Andreas Riel
email: andreas.riel@grenoble-inp.fr

Alan Lelah

Guillaume Mandil

Maud Rio

Serge Tichkiewitch

Feng Zhang

Peggy Zwolinski

CIRP 25th Design Conference Innovative Product Creation An innovative approach to teaching sustainable design and management

Keywords: Sustainable design, eco-design, product lifecycle, design education and learning

For many years the market leadership was driven by offering better price on more product functions and services. The world is currently shifting towards social responsibility thinking. This changes the market behaviour, and leads to more innovations, such as designing new product/service/system using re-usable parts, new electric motor concepts, or inventing new chemical production procedures generating less waste, and many more opportunities. This paper introduces a research valorisation activity that aims at developing a program of training and coaching to prepare students and industry partners to this emerging innovation wave. The most outstanding particularity of this program is that it combines management and engineering aspects of sustainability in a form that empowers trainees to deploy sustainable approaches in practice. The particular target group of design engineers get equipped with fundamentally important sustainability knowledge enabling them to include sustainability considerations in their products and systems design. Eco-design is positioned as a key lever towards achieving sustainable product-service systems.

Introduction

Sustainability has become a global megatrend under the pressure of the climate change, and increasingly stringent resource scarcity. Organisations should better act in a sustainable manner in order to preserve the environment, and, to be socially responsible, while still preserving their economic orientation.

In recent years, important global issues around energy security, unstable fuel prices, greenhouse gas emissions, as well as sustainable procurement, ethical trade, and corporate social responsibility (CSR), have led organisations increase their commitment to moving towards more sustainable products, services, processes, and associated business models. The most popular improvement vectors are in the area of carbon footprint and energy efficiency. The United Nations' Division for Sustainable Development confirms these tendencies in the report [START_REF]Trends in Sustainable Development -Towards Sustainable Consumption and Production[END_REF] by registering a steep increase in the number of companies certified to fulfil the environmental management and social responsibility standards ISO 14001 [START_REF]Environmental management systems --Requirements with guidance for use[END_REF], and ISO 26000 [3] respectively. They also indicate an increase in the number of consumers feeling concerned by social and ecological issues. Revenue figures from fair-trade products have grown to 2.9 billion euros per year despite the recent economic troubles. The same report reveals that governments are also following the sustainability trend by highlighting sustainable procurement policies, both in developed and in developing countries.

The research valorisation work presented in this paper aims at contributing to filling an emerging gap in current education, and in professional training programs. This gap is located in teaching actionable key knowledge in the area of the sustainable design of organisations, products, services, and products. This teaching material is designed in a way that it can actually be applied successfully in both engineering and management activities in order to make them more sustainable. This paper presents the research objectives and approach that are underlying this training program design activity in section 2. Section 3 gives an overview of the trainings' main elements, while section 4 explains the training approach that has been chosen for their implementation. As a conclusion to this paper section 5 presents the upcoming activities to furthermore develop this training program.

Research Objectives and Approach

The larger context of the presented research valorisation activity is a collaborative work of a European project consortium including the following organisations: The main partners' goal is to develop and provide a training program, which combines both the management and engineering aspects of integrating sustainability in organisations, and in their products, services, and processes. It shall also deal with the assessment of an organisations' sustainability performance and provide a practical approach to setting up relevant programs, objectives, and targets in a strategic sustainable manner. Furthermore, the program shall Responsible Managers and Directors of industrial, academic, and governmental organisations will be empowered to develop business strategies leveraging sustainability. They will learn how to successfully deploy them for a better sustainability performance, and for an improved corporate image.

Designers and Managers of products, services, and/or processes will learn essential multi-dimensional sustainability principles, as well as how to take these principles into account in their daily creative work. They will understand that sustainability should be integrated consistently into products, services, and processes by designing from the very earliest phases onwards. Therefore, the designers and managers have significant roles in achieving their organisations' sustainability performance objectives. The training teaches them practical methods and tools to support them in fulfilling these roles.

Master and PhD Students as future young employees in industry and academia should be able to assume key roles in fostering and implementing a sustainability mindset in an organisation. They are increasingly expected to understand sustainability methods and tools, as well as being able to apply those methods and tools in real-life industry and research projects. The training program is therefore designed to be complementary to their studies. It addresses both managerial and practical sustainability topics in a concise and applicable form. This will give them a broad -yet practicalbasis to build upon when assuming their future professional positions.

The training development process goes through the following main stages:

• need-analysis for the training in all the participating partner countries; • development of an EQF-compliant skill card largely based on the results of the needs-analyses; • development of e-learning enabled training materials in English language for all the skill elements of the skill card;

• several pilot training loops all over Europe, and subsequent improvement of both the skill card, and the training materials; • translation of the training materials in all the languages represented in the consortium; • deployment of the training in the e-learning environment of the ECQA; • development and deployment of a pool of test questions and competence recognition criteria for certification; • foundation of an expert committee ensuring the continuous improvement of all project outcomes, as well as supporting the decisions about the certification of prior competencies, and the contracting of education and training organisations.

The whole process will last two years. It has been halfway run through at the time of writing this paper. The results of the need-analysis have been published in [START_REF] Fistis | Leadership in Sustainability[END_REF]. This paper focusses on the identified key subjects that are currently being elaborated in the form of e-learning enabled training materials.

Training Program Design

This section explains the main subjects covered in the training program resulting from our research so far.

Understanding sustainability leadership

To begin with, the program devotes several training elements to shape an understanding of sustainability in the context of different types of organisations, standards, and, regulations that have been developed around it.

Sustainability has become a keyword (or 'buzzword') for product development in several sectors. In this training context the term sustainability should be associated with the notion of the environment. It thereby covers the intersection between challenges, requirements, and constraints imposed by the three dimensions of the environment: society, ecology, and economy (see Fig. 1).

Fig.1. The dimensions of sustainability considered in this research

It is mandatory to ensure that sustainability covers the three main areas shown in Fig. 1. Those areas make up our environment. The concept of environment is defined as "The surroundings or conditions in which a person, animal, or plant lives or operates" [7]. Sustainability is very often reduced to the economic factor only, and often considered for achieving a sustainable profit growth. In the presented training sustainable design fully considers the three areas, including their overlapping regions.

The Bruntland Commission defined in 1987 sustainable development as "development that meets the needs of the present without compromising the ability of future generations to meet their own needs" [START_REF]The Brundtland Report[END_REF]. To achieve sustainable production and consumption, societies have to drastically reduce their consumption of raw material and energy. Accordingly, sustainable design or eco-design denotes product/services/system/process design activities that strongly integrate sustainability considerations into the design process.

The proactive management of environmental aspects converges with enterprise risk management, corporate governance, operational, financial practices, and performance. Therefore international standards are becoming increasingly important for organisations to work with, towards common and comparable environmental management practices. The sustainability standard compliance supports their commitment and performance to the integration of sustainability in their organisations, products, and services. The program explains the scope and key benefits of the ISO standard families 14001 [START_REF]Environmental management systems --Requirements with guidance for use[END_REF], 26000 [3], 31001 [START_REF]Risk management --Risk assessment techniques[END_REF], 50001 [START_REF]Energy management systems --Requirements with guidance for use[END_REF].

Resource Efficiency and Cleaner Production (RECP) methodology

As a first methodological approach towards taking into account rather fundamental sustainability issues in industrial organisations, the training introduces a program jointly driven by the UNIDO and the UNEP to advance sustainable industrial development, consumption, and production in developing and transition countries.

Life-Cycle approaches and eco-design

Sustainability of products and services does not happen by itself; it has to be proactively taken into account in their design. In more technical terms: it has to be "engineered" using methods of integrated engineering design [START_REF] Brissaud | Product models for life-cycle[END_REF]. Hereby, the notion of the product/system life cycle plays a central role. In integrated design the product/system life-cycle comprises all phases that the product/system goes through from the idea to its end-of-life and revival. The principal stages that a product or system typically runs through in its life-cycle are design, manufacturing, distribution, customisation, and the end-of-life. In integrated design closed life-cycle should be preferred. Integrated design targets a "cradle to cradle" approach that goes beyond the often-cited notion of "from cradle to grave".

Taking into account life-cycle issues in integrated product/system design requires developing systematic methods. The Life Cycle Engineering (LCE) approach offers a set of method which are able to optimise a product from an integrated technical, ecological and economic point of view. It is a management practice approach that aims at combining relevant information in one decision support tool. Its objective is to chart an optimum life-cycle for a manufactured product. LCE connects different angles from which one should look at new developments and in particular, it involves technical, costing and ecological points of view.

There is an increasing demand in assessing and reducing the ecological impacts of products. Complex products such as automobiles need effective methodologies and tools to evaluate their environmental impacts without neglecting the technical and cost implications. The consequences of developing new products and services need to be analysed. Companies need to have relevant methods, and powerful tools. Life Cycle Engineering is also an approach that promotes assessing the environmental impacts in conjunction with economic impacts under consideration of technical boundary conditions. The scope of the assessment is usually the whole product life-cycle, including the production, use phase and end of life stages:

• The environmental impacts are assessed according to the ecological (or environmental) Life Cycle Assessment (LCA). It is a quantified multi-criteria and multi-stage impact assessment (ISO 14040 -44). • The economic impacts are assessed according to the Life Cycle Costing (LCC) approach. • Technical boundary conditions are taken into account providing some limitations on the model, thus verifying the technical feasibility. In practice, to be reliable the LCA and the LCC should share the same boundary.

Life Cycle Engineering increasingly aims at providing a complete system of support tools for the product/system lifecycle. The economic objectives of LCE are adding value extension of business areas, reduction of fixed cost, and increase the profitability of products and services.

