Although seeking to develop a general and accurate binding free energy calculation method for protein–protein and protein–ligand interactions has been a continuous effort for decades, only limited successes have been obtained so far. Here, we report the development of a metadynamics-based procedure that calculates Dissociation Free Energy (DFE) and its application to 19 non-congeneric protein–protein complexes and hundreds of protein–ligand complexes covering eight targets. We achieved very high correlations in comparison to experimental binding free energies for these diverse sets of systems, demonstrating the generality and accuracy of the method. Since structures of most proteins are available owing to the recent success of prediction by artificial intelligence, a general free energy method such as DFE, combined with other methods, can make structure-based drug design a widely viable and reliable solution to develop both traditional small molecule drugs and biologic drugs as well as PROTACS. The potential of mean force approaches which rely on physical pathways to move a binding partner relative to another in a complex between bound and unbound states in molecular dynamics simulations were fruitful in analyzing absolute binding free energies of several biological complexes6. Funnel-Metadynamics might be one of the most elegant methods that has been developed so far9, but it was designed only for PLC and not for PPC. Another elegant approach is Free Energy Perturbation (FEP) which generates rigorous relative free energies12, but its applicability was limited to calculation of relative binding free energy changes among congeneric series of ligands, and the relative free energy changes resulting from point mutations of proteins15. In this manuscript, we report the development of a metadynamics-based procedure which brings us much closer to being able to predict protein–protein and protein–ligand binding free energies as well as other types of binding free energies, with demonstrated generality and high accuracy.A general method able to accurately calculate binding affinities of protein–protein complexes (PPC) and protein–ligand complexes (PLC) would be highly empowering for rational design of biologic and small molecule drugs. . It would also offer researchers a theoretical tool to study processes involving molecular recognition and interaction. Although development of such a method has been the focus of continuous effort for decades, only limited successes have been achieved. For example, empirical functions were developed to be used in docking codes, which are critical to making the docking codes work but usually poor in predicting binding free energies. Molecular Mechanics Poisson-Boltzmann/Surface Area methods and other similar methods were sometimes successful in generating binding free energies for specific systemsThe procedure to be reported in this paper generates Dissociation Free Energies (DFE) of molecular complexes such as PPCs and PLCs. We used this procedure to calculate DFE values for 19 non-congeneric PPCs and hundreds of PLCs (eight different protein targets), and we achieved good to excellent correlations in comparison to experimental binding potencies.18, this procedure performs a number of one-way trip runs. In each run, the system undergoes dissociation and does not re-associate due to the immense space available to explore in the dissociated state. In such a run, how long the system stays in the bound (associated) state depends on the free energy well or the intrinsic stability of the complex, while how long it stays in the free (dissociated) state is arbitrary depending on the simulation time. As long as the system stays in the bound state, Gaussian energies will continuously be added into that region. When the accumulation of the Gaussian energies fills up the bound state well, the system exits from the bound state. Therefore, the accumulated Gaussian energies of the bound state provide the raw data from which we can calculate a free energy quantity, namely the DFE of the system. DFE can be understood as a measure of the degree of difficulty of pushing the system out of the bound state region. The Gaussian energies added into the free state do not contribute to this DFE and can be ignored. Giving that a single run inevitably includes errors associated with the random nature of the simulation, multiple runs are performed until the calculated DFE converges to a stable value. According to the metadynamics terminology, the distribution of the accumulated Gaussian energies (functions) is called the negative image of the free energy surface . A Gaussian energy impulse centered at a running value of the CV is periodically added to the total potential energy of the system at every time interval along the progression of the simulation, which forces the CV to increase, leading to the dissociation of the molecules see Fig. a. Unlike20 to do metadynamics runs and we wrote several scripts to perform job launches and post-run data processing including FES regeneration, averaging, DFE calculation and convergence analysis. However, this procedure can be readily reproduced based on other metadynamics programs.We used the Desmond metadynamics module implemented in Maestro suite20. The calculation of DFE and the convergence analysis will be illustrated in more detail below.The calculation of DFE of a complex is composed of the following steps. 1) A first set of independent metadynamics runs are launched. Each run shares the same starting point while using a different random seed, and typically proceeds for 10–40 ns. (2) After completion of these runs, the FES of each run (namely the primitive FES) is calculated  are given in Fig. Denoting the averaged FES using symbol g(D), a nominal partition function Q can be calculated using the following equation:0. Only the region around r0 with energy higher than the minimum point by 4 kcal/mol or less has significant contributions to Q. Therefore, the integration can be done over the entire D region sampled, which would have no significant difference compared to integration over the bound state region. In other words, practically, a can be set to zero and b can be set to the distance where the wall is placed. . One does not have to determine where rb is to calculate Q.It is important to point out that mentclass2pt{minimDFE is calculated from the nominal partition function as,DFE is calculated based on the averaged FES which depends on how many runs are used in the averaging process. To examine if DFE converges into a stable value for a given system, DFE is first calculated using only one run, and then recalculated iteratively by incorporating additional runs into the averaging. An example of a plot of DFE as a function of the number of runs (N) used in the averaging is shown in Fig. 27. The details of the selection are given in the “Nineteen PPCs were selected from the Protein–Protein Interaction Database which provides a collection of crystal structures of PPCs with the corresponding experimental binding affinitiese and dissociation constant Kd of each PPC. The plot of ∆Ge against calculated DFE for the 19 complexes  appears to be an outlier. If this point is excluded from the Least-Square-Fitting, the coefficient of determination R2 and the coefficient of correlation R are 0.84 and 0.92, respectively, indicating a very high correlation. The relationship between DFE and ∆Ge from the Least-Square-Fitting was as follows, with a standard error of 1.61 kcal/mol.The derived averaged FES and convergence plots for these PPCs are given in Supplementary Fig. xes Fig. a indicat2, R and the standard error change to 0.74, 0.86 and 2.06, respectively, which still indicates a high correlation.If the outlier is not excluded, Re, but the individual DFE values differ from the corresponding individual ∆Ge values. This is consistent with the concept that DFE is not the binding free energy by definition, but it is closely coupled with it. The major difference between the concept of DFE and that of binding free energy is that DFE quantifies the work required to dissociate a complex while binding free energy is the free energy difference between the bound and free states. DFE does not require the calculation of the free state explicitly, neither does it require the calculation of the contribution of the standard state concentration28. The ignored terms in DFE may be mostly the constant terms of the binding free energy. More analysis about what are missing in DFE is given in the “The results indicate that DFE is linearly correlated with ∆Ge and DFE was used to convert the DFE values into calculated binding free energies ∆Gc  as well as the total number of runs, chemical time in each run before any extension, and corrections made if any, for each PPC, respectively. Overall, about 13% of the runs could not be considered as completed one-way trips. These runs were either extended to longer simulation times or removed from the calculations according to the process detailed in the “c significantly for any of the complexes; the changes due to the correction are either close to or smaller than 0.5 kcal/mol. In terms of DFE, only three of the 19 complexes showed a difference more than 1 kcal/mol  before and after the correction. The correlation indexes R2 (0.84) and R (0.91) without the correction remained practically the same as those with the correction.The above results were obtained after applying the correction process. Table 10.To investigate the performance of the DFE procedure in predicting protein–ligand binding free energies, we used a published dataset of PLC structures with experimental binding affinities, which had previously been used to benchmark the FEP methode for each target separately, with the Least-Square-Fitting analysis. The derived R2 and standard error for each target are included in Columns 6–7 of Table 2 varied from 0.67 for CDK2 to 0.32 for BACE. R2 for Thrombin was exceptionally low (0.01), however, exclusion of two outliers brought R2 to 0.62. The average R2 across all the eight targets was 0.45 without excluding any points and 0.53 after excluding the two outliers in Thrombin data. Interestingly, the standard error did not follow the trend of R2, but distributed flatly between 0.57 and 1.09 kcal/mol. Thrombin had the smallest standard error. These observations suggest that the prediction errors for different targets are similar and reasonably small, and that the variations of R2 among different targets are mainly due to the different binding free energy spans of the ligand sets. For example, the span for CDK2 is 4.2 kcal/mol while the span for Thrombin is 1.7, causing a higher R2 for the former than for the latter. Likewise, the R2 for these PLCs are generally smaller than the R2 for the PPCs because the latter has a much larger binding free energy span .The derived averaged FES and convergence plots of all the PLCs are given in Supplementary Figs. e derived from Least-Square-Fitting for each target is given in Column 8, Table The linear relationship between DFE and ∆G10 for these same sets of PLCs. A comparison of the R2 and standard error between the two methods in terms of predicting experimental binding free energies is summarized in Supplementary Table The overall predictive power of the DFE method was similar to that of the FEP+ methodThe intermolecular distance used as the CV in the calculation of DFE of a complex needs to be defined such that its forced increase directly leads to the dissociation of the complex without significant conformational changes unrelated to the dissociation. Although it is difficult to differentiate which conformational changes are related to the dissociation and which are not, we took the following precautions in the choice of CVs: (1) we avoided using any residues from flexible loops to define a CV because the Gaussian forces acting upon such a CV could cause conformational changes of the corresponding loops which might not originate from the dissociation process. (2) The CV for a PPC was defined as the distance between the centroids of the backbone heavy atoms of a pair of residues, each from different proteins in the complex. The CV for a PLC was defined as the distance between the centroid of the heavy atoms of a residue of the protein and the centroid of the heavy atoms of the whole ligand in the complex. And (3) the protein residues used to define a CV were tightly packed within the respective proteins and were located within either an α-helix or a β-sheet of multiple strands when possible.c due to the different choices of CV was less than 1.3 kcal/mol, or 0.5 based on the standard deviation of DFE. Given that the standard error of Eq. (We used one PPC 3A4S) to explore the effect of the change of CV definition on DFE, in which we tried seven different CVs by changing the residues defining the CV. For each of the CVs, DFE was respectively calculated with and without the correction process  Conceptually, the DFE of a complex is derived from the sampling of the time period of the entire bound state up to a point where the system exits from the bound state well. (2) The factors captured include at least the interactions between and within the solute molecules, effects stemming from the thermal motions, conformational changes within the sampled period and solvent effects. And (3) the period after the dissociation is not considered, in which the solutes undergo free movements and isolated conformational variations as well as much weaker nonspecific interactions. The free movements of the solutes in the free state are