| The propagation and transformation of laser beams play an important role in laser physics and laser applications. In the last several years, with the development of science and technology, different new type of complex laser beams such as hollow beam, flat-topped beam and laser beam array etc. are required in many applications. The appearance of new complex laser beams made it urgent to study their propagation and transformation in order to meet the requirements in practical applications.Partially coherent light is widely existed in practice. Completely coherent and completely incoherent light does not exist in a strict sense. Partially coherent light has been widely used in material thermal processing by laser beam and inertial confinement fusion by laser etc. The description of partially coherent light and their propagation and transformation have been widely studied for a long time.Since 1993, the fractional Fourier transform was introduced into optics. The fractional Fourier transform has been used widely in signal processing, beam shaping and beam analysis etc. Up to now, however, most efforts have been concentrated to the completely coherent field. Only few works have been done on the fractional transform of the partially coherent beams. In practice, most beams are partially coherent, thus it is necessary to study the fractional Fourier transform of partially coherent beam.Recently, quantum and classical coincidence imaging and diffraction have attracted more and more attentions. There exists a hot debate about whether quantum entanglement is necessary in the coincidence imaging or diffraction and whether a coincidence imaging or diffraction experiment can be realized with classical source. It is of academic significance to find out the physical mechanism of quantum and classical coincidence imaging and diffraction.In chapter 1, the progresses in the field of laser propagation theory, partially coherent laser theory, fractional Fourier transform, coincidence imaging and diffraction are reviewed. The motivations and significance of this paper are presented. In chapter 2, the basic concepts and theories of laser propagation, partially coherent laser, fractional Fourier transform, coincidence imaging and diffraction are introduced, respectively.In chapter 3, a new kind of laser beam called the elliptical Hermite-Gaussian beam (EHGB) is introduced to describe the complex higher-order laser beams by using a tensor method. With the beam parameters properly selected, the EHGB can be used to describe complex laser beams, such as the hollow beam, the beam with non-circular and non-rectangular symmetry and the twisted Hermite-Gaussian beams etc. Using the generalized Collins integral, we derive the propagation formulae of the EHGB passing through axially non-symmetrical and misaligned paraxial optical system. We further introduced the decentered EHGB to describe decentered complex higher-order laser beams, and derive the propagation formulae of the EHGB through complicated optical system. What's more, we also study the propagation properties of the laser beam arrays constructed by coherent and incoherent combination of the EHGBs. The results show that we can obtain the effective laser beam with incoherent combination of the EHGBs.In chapter 4, we introduced the flat-topped beams with circular symmetry and rectangular symmetry firstly defined Gori and Li, respectively. Then as an extension of Gori's definition and Li's definition, we define two kinds of laser beams to describe flat-topped beam with elliptical symmetry. The propagation formulae of the two elliptical flat-topped beams through axially non-symmetrical and misaligned paraxial optical system are derived. By using the derived formulae, the propagation properties of elliptical flat-topped beam passing through free space are calculated and discussed. The results show that elliptical flat-topped beam not only has the general propagation properties of flat-topped beams with circular symmetry and rectangular symmetry, but also has it's own unique propagation properties, i.e., its elliptical spot will rotate during propagation. Our models and methods provide a convenient way to describe flat-topped beam with elliptical symmetry and treat its propagation and transformation.In chapter 5, a new type of hollow beam named hollow Gaussian beam (HGB) is introduced. HGB can be used to describe circular hollow beam whose axial intensity is not equal to zero except in the source plane. The HGB can be expressed as a superposition of a series of Laguerre-Gaussian beam. We derive the propagation formula of HGB through axially symmetrical paraxial optical system, and study the properties of HGB passing through free space. The results show that properties of HGB are very different from the commonly used hollow beams such as TEM 01 * beam and high-order Bessel beam. We further introduced hollow elliptical Gaussian beam (HEGB) to describe the hollow beam with elliptical symmetry. The HEGB can be expressed as a