Abood Mourad
email: abood.mourad@centralesupelec.fr

Jakob Puchinger
email: jakob.puchinger@centralesupelec.fr

Tom Van Woensel
email: t.v.woensel@tue.nl

Combining people and freight flows using a scheduled transportation line with stochastic passenger demands

Keywords: Optimization, freight transportation, pickup and delivery, scheduled lines, heuristic method

published or not. The documents may come

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Introduction

As both people and goods move in the urban environment, a successful integration of their streams has the potential of enhancing the quality of their existing transportation services as well as reducing congestion and pollution levels [1]. This paper considers an integrated system in which a set of freight requests need to be delivered using a fleet of grounded robots where a public transportation service (referred to as scheduled line (SL)) can be used as part of a robot's journey1 . In addition, we consider that passengers and robots (carrying goods) share the same SL capacity where passenger demands are stochastic. Thus, depending on passenger demand realization, a robot might not be able to use the SL service and some recourse actions need to be applied. Thus, we develop a stochastic approach for operating this system and we perform an extensive computational study to analyze its performance and potential benefits.

Problem description

In this problem, a set of autonomous shuttles operate through a fixed-route scheduled line (SL) service for transporting passengers in both directions. This service consists of a set of transfer nodes (i.e. stations) and a set of scheduled lines linking them. Every SL has a capacity and a timetable. In addition, a fleet of grounded, pickup and delivery (PD) robots are located at transfer nodes to serve freight requests. Each PD robot has a capacity and a maximum service distance indicating the maximum distance it can go from a transfer node. Moreover, SLs and PD robots are associated with a shipping cost per one time unit.

Furthermore, a set of freight requests need to be delivered using the fleet of PD robots. Each request is associated with an origin, a destination, pickup and delivery time windows, and a demand quantity. Thus, each request has to be served within its corresponding time windows. Depending on the availability of vacant places, PD robots carrying freight may travel with passengers through SLs. Therefore, delivering a request can be done in either direct or indirect way. In a direct delivery, a request is delivered directly to its final destination by a PD robot without the use of SL (Figure 1a ; request a 1 is picked up at its origin o a1 by a PD robot coming from transfer node s 2 , and delivered to its final destination d a1 before the PD robot returns to transfer node s 3 ). This direct delivery is only feasible if the distance between request origin and destination locations is less than the maximum distance the PD robot can travel. On the other hand, in an indirect delivery, a request is collected by a PD robot, transfered through SL, and delivered afterwards to its final destination by the same PD robot (Figure 1b ; request a 1 is picked up at its origin o a1 by a PD robot, transported through SL from s 2 to s 3 and finally delivered to its final destination d a1 by the PD robot). We assume that a passenger or a PD robot take over one place in a shuttle and that passengers are prioritized. For the sake of simplifying the problem, we assume that each PD robot can serve only one request at a time and that freight quantities are known in advance. On the other hand, passenger demand is only learned upon shuttles' arrivals to SL stations. Thus, the number of available places for PD robots at each SL departure is stochastic which might yield two capacity violation outcomes : (i) PD robot not being able to take the next SL departure due to the high passenger demand at the corresponding station, and (ii) PD robot having to get off the shuttle at an intermediate station in order to give its place to a passenger. When such route failures occur, recourse actions are needed in order to recover feasibility where applying these actions might lead to extra transportation costs compared to original routes.

Solution approach

Similar to [2], we model this problem as a two-stage stochastic problem, where the first-stage aims at defining routes for PD robots carrying freight, and the second-stage involves evaluating these routes over a set of scenarios and computing their recourse costs. Thus, we provide a MIP formulation for the proposed pickup and delivery problem where the overall objective is to minimize the sum of the routing and recourse costs. Then, we propose a sample average approximation (SAA) method along with an Adaptive Large Neighborhood Search (ALNS) algorithm to solve the stochastic optimization problem.

Conclusions

To conclude, our key contributions can be seen in the problem setting we consider, the modeling and solution approach we propose to handle it, and the experimental study we provide to assess its different aspects as well as the potential benefits of such combined systems.

Références

[1] Martin Savelsbergh and Tom Van Woensel. 50th Anniversary Invited Article -City Logistics : Challenges and Opportunities. Transportation Science (2016) 50(2) :579-590.

[2] Veaceslav Ghilas, Emrah Demir, and Tom Van Woensel. A scenario-based planning for the pickup and delivery problem with time windows, scheduled lines and stochastic demands.

  FIG. 1 -Request service modes : direct & indirect delivery

This integrated system was inspired from Toyota new e-Palette concept.


Wessel Reijers

Iris Wuisman

Morshed Mannan

Primavera De Filippi

Christopher Wray

Vienna Rae-Looi

Angela Cubillos Vélez

Liav Orgad

De Filippi

Now the Code runs itself : On-chain and Off-chain governance of blockchain technology

Keywords:

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Introduction

The invention of Bitcoin in 2008 as a new type of electronic cash has arguably brought about one of the most radical financial innovations in the last decade. In the past few years, not only has Bitcoin gained tremendous public attention, but its central architectural backbone-the blockchain-has also spread its wings into a multitude of new fields. Blockchain technologies create a 'trustless' proof mechanism for transactions between users, and allow for the registration of assets and the selfexecution of software code (also known as 'smart contracts') 1 . In particular, the topic of blockchain governance has become central in the blockchain community, notably in the aftermath of the attack on 'The DAO'-a decentralized venture capital fund that was meant to be the world's first fully functioning decentralized autonomous organization (DuPont 2017). We will use The DAO attack as an illustrative case study of some of the key issues that arise in blockchain governance.

The governance of blockchain-based systems usually incorporates a variety of rules and procedures that may be implemented both 'on-chain' and 'off-chain'. Onchain governance refers to rules and decision-making processes that have been encoded directly into the underlying infrastructure of a blockchain-based system. This type of governance defines the rules of interactions between participants through the infrastructure within which these interactions take place; these interactions are solely determined by rules embedded within the underlying blockchain code-the so-called rule of code [START_REF] Filippi | Blockchain and the Law: the Rule of Code[END_REF]. The rules and processes may be layered; meaning that one layer of rules is subject to another. For instance, some rules may allow for infrastructural changes by stipulating the procedures to change other (lowerlevel) rules, and potentially even the (higher-level) rules themselves. On-chain governance cannot be easily avoided or bypassed stricto sensu, because it operates according to a system of rules that have been encoded directly into the system that is responsible for enforcing them. Off-chain governance comprises all other (i.e. nonon-chain) rules and decision-making processes that might affect the operations and the future development of blockchain-based systems. Off-chain governance includes both endogenous and exogenous rules. The former category refers to the rules adopted 1 Since there is a wide literature about the basics of blockchain technology (cf. Grinberg 2012;

Tschorsch and Scheuermann 2016), we will not engage in a technical exposition of how it works. Instead, we will focus on the governance structures adopted by blockchain systems and communities and how these are susceptible to transform social and legal relations between people. by a reference community to ensure the proper functioning and ongoing development of a blockchain-based system (including procedures to implement protocol changes).

The latter category includes all rules imposed by a third-party onto the reference community, e.g. national laws and regulations, contractual agreements, technology standards, and so forth2 .

The ongoing debate about on-chain and off-chain governance turns on the practical question of whether existing rules and decision-making processes governing a blockchain-based system should be changed from the inside or the outside by the reference community, and whether the system should provide for a mechanism to change the governance structure itself. This practical question leads to the more theoretical and normative question of whether an existing set of code-based rules could and should overtake the exercise of human judgment in decision-making, and what are the ethical and political considerations this would entail (De Filippi & Hassan 2018). Addressing these questions brings about a new legal discourse. In an earlier paper on blockchain technology, Reyes argues that distributed ledger technology (DLT), such as blockchain technology, "will change legal discourse about the fundamental elements of legal systems, including substantive law, legal structures, and legal culture" (Reyes 2017, p. 1). The new legal discourse, she claims, "will stand on its own as a new field of legal academic inquiry and area of legal practice" (ibid, p. 63). While this observation seems correct, blockchain technology also brings to the fore ideas that have long been subject to jurisprudential debate. This paper seeks to situate the blockchain discussion within the field of legal philosophy, examining how legal theory can apply in the context of blockchain governance. In particular, the paper explores Carl Schmitt's criticism of a positivist legal order, which he developed in contrast to the account of positivism espoused by Hans Kelsen, and how this criticism is related to the ongoing debates on blockchain governance. Our analysis includes three steps. First, we examine whether a critique of on-chain governance might parallel Schmitt's critique of Kelsen's legal positivism.

Second, we consider to what extent The DAO attack and its aftermath constituted a state of exception that challenged the legal order of Ethereum's internal (on-chain) governance structure. Third, we reflect on Schmitt's argument that the coalescence of private interests is a vulnerability of positivist legal systems, and accordingly posit this as an inherent vulnerability of on-chain governance of existing blockchain-based systems.

