Manipulation of polarization-dependent multivortices with quasi-periodic subwavelength structures

Multiple vortices with different topological charges are formed by the use of two sequential geometric phase elements. These elements are realized by quasi-periodic subwavelength gratings. The first element is a spiral phase element and the second element is a spherical phase element. We provide a theoretical analysis and an experimental demonstration of the formation of the multiple vortices that comprise scalar vortices and a vectorial vortex.     

On the Knowledge Complexity of

We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class . Thus, if the polynomial time hierarchy does not collapse, then -complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The first asserts that statistical zero knowledge is contained in [11, 2], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in [19]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability is less than (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specifically, it is in ). In the standard setting of negligible error probability, there exist PSPACE-complete languages which have sub-linear knowledge complexity. However, if we insist, for example, that the error probability is less than , then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless . In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions , the verifier runs in time polynomial in and and fails with probability at most to detect an additive error in the entropy. We believe that this protocol is of independent interest. Subsequent to our work Goldreich and Vadhan [16] established that the problem of comparing the entropies of two samplable distributions if they are noticeably different is a natural complete promise problem for the class of statistical zero knowledge (). Received January 6, 2000 RID=" " ID=" " This research was performed while the authors were visiting the Computer Science Department at the University of Toronto, preliminary version of this paper appeared in [27] RID="*" ID="*" Partially supported by Technion V. P. R. Found––N. Haar and R. Zinn Research Fund. RID="†" ID="†" Partially supported by the grants OTKA T-029255, T-030059, FKFP 0607/1999, and AKP 2000-78 2.1.     

Periods of automorphic forms

Let be a quadratic extension of number fields and , where is a reductive group over . We define the integral (in general, non-convergent) of an automorphic form on over via regularization. This regularized integral is used to derive a formula for the integral over of a truncated Eisenstein series on . More explicit results are obtained in the case . These results will find applications in the expansion of the spectral side of the relative trace formula.

     

A new method for lower bounds of L-functions

Let $L(s\text{,}\pi \text{,}r)$ be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for $L(s\text{,}\pi \text{,}r)$ when $\mathfrak{R}(s)=1$ using Eisenstein series. This method is applicable even when $L(s\text{,}\pi \text{,}r)$ is not known to be absolutely convergent for $\mathfrak{R}(s)>1$. To cite this article: S.S. Gelbart et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings

We present a novel method for forming linearly polarized axially symmetric beams with various polarization orders that is based on computer-generated space-variant subwavelength gratings. We introduce and experimentally demonstrate that our space-variant polarization state manipulations are accompanied by a phase modification of a helical structure that results from the Pancharatnam-Berry phase. We have verified the polarization properties of our gratings for laser radiation at 10.6-microm wavelength.     

Reduction of laser spot elongation in adaptive optics

Adaptive optics systems measure the wave front to be corrected by use of a reference source, a star, or a laser beacon. Such laser guide stars are a few kilometers long, and when observed near the edges of large telescopes they appear elongated. This limits their utility significantly. However, with more sophisticated launch optics their shape and length can be controlled. We propose to string around the rim of a telescope a number of small telescopes that will add laser beams in the scattering medium to create a compact spot. The method could also be adapted for ocular adaptive optics.     

Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements

Propagation-invariant vectorial Bessel beams with linearly polarized axial symmetry based on quantized Pancharatnam-Berry phase optical elements are described. The geometric phase is formed through the use of discrete computer-generated space-variant subwavelength dielectric gratings. We have verified the polarization properties of our elements for laser radiation at 10.6-microm wavelength and also demonstrated propagation-invariant, controlled rotation of a propeller-shaped intensity pattern through the simple rotation of a polarizer.     

Hydrophilic Chitosan Derivatives: Synthesis and Applications

Polymer alternatives sourced from nature have attracted increasing attention for applications in medicine, cosmetics, agriculture, food, water purification, and more. Among them, chitosan is the most versatile due to its full biodegradability, exceptional biocompatibility, multipurpose bioactivity, and low toxicity. Although remarkable progress has been made in its synthetic modification by using C3/C6 secondary/primary hydroxy (−OH) and the C2 amino (−NH₂) active sites, its solubility under physiological conditions remains limited and has hampered larger-scale adoption. This review summarizes different synthetic methods that increase chitosan‘s hydrophilicity and water solubility by using covalent modifications, namely amino acid addition, quaternary ammonium formation, phosphorylation, and carboxymethylation. We also review several applications for each type of substitution in fields such as cosmetics, medicine, agriculture, and water purification, and provide an outlook and perspective for future modifications and implementations.     

Evolution of the Fermi surface of d-wave superconductors in the presence of thermal phase fluctuations

One of the most puzzling aspects of the high Tc superconductors is the appearance of Fermi arcs in the normal state of the underdoped cuprate materials. These are loci of low energy excitations covering part of the Fermi surface that suddenly appear above Tc instead of the nodal quasiparticles. Based on a semiclassical theory, we argue that partial Fermi surfaces arise naturally in a d-wave superconductor that is destroyed by thermal phase fluctuations. Specifically, we show that the electron spectral function develops a square root singularity at low frequencies for wave vectors positioned on the bare Fermi surface. We predict a temperature dependence of the arc length that can partially account for the results of recent angle resolved photoemission experiments.