We launched an array of nine freely floating submarine seismometers near the Galápagos islands, which remained operational for about two years. P and PKP waves from regional and teleseismic earthquakes were observed for a range of magnitudes. The signal-to-noise ratio is strongly influenced by the weather conditions and this determines the lowest magnitudes that can be observed. Waves from deep earthquakes are easier to pick, but the S/N ratio can be enhanced through filtering and the data cover earthquakes from all depths. We measured 580 arrival times for different raypaths. We show that even such a limited number of data gives a significant increase in resolution for the oceanic upper mantle. This is the first time an array of floating seismometers is used in seismic tomography to improve the resolution significantly where otherwise no seismic information is available. We show that the Galápagos Archipelago is underlain by a deep (about 1900 km) 200–300 km wide plume of high temperature, with a heat flux very much larger than predicted from its swell bathymetry. The decrease of the plume temperature anomaly towards the surface indicates that the Earth’s mantle has a subadiabatic temperature gradient. With the advent of seismic tomography, slight variations observed in the travel times of seismic signals have allowed us to map temperature and compositional anomalies at depth. Ever since the earliest seismograph networks were established more than a century ago, the number and quality of seismic instruments has steadily been increasing. The International Seismological Centre (ISC) currently lists 11,447 reporting stations that are distributed globally. However, fewer than 500 of these (4.2%) are located in the open ocean. Despite efforts by academic groups to distribute very high quality broadband instruments more evenly, a recent list of 909 stations belonging to the Federation of Digital Seismic Stations (FDSN) has less than 10% of its total in the oceans away from the coast . Given that the oceans cover almost 2/3 of the Earth’s surface, this uneven sensor distribution causes problems for seismometry, not in the least for our efforts to image the Earth’s interior and understand the processes operating in the deep mantle. Whereas tomography has been very succesful in mapping shallow subduction anomalies (which are mostly located near dense networks on the coast), the fate of deeper sinking slabs is still controversial. For the mantle plumes that carry hot rock upwards, the situation is even worse, because almost all of them are located in the oceanic domain. Slabs and plumes are considered to be workhorses for heat transport in the mantle, but their exact role in the management of the Earth’s heat budget (as opposed to mantle-wide convection) is still open to many questions5, deploying them in the abundance and for the number of years required to reach sufficient resolution well into the lower mantle is very costly. In this paper we show how a new technology for seismometry in the oceans can bring enhanced resolution in the oceans, and present a first application to the Galápagos hotspot. Instead of OBSs, which reside at their launching location for the duration of an experiment, we deploy Lagrangian drifters equipped with a hydrophone and software to discriminate seismic signals from other acoustic sources, named MERMAIDs (Mobile Earthquake Recording in Marine Areas by Independent Divers) by Simons et al.6 MERMAIDs allow for two-way communication of data and operating commands by satellite, and provide a promising and affordable complement to land and ocean bottom seismometers for P wave tomography8. Our experiment with nine MERMAIDs in the Galápagos region constitutes the first application of this new instrumentation in a network configuration. The primary goal of the experiment was to evaluate the quality of the data and the usefulness of MERMAID deployments for regional tomography. But the target itself is one of only about two dozen deep mantle plumes9, and therefore of significant scientific interest in itself.Though ocean bottom seismographs (OBSs) have been used in efforts to compensate for the paucity of sensors near several oceanic hotspots9 to be 103 kg/s, which conforms to a heat flux of about 0.04 TW. Though relatively modest, it provides a convenient test for the validity of the flux estimates based on swell topography. The basalts on the islands show a high 3He/4He anomaly, pointing to a primordial mantle source at depth, though differences exist between the Sr, Nd, Pb, and Hf isotopic ratios which are more indicative of a plume for basalts from GSC further north than those from central and northeast Galápagos10. There may exist a shallow conduit between the observed top of a plume beneath the islands, and the GSC, an idea also supported by numerical modeling13. A receiver function analysis by Hooft et al.14 showed a thinning of the transition zone, such as to be expected for a hot plume originating in the lower mantle. Montelli et al.15 were the first to image a P-wave anomaly extending from the Galápagos islands down to 1800 km depth, but large differences with later tomographic efforts exist18. The exact origin of the Galápagos heat source is therefore still uncertain. The absence of a He isotope anomaly on the nearby spreading ridges even led Sallarès et al.19 to conclude that the hotspot is caused by a compositional anomaly involving recycled crust, rather than a deep mantle plume.The Galápagos Archipelago is located almost 200 km south of the Galápagos Spreading Center (GSC). The buoyancy flux of the proposed deep Galápagos plume is estimated from the swellThe existence of nearby land stations to the N and E of the archipelago makes this region a good testbed, where even a small number of MERMAIDS to the S and W can provide dense coverage with seismic rays where none existed so far.We launched the first MERMAID on May 7, 2014. The lifetime of these first-generation MERMAIDs, 21 months on average, was limited by the finite battery capacity, and thus determined by the variable activity of the CPU, the number of surfacings and the duration of data transmission after recording a signal in response to a trigger. If a P wave is converted to an acoustic wave at the ocean floor, the signal recorded by the hydrophone is essentially a seismogram, and P wave arrival times can be ‘picked’ to observe delays caused along the wave path. The last such seismogram was received on October 25, 2016 .Their trajectories reflect the turbulent nature of deep ocean currents . A checkerboard test of the increase in resolution is shown in Fig. We carefully estimated the errors in the picked data and performed a separate data analysis for errors in the ISC data (see Methods section), such as to limit the range of possible solutions to those that fit the data to the level expected by the errors, but not less or more. Our preferred model fits the data to the estimated standard errors , one of which is located near the Galápagos. All three persist until the top of the lower mantle (depth 655 km). Anomaly C, just N of Galápagos then continues at 1038 and 1490 km depth and remains weakly visible until 1940 km depth.The three-dimensional image of our preferred solution for the mantle beneath Galápagos is shown in Fig. That this indeed is a continuous plume down to 1900 km depth can be seen from Fig. We have shown that precise onset times of P and PKP waves from moderately strong events can be observed with floating seismographs. Even a limited number of floats can cover a large region, thanks to their continuous movement which serves to avoid duplications in raypaths and limits the redundancy of the data. The average longevity of the MERMAIDS used in this study was rather short at 21 months. This average includes one of the MERMAIDs that lived only six months. The cause of its sudden disappearance is not known. In one rare instance a PP wave triggered data transfer, indicating that we may have missed information by tuning the trigger algorithm to higher frequency P waves and/or by restricting data transfer to the first arrival only and by limiting the length of the transmitted seismograms. The main lessons we learned from this experiment are that significant and affordable new information can easily be obtained with floating seismographs in the oceans, that significant improvements in data yield can be made by prolonging the lifetime of the floats, and also that more data can be retrieved if – besides the automatic transmission following a trigger – data can be requested on command from a buffer. These improvements have since been implemented in a new MERMAID design that uses a spherical float with a three times larger battery capacity. With this increase in lifespan, we expect future MERMAID deployments to spread out over a significantly wider area, depending on the local ocean circulation pattern. This will increase resolution to even greater depths than obtained in this study.The succesful picks of onset times came from earthquakes with magnitudes as low as 5 Fig. . But theThe ability to move with the ocean current sets the floats apart from OBSs. For seismic tomography, this advantage compensates for the lack of shear wave information, though best results can probably be obtained by combining the two types of instruments since their capabilities are so different. Compared to OBSs, the MERMAIDS are very easy to deploy. Both deployment and recovery can be done by one person from small ships, not necessarily scientific vessels. Recovery may not be cost effective in the open ocean as one day’s ship time is usually more expensive than the cost of the instrument itself. However, the increase in battery lifespan of the new version (5–6 years) makes it probable that they will sooner or later drift into coastal areas where retrieval with small ships is affordable. The ability to view the data in quasi-real time, to have clock corrections done by GPS at every surfacing and to be able to manage the data stream automatically is also a time- and cost-savings characteristic that sets the MERMAIDs apart from OBSs. On the other hand, OBSs are free from the cost of data transmission, which depends on the parameters set by users for event triggers and length of seismograms. Transmitting