The purpose of this paper is to study some famous inequalities in Euclidean space.We are able to reveal an elegant relation between the famous Selberg inequality and Bessel inequality in Euclidean space.
We propose a superposed Bessel optical lattice formed by multiple Bessel optical lattices.The static and rotational structures are formed in the presence of a spin-orbit coupling(SOC)interaction in the atomic in Bose...We propose a superposed Bessel optical lattice formed by multiple Bessel optical lattices.The static and rotational structures are formed in the presence of a spin-orbit coupling(SOC)interaction in the atomic in Bose–Einstein condensates are investigated,it is shown that the two structures can be manipulated by adjusting the parameters of the superposed Bessel optical lattices.The results show that the SOC interaction has an important effect on the two structures in the superposed Bessel optical lattices,and the SOC interaction can enhance the robustness of the structures.The Gaussian,toroidal and vortex superposition structures in the superposition lattice are presented,the interference processes in the steady state structures are analyzed,and the effects of SOC interactions on the Gaussian vortex and toroidal vortex structures are investigated,and the angular momentum of the vortex states can be increased by SOC interactions.展开更多
Femtosecond laser processing is an important machining method for micro-optical components such as Fresnel zone plate(FZP).However,the low processing efficiency of the femtosecond laser restricts its application.Here,...Femtosecond laser processing is an important machining method for micro-optical components such as Fresnel zone plate(FZP).However,the low processing efficiency of the femtosecond laser restricts its application.Here,a femtosecond laser Bessel beam is proposed to process micro-FZP,which is modulated from a Gaussian beam to a Bessel annular beam.The processing time for FZP with an outer diameter of 60μm is reduced from 30 min to 1.5 min on an important semiconductor material gallium arsenide(GaAs),which significantly improves the processing efficiency.In the modulation process,a central ablation hole that has an adverse effect on the diffraction performance is produced,and the adverse effect is eliminated by superimposing the blazed grating hologram.Meanwhile,the FZP machined by spatial light modulator(SLM)has good morphology and higher diffraction efficiency,which provides a strong guarantee for the application of micro-FZP in computed tomography and solar photovoltaic cells.展开更多
The spherically layered media theory has wide applications for electromagnetic wave scattering analysis.Due to the involved Bessel functions,the conventional formulations of spherically layered media theory suffer fro...The spherically layered media theory has wide applications for electromagnetic wave scattering analysis.Due to the involved Bessel functions,the conventional formulations of spherically layered media theory suffer from numerical overflow or underflow when the Bessel function’s order is large,the argument is small or the argument has a large imaginary part.The first two issues have been solved recently by employing small-argument asymptotic formulas of Bessel functions,while the third issue remains unsolved.In this paper,the Bessel functions in the conventional formulation of the theory are replaced by scaled Bessel functions which have good numerical properties for high loss media,and stable formulas are derived.Numerical tests show that this approach can work properly with very high lossy media.Also,this approach can be seamlessly combined with the stable computation method for cases of small argument and large order of Bessel functions.展开更多
We propose an approach for generating robust two-dimensional(2D)vortex clusters(VCs)in a Rydberg atomic system by utilizing parity-time(PT)symmetric optical Bessel potential.We show that the system supports novel mult...We propose an approach for generating robust two-dimensional(2D)vortex clusters(VCs)in a Rydberg atomic system by utilizing parity-time(PT)symmetric optical Bessel potential.We show that the system supports novel multicore VCs with four and eight cores,corresponding to topological charges 2 and 4,respectively.The stability of these VCs can be dynamically adjusted through the manipulation of the gain-loss component,Kerr nonlinearities,and the degree of nonlocality inherent in the Rydberg atoms.These VCs are confined within the first lattice well of the Bessel potential,and both the power and width of lights undergo a quasi-periodic breathing phenomenon,which is attributed to the power exchange between the light fields and Bessel potential.Both self-attractive and self-repulsive Kerr interactions can sustain robust VCs within this system.The insights presented here not only facilitate the creation and manipulation of 2D VCs through PT-symmetric potentials but also pave the way for potential applications in optical information processing and transmission.展开更多