On-Chain Governance: "Now the Code Runs Itself"

First, we ask whether a critique of on-chain governance might parallel Schmitt's critique of Kelsen's conception of a positivist legal order. This critique hinges on the question of whether a legal order can be self-sustaining or whether it depends on decision-making by a sovereign authority (Dyzenhaus 1994, p. 10). Offchain governance and on-chain governance fundamentally differ along similar lines: off-chain governance allows for interventions into the blockchain protocol that are not prescribed by the protocol itself, by outside authorities. In other words, with off-chain governance people are in charge of the code, without their actions being determined by it. However, off-chain governance introduces the problem of personal sovereignty (e.g. strong individuals dominating decision-making processes). At first blush, therefore, on-chain governance seems to be the preferable mode of governance for blockchain-based systems because it ensures that no individual or group of individuals can impose their will on the blockchain community at large. Such a mode of governance seems to embrace a number of central premises of legal positivism, notably of the type of positivism espoused by Kelsen. According to Kelsen, a positivist legal order enables a peaceful and legitimate resolution of disputes in a pluralist society, without recourse to external sources (moral or political) to justify its legitimacy. Kelsen believed that laws are valid if promulgated in accordance with the 'basic norm' of the legal order and with the legislative procedure that is authorized by this basic norm (Vinx 2007, pp. 23, 40). He thereby emphasised the content-independence of the law's legitimacy (Hart 1994, p. 36). As Vinx explains, Kelsen denied "a necessary relation between legitimacy and justice because he wanted to attack the idea that positive law is legitimate (…) only as long as its content conforms to some absolute standard of justice external to positive law" (Vinx 2007, p. 23).

Due to its reliance on rational, factual tests to settle disputes, Kelsen's conception of law making and enforcement aims to exclude any notion of private human judgement. This is most evident in Kelsen's conception of a pure legal rule.

Voegelin, who was a doctoral student of Kelsen, condenses Kelsen's "pure legal rule" to a conditional algorithm: "If Mh + E (or Mu + E), then Z => M" from which all codes and statutes derive (Voegelin 1927, pp. 270-271). The first part of the algorithm denotes past occurrences of behaviours, either in terms of performance (Mh) or avoidance (Mu), in conjunction with events (E) that usually occur as a result of said behaviour, and the second part denotes the direction of enforcement by an official authority (Z) against the individual engaging in said human behaviour (M). For instance, if a particular behaviour in conjunction with an event constitutes the operative fact of "theft", then the official authorities are to direct their coercive powers to this operative fact. If the operative facts such as theft are defined explicitly and precisely, the room for individual judgment is narrow. Here, Voegelin argues, the emphasis of the pure legal rule rests on enforcement, whereby behaviour and events only act as conditions for action by official authorities. Thus, it does not matter who makes the laws (or who is the sovereign) as long as the mechanical process of law making and enforcement is operating properly.

Dyzenhaus argues that according to Kelsen's legal theory "the underlying concept is that no individual should be subject to the will of any other individual or group of individuals" (Dyzenhaus 1994, p. 10). In order to make this possible, the extent to which personal judgment is involved in the making and enforcing of a legal order should be reduced to a minimum. Hence, for Kelsen, at the base of a legal system does not lay a sovereign power but a 'basic norm'. To arrive at this concept, Kelsen posits the axiom that a description of what is the case (e.g. facts of nature) cannot account for what ought to be the case (i.e. a norm). For instance, the validity of the norm that one ought not to steal cannot be explained from the description of behaviour that would constitute stealing. Accordingly, Kelsen argues, "the reason for the validity of a norm can only be the validity of another norm" (Kelsen 2005, p. 194). Kelsen refers to the norm underlying a legal system, from which all other norms are derived, as the 'basic norm'. Notably, this basic norm is not posited as a rule within the system, but as a basic presupposition that supports the system from the outside. For instance, the statement that laws should be obeyed because they are issued by the sovereign would, according to Kelsen, need to admit the fundamental notion that obeying the sovereign is the presupposed basic norm (Kelsen 2005, p. 194). Consequently, the key condition for a functional legal system is that its laws are non-contradictory-not that a legitimate sovereign authorises and enforces them (Dyzenhaus 1994, p. 11).

We can trace the logic of on-chain governance along similar lines as those prescribed by Kelsen. A striking commonality between Kelsen's legal theory and the way in which blockchain technologies function is that the validity of transactions in a blockchain-based system is not determined by the content of these transactions, but by their conformity with the consensus protocol, which is determined by a factual and objective mathematical process of verification. For instance, valid blocks with transactions on the Bitcoin blockchain are not added based on their contents (i.e.

based on what is transacted, from whom to whom), but based on the operative fact of whether they comply with an algorithmic test performed by a mining node. All participants in the system must recognise this type of validity for the system to be deemed authoritative. Therefore, it is presumed that the consensus protocol of a blockchain-based system represents a shared concept of authority amongst participants. As such, Kelsen's argument that "by presupposing the basic norm (…) one ought to behave as the constitution prescribes" (Kelsen 2005, p. 202), can be reinterpreted in relation to on-chain blockchain governance: by presupposing the basic norm, one ought to behave as the consensus protocol prescribes. This basic norm is tacitly presupposed, meaning that the blockchain-based system does not itself contain a rule that prescribes it. All other decision-making rules and processes implemented in on-chain governance are derived from this basic norm and ought not to contradict it.

Hence, we can observe a striking similarity between the notion of on-chain governance and Kelsen's notion of a positivist legal order. Under this view, not only are the rules of interaction between participants dictated by the blockchain protocol, but so are the rules defining the operation of the infrastructure within which these interactions take place. Some blockchain-based systems realise this form of positivist on-chain governance by relying on code-based structures not only for the regulation of participants' behaviour, but also for the introduction of changes to the infrastructure within which participants operate. For instance, in the Tezos blockchain, the "seed protocol specifies a procedure for stakeholders to approve amendments to the protocol, including amendments to the amendment procedure itself" (Goodman 2014, p. 1). Notwithstanding the ability of individual stakeholders to change the protocol rules, once they join the system, they agree to be bound by the current rules of the protocol, which ultimately dictates their behaviour. 3 The first rules contained in Tezos' seed protocol are meant to be highly conservative within that particular blockchain ecosystem-displacing the authority of any personalist sovereign that might want to intervene. Hence, on-chain governance realised on a blockchain-based system such as Tezos shows strong similarities with Kelsen's positivist conception of the legal order. However, as is the case with on-chain governance, Kelsen's legal theory has not remained unchallenged. Schmitt criticises Kelsen's conception of the legal order, which, according to him, embodies the apotheosis of the enlightenment project (Dyzenhaus 1994, p. 10). In appropriating Schmitt's criticism, we notably do not intend to accept it4 , but to position it as an important challenge to positivist legal systems. Reflecting on the type of legal positivism that Kelsen's theory represents,

Schmitt states:

"The general validity of a legal prescription has become identified with the lawfulness of nature, which applies without exception. The sovereign, who in the deistic view of the world, even if conceived as residing outside the world, had remained the engineer of the great machine, has been radically pushed aside. The machine now runs by itself" (Schmitt 2005, p. 48).

To problematize the positivist understanding of the legal order, Schmitt claims that the applicability of legal norms presupposes a situation of social normality, because legal norms cannot be applied to systems in a chaotic state. Emergencies that result from chaos-such as military coups, acts of war, natural disasters, financial crises, etc. -could change the situation into abnormality. In conventional public governance, far-reaching decisions need to be taken to deal with these emergencies.

Underlying Schmitt's central points of criticism is the contention that a positivist order-the machine that runs itself-is inherently contradictory as it seeks to replace the role of the sovereign (who can make decisions on fundamental issues) with an impersonal system of rules (e.g. a majoritarian parliamentary democracy) where decisions may be perennially postponed through discussion (Dyzenhaus 1994, pp. 4-5;Schmitt 2005, p. 63). Such a system is impersonal, according to Schmitt, because human decision-making would be subject to an order of rules that excludes private human judgment. However, for a legal order to make sense, Schmitt purports that a situation of social normality must exist and that the sovereign is the one who decides on what constitutes this normal situation (Schmitt 2005, p. 13). His argument rests on his conception of the sovereign as the agent who decides on the exception [START_REF] Schmitt | Political Theology: Four Chapters on the Concept of Sovereignty[END_REF], p. 5)-the very concept that Kelsen sought to dissolve into the legal order (Dyzenhaus 1994, p.10). This conception of the sovereign is most clearly visible in emergency situations in which exceptional decisions are required and any attempt at banishing personal sovereignty is therefore undermined. According to Schmitt, decision-making in such instances does not and cannot be premised on existing legal norms, but instead derives from "principally unlimited authority" (Schmitt 2005, p. 12) that manifests itself when the laws that restrict the sovereign's actions are suspended.

The legal discourse on the state of exception focuses on the conflict between the integrity of the legal order and the effectiveness of a government in a state of emergency. First, a government needs to recognize an emergency that requires a suspension of the legal order. Such an emergency might require the original existing order to be changed, thereby activating the state of exception. Second, a government faces the decision of revealing that certain bodies or individuals have untrammelled authority in contravention of the original order, for example by establishing a court of martial law to preside over civilians, the authority of which is not derived from the original order. Third, a government can decide to resolve the state of exception and stipulate the new order that will ensue from it. Many constitutions of contemporary liberal democracies define a state of exception that, to some extent, suspends the original legal order. Such a state of exception has for instance been in operation during France's "permanent" state of emergency [START_REF] Perolini | France's permanent state of emergency[END_REF]) that was initiated after the terrorist attacks in Paris in 2015. States of emergency are not only declared following political events, but also in the case of economic crises such as the financial crises in Detroit 5 and Puerto Rico. 6 Instead of being dealt with from within the existing legal order, these emergencies entail the delegation of broad, sovereign powers to an executive or designated body to make decisions in the state of exception.

To be sure, rules governing the creation and dissolution of the state of exception might be codified in positive law. Such laws could prescribe what agencies will gain decision-making authority, in what way power of different agencies is to be balanced, and what timespan might apply to the state of emergency. Schmitt's criticism therefore does not amount to the claim that the state of emergency necessarily leads to an undoing of the original positivist legal order. For instance, two years after the state of emergency was announced in France it was dissolved, which means that the original legal order was (partially 7 ) restored. However, Schmitt's central point is that the laws in operation during the time of the state of emergency operate outside of the original legal order (Dyzenhaus 2006, p. 345). In opposition to Kelsen, Schmitt argues that conflict in the state of emergency is resolved through "some personal act of will emanating from outside the law" (Dyzenhaus 1994, p. 11) and that the decision concerning the establishment of an emergency is a personal one.

