Theodoros Vavouras

MOTIVATION TO ENGAGE IN MOOC COURSES

Keywords: Distance open education, massive open online courses, motivation to learning

Massive Open Online Courses (MOOCs) list thousands of course titles with millions of students around the world. Registering the motivational reasons behind this student attitude may help organizers shape courses in a way meeting market needs. In this paper, an attempt is made to record the declared motivational reasons for the participation of people, with different ages, to MOOCs around the world. An internet survey was administered in 91 countries. There were 611 valid answers which were received, in response to the call, and completed questionnaires formulated the sample of the study. The responses showed that interest in lifelong learning and professional development were the strongest motivators on average in our research sample. Also, a strong motivation is the brand name of the institution which offers the MOOC and other personal or family reasons.

INTRODUCTION

Historically, education has diversified of its aims and objectives, a fact that highlights its social adaptation within specific social or other contexts. The use of the internet and the guidelines issued in the last decade by the European Union, the OECD and UNESCO to make education the property of the whole world has led people, who is involved in education, to the creation of MOOCs (Massive OnLine Open Courses).

LITERATURE REVIEW

MOOC

The term MOOC consists of four (4) words: Massive, Online, Open and Courses. The terms that need attention are: Massive, because it refers to the possibility of enrollment in the course with many participants and Open, because it refers to the open possibility of enrollment at any time, of many students (Hollands & Tirthali, 2014, p.27), without previous knowledge or educational background (Τζιμογιάννης, 2015, p. 390), with little or without financial costs for the student. The term also refers to Open Content licensed under a Creative Commons platform on an open-source platform. Constructivism and collaborative learning theories are selected to create MOOCs because they match the characteristics of adult education principles. [START_REF] Ypsilandis | Language Teaching, Language Learning: Current Trends and Practices[END_REF] states that learning is now seen as a process that must be supported. Education models shift the center of gravity from the teacher to the student and activate him to become independent researcher of knowledge towards a heuristic route of self-learning (Λιοναράκης, 2001).

Types of MOOCs

The two main categories proposed by Steven Downes were xMOOCs and cMOOCs (Prpić, Janes, Araz & Tery, 2015). XMOOCs, based on content, have a behavioural approach [START_REF] Kop | Connectivism: Learning theory of the future or vestige of the past? The International Review of Research in Open and Distance Learning[END_REF][START_REF] Kop | The challenges to connectivist learning on open online networks: Learning experiences during a massive open online course[END_REF][START_REF] Yuan | MOOCs and Open Education: Implications for Higher Education[END_REF]. CMOOCs place emphasis on networking and especially on content contributions by the participants themselves.

MOTIVATIONS AND GOALS IN EDUCATION

According to Shunk, Pintrich & Meece (2014) and Κωσταρίδου-Ευκλείδη (2011) motivation is the cause that moves, pushes, or drags the person into action. Researchers have identified two main categories of motivations: internal and external. They have different implications for learning and education in general [START_REF] Ferreira | Motivation and Relationship of the Student with the School as Factors Involved in the Perceived Learning[END_REF]). The research of motivations for learning in online environments is gaining great interest among researchers. The reasons for joining MOOC are not strictly defined and can differ in the same person who enrolls in a different course even at the same time. [START_REF] Shroff | Student e-learning intrinsic motivation: A qualitative analysis[END_REF] found that online learners were more internally motivated than formal students studying on a campus at a formal university. [START_REF] Cho | Self-regulated learning: the role of motivation, emotion, and use of learning strategies in students' learning experiences in a self-paced online mathematics course[END_REF] found that students' internal motivations are positively related to their learning and performance. Within MOOCs, however, participants tend to select parts of the learning environment, following their goals and interests [START_REF] Kizilcec | Motivation as a lens to understand online. learners: Toward datadriven design with the OLEI scale[END_REF]. [START_REF] Wang | Content or platform: Why do students complete MOOCs?[END_REF] found that in online courses, those who complete a course tend to be more interested in the content of the course, while students who do not complete the online courses, tend to be more interested in MOOCs as a kind of learning experience. [START_REF] Barba | The Role of Students' Motivation and Participation in Predicting Performance in a MOOC[END_REF] attended 862 students who continued until the end of the three-week MOOC and found a positive correlation between learner motivation, participation, and performance. [START_REF] Zheng | Understanding Student Motivation, Behaviors, and Perceptions in MOOCs[END_REF] research came up with four broad types of incentives for joining MOOCs: meeting current needs, preparation for the future, satisfaction of curiosity and connecting/meeting other people. A study by research company Qualtrics, with sample of 700 students, states that MOOC students are more attracted to the course topics [START_REF] Carr | MOOC students attracted most by course topics[END_REF]. Apart from the lack of costs, the most frequently mentioned reasons were the subject/topic area of MOOC, professional development, and curiosity about MOOC. From the motivations mentioned so far, we end up to motivations and participation goals that have to do with the interest in learning, social participation, personal reasons, professional reasons, financial reasons, and certification of education.

THE RESEARCH 3.1 Subject, research question

Subject of the research is to record and describe, through a sample, the motivation of enrolled students to participate in open learning programs. The aim of the research is to record and describe the declared motivations for those enrolled in the courses. The research question of the paper is: How were the participants led to their participation in the MOOC courses?

Methodology

After the purpose, the object and the research question were selected, a data collection tool was created in the form of a questionnaire. It was then shared on the internet via google forms. The answers were collected and processed by quantitative analysis methods.

The research tool

The questionnaire was based on questionnaires that were used in motivation surveys on MOOCs. Thus, questions were used from the questionnaires that have already been tested, judged, and used by [START_REF] Hamori | The Drivers of Employer Support for Professional Skill Development in MOOCs[END_REF], [START_REF] Brunton | Giving Flexible Learners a Head Start on Higher Education: Designing and Implementing a Pre-induction Socialisation MOOC[END_REF], research of the Ministry of Labor of England-Department for Business Innovation & Skills (2013), by [START_REF] Henderikx | To Change or Not to Change? That's the Question… On MOOC-Success, Barriers and Their Implications[END_REF], [START_REF] Wang | Content or platform: Why do students complete MOOCs?[END_REF], [START_REF] Hollands | MOOCs: expectations and reality[END_REF], Καραλής (2013) and Καψάλη (2015). The language of the questions was English so that anybody could understand and answer it from all over the world. To collect answers, it was used 'the bar' by [START_REF] Vougiouklis | Questionnaires with the 'bar' in social sciences[END_REF], in a nine-point scale. The questionnaire before being posted was tested by five colleagues-collaborators as a pilot completion and "passed" through literary editing to be short and comprehensive. This provided information on the time needed to be completed, any ambiguities in its writing and special attention was given to the accidental disclosure of personal data. The questionnaire remained online for a period of nine (9) weeks because the MOOC courses have an average duration of 6-13 weeks.

RESULTS

Detection of declared motivations

The answer to the research question of this paper is given to the questions in this part of the questionnaire. Question 12 is about the motivation for interest in learning. It is related to the geographical isolation that the respondents feel. At present time, there seems to be no geographical isolation and this motive exists only for a small part of the sample. Those who "completely agree", feel geographically isolated from educational institutions and are motivated to participate in MOOC, are only 7.43% (Graph-01). Question 23 is in the same group of motivations; the average is: 8.33 which means that learning new things is a powerful motivator. Question 13 is about the motivation for certification and recognition of education. In the statement "Because the course is offered by a well-known university", the average is: 6.17. The brand name of the University seems to be a good motive to enrol in a MOOC (Graph-02).

In the same category of motivations are questions 17 and 18. The statements "To receive a certificate" and "To improve my CV" have answers in average: 4.69 and 5.54, respectively. This shows that it is not a priority to receive a certificate. To improve one's CV, however, is a (not so strong) motive for someone to complete a MOOC as the average value in the agreement statement is 6.0. Economic reasons motivate a person. Relevant findings were in the statement "Because I cannot afford the fees of a formal University" (question-14, average: 4.82). Financial motive can vary with the financial situation of each person, but in many parts of the world education is low cost, while in many areas universities and private education have increased tuition disproportionate to the standard of living of the country. The analysis shows (Table-01) the states with the highest motivation (highest average), they are Vietnam, Ecuador, Guatemala, Panama, Philippines, Malaysia, and Tunisia. In these countries there seems to be a strong financial motive to get involved with MOOCs. On the other hand, there is a very weak economic motive to get involved with MOOCs in the countries with the lowest average (Table-02).

Another financial motivation is: "To raise my salary/income" (question-25). Salary/income is a characteristic of employees and it was deemed necessary to cross tabulate the variable of this question by the factor of employment status. In the results (Table-03) the maximum average value of answers is for the category of unemployed (4.78). The next category are employees (4.29). Self-employed have average 3.94. Retirees are in last place and it looks like that they do not register in the MOOCs to raise financially their income. For those who have not clarified their employment status, the average is 3.00. It seems that there is a financial motive for the unemployed to acquire knowledge while in antithesis retirees are involved in online courses less for financial reasons.

