Omar El Rifai
email: omar.el-rifai@irit.fr

Maelle Biotteau
email: maelle.biotteau@inserm.fr

Xavier De Boissezon
email: deboissezon.xavier@chu-toulouse.fr

Imen Megdiche
email: imen.megdiche@irit.fr

Franck Ravat
email: franck.ravat@irit.fr

Olivier Teste
email: olivier.teste@irit.fr

Blockchain-Based Federated Learning in Medicine

Worldwide epidemic events have confirmed the need for medical data processing tools while bringing issues of data privacy, transparency and usage consent to the front. Federated Learning and the blockchain are two technologies that tackle these challenges and have been shown to be beneficial in medical contexts where data are often distributed and coming from different sources. In this paper we propose to integrate these two technologies for the first time in a medical setting.

In particular, we propose a implementation of a coordinating server for a federated learning algorithm to share information for improved predictions while ensuring data transparency and usage consent. We illustrate the approach with a prediction decision support tool applied to a diabetes data-set. The particular challenges of the medical contexts are detailed and a prototype implementation is presented to validate the solution.

Introduction

Researchers face ethical challenges when handling medical records. Indeed, medical records hold sensitive information about patients that can be prejudicial if leaked. A recent controversy involving unconsenting access to tens of millions of identifiable health records re-sparked an interest in the data ethics debate [START_REF] Schneble | Google's project nightingale highlights the necessity of data science ethics review[END_REF]. As a consequence, medical institutions are reticent in sharing medical records [START_REF] Mehta | Concurrence of big data analytics and healthcare: a systematic review[END_REF]. Researchers go through time consuming procedures to request and use medical data-sets, often at the expense of efficiently advancing research. This situation is exacerbated for data scientists who use large, and heterogeneous data-sets scattered across different sites.

At the forefront of the ethical challenges we find, data privacy, transparency, and usage consent [START_REF] Larson | Building trust in the power of "big data" research to serve the public good[END_REF][START_REF] Mittelstadt | The ethics of big data: current and foreseeable issues in biomedical contexts[END_REF][START_REF] Schneble | Google's project nightingale highlights the necessity of data science ethics review[END_REF]. Data privacy is often thought in terms of identity privacy or confidentiality [START_REF] Mittelstadt | The ethics of big data: current and foreseeable issues in biomedical contexts[END_REF][START_REF] Schneble | Google's project nightingale highlights the necessity of data science ethics review[END_REF]. Traditionally, anonymization techniques have been used for medical data processing [START_REF] Mittelstadt | The ethics of big data: current and foreseeable issues in biomedical contexts[END_REF]. Data transparency is about patients knowing and understanding how and by whom their data are used [START_REF] Schneble | Google's project nightingale highlights the necessity of data science ethics review[END_REF].

And usage consent means that data subjects have the right to decide how and when their data are used. [START_REF] Mittelstadt | The ethics of big data: current and foreseeable issues in biomedical contexts[END_REF][START_REF] Schneble | Google's project nightingale highlights the necessity of data science ethics review[END_REF]. Unfortunately, these points are often glossed over when processing patients' data, and consent is sometimes implicitly assumed if data-sets are anonymized [START_REF] Larson | Building trust in the power of "big data" research to serve the public good[END_REF]. This situation is further complicated in the era of big-data where data-sets are scattered and processed by multiple often noncommunicating parties [START_REF] Leonelli | Locating ethics in data science: responsibility and accountability in global and distributed knowledge production systems[END_REF].

Recently, two technologies emerged independently which address some of the issues highlighted above. Federated Learning (FL) emerged as a paradigm for training Machine Learning (ML) models across decentralized devices and minimizing the risk of exposing sensitive information [START_REF] Kairouz | Advances and open problems in federated learning[END_REF][START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF][START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF]. While the blockchain emerged as a technology which offers unprecedented guarantees of reliability and usage transparency in decentralized settings [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF].

This paper proposes a blockchain-based FL framework whereby the advantages of both technologies are put to use in the medical context. In particular, we propose a Smart Contract (SC) implementation of a coordinating server for a FL algorithm to ensure transparency and usage consent when sharing knowledge.

In Sect. 2, we discuss relevant related work for both FL and the blockchain in health care contexts. Then, in Sect. 3 we describe the problem in detail and explain how our solution can be applied for the medical setting. In Sect. 4, we show and discuss experiment results on a medical data-set for diabetes prediction. Finally, in Sect. 5, we present remaining challenges and open questions.

Health Care Analytics in Distributed Settings

Medical data offers a wealth of potential for improving the quality of care and reducing costs [START_REF] Mehta | Concurrence of big data analytics and healthcare: a systematic review[END_REF]. Nonetheless, medical data, as in other fields, is often "disorganized and distributed, coming from various sources and having different structures and forms" [START_REF] Mehta | Concurrence of big data analytics and healthcare: a systematic review[END_REF]. FL emerged as a response to these settings by providing a way to train ML models in heterogeneous and distributed settings while minimizing data transfers [START_REF] Kairouz | Advances and open problems in federated learning[END_REF][START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF][START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF]. It has been used in health care settings because of the sensitive information of the data handled [START_REF] Xu | Federated learning for healthcare informatics[END_REF].

Likewise the blockchain has been studied in health care as it offers usage guarantees (transparency and immutability) not possible in traditional distributed data architectures [START_REF] Drosatos | Blockchain applications in the biomedical domain: a scoping review[END_REF][START_REF] Mettler | Blockchain technology in healthcare: the revolution starts here[END_REF]. In this section, we look at how these two technologies have been used in health care, and henceforth motivate the introduction of our blockchain-based FL medical decision support model.

Federated Learning Approaches

Although works on distributed computing have been around for decades [START_REF] Li | Learning IoT in edge: deep learning for the internet of things with edge computing[END_REF][START_REF] Li | Federated learning: Challenges, methods, and future directions[END_REF], new contexts have brought up previously unaddressed challenges: instead of evenly, and moderately distributed data-sets, FL approaches deal with uneven, and massively distributed data-sets [START_REF] Kairouz | Advances and open problems in federated learning[END_REF][START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF][START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF]. These contexts have been shown to be applicable in health care settings to "connect all the medical institutions and makes them share their experiences with privacy guarantee" [START_REF] Xu | Federated learning for healthcare informatics[END_REF]. The term FL was coined by Konevcny et al. [START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF] and McMahan et al. [START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF] which proposed a variant to the Stochastic Variance Reduced Gradient Descent (SVRGD) algorithm to solve ML problems in a distributed setting.

Characteristically, participants' raw data never leaves the hosts' devices in FL settings. The only data that is shared with a coordinating server are model parameters of local ML models. These parameters are then aggregated efficiently by a central server and the result sent back to the participants for updating their own models.

A comparison between the FL framework and more traditional computing architectures [START_REF] Kairouz | Advances and open problems in federated learning[END_REF] is shown in Fig. 1. Figure (1a) shows an architecture where all computations are performed on a centralized server. In such an architecture, end devices query the server to use the computation model. The centralized server needs to hold the entire data-sets at the moment of the learning phase for this architecture to work and is typical of a siloed health facility. Figure (1b) shows a distributed computing architecture where multiple servers (sometimes defined as a cloud) share data-sets and the workload. This architecture is common for data-sets that have been anonymized and need to be shared across institutions. Finally, in Figure (1c), the end devices become active participants in the computation and only upload partial information to the server. The new role of the server is then to aggregate the information of the different devices and broadcast back the aggregated information to the end devices. Studies that have used FL in the health care settings have been recently reported in [START_REF] Xu | Federated learning for healthcare informatics[END_REF]. For instance in [START_REF] Brisimi | Federated learning of predictive models from federated electronic health records[END_REF], the authors develop a binary classification problem to predict cardiac-related hospitalizations based on data from electronic health records. They find that their algorithm converges faster than a centralized one at the expense of increased communication costs. In [START_REF] Huang | Patient clustering improves efficiency of federated machine learning to predict mortality and hospital stay time using distributed electronic medical records[END_REF], the authors use a FL-based approach to predict mortality and hospital stay time. They improve on the baseline FL models by first clustering the patients into communities and outperform the baseline FL approaches.