Hence, personal judgement is involved in the state of emergency, which leads to the risk of the original positivist legal order being transformed into a new one. An important historical case in this regard is the use of article 48 on the state of exception in the constitution of the Weimar Republic by its president to dissolve parliament and effectively initiate the transition from a parliamentary democracy to a totalitarian state (Agamben 2005, p. 15). Thus, Schmitt criticises Kelsen's legal positivism through the notion of the state of exception. As such, Schmitt's characterisation of Kelsen's vision of a legal order as a machine that runs itself can be translated into a formulation of on-chain governance: now the code runs itself. Given the similarities that we observed between Kelsen's theory and on-chain governance, we accordingly ask whether a criticism 5 Public Act No. 436 (2013) in Michigan is a recent example of a legislative response to a state of emergency being declared due to financial crises in Detroit -granting an emergency manager extensive powers in achieving a financial rehabilitation plan, including the right to make binding orders on elected officials. 6 The Puerto Rico Emergency Moratorium and Financial Rehabilitation Act (2016) is another example of such a response to impending debt default. For a historical account, Agamben (2005, pp. 10-22) traces the parallels between military and economic emergencies throughout the 20th century. 7 President Macron's decision to replace the state of emergency with a new counterterrorist law has led some commentators to argue that in fact a permanent state of emergency has been put into effect (McQueen 2017). similar to the one Schmitt put forward could apply to the practical operation of blockchain-based systems. In what follows, we first examine a particular event, the DAO attack, which elucidates some of the practical challenges of on-chain governance. This examination will lead us to an interpretation of the DAO attack as a state of exception in the context of on-chain governance.

The DAO Attack

Some of the landmark experiments with blockchain-based governance are the so-called "decentralised autonomous organisations" ("DAOs"). In a nutshell, a DAO is a code-based system with internal capital that lives on the blockchain and operates autonomously, yet relies on individuals to perform certain tasks that the code cannot do. As such, it can be regarded as an algorithmically governed organisation that responds both to automated code-based rules and deliberate human input (DuPont 2017, p. 159). The idea is to have automation at the centre and humans at the edges [START_REF] Buterin | DAs and More: An Incomplete Terminology Guide[END_REF]. One of the first real-world implementations of this concept was created in April 2016, after the company Slock.it developed a decentralized investment fund (called 'The DAO'), which was deployed on the Ethereum blockchain. People that invested money into the fund could directly participate into the governance thereof by selecting the projects that they would like the fund to invest in. The DAO was operated through a set of code-based rules ("smart contracts") to automatically execute payments when certain conditions are met (Norta 2015, p. 1).

The DAO had its own forum ("DAOhub"), where most of the deliberations took place and a few prominent figures from the Ethereum community agreed to act as "curators" for the platform.

The DAO attracted considerable attention from its earliest stages. When it initiated its first round of crowd funding, it raised in 28 days the equivalent of US$150 million worth of ether (the cryptocurrency native to the Ethereum blockchain) from thousands of investors. As one of the largest crowdfunding campaign to date, the popularity of The DAO led the demand for ether to soar, bringing its market capitalization beyond US$1 billion [START_REF] Ryan | The DAO: an experiment in responsibility[END_REF]. However, the backers of The DAO had broader ambitions: their goal was not only to provide a new means for decentralising crowd funding, but also to provide the underlying framework upon which future DAOs could be built (DuPont 2017, p. 158).

Unfortunately, the experiment was short-lived. Several warnings were raised of potential security risks, and the curators initiated a call for a moratorium on funding proposals [START_REF] Mark | A Call for a Temporary Moratorium on The DAO[END_REF]). Yet, shortly after The DAO went live, an anonymous attacker started to drain ether out of the fund by exploiting a vulnerability in the smart contract governing The DAO, which allowed him/her to repeatedly execute withdrawal transactions [START_REF] Daian | Analysis of the DAO exploit[END_REF][START_REF] Pfeffer | The rise of the Dark DAO[END_REF]. A total of 3.6 million ether -worth approximately US$55 million at the time and amounting to about 30% of the total funds raised -were siphoned into another DAO ("The Dark DAO"), created by the hacker for the purpose of the attack. The DAO governance, as codified within the smart contract, lacked a rule on how to deal with this situation. There was a general rule permitting an upgrade of the smart contract code, but it did not provide a means to retrieve the stolen funds. The DAO attack revealed shortcomings not only in the governance of The DAO, but also in that of the Ethereum community as a whole.

It exposed the limitation of on-chain governance that challenged the underlying principles of the Ethereum network, and other blockchain networks (DuPont 2017, p. 157): the factual verification of transactions (based on formal compliance with the protocol, as opposed to their content) and the immutability and irrevocability of such transactions, once they have been recorded onto the blockchain.

The DAO attack triggered fiery discussions within the Ethereum community. 8 Different follow-up steps were discussed and supported by different groups. Several community members advocated 'doing nothing' (echoing the 'code is law' adage). Some wanted to proceed with the freezing of all ether stored in The DAO and child DAOs, while others believed that the only way forward was to perform a hard fork, changing the protocol of the Ethereum blockchain in order to move all funds tied to The DAO (including those drained by the hacker) to a new smart contract that would enable the investors to withdraw their funds. The latter option violated the immutability ethos prevalent in most blockchain-based networks, favouring instead a distributed consensus approach, whereby changes in the blockchain protocol are deemed to be legitimate insofar the community consents to these changes. In the meantime, the attacker wrote an open letter stating that, under the 'code is law' 8 See the relevant Reddit discussions here: https://www.reddit.com/r/TheDao/comments/4oisep/ether_safe_but_dao_cancelled_were_getting_a_re fund/; https://www.reddit.com/r/ethereum/comments/4oiqj7/critical_update_re_dao_vulnerability/ also nicely visualized here: https://dao.consider.it/hard-fork-to-revert-stolen-dao-funds?results=true. Cited 30 July 2018. principle, the ether had been legitimately acquired as per the terms of the smart contract -referring to the following terms of the DAO: "Nothing in this explanation of terms or in any other document or communication may modify or add any additional obligations or guarantees beyond those set forth in The DAO's code (....)

The DAO's code controls and sets forth all terms of The DAO Creation." 9 At the core of these discussions was the question of whether a hard fork was a legitimate response to The DAO attack. A hard fork is a change in the blockchain protocol that is not backward compatible with the previous protocol-thus creating a separate blockchain that people have to voluntarily switch to. 10 Hard forks occur repeatedly in the Ethereum network in order to fix bugs, improve the scalability of the network, or more generally, transition to a superior consensus protocol. Some of these hard forks are required in order to implement a technical fix, others are planned as part of the long-term roadmap of Ethereum [START_REF] Gupta | The Ethereum Launch Process[END_REF][START_REF] Wood | Gav's Ethereum ÐΞV Update V. Ethereum Blog[END_REF], but all are related to a technical issue that needs to be resolved. What made this hard fork exceptional was that it was unplanned, in response to a contentious political and moral issue, rather than a previously acknowledged technical issue. The implementation of this hard fork necessitated a departure from the existing on-chain governance structure of the Ethereum blockchain-one where the order is established, only and exclusively, by the underlying blockchain protocol. As the protocol did not include the possibility of freezing the funds or reversing the contentious transaction to restore the original state of affairs, any attempt at achieving these results would go against the integrity of the system.

Opponents to the hard fork warned of a slippery slope: once it is deemed acceptable to modify a protocol for political, as opposed to technical, reasons, it might compromise the much-touted immutability and reliability of the Ethereum blockchain as a record of truth. Some of the opponents also claimed that such a hard fork was ultimately a 'bailout' intended to protect the core development team, which was 9 See https://pastebin.com/CcGUBgDG. Cited 30 July 2018. It should be noted that the authenticity of this letter is disputed, but as DuPont (2017, p. 174) notes, it nonetheless reflects the view of many in the Ethereum community at the time. 10 Forking is a commonly accepted, though exceptional, practice in open source software development [START_REF] Robles | A Comprehensive Study of Software Forks: Dates, Reasons and Outcomes[END_REF]. Nyman and Lindman argue that it is a "central freedom" in open source licensing (Nyman and Lindman 2013, pp. 7-8). In blockchain-based systems, soft forks involve a temporary split of a blockchain as part of a software protocol upgrade, in which the original blockchain accrues blocks validated by non-upgraded and upgraded nodes (i.e. is backwards compatible) and the forked blockchain accrues blocks only from upgraded nodes which, following the implementation of the soft fork, tries to achieve a majority of hashing power so that the forked chain reflects the truest sequence of events [START_REF] Acheson | Hard Fork vs. Soft Fork[END_REF].

heavily invested in The DAO and thus faced potential conflicts of interest. Instead, it was contended that The DAO should be allowed to 'fail' and that the Ethereum Foundation should not engage in the governance of individual projects such as The DAO. Arguments used by those in favour of taking action were mostly grounded in political considerations: humans should still have the final say based on social consensus, the exploit was significant enough to justify action, and, lastly, taking action would keep regulators out of the debate. Some conceded that 'flexible and pragmatic' governance was desirable and that full decentralization, autonomy and algorithmic authority could be realised once Ethereum had reached maturity (DuPont 2017, pp. 168-169).

