Dipole Source Research Articles

A New Iterative Algorithm for Magnetic Motion Tracking.

Motion analysis is of great interest to a variety of applications, such as virtual and augmented reality and medical diagnostics. Hand movement tracking systems, in particular, are used as a human-machine interface. In most cases, these systems are based on optical or acceleration/angular speed sensors. These technologies are already well researched and used in commercial systems. In special applications, it can be advantageous to use magnetic sensors to supplement an existing system or even replace the existing sensors. The core of a motion tracking system is a localization unit. The relatively complex localization algorithms present a problem in magnetic systems, leading to a relatively large computational complexity. In this paper, a new approach for pose estimation of a kinematic chain is presented. The new algorithm is based on spatially rotating magnetic dipole sources. A spatial feature is extracted from the sensor signal, the dipole direction in which the maximum magnitude value is detected at the sensor. This is introduced as the "maximum vector". A relationship between this feature, the location vector (pointing from the magnetic source to the sensor position) and the sensor orientation is derived and subsequently exploited. By modelling the hand as a kinematic chain, the posture of the chain can be described in two ways: the knowledge about the magnetic correlations and the structure of the kinematic chain. Both are bundled in an iterative algorithm with very low complexity. The algorithm was implemented in a real-time framework and evaluated in a simulation and first laboratory tests. In tests without movement, it could be shown that there was no significant deviation between the simulated and estimated poses. In tests with periodic movements, an error in the range of 1° was found. Of particular interest here is the required computing power. This was evaluated in terms of the required computing operations and the required computing time. Initial analyses have shown that a computing time of 3 μs per joint is required on a personal computer. Lastly, the first laboratory tests basically prove the functionality of the proposed methodology.

A theoretical analysis of the logging-while-drilling dipole acoustic reflection measurement

Post-drilling wireline acoustic single-well imaging technology can now detect geological structures tens of meters away from boreholes. Further development of this single-well imaging technology in the logging-while-drilling (LWD) environment will have significant values in real-time applications such as geosteering and reservoir navigation. Based on the wireline imaging application, we propose a new method for the LWD application. In wireline imaging, the four-component (4C) dipole acoustic data are azimuthally rotated to scan the reflectors around the borehole. In LWD, azimuthal scanning is achieved by drilling rotation such that the 4C dipole system in the wireline is replaced by a one-dipole-source and two-receiver LWD system, where the two receivers are mounted on opposite sides of the drill collar. For the LWD application, we first developed the theory for LWD dipole shear-wave reflection imaging and validated the theory using 3D finite-difference waveform modeling. Using the analytical solution, we analyzed the far-field radiation directivity of an acoustic LWD dipole source and the effect of drilling rotation on the shear-wave reflection imaging using the LWD acoustic system. The LWD analysis results show that, for fast formations, the SH-wave is the dominant component for imaging, whereas for slow formations, the P-wave becomes important and can be used for imaging. Our results also indicate that the reflection data acquired by the system are affected by the speed of drilling rotation. The take-off azimuth at the wave radiation may be different from the incident azimuth at the wave reception. Knowing the rotation speed, this azimuth difference can be corrected. A further advantage of using the oppositely mounted receivers is that the reflected wave arrives earlier (later) at the front (back)-side receiver; thus, the arrival time difference between the receivers can be used to eliminate the 180°-azimuth ambiguity of dipole acoustic imaging. For reflection imaging, using the proposed LWD system configuration, we tested its azimuth sensitivity and validated its 180°-ambiguity solution using synthetic LWD and field wireline dipole data. The results of this work, therefore, provide a theoretical foundation for the development of the LWD acoustic reflection imaging system.

A study on aerodynamic pass-by noise based on equivalent source method

Pass-by noise generated by high-speed trains is an important influencing factor on the surrounding environment and is getting more and more attention with the increase of speed. In the past, the aerodynamic pass-by noise prediction was usually conducted with the wind tunnel mode, ignoring the Doppler effect. In this paper, an equivalent source method considering the Doppler effect is proposed to study the aerodynamic pass-by noise. A cylinder case is considered first to verify the effectiveness of the method. Then a DSA 380 pantograph of a Chinese high-speed train is considered. The pantograph is divided into three parts, and each part is equivalent to a monopole or dipole sound source. Results show that the upper part of the pantograph is closer to a dipole source, while the middle and lower parts are closer to a monopole source. When the measurement point is directly above the pantogragh, the contribution of the upper part is the largest, while for both sides measurement points of the pantograph, the contribution of the upper part is the smallest.