However, in the words of the authors in [START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF]: "Clearly, some trust of the server coordinating the training is still required [when using FL]". That is, server-side computations often suffer from a lack of transparency and it is difficult for users to verify the computations performed [START_REF] Santhosh | Towards trustless prediction-as-a-service[END_REF]. Specifically, cloud computing has been argued to decrease the sovereignty of users over their data and models [START_REF] De Filippi | Cloud computing: Centralization and data sovereignty[END_REF].

Blockchain with Health Care Applications

The blockchain was invented to solve the consensus problem in a decentralized, trust-less network [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF]. That is, given a distrusting network of peers, the blockchain provides a mathematically robust way of verifying that data stored on our device are identical with data held by other peers. Initially designed with a financial application in mind, the blockchain quickly evolved to different domains [START_REF] Janssen | A framework for analysing blockchain technology adoption: integrating institutional, market and technical factors[END_REF].

In particular, the blockchain has been heavily applied in the medical domain whereby the properties it possesses (decentralization, immutability, and transparency) are core issues [START_REF] Agbo | Blockchain technology in healthcare: a systematic review[END_REF][START_REF] Drosatos | Blockchain applications in the biomedical domain: a scoping review[END_REF][START_REF] Mettler | Blockchain technology in healthcare: the revolution starts here[END_REF]. One such application which has been studied extensively is Electronic Health Records (EHR) management [START_REF] Halamka | MedRec" prototype for electronic health records and medical research data[END_REF][START_REF] Dagher | Ancile: privacy-preserving framework for access control and interoperability of electronic health records using blockchain technology[END_REF][START_REF] Leeming | A ledger of me: personalizing healthcare using blockchain technology[END_REF]. EHR management is inherently decentralized as stakeholders are distributed between patients, medical institutions and government institutions in some cases. Blockchain-based solutions, allow the different stakeholders to manager EHR transparently while guaranteeing fairness and usage (records access) consent [START_REF] Leeming | A ledger of me: personalizing healthcare using blockchain technology[END_REF].

Discussion

Both the blockchain and FL algorithms address important ethical challenges and have been successfully used in many health care settings [START_REF] Gordon | Blockchain technology for healthcare: facilitating the transition to patient-driven interoperability[END_REF][START_REF] Xu | Federated learning for healthcare informatics[END_REF]. Indeed, by reducing the data that is shared when training models, FL algorithms reduce the risks of exposing sensitive patients data and hence address the privacy issue. Similarly, the blockchain addresses transparency and usage consent when dealing with medical records (c.f Appendix A.2).

Few papers have integrated ML models directly with the blockchain. Among those, Wang et al [START_REF] Wang | A unified analytical framework for trustable machine learning and automation running with blockchain[END_REF] for instance, have used the blockchain as a platform for hosting ML models, guaranteeing algorithmic correctness and usage traceability. Harris et al. [START_REF] Harris | Decentralized and collaborative AI on blockchain[END_REF] use the blockchain as a collaborative training platform for ML models. That is, the platform hosts ML models written in SCs and encourages data uploads from different users. Finally, the authors in [START_REF] Weng | DeepChain: auditable and privacy-preserving deep learning with blockchain-based incentive[END_REF] propose a domain agnostic setting where FL and blockchain technologies are successfully integrated. Indeed, a FL platform is used to train a deep learning network and instead of a centralized server, a SC is setup for federating the computations. The setup is said to provide "data confidentiality, computation auditability, and incentives for parties to participate in collaborative training" [START_REF] Weng | DeepChain: auditable and privacy-preserving deep learning with blockchain-based incentive[END_REF]. We bring these ideas for the first time to the medical context where privacy, transparency and usage consent are primordial. Indeed, medical records often fall under strict regulations such as the European General Data Protection Regulation (GDPR) and the suitability of such solutions needs to be investigated. Accordingly, we present in the next section, a medical decision support tool in a blockchain-based FL framework.

As discussed in previous sections, FL algorithms reduce the data communicated between participants and reduces the risks of exposing sensitive patients information compared to a cloud-based approach. Additionally, using SCs instead of a federating server, we can add transparency and usage guarantees to the setup. Specifically, at each federation round, the SC collects the values from the different participants and returns the aggregated parameters for participants to update their ML model in a completely automated way. Participants can also at any time verify the correct execution of this step in a completely transparent manner given the open and distributed execution of SCs. For this work, we choose to train an Artificial Neural Network (ANN) as previous results are available for comparison 1 . That said, other ML algorithms can easily be used in FL settings [START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF].

Let i be an index for the n different facilities that choose to participate in a collaborative ANN prediction model. The input to the model includes relevant patients' characteristics for the prediction model and the output is a binary variable. Each participant locally trains their ANN with weight parameters w and biases b [START_REF] Nielsen | Neural Networks and Deep Learning[END_REF] before sending the parameters to the federation SC. The loss function of such a ANN is of the following form.

f (w, b) def = 1 2n x y(x) -a 2 (1)
Where x are the different data-set samples and y(x) corresponds to the output of our model for the particular input x. In order to train the model, the cost function [START_REF] Nielsen | Neural Networks and Deep Learning[END_REF] needs to be minimized so that the difference between y(x) and the actual output a (indicating if the patient of sample x is diabetic or not) is minimized. Of course, more complex objective function are possible but this is not the focus of this work. The overall architecture of our system is illustrated in Fig. 2. At each round τ , medical facilities send their trained parameters to a "Federating" SC uploaded on a blockchain network. The SC then aggregates the different parameters and sends back the result to the facilities for them to train the individual models again. As the aggregated parameters incorporate information from the different facilities, they have been shown to achieve near-optimal accuracy [START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF].

Let ω represent both the weights and the biases, we use the formulation of Mc Mahan et al [START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF] in a federated setting and define the following loss:

min ω∈R d f (ω) where f (ω) def = 1 n n i=1 f i (ω) (2)
The objective in Eq. 2 is to minimize the overall loss function defined as the average of the participants' individual loss functions. The algorithm for minimizing the loss functions can be separated into "client" steps and "server" steps as defined in [START_REF] Konečnỳ | Federated optimization: distributed optimization beyond the datacenter[END_REF][START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF]. In our setting, the clients are the different facilities i and the "server" is the SC with access to all the local updates. The clients perform a normal gradient descent process and then send their parameters ω to the SC which proceeds with a weighted average of the different client parameters as in Eq. 3.

ω τ +1 = i n i n ω τ (3)
Next, we investigate in Sect. 4 some experiments to validate the solution explained in this section to a real-world medical problem.

Experiments

In this section we evaluate our solution using a diabetes data-set from the American National Institute of Diabetes and Digestive and Kidney Diseases available online 2 . First, we validate our ANN model's capacity to predict diabetes by using it in a centralized setting. Then, we test the FL setting by distributing the medical records between 15 participants and having the participants collaborate. Finally, we implement a small blockchain prototype based on Ethereum to validate feasibility. For all the tests, we use ANN consisting of 2 hidden layers with 32 and 16 neurons respectively and a binary output layer to indicate whether the patients is predicted diabetic or not. Also, we use 80% of the data for training the ANN and the remaining 20% for testing.

