The Pinching Theorem of Global Umbilic Submanifolds in a Riemannian Manifold with Quasi Constant Curvature

We study the global umbilic submanifolds with parallel mean curvature vector fields in a Riemannian manifold with quasi constant curvature and get a local pinching theorem about the length of the second fundamental form.

Understanding Lens Aberrations

In most textbooks, lens aberrations are usually described in the briefest possible manner, without any attempt for their proper derivation. At the same time, monographs which do go into more detail are often inaccessible to most students and non-specialists interested in deeper understanding of this topic. This article tries to fill this gap and provide an introduction to what happens when basic formulas of Geometrical Optics are extended by third-order terms in Taylor’s expansion of sin (α). The presentation is accessible to most undergraduate students as it requires only some knowledge of basic calculus and planar geometry. The resulting five aberrations are then described in detail, including a novel derivation of the exact shape of coma. A simple Mathematica program is included to facilitate numerical exploration of the magnitude of the resulting aberrations for various optical systems.

A level set based segmentation approach for point-sampled surfaces

Segmenting a complex 3D surface model into some visually meaningful sub-parts is one of the fundamental problems in digital geometry processing. In this paper, a novel segmentation approach of point-sampled surfaces is proposed, which is based on the level set evolution scheme. To segment the model so as to align the patch boundaries with high curvature zones, the driven speed function for the zero level set inside narrow band is defined by the extended curvature field, which approaches zero speed as the propagating front approaches high curvature zone. The effectiveness of the proposed approach is demonstrated by our ex- perimental results. Furthermore, two applications of model segmentation are illustrated, such as piecewise parameterization and local editing for point-sampled geometry.

Proton loss of inner radiation belt during geomagnetic storm of 2018 based on CSES satellite observation

The proton distribution in inner radiation belt is often affected by strong geomagnetic storm disturbance.Based on the data of the sun-synchronous CSES satellite,which carries with several high energy particle payloads and was launched in February 2018,we analyzed the extensive proton variations in the inner radiation belt in a wide energy range of 2 MeV-220 MeV during 2018 major geomagnetic storm.The result indicates that the loss mechanism of protons was energy dependence which is consistent with some previous studies.For protons at low energy 2 MeV-20 MeV,the fluxes were decreased during main phase of the storm and did not come back quickly during the recovery phase,which is likely to be caused by Coulomb collision due to neutral atmosphere density variation.At higher energy 30 MeV-100 MeV,it was confirmed that the magnetic field line curvature scattering plays a significant role in the proton loss phenomenon during this storm.At highest energies>100 MeV,the fluxes of protons kept a stable level and did not exhibit a significant loss during this storm.

Generation of biaxially accelerating static Airy light-sheets with 3D-printed freeform micro-optics

One-dimensional Airy beams allow the generation of thin light-sheets without scanning,simplifying the complex optical arrangements of light-sheet microscopes(LSMs)with an extended field of view(FOV).However,their uniaxial acceleration limits the maximum numerical aperture of the detection objective in order to keep both the active and inactive axes within the depth of field.This problem is particularly pronounced in miniaturized LSM implementations,such as those for endomicroscopy or multi-photon neural imaging in freely moving animals using head-mounted miniscopes.We propose a new method to generate a static Airy light-sheet with biaxial acceleration,based on a novel phase profile.This light-sheet has the geometry of a spherical shell whose radius of curvature can be designed to match the field curvature of the micro-objective.We present an analytical model for the analysis of the light-sheet parameters and verify it by numerical simulations in the paraxial regime.We also discuss a micro-optical experimental implementation combining gradient-index optics with a 3D-nanoprinted,fully refractive phase plate.The results confirm that we are able to match detection curvatures with radii in the range of 1.5 to 2 mm.

Large-range alignment-free distributed-cavity laser based on an improved multi-lens retrorefleetor

A multi-lens retroreflector with field curvature compensation was designed and used in an alignment-free distributed-cavity laser with a long working distance for resonant beam charging applications.The multi-lens design,which makes use of off-the-shelf components,also allows a large field of view[FoV]without requirement of large element aperture.By implementing this design,an end-pumped 1063 nm Nd:GdVO4 laser could deliver over 5 W continuous-wave output power over a large range of working distances(1-5 m)and with ±30°receiver FoV under an incident diode pump power of 16.6 W.The output power fluctuation was less than 10%when moving and tilting the receiver over such a large range,without requiring any realignment of the cavity.

On Submanifolds of the Unit Sphere with Vanishing Mobius Form and Parallel Para-Blaschke Tensor

The para-Blaschke tensor are extended in this paper from hypersurfaces to general higher codimensional submanifolds in the unit sphere S^(n),which is invariant under the Mobius transformations on Sn.Then some typical new examples of umbilic-free submanifolds in Snwith vanishing Mobius form and a parallel para-Blaschke tensor of two distinct eigenvalues,D_(1) and D_(2),are constructed.The main theorem of this paper is a simple characterization of these newly found examples in terms of the eigenvalues D_(1) and D_(2).