These different views were gauged through opinions expressed on online forums such as Reddit, private discussions held with large exchanges and miners, as well as through the votes cast by ether and DAO token holders. Eventually, the Ethereum core development team released new software with an upgrade of the protocol: 89% of the miners voted in favour of the hard fork, whereas 11% decided to remain on the original blockchain. This resulted in the creation of two separate blockchains: 'Ethereum Classic' run in accordance with the original protocol, alongside the new 'Ethereum' blockchain, governed by an upgraded protocol. The reverberations of this event were felt for a long time, with the price of ether taking close to a year to recover.

Having outlined the course of events in the wake of the DAO attack, we argue that it is similar to the state of exception as conceptualised by Schmitt in several ways. First, a situation of social normality was disrupted by the act of a single agent who radically altered the distribution of funds within the system, thereby initiating a situation similar to, for instance, a financial crisis. This rupture from the anticipated course of events caused by The DAO attack is comparable to political emergencies in a situation of abnormality. Second, the new situation of abnormality brought about the necessity for a decision but could not be resolved by means of existing on-chain procedures or accepted off-chain processes (e.g. a soft fork). It therefore necessitated recourse to measures that were not covered by the existing legal order, by the existing rules and procedures on the blockchain protocol. Third, the state of exception had to be declared in order to be put into effect, which happened when a group of Ethereum developers in a closed communication channel developed the strategy to get the major cryptocurrency exchanges to halt trading activities (Dupont 2017, p. 163). Fourth, the state of exception effectively resulted in a new legal order that was implemented through the hard fork.

We should also, however, observe some important differences between the DAO Attack and the state of exception as discussed by Schmitt. First, in states. Even though one might speak of the validating nodes in a blockchain-based system as a 'parliament of miners', the system does not provide for any public offices such as those of president or judge. Notwithstanding these differences, however, the similarities seem to warrant the claim that the state of exception is a challenge to the notion of on-chain governance, especially considering potential future developments of blockchain-based systems.

The Coalescence of Private Participants

Thus far, we have argued that (1) on-chain governance is similar to Kelsen's conception of the positive legal order and (2) the DAO attack shows that the state of exception poses a challenge to on-chain governance. However, we have not yet elucidated a more general insight for the governance of blockchain-based systems.

One might argue that the DAO attack and its response, even while having been in a sense a state of exception, constituted a one-off event and did not expose any inherent vulnerability in on-chain governance. We return to Schmitt's critique of the positivist legal order in order to argue against such a view and to flag the inherent vulnerability of on-chain governance to the coalescence of private participants, which became evident in the wake of the DAO attack.

A central point of Schmitt's critique, springing from his notion of politics, concerns the vulnerability of positivist legal systems to the "growth of private powers"11 (Dyzenhaus 1994, p. 13). Schmitt argues that Kelsen's legal positivism equates legitimacy (i.e. the justification of a legal order) with formal legality (i.e. with validity of legal prescriptions), which does not account for the content of the rules and therefore lacks a substantive conception of the common good (Dyzenhaus 1994, p. 12). 12 Schmitt attributes this procedural conception of legitimacy to parliamentary, liberal democracy. He contends that the concrete liberal order cannot accept any substantive notion of the common good as the basis of its legitimacy, but can instead only rely on a procedural conception of how a shared understanding of the common good is to be arrived at, for instance through structured debate and voting in parliament. In other words, private actors in a liberal democracy cannot appeal to a higher notion of the common good, but only to the procedures that define their interactions in a decision-making process. Even though a blockchain-based system is far from the equivalent of a liberal parliamentary democracy, it shares this particular feature in that a conception of the common good of its reference community can solely be based on a procedural conception of legitimacy.

According to Schmitt, this feature of positivist legal systems provides a fertile ground for powerful, competing private interests to arise. The inherent contradiction in the legal positivist system, according to Schmitt, is that all private participants who are subject to a particular legal order also expect that those who decide subject themselves to the same legal order. The content of this legal order-which consists only of formal legality-is therefore empty and can only result from negotiations between private participants. Because the legal system is incapable of distinguishing between right and wrong (i.e. incapable of being grounded in a substantive notion of the common good), the capacity to do so is delegated to private participants. They can organise themselves into what Schmitt designates as civil society, which encompasses non-state actors such as civil society organisations (CSOs) and corporations. A crucial point is that private participants will have to coalesce in order to provide substance for the legal order. In liberal democracies, this happens through the use of distinctive liberties such as freedoms of enterprise, of association and of assembly that mobilise powerful corporate lobby groups, CSOs and trade unions to promote private interests.

Schmitt insists that this dynamic of coalescence of private participants and growth of private power is inherent to parliamentary liberal democracies and ultimately poses a risk of subverting the legal order, notably in a state of exception when a decision based on a substantive notion of the common good is called for.

To what extent might on-chain governance of blockchain-based systems be susceptible to the vulnerability of coalescence of private participants in a similar way as are liberal parliamentary democracies, at least according to Schmitt? Considering the aftermath of the DAO attack, it appears that Schmitt's criticism has some currency. First, participants in the Ethereum community initially shared some of the commitments of legal positivism by conflating formal legality (i.e., the valid state of a blockchain-based system) with legitimacy (i.e., the justification of the authority of the system). This prevented the emergence of any grounded conception of the common good and reserved the decisions between right and wrong exclusively to individuals or particular groups in these communities. The sovereign was thereby replaced by a burgeoning civil society (Dyzenhaus 1994, p. 10), comprising private actors such as Ethereum users, miners, mining pools, exchanges, the Ethereum foundation, and so forth. Individuals and groups in such a system compete with one another and use tactics to gain prominence and power. 13 Reflecting on this tendency of growing private powers in the context of the theory of the firm, [START_REF] Wright | Proof of Work as it Relates to the Theory of the Firm[END_REF] argues that all proof of work (and proof of stake) systems-the mechanisms that make the machine run itself-will ultimately lead to corporate consolidation or to plutocracy. This is because public blockchains are not governed according to a 'one-person, one-vote' principle. Instead, voting rights are allocated in proportion to the number of tokens or hashing power that each individual has. A plutocracy implies government or rule by the wealthy, and consequently favours private interests over the common good. Several contemporary societies implement some form of plutocracy (the United States is arguably an example), but blockchain-based systems are distinct in that they do not offer a mixed form of 13 Looking at the most prominent blockchain networks-Bitcoin and Ethereum-this pattern indeed seems to be present. Both these networks suffer from a concentration of voting, or 'mining', power, with a few participants who own large portions of the available assets. In the Ethereum community, ether holders, Ethereum developers, miners and exchanges all participate in the governance of the network in order to promote their own interests, without much clarity as to which party has decisive influence (ConsenSys 2016). decision-making (e.g. partially democratic and partially plutocratic) and implement an exclusively plutocratic governance structure, at least as long as this structure is limited to the workings of on-chain governance.

As an initial response, one might simply accept this situation and embrace plutocracy as an inevitable status quo for these communities. However, as Schmitt argues, such a status quo can be radically altered in the case of an unforeseen state of exception, which calls for an arbitrary decision to be made and responded to. In such a case, one or more private actors might arise as the sovereign by deciding on a particular course of action, thereby overriding the interests of other private actors. In the case of the DAO attack, the development team, the Ethereum Foundation and the super-nodes that were heavily invested in the DAO had a significant influence in the discussions regarding the response to the attack. Moreover, personal sovereignty within a plutocracy negates the basic principle of blockchain-based systems, according to which no individual or group shall be allowed to impose their will on the community. As such, we can equate Schmitt's claim that a positivist legal system bears the risk that the liberal principles that support it are contradicted, with the notion that an on-chain governance system faces, in turn, the risk that its supporting principles are contradicted.

We claim that such a contradiction is precisely what happened in the wake of the DAO attack. Core developers and members of the Ethereum Foundation preliminarily discussed options for resolving the attack behind closed doors. Some of the most prominent voices of the Ethereum community took the lead in publicly presenting possible options, often without accounting for the interests of all stakeholders. The Ethereum Foundation set the parameters for polling community opinion 14 and the core developers issued the specifications for updating the participants' software client-all in the name of the community. While the community could vote on what the default option should be, it was the Foundation that set the block number by which votes would be tallied. If we unpack the decisions that were taken when the emergency arose, we can see how a particular group of private participants decided to: recognize the factual existence of an emergency; reveal the extent of authority they possess in contravention of the existing order [i.e., decentralized consensus]; resolve the state of exception within a set time frame, and prefigure the order that will exist following the termination of the state of exception.

In line with Schmitt, we argue that prior to the hard fork, it was already the case that core developers and the Ethereum Foundation acted as the sovereign of the order created on the Ethereum blockchain. It could be argued that the locus of sovereign power and the exercise of sovereign powers are divided. After all, the majority of nodes needed to actively upgrade their software client for the hard fork to take effect. However, individual participants that upgraded their client, far from exercising their sovereign will, were legitimising the decision that had been made on their behalf during the state of exception. This is an illustration, in the blockchain space, of Rossiter's claim that, while most democracies provide an elected representative body with the final authority in deciding to initiate a 'constitutional dictatorship' pursuant to an emergency, it is usually the parties that wield emergency powers who decide whether an emergency exists in the first place (Rossiter 1948, p. 299). We might clarify this point by drawing an analogy with Roman law. When the Roman Senate sensed danger to the Roman Republic, it could issue a decree declaring a tumultus (e.g. insurrection, foreign war) and call upon everyone, from consuls to ordinary citizens, to take measures to protect the Republic (Agamben 2005, p. 42). 15 By following through, the citizenry gave democratic backing to a sovereign decision, and thereby to the particular entity that made the decision. This political situation concentrates power more radically than plutocracy, as it reveals that a particular group can act beyond the legal order by exerting its will on the community.