Achieving Ideal Magnetic Light Emission with Electric-Type Emitters.

Optical magnetic dipole (MD) emission predominantly relies on emitters with significant MD transitions, which, however, rarely exist in nature. Here, we propose a strategy to transform electric dipole (ED) emission to a magnetic one by elegantly coupling an ED emitter to a silicon nanoparticle exhibiting a strong MD resonance. This emission mode transformation enables an artificially ideal magnetic dipole source with an MD purity factor of up to 99%. The far-field emission patterns of such artificial MD sources were experimentally measured, which unambiguously resolved their magnetic-type emission origin. This study opens the path to achieving ideal magnetic dipole emission with nonmagnetic emitters, largely extending the availability of magnetic light emitters conventionally limited by nature. Beyond the fundamental significance in science, we anticipate that this study will also facilitate the development of magnetic optical nanosource and enable potential photonic applications relying on magnetic light emission.

Toggling near-field directionality via manipulation of matter's anisotropy.

Near-field directional excitation of dipolar sources is crucial for many practical applications, such as quantum optics, photonic integrated circuits, and on-chip information processing. Based on theoretical analyses and numerical simulations, here we find that the near-field directionality of circularly polarized dipoles can be flexibly toggled by engineering the anisotropy of the surrounding matter, in which the dipolar source locates. To be specific, if the circularly polarized dipole is placed close to the interface between a hyperbolic matter and air, the main propagation direction of excited surface waves would be reversed when the location of the dipolar source is changed from the air region to the hyperbolic-matter region. The underlying mechanism is that the spatial-frequency spectrum of evanescent waves carried by the dipolar source in a homogeneous surrounding matter could be flexibly reshaped by the matter's anisotropy, especially when the isofrequency contour of the surrounding matter changes from the circular shape to the hyperbolic one.

Directional excitation of structural vibrations due to correlated sources

Dynamical Energy Analysis (DEA) is an approach used to compute the vibro-acoustic response of complex structures to externally applied high-frequency excitation. DEA tracks the global energy flow in a structure in terms of a local ray tracing approach implemented on meshes. Due to its in-built directional set-up, the approach is ideally suited for modelling the response of structures to directional excitation. Examples thereof are dipole sources or - more generally - correlated and distributed source excitation such as due to a Turbulent Boundary Layer (TBL) excitation across an aircraft wing during flight. We will demonstrate here how such directional sources are implemented in a DEA setting. TBL induced sources or general correlated sources can be described in terms of appropriate force or wave correlation functions (CF). Using Wigner-transformation, these CFs can be associated with a directional energy flow source. Vibrational energy field results are presented for a pair of phase-locked point forces. Results are also presented for 9 point-sources, which demonstrate the ability to channel the energy flow away from the sources by carefully choosing their relative phases. An implementation of this approach within DEA is also presented, with results demonstrating strong agreement with direct calculations.

Doubly special relativity as a nonlocal quantum field theory

In this work, we explore the quantum theories of the free massive scalar, the massive fermionic, and the electromagnetic fields in a doubly special relativity scenario. This construction is based on a geometrical interpretation of the kinematics of this kind of theory. In order to describe the modified actions, we find that a higher (indeed infinite) derivative field theory is needed, from which the deformed kinematics can be read. From our construction, we are able to restrict the possible models of doubly special relativity to those that preserve linear Lorentz invariance. We quantize the theories and also obtain a deformed version of the Maxwell equations. We analyze the electromagnetic vector potential for both an electric pointlike source and magnetic dipole. We observe that the electric and magnetic fields do not diverge at the origin for some models described with an anti–de Sitter momentum space, but they do for a de Sitter space in both problems. Published by the American Physical Society 2024

Super-resolution localization and orientation estimation of multiple dipole sound sources: From a maximum likelihood framework to wind tunnel validation