independent of the solute molecular interactions, contributing a constant term to the binding free energy. The nonspecific interactions of the system after its exit out of the bound state are solute-dependent but may be much weaker relative to the bound state interactions and the entropic term of the free movements of the free state. The role of the conformational changes or dynamics of solutes in the free state (before one solute finds an entrance into another) is not captured in DFE and may be significant for some systems. More studies may be useful to clarify this point in the future. However, the role of the conformational changes or dynamics for the period between the entrance and the “endpoint” of the bound state is accounted for in the DFE procedure.Although the DFE procedure samples the free energy surface based on the dissociation events, the free energy surface obtained would be the same as that obtained from the association events for the bound state region. This is because binding is a reversible process. An event that travels from the free state to the bound state (namely association event) would sample the same bound state free energy well as an event from the opposite direction (dissociation event). This is true for both real and simulated events. Based on this reasoning, the calculated free energy should have captured the effects of part of both on- and off- processes of binding, the part that covers the bound state phase.In general, the averaged FES has a primary minimum corresponding to the bound state significantly deeper than other regions along the CV. As a result, the derived DFE primarily depends on the bottom of the minimum . The sampling beyond this bottom region does not contribute to the DFE significantly. In the other words, in addition to the free state region being unimportant for DFE, the transition state and bound state regions beyond the bottom of the minimum are also unimportant. This implies that after the system moves away from the bottom region, even though the molecules may retain much of the native interactions and/or gain some nonnative interactions, the simulation of these regions does not contribute to the DFE significantly. This is counterintuitive but true, and it is one of the reasons for the robustness of the DFE calculation. However, this does not mean that some parts of the native interactions are not needed for DFE. Every part of the native interactions contributes to the free energy values of the bottom region of the minimum and thus contributes to the DFE.6 and between DFE and other metadynamics methods9 are given as follows. (1) In the potential of mean force methods, a complex is dissociated by forcing one of the binding partners to move relative to the other along a user-defined path. Such a steered movement is achieved by dividing the path into n windows and using restraining forces to gradually move the system from one window to the next. In comparison, a DFE run does not require a user-defined path: the system moves in an unrestrained molecular dynamics simulation governed by the total energy which is the sum of the normal molecular energy and the added Gaussian functions. Gaussians are periodically added to wherever the system is in an unrestrained simulation rather than to pre-defined locations. The system is driven out of the bound state well by the accumulation of the Gaussian energies via any pathways that it can find at random in the simulation. (2) Unlike the potential of mean force methods, a DFE simulation does not require any restraints or constraints on any translational, rotational or conformational variables of the molecules. (3) Unlike other metadynamics methods, a DFE simulation does not use comprehensive collective variables but uses a single physical distance as the collective variable. And (4) the DFE procedure achieves convergence despite the above three simplifications because it uses one-way trip simulations in which the dissociated partners are not required to re-associate, and it reaches the convergence by averaging multiple independent runs. In the traditional methods, the dissociated partners are required to re-associate (backward trips) for convergence, which poses a problem that the partners cannot find the original native interaction mode if their orientations and conformations are not restrained. Thus, there is a need of imposing system-dependent restraints or constraints and using comprehensive collective variables in the traditional methods. In contrast, the DFE method is free of these complications and can be performed much more easily.Some comparisons between our DFE method and the traditional potential of mean force methodsOne drawback of this method is that it does not predict binding modes. One remedy is to combine DFE with docking using DFE to select binding modes and generate more accurate estimates of binding free energies.30 methodology has removed a major obstacle hampering structure-based drug design. The combined use of DFE with artificial intelligence31, docking33 and de novo design36 could overcome the remaining barriers and make rational drug design a widely viable and reliable approach. Several important issues such as hit identification, potency, selectivity and off-target toxicity can be addressed using this type of structure-based approach. This applies both for small molecule drugs and biologic drugs. And finally, DFE can also be used to study protein–protein interactions of trimer complexes in the design of PROTACS41.The recent success of prediction of protein structures of almost the entire human proteome with high accuracy with AI-driven AlphaFold42 (version 2019-3 with OPLS3e force-field43) to generate raw data, and by processing the raw data to obtain DFE using our own scripts. It is composed of the following steps.44 is used to add hydrogen atoms, patch end groups, add missing side chains and missing loops, assign protonation states of histidine, aspartate and glutamate at pH 7.045 and optimize the polar hydrogen orientations in crystal water molecules and proteins.Protein preparation. The crystal structure is imported into Maestro GUI. The Protein Preparation Wizard panelSimulation system setup. The complex is solvated in a cubic solvent box of which the size is set to be about the sum of the largest dimension of the complex and 35 Å. The solvents are composed of water molecules, sodium ions and chloride ions with a salt concentration of 0.15 M. Any potential ligands are checked with Force Field Builder module. If dihedral angles with suboptimal parameters exist, the corresponding parameters will be generated using this module which fits the force-field energy against the quantum mechanical energy for the related fragments.Preparation of metadynamics input files. The solvated complex is loaded into the Metadynamics panel. The CV distance is defined. The Gaussian width and height are set to 0.05 Å and 0.01 kcal/mol, respectively. A wall at the CV value equal to the sum of the largest dimension of the complex and 30 Å is set to prevent the CV from going too far. The time interval between which a Gaussian is injected is set to 0.09 picoseconds (ps). The simulation temperature and pressure are set to 310 K and 1.01325 bar, respectively. The simulation time is set to the desired time . This step generates three input files , a command file (sh) and a force-field directory opls_dir.Generation and submission of multi-replica jobs. This step is done using one of our scripts. The initial input directory and the input files therein are replicated into multiple replicas which differ from each other only by the random seeds used. A typical scenario is 50 replicas with random seed changing from 2007 to 2056. Each replica is launched into a different instance of type p3.xlarge in an AWS virtual cluster, and each replica employs one GPU of type V100 and one CPU.20 run of each replica. Each replica run undergoes multiple stages as instructed in the msj file as follows. (a) Brownian dynamics of NVT ensemble: temperature 10 K, small timesteps, restraints on solute heavy atoms and 100 ps. (b) Berendsen dynamics of NVT ensemble: temperature 10 K, small timesteps, restraints on solute heavy atoms and 12 ps. (c) Berendsen dynamics of NPT ensemble: temperature 10 K, restraints on solute heavy atoms and 12 ps. (d) Berendsen dynamics of NPT ensemble: temperature 310 K, restraints on solute heavy atoms and 12 ps. (e) Berendsen dynamics of NPT ensemble: temperature 310 K, no restraints and 500 ps. (f) Production. Martyna-Tobias-Klein dynamics of NPT ensemble with metadynamics is performed at a temperature of 310 K without restraints up to the targeted simulation time. This generates a kerseq file recording all the Gaussian functions infused into the system along the simulation time, the trajectory file and other output files.Multisim16. D is a spatial variable that covers the full range of the CV variations.Calculation of primitive FES from each replica run. The expression of the Gaussian energy function G(D) that is added to the system at a time point t isThe free energy value at a CV point is the negativity of the sum of all the Gaussian energies at that point. The corresponding distribution of the free energy versus the CV is the so-called FES. Although the module automatically generates a FES file at the end of each run, a new FES is generated by dividing the sampled region of CV into 80 equal intervals and calculating the free energy at each division point by summing up the values of all the Gaussian energies at that point, based on the Gaussian functions recorded in the kerseq file. The resulted FES file contains an 80 × 2 table in which each row is composed of a CV value and the corresponding free energy value, and there are 80 such rows.1 (D1 < D) and D2 (D2 > D) and the free energy values at D1 and D2 are g1 and g2, respectively, the free energy value g at D is as below.Calculation of average FES. This step is done using one of our scripts with an algorithm as follows. The primitive FES files of all the runs are scanned to find the lowest and highest values of CV that have been visited. The segment of CV between the lowest value and the highest value is divided into equal intervals with the interval size equal to the smallest of the different interval sizes of different primitive FES. The free energy value at each division point (based on this new division) for each primitive FES is calculated based on the two closest points in the previous primitive FES file using a linear interpolation scheme. Assuming that, for a division point at CV value D in the new division, the closest points to D in the old primitive FES file correspond to DAfter the recalculation of each primitive FES for the common division scale above, the average free energy value at each division point of CV is calculated by averaging the free energy values from all the primitive FES of replicas.0 and the boundary rb between the bound state and the free state are determined based on the average FES. A one-way trip run is a run that the CV travels from the bound state region (< rb) to the region outside of it (> rb), and it does not come back as reflected in the CV versus time plot .The calculation of DFE of a complex consists in a multi-step procedure by employing the Desmond module implemented in Maestro Drug Discovery Suite10 on a per ligand basis.The time of calculation of DFE of a complex is dominated by the metadynamics simulation time, since the time to run the scripts for post-run data processing is insignificant. Assuming that one deploys sufficient instances to allow all the runs of a complex to go in parallel, and that each run has a length of about 10–40 ns, the whole process takes about 1–4 h on p3.xlarge/V100 instances . In the calculations of PLCs, the computer time of the DFE was about the same as that of the FEP+27. We carefully analyzed the full set and removed the complexes where the protein–protein interfaces met the following conditions: (1) consisted of more than two proteins, (2) were proximal to the truncated termini of protein constructs used in the structures, (3) involved coordinated metal ions, and (4) were close to missing loops of the proteins. In addition, the complexes that contain ATP or GTP analogs were also removed since the experimental interaction geometries around these groups were usually difficult to be maintained by constraint-free simulations due to the existence of coordinate bonds and strong charge-charge interactions. From the remaining structures, we randomly selected 19 complexes. These complexes spanned the full experimental affinity range from micromolar to femtomolar Kd about uniformly and were structurally diverse in size and nature of interactions. Most binding interfaces consisted of a mixture of hydrophobic interactions, hydrogen bonds and salt bridges. Our selected complexes included the ones interacting mostly through the helical packing or beta-sheet interfaces, as well as the ones interacting via a mixture of different secondary structure elements. Thus, we selected a reasonably diverse subset of protein complexes to test the performance of DFE method. The PLC structures and the binding data were taken from reference10 without exception.A small subset of PPCs were selected from the Protein–Protein Interaction Affinity Database 2.0 which contains 179 high quality PPC crystal structures with the corresponding experimental binding constants, the so-called “affinity benchmark” that the researchers had prepared to validate various protein–protein docking programsSupplementary Information.


