Kevin Nadaud

Raphaël Gillard
email: raphael.gillard@insa-rennes.fr

Erwan Fourn

Hartmut W Gundel

Equivalent circuit of a reconfigurable triple-slot reflectarray cell

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Reflectarrays are widely studied because they combine the advantages of reflector antennas and array antennas. In order to increase bandwidth and reduce losses, multiple-resonator cells have been proposed, using various topologies such as stacked patches [START_REF] Encinar | Design of a 1-metre reflectarray for dbs application with 15 % bandwidth[END_REF], coupled dipoles [START_REF] Perez-Palomino | Design and experimental validation of liquid crystal-based reconfigurable reflectarray elements with improved bandwidth in F-band[END_REF] or coupled slots [START_REF] Makdissy | Triple-slot phase-shifting cell loaded with capacitances for reflectarray applications[END_REF]. All these passive cells involve many geometrical parameters, which makes their optimization process complex and time-consuming as it relies on full-wave simulations.

In the case of reconfigurable cells, the situation is even trickier as successive steps have to be completed. First, the passive distributed structure of the cell has to be designed (as in the case of passive cells). Then, lumped reconfigurable elements have to be implemented to control the reflected phase within a prescribed range. Finally, a redesign of the initial passive distributed structure is usually necessary to take into account the biasing lines and other topological arrangements resulting from the integration of these tuning elements. Although the compression approach can advantageously be used [START_REF] Kunisch | The compression approach: A new technique for the analysis of distributed circuits containing nonlinear elements[END_REF] for tuning lumped elements, the design and redesign of the passive distributed structure still require numerous full-wave simulations.

On the other hand, efficient synthesis processes have been developed in the field of frequency selective surfaces (FSS) [START_REF] Al-Joumayly | A new technique for design of low-profile, second-order, bandpass frequency selective surfaces[END_REF][START_REF] Behdad | A generalized synthesis procedure for low-profile, frequency selective surfaces with odd-order bandpass responses[END_REF] which replace the full-wave simulations by circuit-based approaches.

First, an equivalent circuit for the passive distributed structure is derived using a limited number of full-wave simulations. This provides a direct correspondence between the geometrical parameters and the related circuit elements. Then, any further modification of these geometrical parameters can be analyzed at the circuit level without any additional full-wave simulation. This definitively makes the optimization process simpler and much faster. In [START_REF] Tahir | Optimisation of mems-controlled reflectarray phase shifter cell[END_REF][START_REF] Tahir | Equivalent electrical circuit for designing memscontrolled reflectarray phase shifters[END_REF], an equivalent circuit has been proposed for a slot-loaded patch reflectarray cell. It permits to optimize the number and the placement of the lumped components used for changing the reflected phase. Nevertheless, this equivalent circuit does not permit to obtain the lengths of the slots or of the patch. In [START_REF] Hum | Modeling and design of electronically tunable reflectarrays[END_REF] a model is given for a patch loaded with varicap diodes but it has only one resonator, leading to a high frequency dispersion.

In this paper, we present an equivalent circuit for the reflectarray triple-slot cell described in [START_REF] Makdissy | Triple-slot phase-shifting cell loaded with capacitances for reflectarray applications[END_REF].

Although this cell has been proved to be a good candidate for both, large bandwidth and reconfigurability, no simple process has been reported yet to optimize its dimensions. The proposed circuit allows predicting the phase response when the slots length (the most influent parameters) and the distance to the ground plane change. Only a limited number of simulations is needed, which is essential for fast optimization. In a first time, the interest of this approach is validated for the design of a passive cell. Then, we show that it can be extended to the analysis of a reconfigurable cell by combining the equivalent circuit of the passive distributed structure with lumped elements that represent the controllable tuning components. Experimental validations at the cell level have already been published [START_REF] Makdissy | Triple-slot phase-shifting cell loaded with capacitances for reflectarray applications[END_REF][START_REF] Nadaud | A simple Phase-Shifting cell for reflectarray using a slot loaded with a ferroelectric capacitor[END_REF][START_REF] Nadaud | A triple-slot active reflectarray cell using a ferroelectric capacitor[END_REF] for quite similar structures and will not be repeated here for the sake of brevity. We rather focus on the comparison of the results provided by our new equivalent circuit with those resulting from full-wave simulations, with the clear goal to replace those ones.

Finally, the utility of the equivalent circuit is demonstrated by simulating a reflectarray with a steered beam.

Note that the proposed equivalent circuit has similarities with that used in [START_REF] Eshrah | Analysis of waveguide slot-based structures using wide-band equivalent-circuit model[END_REF][START_REF] Eshrah | Load-independent equivalent circuit model for transverse waveguide slots[END_REF] for slotted waveguides. However, our formulation takes mutual coupling between slots into account. Additional elements are also included to provide a realistic modelling when active components are integrated in the slot.

Studied topology and associated equivalent circuit

The general studied topology (Fig. 1) consists of three slots in a metallization printed on a dielectric substrate (permittivity 2.17, loss tangent 9 × 10 -4 and thickness 1.6 mm). This printed circuit is suspended in a square waveguide at a distance h air from the short-circuited termination (Fig. 2). This waveguide configuration is usually used for single cell measurement (as in [START_REF] Nadaud | A triple-slot active reflectarray cell using a ferroelectric capacitor[END_REF] for instance).

Moreover, reflectarray cells (especially active ones) are sometimes embedded in metallic cavities to limit mutual coupling. In such situations, the waveguide between the printed circuit and the shortcircuit plane is really present [START_REF] Cheymol | A X-band electronically scanned reflectarray antenna for space telemetry[END_REF]. The section of the square waveguide used here is 35 × 35 mm 2 providing a [4.5, 6.5] GHz frequency band.

The reflected phase can be controlled by changing only the length of the central slot or loading it by a tunable capacitor [START_REF] Makdissy | Triple-slot phase-shifting cell loaded with capacitances for reflectarray applications[END_REF]. The other parameters, especially the length of the external slots L s2 , allow controlling the frequency dispersion and the accessible phase range. Note that both external slots have the same length in order to maintain the cell symmetry. L s1 and L s2 are definitively the most influent parameters regarding the phase response of the cell and are the only ones to be varied in this study. This choice is also consistent with the fact that, in a realistic reflectarray synthesis, the number of tunable parameters has to be small to keep the optimization process affordable. The variation ranges for L s1 and L s2 have been chosen in order to approach a full 360°phase range at center frequency, which is the usual goal when designing a reflectarray cell. In the considered examples, W s1 and W s2 are set to a quite large value (5 mm) in order to achieve smooth phase states [START_REF] Moustafa | The phoenix cell: A new reflectarray cell with large bandwidth and rebirth capabilities[END_REF]. The distance between adjacent slots e f is set to 2 mm to keep enough space for a future biasing circuit. Nevertheless, the discussed methodology is also valid in the case of any other values.

The proposed equivalent circuit consists of a shunt admittance (for the printed circuit) backed by a short-circuited transmission line of length h air . The admittance is composed of three shunt resonators, one for each slot (Fig. 3), which are coupled with inductive (M c ) and capacitive (C c ) elements. Both types of coupling are necessary to correctly take into account all interactions between the different slots [START_REF] Hong | Microstrip Filters for RF/Microwave Applications[END_REF]. The propagation characteristics of the transmission line backing the printed circuit are calculated using simple rectangular waveguide theory [START_REF] Pozar | Microwave Engineering[END_REF]. The lumped Ansys HFSS commercial software. To do so, the printed circuit is simulated as a 2-port in the waveguide (Fig. 2). One reference plane is placed on each face of the dielectric substrate. The simulated 2 × 2 scattering matrix is then imported into a circuit simulator (Keysight ADS) and the elements of the shunt admittance are tuned until its scattering parameters match those from the full-wave simulation.

Then, the reflected coefficient from the whole cell (including the short-circuited waveguide) can be obtained directly with the circuit simulator simply by loading the obtained admittance with the appropriate length of short-circuited transmission line. As h air can be varied in the circuit simulation itself, the modification of this parameter does not require any additional full-wave simulation, which is a first advantage of the equivalent circuit. The second advantage, as described in the next section, is the possibility to express the elements forming the admittance from the variable geometrical parameters (here L s1 and L s2 ). Once these expressions have been derived, which requires only 2 or 3 full-wave simulations, the response of the cell for any value of L s1 and L s2 can be obtained directly from the circuit simulator, without any new full-wave analysis.

Derivation of the equivalent circuit

General Process

In this section, the derivation of the elements of the equivalent circuit from full-wave simulations is described in details. As a first approximation, we consider that C 1 and L 1 only depend on L s1 while C 2 and L 2 only depend on L s2 . C c and M c are supposed to be independent from both L s1 and L s2 . As will be seen from the obtained results, this assumption proves to be sufficient. Moreover, L i and C i are linked by the following relations:

f i = 1 2π √ C i L i and Z i = π 2 L i C i , (1)
where f i and Z i stand for the resonant frequency and the characteristic impedance of slot i. Since the resonant frequency of a transmission line resonator varies with the inverse of its length, we can express f i as:

f i = 1 α i L si + β i , (2)
where α i and β i are unknown parameters to be determined. Note that α i is supposed to be close to 2 √ ε reff /c for a half wavelength slot (with c the speed of light and ε reff the effective relative permittivity) and β i accounts for the imperfect short circuits at both ends of the slot. More precisely, it represents possible reactive effects which may be responsible for resonance deviations. Finally, using ( 1) and ( 2), C i and L i can be related directly to the length of the slot as:

C i = α i L si + β i 4Z i and L i = Z i (α i L si + β i ) π 2 . ( 3
)
The derivation of the electric circuit then consists in determining

C c , M c , α 1 , β 1 , Z 1 , α 2 β 2 and
Z 2 from a reduced number of full-wave simulations. Once this has been done, C i and L i can be calculated for any new value of L si using (3). This allows using the electric circuit (instead of a new EM simulation) for characterizing the phase response of the cell whatever the value of the slots length.