In the first experiment the centralized model is trained for 50 epochs and the accuracy of the trained model on the testing data-set is evaluated after each epoch. Figure 3 illustrates the results obtained after running the experiment 10 times with different data distribution scenarios between the test and validation set. For each scenario, we randomly select 80% of the entries for training and leave the remaining for testing. We note that the initial data distribution markedly impacts the accuracy of the final model. This clearly indicates an influence of the training data-set and corroborate the idea of benefiting from a collaborative setting. Next, to test the FL model, we distribute the data-set among 15 participants to simulate a mid-sized collaboration scenario. After each 10 local epochs, the participants all share their model's parameters (weights and biases) with a centralized federation server. The data is distributed among the participants in a randomized way. Some example results are shown in Fig. 4. The blue line represents the accuracy with the federated parameters while the grey line is a baseline model without federation. The results show that individual models' accuracy are lower than in the centralized setting with an average accuracy of 73% (against 76% in the centralized setting) but all local models benefit from the data aggregates of the FL system with their accuracy improving between 1% to 5%. We choose to illustrate the results for 3 participants only as they had the most characteristic behavior, but other participants have similar trends. By having the ANN implemented in a FL setting, patients' raw medical records are protected from attackers as they never leave the medical facilities. However, some studies have shown that inference attacks are possible on aggregated data. That is, some unintended information can be leaked from model parameters only. Possible mitigation strategies are discussed in Sect. 5.

Finally, with the above setup, participants have no visibility on the computations performed on the federating server. To add transparency and facilitate usage consent, we implement the federating algorithm in a SC using the Ethereum blockchain [START_REF] Wood | Ethereum: a secure decentralised generalised transaction ledger[END_REF] and several open source tools (c.f Appendix A.4). Our implementation is openly accessible on this link https://github.com/n-vcs/solidity-fl. By having the SC deployed instead of a federating server, participants can verify how the parameters submitted are used and track all the transactions that are happening. Of course, a full-fledged implementation is needed to solve setting-specific issues that are not addressed in this paper due to space limitation. Clinical Relevance: Promptly testing for diabetes can help patients receive timely and accurate treatment and help prevent misdiagnosis. Furthermore, training the model in the setting explained here allows medical centers to transparently and securely share knowledge with other facilities. This knowledge sharing creates more robust models which are resilient to environment-specific biases.

Conclusion and Future Work

In this work, we presented a blockchain-based FL solution to a diabetic prediction model. Our solution capitalizes on the benefits of both the blockchain and FL algorithms and caters to the particularities of a medical context. We provided a prototype implementation and presented tests to validate the relevance of the solution in the medical context. However, although no raw training data ever leaves the participants' devices in FL settings, the updates sent to the coordinating server may still contain private information. Indeed, FL algorithms have been shown to be vulnerable to inference attacks [START_REF] Weng | DeepChain: auditable and privacy-preserving deep learning with blockchain-based incentive[END_REF]. These types of attacks can deduce information about the training population which was not intended by the model.

To deal with this problem, privacy preserving protocols have been developed to prevent leakage of sensitive data. For instance, differential privacy solutions, Differential Privacy [START_REF] Dinur | Revealing information while preserving privacy[END_REF]and Homomorphic Encryption [START_REF] Gentry | A Fully Homomorphic Encryption Scheme[END_REF] have been shown to be relatively useful against inferential attacks [START_REF] Yang | Federated machine learning: concept and applications[END_REF]. However, even without revealing any details of the model, it is quite hard to achieve perfect privacy [START_REF] Shokri | Membership inference attacks against machine learning models[END_REF]. Against this grim outlook, it is however essential to be aware of the risks of each particular settings and measure them against the potential benefits.

Fig. 1 .

 1 Fig. 1. An illustration of the different computing architectures. A gearwheel indicates where the main computation tasks is executed and dashed lines indicate that minimal information is transferred.

Fig. 2 .

 2 Fig. 2. An illustration of the proposed system. The blockchain is used instead of the coordinating server in the original FL architecture.

Fig. 3 .

 3 Fig. 3. The prediction accuracy of our ANN in a centralized setting for different training and testing scenarios.

Fig. 4 .

 4 Fig. 4. Resulting accuracy for three participants in a FL setting. (Color figure online)

https://www.kaggle.com/uciml/pima-indians-diabetes-database.

https://github.com/n-vcs/solidity-fl.

Acknowledgements. This work was supported by a grant from the Roche Institute 2018.

Appendix A The Blockchain Technology: Key Concepts and Implementation

This appendix explains some key concepts related to the blockchain as well as technical implementation details relevant to the present work.

A.1 Main Characteristics

The blockchain is a distributed ledger technology managed by a network of peers. Data on the blockchain are visible and duplicated across participants. New data are added to the blockchain through a consensus protocol. As such, the blockchain is decentralized, immutable and transparent by design.

A.2 Smart Contracts (SCs)

SCs are set of instructions, specified in digital form and executed when predefined conditions are met. Contrary to regular software, SCs benefit from the main characteristics of the blockchain and can help attain transparency and usage consent.

As opposed to centralized algorithms, SCs allow data owners to verify, at any time, the implementation of the FedAvg algorithm and replicate the results locally. Additionally, usage consent can be facilitated through SCs by logging data ownership certification directly on the blockchain as described in [START_REF] Leeming | A ledger of me: personalizing healthcare using blockchain technology[END_REF].

A.3 Consensus and Incentive Mechanism

A consensus protocol is at the heart of the blockchain' mechanism. The Ethereum blockchain uses Proof-of-Work (PoW) by default. PoW relies on computational power to validate transactions or execute SCs. In our context, Medical institution are natural candidates for running the network as they will benefit from the resulting model. But to keep the network alive, nodes needs to be incentivized. As such, a possible set-up proposed in [START_REF] Weng | DeepChain: auditable and privacy-preserving deep learning with blockchain-based incentive[END_REF] is to have a reward mechanism for participants based on their data contribution.

A.4 Current Set-up

For the prototype 3 an Ethereum blockchain is set-up on the back-end using the Truffle suite (Ganache and the Truffle Development Environment). Also, the FedAvg [START_REF] Mcmahan | Communicationefficient learning of deep networks from decentralized data[END_REF] algorithm and a basic ANN are developed using Python's scientific computing library (Numpy). The front-end is developed in React. Data for each participant are stored locally in a MongoDB database.


L Fedeli

A Formenti
email: arianna.formenti@polimi.it

A Pazzaglia

F M Arioli

A Tentori

M Passoni

Enhanced laser-driven hadron sources with nanostructured doublelayer targets

Keywords: laser-driven ion acceleration, ultra-intense lasers, neutron sources, nanostructured targets, particle-in-cell simulations, Monte Carlo simulations

Laser-driven ion sources are approaching the requirements for several applications in materials and nuclear science. Relying on compact, table-top, femtosecond laser systems is pivotal to enable most of these applications. However, the moderate intensity of these systems (I10 19 W cm -2 ) could lead to insufficient energy and total charge of the accelerated ions. The use of solid foils coated with a nanostructured near-critical layer is emerging as a promising targeted solution to enhance the energy and the total charge of the accelerated ions. For an appropriate theoretical understanding of this acceleration scheme, a realistic description of the nanostructure is essential, also to precisely assess its role in the physical processes at play. Here, by means of 3D particle-in-cell simulations, we investigate ion acceleration in this scenario, assessing the role of different realistic nanostructure morphologies, such as fractal-like foams and nanowire forests. With respect to a simple flat foil, the presence of a nanostructure allows for up to a×3 increase of the maximum ion energy and for a significant increase of the conversion efficiency of laser energy into ion kinetic energy. Simulations show also that the details of the nanostructure morphology affect both the maximum energy of the ions and their angular distribution. Furthermore, combined 3D particle-in-cell and Monte Carlo simulations show that if accelerated ions are used for neutron generation with a beryllium converter, double-layer nanostructured targets allow to greatly enhance the neutron yield. These results suggest that nanostructured double-layer targets could be an essential component to enable applications of hadron sources driven by compact, table-top lasers.

