Patrick Dular

Patrick Kuo-Peng

Mauricio V Ferreira Da Luz

Laurent Krähenbühl

MODEL REFINEMENTS OF TRANSFORMERS VIA A SUBPROBLEM FINITE ELEMENT METHOD

A progressive modeling of transformers is performed via a subproblem finite element method. A complete problem is split into subproblems with different adapted overlapping meshes. Model refinements are performed from ideal to real flux tubes, 1-D to 2-D to 3-D models, linear to nonlinear materials, perfect to real materials, single wire to volume conductor windings, and homogenized to fine models of cores and coils, with any coupling of these changes. The proposed unified procedure efficiently feeds each subproblem via interface conditions, which lightens mesh-to-mesh sources transfers, and quantifies the gain given by each refinement on both local fields and global quantities.

Introduction

For efficient and accurate numerical modeling of transformers, an innovative step-by-step methodology is developed. It is based on a finite element (FE) subproblem (SP) method (SPM) with magnetostatic and magnetodynamic problems solved in a sequence on different adapted meshes [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF], from simple 1-D models up to accurate 3-D models, in a large frequency range, of the magnetic circuits and their windings (stranded or massive coils). Each step of the SPM aims at improving the solution obtained at previous steps via any coupling of the following changes, defining model refinements: change from ideal to real (with leakage flux) flux tubes [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF], change from 1-D to 2-D to 3-D [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF], change of material properties [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF] (e.g., from linear to nonlinear), change from perfect to real materials [START_REF] Dular | Perfect conductor and impedance boundary condition corrections via a finite element subproblem method[END_REF], change from single wire to volume conductor windings [START_REF] Dular | Perfect conductor and impedance boundary condition corrections via a finite element subproblem method[END_REF], [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF], and newly developed change from homogenized [START_REF] Krähenbühl | Homogenization of lamination stacks in linear magnetodynamics[END_REF] to fine models (cores as lamination stacks and coils as wire or foil windings, with the details affecting their high frequency behaviors). The methodology involves and couples numerous techniques that have been developed by the authors and, up to now, only applied for simplified test problems [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF]. It can also help in education with a progressive understanding of the various aspects of transformer design.

Progressive Magnetostatic and Magnetodynamic Models

Methodology

Transformers are made of magnetic regions, defining magnetic circuits, and active (connected to external electric circuits, i.e., windings or coils) and passive (not fed by circuit, e.g., tank) conducting regions. Such systems are planned to be studied with the following methodology, defining sequences of adequate changes/corrections as these listed in the introduction. Magnetic circuits -For magnetic cores with possible air gaps, the analysis is first focused on ideal flux tubes neglecting the leakage fluxes and core losses, before considering the coil conductors with their details. A preliminary equivalent magnetic circuit can define a 0-D model. Some of its branches can then be progressively refined via the consideration of their actual geometries, in 1-D, 2-D and 3-D, via SPs of associated dimensions [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF]. Such SPs consider changes of boundaries of domains that can be either extended or connected together. Each dimension fixes some boundaries through which some assumptions on magnetic flux are considered via some boundary conditions (BCs). A higher dimension modifies such BCs via interface condition (IC) surface sources (SSs) [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF]. Changes of material properties can be considered via volume sources (VSs) or SSs when adding, removing, changing or moving some regions [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF]. Nonlinear behaviors of materials can naturally be taken into account as a change from linear to nonlinear material properties. Stranded and massive conductors -Once the real flux tubes are tackled, the actual geometry of the windings has to be considered, instead of magnetomotive force (MMF) or flux sources. Simultaneously with the IC sources allowing the ideal flux tubes to become permeable [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF], the windings are progressively defined at various levels of precision for their geometry and the distribution of the current they carry. They can progress from 1-D to 2-D to 3-D, from single wire to volume FE geometries, and from stranded (uniform current density distributions in their cross sections) to massive conductors, to improve the local field distributions and to accurately render skin and proximity effects [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF]- [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF]. The global inductive and resistive behaviors are improved as well. Passive conductors (e.g., transformer tank, magnetic shield) can similarly be added and their models improved. Juxtaposition of thin or wired regions -Lamination stacks in magnetic cores, and wire and foil windings are first classically considered via homogenization models [START_REF] Krähenbühl | Homogenization of lamination stacks in linear magnetodynamics[END_REF] before being locally corrected (in critical regions of interest; in particular for accurate eddy current losses), with an FE SPM derived from [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF] and applied in certain thin or wired regions separately with their details (e.g., single conductors with rectangular cross-section and possible oil ducts for cooling; which is usually unfeasible in a direct 3-D approach), coupled to the remaining regions kept homogenized. Any change that gives a significant effect on the previously solved SPs has to be considered as a source for these. This defines series of corrections on both magnetic circuits and conductor models.

Subproblem Method -Sequence of Subproblems

A canonical magnetodynamic problem p, to be solved at step p of the SPM, is defined in a domain Ω p , with boundary ∂Ω p = Γ p = Γ h,p ∪ Γ b,p . The eddy current conducting part of Ω p is denoted Ω c,p and the non-conducting one Ω c,p C (superscript C means "complementary to Ω c,p in Ω p "), with Ω p = Ω c,p ∪ Ω c,p C . Stranded conductors, when homogenized, belong to Ω s,p ⊂ Ω c,p C . Magnetic field h p and electric current density j p traces are subject to BCs on surfaces Γ h,p , whereas magnetic flux density b p and electric field e p traces are subject to BCs on surfaces Γ b,p , i.e.,

n × h p | Γ h,p = h s,p , n ⋅ j p | Γ b,p = j s,p , n ⋅ b p | Γ b,p = b s,p , n × e p | Γ
] γ p = [h s,p ] γ p , [n ⋅ j p ] γ p = [j s,p ] γ p , [n ⋅ b p ] γ p = [b s,p ] γ p , [n × e p ] γ p = [e s,p ] γ p . (2a-b-c)
The fields are related via the material relations

h p = µ p -1 b p in Ω p , j p = σ p e p in Ω c,p , j p = j s,p in Ω s,p ⊂ Ω c,p C , (3a-b-c)
where µ p is the magnetic permeability and σ p is the electric conductivity, both region-wise defined, and j s,p is a given source current density in Ω s,p .

With the magnetic vector potential a p and electric scalar potential v p defined via

b p = curl a p , e p = -∂ t a p -grad v p = -∂ t a p -u p , (4a-b)
the Gauss and Faraday equations are strongly satisfied. The a p weak formulation of the magnetodynamic problem is then obtained from the weak form of the Ampère equation, i.e. [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF],

(µ p -1 curl a p ,curl a ') Ω p -( j s, p ,a ') Ω s, p +(σ p ∂ t a p ,a ') Ω c, p +(σ p u p ,a ') Ω c, p +< n× h p ,a ' > Γ h, p \γ p +< [n× h p ] γ p ,a ' > γ p =0 , ∀a ' ∈ F p 1 (Ω p ), (5)
where F p 1 (Ω p ) is a curl-conform function space defined on Ω p , gauged in Ω c,p C , and containing the basis functions for a p and for the test function a' (at the discrete level, this space is defined by edge FEs; the gauge can be based on the tree-co-tree technique); ( • , • ) Ω and < • , • > Γ denote a volume integral in Ω and a surface integral on Γ, respectively, of the product of their field arguments.