Validation with variable internal slot and fixed external slots

For the sake of simplicity, we first present the derivation process where only the length of the central slot L s1 varies (L 2 and C 2 are supposed to be constant in this case). Two full-wave simulations are run. They correspond to two printed circuits in the waveguide (Fig. 2), only differing in the length of the central slot (L s1 and L s1 for the first and second simulation respectively). Then, two equivalent shunt admittances are tuned simultaneously so that the corresponding S parameters match those of the two simulated structures in the [4.5, 6.5] GHz frequency band. Only C 1 and L 1 differ from one admittance to another: we name C 1 and L 1 (respectively C 1 and L 1 ) the values associated to L s1 (respectively L s1 ). The tuning process also provides C c , M c , C 2 and L 2 but it assumes they are identical for both admittances. Finally, α 1 and β 1 can be derived from (3) using:

α 1 = 2π C 1 L 1 -C 1 L 1 L s1 -L s1 , (4)
β 1 = 2π C 1 L 1 -L s1 α 1 , (5)
Once the elements forming the shunt admittance have been identified, the full equivalent circuit of the reflectarray cell can be constructed by adding the short-circuited transmission line. The reflected phase obtained from full-wave simulations and the results from the equivalent circuit simulations are compared in Fig. 4a. The extraction of the parameters has been carried out using and Z 1 = 99.7 Ω.

L s1 =
As can been seen in Fig. 4a, the agreement is very good even though L s1 varies from 16 to 24 mm, which is a larger range than the initial [L s1 , L s1 ] interval. Fig. 4b shows the difference between the full wave simulation and the equivalent circuit. For L s1 less than 22 mm, the maximum phase difference is 12°. For the larger lengths (L s1 = 23 mm or 24 mm), the phase difference is much higher at low frequencies. This is due to rapid variation of the phase in that region. Indeed, this corresponds to a high frequency dispersion region (involving resonance phenomena) where the cell is not supposed to be used. For the central frequency (5.5 GHz) the maximum error is 5°. Finally, the average phase difference for all lengths L s1 over the total frequency band is 4.11°, which provides a sufficient accuracy for reflectarray design.

Validation with variable internal and external slots

Now we consider L s2 can also vary, which means that α 2 and β 2 have to be determined. The derivation process described in section 3.2 can be extended straight forwardly. The new process relies on three full-wave simulations involving the following length couples (L s1 , L s2 ), (L s1 , L s2 ) and (L s1 , L s2 ). Then, three admittances (instead of two) are tuned simultaneously. The first two full-wave simulations, involving (L s1 , L s2 ) and (L s1 , L s2 ) are the same as in section 3.2. As already Table 1. Extracted coefficients for passive and active cases. and L 2 for these configurations. The third full-wave simulation (L s1 , L s2 ) leads to a different value of C 2 and L 2 , referred to as C 2 and L 2 . Finally, the whole tuning process allows determining α 2 and β 2 using (4) and [START_REF] Al-Joumayly | A new technique for design of low-profile, second-order, bandpass frequency selective surfaces[END_REF]. Of course, it also provides all the other parameters (as in section 3.2). Note that M c and C c are kept identical for all three admittances since we assume that mutual coupling does not change significantly when the slots length are varied. This assumption is reasonable as the required variation for L si (in order to achieve a full 360°range) is small. Also, since we for all lengths L s1 over the total frequency band, is below 10°for a variation of the external slot of ±3.5 mm. This difference is attributed to two main causes. Firstly, error logically gets larger when the considered L si is outside the [L si ,L si ] extraction range, because extrapolation is required.

α 1 (Hz -1 m -1 ) β 1 (Hz -1 ) Z 1 (Ω) α 2 (Hz -1 m -1 ) β 2 (Hz -1 ) Z 2 (Ω) M c (nH) C c (fF) L p (nH) C p (
Secondly, it is also partly due to the fact we assume that coupling is independent from L si .

Extension to the active case

The circuit presented in the first part of this communication is now upgraded for the study of a reconfigurable cell. In the considered example, reconfiguration is achieved by loading the internal slot with a controllable capacitance C (as in [START_REF] Nadaud | A triple-slot active reflectarray cell using a ferroelectric capacitor[END_REF]). A typical implementation is shown in Fig. 7a.

Note that two narrow metallic strips are used for the connection of the lumped element (variable capacitance) in the slot center.

The equivalent admittance is modified as shown in Fig. 7b: a capacitance C p and an inductance L p have been added to represent the capacitive effect of the gap supporting the lumped element and the inductive effect of the metallic strips respectively.

The extraction of the circuit elements using full-wave simulations is very similar to these described in the previous sections. In the full-wave simulation, a lumped port (Fig. 7a), corresponding to the third port in the model (Fig. 7b), is added for the extraction of the equivalent circuit. This third port is similar to those used in the compression approach [START_REF] Kunisch | The compression approach: A new technique for the analysis of distributed circuits containing nonlinear elements[END_REF] and will provide the place to connect the variable capacitor during the post-processing, after the equivalent circuit extraction.

Before that, as in 3.3, three full-wave simulations, involving three different combinations of L s1

W s1

Internal port The extracted values are given in Tab. 1.

W s2 W l L l L s1 L s2
Fig. 9 shows that the proposed equivalent circuit still provides satisfactory results when the air gap between the substrate and the short circuit is modified. The accuracy is better than 7°in all cases.

Reflected phase (°)

Application of the equivalent circuit

To demonstrate the capabilities of the proposed equivalent circuit, it has been used to synthesize a canonical reflectarray with a steered beam. In order to make a comparison with full-wave simulation feasible, it consists of an 8 × 6 array with λ 0 /1.56 inter-element spacing (35 mm at 5.5 GHz)

as shown in Fig. 10. For the sake of numerical validation, it is illuminated by a plane wave under normal incidence. The length of the central slot varies with the y coordinate so that the beam is steered at θ = 10°in the (yOz) plane. The necessary phase difference between adjacent cells is 40°and the equivalent circuit is used for determining the corresponding length of the central slots.

For the presented example, the length of the external slot L s2 has been fixed to 29 mm and is kept constant for all the cells.

Using the equivalent circuit, the total time to synthesize the array (including the construction of the equivalent circuit itself) is less than 5 minutes on a standard PC workstation (16 GB RAM, Intel® Core™ i7-4790 @ 3.6 GHz). Using full-wave simulations instead would dramatically increase the computation burden (especially in the case of an even larger array). The values of the elements in the equivalent circuit are those given in Table 1. The needed lengths to satisfy the phase-law are given in Table 2. To verify the synthesis approach using the equivalent circuit, a full-wave simulation of the complete array has been done using HFSS (which takes 2 hours). and the array with the lengths obtained previously. This clearly demonstrates that the beam has been steered properly.

Conclusion

In this paper, an equivalent circuit is proposed for a triple-slot phase-shifting cell. This model describes precisely the behavior of the cell and allows predicting the reflected phase while needing only a limited number of full-wave simulations. Both the lengths of the slots and the distance between the cell and the end of the waveguide can be modified. An upgraded version of the equivalent circuit has also been proposed which enables the integration of lumped elements for an active control of the cell. Finally, the equivalent model permits to optimize the slots length of the distributed passive structure with a limited number of full-wave simulation. The comparison with full-wave simulations clearly demonstrates the capability of the equivalent circuit approach to synthesize a reflectarray with a steered beam.

Figure 1 .

 1 Figure 1. Schematic view of the studied cell. a top view b 3D view

Figure 2 .Figure 3 .

 23 Figure 2. Schematic view of the waveguide containing the cell simulated with HFSS.

Figure 4 .

 4 Figure 4. Reflected phase and phase difference as a function of frequency obtained from full wave HFSS simulation (+) and predicted with the proposed equivalent circuit (-). L s2 = 27 mm, L s1 varies from 16 to 24 mm in steps of 1 mm. a Reflected phase. b Phase difference.

Figure 5 .

 5 Figure 5. Reflected phase simulated with HFSS (+) and predicted with the equivalent circuit (-) for variable L s1 with a step of 1 mm. a L s2 = 25 mm b L s2 = 31 mm

Figure 6 .

 6 figure, the prediction of there is a quite good agreement. To better evaluate the validity of the model, the average phase difference between the full-wave and the circuit simulations is reported in Fig.6as a function of the slot length L s2 . This difference, which is the average difference

Figure 7 .

 7 Figure 7. Schematic view of the studied cell with an internal port (a) and the proposed equivalent circuit (b).

Figure 8 .ReflectedFigure 9 .

 89 Figure 8. Simulated reflected phase with HFSS (+) and predicted with the proposed equivalent circuit (-) for different values of the capacitance C in steps of 10 fF and different couples (L s1 , L s2 ). a (L s1 , L s2 ) = (14 mm, 27 mm), average phase difference: 7.6°b (L s1 , L s2 ) = (12 mm, 25 mm), average phase difference: 6.8°c (L s1 , L s2 ) = (14 mm, 25 mm), average phase difference: 11.4°1

Figure 10 .

