統計 53
4.2.4三種演算法之抗噪性討論 57
4.2.5類神經網路演算法之訓練時間 59
4.2.6一個完整實驗案例討論 61
五、結論與未來展望 63
參考文獻 65

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