seismograms of length 200–250 s, the cost of sending data per Iridium satellites, using Rudics protocol, was about $ 100 per float per month in our experiment. Transmission costs also depend on the appetite of the user, who defines sampling rate and duration.Despite the limitations in this first experiment, our tomographic results are of interest for geodynamics. We consider our error estimates reliable, both for the picked data and for the ISC delays, which limits the range of allowable damping parameters, and which therefore reduces the uncertainty in anomaly amplitudes to a useful level  flux contribution of the sinking slabs from the observed mantle heat flux1, and would imply that there is no other heat exchange between lower and upper mantle besides that carried by slabs and plumes. However, this is just a very rough calculation and to answer this question more definitely, we need to have more accurate flux estimates and for many more plumes. Sukhovich et al.8 calculated that a global fleet of 1000 MERMAIDs, operating for 5 years in the oceans, would be able to image plumes in mid-mantle with the resolution needed to estimate their widths and P-velocity anomaly with high precision.We tried an order-of-magnitude calculation for the flux into the transition zone, assuming an average rise of Although an almost vertical continuation between 1900 km depth and the core-mantle boundary at 2890 km seems visible in the section along longitude in Fig. et al.24 to be the source of enrichment in primordial 3He and 4Ar for deep mantle plumes. Our observations indicate that the Galápagos plume is ultimately fed by heat produced in such a reservoir, by high flux from the core, or – most likely – by both.It may be a coincidence, but the depth of about 1900 km for the source of the upwelling matches the depth of the top of a mantle reservoir proposed by Kellogg 25.We conclude that Galápagos is fed by a 1900 km deep, 200–300 km wide, almost vertical plume beneath 0 °N, 91 °W with a large temperature anomaly, which is separate from a stronger upper mantle upwelling beneath the East Pacific Rise further west. We note that the observed width of the Galápagos plume is consistent with the size of upwellings needed to explain the distribution of isotope ratios in the uranium series in comparable hotspots21. If the P wave has sufficient energy for frequencies above about 0.5 Hz , the onset is clearly visible even if the microseismic noise is high. We correlate onsets ‘by eye’ between MERMAIDS and nearby land stations. For shallow events, the frequency bands of P and noise largely overlap, and even strong events with magnitudes >6.5 can easily be missed by the trigger or their onsets can be difficult to read. For these data we developed a filter to remove much of the microseismic noise. To avoid any pre-causal artefacts it does not use FFT, but fits instead a Fourier series of finite length to the time signal of T seconds:ω0 = π/T and nmax depends on the noise spectrum but is typically such that frequencies are limited to the 0–0.4 Hz band. We then subtract sm(t) from the observed seismogram s(t) before picking the arrival. An example is shown in Fig. Microseismic noise impedes the use of cross-correlation for accurate measurement of delays, except for a few very strong arrivals. We therefore resorted to the classical ‘picking’ of the onset of P and PKP wave arrivals. Since they follow a minimum-time path, the travel time for such onsets can be exactly computed with ray theory26, but these cases are rare . The depth of the float is accurately determined by the pressure sensor with an accuracy of a few meters and does not contribute to the over-all data error.There are, however, indirect data errors that must also be taken into consideration. The first one is caused by errors in the float location. When the float records the signal it is deep beneath the surface and out of reach of GPS, so that we have to estimate its recording location from float drift measurements, after it surfaces. The associated location error is equivalent to a timing error in the onset – the steeper the incident ray, the smaller is this error. We measure the surface drift (median: 1.35 km/h) using two or more GPS fixes while the MERMAID transmits its data to the Iridium satellite network. This enables us to pinpoint the location of the float’s arrival at the surface very precisely. We measure the abyssal drift assuming a constant velocity between the current position and the previous surfacing, and thus determine the float’s position at cruising depth at the time of earthquake detection and recording. The correction for the total drift in the water column during ascent is typically less than 400 m. The equivalent time correction for a mantle P wave is less than 0.05 s, and the expected error even smaller. If the float did not rise immediately to the surface the error can be about three times as large27 which averages over cells of 2° × 2° in longitude and latitude. This is much larger than the width of the Fresnel zone of a 1 Hz P wave at the ocean bottom, which varies with the bathymetry but is typically only a few km, and introduces an error in the length of the water column and thus in the crustal correction. By comparing the bathymetry of CRUST2.0 with that of ETOPO128 on a 10 km grid where the ocean is deep enough to host a MERMAID we estimate that 78% of the bathymetry errors are less than 300 m, or equivalent time errors <0.15 s.The tomographic inversion software includes a correction for crustal structure based on model CRUST2.0σe.The variance in delays caused by location and bathymetry must be added to the variance 29, at the time of our analysis available for the period 2000–2003. Events with less than 180° azimuth coverage, or fewer than 15 stations reporting were rejected. The ISC data set contains few raypaths crossing the upper part of the mantle under the Galápagos, hence the MERMAID data were crucial in providing the regional illumination shown in Fig. d is to a high approximation linearly related to the vector of model perturbations and source corrections m:Most of the picked P waves have part of their wave path outside the region of interest, and we must therefore prevent that distant anomalies elsewhere influence the imaging beneath the Galápagos. For this reason we embed the MM data in a global inversion of P and PKP delays reported by the ISC. We selected the events from the recently revised ‘EHB’ data set that minimizes mislocation bias21. To determine the standard error in the ISC data, we adapt the method of Voronin et al.30, and organize the data in event clusters . The tomographic submatrices (Ac) for these clusters contain highly redundant information and if the data are projected on the eigenvectors of the matrix σe as well as the standard devation σm of the true delays, caused by the Earth’s heterogeneity. The variance of the datum projected on the i–th eigenvector with eigenvalue This system is scaled to unit prior variance for both data and model elementsσe and σm are dimensionless and of order 1, and the distribution of projected data with 31 for the ISC P-delays. It is at the low end of the 0.4–0.9 s range found by Gudmunsson et al.32 for errors at teleseismic distances only.Because of the scaling to unit variance in the data and unit prior variance of the model parameters that we applied to , σe and ays Fig. , which iσe. This reduced the total data set to 1,579,503 data (9.1% of the MM data were rejected). Delay times are defined with respect to the predicted times for model AK13533. To parameterize the P velocity anomalies in the mantle we used a cubed sphere with 37 layers and 6 mantle sections of 128 × 128 voxels for a total of 3,637,248 voxels34. The average horizontal voxel size at the surface is 72 km. Most layers are 90 km thick, though we accommodate discontinuities by halving some layers. The linear system is augmented by regularization constraints and solved using a parallel version of LSQR21.This error estimation requires clusters of data that are large, and may not be representative for delays from very isolated events, which we therefore decided not to use. After these selections, we used 1,539,758 P delays. We also selected 237,166 PKP delays. The PKP data set is too small to estimate their errors reliably from event clusters and we assigned a standard error of 1 s to PKP delays, which includes delays acquired in the core, which is not included in the tomographic model. After a first inversion run with very low damping, we rejected 11.3% of data that could not be fitted within 3χ2 is exactly the misfit imposed by normally distributed errors, thus considerably reducing the uncertainty in the amplitudes of anomalies that often hampers tomographic images (see next section). Most importantly, this allowed us to get temperature estimates for the plume at depth that still have a large uncertainty, but that are more precise than can be inferred from more arbitrarily damped tomographic solutions.The large number of voxels in the cubed Earth calls for a regularization dominated by smoothing. Knowledge of the errors allowed us to calibrate the regularization such that the misfit m = Sy and first solving for a minimum norm solution y, where S is one step in a relaxation scheme that solves yi over its neighbours – this is essentially a discretized version of a scheme originally proposed by VanDecar and Snieder35, extended to three dimensions). Since we consider our estimates of standard errors to be reliable, we solve for a data fit that has a reduced χ2 close to 1. In summary, the system we solve is:In addition to a norm damping, we bias the the solution towards smoothly varying models by minimizing absolute spatial derivatives. We precondition the system  by substI is the unit matrix and D differences model parameters but not the source corrections21. εn and εs are weights for norm damping and smoothing, respectively. Except for two alternative inversions (see below) we kept the ratio εn/εs = 0.05. After 2500 iterations of LSQR the data misfit agreed to 4 figures with its final value, but it required 8500 iterations to converge to the optimal fit of the regularization constraints. The resulting model for the mantle under the Galápagos has a The checkerboard resolution test shown in Fig. For the tests in Fig. εn/εs ratio gives the model shown in Fig. Finally, to test the sensitivity of the solution to the damping, we performed three more inversions of the real data. Figure Supplementary figures