 10 Figure 10. Schematic view of the proposed reflectarray. The central slot lengths vary from column 1 to 8 but are identical into each column.

Figure 11 .

 11 Figure 11. Directivity and 3D radiation pattern of the reflectarray with identical column and with the calculated phase-law. a Directivity in the (yOz) plane (H-plane). b 3D radiation pattern.

Table 2

 2 Length of the central slot for each column and reflected phase difference with the first column. presents the directivity of the reference array (with all central slots measuring 20 mm long)

	Column Length of the central slot Relative phase
	1	16 mm	0°2
		18.57 mm	40°3
		19.44 mm	80°4
		19.97 mm	120°5
		20.41 mm	160°6
		20.89 mm	200°7
		21.58 mm	240°8
		23.22 mm	280°F
	ig. 11

Acknowledgements

The authors would like to acknowledge the CNRS (Centre National de la Recherche Scientifique) and the French Region "Pays de la Loire" for the financial support.


Mehdi Terdjimi
email: mehdi.terdjimi@liris.cnrs.fr

Lionel Médini
email: lionel.medini@liris.cnrs.fr

Michael Mrissa
email: michael.mrissa@liris.cnrs.fr

Towards a Meta-model for Context in the Web of Things

Keywords:

The Web of Things (WoT) uses Web technologies to engage connected objects in applications. Building context-aware WoT applications requires modeling and reasoning about context. In this paper, we overview different context modeling approaches related to the WoT, before studying the architecture of WoT applications. We then propose a multi-level, multi-dimensional and domain-independent context meta-model to help WoT applications identify, organize and reason about context information.

Introduction

The Web of Things (WoT) applies to various domains such as homes, enterprises, industry, healthcare, city or agriculture. It builds a Web-based uniform layer on top of the Internet of Things (IoT) to overcome the heterogeneity of protocols present in the IoT networks. Today's WoT applications (WoT apps) need relevant context models to exhibit context-adaptive behavior. Basically, a WoT app provides added value by combining access to connected objects and external data sources (i.e. Web services). It requires an accurate description and exploitation of WoT apps context, hence justifying the need for context models. As a consequence of the diversity of use-cases and applications, numerous domain-specific models relying on different formalisms and reasoning mechanisms have been designed [START_REF] Perera | Context aware computing for the internet of things: A survey[END_REF]). However, a single context modeling framework for WoT apps is still missing. In this paper, we propose a context meta-model as a first building block for a framework that allows reusing the same reasoning tools in the diversity of WoT use-cases. In Section 2, we study general definitions for context and the major existing context models. In Section 3, we draw conclusions on existing work and study the different parts of WoT apps to identify the elements that will contribute to build our meta-model. In Section 4, we discuss our solution according to a set of evaluation criteria. In Section 5, we summarize our results and give perspectives for future work.

Describing Context: Related Work

While [START_REF] Schilit | Disseminating active map information to mobile hosts[END_REF] define context-aware computing as "the ability of applications to discover and react to changes in the environment", context can be seen as the information thats answers to the Where, Who, When and What questions as discussed in [START_REF] Abowd | Towards a better understanding of context and context-awareness[END_REF]. [START_REF] Dey | Understanding and using context[END_REF] defines context as "any information that can be used to characterize the situation of an entity. An entity is a person, place, or object that is considered relevant to the interaction between a user and an application, including the user and applications themselves". [START_REF] Chaari | Adaptation des applications au contexte en utilisant les services web[END_REF] define a situation as an instance of a set of contextual attributes. Context information described in [START_REF] Schilit | The parctab mobile computing system[END_REF] include environmental and geospatial information, such as location, co-location (i.e. what is nearby), time, etc.. Most work [START_REF] Schmidt | Ubiquitous computing-computing in context[END_REF]; [START_REF] Zimmermann | An operational definition of context[END_REF]; [START_REF] Abowd | Towards a better understanding of context and context-awareness[END_REF] structure context as multi-dimensional views. Some work describe context with a specific focus, such as privacy in [START_REF] Dey | The conference assistant: Combining context-awareness with wearable computing[END_REF], or computing resources (availability, remaining battery power) and network information (types of connection, services in reach, distances, disconnection rates) in [START_REF] Gold | Use of context-awareness in mobile peerto-peer networks[END_REF]; [START_REF] Mascolo | Mobile computing middleware[END_REF]; [START_REF] Musolesi | Car: context-aware adaptive routing for delay-tolerant mobile networks[END_REF].

Network context is often combined with other context elements such as user profile or preferences in [START_REF] Wei | Context-aware handover using active network technology[END_REF]; [START_REF] Raverdy | Efficient context-aware service discovery in multi-protocol pervasive environments[END_REF]; [START_REF] Yu | Supporting context-aware media recommendations for smart phones[END_REF]. Users are of major importance in context-aware applications as shown in [START_REF] Cao | Context-aware query suggestion by mining click-through and session data[END_REF][START_REF] Cao | Context-aware query classification[END_REF]; [START_REF] Xiang | Contextaware ranking in web search[END_REF]. In [START_REF] Brézillon | Misuse and nonuse of knowledge-based systems: The past experiences revisited[END_REF]; [START_REF] Brézillon | Context in Artificial Intelligence: II. Key elements of contexts[END_REF]; [START_REF] Bucur | Representing context in an agent architecture for context-based decision making[END_REF] the user and its environment have a central place in the context model together with its interaction and domain-specific information. [START_REF] Brézillon | Contextual knowledge sharing and cooperation in intelligent assistant systems[END_REF] use context for helping users with decision making, later with different knowledge sources as in [START_REF] Brézillon | Representation of procedures and practices in contextual graphs[END_REF] and with the notion of situation as in [START_REF] Bazire | Understanding context before using it[END_REF]. Context information may be domain-specific as mentioned in [START_REF] Munnelly | An aspect-oriented approach to the modularisation of context[END_REF], or related to the application architecture as discussed in [START_REF] Truong | ESCAPEan adaptive framework for managing and providing context information in emergency situations[END_REF][START_REF] Truong | Incontext: A pervasive and collaborative working environment for emerging team forms[END_REF]; [START_REF] Kirsch-Pinheiro | Representing context for an adaptative awareness mechanism[END_REF]. More recently [START_REF] Coppola | The context-aware browser[END_REF] include meta-information about context (Probability, Importance, Description and Name).

Modeling Context for Web of Things applications

The related work presented above highlights the diversity and complexity of context. Each work propose a unique combination of device, user, network, and application context elements. The traditional semantic heterogeneities can be found between context models, i.e. polysemy, heteronymy, and so on. In the following, we discuss these approaches and then present our arguments for a context meta-model.

A Context Meta-model for the Web of Things

In our work as part of the ANR ASAWoO project1 , physical object functionalities and applications are exposed as RESTful services. Based on the context model and instances, the framework should be able to activate or deactivate functionalities or applications. The latter might not be available due to physical constraints, security, privacy, or other policies, which can vary according to the use-case.

Hence, we design a context meta-model for WoT apps to control the instantiation of domain-specific models and to reuse the same reasoning mechanisms. We rely on the different context information studied in the literature to justify our choices, and organize our meta-model into levels that characterize the different parts of a WoT app. As depicted in Fig. 1, mobile and distributed clients communicate with the WoT app, which itself communicate with physical objects and data sources. A client could be a user (via a smartphone or computer), or another software application to allow for composite applications.

According to Fig. 1, we organize context information in four levels. The Physical level (1) describes context information about physical objects, including their internal states and information coming from their sensors that describe the objects' environment. The Application (2) level describes the application architecture, its state, its configuration, as an application can rely on different architectural paradigms, such as components or services, and can be either locally stored or distributed. The Communication (3) level describes the context of links between the application and clients, physical objects and data sources. It includes information such as the state of the network, bandwidth, latency, or the type of connection, such as wired or wireless. The Social (4) level characterizes the client's environment. It includes cognitive aspects such as user roles withing an organization, user behavior, and types of users (human, software). Our meta-model presents the following advantages. First, it allows WoT app developers for high flexibility with an unrestricted numbers of dimensions to be created. Second, all the models will follow the same description language (DL) thus allowing the reasoning process to always be effective. Third, the level/dimension view can be reused differently according to a WoT app settings, or even between several WoT apps. We illustrate the latter advantage in the section below with two use cases that rely on different models, and one use case that uses different combinations of levels/dimensions from the same model.

Illustration with different use-cases

We illustrate our meta-model with two use-cases that use specific context models. The first one is temperature regulation that needs to adapt to the user presence and settings. Its context model involves four dimensions that contains geolocation and network information, as well as time, actuator states, and user home preferences as shown in Fig. 3. Depending on the user settings, the application can rely different combinations of levels and dimensions to enable context-awareness. Typical users would rely on a model that includes both home and enterprise settings (plain squares in Fig. 3); children and elderly people would require different levels and dimensions (dashed squares in Fig. 3). The second use-case concerns agriculture. It involves drones that scatter bugs on plantations to protect them from threatening insects, an automatic watering system, and robots that pull off weeds from plantations. Each of them have their own context model, to provide the whole system with safe operation. According to the type of devices, level and dimension combinations to describe the context models can also differ. For example, drones robots only require a day/night granularity whereas the watering system requires hourly time slots. As well, the watering system do not use location information due to its immobility.