Life Cycle Assessment (LCA) is a technique for assessing the potential environmental aspects and potential aspects associated with a product (or service) [START_REF]Environmental management --Life cycle assessment --Principles and framework[END_REF], by: • defining the goal and scope of the study,

• compiling an inventory of the included inputs and outputs material and energy flows (within the scope), • evaluating the potential environmental impacts associated with those inputs and outputs (using appropriate calculation methods), and • interpreting the life cycle impact results regarding the objectives of the study.

When conducting a LCA, the environmental impact of the design/development stage is usually excluded, since it is often assumed not to contribute significantly. It is obvious that the decisions made in the design/development stage highly influence the environmental impacts of the product during its life-cycle stage. The design of a product strongly predetermines its behaviour in the subsequent phases (e.g., the design of an automobile more or less determines the fuel consumption and emissions per kilometre driven in the use phase and has a high influence on the feasible recycling options in the end-of-life phase).

The high interdependency between design/development and the other life-cycle stages is vital to understand to practice eco-design. If the main aim of conducting a life cycle assessment is to address recommendations to product designers to improve the environmental performance of goods and services. The life cycle study should therefore be carried out as early as possible in the design process, as well as concurrently to the other design procedures. This approach is applied analogously to the designing or to the improvement of a process within a life-cycle of a product, especially if several interactions with other processes or life-cycle stages are occurring (mainly the case).

Life Cycle Costing (LCC), also called Whole Life Costing, is a technique to establish the total cost of ownership. It is a structured approach that addresses all the elements of this cost and can be used to produce a spent profile of the product or service over its anticipated life-span. The results of an LCC analysis can be used to assist management in the decisionmaking process where there is a choice of options. The accuracy of LCC analysis diminishes as it projects further into the future, so it is most valuable as a comparative tool when long term assumptions apply to all the options and consequently have the same impact. The life cycle cost of an item is the sum of all funds expended in support of the item from its conception and fabrication through its operation to the end of its useful life.

Product Structure Assessment (ProSA) is used to assess a product's features in terms of disassembly, recycling and reuse from the viewpoint of cost optimisation. It gives the inputs for calculating the end-of-life costs related to these activities.

In terms of tools that support LCE, there are four general classes of information technology (IT) systems in modern enterprise infrastructures:

• SCM (Supply Chain Management) -for the management of the supply chain; • CRM (Customer Relationships Management) -the systems that store information about the history of customers' interaction with the company (business offers, purchases, services, etc.); • PPC (Production Planning and Control) -for the management of manufacturing system (purchases, orders, production planning and scheduling); • PLM (Product Lifecycle Management) -the class of IT systems that collect and manage the data that emerges during product life-cycle.

All the mentioned methods and tools form an integral part of the training program. Their theory is explained concisely, while their practical application is shown using real-world case studies.

Deployment of eco-design and sustainability

Knowing about the necessity for sustainability thinking and acting form the necessary basis are mandatory for successfully implementing sustainability in an organisation. Mastering methods and tools for doing so are also indispensable. This implementation is typically a process that covers several stages of maturity. It reaches from opposition and ignorance, over risk and cost considerations, towards realising competitive advantages thanks to sustainability, and finally the total transformation in order to re-interpret the nature of the organisation to an integral self-renewing element of the whole society and its ecological context [START_REF] Downing | Building sustainable strategic management to whose profit?[END_REF]. Organisation leaders have to realise the added value of investing time and effort into leading their organisations through these different stages of sustainability deployment, and they have to be ready to face the risks associated with the transformation process.

A survey published in July 2007 by McKinsey & Company of CEOs that have signed up for the UN Global Compact identified a broad range of areas where CEOs were having problems in driving greater responsibility and sustainability [START_REF] Mckinsey | Shaping the New Rules of Competition: UN Global Compact Participant Mirror[END_REF]. The survey found that more than 90 per cent of the CEOs said they were doing more than they did five years ago to incorporate environmental, social and governance issues into their strategy and operations. 72 per cent of them said they thought corporate responsibility should be embedded fully into strategy and operations through their firms. However, only 50 per cent actually did so. Although 59 per cent of them said corporate responsibility should be embedded into their global supply chains, only 27 per cent of them thought they were doing it. The problem areas identified by the CEOs included board oversight, stakeholder engagement and linking sustainability issues thoroughly into business purpose and strategy. The survey found that these committed CEOs regarded competing strategic priorities as the biggest barrier to implementing a systemic approach to sustainability across their companies. The complexity of implementing these issues across different business functions also posed problems, as did the lack of recognition from financial markets.

The training deals with approaches that facilitate the transformation process. One key lever of introducing sustainable design and thinking into a business organisation is to make sustainability an engine of innovation of products, services, and processes [START_REF] Johannessen | The future of value creation and innovations: Aspects of a theory of value creation and innovation in a global knowledge economy[END_REF]. While nowadays in many organi sation s sustainability measures are still considered as a "necessary must", sustainability leaders have understood turning sustainability into a competitive advantage rather than an additional burden. The training elaborates several examples from different sectors to point this out. Another key lever for sustainable value creation by innovation is linked to the paradigm of Service Innovation. Industrial Product-Service Systems (IPS²), or simply Product-Service Systems (PSS), focus on maximising value creation by complete and integrated solutions consisting of addedvalue services enabled by and co-designed with products [START_REF] Aurich | Life cycle oriented design of technical Product-Service Systems[END_REF][START_REF] Trevisan | Service Delivery and Co-Creation to support Value and Sustainability in PSS design[END_REF]. PSS design offers the opportunity of integrating business model design and eco-design, which thereby both converge in a life cycle approach. This approach integrates service delivery and the co-creation process together to support value creation and sustainability from the points of view that each stakeholder group involved in the product/service system life cycle has. The focus is thus on the creation of value all through the life cycle rather on profit generation for the producers and vendors [START_REF] Brady | Creating value by delivering integrated solutions[END_REF]. The training largely elaborates on this aspect, and explains numerous practical case studies.

Social responsibility

Corporate social responsibility (CSR) refers to companies taking responsibility for their impact on society. CSR is defined by the European Commission as "the responsibility of enterprises for their impacts on society" [START_REF]A renewed EU strategy 2011-14 for Corporate Social Responsibility[END_REF]. The Commission encourages that enterprises "should have in place a process to integrate social, environmental, ethical human rights and consumer concerns into their business operations and core strategy in close collaboration with their stakeholders" [START_REF]A renewed EU strategy 2011-14 for Corporate Social Responsibility[END_REF]. As evidence suggests, CSR is increasingly important to the competitiveness of enterprises. It can bring benefits in terms of risk management, cost savings, access to capital, customer relationships, human resource management, and innovation capacity.

The training explains CSR and its principal challenges, however it particularly focusses on stakeholder involvement, communication challenges for sustainability, as well as social impact assessment.

Training Approach

Figure 2 shows the training approach applied in the training program. Students study the training material per unit within an e-learning environment. They can contact trainers for questions comments, etc. via the forum which are associated with each learning element of a particular learning unit. As the training material is strictly focused on principles and core messages, students then have to do practical exercises in order to show that they understand these principles, and that they are able to apply them. Exercises are also done as homework, and must be uploaded to the respective discussion forums, where they will be commented by trainers, and other course participants.

In order to explain exercises and trainer comments, students are put together in groups of up to eight participants, who meet via telecommunication facilities with trainers. After a summary-style introduction to the studied learning units done by the trainers, every student is asked to explain his/her exercise. Trainers and other students comment on this explanations and ask questions. This facilitates knowledge and experience exchange among the entire group. After this on-line discussion session, students are obliged to upload an improved version of their exercises.

The complete training process can also happen on-site, i.e., in the form of training seminars given in classical form as classroom sessions. The complete training program can be delivered in five to ten seminar days, mainly depending on the amount and level of detail of exercises. There can also be mixture of both, e.g., the exercise discussion sessions could be done on-site rather than remotely. Thanks to the modularity of the training units it is possible to tailor session schedules to specific needs while still enabling an independent and neutral certification by the ECQA.

Conclusion

For many years the leadership on the market has been driven by offering more functions and services at a better price for maximum financial profit. Under the increasing pressure of global social and ecological challenges, the economic world is undergoing a shift in thinking towards assuming social responsibility by the environmentally responsible design of products and services for their entire life cycles. This shift is changing the market behaviour, confronting organisations with new challenges in terms of value understanding and value creation through innovative product-service systems and business models [START_REF] Ueda | Value creation and decision-making in sustainable society[END_REF].

Education and professional training for knowledge and awareness creation is an important lever to facilitate fundamental shifts in thinking in organisations. The work presented in this paper aims at filling a gap in existing environmental sustainability education and training programs in terms of covering both the "why's" and "how's" to contribute to a more sustainable behaviour and performance by the very design of both organisations and systems of products, services, and processes.

The chosen approach is based on an e-learning enabled training program designed in a way that it can be used both in academic and professional training environments, be continually extended and nonetheless certified on a European level. The training program is practically oriented, considering for all technical elements exercises and best practice examples besides theoretical inputs. The concept of sustainable value creation thanks to the integrated design of product-service systems with the total incorporation of environmental sustainability considerations in all design steps and activities takes a central position in this program. However, teaching a design concept without also addressing approaches towards creating the necessary management attention and support, would not be sufficient for a training that claims to facilitate the future leaders in sustainability. Therefore, a considerable part of the training is dedicated to managerial approaches towards capturing the added value of acting environmentally responsible in heavily competitive economic environments.

With the competencies reached by the trainees with different responsibilities in organisations it is expected that the paradigm shift towards designing product-service systems and their associated processes with sustainable value creation as a fully integrated design objective will be facilitated. The most competitive organisations will be those who integrate concrete social and environmental performance objectives in their business strategy and drive innovation on their basis.