This paper investigates the identification of multiple dipole sound sources using sound pressures measured from a microphone array. The problem is addressed in the maximum likelihood (ML) framework, where the locations, orientations, and powers of multiple dipole sound sources are unknown parameters to be estimated. By the consistency property of ML, the estimated parameters converge to their actual values, which implies an asymptotically perfect spatial resolution, if a sufficiently high signal-to-noise ratio can be achieved. In order to reduce the dimension of the optimization problem of ML, the contribution of each dipole source to the measured pressures is assumed to be a latent variable and the ML problem is equivalently solved via the expectation–maximization (EM) algorithm, which iteratively and sequentially updates each source contribution and the associated sound source parameters. The number of sound sources can also be determined by the model selection approaches which add a penalty of model dimension to the ML objective function. The proposed method is assessed via a laboratory experiment where the sound field is produced by dipole speakers and a wind tunnel experiment where airframe aerodynamic noise is generated at a high Reynolds number. Experimental results show that the proposed method outperforms existing approaches in the sense of higher spatial resolution, more accurate localization, and the capacity to identify the orientations of multiple dipole sound sources.

Locating Dipole Source Using Self-Propelled Robotic Fish With Artificial Lateral Line System

Locating Dipole Source Using Self-Propelled Robotic Fish With Artificial Lateral Line System

Self‐Potential Tomography Preconditioned by Particle Swarm Optimization—Application to Monitoring Hyporheic Exchange in a Bedrock River

AbstractA self‐potential (SP) data‐inversion algorithm was developed and tested on an analytical model of electrical‐potential profile data attributed to single and multiple polarized electrical sources. The developed algorithm was then validated by an application to SP‐monitoring field data measured on the floodplain of East Fork Poplar Creek, Oak Ridge, Tennessee, to image electrical sources in areas conducive to preferential flow into the flood plain from the bedrock‐lined riverbed. The algorithm combined stochastic source‐localization by particle‐swarm‐optimization (PSO) of electrical sources characterized by simplified geometries with source tomography by regularized weighted least‐squares minimization of a quadratic objective function. Prior information was incorporated by preconditioning the tomography algorithm by PSO results. Variable percentages of random noise were added to analytical‐model data to evaluate the algorithm performance. Results indicated that true parameters of single‐source models were inverted and approximated with small residual error, whereas inversion of analytical‐model data representing multiple electrical sources accurately approximated the locations of the sources but miscalculated some parameters because of the non‐uniqueness of the inverse‐model solution. Source tomography applied to analytical model data during testing produced a spatially continuous parameter field that identified the locations of point‐scale synthetic dipole sources of electrical current flow with varying degrees of accuracy depending on the prior information incorporated into the tomography. When applied to SP‐monitoring field data, the algorithm imaged electrical sources within a known fault that intersects the bedrock riverbed and flood plain of East Fork Poplar Creek and depicted dynamic electrical conditions attributed to hyporheic exchange.

An Approach to Calculate Source Locations for Thicker Plates when Acoustic Emission Signals are Dominated by Trailing Waves

In many acoustic emission (AE) applications, signal arrival times from an array of sensors are coupled with an appropriate velocity to calculate the source location of AE events. To date, the AE signals present in these applications are Lamb and/or Rayleigh waves. In a previous publication [1], finite element modeled (FEM) AE signals in thicker plates were dominated by trailing wave (TW) packets, when no Rayleigh wave was present. The typical character of the TW dominated signals was described as a spaced-in-time train of short-in-time wave packets. Further, it was pointed out, that when TW dominate the signals, it is not clear how arrival times for source location calculations might be obtained along with an appropriate propagation velocity. The goal of this work is to develop a source location approach for cases of dominant TWs in the AE signals in thicker plates. The FEM-signal database from the previous publication was used for this study. The database consisted of in-plane dipole sources at five different depths and four different steel plate thicknesses (50, 75, 100 and 125 mm). Bottom-surface out-of-plane displacement signals versus time for propagation distances from 250 to 1500 mm were examined. The FEM signals were frequency filtered to examine a broadband case of 80 to 500 kHz and a narrowband case of 100 to 300 kHz. To study the potential source location accuracy, a total of seven four-sensor arrays were employed. The study results describe a process to select appropriate arrival times and obtain a propagation “velocity.” This information was used to calculate source locations for the sensor arrays. The location errors were shown to be less than the plate thickness.