Evaluation

We build on the methodology proposed in Gómez-Pérez (1998) to evaluate our meta-model according to the following steps: (S1) Purpose and scope, (S2) Intended uses, (S3) Intended users, (S4) Requirements, (S5) Competency Questions (CQs), and (S6) Validation of CQs, as shown in Fig. 4. We refer to the metrics proposed in [START_REF] Hlomani | Approaches, methods, metrics, measures, and subjectivity in ontology evaluation: A survey[END_REF]

Conclusion and perspectives

In this paper, we build a meta-model that combines levels and dimensions to help WoT apps identify, organize and reason about context information. Our meta-model aims at enhancing the reusability of reasoning mechanisms across application domains while leaving flexibility for developers to design their application-specific context models.

As future work, we aim at optimizing the reasoning process for WoT apps to handle fast-paced data streams as context sources, as well as to provide dynamic reconfiguration and adaptation for their components.

Figure 1 :

 1 Figure 1: A typical WoT app architecture and its context dimensions

Fig. 2

 2 Fig.2shows an UML representation of our meta-model. The Model class aggregates the four levels presented above, which themselves compose the dimensions. For each level, the Dimension class contains a name URI that provides the ontological concept that describe the value object.

Figure 2 :

 2 Figure 2: UML representation of our meta-model

Figure 3 :

 3 Figure 3: An example of Multi-level context model

Figure 4 :

 4 Figure 4: Evaluation of our meta-model.

  in (S6). Does the DL language allow for complete and sound results in a finite time? (CQ3) Does the DL language provide the logical constructs required by the reasoner? (CQ4) Are the number of levels sufficient to describe a WoT app context ? (CQ5) Are the levels redundant or overlapping? (CQ1) The meta-model allows any dimension to be created for complete coverage. (CQ2) The DL used (OWL) provides 3 profiles (EL, QL, RL) that are able to answer any reasoning problem (conjunctive query answering, class expression subsumption...) in a finite time. (CQ3) OWL provides expressive relationships (object, datatype properties) and concepts (classes, individuals, data values) allowing the reasoner to correctly answer queries. (CQ4) Levels are based both on related work and a typical WoT app architecture, thus covering the needed context information. (CQ5) Our levels do not overlap by design.

	(S1)	The scope of our work includes smart home, smart farms… Our purpose is to provide context-awareness with reusable and relevant models.
	(S2) Any WoT scenario (see Section 3.2).
	(S3) WoT apps developers.
	(S4) Context information coverage (Location, Time, User Preferences...).
		(CQ1) Does the model cover required context information?
	(CQ2) (S5)
	(S6)

http://liris.cnrs.fr/asawoo/

Acknowledgement. This work is supported by the French ANR (Agence Nationale de la Recherche) under the grant number <ANR-13-INFR-012>.


Dominique Béréziat

Isabelle Herlin

Coupling dynamic equations and satellite images for modelling ocean surface circulation

Keywords: Dynamic model, Optical flow, Data Assimilation, Satellite image, Ocean circulation

Satellite image sequences visualise the ocean surface and allow assessing its dynamics. Processing these data is then of major interest to get a better understanding of the observed processes. As demonstrated by state-of-the-art, image assimilation permits to retrieve surface motion, based on assumptions on the dynamics. In this paper, we demonstrate that a simple heuristics, such as the Lagrangian constancy of velocity, can be used and successfully replace the complex physical properties described by the Navier-Stokes equations for assessing surface circulation from satellite images. A data assimilation method is proposed that adds an acceleration term a(t) to this Lagrangian constancy equation, which summarises all physical processes other than advection. A cost function is designed that quantifies discrepancy between satellite data and model values. This cost function is minimised by the BFGS solver with a dual method of data assimilation. The result is the initial motion field and the acceleration terms a(t) on the whole temporal interval. These values a(t) model the forces, other than advection, that contribute to surface circulation. Our approach was tested on synthetic data and with Sea Surface Temperature images acquired on Black Sea. Results are quantified and compared to those of state-of-the-art methods.

Introduction

Satellite image sequences permit to visualise oceans' surface and their underlying dynamics with a high spatial resolution. Processing these images is then of major interest for a better understanding of the observed processes and the forecast of extreme events. As demonstrated by state-of-the-art, image assimilation allows to retrieve surface motion from image sequences, using heuristics on the dynamics [START_REF] Papadakis | Dynamically consistent optical flow estimation[END_REF][START_REF] Titaud | Assimilation of image sequences in numerical models[END_REF]. Among those heuristics, the shallow water model [START_REF] Vallis | Atmospheric and oceanic fluid dynamics[END_REF] has been proven to be suitable for representing the surface circulation of closed seas, such as Black Sea [START_REF] Oguz | The upper layer circulation of the Black Sea: Its variability as inferred from hydrographic and satellite observations[END_REF]. These shallow water equations have also been successfully used to estimate the upper layer circulation of Black Sea from Sea Surface Temperature (SST) images [START_REF] Huot | Estimating apparent motion on satellite acquisitions with a physical dynamic model[END_REF][START_REF] Korotaev | Retrieving ocean surface current by 4-D variational assimilation of sea surface temperature images[END_REF] with a data assimilation method.

In this paper, we propose to learn the surface dynamics from SST image acquisitions with a data assimilation method applied to an image model derived from the shallow water equations. In the shallow water model, surface circulation is characterised by the horizontal velocity, that is advected by itself and subject to geophysical forces such as Coriolis, Earth gravity and viscosity. The advection process is kept in the image model, but all other components are summarised in a global term, denoted a (letter a stands for acceleration) that is estimated by our approach. Adding this term to the advection is similar, from a mathematical point of view, to the weak data assimilation framework [START_REF] Dee | Bias and data assimilation[END_REF][START_REF] Sasaki | Some basic formalisms in numerical varational analysis[END_REF][START_REF] Trémolet | Accounting for an imperfect model in 4D-Var[END_REF][START_REF] Valur Hólm | Lectures notes on assimilation algorithms[END_REF]. A data assimilation technique is then designed to compute the solution: a cost function is constructed, whose control variables are the motion field at the first acquisition date and the acceleration values a(t), at all dates of the acquisition interval. The minimum of the cost function is obtained thanks to optimal control techniques [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF] and it is computed with the BFGS solver [START_REF] Zhu | L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization[END_REF].

Section 2 provides notations that are used in the remaining of the paper and the mathematical description of the proposed approach for modelling the dynamics of the ocean's upper layer. The data assimilation method is outlined in Section 3. It corresponds to a weak formulation, where the non advective terms are summarised as an additional term in the evolution equation. The implementation is shortly described in Section 4, in order to permit that readers apply the method by themselves. Validation on synthetic data and results on SST image sequences acquired over Black Sea by NOAA-AVHRR sensors are displayed and quantified in Section 5.

Problem statement

Image data are acquired on a bounded rectangle of R 2 , named Ω, and on a temporal interval [0, T]. Let define A = Ω × [0, T] the corresponding spacetime domain, on which the dynamics is modelled. A point x ∈ Ω is defined as x = x y T and the motion vector at point x and date t ∈ [0, T] is written w(x, t) = u(x, t) v(x, t) T . At each date t, the motion field on the domain Ω is written as w(t). N Sea Surface Temperature acquisitions are available at dates

t i , i = 1 • • • N . They are denoted T (t i ) with pixels values T (x, t i ).
A state vector X is defined on A. It includes the two components u and v of the motion vector w(x, t) and a pseudo-temperature value T M (x, t), which has properties similar to those of the Sea Surface Temperature function:

X(x, t) = w(x, t) T T M (x, t) T (1)
The index M in T M reminds that this is a component of the M odel state vector. At the end of the data assimilation process, the discrepancy between the pseudotemperature T M and the satellite acquisitions T has to be small. The heuristics on dynamics, used in the paper, are derived from the shallow water equations that express the principles of mass and momentum conservation [START_REF] Vallis | Atmospheric and oceanic fluid dynamics[END_REF]. Circulation of the upper ocean is represented by the 2D velocity w = u v T and the thickness h of the mixed layer. In our model, the pseudotemperature T M is transported by the motion field. This provides the following set of equations:

∂u ∂t = -u ∂u ∂x -v ∂u ∂y + f v -g ∂η ∂x + K w ∆u (2)
∂v ∂t = -u ∂v ∂x -v ∂v ∂y -f u -g ∂η ∂y + K w ∆v (3)
∂η ∂t = - ∂(uη) ∂x - ∂(vη) ∂y -h m ∂u ∂x + ∂v ∂y (4)
∂T M ∂t = -u ∂T M ∂x -v ∂T M ∂y (5)
with η the thickness anomaly η = h -h m , h m the average value of h, f the Coriolis parameter, K w the viscosity and g = g(ρ 0 -ρ 1 )/ρ 0 the reduced gravity. ρ 0 corresponds to the reference density and ρ 1 to the average density of the mixed layer.

As explained in the introduction, we propose to group all geophysical forces that do not correspond to advection in a unique term, that corresponds to the acceleration and is denoted by a. The variable η is then considered as an hidden variable of the system. In such way, System (2, 3, 4) reduces to:

∂u ∂t = -u ∂u ∂x -v ∂u ∂y + a u ( 6
)
∂v ∂t = -u ∂v ∂x -v ∂v ∂y + a v (7)
where a = a u a v T expresses the discrepancy to the Lagrangian constancy of velocity:

dw dt = ∂w ∂t + (w.∇)w = a (8)
From Equations ( 2) and (3), we get:

a u = f v -g ∂η ∂x + K w ∆u (9)
a v = -f u -g ∂η ∂y + K w ∆v (10)
where η verifies Eq.(4). Our approach estimates w(0) and the acceleration term a(t) at each date t ∈ [0, T], thanks to the data assimilation process summarised in Section 3. Deriving the values a(t) permits describing empirically the physical processes generating the image sequence.

