| With rapid advances in pulsed laser techniques, ultrashort few-cycle, even single cycle, pulsed beams have been produced in laboratory. On the other hand, the correlation properties of pulsed beams from different kind of resources are very different. For instance, the pulsed beams from a model-locked laser and from truncation of a continuous wave have very different properties, even though they may have same intensity profiles. The propagation of such kind ultrashort pulsed beams has attracted much attention. The present dissertation is mainly devoted to studying the propagation properties of ultrashort pulsed beams propagating through paraxial optical systems. The main results achieved in this diseration are summarized as follows:1. The M~2 factor of partially coherent ultrashort pulsed beams has been dtudied. Starting from the space-time Wigner distribution function and taking the Gaussian Schell-Model (GSM) pulsed beam as a typical example, an explicit expression for the M~2 factor of the GSM pulsed beam through a paraxial system without dispersive effects has been derived. The results show that the M2 factor depends on the pulse bandwidth (or pulse duration), spatial correlation degree and average wavelength. Furthermore, the M2 factor of chirped GSM pulsed beams is additionally dependent on the chirped parameter.On the other hand, the transformation and focusing properties of the GSM pulsed beams by a dispersive lens has been studied. Numerical result shows that, compared to the spatially fully coherent case, the spatial correlation degree leads to an increase of the width and a decrease of the peak value of the energy density distribution at the focal plane.2. The propagation properties of partially coherent pulsed beams in temporal demain have been studied. By using the Wigner distribution function, an analytical propagation formula of the correlation function for partially coherent ultrashort pulsed beams in temporal domain through optical system with dispersion is derived. It is shown that both the intensity and pulse width are dependent on the correlation width and the temporal ABCD matrices of the dispersive optical system. For the partially coherent ultrashort pulsed beams in temporal domain propagating in a simple optical system consisting of an ideal chirper and a dispersive media, the pulse width changes from decreasing, reccovering, to increasing. The propagation distance where the pulsed beam recovers its pulse width is dependent on the chirp parameter, the group-velocity dispersion (GVD) parameter, and the correlation width. Due to the effects of the correlation width, the maximum propagation distance where the pulsed beam recovers its pulse duration is always greater than the corresponding dispersion length.3. A model of isodiffracting hollow Gaussian pulsed beams (HGPBs) has been introduced and the corresponding propagation equation has been derived by making use of the Fourier transform method. Isodiffracting Gaussian pulsed beams can be treated as a special case of the result. The propagation properties of the HGPBs in free space are studied. It is found that for high orders of HGPBs there are symmetrical temporal profiles on axis and there are dark rings in the amplitude distribution area off axis. At the positions of the dark rings, the temporal profiles of the pulsed beams are distorted. In addition, both the spectral intensity and energy density distribution of the HGPBs are studied.4. Starting from the Fresnal diffraction integral, and considering only the one-dimensional field distribution, an analytical propagation equation of ultrashort Gaussian pulsed beams diffracted by a rectangular aperture is derived. It is shown that the spectral intensity distribution is not only shifted but also split into two or more parts, which are dependent on both the diffraction angle and the truncation parameter. On the other hand, the time intensity distribution splits gradually into two symmetrical parts with increasing local time.
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