Jolita Ralyté
email: jolita.ralyte@unige.ch

Abdelaziz Khadraoui
email: abdelazis.khadraoui@unige.ch

Michel Léonard
email: michel.leonard@unige.ch

From Information Systems to Information Services Systems: Designing the Transformation

Keywords: Information service, information services system, service-oriented paradigm, IS evolution. 1

Service-orientation is currently considered as a promising paradigm to deal with the complexity, interoperability and evolution of enterprise Information Systems (IS), which are the foremost preoccupation in today's enterprises. However, the shift from a conventional IS architecture to a serviceoriented one is not an easy task despite of the various service design approaches proposed in the literature. In this paper we promote the concepts of information service and Information Services System (ISS) and we present three different ways to design an ISS taking into account enterprise legacy IS and/or from scratch. We illustrate the three approaches with examples taken from industrial projects and case studies.

Introduction

Sustainability and evolution of legacy Information Systems (IS), including their maintenance, extension with new components, interoperability with other systems and applications, is one the foremost preoccupation in today's enterprises, not only at technical but also at information and business strategic levels. In addition, the need for inter-organizational and networked information systems, cloud platforms and services systems is growing because of the enterprise business models transformations into networked and service-oriented ones. At the same time, it is unthinkable to replace existing IS by the new ones for each enterprise business and/or organizational change. Legacy IS has to evolve together with enterprise changes, and this evolution can take different forms: integration of new components from the market or custom-made, development of services on top of the existing IS, establishment of interoperability between two or more IS, etc.

In this context, service-oriented approaches emerge as prospective ones to deal with IS fragmentation, interoperability and evolution problems [START_REF] Abbass | Accessing Enterprise Information Systems in a Service-Oriented Architecture[END_REF][START_REF] Chua | Adoption of Service-Oriented Architecture by Information Systems[END_REF][START_REF] Ralyté | Information Systems Evolution: A Process Model for Integrating New Services[END_REF] as well as to support inter-organizational IS development [START_REF] Le Dinh | A Conceptual Framework for Designing Service-Oriented Inter-organizational Information Systems[END_REF][START_REF] Le Dinh | A Conceptual Framework for Service Modelling in a Network of Service System[END_REF][START_REF] Ralyté | Applying Transdisciplinarity Principles in the Information Services Co-creation Process[END_REF]. Modularity, reusability and evolution are considered as the main values that service-oriented paradigm brings to the IS domain. To reach modularity, an IS has to be composed of a collection of inter-related and autonomous components. The notion of service, and in particular information service (see section 2.1), is introduced to cope with IS modularity. Modularity is the necessary basis to ensure incremental and evolutionary IS development and therefore a prerequisite for legacy IS evolution with new components/services. It also allows to avoid chaotic IS fragmentation. Reusability, as opposed to the more expensive "from scratch" development, means the reuse of legacy data and applications to provide new business functionalities -new services. Finally, the evolution principle consists in the ability to easily replace an existing component/service by a new one.

The literature review demonstrates the advent of proposals to redesign conventional IS architectures into the service-oriented ones [START_REF] Chua | Adoption of Service-Oriented Architecture by Information Systems[END_REF][START_REF] Haesen | Designing service-oriented information systems architectures[END_REF][START_REF] Le Dinh | A Conceptual Framework for Designing Service-Oriented Inter-organizational Information Systems[END_REF][START_REF] Le Dinh | A Conceptual Framework for Service Modelling in a Network of Service System[END_REF][START_REF] Ralyté | Information Systems Evolution: A Process Model for Integrating New Services[END_REF][START_REF] Spohrer | The Service System is the Basic Ab-straction of Service Science[END_REF]. The notion of information services system (ISS) introduced in [START_REF] Arni-Bloch | MISS: A Metamodel of Information System Service[END_REF] (see section 2.2) appears as a natural evolution of the IS concept. Similarly to IS, ISS emphasizes the value of information, its creation, management and sharing, while improving its modularity, agility and interoperability.

The research question considered in this paper is how enterprise information systems could evolve from conventional to the service-oriented ones. In particular, we aim to identify and explore different ways to lead the shift from traditional information systems towards information services systems. Based on the related literature survey, our previous works [START_REF] Arni-Bloch | MISS: A Metamodel of Information System Service[END_REF][START_REF] Arni-Bloch | Service-Driven Information Systems Evolution: Handling Integrity Constraints Consistency[END_REF][START_REF] Khadraoui | A Framework for Compliance of Legacy Information Systems with Legal Aspect[END_REF][START_REF] Khadraoui | Laws-Based Ontology for e-Government Services Construction. Case Study: The Specification of Services in Relationship with the Venture Creation in Switzerland[END_REF][START_REF] Khadraoui | Service Specification upon Multiple Existing Information Systems[END_REF][START_REF] Ralyté | Information Systems Evolution: A Process Model for Integrating New Services[END_REF] and various projects realized in collaboration with the Information Technology Center of the State of Geneva (some of them are reported in [START_REF] Khadraoui | A Framework for Compliance of Legacy Information Systems with Legal Aspect[END_REF][START_REF] Khadraoui | Laws-Based Ontology for e-Government Services Construction. Case Study: The Specification of Services in Relationship with the Venture Creation in Switzerland[END_REF][START_REF] Khadraoui | Service Specification upon Multiple Existing Information Systems[END_REF]), we present in this paper three different but complementary approaches to design information services systems based on enterprise legacy IS and/or from scratch. The three approaches have been tested in various projects and case studies that we mention in this paper.

The rest of the paper is organized as follows: in the next section we define the notions of information service and information services system. Then, in section 3 we present and illustrate three approaches supporting ISS design. We discuss the related work and how our contribution complements it in section 4. Finally, section 5 concludes the paper and highlights future perspectives.

2

Information Services and Services Systems

Information Service

In the domain of enterprise information systems, the notion of service was introduced as a potential means to improve legacy IS agility and evolution and to facilitate IS interoperability. It is built upon the concept of IS component [START_REF] Turki | IS Components with Hyperclasses[END_REF], which is defined over classes, methods, integrity rules, processes, roles and events that constitute a semantic unit where several actors aim to achieve a common goal. In order to fit the IS context, a service is expected to support inter-organizational and/or intraorganizational business activities trough a collaborative creation, transformation and transmission of information. This type of service is named an information service [START_REF] Arni-Bloch | MISS: A Metamodel of Information System Service[END_REF][START_REF] Arni-Bloch | Service-Driven Information Systems Evolution: Handling Integrity Constraints Consistency[END_REF] and is defined as "a component of an information system representing a well defined business unit that offers capabilities to realize business activities and owns resources (data, rules, roles) to realize these capabilities". Therefore, an information system can be seen as built of a collection of interoperable information services. The metamodel of the information service is an extension of other computational services metamodels such as: WSPER [START_REF] Dubray | Wsper an abstract SOA framework[END_REF] or the model proposed in [START_REF] Andrikopoulos | Managing the Evolution of Service Specifications[END_REF]. The particularity of the information service definition is that it requires to make the service transparent, i.e. it explicitly distinguishes four interrelated information spaces -static, dynamic, rule and role -that are necessary to define service resources and capabilities in the organizational context. We claim that in the IS context it is not sufficient to consider services as black boxes with only interface part available for their selection and composition purposes. It is essential to make explicit the information concerning service structure, processes, rules and roles and to be able to identify those shared with other services. Fig. 1 shows the simplified metamodel of the information service where only the main concepts are represented (see [START_REF] Arni-Bloch | MISS: A Metamodel of Information System Service[END_REF] for the detailed version). The static space of the service defines its data structure in terms of classes and relationships between classes. The notion of Hyperclass (introduced in [START_REF] Turki | Hyperclasses: towards a new kind of independence of the methods from the schema[END_REF] to specify information system components) is used to represent complex domain concepts by putting together the corresponding set of classes. Classes are linked only via existential dependencies and specialization relationships. An existential dependency is materialized via an attribute with mandatory and permanent constraints. The dynamic space defines service capabilities in terms of actions that can be executed by the service and their effects on service classes. An action is triggered by an event that occurs in the service information space and is described by a process to be executed, which can be a simple function or a more complex interaction involving several actors. An action produces one or more effects on the static space (e.g. create an object of a class, modify an attribute). The notion of effect is used to characterize the result of the action and allows to evaluate the impact of the action on the rule space. The rule space deals with service regulation policies, which are formalized as pre-, postconditions on service actions and integrity constraints on service data. An integrity constraint has a scope, which includes all the effects that could transgress the rule. Such an effect is called a risk of the rule. Finally, the role space defines service actors, the roles they play in the organization and the rights and responsibilities they have on service actions.

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The metamodel presented above represents a foundation for engineering information services and services systems. In particular, it supports information services definition, composition, identification of the overlap between information services, and also new services integration into an information services system.

Information Services System

According to [START_REF] Spohrer | Steps Toward a Science of Service Systems[END_REF][START_REF] Spohrer | The Service System is the Basic Ab-straction of Service Science[END_REF], a service system is a configuration of people, technology, shared information (such as language, processes, metrics, prices, policies, and laws) and other resources that interact with other service systems to create a mutual value. Spohrer et al., [START_REF] Spohrer | Steps Toward a Science of Service Systems[END_REF] also say "service systems comprise service providers and service clients working together to coproduce value in complex value chains or networks". In their vision, there is a clear separation between the service provider and its client. However, this distinction is not so obvious from the information systems point of view. An actor of an information service can be allowed to provide and to consume service information and capabilities. In this case the term "service prosumer" fits better the situation. Though, the notion of value coproduction is also key in the domain of information systems.

In [START_REF] Spohrer | Steps Toward a Science of Service Systems[END_REF], the authors claim that the main difference between service systems and computational systems is people -unlike computational system components, the behavior of people doing work in service systems cannot be easily modeled and simulated which can create risk but also generate innovations. We agree with these authors and we take into account this issue. We argue that in Information Services Systems (ISS) people are not considered as independent system components but rather as actors enabling capabilities of information services. Their behavior is specified through the organizational roles and responsibilities involved in the information services enactment (provision and/or consumption of information resources) and governed by a set of rules implemented in the services. We also think that the unpredicted behavior of actors could be explored as a potential source of ideas for future business innovations. It could be captured in the form of initiatives. As service systems, information services systems aim to stay dynamic and open in order to enable innovation and facilitate their evolution. They have to take a risk to give people some liberty to informally and formally change rules and policies.

In our approach, an ISS is seen as a collection of interoperable information services. It aims to transform an integrated and rather rigid IS architecture into a more flexible, modular and sustainable one where services can be modified or replaced and new services can be integrated.

Designing Information Services Systems

There are certainly many different ways to design Information Services Systems (ISS). In this work we focus our attention on the evolution of conventional enterprise IS into the service-oriented ISS and we discuss different transformation situations and approaches. In particular, we identify and illustrate three approaches taking into consideration different organizational contexts and ISS design situations, and legacy IS reuse:

• Services upon legacy IS. This approach aims to bring some flexibility and modularity to the rather monolithic and fragmented legacy enterprise information systems without inflicting to them any major transformation. Indeed, in this approach services are created upon legacy IS. They utilize resources existing in various IS (their data, processes, rules, responsibilities) and provide some added value to the service users. The approach consists in identifying for each new service the existing resources that are potentially scattered in different IS and to guarantee that the execution of the service will keep these legacy IS in a consistent state, i.e. will ensure data consistency and will not violate their rules and responsibilities. • Fully service-oriented ISS. This approach, in the contrary, considers an information system as a composition of information services and for that needs a more deep transformation of the existing IS into a services-oriented one. Indeed, the approach consists in "decomposing" an IS into a collection of information services and defining the overlap (common data, activities, roles, rules) between them. Information overlap management is the most important difficulty when including new services into an existing ISS. • Information kernel-based ISS. This approach proposes an intermediate architecture based on a core IS and information services as its extensions. The core IS captures the kernel information -the invariant data, processes and rules, while information services offer capabilities for business activities that are subject to change. In this type of architecture the main challenges are (1) the definition of the information kernel, which is formalized as a collection of kernel services, and (2) the preservation of this kernel when adding new services to the ISS.

