An ultra-wideband linear polarization converter based on a reflecting metasurface is presented. The polarizer is composed by a periodic arrangement of miniaturized metallic elements printed on a grounded dielectric substrate. In order to achieve broadband polarization converting properties, the metasurface is optimized by employing a genetic algorithm (GA) which imposes the minimization of the amplitude of the co-polar reflection coefficient over a wide frequency band. The enhanced angular stability of the polarization converter is due to the miniaturized unit cell which is obtained by imposing the maximum periodicity of the metasurface in the GA optimization process. The pixelated polarization converter obtained by the GA exhibits a relative bandwidth of 102% working from 8.12 GHz to 25.16 GHz. The analysis of the surface current distribution of the metasurface led to a methodology for refining the optimized GA solution based on the sequential removal of pixels of the unit cell on which surface currents are not excited. The relative bandwidth of the refined polarizer is extended up to 117.8% with a unit cell periodicity of 0.46 mm, corresponding to λ/20 at the maximum operating frequency. The performance of the proposed ultra-wideband polarization metasurface has been confirmed through full-wave simulations and measurements. They are also used to perform remote environmental monitoring2, or to realize microwave devices such as circulators and isolators2. Moreover, polarization converters are widely applied in optical instrumentation. Typically, they are realized with optical gratings and dichroic crystals3. A very effective way to manipulate several basic properties of the electromagnetic (EM) wave  is the use of metasurfaces. The latter are artificial surfaces designed to obtain particular electromagnetic or optical properties that cannot be found in nature. A metasurface is a single or multi-layer structure where periodic metallic elements with dimensions smaller than the operational wavelength are included. In this way, near the interface, anomalous reflections and refractions occur, which can be explained by the generalized Snell’s Law4. The scattering properties of the composite material can be manipulated through the choice of substrate material, the size and the geometry of the surface. Metasurfaces have been considered as an effective solution for the design of different devices and applications such as lens8, broadband low-profile antennas9, invisible cloak14, polarizers17, and super-resolution imaging18. In particular, metamaterials with negative permittivity and with negative refracting index have been employed for the design of novel superlenses20. Recently, polarization converters based on metasurfaces have been intensively studied because of their capability of manipulating the polarization of EM waves. Several examples of linear to linear22, linear to circular25 and multifunctional27 polarization converters have been proposed in the literature. In addition, different types of polarization converters based on anisotropic and chiral metasurface have been proposed30. However, these polarization converters suffer from a narrow bandwidth which could be a limitation in their practical application. Many studies have been devoted to techniques for the expansion of the bandwidth. A possible approach is to design reconfigurable metasurfaces which can be controlled with p-i-n31 or varactor32 diodes. An alternative approach is to stack periodic frequency selective surfaces in a multilayer configuration33, in which complementary circular symmetric split-ring resonators are employed in a multilayer structure. Another approach is to generate multiple plasmon resonances34 by electric and magnetic resonances. In this way, it is possible to achieve the bandwidth expansion of crosspolarization reflection on the metasurface35. Recently, metasurfaces based on galinstan27 and graphene36 have been proposed for the design of broadband polarization converters. In ref.21, a double V-shaped metasurface operating from 12.4 GHz to 27.96 GHz is printed on a FR4 substrate with a thickness of 1.6 mm. In this case, the expansion of the bandwidth is based on the generation of multiple EM resonances. This polarization converter can change a linearly polarized wave over a relative bandwidth of 77.1%. An ultra-wideband reflective linear cross-polarization converter based on anisotropic metasurface has been presented by Jia-Liang Wu et al.37. The unit cell is composed by a square-shaped resonator with diagonal and metallic ground sheet separated by dielectric substrate with a thickness of 3 mm. This polarization converter exhibits a relative bandwidth of 91.2% working from 7.57 GHz to 20.46 GHz.The manipulation of the electromagnetic wave polarization state is a desirable characteristic in several electromagnetic applications. Common applications are related to microwave communications and antennas40 and leaky wave antennas41. In addition, this kind of devices are widely used for military applications in the microwave regime for the design of stealth surfaces. In these cases, wideband polarization converters (as the polarizer presented in the paper) are needed. In particular, the reduction of the Radar Cross Section (RCS) of targets relies on wideband reflecting polarization rotators45 thus exploiting the scattering diffusion phenomenon44. Polarization converting surfaces are also a fundamental building block in several microwave and optical devices48.The presence of the ground plane in reflective metasurfaces allows for the reflection of almost all electromagnetic energy impinging on the surface. For this reason, this kind of metasurface is employed for applications in which a high efficiency level is required. Polarization converters are employed in several applications in microwave regime. Specifically, in the field of microwave antennas, polarization converters are particularly desirable for the design of reflectarray antennas50. The relative bandwidth has been extended up to 117.8% with a unit cell periodicity of 0.46 mm, that is λ/20 at the maximum operating frequency. In addition, the miniaturized size of the unit cell allows to obtain robust polarization conversion performance with respect to the elevation angle of the impinging wave.In this paper, an ultra-wideband polarization converter based on a single-layer reflective metasurface is presented. This device is able to perform a linear-to-linear polarization conversion for an orthogonal incidence of the EM wave. The aspects that most distinguishes this polarization converter from all others proposed in literature are its operative bandwidth and the compact size of the periodic elements. In fact, this polarization converter is able to work from 8.12 GHz to 25.16 GHz with a relative bandwidth of 102%. In addition, in order to further improve the performance of the polarization converter, the topology of the metasurface has been modified with a refinement procedureThe paper is organized as follows. Section II shows the theoretical background of the scattering from a grounded metasurface and the working principle of the proposed polarization converter. In section III, the design and the refinement of the unit cell topology is addressed and simulated results are shown. Finally, in section IV the measures of the fabricated prototype are shown.4. By properly designing the surface, it is possible to exploit the anomalous interaction of the EM wave with the interface in order to achieve a non-conventional behaviour of the reflecting surface. In particular, for a reflective linear polarization converter, the metasurface needs to reflect the two components of the impinging electric field with the same amplitude and with a phase difference equal to 180°. In Fig. The interaction mechanism of an EM wave impinging on an interface when the interface is formed by a periodic arrangement of metallic elements are regulated by generalized Snell’s lawy-polarized wave impinges orthogonally on the surface of the grounded periodic surface and the electric reflected field is directed along the x-axis. The working principle of the polarization converter can be explained by considering the additional coordinates system originated from the two perpendicular symmetric axes u and v, which are 45° rotated with respect to the y-axis shown in Fig. y-polarized field into u and v components and to write the incident electric field as follows.Ru and Rv in the uv-coordinates system:x-axis only if the following conditions are met:Ru ≠ Rv, the metasurface needs to be anisotropic. In addition, since the magnitude of Ru and Rv must be equal to 1.0, a dielectric substrate with a very low tangent loss is required.In this case, a The most challenging task of a wide-band polarization converter design is to find the proper combination of thickness, number of layers, substrate materials and unit cell topology of the metasurface matching the conditions (3) and (4) over a wide frequency range.y-polarized wave which travels from the medium 1 to the medium 2 is indicated as Ryy12 (Tyy12). Similarly, the cross-polar reflection (transmission) coefficient for a y-polarized wave which travels from the medium 1 to the medium 2 is Ryy12 (Tyy12). In this analysis, the free space is referred as medium 1 and the dielectric substrate is referred as medium 2. The propagation constant inside the dielectric substrate with electric permittivity ε is k0 = free space propagation constant and d is the substrate thickness). When the EM wave impinges on the interface, it is partially reflected and partially transmitted. The transmitted wave travels in the dielectric substrate and is reflected by the ground plane, which is characterized by a reflection coefficient equal to −1. When the two waves impinge on the interface, they have experienced a phase rotation equal to y-polarized impinging field can be immediately reflected in cross-polarization by the metasurface but some of the signal penetrates the first layer and start bouncing between the metasurfaces and the ground plane. The part of the signal which penetrates the first interface can have two polarization states (x and y polarization). This signal can emerge in cross-polarization if suitable combinations of reflections verifies. For instance, the signal can penetrate the first layer remaining y-polarized, but it can be converted as it impinges the metasurface from the bottom. Equivalently, the transmitted signal can penetrate the first interface being converted in x-polarization and emerge after one bounce in the ground plane. It can be demonstrated that an infinite set of these contributions exist and can sum up coherently or not. Each of these terms can be written in the form of geometric series. Beside the rigorous analytical analysis, it is worth underlining that the total reflection of the composite surface can be written as a summation of terms of different orders. The number of terms needed to correctly estimate the backscattered field depends on the reflectivity of the surface. In the case of metasurface reported in Fig. Ryy12) up to fourth order In order to provide a more comprehensive explanation of the working principle of a reflective metasurface, the effect of the ground plane will be analysed. Indeed, it is worth highlighting that the anisotropic behaviour of the reflecting surface does not depends only on the unit cell topology, but it is affected by the interaction between the periodic surface