Wave Propagation Control Based On Coordinate Transformation And Its Fractional Fourier Analysis

Controlling wave propagation in different media is an important research direction formathematics and physics. As an important method in modern signal processing, thefractional Fourier transform (FRFT) is not only the design consequence of control the wavepropagation, but also an important tool in analyzing the scattering effect of wavepropagation control. The close relationship between signal processing and wavepropagation shows that it is of great significance to research the control wave propagationfrom the perspective of signal processing method, in order to better serve the modern signalprocessing technology.Supported by the National Natural Science Foundation of China, and built upon thestudy of wave propagation control theory, this dissertation explores its application in thedesign of signal processing devices, and utilizes FRFT to examine wave control effectsfrom the perspective of signal processing, which provides a basis for optimized designs.The results of this paper provide an effective method for the design of medium with signalprocessing functions, and also a theoretical basis and practical reference for thedevelopment of wave propagation control system. The main contents and innovation pointsof the research are as following:1. Research on the various conversion relationships of the coordinate transformationdesign, and presentation of a new way to obtain the transformation relationshipthrough spatial deformation.First of all, taking the substitution method as an example, it analyzes the changingvariable method of typical coordinate transformation, and points out that this method is themathematical interpretation of the control equation’s form invariance under differentcoordinate system, together with the problems of the interpretation. Then it proposes thephysical interpretation of control equation form invariance, elaborating the significance ofthe interpretation.Correspondingly, based on the physical interpretation, it provides a method to obtaintransformation relationship of the physical quantities. This method, which is calledconstraint method, is built on the space deformation perspective that can obtain thetransformation relationship. It views transformation as rotation and stretch of pace differentelements. The transformation of material parameters and the field can also be described bymeans of infinitesimal deformation. The constraint relations after deformation, such asgeometric constraints, constraints of conservation of energy, are used to solve the equationsto obtain the results of various physical quantities transformation. The transformation ofmaterial parameters and field quantity can also be described with the help of elementsdeformation. Using the geometric constraint of elements after deformation, and energyconservation constraint, we could obtain the transformation result of each physical quantityafter substituting the constraint into the control equation. Related applications in the nextchapters show that the proposed method is more convenient to calculate the transformationrelations, and can reduce medium scattering effect very well. This method can not only realize the design of medium parameters, but also can be extended to more general controlin the physical domain.2. The design and application of fractional Fourier transform lensBased on the above-mentioned physical interpretation and coordinate transformationmethod, the research proposes a FRFT lens design based on transformation material, ie., anoptical Fourier Transformation lens with enhanced input spatial frequency scope, and aFRFT lens design of arbitrary order. This design can be realized with isotropic materials. Itsmedium parameters are simple and have great engineering value. Furthermore, withbroadened input frequency domain to the designed lens, it is therefore equipped with goodencryption effect, capable of encrypting and decrypting image information.3. The scattering analysis of wave propagation control based on fractional FouriertransformCompared with change variable method design, the constraint method is designed togreatly reduce the scattering effect of wave propagation control. Firstly, the analysis anddesign of change variable method and the constraint method are presented, providing goodexamples for FRFT.Then, focusing on the elastic wave concentrator and approximate directional cloak, thequantifies analysis of the scattering effect of different situation is presented. Consideringthe spatial frequency change caused by the scattering effect, the FRFT method is presentedto analyze the scattering phenomenon. In this way, we can establish the correspondencesbetween fractional transformation orders and the space-frequency changes, which providesreference for the evaluation of the two design methods.