Research On Tightly Focusing Property Of Hybrid Order Vortex Beams And The New Technology Of Detecting Topological Charge

Optical vortex beams(VB)carry helical phase structure exp(il?),where l is topological charge(TC).Because of the phase singularity,the intensity of the VB appears a hollow ring distribution.It has been proved that the helical phase associates with orbital angular momentum(OAM)of light,and the polarization relates to spin angular momentum(SAM)of light.Owing to the distinct properties of intensity,phase,and angular momentum(AM),VB have drawn much attention for widespread application in the fields of optical coding,super-resolution,optics plasmon,and laser processing.Rencently,because the polarization and phase of light become more complicated,hybrid order Poincaré sphere(HPS)was proposed to describe the hybrid order VB.However,the present articles have not reported the tightly focusing of HPS beams,eapecially the systematic discussion of the influence factors on its focusing properties.Moreover,the present methods on detecting TC have the problems of complicated systems,indistinguishable intensity patterns,limited capacity,and so on.Therefore,to solve the above problems,we carry out the following studies:Firstly,we systematically study the effects of helical phase and beams waist parameter on the tightly focusing properties of radially and azimuthal polarized.We explore the relation between focusing intensity and beams waist parameter of radially polarized vortex beams,and find that small beams waist parameter is beneficial to the super-resolution spot.In the case of high beams waist parameter,the focal intensity of radially polarized VB will be distorted in transverse and split in longitudinal direction.Considering the azimuthal polarized vortex beams,we find that multi-split of focal intensity along longitudinal direction can be fulfilled with high beam waist parameter.Meanwhile,the focal intensity distribution becomes non-symmetrical as well as expansion.Therefore,appropriate beams waist must be chosen for application.The results obtained in this paper can be applied in many fields.We theoretically study the tightly focusing properties of HPS beams for the first time.We establish the tightly focusing theory of HPS beams and analyze the focusing properties,which include intensity,phase distribution,defocus intensity,and SAM density.We find that the intensity distribution of the focus is petal-shaped if the two TCs have opposite sign,where the number of intensity lobes in the focal plane is ?m-n +2?.When only changing the TC n,we find that the hollow radius of focal intensity is almost unchanged,which leads to merged lobes until ring shaped intensity.Furthermore,comparing with the change of the longitudinal fields,we obtain the ciritical point for judging the disappearance of petal-shaped intensity.Meanwhile,the OAM states of the longitudinal component annihilate when m =-n;otherwise,they split into two parts with topological charges m +1 and n-1.We analyze a generalized case of HPS beams,and demonstrate that the focal intensity and phase distribution are also influenced by the coefficients.Further,we investigate the defocus intensity and SAM density distribution of petal-shaped HPS beams.The results reveal that the helical structure appears with the defocus increases,and the rotation of helical strcture depends on the sign of m +n.We study the SAM density distribution of tightly focusing field and find that the density not only depends on the change of two TCs,but also influenced by the amplitude ratio of two beams.Therefore,spatially changed SAM density can be fulfilled by modulating the TC and amplitude ratio.The results obtained may pave a new application of HPS beams in many fields.We propose a new method for detecting the topological charge(TC)value of VB by the tightly focusing intensity after polarization modulated.Firstly,we establish the theory.Then,we explain the work principle of the detect systems.Finally,simulations are performed to verify the effectivity of the method.The results reveal that the tightly focusing intensity represents a smallest solid spot when the TC value of VB equals the polarization azimuthal index.Based on this property,we can quikly determine arbitrary states of VB by the focal smallest solid spot,especially in the case of higher charge.The proposed method has the characters of clear principle,simple system,novelty and fast,unlimited capacity.We propose the method for simultaneously distinguishing the magnitude and sign of TC based on the defocus intensity.We find that the defocus intensity of VB after high order polarization modulation would present helical structure,and the rotation direction of the helical structure is related to the sign of TC.Of which clockwise corresponds the positive TC,whereas corresponds to the negative TC.The helical structure of defocus intensity holds a small solid spot when the topological charge(TC)value of VB matches the polarization order number,which can be regarded as the mark to distinguish the magnitude of TC.Therefore,the defocus intensity of VB via polarization modulation have detected the full information of TC,and the method also can fufill the detection of arbitrary polarized VB.