Personal motivations are detected by the questions: 15, 16, 19 and 20. The statement "I am registering because of curiosity about the Online Course" (question-15) is strongly agreed by 31.57% of the respondents. Curiosity seems to be a strong motivator for the student. Question 16, "To be an example to my children" has average 3.51. This question is related to the marital status. Married with children have stronger motivation. Their average is 4.19, while for married with no children it is slightly lower (3.75). Those who are not in a relationship (of marriage) have average 2.78 in answers. Those who did not clarify their marital status answered with average 3.83. It can be seen from the distribution of the answers that the family and in particular the upbringing of children gives an additional motivation for acquiring knowledge and engaging in online lessons in antithesis to those who do not have children.

The question/statement "To get away from my problems", finds 55.6% of the total sample strongly disagree. This incentive works but is not strong for most people, as only 3.8% completely agree. The last question of personal motivation is the 20th "To use my free time better". The average of the answers is 5.97. It seems that the optimal use of free time is a strong personal motivation. Motivation for social participation is expressed in questions 21, "To improve my social network" and 22, "To meet new people". Regarding the improvement of the social network in the answers, the average is 4.12. Meetings with new people have average of answers 3.83. It is statistically obvious that the increase of social participation is not a strong motivation. The last group of motivations is the professional upgrade. The relevant questions are 24 "To be more efficient in my work" and 26 "To keep my current job". These questions are related to the working status and will therefore be correlated with the working condition variable of the sample. In question 24 the average is 6.92. Regarding the distribution of averages (Table-04), according to the employment status, the rank is: employees, unemployed, self-employed and retired.

The Distribution of averages according to the working status in question 26 is: employees 3.46, unemployed 2.69, self-employed 2.82 and retirees 2.05. Work efficiency (question-24) is a motive to engage with MOOCs primarily for employees, while retaining current work is not for retirees.

Demographic profile of research participants

From the review of statistical findings, we can sketch the profile of the participants. The percentage of women who took part in the survey was 57.8% and men 41.7%. The age of the sample is on average 43.3 years. The age distribution does not change significantly from men to women. The nationalities were from a total of 91 countries, we single out the countries of the top three in descending order: America, India, France. It seems that the majority of those involved in MOOCs have a high level of education, as 47.2% of the sample has a master's degree (Graph-03). Married with children are 31%, while single are 40% of the sample. In the use of technology, the majority is Advanced Users. They are completely autonomous users; they are 41.7% of the subjects of the sample. The technical means with which the trainees connect with the learning platforms is the laptop (52.7%) (Graph-04). The correlation between age and equipment shows that as the age decreases, the devices with which the sample use to connect to the MOOCs, are getting smaller and flexible. The number of MOOCs involved, registered, and completed by each, is three (mode value). The active workers are 81.2% of the sample and they work in average 31.36 hours/week. Survey participants state that they read during the day after 21:00 at a rate of 43.4% (max. value). On average they read 6.87 hours per week.

RESEARCH RESULTS -DISCUSSION

The typical MOOC learner is a woman from America, 43.3 years old. She holds a master's degree, and she is married. She works as an employee with a schedule of 31.3 hours per week and she is familiar with technology. She connects to the MOOC classes with a laptop and reads 6.87 hours per week during the evening hours. From the average in every question came up the declining classification of declared motivations to engage with MOOCs in Table-05. The color code on the lines indicates a motivation category as follows: yellow: interest in learning, green: professional upgrade, light blue: personal/family reasons, white: education certification/recognition, orange: financial reasons, gray: social participation. The averages listed in the table below are in the decimal range.

PROBLEMS -LIMITATIONS

In the letter-call for the completion of the questionnaire, the participants were explicitly asked to provide honest answers because only in this way could the research give correct and objective results. It was possible to exclude occasional or professional participants by using databases from social networks as well as by using the discussion forums of the MOOCs. Thus, with great certainty, the users who responded to the research tool belonged to the target population of the research. The online sample was randomly identified, and emphasis was placed on participants' satisfaction with the research process. The last question in the questionnaire asks participants if they wish to receive a copy of the survey results via email. The purpose of the question is twofold, on the one hand this way the sample is identified as the email addresses are real and can be identified to a large extent, while on the other hand there is the reward of participants by sending and communicating the results of the survey. The identification rate exceeded 45% of the sample and at the same time gave a sense of reward to the participants. Regarding the research sample it should be said that there is a limitation in terms of representativeness in countries from which the answers were limited. Thus, for countries with less than twenty participations, the results cannot be reduced to the general population of each country and there is a limitation of the results to their generalization.

FUTURE RESEARCH

An issue that has arisen is the recognition of MOOC courses by academic institutions or even by employers. The courses that can be found by thousands in the platforms are pieces or even whole university courses. They are taught by professors who also teach in typical University Institutions. The continuation of research in the field of open online learning, would be by exploring the possibility of recognizing academic or professional the studies in open online courses. Online open education seems to be the future of education, especially after the Covid-19. The use of MOOC programs (or as they will be called in the future) for academic and professional purposes should be adopted so that education becomes the property of every citizen of the earth without racial, demographic, geographical, economic, and cultural barriers.

In the following sentences please tick the number according to your opinion (1=strongly disagree and 9=strongly agree) I enrolled to the MOOC: 12.

Because I am isolated geographically from education institutes 13.

Because the course is offered by a well-known University 14.

Because I cannot afford the fees of a formal University 15.

Because of curiosity about the Online Course 16.

To be an example to my kids 17.

To get a certificate 18.

To enhance my CV 19.

To get away from my problems (personal/family/workspace) 20.

To use my free time better 21.

To improve my social network 22.

To meet new people 23.

To learn new things 24.

To be more efficient in my work 25.

To raise my salary/income 26.

To keep my current job During Mooc 27.

I attended: 1 Week, 2 Weeks, 3 Weeks, 4 Weeks, 5 Weeks or more, I completed the Course 28.

My study was mostly: In the morning, Midday, Afternoon after 21:00 29.

How long on average did you spend on the MOOC per week? . Answer the next question only if you haven't completed the Course you're enrolled by now 30.

Are you going to complete the Course? Yes, No Barriers MOOC related In the following sentences please tick the number according to your opinion (1=strongly disagree and 9=strongly agree) 31.

My student experience was satisfactory 32.

The support tools were user friendly (chat, email, forum, fb group etc.) 33.

The videos of the MOOC were helpful 34.

The length of the MOOC was long 35.

The MOOC's language is my native language 36.

It is easy to me to attend the course in this language Barriers non-MOOC related In the following sentences please tick the number according to your opinion (1=strongly disagree and 9=strongly agree) 37.

The courses matched my interests 38.

I got support from my family 39.

I got support from my managers 40.

I have insufficient time 41.

My family and work commitments are demanding 42.

Prior to this MOOC I had got sufficient academic background 43.

In the meantime I had health issues After MOOC In the following sentences please tick the number according to your opinion (1=strongly disagree and 9=strongly agree) 44.

I feel I can become a more flexible learner after this course 45.

I am more effective in time management now 46.

The course helped me to interact with other flexible learners 47.

I would recommend this course to others Notes you'd like to write about Moocs 48. ____________________________ 49.

In case you want me to inform you about my survey you can write your email here


Kevin Hoarau
email: kevin.hoarau@univ-reunion.fr

Pierre Ugo Tournoux
email: pierre.tournoux@univ-reunion.fr

Tahiry Razafindralambo
email: tahiry.razafindralambo@univ-reunion.fr

BML: An Efficient and Versatile Tool for BGP Dataset Collection

Keywords: BGP Anomaly, Machine Learning, Dataset

I. INTRODUCTION

The Border Gateway Protocol (BGP) is the routing protocol standard on the Internet. A failure of the protocol could impact any service relying on the Internet. Such failures, namely BGP anomalies, happen for several reasons ranging from hardware failures to malicious attacks [START_REF] Al-Musawi | BGP anomaly detection techniques: A survey[END_REF]. BGP anomalies and their detection are studied using BGP data collection projects such as [START_REF] Routeviews | Routeviews -university of oregon route views project[END_REF], [START_REF]Routing information service (RIS)[END_REF]. The study of BGP traces is fundamental to the understanding of the protocol and anomalies that could arise.

BGP data are stored as logs at different collecting points on the Internet and must be pre-processed to be useful. The BGP data pre-processing is time consuming due to the amount of data. Moreover, the data pre-processing method is always influenced by the anomalies to be studied or the period to be observed. There is no systematic and standard way to preprocess BGP data to extract useful information or to be used in machine learning tools.

