Generation of an elliptic hollow beam using Mathieu and Bessel functions (original) (raw)
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Hollow Gaussian beams (HGB) are a special class of doughnut shaped beams that do not carry orbital angular momentum (OAM). Such beams have a wide range of applications in many fields including atomic optics, bio-photonics, atmospheric science, and plasma physics. Till date, these beams have been generated using linear optical elements. Here, we show a new way of generating HGBs by three-wave mixing in a nonlinear crystal. Based on nonlinear interaction of photons having OAM and conservation of OAM in nonlinear processes, we experimentally generated ultrafast HGBs of order as high as 6 and power >180 mW at 355 nm. This generic concept can be extended to any wavelength, timescales (continuous-wave and ultrafast) and any orders. We show that the removal of azimuthal phase of vortices does not produce Gaussian beam. We also propose a new and only method to characterize the order of the HGBs. The dark hollow beams (DHB) are identified with their characteristic doughnut intensity distribution, a dark center enclosed by a bright ring in the beam cross section. Like conventional DHBs such as optical vortices 1 , higher order Bessel 2 and Mathieu 3 beams, HGBs 4,5 also have doughnut intensity profile but do not carry any OAM. In addition to the vast applications ranging from atomic optics to plasma physics 6–15 , HGBs have also attracted a great deal of scientific interest in understanding its propagation and transformation dynamics 5,16,17,18. Generation of HGBs have been realized using linear optical elements in different methods such as spatial filtering 19 , geometrical optics 20 , fibers 21 , spatial-light-modulator 16 , and Laguerre-Gaussian (LG) beam transformation 22. However, nonlinear generation processes enable HGBs to have a new wavelength across electromagnetic spectrum and also to have high output power and higher order in all timescales as required for most of the applications 6–15. The HGBs have a similar functional form 5 as that of optical vortex beams except the azimuthal phase term exp(-ilθ) where, l is the topological charge or OAM mode of the vortex. Therefore, HGBs can be generated by removing the azimuthal phase term of the vortices 22. However, due to the absence of suitable technique to characterize HGBs in terms of its order, one cannot be sure about the order of the HGB generated in ref. 22. Given that the nonlinear frequency conversion processes 23–25 satisfy OAM conservation 26,27 , one can in principle, remove the azimuthal phase term of the generated beam through annihilation of OAM modes of the interacting beams in three wave-mixing process. As a proof of principle, here we report, for the first time to the best of our knowledge, nonlinear generation of HGBs. Based on sum frequency mixing of two OAM carrying ultrafast beams at 1064 nm and 532 nm having equal OAM orders but of opposite helicity in a nonlinear medium, we have generated HGBs of order as high as 6 and output power as much as 180 mW at 355 nm. It is a generic concept and can be extended to any wavelength and timescale. In addition, by controlling the sign of the helicity of OAM modes one can generate higher order optical vortices at desired wavelengths. We also propose a new and (at present,) only method to characterize the order of HGBs. On contrary to the common belief 28 , the present study show that the removal of azimuthal phase of vortices does not produce Gaussian beam.