We often need to interact with targets that move along arbitrary trajectories in the 3D scene. In these situations, information of parameters like speed, time-to-contact, or motion direction is required to solve a broad class of timing tasks . There is a large body of literature addressing how we estimate different parameters when objects move both in the fronto-parallel plane and in depth. However, we do not know to which extent the timing of interceptive actions is affected when motion-in-depth (MID) is involved. Unlike previous studies that have looked at the timing of interceptive actions using constant distances and fronto-parallel motion, we here use immersive virtual reality to look at how differences in the above-mentioned variables influence timing errors in a shooting task performed in a 3D environment. Participants had to shoot at targets that moved following different angles of approach with respect to the observer when those reached designated shooting locations. We recorded the shooting time, the temporal and spatial errors and the head’s position and orientation in two conditions that differed in the interval between the shot and the interception of the target’s path. Results show a consistent change in the temporal error across approaching angles: the larger the angle, the earlier the error. Interestingly, we also found different error patterns within a given angle that depended on whether participants tracked the whole target’s trajectory or only its end-point. These differences had larger impact when the target moved in depth and are consistent with underestimating motion-in-depth in the periphery. We conclude that the strategy participants use to track the target’s trajectory interacts with MID and affects timing performance. An additional factor that may influence performance when targets move around us in three-dimensional (3D) space, creating motion-in-depth (MID), is the changing distance of the object with respect to the observer. These changes affect the retinal speed of the object even when the 3D motion is constant. Yet, to properly interact with moving targets speed estimates should remain approximately constant.In daily-life we often need timing our actions with moving objects that can travel at different speeds and in different directions . In order to succeed in tasks that require synchronizing actions with particular times or locations of moving targets, people would certainly benefit from ascertaining parameters like the target’s speed, time-to-contact (hereafter TTC) or motion direction. Knowledge of such parameters can help people better predict future positions of the object and plan timing responses in advance. Moreover, information such as velocity can be used to update targets’ changing positions8 or following parabolic flights on the frontal plane10 have been extensively used. Thus, despite the relevance of MID for daily life interceptive timing  the effects of different 3D trajectories on interceptive timing, i.e., when a response is required within a particular time window, have been mostly overlooked by previous studies but, see12 for some exceptions in coincidence timing and motion extrapolation tasks respectively.The use of visual fronto-parallel motion, which does not contain a depth velocity component, has predominated in studies aiming at understanding how we time our motor actions. Consequently, targets moving in linear motion14, and prone to individual differences15. This is probably because there is an overabundance of available depth cues16 that makes predictions on timing performance more difficult. When MID is involved, typical monocular cues include optical expansion17, shading and texture gradients19, kinetic depth cues and motion parallax, among others. The main binocular cues for MID include the change of retinal disparity, and inter-ocular velocity differences22. However, all these monocular and binocular retinal cues need the support of extra-retinal signals23 for the estimation of 3D motion, that is MID. In addition to this diversity of cues, the perceived speed of MID depends on which part of the retina is stimulated24 and the perceived spatial 3D trajectories or motion extent are affected by well known biases28. The variability in the perception of MID, including speed and direction, makes it worth studying the performance of response timing when dealing with objects moving in depth. While most of the studies interested in MID have used perceptual judgements of speed or direction, timing performance has not been tested and it is difficult to predict due to the large number of MID cues and individual differences in processing them. An exception of interceptive timing tasks with motion other than linear or fronto-parallel are all the studies involving parabolic motion in depth , which have been motivated by testing specific models usually involving internalized variables like Gravity or physical size35 rather than the effects of MID themselves.Several monocular and binocular cues both retinal and extra-retinal contribute to the perception of speed and direction in MID. Compared to lateral motion, speed discrimination thresholds for MID are usually higherβ) that could be 0, 10, 20, 30 or 40 deg. A β = 0 deg  to recover MID or the different approaching distances often depends on the level of experience of the participant with VR settings37 and that non-experienced users would rely more on monocular cues like optical expansion  used by participants in any way, and so was it the one resulting from their own natural behavior or choices in exploring ways of solving the task. By using different tracking strategies people can exploit more foveal or peripheral information. This is important because some relevant cues for MID are perceived differently in the fovea than in the periphery. For example, speed of MID is underestimated in the periphery24, suggesting a contribution of inter-ocular velocity differences in speed perception in depth40.In this study we used immersive virtual reality (VR) to look at whether and if so, how, performance  in a shooting task differs across multiple 3D movement trajectories in a shooting task. Figure Fourteen subjects  participated in the experiment. Twelve of them were right-handed and two were left-handed as by self-report. All of them had normal or corrected to normal vision (with contact lenses), and none had evident motor abnormalities. The study was approved by the ethical committee of the University of Barcelona (IRB00003099) and followed the requirements of the Helsinki convention. All subjects gave written informed consent to take part on the study.The experiment was run by an Intel i7-based PC . The stimuli were rendered by an NVIDIA GeForce GTX 1070 and sent for display to an HTC Vive Pro head-mounted display (HMD) at 90 Hz per eye (2880 × 1600). The field of view (FOV) of the HMD is 100°  by 110°  for each eye. The head position in 3D coordinates  and the angular information  were recorded by two SteamVR BASE STATION 2.0 at 90 Hz. Participant’s responses were registered through one of the headset controllers.β) between the target and the initial position of the observer  in Fig. β in Fig. The virtual scene and stimuli were created with Unity software . Moving stimuli or targets consisted of a red sphere of 22 cm of diameter  that appeared at one of two possible initial positions see Fig. A. After β angles and targets’ speeds.The target moved at the designated speed towards a shooting location denoted by a violet sphere (same size of the target) that was located at a distance of mentclass2pt{minimEach trial started when the participant entered a white square (50 × 50 cm) rendered on the floor at position   when the target was not there, it turned into a red circle. Pilot sessions showed that the task was quite difficult (less than 20% hits) in terms of getting positive feedback based on the overlapping of target and shooting location. We thus relaxed the temporal constraints to increase the proportion of trials in which positive feedback was provided in order to keep participants motivated. Therefore, we used twice the size of the target as a boundary to define the overlap with the shooting location and give feedback. Participants never reported being aware of this manipulation after questioning them at the end of the whole experiment. Furthermore, participants found the task very engaging and motivating.delayed feedback and immediate feedback) that were run in counterbalanced order across participants. The difference between these two conditions consists in the temporal delay to obtain the feedback about the success after the motor response.The study had two blocked conditions (named β=0° was 0.111 s longer than when the β = 40°. Participants were not explicitly told about the delays.In this condition we examined performance in the shooting task when the feedback of the action was delayed. The shooting task was implemented by triggering a bullet with a gun that was rendered at the same position of the hand-held controller. The bullet was represented by a pink sphere (diameter of 4.4 cm) that travelled at 55 m/s. This speed will introduce a delay between the moment at which the participant pulls the trigger and the moment at which the bullet reaches the target’s path when it is within the shooting location . The delay in reaching the designated shooting location when the trajectory had an angle 42. In that case, when participants pulled the trigger they briefly saw a pink line extending from the gun onwards. By using a laser, the differential delayed impact between the different trajectory angles disappeared. That means that the laser immediately intersected the target’s path.In this other condition, we replaced the bullet by a simulated instant laser transforming the task in a typical coincidence timing taskIn both conditions targets moved on a fine-grained grass textured background that did not provide any landmark or familiar size cue see Fig. B. Each cimmediate feedback condition would allow to see whether any systematic error in the delayed feedback condition is only due to the differential time that it took for the bullet to intersect the different trajectories (β angles) that the target could move along. The second reason is theoretical. It has been reported that timing a response with a single press  favours the use of spatio-temporal variables 43 over positional information. In our case, this could result in using the target speed differently when updating the position of the target. This reason also motivates using different target speeds in addition to the fact that using a single speed usually makes participants use positional or traveled distance information.The motivation underlying the use of these conditions is twofold. The first reason is methodological. The use of the immediate feedback condition, the time at which the trigger is pulled is the same as the time at which the laser crosses the target’s path. Spatial errors were defined as the difference in position between the laser , at the moment the laser (or bullet) crossed the target’s path. Temporal errors were defined as the time between when the laser (or bullet) crossed the target’s path and when the target’s reached that position. In addition to these variables, we also recorded the orientation and position of the head. Since we did not record eye movements we have to infer, and consequently be cautious, where participants might be looking by checking the head orientation in combination with the field of view of the HMD.We registered the time at which participants pulled the trigger in both conditions and the temporal  error in each trial. In the 44, except for the trials in which subjects missed the shooting