and the ground plane. In fact, treating the periodic arrangement of metallic elements as a partially reflecting surface, the working principle of a reflecting metasurface can be explained with the interference theory. In light of this, the backscatterd field can be interpreted as a sum of infinite contributions of electric field transmitted through the metasurface interface and reflected by the ground plane. In the following analysis, the co-polar reflection (transmission) coefficient for a 51 and with the interference theory approach have been respectively reported in Fig. The co-polar and cross-polar reflection coefficient of the optimized unit cell calculated by using an in-house developed Periodic Method of Moments code (PMM)Ryy21 and Ryx21, as a function of the frequency for the optimized unit cell calculated with the PMM and with the interference theory approach extended up to the fourth order, are shown in Fig. The From both figures, it is evident that increasing the order of the interference theory analysis, the agreement with the PMM calculated curve increases. In particular, considering only the zero-th order term (metasurface without ground plane), the deep nulls of the co-polar reflection are not present. This analysis highlights as the desired operation of the polarization converter is obtained when the interaction between the metasurface and ground plane is controlled.54.The aim of this work is to design a linear polarization converter based on a reflective metasurface. As shown in the previous section, the design of the device is based on the choice of the metasurface parameters, which are able to satisfy the conditions (3) and (4) over a wide frequency band. A possibility to design these kind of devices is to use canonical shapes. However, in the latter case is difficult to enlarge the bandwidth above 90% and preserve at the same time a compact unit cell size. Indeed, if the unit cell is not sufficiently compact, grating lobes can emerge at the highest operation frequency of the wideband polarization converter. For this reason, a numerical code based on a genetic algorithm (GA) has been employed for the design of the metasurfacenf = 21 frequency points:The basic periodic cell is discretised in a 16 × 16 binary pixel matrix in which 1 or 0 represents whether a pixel is metallized or not. The GA selects the topology of the pixel matrix on the basis of the objective function defined as consists in minimizing the reflection coefficient of the fundamental Floquet harmonic over a number of frequency points within the desired frequency band. The goodness of the solution is evaluated by using the following fitness function calculated on In order to ensure a fast convergence of the algorithm, the initial population is obtained from the best chromosomes of several trial populations. The following generations are created by selecting only new chromosomes which present fitness values lower than the older chromosomes. At least two metallized contiguous pixels are needed in order to assign a roof-top basis function. In this work, a specific formulation of the GA based on the weighted roulette wheel selection scheme is applied to the design of the metasurface in combination with the PMM code. At the end of the optimization process, the unit cell topology is cleaned from isolated pixels on which is not possible to define a basis function. Afterwards, a refining procedure is run in order to further optimize the polarization reflecting surface and checking the usefulness of the unit cell pixels. The GA algorithm is run about 15 time and each GA run required roughly 4 hours. The GA is stopped after 200 generations or if the fitness function is stable for more than 50 generations. The estimated total computational time is of the order of two or three days (it mainly depends on the number of times that the GA is run). Being a stochastic process, it is not guaranteed that the global best solution is found with a precise criterion but, for the best of our experience, running the GA algorithm more than 15 times does not provide a better solution than the one selected with the first 15 runs. On the other hand, the refinement process based on the sequential removal of the pixels of the unit cell required few minutes.The main advantage of the Periodic Method of Moments (PMM) with respect to other fullwave methods, relies in the computational time required to the simulation of a Frequency Selective Surface. In particular, the PMM is able to perform a full-wave simulation of a very complex structure (e.g. multilayer stackup composed of multiple lossy dielectric and FSS layers) over of hundreds of frequency points in few minutes.x, y-periodicity (T) chosen by the GA is equal to 4.6 mm and the thickness d is equal to 3 mm. Once the unit cell geometry has been selected, a parametric analysis has been performed in order to optimize the thickness of the air layer. In Fig. Rxy21) as a function of the frequency is shown for the values of the substrate thickness d equal to 2 mm, 3 mm and 4 mm. As it is evident from Fig. d = 3 mm, the reflection coefficient Rxy21 is higher than −1dB over the frequency band ranging from 8.12 GHz to 25.16 GHz. The operative bandwidth is 17.04 GHz, which corresponds to a relative bandwidth equal to 102%. With a thinner substrate, a 700 MHz bandwidth reduction occurs at low frequency. Moreover, the curve drops below −1dB around 12.5 GHz. Conversely, by increasing the substrate thickness, a reduction of the bandwidth at high frequency is present. In particular, when d = 4 mm, the highest working frequency is reduced to 23 GHz. These considerations justify the choice d = 3 mm. With a substrate thickness equal to 3 mm, the performance of the metasurface has been analysed for different permittivity values of the dielectric substrate. In Fig. Rxy21 as a function of the frequency is shown for the values ε of the losses less substrate equal to 1, 1.6, 2.3 and 3. As shown in the figure, increasing the dielectric permittivity, the highest working frequency of the polarization converter decreases.As demonstrated in the previous section, in order to convert the polarization of the impinging electric field, a low loss dielectric substrate is needed. For this reason, in the optimization process of the unit cell, an air spacer has been employed as substrate of the metasurface. In order to obtain a polarization converter with stable performance with respect to the elevation angle, a unit cell of miniaturized dimensions is required. For this purpose, in the GA optimization process, the maximum periodicity of the metasurface is imposed. In Fig. Ryy21 and Ryx21 calculated by the PMM and by HFSS are shown. It is apparent that there is a good agreement between the two numerical methods in the evaluation of the copular reflection coefficient. The simulations show a relative operative bandwidth of the polarization converter equal to 102%. A shift between the co-polar reflection coefficients provided with the two numerical methods is revealed between 8 GHz and 23 GHz. On the contrary, there is good agreement on the bands 4 GHz-8 GHz and 23 GHz-30 GHz. Moreover, the phase of the reflection coefficient along the u and v directions (ϕ(Ru), ϕ(Rv)) and phase difference Δ(f) = ϕ(Ru) − ϕ(Rv) as a function of the frequency calculated by the PMM are reported in Fig. In the light of the presented parametric analysis, an air dielectric substrate with a thickness of 3 mm has been adopted in the final design of the metasurface. In order to evaluate the performance of the proposed design, full-wave simulations have been performed by the PMM and by Ansys Electronic v. 16 (HFSS). In Fig. In order to better understand the physical mechanism of the metasurface obtained at the end of the optimization process, the surface currents distributions on the unit cell are presented. In Fig. As mentioned in the previous section, the proposed polarization converter operates from 8.12 to 25.16 GHz. As it is evident from Fig. Ryx. In particular, by indicating with Δ the value of a fixed threshold, the single pixel is removed on the basis of the following condition:Firstly, one pixel is removed from the unit cell provided by the GA and a full-wave simulation of the new unit cell is performed by the PMM code. In order to understand whether the pixel removed is necessary or not to the metasurface, the reflection coefficient nf frequency points, the magnitude of the difference between the two reflection coefficients is calculated. If the sum of the difference is above Δ, it means that the polarization conversion capabilities of the metasurface are compromised by the removal of the pixel. Consequently, the pixel is re-inserted in the unit cell. Conversely, if the difference is above the threshold, the pixel is permanently removed. This procedure is applied to each pixel of the original unit cell. In the case of the proposed polarization converter, the threshold was set to 0.1 dB. The unit cell topology obtained at the end of the refinement process is reported in Fig. For each one of the Ryx| as a function of the frequency for the optimized and refined unit cell is shown. The highest working frequency of the refined polarization converter is extended up to 32.84 GHz. On the other hand, a 370 MHz increase of the lowest working frequency has found thus shifting the starting working frequency to 8.49 GHz. The final effect of the refinement process is a 15.8% increment of the relative bandwidth thus resulting in 117.8% overall bandwidth. It is worth underlining that the topology of the refined unit cell is highly dependent on the value of Δ. In particular, the stricter the condition on the tolerable error the lower is the number of removed pixels.In Fig. The surface currents have been calculated for the refined unit cell at 4, 6, 8, 16, 26, 34 and 36 GHz and are reported in Fig. As in the case of the original unit cell topology, the surface currents are not excited outside of the working band of the polarization converter Fig. . DiffereIn order to have an easier and more complete overview of the state of the art, the performance and the geometry in terms of the most remarkable examples of reflective linear polarization converter presented in literature have been reported in Table T is at least 50% smaller than the average periodicity of the other analysed solutions. The miniaturization of the proposed metasurface is evident in Fig. 55.The proposed polarization converter is highly miniaturized. Its periodicity Ryx| of the refined metasurface as a function of the frequency for different elevation angles  of the incident wave is shown.The direct consequence of the miniaturized topology is the polarization conversion stability of the metasurface with respect to the elevation angle of the incoming wave. This is evident in Fig. θ = 40° and there is no grating lobe onset in the operating frequency range even at oblique incidence because of the unit cell miniaturization. In addition, the proposed solution exhibits smallest thickness (d/λin) with respect to the wavelength at the lower working frequency (λin). Moreover, it is remarkable to say that the relative bandwidth of 117.8% of the best solution is higher than any other similar device presented in the literature so far.The numerical results demonstrate that the polarization converter exhibits robust performance up to T of 9.2 mm.In order to validate the numerical results presented in the previous section, some prototypes have been fabricated and