Nonasymptotic Homogenization of Periodic Electromagnetic Structures: Uncertainty Principles

We show that artificial magnetism of periodic dielectric or metal/dielectric structures has limitations and is subject to at least two "uncertainty principles". First, the stronger the magnetic response (the deviation of the effective permeability tensor from identity), the less accurate ("certain") the predictions of any homogeneous model. Second, if the magnetic response is strong, then homogenization cannot accurately reproduce the transmission and reflection parameters and, simultaneously, power dissipation in the material. These principles are general and not confined to any particular method of homogenization. Our theoretical analysis is supplemented with a numerical example: a hexahedral lattice of cylindrical air holes in a dielectric host. Even though this case is highly isotropic, which might be thought as conducive to homogenization, the uncertainty principles remain valid.Comment: 11 pages, 5 figure

Can photonic crystals be homogenized in higher bands?

We consider conditions under which photonic crystals (PCs) can be homogenized in the higher photonic bands and, in particular, near the $\Gamma$-point. By homogenization we mean introducing some effective local parameters $\epsilon_{\rm eff}$ and $\mu_{\rm eff}$ that describe reflection, refraction and propagation of electromagnetic waves in the PC adequately. The parameters $\epsilon_{\rm eff}$ and $\mu_{\rm eff}$ can be associated with a hypothetical homogeneous effective medium. In particular, if the PC is homogenizable, the dispersion relations and isofrequency lines in the effective medium and in the PC should coincide to some level of approximation. We can view this requirement as a necessary condition of homogenizability. In the vicinity of a $\Gamma$-point, real isofrequency lines of two-dimensional PCs can be close to mathematical circles, just like in the case of isotropic homogeneous materials. Thus, one may be tempted to conclude that introduction of an effective medium is possible and, at least, the necessary condition of homogenizability holds in this case. We, however, show that this conclusion is incorrect: complex dispersion points must be included into consideration even in the case of strictly non-absorbing materials. By analyzing the complex dispersion relations and the corresponding isofrequency lines, we have found that two-dimensional PCs with $C_4$ and $C_6$ symmetries are not homogenizable in the higher photonic bands. We also draw a distinction between spurious $\Gamma$-point frequencies that are due to Brillouin-zone folding of Bloch bands and "true" $\Gamma$-point frequencies that are due to multiple scattering. Understanding of the physically different phenomena that lead to the appearance of spurious and "true" $\Gamma$-point frequencies is important for the theory of homogenization.Comment: Accepted in this form to Phys. Rev. B. Small addition in Sec.V (Discussion) relative to previous version. The title to appear in PRB has been changed to "Applicability of effective medium description to photonic crystals in higher bands: Theory and numerical analysis" per the journal policy not to print titles in the form of question

Trefftz Approximations in Complex Media: Accuracy and Applications

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential convergence with respect to the size of the basis set. We highlight two separate examples of that in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one -- somewhat surprisingly -- to well-posedness of random matrices.Comment: 31 pages, 11 figure

Mesh shape and anisotropic elements : theory and practice

The relationship between the shape of finite elements in unstructured meshes and the error that results in the numerical solution is of increasing importance as finite elements are used to solve problems with highly anisotropic and, often, very complex solutions. This issue is explored in terms of a priori and a posteriori error estimates, and through consideration of the practical issues associated with assessing element shape quality and implementing an adaptive finite element solver

(E)-2-Methyl-5-(thiophen-2-ylmethyl­idene)cyclopentan-1-one

The exocyclic C=C double-bond in the title compound, C11H12OS, has an E configuration. The methyl-bearing C atom in the cyclo­pentane ring is disordered over two positions with a site-occupation factor of 0.899 (8) for the major occupied site

Comparison of large-angle production of charged pions with incident protons on cylindrical long and short targets