Information Services upon Legacy IS

Even though enterprises are constantly seeking to renew the range of their IS to be compliant with their business evolution, replacing or transforming legacy IS can be very expensive and error-prone or even impossible. Introducing a service-oriented layer upon multiple legacy systems is considered here as a potential solution to limit the transformation of legacy IS and at the same time to bring some flexibility and agility to them. In this approach we consider that several existing IS and enterprise applications are information providers to the newly defined services. The identification and selection of services to be developed has to be done prior to the application of this approach. The key step of this approach consists in the construction of a common base on top of the existing IS. This common base should help to offer each service the access to the precise and consistent information distributed in various IS and to guarantee service compliance with the existing IS and with the enterprise legal frame, which is a composition of laws and regulation policies that govern enterprise activities. Several aspects have to be considered in the construction of this common base. In particular, we need to specify the legal frame that has to be respected by the new services and to define the organizational contexts for each identified service. The approach was introduced first in [START_REF] Khadraoui | Service Specification upon Multiple Existing Information Systems[END_REF]; its overview is shown in Fig. 2a. The process model of the approach is composed of four main steps that are expressed as four engineering intentions with one or several strategies to achieve each of them (see Fig. 2c). It is represented by using Map [START_REF] Rolland | A Multi-model Vew of Process Modelling[END_REF] process modeling formalism, which provides a representation system based on a non-deterministic ordering of intentions and strategies in the form of a labeled directed graph. Each step is detailed below. Step 1: Construct a frame of reference means for each service to define: (1) its organizational contexts and the existing IS containing the information necessary for the service execution, (2) the information necessary for the service implementation in the form of a conceptual model, (3) the roles responsible for its execution. The frame of reference is constructed by instantiating the Service Analysis Metamodel shown in Fig. 2b. This metamodel defines the key concepts necessary to link an information service to the existing IS that provide data necessary to the service execution. A service can be defined on one or several Organizational Contexts that describe business rules, legal constraints and the capability of the organization to enforce laws and policies. The notion of Hyperconcept is used to specify how the organizational contexts are formalized in different information systems. In fact, a hyperconcept is a complex concept (composed of several sub-concepts) representing a semantic unity in the domain of analysis. Finally, the Role concept represents the responsibilities and authorizations to execute IS activities in a particular organizational context.

Step 2: Select the candidate informational elements for the common base by analyzing the information collected in the frame of reference. This analysis can be done in three different and complementary ways: by organizational context analysis, by organizational role and/or by hyperconcepts analysis respectively. Typically, if a hyperconcept representing a part of the service organizational context is implemented in more than one IS, the data representing it is candidate to the common base. The same applies to the roles related to the service organizational context.

Step 3: Construct a common base by collaborative decision making where all project stakeholders (business practitioners, IS architects, database architects, etc.) are invited to discuss on several formalized alternatives and to choose the elements (e.g. service related roles) to be implemented in the common base.

Step 4: Adapt the existing IS by adding new elements (e.g. adding new role) or transforming the existing ones (e.g. changing a business rule) in order to guarantee that each legacy IS is interoperable with the new common base. The number and complexity of identified transformations indicate the weight of the impact of the new service creation on the legacy IS, and this impact has to be minor.

This approach was applied in collaboration with the Information Technology Center of the State of Geneva (Switzerland) to design new services for e-administration based on five existing independent information systems (see Fig. 3). Three of these IS operate at the cantonal level: (1) the Commercial Register (RC) that allows to build and identify all legal entities in the State of Geneva and to register their associated legal events, (2) the Tax IS (R-Fisc) that stores the taxation data about businesses at the cantonal level, and (3) the Geneva Business Repertory (REG) that contains administrative information on businesses and companies located in the canton of Geneva and makes this information available for administrative purposes and for the applicable dissemination in public and private sectors. These cantonal information systems have to interact with two similar IS in use at the federal level: the Federal Commercial Register (RCF) and the Federal Business Repertory (REE). Several information services were identified for this project. For example, one of them concerned the "Transmission of business statistical data" and should help companies to transmit their data to the cantonal and federal offices in order to build statistics. The organiza-tional context of this service is based on two laws on public statistics, one at the cantonal level (LStat) 1 and the other at the federal level (LSF) 2 , and the data related to this context is available in two information systems -the REG and REE. The construction of the common base consisted in the selection of elements (data, roles, rules) common to both concerned IS and necessary for the service execution, and also the creation on new elements such as new roles for the execution of service activities. The main difficulties encountered in this project were related to the common base management, in particular to decide who will be responsible for this common base and how to manage the existing flows between the two IS (REG and REE).

Fully Service-Oriented ISS

The second approach for ISS engineering considers the enterprise IS as a composition of information services where each service provides a support for a particular business or administration activity. It differs from other service-oriented approaches in many perspectives. For example, the granularity level of information services is quite large -a service represents a full business unit not only a simple action or application.

A service can represent an inter-organizational collaboration and value exchange between organizations as well as, and in particular, inter-organizational activities. Indeed, most of the existing service-oriented approaches deal with the design of different computational services (e.g. web, mobile, cloud) and their supporting platforms in a pure "service customer vs. provider" frame. Up to now, only little attention has been paid to the intra-organizational IS context and legacy IS transformation into the service-oriented one. In [START_REF] Arni-Bloch | Service-Driven Information Systems Evolution: Handling Integrity Constraints Consistency[END_REF][START_REF] Ralyté | Information Systems Evolution: A Process Model for Integrating New Services[END_REF] a service-oriented and situation-driven approach for legacy IS evolution has been proposed. In this paper, we argue that this approach can be used for ISS engineering in an incremental way by progressively integrating new information services into an ISS. As said above, in this approach an IS is seen as a collection of information services, at least at the conceptual level. The main difficulty of this approach is in the fact that services composing the ISS are not totally independent components, they inevitably overlap, which is mainly because of the data sharing. In fact, the overlap between information services can exist in the four information spaces (static, dynamic, rule and role) (see Fig. 1). The construction of an ISS consists in integrating progressively new services into the existing ISS as shown in Fig. 4a. Of course, each new integration crates new overlap situations -some data, activities, roles and rules can be shared between the existing and new services. Therefore, this approach is based on the analysis and resolution of the overlap between legacy and new services by preserving the legacy ones as much as possible. The service integration approach is articulated in five main intentions with several strategies available to reach each of them as shown in Fig. 4b. Step 1: Identify and characterize the overlap between the four information spaces of the new service and the services already existing in the ISS. An overlap report has to be elaborated for each space with the objective to identify similar elements and to evaluate if they need some structural or semantic unification or not. For example, two services can contain the same class (e.g. Student) with the same or different structure (e.g. the attribute Birthdate is missing in one of the classes), which represents a static space overlap. An example of the functional overlap would be when two services are producing the same effect on a class in overlap, as for instance updating an attribute or creating new objects of this class (e.g. create a new student). The rule overlap appears when the same class is governed by different rules in different services (e.g. the deadline for registering new students exist in one service but not in the other). Finally, the role space overlap means the existence of the same or similar roles in two services (e.g. Students' administrator).

Step 2: Validate the overlap conformity for each couple of overlapping elements (classes, actions, rules and roles respectively) identified and characterized in the overlap report. That means to check if they conform to each other, i.e. can be substituted one by another. In the static space two classes are considered as conforming if they have the same name, their sets of attributes and sets of methods are identical and they have the same super-classes. If two overlapping elements are not conforming, one of them (generally the element of the new service) has to be adapted in the next step.

Step 3: Settle the overlap conformity for each couple of non-conforming overlapping elements according to the type of the identified disparity. The disparities can be of semantic nature (the same name but different meaning or vice versa) or of structural nature (different set of attributes in similar classes). Therefore, it can be necessary to modify the concerned classes, actions, rules and roles by semantic unification (i.e. renaming one of the elements) or by structural unification (i.e. transformation of the element structure) accordingly.

Step 4: Organize the overlap means to clarify the relationships between the new service and the legacy ones. Depending on the situation, it can be necessary to adapt service responsibilities on common elements -to determine the effects (create, update, delete, etc.) that the service is still authorized to realize on the shared data to be compliant with the legacy services. In order to clarify the visibility of effects on common elements it is important to define a cooperation protocol for each of them. Finally, to make sure that the new service respects the regulation policy of the legacy ISS, we need to guarantee that validity of rules, and especially of integrity constraints, defined on legacy services will not be violated by the new service after its integration.

Step 5: Consolidate the integration by revising the four service spaces and adding if necessary missing elements in each of them in order to ensure that the obtained integration is valid. In fact, the integration of a new service can generate new situations that did not exist before, neither in the new service nor in the legacy ones, and addition of new actions, rules, roles and/or authorizations can be necessary to face these situations.

In order to validate this approach, we have realized a case study that concerned the extension of our University Students Management System with new information services. In particular, we have considered the Diploma Management Service (DMS) as an existing information service and the Online Registration Service (ORS) as the new service to be integrated in the existing ISS. The DMS provides several diploma management capabilities such as: to create the curriculum of each diploma by defining its courses and linking them to their lecturers, to manage students' registration to different diplomas and to the corresponding courses, to manage examination results, etc. The new service, the ORS, enables students' registration by providing a web interface for this purpose. A candidate can create a university registration request on-line by filling the registration form and by uploading different required documents. Then, the students' administrator validates the on-line created registration request and asks for additional documents if necessary. He/she is responsible for recording the candidate as a student and for registering him/her to the selected diploma.

Because, the two services have been developed independently, it is obvious that some information overlap between them is inevitable (e.g. class Person, class Student, activity UpdatePerson, rule RgistrationToDiploma.RegistrationDate < Diploma.RegistrationLimitDate, role Student, role Diploma Manager, etc.). These shared elements (identical or similar) constitute the information overlap between the existing and the new service and the integration process was to make this overlap consistent in order to enable reliable new service exploitation within the ISS and the non-violation of the already existing ISS consistency.

Information Kernel-Based ISS

The third approach for ISS engineering proposes to use a mixed architecture where a kernel IS is extended with a variety of information services. In fact, in this approach we consider that the core and invariant information (including data, processes, rules and roles) can be found in the laws and other regulation policies governing enterprise activities, and should be captured in the kernel of the IS independently of the information services that could extend this IS later. This approach is especially adapted for the ISS development in public and governmental sectors. We argue that legal documents include precise definitions of concepts, rules and constraints governing the institutional activities and represent a rich source of knowledge for the ontological information extraction the information kernel conceptualization. Moreover, the use of laws permits enhance the adequacy and compatibility of an institutional IS with the corresponding institution activities and to construct a stable information kernel as a basis for the sustainable ISS development. Therefore, this approach has two parts: (1) the ontological model construction based on the analysis of legal sources and (2) its mapping into the conceptual model representing the kernel IS as shown in Fig. 5a. This kernel IS then can be extended with new services following Fully Service-Oriented ISS engineering approach presented above. Formally, the process model of the approach is defined in terms of three main intentions and several strategies to achieve each of them as depicted in Fig. 5c. The first two intentions, Identify Hyperconcepts and Build Hyperconcepts, deal with the ontological model construction in terms of a collection of interrelated hyperconcepts while the third one, Construct Kernel Information Services, defines the kernel IS as a collection of kernel information services.