Fractional orbital angular momentum(OAM) vortex beams present a promising way to increase the data throughput in optical communication systems. Nevertheless, high-precision recognition of fractional OAM with different...Fractional orbital angular momentum(OAM) vortex beams present a promising way to increase the data throughput in optical communication systems. Nevertheless, high-precision recognition of fractional OAM with different propagation distances remains a significant challenge. We develop a convolutional neural network(CNN)method to realize high-resolution recognition of OAM modalities, leveraging asymmetric Bessel beams imbued with fractional OAM. Experimental results prove that our method achieves a recognition accuracy exceeding 94.3% for OAM modes, with an interval of 0.05, and maintains a high recognition accuracy above 92% across varying propagation distances. The findings of our research will be poised to significantly contribute to the deployment of fractional OAM beams within the domain of optical communications.展开更多
Orbital angular momentum(OAM), as a new degree of freedom, has recently been applied in holography technology.Due to the infinite helical mode index of OAM mode, a large number of holographic images can be reconstruct...Orbital angular momentum(OAM), as a new degree of freedom, has recently been applied in holography technology.Due to the infinite helical mode index of OAM mode, a large number of holographic images can be reconstructed from an OAM-multiplexing hologram. However, the traditional design of an OAM hologram is constrained by the helical mode index of the selected OAM mode, for a larger helical mode index OAM mode has a bigger sampling distance, and the crosstalk is produced for different sampling distances for different OAM modes. In this paper, we present the design of the OAM hologram based on a Bessel–Gaussian beam, which is non-diffractive and has a self-healing property during its propagation. The Fourier transform of the Bessel–Gaussian beam is the perfect vortex mode that has the fixed ring radius for different OAM modes. The results of simulation and experiment have demonstrated the feasibility of the generation of the OAM hologram with the Bessel–Gaussian beam. The quality of the reconstructed holographic image is increased, and the security is enhanced. Additionally, the anti-interference property is improved owing to its self-healing property of the Bessel-OAM holography.展开更多
We investigate the ground states of spin-orbit coupled spin-1 Bose-Einstein condensates in the presence of Zeeman splitting.By introducing the generalized momentum operator,the linear version of the system is solved e...We investigate the ground states of spin-orbit coupled spin-1 Bose-Einstein condensates in the presence of Zeeman splitting.By introducing the generalized momentum operator,the linear version of the system is solved exactly,yielding a set of Bessel vortices.Additionally,based on linear solution and using variational approximation,the solutions for the full nonlinear system and their ground state phase diagrams are derived,including the vortex states with quantum numbers m=0,1,as well as mixed states.In this work,mixed states in spin-1 spin-orbit coupling(SOC)BEC are interpreted for the first time as weighted superpositions of three vortex states.Furthermore,the results also indicate that under strong Zeeman splitting,the system cannot form localized states.The variational solutions align well with numerical simulations,showing stable evolution and meeting the criteria for long-term observation in experiments.展开更多
A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to th...A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to this particular case I1/6(x)and I1/7(x). In this way, it shows how to apply the technique to any particular orderν, in order to obtain an approximation valid for any positive value of the variable x. In the present method power series and asymptotic expansion are simultaneously used. The technique is an extension of the multipoint quasirational approximation method, MPQA. The main idea is to look for a bridge function between the power and asymptotic expansion of the I1/6(x), and similar procedure for I1/7(x). To perform this, rational functions are combined with hyperbolic ones and fractional powers. The number of parameters to be determined for each case is four. The maximum relative errors are 0.0049 for ν=1/6, and 0.0047 for ν=7. However, these relative errors decrease outside of the small region of the variables, wherein the maximum relative errors are reached. There is a clear advantage of this procedure compared with any other ones.展开更多
The Bessel beam,characterized by its unique non-diffracting properties,holds promising applications.In this paper,we provide a detailed introduction and investigation into the theory and research of the Bessel beam,wi...The Bessel beam,characterized by its unique non-diffracting properties,holds promising applications.In this paper,we provide a detailed introduction and investigation into the theory and research of the Bessel beam,with a special focus on its generation and applications in the near-field region.We provide an introduction to the concepts,properties,and foundational theories of the Bessel beam.Additionally,the current study on generating Bessel beams and their applications is categorized and discussed,and potential research challenges are proposed in this paper.This review serves as a solid foundation for researchers to understand the concept of the Bessel beam and explore its potential applications.展开更多
