Fundamental Mode Approximation Research Articles

Experimental validation of the fundamental mode approximation for stacked metasurfaces and its application to the treatment of arbitrary period ratios

We experimentally realize a series of incommensurable metasurface stacks that transition from near-field coupling to a far-field regime. Based on a comparison between a semi-analytic model and measurements, we, furthermore, present an experimental study on the validity of the fundamental mode approximation (FMA). As the FMA is a condition for the homogeneity of a metasurface, its validity allows for strong simplification in the design of stacked metasurfaces. Based on this, we demonstrate a method for the semi-analytic design of stacked periodic metasurfaces with arbitrary period ratios. In particular, incommensurable ratios require computational domains of impractically large sizes and are usually very challenging to fabricate. This results in a noticeable gap in parameter space when optimizing metasurface stacks for specific optical features. Here, we aim to close that gap by utilizing the principles of the FMA, allowing for additional parameter combinations in metasurface design.

EXPERIMENT OF UNIQUE COMBINATION NUMBER DUE TO THE THIRD-ORDER NEUTRON-CORRELATION

From zero-power reactor noise measurement, the second- and third-order neutron correlation factors Y and y3 can be evaluated by analyzing mean, variance, the third-order central moment of neutron count data. Theoretically, it is expected that the neutron-correlation ratio y3/Y2 converges to the unique combination number “3” at a near-critical state in an arbitrary system without depending on the fissile material and the neutron-energy spectrum of core, as the neutron counting gate width T increases sufficiently. Thus, the information about the difference between y3/Y2 and “3” has the potential to judge whether a target unknown system is critical or not and to roughly guess the absolute value of subcriticality. In this study, the detector dead-time effect on y3/Y2 is theoretically investigated based on the heuristic method using the single-, pair-, and trio-detection probabilities with the fundamental mode approximation. As a result, it is clarified that the saturation value of y3/Y2 converges to “3” independent of the dead time, when a target system is a critical state. For validation, actual experimental results are presented for a non-multiplication system driven by 252Cf spontaneous source, and shallow and deep subcritical systems at Japanese experimental facilities (UTR-KINKI and KUCA) under the shutdown state. Consequently, it is demonstrated that y3/Y2 shows a significant difference from “3” in the non-multiplication system. In the case of subcritical systems driven by inherent neutron sources, it is confirmed that the ratios y3/Y2 are close to the unique combination number “3,” and the slight difference from “3” is measurable by the long-time reactor noise measurement for the deep subcritical system.

Subcriticality measurement using time-domain decomposition-based integral method for simultaneous reactivity and source changes

ABSTRACT To estimate the subcriticality in dollar units for an arbitrary state-change, the time-domain decomposition-based integral method (TDDI) is proposed using the point kinetics theory based on the fundamental mode approximation. In a general transient subcritical system, reactivity, neutron source intensity, and point kinetics parameters can vary simultaneously. Furthermore, the state-change may not necessarily be a stepwise change. For such a transient, the TDDI method can estimate the subcriticality after the transient using only the time variation of the neutron count rate. Therefore, the proposed method is useful to approximately estimate the subcriticality in a system where a detailed core configuration is unknown. To investigate the applicability of the TDDI method, transient experiments with simultaneous reactivity and source changes or to two successive safety rods dropping were performed at the Kindai University Training and Research Reactor (UTR-KINKI). By comparing with reference values using excess reactivity and control rod worth, it was validated that the subcriticality values obtained by the TDDI method better agree with the reference values than the previous integral method.

Comparison of theoretical formulae and bootstrap method for statistical error estimation of Feynman-α method

This paper discusses the statistical error of the variance-to-mean ratio, or the Y value in the Feynman-α method, from a single measurement of reactor noise. As a theoretical approach, two practical theoretical formulae are derived to estimate the statistical error of Y: one is based on the propagation of uncertainty with unbiased estimators for the third- and fourth-order central moments; the other is a simplified formula that reuses the Y value under the fundamental mode approximation, where the subcriticality is approximately less than 10,000 pcm. As a numerical approach, the bootstrap method is improved to efficiently estimate the correlations of Y between different counting gate widths, or covariance matrix ΣY, due to the bunching method. Through an actual reactor noise experiment at the Kyoto University Criticality Assembly, the statistical errors of Y using the theoretical formulae and the bootstrap method are validated by comparing the reference statistical errors that are estimated from the multiple experiments of reactor noise. Furthermore, the impact of ΣY on the statistical error of the prompt neutron decay constant α is numerically investigated. Consequently, in the case of this experimental analysis, it was confirmed that the bootstrap method with the correlations of Y seems to be slightly better from the viewpoint of the coverage probability of the estimated confidence intervals of α, although the fitting error method without the correlation of Y could be useful for the order estimation of the statistical error of α.

