隨著孤立埃秒脈衝產生技術的發展，可見至近紅外光之單週期寡載波之重要性日益提升。因此，以非線性方法產生超連續光譜並壓縮該脈衝至轉換極限已然成為此領域的必備技術。目前已存在許多量測脈衝完整資訊的方法，其中，「頻率解析光柵」或「修正場自相關干涉量測法」之類以二階非線性現象作為原理的量測技術擁有較其他方法更高的靈敏度。由於在某些實驗架構中，「修正場自相關干涉量測法」使用厚非線性晶體來達成頻率轉換與濾波，其靈敏度更是有明顯的優勢。
在此研究前，群相速不匹配被認為是在「修正場自相關干涉量測法」中選擇非線性晶體的唯一限制，並因此訂定了小於百分之五脈衝展寬作為標準。在此一研究中，我們發現了另一個有更加深遠影響的機制並稱之為「基頻相位不匹配」。在發展「修正場自相關干涉量測法」理論時，為了得到解析解而使用了「轉換函數」模型，此一機制因而沒有被發現。然而隨著脈衝頻寬逐漸展寬，此一模型的有效程度便逐步降低。在此論文中，我們描述「基頻相位不匹配」的數學模型並以數值方法分析其對「頻率解析光柵」與「修正場自相關干涉量測法」之脈衝重建結果的影響。我們的模擬顯示，在考慮「基頻相位不匹配」的情形下，當使用同樣厚度之非線性晶體，「修正場自相關干涉量測法」得到相較於「頻率解析光柵」更低的錯誤率。最後，我們用數值方法說明「基頻相位不匹配」確實是造成「修正場自相關干涉量測法」解析錯誤的其中一個主要成因。

Single-to-sub-cycle pulses in the visible-to-infrared regime have gain importance in generating isolated attosecond bursts. Therefore, generating octave-spanning spectra and compress them to transform-limited pulses have become a routine work. There are many existing techniques that completely characterize ultrashort pulses. Among them, techniques that utilize second-order nonlinearity like second-harmonic generation frequency-resolved optical gating (SHG-FROG) and modified interferometric field autocorrelation (MIFA) are known for their high sensitivity. MIFA especially has the edge since in some setups, thick nonlinear crystals are used to not only perform second-harmonic generation but also act as a spectral filter.
Prior to this work, we believed the choice of nonlinear crystal in MIFA is only limited by group velocity dispersion (GVD) and set the criterion of 5% full width at half maximum pulse-width expansion as tolerance. However, we’ve discovered in the work that there exists a more profound limitation, and coined the name “fundamental phase-mismatch.” This effect was neglected because of the use of transfer function model while developing the formulation of MIFA, the model loses its validity as spectral bandwidth increases. In this thesis, we describe in detail the formulation of fundamental phase-mismatch and analyze numerically how it influences the retrieve results of SHG-FROG and MIFA. Our simulation shows even in the presence of fundamental phase-mismatch, MIFA performs superior to SHG-FROG in terms of retrieve error with the same thickness of nonlinear crystal used. Finally, we show numerically that fundamental phase-mismatch is indeed a primary factor of MIFA retrieve error and give numbers on how error changes with different thicknesses of nonlinear crystal used.

Acknowledgements i
Abstract ii
摘要 iii
Table of Contents iv
List of Abbreviations v
1. Introduction 1
2. Theory of Pulse Measurement Techniques via Second-Harmonic Generation 3
2.1 SHG-FROG 3
2.1.1 Mathematical Model and Experimental Setup 3
2.1.2 FROG Algorithm: PCGP 5
2.2 MIFA 10
2.2.1 Mathematical Model and Experimental Setup 10
2.2.2 Spectral Phase Retrieval Algorithm 13
3. Theory of Fundamental Phase-Mismatch in Second-Harmonic Generation 16
3.1 Second-Harmonic Generation of Ultrashort Pulses 16
3.2 Transfer Function Model 21
3.3 Fundamental Phase-Mismatch 23
3.4 Simulation of FROG Trace in the Presence of Fundamental Phase-Mismatch25
3.5 Simulation of MIFA Trace in the Presence of Fundamental Phase-Mismatch28
4. Pulse Reconstruction of MIFA and SHG-FROG in the Presence of Fundamental Phase-Mismatch 30
4.1 MIFA vs. SHG-FROG 30
4.1.1 How Phase-Mismatch Affects SHG-FROG and MIFA Traces 30
4.1.2 MIFA vs. FROG with Different Thicknesses of BBO 33
4.2 MIFA 38
4.2.1 Definition of R 38
4.2.2 Errors of MIFA with Different Thicknesses of BBO 39
5. Conclusion and Perspectives 43
References 45

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