| Among the atomic clocks, optical clock is considered as the most accurate one. Today optical clocks based on optical transitions have been widely accepted as the most important candidate to replace Cs microwave Frequency standards for a more accurate definition of SI second. To realize an optical clock, three techniques are mainly involved, i.e., the laser cooling and trapping technique which can drop the temperature of the atoms or ions system to mK even μK, the ultra-narrow linewidth laser (ULL) stabilization technique which is used to detect the optical transitions with high spectrum purity, and the optical frequency comb (OFC) technique based on femtosecond mode-locked laser. Presently, the instability of the state-of-the-art optical clock is about an order larger than that desired by theory. One of the main limititations is due to the frequency instability of its local oscillator, that is, the ULL. Thus, developing an ULL with more stable frequency is a critical route to improve the stability of the optical clock. Furthermore, optical frequency comb based on a mode-locked femtosecond laser, working as an optical frequency synthesizer chain, can be used to measure the output frequency of optical clock and obtain high precision spectroscopy. In the frequency-domain, in order to extend the frequency bandwidth of the OFC, one should investigate many kinds of frequency conversions, such as sum frequency generation (SFG), etc. In the time-domain, the OFC can be lately applied to the active "coherent control" area. Therefore, it is very important and valuable to investigate the working principle, character, and application of the OFC in both the time-and frequency-domain.This work can be mainly divided into two parts, i.e., experimental and theoretical parts. In experiment, we develop an ULL at698nm. Pound-Drever-hall (PDH) method is utilized to achieve a Hz level linewidth laser with two-steps frequency stabilization. With the first frequency stabilization we achieved a1.1kHz linewidth laser relative to the primary frequency reference (pre-stable Fabry-Perot cavity or PFC in abbreviation) as planned. While for the second frequency stabilization, the setup was built and is now under optimization to achieve the expected1Hz linewidth laser relative to the advanced frequency reference (ultra-stable Fabry-Perot cavity or UFC). In theory, we presented a novel vibration-insensitive Fabry-Perot (F-P) cavity design; we studied the bandwidith extension technique of OFC i.e., SFG method; we also investigated the coherent control method during the interaction between a femtosecond laser pulse and an atom. At last, we reported how to achieve ultrahigh resolution spectroscopy using an OFC. Specified contents are shown as follows,1. Development of an ULL at698nm(1) A finesse of-1000F-P cavity is designed and prepared as the PFC, while a finesse of-560000F-P cavity is used as the UFC. Then, these two F-P cavities are placed in two vacuum systems with pressure of-10-6Pa to reduce the influence of air fluxion on the stability of cavities's length.(2) The frequency stability of the ULL mainly depends on the length stability of the UFC, while the length stability of the UFC is mainly influenced by environmental temperature and mechanic vibration. In this thesis, we adopt four steps to weaken the influence of these two factors, which are sketched as follows,i. The UFC is made of ULE (ultra low expansion), and a temperature-controlled box is prepared with multi-layer thermal-and voice-isolations.ii. Temperature-controlled system for the UFC is developed to reduce the impact of the environmental temperature change on the length stability of the UFC.iii. Active vibration isolator is used to weaken the high frequency vibration.iv. Low frequency vibration is reduced by simply optimizing the supports of UFC.(3) Mode-matching between the coupled laser mode and the PFC cavity mode as well as the UFC cavity mode are investigated and performed by using a set of mode-matching lens, to obtain the error signal of high signal-to-noise ratio for frequency stabilazation.(4) With respect to the PFC1.1kHz linewidth laser has been achieved, while680Hz linewidth laser relative to the UFC has been obtained by direct frequency stabilazation with UFC.(5) Based on PFC and UFC, two-steps frequency stabilazation are used to experimentally achieve Hz level ULL output. At present, two-steps frequency stabilazation are preliminarily achieved, and many efforts such as optimizing locking parameters and reducing the servo loop noises are to be made to obtain better result.2. Based on Poisson effect and geometry symmetry, a F-P cavity with new placement is designed using finite element method. Through extensive simulations the optimal supports were obtained. The results are close to the best results achieved in the world. Furthermore, a portable F-P cavity is being designed for application in the space optical clock project.3. Based on the dressed state formalism and numerical calculation, we investigated the SFG method when mixing three femtosecond lasers. We also showed the physical essence of adiabatic SFG. We demonstrated that the adiabatic and non-adiabatic processes play important roles in SFG. We presented another method to calculate the frequency response bandwidth of the generated light pulse. At last, we described the physical processes of SFG in the frequency-domain, which resembles the diffraction phenomena induced by the single slim or double slims. This study has valuable reference for broad bandwidth SFG technique of OFC.4. We investigated the interference and diffraction effect when using a tailored femtosecond laser pulse drives an atom. By modulating the phase and amplitude of a femtosecond laser pulse, we theoretically achieved the quantum coherent control for a wave function related to the atom transition. Furthermore, using a given wave function as a reference one, the interference can occur when another wave function is generated by exciting the same atom with another femtosecond laser pulse. It is shown that the interference can be controlled by modulating the phase of the second laser pulse. Based on the holography we presented a new method to measure an unknown wave function. This work provides guidelines for application of OFC in quantum coherent control.5. In the time-and frequency-domain, we investigated the physical mechanism to achieve ultrahigh resolution spectroscopy, theoretically demonstrated the two-photon transition by driving a87Rb atom with an OFC. It is shown that the ultra-high resolution spectroscopy of87Rb atom can be achieved by independently changing the two parameters of OFC.
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