Fast and reliable method to estimate losses of single-mode waveguides with an arbitrary 2D trajectory.

Photonic wire bonds, i.e., freeform waveguides written by 3D direct laser writing, emerge as a technology to connect different optical chips in fully integrated photonic devices. With the long-term vision of scaling up this technology to a large-scale fabrication process, the in situ optimization of the trajectory of photonic wire bonds is at stake. A prerequisite for the real-time optimization is the availability of a fast loss estimator for single-mode waveguides of arbitrary trajectory. Losses occur because of the bending of the waveguides and at transitions among sections of the waveguide with different curvatures. Here, we present an approach that resides on the fundamental mode approximation, i.e., the assumption that the photonic wire bonds predominantly carry their energy in a single mode. It allows us to predict in a quick and reliable way the pertinent losses from pre-computed modal properties of the waveguide, enabling fast design of optimum paths.

A comparison of Monte Carlo methods for neutron leakage at assembly level

The treatment of neutron leakage at assembly level is an essential step in the generation of few-group cross section data for reactor calculations. This work compares methods used for the characterization of assembly leakage in the framework of Monte Carlo cross section generation, using nearly-critical colorset configurations in order to obtain approximate reference values. The B1 fundamental mode approximation and the albedo-search method are compared in terms of eigenvalue, multi-group homogeneous scalar fluxes and maximum pin power differences for several cell types. The performance of a novel heterogeneous leakage method is also tested. Whereas B1 corrections provide the best agreement in homogeneous fluxes, the new method furnishes the best consistence in terms of pin powers, whilst providing rich spectral information and reducing computational overheads associated with colorset calculations. Future investigations are suggested in order to test the performance of these methods in the calculation of few-group, leakage-corrected assembly cross sections and discontinuity factors.

Higher harmonic analyses of the 252Cf source driven noise analysis method

A theoretical formula has been derived to reconstruct the ratio of spectral densities in the 252Cf source driven noise analysis (CSDNA) method from the higher harmonic eigenvalues and eigenfunctions of the α-mode neutron transport equations. The formula closely reproduces each power spectral density and the ratio of spectral densities calculated by Monte Carlo simulations, thereby verifying the theoretical formula. The reactivity or keff of a subcritical system is related to the ratio of spectral densities by the fundamental mode approximation in which the higher harmonic modes are neglected. However, the ratio of spectral densities measured in the CSDNA method yields an ambiguous reactivity or keff that depends on the locations of detectors due to the effect of the higher harmonics. A more elaborate method developed by Mihalczo and Valentine et al. infers an “experimental” keff using a measurement and accurate Monte Carlo simulation of CSDNA. This paper discusses the uniqueness of the inferred “experimental” keff.

Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study

The availability of novel technologies that enable the fabrication of bulk metamaterials made it necessary to develop a theoretical language to discuss their properties appropriately. Here we use the fundamental Bloch mode approximation to explore bulk properties of metamaterials that operate at terahertz frequencies. We demonstrate that this approximation is valid in the case of terahertz meta-foil metamaterials. While relying on theoretical and experimental results, we show that the fundamental Bloch mode approximation can explain minuscule details that are unique to the bulk metamaterial. Various applications such as cylindrical lenses, terahertz cloaks, and sensors will benefit from our analysis.

Diffusion coefficients for LMFBR cells calculated with MOC and Monte Carlo methods

The present work discusses the calculation of the diffusion coefficient of a lattice of hexagonal cells, with both “sodium present” and “sodium absent” conditions. Calculations are performed in the framework of lattice theory (also known as fundamental mode approximation). Unlike the classical approaches, our heterogeneous leakage model allows the calculation of diffusion coefficients under all conditions, even if planar voids are present in the lattice. Equations resulting from this model are solved using the method of characteristics (MOC). Independent confirmation of the MOC result comes from Monte Carlo calculations, in which the diffusion coefficient is obtained without any of the assumptions of lattice theory. It is shown by comparison to the Monte Carlo results that the MOC solution yields correct values of the diffusion coefficient under all conditions, even in cases where the classic calculation of the diffusion coefficient fails. This work is a first step in the development of a robust method to calculate the diffusion coefficient of lattice cells. Adoption into production codes will require more development and validation of the method.