The future research activities linked to this new training program include an exhaustive assessment of trainings delivered to different target groups and in different countries over the next two years, as well as the implementing of a continuous improvement cycle during the same period.

Fig. 2 .

 2 Fig. 2. The training approach

  The higher objective of this collaboration is to establish a new certifiable skill set around sustainability engineering and management on the European level. The training program therefore aims at addressing a wide range of target groups, as achieving sustainability requires the collaboration, and the interaction of several different stakeholder groups.

	be certified by the influential European Certification and
	Qualification Association ECQA. This certification provides
	numerous added values to the training:
	• training architecture and modular ISO 17025 [4] compliant
	certification according to the European Qualification
	Framework EQF [5];
	• assurance of a high quality level of the training materials
	and certification procedures;
	• clear certification procedures for trainers and trainees;
	• learning and exam portals provided and driven by the
	ECQA;
	• sustainable and continuous development managed by a Job
	Role Committee composed of representatives from
	academia, and from industry;
	• international promotion of the program.

Acknowledgements

This work is financially supported by the European Commission in the Leonardo da Vinci Lifelong Learning Program under the project number 2013-1-RO1-LEO05-28771. This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.


Point-to-hyperplane RGB-D Pose Estimation: Fusing Photometric and Geometric Measurements

Abstract-The objective of this paper is to investigate the problem of how to best combine and fuse color and depth measurements for incremental pose estimation or 3D tracking. Subsequently a framework will be proposed that allows to formulate the problem with a unique measurement vector and not to combine them in an ad-hoc manner. In particular, the full color and depth measurement will be defined as a 4vector (by combining 3D Euclidean points + image intensities) and an optimal error for pose estimation will be derived from this. As will be shown, this will lead to designing an iterative closest point approach in 4 dimensional space. A kdtree is used to find the closest point in 4D-space, therefore simultaneously accounting for color and depth. Based on this unified framework a novel Point-to-hyperplane approach will be introduced which has the advantages of classic Point-toplane ICP but in 4D-space. By doing this it will be shown that there is no longer any need to provide or estimate a scale factor between different measurement types. Consequently, this allows to increase the convergence domain and speed up the alignment, whilst maintaining the robust and accurate properties. Results on both simulated and real environments will be provided along with benchmark comparisons.

I. INTRODUCTION

Color and depth images acquired from RGB-D sensors are increasingly useful, especially in robotics for computing visual odometry, performing autonomous navigation and reconstructing 3D environments. One of the most fundamental problems is estimating the pose that relates measurements obtained from a moving sensor at different times. Some recent approaches have combined both measurements together in a limited hybrid manner.

The problem of pose estimation from color or depth images have each been individually studied in the computer vision and robotics literature. Classically, color and depth measurements have been used separately in imagebased and geometric-based pose estimation. In the case of depth images, the well known Iterative Closest Point (ICP) algorithm prevails [START_REF] Besl | A method for registration of 3-d shapes[END_REF] and in particular the point-to-plane ICP algorithm is especially efficient and robust [START_REF] Chen | Object modelling by registration of multiple range images[END_REF]. On the other hand, color images have been used to estimate the pose of the camera using direct and dense error functions based on view synthesis [START_REF] Andrew I Comport | Accurate quadrifocal tracking for robust 3d visual odometry[END_REF]. The latter will be referred to here as image-based approaches. It can be noted that featurebased image approaches first extract geometric information from the image before performing estimation on a geometric error. Feature-based approaches are a sub-part of the direct approach and won't be detailed here. Whilst color and depth based pose estimation have been studied separately, similar solutions have been used for both using a non-linear iteratively re-weighted least squares (IRLS) method. The general IRLS pipeline for pose estimation, follows the common strategy across different measurement types:

1) Transform/warp the current measurement onto a reference frame using the last pose estimate. 2) Find the closest points between the two datasets.

3) Determine an error between the two datasets (and robust weights). 4) Estimate an incremental update on the pose. 5) Repeat to 1. until convergence. Recently, several strategies have combined color and depth measurements together in different ways and attempt to retain the respective benefits of each. The advantages of using both include increased efficiency, accuracy and robustness. In [START_REF] Zhou | Color map optimization for 3d reconstruction with consumer depth cameras[END_REF] a recent survey of the real-time performance of these approaches is provided. Image-based approaches alone are dependant on texture in the images to constrain all degrees of freedom. For example, a wall with only horizontal lines would be degenerate. ICP approaches require sufficient geometry and are, for example, degenerate in the case of a movement parallel to a flat wall. In this paper, those approaches which combine depth and color for robust and accurate pose estimation will be referred to as hybrid approaches (See Fig. 1).

Amongst the various hybrid methods, those of most interest are those that minimize a photometric and geometric error simultaneously in real-time [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Whelan | Robust real-time visual odometry for dense rgb-d mapping[END_REF]. The two main differences in the proposed approaches are categorized as:

• How the closest points are determined between different RGB-D measurements. • How the joint optimization is performed.

The aforementioned hybrid approaches are somewhat ad-hoc because they do not necessarily consider the color and depth simultaneously when computing closest points. Furthermore, in the optimization stage they simply combine the classic ICP and image-based approaches by minimizing both error types simultaneously. This, however, requires the definition or estimation of a tuning parameter λ which weights the respective contribution of each different measurement type.

First, consider a fused version of the closest point search in Step 2 which is required for both ICP and image-based approaches. In the case of ICP alone, the closest points are often obtained by performing a kd-tree (k-dimensional) for nearest neighbours search. Alternately, in the image-based approaches the image warping function finds the closest color values by view interpolation (nearest neighbour, bi-linear, bi-cubic,...) directly in image space. Of the recent hybrid approaches [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Whelan | Robust real-time visual odometry for dense rgb-d mapping[END_REF], each performs the closest point searching separately for both color and depth and no fused information is considered. Methods that consider both color and depth in the closest point matching stage include [START_REF] Edie | registration and integration of textured 3d data[END_REF], [START_REF] Morency | Stereo tracking using icp and normal flow constraint[END_REF], [START_REF] Michael Korn | Color supported generalizedicp[END_REF]. The former and later approach use 3 channels of color and differ in the color spaces used while [START_REF] Morency | Stereo tracking using icp and normal flow constraint[END_REF] considers only greyscale information. Finding the closest points using both color and depth increases the accuracy of finding the true nearest neighbour, however, this requires an efficient search in 4-space (3D points + intensity) or higher dimensional space. Now consider the joint optimization problem of the IRLS algorithm that minimizes both a fused ICP and image-based error. The large majority of classic approaches involve simply stacking the two error functions and minimizing the resulting joint error simultaneously [START_REF] Edie | registration and integration of textured 3d data[END_REF], [START_REF] Morency | Stereo tracking using icp and normal flow constraint[END_REF], [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Whelan | Robust real-time visual odometry for dense rgb-d mapping[END_REF], [START_REF] Michael Korn | Color supported generalizedicp[END_REF]. All except [START_REF] Edie | registration and integration of textured 3d data[END_REF] perform ICP point-to-plane combined with the image-based approach. The drawback of these approaches is that they require the definition of a tuning parameter λ which weights the respective contribution of each measurement. These methods, then vary in how this tuning parameter is determined. In [START_REF] Edie | registration and integration of textured 3d data[END_REF], λ is computed by estimating the ratio between the minimum and maximum values of both, color space and geometric errors and the best value is chosen experimentally in this range. [START_REF] Morency | Stereo tracking using icp and normal flow constraint[END_REF] proposes interestingly an adaptive λ which is varied using a sigmoidal function which favors the ICP approach far from the solution and the image-based approach close to the minimum. This has the benefit of faster convergence and more accuracy of the solution. In [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], the scale factor is automatically estimated as the ratio between the Median Absolute Deviations (MAD) of the color error and the median of the depth error (i.e. their relative robust variance). In [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Steinbruecker | Large-scale multiresolution surface reconstruction from rgb-d sequences[END_REF], λ is estimated by computing the covariance of the residuals for each point individually assuming a tdistribution of the error. This improved the convergence rate, however, is computationally expensive to iteratively compute a λ for each pixel.

The aim of this paper is to propose a unified framework for fusing both image-based and ICP strategies for pose estimation at each stage of the IRLS process. As will be shown, this leads to a novel Point-to-hyperplane ICP approach in 4 dimensions (3D + Intensity) which could easily be extended to greater dimensions (for example color RGB). This formulation also naturally leads to a fused closest point search strategy that exploits both color and geometric information simultaneously. The approach used in the paper to find the closest points in 4D space uses a kd-tree, however, alternative search strategies could also be used. In practice, and for computational efficiency, the ANN (Approximate Nearest Neighbour) [START_REF] Muja | Scalable nearest neighbor algorithms for high dimensional data[END_REF] algorithm is used. Furthermore, the kd-tree can be built only once from the reference image before the iterative loop, therefore maintaining efficiency.

The paper is organized as follows. Section II briefly explains the classic hybrid approach that jointly minimizes intensities and Point-to-plane ICP. In Section III a novel Point-to-hyperplane approach is introduced. Section IV provides the implementation details common to all the methods that were evaluated. Finally, simulated and real experimental results with benchmarks are presented in Section V.

II. JOINT METHOD FOR COMBINING GEOMETRIC AND PHOTOMETRIC APPROACHES

Pose estimation from hybrid methods is achieved by fusing the geometric and photometric optimization functions and minimizing the errors simultaneously. The main feature of hybrid methods for estimating the camera poses, is that they constrain the pose estimation better and can converge faster than using the techniques alone.