Introduction

Ion acceleration driven by ultra-intense laser pulses is a well established reasearch topic [START_REF] Daido | Review of laser-driven ion sources and their applications[END_REF][START_REF] Macchi | Ion acceleration by superintense laser-plasma interaction[END_REF][START_REF] Schreiber | Invited review article: 'hands-on' laser-driven ion acceleration: a primer for laser-driven source development and potential applications[END_REF]. In state-of-the-art Petawatt-class laser facilities, few nanocoulomb, relatively collimated, broad spectrum ion bunches can be accelerated up to a ∼100 MeV [START_REF] Kim | Radiation pressure acceleration of protons to 93 MeV with circularly polarized petawatt laser pulses[END_REF][START_REF] Wagner | Maximum proton energy above 85 MeV from the relativistic interaction of laser pulses with micrometer thick CH 2 targets[END_REF][START_REF] Higginson | Near-100 MeV protons via a laser-driven transparency-enhanced hybrid acceleration scheme[END_REF] cut-off energy. Target normal sheath acceleration (TNSA) [START_REF] Wilks | Energetic proton generation in ultra-intense laser-solid interactions[END_REF] is arguably the most robust and widely studied ion acceleration scheme. In TNSA an ultra-intense laser pulse interacts with a thin solid foil, heating the electrons of the target up to several MeV energies. The expanding electron cloud generates an intense electric field at the back side of the target. This electric field is then responsible for the ion acceleration process. Historically, the foreseen societal application in radiotherapy [START_REF] Bulanov | Feasibility of using laser ion accelerators in proton therapy Plasma[END_REF][START_REF] Linz | What will it take for laser driven proton accelerators to be applied to tumor therapy?[END_REF] has been a preeminent driving force behind research efforts on laser-driven ion acceleration. However, existing laser-driven ion sources still fall short of the stringent requirements in terms of energy, monochromaticity and stability [START_REF] Linz | Laser-driven ion accelerators for tumor therapy revisited[END_REF]. Indeed, up to now, laser-driven ion sources have been routinely used only as a diagnostic tool for transient electromagnetic fields in laser-plasma interaction experiments [START_REF] Borghesi | Electric field detection in laser-plasma interaction experiments via the proton imaging technique[END_REF][START_REF] Mackinnon | Proton radiography of a laser-driven implosion[END_REF][START_REF] Borghesi | Laser-driven proton acceleration: source optimization and radiographic applications Plasma[END_REF].

Especially in recent years, materials and nuclear science have emerged as a promising domain of application of laser-driven ion sources. Proof-of-principle experiments [START_REF] Barberio | Laser-accelerated proton beams as diagnostics for cultural heritage[END_REF] and numerical works [START_REF] Mirani | Ion beam analysis with laser-driven proton beams[END_REF][START_REF] Barberio | Laser-PIXE using laser-accelerated proton beams[END_REF][START_REF] Passoni | Superintense laser-driven ion beam analysis[END_REF] support the feasibility of laser-driven ion beam analysis for non-destructive materials characterization. Laser-driven ion sources have been considered to test electronic components in a harsh radiation environment [START_REF] Hidding | Laser-plasma-based space radiation reproduction in the laboratory[END_REF], for thermal stress testing [START_REF] Barberio | Laser-accelerated particle beams for stress testing of materials[END_REF], to study ultra-fast dynamics in irradiated materials [START_REF] Dromey | Picosecond metrology of laser-driven proton bursts[END_REF][START_REF] Taylor | Probing ultrafast proton induced dynamics in transparent dielectrics Plasma[END_REF], and for materials synthesis [START_REF] Barberio | Laser-plasma driven synthesis of carbon-based nanomaterials[END_REF][START_REF] Barberio | Laser-generated proton beams for high-precision ultra-fast crystal synthesis[END_REF]. Ultrashort pulsed neutron sources driven by laser-accelerated ions [START_REF] Lancaster | Characterization of 7Li(p, n)7Be neutron yields from laser produced ion beams for fast neutron radiography[END_REF][START_REF] Petrov | Generation of high-energy (>15 MeV) neutrons using short pulse high intensity lasers[END_REF][START_REF] Maksimchuk | Dominant deuteron acceleration with a high-intensity laser for isotope production and neutron generation[END_REF][START_REF] Zulick | Energetic neutron beams generated from femtosecond laser plasma interactions[END_REF][START_REF] Kar | Beamed neutron emission driven by laser accelerated light ions[END_REF][START_REF] Alejo | Recent advances in laser-driven neutron sources[END_REF][START_REF] Mirfayzi | Experimental demonstration of a compact epithermal neutron source based on a high power laser[END_REF][START_REF] Brenner | Laser-driven x-ray and neutron source development for industrial applications of plasma accelerators Plasma[END_REF][START_REF] Favalli | Characterizing laser-plasma ion accelerators driving an intense neutron beam via nuclear signatures[END_REF] have been investigated for applications such as fast neutron spectroscopy [START_REF] Higginson | Laser generated neutron source for neutron resonance spectroscopy[END_REF] and radiography [START_REF] Roth | Bright laser-driven neutron source based on the relativistic transparency of solids[END_REF].

What make these applications particularly attractive are the requirements for the ion source. Energies of few MeVs are perfectly suitable for several ion beam analysis techniques [START_REF] Verma | Atomic and Nuclear Analytical Methods: XRF, Mössbauer, XPS, NAA and Ion-Beam Spectroscopic Techniques[END_REF] or to generate neutrons with a lithium [START_REF] Macklin | Study of the T(p, n)He 3 and Li 7 (p, n)Be 7 reactions[END_REF] or beryllium [START_REF] Marion | Excited states in B 10[END_REF] converter. Moreover, for selected applications, the inherently broad energy spectrum of a laser-driven ion source [START_REF] Macchi | Ion acceleration by superintense laser-plasma interaction[END_REF] is not detrimental [START_REF] Mirani | Ion beam analysis with laser-driven proton beams[END_REF][START_REF] Passoni | Superintense laser-driven ion beam analysis[END_REF] and could even be beneficial [START_REF] Hidding | Laser-plasma-based space radiation reproduction in the laboratory[END_REF]. Maximum ion energies of ∼1 MeV have been recently demonstrated even with commercial, sub-terawatt laser systems [START_REF] Morrison | Mev proton acceleration at kHz repetition rate from ultra-intense laser liquid interaction[END_REF]. These lasersystems are truly table-top and can operate at a high repetition rate (kiloHertz). Relying on these systems could make laser-driven ion sources portable and competitive with conventional accelerators, paving the way for their widespread adoption in materials and nuclear science.

Over the years, several advanced ion acceleration schemes have been proposed to enhance the properties (maximum energy, angular distribution, total charge K) of the accelerated ions [START_REF] Macchi | Ion acceleration by superintense laser-plasma interaction[END_REF][START_REF] Borghesi | Summary of working group 2: ion beams from plasmas 2nd European Advanced Accelerator Concepts Workshop-EAAC 2015[END_REF][START_REF] Flacco | Summary of working group 2: ion beams from plasmas 3rd European Advanced Accelerator Concepts Workshop (EAAC2017)[END_REF], which could be very beneficial for several of the aforementioned applications. In some scenarios where the ion energy is a critical issue, the use of advanced ion acceleration strategies could even enable a particular application on a given laser system. Neutron conversion is emblematic of this last case, since it is crucial to reach at least the energy threshold for the nuclear reaction.

