The impact of the ethnical background and the number if siblings on the scores of Mathematics Anxiety: A study on Mathematics Anxiety of undergraduate students of mathematics and engineering

Mathematics Anxiety (MA), the ‘phobia of numbers’, is related to poor performance in Mathematics. There are numerous studies that discuss a wide range of factors affecting Mathematics Anxiety in students at primary and secondary schools. Furthermore, there are some studies looking into MA in students of Psychology, Engineering and Nursing at a Higher Education level, see, for example (Alves et al. 2016; McMullan et al. 2012) and more references therein. However, we believe that this is the first work on MA in undergraduate students of Mathematics. Consequently, our purpose is to determine whether factors such as gender or ethnicity affect MA. Our main results are that there are significant differences between male and female students; there is a significant difference among students with three siblings or more, compared to students who have two siblings or less. Finally, we discuss the significant difference between the gender of the main family figure providing Mathematics support amongst students with a British and Non-British background

Multiscale analysis of subwavelength imaging with metal-dielectric multilayers

Imaging with a layered superlens is a spatial filtering operation characterized by the point spread function (PSF). We show that in the same optical system the image of a narrow sub-wavelength Gaussian incident field may be surprisingly dissimilar to the PSF, and the width of PSF is not a straightforward measure of resolution. FWHM or std. dev. of PSF give ambiguous information about the actual resolution, and imaging of objects smaller than the FWHM of PSF is possible. A multiscale analysis of imaging gives good insight into the peculiar scale-dependent properties of sub-wavelength imaging.Comment: 3 pages, 5 figures

Tight focusing of plane waves from micro-fabricated spherical mirrors

We derive a formula for the light field of a monochromatic plane wave that is truncated and reflected by a spherical mirror. Our formula is valid even for deep mirrors, where the aperture radius approaches the radius of curvature. We apply this result to micro-fabricated mirrors whose size scales are in the range of tens to hundreds of wavelengths, and show that sub-wavelength spot sizes can be achieved. This opens up the possibility of scalable arrays of tightly focused optical dipole traps without the need for high-performance optical systems.Comment: 8 pages, 5 color figures, 1 .sty file; changes made in response to referee comments; published in Optics Expres

Polarization-Current-Based FDTD Near-to-Far-Field Transformation

A new near-to-far-field transformation algorithm for three-dimensional finite-different time-domain is presented in this article. This new approach is based directly on the polarization current of the scatterer, not the scattered near fields. It therefore eliminates the numerical errors originating from the spatial offset of the E and H fields, inherent in the standard near-to-far-field transformation. The proposed method is validated via direct comparisons with the analytical Lorentz-Mie solutions of plane waves scattered by large dielectric and metallic spheres with strong forward-scattering lobes.Comment: 8 pages, 2 figures. Submitted to publis

A statistical analysis on the factors influencing mathematics anxiety in undergraduate students of mathematics and engineering

Mathematics Anxiety (MA), the ‘phobia of numbers’, is related to poor performance in Mathematics. There are numerous studies that discuss a wide range of factors affecting Mathematics Anxiety in students at primary and secondary schools. Furthermore, there are some studies looking into MA in students of Psychology, Engineering and Nursing at a Higher Education level, see, for example (Alves et al. 2016; McMullan et al. 2012) and more references therein. However, we believe that this is the first work on MA in undergraduate students of Mathematics. Consequently, our purpose is to determine whether factors such as gender or ethnicity affect MA. Our main results are that there are significant differences between male and female students; there is a significant difference among students with three siblings or more, compared to students who have two siblings or less. Finally, we discuss the significant difference between the gender of the main family figure providing Mathematics support amongst students with a British and Non-British background

Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers

We demonstrate numerically the diffraction-free propagation of sub-wavelength sized optical beams through simple elements built of metal-dielectric multilayers. The proposed metamaterial consists of silver and a high refractive index dielectric, and is designed using the effective medium theory as strongly anisotropic and impedance matched to air. Further it is characterised with the transfer matrix method, and investigated with FDTD. The diffraction-free behaviour is verified by the analysis of FWHM of PSF in the function of the number of periods. Small reflections, small attenuation, and reduced Fabry Perot resonances make it a flexible diffraction-free material for arbitrarily shaped optical planar elements with sizes of the order of one wavelength.Comment: 5 pages, 4 figure

Perturbation theory for anisotropic dielectric interfaces, and application to sub-pixel smoothing of discretized numerical methods

We derive a correct first-order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted. Most previous perturbative methods give incorrect results for this case, even to lowest order, because of the complicated discontinuous boundary conditions on the electric field at such an interface. Our final expression is simply a surface integral, over the material interface, of the continuous field components from the unperturbed structure. The derivation is based on a "localized" coordinate-transformation technique, which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking a limit in which a coordinate perturbation is infinitesimally localized around the boundary. Not only is our result potentially useful in evaluating boundary perturbations, e.g. from fabrication imperfections, in highly anisotropic media such as many metamaterials, but it also has a direct application in numerical electromagnetism. In particular, we show how it leads to a sub-pixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media, in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes.Comment: 10 page

First-principles method for high-QQ photonic crystal cavity mode calculations

We present a first-principles method to compute radiation properties of ultra-high quality factor photonic crystal cavities. Our Frequency-domain Approach for Radiation (FAR) can compute the far-field radiation pattern and quality factor of cavity modes $\sim 100$ times more rapidly than conventional finite-difference time domain calculations. It also provides a simple rule for engineering the cavity's far-field radiation pattern

Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure

It has recently been predicted that a conical singularity (= Dirac point) in the band structure of a photonic crystal produces an unusual 1/L scaling of the photon flux transmitted through a slab of thickness L. This inverse-linear scaling is unusual, because it is characteristic of radiative transport via diffusion modes through a disordered medium -- while here it appears for propagation of Bloch modes in an ideal crystal without any disorder. We present a quantitative numerical test of the predicted scaling, by calculating the scattering of transverse-electric (TE) modes by a two-dimensional triangular lattice of dielectric rods in air. We verify the 1/L scaling and show that the slope differs by less than 10% from the value predicted for maximal coupling of the Bloch modes in the photonic crystal to the plane waves in free space.Comment: 4 pages, 7 figures. Figure adde

Whispering-gallery modes and light emission from a Si-nanocrystal-based single microdisk resonator

We report on visible light emission from Si-nanocrystal based optically active microdisk resonators. The room temperature photoluminescence (PL) from single microdisks shows the characteristic modal structure of whispering-gallery modes. The emission is both TE and TM-polarized in 300 nm thick microdisks, while thinner ones (135 nm) support only TE-like modes. Thinner disks have the advantage to filter out higher order radial mode families, allowing for measuring only the most intense first order modal structure. We reveal subnanometer linewidths and corresponding quality factors as high as 2800, limited by the spectral resolution of the experimental setup. Moreover,we observe a modification of mode linewidth by a factor 13 as a function of pump power. The origin of this effect is attributed to an excited carrier absorption loss mechanism.Comment: 5 pages, 5 figure