The pose will be defined here as the homogeneous pose matrix T (x) ∈ R 4×4 which depends on a minimal parameterization of 6 parameters which are defined here as the linear and angular velocity x = [υ, ω] ∈ R 6 , respectively. The homogeneous transformation matrix can be decomposed into rotational and translational components T(x) = (R(x), t(x)) ∈ SE [START_REF] Besl | A method for registration of 3-d shapes[END_REF]. The relationship between both is given by the exponential map as T(x) = e [x] , with the operator [•] as:

[x] = [ω] × υ 0 0 (1)
where [•] × is the skew symmetric matrix operator. The hybrid approach used to estimate the pose is depicted in Fig. 1(a). It defines an error function that minimizes the joint error between subsequent RGB-D image frames (see [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF] for more detail) such as:

e H i = ρ i   λ RR(x)N * i P m i -Π 3 TT (x) P * i I i w( TT(x), P * i ) -I * i (p * )   ∈ R 4 (2)
where the first row of ( 2) is the Point-to-plane ICP error with the projective data association and the second row is the photometric term. The superscript * identifies the reference measurements, Π 3 = [1, 0] ∈ R 3×4 is the projection matrix, N * i ∈ R 3 is the surface normal for each homogeneous 3D point P * i ∈ R 4 . The closest point P m i can be obtained by linearly interpolating the warped pixel coordinates into the current depth map as in [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF]. The geometric warping function w(•) projects a reference 3D point P * i ∈ R 3 onto the current image plane. The closest image intensity is then found by interpolation of the current intensity function at the warped pixel coordinates to obtain the corresponding intensity as:

I w i (p * i ) = I i (p w i ) ∈ Z.
The 3D point is computed by the back projection function as 3 , where K ∈ R 3×3 is the calibration matrix which contains the intrinsic parameters of the camera, and Z i ∈ R + is the metric measurement for each pixel coordinate

P i = K -1 p i Z i = X i Y i Z i ∈ R
p i = [u i v i 1] ∈ R 3 of the depth image.
The given non-linear error in ( 2) is minimized iteratively using a Gauss-Newton approach to compute the unknown parameter x with increments given by:

x = -(J WJ) -1 J W λ e G e I (3)
where J = [J I J G represents the stacked Jacobian matrices obtained by derivation of the stacked photometric and geometric error functions (e I and e G respectively), and the weight matrix W contains the weights ρ i associated to each set of coordinates obtained by M-estimation [START_REF] Huber | Robust statistics[END_REF]. The photometric Jacobian J I is computed using the efficient second order minimization method (ESM) [START_REF] Benhimane | Real-time image-based tracking of planes using efficient second-order minimization[END_REF]. The pose estimate T(x) is computed at each iteration and is updated incrementally as T ← TT(x) until convergence.

The parameter λ is a constant that scales the relative error distributions. As mentioned in the introduction, many methods have been proposed to estimate this parameter ranging from manual tuning to more complex estimation. Manually fixing λ is not optimal nor efficient and estimating the parameter requires extra computational cost. For the purposes of this paper, λ has been estimated as in [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF]. In the following section, is going to be seen that λ is not required if we consider a Point-to-hyperplane approach.

III. POINT-TO-HYPERPLANE METHOD

As mentioned in the introduction, the objective of this paper is to perform both closest point matching and minimization using a 4-vector containing color and depth. Since 4D space has an additional degree of freedom, the normal obtained for the 3D point-to-plane method will be orthogonal to a surface in 4D which spans both geometry and color. This surface will be referred in this paper as hyperplane. The 4-vector is defined as M i = P i I i ∈ R 4 , where the 3D Euclidean point P i is fused with its associated greyscale intensity I i in a single measurement vector.

Two measurement vectors obtained at different views of the same scene are generally not in correspondence. The fused error can then be defined as a 4D IRLS problem between two point clouds. The hybrid error function (case 1) is defined such as:

e H i = ρ i N * i (M * i -M m i ) ∈ R 4 = ρ i N * i P * i -Π 3 TT (x) P m i I * i -I i w( TT(x), P * i ) ∈ R 4 (4)
where M * i is the reference 4D point, M m i corresponds to the warped closest points to image according to the unknown transformation TT(x). This is similar to (2) except that the normal is computed in 4D and the closest points can be determined in 4D. Also, note that the current measurements in ( 4) are all warped to the reference frame so that it is no longer necessary to rotate the normals as in (2). This formulation seems better than in [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF] and the computational time is negligible.

Alternately, it is also possible to find the closest points in the current 4-vector (case 2) that solve the optimization problem as:

e H i = ρ i N m i (M m i -M w i ) ∈ R 4 (5)
where M w is the warped 4-vector. Solutions in both cases will be considered, but first lets consider the 4D normal.

Mathematically, at least four 4D-points are needed to compute three vectors that will be used to perform a threeway cross product to obtain the 4D-normal. The cross product does not, however, account for the uncertainty of the points in its computation. Instead, a Principal Component Analysis (PCA) can be computed on the covariance matrix of the nearest neighbours surrounding each point. In that case the smallest eigenvalue corresponds to the surface normal [START_REF] Segal | Generalized-icp[END_REF]. An alternative solution to compute normals fast and accurately is given in [START_REF] Badino | Fast and accurate computation of surface normals from range images[END_REF]. Considering that N m i is directly obtained from N * i , it is only necessary to compute the normals once for the reference image as in the case of the inverse compositional algorithm for image-based registration. The covariance matrix, used in computing the 4D-normal, allows to project and weight the errors such that the Pointto-hyperplane error is invariant to any tuning parameter λ (See [START_REF] Fernando | A proof that fusing measurements using point-to-hyperplane registration is invariant to relative scale[END_REF] for the proof).

Therefore, the error functions ( 4) or ( 5) can be minimized iteratively without scaling the geometric and photometric error such as:

x = -(J WJ) -1 J We H (6)
where e H is the stacked hybrid error

N * i (M * i -M m i ) or N m i (M m i -M w i
), and the weights ρ i are stacked in W. As presented above in (4) and ( 5) it is possible to find the closest points either in the reference frame or the current frame. Consider first case 1. Computing in the reference frame has the advantage that several parameters can be precomputed only once and not at each iteration. In that case the reference normals can be precomputed along with a quick search strategy for finding closest points such as a kd-tree. Consider now case 2. When finding the closest points in the current image it is not possible to recompute the kd-tree for each new image because it is computationally too expensive. In this case it is possible to consider approximating the closest point by simply search for the closest point in the image (as is done in ( 2)). In this case nearest neighbour, bilinear or bi-cubic interpolation can be performed [START_REF] Kerl | Dense visual slam for rgb-d cameras[END_REF], [START_REF] Meilland | On unifying key-frame and voxelbased dense visual SLAM at large scales[END_REF], [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF], [START_REF] Whelan | Robust real-time visual odometry for dense rgb-d mapping[END_REF]. In the case where no depth is provided by the RGB-D sensor for a pixel, it is rejected, unless it is possible to interpolate a depth value from neighboring pixels. Another strategy often used in the ICP literature (see for example [START_REF] Druon | Color constrained icp for registration of large unstructured 3d color data sets[END_REF], [START_REF] Henry | Experimental Robotics: The 12th International Symposium on Experimental Robotics, chapter RGB-D Mapping: Using Depth Cameras for Dense 3D Modeling of Indoor Environments[END_REF]), is to compute closest point matching only in the first iteration of the IRLS minimization loop. This allows to avoid to much computational complexity while obtaining the benefits of finding the closest points. In this paper a classic kd-tree with an ANN algorithm is considered so that large displacements will convergence to a solution. To estimate the closest points, the optimized search function of the FLANN library [START_REF] Muja | Scalable nearest neighbor algorithms for high dimensional data[END_REF] is employed to find the true nearest neighbours in 4D 1 . In a local tracking situation, faster projective data association techniques can be employed.

IV. IMPLEMENTATION DETAILS

In this section, some parameters considered for the experiments will be established. The experiments were done for real and synthetic RGB-D greyscale images in MATLAB. A multi-resolution pyramid was used to improve the computational efficiency. The images were warped at the second level of the pyramid (resolution 160 × 120).

There are two convergence criteria that stop the iterative loop for real and simulated experiments. The first one is a maximum number of iterations, which is established as 200, and the norm of the estimated rotation and translation. If the transformation matrix T(x) gets closer to the identity matrix, then the iterative loop stops. The parameters used to determine this second break are norm(x R ) < 1 × 10 -6 for rotation and norm(x t ) < 1 × 10 -5 for translation. 1 The legend NN4D is employed throughout the paper to indicate that the closest points were estimated in the first iteration only by finding the nearest neighbours with a kd-tree in the 4-vector. To reject outliers, M-estimators were employed. They are more general because they permit the use of different minimization functions not necessarily corresponding to normally distributed data. In this paper, the Huber influence function was used for this purpose.

The normal in 4D is adjusted to a 3 × 3 window that perform the PCA algorithm to find the covariance matrix of the fused error, leading to find the smallest eigenvalue which corresponds to the normal parameter.

All the experiments were validated on a workstation with Ubuntu 14.04, Intel Core i7-4770K and 16 GB RAM.

V. RESULTS

During the experimental part, it was seen that the parameter λ does not change the performance or accuracy of the pose estimation when the Point-to-hyperplane approach is used, but it improves the performance of other hybrid approaches if it is well estimated. An introduction to the improvement can be seen in Fig. 4, where the tracking trajectories estimated by 4 strategies are shown. In the first column the groundtruth obtained by an external motion capture system is shown, the column in the middle shown the results of the direct approach + 3D point to plane algorithm and the last column shown the result of the Point-to-hyperplane method. This difficult 360 degree sequence with motion blur clearly shows that the proposed method can achieve more robust estimations. The λ parameter was set initially using the automatic approach from [START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF] and was not tuned individually for each frame of the sequence. Therefore, the method could be improved with strategies as loop closure detection algorithms and keyframe detectors. [START_REF] Handa | A benchmark for rgb-d visual odometry, 3d reconstruction and slam[END_REF], [START_REF] Sturm | A benchmark for the evaluation of rgb-d slam systems[END_REF]. 1) Simulated environment -Tracking: A random transformation is applied to a reference RGB-D image. The new synthesized image is considered as the current image and the methods finds the alignment between each new image and the reference. The motivation for using synthetic data is that the generated images provide a groundtruth for evaluation, since the correspondences between the transformed views are known. The averages shown in Table II demonstrates that the Point-to-hyperplane method improves the number of iterations and computational time 2 .