Among the advanced ion acceleration schemes, the use of foils coated with a low-density, near-critical, nanostructured layer [START_REF] Zani | Ultra-low density carbon foams produced by pulsed laser deposition[END_REF][START_REF] Nagai | A review of low density porous materials used in laser plasma experiments[END_REF] as a target for TNSA is emerging as a promising strategy [START_REF] Yogo | Laser ion acceleration via control of the near-critical density target[END_REF][START_REF] Willingale | Characterization of high-intensity laser propagation in the relativistic transparent regime through measurements of energetic proton beams[END_REF][START_REF] Nakamura | Interaction of high contrast laser pulse with foam-attached target[END_REF][START_REF] Sgattoni | Laser ion acceleration using a solid target coupled with a low-density layer[END_REF][START_REF] Passoni | Energetic ions at moderate laser intensities using foam-based multi-layered targets Plasma[END_REF][START_REF] Prencipe | Development of foam-based layered targets for laser-driven ion beam production Plasma[END_REF][START_REF] Passoni | Toward high-energy laser-driven ion beams: nanostructured double-layer targets[END_REF][START_REF] Bin | Enhanced laser-driven ion acceleration by superponderomotive electrons generated from near-critical-density plasma[END_REF][START_REF] Ma | Laser acceleration of highly energetic carbon ions using a double-layer target composed of slightly underdense plasma and ultrathin foil[END_REF]. Here, 'nearcritical' means having an electron density close to the critical one, n c =π m e c 2 /λ 2 e 2 (where m e is the electron mass, λ is the laser wavelength and e is the elementary charge). n c marks the density threshold for electromagnetic wave transparency. Ultra-intense laser interaction with a plasma having a near-critical density is characterized by a strong coupling and an efficient energy absorption [START_REF] Pukhov | Relativistic magnetic self-channeling of light in near-critical plasma: three-dimensional particlein-cell simulation[END_REF][START_REF] Borghesi | Relativistic channeling of a picosecond laser pulse in a near-critical preformed plasma[END_REF][START_REF] Robinson | Absorption of circularly polarized laser pulses in near-critical plasmas Plasma[END_REF]. The low-density layer can either be a controlled pre-plasma or a solid nanostructured material with ultra-high void fraction. Both solutions allow for a high absorption efficiency of the laser energy by the hot electrons population, which results into more ions accelerated at higher energies with respect to a simple flat foil [START_REF] Cialfi | Electron heating in subpicosecond laser interaction with overdense and near-critical plasmas[END_REF][START_REF] Fedeli | Structured targets for advanced laser-driven sources Plasma[END_REF][START_REF] Batani | Effects of laser prepulses on laser-induced proton generation[END_REF]. Nevertheless, pre-plasmas-usually generated via target pre-expansion induced by the laser pedestal or ns-duration pulses [START_REF] Kaluza | Influence of the laser prepulse on proton acceleration in thin-foil experiments[END_REF][START_REF] Flacco | Dependence on pulse duration and foil thickness in high-contrast-laser proton acceleration[END_REF]-is affected by poor stability and control, since large deviations may arise depending on the specific laser facility. Indeed, within the literature on laser-driven ion acceleration, the pre-expansion of the target was reported to lead to very different results: ranging from strong enhancement [START_REF] Gray | Laser pulse propagation and enhanced energy coupling to fast electrons in dense plasma gradients[END_REF], to mild enhancement [START_REF] Batani | Effects of laser prepulses on laser-induced proton generation[END_REF] or even a reduction [START_REF] Ceccotti | Proton acceleration with high-intensity ultrahigh-contrast laser pulses[END_REF] of the effectiveness of the acceleration process. Conversely, a nanostructured low-density layer can be produced with tunable and finely controlled properties, which, combined with an in-depth characterization, should lead to more reproducible experimental conditions. For a given envisaged application of laser-driven ion sources, the enhancement due to the advanced targetry would allow to dramatically reduce the requirements on the laser driver. Cheaper, more compact and higher repetition rate lasers could be used for applications which otherwise would have required more powerful, more expensive and definitely non-portable systems.

An accurate theoretical understanding of laser-driven ion sources based on double-layer targets (DLTs) is essential to guide experimental investigations and ultimately to enable applications of these sources. However, the presence of a nanostructured layer poses considerable modeling challenges. Indeed,the exceptional temporal contrast of modern day laser facilities [START_REF] Yu | Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirpedpulse amplification ti:sapphire laser Opt[END_REF][START_REF] Schramm | First results with the novel petawatt laser acceleration facility in dresden 8th[END_REF][START_REF] Obst | On-shot characterization of single plasma mirror temporal contrast improvement Plasma[END_REF] could allow a nanostructure to survive long enough to affect the interaction. In previous numerical works [START_REF] Fedeli | Parametric investigation of laser interaction with uniform and nanostructured near-critical plasmas[END_REF][START_REF] Fedeli | Ultra-intense laser interaction with nanostructured near-critical plasmas[END_REF], we observed that a detailed modeling of the nanostructure morphology is crucial for a reliable description of the physical processes at play.

In this work, by means of fully 3D particle-in-cell simulations, we model an experimental scenario of enhanced laser-driven ion acceleration where the target is a solid foil coated with a low-density nanostructured layer. A realistic morphology is used for the nanostructured layer and the laser parameters are those of a typical 30-50 Terawatt-class system. We consider two realistic morphologies for the low-density nanostructured coating (a fractal-like foam and a forest). We also couple the results of our particle-in-cell simulations to Geant4 [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF] Monte Carlo (MC) code, in order to simulate the conversion of laser-accelerated ions into neutrons [START_REF] Tentori | Laser-driven neutron sources: a first numerical investigation Master's[END_REF][START_REF] Arioli | Compact laser-driven neutron sources: a theoretical investigation Master's[END_REF], which is a promising application of laser driven ion acceleration. DLTs having a uniform near-critical layer with the same average density of the nanostructured layers are considered as well for comparison. Figure 1 illustrates the general setup and the numerical framework adopted in this work.

Our simulations show that the presence of a nanostructure allows for a significant increase of the maximum ion energy and of the total number of accelerated ions, with the details of the nanostructure morphology affecting both the maximum energy and the angular distribution of the accelerated ions. This translates directly into more neutrons being emitted per laser shot.

Methods

Our work is based on two numerical tools: particle-in-cell and MC simulations. This section describes in detail the methods and codes used in this work and the parameters of the simulations. In the following we will consider a laser wavelength λ=800 nm, as for a Ti:sapphire laser system.

Particle-in-cell simulations

Particle-in-cell simulations are a standard numerical technique to investigate laser-plasma interaction [START_REF] Birdsall | Plasma Physics Via Computer Simulation[END_REF][START_REF] Arber | Contemporary particle-in-cell approach to laser-plasma modelling Plasma[END_REF]. For our 3D simulations we relied on the open source, massively parallel piccante code [START_REF] Sgattoni | Optimising PICCANTE-an open source particle-in-cell code for advanced simulations on Tier-0 systems[END_REF]. The numerical box was 100λ×60λ×60λ, with periodic boundary conditions. The spatial resolution was 40 points per λ (ppλ) in each direction, while the temporal resolution was 98% of the Courant condition. The total duration was 80 λ/c. The laser beam had a Gaussian spatial profile with a waist equal to 5 λ and a cos 2 temporal profile with a FWHM for the fields equal to 15 λ. These parameters model an idealized, high contrast, Ti:sapphire laser system with tens of TW power. The beam was linearly polarized and the angle of incidence was 0°. The normalized intensity was a 0 =4 (a 0 =eA/m e c 2 , where e is the elementary charge, A is the vector potential, m e is the electron mass and c is the speed of light).