In this paper wavelet functions are introduced in the context of q-theory. We precisely extend the case of Bessel and q-Bessel wavelets to the generalized q-Bessel wavelets. Starting from the （q,v）-extension （v = ...In this paper wavelet functions are introduced in the context of q-theory. We precisely extend the case of Bessel and q-Bessel wavelets to the generalized q-Bessel wavelets. Starting from the （q,v）-extension （v = （α,β）） of the q-case, associated generalized q-wavelets and generalized q-wavelet transforms are developed for the new context. Reconstruction and Placherel type formulas are proved.展开更多
本文先研究了一类变型Bessel方程的通解,由此进一步研究求解复合变型Bessel方程边值问题,发现这类微分方程边值问题的解在不同的边界条件下具有相似的结构,且其解是由边界条件的系数和相似核函数决定的,由此提出了相似构造法。该方法的...本文先研究了一类变型Bessel方程的通解,由此进一步研究求解复合变型Bessel方程边值问题,发现这类微分方程边值问题的解在不同的边界条件下具有相似的结构,且其解是由边界条件的系数和相似核函数决定的,由此提出了相似构造法。该方法的主要步骤是首先求出方程的基础解系,构造引解函数;再利用引解函数和初边值条件、交界面条件的系数构造内区核函数和外区核函数;最后根据核函数和边值条件的系数得到微分方程组的相似结构解。利用相似构造法在求解微分方程的初边值问题时,能够极大地简化求解过程,便于试井软件的编写以及分析相应的参数。In this paper, the general solution of a class of variant Bessel equations is studied, and then the boundary value problem of the composite variant Bessel equation is further studied, and it is found that the solution of the boundary value problem of this kind of differential equation has a similar structure under different boundary conditions, and its solution is determined by the coefficient and similar kernel function of the boundary condition, so the similarity construction method is proposed. The main steps of this method are to first find the basic solution system of the equation and construct the induction function. Then, the coefficients of the induction function, the initial boundary value condition and the interface condition are used to construct the inner and outer kernel functions. Finally, according to the coefficients of the kernel function and the boundary value condition, the similar structural solutions of the differential equations are obtained. The similarity construction method can greatly simplify the solution process when solving the initial boundary value problem of differential equations, which is convenient for the compilation of well testing software and the analysis of corresponding parameters.展开更多
基金Supported by Shandong Provincial Natural Science Foundation(ZR2024MA053).
文摘The purpose of this paper is to study some famous inequalities in Euclidean space.We are able to reveal an elegant relation between the famous Selberg inequality and Bessel inequality in Euclidean space.
基金supported by the Longdong University Doctoral Fund Program Projects(Grant Nos.XYBYZK2227,XYBYZK2219).
文摘We propose a superposed Bessel optical lattice formed by multiple Bessel optical lattices.The static and rotational structures are formed in the presence of a spin-orbit coupling(SOC)interaction in the atomic in Bose–Einstein condensates are investigated,it is shown that the two structures can be manipulated by adjusting the parameters of the superposed Bessel optical lattices.The results show that the SOC interaction has an important effect on the two structures in the superposed Bessel optical lattices,and the SOC interaction can enhance the robustness of the structures.The Gaussian,toroidal and vortex superposition structures in the superposition lattice are presented,the interference processes in the steady state structures are analyzed,and the effects of SOC interactions on the Gaussian vortex and toroidal vortex structures are investigated,and the angular momentum of the vortex states can be increased by SOC interactions.
基金Projects(51875584,51875585,51975590)supported by the National Natural Science Foundation of China。
文摘Femtosecond laser processing is an important machining method for micro-optical components such as Fresnel zone plate(FZP).However,the low processing efficiency of the femtosecond laser restricts its application.Here,a femtosecond laser Bessel beam is proposed to process micro-FZP,which is modulated from a Gaussian beam to a Bessel annular beam.The processing time for FZP with an outer diameter of 60μm is reduced from 30 min to 1.5 min on an important semiconductor material gallium arsenide(GaAs),which significantly improves the processing efficiency.In the modulation process,a central ablation hole that has an adverse effect on the diffraction performance is produced,and the adverse effect is eliminated by superimposing the blazed grating hologram.Meanwhile,the FZP machined by spatial light modulator(SLM)has good morphology and higher diffraction efficiency,which provides a strong guarantee for the application of micro-FZP in computed tomography and solar photovoltaic cells.