Eqs. ( 8) and ( 5) are further contracted in an evolution model M of the state vector X:

∂X ∂t

+ M(X) = a 0 ( 11
)
An observation equation links the state vector to the observed Sea Surface Temperature images acquisitions T :

HX = T + ε R (12)
The observation operator H projects the state vector into the space of image observations and consequently: HX = T M . The term ε R (x, t) models the acquisition noise and the uncertainty on the state vector value. This last comes from the approximation of the model and from the discretization errors.

Some approximate knowledge of the value X(0) could be available and named background X b . However, the result of the state vector at date 0 is not exactly equal to that background value and a term ε B is therefore introduced:

X(x, 0) = X b (x) + ε B (x) (13)
The variables ε R and ε B are supposed independent, unbiased, Gaussian and characterised by their respective covariance matrices R and B.

Eqs. [START_REF] Papadakis | Dynamically consistent optical flow estimation[END_REF][START_REF] Sasaki | Some basic formalisms in numerical varational analysis[END_REF][START_REF] Sun | Secrets of optical flow estimation and their principles[END_REF] summarise the whole knowledge that is available for modelling the surface dynamics. This knowledge is processed by our approach thanks to the data assimilation algorithm that is shortly described in the next section.

Data Assimilation

In the data assimilation scientific community, an approach, named weak 4D-Var, has been defined in order to obtain the solution X that solves System [START_REF] Papadakis | Dynamically consistent optical flow estimation[END_REF][START_REF] Sasaki | Some basic formalisms in numerical varational analysis[END_REF][START_REF] Sun | Secrets of optical flow estimation and their principles[END_REF]. A cost function is first designed:

J[ε B , a] = ε B , B -1 ε B + t γ ∇a(t) 2 + t HX(t) -T (t), R -1 (HX(t) -T (t)) (14)
where •, • denotes the canonical inner product in an abstract Hilbert space on which the state vector is defined, with norm . 2 and ∇a 2 = ∇a u , ∇a u + ∇a v , ∇a v . The function J is then minimised with control on ε B and on the values of the acceleration term a.

The first term of J comes from Eq. ( 13) and expresses that the value X(0) at date 0 should stay close to the background value X b . It should be noted that the control on ε B is equivalent to the control on initial condition as ε B (x) = X(x, 0) -X b (x). The second term constrains the acceleration term a(t) to be spatially smooth. The last term, coming from Eq. ( 12), expresses that the pseudotemperature value T M has to be close to that of satellite acquisitions at the end of the assimilation process.

The gradient of J is derived with the calculus of variations, as given in [START_REF] Lions | Optimal Control of Systems Governed by Partial Differential Equations[END_REF]. Its two components are:

∂J ∂ε B [ε B , a] = 2 B -1 ε B + λ(0) (15) ∂J ∂a(t) [ε B , a] = 2 (-γ∆a(t) + λ(t)) (16)
with λ(t) being the adjoint variable, that is computed backward in time with the two following equations:

λ(T) = 0 (17a) - ∂λ ∂t + ∂M ∂X * λ = H T R -1 (HX -T ) (17b)
The adjoint operator ∂M ∂X * verifies the following property. For all functions η and λ of the studied Hilbert spaces, it comes:

Zη, λ = η, Z * λ . (18)
Proof: For sake of simplicity, we suppose in this proof that Eq. ( 11) is written as ∂X ∂t + M(X) = a, a denoting simultaneously the acceleration involved in motion evolution and a 0 .

The state vector and the functional J depend on ε B and a(t). Let δJ and δX be the perturbations on J and X obtained if ε B and a(t) are respectively perturbed by δε B and δa(t).

From the definition of J, we obtain:

δJ 2 = δε B , B -1 ε B + t γ ∇δa(t), ∇a(t) + t δX(t), H T R -1 [HX(t) -T (t)] (19)
The evolution equation of X, Eq. ( 11), gives:

∂δX(t) ∂t + ∂M ∂X δX(t) = δa(t) (20)
and that of background, Eq. ( 13):

δX(0) = δε B (21)
Eq. ( 20) gives, after multiplication by λ(t) and integration on the space-time domain, the following equality:

t ∂δX(t) ∂t , λ(t) + t ∂M ∂X δX(t), λ(t) = t δa(t), λ(t) (22)
Integration by parts is applied on the first term and the definition of adjoint operator is used in the second one in order to obtain:

δX(T), λ(T) -δX(0), λ(0) - t δX(t), ∂λ(t) ∂t + t δX(t), ∂M ∂X * λ(t) = t δa(t), λ(t) (23)
From Eq.(17a), it comes that δX(T), λ(T) has a null value. From Eq. ( 21) it comes that δX(0), λ(0) is equal to δε B , λ(0) . Eq. ( 17b) is then used to obtain:

-δX(t), ∂λ(t) ∂t + δX(t), ∂M ∂X * λ(t) = δX(t), H T R -1 (HX(t) -T (t))
(24) and rewrite Eq. ( 23) as:

t δX(t), H T R -1 (HX(t) -T (t)) = δ B , λ(0) + t δa(t), λ(t) (25)
From this and Eq. ( 19), we derive:

δJ 2 = δ B , B -1 B - t γ δa(t), ∆a(t) + δ B , λ(0) + t δa(t)λ(t) (26)
and obtain the gradient of J, as written in Eqs. [START_REF] Titaud | Assimilation of image sequences in numerical models[END_REF][START_REF] Trémolet | Accounting for an imperfect model in 4D-Var[END_REF].

The cost function J is minimised with an iterative steepest descent method. At each iteration, the forward time integration of X is done, according to Eq. [START_REF] Papadakis | Dynamically consistent optical flow estimation[END_REF]. This forward time integration provides the value of J. Then a backward time integration of λ, according to Eqs. (17a) and (17b), computes the value of ∇J. An efficient solver, described in [START_REF] Byrd | A limited memory algorithm for bound constrained optimization[END_REF][START_REF] Zhu | L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization[END_REF], is used to perform the optimisation given values of J and ∇J. To our knowledge, [START_REF] Dimet | Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects[END_REF] is the first paper of the literature that describes the use of such method for estimating the initial state vector value.

Numerical implementation

Time integration of Eq. ( 11) relies on an explicit Euler scheme. The space discretization of motion advection, described by the two following equations:

∂u ∂t + u ∂u ∂x + v ∂u ∂y = 0 (27) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = 0 ( 28
)
involves a source splitting method [START_REF] Wolke | Implicit-explicit Runge-Kutta methods applied to atmospheric chemistry-transport modelling[END_REF], as explained below.

Given an integration interval [t 1 , t 2 ], Eqs. (29) (where n stands for non linear advection) and (30) (where l stands for linear advection) are first independently integrated:

∂u n ∂t + u n ∂u n ∂x = 0 t ∈ [t 1 , t 2 ] ( 29
)
∂u l ∂t + v ∂u l ∂y = 0 t ∈ [t 1 , t 2 ] ( 30
)
with u n (x, y, t 1 ) = u l (x, y, t 1 ) = u(x, y, t 1 ). Then u(x, y, t 2 ) is obtained as:

u(x, y, t 2 ) = u l (x, y, t 2 ) + u n (x, y, t 2 ) -u(x, y, t 1 )
The linear advection involved in Eq. ( 30) is discretized by a first-order upwind scheme, as described in [START_REF] Hundsdorfer | An efficient horizontal advection scheme for the modeling of global transport of constituents[END_REF]. The nonlinear advection of Eq. ( 29) is first rewritten in a conservative form:

∂u ∂t + ∂ ∂x 1 2 u 2 = 0 (31)
and approximated by a first-order Godunov scheme [START_REF] Leveque | Numerical Methods for Conservative Laws[END_REF].

The backward time integration of the adjoint variable λ involves the adjoint operator ∂M ∂X * (see Eq. ( 17b)). In order to be accurate, the method requires the adjoint of the discrete model and not the discretization of the continuous adjoint. For that reason, the discrete adjoint operator ∂M ∂X * is derived with the automatic differentiation software Tapenade described in [START_REF] Hascoët | The Tapenade Automatic Differentiation tool: Principles, Model, and Specification[END_REF].

The background used in Eq. ( 13) is defined as null for motion and as the first image of the studied sequence for the pseudo-temperature. As we have no information on motion, we do not want to constrain w(0) to stay close to that null value. For that reason, the first term of J reduces to ε B T , B -1

T ε B T . B T and R are taken diagonal with standard deviation values corresponding to 25% of the image brightness range. γ is given a value as small as possible: smaller values would not verify the Courant-Friedrichs-Lewy condition during the advection of the state vector.

Results

For sake of simplicity, we name in the following AM, or Acceleration Model, the model described in Sections 2 and 3: it includes the advection and the acceleration term a summarising the Coriolis, gravity and viscosity forces. We denote ME the Motion Estimation method that is obtained if suppressing a(t) in Eq. [START_REF] Papadakis | Dynamically consistent optical flow estimation[END_REF]. In that case, the second term of the cost function J, in Eq [START_REF] Suter | Motion estimation and vector splines[END_REF], vanishes and minimisation is only controlled by ε B .

The approach was experimented on a data base of Sea Surface Temperature sequences, acquired over Black Sea by NOAA-AVHRR sensors. The resolution of the AVHRR data is 1.1 km at nadir. Such image sequences are moreover able to display the dynamics of the submesoscale phenomena like eddies, jets, filaments and mushroom-like structures with a rather high temporal frequency. Our approach computes the surface motion fields and the trajectories of these structures by assimilating images. This high-level information is critical for improving the forecast of 3D ocean models such as NEMO 5 when they are tuned to resolve submesoscale processes.

