In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré in...In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré inequality and,in particular,the explicit relation between the constant and the geometric characters of the domain.展开更多
The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)...The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)and hybrid-order Poincarésphere(HyOPS)has provided a systematic description of VVB.However,the generation of arbitrary VVBs on a HOPS and a HyOPS via a metasurface lacks a unified design framework,despite numerous reported approaches.We present a unified design framework incorporating all design parameters(e.g.,focal lengths and orders)of arbitrary HOPS and HyOPS beams into a single equation.In proof-of-concept experiments,we experimentally demonstrated four metasurfaces to generate arbitrary beams on the fifth-order HOPS(nonfocused and tightly focused,NA 0.89),0-2 order,and 0-1 order HyOPS.We showed HOPS beams’propagation and focusing properties,the superresolution focusing characteristics of the first-order cylindrical VVBs,and the different focusing properties of integerorder and fractional-order cylindrical VVBs.The simplicity and feasibility of the proposed design framework make it a potential catalyst for arbitrary VVBs using metasurfaces in applications of optical imaging,communication,and optical trapping.展开更多
<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>It is well known that Einstein published in June 1905...<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>It is well known that Einstein published in June 1905 his theory of Special Relativity (SR) without entirely based on space-time Lorentz Transformation (LT) with invariance of Light Velocity. It is much less known that Poincaré published, practically at the same time, a SR also based entirely on LT with also an invariant velocity. However, according to Poincaré, the invariant is not only that of light wave but also that of Gravific Wave in Ether. Poincaré’s Gravific ether exerts also a Gravific pressure, in the same paper, on <i>charged </i>(e) Electron (a “Hole in Ether” according to Poincaré). There are thus two SR: That of Einstein (ESR), without ether and without gravitation, and that of Poincaré (PSR), with Electro-Gravific-Ether. The crucial question arises then: Does “SPECIAL” Poincaré’s (e)-G field fall in the framework of Einstein’s GENERAL Relativity? Our answer is positive. On the basis of Einstein’s equation of gravitation (1917) with Minkowskian Metric (MM) and Zero Constant Cosmological (CC) we rediscover usual Static Vacuum (without <i>charge e </i>of electron). On the other hand with MM and <i>Non-Zero </i>CC, we discover the gravific field of a Cosmological Black Hole (CBH) with density of dark energy compatible with expanding vacuum. Hawking’s Stellar Black Hole (SBH) emits outgoing Black Radiation, whilst Poincaré’s CBH emits (at time zero) incoming Black Radiation. We show that Poincaré’s G-electron involves a (quantum) GRAVITON (on the model of Einstein’s quantum photon) underlying a de Broglie’s G-Wave. There is therefore a Gackground Cosmological model in Poincaré’s basic paper which predicts a density and a temperature of CBR very close to the observed (COBE) values. </p>展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.12001170 and 11601124)Innovative Funds Plan of Henan University of Technology(Grant No.2021ZKCJ11).
文摘In this paper,we explicitly establish Poincaréinequality for 1≤p<∞ over simple geometric domains,such as segment,rectangle,triangle or tetrahedron.We obtain sharper bounds of the constant in Poincaré inequality and,in particular,the explicit relation between the constant and the geometric characters of the domain.
基金the Chinese Scholarship Council is acknowledged.This work was supported by the UK’s Engineering and Physical Sciences Research Council(Grant Nos.EP/V000624/1,EP/X03495X/1,EP/X041166/1,and EP/T02643X/1)the Royal Society(Grant No.RG\R2\232531).
文摘The unique phase profile and polarization distribution of the vector vortex beam(VVB)have been a subject of increasing interest in classical and quantum optics.The development of higher-order Poincarésphere(HOPS)and hybrid-order Poincarésphere(HyOPS)has provided a systematic description of VVB.However,the generation of arbitrary VVBs on a HOPS and a HyOPS via a metasurface lacks a unified design framework,despite numerous reported approaches.We present a unified design framework incorporating all design parameters(e.g.,focal lengths and orders)of arbitrary HOPS and HyOPS beams into a single equation.In proof-of-concept experiments,we experimentally demonstrated four metasurfaces to generate arbitrary beams on the fifth-order HOPS(nonfocused and tightly focused,NA 0.89),0-2 order,and 0-1 order HyOPS.We showed HOPS beams’propagation and focusing properties,the superresolution focusing characteristics of the first-order cylindrical VVBs,and the different focusing properties of integerorder and fractional-order cylindrical VVBs.The simplicity and feasibility of the proposed design framework make it a potential catalyst for arbitrary VVBs using metasurfaces in applications of optical imaging,communication,and optical trapping.
文摘<p align="justify"> <span style="font-family:Verdana;"></span><span style="font-family:Verdana;"></span>It is well known that Einstein published in June 1905 his theory of Special Relativity (SR) without entirely based on space-time Lorentz Transformation (LT) with invariance of Light Velocity. It is much less known that Poincaré published, practically at the same time, a SR also based entirely on LT with also an invariant velocity. However, according to Poincaré, the invariant is not only that of light wave but also that of Gravific Wave in Ether. Poincaré’s Gravific ether exerts also a Gravific pressure, in the same paper, on <i>charged </i>(e) Electron (a “Hole in Ether” according to Poincaré). There are thus two SR: That of Einstein (ESR), without ether and without gravitation, and that of Poincaré (PSR), with Electro-Gravific-Ether. The crucial question arises then: Does “SPECIAL” Poincaré’s (e)-G field fall in the framework of Einstein’s GENERAL Relativity? Our answer is positive. On the basis of Einstein’s equation of gravitation (1917) with Minkowskian Metric (MM) and Zero Constant Cosmological (CC) we rediscover usual Static Vacuum (without <i>charge e </i>of electron). On the other hand with MM and <i>Non-Zero </i>CC, we discover the gravific field of a Cosmological Black Hole (CBH) with density of dark energy compatible with expanding vacuum. Hawking’s Stellar Black Hole (SBH) emits outgoing Black Radiation, whilst Poincaré’s CBH emits (at time zero) incoming Black Radiation. We show that Poincaré’s G-electron involves a (quantum) GRAVITON (on the model of Einstein’s quantum photon) underlying a de Broglie’s G-Wave. There is therefore a Gackground Cosmological model in Poincaré’s basic paper which predicts a density and a temperature of CBR very close to the observed (COBE) values. </p>