We present a finite difference time domain (FDTD) model for computation of A line scans in time domain optical coherence tomography (OCT). The OCT output signal is created using two different simulations for the reference and sample arms, with a successive computation of the interference signal with external software. In this paper we present the model applied to two different samples: a glass rod filled with water-sucrose solution at different concentrations and a peripheral nerve. This work aims to understand to what extent time domain OCT can be used for non-invasive, direct optical monitoring of peripheral nerve activity. Optical coherence tomography is a low coherence interferometric technique that has been first used in 1991 to examine the peripapillary region of the retina , 3 or co® Xeon® X5680 CPU).To this day there are several FDTD models for OCT imaging, and the ones currently known in literature focus mostly on trying to simulate big or three dimensional environments in a reasonable  time. Tin vivo experiments.During the years, different techniques have been developed to record neural activity, varying in the level of invasiveness. Non invasive techniques—e.g. EEG  or fMRI —measure activity at a global level. On the other hand, invasive techniques—e.g. penetrating micro electrodes—are capable of a much higher spatial resolution. At this point in time, it is necessary to establish techniques that can give the best of the two worlds, providing both high resolution and low invasiveness. Optical recordings have been gaining momentum in the past 50 years and the advent of calcium sensitive dyes and genetically encoded indicators has revolutionised neural recordings. However both techniques exhibit either phototoxicity or very low brightness and this limits their usage in Optical recordings for neural activity started in the 1950s  and procThe FDTD method is one of the simplest of the computational electrodynamics techniques and it can solve a broad range of problems in a very accurate way. It was first developed by Yee  in 1966,x0 but the function is sampled at the neighbouring points x0 + h and x0 − h.FDTD is a grid-based differential numerical modelling method where both the spatial and temporal derivatives that appear in Maxwell’s equations are discretised using a central-difference approximation. Central-difference approximations exist in several orders, this work uses a second order one:The algorithm used in this work is the Yee algorithm  = —and the derivatives in Ampere’s and Faraday’s laws are replaced with finite differences.Space and time are discretised—for each point in space and time . In both cases the refractive indexes of the different layers of the simulation domain are considered to be constant over the range of frequencies considered, as the dispersion in water for the narrow range of frequency considered can be assumed insignificant [Two different samples have been studied using this computational model: a glass rod filled with sucrose solutions at different concentration and part of a myelinated nerve made of one single fascicles of axons , glass (nw@850nm = 1.5098) and sucrose solution.The simulation environment consists of three layers: water .Xenopus laevis’ sciatic nerve presents an external surface made of loose epineurial tissue, we have assumed a high fluid content (40%), with the connective fibres making up 70% of the dry mass and the lipids 30% [The epineurium is connective tissue composed of ground substance (n = 1.345), lipids (n = 1.45) and tissue fibres (n = 1.43 fully hydrated) in variable proportions. Since the It was not possible to obtain a specific value for the elastic fibres, so we have considered the value of the fibrils inside the human sclera (n = 1.41).Axons, in our simulations, are made of axoplasm and axonal membrane which consist of water, lipids and proteins (n = 1.5). Considering the percentages observed in  and a voMyelin is composed of 40% water n = 1.3290), and the rest is 70 to 85% lipids and ∼ 15 to 30% proteins [90, and tthe endoneurial fluid has been considered to be like the cerebrospinal fluid (n = 1.335) since they perform the same physiological function and is reasonable to assume they have the same physical properties.The values used for each different tissue type are reported in Since the time an electromagnetic wave needs to travel a few centimetres in air and water is on the nanosecond scale, it is possible to simulate the scattering change undergone by neurons by running different simulations changing discretely the value of the refractive index of the sample.To obtain value for the backscattered light change, we have first developed a ray tracing model to study what kind of change in refractive index would result in a forward scattering change as observed in . We haveAs mentioned earlier, we have decided to remove from the simulation domain the whole splitter/combiner part and to obtain the interference between the reference and sample arms using an external software (Matlab R2017a). The FDTD simulations output the amplitude of the electric field in a preselected point every ten time steps. For each of the saved steps the signals obtained from the nerve and mirror simulations are summed and squared to obtain the intensity interference pattern  noise Click here for additional data file.S2 VideoThe first white vertical line represents the interface water-epineurium and the second the interface epineurium-nerve fibres.(MP4)Click here for additional data file.


Stochastic models are of fundamental importance in many scientific and engineering applications. For example, stochastic models provide valuable insights into the causes and consequences of intra-cellular fluctuations and inter-cellular heterogeneity in molecular biology. The chemical master equation can be used to model intra-cellular stochasticity in living cells, but analytical solutions are rare and numerical simulations are computationally expensive. Inference of system trajectories and estimation of model parameters from observed data are important tasks and are even more challenging. Here, we consider the case where the observed data are aggregated over time. Aggregation of data over time is required in studies of single cell gene expression using a luciferase reporter, where the emitted light can be very faint and is therefore collected for several minutes for each observation. We show how an existing approach to inference based on the linear noise approximation (LNA) can be generalised to the case of temporally aggregated data. We provide a Kalman filter (KF) algorithm which can be combined with the LNA to carry out inference of system variable trajectories and estimation of model parameters. We apply and evaluate our method on both synthetic and real data scenarios and show that it is able to accurately infer the posterior distribution of model parameters in these examples. We demonstrate how applying standard KF inference to aggregated data without accounting for aggregation will tend to underestimate the process noise and can lead to biased parameter estimates. Stochastic differential equations (SDEs) are used to model the dynamics of processes that evolve randomly over time. SDEs have found a range of applications in finance  is widely used to describe the evolution of biochemical reactions inside cells stochastically , and the stochastic part is given by Q(t). Subsequently, we have the following ODEs:We are now interested in the integral of Eq. , as thisThe classical filtering problem is concerned with the problem of estimating the state of a linear system given noisy, indirect or partial observations Kalman . In our First, we consider the case where observations are taken from the process Prior: Predictive distribution: Posterior or Update distribution: The predictive distribution is given by Following the Kalman filter (KF) methodology, we need to define the following quantities:-1] Eqs.  and 7)  Prior: \We consider now the case where the state The predictive distribution of our system is given by the ODEs , 10),  an an\documthe ODEs  with theTo compute the posterior distribution We first investigate the effect of integration in a one-dimensional, zero-mean OU process of the following form:The OU process is a special case of a linear SDE (Eq. ), since l of Eq.  is givenWe are interested in inferring the parameters a et al. .Since we are going to use the normalised observations from We have generated aggregated data from the integral of an OU process with We tested inference using KF1 with normalised data and KF2 with aggregated data. Results of parameter estimation using a standard random walk MH algorithm are presented in Table It is of interest to investigate the inferred stationary variance of the OU process using KF1 and KF2. We have plotted the inferred stationary variances obtained by the MH for both KF1 and KF2 in Fig. In this section, we have looked at an example of inferring the parameters of an SDE using aggregated data, and we have found that to obtain accurate results we need to explicitly model the aggregated process. As the observation intervals become larger, there is a greater mismatch between KF1 and KF2. In the next two sections, we will look at examples of more complex stochastic systems that must be approximated by the LNA and compare again inference results using KF1 and KF2.We are now going to look at a system of two species that interact with each other according to three reactions The LNA can be used to approximate the dynamics and the resulting ODEs can be found in “Appendix The Gillespie algorithm was run for 20 min. Data were aggregated and collected every 2 min resulting in 10 observations per sample. To infer the parameters, we assumed that we had 40 independent samples available. Since we assumed independence between the samples, we worked with the product of their likelihoods. In the ideal case of having complete data of a stochastic kinetic model the likelihood is conjugate to an independent gamma prior for the rate constants . Code reproducing the results of the experiments can be found on GitHub