A complete problem is aimed at being solved as a sequence of SPs, involving models for magnetic and conducting regions of progressive accuracy and calculating successive additive corrections. A particular studied domain with an adapted FE mesh is defined for each SP. In the following, region names with no SP index p are regions of the complete problem, to be involved in particular SPs.

Sequenced Models

Magnetic Circuits -Ideal Flux Tubes

The magnetic (conducting or not) core regions Ω m ⊂ Ω p , possibly extended with air gaps Ω g ⊂ Ω m , can first be considered as ideal flux tubes (IFTs), i.e., with no leakage flux (Fig. 1). Their boundaries Γ m = ∂Ω m are then considered as perfect magnetic walls (usually made of non-connected portions Γ m,i in 2-D, with i the portion index). The actual current source regions Ω s exterior to Ω m are thus idealized as perfectly wounded current sheets around Ω m .

For the so-defined SP p ≡ IFT, the studied domain is limited to Ω p ≡ Ω m , with a BC on Γ m ⊂ Γ b,p fixing a zero normal trace of the magnetic flux density b p (its zero trace is also fixed on the possible remaining zero-flux symmetry planes Γ b,p \ Γ m , together with n × e p = 0 on portions of Γ m also in ∂Ω c,p ). In terms of a p , one has the equivalent essential BCs

n ⋅ b p | Γ m = 0 ⇔ n × a p | Γ m = n × grad w p | Γ m , (6a-b)
where w p is a multivalued surface scalar potential defined on Γ m . In 3-D, this scalar potential undergoes a constant jump through each of the cut lines making Γ m simply connected [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF]. This can be defined via discontinuous components w disc,cl,p acting only on one side of each cut line cl (at the discrete level, in a surface layer of surface FEs on one side of each cut line, with w disc,cl,p varying from a constant on cut line cl to zero on the other boundary of the layer), to be added to the continuous component w cont,p to give w p . In 2-D, such constant jumps come down to the definition of a constant a p (a kind of floating potential when unknown) on each non-connected portion Γ m,i of Γ m (Fig. 1). The constant jumps are directly (strongly) related to the unknown magnetic fluxes flowing in Ω m and are weakly related to the MMFs F cl along each cut line cl via the weak formulation [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF]. Indeed, when tested with the non-local jump test functions a' surf,cl = grad s w' disc,cl,p (grad s is the surface gradient) [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF], [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF], it gives, if Ω p ⊂ Ω c,p C for simplification,

(µ p -1 curl a p ,curl a ' surf,cl ) Ω p +< n× h p ,a ' surf,cl > Γ m = 0 , with < n× h p ,a ' surf,cl > Γ m ≡ F cl . (7a-b)
If domain Ω p is not a massive conductor, i.e., Ω p ⊂ Ω c,p C , the required gauge condition on a p in Ω c,p C , thus also on Γ b,p , allows to fix potential w cont,p to zero on Γ m ⊂ Γ b,p . However, if Ω p ⊂ Ω c,p , potential w cont,p must be kept unknown on Γ m , because a p is not gauged there.

Each Γ m,i can be considered as the boundary of a slot in a device. The related MMF gathers all the current sources in the slot, for all coils, e.g., primary and secondary coils in a transformer. A slot can be generalized to represent the exterior region, including coils as well (Fig. 1, right).

Ω m Γ m,2 Ω s n X Y Ω g Ω s Γ m,1 Ω m Ω g Ω s Γ m,2 Γ m,1 n Γ h Γ b X Y X Z Ω s Γ b Γ b Ω m Γ m
Fig. 1. Shell type transformer: example of a magnetic region Ω m , including a possible air gap Ω g , first considered without leakage flux via perfect magnetic walls BCs on Γ m,1 and Γ m,2 (left); three symmetry planes allow to reduce the studied domain (right); the end windings need to be considered in 3-D (bottom right).

Stranded and Massive Conductors -Real Flux Tubes with Leakage Fluxes

FE models of the windings are necessary for the accurate determination of their characteristics (e.g., impedances, losses, forces, leakage flux). These can define additional SPs aiming at correcting IFT models, with associated progressive geometrical or physical models. An IFT solution q is by construction discontinuous through the considered flux walls Γ m , with a non-zero trace n × h q | Γ m + on Ω m -side Γ m + and a zero trace n × h q | Γ m -on the other side Γ m -(or nonzero too in case Ω m C ⊂ Ω p , which can be a useful extension) (Fig. 2); trace n ⋅ b q is however continuous because zero on both sides. Its correction via an SP p ≡ RFT (real flux tube), that consists in changing the flux wall to a permeable surface allowing leakage flux, can be done by suppressing the discontinuity of n × h q through the considered flux walls Γ m and by simultaneously adding the actual windings (or some simplified models to be further corrected). The opposite of this discontinuity thus serves as an IC-SS [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF] for [n × h p ] Γ m weakly defined in [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF], whereas n ⋅ b p and thus n × a p remain continuous, i.e.,

[n × h p ] Γ m = -[n × h q ] Γ m = -n × h q | Γ m + , [n × a p ] Γ m = 0 . (8a-b)
In practice, at the discrete level, the studied domain Ω p for SP p ≡ RFT includes the windings with their neighborhood, thus also comprising the magnetic cores previously considered via an IFT model (Fig. 2). At this step, if a adequately fine mesh is required for the windings and their vicinity, a coarse mesh can usually be considered for the magnetic core, in which the main flux has already been taken into account with SP q ≡ IFT with a mesh allowing a good accuracy level. Indeed, the leakage flux, flowing in some parts of the magnetic cores, usually slightly modifies the IFT flux. If it is not the case, an additional SP can correct the flux distribution in the cores. The core could even be considered as a perfect magnetic material (PMM) in a prior SP r ≡ CORE-PMM, defined in Ω r ≡ Ω \ Ω m with BC n × h r | Γ m -= 0, the solution of which giving a non-zero trace n × a r | Γ m -. SP p ≡ RFT would then combine the solutions of SPs q and r, in complementary domains Ω q and Ω r , as SSs for ICs, with no need to include the windings already considered in SP r, i.e.,

Ω m Ω g n Γ h Γ b X Y Ω m Ω g Ω s n Γ h Γ b X Y Γ m + Γ m - n + n - Γ m SP q ≡ IFT SP p ≡ RFT IC-SS: [n×h p ] Γm = -[n×h q ] Γm Γ m Ω p
[n × h p ] Γ m = -[n × (h q +h r )] Γ m = -(n × h q | Γ m + -n × h r | Γ m -) = -n × h q | Γ m + , (9a) [n × a p ] Γ m = -[n × (a q +a r )] Γ m = -(n × a q | Γ m + -n × a r | Γ m -) . (9b)
A correction from one dimension (SP q) to the next one (1-D to 2-D or 2-D to 3-D) in an SP p uses again SS-ICs (8a-b) or (9a-b). Indeed, a lower dimension model actually defines some perfect magnetic walls that are made permeable when increasing the dimension [START_REF] Dular | Finite element magnetic models via a coupling of subproblems of lower dimensions[END_REF].