location . The temporal error between when the bullet reached the designated shooting location and when the target did so. The trials in which the designated shooting location was missed (3.9%) were excluded from the analysis. The temporal errors and the proportion of hits are the dependent variables we use to define performance.Since we instructed participants to hit the target at the shooting location defined by the violet sphere, we only had temporal errors associated to responsestrajectory or the end-point. In order to automatically detect these strategies we used a threshold in the change of the yaw angle (rotation of the head around the y-axis) to categorize the trials into one of these two strategies. A trial was considered as tracking the trajectory if the change in the yaw angle was larger than 20° during the target motion, that is if the head rotated more than 20° during the target motion and the rotation was consistent with the head tracking the target motion. Otherwise, it was considered as tracking the end-point. The 20° threshold value was chosen after manually checking a sample of 100 trials across different participants and obtaining a 99% accuracy in classifying the trials as following one or the other tracking strategy.We also analyzed the head position and orientation during the target’s motion and categorized every trial into one of two possible groups depending on the head movement. A preliminary visual inspection of head orientation  during the target movement in every trial revealed that two possible strategies were used: participants tracked with their heads either the delayed vs immediate) have any significant effect on the temporal error. β). ANOVAs were conducted on the LMM outputs when the analyses involved the tracking strategy, since this is a post-hoc variable depending on individual strategies and would likely lead to an unbalanced design, which is better handled by LMM.Generally, we performed ANOVAs to test whether approaching angle, target speed and feedback condition .From previous literature in TTC and interceptive timing, we know that people tend to commit later temporal errors for faster speeds for both lateral motion46 and that eye movements  can help refine an internal representation of the target motion47; but this has been shown mainly in 2D or lateral motion. However, tracking the moving object does not necessarily lead to more accurate timing responses or better interceptive performance. For example, using real ball 3D catching48 different eye movement strategies led to similar performance. Similarly, studies have shown that errors in an manual interception task (2D motion) did not depend on the quality of pursuit49. In these studies, motion was predictable. One key factor for tracking the target to improve task performance seems to depend on the predictability of the target50. Pursuit would help estimate speed in cases where motion needs to be extrapolated after an occlusion before the moment of interest52. Since motion is predictable in our experiment we do not have strong predictions concerning different possible tracking strategies. However, target speed estimates could benefit from tracking the trajectory but the effect on the interception error, like more anticipatory responses, would be mediated by how much weight is given to speed when updating the position of the target2.Based on previous work, the effect of tracking the target  on timing errors is more controversial. It is known that the speed of targets is better estimated when one directs the gaze at them than when fixatingdelayed feedback condition. This figure illustrates the two tracking strategies followed by the participants. The participant in the upper panels  and by #8 in the immediate feedback condition (yellow bars). Other participants  predominantly used the end-point tracking strategy. In order to give an idea of the number of trials for the proportions shown in Fig. delayed feedback condition, while the frequencies in the immediate feedback condition were 25 and 685 trials for end-point and trajectory tracking respectively. The number of total trials included in each condition is similar for all participants.Figure β. Each participant is shown on a different panel. There is a very clear and consistent trend in the temporal error: the error gets more positive (earlier responses) with larger values of the angle β. This trend is much clearer in the delayed feedback condition . The dashed lines denote what would be the predicted slope of the temporal error if participants had not taken into account the temporal differences of the bullet intersecting the target’s paths for the different angles. Except for participants #6 and #8, the estimated slope of the temporal error is smaller than that predicted if the differential delay had not been considered. However, the immediate feedback condition also shows a significant trend of the temporal error despite the laser intersecting the target’s path at the same time as the trigger is pulled .Figure β angles and the two feedback conditions. By visual inspection one can directly see clear differences in the pattern of temporal errors between the two feedback conditions. As expected, faster speeds provoke later temporal errors and the effect of speed on the temporal error is larger in MID .We used three different target speeds and wanted to know whether they affected the temporal error. Figure β as within subjects variables. Both feedback condition (β angle (β) was significant (β angle gets larger (double interaction) and that this trend is much stronger in the delayed feedback condition (triple interaction). The larger effect of target speed on the temporal error in the delayed feedback condition suggests that speed can be used differently in the two conditions. Since the final temporal error is basically determined at the moment of the response, it will be worth looking at the position of the target when the response was triggered .In order to test the significance of the effects of all these independent variables, we ran an full ANOVA on the temporal error with speed, feedback condition and β in the delayed feedback condition being larger than the slope in the immediate feedback condition. Importantly, the estimated slope of β (0.51 ms/°) in the immediate feedback was significantly larger than zero (t(13) = 5.42, p < 0.001), which denotes a significant contribution of the β angle to the temporal error in this condition. To further test whether this effect is significant we compared two nested linear models in the immediate feedback condition: a null model with only the intercept and a model with β. The model that includes β was better than the null model  are also clear and suggest that the temporal bias is caused by the angle of approach rather than by whether the differential delays across trajectory angles are considered.The effect of the angle trajectory on the observed bias in the temporal error is clear and very consistent across participants. The larger effect in the see Fig. . Howeverβ × tracking strategy failed to reach significance too . The model included the β angle, the target speed and the tracking strategy as fixed effects, and participants were considered as random effect.Figure delayed feedback condition, the ANOVA yielded very significant main effects of approaching angle (β)  = 8.59, p < 0.001). However, the positional difference between speeds becomes smaller as the MID increases  as can be noted by the separation between the medians of the distributions for the different speeds . In addition, when targets moved in depth (larger angles) people responded  when the distance between the target and the shooting location was smaller. This is consistent with compensating for the travel delay of the bullet. Interestingly, however, the rate at which the distance was shortened with β in the delayed feedback condition, was different depending on the tracking strategy. When tracking the end-point participants responded, as β increased, when the target got progressively closer at a rate of 1.18 cm per deg of β . This rate  was smaller (0.3 cm/°) when participants tracked the trajectory  consistent with the significant interaction β × tracking strategy reported above. This different pattern between the two strategies can be explained by participants perceiving MID differently in the fovea than in the periphery. More concretely, the shorter distance of the target to the shooting location when participants tracked the end-point and MID was present is consistent with underestimating the speed in the periphery24.In the immediate feedback condition. Figure β angle and the effect of the tracking strategy is also different. Unlike in the delayed feedback condition, the average distance of the target to the shooting location increased with β rather than decreasing. Note the progressive rightward shift of the vertical lines  in the immediate feedback condition. When tracking the end-point, the distance to the shooting location increased at an average rate of 0.17 cm/° . Like in the delayed feedback condition, the distance increased with β at a larger rate (0.28 cm/°) when participants tracked the trajectory consistently with the significant interaction between β and tracking strategy reported above for the immediate feedback condition. This is again consistent with participants perceiving slower the speed in depth in the periphery, or faster when they tracked the trajectory . The difference, however, is smaller (end-point strategy: 0.17 cm/° vs trajectory strategy: 0.28 cm/°) than in the delayed feedback condition. The fact that we obtain smaller changes of distance between the target and the shooting location with β in the immediate feedback condition can be explained by the fact that the approaching angle β does not introduce any travel delay. However, the differences between tracking strategies cannot be explained by these delays, but are consistent with previously reported MID effects on perceived speed.A slightly different pattern is observed in the delayed feedback condition the width of the distributions seems to narrow as β increases. This effect of the angle of approach is not clearly noticeable in the immediate feedback condition; the width of the distributions does not seem to change with β. After computing the SD of the individual distributions across feedback condition, angle of approach and target speed, we conducted an ANOVA on the SD of distance distribution with these variables as independent variables. The main effect of β was significant in the delayed feedback condition . The positional SD then decreased at a rate of 0.243 cm/°. The target speed had a very significant effect in both conditions (delayed and immediate feedback conditions respectively). This is expected and well known in the interception literature as increasing speed makes positional judgements more difficult53.In addition to the mean distances between the target and the shooting location at the moment at which the trigger is pulled, Fig. β) leads to more anticipatory responses. This pattern is present in the two studied feedback conditions: delayed and immediate feedback. Although part of this trend can be accounted for by the travel delay of the simulated bullet , in the immediate feedback condition there is no such delay and the trend is very significant and consistent across participants too. Apparently, participants failed to fully recover the 3D speed when MID was present, in a similar way as failure of velocity constancy has been shown54 when there is a lack of familiar-size cues that could have otherwise helped scale the changing distance across the different angles, specially in VR settings55. This effect in an interceptive timing task