文摘The spherically layered media theory has wide applications for electromagnetic wave scattering analysis.Due to the involved Bessel functions,the conventional formulations of spherically layered media theory suffer from numerical overflow or underflow when the Bessel function’s order is large,the argument is small or the argument has a large imaginary part.The first two issues have been solved recently by employing small-argument asymptotic formulas of Bessel functions,while the third issue remains unsolved.In this paper,the Bessel functions in the conventional formulation of the theory are replaced by scaled Bessel functions which have good numerical properties for high loss media,and stable formulas are derived.Numerical tests show that this approach can work properly with very high lossy media.Also,this approach can be seamlessly combined with the stable computation method for cases of small argument and large order of Bessel functions.
基金Project supported by the National Natural Science Foundation of China(Grant No.62275075)the Science and Technology Research Program of the Education Department of Hubei Province,China(Grant No.B2022188)+2 种基金the Natural Science Foundation of Hubei Province,China(Grant No.2023AFC042)the Training Program of Innovation and Entrepreneurship for Undergraduates of Hubei Province,China(Grant No.S202210927003)the Medical Project of Hubei University of Science and Technology(Grant No.2023YKY08)。
文摘We propose an approach for generating robust two-dimensional(2D)vortex clusters(VCs)in a Rydberg atomic system by utilizing parity-time(PT)symmetric optical Bessel potential.We show that the system supports novel multicore VCs with four and eight cores,corresponding to topological charges 2 and 4,respectively.The stability of these VCs can be dynamically adjusted through the manipulation of the gain-loss component,Kerr nonlinearities,and the degree of nonlocality inherent in the Rydberg atoms.These VCs are confined within the first lattice well of the Bessel potential,and both the power and width of lights undergo a quasi-periodic breathing phenomenon,which is attributed to the power exchange between the light fields and Bessel potential.Both self-attractive and self-repulsive Kerr interactions can sustain robust VCs within this system.The insights presented here not only facilitate the creation and manipulation of 2D VCs through PT-symmetric potentials but also pave the way for potential applications in optical information processing and transmission.
基金supported by the National Natural Science Foundation of China (Grant Nos.12174338 and 11874321)。
文摘Fractional orbital angular momentum(OAM) vortex beams present a promising way to increase the data throughput in optical communication systems. Nevertheless, high-precision recognition of fractional OAM with different propagation distances remains a significant challenge. We develop a convolutional neural network(CNN)method to realize high-resolution recognition of OAM modalities, leveraging asymmetric Bessel beams imbued with fractional OAM. Experimental results prove that our method achieves a recognition accuracy exceeding 94.3% for OAM modes, with an interval of 0.05, and maintains a high recognition accuracy above 92% across varying propagation distances. The findings of our research will be poised to significantly contribute to the deployment of fractional OAM beams within the domain of optical communications.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.62375140 and 62001249)the Open Research Fund of the National Laboratory of Solid State Microstructures (Grant No.M36055)。
文摘Orbital angular momentum(OAM), as a new degree of freedom, has recently been applied in holography technology.Due to the infinite helical mode index of OAM mode, a large number of holographic images can be reconstructed from an OAM-multiplexing hologram. However, the traditional design of an OAM hologram is constrained by the helical mode index of the selected OAM mode, for a larger helical mode index OAM mode has a bigger sampling distance, and the crosstalk is produced for different sampling distances for different OAM modes. In this paper, we present the design of the OAM hologram based on a Bessel–Gaussian beam, which is non-diffractive and has a self-healing property during its propagation. The Fourier transform of the Bessel–Gaussian beam is the perfect vortex mode that has the fixed ring radius for different OAM modes. The results of simulation and experiment have demonstrated the feasibility of the generation of the OAM hologram with the Bessel–Gaussian beam. The quality of the reconstructed holographic image is increased, and the security is enhanced. Additionally, the anti-interference property is improved owing to its self-healing property of the Bessel-OAM holography.