Volume Regions -Changes of Geometrical and/or Physical Properties

When progressive geometrical models or changes of physical properties are considered for a region, e.g., from a simplified winding geometry in a SP q (via a Biot-Savart or FE model) to the actual FE volume one Ω vol,p in an SP p, or when changing a portion of a flux tube to an air gap, each change can be done via the following SP p ≡ VOL-REGION-FE. The general key is to consider such an SP p as the superposition of two SPs, pa and pb, actually simultaneously solved (Fig. 3). SP pa aims at removing the approximate solution q in the newly defined region Ω vol,p while keeping it unchanged outside, simultaneously, in FE SP pb, with the consideration of the actual properties µ p and σ p in Ω vol,p . SP pa exactly cancels out, with no need of any solving, the VSs that would be needed [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF]. It rather leads to trace discontinuities of its direct solution through Γ vol,p given by the corresponding traces of solution q on Γ vol,p as IC-SSs, i.e.

(inner minus outer field traces; normal n exterior to Ω vol,p ),

[n × h pa ] Γ vol,p = -n × h q | Γ vol,p + , [n × a pa ] Γ vol,p = = -n × a q | Γ vol,p + . (10a-b)
Because SP pb defines a continuous solution through Γ vol,p , the trace discontinuities (10a-b) remain valid for the resulting SP p, i.e.,

[n × h p ] Γ vol,p = -n × h q | Γ vol,p + , [n × a p ] Γ vol,p = = -n × a q | Γ vol,p + . (11a-b) SP p µ p =µ q σ p =σ q SP q SP pa SP pb + IN OUT h pa = -h q b pa = -b q h pa = 0 b pa = 0 Ω vol,p Γ vol,p IN OUT Ω vol,p µ p σ p n h q b q h q b q Ω p Ω q Ω p h q +h pa = 0 b q +b pa = 0 ! No VS
No change outside approximate model ∂Ω q ∂Ω p Fig. 3. FE correction (SP p) of an approximate solution (SP q): SP p is split into SPs pa and pb, simultaneously solved, SP pa removing the approximate solution q inside the region Ω vol,p ⊂ Ω p to be geometrically improved and SP pb considering the improved model with its actual geometry and properties.

Note that in all cases with ICs, the studied domain covers both sides of the involved interface, that are therefore naturally coupled. Especially for Γ vol,p , this is absolutely different than limiting Ω p to Ω vol,p with a classical BC on Γ vol,p , fixing either n × h p or n × a p equal to the corresponding trace of the solution of SP q. This solution should then be more or less the true solution on Γ vol,p , which is in general not true. The corrections, supported by different adapted meshes, are usually of local nature, which allows some simplifications of the outer regions, possibly replaced by equivalent circuits extracted from previous SPs. Each volume correction gives the total field in Ω vol,p , is discontinuous through Γ vol,p and usually quickly decreases outside, which allows the use of a coarse mesh in the outer region. A volume correction can also be applied in the proposed unified form for a change from a linear to a nonlinear property, and for a newly added region (e.g., tank or shield) in a field solution of an SP q, thus acting as a source field in an SP p.

Juxtaposition of Thin or Wired Regions -Homogenization to Fine Models

A homogenization model of core lamination stacks or windings is usually based on assumptions that neglect fringing effects at lamination or winding borders [START_REF] Krähenbühl | Homogenization of lamination stacks in linear magnetodynamics[END_REF]. Some local corrections can be done via SPs for a certain number of laminations or winding turns (Fig. 4), at some particular positions, e.g., where significant edge effects occur. For each correction SP p ≡ VOL-FINE-FE, the studied domain Ω p comprises the candidate thin or wired regions for correction, now defined with their actual volumes Ω c,p surrounded by their insulating layers Ω nc,p , defining Ω vol,p = Ω c,p ∪ Ω nc,p , that can be finely meshed, and the other thin or wired regions kept homogenized, so coarsely meshed, together with a well chosen neighborhood.

Because SP p ≡ VOL-FINE-FE implies changes of both geometrical and physical properties, it can directly be defined via the method previously developed for SP p ≡ VOL-REGION-FE. Each correction SP is thus again defined via IC-SSs, with the interesting advantage to correct the solution in the actual laminations or windings in Ω vol,p while keeping a coupling with the still homogenized region exterior to Ω vol,p . Here also, using classical BCs on Γ vol,p instead of ICs would generally lead to inaccurate corrections. From the newly calculated total field in Ω vol,p , a more accurate determination of losses can be obtained.

Corrections of other additional regions can use, as IC-SSs, the updated outer solutions from previous corrections, i.e., the homogenized solution plus the results of these corrections. This is of importance when significant proximity effects occur between laminations or windings.

SP q

Homogenized solution: ICs of type (a) are of weak nature, being involved in a surface integral term in weak formulation [START_REF] Dular | Subproblem finite element refinement of inductors from wire to static and dynamic volume models[END_REF], defined on a double layer surface γ p . In all cases, this term becomes

h q , b q Ω q Ω p
< [n× h p ] γ p ,a ' > γ p =< -n× h q ,a ' > γ p + , with γ p ≡ Γ m or Γ vol,p , (12)
and is non-zero only for test functions a' with a non-zero trace on γ p . The right hand side of (12) is to be weakly expressed from (5) now applied to SP q, i.e., from volume integrals usually limited to

< -n× h q ,a ' > γ p + = (µ q -1 curl a q ,curl a ') Ω p =Ω q . ( 13
)
At the discrete level, the volume integral in ( 13) is limited to the single layer E TL,p of FEs touching γ p , on side Ω m or Ω vol,p , because it involves only the associated trace n × a'| γ p + having a contribution only in E TL,p . The source a q , initially in the mesh of SP q, has to be projected in the mesh of SP p only in E TL,p . This significantly decreases the computational effort of the projection process. An L 2projection is used [START_REF] Dular | Perturbation finite element method for magnetic model refinement of air gaps and leakage fluxes[END_REF]- [START_REF] Dular | A finite element subproblem method for position change conductor systems[END_REF]. ICs of type (b) are of strong nature, in the sense that they are to be directly defined in function space F p 1 (Ω p ) via a fixed discontinuity of the tangential component of a p , obtained from the previous solution(s). This can be done via a discontinuous component a disc,p of a p , with support only on one side of the interface, added to the classical continuous component. At the discrete level, this consists in using for a disc,p additional edge basis functions only on γ p that are by definition non-zero only in the previously defined FEs layer E TL,p . The result of the projection of a q in E TL,p can also be used to fix a disc,p . The discontinuity can simply be zero, like in (8b).

Application

The validation and quantification of various SP steps are done on a shell type transformer problem, with a magnetic core Ω m (6 cm × 5 cm × 2 cm, leg width 1 cm, relative permeability 500), a possible air gap (0 to 3 mm), and primary (thickness 4 mm, height 2.7 cm, gap with core center leg 1 mm, 240 turns, fixed voltage at phase 0 at 50 Hz) and secondary coils (thickness 3 mm, height 2.7 cm, gap with core primary coil 1 mm, 24 turns, open or short-circuit tests) in Ω s . Symmetry conditions allow to reduce the studied domain for 1-D, 2-D and 3-D analyses (Fig. 5). The magnetic core with a possible air gap is first considered as an IFT in 1-D, 2-D and 3-D in an open-circuit test, before some correction RFT SPs, applying in various ways some developed SP tools for dimension and model changes; also for material property changes when introducing an air gap; various other correction SPs are possible as theoretically presented and will be further presented. The transformer is also considered in a short-circuit test, in 1-D, 2-D and 3-D. Studied domains and some significant results are illustrated in Fig. 5, with comparisons of the relative corrections on a particular global quantity, the inductance of the primary coil, that are summarized in table 1.

