| Quantum entanglement is a fundamental concept in quantum physics. It stimu-lates a deep concern about quantum mechanics. Meantime, the entanglement is al-so a key source for the practical quantum technologies such as quantum information processing, quantum communication and quantum imaging etc. So the entanglement plays a key role for the development of quantum optics. It becomes a primary task to engineer the entangled photon source for the practical use. This thesis focuses on the the physical process of entanglement generation and find new way for manipulating the entanglement by using new types of nonlinear optical materials. Based on the cold atom ensemble, homogenous bulk crystal and optical superlattice, we made a compre-hensive study on the frequency-time, momentum-position correlation of the entangled photon pair. New types of entangled photon states are explored as well as the possible applications are discussed. The main results of our thesis are listed as the following:1.Generation of narrowband biphotons with positive frequency-correlation from the four-wave mixing process is analyzed through manipulating optical properties of light propagation inside a cold atomic ensemble. In contrast with the spontaneous parametric down-conversion case, we find a much looser condition which allows pro-ducing such frequency-correlated paired photons. We propose a two-photon temporal conditional correlation measurement to directly detect these photons because of the narrower width of entangled photon pair. In addition, our results provide another ev-idence of observing backward pulse propagation from the viewpoint of two-photon correlation measurement.2.Based on type-II noncritical phase-matching, the spatial entanglement of two-photon state from spontaneous parametric down-conversion is studied under the fo-cused Gaussian pump theoretically. It is found that by manipulating the focused pump, we can engineer the two-photon momentum to be strongly anticorrelated, n- early uncorrelated or strongly correlated. And the corresponding position correlation at the output surface of nonlinear crystal shows a strong positive correlation, near uncorrelation or strong anticorrelation. The degree of spatial entanglement is stud-ied by employing the entanglement entropy and it shows a dependence on the focal parameter ξ which is defined as the ratio between the crystal length to the pump's confocal length. When ξ=1.32, the entanglement entropy approximates to its min-imal value0.47. When ξ>>1.32, two-photon state with correlated-momentum and anticorrelated-position can be engineered with a high degree of entanglement. This state is just reverse to that of the Einstein-Podolsky-Rosen state which is usually gen-erated under ξ<<1.32and the possible applications of such state are also investigated. This work can give applicable references to the spatial control of entangled two-photon source in experiment, including the precise measurement of spatial correlation and the transformation of spatial entanglement for developing new quantum technologies.3. We propose to generate the counterpart of Einstein-Podolsky-Rosen state which possesses correlated-momentum and anticorrelated-position directly from spontaneous parametric down conversion by engineering quadratic nonlinear photonic crystals. The crystal supplies a set of gradually varying reciprocal vectors to fulfill the phase-matching condition. And on the other hand it and the plane wave pump together can be considered as an effective focusing pump. The "two-photon spherical wave" is pro-duced and the photon pair exit from the back face with centrosymmetric positions and identical momentum. A new type of erect quantum imaging is deduced because of the new momentum-position correlation. This work may bring sharp focus on quantum fundamentals and enables new types of quantum technologies.
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