Step1: Identify the hyperconcepts that correspond to the different ontological information spaces related to the organization activities. As depicted in Fig. 5b representing the metamodel for the kernel IS construction, we use the notion of the hyperconcept to capture the information related to a fragment of any legal and/or ontological source named here an Ontological Fragment. A hyperconcept is composed of a set of concepts extracted from ontological fragments and represents an ontological unity with precise semantics. Ontological Business Rules can be extracted from one or several ontological fragments and are related to one or more hyperconcepts. Besides, we use the notion of Ontological Role, which represents a set of responsibilities and permissions to perform business activities in the organization. Each ontological role is defined in at least one ontological fragment and is valid in the context of one or more hyperconcepts. The identification of hyperconcepts can be done in two complementary ways: (1) by selecting and analyzing different ontological fragments and (2) by analyzing the organization and identifying its ontological roles. An ontological fragment is selected if it is considered as stable and invariant for the lifespan of the organization and of the ISS under construction. It is obvious, that this type of decision requires some experience and risk management abilities. Step2: Build the hyperconcepts by extracting concepts and business rules from the selected ontological fragments and refine their structure with a set of validation criteria based on the ontological level metamodel (Fig. 5b).

Step 3: Construct information services composing the kernel of the ISS. As shown in Fig. 5b, a Kernel Information Service is based on one or more hyperconcepts from the ontological level. The structure of the kernel information service is formalized by using information service metamodel shown in Fig. 1. We use a collection of mapping rules to extract from the ontological model the four information spaces (static, dynamic, rules and roles) of each kernel information service.

This approach was applied in several e-government projects (e.g. [START_REF] Khadraoui | Laws-Based Ontology for e-Government Services Construction. Case Study: The Specification of Services in Relationship with the Venture Creation in Switzerland[END_REF][START_REF] Khadraoui | A Framework for Compliance of Legacy Information Systems with Legal Aspect[END_REF]) in collaboration with the Information Technology Center of the State of Geneva (Switzerland). One of the projects concerned the development of the kernel information services system for managing the prescription and delivery of narcotics for the treatment of the addicted people. We have used the Geneva law K 4 20.063 as a legal source of knowledge to build and to implement the kernel ISS supporting the activities of narcotics prescription, distribution and administration. Indeed, this law describes the procedure that medical doctors have to follow in order to record a request of authorization allowing to prescribe a narcotic for the treatment of a dependent person (drug addict). For example, the law says that a doctor has to obtain an authorization from the cantonal doctor before the prescription of any narcotic to a patient. Then the pharmacist, on the basis of the authorization delivered by the cantonal doctor, provides the doctor, or directly the patient, the drug prescribed. From the law text a set of invariant concepts {Doctor, Patient, Authorization, Cantonal Doctor, Prescription, Drug, etc.} can be identified. The law also describes how the drug has to be distributed and administered by enumerating different cases where a patient can benefit of the administration of narcotics. Therefore, the law also defines business rules and constraints (ontological business rules) to be respected. Finally, the law allows to identify organizational roles (ontological roles) such as Doctor, Pharmacist, Cantonal Doctor, etc. A kernel ISS was built directly from this law.

Related Work and Discussion

Most of the current service-oriented approaches, like SOA [START_REF] Erl | SOA Principles of Service Design[END_REF][START_REF] Krafzig | Enterprise SOA: Service-Oriented Architec-ture Best Practices[END_REF][START_REF]Reference Model for Service Oriented Architecture 1.0, Oasis Standard[END_REF] instruct how to develop service-oriented software systems from scratch and do not pay much attention to the legacy software, and in particular information systems, reuse and evolution. Moreover, the compliance of services to the laws and enterprise regulation policies is not considered. In the context of the IS engineering, these approaches can only recommend to rebuild enterprise IS in terms of autonomous services that could be composed in different ways. Such services have to be elaborated from scratch in order to avoid any overlap between them. This type of development represents a rather extreme solution and is not adapted to the legacy IS evolution. The lifecycle of an IS is a continuous incremental and evolutionary process and it is not possible to rethink the entire IS at each iteration in order to guarantee the autonomy and correctness of the existing and new services.

Up to now, we found relatively few attempts to define service-oriented architectures for information systems. We can mention the work of Chua [START_REF] Chua | Adoption of Service-Oriented Architecture by Information Systems[END_REF] who discusses how service-oriented design should be applied in an organization in order to adopt SOA. Haesen [START_REF] Haesen | Designing service-oriented information systems architectures[END_REF] presents a research plan for developing an approach to design SOA for IS. Le Dinh and Nguyen-Hgoc [START_REF] Le Dinh | A Conceptual Framework for Designing Service-Oriented Inter-organizational Information Systems[END_REF] propose a conceptual framework for designing service-oriented inter-organizational IS. Thomas and Brocke [START_REF] Thomas | A value-driven approach to the design of serviceoriented information systems-making use of conceptual models[END_REF] present a valuedriven approach to design service-oriented IS; the approach is based on business process modeling and cost/benefit analysis to determine whether the introduction of SOA justifies the effort. Lo and Yu [START_REF] Lo | From Business Models to Service-oriented Design: A Reference Catalog Approach[END_REF] propose a reference catalogue approach to design an SOA system. This approach uses the i* modeling technique adapted to the serviceoriented business modeling for the selection of reference business models from the catalogue and their adaptation to the particular case. Estrada et al. [START_REF] Estrada | Extending Organizational Modeling with Business Services Concepts: An Overview of the Proposed Architecture[END_REF] introduce a service-oriented organizational model in order to reduce the mismatch between business models and service-oriented designs. Most of these approaches consider SOA for IS at business level rather than at information management and implementation level, and only from the customer-provider perspective -how the organization could offer services to its customers. The intra-organizational perspective of a service as a support for internal enterprise activities and information exchange is not really considered. Besides, the evolution of legacy IS into the service-oriented ones is not their major preoccupation.

Unlike the contributions mentioned above, the three approaches for ISS engineering summarized in this paper aim to support service-oriented inter-and intraorganizational ISS development taking into consideration legacy IS and their evolution. Each of the three approaches is dedicated to a particular situation but together they are considered as complementary ones.

The first approach, named information services upon legacy IS, deals with the situation where multiple legacy IS have to continue to operate and cannot be transformed drastically. It helps to preserve existing IS by creating information services upon them via a common base capturing their overlap. The main difficulties of this approach are related to the number and size of existing IS that have to be analyzed and used by the information services, the organizational changes (e.g. new roles, new responsibilities) entailed by the new services, and the data opening and sharing.

The second approach, named fully service-oriented ISS, is dedicated to support incremental ISS construction as well as legacy IS evolution by extending them with new information services. This approach is based on the analysis and resolution of the overlap between the legacy information services and the new services and helps to preserve the legacy ones as much as possible thanks to the integration impact analysis.

Finally, the third approach, named information kernel-based ISS, aims to construct a stable and invariant basis for an ISS -the kernel information services -that could then be extended with more volatile services. According to this approach, the information stability can be identified from the legal sources (laws and other regulation policies) governing enterprise activities and therefore this approach is especially suitable for the ISS engineering in public administration and governmental sectors such as e-government services. The kernel information services constitute a reference and a foundation for the ISS designers. They help to understand which information is invariant and stable during the ISS lifecycle and to design new services to be added to the ISS. Thus, various situations of services interoperability are identified, discussed and settled at early development stages.

Conclusion

Service-oriented paradigm seems to be well adapted to deal with the complexity, interoperability and evolution of enterprise legacy IS, and the most promising one to consider the development of the next generation information services systems. How-ever, the literature dealing with this topic is still modest. In this paper we aim to demonstrate how IS-specific service-oriented architectures can be elaborated, in particular to cope with the legacy IS evolution. The three approaches presented in this paper overview our work in the domain of service-oriented IS development where we introduce the notions of information service and information services system (ISS).

We aim to demonstrate that depending on the enterprise legacy IS situation and its evolution strategy the ISS development approach and the ISS architecture will be different. For example, if there is a need to open the access to the resources of multiple existing IS but the requirement to preserve these IS untouched is very strong, the approach for defining services upon legacy IS helps to deal with such situation. In the contrary, if a new services system has to be developed from scratch in order to support some new business or to computerize services that until now were provided manually or with the help of simple communication techniques like e-mail and/or fax, the information kernel-based approach seems to be the most appropriate one. Finally, the fully service-oriented approach is particularly helpful to deal with the evolution of an IS which is already designed as a composition of services. Indeed, it provides guidance for extending existing ISS with new services. The notion of information overlap is recurrent in the three approaches and it is also original in comparison with the conventional SOA, which considers software services as completely autonomous and independent modules. In the context of IS this type of autonomy is not reachable because different information services have to share data, roles and rules governing enterprise business. The three approaches have been applied in various case studies and collaborative projects in the sector of public administration. Our future preoccupation is to apply them in different industrial settings and to extend them with additional strategies and techniques.

Fig. 1 .

 1 Fig. 1. Simplified metamodel of the information service

Fig. 2 .

 2 Fig.2. The approach "Information services upon legacy IS"

Fig. 3 .

 3 Fig. 3. Example of Service Organizational Context Analysis

  model for legacy IS extension with new services

Fig. 4 .

 4 Fig. 4. The approach "Fully service-oriented ISS"

Fig. 5 .

 5 Fig. 5. The approach "Information kernel-based ISS"

-onal% Role% Rule%

	Dependency% Existen-al% Sta-c%Space%	DependantCl++ ReferenceCl++ AAribute%	A4rInClass+	EffectOnClass+	Effect%	Process% Dynamic%Space%
				ClassInHclass+	Event%	Trigger+
	Specializa-on%
			ServiceRule+
							Authoriza-on%
	Context%
	Scope%	Integrity% Constraint%	Condi-on%	OrgFunc(on+	Actor%
				PreCondi(on+
		Risk%		PostCondi(on+
	Rule%Space%						Role%Space%
	Legend:++ Class%	Existen(al+Dependency+	Specializa(on+

http://www.geneve.ch/legislation/rsg/f/s/rsg_B4_40.html

http://www.admin.ch/ch/f/rs/431_01/index.html

Today Geneva Law K

[START_REF] Rolland | A Multi-model Vew of Process Modelling[END_REF].06 is abrogated and replaced by the new law K 4 20.02 http://www.geneve.ch/legislation/rsg/f/rsg_k4_20p02.html


Jean-Michel André

Philippe Jonnard

Ultra-short and ultra-intense x-ray free-electron laser single pulse in one-dimensional photonic crystals

Keywords: X-ray free electron laser, one-dimensional photonic crystal, non linear processes

The propagation within a one-dimensional photonic crystal of a single ultrashort and ultra-intense pulse delivered by an x-ray free-electron laser is analysed with the framework of the time-dependent coupled-wave theory in non-linear media. It is shown that the reflection and the transmission of an ultra-short pulse present a transient period conditioned by the extinction length and also the thickness of the structure for transmission. For ultra-intense pulses, non-linear effects are expected: they could give rise to numerous phenomena, bi-stability, self-induced transparency, gap solitons, switching, …, which have been previously evidenced in the optical domain.

Introduction

The purpose of this work is to analyse different aspects of the time-domain propagation of an ultra-intense and/or ultra-short single pulse delivered by an x-ray free-electron laser (X-FEL) within a Bragg structure. In the hard x-ray domain, the Bragg structures firstly implemented were the natural crystals and the steady-state (time independent) diffraction by crystals was studied in details within the framework of the so-called dynamical theory of diffraction [START_REF] Authier | Dynamical Theory of X-Ray Diffraction International Union of Crystallography Monographs on Crystallography[END_REF]. Since the seventies, synthetic Bragg crystals based on periodic multilayer stacks, sometimes called multilayer interferential mirrors (MIMs) have been developed for the soft-x-ray domain for which natural crystals were unusable; the time-independent diffraction by these multilayer structure have also been extensively studied [START_REF] Pardo | [END_REF].