Factors Influencing Radiation Sound Fields in Logging While Drilling Using an Acoustic Dipole Source

With the increasing number of complex well types in the development stage of oil and gas fields, it is becoming increasingly urgent to use remote detection logging while drilling (LWD) to explore the geological structures in a formation. In this paper, the feasibility and reliability of the dipole remote detection of logging while drilling are demonstrated theoretically. For this purpose, we use an asymptotic solution of elastic wave far-field displacement to derive the calculation formula for the radiation pattern and energy flux of an LWD dipole source. The effects of influencing factors, including the source frequency, formation property, drill collar size, and mud parameter, on the radiation pattern and energy flux are analyzed. The results show that the horizontally polarized shear wave (SH-wave) has a greater advantage in imaging the reflector compared with the cases of the compressive wave (P-wave) and vertically polarized shear wave (SV-wave), which indicates the dominance of the SH-wave in dipole remote detection while drilling. The optimal source excitation frequency of 2.5 kHz and inner and outer radii of the drill collar of 0.02 and 0.1 m, respectively, should be considered in the design of an LWD dipole shear wave reflection tool. However, the heavy drilling mud is not conducive to remote detection during logging while drilling. In addition, the reflection of the SH-wave for the LWD condition is simulated. Under the conditions of optimal source frequency, drill collar size, and mud parameters, the reflection of the SH-wave signal is still detected under the fast formation.

Improved Dipole Source Localization from Simultaneous MEG-EEG Data by Combining a Global Optimization Algorithm with a Local Parameter Search: A Brain Phantom Study.

Dipole localization, a fundamental challenge in electromagnetic source imaging, inherently constitutes an optimization problem aimed at solving the inverse problem of electric current source estimation within the human brain. The accuracy of dipole localization algorithms is contingent upon the complexity of the forward model, often referred to as the head model, and the signal-to-noise ratio (SNR) of measurements. In scenarios characterized by low SNR, often corresponding to deep-seated sources, existing optimization techniques struggle to converge to global minima, thereby leading to the localization of dipoles at erroneous positions, far from their true locations. This study presents a novel hybrid algorithm that combines simulated annealing with the traditional quasi-Newton optimization method, tailored to address the inherent limitations of dipole localization under low-SNR conditions. Using a realistic head model for both electroencephalography (EEG) and magnetoencephalography (MEG), it is demonstrated that this novel hybrid algorithm enables significant improvements of up to 45% in dipole localization accuracy compared to the often-used dipole scanning and gradient descent techniques. Localization improvements are not only found for single dipoles but also in two-dipole-source scenarios, where sources are proximal to each other. The novel methodology presented in this work could be useful in various applications of clinical neuroimaging, particularly in cases where recordings are noisy or sources are located deep within the brain.

