No Diffraction Mathieu Beam And Mathieu - Gaussian Beam Of The Focal Shift And Focal Switch

"Nondiffraction beam" refers to the beam which has a small centralspot diameter and central spot does not change with the transmissiondistance. Therefore, it is widely applied in laser precise collimation, laserstiletto, measurement of small objects, navigation and distancemeasurement. Following "Bessel nondiffraction beam" which was firstproposed by J. Durnin from University of Rochester in 1987, J. C.Gutiérrez-Vega and his group of the Photonics and Mathematical Opticsin Monterrey university of México proposed another kind ofnondiffraction beam—Mathieu beam at the beginning of this century.There has been much interest in this new beam which has already beenresearched in some papers. The calculation of Mathieu beam is verycomplex, and so far, there is no research on the properties of beamsfocusing. It is the existence of focal shift and focal switch in beamsfocusing that makes the actual work about focusing more complex andmore significant. Therefore, this thesis is devoted to the study of the focalshift and focal switch of Mathicu beams and Mathieu-Gaussian beams,and the main work is concluded as follows:(1)Using the expanded form of complex Gaussian function, weinvestigate the focal shift and focal switch of Mathieu beams passingthrough an aperture-lens separated optical system. The conditions andaffecting factors of the occurrence of the focal switch were discusseddetailedly by analyzing the numerical values. We conclude that thereexists a critical beam parameter b_c, and the phenomenon of focalswitch will appear when the beam parameter b＞b_c.(2)We study the focal shift and focal switch of Mathieu-Gaussianbeams passing through an aperture-lens separated optical system,using the method which is to expand Mathieu function to the sum ofBessel functions. We discuss the influence of truncation parameter a,beam parameters b, Fresncl number N_F and relative incidcnt distances/f on the focal switch, find that the focal switch ofMathicu-Gaussian beams can take place when relative incident distance demand of s/f=1, and get the range of parameters a and bwhere the focal switch exists.(3)Starting from Collins formula, the focal shift and focal switcheffect of Mathieu-Gaussian beams focused by a thin lens withoutaperture is studied in detail. It has deduced the simple cubic equationwith the maximum light strength and the relative transition formula. Itshows that when the beam parameter b＞2^1/2, focusedMathieu-Gaussian beam will appear the focal switch. The studyshows that the focusing properties of Mathieu-Guassian beam andBessel-Gussian beam are similar greatly.