In this paper, we consider a MIM but some conclusions could be drawn in a similar way to the case of a natural crystal diffracting in the Bragg (reflection) or Laue (transmission) geometry where only one dimension is involved. Thus, hereafter we use the generic term, one-dimensional photonic crystal (1D-PC) to refer to the two types of Bragg structures crystal or MIM. The term distributed-feedback structure (DFB) has also been commonly used in the literature to refer to this kind of structure. Let us outline that the terminologies PC and DFB are rarely used for the x-ray range but are more common in the long wavelength domain (optical range, …). Indeed most of the effects that we describe in this paper for the x-ray domain can be found in the whole electromagnetic spectrum and in particular the non-linear (NL) processes have been studied in details in the optical domain mainly in the context of optics of fibers and waveguide Bragg gratings, so that a large part of Section 4 is borrowed from the numerous studies published on these topics. The objective of the present work is mainly to inform the community of X-FEL users about some linear and NL effects that could be evidenced with X-FEL radiation pulses.

The X-FEL facilities are now able to deliver single ultra-intense and ultra-short pulses of coherent radiation; considering for instance FERMI facility, the photon beam parameters for FEL1 are 100 J per pulse with an estimated pulse length (FWHM) less than 150 fs. Ultra-short means that the duration of the pulse is less that the relaxation times of the media, ultra-intense means that non-linear optical effects are expected and finally coherence means that the photons in a pulse have fixed phase relationships forming a single mode. All these unique features (high brightness, short time duration, temporal and spatial coherence) open the way to the observation of coherent non-linear (NL) processes. The induced polarization P of the materials is given as a power-series expansion of the incident electric field E, in cgs units, 𝑷 = 4𝜋 ∑ 𝜒 (𝑖) 𝒊≥𝟏 |𝑬| 𝑖-1 𝑬 (1) 𝜒 (𝑖) being the susceptibility of the i th order. Generally, the even order terms are null for any centro-symmetric system so that the lowest order NL processes is of the third order. One among them, the most famous one is the Kerr effect for which the refractive index change is proportional to the square of the applied electric field or electric field intensity. Recently x-ray reflectivity enhancement in titanium has been reported [START_REF] Bencivenga | [END_REF]) by a team using ultrafast XUV radiation at the TIMEX endstation of the EIS beamline of FERMI X-FEL facility; the experimental results have been interpreted as a dependence of the plasma frequency with respect to the energy density E ̅ ~|𝑬| 2 which leads in the framework of the Drude refractive index model to variation of reflection upon the deposited energy density. The dependence of the refractive index upon E ̅ can be regarded has a special case of Kerr effect known as refractive index intensity dependence which can give rise to numerous NL effects observed and expected especially in the low energy domain of the electromagnetic spectrum: self-focusing, selfphase modulation, spatial solitons [START_REF] New | Introduction to Nonlinear Optics Cambridge[END_REF], NL surface polaritons [START_REF] Leung | [END_REF]. Intensity-dependent refractive index is also at the origin, in Bragg structure, of various effects such as gap solitons [START_REF] Chen | [END_REF]Mills & Trullinger, 1987;[START_REF] Martijn De Sterke | Progress in Optics[END_REF] connected to self-induced transparency (Aceves & Wabnitz, 1989), pulse compression in Bragg optical fiber [START_REF] Winful | [END_REF], bi-stability in NL distributed feedback structure (Winful et al., 1979).

Time-dependent diffraction by Bragg structure have comparatively received poor attention: Chukhovskii and Förster have considered time-dependent diffraction by crystal (Chukhovskii & Förster, 1995) and more recently their approach has been extended to one-dimensional photonic crystal (André & Jonnard, 2015). Let us mention that the response of Bragg structures, MIMs (Ksenzov et al., 2008[START_REF] Ksenzov | [END_REF]Bushuev & Samoylova, 2011) and crystals (Shastri et al., 2001;Bushuev, 2008) to X-FEL sources, have been treated by means of methods implemented for the frequency domain.

We lead our investigation in the framework of the time-dependent coupled-wave, TDCW, analysis extended to NL materials. In Section 2 we establish the system of coupled-wave equations taking into account the NL response of the media. In Section 3 we consider the low-intensity regime, LIR, where NL behaviour can be neglected; we show that the indicial response of a MIM displays a transient period determined by the extinction length of the Bragg structure in terms of reflection and by the extinction length combined with the thickness of the structure in terms of transmission. In Section 4 we consider the high-intensity regime, HIR, with NL effects first in the steadystate case where bi-stability can be envisaged, then in the time-dependent case where we consider the possibility to observe Bragg solitons associated with a kind of selfinduced transparency of the Bragg structure.

2.Time-dependent coupled-wave theory in non-linear 1D-PC

We consider the propagation of an ultra-short and ultra-intense (generally coherent) single pulse such as the one delivered by an X-FEL in a 1D-PC as shown in Figure 1. The figure also gives the geometry of the problem and some notations. We consider a periodic stack of N bilayers. The bilayer is made up of a material a with dielectric susceptibility 𝜒 𝑎 and material b with dielectric susceptibility 𝜒 𝑏 with layer thickness da =  d and db = (1 -)d respectively. The incoming radiation with a wave vector k = (𝑘 𝑥 , 𝑘 𝑧 ) in the plane (x, z) strikes the multilayer structure under a glancing angle . A Cartesian orthogonal reference frame (𝐱 ̂, 𝐲 ̂, 𝐳 ̂) is used. L is the total thickness of the stack equal to Nd. The reciprocal vector 𝒈 = 2 π 𝑑 𝐙 ̂ is orthogonal to the stratification planes. The electric field 𝑬(𝑥, 𝑧, 𝑡) of the X-FEL single pulse is modelled by a quickly varying carrier with frequency   𝐸 0 (𝑧, 𝑡) and we write it as follows (assuming a s-polarization case): 𝑬(𝑥, 𝑧, 𝑡) = 𝐸 0 (𝑧, 𝑡) 𝑒 𝑖 (𝑘 𝑥 𝑥 -𝜔 𝑡) 𝒚 ̂

(2) We assume that the polarization of the media follows instantly the change of the electric field but that the media have a NL behaviour; more precisely we consider that the medium is subject to the special case of optical Kerr effect for which the refractive index is intensity-dependent. Consequently we write the polarisation 𝑷 as follows:

𝑷(𝑥, 𝑧, 𝑡) = 𝜒(𝑧, 𝐸 0 (𝑧, 𝑡) ) 𝑬(𝑧, 𝑡) = ( 𝜒 (1) (𝑧) + 𝜒 (3) (𝑧) |𝐸 0 (𝑧, 𝑡)| 2 ) 𝐸 0 (𝑧, 𝑡) 𝑒 𝑖 (𝑘 𝑥 𝑥 -𝜔 𝑡) 𝒚 ̂

(3) that corresponds to a third order NL. The susceptibilities 𝜒 (𝑛) (𝑧) do not depend on time while the total susceptibility 𝜒 depends on time through 𝐸 0 (𝑧, 𝑡), but they vary periodically along the z direction. Consequently they can be expanded in Fourier series:

𝜒 (𝑛) (𝑧) = 𝜒̅ (𝑛) + ∑ Δ𝜒 (𝑛) 𝑢 𝑝 𝑒 𝑖 𝑝𝑔 𝑧 +∞ 𝑝=-∞ (4a) with 𝜒̅ (𝑛) = 𝜒 𝑎 (𝑛) 𝛾 + 𝜒 𝑏 (𝑛) (1 -𝛾) (4b) Δ𝜒 (𝑛) = 𝜒 𝑎 (𝑛) -𝜒 𝑏 (𝑛) ≪ 𝜒̅ (𝑛) (4c) 𝑢 𝑝 = 𝛾 sinc(𝜋𝑝𝛾) 𝑒 -𝑖𝜋𝑝𝛾 (4d) 𝑔 = 2π 𝑑 (
4e) In our study we restrict to the Kerr NL and consequently we keep only the terms 𝜒 ̅ (1) , 𝜒 ̅ (3) , Δ𝜒 (1) and Δ𝜒 (3) .

k q z x a b
In the 1D-PC, the electric field envelope can be written as the superposition of two contra-propagating waves along the z-axis: 𝐸 0 (𝑧, 𝑡) = 𝐹(𝑧, 𝑡) 𝑒 + 𝑖 𝜅 𝑧 + 𝐵 (𝑧, 𝑡) 𝑒 -𝑖 𝜅 𝑧 (5a) with 𝜅 = 𝑘 𝑠𝑖𝑛𝜃 (5b) and using the following auxiliary amplitude terms

ℇ + (𝑧, 𝑡) = 𝐹(𝑧, 𝑡) exp [-𝑖 ( 𝑝 𝑔 2 -𝜅) 𝑧 ]
(6a) and

ℇ -(𝑧, 𝑡) = 𝐵(𝑧, 𝑡) exp [+𝑖 ( 𝑝 𝑔 2 -𝜅) 𝑧 ]
(6b) In the framework of the two-wave theory where only the 0 th order and the pth Fourier term are strongly coupled (i.e. in the vicinity of the pth Bragg resonance), it can be shown that in the slow-varying approximation both in space and in time, that the

column amplitude vector ℇ ̅ (𝑧, 𝑡) = ( ℇ + (𝑧, 𝑡) ℇ -(𝑧, 𝑡) ) and its complex conjugate ℇ ̅ * (𝑧, 𝑡) = ( ℇ + * (𝑧, 𝑡) ℇ - * (𝑧, 𝑡)
) obey the following system of partial differential equations (PDEs) forming the so-called non-linear coupled mode equations with loss or gain:

𝜕 𝑧 ℇ ̅ (𝑧, 𝑡) = 𝒯 ̅ 𝜕 𝑡 ℇ ̅ (𝑧, 𝑡) + 𝑖 ℳ ̅ ℇ ̅ (𝑧, 𝑡) + 𝒩 ̅̅̅̅ ℇ ̅ (𝑧, 𝑡) + 𝒩 𝑐 ̅̅̅̅ ℇ ̅ * (𝑧, 𝑡) (7
) where ℳ ̅ is the propagation matrix in space given by

ℳ ̅ = ( -𝛼 𝐾 + 𝐾 -𝛼 ) (8a) with 𝛼 = 𝑝 𝑔 2 + 𝑘 2 𝜅 2𝜋 𝜒̅ (1) -𝜅 (8b) 𝐾 + = - 𝑘 2 𝜅 2𝜋 Δ𝜒 (1) 𝑢 𝑝 (8c) 𝐾 -= 𝑘 2 𝜅 2𝜋 Δ𝜒 (1) 𝑢 -𝑝 (8d) 𝒯 ̅ is the propagation matrix in time: 𝒯 ̅ = ( - 1 𝑐 𝑠𝑖𝑛𝜃 0 0 1 𝑐 𝑠𝑖𝑛𝜃 ) ( 9a
)
𝒩 ̅̅̅̅ is the NL matrix corresponding to the average NL term:

𝒩 ̅̅̅̅ = ( -Γ 𝑎 Λ + (𝑧, 𝑡, 𝑆, 𝑋) Γ 𝑝∆ Λ -(𝑧, 𝑡, 𝑆, 𝑋 ) -Γ 𝑝∆ Λ + (𝑧, 𝑡, 𝑆, 𝑋 ) Γ 𝑎 Λ -(𝑧, 𝑡, 𝑆, 𝑋 ) ) (9b) with Λ + (𝑧, 𝑡, 𝑆, 𝑋) = 𝑆 |ℇ + (𝑧, 𝑡)| 2 + 2 𝑋|ℇ -(𝑧, 𝑡)| 2 (9c) Λ -(𝑧, 𝑡, 𝑆, 𝑋 ) = 𝑆|ℇ -(𝑧, 𝑡)| 2 + 2 𝑋 |ℇ + (𝑧, 𝑡)| 2
(9d) The quantity S corresponds to the self-phase, SP, modulation term while the quantity X is for the cross-phase, CP, modulation; moreover the term Γ 𝑎 corresponds to the average NL term while the term Γ 𝑝∆ is associated with the pth Fourier component of the NL term of the dielectric susceptibility Δ𝜒 (3) 𝑢 𝑝 .