The communities surrounding a blockchain-based system should not be conflated with traditional political communities, or citizens of a particular nation state. What currently sets blockchain-based systems apart from nation states is their much more radically voluntary character: all participants are free to leave or to implement a hard fork in order to establish a new voluntary community. The possibility of exiting the original community at little to no cost diminishes the need to voice dissatisfaction and resolve problems within the same political community (Hirschman 1972, p. 43). In the DAO attack, those rejecting the hard fork have effectively "left" the Ethereum community by selecting the Ethereum Classic blockchain as the authoritative blockchain. However, community members are free to participate in both blockchain-based systems (Ethereum and Ethereum Classic) should they see any value in doing so. We may thus discern two communities with potentially overlapping membership. Only the community members who decided to participate only in the original network and boycott the upgraded network might be said to have retained their original ideology.

However, two issues should be kept in mind. First, dealing with states of exception in blockchain-based systems will become an increasingly contentious and problematic matter when the stakes of the participants become higher. In the event that a hard fork would decimate the wealth of all participants who oppose a particular state of emergency, the existing strategy could hardly be conceived of as a voluntary response. Second, the state of exception will become more pressing once open blockchain architectures become increasingly used in commerce or by governments (for instance when these systems become general-purpose technologies or provide basic public goods). If and when utopian visions of cloud communities [START_REF] Orgad | Cloud Communities : The Dawn of Global Citizenship? EUI Working Paper RSCAS[END_REF])-such as those expressed by the Bitnation project (Atzori 2015)become a reality and refugee communities engage in self-governance by providing themselves with identity services or property right regimes, any violation of the onchain governance structure could create crises and conflicts in the realm of off-chain governance (e.g., threats between participants, physical violence). Even though the second point, in particular, refers to a speculative state of affairs, the pace of technological change and its consequential impact urge us to discuss these issues.

Mechanisms should be put in place to discourage excessive reliance on the exit strategy, and to implement instead more meaningfully and cost-effective ways to express the community's diverse voices.

Concluding Remarks

This article has shown that the ideal of on-chain governance has striking similarities with Kelsen's positivist notion of the legal order, which presupposes that no individual or group of individuals should be allowed to enforce their will on others and that individual sovereignty should be minimised in the decision-making process.

At first blush, blockchain-based systems seem to provide the technological apparatus for realising Kelsen's vision of a content-independent understanding of law, where formal legality is equated with legitimacy and personal sovereignty is dissolved into the legal order. However, in doing so, blockchain-based systems become vulnerable to the rise of private interests using off-chain mechanisms to usurp the system of onchain governance. This risk becomes most apparent during states of exception, such as the DAO attack discussed above, when personal authority asserts itself. This case indicates that while the 'rule of code' may be formalistically followed within a particular on-chain order, sovereignty asserts itself through off-chain mechanisms during the state of exception. In light of these traits, extant blockchain governance regimes need to be carefully reconsidered and aligned with the ideology of their relevant communities.

Future research can consider the steps that blockchain communities may take to resolve states of exception, in a manner that is consonant with their respective ideologies. Such research could, for instance, aim to examine the role of the Ethereum foundation in both proclaiming the state of exception and acting upon it in the event of the DAO attack. A promising way to approach the difficulty of institutionalising the state of emergency for blockchain-based systems would be to consider [START_REF] Rossiter | Constitutional Dictatorship: Crisis Government in the Modern Democracies[END_REF] model of "constitutional dictatorship", which confronts states of exceptions in a manner that preserves, rather than threatens, liberal democracy and civil rights.

Other avenues of research to consider would be to reflect on Heller's notion of "ethical fundamental principles of law", as discussed in a recent article by Dyzenhaus (2015, p. 353), or the political philosophy of Strauss, which was largely a response to Schmitt's devastating criticism of liberal ideology and which called for the recovery of a classical perspective on politics (Howse 1997, p. 92).

  contrast to liberal democracies the Ethereum community had not developed any positive laws or on-chain rules and procedures that would regulate a potential state of exception. No procedures existed that regulated which agents should have the authority to decide on the exception, what delegation of powers should apply, and what time schedule should be in effect. Second, the parties involved in the DAO Attack differ substantially from those involved in the managing of states of emergencies in modern

Because of these specificities, most blockchain-based systems rely on off-chain governance only in exceptional situations when on-chain governance fails or is unable to process a certain decision-for example, when a protocol change is required to improve a network's functionalities or fix technical issues that would otherwise place the whole network in jeopardy.

This also accords with the way Lessig imagined the code as law, that the limitations placed by code (as opposed to simply the limitation of code) intrinsically binds the behaviour of participants[START_REF] Lessig | Code version 2.0. Basic Books[END_REF]. As he argued in an early paper on constitution-making in cyberspace, "While regulation in real space is primarily regulation that relies upon the cooperation of the individuals who live under the regulation, regulation in cyberspace can be something different. The code in cyberspace -the software -can enforce its control directly."(Lessig 1996, p. 899).

In line withDyzenhaus (1994, p. 19), we explicitly distance ourselves from accepting Schmitt's critique -taking heed of Schmitt's highly problematic embrace of Nazism -but instead use it as a productive critique with which positivist attempts to construct a legal order should be concerned.

This problem is not unique to the blockchain ecosystem. The concept of a 'benevolent dictator' and an oligarchy among co-developers in open-source software development projects has been discussed since the 1990's[START_REF] Raymond | Homesteading the Noosphere[END_REF].

[START_REF] Reijers | Governance in Blockchain Technologies & Social Contract Theories[END_REF] argue, the sum of individual "wills" in blockchain-based systems, represented by the nodes, do not allow for the conception of a common will, or common good, as it was put forward by the social contract theory of Rousseau.

Although community votes were tallied within 12 hours of voting being opened, they accounted for only 5% of ether holders(Santos 2018, pp. 38-44).

The decree declaring tumultus was distinct from the decree proclaiming a iustitium -or a standstill of the law -that could follow.


A Rudenko

C Mauclair

F Garrelie

R Stoian

J P Colombier

Light absorption by surface nanoholes and nanobumps

This paper deals with a numerical investigation of the energy deposition induced by ultrafast laser interaction with nanostructures. We calculate and analyze the intensity near-field reactive and radiative patterns resulted from the interference of the incident light with light scattered by individual subwavelength holes and bumps on the surface of metallic and dielectric materials. The role of light polarization, optical material properties, collective effects and nature of the imperfections in localized energy absorption is elucidated. The results open new perspectives in precise light manipulation by surface inhomogeneities and well-controlled surface nanostructuring by ultrashort laser.

Introduction

Light interaction with basic surface nanostructures, such as holes and ridges, lies at the heart of subwavelength optics, signal processing on the nanoscale, nanophotonics and nanoplasmonics. The presence of these tiny imperfections has been proved to modify the optical properties of the pristine material, contributing to strongly enhanced transmission, anomalous optical absorption, wavelength filtering, and surface plasmon wave excitation on metal surfaces [START_REF] Ebbesen | Extraordinary optical transmission through sub-wavelength hole arrays[END_REF][START_REF] Genet | Light in tiny holes[END_REF][START_REF] Le Perchec | Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light[END_REF][START_REF] Gramotnev | Plasmonics beyond the diffraction limit[END_REF][START_REF] Coenen | Optical properties of single plasmonic holes probed with local electron beam excitation[END_REF][START_REF] Kuchmizhak | Laser printing of resonant plasmonic nanovoids[END_REF][START_REF] Wang | Single-step laser plasmonic coloration of metal films[END_REF][START_REF] Hu | Lasersplashed three-dimensional plasmonic nanovolcanoes for steganography in angular anisotropy[END_REF][START_REF] Guay | Topography tuning for plasmonic color enhancement via picosecond laser bursts[END_REF]. Efficient light manipulation on the nanoscale is hence possible nowadays by using optical devices, where diffraction gratings or nanoslits, periodic arrangements of holes or nanoparticles serve as general counterparts. Control over the optical response of such devices requires precise knowledge of the interference patterns and field distributions in the vicinity of individual nanoobjects. Even if the near-field optical microscopy is already able to reconstruct the complete interference maps with nanoscale resolution [START_REF] Yin | Surface plasmons at single nanoholes in Au films[END_REF][START_REF] Permyakov | Mapping electromagnetic fields near a subwavelength hole[END_REF], the understanding of the involved electromagnetic phenomena is possible only via rigorous theoretical or numerical analysis.

The energy absorption maps on the surfaces are particularly required to understand laser interac-tion with matter, because the patterns of maximum absorbed energy define precisely the thermoaffected zones, where the material modification such as ablation is likely to take place after the energy deposition [START_REF] Obara | Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates[END_REF][START_REF] Obara | Plasmonic and mie scattering control of far-field interference for regular ripple formation on various material substrates[END_REF][START_REF] Murphy | Laser-induced periodic surface structure formation resulting from single-pulse ultrafast irradiation of Au microstructures on a Si substrate[END_REF][START_REF] Enami | Electron excitation effect on scattering near-field and farfield ablation material processing by femtosecond laser irradiation[END_REF][START_REF] Zhou | The influences of surface plasmons and thermal effects on femtosecond laser-induced subwavelength periodic ripples on Au film by pump-probe imaging[END_REF][START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF][START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF][START_REF] Liu | Ultrafast imaging on the formation of periodic ripples on a Si surface with a prefabricated nanogroove induced by a single femtosecond laser pulse[END_REF]. This way, the laseraffected surface can be efficiently treated pulse by pulse. The modifications strongly influence the surface roughness and the energy deposition below the surface [START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF][START_REF] Bloembergen | Role of cracks, pores, and absorbing inclusions on laser induced damage threshold at surfaces of transparent dielectrics[END_REF], therefore, it appears crucial to predict in which way the absorbed energy profiles change. Self-organization of polarizationdependent periodic surface nanostructures is among the impressive phenomena induced by multipulse ultrashort laser irradiation [START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF][START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF][START_REF] Skolski | Laser-induced periodic surface structures: Fingerprints of light localization[END_REF][START_REF] Zhang | Coherence in ultrafast laser-induced periodic surface structures[END_REF].