measured. As shown in Table h = 150 μm is accommodated on a layer of Rohacell HF31 (εr = 1.05 − j0.0017) with thickness d = 6 mm. As shown in Fig. With the purpose of validating the working principle of the proposed polarization converter, a preliminary prototype has been fabricated. The resonant elements are is inkjet-printed on a thin sheet of NB-TP-3GU100 Mitsubishi paper. The silver nanoparticle ink is deposited with a conventional piezoelectric inkjet printer (Brother DCP-J152W). The thin dielectric film with thickness Ryx| when Rs = 0.3 Ω/sq. It has to be underlined that there is no dramatic degradation of the performance even if non-perfect metallic pattern is employed in the design of the polarization converter. The GA-optimized and refined polarization converter has been also fabricated with the photolithographic process. The adopted stack-up is the same of the one shown in Fig. r = 3.5-j0.007) with thickness h = 75 μm. The GA-optimized and refined prototype fabricated with the photolithographic process are shown in Fig. As shown in Fig. Ryx| and |Ryy| as a function of the frequency for the GA-optimized and refined polarization converter are reported in Fig. The simulated and measured |An optimization procedure for the design of ultra-wide band reflecting metasurfaces for polarization conversion has been presented. The pixelated unit cell topology has been optimized with a GA in conjunction with a full-wave code based on a Periodic Method of Moments. With the aim of obtaining a polarization converter with robust performance with respect to the elevation angle of the impinging wave, a miniaturized metasurface is designed. The miniaturized unit cell is obtained limiting the maximum periodicity of the metasurface in the optimization process. The polarization converter obtained at the end of the optimization process exhibits a relative bandwidth of 102%. From the analysis of the surface current distribution it was observed that some pixels of the unit cell were unnecessary. Consequently, an innovative approach aimed at removing the useless pixels have been proposed. The polarization converter obtained at the end of the refinement process exhibits a relative bandwidth of 117.8%. The numerical results are confirmed by measurement performed on scaled prototypes of both optimized and refined polarization converter.The measurements of the cross-polar reflection coefficient of the metasurfaces have been performed with one dual-polarized horn antenna (Flann DP240) operating between 2 GHz and 18 GHz. The adopted VNA is the Keysight ENA E5071C operating up to 20 GHz. The VNA is connected to the horn antenna with a Huber-Suhner cable (SUCOFLEX_102) with length equal to 2 m. The cable introduces acceptable losses up to 18 GHz. The measurements have been performed at normal incidence by using a single horn antenna.


Physical properties of crystalline materials often manifest themselves as atomic displacements either away from symmetry positions or driven by external fields. Especially the origin of multiferroic or magnetoelectric effects may be hard to ascertain as the related displacements can reach the detection limit. Here we present a resonant X-ray crystal structure analysis technique that shows enhanced sensitivity to minute atomic displacements. It is applied to a recently found crystalline modification of strontium titanate that forms in single crystals under electric field due to oxygen vacancy migration. The phase has demonstrated unexpected properties, including piezoelectricity and pyroelectricity, which can only exist in non-centrosymmetric crystals. Apart from that, the atomic structure has remained elusive and could not be obtained by standard methods. Using resonant X-ray diffraction, we determine atomic displacements with sub-picometer precision and show that the modified structure of strontium titanate corresponds to that of well-known ferroelectrics such as lead titanate. 3 using a resonance X-ray scattering technique.It is a challenge to measure changes in the crystal structures in picometer scale and the associated phase. Here the authors demonstrate the lattice expansion and polar distortions of oxygen deficient SrTiO Next to high beam intensity, low emittance, and short pulse durations, one of the main benefits of synchrotron light sources is the wide range of accessible photon energies from visible light to hard X-rays. This degree of freedom allows to perform spectroscopy in the X-ray regime (XAS) but also to enhance traditional X-ray scattering methods. This is based on the variation of the complex atomic scattering for photon energies close to electronic transitions in a certain chemical species and therefore referred to as anomalous or resonant X-ray scattering.2. The variation in its real and imaginary part is used for contrast enhancement of atoms with similar3 or even equal4 number of electrons, or to solve the phase problem in crystallography6. Small oscillations of the atomic scattering amplitude beyond absorption edges are used for site-selective study of coordination geometry7. Other methods rely on the dependence of the scattering amplitude on the X-ray polarization and wave vector. These make the resonant atom an anisotropic scatterer, which can result in additional forbidden reflections8 and allows to study different origins of reduced symmetry such as spin, charge order, orbital order, or point defects12.Today, the field of resonant X-ray diffraction (RXD) spans a set of techniques based on the different dependencies in the resonant atomic scattering amplitudeIn this work, we describe a way to enhance the sensitivity of RXD to atomic displacements in the picoscale. In contrast to those above, this our approach employs the energy dependence of atomic scattering amplitudes to vary the structure factor aiming at a suppression of a selected Bragg reflection through the destructive interference of waves scattered by different chemical species. As a result, the structure factor is highly sensitive to a slight dynamic or static displacement of atoms. Moreover, it allows to refine the positions of atoms by analyzing the energy dependence of a few of these reflections. This way, normalization problems are avoided and a structure determination becomes possible in many cases where conventional X-ray crystallography cannot yield this information, e.g., for thin films or heterogeneous samples.3, STO) both in cubic and polar state. STO receives a lot of attention as a basis for a wide variety of complex oxide electronics. Despite being a wide-bandgap semiconductor with centrosymmetric crystal structure under normal conditions, it can acquire magnetic or ferroelectric properties, become conducting and even superconducting via doping, strain or defect manipulation17. Furthermore, it was found that single crystals of centrosymmetric STO undergo a structural transition under static external electric field18. This phase transition is quite unusual as it is triggered by a redistribution of oxygen vacancies in the crystal. It leads to a distortion of unit cells which is stabilized by the electric field and eventually results in the formation of a transformed layer at the anode side. This modified structure is therefore referred to as migration-induced, field-stabilized, polar (MFP) phase19 and is connected to the substrate by a strain gradient. A loss of centrosymmetry in the MFP layer was concluded due to the appearance of additional Raman modes as well as piezo- and pyroelectricity21. On that basis, a transition from space group P4mm was predicted20, but no information on the atomic positions for the MFP phase could be obtained so far. A structure analysis using conventional methods is not feasible due to the restricted geometry and the overlap of reflections from different layers. This problem is avoided using the outlined resonant RXD method. To test it, we characterize the initial, cubic state of STO and obtain the anisotropic mean square atomic displacements . Based on that, a refinement of atomic positions corresponding to polar displacements in the MFP phase is performed.We apply this method to strontium titanate  are often too large to give a reliable prediction of the spectral shape near destructive interference. To reach the aspired resolution in atomic positions below 1 pm, this sensitivity and its correlations with static atomic displacements is thus discussed in detail and improved anisotropic atomic displacement parameters (ADPs) are determined before the refinement of atomic positions. Other factors influencing the results, such as site occupancy and valence state, are usually known but can also be disentangled based on the scattering vector dependence of the Debye–Waller factors.F ≈ fSr − fO − fTi. Usually, these reflections show a substantial intensity since strontium is the dominating scatterer. Near the strontium K absorption edge, however, the real part of the strontium scattering amplitude is strongly reduced, causing the structure factor to approach zero and leading to the pronounced minimum in intensity.In this work, using several reflections, we vary the weighting of the different atomic scattering amplitudes to gain additional information allowing us to refine all free parameters. We found suitable conditions to perform such measurements for Bragg reflections where two of the Miller indices are even and the other is odd with respect to the pseudocubic setting. If atomic displacement is neglected, the structure factor for this set of reflections is composed of the simple amplitude difference 3 where all average atomic positions are known. The experiments involve a measurement of the Bragg intensity as a function of energy while the crystal is kept in the maximum of the Bragg condition by adjusting the scattering angles resulting in a fixed momentum transfer on the photons.In order to test the method but also to get a good starting model for the characterization of structural modifications, we characterized the thermally induced atomic displacement for the initial (cubic) state of SrTiOK edge and for different temperatures. A clear trend can be seen and reflects that the Debye–Waller factors of the different atoms do not scale proportionally with temperature. This fairly corresponds to the changes initially calculated . For energies beyond the edge at ≈ 16,100 eV, it was necessary to include an absorption correction and the resonant fine structure.A high sensitivity of these RXD spectra to thermal displacement is evident in Fig. ted Fig. , bottom 3, oxygen occupies a noncubic Wyckoff position leading to two different oxygen ADPs necessary to describe the measured intensity correctly—one parallel and one perpendicular to the Ti–O bond. The results of least-squares refinement displayed in Fig. 23 and taking into account a decomposition of the Debye temperature Θ for oxygen into a parallel and a perpendicular part following Potzel et al.24. For this reason, it was necessary to record spectra of additional reflections to ensure an unambiguous refinement of the ADPs. Table In cubic SrTiO25 and the absolute values are within the error margins. This is remarkable as the results were obtained by different methods. The errors shown in Table Despite its lower mass, titanium shows a smaller thermal vibration than strontium and the oxygen oscillations along the Ti–O bond are much smaller than perpendicular to it. Both observations are in good agreement with literature results3, it is possible to characterize the