If the field distribution is already accurate for some simplified 1-D and/or 2-D models, e.g., in the open-circuit test, it is not the case in the short-circuit test, in particular with relative corrections from 2-D to 3-D (significant 3-D effects) that can reach hundreds of percents for impedances if no care is given. Also, the accuracy improvement with a higher dimension model is clear (e.g., up to 35%) when increasing the air gap.

Conclusion

A methodology for the progressive FE modeling of transformers has been developed. Models of different accuracy levels (IFT to RFT, 1-D to 3-D, addition of geometrical details, etc.), possibly coupled, are sequenced in the frame of the SPM, with successive additive corrections supported by different adapted meshes. The way the sources act at each correction step, up to the full models with their actual geometries, has been given a particular care and generalized. For all the considered corrections, the sources are always applied via IC-SSs, thus only needed in layers of FEs along boundaries, which strongly lightens the required mesh-to-mesh projections between SPs. Corrections of fields and global quantities can be obtained step by step with a clear view on their significance, to justify their usefulness or not.

Fig. 2 .

 2 Fig. 2. From IFT model (left) to RFT model (right) with leakage flux via an IC-SS.

Fig. 4 .

 4 Fig. 4. Local corrections (SP p) of an homogenized solution (SP q) for some laminations (e.g., 2 outer laminations) (left) or winding turns (from uniform to non-uniform current density; right).

Fig. 5 .

 5 Fig. 5. Shell type transformer with a sequence of 1-D, 2-D and 3-D models with IFT and RFT open and short-circuit tests (arrows: current density, pyramids: magnetic flux density, black lines: field lines (2-D); 2-D and 3-D edge effects are progressively corrected).

with n the unit normal exterior to Ω p and h s,p , j s,p , b s,p and e s,p some given SSs; note that the mag- netodynamic problem only defines e p in

  Ω c,p , with Γ e,p ≡ ∂Ω c,p ∩ Γ h,p . When zero, the SSs define homogeneous BCs, in particular usually applied on symmetry planes. Some BCs can result in ICs, i.e., fixed discontinuities[ ⋅ ] γ p = ⋅ | γ p + -⋅ | γ p -,when applied on some double layer surfaces γ p + and γ p -

	e,p ⊂ Γ b,p = e s,p , (1a-b-c-d)
	that are geometrically defined as a single surface γ p ⊂ Γ h,p ∪ Γ b,p , with the thin region in between
	exterior to Ω p [3]-[5], i.e.,
	[n × h p

TABLE 1 .

 1 CORRECTIONS OF GLOBAL QUANTITIES: INDUCTANCE OF PRIMARY COIL (CORRECTED VALUES AND RELATIVE CORRECTIONS IN % WITH RESPECT TO PREVIOUS DIMENSION AND RFT MODEL AFTER IFT MODEL)

	Air gap	Dim.	IFT open-circuit test	RFT open-circuit test	RFT short-circuit test
	(mm)		Inductance (mH)	Inductance (mH)	Inductance (mH)
	0	1-D	120.6		121.1	0.1629
		2-D	140.7	(+16.6%)	141.2 (+16.6%, IFT+0.36%)	0.3452 (+212%)
		3-D	140.7	(+0.0%)	141.6 (+0.23%, IFT+0.36%)	0.9664 (+280%)
	1	1-D	23.39		23.47	-
		2-D	24.05	(+2.8%)	25.79 (+9.9%, IFT+7.2%)	0.4104
		3-D	24.05	(+0.0%)	27.58 (+6.9%, IFT+14.7%)	1.521	(+371%)
	3	1-D	8.953		8.986	-
		2-D	9.048	(+1.1%)	10.62 (+18.2%, IFT+17.4%)	0.5086
		3-D	9.048	(+0.0%)	12.26 (+15.4%, IFT+35.5%)	2.058	(+404%)


B Oksengorn
email: boris.oksengorn@limhp.univ-paris13.fr

Keywords: Coherent microwave radiation emitted during PACS: 42.65-k, 42.65.Dr, 42.65.Sf SRS, hydrogen, microwave radiation, parametric instabilities --------------

Coherent microwave radiation emitted during the process of stimulated Raman scattering in weakly compressed hydrogen : experimental studies, together with an attempt to interpret this microwave emission

Boris Oksengorn

To cite this version: Boris Oksengorn. Coherent microwave radiation emitted during the process of stimulated Raman scattering in weakly compressed hydrogen : experimental studies, together with an attempt to interpret this microwave emission. 2006.

I-Introduction

-First of all, we want to present some qualitative results, obtained a long time ago, when studies of radiations emitted in the infrared region, during SRS experiments performed with a Q-switched ruby laser in weakly compressed hydrogen, had been carried out.

As a matter of fact, using a filter of silicon after the pressure cell, we had observed fortuitously a new kind of radiation provided by our infrared detector, a high-speed Ge:Au one working at the liquid nitrogen temperature and sensitive to infrared radiations up to 9 µm, and supplied by the manufacturer Santa Barbara Research Centre, Goleta, Cal.(USA), as we are going to show it below. It is truthful that possibilities of common artefacts had existed, due to the radiations emitted by the flash-lamp of our Q-switched ruby laser, and by the electric discharges in the laser equipment, or by the plasma induced in hydrogen by the laser field. Consequently, different tests were carried out to show that we had observed effectively a new type of radiation, as follows: 1) the signal obtained by means of our Ge:Au detector had a shape which is very different from those of the laser, Raman and flash-lamp pulses (see Fig. 2); 2) at the pressure of one bar no stimulated Raman lines had been observed, and no signal detected with our Ge:Au detector; 3) with argon gas instead of hydrogen, no signal had be seen, originating from the light emitted by the laser-induced plasma; 4) by means of an aperture of 3 mm diameter laid on the Ge:Au detector, only infrared radiations due to SRS in weakly compressed hydrogen had been observed; 5) with a black paper before the normal aperture of 2 cm diameter, only the new radiation had been observed; 6) in liquid water and gaseous ammonia, absorptions of this radiation were detected.

Afterwards, new experiments have been carried out more recently, to yield quantitative data as a function of several parameters, published in a short paper [1].