Light scattering by surface inhomogeneities has been intensively studied theoretically. For instance, the expanded Mie solutions [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF][START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF] were obtained for the scattered fields of a single subwavelength spherical nanoparticle near and on the perfectly conducting surface [START_REF] Bobbert | Light scattering by a sphere on a substrate[END_REF][START_REF] Videen | Light scattering from a sphere on or near a surface[END_REF]. However, the analytical solution is complicated in case of metal or lossy interface, as well as the arbitrary sizes of the inhomogeneity (hole or bump) due to the non-trivial boundary conditions at the sphere (or half-sphere) and plane surface, which should be satisfied simultaneously. The charge accumulation at the edges of the imperfection results in the phase retardation of the scattered wave, and, therefore, different spatial energy localization [START_REF] Weiner | Phase shifts and interference in surface plasmon polariton waves[END_REF][START_REF] Liu | Symmetry in the elementary scattering of surface plasmon polaritons and a generalized symmetry principle[END_REF][START_REF] Klick | Amplitude and phase of surface plasmon polaritons excited at a step edge[END_REF]. Only the semianalytical dipole approximations for the scattered fields are known in this case [START_REF] Lalanne | Interaction between optical nano-objects at metallo-dielectric interfaces[END_REF][START_REF] Nikitin | Surface electromagnetic field radiated by a subwavelength hole in a metal film[END_REF], whereas the general problem requires calculation of Sommerfeld's integrals over all diffraction modes [START_REF] Collin | Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies[END_REF]. The alternative way consists in treating the electromagnetic problem numerically based on the complete solution of three-dimensional Maxwell's equations.

In this paper, we investigate numerically the energy absorption below dielectric and metallic surfaces with a single or several holes or ridges and analyze the influence of light polarization, optical properties of material, and arbitrary shape of nanoobjects on the resulting scattered radiative and nonradiative fields. For this, we solve a full system of Maxwell's equations by Finite-Difference Time-Domain approach [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media[END_REF][START_REF] Inan | Numerical electromagnetics: the FDTD method[END_REF], which was successfully used for scattering, transmission and absorption data analysis in case of subwavelength individual, periodic and randomly distributed holes and bumps on the surface [START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF][START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF][START_REF] Skolski | Laser-induced periodic surface structures: Fingerprints of light localization[END_REF][START_REF] Zhang | Coherence in ultrafast laser-induced periodic surface structures[END_REF][START_REF] Liu | Symmetry in the elementary scattering of surface plasmon polaritons and a generalized symmetry principle[END_REF][START_REF] Kihm | Separation of surface plasmon polariton from nonconfined cylindrical wave launched from single slits[END_REF][START_REF] Irannejad | Finite difference time domain study of light transmission through multihole nanostructures in metallic film[END_REF]. We focus on two different materials, widely used in industry: unexcited fused silica glass (dielectric) and stainless steel (metal).

Modeling details

The system of three-dimensional Maxwell's equations is solved

       ∂ E ∂t = ∇ × H 0 ∞ - 1 0 ∞ J ∂ H ∂t = - ∇ × E µ 0 , ( 1
)
where E is the electric field, H is the magnetizing field, and J is the electric current density, µ 0 is the permeability of free space, 0 is the free space permittivity, and ∞ is the value of material permittivity at infinite frequency. Drude model is incorporated to model the properties of the materials. The equation for J current is solved to include the dispersion of the optical properties in (1) via the auxiliary differential equation method [START_REF] Inan | Numerical electromagnetics: the FDTD method[END_REF] ∂ J ∂t

+ J/ν = 0 ω 2 pl E, (2)
where ω pl is the plasma frequency, and ν is the collision frequency. At the edges of the grid, absorbing boundary conditions related to convolutional perfect matched layers (CPML) are set to avoid nonphysical reflections [START_REF] Roden | Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media[END_REF]. Laser wavelength is fixed to be λ = 800 nm. The refractive index and the extinction coefficients for stainless steel are taken n = 2.57 and k = 3.12, whereas the refractive index of fused silica glass is n = 1.45. For stainless steel, the effective values for the collision frequency ν = 9.15 • 10 15 Hz and for the plasma frequency ω pl = 19.2 • 10 15 Hz are deduced from Drude formula approximation for real and imaginary parts of permittivity and the known refractive indices as ν = 2nkω(1 -n 2 + k 2 ) and

ω 2 pl = (ω 2 + ν 2 )(1 -n 2 + k 2 )
, where ω = 2πc/λ is the laser frequency. The dispersion-less model is used for unexcited glass with ∞ = n 2 . The influence of transient optical properties on the energy distribution under strong excitation of dielectrics is beyond the scope of this article, but was previously addressed in Refs. [START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF][START_REF] Rudenko | From random inhomogeneities to periodic nanostructures induced in bulk silica by ultrashort laser[END_REF]. Holes and bumps are half-spheres with radius R = 20 nm. Both holes and bumps are centered at (0, 0, 0) positions. An ultra-short Gaussian pulse is introduced as the linear-polarized electric field excitation source

E x (t, r, z) = w 0 w(z) exp - (t -t 0 ) 2 τ 2 • •exp - r 2 w(z) 2 -ikz -ik r 2 2R(z) + iς(z) , (3)
where τ = 80 fs is the pulse width at half maximum (FWHM), t 0 = 160 fs is the time delay, w 0 = 10µm is the beam waist radius, w(z) = w 0 1 + ( z z R ) 2 is the Gaussian's beam spot size, z R = πw 0 2 n 0 /λ is the Rayleigh length, r = x 2 + y 2 is the radial distance from the beam's waist, R(z

) = z[1 + ( z R z ) 2 ] is
the radius of curvature of the beam's wavefront, and ς(z) = arctan( z z R ) is the Gouy phase shift. The effects of the finite spectral bandwidth in case of extremely short laser pulses τ < 40 fs is also investigated in the article, considering temporallydependent dispersion ∆λ.

Light propagates along 0z axis, whereas the light polarization is along 0x axis. This way, x0z is the propagation plane and x0y is the transverse plane perpendicular to light wave-vector k. In most cases, the time-integrated intensity distribution inside material (stainless steel or glass) I = We define different regimes of scattering as a function of distance from the object under consideration as indicated in Fig. 1 in case of a single nanohole on metal surface. The closest near-field region includes non-radiative patterns, for which the electric field is confined to the edges of the nanoobject. The boundary between radiative and non-radiative fields classically corresponds to the distance r = λ/2π, where the contributions with amplitudes decaying proportional to 1/r and 1/r 3 of an isolated electric dipole are equal [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF]. The intermediate region λ/2π < r < 2λ is the transition zone between near-field and far-field radiative scattering.

The electric fields of both non-radiative and radiative patterns are directly accessed via FDTD, without applying the near-to-far-field transform. In this work, we are mainly interested in absorption on surfaces irradiated by a focused laser pulse but not in reflection characteristics. Therefore, the analysis is limited to the region of few laser wavelengths, where the absorption is the highest, including both non-radiative and radiative interference patterns. The scattered fields at distances farther than few wavelengths have repetitive behavior and can be intuitively reproduced by taking account the decay of the surface wave and the wavefront transformation from quasi-spherical to planar or by applying near-to-far-field transformations [START_REF] Inan | Numerical electromagnetics: the FDTD method[END_REF][START_REF] Pors | Quantum emitters near layered plasmonic nanostructures: Decay rate contributions[END_REF][START_REF] Yang | Near-to-far field transformations for radiative and guided waves[END_REF]. Fig. 2 shows the intensity distribution on a glass surface in the propagation (a, b) and in the transverse planes (c, d). In case of a bump on dielectric surface, the energy is equally deposited on the top of the bump and on both edges of the equatorial plane intersecting the surface. Therefore, the peaks of the nanometric bumps are likely to be erased by laser ablation. The local field enhancement results in the maxima parallel and minima perpendicularly to light polarization in the transverse plane. In case of a nanometric hole on dielectric surface, the energy is mostly deposited inside the hole. In this case, the maxima of local field are perpendicular to laser polarization. This way, the material ablation with holes leads to more pronounced deeper holes and the reinforcement of polarization dependency of the absorbed energy below dielectric surface. The polarization dependency of the intensity maxima can be understood by a simple analogy with Mie theory. In fact, hole on glass surface owns the properties of matter with smaller refractive index inside the material with greater one. In contrast, the properties of bump below the surface result from the continuous prolongation of the electric field above the surface, where a particle of glass (bump) is surrounded by air, therefore, matter with larger refractive index embedded in medium with smaller refractive index. Regarding Fig. 2(c Fig. 3 demonstrates the intensity distribution below a metallic surface. One of the principle difference with glass is that the energy is no more deposited on the top of the metallic bump but only on the surface dominantly in the direction parallel to laser polarization. This result indicates that the laser ablation not necessarily erases the bump, but rather yields a more pronounced imperfection. The formation mechanisms of laser-induced bump-like bubble crescents and cone-shaped nanovolcanoes on thin titanium films have been recently investigated and the dominant vertical modes of the structures at normal incidence irradiation have been successfully applied to color marking [START_REF] Hu | Lasersplashed three-dimensional plasmonic nanovolcanoes for steganography in angular anisotropy[END_REF]. The energy is still deposited mostly inside the nanometric hole, and similar dependencies of local field maxima perpendicular to laser polarization for stainless steel are revealed in Fig. 3(b,d). Therefore, the material ablation favors the formation of deep ellipsoidal holes elongated perpendicular to laser polarization. The light interaction with non-symmetrical imperfections is investigated further in the manuscript.