superposition of a series of elliptical Hermite-Gaussian beam, and it has its own unique propagation properties. The propagation formulae of HEGB through axially non-symmetrical and misaligned paraxial optical system are derived. Our methods provide a convenient way to describe hollow laser beams with circular symmetry and elliptical symmetry, and the HEG and HEGB can be used conveniently to analyze atoms guiding and atoms trapping.In chapter 6, a new 4×4 complex curvature tensor M-1 is introduced to describe partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams. The generalized diffraction integral formulae for partially coherent beam through axially non-symmetrical and misaligned paraxial optical system are derived in spatial domain and spatial-frequency domain, respectively. The propagation and transformation formulae for the cross-spectral density of twisted anisotropic GSM beam through complicated optical system were derived. The propagation law of M-is also derived which may be called partially coherent tensor ABCD law. By using the derived formulas, we calculated the propagation properties and spectral shift of twisted anisotropic GSM beam passing in free space and through a thin lens. We also derive an analytical propagation formula for the cross-spectral density of twisted anisotropic GSM beams through dispersive and absorbing media, and study the evolution properties and spectrum properties of twisted anisotropic GSM beams in dispersive and absorbing media. We further studied the propagation properties of decentered twisted anisotropic GSM beam and partially coherent laser beam array consisted of superposition of decentered twisted anisotropic GSM beam. What's more, we introduced several kinds of mathematical models to describe partially coherent beam with circular symmetry, elliptical symmetry and rectangular symmetry, and derived the propagation formulae of partially coherent flat-topped beams through complex optical system. The propagation properties of partially coherent flat-topped beams are studied in detail.In chapter 7, we studied the fractional Fourier transform (FRT) of partially coherent beam in spatial domain and spatial-frequency domain. The transformation formulae for partially coherent twisted anisotropic GSM beam passing through the FRT system in spatial domain and spatial-frequency domain are derived, and the equivalent tensor ABCD law for the FRT of twisted anisotropic GSM beam is also derived. The connections between the FRT and the diffraction integral formulae for partially coherent beams through axially non-symmetrical and misaligned paraxial optical system are discussed. We also introduced the tensor method to describe the beam coherence-polarization matrix of partially coherent and partially polarized GSM beam, and studied the transformation properties of partially coherent and partially polarized GSM beam. What's more, we extended the definition of FRT to partially coherent optical pulses, and derived the transformation formula for the FRT of partially coherent GSM pulse. Our method provides a convenient way to treat the FRT of partially coherent twisted anisotropic GSM beam, partially coherent and partially polarized GSM beam and partially coherent GSM pulse.In chapter 8, by using the two-photon theory and Collins formula, we analyze the quantum coincidence imaging and diffraction, and derive the quantum coincidence imaging and diffraction formulae, which are well coincident with the experimental results. We also studied the coincidence imaging and diffraction with coherent beam by using coherent optical theory, our results can well simulate the experimental results of coincidence imaging and diffraction with coherent beam recently reported by Bennink et al.. What's more, we studied the coincidence imaging and diffraction with incoherent and partially coherent light, and derived the coincidence imaging and diffraction formulae. Our results clearly show the different physical natures of above three different coincidence imaging and diffraction, and can resolve the current debate. Recently, the group of Y. H. Shih has experimentally verified our results about the coincidence imaging and diffraction with incoherent and partially coherent light.In chapter 9, we proposed the concepts of quantum and classical coincidence fractional Fourier transform (FRT), and designed the optical systems to realize the coincidence FRT with entangled photon pairs, and the coincidence FRT with incoherent and partially coherent light. The differences between the quantum and classical coincidence FRT are discussed, and the influence of the source on the quality and visibility of coincidence FRT are calculated in detail. We further proposed the concepts of quantum and classical coincidence sub-wavelength FRT, and designed the optical systems to realize the coincidence sub-wavelength FRT with entangled photonpairs, and the coincidence FRT with incoherent and partially coherent light.
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