Many approaches based on statistical features or graph features have been developed in the field of BGP anomaly detection. These features describing the behavior of BGP are extracted from the BGP data. However, the extraction of the features of interest is strongly related to the pre-processing method. Studying different features may lead to different time consuming pre-processing method. A unique pre-processing method should allow the extraction of all important features at once and therefore eases the study of BGP anomalies.

The authors of [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF] developed a tool to build a BGP dataset. This tool focuses on generating datasets of the most commonly used BGP statistical features but cannot generate BGP's graph structure and embedings as used in [START_REF] Goyal | Dyngem: Deep embedding method for dynamic graphs[END_REF]- [START_REF] Sanchez | Comparing machine learning algorithms for bgp anomaly detection using graph features[END_REF]. Their tool also lacks versatility as it requires to restart the lengthy process of data collection in order to modify the generated features. Therefore, the tool developed in [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF] is not suitable for machine learning (ML) approaches for anomaly detection in BGP as the ones described in [START_REF] Goyal | Dyngem: Deep embedding method for dynamic graphs[END_REF]- [START_REF] Sanchez | Comparing machine learning algorithms for bgp anomaly detection using graph features[END_REF]. The tool developed in this paper, called BML, allows the extraction of BGP synthetic features (as in [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF]), BGP's graph structure and provides the users with the opportunity to craft their own features.

This paper focuses on the construction of BGP datasets to extract synthetic features and graph structure. We propose BML, a tool that automates the BGP data collection and preparation processes. BML only needs a set of timestamps that may relate to relevant BGP events and define the parameters that affect the trade-off between completeness, collection time and the storage footprint of the resulting dataset. Once the dataset is collected, the user can define and apply several data transformation functions to refactor and exploit the data.

We illustrate the usefulness of BML with a simple use case where we analyze the BGP behavior during a large scale anomaly that caused significant disruptions on the Internet and mainly in Japan. The results shows that the tool only needs a few minutes on a low-end computer and less than 30 lines of Python code to collect the relevant BGP data and extract the 32 BGP synthetic features and 14 BGP's graph features.

The remaining of this paper is structured as follows: section II provides a short description of BGP, its anomalies and a state of the art. Section III describes our tool (BML). Section IV is dedicated to a use case study, using BML. Finally, section V concludes and discusses our contribution.

II. BACKGROUND

A. BGP in a nutshell

The Internet consists of Autonomous Systems (ASs) interconnected by Border Gateway Protocol (BGP). Most of the ASs are Internet Service Providers (identified by an ASN) that own IP prefixes [START_REF] Gao | On inferring autonomous system relationships in the internet[END_REF]. ISPs operate BGP routers that maintain TCP connections with a set of BGP neighbors to exchange routing information with other ASes. Traffic is sent through

Feedback

Step 1: Selection of periods of interest

Step 2: BGP data collection

Step 3: Generation of routes snapshots sequences

Step 4: Data transformation

Step 5: Machine learning, time series analysis, statistical pattern recognition, ...

Our proposal

Fig. 1: Iterative workflow of BGP anomaly study.

routes learned by BGP. BGP incrementally updates its routes. When using BGP, a route to an IP prefix is identified by the set of ASes (namely the AS-PATH) that participate in the traffic forwarding which avoids routing loops [START_REF] Rekhter | A border gateway protocol 4 (bgp-4)[END_REF].

B. BGP Anomalies

BGP anomalies are failures or malfunctions of the routing protocol. They can be the result of a protocol vulnerability [START_REF] Butler | A survey of bgp security issues and solutions[END_REF], a configuration error [START_REF] Mahajan | Understanding bgp misconfiguration[END_REF], an external event such as a hardware failure [START_REF] Cowie | Impact of the 2003 blackouts on internet communications[END_REF] or a worm spread [START_REF] Wang | Observation and analysis of bgp behavior under stress[END_REF]. Anomalies can cause instability or overload on the BGP routers. They can lead to invalid network topologies [START_REF] Gao | On inferring autonomous system relationships in the internet[END_REF] that may result in the unreachability of some prefixes. They can also be used to impersonate some prefixes or for traffic interception [START_REF] Ballani | A study of prefix hijacking and interception in the internet[END_REF]. To alleviate the effect of BGP anomalies, several works have been done on the detection of BGP anomalies [START_REF] Al-Musawi | BGP anomaly detection techniques: A survey[END_REF] using time series analysis, statistical pattern recognition or machine learning.

C. State of the art for BGP data collection

RouteViews [START_REF] Routeviews | Routeviews -university of oregon route views project[END_REF] and RIPE RIS [START_REF]Routing information service (RIS)[END_REF] projects have been collecting and archiving BGP data from different collectors distributed across the world since 2000. The collection of BGP routing information is the cornerstone for any analysis of the BGP protocol. Each of these collectors receives BGP updates from all its neighbors and updates its Routing Information Base (RIB) accordingly. Data from RouteViews and RIPE RIS projects require cleaning and time rearrangement from the different collectors. The BGPStream framework [START_REF] Orsini | Bgpstream: a software framework for live and historical bgp data analysis[END_REF] has been designed to sanitize data from BGP routing information.

Once sanitized, the BGP data need transformation to extract useful information. In the literature, a widely adopted BGP data transformation is feature extraction. These features falls into different categories: i) volume features that captures BGP's stability and AS-PATH features that capture topological changes [START_REF] Ding | Detecting bgp anomalies using machine learning techniques[END_REF]- [START_REF] De Urbina Cazenave | An anomaly detection framework for bgp[END_REF] and ii) graphs features that capture the underlying graph structure of BGP [START_REF] Sanchez | Comparing machine learning algorithms for bgp anomaly detection using graph features[END_REF]. However, none of the work from the literature provide an systematic approaches to extract all of the features from BGP data.

In the literature, each contribution develops its own tool to extract BGP information. On one hand, in [START_REF] Goyal | Dyngem: Deep embedding method for dynamic graphs[END_REF]- [START_REF] Sanchez | Comparing machine learning algorithms for bgp anomaly detection using graph features[END_REF], BGP's graph structure and embedings are used to study BGP behavior. In these papers, the extraction processes are developed

P x = [t a x , t b x ]
Priming data:

P m x = [t a x -D pm , t b x ]
Fig. 2: BGP data collection.

for the specific purpose of each paper and do not allow easy reuse. Moreover, these papers do not extract volume features and AS-PATH features (synthetic features). On the other hand, Fonseca et al. in [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF] developed a tool for the construction of BGP datasets. Their representation of BGP's structure and dynamic is limited to synthetic features. The tool described in [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF] is the closest to ours but does not capture the graph structure of BGP especially needed for machine learning (ML) models. Unlike other tools in the literature, BML can change the data representation and features without recollecting the data. BML developed in this paper fills the gap in the literature by providing both the extraction of volume features, AS-PATH features and the extraction of BGP's graphs features but also allows the user to define its own features.

III. OUR PROPOSAL

Our proposal (BML) allows BGP specialists and machine learning practitioners to focus on developing and evaluating BGP or ML models by automating part of the BGP workflow (see Figure 1). Once the period of interest is defined in step ➊, in step ➋ BML automatically collects the data with a specific data structure (see Section III-A for more details). For each period of interest, BML extracts periodic snapshots of the BGP routes and produces a lossless yet storage efficient representation in step ➌ (see Section III-B for more details). During step ➍, BML proposes helper functions to ease the development of a data transformation that fits the requirements of the user or an ML model (see Section IV for an example). The output of this transformation function can be interpreted or used to train and test a ML model (step ➎).

The workflow presented in Figure 1 is iterative and relies on a try-and-error methodology. Our goal is to speed up the iterations of the model by ensuring that: first, constructing and updating a dataset only requires to define periods of interest and a set of parameters, and second, a change in an ML model and/or the data transformation does not require to recollect the BGP data.

A. BGP data collection

In BML, an event that is worth including in the dataset is called a period of interest P x = [t x a , t summarized in Table I). All of the BGP updates from the selected collectors (section II-C) whose timestamp lies in [t x a , t x b ] interval will be included in the dataset. However, at t x a there is obviously not enough information to reflect the actual BGP Routing Information Base (RIB). Indeed, as the BGP protocol is incremental, we face a cold start as the routing updates collected between t x a and t x a + ǫ do not cover all the routes nor they allow to reproduce BGP's graph at time t x a + ǫ. To overcome this issue, we introduce a priming data collection period P m x = [t x a -D pm , t x a ] where BGP data is collected before the period of interest (see Figure 2). BML can also leverage the most recent RIB dump available on the collector augmented with the updates received until t x a . However, the use of RIB dumps comes with twofold increase in the execution time and should only be used if relevant.

After the priming data collection, the BGP updates whose timestamp falls within [t x a , t x b ] are also collected. 1) Data storage: The priming period is stored as a route snapshot obtained from algortihm. 1. This representation allows a reduction of the storage space. The updates collected during the period of interest cannot be stored in such route dumps as their temporal dimension needs to be preserved. The default priming period is set to 10h and needs an average of 20MB while the period of interest requires 75MB of storage per hour.