TABLE I RELATIVE POSE ERROR (RPE) AND ABSOLUTE TRAJECTORY ERROR (ATE) FOR THE SIMULATED AND REAL DATASET

2) Real environment -Visual odometry: The methods were evaluated on the ICL-NUIM RGB-D benchmark dataset [START_REF] Handa | A benchmark for rgb-d visual odometry, 3d reconstruction and slam[END_REF] and on full benchmarks freiburg 1 sequences from TUM [START_REF] Sturm | A benchmark for the evaluation of rgb-d slam systems[END_REF]. Therefore, in order to evaluate the estimated trajectories frame-to-frame, the online tool was used with the default settings to evaluate the ATE (Absolute Trajectory Error) and RPE (Relative Pose Error), which are compared alongside an accurate groundtruth trajectory in Table I. Some examples of the improvement are shown in Fig. 2. A demonstration of the performance ot the proposed method can be seen in the video attachment of this paper.

VI. CONCLUSION

A novel Point-to-hyperplane strategy was proposed based on a 4D-vector. Two main advantages of this unified framework are underlined. First, it is shown that the sensor pose can be estimated by minimizing the combined error without computing a scale parameter between them. Second, it is shown that nearest neighbour techniques can be used in 4D-space to improve the convergence rate and domain for IRLS pose estimation. Experimental results and analysis are provided which compare two variants of the new Pointto-hyperplane approach with the classic hybrid approach. The results show improved computation time and faster convergence on well known benchmarks. Future work will be dedicated to testing the new approach in a simultaneous localization and mapping context which will provide for interesting comparisons with the map reconstruction. The framework would also apply directly to volumetric representations by modifying the projective error function accordingly. We will also extend the strategy to higher dimensions by using color and other measurement to increase robustness.

  Fernando I. Ireta Muñoz 1 and Andrew I. Comport 2

Fig. 1 .

 1 Fig. 1. Two hybrid-based approaches to estimate the unknown pose x between two sequent RGB-D frames. (a) A direct approach for the color images I and I * , and a point-to-plane algorithm for the geometric cloud of points P and P * is used. (b) Proposed method: a matching stage considers the 4-vector for minimizing the integrated error with the point-to-hyperplane algorithm, which computes the normals N in 4D space. The scale factor λ is no longer needed.

Fig. 2 .

 2 Fig.2. Examples of the Absolute Trajectory Error evaluation, the first and third column show the results obtained by methods that combine the direct approach with the geometric Point-to-plane method. The second and fourth column for the point-to-hyperplane method. The trajectories presented here are obtained for the simulated (a)(b) lvr/traj0, (c)(d) lvr/traj2,[START_REF] Handa | A benchmark for rgb-d visual odometry, 3d reconstruction and slam[END_REF] and the real (e)(f) fr1/xyz, (g)(h) fr1/room[START_REF] Sturm | A benchmark for the evaluation of rgb-d slam systems[END_REF] sequences. More sequences are shown in[START_REF] Fernando | A proof that fusing measurements using point-to-hyperplane registration is invariant to relative scale[END_REF].

Fig. 4 .

 4 Fig. 4. Example of 4 estimated camera trajectories between a pair of images with a random pose in a simulated environment. The green and red dot indicate the initial and the final pose. The Point-to-hyperplane methods improve the Point-to-plane + direct methods, obtaining more direct trajectories when NN4D are estimated. 1000 synthesized images with a random pose were equally tested, obtaining a similar performance.

  Fig.3. 3D reconstruction of sequences fr1/room and fr1/360 (first and second row, respectively). In the first column the groundtruth obtained by an external motion capture system is shown, the column in the middle shown the results of the direct approach + 3D point to plane algorithm and the last column shown the result of the Point-to-hyperplane method. This difficult 360 degree sequence with motion blur clearly shows that the proposed method can achieve more robust estimations. The λ parameter was set initially using the automatic approach from[START_REF] Tykkälä | Direct Iterative Closest Point for Real-time Visual Odometry[END_REF] and was not tuned individually for each frame of the sequence. Therefore, the method could be improved with strategies as loop closure detection algorithms and keyframe detectors.

  IT CAN BE SEEN THAT THE POINT-TO-HYPERPLANE METHODS (3 & 4) IMPROVE THE HYBRID METHODS THAT COMBINE THE DIRECT APPROACH AND THE GEOMETRIC POINT-TO-PLANE (1 & 2) IN THE MAJORITY OF DATASET FOR THE RPE TRANSLATIONAL EVALUATION AND IN ALL DATASET FOR RPE ROTATIONAL EVALUATION. NOTE THAT THE ONLY DIFFERENCE BETWEEN 1 & 2 AND 3 & 4 IS THE KD-TREE MATCH ON THE FIRST ITERATION, WHICH ONLY AFFECTS COMPUTATION TIME AND NOT ACCURACY.

  THE REMAINING ITERATIONS USE THE SAME ERROR FUNCTION.

	Sequence Method	RPE translational (m) RMSE MEAN STD	RPE rotational (deg) RMSE MEAN STD	ATE (m) RMSE MEAN	STD
	fr1/xyz	1 & 2 3 & 4	0.033 0.021	0.030 0.019	0.014 0.008	2.025 1.106	1.741 0.998	1.034 0.477	0.095 0.045	0.087 0.038	0.039 0.024
	fr1/rpy	1 & 2 3 & 4	0.062 0.038	0.050 0.032	0.037 0.020	3.161 2.820	2.887 2.652	1.288 0.959	0.136 0.035	0.115 0.032	0.072 0.015
	fr1/360	1 & 2 3 & 4	0.146 0.152	0.118 0.114	0.086 0.100	4.171 3.159	3.844 2.859	1.621 1.343	0.520 0.322	0.484 0.296	0.188 0.125
	fr1/room	1 & 2 3 & 4	0.076 0.056	0.060 0.047	0.048 0.030	3.285 2.673	2.912 2.329	1.520 1.313	0.434 0.174	0.404 0.152	0.158 0.086
	fr1/desk	1 & 2 3 & 4	0.047 0.044	0.039 0.036	0.027 0.025	2.826 2.309	2.503 2.027	1.312 1.106	0.108 0.071	0.104 0.067	0.029 0.023
	fr1/desk2	1 & 2 3 & 4	0.058 0.060	0.051 0.051	0.027 0.031	3.483 3.026	3.026 2.641	1.725 1.478	0.189 0.133	0.174 0.116	0.075 0.065
	fr1/floor	1 & 2 3 & 4	0.094 0.080	0.038 0.051	0.086 0.062	4.660 3.909	1.953 1.915	4.231 3.408	0.772 0.473	0.666 0.405	0.391 0.244
	fr1/plant	1 & 2 3 & 4	0.106 0.055	0.067 0.043	0.082 0.034	3.941 2.130	3.223 1.947	2.268 0.864	0.324 0.101	0.296 0.093	0.132 0.037
	fr1/teddy	1 & 2 3 & 4	0.096 0.070	0.081 0.056	0.051 0.043	3.410 2.287	3.021 1.954	1.583 1.187	0.615 0.169	0.553 0.158	0.271 0.059
	lvr/traj0	1 & 2 3 & 4	0.001 0.002	0.001 0.001	0.001 0.002	0.044 0.042	0.035 0.026	0.027 0.033	0.128 0.050	0.114 0.046	0.057 0.019
	lvr/traj1	1 & 2 3 & 4	0.002 0.001	0.001 0.001	0.001 0.001	0.048 0.021	0.041 0.017	0.024 0.013	0.114 0.041	0.104 0.032	0.046 0.026
	lvr/traj2	1 & 2 3 & 4	0.002 0.001	0.001 0.001	0.001 0.001	0.044 0.024	0.039 0.019	0.021 0.014	0.074 0.039	0.067 0.036	0.030 0.016
	lvr/traj3	1 & 2 3 & 4	0.002 0.001	0.001 0.001	0.001 0.001	0.070 0.044	0.053 0.027	0.045 0.035	0.218 0.080	0.202 0.066	0.082 0.045

TABLE II AVERAGES

 II IN TIME AND IN NUMBER OF ITERATIONS UNTIL CONVERGENCE FOR 1000 SYNTHESIZED IMAGES AT RANDOM POSES.

	Method	# Iterations Time (sec)
	1) Intensity + point-to-plane	65.65	0.61
	2) Intensity + point-to-plane (NN4D)	34.23	0.35
	3) Point-to-hyperplane	53.24	0.56
	4) Point-to-hyperplane (NN4D)	12.83	0.16

ireta@i3s.unice.fr

Andrew.Comport@cnrs.fr

The time shown does not consider the computation of normals nor construction of the kd-tree. A more detailed computational analysis can be found in[START_REF] Fernando | A proof that fusing measurements using point-to-hyperplane registration is invariant to relative scale[END_REF].

supported by the European H2020 project: COMANOID, and CONACYT, México.