The macroscopic tensile strength of a panel containing a centre-crack or a centre-hole is predicted, assuming the simultaneous activation of multiple cohesive zones. The panel is made from an orthotropic elastic solid, and the stress raiser has both a tensile cohesive zone ahead of its tip, and shear cohesive zones in an orthogonal direction in order to represent two simultaneous damage mechanisms. These cohesive zones allow for two modes of fracture: (i) crack extension by penetration, and (ii) splitting in an orthogonal direction. The sensitivity of macroscopic tensile strength and failure mode to the degree of orthotropy is explored. The role of notch acuity and notch size are assessed by comparing the response of the pre-crack to that of the circular hole. This study reveals the role of the relative strength and relative toughness of competing damage modes in dictating the macroscopic strength of a notched panel made from an orthotropic elastic solid. Universal failure mechanism maps are constructed for the pre-crack and hole for a wide range of material orthotropies. The maps are useful for predicting whether failure is by penetration or kinking. Case studies are developed to compare the predictions with observations taken from the literature for selected orthotropic solids. It is found that synergistic strengthening occurs: when failure is by crack penetration ahead of the stress raiser, the presence of shear plastic zones leads to an enhancement of macroscopic strength. In contrast, when failure is by crack kinking, the presence of a tensile plastic zone ahead of the stress raiser has only a mild effect upon the macroscopic strength. Macroscopic notches in a panel induce stress concentrations and reduce its load-carrying capacity. The strength reduction depends upon the degree of material non-linearity and on the failure path. Recall that a circular hole in an infinite elastic sheet has an elastic stress concentration factor of 3  tensile damage ahead of the notch and (ii) shear damage orthogonal to the notch but originating at the notch tip.Consider for example, the case of unidirectional long fibre composites with a ceramic or polymeric matrix. Typically, these composites are stiff and strong along the fibre direction in comparison to the transverse direction. Natural woods also fall into this class. Depending on the notch geometry and material properties of the constituents , damage at the notch tip can occur by: (i) matrix cracking, (ii) splitting along the fibre direction, (iii) cracking transverse to fibre direction Zok , (iv) fiE as Young’s modulus of the composite, assumed isotropic. Then, dimensional analysis demands that the macroscopic net-section tensile strength a, scales asf taking the limiting values of a in relation to a material length scale The transition of notch behaviour from brittle to ductile for fibre-reinforced ceramic-matrix composites has been addressed in part by Suo et al. , He et aCottrell  and now Cottrell  for metaThe present study builds upon the recent study of Noselli et al.  wherein tensile band behaves as a bridged mode I crack. The shear traction on the tensile band vanishes due to symmetry in both geometry and loading. The tensile band is idealized as rigid, perfectly plastic and of strength shear band is assumed to behave as a bridged mode II crack. It has a strength Consider a mode I crack in an orthotropic elastic solid, with a tensile band directly ahead of the crack tip and two shear bands orthogonal to the cracking plane, as shown in Fig. mentclass2pt{minimmentclass2pt{minimboth the interface toughness ratio Recall that the purely energetic criterion Eq. , as propi et al.  predictsmentclass2pt{minima et al.  considermentclass2pt{minima and minor axis 2b. Two notch geometries are considered in detail in the present study: a sharp crack  a penetration mode of failure (when the tensile zone fails first) and (ii) a kinking mode of failure (when the shear zone fails first). The dominant mode depends upon (i) the relative strength and toughness of the competing zones  calculations, an initial stiffness Recall that the elastic constitutive relation of a general anisotropic solid has the following vector-matrix Voigt form in the Cartesian frame entclass1pt{minimaAdditional simplification can be achieved as follows. Suo et al.  has showP is the sum of the reaction forces per unit thickness on the top edge. The relevant notch tip displacements are indicated in the inset of Fig. The FE model of the problem, as defined in Fig. a / H and a / W. Predictions are shown in Fig. It is instructive to consider the response of a centre-cracked panel with tensile and shear cohesive zones for the case Three distinct behaviours emerge, depending on the assumed value of Now consider the choice a et al. . As \docAn isotropic solid with An intermediate orthotropic solid with A strongly orthotropic solid with First, note from Fig. For completeness, the sensitivity of the magnitude of Recall from the Introduction that the mode of fracture and consequently the corresponding net-section tensile strength of the notched panel are governed by the attainment of a critical value of a. We know from Tankasala et al. ((i)penetration mode is active when the tensile band fails first: E is defined as The entclass1pt{minima(ii)kinking mode occurs when the shear band fails first: The entclass1pt{minimaThe non-dimensional functions First consider the case of a sharp crack of semi-length a et al.  that a tK-field exists in the vicinity of the crack tip. In this regime, the failure strength penetration fracture toughnessK-calibration factor Y(a / W) on the right side of , see Fig. Now consider the alternative failure mechanism of kinking, see Fig. a exceeds that for a sharp crack of semi-length a. However, at small It is instructive to compare the notch tensile strength for a sharp crack and for a circular hole for various degrees of orthotropy, see Fig. Second, consider Now consider failure by kinking, with The dominant failure mechanism of penetration versus kinking is dictated by the lower value of fracture strength associated with each mode, recall  and 10)10). UponScaling for the toughness axis: For plane elasticity problems, Suo et al. The rescaling functions  and 20)20) for \(ii)For the case of a circular hole, the rescaling functions (iii)When (iv)The threshold value of (v)Penetration is encouraged by a greater level of orthotropy for both the hole and the crack. This behaviour is consistent with the previous findings of Tankasala et al.  for a seo et al.  showed to et al.  for craco et al. ) was geng et al. . A regreg et al.  for \doca et al.  for the There is a growing literature on the competition between cracking modes in both synthetic composites, and natural composites such as woods. We proceed to compare the predictions of the current study with the observations from the literature, first, for ceramic composites, and then, for natural hardwoods.Ceramic–ceramic, long fibre composites such as SiC–SiC and oxide–oxide composites are beginning to enjoy use in the hot section of aerospace gas turbine engines due to their high temperature capability and low density compared to nickel-based superalloys. The high toughness of these materials is achieved through microstructural design such that crack kinking along the fibre direction occurs instead of penetration of the fibres.Consider, for example, the case of a plain weave of SiC fibres in a SiC matrix. This composite is approximately isotropic in in-plane elastic properties. Typically, the strength and toughness ratios relevant to the present study are nd Lamon . Now pland Lamon . Evans and Lamon  also obsSecond, consider a composite such that  and Zok . Again, g et al. ; this isThird, consider the highly anisotropic case of a unidirectional carbon fibre in epoxy lamina such that mentclass2pt{minimNatural woods are cellular solids with hexagonal prismatic cell walls with the relative density In this section, we develop case studies for various natural hardwoods containing a finite pre-crack to assess the competition between penetration (across the grain) and kinking . Assume that a pre-crack is aligned with the radial direction of the wood nd Ashby ; these aentclass1pt{minimaIn order to deploy the failure mechanism map of Fig. low from  as21\doco et al.  as24\doco et al.  has the The interface toughness ratio ows from  and 25)\documenty et al. . In doin, recall . Upon pl, recall .simultaneous development of one or more inelastic damage zones at the crack tip. In the present study, we have investigated the macroscopic tensile strength of a panel with a centre crack or circular hole associated with the simultaneous activation of tensile and shear damage zones. The simple approach adopted herein of two simultaneous damage zones is able to account for several experimentally observed phenomena. These include: (i) the hole size effect, (ii) sensitivity to notch acuity, and (iii) the role of elastic orthotropy. The role of the relative penetration to kinking strength and relative toughness in promoting failure via extension by penetration versus splitting is quantified in failure mechanism maps for a sharp crack and a circular hole.Many synthetic and natural composites are orthotropic and contain interfaces of low strength and low toughness such that damage zones at the notch tip are on the order of the notch size , for the full range of orthotropy studied.With increasing orthotropy, the penetration mode is promoted for any given flaw size (crack or hole).Commonly, the strength ratio The cohesive zone model gives insight into the observed failure modes of notched fibre composites and natural hardwoods. However, we emphasize that the theory holds for a much broader range of orthotropic solids such as drawn polymers and anisotropic ceramic crystals.We also emphasize that the scope of the present study is limited to the case when the kink band deforms and fails in shear. The nature of the cohesive zone will depend upon the micromechanism of failure in any material system. The competition between kinking and penetration will be affected by the choice of the cohesive zone law for the kink bands, and this could be explored in a future study.The main findings of the present study are as follows.


