对于自由电子-光的量子相互作用机理的探索，为未来光电系统的发展提供了新的可能和方法。考虑自由电子波动性和光的量子性而发展起来的光致近场电子显微术（Photon-induced Near-field Electron Microscopy，PINEM），在新型自由电子辐射源、光场对电子波函数的调控、电子修饰光量子态等方面展现了重要的应用潜能。但是至今为止针对PINEM效应的研究只限于单光子过程，自由电子与双光子作用的过程是否存在，以及该过程的物理机理及规律尚未有研究；同时，自由电子的Kapitza-Dirac（KD）效应、广义KD效应仍缺乏基于量子电动力学的解释及全量子化的解析结果，它们与电子-双光子作用和PINEM的关联尚未明晰。本论文针对以上物理问题开展了理论研究工作，取得了如下研究成果。首次建立了同时考虑电子与单光子和双光子相互作用的PINEM全量子理论模型，推导出包含双光子过程的散射算符S ?，并新定义了表征电子-双光子作用强度的二阶量子耦合常数g_(qu,ij)^((2))和g_(p,ij)，发现其不仅与光场纵向分量有关，还与场横向分量有关，从而提出了准连续域束缚态硅超表面结构用于增强电子-双光子作用，计算获得g_qu^((2))可高达~0.099，比此前PINEM实验中的结构提高了8个量级。分析了包含双光子过程的散射算符S ?，将传统PINEM效应、KD效应和广义KD效应统一到同一个散射算符S ?表达式中。基于此分析了KD效应，获得了电子横向动量概率分布的解析表达式，并揭示了S ?中的-g_p (a ?a ̂^?+a ̂^? a ̂ )项是KD效应相关项；同时分析了广义KD效应实验，推导了与光行波场作用后的自由电子能量概率分布解析式，与已报道的实验数据拟合的方差仅~0.0124，优于该报道中利用Schr?dinger方程数值计算的结果，精确地解释了广义KD效应的实验结果。理论阐明了电子-光子相互作用不仅受量子耦合常数g_qu和g_qu^((2))大小的影响，也受其相位的影响。计算结果表明，单光子作用和双光子作用两个过程之间发生了量子干涉；通过与强激光泵浦的近场作用，电子能谱包含了光场的全矢量和相位信息；双光子过程使得电子与光子发生更强纠缠、且纠缠度随g_qu、g_qu^((2))的相位变化，并且电子可以诱导两个相位锁定的光子态产生纠缠。本论文电子-双光子相互作用的理论研究成果，为深入理解和操纵电子-光相互作用中的非线性过程奠定了基础。

The exploration of the quantum interaction mechanism of free electrons and light provides new possibilities and methods for the development of future optoelectronic systems. Photon-induced Near-field Electron Microscopy (PINEM), which is developed by taking into account the wave nature of free electrons and the quantum nature of light, has shown important application potential in new free electron radiation sources, the regulation of electron wave functions by light fields, and modification of light quantum states by electrons. However, the research on the PINEM effect has been limited to the single-photon process so far. Whether the process of free electron-two-photon interaction exists, as well as the physical mechanism and law of this process have not been studied. At the same time, the Kapitza-Dirac (KD) effect and generalized KD effect of free electrons still lack explanations based on quantum electrodynamics and fully quantized analytical results, and their relationship with electron-two-photon interaction and PINEM is not yet clear. This thesis carried out theoretical research on the above physical problems and achieved the following research results.This thesis establish the PINEM full quantum theoretical model that considers the interaction simultaneously between electrons and the single-photon as well as two-photon for the first time. The scattering operator S ? that includes the two-photon process is derived, and the second-order quantum coupling constants g_(qu,ij)^((2)) and g_(p,ij) characterizing the intensity of the electron-two-photon interaction are newly defined. It is found that g_(qu,ij)^((2)) and g_(p,ij) are not only related to the longitudinal component of the light field, but also to the transverse component of the light field. Thus, a quasi bound state in the continuum silicon metasurface structure is proposed to enhance the electron-two-photon interaction. The calculated g_qu^((2)) is as high as ~0.099, which is 8 orders of magnitude higher than the structure in the previous PINEM experiment.The scattering operator S ? involving two-photon processes is analyzed, and the traditional PINEM effect, KD effect and generalized KD effect are unified into the same scattering operator S ? expression. Based on this, the KD effect is analyzed, and an analytical expression for the probability distribution of electron transverse momentum is obtained, and it is revealed that the -g_p (a ?a ̂^?+a ̂^? a ̂ ) term in S ? is a KD effect related term. At the same time, the generalized KD effect experiment is analyzed, and an analytical expression for the probability distribution of free electron energy after the interaction with the traveling wave light field is derived. The variance of the fitting with the reported experimental data is only ~0.0124, which is better than the result of numerical calculation using the Schr?dinger equation in the previous report, and accurately explains the experimental results of the generalized KD effect.The theory explains that the electron-photon interaction is not only affected by the magnitude of the quantum coupling constants g_qu and g_qu^((2)), but also by their phase. The calculation results show that quantum interference occurs between the single-photon process and the two-photon process; through interacting with the near field pumped by strong laser, the electron energy spectrum contains the full vector and phase information of the light field; the two-photon process makes the electron and photon more strongly entangled, the degree of entanglement changes with the phase of g_qu and g_qu^((2)), and the electron can induce entanglement between two phase-locked photon states.The theoretical research results of the electron-two-photon interaction in this thesis lay the foundation for a deeper understanding and manipulation of nonlinear processes in electron-light interactions.