(Invited) Simple and Optimized Silicon Nano-Antennas for Quantum Emitter Control

Improving the brightness and efficiency of single photon sources by the mean of optically resonant nanostructures is a major stake for the development of efficient nanodevices for quantum communications.For a couple of decades, these resonant nanostructures have mainly been made of noble metal that sustain strong localized resonances (LSPR) that can be used to manipulate, concentrate or redirect visible light. Such properties have led to numerous actual or potential applications in integrated optics, sensors, nonlinear optics, field-enhanced spectroscopies, or photovoltaics. Recently, an alternative emerged with high refractive index dielectric nanostructures, which offer the same range of applications as plasmonics by manipulating Mie optical resonances instead of LSPR [1].These resonances can be efficiently tuned by modifying the size, shape, and material of those nanostructures (e.g. silicon, n ~ 4) [2]. Furthermore, high index dielectric nanostructures offer several key advantages when compared to their metallic counterparts: absorption losses are far weaker for wavelengths longer than the direct band gap, access to semiconductor (CMOS) technology for nanostructure fabrication, and presence of intrinsic strong magnetic and electric resonances, providing an unique opportunity to spatially separate and redistribute the energy of the magnetic and electric parts of the electromagnetic field in the near field, otherwise inextricably connected in the far field [3].We discuss here the effect of simple high-index dielectric nanoantennas on the spontaneous emission of model quantum emitters. First, we accurately positioned arrays of nanodiamonds hosting NV colored centers in the gap of silicon dimer nanoantennas, using atomic force microscopy (AFM – [4]) nanoxerography [5]. The NV center is an ideal model system exhibiting single photon emission properties at room temperature, which is well adapted for proof of concept experiments in quantum nano-optics. Since the local density of photonic states (LDOS) is modified by the nanostructure, we show by the mean of time-resolved photoluminescence acquisitions that the photodynamics of these quantum emitters can be enhanced by the coupling to the nanoantenna, down to the single photon emission regime [5]. Our experimental results are in good agreement with multipolar analysis and numerical simulations based on the Green Dyadic Method - GDM.Second, in order to go a step further, we explore the control of the directive emission from quantum dipolar sources by the mean of complex silicon antennas. The latter, made of a given number of building (nano)blocks, are optimized by an Evolutionary Algorithm (EA) coupled to full-field electrodynamical GDM simulations. The geometries obtained allow to tailor the emission direction of a single dipolar source, maximizing the intensity of the emitted light within a given solid angle. Our numerical and preliminary experimental results demonstrate the efficiency of such compact EA-optimized antennas for the control of quantum nanosources. Acknowledgements : This work was supported by Programme Investissements d’Avenir through the grants ANR-10-LABX-0037-NEXT (MILO) and NanoX n° ANR-17-EURE- 0009 (Q-META), by the HiLight ANR project ANR-19-CE24-0026, by LAAS-CNRS micro and nanotechnologies platform member of the French Renatech network, by the CNRS and INSA (PhD grant), by the Région Midi-Pyrénées via the Institute for Quantum technologies in Occitanie (IQO), and by the computing facility center CALMIP of the University of Toulouse under grants P12167 and P1107. References : Kallel et al, Tunable enhancement of light absorption and scattering in Si1xGex nanowires, Phys. Rev. B 12: 085318, 2012.P.R. Wiecha et al., Strongly directional scattering from dielectric nanowires, ACS Photonics 4: pp 2036–2046, 2017.Montagnac et al., Control of light emission of quantum emitters coupled to silicon nanoantenna using cylindrical vector beams, Light : Science and Applications 12 (1), 239, 2023.Humbert et al., Versatile, rapid and robust nanopositioning of single-photon emitters by AFM-nanoxerography, Nanotechnology 33, 215301, 2022.Humbert et al., Large-scale controlled coupling of single-photon emitters to high index dielectric nanoantennas using AFM nanoxerography, Nanoscale 15, 599-608, 2023. Figure 1

Biophysical fluid dynamics in a Petri dish

The humble Petri dish is perhaps the simplest setting in which to examine the locomotion of swimming organisms, particularly those whose body size is tens of microns to millimeters. The fluid layer in such a container has a bottom no-slip surface and a stress-free upper boundary. It is of fundamental interest to understand the flow fields produced by the elementary and composite singularities of Stokes flow in this geometry. Building on the few particular cases that have previously been considered in the literature, we study here the image systems for the primary singularities of Stokes flow subject to such boundary conditions—the Stokeslet, rotlet, source, rotlet dipole, source dipole, and stresslet—paying particular attention to the far-field behavior. In several key situations, the depth-averaged fluid flow is accurately captured by the solution of an associated Brinkman equation whose screening length is proportional to the depth of the fluid layer. The case of hydrodynamic bound states formed by spinning microswimmers near a no-slip surface, discovered first using the alga , is reconsidered in the geometry of a Petri dish, where the power-law attractive interaction between microswimmers acquires unusual exponentially screened oscillations. Published by the American Physical Society 2024

Flow structure around a microswimmer at fluid–fluid interface

Living organisms such as bacteria and algae often form biofilms at air–liquid and/or liquid–liquid interfaces. Therefore, it is important to understand the hydrodynamic interaction between the fluid–fluid interface and microorganisms. In the present study, a squirmer model is employed to study the flow field structures around a symmetrically trapped spherical microswimmer at an interface separating two fluids with viscosity contrast. In these simulations, Reynolds ( Re ) and Capillary ( Ca ) numbers are very small, and hence the contribution of inertia and interface deformations are neglected. The squirmer model is validated against the analytical solution obtained by considering a source dipole. It is observed that the flow structure and vorticity distribution around the microswimmers are strongly influenced by the squirmer parameter (β) and viscosity contrast (λ). Furthermore, the interplay between force-dipole and source-dipole along with viscosity contrast leads to a range of flow structures such as symmetric four-lobe and asymmetric quadrupolar flow fields. This flow structure asymmetry around the swimmer is quantified with different steady state orientations (φ). The effect of flow profile on the passive tracer particle transport is explored in terms of trajectories. Specifically, larger values of φ result in more profound curly trajectories.