The above results remain qualitatively accurate for sizes of holes and bumps smaller than the wavelength of light R < λ 2π . 4(d)). The striking difference is revealed, however, in the distances between the intensity maxima positions and the emission centers. This fact is related to different phase retardation of surface waves scattered by holes and bumps. This way, the first far-field maxima (to differ between the local near-field) are on the greater distance from the hole than from the bump for glass surface and from the bump than from the hole for metal surface. Note that the difference is related directly to charge accumulation on the interface of imperfection and, therefore, to local near-field distributions. For glass, the interference patterns represent spherical standing waves with λ/n spacings in Fig. 4(a,b), whereas for metal the symmetry is broken due to additional contribution of surface plasmon wave in the direction parallel to the electric field with spacings ≈ λ in Fig. 4(c,d). The absorbed energy maps, however, do not allow us to separate the contribution of surface plasmon wave from so-called quasi-cylindrical wave [START_REF] Lalanne | Interaction between optical nano-objects at metallo-dielectric interfaces[END_REF]. This fact will be discussed further in the article. In case of holes on thin metallic films, the near-field optical microscopy was applied to observe the radiative interference patterns similar to ones calculated in Fig. 4(d) [START_REF] Yin | Surface plasmons at single nanoholes in Au films[END_REF][START_REF] Permyakov | Mapping electromagnetic fields near a subwavelength hole[END_REF].

Radiative and non-radiative responses

𝑘 𝑘 𝐸 𝐸 c) d) X[µm] X[µm]
Interestingly, Fig. 4 evidences the symmetry principle of scattering by nanoholes and nanobumps of equivalent size. If we superpose the interference patterns either in Figs. 4(a,b) or in Figs. 4(c,d), we can note that they are more pronounced for hole for locations where they are less pronounced for bump. The presence of intrinsic imperfections also influences the field distribution below the surface [START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF]. The response of such intrinsic voids is completely different from surface holes and bumps and can be understood by Mie theory for a spherical nanoparticle in media. The resulting interference patterns calculated by FDTD are shown in Fig. 5 for intrinsic voids inside but near metal and glass surfaces. The non-radiative fields are qualitatively similar for the imperfections of different nature. Regarding the radiative field, the maxima of the absorbed energy correspond to the positions in the direction perpendicular to polarization in both cases of metal and glass surface. The spherical wave has, however, different decays as the distance from the emission center changes. The surface wave decays faster in case of metal due to non-negligible extinction coefficient and significant losses. This way, only the first intensity maxima is visible. In case of glass, the decay of periodic patterns with characteristic λ/n spacings is only due to radiative losses and is similar to the optical response of a hole on a glass surface. Some additional information can be accessed via FDTD by analyzing the electric field distributions in the propagation plane. Recently, the nature of surface waves responsible for the electric field maxima has been intensively debated in the literature [START_REF] Lalanne | Interaction between optical nano-objects at metallo-dielectric interfaces[END_REF][START_REF] Nikitin | Surface electromagnetic field radiated by a subwavelength hole in a metal film[END_REF]. The semi-analytical approaches considering the scattered fields of a dipole on metal-air interface indicate that the total scattered field is the sum of surface plasmon and quasi-cylindrical wave contributions [START_REF] Lalanne | Interaction between optical nano-objects at metallo-dielectric interfaces[END_REF][START_REF] Nikitin | Surface electromagnetic field radiated by a subwavelength hole in a metal film[END_REF]. The behavior of surface plasmon wave near metal-air interface is well-known, particularly the strong exponential decay with the distance from the interface in the air and in the metal. In contrast, the quasi-cylindrical wave decays more slowly and can still propagate in the air. This way, the total electric field distribution in Fig. 6(a) shows that both of waves exist on metal-air interface and are phase-shifted. The nature of this phase-shift has been previously discussed in terms of surface wave retardation due to charge accumulation at the edges of hole or bump [START_REF] Weiner | Phase shifts and interference in surface plasmon polariton waves[END_REF]. In case of glass surface, there is no mismatch in Fig. 6(b), because the condition of surface plasmon wave excitation is not satisfied, therefore, only the quasicylindrical surface wave contributes to the total scattered electric field.

Nature of the surface waves

Interestingly, different surface topography results in different shifts and electric field maxima positions regarding the emission center. Fig. 6 clearly demonstrates the symmetry principle both for metal and glass surface imperfections. Disregarding the fact that the amplitude of scattered field from the hole is greater than for the bump, the electric fields of opposite imperfections are out of phase by π and, therefore, compensate each other. The shorter the femtosecond pulse is, the larger the spectral bandwidth, defined as ∆λ = λ ∆ω ω = 2 ln 2 πcτ λ 2 for a Gaussian pulse. This way, one obtains ∆λ ≈ 94.08 nm for τ = 10 fs with bandwidth centered to λ = 800 nm. The other two factors that potentially influence the interference patterns are the changes in refractive indices due to the presence of multiple wavelengths and the pulse duration itself -the number of optical cycles (one optical cycle is ≈ 2.67 fs). We investigate the influence of these factors on the basic interference patterns from a nanohole on metal surface. Firstly, we check that the refractive index changes (the refractive index dependence on wavelength is taken from Ref. [START_REF] Johnson | Optical constants of transition metals: Ti[END_REF] for Fe) do not play a significant role even quantitatively. Then, we verify that the similar interference patterns with similar decay and amplitude can be obtained for one or two-cycle pulses without taking into account the dispersion and keeping unchanged the fluence. Hence, the limited distance of scattered wave propagation from the object due to short pulses does not affect the radiative interference patterns up to few microns from the object. Finally, we investigate the impact of dispersion λ ± ∆λ, by simulating Gaussian pulse with temporally dependent wavelength. The results for 10 fs, 20 fs and 40 fs are shown in Fig. 7 and compared with the interference patterns in case of fixed wavelength equal to λ = 800 nm. The fluence is kept unchanged for different pulse durations. The radiative interference patterns are similar in case of 40 fs pulse, but change quantitatively and become less pronounced for shorter pulses of 10 fs and 20 fs in Fig. 7(a,b). Therefore, the temporal pulse broadening results in a smaller amplitude produced by the central wavelength and lower absorbed energy on the surface. The visible interference patterns in Fig. 7(a,b) are still the result of the interaction mostly with central wavelength of laser pulse, but the patterns are more dispersed and less pronounced due to the presence of other shifted wavelengths. Furthermore, both quasi-cylindrical and surface plasmon waves are affected, because the intensity is reduced in both perpendicular and parallel direction to laser polarization. The influence of finite spectral bandwidth on resonant surface plasmon excitation by prism coupling technique was previously measured and investigated [START_REF] Yalcin | Spectral bandwidth and phase effects of resonantly excited ultrafast surface plasmon pulses[END_REF]. Fig. 4 indicates that the polarization-dependent orientation of the maxima of radiative patterns are mostly subject of the material properties, whereas the nature and the arbitrary size or shape of the imperfection do not play a key role. However, this re- mains true only if the size of hole or bump is smaller than wavelength and its shape is symmetrical. For instance, Fig. 8 shows that the maxima of the electric field in the direction parallel to the electric field polarization on metal surface or in the direction perpendicular to the electric field on glass surface can be significantly enhanced if the imperfection has an ellipsoidal shape and a length approaching to laser wavelength. For instance, the closest intensity maxima to imperfections switch from parallel (typical to subwavelength symmetrical holes or bumps as in Fig. 4(a,b)) to perpendicular positions on glass surface in Fig. 8(a,b). In contrast, the closest intensity maxima switch from perpendicular (as in Fig. 4(c,d)) to parallel positions on metal surface as shown in Fig. 8(c,d). These features have consequences in multipulse irradiation, where evolving surface topography or deformation centers influence the feedback process [START_REF] Rudenko | Spontaneous periodic ordering on the surface and in the bulk of dielectrics irradiated by ultrafast laser: a shared electromagnetic origin[END_REF]. The introduction of artificial supra-wavelength scattering objects on the surface, such as elongated nanoridges and nanoslits, affects not only the dominant pattern configuration, but also results in more regular planar wavefront of the scattered waves [START_REF] Obara | Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates[END_REF][START_REF] Obara | Plasmonic and mie scattering control of far-field interference for regular ripple formation on various material substrates[END_REF][START_REF] Liu | Ultrafast imaging on the formation of periodic ripples on a Si surface with a prefabricated nanogroove induced by a single femtosecond laser pulse[END_REF].

4 2 0 -2 -4 X [µm] 4 2 0 -2 -4 X [µm] 4 2 0 -2 -4 X [µm] 4 2 0 -2 -4 X [µm] a) b) c) d) 𝑬 𝑬 𝑬 𝑬

Elongated and random topographies

𝐸 𝑘 3 2 -2 1 0 -1 -2 1 3 2 -3 0 -1 X[µm] a) Glass Bump -3 𝐸 𝑘 3 2 -2 1 -3 -1 -2 1 3 2 -3 0 -1 X[µm] b) Glass Hole 𝐸 𝑘 -2 1 3 2 -3 0 -1 X[µm] c) Metal Bump 𝐸 𝑘 3 2 -2 1 0 -1 -2 1 3 2 -3 0 -1 X[µm]
Our calculations indicate, however, that the origin of these waves is mixed depending on the structure geometry and, therefore, on Mie scattering or previously mentioned quasi-cylindrical waves on the surface, optical properties of the media (metal or glass), and, where the conditions are satisfied, additional contribution of surface plasmon wave for metal surfaces, as proposed in Refs. [START_REF] Obara | Direct observation of surface plasmon far field for regular surface ripple formation by femtosecond laser pulse irradiation of gold nanostructures on silicon substrates[END_REF][START_REF] Liu | Ultrafast imaging on the formation of periodic ripples on a Si surface with a prefabricated nanogroove induced by a single femtosecond laser pulse[END_REF]. Similar geometrical effects from more complexed structures such as microtriangles and microsquares with imprinted patterns oriented both perpendicular and parallel to laser polarization were observed experimentally [START_REF] Murphy | Laser-induced periodic surface structure formation resulting from single-pulse ultrafast irradiation of Au microstructures on a Si substrate[END_REF].