2) Data collection parameters: The following parameters cannot be modified after the data collection. They have a significant impact on the execution time, storage and data quality, and as such, they need to be chosen carefully : i) List of periods of interest: the bound of each period of interest should be t x bt x a specified as well as the duration of the priming period D pm which should be large enough for to prevent cold start while considering its impact on the dataset size and collection time. ii) List of collectors: this parameter highly depends on the use case as in some cases, the location of the collectors may be important.

Algorithm 1: Routes snapshot

Input: U an ordered set of updates, R a dictionary of routes Output: R updated dictionary of routes begin foreach update u in U do if u type = announcement then R[u pref ix , u collector , u peer ] ←u as-path end else if u type = withdrawal then R[u pref ix , u collector , u peer ] ←-∅ end end end

B. Data transformation

the data is collected it must be transformed to fit the goal of practitioners and the ML model requirement. Using algorithm 1, BML first generates a route snapshot every ∆ seconds for the period of interest

P x = [t x a , t x b ] resulting in a set of t x b -t x a ∆
routes snapshots denoted as S x = {S t x a , S t x a +∆ , ..., S t x b }. An example of such a snapshot is given in Figure 3.

To complete the data transformation, a feature specific data transformation function T (.) is applied to the sequence of routes snapshot S x = {S t x a , S t x a +∆ , ..., S t x b }. The result is the representation sequence T x = T (S x ). Alternatively, the data transformation can take as input the update sequence instead of the corresponding routes snapshot. This might be required for some transformation such as the computation of the number of routes announcements made within the period of interest.

1) Examples of data transformation functions: Various features and data representation have been or may be used by ML models for BGP analysis: { "195.252.70.0/24":{ "rrc19":{ "37468":"37468 5483 6700", "37697":"37697 37468 8400 6700 6700", "57695":"57695 328383 327782 37100 8400 6700 6700", "37640":"37640 8400 8400 8400 8400 8400 6700 6700" }, "rrc00":{ "202365":"202365 49697 5483 6700", "396503":"396503 6939 5483 6700", ... }, ... } Fig. 3: Sample of a JSON routes snapshot which maps every (prefix , collector , peer ) triples observed to an AS-PATH {AS 0 , AS 1 , ..., AS j }. We can see that the collector "rrc19" has four routes to the prefix "195.252.70.0/24" and that one of these routes is announced by its peer "37468" with the AS-PATH "37468 5483 6700". • Statistical features: e.g. number of prefixes or routes, average hop count of the routes [START_REF] Cheng | Multi-scale lstm model for bgp anomaly classification[END_REF], [START_REF] De Urbina Cazenave | An anomaly detection framework for bgp[END_REF], [START_REF] Li | An internet routing forensics framework for discovering rules of abnormal bgp events[END_REF]. • Graphs: as BGP reflects the internet topology, a route snapshot can easily be represented as a graph whose structure may differ depending on the ML model [START_REF] Goyal | Dyngem: Deep embedding method for dynamic graphs[END_REF]- [START_REF] Han | Partially supervised graph embedding for positive unlabelled feature selection[END_REF].

• Graph features: while many ML models can't exploit graph structured data, they can consume synthetic metrics derived from BGP graphs such as the centrality metrics [START_REF] Sanchez | Comparing machine learning algorithms for bgp anomaly detection using graph features[END_REF] or adjacency matrix.

2) Data transformation parameters:

The data transformation step requires only two inputs from the user :

• The snapshotting interval ∆: defines the interval between each route snapshot in each period of interest. • The data transformation function T (.): takes as input a routes snapshot and produces a representation adequate for the ML model. Note that if these parameters need to be modified, the practitioner only needs to re-execute the data transformation step and not the data collection steps. This allows quick iterations of the BGP workflow and gains in productivity.

IV. USE CASE

In this section, we illustrate how BML may benefit ML practitioners and other BGP specialists by studying the BGP anomaly induced by the Google leak 1 that occurred on the 25th of August 2017. At 03:22 UTC, the AS 15169 owned by Google accidentally started to announce several routes to prefixes it does not own and thus became a transit provider. This error caused significant disruptions on the Internet and mainly in Japan.

BML is configured to collect the period of interest P=[03:00;04:00] with a default priming period of 10 hours from the RIPE RIS collector rrc06 which is located in Japan. BML integrates 46 predefined transformation functions including 32 statistical features that are mostly used in the anomaly detection literature (please refer to Table II). The entire pipeline of the dataset definition, collection, and transformation takes only 20 lines of Python code as shown in Figure 4. The code shows how simple it is to extract features with BML since the variables in this code are mostly descriptions.

During our test case, the data collection took 4 minutes and 21 seconds to complete 2 . We set ∆ = 1 minute. The 1 https://bgpmon.net/bgp-leak-causing-internet-outages-in-japan-andbeyond/ 2 During our test case, we used a computer with the following specification: 3.7GHz/8 cores processor with 64 Gb of RAM running Ubuntu 18.04 from BML.data import Dataset from BML.transform import DatasetTransformation from BML import utils ################# # Data collection folder = "dataset/" dataset = Dataset(folder) dataset.load({ "PrimingPeriod": 10 * 60 * 60, # 10 hours of priming data "IpVersion": [START_REF] Fonseca | Bgp dataset generation and feature extraction for anomaly detection[END_REF], # only IPv4 routes "Collectors": ["rrc06"], # rrc06: at Otemachi, Japan "PeriodsOfInterests": [{ "name": "GoogleLeak", "label": "anomaly", "start_time": utils.getTimestamp(2017, 8, 25, 3, 0, 0), # August 25, 2017, 3:00 UTC "end_time": utils.getTimestamp(2017, 8, 25, 4, 0, 0) # August 25, 2017, 4:00 UTC }] }) # run the data collection utils.runJobs(dataset.getJobs(), folder+"collect_jobs")

##################### # Data transformation
# features extraction every minute datTran = DatasetTransformation(folder, "BML.transform", "Features", 1) # run the data transformation utils.runJobs(datTran.getJobs(), folder+"transform_jobs") Fig. 4: BML code to collect the data of the Google leak use case.

changes of the settings regarding the data transformation e.g. the snapshotting interval ∆ or the features, only take a few seconds since BML will not recollect the data.

Figure 5 shows a plot of the 32 statistical features re-scaled using min-max normalization. Among the plotted features, nb A and nb W are respectively the number of route announcements and withdrawals received during the snapshotting interval. First, between 03:22 UTC and 03:27 UTC a lot of announcements are received due to the routes leak. Then, between 03:31 UTC and 03:36 UTC a second wave of withdrawals and announcements arrived due to Google team fixing the error. While this work doesn't focus on feature evaluation, the data generated by BML clearly shows that these features are relevant for anomaly detection.

BML also implements data transformation that generates a graph structure on which are computed a total of 14 graph features. Using ∆ = 1 minute, this transformation took 7 minutes and 47 seconds to complete 2 . The computation of these graph features are much more complex compared to the statistical features. Figure 6 shows a plot of these features rescaled using min-max normalization. The impact of the BGP anomaly seems to be better rendered with the graph features than with the statistical features. Modern ML models tend to rely on multiple features. To illustrate the relevance of BML in this context, we computed a principal component analysis (PCA) on all of the 32 statistical features (resp. all of the 14 graph features) which are then plotted on two dimensions. Figure 7 shows the 2d projection of the PCA which allows to visualize the separability of the data during the normal behavior of BGP and during an anomaly. The data within the interval 03:22 UTC and 03:37 UTC are labeled as anomalous. We can see that the non-anomalous data tend to be projected close to each other whereas the anoma-lous data are more randomly distributed. The non-anomalous projection generated from the graph features appears to be more compact than the one generated from the statistical features. With the ability of BML to extract several features, the development of ML models relying on multiple features is simplified.

A major benefit of BML is the ability for a user to easily define its own data transformation function. To demonstrate this functionality of BML, suppose we want to compute the number of announcements received during an interval where the Google AS (AS 15169) is in the AS-PATH. As shown on Figure 8 this transformation can be implemented in only 10 lines of Python code. We computed this transformation every minute and it took only 7 seconds to complete 2 . Figure 9 shows a plot of this transformation and we observe a similar pattern as the statistical features.

V. PERSPECTIVE AND CONCLUSION

This paper introduced BML 3 , a tool for BGP dataset's collection used to study BGP anomalies. BML eases the development of machine learning models by automating the dataset collection for BGP researchers and allowing the MLpractitioners to focus on the training and optimization of their models. While similar tools were limited to the collection of statistical features dataset, BML stores the data in a loss-less yet affordable data structure. BML then allows to generate 32 of the most used statistical features and 14 graph features. Users can also leverage the data transformation helper function to create their custom features without the need to recollect the data. We illustrated BML's capability on the BGP anomaly induced by the Google leak of 2017. It included a principal component analysis that showed that machine learning tools using multiple features may be promising, emphasizing the need to easily generate a wide variety of features in a short amount of time. Tools such as BML may also help researcher to share datasets and promote reproducibility and replicability.

