Helmholtz Reciprocity Principle Research Articles

Boundary element fast multipole method for modeling electrical brain stimulation with voltage and current electrodes

Objective. To formulate, validate, and apply an alternative to the finite element method (FEM) high-resolution modeling technique for electrical brain stimulation—the boundary element fast multipole method (BEM-FMM). To include practical electrode models for both surface and embedded electrodes. Approach. Integral equations of the boundary element method in terms of surface charge density are combined with a general-purpose fast multipole method and are expanded for voltage, shunt, current, and floating electrodes. The solution of coupled and properly weighted/preconditioned integral equations is accompanied by enforcing global conservation laws: charge conservation law and Kirchhoff’s current law. Main results. A sub-percent accuracy is reported as compared to the analytical solutions and simple validation geometries. Comparison to FEM considering realistic head models resulted in relative differences of the electric field magnitude in the range of 3%–6% or less. Quantities that contain higher order spatial derivatives, such as the activating function, are determined with a higher accuracy and a faster speed as compared to the FEM. The method can be easily combined with existing head modeling pipelines such as headreco or mri2mesh. Significance. The BEM-FMM does not rely on a volumetric mesh and is therefore particularly suitable for modeling some mesoscale problems with submillimeter (and possibly finer) resolution with high accuracy at moderate computational cost. Utilizing Helmholtz reciprocity principle makes it possible to expand the method to a solution of EEG forward problems with a very large number of cortical dipoles.

Laboratory simulations of planetary surfaces: Understanding regolith physical properties from remote photopolarimetric observations

We present reflectance and polarization phase curve measurements of highly reflective planetary regolith analogues having physical characteristics expected on atmosphereless solar system bodies (ASSBs) such as a eucritic asteroids or icy satellites. We used a goniometric photopolarimeter (GPP) of novel design to study thirteen well-sorted particle size fractions of aluminum oxide (Al2O3). The sample suite included particle sizes larger than, approximately equal to, and smaller than the wavelength of the incident monochromatic radiation (λ = 635 nm). The observed phase angle, α, was 0.056 o < α < 15°. These Al2O3 particulate samples have very high normal reflectance (> ∼95%). The incident radiation has a very high probability of being multiply scattered before being backscattered toward the incident direction or ultimately absorbed. The five smallest particle sizes exhibited extremely high void space (> ∼95%).The reflectance phase curves for all particle size fractions show a pronounced non-linear reflectance increase with decreasing phase angle at α∼ < 3°. Our earlier studies suggest that the cause of this non-linear reflectance increase is constructive interference of counter-propagating waves in the medium by coherent backscattering (CB), a photonic analog of Anderson localization of electrons in solid state media.The polarization phase curves for particle size fractions with size parameter (particle radius/wavelength) r/λ < ∼1, show that the linear polarization rapidly decreases as α increases from 0°; it reaches a minimum near α = ∼2°. Longward of ∼2°, the negative polarization decreases as phase angle increases, becoming positive between 12° and at least 15°, (probably ∼20°) depending on particle size. For size parameters r/λ > ∼1 we detect no polarization.This polarization behavior is distinct from that observed in low albedo solar system objects such as the Moon and asteroids and for absorbing materials in the laboratory. We suggest this behavior arises because photons that are backscattered have a high probability of having interacted with two or more particles, thus giving rise to the CB process.These results may explain the unusual negative polarization behavior observed near small phase angles reported for several decades on highly reflective ASSBs such as the asteroids 44 Nysa, 64 Angelina and the Galilean satellites Io, Europa and Ganymede. Our results suggest these ASSB regoliths scatter electromagnetic radiation as if they were extremely fine grained with void space > ∼95%, and grain sizes of the order < = λ. This portends consequences for efforts to deploy landers on high ASSBs such as Europa. These results are also germane to the field of terrestrial geo-engineering, particularly to suggestions that earth's radiation balance can be modified by injecting Al2O3 particulates into the stratosphere thereby offsetting the effect of anthropogenic greenhouse gas emissions.The GPP used in this study was modified from our previous design so that the sample is presented with light that is alternatingly polarized perpendicular to and parallel to the scattering plane. There are no analyzers before the detector. This optical arrangement, following the Helmholtz Reciprocity Principle (HRP), produces a physically identical result to the traditional laboratory reflectance polarization measurements in which the incident light is unpolarized and the analyzers are placed before the detector. The results are identical in samples measured by both methods. We believe that ours is the first experimental demonstration of the HRP for polarized light, first proposed by Helmholtz in 1856.