The HARP collaboration has presented measurements of the double-differential pi+/pi- production cross-section in the range of momentum 100 MeV/c <= p 800 MeV/c and angle 0.35 rad <= theta <= 2.15 rad with proton beams hitting thin nuclear targets. In many applications the extrapolation to long targets is necessary. In this paper the analysis of data taken with long (one interaction length) solid cylindrical targets made of carbon, tantalum and lead is presented. The data were taken with the large acceptance HARP detector in the T9 beam line of the CERN PS. The secondary pions were produced by beams of protons with momenta 5 GeV/c, 8 GeV/c and 12 GeV/c. The tracking and identification of the produced particles were performed using a small-radius cylindrical time projection chamber (TPC) placed inside a solenoidal magnet. Incident protons were identified by an elaborate system of beam detectors. Results are obtained for the double-differential yields per target nucleon d2 sigma / dp dtheta. The measurements are compared with predictions of the MARS and GEANT4 Monte Carlo simulations.Comment: 43 pages, 20 figure

Forward production of charged pions with incident π±\pi^{\pm} on nuclear targets measured at the CERN PS

Measurements of the double-differential $\pi^{\pm}$ production cross-section in the range of momentum 0.5 \GeVc \leq p \le 8.0 \GeVc and angle 0.025 \rad \leq \theta \le 0.25 \rad in interactions of charged pions on beryllium, carbon, aluminium, copper, tin, tantalum and lead are presented. These data represent the first experimental campaign to systematically measure forward pion hadroproduction. The data were taken with the large acceptance HARP detector in the T9 beam line of the CERN PS. Incident particles, impinging on a 5% nuclear interaction length target, were identified by an elaborate system of beam detectors. The tracking and identification of the produced particles was performed using the forward spectrometer of the HARP detector. Results are obtained for the double-differential cross-sections ${{\mathrm{d}^2 \sigma}}/{{\mathrm{d}p\mathrm{d}\Omega}}$ mainly at four incident pion beam momenta (3 \GeVc, 5 \GeVc, 8 \GeVc and 12 \GeVc). The measurements are compared with the GEANT4 and MARS Monte Carlo simulationComment: to be published on Nuclear Physics

Large-angle production of charged pions by 3 GeV/c - 12 GeV/c protons on carbon, copper and tin targets

A measurement of the double-differential $\pi^{\pm}$ production cross-section in proton--carbon, proton--copper and proton--tin collisions in the range of pion momentum 100 \MeVc \leq p < 800 \MeVc and angle 0.35 \rad \le \theta <2.15 \rad is presented. The data were taken with the HARP detector in the T9 beam line of the CERN PS. The pions were produced by proton beams in a momentum range from 3 \GeVc to 12 \GeVc hitting a target with a thickness of 5% of a nuclear interaction length. The tracking and identification of the produced particles was done using a small-radius cylindrical time projection chamber (TPC) placed in a solenoidal magnet. An elaborate system of detectors in the beam line ensured the identification of the incident particles. Results are shown for the double-differential cross-sections at four incident proton beam momenta (3 \GeVc, 5 \GeVc, 8 \GeVc and 12 \GeVc)

Absolute Momentum Calibration of the HARP TPC

In the HARP experiment the large-angle spectrometer is using a cylindrical TPC as main tracking and particle identification detector. The momentum scale of reconstructed tracks in the TPC is the most important systematic error for the majority of kinematic bins used for the HARP measurements of the double-differential production cross-section of charged pions in proton interactions on nuclear targets at large angle. The HARP TPC operated with a number of hardware shortfalls and operational mistakes. Thus it was important to control and characterize its momentum calibration. While it was not possible to enter a direct particle beam into the sensitive volume of the TPC to calibrate the detector, a set of physical processes and detector properties were exploited to achieve a precise calibration of the apparatus. In the following we recall the main issues concerning the momentum measurement in the HARP TPC, and describe the cross-checks made to validate the momentum scale. As a conclusion, this analysis demonstrates that the measurement of momentum is correct within the published precision of 3%.Comment: To be published by JINS