Polarization dependent devices realized by using asymmetrical hole array on a metallic film

A method is brought forward for realizing polarization dependent devices by employing sub-wavelength asymmetrical hole array on a metallic film. Based on the fundamental mode approximation, the phase retardations of rectangular hole for two orthogonal polarization incident waves are analyzed and calculated. Using rectangular hole array, a bifocal-polarization lens for the infrared radiation with 10.6 microm wavelength is designed. Its focal lengths for x- and y- polarized incident wave are examined by the finite difference time domain (FDTD) method and the Rayleigh-Sommerfeld diffraction integrals and the obtained results agree well with the designed values.

Structured lens formed by a 2D square hole array in a metallic film

A method is proposed to modulate phase using variant square holes in a metallic film based on a fundamental mode approximation model. Phase retardation through square holes in a subwavelength scale in a thin metal film has been analyzed and calculated. Based on the model, a structured lens with a numerical aperture of 0.583 and a focal length of 240 microm formed by a 2D square hole array in a metallic film is designed. Numerical simulation using the finite-difference time-domain method is carried out, and the results agree with the theoretical analysis. A focal spot close to the diffraction limit can be obtained.

Component Mode Synthesis Methods for 3-D Heterogeneous Core Calculations Applied to the Mixed-Dual Finite Element Solver MINOS

This paper describes a new technique for determining the pin power in heterogeneous three-dimensional calculations. It is based on a domain decomposition with overlapping subdomains and a component mode synthesis (CMS) technique for the global flux determination. Local basis functions are used to span a discrete space that allows fundamental global mode approximation through a Galerkin technique. Two approaches are given to obtain these local basis functions. In the first one (the CMS method), the first few spatial eigenfunctions are computed on each subdomain, using periodic boundary conditions. In the second one (factorized CMS method), only the fundamental mode is computed, and we use a factorization principle for the flux in order to replace the higher-order eigenmodes. These different local spatial functions are extended to the global domain by defining them as zero outside the subdomain. These methods are well fitted for heterogeneous core calculations because the spatial interface modes are taken into account in the domain decomposition. Although these methods could be applied to higher-order angular approximations—particularly easily to an SPN approximation—the numerical results we provide are obtained using a diffusion model. We show the methods’ accuracy for reactor cores loaded with uranium dioxide and mixed oxide assemblies, for which standard reconstruction techniques are known to perform poorly. Furthermore, we show that our methods are highly and easily parallelizable.

Space and energy dependent theoretical formula for the third order neutron correlation technique

We have studied measurement techniques of subcriticality by using the neutron correlation techniques. Among various techniques, we have paid attention to the third order neutron correlation technique, which was proposed by Furuhashi originally and utilizes the third order fluctuations of neutron counts. By using this technique, we can obtain the absolute value of subcriticality with the aid of factorial moments of fission neutron ν and external source neutron q, without prior knowledge of the prompt neutron decay constant at a reference state. However, in order to apply the third order neutron correlation technique to actual experiments, we must consider the spatial and neutron energy effects in this technique. For this purpose, we derived more generalized theoretical formulas of the third order neutron correlation technique that took account of the spatial and neutron energy effects by virtue of the α-eigenfunction expansion. Making use of the newly derived formulas, we propose the practical formulas and the experimental procedures based on the fundamental mode approximation.

The pulsing CPSD method for subcritical assemblies driven by spontaneous and pulsed sources

Stochastic neutron transport theory is applied to the derivation of the two-neutron-detectors cross power spectral density for subcritical assemblies when external pulsed sources are used. A general relationship between the two-detectors probability generating functions of the kernel and the source is obtained. This relationship considers the contribution to detectors statistics of both the pulsed source and the intrinsic neutron one. An expansion in α-modes is derived for the final solution, which permits to take into account the effect of higher harmonics in subcritical systems. Further, the expression corresponding to the fundamental mode approximation is compared with recent results from experiments performed under the MUSE-4 European research project.