The approach is first validated on two synthetic experiments. Then, results on four satellite image sequences are displayed and discussed in the remaining of the section.

Synthetic experiments

Given initial conditions at time 0, displayed in Fig. 1, and given acceleration values a ref (t), Eq. ( 11) is integrated in time and produces the ground-truth on motion and image fields. These initial conditions are similar to those on real data described in the next subsections. motion estimation method of Sun et al. [START_REF] Sun | Secrets of optical flow estimation and their principles[END_REF]. The estimated motion and acceleration fields are respectively denoted w e and a e . Table 1 discrepancy between results and ground truth. They concern the norm w and orientation θ of motion. This table shows that AM estimates motion field with a quality comparable to ME: it demonstrates that the acceleration estimated by AM is almost null. Statistics on acceleration confirm this property as the average value of a e is 3.6 × 10 -5 on the whole temporal interval.

provides statistics on the

In a second experiment, the acceleration a ref (t) is null for t = 41 and equal to the vertical constant 9 × 10 -2 at t = 41. Results are displayed on Fig. 4. Table 2 provides statistics on the discrepancy between motion results and ground-truth. Statistics on the norm of acceleration a e (t) are displayed on Fig. 5. From Table 2, it can be seen that AM gives a better estimation of velocity than Sun. As the simulation generating image data includes an acceleration, ME fails to estimate an accurate velocity. It can be seen on Fig. 5 that our approach computes an acceleration having a maximal norm value at date 41. This means that AM correctly localises the acceleration in time. Moreover, the average orientation of a e (41) is 90.4 • with a standard deviation of 11.4 • against 90 • for the ground truth. That further demonstrates the accuracy of the estimation.

Satellite experiments

In a first experiment, we analyse the ability of the proposed method to correctly estimate motion, which is a natural consequence of a correct assessment of geophysical forces. For that purpose, AM results are compared with those obtained by the motion estimation method of Sun et al. [START_REF] Sun | Secrets of optical flow estimation and their principles[END_REF]. The satellite sequence is displayed on Fig. 6. Acquisition dates are at 30 min, 6 hours, 15 hours, and 30 hours after the beginning of the studied temporal interval. Two gyres are clearly visible on these data. Motion results w(0), obtained by Sun and AM, are displayed on Fig. 7. AM successes to capture the two gyres while Sun's method fails. As w(0) is obtained from the analysis of the whole image sequence, its correct estimation means that the physical processes involved in a(t) are correctly assessed by the model. It also means that the non advective geophysical forces may be correctly described with the unique term a(t). In a second experiment, the capability of AM to track features or points of interest on the whole image sequence is examined. If the method provides accurate tracking results, this means that motion is correctly estimated on the studied temporal interval and properly transports the observed structures. A sequence of four SST images acquired on October 8 th 2005 is displayed on Fig. 8. They are acquired 30 min, 10 hours 15 min, 12 hours and 15 hours 30 min after the beginning of the studied interval. Nine characteristic points are defined in white on the first observation. Points are additionally surrounded by a coloured circle that helps to discriminate them on the following observations. These points are considered as characteristic, because they sample the various types of trajectories that can be observed on the sequence. On observations 2 to 4, the position of these nine points obtained with Sun's method are displayed in red while those obtained with AM are in blue. On the fourth acquisition, in the "light pink circle" on the upper right, the point obtained with Sun's method is outside of the image domain. Looking at the trajectories, it can be observed that Sun's algorithm fails to track these characteristic points, due to a wrong estimation of motion. Another sequence of five SST images, acquired on July 27 th 2007 is displayed on Fig. 9. The sensor acquired data 30 min, 8 hours 15 min, 13 hours, 22 hours 30 min and 24 hours 30 min after the beginning of the studied interval. Seven characteristic points are defined in white on the first observation. On the next ones, positions obtained with Sun's method are displayed in red while those obtained with AM are in blue. At the second date, two points are at the same position with Sun and AM: only the red point is visible as the blue one is hidden behind it. On the fourth observation, one red point has disappeared from the display as it is located outside of the image domain. On the last frame, the colour of the ellipse surrounding each couple of points gives an additional information on the quality of the result: a blue ellipse means that our method gives the best result while the white one means that both methods are equivalent. Again, Sun's motion results fail to track characteristic points on these data as physical processes are not correctly assessed by the underlying image model.

Last, AM is also compared with the optical flow estimation of Suter [START_REF] Suter | Motion estimation and vector splines[END_REF], that is dedicated to fluid flows, on a sequence of five images acquired on May Sun's methods are both only constrained by grey level values and do not rely on the underlying dynamics. However, Suter's algorithm provides better result than Sun's method, because it is specifically designed for fluid flows motion. In particular, it correctly assesses rotational motion. From left to right in Subfigure 10(e): AM gives the best result for the first, third, fourth and fifth points (blue ellipses). For the second and sixth (white ellipses) points, it is not possible to determine which one from Suter and AM provides the best result. The same conclusion is valid for all studied image sequences.

Conclusion

This paper describes how to learn the ocean surface dynamics from an image model, AM, that summarises the shallow water equations by an advection term and an acceleration term a. This last represents physical processes such as the Coriolis force, the gravity force and the viscosity. A data assimilation algorithm was designed for AM that estimates the velocity field at the first acquisition date and the acceleration a(t) at each date of the studied interval. The function a(t) is of major importance for correctly assessing the hidden physical processes and accurately estimating motion on the whole image sequence. The method has been quantified on synthetic data and illustrated on several SST sequences of Black Sea. On these last experiments, a display was given on motion fields and on the tracking of characteristic points. Moreover, the approach was compared with state-of-the-art optical flow algorithms. The conclusion is that modelling the acceleration, even as a simple unique term, improves motion estimation and allows tracking of structures. The short-term perspectives of this research work will be to compare the acceleration term a(t) with forces involved in the shallow water model, in order to further validate the ability of the empirical model to assess geophysical processes.

Fig. 1 :

 1 Fig. 1: Initial conditions on motion and image. The arrow representation is superposed to the coloured one.

Fig. 2 :

 2 Fig. 2: Image observations.

Fig. 3 :

 3 Fig. 3: First line: Groundtruth, Sun -Second line: ME, AM.

Fig. 4 : 2 :Fig. 5 :

 425 Fig. 4: First line: groundtruth, Sun -Second line: ME, AM.

Fig. 6 :

 6 Fig. 6: Left to right: SST images acquired on October 19 th 2007 over Black Sea.

Fig. 7 :

 7 Fig. 7: Motion results computed by Sun (left) and AM (right) at first observation date.

Fig. 8 :

 8 Fig. 8: Tracking of characteristic points. Results of Sun correspond to red points, those of AM to blue points.

Fig. 9 :

 9 Fig. 9: Tracking of characteristic points. A blue ellipse on (d) expresses that AM is the best while the white ellipse expresses that results are equivalent.

(

  14 th 2005. The acquisitions were obtained 30 min, 2 hours and 45 min, 5 hours and 15 min, 7 hours and 15 min, and 16 hours and 15 min after the beginning of the studied interval. As previously, six feature points are chosen on the first observation and displayed on the upper image of Fig. 10. Their final position on the fifth observation is given in the lower part of the same figure. Suter's and (a) Characteristic points on the first observation (b) Observation 2

Fig. 10 :

 10 Fig. 10: Sun: red, Suter: green, AM: blue.

Table 1 :

 1 |θe -θ ref | | w ref -we |/ w ref Statistics on motion errors for Sun, ME (without a) and AM (with a).

	method mean	mean
	Sun	16.1	43%
	ME	3.2	9%
	AM	3.5	10%

Acknowledgements

Data have been provided by E. Plotnikov and G. Korotaev from the Marine Hydrophysical Institute of Sevastopol, Ukraine.


Hervé Cercellier
email: herve.cercellier@neel.cnrs.fr

Hadrien Grasland

H Q Luo

X Y Lu

D L Gong

Z. -S Wang

Thierry Klein

Dynamically current-driven de Gennes -St James states in Metal-Superconductor junctions

We investigate the conductance of a Normal-Normal'-Superconductor (NN'S) junction, in which current injection destroys superconductivity in a small region N' of the superconductor, with a size varying with the applied voltage V. Voltage-dependent de Gennes -Saint James (dGSJ) bound states appearing in the N' slab lead to two distinct sets of conductance oscillations. We show that this effect significantly alters the conductance of systems for which κ 2 vF ∼ 10 9 m/s such as pnictides (κ and vF being the Ginzburg number and the Fermi velocity, respectively), and we discuss their consequences on the identification of the bosonic modes of strongly coupled superconductors.

Current-voltage I(V) characteristics of junctions are powerful tools to study the coupling of charge carriers with the various excitations of a solid. In particular the second derivative of the differential conductance is proportional to the Eliashberg function, α 2 F (ω) 1 , that quantifies the interaction of electrons with the bosonic excitations of a given system. This quantity has hence been extensively used to study a large variety of bosonic modes, such as phonons in normal metals or magnons and crystal field splittings [START_REF] Khotkevich | Atlas of Point Contact Spectra of Electron-Phonon Interaction in Metals[END_REF] . Likewise, in the strong-coupling limit, the Eliashberg theory of superconductivity predicts an energy-dependent order parameter ∆(ω), which exhibits strong modulations at energies for which α 2 F (ω) is large 3 , leading to modulations of the single-particle excitation spectrum (the so-called tunneling density of states, tDOS). One of the first and most famous examples is the tDOS of Pb, which exhibits two dip-hump features related to the two phonon modes responsible for most of the pairing interaction 4 . More recently, a well-defined satellite structure was evidenced in the tDOS of pnictides 5,6 . The energy and temperature dependence of this satellite closely match the behavior of a magnetic mode observed by neutron scattering, commonly accepted to be involved in the unconventional s ± gap symmetry 7 .

