Microwave Experiments Simulating Integrable,Almost Integrable And Chaotic System

We use scalar Helmholtz equation describe the electric field of flat(microwave billiard)two-dimensional resonators,that is the electric field is perpendicular to the top and bottom plate,the Helmholtz equation is mathematically identical to the Schrodinger equation of a quantum billiard.Thus,we can use the equivalence of them to simulate the problems that relate to two-dimensional quantum billiard.Ex-periments on the statistical properties of energy spectrum and electric field strength of resonators with different boundaries are the main tools for investigating quan-tun chaos.we use a microwave resonator with the shape of a 60° circle sector,the classical dynamics is integrable.To observe a transition from integrable to almost integrable to chaotic,we insert cylinders of increasing size.The system is almost integrable in low frequency range for size of cylinder which are small compared to the size of resonator where diffractive orbits occur and the properties are similar to those of a certain class of pseudointegrable systems.So we do the simulation and experiment:The classical dynamic of sector is integrable,the classical dynamic is almost integrable when we add a cylinder as the point-like scatter into the sector cavity of which the size is small compared to the size of Acircle/Asector=3/800 and the classical dy-namic is chaotic as we put three cylinders into the sector cavity.For these three sys-tems,we measure the resoance spectrum and determine the resonance frequency in order to analyze the properties of resonance spectrum fluctuations,the experimen-tal results show that the transition from Poisson distribution to Semipoisson distri-bution to GOE(Gaussian orthogonal ensemble of random matrix)distribution is a direct embodiment of the variation from classical integrable to almost-integrable up to chaotic.We built perturb ation-body equipment to measure the electric field in resonators of arbitrary shape and measure the electric field of three systems,when we add one cylinder or three cylinders,we find that the distribution of electric field of the system tend to chaotic with the increase of frequency.Secondly,we use CST(Computer Simualtion Technology)electromagnetic field simulation software simulate the scattering matrix of the titled stadium bil-liard when magnetized ferrite is added into it,we get that the scattering matrix element Sab transmits from antenna a to b is different from Sba,To quantify the time-invariance breaking,we calculate the cross-correlation function of S12(f)andS21*(f),the results indicate the magnetized ferrite partially breaks the time in-variance of the system.We also simulate sector cavity with magnetized ferrite is added to the boundary,simulate electric field and magnetic field at the frequency which breaks the time invariance.We will design an experiment which the ferrite placed adj acent to a sector cavity boundary to investigate the effect of break time invariance about electric field.At last,we also use CST simulate the electric field of tilted stadium billiard to study the amplitude distribution and the properties of the spatial-autocorrelation function.The amplitude distribution of the electric field can be well described by the Gaussian distribution,and the average value of its spatial autocorrelation function in all directions accords with the theoretical zeroth order Bessel functions.