J Degallaix
email: j.degallaix@lma.in2p3.fr

J Komma

D Forest

G Hofmann

M Granata

D Heinert

C Schwarz

R Nawrodt

L Pinard

C Michel

R Flaminio

G Cagnoli

Measurement of the optical absorption of bulk silicon at cryogenic temperature and the implication for the Einstein Telescope

Keywords: silicon, Einstein Telescope, optical absorption PACS numbers: 78.40.Fy, 04.80.Nn, 95.55Ym

We report in this article on the measurement of the optical absorption of moderately doped crystalline silicon samples at 1550 nm, which is a candidate material for the main optics of the low temperature interferometer of the Einstein Telescope (ET). We observe a nearly constant absorption from room temperature down to cryogenic temperatures for two silicon samples presenting an optical absorption of 0.029 cm -1 and 780 ppm cm -1 , both crystals doped with boron. This is in contradiction to what was assumed previously-a negligible optical absorption at low temperature due to the carrier freezeout. As the main consequence, if the silicon intrinsic absorption can not be lowered, the cross section of the mirror suspension of the ET must be increased to be able to carry away the excess heat generated by the partially absorbed laser beam during the operation of the interferometer.

Introduction

The Einstein Telescope (ET) is a European project aimed at directly detecting gravitational waves from astrophysical sources on a daily basis [START_REF] Punturo | The Einstein telescope: a third-generation gravitational wave observatory[END_REF]. This detector is the planned successor of the Advanced LIGO and Advanced Virgo interferometers currently in their installation phase and early commissioning [START_REF] Losurdo | Ground-based gravitational wave interferometric detectors of the first and second generation: an overview[END_REF]. Those advanced detectors are expected to achieve the first direct detection of a gravitational wave within the next five years and hence open a new window to study the universe, complimentary to our current vision based on the electromagnetic spectrum.

To greatly increase the detection rate in the following decade, the ET has been designed to be ten times more sensitive than the advanced detectors [START_REF] Punturo | The Einstein telescope: a third-generation gravitational wave observatory[END_REF], probing a volume of space a thousand times larger. To achieve this goal, this observatory will be composed of two complementary interferometers sharing a common underground infrastructure. One high frequency interferometer (ET-HF) will be at room temperature with fused silica mirrors similar to the advanced detectors. The low frequency interferometer (ET-LF) will be more innovative using silicon mirrors operating at cryogenic temperatures.

Since the cavity input test masses made of silicon will be used in transmission, the optical absorption is an essential parameter which determines the amount of heat generated in the optics, which is equal to the amount of heat to be removed through the suspensions in order to keep the mirrors at their operating cryogenic temperature [START_REF] Tomaru | Maximum heat transfer along a sapphire suspension fiber for a cryogenic interferometric gravitational wave detector[END_REF]. Additionally it seems unlikely that high purity silicon (or even intrinsic silicon having no significant doping/impurities) will be available in large enough pieces being suited for ET. Thus the focus of this work lies on moderately doped silicon that can be provided by Czochralski growth of single crystals; by 'moderately' we mean samples with resistivities ranging from few Ω cm to few kΩ cm.

Contrary to the low temperature regime, the optical absorption of bulk silicon at room temperature and in the near infrared is well understood and characterized [START_REF] Soref | Electrooptical effects in silicon[END_REF][START_REF] Degallaix | Bulk optical absorption of high resistivity silicon at 1550 nm[END_REF]. The optical absorption is proportional to the amount of free carriers (the proportionality coefficient been simply called the free carrier coefficient). At room temperature the concentration of the free carriers is itself equal to the concentration of dopant impurities as every atom of dopant is ionized and gives either one electron in the conduction band (n-type) or one hole in the valence band (p-type). In the case of moderately doped samples all dopants are ionized at room temperature. Thus, the concentration of free carriers is equal to the concentration of doping elements.

At cryogenic temperatures, below 40 K, the thermal energy is not enough to ionize the impurities and hence the amount of free carrier drops and rapidly becomes very small. It was reasonably expected that the optical absorption of such samples also drops rapidly proportionally to the free carrier concentration. However, we demonstrate in this paper that although the concentration of free carriers drops as expected the optical absorption nearly stays constant throughout a wide temperature range from 300 K down to 5 K.

The measurements are done using two independent techniques based on photodeflection and calorimetry. Using a third technique (based on spectrophotometric measurement) Spitzer and Fan [START_REF] Spitzer | Infrared absorption in n-type silicon[END_REF] already showed no change in absorption for highly doped silicon wafers from room temperature down to 5 K. This article is organised as follows: first the two techniques used to measure the optical absorption are explained; second, the experimental setup as well as the silicon samples tested are described and third the experimental results are presented. Finally, the consequences of theses measurements for the ET are derived.

Measurement techniques and experimental setups

In this section, the two experimental techniques used to derive the optical absorption of bulk silicon are presented. The two techniques applied in this paper-the mirage effect and the heating of the sample by light-are sensitive enough to measure the absorption of moderately doped bulk silicon, the expected material of the ET cryogenic mirror. For samples with higher optical absorption (larger than 0.1 cm -1 ), a direct measurement of the transmission is enough to derive the absorption [START_REF] Spitzer | Infrared absorption in n-type silicon[END_REF].

After the measurement principles are summarized, the experimental setups as well as the silicon samples which have been measured are explained in detail.

The photothermal deflection technique

The photodeflection technique based on the mirage effect is one of the most sensitive methods to measure low optical absorption in materials [START_REF] Amer | Photothermal deflection spectroscopy and detection[END_REF]. Using this very same technique, it has already been shown that pure silicon can present an absorption as low as 5 ppm cm -1 at room temperature [START_REF] Degallaix | Bulk optical absorption of high resistivity silicon at 1550 nm[END_REF].

The typical setup of a photodeflection bench is composed of two lasers: one high power laser called the pump laser and one low power laser, the probe laser. The pump laser is focused on the material and so is partially absorbed within the sample. The absorbed optical power, converted to heat creates a gradient of temperature. The gradient of temperature induces a gradient of refractive index via the thermo-optic coefficient. The probe beam is aligned to intersect the pump beam in the sample and is then deflected by the gradient of refractive index. The deviation of the probe beam is read out by a quadrant photodetector and is directly proportional to the amount of power absorbed in the sample and hence to the optical absorption.

In order to increase the signal-to-noise-ratio of the deviation method, the pump beam is intensity modulated and the deviation is then detected by a lock-in amplifier. The calibration which links the deviation amplitude to the optical absorption is performed with a sample with a high optical absorption that was directly measured by a spectrophotometer.

Calorimetric measurement

A calorimetric measurement technique determines the temperature increase of a sample under illumination by a laser of wavelength under study [START_REF] Tomaru | Cryogenic measurement of the optical absorption coefficient in sapphire crystals at 1.064 μm for the large-scale cryogenic gravitational wave telescope[END_REF]. The temperature increase is directly linked to the optical absorption if the heat capacity of the setup as well as all thermal losses are known. Calorimetric techniques using lasers as a light source are routine techniques allowing the measurement of absorptions below 1 ppm cm -1 for high purity optical elements or fiber materials [START_REF] Willamowski | Measuring the Absolute Absorptance of Optical Laser Components[END_REF][START_REF] Stone | Calorimetric measurement of absorption and scattering losses in optical fibers[END_REF].

A laser beam is passed through a sample and a temperature sensor is measuring the increase in temperature under illumination. Care must be taken to avoid any scattered light to hit the sensor directly. This method probes the integral optical absorption of the sample-both possible surface absorption as well as intrinsic bulk absorption. By changing the length of the sample it is possible to disentangle these effects.

In order to reduce extrinsic surface absorption effects great care was taken to clean the samples. A sophisticated cleaning technique available from the semiconductor industry was used and described in section 3.1.

The calorimetric technique becomes especially very sensitive at cryogenic temperatures as the heat capacity of the samples rapidly drops below 50 K following Debye's law (∝T 3 ).

Description of the setup at LMA

The photodeflection bench used for the measurement presented in this article is composed of a 30 W pump laser at a wavelength of 1550 nm which is the same wavelength planned for ET and a 1 mW probe laser at 1310 nm. The wavelength of the probe beam is selected to be in the transparency region of the silicon and close enough to 1550 nm to ensure a similar thermorefractive coefficient [START_REF] Frey | Temperature-dependent refractive index of silicon and germanium Astron[END_REF]. The pump beam is focused in the silicon sample with a measured waist radius of 0.7 mm (measured with the knife edge technique at the position of the sample) resulting in power densities of maximum 4 kW cm -2 .

The tested silicon sample is located in a continuous flow cryostat linked to a liquid helium reservoir. The cooling from room temperature to 10 K is achieved within 24 hours with the sample mounted to a 20 cm diameter cryogenic breadboard via a holder made of copper. A calibrated temperature sensor (Cernox-1070-BO) is directly attached to the sample to continuously record the temperature of the silicon substrate, as shown in figure 1(a). Several optical glass windows with a diameter of 2 inches have been installed to ensure the correct injection and extraction of the pump and probe laser beams. Further details on the cryostat performance can be found in a previous article [START_REF] Degallaix | The new cryogenic facility at LMA[END_REF].

For the LMA cryogenic run described in this article, the tested silicon sample has a diameter of two inches and is 20 mm long. The sample is doped with boron atoms, giving an averaged resistivity of 2.4 Ω cm (concentration of boron atoms of × 5.7 10 15 cm -3 ). The optical absorption at room temperature due to the free carriers was measured to be 0.029 cm -1 at 1550 nm.

Description of the setup at the FSU Jena

The setup at the FSU Jena consists of a 5 W CW 1550 nm fiber laser system (NKT Photonics Koheras Adjustic seed laser and Boostik HPA amplifier) and a custom-made bath cryostat [START_REF] Nawrodt | A new apparatus for mechanical Q-factor measurements between 5 and 300[END_REF] for measurements between room temperature and 5 K. The laser beam has a measured waist radius of 0.9 mm inside the sample resulting in a power density of 0.4 kW cm -2 . The sample under investigation is a float zone 100 Ω cm silicon single crystal having a diameter of 150 mm and a thickness of 86 mm as shown in figure 1(a). The concentration of boron atoms is about 1 × 10 14 cm -3 .