is consistent with more anticipatory errors for trajectories approaching the observer (MID) than for fronto-parallel ones in a motion prediction task reported previously12.In the present study we have shown that MID has an effect on the temporal errors people make in an interceptive timing task (shooting task in our case) that is very consistent across participants and conditions. When compared with fronto-parallel motion, performance when MID is present shows earlier final temporal errors. This conclusion is based on the fact that increasing the angle of approach  first and then processing binocular motion56. Previously, these biases have been shown in trajectory, extent and speed28 and, to our knowledge, we now show that timing responses can also be affected by MID.Accurate perception of MID (both speed and direction) is a complex process that requires integrating extra-retinal  and retinal information so that 2D information from the retinae are properly mapped to a 3D layout, but this transformation can be incomplete58. However, independently of the tracking strategy, MID increased the difference between speeds. We know that accurate speed estimation does not bode well with 3D motion. Not only is speed in depth underestimated with respect to lateral movement61, but the bias affecting perceived speed in depth might have more effect on larger speeds14 under a slow prior interpretation62. The difference between speeds is less clear in the immediate feedback condition. In this condition, the rate of change of the optical gap between target and interception zone becomes an additional relevant variable63 that is less affected by possible incomplete mappings from 2D retinal information to a 3D motion and could have been exploited by the participants.The speed condition also leaves us with some differences that need further discussion. Faster speeds provoke later temporal errors than slower ones as has been reported in previous studies64 whose internal state could be updated by velocity in a Kalman filter-like way2. It could then be that the response is mainly spatially controlled in the delayed feedback condition, while velocity would have to be used as well in the immediate feedback condition since it becomes a task-relevant variable at the very same moment of the response. This is so because the target speed constraints more clearly the temporal window in which a response has to be executed. Although this interpretation is speculative, there is previous evidence from the TTC literature that is consistent with it. When a coincidence timing task using a single key press was used to estimate the TTC of looming objects participants heavily weighted the rate of expansion, an optical variable which contains spatio-temporal structure42. However, in a different study43, where the interceptive action did not consist of a key press and movement time could be modulated, participants relied on the visual angle and much less on the rate of expansion.Another possibility is that velocity and position are used differently in these two feedback conditions. Some studies have proposed that response initiation in interceptive timing tasks would be controlled by positional task-relevant variablesβ). People that mainly looked at the end-point showed a trend to respond later, which would be consistent with them perceiving the speed in depth more slowly  did not undermine performance and did not significantly affect the temporal error. On the contrary, even increased the proportion of hits in the immediate feedback condition significantly because more anticipatory responses when tracking the target reduced the number of hits. Unlike previous studies showing a benefit for tracking52, in our study the target trajectory was predictable from exploiting the sensory input and the contribution of extra-retinal signals to speed estimates could have been less relevant47. This interpretation based on predictability of target motion50 bodes well with studies in which timing performance with 3D trajectories was not affected by gaze strategies48 or the moment at which the trajectory was seen as long as it was not too late to prepare the response65.Another factor that seems to contribute to this incomplete transformation between 2D images and the 3D motion is the tracking strategy. Although we found no significant differences on the final temporal errors due to the tracking strategy participants used, the strategy did affect the response time through its interaction with the angle of approach, that is, when a MID component became more prominent .In spite of the trajectories being predictable and knowing the shooting location in advance, some participants did track the trajectory with the head leading to a trend of more anticipatory responses in both feedback conditions. It is known that keeping the eyes on the moving target enhances the prediction of visual motionWe have reported that MID affects the temporal performance in an interceptive task: the angle of approach of a target with respect to an observer affects the moment of action response in systematic ways. The temporal requirements of the task defined by the type of action effect (delayed or immediate) need to be considered regarding target speed and the amount of MID present in the target trajectory.


The evolution of dense plasmas prior to the arrival of the peak of the laser irradiation is critical to understanding relativistic laser plasma interactions. The spectral properties of a reflected laser pulse after the interaction with a plasma can be used to gain insights about the interaction itself, whereas the effect of holeboring has a predominant role. Here we developed an analytical model, describing the non-relativistic temporal evolution of the holeboring velocity in the presence of an arbitrary overdense plasma density and laser intensity profile. We verify this using two-dimensional particle-in-cell simulations, showing a major influence on the holeboring dynamic depending on the density profile. The influence on the reflected laser pulse has been verified during an experiment at the PHELIX laser. We show that this enables the possibility to determine the sub-micrometer scale length of the preplasma by measuring the maximum holeboring velocity and acceleration during the laser-plasma interaction. Experimental measurements involving multiple laser and plasma parameters are useful in understanding the relativistic laser-plasma interactions. Here the authors extend the model of holeboring for arbitrary profiles of laser pulse and plasma scale lengths. Despite numerous efforts both theoretically and experimentally, the gap remains, showing a clear deficiency in the understanding of the laser-plasma interaction at relativistic intensities. While many high-profile results have been published including proton energies approaching 100 MeV7, further progress requires a more detailed understanding of the interaction. For instance, the evolution of the target during the last instants, before the interaction with the peak intensity begins, requires a more thorough treatment, and subsequently one should control it to reach the ideal conditions for the desired outcome.One of the outstanding questions in relativistic plasma physics is the obvious discrepancy between the kinetic energy of protons accelerated by short laser pulses predicted by theory and simulations more than 10 years ago8.The interaction of short, intense laser pulses with sub-micrometer-thick targets is a field that has been numerically explored with particle-in-cell (PIC) codes extensively. For the thinnest targets, the experimental challenge is to achieve the optimal conditions identified in simulation results. One complication arises, because PIC codes typically simulate the interaction of a laser pulse with a solid target on timescales limited to few picoseconds and therefore cannot adequately describe the evolution of the target in the rising edge of the pulse on the picosecond to nanosecond scale. This requires the initial plasma parameters to be treated as a simulation input. This ad-hoc treatment is the response to the temporal contrast of the laser, that causes the light intensity to reach the ionization threshold at the target surface tens of picoseconds to a few nanoseconds ahead of the pulse peak, and to trigger an uncontrolled preplasma expansion. This uncertainty of the input state has long been considered one of the key reasons for the discrepancy between the predictions given by the PIC codes and the experimental observations10, only allowing for the measurements of long-scale preplasmas. By contrast, short scale lengths can be measured by frequency domain interferometry, when using a probe laser pulses with a duration in the tens-of-femtosecond range12. However, the necessity for a short-pulse probe beam restricts this method from being used at facilities with an intermediate pulse duration in the hundreds of femtoseconds.To benchmark existing PIC simulations with experiments, a precise monitoring of plasma properties like its scale length, temperature and density distribution is necessary. Interferometry or shadowgraphy were used successfully in many experiments before to characterize the preplasma expansion. Their drawback is given by the limited spatial resolution in the micrometer range13 in 1994 and Zepf et al.14 in 1996, relies on measuring the spectral change of a laser pulse which is back-reflected from the plasma during the interaction, which used the holeboring model of Wilks et al.15. This method was further developed to include a time resolved measurement by Sauerbrey16 in 1996 and Haessner et al.17 in 1998, and later on, at even higher intensities, to time-resolve the onset of relativistic transparency18. The work by Kingham et al. was crucial for these measurements, who showed the correlation between the movement of the critical density surface and the introduced phase shift of the reflected laser pulse during the interaction19. To use such time-resolved information about the displacement of the critical plasma density to get insight into the properties of the preplasma, a good knowledge about the behavior of the holeboring dynamics is crucial. An important step for this has been done by Iwata et al. who recently introduced the concept of an upper plasma density limit for holeboring when a constant laser intensity is applied to a exponentially growing plasma density20. According to this model, the laser pulse that interacts with the critical plasma density at first induces a steepening of the plasma density profile at the interaction point. This in turns slows down the holeboring velocity and a commensurate hole boring time is analytically derived.A non-invasive method to gather information about the laser-plasma interaction, performed by Kalashnikov et al.In this paper, we extend the previous work from Iwata et al., by showing that the dynamics of holeboring is completely dominated by what happens in the preplasma region of solid targets irradiated by short-pulse lasers. First, we derive a general formulation of the holeboring velocity for any arbitrary temporal intensity and preplasma profiles. This significantly changes the picture of the non-relativistc holeboring dynamics during the interaction of a intense laser pulse with an overdense plasma, especially in the presence of a steep density profile with scale lengths in the micrometer to sub-micrometer regime. We are able to reproduce the results from 2-D PIC simulations in terms of the maximum holeboring velocity and acceleration for the specific case of the interaction of a Gaussian laser pulse with an exponential density profile of constant scale length. The maximum holeboring acceleration and velocity shows an