基金supported by the Guangdong Basic and Applied Basic Research Foundation(Grant No.2023A1515110198)the Natural Science Foundation of Guangdong Province,China(Grant Nos.2024A1515030131 and 2021A1515010214)+2 种基金the National Natural Science Foundation of China(Grant Nos.12274077,11905032,and 12475014)the Research Fund of the Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology(Grant No.2020B1212030010)the Israel Science Foundation(Grant No.1695/22).
文摘We investigate the ground states of spin-orbit coupled spin-1 Bose-Einstein condensates in the presence of Zeeman splitting.By introducing the generalized momentum operator,the linear version of the system is solved exactly,yielding a set of Bessel vortices.Additionally,based on linear solution and using variational approximation,the solutions for the full nonlinear system and their ground state phase diagrams are derived,including the vortex states with quantum numbers m=0,1,as well as mixed states.In this work,mixed states in spin-1 spin-orbit coupling(SOC)BEC are interpreted for the first time as weighted superpositions of three vortex states.Furthermore,the results also indicate that under strong Zeeman splitting,the system cannot form localized states.The variational solutions align well with numerical simulations,showing stable evolution and meeting the criteria for long-term observation in experiments.
文摘A general technique to obtain simple analytic approximations for the first kind of modified Bessel functions. The general procedure is shown, and the parameter determination is explained through the applications to this particular case I1/6(x)and I1/7(x). In this way, it shows how to apply the technique to any particular orderν, in order to obtain an approximation valid for any positive value of the variable x. In the present method power series and asymptotic expansion are simultaneously used. The technique is an extension of the multipoint quasirational approximation method, MPQA. The main idea is to look for a bridge function between the power and asymptotic expansion of the I1/6(x), and similar procedure for I1/7(x). To perform this, rational functions are combined with hyperbolic ones and fractional powers. The number of parameters to be determined for each case is four. The maximum relative errors are 0.0049 for ν=1/6, and 0.0047 for ν=7. However, these relative errors decrease outside of the small region of the variables, wherein the maximum relative errors are reached. There is a clear advantage of this procedure compared with any other ones.
文摘The Bessel beam,characterized by its unique non-diffracting properties,holds promising applications.In this paper,we provide a detailed introduction and investigation into the theory and research of the Bessel beam,with a special focus on its generation and applications in the near-field region.We provide an introduction to the concepts,properties,and foundational theories of the Bessel beam.Additionally,the current study on generating Bessel beams and their applications is categorized and discussed,and potential research challenges are proposed in this paper.This review serves as a solid foundation for researchers to understand the concept of the Bessel beam and explore its potential applications.
文摘In this paper wavelet functions are introduced in the context of q-theory. We precisely extend the case of Bessel and q-Bessel wavelets to the generalized q-Bessel wavelets. Starting from the （q,v）-extension （v = （α,β）） of the q-case, associated generalized q-wavelets and generalized q-wavelet transforms are developed for the new context. Reconstruction and Placherel type formulas are proved.
文摘本文先研究了一类变型Bessel方程的通解,由此进一步研究求解复合变型Bessel方程边值问题,发现这类微分方程边值问题的解在不同的边界条件下具有相似的结构,且其解是由边界条件的系数和相似核函数决定的,由此提出了相似构造法。该方法的主要步骤是首先求出方程的基础解系,构造引解函数;再利用引解函数和初边值条件、交界面条件的系数构造内区核函数和外区核函数;最后根据核函数和边值条件的系数得到微分方程组的相似结构解。利用相似构造法在求解微分方程的初边值问题时,能够极大地简化求解过程,便于试井软件的编写以及分析相应的参数。In this paper, the general solution of a class of variant Bessel equations is studied, and then the boundary value problem of the composite variant Bessel equation is further studied, and it is found that the solution of the boundary value problem of this kind of differential equation has a similar structure under different boundary conditions, and its solution is determined by the coefficient and similar kernel function of the boundary condition, so the similarity construction method is proposed. The main steps of this method are to first find the basic solution system of the equation and construct the induction function. Then, the coefficients of the induction function, the initial boundary value condition and the interface condition are used to construct the inner and outer kernel functions. Finally, according to the coefficients of the kernel function and the boundary value condition, the similar structural solutions of the differential equations are obtained. The similarity construction method can greatly simplify the solution process when solving the initial boundary value problem of differential equations, which is convenient for the compilation of well testing software and the analysis of corresponding parameters.