Nevertheless, a more detailed description of these experimental studies is given in Section II of the present paper. Furthermore, in Section III is presented an attempt to interpret this new type of radiation, appearing together with SRS in hydrogen. In Section IV a large discussion is given about similarities and differences between the physical process studied in this paper and others nonlinear phenomena. Finally, Section V is devoted to some remarks about the experimental SRS gains, which had been found above the SRS threshold, very much greater than the theoretical ones [2]. II-Experimental set-up and results.-For our new experiments a Nd:Yag pulsed laser has been used, the oscillator of which was Q-switched by a Pockels cell, followed by an amplifier. The laser beam was linearly polarised, monomode transverse, and almost longitudinally monomode, with 2 or 3 modes only, by using an intracavity FP glass disk. The laser pulse had a spectral width of 0.08 cm -1 , and its duration was equal to 15 ns. Besides, the laser power at λ=1.064 µm was equal to a maximum of 30 MW, reduced to 10 MW at λ =532 nm by frequency-doubling, and the laser working was effected at the repetition rate of 10Hz.. The experimental set-up is described on Fig. 1: the laser beam was focused by means of a lens of 50 cm focal length in the high-pressure optical cell of 1m length and 2 cm inner diameter. At the end of the cell two filters were put to stop all visible and infrared radiations:

one was a silicon disk of 5 mm thickness, the other a polyethylene rod of 20 cm length and of 2 cm diameter, which was put in a Cu tube. The efficiency of this filter system had been checked by means of a InSb detector, cooled at liquid nitrogen temperature, the sensitivity of which was 100 times higher than that of the Ge:Au detector in the infrared region of 5 µm:

no signal was detected during SRS in hydrogen. Behind the filter system a Cu mirror was set for reflecting the microwave radiation to the Ge:Au detector. Finally, data were collected by means of a pulse analyser or a digital oscilloscope.

On Fig. 2 is displayed the signal observed with the Ge:Au detector during a SRS experiment; it is strongly asymmetrical, with a short rise time estimated to 7 ns, the Stokes pulse duration being equal to 10 ns; the exponential decay of this signal has a characteristic time of 65 ns. Moreover, in the inset is shown a signal which is very similar to the previous one, and observed in experiments carried out by Manenkov et al., concerning thermal breakdown of excitons by a microwave field; these particles were produced previously by laser excitation of free carriers in a pure germanium disk cooled at 1.3 K, followed by a binding process [3]. In addition, Keldysh et al. had improved their experiments, and had obtained for the rise and fall-off times of the breakdown pulses the values of 100 and 200 ns, respectively [4]. Accordingly, we think that in our SRS experiments with hydrogen a coherent emission of microwave pulses was also detected, because the sensitivity of our detector to microwave radiation might be due to the fact that under the influence of the polarisation field, induced by a dc 40 V voltage, an equilibrium between free carriers, excitons and / or exciton complexes might appear in the Ge:Au disk, cooled at 77 K. After that, the heating of free carriers by the assumed microwave radiation would be the cause of the exciton breakdown, giving rise to the increasing part of the signal, whereas its decreasing part should correspond to the characteristic time of exciton recombination. Besides, experiments had been performed to study the microwave intensity as a function of the gas pressure. On Fig. 3 is shown this dependence, in the case of a SRS experiment with excitation by the 1.064 µm laser beam, while in the inset this dependence is depicted for a shorter region of the gas pressure, when the SRS experiment was performed by means of a 532 nm laser excitation. It can be seen that the microwave intensity is strongly increasing in the very low pressure range. Afterwards, a maximum appears in the 8-10 bar region, followed by an exponential decreasing up to about 20 bar; at higher pressure the signal intensity is nearly constant. Moreover, the exponential behaviour of the decreasing part is confirmed on Fig. 4, where the neperian logarithm of the signal intensity follows a linear variation with the number density of hydrogen. In addition, the microwave intensity, measured at the pressure of about 10 bar and with the 532 nm laser excitation of the signal, follows a linear variation with the product of the laser and Stokes beam powers, as shown on Fig. 5. This linear relationship should point out that the process of coherent microwave emission in hydrogen should be due to a nonlinear optical phenomenon, like a parametric amplification. Moreover, on Fig. 6 the evolution of the normalized intensity of the microwave signal is shown as a function of the distance between the detector, set at 1m from the end of the high pressure cell, and the propagation axis of the laser beam. One can see that the signal may be detected during a large displacement of the detector, and that its profile is modulated, probably by diffraction effects.

On the other hand, we had determined the wavelength of the microwave radiation by using a home-made wavemeter, with which a wavelength of about 1.4 cm had been measured, corresponding to a frequency of 20 GHz. Moreover, we had attempted to determine the microwave power by means of a detector manufactured by Spacek Labs. Inc.

(model DK-2); the sensivity of this detector was 2.5 V/mW in the K band (18 to 26 GHz). An average of 6 nW had been obtained, which corresponds for the peak power to an order of magnitude of 10 µW. However, this power is not the true value emitted during the SRS process, because different effects are not taken into account during the propagation of the microwave emission, as reflections, absorptions, diffractions and scatterings.

Finally, during a recent SRS experiment with hydrogen , we have been able to detect a backscattered microwave radiation, the maximum intensity of which has been found to be at the same pressure of about 10 bar, as in the case of forward scattering. nonlinear refractive index changes in optical solids, and had obtained a rather good agreement between experimental data and their empirical relationships [7].

III-

On the other hand, we assume that the quasiparticle system is an intermediate state which should have a very short lifetime τ R , of about 10 -13 -10 -12 s, compared to the laser pulse duration. Besides, we admit that the main interaction between these quasiparticles is of coulombian type, yielding a macroscopic electric field, longitudinally polarised. Moreover, it is assumed that the stationary wave associated to the quasiparticle system is modulated at the angular frequency ω R of the quasiparticle vibration, which should be very close to the angular frequency ω v of the isolated molecular vibration, but slightly higher than the latter.

Furthermore, another important hypothesis should be assumed: the quasiparticles are distorded by the electrostriction effect due to the laser field, what gives infrared active their vibrational mode, together with the advent of an induced dipole moment and an anomalous dispersion of the refractive index at the frequency ω R ; another effect of this distortion leads to the lack of the centre of inversion for the quasiparticle system, which gets a "piezoelectric" character [8].

Afterwards, we will study the interaction between the dielectric medium and the laser light, by assuming that the wave associated to the quasiparticle system, and being stationary compared to the velocity of light in the medium, may be split into two progressive waves, propagating the one to forward, the other to backward, with a phase velocity very close to the light velocity in vacuum c, but higher than the light velocity in the dielectric medium. Thus, the stimulated processes, like Stokes and microwave emissions, should be distinguished by a Cherenkov-type effect [9].

We assume also that the "piezoelectric" character of the quasiparticle system can be depicted by two parameters, induced by the laser field: one is R ind.

C , which denotes the atomic displacement, whereas the other is E ind.