A temperature sensor (Cernox CX1070-SD) is directly attached to the sample by means of varnish (VGE 7031). Additionally, an electrical heater is attached to the sample for calibration purposes. Using such a heater avoids the need of a careful treatment of heat losses in the setup. Applying an electric heating power of 1 mW results in the same heating as is caused by 1 mW of absorbed optical power-independent of all heat losses through the mount, wires or even radiation. These heat losses are kept small by avoiding too much heating of the sample as all heat losses are driven by temperature differences. Typically, no temperature difference larger than 100 mK was used in the experiments.

The sample is mounted inside the cryostat and a weak thermal link is established between the sample and the cold plate. For initial cooling He gas can be applied to the probe chamber allowing an efficient and fast cooling of the sample. The laser beam is aligned in a way that the sample is illuminated under the Brewster angle to avoid unwanted reflections. By means of a set of mirrors the laser beam is then aligned in a way that it leaves the cryostat through the same window as it enters. This technique is needed as the cryostat is only equipped with one window.

Experimental investigations

The optical absorption measured at low temperature is summarized in the following sections and compared to the values found at room temperature.

Investigation of a possible surface absorption

Optical absorption measurements done before on a monolithic cavity [START_REF] Khalaidovski | Indication for dominating surface absorption in crystalline silicon test masses at 1550 nm[END_REF] reported a high surface absorption. Also, the calorimetric measurements will be extremely sensitive to such absorptions-either being caused by intrinsic surface effects or simply dirt on the surface. Thus, a special run was initially carried on to check the cleaning procedure used for the samples. All samples have been cleaned using a hot mixture of sulfuric acid (96%) and hydrogen peroxide (35%). This mixture does not attack silicon but efficiently removes all organic contamination from the surface. After this cleaning process all samples are given a methanol rinse using ultra pure solvents.

As the surface is the potential origin of unwanted absorption thin wafers have been used for testing surface effects. These wafers are made from high purity material (typically between 10-100 kΩ cm) whose intrinsic absorption is expected to be below 4 ppm cm -1 [START_REF] Degallaix | Bulk optical absorption of high resistivity silicon at 1550 nm[END_REF]. Such samples have been used to refine the cleaning procedure. A minimum average absorption over at least ten subsequent runs of ± (12 9) ppm has been observed. As all absorption values reported in this paper are well above 100 ppm cm -1 any influence of the surface to our measurements can be excluded. Additionally, these results make the hypothesis of potential intrinsic strong surface absorptions in silicon very unlikely [START_REF] Khalaidovski | Indication for dominating surface absorption in crystalline silicon test masses at 1550 nm[END_REF].

Freeze-out of free carriers

As the optical absorption of silicon is dominated by free carrier absorption at room temperature, it is an interesting step to investigate the change of free carriers during cooling. In order to do so, a moderately doped silicon wafer sample (1-10 Ωcm, p-type boron-doped) was contacted in a four point geometry as shown in figure 2. Four aluminium pads were evaporated onto the surface of the sample forming electronic contacts. Low contact resistivity Ohmic contacts have been achived by a careful annealing of the sample. The four contact pads have been attached to wires of a cryogenic probe station by means of indium. The two outer contacts have then be used to feed a constant current through the sample while the inner contacts have been used to measure the voltage created by the current. An Agilent B2901A SMU has been used to apply the current and measure the voltage. The finite geometry of the sample in use (typically 10 mm ×10 mm) has been taken into account numerically by means of a FEA using COMSOL .

Figure 3 shows the results obtained from a voltage-current-measurement over a wide range of temperatures from 5 K to 300 K. It can be seen that the resistivity starts at around 8 Ω cm at room temperature, decreases to about 1 Ω cm at 80 K and then steeply rises to more than 10 MΩ cm at cryogenic temperatures. Here, the measurement is limited by the finite input resistance of the pre-amplifier in use.

The resistivity ρ of a semiconductor is given by [START_REF] Grundmann | The Physics of Semiconductors: An Introduction Including Nanophysics and Applications[END_REF] ρ = -+ enμ epμ

1 (1) n p
with the mobility of the electrons μ n and the holes μ p and the density of electrons n and holes p. In the case studied here holes are mainly carrying the current (as the sample is p-doped). Thus, the concentration of electrons can be approximated to be zero. The observed behaviour of the resistivity by changing the temperature cannot be described by the temperature dependence of the mobility μ p (see e. g. [START_REF] Jacoboni | A review of some charge transport properties of silicon Solid-State Electron[END_REF]) thus the change in hole concentration is the only 3. Measured values of the resistivity of the silicon sample (1-10 Ω cm, boron doped) under investigation in a wide temperature range from 5 K to 300 K. explanation for the measured trend presented in figure 3. The concentration of free carriers is therefore strongly reduced by cooling below 50 K.

Bulk absorption measurement at LMA

The photodeflection technique presented in section 2.1 has been very successful at room temperature however it remains an indirect measurement where physical parameters of the substrate play an important role. For example, the deviation of the probe beam is different for two samples presenting the same optical absorption if they have different thermal conductivities or thermo-refractive coefficient.

The same phenomenon happens when the temperature of the silicon sample changes. The measured deviation will change not only because of a possible change in the absorption but also since the specific heat, the thermal conductivity and the thermo-refractive coefficient depend on temperature. We can already predict that the deviation signal for a constant optical absorption will decrease with temperature as the thermooptic coefficient decreases and the thermal conductivity increases for silicon substrate.

More precisely, for silicon the thermo-refractive coefficient will decrease by a factor 100 from room temperature to 20 K [START_REF] Komma | Thermo-optic coefficient of silicon at 1550 nm and cryogenic temperatures[END_REF], hence reducing the gradient of refractive index responsible for the deviation of the probe beam. That would be the major effect, the change in thermal conductivity [START_REF] Thompson | Thermal conductivity of silicon at low temperatures[END_REF] and specific heat having a smaller effect [START_REF] Flubacher | The heat capacity of pure silicon and germanium and properties of their vibrational frequency spectra Phil[END_REF]. That statement is valid for our setup and dependent on the modulation frequency of the pump beam as well as the size of the pump laser beam. So we already expect that the deviation signal will decrease by a factor 100 between room temperature and cryogenic temperature if the absorption is constant. So, for this reason, we tested a sample with relatively high optical absorption (0.029 cm -1 ) in order to have large enough deviation signal from 300 K to 10 K (the SNR of the deviation signal is around 2 × 10 3 at room temperature).

The theoretically predicted deviation signal as a function of temperature is shown as the dashed line in figure 4, and assuming a constant absorption for the silicon sample. On that figure, the deviation is normalized to be 1 at room temperature. The measured deviation, also normalized to the value at room temperature, is also added to the figure.

The deviation has been measured at different pump powers to rule out nonlinear effects. Also near room temperature, high pump power is used to generate a noticeable rise in the temperature of the sample in order to still derive the absorption also from calorimetric measurement. The deviation measurement were also normalized with the pump power to compare all the data points to the same theoretical curve.

At low temperature, below 50 K, we can notice from graph 4 that the measured deviation matches the theoretical deviation from constant absorption within 20%. This result indicates that the optical absorption is the same at room temperature and low temperature for bulk silicon.

At the same time as the measurement is done with the photodeflection technique, the temperature of the sample is monitored. In that way, the heating due to the pump laser beam is recorded. Such a plot of the temperature evolution is shown in the left part of the figure 5.

To derive the heating power from the temperature rise a fit is performed. The fitting function is given by solving the heat equation in the simple case of assuming an isothermal behaviour of the sample. This assumption is valid for silicon since the thermal conductivity is very high in the temperature range under investigation (with the thermal conductivity being always larger than 150 W (m K) -1 ). Such fits are presented in figure 5. The fit is remarkably good, considering the simplicity of the model and that only two parameters are fitted: the absorbed optical power and the heat lost by conduction through the mount of the sample.

From the fit, at low temperature (22 K), the heating power absorbed by the sample is found to be 0.11 W for an input laser of 2 W, giving an optical absorption of 0.030 cm -1 . At room temperature we found an absorbed power of 0.44 W for an input power of 8 W which gives the same absorption as at low temperature. The uncertainty on such measurement is within five percent coming from the uncertainty on the specific heat (especially at low temperature) and on the measured laser power. The measured absorption at room temperature derived from the calorimetric measurement is within 2% of the value found by spectroscopic measurement.

The results from the calorimetric measurement suggest that no decrease (nor increase) in the optical absorption of silicon should be expected by cooling the substrate from room Example of a temperature rise at cryogenic temperature (a) and room temperature (b) when the pump laser is switched on for few minutes. For the left plot, the pump (heating) laser is switched on at 94 s and switched off at 164 s and for the right plot the times are respectively 48 s and 193 s. The laser power is set to 2 W at cryogenic temperature but to 8 W at room temperature in order to create a noticeable temperature rise. temperature to 20 K. That is the same conclusion as the measurement based on the photodeflection technique.

Bulk absorption measurement at the FSU Jena

Figure 6 represents a typical set of measured values at cryogenic temperatures. The temperature of the sample is first stabilized and recorded for a given time of a few minutes. Then the laser illuminates the sample for a certain time interval with a certain laser power. Illumination time and laser power are adjusted in a way that the resulting temperature increase can be easily measured and is not too large. Under this assumption the thermal properties of the sample can be assumed to be constant. Typical temperature differences of a few 10 mK are used. After the thermal equilibrium is reached again an electrical heater is used to measure the temperature increase for a given electrical heating power. The power is roughly adjusted to give a similar temperature increase during this period. This method allows to eliminate unknown parasitic heat fluxes to the surroundings.