It is worth noting that BML is the first data collection tool that allows to generate graph representation. These have recently been shown promising to enhance the performance of BGP anomaly detection models and our evaluation on Google's BGP leak showed similar insight.

Fig. 5 :

 5 Fig. 5: Statistical features during Google leak.

Fig. 6 :

 6 Fig. 6: Graph features during Google leak.

Fig. 7 :Fig. 8 :

 78 Fig. 7: 2D projection of features using PCA (Principal Component Analysis).

TABLE I :

 I Table of notations.

TABLE II :

 II Features extracted by BML (by default).

This project has received funding from the Région Réunion and the European Union -European Regional Development Fund (ERDF) as part of the INTERREG V -2014-2020 program.


C Fischer
email: fischeu@uni-muenster.de

K Tanaka

G W Burr
email: burr@almaden.ibm.com

T Grosjean
email: thierry.grosjean@univ-fcomte.fr

T Maletzky
email: maletzk@uni-muenster.de

U C Fischer

Superfocussing in a metal-coated tetrahedral tip by dimensional reduction of surface-to edge-plasmon modes

Keywords: 07.79.Fc, 78.67.-n, 42.82.Gw

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Metal-coated dielectric tetrahedral tips (T-tip) have long been considered to be interesting structures for the confinement of light to nanoscopic dimensions [START_REF] Yatsui | Plasmon waveguide for optical far/near-field conversion[END_REF][START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy[END_REF][START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy at 30 nm resolution[END_REF][START_REF] Fischer | Controlling light confinement by excitation of localized surface plasmons[END_REF], and in particular as probes for scanning near-field optical microscopy (SNOM) [START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy[END_REF][START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy at 30 nm resolution[END_REF]. Metal-coated glass fragments of a tetrahedral shape have been used to image the photonic nanopatterns related to the surface plasmon excitation of metal nanostructures down to a resolution of 30 nm [START_REF] Maas | Imaging of photonic nanopatterns by scanning near-field optical microscopy[END_REF][START_REF] Maas | Photonic nanopatterns of gold nanostructures indicate the excitation of surface plasmon modes of a wavelength of 50-100 nm by scanning near-field optical microscopy[END_REF]. Even higher resolution, on the order of 10 nm, was obtained in SNOM images of the granular structure of metal films [START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy at 30 nm resolution[END_REF][START_REF] Heimel | SNOM/STM using a tetrahedral tip and a sensitive current-to-voltage converter[END_REF][START_REF] Fischer | Super-resolution scanning near-field optical microscopy[END_REF] and of the domain structure of dye monolayers adsorbed to a metal substrate [START_REF] Bortchagovsky | Dual wavelength snom imaging of monolayers of j-aggregated dye molecules[END_REF]. Recently a T-tip was also successfully used as a probe for tip-enhanced Raman spectroscopy [START_REF] Klein | Near-field Raman spectroscopy using a tetrahedral snom tip[END_REF][START_REF] Bortchagovsky | Surface plasmon mediated tip enhanced Raman scattering[END_REF].

Metallic wedges have recently received attention as waveguiding structures [START_REF] Veronis | Guided subwavelength plasmonic mode supported by a slot in a thin metal film[END_REF][START_REF] Novikov | Channel polaritons[END_REF][START_REF] Gramotnev | Single mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface[END_REF][START_REF] Bozhevolnyi | Channel plasmon subwavelength waveguide components including interferometers and ring resonators[END_REF][START_REF] Pile | Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding[END_REF][START_REF] Moreno | Guiding and focusing of electromagnetic fields with wedge plasmon polaritons[END_REF], due to their tight modal confinement and the prospect for device miniaturization beyond the optical wavelength limit. Metal-coated dielectric wedges are also an integral part of a metal-coated T-tip. Such wedges serve an important but not yet well-understood role within a T-tip, linking the incident beam within the dielectric tip to the strongly localized excitation at the sharp apex of the tip. There are a number of different surface plasmon modes that can be produced on a metal-coated dielectric wedge depending on the exact configuration of the metal coating. Slit or gap modes [START_REF] Veronis | Guided subwavelength plasmonic mode supported by a slot in a thin metal film[END_REF][START_REF] Tanaka | Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide[END_REF] can exist if the metal film coating the tip is discontinuous across the edge separating two neighboring triangular faces. In the case of a continuous metal film, plasmon channel modes [START_REF] Novikov | Channel polaritons[END_REF][START_REF] Gramotnev | Single mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface[END_REF][START_REF] Bozhevolnyi | Channel plasmon subwavelength waveguide components including interferometers and ring resonators[END_REF] can exist at the glass-metal interface of the metal-coated edge. And, as a third possibility, edge modes [START_REF] Pile | Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding[END_REF][START_REF] Moreno | Guiding and focusing of electromagnetic fields with wedge plasmon polaritons[END_REF] can be excited at the air-metal interface of a fully metal-coated edge.

Here we propose that the edge modes at the air-metal interface are responsible for the efficient confinement of a beam of light incident into a metal-coated T-tip polarized in the plane of incidence (the plane between the incident beam and one edge of the tip, referred to afterwards as the K1 edge). Pile et al. [START_REF] Pile | Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding[END_REF] calculated dispersion relations for edge modes on a solid metal wedge. These modes have a wavelength that is shorter than the wavelength of surface plasmon modes of a 2D metal surface, yet longer than the wavelength of a plane wave in glass of refractive index 1.5. Therefore it is conceivable that similar modes could exist on the edge of a metal-coated dielectric wedge. These modes could then be excited, in analogy to the conventional Kretschmann configuration for metal-dielectric thin-film stacks, by phasematching a beam of light incident at an angle onto an edge in an Attenuated Total Reflection (ATR) configuration.

However, such an incident beam, polarized in the plane of incidence, interacts not only with the metal-coated edge but also with the neighboring metal-coated faces of the wedge (see Fig. 1). If this incident beam is oriented so as to phasematch the wavevector of the edge mode, the resulting incidence angle is then smaller than the angle that would lead to resonant excitation of the surface plasmon modes on the neighboring faces. However, damped 2D evanescent modes will still be excited on these faces, which can then converge towards the K1 edge and contribute to the excitation of the 1D edge mode. The resulting edge mode will travel along the K1 edge and lead to localized (0D) excitation at the apex of the tip. In this sense the confinement of the incident light to a highly localized excitation of the tip can be considered as a superfocussing of light by a dimensional reduction of surface plasmon modes, from damped 2-dimensional face modes to a 1D edge mode, to a particle-like 0-dimensional resonance. The particle can either be part of the metal-coated tip, a local gap-plasmon mode formed between the tip and a metal surface in close proximity, or a resonant dipolar structure such as a nearby dye molecule. The glass structure is coated with a gold film of a thickness of 40 or 50 nm as specified in the text with a dielectric constant ε m =-3.515 -i2.9 at 514 nm and ε m =-13.2 -i1.08 at 633 nm, respectively. The angle between the x-axis and the K1 edge is α = 45°, so that the angle between z-axis and K1 is also 45°. For FDTD calculations, the size of the tetrahedral tip is given by B x = 6.2µ m , B y = 5.5 µm, and B z = 3.1 µm. A dipole source is placed at a distance of 10 nm below the tip. For VIE calculations, a Gaussian beam is incident along the z-axis into the tip with an extent defined by B x = 2276 nm, B y = 2629 nm, and B z = 1068 nm. The tip-apex is placed at the origin of the coordinate-system This concept of superfocussing is similar to those suggested by Bouhelier et al. [START_REF] Bouhelier | Plasmoncoupled tip enhanced near-field microscopy[END_REF] for the case of a tapered metal-coated cylindrical structure and by Keilmann [START_REF] Keilmann | Surface polariton propagation for scanning near-field microscopy[END_REF]for a metal-coated curved circular structure. However, these structures do not have a metal covered edge as a link between 2D surface plasmon modes and the excitation at the apex of the tip. An alternate concept of superfocussing was suggested by Li et al. [START_REF] Li | Self similar chain of metal nanospheres as an efficient nanolens[END_REF], who considered a superfocussing structure involving coupled resonances between nanoparticles of different sizes. However in our case, in addition to coupling between resonant structures, there is phasematching between the incident beam, the damped surface plasmon modes, and the edge mode. We consider this sequential phase matching as an important property for efficient coupling between the incoming plane wave and the localized excitation.

