Propagation Analysis Of Curved Trajectory Beams In Phase Space

The curved-trajectory accelerating beams are a kind of novel structured light fields,of which intensity distribution shifts transversely with propagation and the lobes propagates along the curve traj ectory.Besides,the curved accelerating beams generally have unique weak diffraction and self-healing characteristics.Since Siviloglou et al introduced the non-spreading accelerating Airy wave packets in the context of quantum mechanics into the optical domain and generated the accelerating beams experimentally in 2007,the curved-trajectory accelerating beams have become one of the important research fields of manipulation of light fields in recent years.The curved-trajectory accelerating beams with unique characteristics have many important applications in particle manipulation,atmospheric communication,high resolution imaging,plasmon and so on.Although many achievements have been made in the generation methods,control methods and applications of curved-trajectory beams in recent years,the understanding of the unique propagation characteristics of these beams are still not deep enough.The traditional analysis methods of curved beams are limited to spatial domain or frequency domain,while the phase space representation describes both the position and the spatial frequency of light rays.It's widely used in quantum mechanics,optics,acoustics,signal processing and image processing.In this thesis,we study the propagation characteristics of the curved-trajectory accelerating beams based on the optical phase space method.We propose and explore the non-paraxial Airy-like beams and paraxial hyperbolic trajectory acceleration beams and study the relations of propagation characteristics between the hyperbolic trajectory accelerating beams and the Hermite-Gaussian beams.The main research contents are as follows:(1)By analyzing the initial rays' propagation of the paraxial Airy beams based on ray optics method,we derive the phase space curves of Airy beams and their evolution with propagation.Then,we calculate the variation of ray density and phase shift during the propagation of the beams.Finally,we obtain the propagation equation of the paraxial Airy beams which is completely consistent with the results obtained by the paraxial wave equation or the scalar diffraction integral formula.The phase space representation method can explain the physical significance of each term in Airy beam solutions,which can help us to understand the self-accelerating and non-diffracting characteristics of the Airy beams more clearly.(2)We propose a new kind of Airy-like beams that can keep self-accelerating along arbitrary parabolic trajectory under non-paraxial conditions.Through analyzing the phase space curves of Airy-like beams,we derive the explicit expression of the initial phase function of the Airy-like beams and introduce an amplitude function to ensure the changeless intensity of the beams with propagation.Besides,we compare the initial phase and amplitude distribution of the Airy-like beams with usual Airy beam and use the angular diffraction formula to numerically calculate the propagation of those two beams.The results show that the phase space curves and real space propagation of usual Airy beams are the approximate form of the Airy-like beams under paraxial conditions.Unlike usual Airy beams,the Airy-like beams can maintain non-diffraction and transverse self-acceleration characteristics under non-paraxial conditions.(3)According to the pre-designed hyperbolic caustic,we derive the initial phase function and the amplitude function of hyperbolic accelerating beams.and discuss the relationship between hyperbolic accelerating beams and Hermite-Gaussian beams.The results show that hyperbolic accelerated beams are the larger class of beams than Hermite-Gaussian beams.The phase space curves of hyperbolic accelerating beams are ellipses which rotate and deform as they propagate.When the bending parameter is an integer,the hyperbolic accelerating beams have almost the same initial complex amplitude distribution and propagation characteristics as the Hermite-Gaussian beams.Although the proposed approximation complex amplitude function is more complex than the commonly used Hermite-Gaussian beams,they represent the local amplitude,wave vector and internal ray structure of these beams clearly.So that we can have a clearer geometric understanding of these accelerating beams.