Impalpable breast cancers require precise pre‐operative lesion localisation to minimise re‐excision rates. Conventional techniques include hookwire insertion using stereotactic guidance. Newer techniques include the use of tomosynthesis guidance and the use of iodine‐125 seeds. This study compares the accuracy of lesion localisation with hookwire or seed insertion using prone stereotactic or upright tomosynthesis guidance.This registered quality improvement activity did not require formal ethics approval. The post‐localisation images for 116 lesions were reviewed. The distance from the lesion or breast biopsy marker to the hookwire or seed was measured on post‐insertion mammograms. The relative placement accuracy of hookwire or seed using prone stereotactic or upright tomosynthesis guidance was compared. A lesion to seed or wire distance > 10 mm was considered technically unsatisfactory.94.8% of the seeds and wires inserted via prone stereotactic guidance were accurately placed, compared with 89.6% of those inserted via upright tomosynthesis. There were twice as many technically unsatisfactory insertions under upright tomosynthesis guidance. The majority of the unsatisfactory insertions using upright tomosynthesis occurred when the lesion was at or below the level of the nipple and the insertion was performed craniocaudally.The degree of accuracy of pre‐operative localisation of impalpable breast lesions is significantly higher with the use of prone stereotactic rather than upright tomosynthesis guidance. This was most evident with the placement of I‐125 seeds, and in cases where the target lesion was located below the level of the nipple. Mammographic screening programmes, combined with improved imaging techniques, have resulted in the identification of a large number of impalpable breast lesions, which require localisation under imaging guidance for successful surgical excision.Historically, the most common method for pre‐operative localisation of impalpable lesions has been via the insertion of a hookwire under ultrasound or mammographic guidance.European guidelines indicate a minimum 90% of localising wires should be within 10 mm of the lesion in any plane,At our institution, pre‐operative localisation, using either a hookwire or a radioactive seed, is performed preferentially under ultrasound guidance if the lesion or accurately placed biopsy marker clip is sonographically visible. Mammographic guidance, using either conventional stereotactic or tomosynthesis control, is undertaken if the lesion/clip cannot be sonographically visualised. Pending discussion with the surgeon, a distance between the hookwire or radioactive seed of greater than 10 mm, is usually considered unacceptable and prompts insertion of a further hookwire.Procedures performed using tomosynthesis guidance are being increasingly used in clinical practice as they are quicker and easier to perform than those using conventional grid or stereotactic guidance.The purpose of this study was to compare the use of conventional prone stereotactic guidance with upright tomosynthesis guidance for pre‐operative lesion localisation, to determine whether there is any difference in the accuracy of these two methods.n = 130) who underwent mammographically guided pre‐operative localisation at our institution between 1 July 2015 and 31 December 2017. Ten patients who required bracketing wires or seeds, three who had no visible residual lesion where the position of the biopsy marker was deemed unacceptable and one for whom post‐wire mammograms were unavailable on the PACS system were excluded, leaving 116 patients in the study.This retrospective review was registered as a clinical audit and quality improvement activity , not requiring ethics approval. Data were gathered from all patients   randomised controlled trial (ACTRN12613000655741) in which participants were randomised to undergo hookwire localisation or ROLLIS. For the remaining patients, the choice between the use of wire or seed was dependent on the availability of localisation booking slots in relation to the date of surgery. Patients who underwent hookwire localisation had their procedure on the day of surgery, while ROLLIS patients had their seed inserted up to eight days prior to surgery.For stereotactic guided wire or seed localisation, the patient was positioned on a prone table  according to the approach determined by the performing radiologist. A scout image was obtained to confirm the position of the target lesion, and then, a stereotactic pair was taken for targeting purposes. If using a wire, the needle was advanced 10–15 mm beyond the target, to adjust for the position of the thickened part of the wire, while the needle containing the seed was placed at the centre of the lesion. Once the needle was in position, a second stereotactic pair was taken to confirm position prior to deployment.For tomosynthesis‐guided wire or seed localisation, a Selenia Dimensions system (Hologic) was used. The patient was positioned on a dedicated biopsy chair, either seated or in a lateral decubitus position. The approach taken for localisation, either CC or lateral, was at the discretion of the performing radiologist.Following the localisation procedure, all patients underwent two‐view mammography to document the position of the wire or seed in relation to the marker clip. Positioning was considered acceptable if the seed or thickened part of the wire was within 10 mm of the centre of the lesion or marker clip.The accuracy of placement of the wire or seed in relation to the lesion was assessed as per Taylor et al.Lesion localisation was categorised as follows: SD 0–1 mm: optimal Figs.  and 3, SIn cases in which the position of the seed or wire was deemed suboptimal by the performing radiologist, a second localisation using a wire was undertaken Fig. .An intra‐operative specimen X‐ray was taken in all cases to verify the removal of the target lesion and the seed or hookwire.Patients were aged between 29 and 79 years, with an average of 57.9 years. The number undergoing prone stereotactic and upright tomosynthesis‐guided localisation and the localisation device used are listed in Table Of 58 localisations performed using prone stereotactic guidance, a hookwire was used in 46 and a radioactive seed in 12. Optimal placement of the wire or seed was achieved in 33 of the 58 cases, with positioning considered excellent or satisfactory in further 22 cases, giving an acceptable positioning rate of 94.8% for conventional stereotactic guidance.For the 58 patients in the upright tomosynthesis group, 37 cases were localised using a hookwire and 21 cases were localised using a radioactive seed. Optimal positioning was achieved in 17 of these cases, and excellent or satisfactory positioning in a further 35 cases, giving an acceptable positioning rate of 89.6%.U‐test was used to test the level of significance of this difference. The resulting P‐value of 0.0005 is highly statistically significant.The mean SD value for localisations performed using upright tomosynthesis (4.92) was found to be larger than for those undertaken using prone stereotactic guidance (2.26). In view of the non‐parametric nature of the data, the Mann–Whitney There were three technically unsatisfactory localisations carried out under prone stereotactic guidance. In the case involving a hookwire, the thickened part of the wire was measured to be 15.1 mm deep to the lesion. A second wire was not deemed necessary, and the lesion was successfully excised. In the first seed case, the seed was located 12.7 mm short of the lesion. The radiologist inserted a wire from the same direction, which was placed 2.2 mm from the lesion. In the other case, the seed was 17.6 mm deep to the lesion. A further wire was inserted from the same direction lying 1.8 mm from the lesion.There were six technically unsatisfactory localisations under upright tomosynthesis guidance, four hookwires and two seeds. For the unsatisfactory wire localisations, in the first case the wire was measured to be 10.8 mm short of the lesion and a second wire was accurately deployed also using tomosynthesis. In the second case, the initial wire, inserted using a craniocaudal approach, was extremely short of the lesion, at a distance of 32.5 mm. A second wire, inserted under stereotactic guidance and using the opposite approach of caudal cranial, was successfully placed at a distance of 4.3 mm from the lesion. The third case involved the wire being 19 mm deep to the lesion. After discussion with the surgeon, insertion of a further wire was not considered necessarily determined it was not necessary due to the position of the lesion and the planned surgery. In the fourth case, the wire was 13.7 mm short of the lesion, with the hook of the wire rather than the thickened portion adjacent to the targeted microcalcifications, and the surgeon did not require a second wire to be inserted.In the first case of a failed seed, the seed was measured to be 25.1 mm short, and a second wire was inserted under stereotactic guidance. In the final case, the seed was deployed 20.0 mm lateral to the target lesion of the residual calcification and 12.0 mm lateral to the breast biopsy marker deployed at the time of the biopsy in the CC view. On the lateral view, the seed was seen to be 5.0 mm inferior to the residual calcification Fig. . This coIn five of the six cases of unsatisfactory pre‐operative localisation undertaken using upright tomosynthesis guidance, the lesion was noted to lie at or below the level of the nipple on the pre‐procedural lateral mammogram.Tomosynthesis‐guided procedures, including pre‐operative lesion localisation, have been shown to be quicker and easier to perform than those using stereotactic methods,The 2006 European guidelines for pre‐operative localisation of impalpable breast lesions using hookwires state that 95% of wires should lie within 10 mm of the lesion.Mucci et al.Prone tables with tomosynthesis capabilities are now available to purchase and may alleviate some of the problems we discovered with performing pre‐operative lesion localisations upright under tomosynthesis guidance. However, it should be noted that the cost difference between an add‐on biopsy device for an existing upright mammography machine and a stand‐alone prone biopsy table is significant, and some centres may not have either the funds or the space available to choose between the two options.In five of the six technically unsatisfactory cases of upright tomosynthesis‐guided localisation, the lesion was seen to be at or below the level of the nipple on the lateral mammogram, with the approach chosen for localisation being craniocaudal from above. This highlights the importance of choosing the appropriate approach for insertion of the localising device, with the aim being to traverse the shortest distance through the breast wherever possible. This helps to minimise the movement of the wire or seed due to the accordion effect when the compression is released from the breast.An interesting observation from this study is that seeds placed using prone stereotactic guidance were either in the lesion itself or greater than 10 mm away. Conversely, seeds placed using upright tomosynthesis guidance were found to have a spread of acceptable locations, with most being situated within 5 mm of the target lesion, not actually in the lesion itself.A possible explanation for this finding is the difference in the method used to determine the depth with each technique. With stereotactic guidance, accurate calculation of the depth of the lesion is dependent upon the radiologist selecting the same target on the pair of localisation images. The tip of the needle is then placed at this depth, and the seed is deployed into the centre of the lesion. If the same portion of the lesion has not been accurately selected on targeting, the depth measurement will be inaccurate, and the seed deployed outside the lesion. With tomosynthesis guidance, the deployment depth is judged by subtracting the lesion depth from the overall number of tomo slices. However, with tomosynthesis an additional five slices are reconstructed on the compression paddle side automatically in order to ensure the entire breast is imaged.Schrading et al.Five of the nine technically unacceptable localisations were performed by consultants, and the other four were performed by registrars or fellows. It is standard practice in our facility for the consultant to perform the localisation if it is likely to be complicated or difficult. Each of our five consultants had one failed localisation; however, three of the four failures performed by non‐consultants were performed by the same breast fellow. This suggests a possible problem with the technique used by this operator, with the introduction of bias into the results. If these cases are removed, the rates for acceptable localisation using stereotactic and tomosynthesis guidance are 96.5% and 93%, respectively.The limitations of our study include a small sample size and that we are reporting the experience of a single institution. Although retrospective in nature, the inclusion of a consecutive series of patients minimises potential selection bias.The degree of accuracy of pre‐operative localisation of impalpable breast lesions is higher with the use of prone stereotactic rather than upright tomosynthesis guidance. This was most evident with the placement of I‐125 seeds, and in cases where the target lesion was located below the level of the nipple.No authors have any disclosures or conflicts of interest.