Natural source-field induced polarisation exploration of an iron-oxide copper-gold (IOCG) deposit under thick cover

Induced polarisation (IP) is a common geophysical method of exploration for disseminated sulphides. However, in areas with deep (>200 m) and conductive (10 Ωm or less) cover, the technique is less successful as it requires a significant transmitter source and large-offset dipoles. An alternative approach is to use natural-variations in Earth’s external magnetic field as the polarising source of the signal, known as natural-field IP. This paper presents a study of extracting IP information from electrical dipole observations during a broadband magnetotelluric (MT) program with a 9 km by 9 km grid of 100 sites above the Vulcan IOCG deposit beneath 750 m of cover, ∼40 km north of the Olympic Dam IOCG mine in South Australia. Inter-site transfer functions between 95 sites and five reference sites at the southeast and southwest corners of the grid were computed to determine a phase shift between horizontal electric fields in the bandwidth of 1–100 s period. Phase shifts of up to – 5 degrees were centred on a region of brecciated hematite where drill holes intersected pyrite, and an inferred fault-zone from passive seismics that marks the boundary between upper crust that is resistive (>100 Ω.m) with high magnetic susceptibility to the south of the fault, and a region of conductive crust (<10 Ω.m) which is low magnetic susceptibility. Our study suggests that the natural-field IP method can identify regions of polarizable minerals beneath deep cover where artificial power sources cannot be feasibly deployed. Such surveys are cheaper, safer, and easier to deploy as only receivers are required, and 3D coverage can be obtained as the source-field is of much larger dimension than the survey array. In addition, the natural-field IP signals are observed as part of the MT program so that both electrical resistivity and polarisation parameters can be determined.

Perfect active absorption of water waves in a channel by a dipole source

This study investigates the potential use of an active device to efficiently absorb water waves propagating in a channel. The active device comprises a dipole source consisting of two sources in quasi-opposition of phase. We explore the feasibility of this approach to achieve perfect absorption of guided waves through interference phenomena. To accomplish this, we establish the law governing the waves emitted by the dipole source to optimize the absorption of specific incident waves. The validity of this law is demonstrated through numerical simulations and laboratory experiments, encompassing both the harmonic and transient regimes of the experimental set-up.

Near-field scattering prediction and analysis

ABSTRACT To overcome the shortcomings of traditional scattering evaluation methods in near-field scattering prediction, a novel fast scattering calculation algorithm suitable for a near-field region is proposed in this paper. Based on physical optics, combined with locally modified approximate Green’s function, the integral expressions of near-field electromagnetic scattering radiated by a plane wave and a dipole source are derived in a frequency domain. The simulation results show that the modified algorithm not only retains the simplicity and efficiency of the traditional code, but also can be directly used for near-field scattering prediction.

Full Vector Inversion of Magnetic Microscopy Images Using Euler Deconvolution as Prior Information

AbstractPaleomagnetic data is collected from bulk samples, containing a mixture of stable and unstable magnetic particles. Recently, magnetic microscopy techniques have allowed the examination of individual magnetic grains. However, accurately determining the magnetic moments of these grains is difficult and time‐consuming due to the inherent ambiguity of the data and the large number of grains in each image. Here we introduce a fast and semi‐automated algorithm that estimates the position and magnetization of dipolar sources solely based on the magnetic microscopy data. The algorithm follows a three‐step process: (a) employ image processing techniques to identify and isolate data windows for each magnetic source; (b) use Euler Deconvolution to estimate the position of each source; (c) solve a linear inverse problem to estimate the dipole moment of each source. To validate the algorithm, we conducted synthetic data tests, including varying particle concentrations and non‐dipolarity. The tests show that our method is able to accurately recover the position and dipole moment of particles that are at least 15 μm apart for a source‐sensor separation of 5 μm. For grain concentrations of 6,250 grains/mm3, our method is able to detect over 60% of the particles present in the data. We applied the method to real data of a speleothem sample, where it accurately retrieved the expected directions induced in the sample. The semi‐automated nature of our algorithm, combined with its low processing cost and ability to determine the magnetic moments of numerous particles, represents a significant advancement in facilitating paleomagnetic applications of magnetic microscopy.