C ,which is an electro-optic coefficient. These parameters are tensors of third rank, and can give rise to a nonlinear coupling of anharmonic origin, between light and matter waves, as shown by Merten et al. in the case of piezoelectric cubic solids [10][11][12][13]. In our case, we propose that the coupled mode photon-quasiparticle might be so-called "anharmonic longitudinal polariton" [12]. Furthermore, taking into account only the coupling with the Stokes wave in the parametric approximation, and writing: 1) R z for the longitudinal component of the normal coordinate R of a quasiparticle, the z axis being the laser and Stokes wave propagation axis; 2) P z for the longitudinal component of the polarisation density, associated to the quasiparticle waves; 3) ( S x P ) * for the transverse component of the complex conjugate of the polarisation density associated to the Stokes wave, the x axis being the direction of polarisation for the laser and Stokes waves, we are able to obtain the three following equations, according to ref. [10][11][12][13]:

- 2 R ω R z = BE z + R ind. C L x E ( S x E ) * (1a) P z = [( 2 R n -1)/4π] E z + E ind. C L x E ( S x E ) * (1b) ( S x P ) * =[( 2 L n -1)/4π]( S x E ) * + E ind. C L x E E z + R ind. C R z (1c)
In these equations the wave damping and the dispersion of the refractive index are neglected, excepting the anomalous dispersion at the frequency ω R ; the parameter B is the coefficient of the mechanical-electric linear coupling, and is a scalar which we assume to be a function of the anharmonicity of the quasiparticle vibration; E z is the longitudinal electric field, L

D z = (1) z ε E z + ) 2 ( zx ε ( S x E ) * (2a) ( S x D ) * = ) 2 ( zx ε E z + ) 3 ( zx ε ( S x E ) * (2b)
where the dielectric functions )

1 ( z ε , ) 2 ( zx ε and ) 3 ( zx ε are functions of the frequency ω R , as follows ) 1 ( z ε = 2 R n -4πB 2 / 2 R ω ) 2 ( zx ε = 4π L x E ( E ind. C -B R ind. C / 2 R ω ) ) 3 ( zx ε = 2 L n -[4π( R ind. C ) 2 / 2 R ω ]( L x E ) 2
Besides, D z and ( S x D ) * must fulfil Maxwell's equations, which lead to the following relations

- ) 1 ( z ε E z - ) 2 ( zx ε ( S x E ) * = 0 (3a) - ) 2 ( zx ε E z + ( 2 L n - ) 3 ( zx ε ) ( S x E ) * = 0 (3b)
The solution for this system of two equations exists, only if the determinant of its coefficients vanishes, leading to the relation

) 1 ( z ε ( 2 L n - ) 3 ( zx ε ) + ( ) 2 ( zx ε ) 2 = 0 (4)
From the solution of this equation ( 4) it is possible to determine the ratio C / E ind.

C = 4πB/ 2 R n ± i 2π 1/2 R ω / R n ( 5
)
where it can be seen that these two parameters are complex quantities, while in the case of piezoelectric cubic solids, they are only real.

It is also possible to calculate the dispersion functions for the coupled waves by means of the effective dielectric function, given by

ε eff = [ ) 3 ( zx ε ) 1 ( z ε -( ) 2 ( zx ε ) 2 ] / ) 1 ( z ε ε eff = 2 L n -(32 π 2 -2 R n )  E ind. C  2  L x E  2
The results for the dispersion functions can be represented by complex-conjugate values of the wave vectors, as follows

K 1 = K L -i L n [1-16 π 2 ( 2 L n 2 R n ) -1  E ind. C  2  L x E  2 ] (ω L -ω R )/c K 2 = K L + i L n [1-16 π 2 ( 2 L n 2 R n ) -1  E ind. C  2  L x E  2 ] (ω L -ω R )/c
According to ref. [10][11][12][13], the imaginary parts of these relations should correspond to an exponential amplification for the two progressive waves associated to the quasiparticle system, one to forward and the other to backward.

Moreover, from equations (3a) and ( 5) it is possible to obtain the following relation between the longitudinal electric field E z , and the laser and Stokes fields

E z = -4π -2 R n E ind. C L x E ( S x E ) * ± i 8 π 3/2 -3 R n B 1 - R ω E ind. C L x E ( S x E ) * (6)
It is interesting to note that this longitudinal field induces a polarisation P z , to which two progressive waves may be associated, with the same phase velocity as assumed above. On the other hand, it can be seen that a part of these longitudinal polarisation waves are in phase quadrature, and are mainly proportional to the imaginary part of the electro-optic coefficient E ind.

C , as given by trans.

P z = -2 R n i ( E ind. C ) " L x E ( S x E ) * (7)
Consequently, we assume that this transverse part of these longitudinal polarisation waves should be the origin of the coherent microwave radiation, observed in our SRS experiments with weakly compressed hydrogen.

Besides, these two progressive polarisation waves should have theoretically an infinite wavelength, but on the physical point of view, this wavelength must be finite and very large compared to the light wavelengths. This is obtained by taking into account of an anomalous relativistic Doppler effect in a dielectric medium, as proposed in ref. [9]. Then, the angular frequency ω o of the microwave emission is related to the frequency ω R by the following equation

ω o = (1-β 2 ) 1/2 ω R ( 8
)
where β is the ratio between the phase velocity V o of the polarisation waves and c; this Furthermore, the microwave radiation, which is assumed to be of Cherenkovtype, should be emitted inside the Cherenkov cone, simultaneously with the SRS process, at least to all appearances, as predicted by Ginzburg et al. in their theory [14][15]; our experimental results, shown in Fig. 6, seem to agree with their assumption. On the other hand, a comparison might be made between our results about a coherent microwave emission by SRS in hydrogen , and observations of forward conical emissions in barium or sodium dense vapours illuminated by dye lasers at a slightly shorter wavelength than those of the resonance transitions; these emissions are picked at a longer wavelength, with same shifts from the resonance transitions, as those used for their excitation [19][20][21][22]. As a matter of fact, these last phenomena have been interpreted as Cherenkov-type processes, the origin of the conical emissions being at the surface of self-focusing filaments [20][21][22]. However, in our case we propose also to take into account of a Cherenkov-type process, but the microwave radiation must be emitted in volume and inside the Cherenkov cone (see Fig. 5), without self-focusing and filaments in hydrogen. Moreover, Golub et al.

have calculated the refractive indices at different detunings to the red of the resonance transition, and have found that these parameters show anomalous dispersion profiles; they have also calculated the half-angles of the Cherenkov cones, given by the ratio n L /n C where n C is the maximum of the anomalous profile, and have observed a good agreement with the experimental values [21]. On the other hand, in our case we assume that the half-angle of the Cherenkov cone is given by the ratio 1/n R , since the process of microwave emission takes place exactly at the resonance frequency ω R .

Besides, we think that our model might be drawn somewhat to the interpretation proposed by Le Quéau et al. for explaining observations of a coherent auroral kilometric radiation [23][24]. As a matter of fact, these authors have proposed a model based on the maser synchrotron instability, as follows: they assume that electron beams arising from the solar wind are weakly accelerated to several keV by an electric field, in a region of the magnetosphere where this field is parallel to the terrestrial magnetic field force lines, and have a spiral motion around the magnetic field lines. Therefore, if the electron distribution function has a positive slope between two values of the perpendicular electron velocity, which should correspond to a "population inversion" having a short lifetime of about 1 ms, an energy exchange between particle and wave is possible, showing a peak when the following resonance condition is fulfilled:

ω X = (1-V 2 /c 2 ) 1/2 ω c
where V is the total electron velocity, ω X the angular frequency of the coherent kilometric radiation and ω c the cyclotron frequency of electrons.

As can be seen, there are several similarities between this last model and our own; however, one of the main differences is due to the fact that in the former the Doppler effect is weakly relativistic, while in our model it is necessary to take into account of an ultrarelativistic Doppler effect.

