We present a simple method to measure the topological charges of optical vortices with multiple singularities. Using a cylindrical lens, a vortex beam can decay into a light field distribution with multiple separated ...We present a simple method to measure the topological charges of optical vortices with multiple singularities. Using a cylindrical lens, a vortex beam can decay into a light field distribution with multiple separated dark holes, whose number just equals the topological charge of the input beam. This conclusion is then verified via experiments and numerical simulations of the propagation of vortex beams with multiple singulaxities. This method is also reliable to measure the topological charges of broadband vortex beams with different distributions of singularities, which does not resort to multiple beam interferometrie experiments.展开更多
Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i ...Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.展开更多
The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms.One meets difficulties when trying to extend these definitions to vector f...The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms.One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincare flow rather than the tangent flow in order to study the properties of the derivative.In this paper,we define the notion of singular domination,an analog of the dominated splitting for the linear Poincar´e flow which is robust under perturbations.Based on this,we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz(2017).The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.展开更多
基金Supported by the National Basic Research Program of China under Grant No 2012CB921900the National Natural Science Foundation of China under Grant Nos 61377035 and 11404264the Fundamental Research Funds for the Central Universities under Grant No 3102014JCQ01085
文摘We present a simple method to measure the topological charges of optical vortices with multiple singularities. Using a cylindrical lens, a vortex beam can decay into a light field distribution with multiple separated dark holes, whose number just equals the topological charge of the input beam. This conclusion is then verified via experiments and numerical simulations of the propagation of vortex beams with multiple singulaxities. This method is also reliable to measure the topological charges of broadband vortex beams with different distributions of singularities, which does not resort to multiple beam interferometrie experiments.
基金supported partly by the National Natural Science Foundation of China (10771219)the Science Foundation of the SEAC of China (07ZN03)
文摘Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.
基金The first author was supported by the European Research Council(Grant No.692925)The third author was supported by National Natural Science Foundation of China(Grant Nos.11671288,11822109 and 11790274)The fourth author was supported by the starting grant from Beihang University and the European Research Council(Grant No.692925).
文摘The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms.One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincare flow rather than the tangent flow in order to study the properties of the derivative.In this paper,we define the notion of singular domination,an analog of the dominated splitting for the linear Poincar´e flow which is robust under perturbations.Based on this,we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz(2017).The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.