The matrix 𝒩 𝑐 ̅̅̅̅ affecting ℇ ̅ * (𝑧, 𝑡) is:

𝒩 𝑐 ̅̅̅̅ = Γ 𝑚 ( ℇ -(𝑧, 𝑡) 2 0 0 -ℇ + (𝑧, 𝑡) 2 ) (10)
The coefficient Γ 𝑚 is also related to the 3 rd order dielectric susceptibility. It contributes to the wave mixing process. If the polarization is stationary according to our assumption, one can show that in our geometry:

S = X = 1 ; Γ 𝑎 = Γ 𝑚 ∝ 1 + 4𝜋 𝜒 ̅ (3) (11)
The loss (or gain) has been taken into account in the above theory by introducing an imaginary part in the dielectric constants. Generally in the x-ray domain, materials are absorbing; nevertheless it has been reported stimulated emission in solid (silicon and magnesium oxide) under X-FEL excitation (Beye et al., 2013;Yoneda et al., 2015;[START_REF] Jonnard | EUV stimulated emission from MgO pumped by FEL pulses[END_REF], so that the case of medium with gain (lasing medium) needs to be considered too. Let us mention that X-UV lasing in 1D-PC with pumping by X-FEL, forming a so-called distributed feedback laser, has been recently examined (André et al., 2014).

Low-intensity regime: transient response

In this section we assume that the intensity of the X-FEL radiation is low enough so that no NL behaviour occurs: it is the low-intensity regime, LIR. We will see that, even if the response of the 1D-PC to the pulse is then purely linear, notable effects occur when the time width of the pulse is very short. First we will consider the so-called impulse and indicial response of the 1D-PC in terms of reflection and transmission; the impulse response corresponds to an incident Dirac- pulse while the indicial response corresponds to an incident step-like Heaviside-Θ signal in terms of time dependence. The 1D-PC is assumed to work at the Bragg resonance.

Under LIR, the system of PDEs, Eq.( 7), is linear and can be reduced by means of the following characteristic coordinates 𝑣, 𝑤:

𝑣 = 1 2 (𝑐 𝑡 𝑠𝑖𝑛𝜃 -𝑧) (12a) 𝑤 = 1 2 (𝑐 𝑡 𝑠𝑖𝑛𝜃 + 𝑧)
(12b) and reduced field amplitudes defined by: ℇ + (𝑧, 𝑡) = 𝑒𝑥𝑝(-𝑖 𝛼 𝑐 𝑡 𝑠𝑖𝑛𝜃 ) 𝑓 ̃(𝑣, 𝑤) (13a) ℇ -(𝑧, 𝑡) = 𝑒𝑥𝑝(-𝑖 𝛼 𝑐 𝑡 𝑠𝑖𝑛𝜃) 𝑏 ̃(𝑣, 𝑤) (13b) to the following hyperbolic second order PDE ℒ (𝑣,𝑤) [𝑓 ̃(𝑣, 𝑤) ; 𝑏 ̃(𝑣, 𝑤)] = 0 (14) where ℒ (𝑣,𝑤) is the differential operator defined by:

ℒ (𝑣,𝑤) = ( 𝜕 2 𝜕𝑣𝜕𝑤 + 𝜋 2 Λ 2 )
(15) with Λ 2 , the quantity related to the coupling constants 𝐾 + , 𝐾 -by:

Λ 2 = - 𝜋 2 𝐾 + 𝐾 - (16)
The quantity Λ is the extinction length of the dynamical theory of diffraction. For given boundary conditions, the PDE given by Eq.( 14) can be solved by implementing Riemann's method [START_REF] Courant | Partial Differential Equations[END_REF]. This method requires an integration contour in the characteristic coordinate plane shown in Figure 2.

Figure 2: 1D-PC geometry in the characteristic coordinate reference frame (v,w); the front surface is given by w = v (line PQ) while the rear surface is given by w = v + L (line RT).

Application of Riemann's method leads to write the backward propagating (reflected) field as follows:

𝑏 ̃(𝑣, 𝑣) = -𝑖 𝐾 -Λ 𝜋 ∫ 𝐽 1 [ 2 𝜋 Λ (𝑣 -𝑣 ′ )] (𝑣 -𝑣 ′ ) 𝑓 ̃(𝑣 ′ , 𝑣 ′ ) 𝑄 𝑃 𝑑𝑣 ′
(17) where 𝑓 ̃(𝑣 ′ , 𝑣 ′ ) corresponds to the incoming (forward propagating) wave at the front surface (z =0). 𝐽 1 is the Bessel function of the first kind. From Eq.( 17) one can deduce the

R P T Q w v L front rear
impulse in terms of reflection coefficient. The impulse incident reduced field amplitude 𝑓 ̃(𝑣, 𝑣) δ can be written as:

𝑓 ̃(𝑣, 𝑣) δ = 𝑒𝑥𝑝(+ 𝑖 2 𝛼 𝑣 ) √2𝜋 𝑠𝑖𝑛𝜃 2 𝛿 [ 𝑥 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛𝜃 2 -𝑣 ]
(18) where 𝛿 stands for the Dirac peak. Inserting Eq.( 18) in Eq.( 17) and performing the integration gives for the reduced diffracted field 𝑏 ̃(𝑣, 𝑣) δ under the incidence of a Dirac pulse:

𝑏 ̃(𝑣, 𝑣) δ = -𝑖 𝑠𝑖𝑛𝜃 𝐾 -Λ 𝜋 𝑒𝑥𝑝(+ 𝑖 𝒶 𝑥 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛𝜃 ) √2𝜋 𝐽 1 [ 𝜋 Λ 𝑠𝑖𝑛𝜃(𝑐 𝑡 -𝑥 𝑐𝑜𝑠 𝜃)] 𝑠𝑖𝑛𝜃(𝑐 𝑡 -𝑥 𝑐𝑜𝑠 𝜃 ) (19
) that is, for the diffracted field: Following a similar calculation it is possible to show that forward propagating transmitted field 𝑓 ̃(𝑣, 𝑤) ] 𝑇 is given by:

ℇ -(𝑧 = 0, 𝑇) δ = 𝑖 𝑠𝑖𝑛𝜃 𝐾 -Λ 𝜋√2𝜋 𝐽 1 [𝜁(𝑇)] 𝜁(𝑇) Θ(𝑇) ; 𝜁(𝑇) = 𝜋 Λ 𝑠𝑖𝑛𝜃 𝑐 𝑇 (
𝑓 ̃(𝑣, 𝑤) ] 𝑇 ≈ - 2 𝜋 2 𝐿 Λ 2 ∫ 𝑓 ̃(𝑣 ′ , 𝑣 ′ ) 𝐽 1 [ 2 𝜋 Λ √(𝑣 + 𝐿 -𝑣 ′ )(𝑣 -𝑣 ′ )] 2 𝜋 Λ √(𝑣 + 𝐿 -𝑣 ′ )(𝑣 -𝑣 ′ ) 𝑄 𝑃 𝑑𝑣 ′
(23) To determine the impulse response in terms of transmission, we insert in Eq.( 22) the expression of the incident pulse given by Eq.( 18) as for the reflection case and we perform the integration as for the reflection geometry. It results that the impulse response in terms of transmission 𝑇 ̂𝛿(𝑡) is ℇ + (𝑧 = 𝐿, 𝑇) δ :

ℇ + (𝑧 = 𝐿, 𝑇) δ = 𝜋 2 𝐿 𝑠𝑖𝑛𝜃 Λ 2 𝐽 1 [𝜉(𝑇)] 𝜉(𝑇) Θ(𝑇); 𝜉(𝑇) = 𝜋 Λ √ 𝑐 𝑇 ( 2 𝐿 𝑠𝑖𝑛𝜃 + 𝑐 𝑇 𝑠𝑖𝑛𝜃 2 ) (
Ζ ̂𝛿(𝑠) being the transfer function that is the Laplace transform of the impulse response Ζ ̂𝛿(𝑡)(≡ 𝑅 ̂𝛿(𝑡) or 𝑇 ̂𝛿(𝑡)). Ζ ̂𝛿(𝑡) can be determined from Ζ ̂Θ(𝑡) by Strejc's method [START_REF] De Larminat | Analysis and Control of Linear Systems London[END_REF]. By performing an inverse Laplace transform of Eq.( 26), one determines the temporal response (André & Jonnard, 2015). We now illustrate the previous theory in the case of a MIM consisting in a stack of N = 7 Ti/Si bilayers; the period d is equal to 70 nm and the  ratio is equal to 0.5. The photon energy of the incident XFEL radiation is 20 eV (=62 nm) diffracted at  = 60°. The choice of the Ti/Si system is not governed by the objective of optimizing the reflectance of the MIM but by the fact that it has been shown, see [START_REF] Bencivenga | [END_REF], that titanium has a NL behaviour at the photon energy of interest. The temporal profile XFEL pulse is modelled by a sine-squared function, that is the envelope of the incident pulse varies as sin 2 (𝜋 𝑡 𝜏 ) in the time interval [0, 𝜏] and 0 outside.

The NL dielectric constant (square of the refractive index) of titanium is assumed to be given by Drude's formula modified to account for the intensity dependence [START_REF] Bencivenga | [END_REF]:

𝜖 𝑁𝐿 𝑇𝑖 = 1 - 𝜔 𝑝0 2 ( 1 + 𝐴 E ̅ ) 𝜔 2 + 𝑖 𝜔 𝛾̅ ≈ 𝜖 𝐿 𝑇𝑖 + ∆𝜖 𝑇𝑖 E ̅ (27a) 𝜖 𝐿 𝑇𝑖 = 1 - 𝜔 𝑝0 2 𝜔 2 + 𝑖 𝜔 𝛾 ̅ ≈ 1 - 𝜔 𝑝0 2 𝜔 2 + 𝑖 𝛾̅ 𝜔 𝑝0 2 𝜔 3 ; ∆𝜖 𝑇𝑖 = -𝐴 𝜔 𝑝0 2
𝜔 2 (27b) 𝜔 𝑝0 is the plasma frequency at zero energy density (E ̅ = 0) equal to 17.7 eV, A being an empirical constant equal to 7.5 10 12 m 3 /J. Following [START_REF] Bencivenga | [END_REF] the damping term 𝛾̅ is taken to be 0.1 𝜔 𝑝0 as for aluminum (Ujihara, 1972) since a reference value is not available in the literature for Ti. Numerically at 20 eV, one has:

𝜖 𝐿 𝑇𝑖 = 0. 217 + 𝑖 0.069 = (0.474 + 𝑖 0.087) 2 ; ∆𝜖 𝑇𝑖 = -5.87 10 12 For silicon, from [START_REF] Palik | Handbook of Optical Constants of Solids Boston[END_REF], at 20 eV:

𝜖 𝐿 𝑆𝑖 = (0. 567 + 𝑖 0.083) 2

Figure 4 displays the indicial response in terms of reflectance for the Ti/Si MIM under consideration, for different glancing angles close to the Bragg angle ( = 60°) obtained by solving Eq.( 7), (André & Jonnard, 2015). It appears that the transient time is around 6.5 fs to reach 90% of the stationary indicial response. The temporal response in terms of peak reflectance (that is for resonance at Bragg angle  = 60°) to a single pulse for several time widths 𝜏 (10, 50, 100 fs) is shown in Figure 5. It can be seen that for the shortest incident pulses, the reflectance does not have enough time to reach its steadystate value. For the longest pulses the response duration is quite symmetrical and lasts about the time of the pulse. For the shortest pulse, the response is asymmetrical and lasts at least twice the time of the pulse.

