Some New Operators In Quantum Optics Corresponding To Classical Fresnel Transformation

One of the important parts in classical optics (Fourier optics) is the Fresnel diffrac-tion and its Collins formula. Scholars of China Fan Hong-yi et.al firstly studied the quantum correspondence of Fresnel diffraction by using the coherent state. By using the Integral technique within ordered product (IWOP) of operators they constructed a quantum Fresnel operator to realize the Fresnel transformation in quantum optics and it corresponds to the classical Collins formula. Since the introduction of the quantum Fresnel transformation, it has been widely applied to the discussion of the relation be-tween the classical optics and the quantum optics and the relations with other optical transformations. In this paper will use the view of quantum optics to resurvey some classical optical transformation by using the squeezed coherent state and coherent en-tangled state representation to put forward some new unitary operators (e.g. the single mode and two mode generalized Fresnel operators, the Fresnel-Hadamard complimen-tary operator) in quantum optics corresponding to the classical Fresnel transformation and give their physical meanings. The classical correspondence of these new unitary operators may provide probability for some new transformations in classical optics. The starting point of our work is based on the below considering:the Fresnel operator given by Fan Hong-yi et.al is derived from a coherent's moment in phase space from point to point sz-rz*, where ss*- rr*= 1. According to the intuitional analysis, a coherent state is graphically corresponding to a small round. The quantum Fresnel transformation indicates a small round moving to another small round. The condition ss*-rr*=1 guarantees the transformation is symplectic and also guar-antees satisfying the Liouvil theorem (the invariability of phase volume). According to that the coherent state is a special case of squeezed coherent state and thinking that the squeezed coherent state has been got widely used in quantum optics and quantum information, the generalized Fresnel operator we introduce in this paper is just based on the moment of squeezed coherent state(graphically corresponding to a ellipse in phase space) and by using the IWOP technique. Whereas the filtering of the theory of quantum entanglement to quantum optics, based on the coherent entangled state, we introduce a Fresnel-Hadamard complementary operator, which can play the roles of Hadamard transformation and Fresnel transformation respectively for the two output fields of and of a beamsplitter (The two input fields of the beamsplitter is a1 and a2).My Ph.D dissertation is arranged as following:In chapter one, we introduce some background knowledge of our work, the Fres-nel diffraction formula in classical Fourier optics and the deriving process to obtain the Collins formula in classical frame.In chapter two, in order to bridge the classics to quantum, we introduce some background knowledge. Firstly, we introduce some basic representation like the co-ordinate, momentum, particle number and the coherent state representations, then we introduce the background and meaning of the Integral technique within ordered product (IWOP) of operators which really has the above bridge function.In chapter three, we introduce the quantum operator corresponding to the classi-cal Fresnel transformation, i.e. the Fresnel operator constructed by Fan et.al, which is based on the coherent state representation in quantum optics and the IWOP technique. According to the intuitional analysis in phase space, the Fresnel operator corresponds to the moving transformation of the representing point(a round with h/2 area) of co-herent state in phase space. we also introduce the quantum operator corresponding to the classical Fresnel transformation, i.e. the two-mode Fresnel operator constructed by Fan et.al, which is based on the two-mode coherent state representation in quantum optics and the IWOP technique.In chapter four, based on the squeezed coherent state representation in quantum optics and the IWOP technique, the quantum operator corresponding to the classical Fresnel transformation, i.e. the generalized Fresnel operator is introduced. According to the intuitional analysis in phase space, the generalized Fresnel operator corresponds to the moving transformation of the representing point(an ellipse with h/2 area) of squeezed coherent state in phase space.In chapter five, based on the two-mode squeezed coherent state representation we introduced the two-mode generalized Fresnel operator corresponding to the two-mode Fresnel operator.In chapter six, based on the coherent entangled state, we find a Fresnel-Hadamard complementary operator. For the optical fields and considered as two out-put fields after the two mode optical fields a1 and a2 passing through a beamsplitter, the Fresnel-Hadamard complementary operator can play the roles of Hadamard trans-formation and Fresnel transformation respectively. Above discussion indicates that from the new representation in quantum optics and using the Integral technique within ordered product (IWOP) of operators we can find new optical transformations.Finally, we give some conclusions and expectations.