structural changes in the strained MFP layer. The MFP phase is formed under influence of a [001] oriented, static electric field of about 1 MV m−1 in a few micrometer thick layer on the anode side of the crystal  for each studied reflection hkl. At these positions of reciprocal space, we then measured the energy dependence of Bragg intensity for several reflections in a similar region as before the electroformation. The resulting data along with calculations are shown in Fig. As its strain in equilibrium depends on the applied electric field strength, crystal thickness, as well as sample history, there is no prototype of the MFP phase. Furthermore, there is a continuous decrease of strain towards the cubic bulk SrTiOsee Fig.  correspo3. The validity is further discussed with aid of absorption spectroscopy measurements in Supplementary Note As it is on the order of the penetration depth of the X-rays, the refinement also includes the thickness of the probed, strained region using Eq. 3 and PbTiO3 with space group P4mm. In comparison, the absolute displacements of atoms from high symmetry positions are significantly smaller in the present case of the MFP phase. However, similar to PbTiO3, the displacement of titanium is roughly a factor of three smaller than those of oxygen. The relative uncertainties in the refined results are much smaller than in the case of Debye temperatures. Even though the refinement was based on iterating through all combinations of ADPs within the confidence volume, the error did not propagate proportionally indicating a stronger sensitivity to the rather small static displacement. Figure z-direction and away from their original high-symmetry positions meaning that the ratio of the shifts is again known more precisely than one would estimate based on Table The results of the refinement of internal structure parameters are summarized in Table 3 crystal via a strain gradient. An open question is, how the displacement evolves with strain or whether a polar phase is only formed in the thin layer of constant strain closest to the anode which is associated with the additional peak in the diffraction profile. In the latter case, the region of continuous strain gradient, which is seen as a plateau between the peaks  corresponding to a value of 5.9 · 10−7 per unit cell. For the 007 reflection, this would result in a Qz ≈ 0.0035 r.l.u. . This estimation has been done for the position of highest strain. For lower strain, the broadening of Bragg peaks will decrease and the depth resolution of the measurement will improve. Therefore, the sampling we used in this work was always below the limits of resolution.Regions of the sample having different strain in see Fig.  near theQ energy scans corresponding to the four selected strain values are presented in Fig. Qz/Qz = −0.04% , the energy dependence shows an enormous response which is enhanced with larger strain. The changes in RXD curves saturate when further increasing the strain and this saturation is reached earlier for the 205 reflection. The legend in Fig. Qz-scans near the 007 reflection for two selected energies.The fixed 29, where the lattice strain was caused by atomic displacements due to electric field and elastic vibrations due to the short rising time.The strong response seen in Fig. z/c of titanium and oxygen in field direction as they were experimentally found for the MFP phase according to Table ptot are given in Table 30 in Perdew-Burke-Ernzerhof (PBE) parameterization31 was employed as implemented in the VASP code32. Total energies have converged better than 10−10 eV with a maximum kinetic energy of 450 eV for the plane-wave basis set and Γ-centered 12 × 12 × 12 Monkhorst–Pack k-point meshes with spacings < 0.02 × 2πÅ−1. For the evaluation of strain, we kept relative atomic positions fixed and relaxed the cell geometry within the space group to forces less than 10−5 eV Å−1. The change in polarization has been assessed within the framework of a Berry-phase treatment33.Electronic structure calculations using density functional theory (DFT) were performed to validate the relation between strain and polar displacements, as well as to estimate the macroscopic polarization resulting from these displacements. The effects caused by the external electric field have been emulated by imposing relative atomic displacements Δc-direction of 19.During the measurements on the structure of the MFP phase, we selected a strain value in 3 and presented a resonant RXD method to obtain these even for inhomogeneous samples. The method involves adjusting the X-ray wavelength—and thus the scattering amplitude of the resonant atoms—to tune destructive interference of the partial waves from different scatterers. To which extent this can be achieved strongly depends on the atomic positions, giving an access to reveal slight displacements of the atoms. In the present case, a sensitivity to changes of average positions at the level of 1 pm and below has been demonstrated. Prerequisite for a careful analysis of the data was a refinement of temperature induced atomic displacement parameters of cubic SrTiO3. We could confirm directly that SrTiO3 forms a polar structure under electric field, which is of the same type as ferroelectric BaTiO3 at room temperature. This is remarkable, as, in contrast to BaTiO3 and PbTiO3, SrTiO3 does not form this phase when cooling to low temperatures: the beginning of a paraelectric to ferroelectric phase transition was observed at 40 K but remains incomplete down to 0 K34. Yet, it is known that other external influences, such as strain, can stabilize this ferroelectric phase even at room temperature15. Using DFT calculations, we could validate the relation between strain and atomic displacement that was experimentally found for the MFP structure and furthermore give an estimate for the resulting electric polarization of the unit cell. We have shown that the magnitude of polar displacement exhibits a nonlinear relation to the strain. The recently found indications for an electrostrictive origin of piezoelectricity in the MFP phase21 would suggest a quadratic dependence of the strain on polarization. In our case, however, the polarization dependence of strain appears to follow an even higher power law  the polarity of the structure38 with an electric field in order to measure the Friedel pair contrast. This is not always feasible or may alter the material as in the present case. Furthermore, in all cases the impact of Debye-Waller factors was neglected and the obtained resolution was much lower than in our approach, which exploits destructive interference.The employed resonant suppression of Bragg reflections can be universally combined with all diffraction based X-ray techniques as long as the photon energy of the X-ray source is tunable. There are similar attempts to extract structural information reported in literature, all subject to substantial limitations compared to our development. To avoid ambiguities in the refinement, fixed structural models were presupposed or only one structural parameter was assumed to be unknownIn Supplementary Note E is given by39F denotes the structure factor and A describes absorption. Further dependencies on E, such as in Lorentz and polarization correction, are usually weak and can be approximated as linear. The momentum transfer vector Q defines the Bragg reflection. In Bragg geometry, the absorption factor can be written as40t being the thickness of the probed crystalline layer, μ the linear absorption coefficient, and α and β respectively referring to the angle of the incidence and exit of the X-ray beam with respect to the crystal surface. In the independent-atom model, the structure factor is expressed as a weighted sum of the scattering amplitudes fj of all atoms present in the crystallographic unit cell. Real crystals exhibit a certain degree of disorder due to lattice vibrations and defects. Therefore, an averaging over all unit cells is necessary, which is commonly parameterized with the occupancy of the crystallographic site oj and the Debye-Waller factor r of atom j:Within the kinematic theory of diffraction from crystals, in the absence of extinction effects, the dependence of the diffracted intensity of a Bragg reflection on the photon energy uj of the atom j in the direction of Q via Uij41. Neglecting the polarization and wave-vector dependence of the dispersion corrections f′ and f″, the scattering amplitude can be written asThe Debye–Waller factor is a result of harmonic approximation of lattice vibrations due to temperature and is related to the mean square of the atomic displacement In resonant diffraction, it is common to investigate Bragg reflections where the structure factor is zero due to destructive interference of scattered waves coming from atoms of the same species. Under resonant conditions, these atoms may have different scattering amplitudes due to their different chemical or magnetic environment lifting the destructive interference and resulting in non-zero intensity and therefore new, previously forbidden reflections. In this work, we went the opposite way and tuned the scattering amplitude of distinct elements to cancel out each other. This can only be true in a small range of photon energy, since the scattering amplitude of each element shows a different energy dependence. It also strongly depends on any contribution to the structure factor such as occupancy, shifts in the atomic positions, temperature-induced dynamic displacement, defects, or valence state.u be the small displacement of a selected atom j from the ideal position (u = uj to improve legibility). To point out the contributions of displacement, the structure factor can be approximated, assuming it is close to zero, asLet One can see that the term of the mean displacement carries the imaginary unit resulting in the orthogonal displacement of the complex structure factor for the two contributions that was seen in Fig. K edge  for a set of reflections where exactly one Miller index is odd. On the other hand, the post-edge region contains valuable information about the direction of a considered atomic displacement since a significant imaginary part is typically added to the scattering amplitude. Owing to variations in the Q-dependent scattering amplitudes, the minimum is located at different energies for different reflections which defines the energy range for each measurement. Energy-dependent RXD data have been collected in quasi-symmetric geometry  at the beamlines BM2042 and BM2843 of the European Synchrotron Radiation Facility (ESRF) using the diffractometers in four-circle geometry. The preparation of samples  has been described in detail elsewhere19. To facilitate the formation of the MFP phase, the photon flux had to be reduced such that the leakage current was about 1 μA or lower depending on the formation state. For the study of influences from thermal displacement, a continuous-flow cryostat was used at BM20 to cool the crystals down to a minimum temperature just above the cubic to tetragonal phase transition at 105 K44.We found that the Bragg intensity has a minimum in the pre-edge region of the Sr-f′, f″ far off the edge have been taken from the Sasaki database45, whereas the near edge fine structure in f′, f″ has been obtained for strontium by means of XAS measurements on SrTiO3 powder at the Sr-K edge and subsequent application of Kramers-Kronig relations following Lucarini et al.46. The nonresonant atomic scattering factors f0(Q) have been taken from the International Tables for Crystallography47. Additional peaks appearing in the energy dependent RXD scans are attributed to Umweganregung48 and have been filtered out through multiple scans for small rotations about the Bragg vector Q where feasible.The fit of the energy dependent Bragg intensities was based on Eq. 49. Software routines used for data modeling are available online50.Raw data were generated at the ESRF and can be accessed from the Open Science FrameworkSupplementary InformationPeer Review File