Point-Contact Spectroscopy (PCS) is a technique in which a microscopic Normal Metal-Superconductor (NS) junction is created, for example by making a nanoscale contact between a metallic tip and the studied superconductor (the so-called spear-anvil technique which is used in our setup). At the NS interface, the superconducting pairing potential ∆ is weakened by the proximity of the normal metal, which results in the formation of coherent bound states, first described by de Gennes and Saint-James 8 , and experimentally observed in various systems 9 . Experimentally, it turns out that in most PCS measurements the proximity effect can be neglected. However, in the case of pnictides, unusually large lifetime broadenings must be introduced to fit the conductance peaks, hindering a detailed analysis of the gap symmetry. Moreover, many PCS spectra exhibit non-negligible modulations at voltages in the range 20-60 meV which are poorly accounted for 6,10,11 . We show here that injection effects actually have to be taken into account in the modeling of the metal -superconductor junction in systems for which κ 2 v F ∼ 10 9 m/s such as pnictides (κ and v F being the Ginzburg number and the Fermi velocity, respectively). Indeed, in this case, a small normal region (N') develops at the NS interface, forming a NN'S junction. The size of this N' region is dynamically driven by the applied current (and is hence voltage-dependent) significantly altering the electronic structure of the superconductor on a typical distance on the order of the superconducting coherence length.

The model is based on the Bogoliubov-De Gennes equations in a simplified 1D geometry, following the approach of Hahn 12,13 . The normal metal (superconductor) occupies the y<0 (y>0) half-space (see Fig. 1). As discussed below, a metallic slab (with effective mass and Fermi velocity similar to those of the superconductor normal state) can develop in the superconductor on a size L due to injection effects and the one-electron potential will hence be given by V (y) = V N + Hδ(y) + (V S -V N )Θ(y), with δ(y) and Θ(y) the Dirac and Heaviside distributions, respectively. Finally, the pairing potential ∆(y) can then be taken as ∆(y) = ∆Θ(y -L), where ∆ is constant and isotropic.

The transmission coefficient of the NN'S junction will be given by (see Eq.( 18) in 12 ) :

T (ε, λ) = τ N (1 + τ N x -(1 -τ N )x 2 ) 1 -2(1 -τ N )x cos[2(α -φ)] + (1 -τ N ) 2 x 2
where τ N = 1/(1 + Z 2 ef f ) (Z ef f being the effective transparency of the barrier), α = 2λε/π with ε = E/∆ being the normalized energy in unit of the superconducting gap and λ = L/ξ the renormalized length of N' normal slab in unit of the coherence length ξ, We assume that a dynamically current-driven N' normal state develops at the NS interface for 0 < x < L(V ). The superconducting pairing potential is destroyed in this region and becomes finite for x > L (right scale) whereas the one-electron potential is discontinuous at the x = 0 (left scale).

[x, φ] = [(ε - √ ε 2 -1) 2 , 0] if ε 1 [1, arccos ε] if ε < 1 N N' S V N V S V (
and the differential conductance of the junction by :

dI N S dV = ∂I N S ∂V λ + dλ dV ∂I N S ∂λ V
where

I N S (V ) = -∆ eR N eV /∆ 0 T (ε,λ)
τ N dε and the second term explicitly takes into account the dynamical change of the slab width λ. The calculation is straightforward and, introducing the renormalized voltage Ṽ = eV /∆, one finally obtains

G( Ṽ ) = T [ Ṽ , λ( Ṽ )] τ N + 1 τ N dλ d Ṽ Ṽ 0 dT dλ dε = G 0 ( Ṽ )+G d ( Ṽ )
where G 0 corresponds to a standard "static" conductance channel, which is non-zero in all possible geometries, and G d a "dynamical" channel which exists only when the slab width changes with voltage. The differential conductance of the junction is then fully determined by the Blonder-Tinkham-Klapwijk (BTK) barrier 14 (τ N ) and the voltage dependence of the slab width, λ( Ṽ ). The fixed L geometry (G(V ) = T (eV, L)/τ N ) has been thoroughly studied in the literature and we will rather focus on the case for which the slab width L changes with applied voltage due to high current injection that will drive the junction out of equilibrium (see discussion below). Note that the former equations have been derived at T=0 and, at finite temperature, the differential conductance is given by the standard convolution by the Fermi function.The temperature hence acts as a low-pass filter on the differential conductance, strongly smearing out oscillations with periods smaller than a few k B T . Fig. 2a displays the G 0 and G d components of the differential conductance at T=0, calculated for a moderate barrier Z ef f = 1 and a linearly growing slab of width L = αeV with a typical α value on the order of ξ/∆ i.e. λ = β Ṽ with β = 1. For this Z ef f value, G 0 exhibits a low-bias peak and an oscillating behavior for Ṽ > 1 with a period and amplitude that decrease with increasing voltage. As expected, G 0 tends towards 1 at high biases. Similarly, G d also exhibits a low-bias peak and oscillations which coincide with the G 0 ones but, the highbias G d baseline is centered around zero conductance, as it is mainly a correction to the excess current caused by the dynamical change of the N' slab. Besides, a second, low-frequency oscillation is visible, with a slowly decreasing amplitude.

In the standard BTK model (i.e. no N' slab), the conductance is featureless except for the quasiparticle peak located exactly at the gap energy ( Ṽ = 1, see inset of Fig. 2a), but it is important to note this peak is shifted to lower biases in the presence of the N' slab (either of constant width or for a dynamically increasing slab). Indeed, the De Gennes-Saint James (dGSJ) subgap states present in the N' slab 8 give rise to oscillations in the differential conductance (cosine term in the transmission coefficient) and, in presence of the slab, G is maximum when α-φ = 0 [π]. A first subgap state is hence obtained for Ṽ = cos(2λ Ṽ /π) ≤ 1 and the corresponding broad conductance peak is a signature of the discontinuity in the dGSJ density of states and not of a true quasipartic-ule peak anymore (see inset of Fig. 2a). The peak position still gives directly the energy position of the dGSJ feature for constant L, but does no longer correspond to any particular energy in the case of a dynamically growing N' slab. It is hence important to note that the differential conductance can no longer be directly related to the density of states of the junction as the electronic structure of the slab dynamically evolves with the applied voltage.

For Ṽ > 1, the conductance maxima occur for voltages Ṽ = nπ 2 /2λ( Ṽ ), which correspond to the well-known Rowell-McMillan (RMM) oscillations 15 . Those oscillations are periodic for a constant L value (see Fig. 2b) and are present in both G 0 and G d for the dynamically current-driven slab with a quasi-period that decreases as λ increases. The low-frequency oscillations visible in G d have a more subtle origin : they correspond to the entrance of a new dGSJ bound state below the superconducting gap and are defined by : λ( Ṽ ) = nπ 2 /2. As the slab size increases, the total number of subgap states inside the slab increases, with sudden jumps appearing when a new state enters. The differential conductance of a dynamically growing NN'S junction will hence show two kinds of characteristic features, both related to the existence and the evolution of dGSJ bound states inside the N' region (see Fig. 2b). First,the quantized nature of the dGSJ spectrum will give rise to broad subgap states and non-periodic RMM type oscillations above the gap. Second, the modulation of the total number of bound states inside the N' region as it grows with voltage will lead to a low-frequency, slowly decaying oscillation.

The effect of the interface effective barrier Z ef f on the differential conductance of the dynamical NN'S model is displayed in Fig. 3. As expected from the BTK model, the zero-bias conductance decreases with increasing Z ef f , as the probability of Andreev reflexion at the NN' interface drops, and the shape of the conductance becomes more and more tunnel-like. The oscillations vanish for Z ef f = 0, as the differential conductance tends towards the "bare" BTK prediction. Indeed, in this case, the charge carriers can cross the NN' interface without being scattered or reflected, and no quasiparticle interference can occur : the NN'S sandwich behaves as an effective NS junction. For increasing Z ef f quasi-particule interferences become more and more effective, resulting in sharper subgap peaks and RMM oscillations of higher amplitude. As shown in Fig. 3b, the low frequency oscillations increases for Z ef f 1 but are hidden by the RMM signal for higher barrier strengths. For Z ef f = 10 the low-frequency signal is completely invisible compared to the 1/ε 2 envelope function of the RMM oscillations.

Finally, the influence of the grow rate β = dλ/d Ṽ is displayed in the inset of Fig. 3a. As discussed above, for β ≤ 4 the main low biais conductance peak shifts towards lower energies with increasing β and do not correspond to the quasiparticle peak anymore. For higher β values, several dGSJ subbands appear before the gap voltage is reached and the differential conductance exhibits n subgap peaks. Correspondingly, the quasi period of the high bias oscillations decreases with increasing β.

Let us now discuss the two main mechanisms that can destroy superconductivity in the vicinity of the NN' contact . The model will be derived for the case of a point contact between the N and N' regions, and a hemispherical N' region (which we call a "normal bubble"). It turns out, that the dGSJ states for a s-wave superconductor in spherical symmetry have the same characteristic energies and lengths as the 1D case 16 . Besides, taking into account the 2D or 3D nature of the junction often leads to results that can be nearly reproduced by a 1D model with an effective barrier strength 2 . Apart from some amplitude deviations, we then expect the 1D model to be a fair approximation of a more realistic 3D point-contact.