unambiguous correlation to the laser intensity and present preplasma scale length. To verify this behavior, we performed an experimental campaign at the PHELIX laser, in which we applied time-resolved Doppler spectroscopy on the light reflected by the target to follow the hole boring dynamics, confirming the findings. In addition, we show that one can infer the preplasma scale length from a measurement of the maximum holeboring acceleration and velocity, in a time-integrated manner.v(t) in the presence of an arbitrary plasma density profile ne at the interaction point, whose position defines the critical surface. The necessity to account for an exponential density profile has been discussed by Kemp et al.21 and used to calculate the position of the critical surface during the interaction, which has also been used by Iwata et al.20 to calculate the holeboring duration and the plasma density cutoff. Our derivation is kept more general to include an arbitrary density and laser intensity temporal profile, while assuming that the preplasma is already formed at the arrival of the laser pulse. This is obtained by solving the equation of the pressure balance niMiv2 = cos(θ)I/c22, with a spatial and temporal dependent electron density distribution ne = Zni, the laser intensity I(t) and extended by the influence of the incidence angle θ. We assume a constant ion mass Mi and corresponding charge state Z during the interaction. Note that the charge state is the charge state that yielded the plasma expansion before the interaction begins. When the hole boring process has started, further ionization of the target ions plays a negligible role, as the hole boring models only considers a momentum transfer from the laser to the ions. It has to be noted that the effect of plasma pressure increase due to plasma heating is neglected. The further assumption that we use to solve this equation is given by the electron density, which has to fulfill the relation nc. With the definition of the time-varying scale length To improve our understanding of the holeboring dynamics, we developed an analytical description for the holeboring velocity Lc(t) = Lc for the initial profile at the turning point. This results in:Equation  shows th22 for a constant plasma density. This shows that the impact on the holeboring is only given for short scale length plasmas, in the sub-micrometer regime, and is therefore not in conflict with previous models or measurements14.If the scale length of the plasma tends towards infinity, the scale length dependence vanishes and the velocity can again be described by the description of Kruer et al.Not only the holeboring velocity increases for increasing preplasma scale lengths and laser intensities, but also the holeboring acceleration is strongly influenced by these properties, which can simply be calculated by the temporal derivative of Equation .Lc in the sub-micrometer range and laser intensities ranging from 1018 W cm−2 to 1021 W cm−2 with a laser pulse duration of Δt = 250 fs and a linear polarization. The holeboring velocity is extracted from the simulation data via spectral analysis (Doppler spectroscopy) of the reflected pulse. A Fourier analysis of the electric field yields the instantaneous wavelength of the laser pulse, which relates to the holeboring velocity, with the assumption of negligible self-phase-modulation (SPM)23.To assess the validity of our one-dimensional model, we performed 2-D PIC simulations. The simulations are performed for an exponential density profile with a certain scale length As the laser intensity rises during the interaction, the holeboring velocity and acceleration increases, which, according to the analytical description, should depend on the preplasma scale length. As a verification, we first monitored the maximum holeboring acceleration for different maximum laser intensities and preplasma scale lengths, which is visible in Fig. t = 250 fs, different peak intensities and different preplasma scale lengths, for an incidence angle of θ = 0∘. The results of the calculated maximum holeboring acceleration for different laser intensities is given by the dashed colored lines in Fig. For a direct comparison, we calculated the maximum holeboring acceleration for a Gaussian laser intensity distribution with a pulse duration of ΔThis indicates that the preplasma scale length can be retrieved using our model by measuring the maximum holeboring acceleration and the incoming laser intensity distribution.In addition we monitored the maximum red shift of the spectrum which corresponds to the maximum holeboring velocity, again under the assumption of negligible SPM. We extracted the maximum wavelength of the complete spectrum, averaged by the value at 1 and 10% of the maximum spectral intensity, for the different peak laser intensities and plasma scale lengths, which are shown by the colored dots in Fig. n0 and nc and the ion sound speed cs, containing the electron temperature of the plasma24.In combination with this, we calculated the maximum wavelength, using the maximum holeboring velocity from Equation  and addiTaking this initial expansion velocity into account, the dashed colored lines represent the analytical maximum wavelength of the spectrum, which again shows the same behavior as the 2-D PIC simulations. This shows that a measurement of the maximum holeboring velocity grants a possibility to determine the plasma scale length with a high precision. It is possible to directly determine these quantities with either a time-resolved or time-integrated measurement of the reflected spectrum during the laser plasma interaction.25, using a plasma mirror (PM) to enhance the contrast and therefore reduce the preplasma formation. The reflected spectra of the s-polarized pulse were measured and the maximum wavelengths, taken at 1 and 10%  of the spectral intensity, were extracted for different target thicknesses. The averaged results and the corresponding uncertainty  are given in Fig. To verify this behavior and use the derived equation to determine the preplasma scale length, we conducted an experiment at the Petawatt High-Energy Laser for Heavy-Ion eXperiments (PHELIX) at the GSI Helmholtzzentrum für Schwerionenforschung GmbHThe spectra reflected by the target are substantially different for the two configurations, indicating that the preplasma conditions strongly influences the motion of the critical density. The spectral cutoff is shifted by (19 ± 6) nm and 2. However, the analytical description and correlation to the scale length holds true for the incidence angles considered here, when taking the change of the reflection point into account, which is further described in Supplementary Note It also has to be noted that the direct correlation between the critical density movement and the phase shift of the reflected laser pulse is not necessarily given for large incidence angles and a p-polarized laser pulseThis technique in combination with the derived analytical description of the holeboring velocity and acceleration seems to offer a powerful tool for the determination of the interaction dynamics in general and more particularly the preplasma properties. By extracting the acceleration from the instantaneous frequency of the reflected pulse, the maximum wavelength of the spectrum and the expansion velocity, which is given by the initial blue shift of the laser pulse, the preplasma scale length can be calculated and the temperature of the plasma estimated. This allows the determination of the preplasma properties on the picosecond time-scale before the laser peak intensity is reached, which helps to improve our knowledge about the laser-plasma interaction.20 W cm−2 with maximum pulse energies of 180 J on target and a pulse duration of (500 ± 75) fs using a F/1.7 focusing parabola with s-polarization. The system ensures a high temporal contrast on the nanosecond-to-100-picosecond time scale of 10−12 27. From 100 ps on, the pulse intensity increases with a factor of 3.55 every 10 ps. The contrast was additionally improved by four orders of magnitude using a double plasma mirror (PM), which additionally steepened the rising slope of the laser pulse switch on time. The intensity after the PM dropped to ≈1 × 1020 W cm−2, at an angle between laser and target of 30 ∘. The used foil targets consist of polystyrene (CH) with a varying thickness between 35 nm and 1 μm.The experiment was conducted at the PHELIX facility. During this beamtime, the system provided intensities of up to 2.4 × 10The schematic experimental setup is given in Fig. 28, and a 1 ω time-integrating spectrometer  for comparison. The time-ambiguity of the SHG-FROG was overcome by a comparison to the analytical description of the holeboring velocity. This shows that the maximum red shift occurs prior to the maximum of the back-reflected intensity. This enables the possibility to flip the time of the measurements accordingly. The spectra and FROG-traces were recorded for several shots for the two different temporal contrast options and varying target thickness.A small part of the beam with a full opening angle of 3° in specular direction of the reflected beam was image-relayed to the diagnostics, which consisted of a time-resolved spectrometer based on Frequency-Resolved Optical Gating (FROG), which has been built and commissioned at our facility29 on the Kronos cluster at the GSI Helmholtzzentrum für Schwerionenforschung GmbH. For the simulation, a laser pulse with a central wavelength of 1053 nm, a pulse duration of Δt = 250 fs (FWHM) and varying peak intensity ranging from 1018 to 1021 W cm−2 was incident on a solid-density target normal to the target surface, with an initial peak electron density of 4 × 1023 cm−3 and a thickness of 1 μm. The laser was focused to a spot size of 6 μm (FWHM) at the target front surface, initially located at x = 0 μm. The simulations were performed with a spatial resolution of 12 nm in the direction of the initial laser propagation, and 19.5 nm perpendicular to it. The grid size was 27,500 × 8912 cells, which corresponds to a simulation box ranging from −320 μm to 10 μm in x- and −80 μm to 80 μm in y-direction. This simulation contained ≈2.2 × 109 particles. The laser pulse entered the simulation box with an intensity ratio of 10−4 at the simulation time t = 0 fs and hits the target at t = 1150 fs. To account for different expansion states of the target, the initial electron temperature was varied between 10 eV and 7 keV, whilst assuming a fully ionized CH2 target. Due to the time difference between simulation start and the arrival of the pulse at the target, a pre-expansion occurs, increasing the scale length in dependence on the initial temperature. This leads to an exponential density profile with scale lengths of 0.04 μm, 0.269 μm, 0.44 μm, and 0.739 μm for the initial electron temperatures of 10 eV, 0.5 keV, 2 keV, and 7 keV, respectively.The simulations were run with the 2-D PIC code EPOCHThis overdense plasma deflects the laser towards the incoming direction, and we analyze the reflected radiation on the reflection axis. When the electromagnetic pulse propagates again in the vacuum region, an unambiguous space-time transformation is performed to obtain a temporal representation of the light pulse.