Over the past decades, clinical guidelines have become an increasingly important aspect of clinical practice and every day international guidelines influence the clinical decisions we make at the patient’s bedside. We define evidence-based medicine as a conscientious, explicit and judicious use of current best evidence in the decision-making process in the care of individual patients .As doctors we realise that guidelines should only be used as a starting point in our clinical decision-making. In daily practice, we must incorporate the cultural, social and economic contexts of the patient for the treatment to be a success. Patient characteristics appear to exert influence: for instance, co-morbidity in a patient appears to reduce the chance that guidelines are followed . It has In the prevention of coronary heart disease and cardiovascular disease we focus on reducing the overall absolute risk of disease rather than on managing individual risk factors. Here, cultural, social and economic contexts are now more important than ever. Estimating the total cardiovascular risk is a crucial part of our clinical practice with the obvious statement being that those at highest risk from a cardiovascular disease event gain most from preventive measures. This can only be achieved with a holistic approach.Meanwhile, populations across the world are becoming increasingly ethnically diverse due to globalisation. Health risks seem to differ between ethnic groups. Most large population-based cohort studies have excluded ethnic minority populations . The NetIn this issue, Perini et al. describe an elegant observation from the Dutch HELIUS  study that was designed to unravel potential mechanisms underlying the impact of ethnicity on communicable and non-communicable diseases . The autWe applaud the researchers for using the HELIUS study to get more data on the colourful practice that we face every day. It is a beautiful observation to describe different cardiovascular risk profiles per ethnicity. If we want to take this observation to the next level, however, more questions arise. Are these observations due to genetics or due to lifestyle? Did the participants have the same access to the health care system? What is their level of education and participation in society? It is only when we find the answers to these questions that we can really make a difference.After all, the prevention of cardiovascular disease relies on the reduction of the overall absolute risk of disease rather than management of individual risk factors. It appears that there is a lot to gain in specific ethnic minorities. The next step should be that we determine how we can improve prevention of cardiovascular disease in these groups. We certainly will need to individualise our prevention programs for ethnic-specific prevention strategies.But even a categorisation based on the ethnicity of patients is an oversimplification of the truth. Every day, we are faced with different views on health, lifestyle and the quality of life in general, regardless of background, with no guidelines or SCORE calculators to help us incorporate these different dimensions in clinical decision-making. So, is it possible to create international standards that will fit the cultural, social, and economical contexts of each unique patient? Fortunately, we as doctors can still be of added value in the individual patient care in daily practice.


Coupling electron beams carrying information into electronic units is fundamental in microelectronics. This requires precision manipulation of electron beams through a coupler with a good focusing ability. In graphene, the focusing of wide electron beams has been successfully demonstrated by a circular p-n junction. However, it is not favorable for information coupling since the focal length is so small that the focal spot locates inside the circular gated region, rather than in the background region. Here, we demonstrate that an array of gate-defined quantum dots, which has gradually changing lattice spacing in the direction transverse to propagation, can focus electrons outside itself, providing a possibility to make a coupler in graphene. The focusing effect can be understood as due to the gradient change of effective refractive indices, which are defined by the local energy band in a periodic potential. The strong focusing can be achieved by suitably choosing the lattice gradient and the layer number in the incident direction, offering an effective solution to precision manipulation of electron beams with wide electron energy range and high angular tolerance. Graphene has exceptional electrical properties, including high electrical conductivity, high carrier mobility, ballistic transport, etc.234579141516Both the focusing and the focal imaging mentioned above are achieved based on parallels between optics and electronics in monolayer graphene. Quantum Fabry-Pérot interferences for electrons also has been realized in graphene2015222324In the past decades, the exploitation for artificial optical structures has largely stimulated the development of photonics and optics. Photonic crystals, as a typical example, have shown a powerful ability in manipulating the flow of light for their abundant frequency-wavevector dispersion relation2728293032E close to the Dirac point, the Hamiltonian can be written asIn the scattering problem of electron waves with energies R − r) is the Heaviside function with R the radius of the circular potential, V is the applied constant bias of the gated region and σ =  are Pauli matrices. We employ reduced units, i.e., ħ = 1 and the Fermi velocity vF = 1 throughout. The focusing is realized with an array of gate-defined quantum dots which share an identical radius R = 1 in graphene and are applied a constant bias V = 5. Here all quantities with displacement dimension are in unit of the radius R since it is set to be 1. At the same time, the incident electron beam has unit electron density at the beam center for the convenience of comparison when the electron density is plotted in all figures. The unmodified structure is a square-lattice crystal with lattice constant d = 5. In order to achieve the focusing effect, a lattice gradient in the transverse direction (perpendicular to the incident wave vector) is added to the original. We study an nt × nl array which is illuminated by a Gaussian beam along the +y direction, where nt and nl denote the layer number of quantum dots in the transverse (x) and longitudinal (y) direction, respectively. In the two edges of the array, there is the equal lattice spacing in the x and y direction and then the lattice spacing in the x direction will linearly increase from the two edges to the center with the increment g = di+1 − di, as is schematically shown in where Θ line d = 7. One can see that the energy band exhibits a noticeable variation when the lattice constant is changed. For the dispersion relation of light in photonic crystals, we have dω/dk ∝ 1/n where n is the ERI and thus the slope of the optical band is related with the ERI35neff for electron waves in graphene with dE/dk ∝ 1/neff based on the same principle. In E ranging from 0.3 to 0.5, so their different slopes illustrate that the larger lattice constant produces the larger ERI. This means that the gradient of lattice constant will lead to the gradient of the ERI neff. Thus the array probably satisfies the requirement of the principle of equal optical path due to the ERI gradient and has ability to focus electron waves.The focusing of electron waves can be similarly understood as that of light through optical lenses. In optics, an optical system for focusing satisfies the principle of equal optical path according to Fermat’s principleg = 0.4 and nt = 20 unchanged and see the effect of the thickness, i.e., the layer number in the longitudinal direction nl, on focal length and the electron density at focal spot. nl enhancing from 3 to 18. It seems that the more the layer number the electron wave travels is, the more strongly it is focused. In order to verify the assumption, we plot the full width at half maximum (FWHM) along the x-directional line at focal plane to characterize the size of focal spots in nl increasing, showing that the electron wave is better focused with the nl increasing. nl increases. Namely, the focus spot is more and more close to the array with the layer number increasing. This means that the array with larger thickness is more like an optical converging lens with smaller radius of curvature. One can also note that the focal spot always locates outside the array for arbitrary nl, meeting the requirement of a coupler. To illustrate these features of the flat lens, we plot the electron density distribution in a large spatial region in nl = 8 and nl = 16, respectively. We can find more details about the focusing from them. First, the electron wave is hardly reflected by the arrays. The fact that all of waves can arrive at the transmission side ensures that our design has a high focusing efficiency. Second, a strong focusing shown in nl larger than 16.Just like an optical lens, the focussing property of electron waves is also related with the geometrical property of the array, such as its thickness and the gradient. We first keep the increment g is changed from 0.2 to 1.0 and the layer number in the longitudinal direction nl = 6 and in the transverse direction nt = 20 remains unchanged. One can see that the electron density at focal spot tends to achieve the maximum value when the increment g is in the vicinity of 0.3 and the FWHM has the minimum value at the same time, implying that a good focusing occurs. This demonstrates that the increment g = 0.3 is an optimal value of the gradient for the array to product the focusing behavior. To understand the existence of the optimal value, we plot the electron density distribution in a spatial region for the arrays with the increment g = 0.2 and g = 1.0 in g is far away from 0.3. In the language of light, too small change of ERI is not enough to deflect the wave, while too large change will lead to the scattering of waves in other directions  can be expanded as follows,In a system of multiple gated potentials, the incident wave that strikes the surface of potential j consists of two parts: (1) the initial incident wave and (2) the scattered waves of all other potentials according to multiple scattering theory39whereaj)( = Spj) into n = nc. The resultant error incurred due to truncation is assumed to be insignificant. In our problem, the minimum nc should be set to be 5 in order to guarantee convergence. The calculation for any system of multiple gated potentials in this paper usually can be completed within several minutes in a personal computer system.In practical numerical calculation, the series expansion of How to cite this article: Tang, Y. et al. Flat-Lens Focusing of Electron Beams in Graphene. Sci. Rep.6, 33522; doi: 10.1038/srep33522 (2016).