Other superfocussing metal structures have been suggested as well, such as a metal cone [START_REF] Babadjanyan | Superfocusing of surface polaritons in the conical structure[END_REF][START_REF] Stockman | Nanofocussing of optical energy in tapered plasmonic waveguides[END_REF] and a metalcoated dielectric wedge [START_REF] Kh | Superfocussing of a surface polariton in a wedge-like structure[END_REF][START_REF] Pile | Adiabatic and non adiabatic nanofocusing of plasmons by tapered gap plasmon waveguides[END_REF][START_REF] Gramotnev | Adiabatic nanofocussing of plasmons by sharp metallic grooves: Geometrical optics approach[END_REF]. These designs are based on a reduction of the wavelength of the surface plasmon modes with the taper of the structure. An incoming optical wavefront induces an end-fired excitation of surface plasmon modes at the wide end of the tapered structure. As surface plasmons travel along the tapered structure, the wavelength of the surface plasmon mode decreases continuously, leading to a highly localized excitation at the tip or small gap at the end of this structure. In comparison to the wedges within our T-tip, the metal-coated dielectric wedges discussed in [START_REF] Kh | Superfocussing of a surface polariton in a wedge-like structure[END_REF][START_REF] Pile | Adiabatic and non adiabatic nanofocusing of plasmons by tapered gap plasmon waveguides[END_REF][START_REF] Gramotnev | Adiabatic nanofocussing of plasmons by sharp metallic grooves: Geometrical optics approach[END_REF] are nearly identical, except that the polarization of the incident beam is chosen to be perpendicular to the plane of incidence (the opposite of the T-tip case). The superfocussing effect consists in the generation of a highly confined plasmon mode of an extremely small wavelength in a narrow gap between two metal faces. In our case, where no plasmon modes are excited with wavelengths that decrease with the progressive taper, the light confinement is generated at the metal-air interface. Finally, we mention that a complex structure for the confinement of a surface plasmon wave to edge modes on a solid metal wedge was proposed by Moreno et al. [START_REF] Moreno | Guiding and focusing of electromagnetic fields with wedge plasmon polaritons[END_REF]. As before, the superfocussing effect is based on a gradual decrease of the wavelength of plasmon modes by an adiabatic introduction of the wedge shape.

Here we present detailed numerical investigations on the mechanism of how a beam of light incident into the metalcoated tetrahedral tip can generate a highly confined electric field at the apex of the tip. In Sect. 2, numerical investigations using the Finite-Difference Time-Domain (FDTD) method are used to explore the operation of the T-tip in extraction mode. A dipole source in close proximity to the apex of the tip excites the tip, revealing the field evolution in the tip, the resulting edge and face modes on the metalcoated surfaces, and the coupling from these modes into highly directional radiation into the dielectric interior of the tip. These results are the starting point for the illuminationmode numerical investigations of Sect. 3, where the volume integral method is used to compute the fields in the vicinity of the tip and in particular at the apex of the tip. These numerical calculations give a detailed account of how a highly confined excitation at the apex of the tip can be efficiently generated. In a following paper, the optical properties of a metal-coated tetrahedral tip will be compared to the numerical results.

Excitation of the T-tip in extraction mode by a dipole at the apex

The FDTD method [START_REF] Taflove | Computational Electrodynamics: The Finite-Difference Time-Domain Method[END_REF] is a well-established method for rigorous simulation of Maxwell's Equations in nanophotonic and nanoplasmonic devices, including near-field optical tips [START_REF] Christensen | Analysis of near-field tip patterns including object interaction using finite-difference time-domain calculations[END_REF][START_REF] Kann | Linear behavior of a near-field optical-scanning system[END_REF]. FDTD has the advantage of great flexibility in the structures and materials that can be simulated, at the expense of a Cartesian grid that can make it difficult to correctly model angled and curved surfaces. However, the advent of workstations with large amounts of memory and parallelized implementation on supercomputers such as BlueGene/L [START_REF] Gara | Overview of the Blue Gene/L system architecture[END_REF] has made it possible to simulate regions of many cubic wavelengths. The model used for the calculations is shown in Fig. 1, consisting of a volume spanning ±3.1µ mi nx (with the K1 edge along +x) and ±2.25 µm in y about the apex of a Tetrahedral-tip. The apex, with a tip-radius equal to the gold thickness of 50 nm, is located at x = y = z = 0. The simulation spans 0.75 µm below the tip in air and terminates 3.15 µm into the body of the tip. The gold layer is simulated using a simple Drude model to implement a dielectric constant of gold ε m =-3.515 -i2.9 at 514 nm and ε m =-13.2 -i1.08 at 633 nm, respectively [START_REF] Johnson | Optical constants of the noble metals[END_REF]. The glass is considered to have a dielectric constant ε 1 = 2.25. The angle between the z-axis and the K1 edge (which projects down onto the +x-axis) and between the z-axis and the opposite face (which contains the projection onto the -x-axis) is 45°.

At z = 3 µm from the tip, the metal surfaces of the tip are terminated by a wide rectangular block of glass, preventing any surface plasmons from unphysically interacting with the Convolutional-PMLs [START_REF] Roden | Convolution PML (CPML): an efficient fdtd implementation of the CFS-PML for arbitrary media[END_REF] that terminate all six boundaries of the simulation volume. The nonuniform grid resolution varies from 25 nm for portions at the periphery of the simulation, to moderate resolution (10 nm in x, y and 5 nm in z) for regions closer than 1.5µ mi nx, y, and +z and closer than 200 nm in -z below the apex. The region immediately around the apex (±200 nm in x and y, ±100 nm in z) is gridded at 2.5 nm. Unphysical reflections due to differences in numerical dispersion between these regions are suppressed [START_REF] Taflove | Computational Electrodynamics: The Finite-Difference Time-Domain Method[END_REF]. The resulting simulations are roughly 550 × 500 × 475 grid cells in size and require 50 minutes of simulation time to converge on a BlueGene/L partition with 512 dual-processor nodes.

To excite the T-tip in extraction mode, a single dipole oriented along a Cartesian direction is placed directly under the tip at a distance of 10 nm and excited with a simple monochromatic sinusoid. Figures 2a and2b show the resulting E x field distributions along y = 0 for dipoles oriented in the x-direction and excitation at a wavelength of 633 nm and 514 nm, respectively. There is a striking difference in the two figures. A noticeable evanescent field extends along the K1 edge (+x side of the structure) at 633 nm (Fig. 2a), whereas this field is nearly absent at 514 nm (Fig. 2b).

In Fig. 2a, there are pronounced features in the field distribution along both the K1 edge and the opposite face (-x side of the structure). Along the K1 edge, a clear edge plasmon mode is visible with a periodicity of 553 nm, smaller than both the vacuum wavelength of 633 nm and the 606 nm periodicity of the 2D surface plasmon mode at the metal-air interface of the opposing face. Furthermore, a beam of light is radiated into the tip starting from the K1 edge, clearly indicating a leaky edge mode. These edge plasmons lead to a strongly directional radiation of light into the glass tip at an angle of 40°with respect to the K1 edge. On the opposing face, evanescent fields extend along both the metalglass and metal-air interfaces, with periodicities of 374 and 606 nm, respectively. 374 nm is the wavelength of a surface plasmon at the metal glass interface of a 50 nm thick gold film. There are also smaller periodicities of evanescent waves which cannot be identified with pure surface plasmon modes. These are most likely produced by beating between different modes. The beam radiated from the K1 edge can also be observed in a z-slice 3.1 µm above the tip, as shown in Fig. 2c. At this plane, the combination of angled incidence and the distributed coupling along the K1 edge produces a directed beam that is displaced from the center of the tip by x = 0.6 µm as shown in Fig. 2c.

We also investigated the excitation of the tip produced by dipoles with different orientations. A z-polarized dipole at 633 nm excites several radiative bulk modes in the tip as showninFig.3a. The edge mode of the K1 edge is strongly excited, and again the edge and 2D surface plasmon modes at the metal-air and the metal-glass interfaces can be identified. However, it is difficult to assign the origin of the radiative modes from the y-slice. A slice at z = 3.1µmofFig.3b shows a nearly but not completely symmetric excitation of all three edges. This is not surprising because the z-direction does not coincide with the axis of symmetry of the tip (the angle of the K1 edge with respect to the z = 0 plane is identical to the face opposite, not to the other two edges). The angle between each edge and the axis of symmetry of the tip is 55°, but the angle between the K1 edge and the z-axis was chosen to be 45°, corresponding to the most typical experimental configuration [START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy at 30 nm resolution[END_REF].