Concerning the target, ions were initialized perfectly cold, while electrons were initialized with a very small temperature (few eV). The substrate-i.e. the flat solid foil or bare target-was 0.5 λ thick, with an electron density of 40 n c . We decided to use an electron density of only 40n c for the substrate because 3D simulations with a more realistic density for a highly ionized target (∼100 n c ) would have required a huge computational effort. However, we performed convergence tests with 2D simulations to assess the appropriateness of our choice for both the solid foil density and the resolution. We let the density of the solid foil vary among 20 n c , 40 n c and 80 n c , while the resolution varies between 40 ppλ and 60 ppλ. The main results of these 2D tests are shown in figure 2, where the protons spectra at time 80λ/c are shown for (a) a simple 0.5 λ-thick solid foil and (b) a DLT modeled with a uniform near-critical layer with 2.14 n c density and 4 λ thickness. Increasing the electron density of the substrate from 20 n c to 80 n c did not result in any appreciable change of the ion energy spectrum for the DLT. Instead, a strong reduction of the maximum ion energy was observed for the simple flat target under the same density variation. Nonetheless, when doubling the density from 40 n c to 80 n c only a slight reduction is obtained. These results confirm that using a substrate density of 40 n c in our simulations should not significantly affect the results presented in this manuscript. Overall, a solid density of 40 n c is found to be a good compromise: largeenough to be overdense, but low-enough to let us resolve the skin depth of the laser with a resolution of 40 ppλ. In any case, to ascertain the goodness of such a resolution value, we also tested a higher resolution of 60 points per λ in 2D geometry and we observed negligible differences in the proton spectra (see figure 2). 40 macro-electrons per cell and 2 macro-ions with Z/A=0.5 per cell were used. The hydrocarbon contaminant layer was 0.05 λ thick with an electron density of 7 n c , a proton density of 1 n c and a fully ionized carbon density of 6 n c (64 macro-particles per cell per species were used). The low-density layer was chosen to be 4 λ thick. Three different morphologies (an idealized uniform plasma, a fractal-like nanostructured foam and a random nanowires forest) and two different average densities (1.5 n c and 2.29 n c ) were considered as well as the case of a simple flat target for comparison, for a total of seven simulations. Both the realistic morphologies considered here are representative of low-density nanostructured materials used in laser-plasma interaction experiments [START_REF] Prencipe | Development of foam-based layered targets for laser-driven ion beam production Plasma[END_REF][START_REF] Passoni | Toward high-energy laser-driven ion beams: nanostructured double-layer targets[END_REF][START_REF] Bargsten | Energy penetration into arrays of aligned nanowires irradiated with relativistic intensities: scaling to Terabar pressures[END_REF][START_REF] Cristoforetti | Transition from coherent to stochastic electron heating in ultrashort relativistic laser interaction with structured targets[END_REF].

The foam layer is generated through an extension of the diffusion-limited cluster-cluster aggregation model (DLCCA), designed to simulate the structure of fractal aggregates, such as colloids and soot [START_REF] Meakin | Formation of fractal clusters and networks by irreversible diffusion-limited aggregation[END_REF] (a detailed description of the method is provided in [START_REF] Fedeli | Ultra-intense laser interaction with nanostructured near-critical plasmas[END_REF]). The building blocks of the fractal aggregates are nanospheres having a local electron density of ∼40 n c , arranged in such a way that the filling factor is 5.7% and 3.8% for the two density values. The radii of the nanospheres are 0.05λ for the higher density foam and 0.047 λ for the lower density foam. Nanowires forests are modeled as a collection of cylinders, with a radius of 0.12 λ and an inclination angle extracted randomly from a uniform distribution between -30°and +30°with respect to laser axis. The nanowires have a local electron density of ∼40 n c and their number is chosen so that the filling factor is either 5.7% or 3.8%. For the nanostructured layers 40 macro-electrons per cell and 2 macro-ions with Z/ A=0.5 per cell were used, while for the uniform layers 4 macro-electrons per cell and 1 macro-ion with Z/ A=0.5 per cell were used. The main properties of the near-critical layers are summarized in table 1. All the considered near-critical layers are coupled to a substrate with 0.5 λ thickness and 40 n c electron density.

We point out that the nanostructures show large local fluctuations of their average density on scales comparable with the laser waist (5λ). Figures 3(a panel (b)), the fluctuations in the random wires are negligible, while those in the foams can be significant (up to a factor of 2).

MC simulations

We simulated the process of proton-to-neutron conversion via MC simulations. To this purpose, we employed the Geant4 (Geometry and Tracking) toolkit [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF], which is widely used to simulate particle transport through matter in a variety of scenarios. Here, we are interested in a reliable description of neutron generation from nuclear reactions induced by moderate energy protons (∼few MeVs). The main physical processes, i.e. the hadronic processes, were included via the G4HadronPhysicsQGSP_BIC_AllHP physics list, which was found to be the most reliable in a previous benchmarking work [START_REF] Tentori | Laser-driven neutron sources: a first numerical investigation Master's[END_REF]. Due to the hadronic nature of the process, the primary bunch is only made of protons, whereas we disregarded electrons and photons coming from the prior laser-plasma interaction process. The proton momentum distribution is obtained from the 3D PIC simulations results. Indeed, 3D PIC simulations allow one to retrieve the full momentum distribution of any species. Therefore, we sampled the accelerated protons momentum distribution from the PIC output with the Inverse Transform Sampling method. Such sampling procedure is used in the MC code to initialize the energy and propagation direction of the primary proton bunch. The proton source is spatially point-like and is positioned 1.5 cm away from the converter. In each case, 10 10 protons were shot. The converter is made of pure 9 beryllium with thickness equal to the range of the most energetic protons (0.009-0.06 cm). The main nuclear reactions at play are (p, n * ).

Results

In this section we present the results of our numerical investigation, with a particular focus on ion acceleration and neutron generation. A brief discussion of the physical processes at play in ultra-intense laser-plasma interaction is provided as well.

Laser absorption into hot electrons

Laser absorption into electron kinetic energy in a near-critical plasma is a complicated, multifaceted process [START_REF] Pukhov | Relativistic magnetic self-channeling of light in near-critical plasma: three-dimensional particlein-cell simulation[END_REF][START_REF] Borghesi | Relativistic channeling of a picosecond laser pulse in a near-critical preformed plasma[END_REF][START_REF] Robinson | Absorption of circularly polarized laser pulses in near-critical plasmas Plasma[END_REF][START_REF] Askar'yan | Magnetic interaction of self-focusing channels and fluxes of electromagnetic radiation: their coalescence, the accumulation of energy, and the effect of external magnetic fields on them[END_REF]. In the case of a nanostructued plasma, the absorption process is further complicated by the inhomogeneities of the density profile, which are likely to be maintained during the interaction, especially because the ion dynamics is slower than the tens-of-fs laser temporal duration [START_REF] Fedeli | Parametric investigation of laser interaction with uniform and nanostructured near-critical plasmas[END_REF][START_REF] Fedeli | Ultra-intense laser interaction with nanostructured near-critical plasmas[END_REF]. In our scenario, the foam targets lead to an increased laser absorption into electrons. The flat target allows for ∼10% conversion of laser energy into electron kinetic energy, while the remaining portion of the laser is mostly reflected. On the other hand, the DLTs allow for a maximum electron absorption of ∼40%-60% (except for the denser homogeneous foam, which is too opaque for the considered pulse, for which it is ∼20%). Depending on the morphology of the nanostructure, the absorption process leads to different features of the hot electron population, as shown in figure 4. In particular, the presence of a near-critical nanostructured layer leads to significantly higher electron energies and a broader angular distribution. Concerning the homogeneous foams instead, the relatively dense 2.29 n c uniform layer is barely transparent for a a 0 =4 laser beam. This means that the absorption efficiency is not significantly higher than for the flat target. The uniform foam with a density of 1.5n c and the generation of hot electrons is improved, but still significantly below what is obtained with the nanostructured foams. These differences translate into different strengths and extensions of the quasi-static electrostatic fields at the back side of the target. Since the ion acceleration process is TNSA-like, these features have a profound influence on the properties of the accelerated ions. However, a detailed study of the absorption process requires an extensive, dedicated, investigation, which is not the focus of this manuscript and has been partially addressed elsewhere [START_REF] Cialfi | Electron heating in subpicosecond laser interaction with overdense and near-critical plasmas[END_REF][START_REF] Fedeli | Parametric investigation of laser interaction with uniform and nanostructured near-critical plasmas[END_REF][START_REF] Fedeli | Ultra-intense laser interaction with nanostructured near-critical plasmas[END_REF].