Finally, let us note that recently it has been possible to obtain coherent radiations of GW power in the far ultraviolet region, owing to a high-gain free electron laser working in a single pass [25][26][27]. In this device the linear electron beam, emitted as short bunches of ps duration, is accelerated to a velocity V', very close to c, and forced to oscillate in the undulator made up of a periodic sequence of alternating transverse magnetic fields.

Then, the resonance wavelength λ ph of the light emission is proportional to the quantity (λ

u /2)[1-(V'/c) 2 ]
, where λ u is the period length between the permanent magnets of the undulator. For instance, in the last experiments, radiation at 32 nm wavelength has been observed, the light pulses having a duration of about 25 fs, while the electron bunch contained a 50 fs long leading spike [27].

Moreover, it is interesting to note that the "normalized emittance" of this free electron laser is proportional to the relativistic factor (V'/c)[1-(V'/c) 2 ] -1/2 , as given in ref. [27];

we feel that this parameter might be somewhat equivalent to the one used below for explaining the physical meaning of the third term in the relation (9), once multiplied by the relativistic factor β 2 (1-β 2 ) -1/2 .

V-Remarks about the SRS gain.-It is well known, when there is a jump at the SRS threshold in a dielectric medium, that the experimental values of the Raman gain for the Stokes wave are very much higher than those calculated by means of the classical SRS theories [START_REF] Shen | The Principles of Nonlinear Optics[END_REF]. These disagreements were explained by many authors, by taking onto account only the process of self-focusing due to the variation of the refractive index at high laser intensity, giving rise to a strong increasing of the laser electric field [START_REF] Shen | [END_REF].

Nevertheless, it seems to us that an additional contribution might be also taken into account, which should be the origin of an enhancement of the Raman gain. As a matter of fact, we can calculate the polarisation density at the Stokes frequency, given by the relation (1c), by taking into account the relations (1a), ( 5) and [START_REF] Kimball | [END_REF]. For obtaining the total polarisation density, we must add also to this computation, the imaginary part of the third order susceptibility at the Raman frequency (fourth term in the relation ( 9)), which is the origin of the exponential gain for the Stokes wave in the classical SRS theories, based only on the approximation of the steady state. Then, the following relation is obtained:

( S x P ) * =[( 2 L n -1)/4π]( S x E ) * -8 π -2 R n  E ind. C  2  L x E  2 ( S x E ) *  i 32 π 5/2 -5 R n B 3 -3 R ω  E ind. C  2  L x E  2 ( S x E ) * + ( ) 3 ( Ram. χ ) "  L x E  2 ( S x E ) * (9)
In this expression the first term denotes the linear part of the Stokes polarisation, which is the source of spontaneous Raman scattering. The second term is a nonlinear part, which depends on the square of the induced electro-optic coefficient, and might be an additional contribution to the amplification of the Stokes wave, effective only for very short times, e.g. during parametric instabilities of about 10 -13 -10 -12 s. Finally, the third term is a in phase quadrature part of the polarisation, and should be a longitudinal component corresponding, once multiplied by the relativistic factor β 2 (1-β 2 ) -1/2 , to the translation energy of the whole medium, which must cancel during instability durations, the momentum associated to the progressive quasiparticle waves, according to the theoretical discussions submitted by de Broglie about some relativistic formulas [30]. detector and propagation axis of the laser beam. Moreover, an estimate of the wavelength had been obtained, the value of which is of the centimetric order of magnitude; an estimate of the peak power of this microwave emission had been also determined to be in the order of magnitude of 0.1-1 mW.

An attempt to interpret the origin of this coherent microwave is proposed, on the basis of a phenomenological model, taking into account of an intermediate quasiparticle state, due to the influence of the vibrational anharmonicity of hydrogen molecules, when the density of vibrationally excited oscillators is high. Besides, we assume that the duration of this intermediate state should be very short (10 -13 -10 -12 s).

On the other hand, it is stated that the electro-mechanical action of the laser light on the quasiparticle system gives rise to a "piezoelectric" character for this system; in addition, a nonlinear coupling of anharmonic type between the Stokes wave and the two progressive waves associated to the quasiparticle system is taken into account, together with the existence of a macroscopic longitudinal electric field. Then, the quasiparticle system might be called an "anharmonic longitudinal polariton" system. A Cherenkov-type effect and an anomalous relativistic Doppler effect are also taken into account.

Therefore, we assume that the source of the coherent microwave radiation is the in phase quadrature part of the longitudinal component of the polarisation waves, associated to the quasiparticle system . On the other hand, one can note that according to our model the microwave radiation should be emitted during very short times, like parametric instabilities which should appear as a continuous series, during the time of propagation of the exciting laser beam in the dielectric medium.

Concerning the SRS processes, recall that above the SRS threshold anomalous large gains had been observed, which had been explained for a long time, only by the influence of self-focusing processes. However, it seems to us that it is necessary also to take into account of an additional contribution arising from the influence of the nonlinear coupling cited above, together with a cooperative effect.

Finally, let us mention that in the case of spontaneous Raman scattering with crystals, observations of new Raman lines over the normal number of lines related to crystal symmetries, and of enhanced Raman scattering, have been correctly interpreted for the first time by Poulet [31]. This author had shown that in the case of piezoelectric cubic solids, these two effects can be accounted for, in the case of the first one by the action of a longitudinal electric field induced by the exciting light beam, and of the second by the influence of the whole medium due to a linear electro-optic coefficient, related to this longitudinal electric field [31]. In other respects, Burstein et al. had been able to interpret experimental results in the semiconductor GaAs, on the basis of the k≈0 coupled plasmon-(LO) longitudinal-optical-phonon collective modes [32]. Furthermore, Birman et al. had propounded a theoretical explanation for the effect of enhanced Raman scattering in crystals, by using the exciton representation as intermediate state; they have shown that "for longitudinal polar optic vibration, there is an additional contribution to the Raman tensor due to the associated electric field" [33].

Fig. 1 -

 1 Fig.1-Experimental set-up : 1=Nd-Yag laser ; 2=focusing lens; 3=high pressure cell;4=silicon disk of 5 mm thickness used as filter; 5=Cu tube; 6=polyethylene rod of 20 cm length; 7=Cu mirror; 8=Ge:Au detector; 9=light detector for triggering; 10=pulse analyser; 11=computer.

Fig. 2 -

 2 Fig.2-Profile of the pulse signal obtained by means of the Ge:Au detector, and due to the absorption of microwave radiation. Inset:the same obtained with pure germanium in ref.[3] (see the text).

Fig. 3 -Fig. 4 -

 34 Fig.3-Variation of the microwave intensity Fig.4-Linear variation of the neperian as a function of hydrogen pressure exciting logarithm of the microwave intensity, laser at λ=1.064 µm). Inset: the same with versus the hydrogen number density. exciting laser at λ=532 nm. The dashed lines The dashed line is only for the eye. are only for the eye.

Fig. 5 -

 5 Fig.5-Linear variation of the microwave intensity, Fig.6-Evolution of the microwave normaas a function of the product of the laser and lised intensity, versus the distance between Stokes beam powers.detector and propagation axis of the laser beam. The arrow might point to the limit of the Cherenkov cone.The dashed line is only for the eye.