Field confinement in the tetrahedral tip by a Gaussian beam incident into the tip

The results of Sect. 2 clearly show how the excitation of the T-tip by a dipole source leads to the excitation of leaky edge modes which are connected to a highly directional radiation into the dielectric interior of the tip at an angle of 40°with respect to the metal-coated edges. These results also give a hint of how to achieve a strongly localized excitation at the apex of the tip in illumination configuration where a focussed beam of light is incident into the tip. By using a volume integral equation method, the development of the field in and around the tip was calculated for a Gaussian beam incident into the tip at a direction of 45°with respect to the K1 edge of the tip. The scattering problem for the metallic structure shown in Fig. 1 can be solved using a volume integral equation, under the assumption of harmonic exp(iωt) time dependence [START_REF] Zwamborn | The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems[END_REF], as

E i (x) = D(x)/ε r (x) -k 2 0 + ∇∇• A(x). (1)
Here, k 0 = ω/c (c is the velocity of light in free space, ω is the angular frequency), D(x) is the total electric flux, E i (x) is the incident electric field, and A(x) is the vector potential, which is expressed by the following volume integral:

A(x) = 1 ε 0 V ε r (x ′ ) -ε 0 ε r (x ′ ) G(x, x ′ )D(x ′ ) d 3 x ′ . (2)
Here, G(x, x ′ ) is the three-dimensional free-space Green's function given by

G(x, x ′ ) = e ik 0 |x-x ′ | 4π |x -x ′ | . (3)
The volume integral region V in (2) represents the entire space, and ε r (x) represents the distribution of permittivity, where ε r (x) = ε m in the metallic coating, ε r (x) = ε 1 in the dielectric region and ε r (x) = ε 0 in free space. Since the region in which |ε r (x) -ε 0 | is nonzero is finite and the integral region V has a finite volume, it is possible to solve (1) numerically by the well-established method of moments. To obtain the solution, the entire region of the problem is divided into small discretized cubes of size δ × δ × δ, and (1) is discretized by the method of moments using roof-top functions as basis and testing functions. The resultant system of linear equations is then solved by iteration using a GMRES iteration with an FFT [START_REF] Zwamborn | The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems[END_REF][START_REF] Tanaka | Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguide[END_REF].

The incident Gaussian beam, which propagates in the negative z-direction, can be expressed as follows [START_REF] Smith | An Introduction to Classical Electromagnetic Radiation[END_REF]:

E i = E 0 W 0 W(z) e x + ie z 1 + γ 2 (z) -1/2 γ(x)e i (z) × exp - x 2 + y 2 W 2 (z) × exp -ik 0 (x 2 + y 2 ) 2R(z) e i (z) e +ik 0 z , (4)
where

W(z)= W 0 1 + γ 2 (z) 1/2 , ( z ) = arctan γ(z), R(z) =-(z -z 0 ) 1 + 1 γ 2 (z) ,γ ( z ) =-2 z -z 0 k 0 W 2 0 , γ(x)=- 2x k 0 W 2 0 .
E 0 is a constant amplitude-factor, W 0 is the radius of the spot size at z = z 0 , and e x , e z are the Cartesian unit-vectors. The validity of the code was checked by confirming that the code gives a reasonably accurate solution compared with the rigorous solution for a dielectric sphere and by confirming the energy conservation for a lossless problem under a paraxial approximation.

Results

The geometry of the model used for the calculations is s h o w ni nF i g .1. The three surfaces of the glass structure are coated by a gold (Au) film of 40 nm thickness. The wavelength is 633 nm, and the relative permittivities of the glass and the gold are given by ε 1 /ε 0 = 2.25 and Fig. 4 The left column shows optical intensity distributions on the xz-plane |E(x,0,z)| 2 and the right column shows those on the yz-plane |E(0,y,z)| 2 .The axis of the incident x-polarized beam is parallel to the z-axis and is centered at y = 0and (a) x =-302 nm, (b) x = 0nm, (c) x =+302 nm. Excitation vacuum-wavelength is λ 0 = 633 nm. The diameter of the Gaussian beam at z = z 0 = 1068 nm is 2W 0 = 0.8λ 0 . The intensity scale range is normalized by the incident intensity. In (c)the intensity saturates at the tip for a better visibility of the intensity distribution in the cross-section ε m /ε 0 =-13.2 -i1.08, respectively. The Gaussian beam is incident perpendicularly to the xy-plane and at an angle of 45°with respect to both the K1 edge and the opposite face of the T-tip. The beam-waist is placed at z = z 0 = 1068 nm if not stated otherwise. The size of the Tetrahedral-tip is given by B x = 2276 nm, B y = 2629 nm, and B z = 1068 nm and is discretized for the computation into cubes of size δ = 10 nm.

Figure 4 shows the distribution of the optical intensity in cross-sections through the T-tip in the (x, y = 0,z)-plane and the (x = 0,y,z)-plane for different conditions. An intense and strongly confined spot at the apex of the tip is only obtained for a displacement of the incident beam away from the central z-axis of the T-tip by about x =+302 nm. In this case, there is an indication of a standing wave phenomenon along the K1 edge with an increasing intensity towards z = 0 nm. This standing wave can arise by the excitation of an edge mode which travels towards the tip and is partly reflected. For the other case, when the beam is displaced to x =-302 nm, a standing wave phenomenon also occurs. Here, however, the intensity of the standing wave is low at the tip apex and increases towards the edge of the simulation volume. This seems to be due to a wave travelling out of the tip which may be reflected at the rim of the tip, which in this model still has a rather limited size. From the data it is difficult to determine the period of the standing wave exactly. It seems, however, to be in the range of 270 and 280 nm, which is consistent with half the wavelength of the edge mode as determined in Sect. 2.

Figure 5 shows the intensity at a distance of 5 nm below the apex of the tip. This figure shows that there is a strongly confined spot of a high intensity when the incident beam is displaced in the x-direction to +302 nm from the center, and that this confined spot is absent when the beam is displaced to the other direction. The intensity is higher by a factor of 15 than the peak intensity in the focus of the incident Gaussian beam. The displacement is necessary to reach such a high intensity. This result shows that the edge plays an important role in the confinement of light to the apex of the tip.

It should be possible to further optimize the intensity at the apex of the tip by further adjustment of the experimental parameters. The results of Sect. 2 show that an angle of incidence of the Gaussian beam closer to 40°(instead of 45°) should lead to a more efficient excitation of the K1 edge mode. Furthermore, the optimal displacement of the beam from the center appears to be larger than 302 nm. We expect that a larger beam diameter would also lead to the accumulation of a higher field intensity. However, the limited size of the tip in these model calculations prevented us from fully optimizing the conditions for the confinement.

Figure 6 shows the optical intensity in the xy-plane at different z-values starting at 5 nm below the tip with z-increments of 101 nm. In Planes #1-4 of Fig. 6, one observes a redistribution of the intensity within the tip that is mainly due to reflections of the incident Gaussian beam at the faces of the tip. There are also evanescent fields outside of the tip which are distributed almost evenly at the faces and edges of the tip. From Plane #5 to Plane #8 it seems that the field is continuously expelled from the inside to the outside of the tip, and a strong evanescent field intensity gradually develops at the K1 edge and not at the faces of the tip. The expulsion of the field from the inside of the tip is connected to a cut-off for the bulk modes at the inside of the metal-coated tip. Evanescent tails corresponding to damped surface plasmon modes excited by the bulk modes in glass at the faces of the tip disappear with a progression towards the apex. The edge mode however develops more and more, so that at the apex only a highly localized excitation remains.

Analogy to plasmon modes of a 1-D metal-insulator-metal sandwich

The rise of the edge mode at the expense of disappearing bulk and surface plasmon modes of the wedge structure is the essential ingredient for the superfocussing effect by a dimensional reduction of surface plasmon modes. In this process, the edge mode is excited by phase-matching from the beam incident at an oblique angle onto the edge. We can study this process of the excitation of an edge mode on a wedge-like structure in terms of surface plasmon modes of a sandwich system of two metal films separated by a dielectric slab [START_REF] Fischer | Controlling light confinement by excitation of localized surface plasmons[END_REF], as indicated schematically in Fig. 7a. The modes of this metal-insulator-metal (MIM) sandwich structure as a function of the thickness d of the dielectric layer give an idea of the evolution of the mode structure in an adiabatically tapered wedge. While this analogy is useful for developing intuition about the complex properties of the strongly tapered 3-D tip, it is important to keep in mind that the simplicity of the 1-D MIM sandwich precludes it from being a completely realistic representation of the full 3-D apex of a T-tip. A ss h o w ni nF i g .7b, the MIM sandwich system supports waveguide modes confined by the dielectric slab and surface plasmon modes at the metal-glass and the metalair interfaces, respectively. Transverse magnetic modes of different symmetry exist which can be classified according to the symmetry of the magnetic field H x . Only the antisymmetric modes are relevant for our wedge configuration because they match the symmetry of the polarization of the incident Gaussian beam (in the plane of incidence corresponding to the plane in the center of the sandwich structure). In the intensity distribution in the yz-plane of Fig. 4c, it is clear that the field intensity in the dielectric tip has a maximum and disappears gradually below a width of 500 nm as the tip gets narrower. This disappearance corresponds to a cut-off of the bulk waves in the dielectric. It can also be seen in this figure that evanescent tails of these bulk modes at the metal-air interface persist to a smaller width of about 135 nm. Decreasing the thickness further, the evanescent tails at the air interface also eventually disappear.