We present a general method of designing optical devices based on optical conformal mapping and rescaling the optical path along a given bunch of rays. It provides devices with the same functionality as those based purely on conformal mapping, but enables to manipulate the refractive index to a great extent—for instance, eliminate superluminal regions of space as well as reduce the refractive index in other regions significantly. The method is illustrated in two examples, a waveguide coupler and a plasmonic bump cloak, and numerical simulations confirm its functionality. Although the method was used initially to design an invisibility cloak to create a concealment area for hiding an object, researchers soon began to adapt it for various applications like carpet cloaking 8, polarization splitting and transforming13, directivity enhancement17, beam expanding19, and waveguide coupling24.Transformation optics (TO) proposed concurrently by Pendry and LeonhardtDespite showing promising degrees of freedom to alter the electromagnetic wave propagation, the material derived by this method is in general anisotropic and non-homogeneous and may require extreme values of permittivity and permeability. These properties essentially limit the application of TO for practical purposes.29. Such dielectric graded index medium can be fabricated by various methods including drilling sub-wavelength holes in a dielectric substrate31, dielectric layered shell deposition34, parallel waveguide plates height modification for mode index alteration37, graded photonic crystals38, electron-beam lithography42 and modification of the guiding silicon substrate height43.To simplify the material, a new branch of transformations was proposed. Conformal and quasi-conformal transformations solve the issue of anisotropy and magnetic medium requirements and eventually result in a non-homogeneous dielectric transformation medium2, waveguide bends28 and directivity enhancers28. Recently conformal transformation has been employed to compress the common circular generalized Maxwell fish-eye lens to an elliptical one44. However, in some cases, a conformal mapping does not exist and one selects quasi-conformal mapping instead to get a dielectric medium. The best-known application of quasi-conformal mapping is the carpet cloak which can also be used for guiding a plasmonic wave over a bump8. In this device, due to conformal module mismatch, a conformal map cannot be found. While conformal mapping leads to zero anisotropy by definition, the anisotropy introduced by quasi-conformal transformation can be made very small45.Conformal TO has been used for designing various electromagnetic elements like unidirectional cloak17 or beam compression21. In addition, employing TO often leads to less-than-unity index in a region of space where expansion occurs. There are three common approaches to deal with this problem. The first solution is to use resonant metamaterials, which, however, limits the bandwidth. The second one is to cut the region of space where the index is low or use unity index instead. This can affect the device performance, especially if that region interacts greatly with the wave15. The third solution is to multiply the whole index profile by a constant such that the index raises above unity. Being the most reasonable approach, matching of the device to the surrounding medium can be lost. When using quasi-conformal TO, even though the amount of anisotropy is minimized by this method, the minimum anisotropy value depends on how abruptly the mapping changes between different regions. For practical purposes, one neglects the small amount of anisotropy to have an isotropic device. It has been shown that this approximation can degrade the performance of the device. For instance, it can create an unwanted lateral shift to the light rays and render the device detectable in the case of carpet cloak46.The designs based on (quasi-) conformal TO can suffer from several drawbacks that are also common among non-conformal TO. Since any alteration to the coordinate lines at the boundaries is translated into changes of refractive index, using (quasi-)conformal TO leads in some cases to index mismatch between the device and the surrounding, meaning that power loss becomes unavoidable. This effect has been observed in devices where the boundary that interacts with the incoming or outgoing wave gets flattened, compressed or expanded to achieve directivity enhancement20. This is problematic if we need to have the desired medium at coupler’s ends. The impedance mismatch issue for a waveguide coupler has been investigated and addressed thoroughly in the literature24. Also, the coupler index at the conjunction with the waveguides will not be uniform, which makes the matching even more difficult21. Additionally, the design leads to less-than-unity index regions in general, which is not favorable. Note that multiplying the index by a constant is not practical here, since it changes the index at the boundaries as well, causing reflections. An interesting solution to manipulate the refractive index called artificial boundary conformal mapping (ABCM) has been proposed for the design of a waveguide crossing47. This method employs boundary modification and optimization to achieve better refractive index at the edges and also enhances the matching and uniformity of the refractive index at the input and output ports. Despite being elegant, using this method will not entirely eliminate the restriction imposed inherently by TO, e.g., the ratio of refractive index on two sides of a coupler being predefined by the compression ratio.Here, we aim to propose a design method to remedy the defects that two devices, namely waveguide coupler, and plasmonic bump cloak, suffer from. A waveguide coupler is a device that can couple modes between two waveguides with different cross-sections and filled with desired dielectric constants. If the coupler is designed purely by TO, it can only couple two waveguides with predefined dielectric constants, governed by the ratio of cross-sectionsHere, a conformal TO is employed to acquire the refractive index. This is done in such a way to have a uniform index at the input and output boundaries, which is essential for the next step. To override the predefined matched dielectric constants for the waveguides and mitigate the less-than-unity index, the optical path is rescaled to match the device to vacuum or other desired dielectrics. At the same time, the rescaling helps to lower the maximum index value. The second device—the plasmonic cloak—has so far suffered from both the presence of less-than-unity index regions, and the anisotropy neglecting defects, the latter being not so prominent. Rescaling the optical path in a similar way as in the case of the waveguide coupler, we propose a conformal design and resolve the less-than-unity index issue, making the device much more practical.u, v, w)] and physical space [with Cartesian coordinates ]. Virtual space is homogeneous (such as vacuum), where the light rays follow straight lines. Physical space is the result of applying a given mapping to the virtual space. The physical space relative permittivity and permeability tensors ε and μ are calculated from the ones of virtual space using the well-known formula1:J = ∂/∂ is the Jacobian matrix.One of the key ingredients of our method is transformation optics. TO relates the electromagnetic fields and the material between two spaces, namely virtual space [with Cartesian coordinates  is conformally mapped to the plane . This way, the transformation is effectively performed only in two dimensions where a plethora of conformal mappings exists based on complex analytic functions. If we define the complex variables W = u + iv and Z = x + iy, we can express the mapping from virtual to physical space in terms of an analytic function Z(W) with the inverse function W(Z). Equation . Importantly, when one restricts the waves to TE polarization , the required optical mapping can alternatively be achieved by a purely dielectric material (μ = 1) with refractive index29In a two-dimensional (2D) conformal scenario, the mapping between virtual and physical spaces is chosen such that Equation  then yieμxx, μyy and εzz, so changing the other components does not affect the wave in any way. One can, therefore, work with a purely dielectric material in the 2D conformal TO, which we will do in the following. Moreover, in many situations this is possible even for TM waves, in particular, if the refractive index does not change significantly on the scale of a wavelength. In such situations the wave has time to adjust to the slowly  changing medium, and polarization effects are not very important. This way, a purely dielectric material of Eq.  has zero first and second