Mapping the polarization distribution of arbitrary vector polarization beam

Optik ◽  
2017 ◽  
Vol 144 ◽  
pp. 124-131
Author(s):  
Xiao Wang ◽  
Feng Yang ◽  
Jianhua Yin
Keyword(s):  
Arbitrary Vector ◽  
Polarization Distribution ◽  
Vector Polarization

scholarly journals Generation and characterization of complex vector modes with digital micromirror devices: a tutorial

2021 ◽  
Author(s):  
XiaoBo Hu ◽  
Carmelo Rosales-Guzmán
Keyword(s):  
Degrees Of Freedom ◽  
Complex Vector ◽  
Arbitrary Vector ◽  
Polarization Distribution ◽  
Research Fields ◽  
Great Relevance ◽  
Quantum Entangled States ◽  
Digital Micromirror Devices ◽  
Novel Applications

Abstract Complex vector light modes with a spatial variant polarization distribution have become topical of late, enabling the development of novel applications in numerous research fields. Key to this is the remarkable similarities they hold with quantum entangled states, which arises from the non-separability between the spatial and polarisation degrees of freedom (DoF). As such, the demand for diversification of generation methods and characterization techniques have increased dramatically. Here we put forward a comprehensive tutorial about the use of DMDs in the generation and characterization of vector modes, providing details on the implementation of techniques that fully exploits the unsurpassed advantage of Digital Micromirrors Devices (DMDs), such as their high refresh rates and polarisation independence. We start by briefly describing the operating principles of DMD and follow with a thorough explanation of some of the methods to shape arbitrary vector modes. Finally, we describe some techniques aiming at the real-time characterization of vector beams. This tutorial highlights the value of DMDs as an alternative tool for the generation and characterization of complex vector light fields, of great relevance in a wide variety of applications.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals Formation of half-period surface relief gratings in azobenzene containing polymer films

2020 ◽  
Vol 126 (9) ◽  
Author(s):  
Joachim Jelken ◽  
Carsten Henkel ◽  
Svetlana Santer
Keyword(s):  
Polymer Film ◽  
Polymer Films ◽  
Surface Relief ◽  
Incident Light ◽  
Glassy State ◽  
Polarized Light ◽  
Half Period ◽  
Polarization Distribution ◽  
Linearly Polarized ◽  
Set Up

Abstract We study the peculiar response of photo-sensitive polymer films irradiated with a certain type of interference pattern where one interfering beam is S-polarized, while the second one is P-polarized. The polymer film, although in a glassy state, deforms following the local polarization distribution of the incident light, and a surface relief grating (SRG) appears whose period is half the optical one. All other types of interference patterns result in the matching of both periods. The topographical response is triggered by the alignment of photo-responsive azobenzene containing polymer side chains orthogonal to the local electrical field, resulting in a bulk birefringence grating (BBG). We investigate the process of dual grating formation (SRG and BBG) in a polymer film utilizing a dedicated set-up that combines probe beam diffraction and atomic force microscopy (AFM) measurements, and permits acquiring in situ and in real-time information about changes in local topography and birefringence. We find that the SRG maxima appear at the positions of linearly polarized light (tilted by 45° relative to the grating vector), causing the formation of the half-period topography. This permits to inscribe symmetric and asymmetric topography gratings with sub-wavelength period, while changing only slightly the polarization of one of the interfering beams. We demonstrate an easy generation of sawtooth profiles (blazed gratings) with adjustable shape. With these results, we have taken a significant step in understanding the photo-induced deformation of azo-polymer films.

scholarly journals Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

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