From the dispersion relations of the sandwich structure of Fig. 7b in this thickness range, the wavelength of the surface plasmon mode at the metal-glass interfaces increases strongly with decreasing thickness of the dielectric layer. In this range, a thickness is reached where these internal plasmon modes become phase-matched to the wavevector of the edge plasmon, so that the surface plasmons at the metalglass interface can now be excited. Decreasing the width further, the phase matching condition is lost again. We conclude that there are, at different widths of the dielectric, different modes which have a phase-matching condition with the edge mode. These modes can couple to and excite the edge modes. This consideration shows how different channels contribute to the excitation of the edge mode.

The characteristic property-that the electric field intensity is expelled from the dielectric as the tip is approached and concentrates at the edge-can be attributed to the final channel. No strong electric field is developed by the antisymmetric plasmon modes in the dielectric because the polarization charges on the opposite faces have the same sign. Furthermore, for a very small thickness of the dielectric, the antisymmetric plasmon modes at the metal-glass interface change their field distribution so that the internal field is expelled to the outside. The plasmon modes at the metalglass interfaces which are excited in the tapered gap of the wedge structure should be related to antisymmetric channel waveguide modes which were described by Novikov and Maradudin [START_REF] Novikov | Channel polaritons[END_REF].

Figure 8 shows similar profiles as Fig. 4 for a wavelength of the incident light of 514 nm. Although a similar field distribution is observed as for 633 nm, the intensity of the electric field close to the apex of the tip is much lower in this case, and no pronounced standing wave phenomenon is observed at the edge. We attribute this difference to the strong damping of surface plasmon modes and edge modes for a gold coated tip at this wavelength as was shown in Sect. 2.

Discussion of the results of the numerical investigations

The numerical results reveal a mechanism of the confinement of light at the apex of a metal-coated dielectric tip of tetrahedral shape. We have shown that edge modes on the metal-coated wedge structures of the tip play an important role. The numerical results reveal that metal-coated edges support leaky edge modes which are connected to highly directive radiative modes in the tip. The process of light confinement can be considered as a superfocussing effect because the intensity of the confined light spot is much higher than the one of the focussed incident beam and the confined spot is much smaller than the focus. The mechanism of superfocussing can be considered as a dimensional reduction of surface plasmon modes. The numerical results reveal that as the incident beam propagates towards the apex of the tip, the electric field is expelled from the dielectric corner of the tip, first to the metal-air interface of the tip and then to an edge of the tip. An edge mode of increasing intensity delivers the energy to the apex of the tip. This process can be understood qualitatively by an interplay of an incident wave with the edge mode and surface plasmon modes on the metal-air and metal-dielectric interfaces. A beam of light is incident into the tip at an angle where it matches the phase of an edge mode. Thus it can directly excite the edge mode. But the incident beam also interacts with different surface plasmon modes of the metal-coated wedge structure of the tip. Where a phase matching condition is met, these modes can be excited by the incident beam, and the surface plasmon modes can in turn excite the edge mode. Thus different pathways exist for the excitation of the edge mode.

The numerical results give us a method for optimizing the parameters for tip excitation in terms of the composition of the tip, its geometry, and the configuration of the incident beam. These parameters can also be varied rather easily in our experimental configuration where the tip is realized by a glass fragment coated with metal [START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy[END_REF][START_REF] Fischer | The tetrahedral tip as a probe for scanning near-field optical microscopy at 30 nm resolution[END_REF]. The type of glass and metal, the thickness of the metal coating as well as the form of the glass fragment can be varied over a wide range. In addition to the conventional coating of the glass fragment with a homogeneous metal film, the coating conditions can be varied so that different faces and edges of the fragment are coated with metal films of different thickness or metal composition. It is also possible to form tips where not all of the faces of the tip are coated or where not all of the edges are coated with metal, which introduces a slit in the metal coating exactly at the edge. Unlike in other configurations where tips are connected to fibres, the incident beam can be adjusted rather freely. It is possible to adjust the numerical aperture of the incident beam as well as its focus and polarization. We can also use radial polarization and structured illumination. This gives us a wide range for an experimental variation of the parameters. Thus the optimization of the tip can be performed by iterative experimental and numerical investigations.

The numerical calculations also show how a dipole source at the apex of the tip leads to a highly directional radiation into the body of the tip. They thus show how the radiation of a molecule into the T-tip can be used as a highly efficient pathway for signal detection in near-field optical microscopy and spectroscopy such as tip enhanced Raman Spectroscopy [START_REF] Klein | Near-field Raman spectroscopy using a tetrahedral snom tip[END_REF][START_REF] Bortchagovsky | Surface plasmon mediated tip enhanced Raman scattering[END_REF]. The radiation into a narrow angular range facilitates the collection of the signal which would otherwise only be possible with optics of a high numerical aperture.

Edge modes are not only of interest in the context of the T-tip which we use as a probe for SNOM. Sharp metal edges are abundant in electronic semiconductor and other microand nanostructures. Thus they could be used to introduce additional nano-photonic functionalities in opto-electronic devices. The T-tip and related structures may well turn out to be well suited for quantum optics with surface plasmons [START_REF] Chang | Quantum optics with surface plasmons[END_REF][START_REF] Chang | Strong coupling of single emitters to surface plasmons[END_REF], as it contains the most important ingredients of such a structure, namely a strong field confinement and an efficient coupling to directed radiation. An important aspect in this context, especially for exploratory purposes to test functionality, is the simple fabrication scheme-just breaking a piece of glass and coating it with metal [2, 3]-as compared to elaborate nanofabrication techniques.

In terms of the numerical methods used here, we want to point out that the numerical FDTD method is not restricted to the case where a dipole excites the tip, nor is the VIE method limited to the inverse case of a beam incident into the tip creating a local excitation at the apex. Both configurations can be treated with either method, and it will be possible to compare the result of the numerical methods and choose the most appropriate method for any particular problem.

Fig. 1

 1 Fig.1Geometry of the T-tip. The body of the T-tip consists of glass of a dielectric constant ε 1 = 2.25. The glass structure is coated with a gold film of a thickness of 40 or 50 nm as specified in the text with a dielectric constant ε m =-3.515 -i2.9 at 514 nm and ε m =-13.2 -i1.08 at 633 nm, respectively. The angle between the x-axis and the K1 edge is α = 45°, so that the angle between z-axis and K1 is also 45°. For FDTD calculations, the size of the tetrahedral tip is given by B x = 6.2µ m , B y = 5.5 µm, and B z = 3.1 µm. A dipole source is placed at a distance of 10 nm below the tip. For VIE calculations, a Gaussian beam is incident along the z-axis into the tip with an extent defined by B x = 2276 nm, B y = 2629 nm, and B z = 1068 nm. The tip-apex is placed at the origin of the coordinate-system

Fig. 2

 2 Fig. 2 Results of FDTD calculations excited by an x-oriented dipole at a distance of 10 nm from the tip. The black arrow indicates the direction of the electric field. (a)Thex-component E x of the electric field is shown for a cross-section through the tip in the xz-plane at y = 0. The vacuum wavelength is 633 nm. The red arrow indicates the

Fig. 3

 3 Fig. 3 Results of FDTD calculations excited by a z-oriented dipole at a distance of 10 nm from the tip. The x-component E x of the electric field is shown. The direction of the electric field is indicated by black arrows, and the orientation of the dipole by red symbols (arrow

Fig. 5 Fig. 6

 56 Fig. 5 Two-dimensional total optical intensity distributions just below the tip on a xy-plane placed at z =-5 nm, i.e., |E(x,y,z =-5nm)| 2 for beam parameters as in Fig. 4 and the x-position of the incident

Fig. 7 1

 7 Fig. 7 1-dimensional metal-insulator-metal sandwich structure. (a) Scheme of the sandwich system. (b) Dispersion relations of the sandwich layer of 2 gold films of a thickness of 50 nm as a function of the thickness of the dielectric layer. Only solid lines correspond to antisymmetric modes. There is a transition of character of

Fig. 8

 8 Fig. 8 For a vacuum-wavelength of λ 0 = 514 nm, the left figure shows the optical intensity distribution on the xz-plane |E(x, 0,z)| 2 and the right figure the one on the yz-plane |E(0,y,z)| 2 .T h ei ncident Gaussian beam is x-polarized and its axis is parallel to the z-axis and placed at y = 0, x =+ 254 nm. The diameter of

Acknowledgement This work was partly supported in part by the German Research Foundation and the Network of Excellence Plasmo-Nano-Devices in the FP6 IST Program of the European Community.