ϕ(t) being the temporal phase of the pulse. The corresponding velocity of the critical density surface v is now calculated by solving the equation of the Doppler shift β = v/c, whereas c corresponds to the speed of light, λs and λ0 being the shifted and central wavelength and the incidence angle θ.The time-resolved information is gathered by calculating the instantaneous frequency 19 W cm−2 can be seen in Fig. Lc = 0.04 μm) and a longer scale length (Lc = 0.739 μm), which corresponds to a cold and hot plasma with Te = 10 eV and Te = 7 keV, respectively. The black solid line corresponds to the intensity of the incoming laser pulse.An example for the resulting instantaneous wavelength shift of the reflected pulse with an initial intensity of 1023. The dashed black lines indicate the region that is used to calculate the maximum holeboring acceleration. The analytical holeboring solution indicates that the maximum acceleration occurs at different interaction times, depending on the laser intensity and plasma scale length. Therefore, the temporal analysis region was chosen such that the maximum acceleration should occur within these limits for all parameters. Since the instantaneous wavelength still contains some oscillations within this temporal regime, which would lead to large spikes when performing a temporal derivative, we perform polynomial fits within this temporal region, ranging from 3rd to 5th order, to get information about the averaged acceleration. The corresponding maximum acceleration values for the different polynomial orders are averaged and their respective uncertainty was included in the peak-to-valley uncertainty presented in Fig. In comparison, the yellow dots show the expected Doppler shift which would be introduced by the movement of the critical density surface, demonstrating a good correlation with the instantaneous Doppler shift of the reflected pulse. The shift can therefore be used as a temporally-resolved measurement of the critical-plasma-density motion during the interaction, under the assumption of negligible self-phase-modulation (SPM)The holeboring velocity is calculated from the balance equation between the ion momentum flux and the laser pressure.Z and mass Mi during the interaction. After performing a temporal derivation of this equation it can be rearranged to a differential equation of the first degree:We assume a constant ion charge state which accepts: k can be determined when assuming that the velocity of the critical density for t → −∞ results in the equation  = v(t) + v0. For this reason a measurement of the acceleration is more favorable, since the constant term is removed during the temporal derivation. In case of a constant plasma scale length, the acceleration can be derived by the temporal derivative of Equation (The integration constant equation  with a cThis equation can be used to find the maximum acceleration during the interaction, which can be correlated to the preplasma properties. The best agreement between the simulation results and the analytical description is given when assuming a constant scale length and neglecting any temperature increase throughout the interaction. An example for this can be found in Supplementary Note Supplementary Information


The development of new methods for generating pulsed electromagnetic microwave radiation is currently an actively developing area of research. Schemes for microwave radiation generation with optical pumping are of great interest. In this paper we propose and experimentally demonstrate principally new method for photonic generation of microwave electromagnetic radiation. This method is based on the use of radiation of charged submicron particles oscillating at their own acoustic frequency. Laser radiation of the optical range implements an effective buildup of acoustic vibrations of submicron particles forming the system under study, according to the Raman mechanism. Such conjugation allows to expand significantly the spectrum of functional capabilities of microwave systems operating in the gigahertz frequency range. Great interest in such systems, as a rule, is associated with the possibility of their application for radar tasks. But their applications have been also actively developing in several other areas, both purely practical, such as the creation of wireless and satellite networks, signal processing and visualization systems, and in tasks related to the solving fundamental physical problems in astrophysics, precision spectroscopy and optomechanics5. For this purpose a lot of methods for effective microwave generation are being developed. Despite the large number of existing classical microwave generators such as the klystron, traveling wave tubes, gyrotrons, backward wave oscillators and others6, development of new types of devices allowing the generation of microwave radiation is a very urgent task today. The focus is on improving the spectral, energy and temporal characteristics of radiation, as well as reducing the cost of the devices themselves. A special place among them is occupied by methods based on using a dual wavelength laser source with two wavelengths separated by a microwave frequency8.One of the most actual problems of modern radio-photonics is the development and creation of systems that make it possible to interface optical circuits with microwave devices9 and currently actively used for the analysis of low-frequency Raman scattering (LFRS)11. For the case of liquid droplets of nano and submicron size, the so-called “droplet model” is used, which gives frequency values somewhat lower than the Lamb model. For experimental verification of the applicability of a model, as a rule, the method of LFRS of light is used, which makes it possible to obtain the values of the natural frequencies of acoustic vibrations considering the matrix in which the particles are located. A stimulated analogue of the LFRS process—stimulated low-frequency Raman scattering (SLFRS)13 of light can also be used to obtain information on the values of acoustic frequencies. SLFRS can also be used to generate acoustic vibrations of the studied system of nano or submicron particles. Note that in the case of SLFRS, the system of nano or submicron particles performs coherent acoustic vibrations, and their phasing is carried out through the re-emission field.The aim of this work was to demonstrate experimentally the possibility of experimental implementation of the generation of pulsed electromagnetic radiation in the microwave range in a system of submicron particles with optical pumping. The main idea, experimentally realized in this work, is to use natural acoustic vibrations of nanoscale and submicron particles with a charge or polarized in an external field as a source of microwave electromagnetic radiation with frequency coinciding with acoustic one. The spectrum of natural acoustic frequencies of submicron particles lies in the gigahertz range, and of nanoscale particles in the terahertz range. To calculate the eigenvalues of the acoustic frequencies in the case of submicron particles, we use the approach proposed in the works of Lamb15.In this work synthetic opal matrices were used as the active medium. Opal matrix is a material with spatial modulation of both optical and acoustic properties on a scale of several hundred nanometers. It has photonic band gaps in the visible spectral range for modes propagating in certain directions. The total photonic band gap with a zero density of photonic states can be realized in inverted opal matrices with a refractive index contrast greater than 2.85. Nevertheless, opal matrices are convenient to use for studies of nonlinear interaction of light radiation with matter due to the possibility to determine easily the position of band gaps and control their properties. The voids between quartz spheres in opal matrices are octahedral and tetrahedral in shape. By filling these cavities with liquids with different refractive indices, one can control the parameters of the photonic band gap (its position in the spectrum and contrast) and increase the efficiency of nonlinear processes due to a change in the density of photon states near the edge of the band gap2 globules synthesized by the modified Stöber method16. The resulting sediment after drying in air was annealed at 125 °C for 1 h and then at 750 °C for 2 h. The sample of the opal matrix had dimensions of 5 × 5 × 4 mm. The (111) plane of the face centered cubic lattice (FCC) of the opal matrices corresponded to the surface of the natural growth of the photonic crystal and was characterized by the absence of visible defects at a scale comparable with the spot of laser radiation on the surface of the sample . Synthetic opal matrices consist of close-packed amorphous silicon dioxide spherical particles with a characteristic size of 200 to 400 nm. We used opal matrices with the diameter of silicon oxide globules close to 270 nm. The standard deviation of the size distribution of globules was about 5%.Three-dimensional opal matrices were grown by the method of slow crystallization of a monodisperse colloid solution of α-SiOSEM image of the surface of the opal matrix used is shown in the inset in Fig. max = 0.3 J, Δν = 0.015 cm−1, divergence 3.5 × 10–4 rad) was used. An experimental setup for determining the frequencies of natural acoustic vibrations of synthetic opal matrices is shown in Fig. SLFRS in opal matrices can be used to determine the values of the frequencies of the natural acoustic vibrations of quartz globules. Under the influence of powerful laser pulses in the visible range, coherent excitation of acoustic vibrations in the system is possible with frequencies lying in the gigahertz range and manifesting themselves in the SLFRS spectra because of modulation of optical radiation by a low-frequency component. For SLFRS excitation ruby laser  which gave possibility to change the range of dispersion from 5 to 30 GHz. Thus, we could register both lines with small frequency shift (0.5 ÷ 2 GHz) and larger frequency shift (10 ÷ 20 GHz).2) SLFRS was excited. Forward and backward scattered first Stokes component with frequency shift about 11.5 GHz corresponding to the quadrupolar mode of the silica sphere was registered. Maximum conversion efficiency of laser radiation into a scattered wave was 43%. When the intensity of the laser pulse reaches a value of 0.03 GW/cm2, lower-frequency components of 0.75 and 1.5 GHz appear in the spectrum of the scattered radiation both in forward and backward directions  gives two types of acoustic modes: a spheroidal (SPH) and a torsional (TOR).D is a lattice displacement vector, μ, λ-Lame’s constants, and ρ is the mass density. The equations obtained for spheroidal modes are10:So, the acoustic spectrum of the sample under investigation which was measured by SLFRS consists of the following spectral lines—11.5 GHz, 0.75 GHz and 1.5 GHz. Let us compare the experimentally obtained values of the natural frequencies of the system with the calculated values. Free isotropic elastic sphere eigenfrequencies definition was realized inThe spheroidal modes occur with a change in volume, while torsional modes occur with a constant density. The eigenfrequencies values are inversely proportional to the particle radius Rl and harmonic n. Only the breathing (l = 0) and quadrupolar (l = 2) spheroidal modes are Raman active16. For our sample the calculated frequencies are shown in Table The eigenfrequencies for both SPH and TOR modes are described by orbital angular momentum quantum number 17. The presence of contacts existing in a real system of a synthetic opal matrix leads to the appearance of lower frequencies in the acoustic spectrum. An increase in the number of contacts leads to an increase in the number of acoustic modes in the system. Using Finite Element Method eigenfrequencies were calculated for the following cases: one free sphere, two connected spheres (1contact), sphere with two contacts and sphere with 12 contacts  as it is shown in Fig. Point out that taking into consideration only eigen vibrations of the free quartz globule the frequencies 11.4 GHz and higher may appear in the spectra of low-frequency Raman scattering of light. Note that it