High-intensity regime HIR: solitons and other non-linear effects

In this part, our purpose is to highlight some NL effects occurring in the highintensity regime where Kerr non-linearity takes place. If one starts from the general CW equation Eq.( 7) to deal with this regime, one has to face with a lot of mathematical problems very difficult to handle, leaving one with only numerical techniques generally computationally intensive and that do not provide the necessary insight into the physics. So we choose to simplify the problem by keeping in the Eq.( 7) only the terms which describe the basic phenomena. In this spirit we restrict our study to the following cases:

-loss and/or gain in the medium are discarded; -the 1D-PC is tuned at the pth Bragg resonance; -wave mixing that is the effect of the matrix 𝒩 𝑐 ̅̅̅̅ in Eq.( 7) is neglected.

NL stationary case

From the study of the time-dependent linear case examined in section 3, one can expect for pulses with time width very large with respect to the characteristic time associated to the extinction length, that the propagation of the pulse in NL media can be satisfactorily described by the time independent (stationary) form of the NL PDEs. It can be shown that in these conditions, the system of hyperbolic PDEs deduced from Eq.( 7) with the above-mentioned assumptions has two conserved quantities depending on ℇ + (𝑧) and ℇ -(𝑧):

-the intensity flow:

|ℇ + (𝑧)| 2 -|ℇ -(𝑧)| 2 = 𝐼 𝑖𝑛 -𝐼 𝑟𝑒𝑓 = 𝐼 𝑜𝑢𝑡 ≡ 𝑇 (see Appendix);
-the real-valued Hamiltonian (non-explicited): and since the degree of freedom (equal to 2) of the PDE system is equal to the number of conserved quantities, by virtue of the Liouville-Arnould theorem, this PDE system is exactly integrable. With the "standard" boundary conditions,

|ℇ + (0)| 2 = 𝐼 𝑖𝑛 , |ℇ -(0)| 2 = 𝐼 𝑟𝑒𝑓 , |ℇ + (𝐿)| 2 = 𝐼 𝑜𝑢𝑡 , |ℇ -(𝐿)| 2 = 0
(28) the integration yields relationships between the transmittance (reflectance) and the input intensity 𝐼 𝑖𝑛 in terms of Jacobi elliptic functions (see Appendix). From the expression of the transmittance deduced from Eq.(A.7), it appears that the 1D-PC can present a bi-stability. This is illustrated by the figure 6 obtained from Eq.(A.12). It shows a typical S-shape response at Bragg resonance in terms of normalized transmittance versus 𝐼 𝑖𝑛 for the value of κ ̅ 𝐿 equal to 2; the negative slope region (between 4.75 and 4.85) is unstable. The Bragg reflectivity associated with the linear component of the dielectric constant is erased until a transparent state (reflectivity close to zero) is reached at a point T as shown in Figure 6. For low value of κ ̅ 𝐿, bistability cannot be achieved by lack of feedback but for larger values of κ ̅ 𝐿, multiple-switching phenomena appear (Winful et al., 1979). NL 1D-PC can give rise according, to the conditions, to multi-stabilities as well as optical limiting (see section 4.2.2 of the paper) and even may undergo chaotic behavior.

NL time-dependent case

Solitons

The physics of the 1D-PC in the linear case is governed by the occurrence of two branches of the dispersion curve in the frequency domain, experiencing anti-crossing owing to the wave coupling. In the contra-propagating case relevant for our problem, the anti-crossing gives rise to a forbidden energy gap. In the LIR considered in the previous section, the Bragg frequency 𝜔 𝑐 lies within the band gap where running waves are forbidden. At HIR we are dealing with in this part, the dielectric constant changes proportionally to the electric field intensity and the branches shift down/up in energy (a kind of blue/red shift of the band gap according to positive/negative NL) so that the frequency 𝜔 𝑐 no longer falls within the forbidden gap but shifts towards a region with allowed traveling waves. This mechanism is sketched in Figure 7. The gap soliton and the optical limiting and switching examined hereafter are related to this mechanism. The dynamics of the system is then governed by of the system of NL PDEs Eq.( 7). A remarkable property of this system is that the entire family of localized solutions can be built analytically forming the "soliton-like" solutions. The adequacy of the term "soliton" is discussed in many specialized works, see for instance [START_REF] Wazwaz | Partial Differential Equations and Solitary Waves Theory Dordrecht[END_REF]; here we merely use the term soliton to mean localized solutions not in the strict mathematical sense of integrability. We first consider the case where SP is neglected (𝑆 = 0). Although one could imagine cases for which 𝑋 = 0, it seems that there are no realistic geometries in which 𝑆 = 0 and 𝑋 = 0 so that the model in these conditions is not relevant directly in our problem. Nevertheless the latter case paves the way to a general solution as we will see hereafter. It has been recognized that the system given by Eq.( 6) with 𝑆 = 0 and 𝑋 ≠ 0 reduces to the massive Thirring model MTM which is integrable and exhibits soliton-like solutions (Martun de [START_REF] Martijn De Sterke | Progress in Optics[END_REF]. The more general solution when the SM term is not neglected (generally X = S) can be built from the soliton-like solutions of the MTM [START_REF] Orfanidis | [END_REF].

We briefly present the mathematical way to the more general (in the sense 𝑆 ≠ 0, 𝑋 ≠ 0 and Im(ε) ≠ 0 but wave mixing terms remain null) soliton-like solution of Eq.( 6). First the column vector ℇ ̅ (𝑧, 𝑡) without SM can be regarded as a Dirac spinor 𝜓 and Eq.( 6) is formally the same than the MTM equation: 𝑖 𝛾 𝜇 𝜕 𝜇 𝜓 = 𝑚 𝜓 -𝑔 𝐽 𝜇 𝜓 ; 𝐽 𝜇 = 𝜓 ̅ 𝛾 𝜇 𝜓 (29) 𝛾 𝜇 being the gamma matrices formed from the Pauli matrices 𝜎 𝜇 . The MTM equation can be reduced to the Sine-Gordon one which is integrable; finally the soliton-like solutions of the MTM are (Aceves & Wabnitz, 1989): 30) represent a two parameter family of solutions of the system Eq.( 7) with Δ and 𝑄 as two free parameters which are respectively obtained from the amplitude and phase of the eigenvalue of the scattering problem connected to MTM; 𝑣 is a dimensionless quantity (|𝑣| < 1) given by: 𝑣 = 1 -Δ 4 1 + Δ 4 (31) that determines the velocity of the soliton. The parameter 𝑄 (0 < 𝑄 < π) gives the location of the soliton in the band-gap and determines its width: 𝑄 = 𝜋 2 corresponds to a soliton with center frequency in the middle of the gap reducing to the slow Bragg soliton of [START_REF] Leung | [END_REF]; the limit 𝑄 → 0 corresponds to soliton-like solutions at the top of the gap reducing to the NL Schrödinger one-soliton solution of the NL PDE (Sipe & Winful, 1988) whereas the limit 𝑄 → π correspond to plane wave solutions. The generalized (i.e. the SM is not neglected) soliton-like solution ℇ ̅ (𝑧, 𝑡) is built from the soliton solutions 𝜓 of the MTM given by Eqs(18):

𝜓 = ( 𝜓 1 𝜓 2 ) (30a) with 𝜓 1 = ± (± κ ̅ 2 Γ 𝑎 ) 1/2 1 Δ sin 𝑄 exp[±𝑖 κ ̅ 𝜁 cos𝑄] sech[ κ ̅ 𝜉 sin 𝑄 ∓ 𝑖 𝑄/2)] (30b)
ℇ ̅ = 𝒜 𝜓 exp (𝑖 ϑ(𝜉)) (32a) with

𝒜 = 1 √1 + 𝑅 + + 𝑅 - (32b) 𝑅 ± = 𝑆 4 X (1 ± 𝑣) 2 1 -𝑣 2 (32c) exp(𝑖 ϑ(𝜉)) = (- 𝑒 2 κ ̅ 𝜉 sin 𝑄 + 𝑒 ∓𝑖 𝑄 𝑒 2 κ ̅ 𝜉 sin 𝑄 + 𝑒 ± 𝑖 𝑄 ) 𝑠 (32d) 𝑠 = 𝑅 + -𝑅 - 𝒜 2 (32e)
The solitons given by Eq.( 32) are called self-transparency solitons. We note that the NL interaction effects appear in Eq.( 32) mainly through the quantities 𝑅 ± via the ratio S 4 X . Figure 8 shows the time evolution of the shape of a slow Bragg soliton formed in the 1D-PC. These soliton-like solutions can be pictured by an effective particle (Martijn de Sterke & Sipe, 1989) with a charge q and a momentum p:

𝐪 = ∫ (ℇ * + ℇ + + ℇ * -ℇ -)𝑑𝑧 +∞ -∞ (33) 𝐩 = -𝑖 ∫ (ℇ * + 𝜕 𝑧 ℇ + + ℇ * -𝜕 𝑧 ℇ -)𝑑𝑧 +∞ -∞
(34) Using Eq.( 7), it comes:

𝜕 𝑧 𝐪 = 2 Im(K) 𝐪 (35) and 𝜕 𝑡 𝐩 = 2 Im(K) 𝐩 (36) It appears that in absence of loss or gain, both momentum and charge will be conserved quantities, otherwise they vary exponentially with time. From these equations it can be shown that the important parameter Q satisfies:

𝜕 𝑡 Q = 2 Im(K) Q (37) From Eq.(37) it follows that the width of the soliton which is determined by Q, depends exponentially on time while its velocity 𝑣 is still undetermined. Detailed analysis concerning the behaviour of gap solitons in 1D-PC made up with NL media with loss and gain can be found in (Martijn de Sterke & Sipe, 1991).

As mentioned previously, the velocity 𝑣 can be regarded as a free parameter; in other words by choosing the conditions appropriately, it should be possible to get very slow solitons, and even stationary solitons, up to solitons moving at about the speed of light. Low velocity (𝑣 ≈ 0) means that energy will be transported very slowly at the manner of the self-induced transparency, SIT, observed in resonant pulse propagation in atoms [START_REF] Mccall | [END_REF]; one can then speak about Bragg SIT. Let us mention the possible existence of multi-soliton-like solutions of the time-dependent NL problem.

We now turn to the study and the conditions of a possible soliton under X-FEL pulse irradiation. First let us outline that the problem of finding the right features of the incident pulse and the appropriate conditions to give rise to a given soliton is far from trivial. For a finite length MIM with 𝒩 unit cells (bilayers), Martijn de Sterke and Sipe (Martijn de Sterke & Sipe, 1989) have given the following criteria to satisfy for the occurrence of a gap soliton:

𝒩 2 Δ𝑛 𝐿 Δ𝑛 𝑁𝐿 ≥ 1 (38) where Δ𝑛 𝐿 is the magnitude of the varying term of the refractive index and Δ𝑛 𝑁𝐿 the maximum NL change in the refractive index. Let us also emphasize that the problem of the soliton stability is a hard task [START_REF] Hwang | [END_REF] out of the domain of this paper.

Optical switching

The dynamics of NL 1D-PC offers the possibility of a large variety of mechanisms very attractive for practical applications: optical limiting and switching, pulse generation or reshaping, … In the following we consider the optical limiting and switching OLS, in further details because of its potential interest in x-ray technology. OLS results from a dynamical change of the NL 1D-PC response to an incident field intensity. To understand the principle of the OLS mechanism, it is convenient to consider the frequency domain. In this one, the width of the forbidden gap (Bragg domain), GW, is proportional to the difference of dielectric constants ∆𝜖 𝒂𝒃 between material a and b. Let us assume that one the material, say a, presents a NL Kerr-like behavior with positive coefficient, i.e. its dielectric constant increases with the incident field intensity E ̅ while the other material as a linear behaviour. Then as E ̅ increases, GW increases too, so that a dynamical widening of the gap occurs. If the carrier frequency of the incident radiation is tuned at the band edge, then a larger domain of frequency of the incident pulse will fall inside the forbidden gap until forbidding the complete propagation of the pulse. This mechanism described in (Scalora et al., 1994) forms the basis of an intensity-driven OLS.