derivatives at x = 0 and x = a and also satisfies t(0) = a and t(a) = aC. The first and second derivatives are set to zero to let the curves CF and DE adapt to their straight-line continuations as smoothly as possible.Our virtual space is a rectangle filled with vacuum shown in Fig. 48. The conformal module (denoted here as M) for a rectangle is simply its aspect ratio. For the quadrilateral shape of physical space, M has to be calculated numerically by solving the following boundary-value problem for a function ϕ in physical space:N is the normal vector to the boundary. The conformal module M is then calculated from the following integral over physical space48:To find the conformal mapping between virtual and physical spaces, we use the fact that a transformation between two quadrilaterals can be conformal only if they share the same conformal moduleM = 1.5. This then sets the aspect ratio of the rectangular virtual space, and has already been taken into account when setting the dimensions of Fig. The calculation can be carried out using MATLAB or COMSOL Multiphysics PDE solvers. For our example of physical space in Fig. u and v28. The Laplace equation and the Dirichlet and Neumann boundary conditions for this case are defined by the following equations:The conformal mapping between the physical and virtual spaces, in this case, cannot be presented by a closed-form formula and should be calculated numerically by solving Laplace equation in physical space for variables The Neumann boundary condition ensures that the mapped virtual space lines maintain the orthogonality to the corresponding boundaries.C = 2 at the left and right boundaries, respectively. This holds true for all kinds of transformation-optical coupler designs20.The refractive index in physical space is then calculated by Eq.  and is sHaving established the conformal mapping between virtual and physical spaces, we now proceed to the second step to solve two problems at the same time: we will match the index on the right-hand side to unity, as well as remedy the less-than-unity index issue.u that corresponds to coordinate u of virtual space. And just as this coordinate u measures the optical path length along the ray in virtual space, the corresponding u measures the optical path length along the ray in physical space. The refractive index then corresponds to the rate at which the optical path changes along the ray:Consider a plane wave propagating in virtual space from left to right. The vertical and horizontal lines in virtual space of Fig. entclass1pt{minimaS(u)  and imagine that it is S rather than u that measures the optical path length along the ray in physical space. The contours of S are the same as those of u, so the phase-fronts are not modified by this change, and therefore rays are not modified either. However, since the rate at which the optical path is growing along the ray has changed, so must do the refractive index. The resulting new refractive index n′ will then beS(u), we can modify the refractive index such that it matches vacuum at the boundaries, and also shift its minimum value above unity to avoid superluminal propagation.Now comes the key step of our construction. We define an increasing function used Eq. . This con. Figure n as a function of u/a at the top boundary of physical space and on the y = 0 line. It is desirable to have unity refractive index on both the left and right sides so that the coupler is matched to the vacuum. We separate the index curve of the top boundary into two parts according to the sign of n − 1, and label the points u0 = −a/2 (left boundary), n = 1) and u2 = 5a/2 (right boundary). The index n is less than one on the segment u0u1  as a polynomial series of degree six. The scaling function value and the first and second derivatives are chosen to be continuous at point u1 for a smooth change of refractive index. Also, the first derivative value at u2 equals n′ unity there, and the second derivative is taken to be zero for smoothness. We apply an optimization for the derivation of polynomial coefficients based on the the above constraints. The index profile n along the line y = 0 is considered during the optimization to prevent the index n′ from dropping below unity inside the coupler domain. Figure S(u) and its first derivative dS/du.As for the interval u ∈ u, u2, we n′ in physical space based on Eq.  = h0cos2 with the height h0 = 0.4a and length of l = 3a. The virtual space lines of constant u and v are illustrated together with their corresponding mapped lines in physical space. Compared to the previous articles in the literature8, the bump here is two times taller and 1.5 times wider relative to the cloak size.Now we turn our attention to the plasmonic bump cloak. For such a device, virtual space is a rectangle and the physical space is a rectangle with a missing bump at the bottom side, see Fig. M for physical space. Having M, we can set the width of virtual space such that the aspect ratio of virtual space equals M. Using Eqs.  and its first derivative dS/du. Using Eq.  of the device is calculated, and plotted in Fig. S are depicted in Fig. The refractive index for this design is shown in Fig. sing Eq. , the refa = 1 m, and microwave operating frequency of 2 GHz. The device is excited with TE1 and TM1 modes, respectively, using a port on the right side. A perfect electrical conductor (PEC) is placed at the top and bottom boundaries to set the proper boundary conditions for TE and TM mode propagation, and the scattering boundary condition is used on the left boundary. The normalized real part and magnitude of electric fields for the TE1 mode are depicted in Fig. 1 mode are illustrated in Fig. We have tested the performance of the coupler and the plasmonic bump cloak designed in the previous sections by multiple numerical simulations in COMSOL. First, the functionality of the waveguide coupler in Fig. Ez for three cases with loss tangent values of tanδ = 0 (lossless), tanδ = 0.001, and tanδ = 0.005. As expected, loss introduces amplitude reduction to the structure. Note that the normalization is done with respect to the lossless case.To include the effect of loss, we have simulated the waveguide coupler in Fig. z-axis in then launched at the narrower side of the coupler. The reduced simulation frequency of 1 GHz is chosen to handle the numerical costs of the simulation. The real part and magnitude of the electric field for this case are represented in Fig. The simulation of a rotationally-symmetric version of the coupler is carried out next. The medium is revolved around its optical axis and the PEC boundaries are removed. A Gaussian wave with polarization along v in Fig. The last simulation for the case of waveguide coupler includes the ray-tracing simulation. This simulation verifies the high-frequency behavior of the device and is based on geometrical optics approximation. As mentioned before, the ray trajectories before the scaling are the mapped lines of constant a = 1 μm is selected to set the device size. An out-of-plane plasmonic magnetic field is launched from the left boundary of the device in Fig. 49. The TM plasmonic wave is excited at the left boundary50.The plasmonic bump cloak is the next simulation case. Parameter Figure Next, the rescaled plasmonic bump cloak in Fig. In conclusion, we have proposed a method of designing optical devices that have the same functionality as previously designed devices, but with much more moderate ranges of refractive index. The key ingredients of the method are the conformal mapping between virtual and physical spaces, and rescaling the optical path along the rays. The method enables to eliminate superluminal regions of space as well as lower the refractive index in other regions to achievable values. It provides a lot of freedom that can be employed to optimize the refractive index profile to match the desired purpose. Numerical simulations confirm our theoretical results very well and show that the devices proposed work very well even for different polarizations.Having said this, we must note that the presented design method is based on rescaling the index for one particular bunch of rays. These rays will be unaffected by the rescaling, but other rays, in principle, will be affected. Therefore our method is useful in situations when light is propagating only in a certain direction; it cannot be used to modify e.g. imaging systems that should focus light coming from different directions. However, we think that there are still many practical situations when the method will be useful—a particular example is a coupler that enables one to connect smoothly waveguides of different sizes without the necessity of high refractive index regions. We hope that our method will find other uses too, for example in designing horn antennas.