The dynamics of the fluctuating electrostatic potential and the plasma density couched in the resistive-drift model at nearly adiabatic state are simulated. The linear modes are unstable if the phase difference between the potential and the density are positive. Exponential growth of the random small perturbations slows down due to the nonlinear E × B flows that work in two ways. They regulate the strength of the fluctuations by transferring the energy from the energy-producing scale to neighboring scales and reduce the cross phase at the same time. During quasi-steady relaxation sporadic appearance of very strong turbulent particle flux is observed that is characterized by the flat energy spectrum and the broad secondary peak in the mesoscale of the order of the gyro-radius. Such boost of the transport is found to be caused by presence of relatively large cross phase as the E × B flows are not effective in cancelling out the cross phase. E × B advection. Turbulence enhances the level of the transport of the particles and the heat much higher than classical collisional dissipation from the hot-and-dense core to the edge. Since there exist various forms of the free-energy source that may be released to turbulence, it is a general practice to study the effect of each source on the transport separately. At the edge region of the confined plasma, the plasma temperature is not as high as the core so that the collisions between the plasma particles are not negligible in the plasma dynamics.In the experiments of the magnetically confined fusion plasmas, the confinement is degraded by the presence of the turbulence. Long-range electromagnetic interactions between the charged particles cause the non-uniform plasma equilibrium linearly unstable. Small perturbations grow exponentially and the plasma becomes turbulent through numerous nonlinear interactions, notably the 1 is a minimal model that is suitable for the study of the edge plasma confined under strong magnetic field. At the equilibrium the electron temperature, which is isothermal, is much higher than the ion temperature and the particle density is non-uniform with a constant gradient length across the radius. After small random disturbance is introduced to the equilibrium, the evolutions of the fluctuating electric potential and the electron density are studied with fixed density gradient. The free energy associated with the non-uniform density is tapped into the plasma by the particle transport across the plane perpendicular to the magnetic field which is then dissipated by the resistive dissipation along the field. The plasma is assumed to be a slab where the Cartesian coordinates y represent the radial and the poloidal positions, respectively. The plasma is linearly unstable if the density perturbation of the resistive-drift wave lags behind the potential perturbation. The linear growth rate is large at the scale of the order of the ion gyro-radius ρs that is associated with the electron temperature. The fact that the stability has a strong dependence on the cross phase between the perturbations has a certain similarity to a damped harmonic oscillator enforced by the external harmonic force. The power supplied to the oscillator is proportional to sin2Δ where Δ is the phase difference between the force and the displacement. The similarity between the two problems ends here because the potential and the density in the plasma are coupled so that they are determined self-consistently by the dynamics. The strength of the perturbation and the particle transport Γ, which is the product of both the particle density n and the E × B velocity vEx, initially grow exponentially, because of the linear instability. With the increase of the fluctuation level, the nonlinear E × B advection becomes important to the evolution of the fluctuations. The energy of the fluctuation becomes saturated and the peak of the energy spectrum moves toward larger scale than ρs. Since Γ depends both on the fluctuation levels of the potential and the density and on the cross phase4 between them, Γ declines at a faster rate than the energy. It is because the cross phase relaxes toward 0 as larger-scale fluctuations carry most energy.Set of Hasegawa-Wakatani equations1, where the fluctuating electron density closely follows the Boltzmann distribution, an interesting result that had not been reported before was discovered during the quasi-steady period of the saturation. It is found that the saturation of Γ does not proceed in monotonic fashion. Instead, the fast-pace increase and decrease of Γ is found to repeat intermittently during the relaxation stage. Complete explanation of the observed non-uniform evolution of Γ would require quantitative information on how the energy and the phase, which are of course coupled to each other, evolve under the E × B energy fluxes. As the analytical evaluation of the nonlinear fluxes is of higher degree of difficulty, we intend to rely on the numerical data to elucidate their impact on the cross phase. A brief account of the results is as follows.While running simulations of the resistive-drift plasma dynamicsE × B nonlinearity clearly controls the behavior of δ during the relaxation, it is natural to separate out its effect on δ. It may be better to divide δ into δ0 and δ1, where δ1 is driven by an advective part Q of the nonlinear E × B energy flux, and δ0 is the rest that includes the linear response. During the course of early development of the energy saturation, δ0 is almost cancelled out by δ1 in the scale where most energy is contained. As a result, Γ decreases continuously in time, until slowly changing Q turns the corner to make δ1 ineffective in blocking δ0 in the mesoscale. Then, a little hump in the spectra of Γ starts to grow, and it becomes larger as δ1 adds to, instead of subtracting from, δ0, as Q changes sign. After Q reaches a peak, it goes on decreasing, to change the sign. Then, the hump in the spectrum becomes lower, and drags Γ down. The same process repeats itself during the quasi-steady phase of the energy saturation.Figure 3. In spite of its importance in the plasma transport by the drift-wave turbulence4 the cross phase has not been studied theoretically as much as the energetics of the fluctuations. For example, the spectrum of the fluctuations was theoretically predicted in refs6 and the energy transfer between the fluctuations was analyzed by Manz7. Naulin8 tried to compute the cross phase by modelling the non-adiabatic response. Camargo9 considered the cross phase in the linearly unstable regime. The role of E × B energy flux Q on the cross phase has been touched upon in the context of the turbulence under sheared flows. Terry11 estimated the suppressions of the turbulence and the cross phase by the flow shear. Experimental evidence that the sheared flow modifies the turbulent transport through the change of the cross phase has been reported13. Impact of the zonal flow on the cross phase was reviewed by Diamond and co-workers14.There exist many reports of the experimental measurements of the cross phaseQ fluxes is not limited to the particle transport. For example, they are equally important in the thermal energy transport because it depends on the cross phase between the electric potential and the pressure. We proceed by describing the evolutions of the energy and the cross phase based on the formalism that was recently developed15, followed by the presentation of the scenario based on the analyses of the numerical simulation, and the summary.The role of n. Through the model one may study the particle transport while the gradient of the background electron density is held fixed with the constant scale length Ln. Since the electron temperature does not fluctuate, the heat transport of the plasma is excluded in the model. Collisions between the ions and the electrons are included so that the steady state is achieved where the free tapped into the plasma is dissipated by the resistive loss. Utilizing the normalizations of the spatial scale ρs, the time Ln/cs, ϕ and n with (Teρs/eLn) and n0ρs/Ln, respectively, where ρs = cs/Ωi, we have:The dynamics of the plasma are pedagogically described by the resistive-drift model that couples the evolutions of the electrostatic fluctuations of the electric potential ϕ and the electron density x direction, the magnetic field along the z axis, and the electron diamagnetic drift along the y. Equation (ρs17.The plasma is assumed to be a slab where the coordinate axes are chosen so that the equilibrium density gradient is in the −Equation  describeEquation  represenThe energy-conservation laws in the Fourier space may be found by multiplying Eqs  and 2)  2) by thk where the modes are most unstable. As the fluctuations grow, the nonlinear energy fluxes develop to slow down the growth of the fluctuations by taking the energy away from the range Let equation  reversesequation  becomes 15: First, Eqs  and at t = 8.5 × 104 (weak transport) is about five in the mesoscale. Yet the ratio of Γk’s at the same times is of the order of 10. Therefore, it agrees with the difference of δk between the times. Note that at t = 6.05 × 104 and 8.25 × 104, when the transport is strong, Γk of the broad secondary peak is well within an order of magnitude of the primary peak around kρs = 0.1. Considering the width of the peaks and the number of modes involved, the secondary peak contributes more than the primary peak to push up the transport at t = (6.05 × 104 and 8.25 × 104). By contrast, the primary peak of K is still two orders of magnitude higher. As a result, Γ fluctuates more than K in time.Figures k1 in the mesoscale is of opposite sign between strong and weak transport: positive in the former, and negative in the latter. As equation . As a result, the cross phase is large in the mesoscale. A secondary broad peak in the mesoscale appears in the spectrum of the plasma particle transport that exceeds the contribution of the primary peak, where most of the energy resides. The energy spectrum becomes flat, instead of decreasing, in the mesoscale, because the particle transport pushes up the plasma energy.Aperiodic manifestation of very strong particle transport of the resistive-drift plasma turbulence at highly adiabatic state is understood by the coupled dynamics between the fluctuation energies and the cross phase. The 20. It is stressed in this work