Heating the sample over a time Δt with a power P results in a temperature increase ΔT of

Δ Δ = P t C T ( 2
)
with C being the net heat capacity of the sample. This is independent whether the power is delivered by a laser or an electrical heater. Using the subscript L for laser and H for heater allows the substitution of the unknown heat capacity C. The absorbed laser power P L is then given by

Δ Δ Δ Δ = P T T t t P . ( 3
) L L H H L H
All input parameters can be easily measured assuming that the heat capacity is the same during laser illumination and electrical heating. Under the experimental constraints used in this study this is always fulfilled. The absorbed laser power P L is linked to the illuminating laser power P 0 , the absorption coefficient α and the optical path of the laser beam inside the sample L by means of Figure 6. Example for a calorimetric measurement run. The sample is heated by a laser (a) and by an electrical heater (b). The electric heating is used for calibration purposes. The laser power was 0.26 W and the electrical heater power 0.73 mW. In both curves the heating time was 30 s.

L L 0 Equations ( 3) and ( 4) are used to determine the optical absorption α. The measurement of the 150 mm bulk sample shows the same absorption at room temperature and at 6 K, which is 780 ppm/cm. An independent cross-check using the photodeflection technique at LMA revealed a similar value for the room temperature value. Thus, also this second sample exhibiting a reduced concentration of dopants shows similar absorption values at 300 K and 6 K. This independently confirms the findings on the sample previously discussed in section 3.3 and suggests that the obtained behaviour might be inherent to silicon.

The measurement has been repeated at least ten times at the different temperatures resulting in a typical statistical fluctuation of 10% at room temperature and 1% at cryogenic temperatures. The higher accuracy at low temperatures is based on the larger temperature difference that needs to be measured due to the smaller heat capacity. Further including the inaccuracy of the laser power sensors as well as of the heater power results in a total relative error of 20% at room temperature and 10% at cryogenic temperatures.

Consequences for ET

To derive the numbers presented in this section, several parameters of the ET have to be taken into account. Relevant parameters are taken from the ET conceptual design study [START_REF]ET science team 2012 Einstein Gravitational Wave Telescope Conceptual Design Study code[END_REF] and are presented below.

The circulating power in the core interferometer is expected to be 65 W (measured on the recycling mirror) and the power circulating in the arm cavity is 18 kW, that is also the power incident to the high reflectivity coating of the input mirror.

The silicon mirror has a cylindrical shape with a diameter of 50 cm and a thickness of 46 cm. The operating temperature is assumed to be 10 K and at the time of writing the design study the absorbed power was estimated to be 20 mW.

Potentially, Czochralski grown silicon having a resistivity between 500 Ω cm and 1000 Ω cm will be available in large enough pieces for optics that can be used in ET-LF [START_REF] Nishi | Handbook of Semiconductor Manufacturing Technology[END_REF].

Here, silicon single crystals with a diameter of 50 cm and thicknesses of about 50 cm are needed. Higher purities (and hence nominally low absorptions [START_REF] Degallaix | Bulk optical absorption of high resistivity silicon at 1550 nm[END_REF]) can be obtained by means of the float zone technique-however, it is not expected that it can be produced in such large pieces within the next years as currently a technical limitation in size to about 200-250 mm exists.

As we have shown in this article, the optical absorption at cryogenic temperatures is roughly the same as at room temperature. Thus, silicon of the purity discussed here will have an absorption of about 300 ppm cm -1 throughout the whole temperature range from 300 K down to 5 K [START_REF] Degallaix | Bulk optical absorption of high resistivity silicon at 1550 nm[END_REF]. This assumption is already conservative accounting for a safety margin in the following estimates.

The most crucial mirror affected by optical absorption is the input test mass of the arm cavities of ET-LF. Here, a laser beam of approximately 65 W will pass through the whole material made of silicon. The total absorption is then given by α = × × P L P (5) abs 0

with the absorption coefficient α, the length of the substrate L and the laser power P 0 . Using the above values, the total absorbed power within the silicon material is around 1 W. This heat needs to be removed by the suspension elements being planned to be fabricated by highly conductive silicon material.

Assuming a suspension length of 1 m, the equilibrium temperature of the mirror was calculated using an FEA model in COMSOL. Here, the temperature dependence of the thermal conductivity of silicon as well as its size dependence was accounted for. As a simplification a single suspension fiber was assumed-a simple scaling using the real number and length of the fiber can be done easily. Figure 7 summarizes the results.

It can be directly seen that the expected operational temperature of 10 K at the test mass for ET-LF is only possible with a total cross section of more than 150 mm 2 . This might result in a very stiff suspension showing bad thermal noise performance [START_REF] Cumming | Silicon mirror suspensions for gravitational wave detectors[END_REF]. Thus, a higher operational temperature for the ET-LF mirrors is preferred which is also compatible with the tolerable mirror thermal noise level. At around 20 K a total cross section of about 25 mm 2 is needed which can be provided by four suspension wires or ribbons.

A further reduction of the temperature of the cold end does not give any benefits for the heat extraction as the thermal conductivity of silicon is small between 2 K and 4 K. This does not lead to a better cooling performance. However, using LHe in the superfluid state might result in a dramatic reduction of mechanical disturbances and could be appealing for the overall performance.

In parallel to the development of suitable cryogenic suspension, research on silicon substrates will continue with the aim to find the silicon with the lowest possible absorption at low temperature and available in sizes required for ET.

The larger thermal absorption in the bulk material relaxes the constraints on the coating absorption significantly. Considering the laser power of 18 kW in the interferometer arm cavities and assuming that the coating is allowed to absorb a tenth of power absorbed in the substrate (i. e. 0.1 W), a maximum coating absorption of 5 ppm (≈0.1 W 18 kW) can be tolerated. A higher tolerable coating absorption has some important consequences since it opens the possibility to coating technology different from the usual alternative layers of fused silica/tantala [START_REF] Harry | Titania-doped tantala/silica coatings for gravitational-wave detection[END_REF]. More specifically, crystalline coating [START_REF] Cole | Tenfold reduction of Brownian noise in high-reflectivity optical coatings[END_REF] or hafnia layers [START_REF] Abernathy | Cryogenic mechanical loss measurements of heat-treated hafnium dioxide[END_REF] to replace tantala may be acceptable. Equilibrium temperatures in the input test mass assuming a realistic optical absorption as well as a temperature dependent thermal conductivity for silicon. The temperature at the cold end of the fiber was assumed to be 4 K, the lengths of the fiber is set to 1 m.

Conclusion

Based on independent measurements on two silicon samples, we have shown that no significant modification in the optical absorption at 1550 nm of bulk crystalline silicon must be expected by cooling the substrate from room temperature to cryogenic temperature. This result is in contradiction to what was assumed for the cryogenic silicon mirrors of the ET, since it was thought that the free carrier freezeout happening at low temperature will guarantee a negligible optical absorption. Further, no significant surface absorption has been observed during our measurement.

Our conclusion implies that a relatively large amount of heat will have to be extracted from the cryogenic mirrors in order to keep the desired temperature of the interferometer if the current silicon absorption cannot be lowered. As a result, in the current design of the suspension the cross section of the fiber holding the mirrors must be significantly increased with the risk of increasing the thermal noise. Further detailed studies are needed to optimise the shape, material selection as well as the optimum operational temperature.

If the heat absorbed within the mirror substrate is relatively high, the constraints on the other potential sources of heat occurring in the mirrors can be relaxed. In particular, the coating absorption could be as high as 5 ppm and more heat from radiation could be tolerated from the room temperature vacuum enclosure.

According to our tests for all practical purposes the absorption can be considered constant between room temperature and low temperature. So, in order to select the suitable silicon quality for the ET mirrors, a measuring campaign for the optical absorption at room temperature is likely to be sufficient without the need of expensive cryogenic runs. Of course, once a particular quality of silicon has been selected, a confirmation of the absorption at low temperature will still be required.

The measured absorption at low temperature is so far unexplained by any theory known by the authors. Research to understand why the absorption is only marginally dependent on the temperature is ongoing.

  Classical and Quantum Gravity Class. Quantum Grav. 31 (2014) 185010 (13pp) doi:10.1088/0264-9381/31/18/185010

Figure 1 .

 1 Figure 1. Photography of the silicon samples in the cryostats at LMA (a) and at the FSU Jena (b). See the text for additional details.

Figure 2 .

 2 Figure 2. (a) 4-probe-setup for measuring the resistance of the silicon sample. (b) Distribution of the electrical potential within the sample to include boundary effects of the finite sized sample using COMSOL.

Figure 4 .

 4 Figure 4. Normalized deviation of the probe beam as a function of the temperature for the photodeflection bench. The dashed curve represents the expected theoretical deviation if the optical absorption is constant with the temperature.

Figure 5 .

 5 Figure 5. Example of a temperature rise at cryogenic temperature (a) and room temperature (b) when the pump laser is switched on for few minutes. For the left plot, the pump (heating) laser is switched on at 94 s and switched off at 164 s and for the right plot the times are respectively 48 s and 193 s. The laser power is set to 2 W at cryogenic temperature but to 8 W at room temperature in order to create a noticeable temperature rise.

Figure 7 .

 7 Figure 7. Equilibrium temperatures in the input test mass assuming a realistic optical absorption as well as a temperature dependent thermal conductivity for silicon. The temperature at the cold end of the fiber was assumed to be 4 K, the lengths of the fiber is set to 1 m.

Acknowledgements

The authors from LMA gratefully acknowledge the support of the European Gravitational Observatory (EGO) and the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Universit de Lyon for its financial support within the program Investissements dAvenir (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR). The support of the German Science Foundation DFG is acknowledged under contract SFB TR7. The EU supports this activity under the APPEC grant ET R&D.