Isochromatic lines as extension of Helmholtz reciprocity principle for effect paints.

Flake-based parameters were recently introduced as a physical concept to predict a series of measurement geometries producing similar reflection data for effect paints. We derive expressions to calculate these so-called isochromatic lines, connecting the two Helmholtz-reciprocal in-plane geometries with a series of out-of-plane geometries. Thus isochromatic lines can be regarded as an extension of the Helmholtz reciprocity principle, which is valid for effect paints. We experimentally studied seven effect paint samples with large angular color variation along the length of four isochromatic lines. A change in illumination angles by up to 75° while following isochromatic lines led to a standard deviation in color parameters of less than two units. When isochromatic lines were not followed, these colorimetric parameters varied by more than 10 units already by change in detection angle of 10°. Therefore the concept of isochromatic lines works well for effect paints.

Path-reversed Auger electron and photoelectron diffraction

We propose a method for the computer simulation of Auger electron and photoelectron diffraction patterns by evaluating the amplitude of propagation paths from the detector to the electron-emitting source, justified by Helmholtz's reciprocity principle. The method offers significant computational advantages over previous schemes, and suggests an easy extension to enable the calculation of a structure-perturbation tensor for rapid crystallographic parameter variation.

Confirmation of Helmholtz Reciprocity Using ScaRaB Satellite Data

The Helmholtz reciprocity principle is often invoked to derive the bidirectional reflectance distribution functions, then the anisotropic function, used for converting satellite radiance measurements into fluxes in Earth radiation budget studies. However, it has seldom been checked by satellite observation. This is only possible using satellites in low but non-Sun-synchronous Earth orbit, and even in this case situations allowing reciprocity verification are rare. The ScaRaB (Scanner for Radiation Budget) radiometer on board Meteor-3-07 provided non-Sun-synchronous observations of the Earth from March 1994 to February 1995. Considering only clear-sky desert scenes which may be presumed to be unchanging, enough viewing/illumination geometry pairs to check the reciprocity principle, using reflectances averaged over a small (1.25°×1.25°) study areas can be obtained. We define a normalized reciprocity departure, which is zero when reciprocity is perfectly respected. The distribution of this parameter shows that the reciprocity principle is verified to better than 5% in 75% of the cases (and 1% in a third of the cases). We also define a parameter analogous to the normalized reciprocity departure, measuring anisotrophy of reflectance for arbitrary viewing/illumination geometry pairs, which would be zero for Lambertian reflection. Comparing the distributions of these parameters with the normalized reciprocity departure, we show that the reciprocal viewing/illumination geometry pairs are very distinct from the arbitrary pairs. This confirmation of the Helmholtz reciprocity principle reinforces our confidence in using it to construct anisotropic reflectance function from measurements for which not all viewing or illumination geometries are sampled.

Helmholtz Reciprocity: its validity and application to reflectometry

Many measurements of reflection properties of materials or components depend for their validity on the Helmholtz Reciprocity Principle. Yet this principle is widely misunderstood, and is stated in textbooks in various forms which have different meanings. This paper examines Helmholtz's own statements of his Reciprocity Principle, which restrict its applicability to corresponding states of polarization for incident and emergent fluxes. This interpretation contrasted with those of more generalised statements, which either allow any incident and emergent states of polarization or which merely require an interchange of unrestricted incident and emergent states of polarization. A comprehensive set of measurements on five matt and two glossy white standard materials has enabled luminance factors to be compared for (0°/45°) and (45°/0°) conditions as well as for the special conditions (0°/57°) and (57°/0°) involving Brewster's Angle for the glossy samples. Each of these nine comparisons has been carried out with all 15 possible combinations of s- and p-plane polarized and unpolarized light. The 135 tests of reciprocity demonstrate that significant and sometimes large discrepancies occur for non-corresponding states of incident and emergent polarizations, whereas no significant discrepancies occur for corresponding states of polarization.