that studying the cross phase between the electrostatic potential and the plasma density, in addition to the energy evolution, is crucial to understanding the particle transport near the adiabatic state in the Hasegawa-Wakatani resistive-drift plasma model. Since the electrons in the fusion plasmas are almost adiabatic, the particle transport may be presumed to be small. Yet, it may intermittently become strong, as indicated in the simulation. Advection of the kinetic energy in the same direction as the ion diamagnetic drift produces large cross phase and the particle transport. In other models of the plasma turbulence, if the transport and the energy relax by different rates from each other in the nonlinear saturation stage, it may indicate that the relaxation of the cross phase between the fluctuations is important in the prediction of the plasma confinement. Analytical work is desirable to supplement the present results, by approximating the advective E × B flux in terms of the energy and the cross phase, to close the dynamic chain between the energy and the cross phase. The evolution of the cross phase is believed to be similarly worked out inside the fusion plasmas. When the zonal flow is present with the turbulence, the advective E × B flux by the non-zonal fluctuations may be negligibly small compared to the advection by the zonal flow. As far as the cross phase is concerned, the latter advection is canceled out and the non-zonal E × B advection may lead to the disparity between the distributions of the fluctuations and the transport. Extensions to the transport of both the particles and the thermal energy in the plasmas with the presence of the zonal flows are in progress.Although the cross phase between the fluctuations is an integral part in the determination of the transport flux, there exist not many works that deal with it on the same footing as the fluctuation energyL = 80π that is evenly divided by N2 grid points, with N = 256. BOUT +  + platform21 is used, employing PVODE with adaptive time stepping to advance in time, and the Arakawa scheme22 for the treatment of the Poisson brackets. For the present work, the periodic boundary conditions are imposed, and the coefficient of the hyper-dissipation is ν = 3 × 10−3, while the adiabaticity parameter α = 10. Initially, the fluctuations are set as n = 0 and θx and θy by 13 other small-amplitude harmonics of box size L in both x and y.Eqs  and 2)  2) are nThe datasets generated during the current study are available from the corresponding author on reasonable request.


Performance and load testing data of a three bladed two stage LENZ type vertical axis wind turbine from the experiments conducted in an open environment condition at Hindustan Institute of Technology and Science, Chennai  are presented here. Low-wind velocity ranging from 2 to 11 m/s is available everywhere irrespective of climatic seasons and this data provides the support to the researchers using numerical tool to validate and develop an enhanced Lenz type design. Raw data obtained during the measurements are processed and presented in the form so as to compare with other typical outputs. The data is measured at different wind speeds prevalent in the open field condition ranging from 3 m/s to 9 m/s. Specifications TableValue of the data•The key focus of this data set on Lenz type VAWT is to provide technical information on this unique type of VAWT in order to quantify its aerodynamic behavior operating at an open field conditions.•The presented data allows the researchers using numerical tool to validate and develop enhanced design based on this comprehensive full-scale measured data.•The complete data set of this open field experimental study is conducted so as to provide a benchmark for future numerical analysis.1p) and coefficient of torque (CT). Further to disclose the influence of tip speed ratio (TSR) on aerodynamic performance, the variations of TSR with Cp and CT at different wind velocities are shown in Lenz type vertical axis wind turbine 22.1(a)Shaft: The shaft is made using solid mild steel material with diameter of 20 mm. The length of the shaft is 2220 mm.(b)Blades: Lenz type airfoil blade profile is made using the material named “plywood” of thickness 12 mm. The dimensional detail of the Lenz type airfoil blade is given in (c)Link: Steel link made of mild steel material is used to transfer the power from the blades to the shaft through the disk.(d)Disk: Two disks made up of mild steel material of 150 mm diameter and 3 mm thick is used to transfer power from the links to the shaft and it is permanently fixed to the shaft.(e)Pulley: A grooved pulley made of Teflon material is used to transfer the force exerted on the rope to the pulley. This force exerted on the rope act as the load applied on the VAWT.The Lenz type vertical axis wind turbine (VAWT) has the following components:0 so as to allow the maximum amount of wind on the blades of VAWT from varying directions. Three blades arrangement is most preferred for the wind turbine rotor because two blades produces noise, rattle, imbalance and it is a poor self-starter at low-wind speeds. Two self-lubricating heavy duty ball bearing with inner diameter of 20 mm is used to fix this Lenz type VAWT with the experimental test rig as seen in 2.2p) and torque Coefficient (CT). In this present study, the airstream is flowing through a swept area or frontal area (A) and the mass flow rate is given as ρAV. The power available in the air is given byThe performance indices of the Lenz type VAWT is measured in terms of power coefficient  and brake torque produced (T) and it is given asshaft) to the power available in the wind (Pavailable) is known as power coefficient (Cp) which indicates the efficiency of conversion.The ratio of shaft power or actual power produced by the turbine (PT) is given as Torque coefficient (Cp is the radius of the pulley attached to the shaft (m), F is the mechanical load applied to the turbine shaft (N), R is the rotational radius of the turbine (m), N is the rotational speed (rpm) and T is the torque applied on the turbine (Nm).where V is the free stream wind speed (m/s), ru is the blade tip speed (m/s).These performance indices are evaluated with respect to tip speed ratio (λ) which is given byp) and torque Coefficient (CT) with respect to tip speed ratio (λ).As the present data is obtained by testing the Lenz type VAWT under open field environmental conditions, a stable wind flow could not be obtained. Therefore, the experiment is repeated in different periods and days to remove any uncertainty in the measurements and data. Mechanical load is applied in a gradual manner on the rotor shaft of the VAWT. At a given load condition, the rotor is loaded for different wind velocities prevalent in the open field conditions and the values of the load and spring balance readings are recorded to calculate power coefficient (CAs can be seen in pmax of 0.045175 at λ = 0.354192 is obtained. Similarly, Cpmax of 0.024813 at λ = 0.35796 and Cpmax of 0.02522 at λ = 0.387566 is obtained for wind velocity 6 m/s and 7 m/s respectively. As seen in p value rises with increase in tip speed ratio up to certain maximum value and later, it droops for further increase in the tip speed ratio.As the load on the VAWT is increased, the rotational speed of the Lenz type VAWT is reduced as can be seen in the data shown in


Neural networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks to naturally finding the most probable error and accounting for correlations between errors. Early in the development of both classical and quantum computers, “threshold” theorems were proven to show the existence of encoding schemes which reliably store and process information in a “logical” set of bits (or qubits), by encoding it redundantly on top of a bigger set of less reliable “physical” bits (or qubits), as long as the error rate on the physical layer is smaller than a fixed threshold5. The vast majority of quantum error correcting codes fall in the class of stabilizer codes 6. They are characterized by the group of stabilizer operators that preserve the logical states . The list of nontrivial stabilizer operator measurements  is called the syndrome of the error. While providing for efficient encoding, linear and stabilizer codes do not necessarily have known efficient decoding algorithms that can deduce from a given syndrome what errors have occurred.Constructing a physical computing machine, whether a classical or a quantum one, requires, inescapably, the implementation of an error correcting mechanism that guards against the noise picked up from the environment and the imperfections in the operations being performed9 permit the creation of quantum codes from classical codes, but they do not guarantee that the decoder that worked for the classical code still works for the quantum one. A particularly interesting example is the class of LDPC codes11 which are high-performing classical codes with efficient decoders, however those decoders do not work for the quantum LDPC codes12.In the general case decoding a stabilizer code is an NP-hard problem. An active area of research is the design of codes with some additional algebraic structure that permits efficient decoders, but still retains high rates  with acceptable distances . Schemes like the CSS approach13. Since the toric code has low-weight local stabilizers, it is also a quantum LDPC code with structure that impedes typical belief propagation algorithms. Our decoder significantly outperforms the standard “minimal-weight perfect matching” (MWPM) decoder15. Moreover, it has comparable threshold with the best renormalization group decoders16. For code-sizes up to 200 physical qubits the decoder is practical and we discuss how to extend our neural network architecture to negate the inefficiencies that kick in at that stage.Here we present a decoding algorithm that can be applied to any stabilizer code — the decoder employs machine learning techniques to “learn” any structures that would make the approximate decoding problem easier than the general NP-hard decoding problem: it “learns” the probability distributions of errors conditioned on a given syndrome and efficiently uses samples from that distribution in order to predict probable errors. The conditional probability distribution is encoded in a deep neural network. The “learning” involves training the neural network on pairs of errors and corresponding syndromes . We test the algorithm on the toric code Fig.  definied17 or convolutional networks for identifying phases of matter18. A preprint on the use of restricted Boltzmann machines to decoding the toric code has been available for a few months as well19, however that architecture does not yet outperform known decoders like MWPM and has been tested only on the Z syndrome on lattices no bigger than 5-by-5. At the time of submission of this manuscript two other related preprints were made available: a fast neural network decoder for small surface codes, that however also does not significantly outperform MWPM20, and a recurrent neural network decoder outperforming MWPM as evaluated on a 17 qubit surface code21. It is worth noting as well that over the last few months work has started on deep learning methods for decoding classical algebraic codes22.Machine learning techniques, specifically neural networks, have been gaining popularity over the last year, in particular with the recent developments in using restricted Boltzmann machines for describing the ground state of many-body systems16 (threshold of 15.2%), and decoders explicitly tuned to correlations between Z and X errors23 . To our knowledge only a renormalization group decoder24 enhanced by a sparse code decoder12 reaches a similar threshold (16.4%). It is worth noting that the sampling procedure in our decoder makes it impractically slow for codes of more than 200 physical qubits, while other decoders remain practical. On the other hand, the neural architecture is versatile enough to be applied to any stabilizer code, unlike the other decoders discussed here, which are limited to only topological codes. The best of both worlds — record threshold and fast decoding — should be achievable if we couple the renormalization decoder of24 with our neural decoder (instead of the currently suggested sparse code decoder12), however this will be applicable only to topological codes. We discuss other ways to avoid the inefficiencies in our decoder without compromising its ability to “learn” to decode any stabilizer code.For testing purposes we trained our neural decoder  stops providing improvements.23.The significant gain in the threshold value relative to some known decoders can be traced to two characteristics of the neural network (discussed in more details in the Methods section). Firstly, the neural network is trained on  pairs generated from the error model, therefore it is optimized directly for producing “most probable error”, not for finding an imperfect proxy like “error with lowest energy” as is the case for MWPM. Secondly  is to be learned from large amount of training data given in the form of pairs ). The input x is set as the value of the input layer of neurons. Each of those neurons is connected through axons with each neuron of the next layer (the first “hidden” layer). Multiple hidden layers of neurons can be connected together in this fashion in order to construct a deeper neural network. The last layer of the network is the output layer - its value represents flearned(x). The value of a neuron  is calculated as a weighted sum of the activation values of the neurons connected to it from the previous layer. That sum is then passed through a non-linear function . This activation value is then further passed on to the neurons of the next layer, where the process is repeated until it reaches the output layer. The weights in the sums (i.e. the strength of connections between the neurons) are parameters which are optimized through stochastic gradient descent in order to minimize the distance between flearned and f calculated on the training data. The choice of activation function, the size of the hidden layers, and the step size for gradient descent  are decided in advance, before training. Current best practices include performing a random search to find the best hyperparameters.Neural networks are particularly efficient tools for function approximationIn the particular case of decoding a stabilizer quantum error correcting code we want to map syndromes to corresponding physical errors, hence, we take the input layer to be the syndrome (obtained from measuring the stabilizers). For instance, for a toric code of lattice size 9-by-9 we have to measure 81 plaquette operators and 81 star operators for a total of 162 input neurons . Similarly, we set the output layer to be the prediction for what physical errors occurred . Using the same example, we have 162 physical qubits and we need to track their eigenvalues under both Z and X operators, requiring a total of 324 output neurons .σ(x) = (1 − ex−)−1∈. The size of the hidden layer was set to four times the size of the input layer. These decisions were reached after an exhaustive search over possible hyperparameters tested on toric codes of distance 3 to 6, and proved to work well for bigger codes as well. The number of hidden layers was varied - deeper networks produce better approximations up to a point of diminishing returns around 15 layers. The step size for the gradient descent (a.k.a. the learning rate) was annealed - gradually lowered, in order to permit rapidly reaching the minimum. The distance measure between training and evaluation data that is being minimized by the gradient descent is their binary crossentropy (a measure of difference between two probability distributions discussed below).To completely define the neural network architecture we set the activation functions of the hidden layers to tanh and the activation of the output layer to the sigmoid function p - the probability of no error happening on a given qubit, or equivalently depolarization rate 1 − p. Under this model, Z, X, and Y (consecutive Z and X) errors had equal probabilities of 1/3(1 − p). For each value of p we trained a new network, however the results showed some robustness to testing a neural network at an error rate different from the one at which it was trained.The training was done over one billion  pairs in batches of 512, taking about a day of GPU wall time for a 5-by-5 toric code. The pairs were generating on the fly, by first generating a sample error from the given error model , and then obtaining the corresponding syndrome by a dot product with the parity check matrix. The error model used for each physical qubit was qubit depolarization, parametrized by qubit fidelity p, the rate at which a Z eigenvalue flips is Pe = 2/3(1 − p) and independently the rate for X flips is the same. In the example of the toric code the rate of non-trivial stabilizer measurements will be the same for Z and for X, namely Ps = 4q3(1 − q) + 4q(1 − q)3 and the variance will be Vs = Ps − Ps2.The performance of the network was improved if we normalize the input values to have an average of 0 and a standard deviation of 1. For a depolarization error rate 1 − At this point we have not discussed yet how to use the fully trained neural network in decoding. A trained network can efficiently evaluate the approximation of the decoding function (from here on referred to as DECODE: syndrome → error), so all Alice needs to do in order to perform error correction on her quantum memory is to measure the syndrome and run the neural network forward to evaluate DECODE(syndrome). However, the neural network is a continuous function and an imperfect approximation, therefore the values in DECODE(syndrome) will not be discrete zeros and ones, rather they will be real numbers between zero and one. A common way to use and interpret those values is to view them as a probability distribution over possible errors, i.e. the i-th value in the array DECODE(syndrome) is a real number between zero and one equal to the probability of the i-th qubit experiencing a flip . This interpretation is reinforced by our use of binary crossentropy as an optimization target during training. In order to deduce what error has occurred we sample this probability distribution. We verify the correctness of the sample by computing the syndrome that the predicted error would cause - if it differs from the given syndrome we resample. This sampling procedure is present in ref. The code for building, training, and evaluating the neural network decoder is publicly available on the authors’ web page, and shell scripts with the parameters for the presented figures are available upon request. Pretrained neural networks can be provided as well.N physical qubits the number of parameters needed to describe a neural decoder of L layers will be 12 for a distance 5 toric code (50 qubits), bigger than 1023 for distance 7 toric code (98 qubits), and bigger than 1037 for distance 9 toric code (162 qubits).On first sight our decoder implementation can look like a look-up table implementation, however we would like to stress the immense compression of data that the neural network achieves. Firstly, one can consider the size of the neural network itself. For a code on Thanks to this compression, to the direct optimization for most probable error, and to the ease of including knowledge of error correlations in the decoding procedure, the algorithm presented here is one of the best choices for decoding stabilizer codes of less than 200 qubits. While we used the toric code for our testing, there is nothing in our design that has knowledge of the specific structure of that code - the neural decoder can be applied to the decoding of any stabilizer code.26 that have the belief propagation as part of their recurrent structure.Due to the probabilistic nature of the sampling, the decoder becomes impractically inefficient for codes bigger than roughly 200 qubits as one can see in Fig. 12 used as part of the renormalization group decoder of 24, hence providing great decoding speed and high threshold values.While this decoder is general and it can be applied to any stabilizer code, one can also design neural network architectures that specifically exploit the lattice structure and translational symmetry of the toric code. For instance, convolutional neural networks are well adapted for processing 2D data. Moreover thanks to the translational symmetry one can envision a decoder that is trained on a fixed patch of the code and it can be used for toric codes of any size. As already mentioned, our decoder can readily replace the sparse code decoderSupplementary Material