Spatial Distribution Depiction Of Bessel Beam

Bessel beam has been widely exploited due to its exclusive intensity distribution and propagating performance.Axicon is frequently employed to obtain a Bessel beam because of its high efficiency to transform a beam.Although there have been a lot of studied works and a lot of breakthrough progress on Bessel beam research,most of the descriptions with respect to light wave field distribution behind axicon still expressed by diffraction integral,especially in Fresnel diffraction area.Because this integral cannot be carried out by means of an analytical expression,the distribution of diffracted field can only be conducted by numerical calculation approximately,which is confined by sampling condition or algorithm,etc.It’s very difficult to execute accurate and quick depiction to light wave field distribution behind axicon.So,researching a novel algorithm,simplifying mathematical expression of 3-D light wave,and decreasing the limitation of numerical calculation will be a significant work in theory.Via the development of research,its final theoretical results not only supply the mathematical support,but also give out the method of design and the evaluation of bottle beam toward Bessel distribution.In this work,starting from the Fresnel diffraction formula in cylindrical coordinate system,by changing it into a convolution form expression,and employing the characteristics of zero-order Hank transform,the analytical expression of Bessel beam converted by an axicon is derived,and its intensity distribution and experimental validation are given.Further prolongation from Fresnel diffraction to Fraunhofer diffraction is executed,and the analytical expression of Fraunhofer diffraction formed by a parallel wave passing through an axicon is presented mathematically,which denotes a bottle beam.Also,when a base transvers mode of Gaussian laser beam is illuminated on the axicon,a bottle beam can be obtained too behind the axicon.Under the conditions of ignoring the effect of axicon’s aperture,a group of analytical expressions concerned with complex amplitude and intensity of converted wave are derived from the diffracting formula,which shows the relationship among the intensity of bottle beam,the Gaussian parameter and the geometrical parameters of axicon.For this meaning,the goal and the task of this research is accomplished.Furthermore,the transforming function of axicon-lens combination toward a laser beam is also analyzed.The analytical expressions about complex amplitude and intensity of converted beam on the focal plan of lens are given,and verified by experiments.The results led from the relevant research,provide a good theoretical foundation to discuss the relation between geometrical parameters of axicon and the diffracted light field distribution behind axicon,and help to design the optical configuration to get a bottle beam.