Enhanced laser-driven ion acceleration

Figure 5 shows the salient features of the accelerated ions obtained with the 2.29 n c foam-coated targets. The results for a simple flat target are reported as well for comparison. With respect to a simple flat target, the presence of the homogeneous foam leads only to a moderate enhancement of the cut-off energy of the accelerated ions (from ∼2 to 3 MeV). On the contrary, the target coated with a fractal-like DLCCA foam attains ∼4.5 MeV and the target coated with a random nanowire forest allows to obtain slightly more than 6 MeV, a 3×increase with respect to the simple flat target. These results are in agreement with what was observed for the electron population, since in a TNSA-like scheme a higher efficiency of laser-to-electrons coupling generally results into higher cut-off energies of the accelerated ions.

Concerning the angular distribution of the accelerated ions, the presence of a nanostructured layer seems to lead to a broader angular spread. Again, this is in agreement with what was observed for the electron population. Moreover, the angular distribution is much less homogeneous, with 'holes' and denser stripes appearing in the 'synthetic radiochromic film' diagnostic. These features are essentially irrelevant for the application of laserdriven proton sources considered here (neutron generation). However, they might be a severe hindrance if such a source is used for proton radiography, where having a smooth angular distribution is essential.

Figure 6 reports also the proton energy spectra obtained with foams having a lower average density (1.5 n c ). The most significant difference concerns the homogeneous foam: the lower density leads to a much greater laser absorption and therefore a higher maximum energy of the ions. For the structured foams, the lower average density has only a mild effect, slightly beneficial for the DLCCA foam and slightly disadvantageous for the nanowires foam. From the point of view of applications requiring stable ion fluxes, this mild dependence on the density could be a considerable advantage. The average density of nanostructured targets might change from sample to sample, due to unavoidable fluctuations in the manufacturing process. Our results suggest that the properties of an ion source based on these targets would remain relatively stable, even with large shot-to-shot variations of the target properties. This is further supported by results obtained in a previous work [START_REF] Fedeli | Ultra-intense laser interaction with nanostructured near-critical plasmas[END_REF], where we considered laser interaction with nanostructured foams having quite different morphologies but the same average density, observing only a mild effect on laser absorption.

Enhanced neutron generation

Figure 6 represents the cross sections for the (p, n * ) reaction in 9 Be, together with the energy spectra of the accelerated ions for all the simulated cases. Figure 7 illustrates the properties of the emitted neutrons in all the simulated cases, while figure 8 summarizes the proton-to-neutron conversion rate (top) and the total neutron yield (bottom). A first crucial observation is that accelerating protons at least up to 2 MeV is absolutely necessary to obtain neutron conversion, since this is the threshold for the nuclear reaction to take place. When the simple flat target is used for ion acceleration, the energy cut-off is barely above the 2 MeV threshold for neutron conversion (see figure 6), with extremely few ions having enough energy to have a non-zero chance of generating a neutron. This leads to a very low p-to-n conversion rate (∼10 -8 neutrons/protons), hence low neutron yield (∼10 2 neutrons/shot), as shown in figures 7 and 8. The presence of a near-critical layer leads to more ions having enough energy for a (p, n * ) reaction (see figure 6). This leads to a very strong enhancement of the p-to-n conversion rate. Assuming that 10 10 protons are accelerated above 2 MeV with the flat foil and taking into account the relative number of protons accelerated to energies above the reaction threshold, we also obtain a very strong enhancement (from 4 to 7 orders of magnitude) of the neutron yield with respect to the simple flat target, as shown in figure 8. Among the DLTs, the denser homogeneous foam is the worst performing case (∼10 6 neutrons/shot), while all the other cases fall within the 10 8 -10 9 neutrons/shot range.

It is also worth highlighting that the cross section for (p, n * ) reaction in 9 Be peaks at ∼6 MeV (see figure 6, gray curve). Therefore, the proton energies obtained with the nanostructured targets are already enough to obtain optimal neutron conversion.

The left panel of figure 7 shows the energy spectra of the neutrons emitted in a 15°cone around the target normal in the forward direction. The distributions are quite broad, with a peak at 1-2 MeV and a cut-off extending up to 4 MeV, which means that the neutron source described here generates fast neutrons. A cut-off of ∼4 MeV is coherent with a maximum ion energy of ∼6 MeV and a negative Q-value of the (p, n * ) reaction of ∼-1.85 MeV. Neutrons are emitted in all directions, as shown in the right panel of figure 7. However, the angular distribution is not uniform: about one order of magnitude more neutrons are emitted in forward direction. This could be beneficial for applications, also possibly mitigating radio-protection concerns.

Discussion

We presented a scheme of an ion source able to accelerate protons to an energy of few MeVs, driven by a moderately intense laser thanks to the use of nanostructured DLTs. We addressed the issue of setting up a description that takes into account non-trivial features, such as the details of the target nanostructure, the effect of a density fluctuation and the direct coupling with a p-to-n converter. Including these features in the description should be crucial if one expects some kind of quantitative agreement with experiments or aims at designing an actual source.

As discussed in the introduction, the few MeVs energy range is suitable for several applications in materials science. There exists a subset of applications requiring only a modest particle flux, such as the so-called proton induced x-ray emission spectroscopy [START_REF] Verma | Atomic and Nuclear Analytical Methods: XRF, Mössbauer, XPS, NAA and Ion-Beam Spectroscopic Techniques[END_REF], an ion beam analysis technique. This technique could be an ideal application of laser-driven ion sources [START_REF] Passoni | Superintense laser-driven ion beam analysis[END_REF], since few 10s of laser shots could be enough to complete a measurement. Our work represents a step forward in making the aforementioned moderate-energy applications of laser-driven ions a reality.

Adding a beryllium converter, we estimate that the laser-driven neutron source described in this work could provide ∼5×10 5 neutrons per laser shot if 10 10 protons are accelerated to energies above the reaction threshold. Adopting targetry solutions analogous to those described in [START_REF] Prencipe | Targets for high repetition rate laser facilities: needs, challenges and perspectives High Power[END_REF][START_REF] Zaffino | Wafer-scale fabrication of target arrays for stable generation of proton beams by laser-plasma interaction[END_REF] should allow operation of such a neutron source in a repetitive regime, for at least ∼10 3 laser shots. Since compact 50 TW laser systems able to attain a repetition rate of ∼10 Hz are commercially available [START_REF] Thales | QUARK 45: Ultrafast Ti:Sa Laser Series[END_REF], we estimate a neutron flux of ∼5×10 6 n s -1 sustained for ∼10 2 s to be within reach of existing technology. This seems doable from the practical point of view. In the most common configuration the target consists of a perforated holder upholding the DLT. In this case, the main limitation comes from the target transverse size and consequently from the number of holes that can be drilled in the holder. No actual limitation comes from having to grow the foam layer on a large substrate (in the worst case different targets may be juxtaposed). In previous experiments holders with ∼200 holes were employed, so that shooting hundreds of times at 1 Hz repetition rate seems doable. After that, a new target needs to be installed.