For a NL behavior with negative coefficient, switching can occur from the linear material in the pump/probe scheme sketched in Figure 9: a strong pump pulse of carrier frequency 𝜔 𝒑𝒑 far below the gap and of intensity E ̅ 𝑝𝑝 inducing the non-linearity and a probe pulse of carrier frequency 𝜔 𝑝𝑟 inside the gap but near the band edge and of intensity E ̅ 𝑝𝑟 ≪ E ̅ 𝑝𝑝 are incident on the 1D-PC. Initially (i.e. without pumping) the probe pulse is not transmitted. For negative coefficient, under pump beam irradiation, the bandwidth GW decreases and over a certain value of E ̅ 𝑝𝑝 , the probe frequency 𝜔 𝑝𝑟 may exit the gap leading to a transmission of the probe beam. This process, which can be encountered with the negative NL coefficient of Ti in our example, may form the basis of an x-ray switch.

Figure 9: Scheme for an optical switching with a NL 1D-PC with negative coefficient. The NL Kerr effect is induced by the pump beam and can allow the transmission of the probe beam. In absence of pump, there is no transmitted probe; this case is illustrated by the cross on the arrows.

Conclusion

NL 1D-PC

Pump wpp

Probe wpr

Transmitted probe

Ultra-short or ultra-intense x-ray pulses delivered by the X-FEL propagating within 1D-PC opens the way to the observation of a lot of phenomena never encountered in the x-ray regime. In fact none of the phenomena considered in this paper have been evidenced until now and their observation is technically challenging. Considering the LIR, the measurement of the intensity of a reflected single pulse remains tricky. Nevertheless techniques as those implemented at the TIMEX end-station in FERMI X-FEL facility to measure the reflectivity [START_REF] Bencivenga | [END_REF] can be envisaged to test our model.

When NL is incorporated into the 1D-PC at HIR, it has been shown that it becomes possible to control dynamically the propagation of ultra-intense x-ray pulse. NL effects are generally studies both experimentally and theoretically for lossless media; in the x-ray domain and in particular for the soft x-ray range, the consequence of absorption on the NL effects are difficult to forecast in the CW theory framework without intensive numerical computation. For this purpose, calculation implementing transfer matrix method extended to Kerr NL in a way similar to (Li et al., 2015) could be envisaged. From the experimental point of view, to test NL effects, techniques borrowed from the optical domain involving pulse compression with cross-phase modulation should be implemented (Martijn de Sterke, 1992).

Figure 1 :

 1 Figure 1: Sketch of the 1D-PC consisting in a periodic stack of alternating a/b bilayers.

  20) where one has introduced the time delay T measured with respect to the diffracted wave plane: 𝑇 = 𝑐 𝑡 -𝑥 𝑐𝑜𝑠 𝜃 𝑐 (21) 𝑅 ̂𝛿(𝑇) = ℇ -(𝑧 = 0, 𝑇) δ is the impulse response in terms of reflection and also the temporal Green function 𝑔 𝑅 (𝑇) for reflection. For time coherent radiation with timedependent causal distribution Ξ (normalized to unity), the indicial response 𝑅 ̂𝛩(𝑡) in terms of reflection coefficient is given by: 𝑅 ̂𝛩(𝑡) = ∫ |𝑔 𝑅 (𝑇)22) allows one to draw a "universal" curve in terms of peak reflectance for the indicial response 𝑅 ̂𝛩(𝑡) versus the reduced time 𝑡 ̅ = 𝑡 𝜋 sin 𝜃 𝑐 Λ , see Figure 3. It appears that the indicial response for reflection is conditioned by the extinction length Λ and that it presents a transient period whose duration is given by a characteristic transient time 𝑡 𝑐 approximately equal to 2 units of reduced time.

Figure 3 :

 3 Figure 3: Indicial response in terms of peak reflectance versus reduced time for a 1D-PC at the Bragg resonance (Bragg angle). The response is normalized to unity.

  24)For time coherent radiation with time-dependent causal distribution Ξ the indicial response 𝑇 ̂𝛩(𝑡) in terms of transmission coefficient is given from the temporal Green function for transmission 𝑔 𝑇 = 𝑇 ̂𝛿:𝑇 ̂𝛩(𝑡) = ∫ |𝑔 𝑇 (𝑇)| 2 +∞ -∞ Ξ(𝑡 -𝑇) 𝑑𝑇 = ∫ |𝑔 𝑇 (𝑇)| 2 𝑡 0 Ξ(𝑡 -𝑇) 𝑑𝑇 (25)The indicial response in terms of transmission is conditioned both by the extinction length Λ and by the thickness L of the 1D-PC.From the indicial responses of the 1D-PC it is possible to calculate the timedependent reflection and transmission of a short pulse. Let 𝐼(𝑡) be the temporal envelope shape of any incident pulse and O(t) the envelope of the corresponding reflected or transmitted pulse. Then the Laplace transform 𝑂(𝑠) of O(t) is related to the Laplace transform 𝐼(𝑠) of 𝐼(𝑡) by means of the convolution theorem: 𝑂(𝑠) = Ζ ̂𝛿(𝑠) 𝐼(𝑠)

Figure 4 :

 4 Figure 4: Indicial response of the Ti/Si MIM versus time in terms of peak reflectance for different glancing angles  red line,  = 60° (Bragg angle); black line,  = 65°; blue line  = 55°.

Figure 5 :

 5 Figure 5: Reflected pulse by the Ti/Si MIM at Bragg resonance versus time for several time widths  of the incident pulse: blue line  f ; black line 

Figure 6 :

 6 Figure 6: Transmittance versus normalized input intensity exhibiting bi-stability of the NL 1D-PC; a quasi-transparent state appears at the point T.

Figure 7 :

 7 Figure 7: Dispersion curves in the frequency () / Bloch wavenumber (K) domain: (a) LIR; (b) HIR and positive NL with a shift-down in frequency; (c) HIR and negative NL with a shift-up in frequency.

  determined by the relative sign of the linear and NL coupling coefficients. Eqs.(

Figure 8 :

 8 Figure 8: Time evolution of the shape of a slow Bragg soliton in arbitrary units, formed in the 1D-PC in the NL regime.

Appendix

With the assumptions mentioned in the main text, the stationary version of the PDE system given by Eq.( 7) becomes at the pth Bragg resonance:

to the transmitted flux is a conserved quantity (i.e. z independent). Indeed setting ℇ ± (𝑧) = |ℇ ± (𝑧)| 𝑒𝑥𝑝(± 𝜙 ± (𝑧)) and subtituting into Eq.(A.1) gives after separation of the real and imaginary parts:

) where 𝜓(𝑧) = 𝜙 + (𝑧) -𝜙 -(𝑧) at the Bragg resonance. Elimination of 𝜓(𝑧) from Eq.(A.3) and Eq.(A.4) gives the constant of propagation 𝑇. Introducing 𝑇 in (A.1) leads to: 𝜌(𝑍 = 2) → ∞ > 𝜌 0 + > 𝐼 𝑜𝑢𝑡 > 0 > 𝜌 0 -(A.8) In this condition, the integral on the right-hand side of Eq. (A.7) can be found by using the method given in [START_REF] Byrd | Handbook of Elliptic Integrals for Engineers and Scientists Berlin[END_REF]. After some algebra:

𝑛𝑑[𝑢(𝑍)/𝑚] being the inverse of the delta amplitude of the Jacobian elliptic (cnoidal) function 𝑑𝑛[𝑢(𝑍)/𝑚]) [START_REF] Byrd | Handbook of Elliptic Integrals for Engineers and Scientists Berlin[END_REF], where the modulus m is given by:

11) The quantity 𝜌(𝑍 = 0) corresponds to normalized input intensity 𝐼 𝑖𝑛 and from Eq.(A.9) one deduces the relation between 𝐼 𝑜𝑢𝑡 and 𝐼 𝑖𝑛 : 𝐼 𝑜𝑢𝑡 = 2 𝐼 𝑖𝑛 1 + 𝑛𝑑 [2 √( κ ̅ 𝐿) 2 + 𝐼 𝑜𝑢𝑡 2 /𝑚] (A.12) In the NL regime, one recovers the well-known expression for the transmission at Bragg resonance:

𝐼 𝑜𝑢𝑡 = 𝐼 𝑖𝑛 𝑠𝑒𝑐ℎ 2 ( κ ̅ 𝐿) (A.13)


Saqib Amin

Usman Zabit

Tassadaq Hussain

Olivier D Bernal

Hardware Implementation of Metric Algorithms for a Self-Mixing Laser Interferometric Sensor

Keywords: Self-mixing, interferometry, real-time, vibration measurement, signal processing I

Self-mixing interferometry (SMI) has been widely used for sensing of diverse vibration, velocity, displacement, biomedical and flow applications. The simplicity of the SMI configuration enables the design of a low-cost, self-aligned and compact sensor with a small optical component count. SMI occurs when a small portion of emitted coherent optical beam is backscattered by the remote target and re-enters the laser cavity, causing interference. Under SMI, the employed laser diode simultaneously acts as a laser source, coherent detector, as well as micro-interferometer. This simple sensor design configuration, however, comes at a price of complex signal processing algorithms, because the SMI signals have rich characteristics as a function of optical feedback level. The purpose of this research work is to implement two of these algorithms, time-frequency signal processing (TFSP) and Consecutive samples based Unwrapping (CSU) in hardware so that real-time displacement and vibration measurements with nanometric precision can be retrieved from the SM sensor in an embedded, autonomous manner. We implemented and tested both algorithms on a Xilinx ZYBO Zynq 7000 development board using VHDL. Results show that our design of CSU algorithm is capable of operating at 432 MHZ minimum clock frequency and latency of 3 clock cycles. Our CSU design consumes 159 slice registers and 215 LUT's, and draws 0.70 watts of onboard power. While the design of TFSP algorithm is capable of operating at 363.3 MHZ clock frequency and latency of 570k clock cycles. TFSP design consumes 5056 slice registers and 5192 LUT's, and draws 0.8 watts of onboard power.

INTRODUCTION

Optical feedback (OF) interferometry [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF] or Self-mixing (SM) interferometry [START_REF] Taimre | Laser feedback interferometry: A tutorial on the self-mixing effect for coherent sensing[END_REF] has been widely studied and used in many diverse applications involving measurements of displacement [START_REF] Donati | Single Channel Self-Mixing Interferometer Measures Simultaneously Displacement and Tilt and Yaw Angles of a Reflective Target[END_REF], velocity [START_REF] Zhao | Self-Mixing Fiber Ring Laser Velocimeter With Orthogonal-Beam Incident System[END_REF], biomedical signals [START_REF] Donati | Self-mixing Interferometry for biomedical signals sensing[END_REF], embedded sensing [START_REF] Zabit | Design and analysis of an embedded accelerometer coupled selfmixing laser displacement sensor[END_REF], and fluid flow rates [START_REF] Lim | Self-mixing flow sensor using a monolithic VCSEL array with parallel readout[END_REF]. Due to a high resolution of SM techniques involving very inexpensive and simple experimental optical setup, these techniques are being increasingly used for practical applications [START_REF] Demir | Evaluation of self-mixing Interferometry performance in the measurement of ablation depth[END_REF], [START_REF] Milesi | Measurement of Local Chest Wall Displacement by a Custom Self-Mixing Laser Interferometer[END_REF].