First, if the local current density exceeds the depairing current density J d in a region of typical size L ≈ ξ, it becomes energetically favorable for the sample to transit into the metallic state in this region. Let us consider a planar NS junction of contact resistance R N and assume that the current is injected in the S electrode through a point contact and spreads isotropically in the superconductor from this point source (this is a good approximation for small Z ef f ). In this approximation the surfaces of constant current densities are hemispheres centered around the injection point and

J ∼ J d for L ∼ V /[2πR N J d ].
Note that we assumed here a linear I(V ) characteristics which leads to an error of at most a factor 2 in the Z ef f → 0 limit (for Ṽ < 1), and becomes exact in the Ṽ 1 limit. The RMM maxima of the differential conductance will then be obtained for Ṽ = (n 2 π 5 ξ 2 R N J d e/2∆) 1/3 . Taking J d (0) ∼ 10 -10 /λ L (0) 2 ξ(0) and ∆(0) = v F /πξ(0) one obtains :

Ṽ ∼ [n 2 10 8 R N /v F κ 2 ] 1/3 F (T /T c )
with κ = λ L /ξ is the Ginzburg parameter and v F being the Fermi velocity. Note that F (T /T c ) ∼ 1 so that Ṽ is only very weakly temperature dependent. Similarly, the low frequency oscillations will be observed for λ = nπ 2 /2 i.e. for Ṽ = [n 2 10 8 R N /v F κ 2 ]. Taking R N ∼ 10 Ω, those out-of-equilibrium effects are hence expected o be observed in systems for which v F κ 2 ∼ 10 9 m/s. Note that in this normal bubble scenario the energy period of the oscillations shrinks with temperature (following the gap) in strong contrast with the ∆(ω) related bosonic spectrum in superconducting tunneling experiments, which shifts in energy with temperature, as the superconducting gap closes.

In standard superconductors (such as Nb) κ ∼ 10, v F ∼ 3.10 5 m/s and 10 8 R N /v F κ 2 ∼ 30; the oscillations are hence barely visible but in 122-pnictides, such as BaFe 2-x (Co,Ni) x As 2 , κ ∼ 100 (see for instance 17 ), v F ∼ 10 5 m/s 18 and 10 8 R N /v F κ 2 ∼ 1 making this system very sensitive to those current induced out of equilibrium phenomena. This system has been previously intensively studied in our group (see [17] and references therein for sample details) and we have performed PCS measurements (with R N ∼ 10 -20 Ω) in BaFe 2-x Ni x As 2 single crystals close to optimal doping. As shown in Fig. 4(a), as expected, clear oscillations are visible in the high bias differential conductivity in very good agreement with our theoretical current-driven N' "bubble" (the low energy spectra present the typical Andreev bump usually observed in those systems, not shown). Those oscillations have been observed in several samples and in all of them the conductance maxima could be ascribed to two quasi periods V max = αn 2/3 with α ∼ 10 meV and α ∼ 16 meV, respectively (see Fig. 4a). Introducing the two gaps of this system (∆ S ∼ 4 meV and ∆ L ∼ 10 meV 6,11 ), one hence gets [10 8 R N /v F κ 2 ] ∼ 0.1 in very reasonable agreement with the electronic parameters of this system 18 . Note that, as discussed above, the presence of those oscillations also leads to a shift (and important smearing) of the low biais peak which then leads for an underestimation of the gap value on the order of ∼ 10% for the small gap and up to ∼ 30% for large gap. This shift/smearing effect can account for the large dispersion of the gap values deduced from PCS measurements 11 . Finally note that, as expected, the position of the observed maxima shrinks with T, nearly following the temperature dependence of the gap (see right panel in Fig. 4a).

The second mechanism which is expected to give rise to an injection-dependent normal bubble is the formation of a non-thermal "hot spot" near the point-contact. In this scenario, multiple scattering of high-energy injected electrons leads to an out-of-equilibrium electron distribution function that does not allow for a superconducting gap to exist on a typical scale that depends on injection current and electron-boson coupling. In order to calculate the λ( Ṽ ) law, numerical calculations were carried out using the formalism introduced by Hahn 19 . The input parameters of the model are the Fermi velocity, the coherence length (or superconducting gap), the density of states at the Fermi energy and the electron-boson coupling Eliashberg function. The λ( Ṽ ) law calculated for typical parameters of the 122 Ni or Co doped pnictides 18 (N 0 = 1.3 10 20 J -1 .nm -3 , ξ 0 = 3nm, ∆ 0 = 4meV ) and the Eliashberg fonction sketched by the dashed-dotted line is displayed in Fig. 4b (dotted line) together with the corresponding calculated differential conductance (red line).

As shown, If α 2 F (ω) exhibits well-defined strongly coupled low-energy peaks λ( Ṽ ), the injected electrons are poorly scattered for low voltages and the normal bubble remains very small. When the injection voltage reaches the first (sharp) bosonic mode, electron scattering becomes very efficient and the normal bubble grows quickly. As the voltage increases, more and more bosonic modes will become available for scattering, resulting in a smoothly growing bubble, with some modulations for voltages matching strongly coupled bosonic modes. This non-trivial bubble growth is then expected to give rise to satellite structures in the differential conductance (see red curve in fig. 4b) opening the possibility to identify the relevant bosonic modes for charge transport even in imperfect, strongly diffusive junctions. Even though most of the single-electron spectroscopic information is lost in such junctions, this work suggests that in real experiments electron-boson coupling can still be probed accurately in "bad" junctions, provided the quasiparticle mean free path is of the order of magnitude of, or slightly larger than the superconducting coherence length. Unfortunately, in our case, the strong oscillations induced by the first mechanism are hindering the clear identifi-cation of those bosonic modes but it is worth noting the enhanced oscillations observed for V ∼ 20 meV which are consistent with the magnetic modes deduced from neutron measurements 7 (see also 6 ).

We have presented here a model for the differential conductance of current-driven NN'S junctions, in which the N' slab is dynamically induced by current injection effects. This slab can be due either to the fact that the injected current density exceeds the depairing current in the vicinity of the injection point or to downscattering of high-energy charge carriers by strongly coupled bosonic modes. Critical current effects are expected to play an important role in systems for which v F κ 2 ∼ 10 9 m/s in very good agreement with PCS spectra performed in BaFe 2-x Ni x As 2 signal crystals. If the electron mean free path is comparable to the Pippard coherence length, enhanced modulations of the differential conductivity are expected to be observed for voltages corresponding to strongly coupled bosonic modes.

This research was performed using funding from the Lab Alliance for Nanosciences and Energies for the Future (Lanef), Institut Néel of the Centre National de la Recherche Scientifique, and Université Grenoble Alpes. The work at IOP, CAS, is supported by NSFC (No.11374011) and CAS (SPRP-B: XDB07020300). This work is supported by the French National Research Agency through Grant No. ANR-12-JS04-0003-01 SUB-RISSYME.

Figure 1 .

 1 Figure 1. Sketch of the NN'S junction studied in this paper.We assume that a dynamically current-driven N' normal state develops at the NS interface for 0 < x < L(V ). The superconducting pairing potential is destroyed in this region and becomes finite for x > L (right scale) whereas the one-electron potential is discontinuous at the x = 0 (left scale).

Figure 2 .

 2 Figure 2. (a) G0 and G d components of normalized differential conductance in a dynamically current-driven NN'S junctions (see text for details) calculated for Z ef f = 1 and a slab growth law λ( Ṽ ) = 1 × Ṽ . Both components exhibit a high-frequency oscillation, whereas a low-frequency oscillation is also present in the dynamical channel G d (see text). Inset : comparison of the low biais spectra in the standard BTK model (no N' slab, black curve), a static N' slab of size L=ξ (blue curve) and a dynamical N' slab of width L(V ) = eV ξ/∆ (Red Curve). Note that the coherence peak is shifted towards low biais voltages in the presence of the slab. (b) high biais zoom the total normalized differential conductance for three same geometries.

Figure 3 .

 3 Figure 3. (a)Effect of the effective barrier parameter Z ef f on the differential conductance of a dynamically current-driven NN'S junctions with λ = 1 × Ṽ . Inset : influence of the N' region growth rate β = dλ/d Ṽ on the differential conductance at T=0. The spectra corresponding to the values displayed on the left axis are displayed as full black lines above the color map. (b) High bias zoom of the differential conductance showing that the amplitude of the low frequency oscillations increases for Z ef f → 1.

Figure 4 .

 4 Figure 4. (a) High bias differential conductivity measurements in a BaFe2-xNixAs2 close to optimal doping between T = 5.5 K and 12.1 K (the different curves have been shifted for clarity). As shown quasi-periodic strong modulations are visible (left) and the position of the maxima roughly scales as the superconducting gap (right). The position of the peaks can be indexed following two sequences scaling as Vmax = αn 2/3 (see text for details) with α ∼ 10 meV (open circles) and α ∼ 16 meV (closed squares), respectively. (b) Calculated λ( Ṽ ) (dotted line) for typical parameters of the 122 pnictides (see text). The red full line is the corresponding calculated differential conductance at T=0. Strong modulations of the differential conductivity are visible in the vicinity of the bosonic modes.

Yu. G. Naidyuk and I. K. Yanson, Point-Contact Spectroscopy, Springer Series in Solid State Science

, 2005;