Computational radio frequency identification (CRFID) sensors are able to transfer potentially large amounts of data to the reader in the radio frequency range. However, the existing EPC C1G2 protocol is inefficient when there are abundant critical and emergency data to be transmitted and cannot adapt to changing energy-harvesting and channel conditions. In this paper, we propose a fast and reliable method for burst data transmission by fragmenting large data packets into blocks and we introduce a burst transmission mechanism to optimize the EPC C1G2 communication procedure for burst transmission when there are critical and emergency data to be transmitted. In addition, we utilize erasure codes to reduce Acknowledgement (ACK) delay and to improve system reliability. Our results show that our proposed scheme significantly outperforms the current fixed frame length approach and the dynamic frame length and charging time adaptation scheme (DFCA) and that the goodput is close to the theoretically optimal value under different energy-harvesting and channel conditions. Currently, miniature and ultra-low power sensors are widely used in urban and indoor areas. They are attached to common items and logistics to be used to track these objects . Such deIn this paper, we focus on mobile CRFID  and consWe propose a method for burst data transmission by fragmenting large data packets into blocks. Then, we dynamically adjust the frame length of every block through an online adjustment strategy at runtime by the feedback of the reader.We introduce a burst transmission mechanism. The core idea is to let a tag occupy all time slots for burst transmission when there are critical and emergency data to be transmitted, which reduces the idle time slot.We utilize erasure codes to reduce acknowledgement frame waiting delay and to avoid retransmission overhead, which improves system robustness and reliability.Under specific energy-harvesting and channel conditions, our proposed scheme is much better than the EPC C1G2 protocol fixed frame length approach and DFCA scheme, and the performance can converge to near optimal. When there are abundant critical and emergency data to be transmitted, our scheme can achieve fast and reliable bulk burst data transmission.In response to the above difficulties, the following are the chief contributions of our work:The rest of the paper is as follows: We discuss related work in As “smart” transponders, UHF-passive RFID tags enhanced with computational and sensing capabilities and CRFIDs are qualified to become fully fledged components of the Internet of Things (IoT). Recent advances in smart systems driven by IoT technologies have opened up great opportunities for the development of backscatter communications. To enable RFIDs to be accessible from/to communicate with any other networked devices in the Internet, Reference  made infIn view of the problems for backscatter communications described in The dynamic adjustment of frame length and erasure code technology has been proven by many literatures to effectively enhance transmission reliability and system goodput under adverse channel conditions. In Reference , they prIn this section, we investigate the fundamental factors underlying the poor performance of backscatter communication and describe the main challenges that need to be addressed to improve the backscatter system efficiency.As shown in Why is the energy available in each discharge cycle difficult to predict? First, if the energy-harvesting condition is too low, the efficiency of storing energy into the capacitor is low . TherefoTherefore, we need to adjust the length of the transmitted data according to the current energy environment. When the energy level is low, we need to shorten the frame length to adapt, but this usually reduces the goodput which is affected by the overhead of each transmission, including preamble, header, and hardware. In order to optimize goodput at the same time, it is important to transfer the data as much as possible given the available energy. Therefore, the problem faced by the node is that it needs to shorten its transmission frame under poor harvest conditions and to scale up to increase goodput when condition permits.The energy-harvesting rate has a significant impact on communication goodput, since higher harvesting rate means that more energy can be used for data transfer. Next, we will explain this phenomenon by looking at how capacitors buffer energy. The charging process of the capacitor can be described the formula of voltage variation across the capacitor:When the harvesting conditions are constant , the number of burst transmission data frames in the In this section, we present the design of our scheme by fragmenting large data packets into blocks. The major goal of our scheme is to achieve fast and reliable burst data transmission and goodput optimization for backscatter communications under dynamic harvesting and channel conditions. The main idea of our solution is to fragment large data packets into blocks and to adjust the frame length and coding redundancy according to the goodput feedback of the reader at runtime, meanwhile controlling the sleep time and the number of transmitted frames to improve the goodput.Our scheme works as follows: CRFID nodes periodically track the voltage of the capacitor and estimate To avoid frequent changes to the frame length, the adjustment parameter After one round of transmission, the reader counts the number of redundant frames Specifically, the CRFID node operates as follows:Step 1: If the CRFID receives Query or QueryRep command from the reader, an RN16 is returned.Step 2: If an ACK of the reader is received and the ACK contains the same 16-bit random number as the RN16, Step 3: The capacitance voltage is tracked. Step 4: Calculate the optimal number of burst transmission frames Step 5: After sleep At the same time, the reader operates as follows:Step 1: The reader sends a Query to request data and returns an ACK if it receives the RN16 of the CRFID. The ACK includes the same random number as in the RN16 and the frame length Step 2: The number of buffered data Step 3: If the data frame of the CRFID is received, let Step 4: Calculate the current goodput Step 5: The reader sends a QueryRep command to request data and returns an ACK frame if receiving the RN16 frame of the CRFID. The ACK contains the same random number as in the RN16 and the transmitting information such as the frame length and the number of redundant frames calculated in Step 4. Then, run Step 2.Pseudo codes of the CRFID and the reader operation procedure are shown in Algorithm 1 and Algorithm 2, respectively.Algorithm 1 The computational radio frequency identification (CRFID) operation procedure.while the buffer id not empty do 2: if receive Query or QueryRep command then 3:  return RN16 4: end 5: if receive the ACK containing the same RN16 then 6:   extract  7: end 8: repeat 9:  track the capacitance voltage and obtain 10:  compute the optimal number of transmission frames 11: until all 12: end 1: Algorithm 2 The reader operation procedure. 2: send the Query or QueryRep command 3: if receive the RN16 from CRFID then 4:   return an ACK with the same RN16 and  5: end 6: if receive the frame from CRFID 7:   if the frame is correct then 8:    9:   else10:  end11:  ifthen12:  compute 13:   ifthen14:     15:   else ifthen16:     17:  end18: end19: compute the number of redundant frames 20: end 1: Initialize The data frame format used is similar to the data frame format of the EPC C1G2 protocol, as shown in The experimental platform is shown in Erasure coding implementations: It is obvious that the redundancy introduced using erasure codes increases energy consumption as its number increases. However, traditional reliability improvement methods, such as data multiplexing or Automatic Repeat Requests (ARQ) [ts (ARQ) , are toor to evaluate these methods. Our results clearly show that erasure coding has the same or less overhead compared to traditional data replication and on-demand retransmission and can provide higher reliability in our scenario. When erasure coding is used, as the code overhead and data rate increase, we observe that the recovery rate increases steadily compared to the other two methods of recovering data. In particular, erasure codes based on Reed–Solomon clearly outperform simple data replication and ARQ, as shown in As a benchmark experiment, we simulated three cases: the normal case, that is, the data packet without encoding; the full copy, that is, the data is sent with an exact copy; and the on-demand retransmission, that is, if the ACK command that the data has been successfully sent is not received, the data will be resent. We use simulation and energy measurement on real hardware platforms to describe the overhead of redundant packets by setting different coding rates Probing energy state: As mentioned earlier, the ADC costs too much energy and should be avoided when tracking the maximum energy-harvesting rate. In our work, instead of measuring the voltage on nodes, we use existing low watermark threshold detectors that already exist on such nodes. This type of detector is very common on passive sensor platforms to control the state of the tag, when it should sleep to avoid interruptions, and when it should wake up to continue operation. Therefore, our algorithm is interrupted when the voltage exceeds the threshold or when the voltage is below the threshold, and this information is used as a one-bit proxy for the actual voltage. The voltage threshold is chosen to be 2 V, which is slightly higher than the minimum voltage of 1.8 V required to operate the microcontroller. This information is entered into the sleep time tracker, which decides how long to wait after exceeding the threshold before starting the transmission. Compared with ADC, this method saves about 100× in energy.To model the changing energy-harvesting and channel conditions, we evaluate the performance with different values of We consider that the CRFID has 7400-bit data in buffer that need to be sent to the reader. Our proposed scheme is simulated to obtain the goodput performance and the corresponding frame length and number of redundant frames, which are compared with the theoretical optimal solution obtained from Reference . The theIn this paper, we achieve fast and reliable bulk burst data transmission by optimizing goodput based on burst transmission when there are critical and emergency data to be transmitted. First, we optimize the EPC C1G2 protocol by introducing burst transmission mechanism and erasure codes and control the optimal number of transmission frames and sleep time of CRFID according to the current energy-harvesting and channel conditions. Then, we fragment large data packets into blocks and design an online adjustment strategy to adjust the frame length and coding redundancy dynamically by the feedback of the reader at runtime. Our results show that our proposed scheme significantly outperforms the current fixed frame length approach and the DFCA scheme, and the goodput is close to the theoretically optimal value under different energy-harvesting and channel conditions.Our proposed scheme in this paper enables passive sensing communication to adapt to dynamic energy-harvesting and channel conditions and to facilitate their application in the field of mobile sensing and pervasive computing. Future research work will consider the issue of data frame retransmission and how to deal with collisions and to ensure data transmission priority to further improve system performance in multi-tag scenario. In addition, this method can be combined with ambient back-scattering technology to make use of all the energy available in the environment such as temperature difference energy supply, mechanical vibration energy, etc. to improve the intelligence of smart devices and to establish a true smart system. That is to say, in the near future, data communication will not be restricted by the outside world, and it can flexibly realize the conversion between different protocols, thereby achieving barrier-free communication between various IoT devices.