Fig. 9 shows the interference patterns as a result of light interaction with randomly distributed holes and bumps below glass and metal surfaces. For simplicity, we choose the identical positions of the imperfections with the same concentration C = 0.5% or the same average inter-object distance d ≈ 1.5µm on x0y transverse plane in all the cases in Fig. 9. The resulting absorption maps are the consequences of the coherent response of individual bumps and holes from Fig. 4, therefore, the features regarding polarization dependency of radiative and non-radiative fields remain qualitatively the same. The coherence here plays an important role as the patterns result from the interference of the same incident field with the scattered surface waves [START_REF] Zhang | Coherence in ultrafast laser-induced periodic surface structures[END_REF]. The coupling effects, however, are almost negligible for the average distances between imperfections exceeding the laser wavelength. This fact is clearly seen from Fourier Transforms (FT) of the corresponding images. The response from radiative fields there corresponds to smaller wavevectors, whereas the non-radiative fields are characterized by more dispersed frequency profiles with larger values of wave-vectors. The radiative field patterns on metal surface are straight perpendicular to laser polarization, whereas the patterns on glass surface conserve more pronounced properties of spherical waves, having the dominant orientation parallel to laser polarization. The average spacings on metal surface correspond to ≈ λ (or k x /k 0 ≈ 1 on FT), whereas the average spacings on glass surface are close to λ/n (or k y /k 0 ≈ 1.5 on FT). Again, the symmetry principle regarding the energy absorption by holes and bumps can be applied for complex systems as in Fig. 9.

The quasi-periodic interference patterns induced by nanoscale surface imperfections have been previously proposed to explain the formation mechanism of laser-induced periodic surface structures with periods of λ/n parallel to electric field polarization on dielectric surfaces, with periods of λ perpendicular to electric field polarization on metal surfaces, as well as periods significantly smaller than laser wavelength [START_REF] Sedao | Self-arranged periodic nanovoids by ultrafast laser-induced near-field enhancement[END_REF][START_REF] Skolski | Laser-induced periodic surface structures: Fingerprints of light localization[END_REF][START_REF] Zhang | Coherence in ultrafast laser-induced periodic surface structures[END_REF]. Our article completes the previous investigations by summarizing the variety of polarization-dependent interference patterns with different orientations depending on the nature of imperfections, their shape and position regarding the surface.

Conclusion

We have numerically investigated the energy deposition below the surface by light interaction with single holes and bumps. This study provides qualitative and quantitative estimations of the scattered radiative and non-radiative patterns below metallic and dielectric surfaces. Among interesting peculiarities of light interaction with imperfections, we can underline the main following observations (1) inverse local non-radiative field response of nanoholes and nanoridges, controllable by light polarization;

(2) inverse radiative response of imperfections depending on optical material properties (metal, dielectric);

(3) inversion of absorbed energy maxima position for surface nanoholes and intrinsic voids close to metal surface;

(4) different phase retardation and position of absorbed energy maxima for metal nanobumps and nanoholes on metal surfaces;

(5) less pronounced interference patterns for femtosecond pulses shorter than 40 fs due to finite spectral bandwidth;

(6) switching of absorbed energy maxima position in case of elongated imperfections of nearwavelength size;

(7) joint interference patterns as a result of collective response of imperfections.

This study provides original ideas towards nanoscale design of surfaces, optimal laser-induced nanostructuring and nanolithography, precise light manipulation by surface imperfections, and fabrication of polarization-sensitive devices.

Highlights:

1) Non-radiative patterns near surface holes and bumps are inversed 2) Far-field maxima and minima positions are inversed for holes and bumps 3) Interference maxima have opposite orientation on glass and metal surfaces 4) Interference maxima qualitatively differ for holes and voids close to surface 5) Less pronounced patterns for short pulses with finite spectral bandwidth 6) Asymmetric shape may switch the absorbed energy maxima positions 7) Multiple holes and bumps generate joint interference patterns on the surface *Highlights (for review)

Figure 1 :

 1 Figure 1: Classification of interference patterns resulted from scattering by a single nanohole on metal surface.

  the figures. This value is proportional to the absorbed energy further transferred to the material via Joule heating. In Figs.4-5and Figs.8-9, the intensity in normalized to the maximum incident value of the electric field E 0 . This way, the value (I -I 0 )/I 0 indicates the intensity enhancement, from which the incident field is extracted for clarity.

Figure 2 :

 2 Figure 2: Energy deposition below glass surface in the propagation plane (a) with a bump R = 20 nm, (b) with a hole R = 20 nm and in the transverse plane on the surface (c) with a bump R = 20 nm, (d) with a hole R = 20 nm.

Figure 3 :

 3 Fig.2shows the intensity distribution on a glass surface in the propagation (a, b) and in the transverse planes (c, d). In case of a bump on dielectric surface, the energy is equally deposited on the top of the bump and on both edges of the equatorial plane intersecting the surface. Therefore, the peaks of the nanometric bumps are likely to be erased by laser ablation. The local field enhancement results in the maxima parallel and minima perpendicularly to light polarization in the transverse plane. In case of a nanometric hole on dielectric surface, the energy is mostly deposited inside the hole. In this case, the maxima of local field are perpendicular to laser polarization. This way, the material ablation with holes leads to more pronounced deeper holes and the reinforcement of polarization dependency of the absorbed energy below dielectric surface. The polarization dependency of the intensity maxima can be understood by a simple analogy with Mie theory. In fact, hole on glass surface owns the properties of matter with smaller refractive index inside the material with greater one. In contrast, the properties of bump below the surface result from the continuous prolongation of the electric field above the surface, where a particle of glass (bump) is surrounded by air, therefore, matter with larger refractive index embedded in medium with smaller refractive index. Regarding Fig.2(c)

Figure 4 :

 4 Figure 4: Absorbed energy in the transverse plane on glass surface with (a) a bump, (b) a hole and on metal surface with (c) a bump, (d) a hole of R = 20 nm. Light interaction exhibit contrasting polarization-sensitive interference patterns (in the micrometer scale) depending on the optical material properties and the nature of imperfections.

Fig. 4

 4 Fig. 4 shows the main features of both radiative and non-radiative fields scattered by subwavelength imperfections on the surface. The interference patterns are more pronounced in perpendicular direction to light polarization on glass surface and in parallel direction on metal surface. In contrast to local near-field, the orientation of the radiative patterns does not depend on the nature of the imperfections (compare Fig. 4(a) with Fig. 4(b) and Fig. 4(c) with Fig.4(d)). The striking difference is revealed, however, in the distances between the intensity maxima positions and the emission centers. This fact is related to different phase retardation of surface waves scattered by holes and bumps. This way, the first far-field maxima (to differ between the local near-field) are on the greater distance from the hole than from the bump for glass surface and from the bump than from the hole for metal surface. Note that the difference is related directly to charge accumulation on the interface of imperfection and, therefore, to local near-field distributions. For glass, the interference patterns represent spherical standing waves with λ/n spacings in Fig.4(a, b), whereas for metal the symmetry is broken due to additional contribution of surface plasmon wave in the direction parallel to the electric field with spacings ≈ λ in Fig.4(c, d). The

Figure 5 :

 5 Figure 5: Absorbed energy below stainless steel surface in the transverse plane with an intrinsic void of R = 20 nm at 50 nm below (a) metal surface and (b) glass surface. The optical response of nanovoids inside the material varies from the response of surface imperfections in Fig. 4.

Figure 6 :

 6 Figure 6: Electric field Ex distribution in the air and below the surface in the propagation plane in case of hole/bump (a) on metal surface and (b) on glass surface. The figures are brought together in order to show the phase difference and the symmetry principle of the response of bumps and holes.

Figure 7 :

 7 Figure 7: Influence of finite spectral bandwidth on the interference patterns in the vicinity of a single nanohole on metal surface. The considered pulse durations and wavelength dispersion are indicated in each sub-figure (a-d).

Figure 8 :

 8 Figure 8: Absorbed energy in the transverse plane on glass surface (a) with a nanoridge of Rx = Rz = 20 nm and Ry = 500 nm or (b) with a nanoslit above the surface Rx = Rz = 20 nm and Ry = 500 nm. Absorbed energy in the transverse plane below metal surface (c) with a nanoridge of Ry = Rz = 20 nm and Rx = 500 nm or (d) with a nanoslit above the surface Ry = Rz = 20 nm and Rx = 500 nm. For clearness, the imperfection positions centered at (0, 0, 0) are indicated by blue ellipsoids. The interference patterns qualitatively differ from the corresponding ones in Fig. 4.

Figure 9 :

 9 Figure 9: Absorbed energy distributions and their Fourier Transforms (FT) in the transverse plane on glass surface with (a) holes, (b) bumps and on metal surface with (c) holes, (d) bumps of R = 20 nm. k 0 = 2π/λ is the wave-vector of propagation in air.

Acknowledgments

This work was supported by IMOTEP project within program 'Investissements d'Avenir' operated by ADEME.