A typical SM interferometric displacement sensor is shown in Fig. 1. SM phenomenon, based on laser feedback, occurs when part of the laser beam is reflected back from moving target and re-enters active cavity of the laser diode (LD). This re-entered laser beam is mixed with the emitted light, producing variations in optical output power (OOP) which can be easily picked up through the monitor photo-diode (PD) enclosed in LD package [START_REF] Donati | Developing self-mixing interferometry for instrumentation and measurements[END_REF]. The shape of OOP strongly depends on optical feedback coupling factor C. C ≤ 1 corresponds to weak feedback regime while 1 < C < 4.3 corresponds to moderate feedback regime [START_REF] Bernal | Classification of laser self-mixing interferometric signal under moderate feedback[END_REF], resulting in characteristic saw-tooth like SM fringes. Even if speckle occurs [START_REF] Zabit | Self-Mixing Laser Sensor for Large Displacements: Signal Recovery in the Presence of Speckle[END_REF], SM operating regime can be made stable [START_REF] Atashkhooei | Analysis and control of speckle effects in self-mixing interferometry[END_REF]. As each fringe corresponds to half wavelength (λ/2) target displacement so λ/2 sensor resolution can be easily achieved by using classical fringe counting [START_REF] Ottonelli | Simultaneous measurement of linear and transverse displacements by laser self-mixing[END_REF].

In order to improve SM sensor resolution beyond λ/2, different phase unwrapping methods (PUMs) have been previously proposed [START_REF] Bes | Displacement measurements using a self-mixing laser Diode under moderate feedback[END_REF][START_REF] Fan | Improving the measurement performance for a self-mixing interferometry-based displacement sensing system[END_REF][START_REF] Bernal | Study of Laser Feedback Phase Under Self-Mixing Leading to Improved Phase Unwrapping for Vibration Sensing[END_REF]. The problem with these PUMs is their need of an iterative routine for joint estimation of C and linewidth enhancement factor (α) parameters, making this approach slow and time-consuming for real-time applications.

In this context, for fast and time critical applications, a frequency domain analysis, proposed by [START_REF] Zabit | Time-frequency signal processing for a Self-Mixing laser sensor for vibration measurement[END_REF], replaced the slow iterative process of conjoint C and α estimation resulting in a time-frequency signal processing (TFSP) approach. This approach not only allowed vibration extraction in a fast manner but also provided comparable sensor resolution as compared with previously proposed PUMs.

Likewise, a still better real-time natured algorithm was proposed in [START_REF] Zabit | Self-mixing sensor for realtime measurement of harmonic and arbitrary displacements[END_REF] based on consecutive sample based unwrapping (CSU). Just as in TFSP, CSU also doesn't require estimation of SM parameters or huge iterative routines, making it suitable for real-time, compact, low cost and self-aligned sensor design. Furthermore, CSU is even faster than TFSP because no FFT or IFFT or any computationally intensive transform is used by it. Thus, CSU is the fastest algorithm to process SM signals for displacement retrieval.

The main objective of this paper is the FPGA-based hardware implementation of these previously proposed CSU and TFSP algorithms. Such an implementation of these metric SMI algorithms paves the way towards a complete digital SM sensor integration on chip, capable of delivering real-time measurements for vibration and displacement applications in an embedded, autonomous manner.

The remainder of this paper is organized as follows: Section II and Section III describe the TFSP and CSU algorithms and their hardware implementation on FPGA respectively. Section IV presents the results and discussion. Finally, Section V provides the conclusions.

II. TFSP AND ITS HARDWARE IMPLEMENTATION

TFSP [START_REF] Zabit | Time-frequency signal processing for a Self-Mixing laser sensor for vibration measurement[END_REF] retrieves target vibration from rough laser feedback phase by means of a frequency domain analysis. TFSP can be divided into three major parts: fast Fourier transform (FFT), Noise Removal and Inverse fast Fourier transform (IFFT), as shown in Fig. 2. By means of FFT, time domain roughly estimated feedback phase from SM signal is converted into frequency domain. Then, in Noise removal block, noise floor as well as frequency components making up the target motion are analyzed. Noise and unwanted frequency components are thus removed from main frequency tones. Finally, IFFT provides clean, noiseless retrieved target displacement with better resolution.

III. CSU AND ITS HARDWARE IMPLEMENTATION

CSU [START_REF] Bernal | Study of Laser Feedback Phase Under Self-Mixing Leading to Improved Phase Unwrapping for Vibration Sensing[END_REF] algorithm retrieves target displacement from SM signal by means of a fast signal unwrapping approach. As the saw-tooth like shape of SM signal fringes belonging to a restricted moderate feedback regime [START_REF] Bernal | Robust Method of Stabilization of Optical Feedback Regime by Using Adaptive Optics for a Self-Mixing Micro-Interferometer Laser Displacement Sensor[END_REF] has a quasi-linear local relation to the target motion so precision beyond λ/2 can be achieved by utilizing this local linearity of SM fringes [START_REF] Bernal | Robust Method of Stabilization of Optical Feedback Regime by Using Adaptive Optics for a Self-Mixing Micro-Interferometer Laser Displacement Sensor[END_REF].

The task thus is simplified to unrolling or unwrapping of all SM fringes at all instances where the laser feedback phase crosses 0 or 2π. Such an unrolling can be achieved through fringe detection followed by fringe counting so as to increase the precision of displacement retrieval well beyond λ/2. Due to its simple and fast nature, CSU has been previously deployed on a micro-controller and used in conjunction with an adaptive-optics based SM displacement sensor prototype enabling robust optical feedback regime stabilization [START_REF] Bernal | Robust Method of Stabilization of Optical Feedback Regime by Using Adaptive Optics for a Self-Mixing Micro-Interferometer Laser Displacement Sensor[END_REF]. CSU algorithm can be divided into three major parts i.e. Fringe detection, Staircase signal formation, and fusion, as shown in Fig. 4. Fringe detection block identifies all fringes contained within SM interferometric signal, while staircase signal formation block provides a staircase signal by counting all the positive and negative fringes in the SM signal. In fusion block, staircase signal is added with scaled version of SM signal to better retrieve the target displacement.

FPGA-based hardware implementation is done using VHDL in Xilinx ISE tool. On hardware, fringe detection block is made up of a subtractor taking derivative of SM signal, followed by thresh-holding on it to provide all the fringes present in SM signal, as shown in Fig. 5 (b).

Staircase signal block consists of a simple fringe counting to produce a staircase signal by continuous addition/subtracting of fringes according to their direction, as shown in Fig. 5 (c). The fusion block consists of an adder and multiplier, it takes SM signal and staircase signal as input, and add them in an appropriate manner after judiciously scaling SM signal to retrieve target displacement from SM signal, as shown in Fig. 5 (d).

IV. RESULTS AND DISCUSSION

ZYNQ 7000 series, 28 nm FPGA board is used for implementation and testing of both the algorithms. This section is further divided into two sections i.e.: TFSP Results and CSU Results.

A. TFSP Results

A total number of 16k samples of SM signal shown in Fig. 3 (a) are used in 32-bit fixed point format for the hardware design of TFSP on FPGA.

Table I shows the timing report of our TFSP design. Our TFSP hardware designed system can operate at a maximum frequency of 363.3MHz, with the latency of 570k cycles. Retrieved target vibration obtained from our FPGA based hardware design of TFSP can be seen in Fig. 3(c). From Fig. 3, it is clear that our designed hardware successfully retrieved the motion from the roughly estimated phase shown in Fig. 3 (b).

B. CSU Results

A total number of 1k samples of SM signal shown in Fig. 5 (a) are used in 16-bit fixed point format while testing the hardware design of CSU on FPGA.

V. CONCLUSION

In This successful FPGA based hardware implementation of these two metric algorithms for SMI is a significant step towards a complete digital SM sensor integration on a chip, which would be capable of delivering nanometric precision for real-time applications of vibration and displacement sensing in an embedded and autonomous manner.

Figure 1 .

 1 Figure 1. Schematic diagram of the embedded real-time self-mixing displacement sensor with an FPGA, requiring only a laser diode package with its built-in photodiode and a focusing lens. A piezoelectric transducer (PZT) is used as moving target. Optical output power variations P(t) at laser diode are processed through FPGA to retrieve the target displacement D(t).

Fig. 3 (

 3 a) shows an SM interferometric signal for a vibrating target illuminated with a laser diode emitting at 785nm. Fig. 3 (b) shows roughly retrieved feedback phase, while Fig. 3 (c) shows the retrieved target vibration from our FPGA based hardware design after applying TFSP on roughly retrieved signal of Fig. 3 (b).

Figure2.

  Figure2. Block diagram of Time-Frequency Signal Processing of roughly retrieved phase from SM interferometric sensor signal.

Figure 3 .

 3 Figure 3. (a) SM interferometric signal composed of 10Hz-15Hz-25Hz-30Hz corresponding to a target vibration for a Laser emitting at 785nm, (b) roughly retrieved target vibration, (c) TFSP retrieved target vibration.

Figure 4 .

 4 Figure 4. Block diagram of Consecutive Sample Unwrapping of SM interferometric sensor signal for displacement retrieval.

Figure 5 .

 5 Figure 5. Displacement retrieval through Consecutive-Samples based Unwrapping. (a) Normalized SM signal, (b) fringe detection, (c) stair-case signal formation by fringe counting, and (d) the retrieved displacement signal obtained by the fusion of signals (a) and (c).

Fig. 5 (

 5 Fig. 5 (a) shows normalized SM interferometric signal of moderate feedback regime produced due to target motion, used as input while implementing and testing CSU design on FPGA. Fig. 5 (b) shows the output of fringe detection block, while Fig. 5 (c) shows the staircase shaped signal achieved from the 2 nd block of CSU involving fringe counting. Fig. 5 (d) shows the retrieved displacement using our hardware designed of CSU algorithm of FPGA.

Fig. 5 (

 5 Fig. 5(d) shows the output displacement result obtained from our FPGA-based hardware design. It is clear that our designed hardware successfully retrieved displacement from the SM interferometric signal presented in Fig. 5 (a).

TABLE I .

 I TIMING SUMMARY OF TFSP DESIGN

	Speed Grade	-3
	Minimum Period	2.752ns
	Maximum Frequency	363.332MHz
	Minimum input Arrival time	1.022ns
	after Clock
	Maximum Output required	0.672ns
	time after Clock

Table

  II shows the device utilization summary. The design consumes 5056 slice registers, and 5192 LUTs on the FPGA, while it consumes 0.756 watts of onboard power.

TABLE II

 II

	.	DEVICE RESOURCES UTILIZATION SUMMARY FOR TFSP
	Slice Logic		Used
	Number of Slice Registers	5056
	Number of slice LUTs	5192
	Number of Occupied Slices	1724
	Number of LUT Flip Flop pairs used	5229
	Number of Bounded IOBs	212

Table

  III shows the timing report of our CSU design. Our CSU FPGA-based design can operate at a maximum frequency of 432MHz, with the latency of 3 clock cycles.

TABLE III

 III

	.	TIMING SUMMARY OF CSU DESIGN
	Speed Grade		-3
	Minimum Period		2.315ns
	Maximum Frequency	432.021MHz
	Minimum input Arrival time	1.205ns
	after Clock
	Maximum Output required	0.562ns
	time after Clock

Table

  IV shows the device utilization summary. The design consumes 159 slice registers, and 215 LUTs on the FPGA while it consumes 0.7 watts of onboard power.

TABLE IV

 IV

	.	DEVICE RESOURCES UTILIZATION SUMMARY FOR CSU
	Slice Logic		Used
	Number of Slice Registers	159
	Number of slice LUTs	215
	Number of Occupied Slices	55
	Number of LUT Flip Flop pairs used	215
	Number of Bounded IOBs	65

  this work, we performed FPGA-based hardware implementation of TFSP and CSU algorithms for self-mixing laser interferometric sensor. The designed hardware algorithms are tested and implemented on a Xilinx ZYBO Zynq 7000 development board. Results show that, our FPGA based hardware design of TFSP can operating at 363 MHZ clock frequency. It consumes 5056 slice registers and 5192 Slice LUTs, and draws 0.8 Watt of static onboard power. Similarly, extracted results from CSU hardware implementation show that, it can operate at 432 MHz and consumes 159 slice registers and 215 LUT's, and draws 0.70 Watts of onboard power.

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