Objective Optical tweezer technology makes the mechanical effect of light used in practice,and it can accurately control microscopic particles.Localized hollow beam has important applications in optical captivity and optical tweezers with its physical characteristics such as barrel light intensity distribution and small dark spot size.The method of producing hollow beams by axicon lens is simple and practicable,and it has high conversion efficiency,which is beneficial for the capture of small particles.In the design of optical systems based on an axicon lens,how to balance the energy loss,beam mass,and adjustment error is worth studying,which will make the research more valuable for engineering applications.Methods In this paper,a pair of axicon lenses was used as a shaper to produce collimated parallel hollow beams.First,the mathematical description of the conical mirror type was given,and then the appropriate configuration of the mirror group was selected according to the requirement of practical application.The expression of light intensity of the hollow beam was obtained according to the law of conservation of energy,and then the diffraction spot and cross-section light intensity distribution of the hollow beam at different propagation distances in free space were simulated based on the Fresnel diffraction integral formula.In addition,the relationship curve of beam quality factor with the ratio of truncation diameter to Gaussian beam radius and the blocking ratio was calculated,so as to obtain good beam quality in practice.The influence of cone-top angle consistency,eccentricity,and tilt error on the wavefront error was analyzed.Finally,near-and far-field measurements of the shaped hollow beam was made.Results and Discussions The axicon lens configuration featuring front concave,rear convex,and adjacent planes can either avoid damage caused by backreflected light back to the laser or meet the same design requirements with shorter air spacing(Fig.2).The hollow beam shaped by the axicon lens can maintain good hollow beam characteristics within the transmission distance of tens of meters,and the inner and outer diameters basically do not change(Fig.4).However,with the increase in the transmission distance,the light intensity of the center(on the axis)is not zero and gradually increases,which is the result of the diffraction transmission(Fig.5).The beam quality factor increases with an increase in the ratio of the truncation diameter to the Gaussian beam radius while the energy truncation loss decreases with an increase in the ratio of truncation diameter to the Gaussian beam radius(Figs.6 and 7).Besides,a larger shielding ratio indicates a larger beam quality factor for both shaped hollow beams and plane wave hollow beams of the same size(Fig.8).Whether it is consistency error,tilt error,or eccentric error of the cone angle,choosing axicon lens with larger cone-top angle can allow larger processing error(Figs.10 and 12).The ratio of the two error sizes under the same root mean square(RMS)value is almost constant(Table 2).When the eccentricity and tilt corresponding to the same RMS value coexist,the wavefront error is almost doubled,and when one of the errors is reversed,about 96.2%of the aberrations can be offset.Therefore,there is a certain equivalent relationship between eccentricity and tilt error on the influence of the wavefront RMS value(Tables 3 and 4).In the Gaussian beam shaping experiment,the far-field beam mass of the Gaussian beam without shaping is 1.10,and that of the hollow beam after shaping is 1.44,which is slightly larger than the theoretical calculation result.This is because the experiment is affected by various aberrations loaded after the beam passes through the collimation and shaping system,processing errors of the axicon lens,air,and dust media(Fig.14).Conclusions In this paper,some basic problems in the process of designing,machining,and assembling the hollow beam shaper of axicon lens groups are studied.Based on the Fresnel diffraction formula,the transmission characteristics of the shaped hollow beam are analyzed,and the hollow beam characteristics can be maintained within a transmission distance of tens of metres.The beam quality factor is calculated.The results show that a greater ratio of truncated diameter to the Gaussian beam radius indicates a greater beam quality factor.The beam quality factor also increases with the increase in the shielding ratio.The relative eccentricity and tilt of the conical mirror group in machining and setting will affect the wavefront error of the beam.The research results show that the design of an axicon lens with a larger cone-top angle allows a larger error tolerance,and the effects caused by the two errors have a certain equivalence,which can be used to offset most of the aberrations in practical applications.It provides a theoretical basis for offsetting eccentric and tilted aberrations in setting work.Finally,the axicon lens group structure based on the study is tested,and the results show that the beam quality factor of the hollow beam after shaping is consistent with the theory.
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