would be possible to observe the torsional modes by Raman scattering if the shape of the particle was slightly asymmetric due to the presence of an odd deformation18. It was shown that the possibility of electromagnetic field generation at phonon frequency at the process of stimulated Raman scattering (SRS) can be realized due to nonlinear polarization modulation by optical phonons. Considering that in the SLFRS process, due to the oscillations of nano or submicron particles, their polarization is modulated, then, by analogy with the SRS process, it is possible to generate electromagnetic radiation at the frequencies of these oscillations. For submicron particles, the frequencies of these oscillations lie in the gigahertz range, which corresponds to the microwave range of the electromagnetic spectrum. For microwave radiation registration the set up presented in Fig. The manifestation of the acoustic eigenfrequencies in the spectrum of SLFRS indicates the possibility obtaining stimulated generation in the microwave range by the mechanism for example specified in the workTitanium-sapphire terawatt femtosecond laser radiation  is focused by a parabolic mirror with focal length f = 500 mm into the opal matrix with dimensions of 5 × 5 × 4 mm. Parabolic mirror has been used because it gave possibility to obtain very sharp focusing of the exciting light on the sample . For microwave radiation registration two types of detectors were used: a wideband horn antenna connected to a spectrum analyzer and Archimedean spiral antenna connected to oscilloscope. We used an oscilloscope with 4 GHz bandwidth, (DSO90404A) and spectrum analyzer with measurement range 9 kHz…40 GHz (8564E) equipped with low noise wideband RF-amplifier 0.3…18 GHz, 20 dB (Anritsu G3H105). Before measurements an antenna’s frequency response in this geometry was characterized  dipole moment can emit at their own vibrational frequencies that satisfy the corresponding selection rules. It is quite difficult to separate the contribution to the radiation process in the microwave range of an oscillating submicron particle with an induced dipole moment from the contribution to the radiation of a particle of the same morphology but with a charge. It is obvious that significantly improve the efficiency of such radiation is possible in the case of the phasing of the oscillations of the particles constituting the investigated system. In the case of molecular systems such phasing is possible by stimulated Raman scattering. In the case of submicron particles while using nanosecond laser pulses the phasing of their acoustic vibrations in gigahertz range can be realized in the process of SLFRS via ponderomotive interaction22. Note that effective direct excitation of intrinsic acoustic vibrations of submicron particles while excited by femtosecond laser pulses may occur by the mechanism of impulsive stimulated Raman scattering (ISRS)25. For efficient generation of electromagnetic radiation by a system of submicron particles, systems with close packing and monodisperse in size are most suitable. Synthetic opal matrices  make it possible to obtain the maximum density of monodisperse particles in size in the range from 200 to 400 nm. Intense acoustic vibrations of quartz globules can lead to the breaking of siloxane bonds that were formed during the sintering operation of samples during their hardening. Breaking the bonds in turn leads to the formation of a charge on the surface of quartz globules. The other way for charge formation on the globules is the well-known effect of charge formation on contacting quartz surfaces26. Thus, the result of the interaction of pulsed laser radiation with a synthetic opal matrix is the formation of a system of charged submicron particles oscillating at the frequencies of their own acoustic vibrations. Such a system is a source of electromagnetic radiation at frequencies corresponding to its eigen acoustic frequencies.Studies of the radiation of charged or being in external electric field submicron or nanoscale particles oscillating at its own acoustic frequency are of great interest both from the point of view of understanding the physics of such fundamental and beautiful physical effects as ball lightning and St. Elmo's lightsIn conclusion, it should be noted that for the first time microwave radiation was generated in synthetic opal matrix via optical pumping. Several natural acoustic frequencies of globules forming synthetic opal coincide with the frequencies of the registered microwave signal. Acoustic frequencies of the sample used were measured experimentally by SLFRS and their values were verified by numerical calculations. In our opinion, the method of obtaining the generation of microwave radiation by optical pumping in systems of submicron and nanoscale particles is very promising for the creation of microwave and terahertz sources. The specific spectral range of the electromagnetic radiation generation is determined by the value of the natural acoustic frequencies of the used nano- and submicron particles, which lie in the range from gigahertz to terahertz, respectively.


Nares Strait is a major pathway along which multi-year sea ice leaves the Arctic, an ice class that has seen a recent dramatic reduction in extent. The winds that blow along the strait play an important role in modulating this ice export as well as in establishing the Arctic’s largest and most productive polynya, the North Water, that forms at its southern terminus. However, its remote location has limited our knowledge of the winds along the strait. Here we use automatic weather station data from Hans Island, in the middle of the strait, to assess the ability of a set of atmospheric renalyses and analyses with a common lineage but with varying horizontal resolution to represent the variability in the wind field. We find that the flow is highly bidirectional, consistent with topographic channeling, with the highest wind speeds from the north and that a model resolution of ~ 9 km is required to capture the observed variability. The wind field at Hans Island is also found to be representative of variability in the flow along much of Nares Strait. The wind field in such a narrow channel is typically ageostrophic and controlled by the pressure gradient in the along-strait direction2. This channeling is enhanced by the common occurrence of low-level temperature inversions in the region3 that serve to inhibit vertical motion4.Nares Strait is the ~ 600 km long and ~ 40-80 km wide strait that connects the Arctic Ocean’s Lincoln Sea to northern Baffin Bay Fig. . The ste5 and the strait is a major pathway for the export of this important ice class out of the Arctic6. The wind field along Nares Strait with its preference for northerly flow plays an important role in the export1. Ice arches that form most winters along Nares Strait can result in the cessation of this ice transport for months at a time7.The oldest and thickest sea ice in the Arctic is situated to the north of Nares Strait8. With an arch present, ice motion along the strait is inhibited and the northerly flow down Nares Strait, that accelerates in the exit region of Smith Sound, is able to remove sea ice thus contributing to the maintenance of the polynya’s open water9.The presence of these arches also contributes to the largest and most productive polynya in the Arctic, the North Water (NOW) that forms at the southern end of Nares Strait Fig. , in the 7 and that the area of the NOW is increasing10. There is also evidence that anomalous winds play a role in the collapse of these arches11. In addition, the Arctic is undergoing a transition towards a younger12 and thinner13 ice pack that is leading to a loss of multi-year ice from the Arctic14. There has also recently been an increase in both ice area and ice volume fluxes along Nares Strait7 that has been suggested to contribute to the loss of multi-year ice from the Arctic6.There is evidence that the ice arches along Nares Strait are becoming less stableThe remote and data sparse nature of the region implies that there is still considerable uncertainty regarding the characteristics of the wind field that limits our ability to fully understand its role in ice export and its contribution to the formation of the NOW as well as the wind’s role in the changes that are occurring in the region.−115. However observations extending back to the 1860s indicate that long-lived gales, events characterized by northerly surface wind speeds in excess of 20 ms−1, are a common occurrence in the Smith Sound region during the winter months16.Limited wind observations from the Smith Sound region confirm the preference for northerly flow with mean wind speeds on the order of 6–8 ms−118. During the winter of 1871–1872, the United States North Polar Expedition’s ship was trapped in ice at an exposed site in Hall Basin and the science party regularly observed wind speeds in excess of 20 ms−1 with a preference for northerly flow19. In the vicinity of Robeson Channel, meteorological observations were made at Fort Conger from August 1881 to August 1883 during the First International Polar Year by members of the United States Expedition to Lady Franklin Bay20. These observations indicated the preference for bidirectional wind flow, either in the NNE or SSW direction, along Robeson Channel. There were numerous events, with both directionalities, where the wind speeds were in excess of 15 ms−120.Wind observations farther north along Nares Strait are even rarer. In April 2005, an ice camp established along the Kennedy Channel was destroyed by northerly winds in excess of 25 mso49′35'N, 66°27′35'W), jointly claimed by both Canada and Denmark, is situated in the middle of the Kennedy Channel, one of the narrower sections of Nares Strait  clearly represents the narrowness of Nares Strait in the vicinity of Hans Island as well as the steep topography along both sides of the strait. The eERA5 cross-section , ~ 30 km (ERA5) and ~ 9 km (ECOA). The common lineage of the models allows one to control for biases arising from difference in model architecture and parameterizationsThe comparison of cross-sections through the reference and model topographies Fig.  clearly −1 for the ERA5 to 2.18 ms−1 for the ECOA; while the slope of the least squares linear fit increased from 0.39 for the ERA5 to 0.8 for the ECOA. The latter characteristic implies that the eERA5 and the ERA5 have a tendency to underestimate the wind speed during high wind speed events to a greater extent than the ECOA.The comparison for the 10 m wind speed Fig.  shows a This may be related to the inability of these models to represent the open water in Nares Strait and its replacement with land that generally has higher surface roughness Fig. . Neverth1. All the models, to a greater or lesser extent, represent the narrowness of the strait and hence the tendency for the flow to be ageostrophic and controlled by the pressure gradient in the along-strait, i.e. meridional, direction. The composite high speed events . Included is the new fifth generation reanalysis from the ECMWF or ERA5 with a horizontal resolution of ~ 30 km, as well as its ensemble version, eERA5 with a horizontal resolution of ~ 60km21.For this study, the instantaneous 6-hourly 10 m winds and 100 m from these three datasets were interpolated to the location of Hans Island and the data was used at those times for which there are observations. A number of different interpolation schemes were used and the results were independent of scheme. For simplicity, the results shown are for a bilinear interpolation scheme. In total, ~ 550 days worth of data was included in this study. The AWS was situated at a height of approximately 170 m above sea level