CLL resonant converters with output short-circuit protection

The use of harmonic excitation of resonant converters as a means of limiting excessive electrical stresses on power switches, and tank components, in the event of over-current or output short-circuit fault conditions is investigated. It is demonstrated that the excitation of the primary tank resonance, by odd harmonics of the square-wave input voltage, facilitates reduced power dissipation for systems that are subject to long periods of standby operation. Due to the inherent limitations of traditional fundamental mode approximation techniques to predict sub-harmonic resonant peaks, an alternative, steady-state, cyclic-averaging methodology, is proposed, that is shown to provide accurate and rapid analysis of the resonances when the converter is operated under short-circuit conditions, thereby allowing the effects of high electrical components stresses to be mitigated, and eliminates the necessity for time-consuming integration-based simulations. Measurements from a prototype converter, along with simulation results from a derived state-variable model and a component-based simulation package, are used to demonstrate the accuracy of the proposed cyclic predictions, and the reduction in losses imparted by the use of harmonic excitation when compared to other previously published techniques for over-current/short-circuit protection.

The effect of spatial inhomogeneity in thermal conductivity on the formation of hot-spots

The steady-state microwave heating of a unit slab consisting of three layers of materials with different thermal conductivities is examined. The governing equations are a damped wave equation derived from Maxwell's equations and a heat-force equation for the temperature. As the primary concern is to investigate the dependence of the steady-state on the thermal-conductivity parameter, a simplifying assumption is made, namely that the electrical conductivity is temperature-independent. Under this assumption, the damped wave equation governing the electric field may be solved separately. An eigenfunction expansion for the problem based on the Galerkin method is described and a fundamental-mode approximation is presented. If this approximation is applied to a unit slab composed of three layers with different thermal conductivities, the hot-spot formation can be addressed and a global steady-state solution is found for the whole domain. Numerical results for some different cases of the three-layer combinations are interpreted to gain some insight in parameter dependence and the position of the low-thermal-conductivity inner layer related to hot-spot formation.

On the effects of higher vibration modes in the analysis of three point bend testing

Influence of higher modes of vibration in the dynamic analysis of impact three point bend specimen based on Euler-Bernoulli and Timoshenko beam theories are investigated in an attempt to predict the oscillatory behavior seen in the measured dynamic SIF history. Forced vibration of the cracked beam is analyzed by normal mode summation method. Contact force history computed using fundamental mode approximation is applied as input forcing function and the computed SIF histories are compared with finite element model and experimental data. Analytical and finite element results show that modes higher than the third have practically no influence on the notched beam response.

On the concept of energy-dependent buckling in multigroup diffusion theory—III

The analysis developed in an earlier paper is applied to a small highly enriched homogeneous fast reactor to evaluate the critical mass of 235U. Numerical results indicate that the energy-dependent buckling analysis gives superior results to the conventional constant-buckling approach, particularly for the fundamental mode approximation. For the fast system investigated the best choices of constant buckling are obtained by selecting from those energy regions with atypically low leakage and high reactivity importance.

A transient heat conduction model for reactor fuel elements

A two region, lumped-parameter model is developed for the treatment of thermal transients in reactor fuel elements. The fuel-region heat condution is treated by a fundamental mode approximation to the time-dependent Green's function. The model also accounts for the thermal resistance of the fuel-cladding gap and the cladding-coolant interface, and for the non-negligible heat capacity of both the fuel and cladding regions. The thermal resistance of the cladding is neglected. The resulting differential equations are similar to those that are obtained were the lumped-parameter model of Tong. The results from the fundamental mode model and the Tong model are compared to exact analytical results that are obtained for a number of highly idealized problems. The fundamental mode model is then used to treat a series of highly idealized loss-of-coolant transients. The fundamental mode model has been incorporated into the FLASH-2 code to treat more realistic accident transients.

Diffusion in Nonlinear Multiplicative Media

The time-dependent behavior of the nonlinear distributions defined by the diffusion equation with several nonlinear source terms is studied. The nonlinear diffusion equation is solved by an eigenfunction-expansion method, which is in principle independent of geometry or number of dimensions. The qualitative time behavior of the distributions and their steady states can be ascertained from a simple analysis of the fundamental mode approximation only. Explicit solutions are presented in one- and two-dimensional geometries.