Our estimation for the neutron flux suggests that it could be comparable with that of other proposed compact laser-driven neutron sources reported in the literature, where for 'compact' we mean relying on tabletop Ti:sapphire lasers not exceeding ∼100 TW. For instance, the interaction of a few mJ laser at kHz frequency with a heavy-water jet [START_REF] Hah | High repetition-rate neutron generation by several-mJ, 35 fs pulses interacting with free-flowing d2o[END_REF][START_REF] Hah | Characterization of a high repetition-rate laser-driven shortpulsed neutron source Plasma[END_REF] has been reported to lead to an average neutron yield of 2×10 5 , while 2×10 6 neutrons per shot have been obtained via photoneutron generation with wakefield-accelerated electrons driven by a 0.5 J, 10 TW laser at ∼10 -2 Hz [START_REF] Jiao | A tabletop, ultrashort pulse photoneutron source driven by electrons from laser wakefield acceleration Matter[END_REF]. Concerning schemes similar to that proposed here, where a target for ion acceleration and a separate converter for neutron generation are used, a yield of ∼10 6 neutrons per steradian per shot can be obtained with few J lasers, but relying on deuterated targets (see [START_REF] Alejo | Recent advances in laser-driven neutron sources[END_REF] and references therein). In any case, it is worth remarking that the estimation of the yield of our source depends on our specific choices for the parameters of the contaminant layer, which are unfortunately not well characterized from the experimental point of view.

Conventional, portable, neutron sources based on deuterium-deuterium or deuterium-tritium fusion reaction can attain neutron fluxes two orders of magnitude higher than the source described here [START_REF] Chichester | Advanced compact accelerator neutron generator technology for active neutron interrogation field work[END_REF]. An improvement of the neutron yield for our source could be obtained using deuterated targets for ion acceleration, a strategy explored in several works [25-28, 32, 34]. Indeed, the 9 Be(d, n * ) cross section is significantly higher than 9 Be(p, n * ) cross section and the energy threshold for the projectile is lower [START_REF] Koning | Modern nuclear data evaluation with the TALYS code system[END_REF]. This could allow to partially close the gap. However, the main strength of laser-driven neutron sources is not the average flux, for which competing with conventional sources seems to be a daunting task. Rather, the main advantages of a laser-driven sources lie in their pulsed nature and in their extremely short duration. A pulsed source could be beneficial to reduce signal-to-noise ratio in several applications (e.g. radiography or spectroscopy), since temporally gated detectors could be synchronized with the laser pulse. Indeed, 'flash neutron radiography' has been suggested in previous works as an application especially suited for laser-driven neutron sources [START_REF] Guler | Neutron imaging with the short-pulse laser driven neutron source at the Trident laser facility[END_REF]. Overall, our modeling approach could be used to guide the design and the optimization of these kinds of applications.

Conclusion

We have performed a numerical simulation campaign of a laser-driven ion acceleration scheme based on an advanced targetry solution, and of a neutron source based on this scheme.

DLTs consisting in a near-critical layer coupled to a solid foil can enhance the efficiency of the ion acceleration process, leading to more ions being accelerated up to higher energies. This results in a strong enhancement of the neutron yield when these ion sources are coupled with a suitable converter. Since one promising way to realize a near-critical layer consists in exploiting low-density nanostructured materials, we assessed the role of different realistic nanostructure morphologies on the ion acceleration process. We found that a realistic modeling of the nanostuctured target is essential to provide reliable indications for the design of experiments and for the development of laser-driven ion sources.

Our results suggest that nanostructured DLTs could be a crucial component to enable applications of hadron sources driven by compact, table-top lasers and our work provides essential tools for the numerical modeling of these schemes.

agreement No 647 554 and INTER grant agreement No 754 916). We also acknowledge Iscra access scheme to MARCONI and GALILEO High Performance Computing machine at CINECA (Casalecchio di Reno, Bologna, Italy) via the projects IRONMAN, PIMIENTO and ELF.

Figure 1 .

 1 Figure 1. Scheme of the numerical framework adopted for this work. (a) Typical parameters of a 50 TW table-top laser system are considered (b) The use of targetry solutions for repetitive operation is considered to estimate total yields over several minutes. (c) Scanning electron microscope picture of a nanostructured low-density foam similar to those used in [48, 49]. (d) 3D particle-in-cell simulations are used to model laser-driven ion acceleration (reproduced with permission from [70]). (e) Laser-accelerated ions are fed into the Geant4 Monte Carlo code to simulate neutron generation in a beryllium converter.

  ) and (b) show the density profiles of the near-critical layers integrated over a transverse region of size 25λ 2 and 100λ 2 , respectively. On the illuminated region (size ∼100λ 2 ,

Figure 2 .

 2 Figure 2. Proton spectra at time 80 λ/c obtained in the 2D PIC convergence tests varying the resolution (40 and 60 points per λ) and the density of the solid foil (20 n c , 40 n c and 80 n c ) for (a) the bare target case and (b) a double-layer target with a homogeneous nearcritical layer of 2.14 n c density and 4 λ thickness.

Figure 3 .

 3 Figure 3. Integrated electron density profile as a function of the depth of the near-critical layer x, which is also the laser propagation direction (the laser hits the target at x = 0). The integration is performed over the transverse regions (a) [-2.5λ, 2.5λ]×[-2.5λ, 2.5λ] and (b) [-5λ, 5λ]×[-5λ, 5λ].

Figure 4 .

 4 Figure 4. (a) Sketch of the TNSA process obtained with 3D PIC simulations at three time steps: at t=0λ/c the laser propagates towards the target, at t=20λ/c a population of relativistic electrons can be identified (red dots), at t=80λ/c the ions from the contaminant layer have been accelerated (blue dots). (b)-(h) The graphs show the projection of the hot electron phase space onto an energy-angle plane (the energy is measured in MeV and the angle is calculated with respect to target normal). (i) Momentum spectra (dN/dp x ) of the electron population for different targets. (j) Electron density (dN/dx) at the back side of the targets. (k) Longitudinal component of the electric field at the back side of the targets. The vertical dashed line in panels (i) and (j) represents the rear target surface. All the graphs are referred to time t=20 λ/c and report results for the various target configurations described in this work.

Figure 5 .

 5 Figure 5. (a) Scheme to clarify the conventions used for the angles in the other panels. (b) Ion energy spectra for the three cases having n e =2.29n c and for the monolayer target for comparison at time 80λ/c. (c) Snapshots of the 3D PIC simulations at time 20λ/c. The zĉ omponent of the magnetic field is shown in blue-red scale, while the electron density is shown in grayscale. (d) 'Synthetic radiochromic films' at time 80λ/c. Each panel shows the angular distribution of the emitted ions for a given energy range (indicated above) and for a given target configuration.

Figure 6 .

 6 Figure 6. The graph shows the cross section for (p, n * ) reaction in 9 Be (gray curve, right axis), together with the energy spectra of the accelerated ions for the different simulated scenarios obtained at 80λ/c (colored curves, left axis).

Figure 7 .

 7 Figure 7. The panel on the left shows the energy distribution of the generated neutrons in all the simulated cases. The emission is integrated in a 15°cone around target normal in forward direction. The panel on the right shows the angular distribution of the emitted neutrons for all the simulated cases. 0°corresponds to the forward direction. The dent at ∼90°corresponds neutrons propagating inside of the beryllium converter.

Figure 8 .

 8 Figure 8. Top panel: conversion rate of protons into neutrons for all the simulated cases. Bottom panel: total neutron yield for all the simulated cases. Here, it is assumed that 10 10 protons are accelerated to energies above 2 MeV with the bare target; the neutron yields obtained with the DLTs are scaled according to combined PIC and MC simulations results.

Table 1 .

 1 Comparison between homogeneous and nanostructured nearcritical layers used in the 3D PIC simulations.

		Near-critical layers
	Morphology	Homogeneous	Foam-like and ran-
			dom wires
	Filling factor	100%	3.8% and 5.7%
	Local electron	1.5 n c and 2.29 n c	40 n c
	density
	Average electron	1.5 n c and 2.29 n c	1.5 n c and 2.29 n c
	density
	Thickness	4 λ	4 λ

Acknowledgments

We thank Francesco Mirani, Politecnico di Milano (Italy), for useful discussions and for his assistance on the setup of the Monte Carlo simulations. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ENSURE grant