  Theory.-First of all, we propose for the slope v o of the linear relationship depicted on Fig.4, the value of which is equal to 6.62 10 -21 cm 3 , the following physical meaning: this parameter should correspond to the volume of a quasiparticle, being composed of a characteristic number of molecules vibrationally excited during the SRS process. An order of magnitude of this number can be calculated by means of the model used by Hirschfelder et al. for a gas, the face-centered cubic lattice gas[START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF]: at low density the free volume occupied by one molecule is given by v=3.18 σ 3 , where σ is the hard sphere diameter of the H 2 molecule, equal to 0.287 nm in the case of a Lennard-Jones potential. By comparing the volume v, equal to 7.52 10 -23 cm 3 to the volume v o of the quasiparticle, an order of magnitude of about one hundred of vibrationally excited oscillators is found to make up one quasiparticle. Our hypothesis might be corroborated by the following points: owing to the excitation by the coherent laser beam, the vibrationally excited oscillators are all in the same phase, and may interact strongly between them, through their anharmonicity. Moreover, if the density of these interacting particles becomes appreciable, what is the case for the Stokes wave intensity above the SRS threshold in the present instance, a cooperative effect should be present, leading to a kind of "localisation" for the excited oscillators, according to the model developed by Kindall et al. for the case of anharmonic solids[START_REF] Kimball | [END_REF]. In other respects, let us note that Boling et al. had used a model based on the anharmonic oscillator to predict

  electric fields of the laser and Stokes waves, respectively; finally, n L and n R are the refractive indices at the laser frequency and the frequency ω R , respectively. Afterwards, it is necessary to introduce the dielectric displacements D z and (

  parameter must be very close from the unity value in our model, what is effectively obtained in the relation given below 1-β =1/2 (ω o /ω R ) 2 ≈1/2 (20 10 9 / 4155 c) 2 ≈ 10 -8 -10 -7

Finally, concerning the

  exponential decreasing of the microwave signal beyond the pressure of 8-10 bar, it is possible to explain this behaviour by assuming that the quasiparticle density is strongly reduced by the increasing effects of dephasing and nonradiative relaxation of the vibrational energy, due to the Van der Waals forces. As a matter of fact, in the pressure range of 3-10 bar the collisional narrowing is the main effect[16], and the intensities of the stimulated Raman waves and of the microwave emission, are both strongly increasing. However, around the pressure of 8-10 bar the influence of the Van der Waals forces becomes sensitive, because the value of the collision rate Ω is equal to about the reciprocal of the average of the quasiparticle lifetime τ R , which corresponds to the relation Ωτ R ≈1; but at higher pressure this equality must be replaced by the inequality Ωτ R >1, which shows that the Van der Waals forces are preponderating.IV-Discussion.-First of all, it seems to us that our model might be drawn to the one developed by Dumartin, who had attempted to interpret his experimental results about the process of Stokes absorption, observed in several liquids during experiments of SRS associated with a broad spectral continuum, when the laser intensity is slightly above the SRS threshold[17][18]. For instance, Fig.7displays the spectrum of the Stokes absorption band of liquid benzene, the bandwidth of which is one order of magnitude larger than the width of the Stokes emission line in spontaneous Raman scattering. To explain this process, Dumartin had shown that near the SRS threshold, the population of vibrationally excited molecules might be sufficiently high during very short times (10 -12 -10 -11 s), compared to the duration of the laser pulse, to assume the existence of a "population inversion", giving rise to very broad Stokes absorption bands[18].

Fig. 7 -

 7 Fig.7-Microdensitometer of the stimulated Stokes absorption band, obtainted in liquid benzene [18]; note the weak stimulated emission inside the strong absorption band.

Conclusion. -

 - In this paper experimental results are presented, concerning observations for the first time, to our knowledge, of a coherent microwave radiation during SRS experiments in weakly compressed hydrogen, concomitant with the emission of stimulated Raman lines. The intensity of this microwave radiation had been studied as a function of the gas pressure, the laser and Stokes beam powers, and the distance between


Melanie Dulong De Rosnay
email: melanie.dulong-de-rosnay@cersa.org

An Actio n -Base d Legal Model for Dyna mic Digital Righ t s Expres sio n

Keywords: ), Rights Expression Language (REL), copyright, legal ontologies, Creative Common s metadata

digital documen ts and data, ie legally or contractually allowed usages or actions. The proposed model associates terminological resources from technology, usages, exclusive rights and public domain. Research aroun d this article takes place in Medialex, an interdisciplinary project aiming at building a transaction ontology to license, access and reuse.

Introduction

This article is a legal contribution to Rights Expression Languages (RELs) requirement s [START_REF] Coyle | Rights Expression Languages -A Report for the Library of Congress[END_REF]. We showed in [START_REF] Dulong De Rosnay | Cognitive interfaces for legal expressions description -Application to copyrighted works, Online sharing and Transactions[END_REF] that legal expression related to works sharing shall enable bi-directional comm unication [START_REF] Mulligan | Implemen ting Copyright Limitations in Rights Expression Languages[END_REF]. We state that the transaction ontology to be developed should express legal constraints and rights in a dynamic manner, coping with the need to update and define a rights expression at each action led on a digital document. Indeed, actions led on a digital document or data imply either a transaction or a modification. Besides, rights expression shall be built according to copyright law granting exclusive rights to rights holders and prerogatives or fair use to end -users [START_REF] Dulong De Rosnay | Cognitive interfaces for legal expressions description -Application to copyrighted works, Online sharing and Transactions[END_REF], and also to contractual users' requirement s, while also matching contract law requirements. Finally, rights expression shall not be only a machine -readable formal language and a lawyer -readable contract, but also human -readable plain text, so that works users (licensors and licensees) can understa nd and negotiate [START_REF] Dulong De Rosnay | Cognitive interfaces for legal expressions description -Application to copyrighted works, Online sharing and Transactions[END_REF] transactions licensing terms . The model presented hereafter describes the development of several terminological resources types to model actions in and on contracts, and the results from their mapping. These resources are plural and associate concepts from DRM technologies (existing RELs categories), law (exclusive rights as positive law, public domain as implicit law) and usages (contracts standard clauses and a use case). We will demonst rate why using these specific resources provides an added value compared to other RELs methodologies, and in which extend they differ from other ontological resources of the domain in terms of accessibility, scope and legal accuracy and logic.

What is Digital Rights Manage m ent?

DRM systems and RELs: Technical and Architectural Norms

Digital Rights Management systems (DRMs) are software technologies aiming at restricting access or usage of digital data, or technologies embedded in hardware players for similar control purposes. DRMs are composed of several bricks including a REL. In order to support semantic interoperability between operating systems, players and other DRM tools, RELs have been standar dize d [START_REF]ISO/IEC FDIS 21000 -5, MPEG-21 Rights Expression Language (REL)[END_REF][START_REF]Open Digital Rights Language[END_REF] and provide syntax and semantic to express restrictions and licenses to be conveyed to the end -user through a technical information measure or implemente d by a technical protection measure.

Market norms: Digital Business Manage ment

We claim that the term « Rights » used in common language and within « DRM » can be misleading. Indeed, not legal « rights » (reproduction, distribution…) are digitally represente d, but technological acts or usages (send, print…) allowed by business models. This information can be considered as an electronic contract offer. This private ordering may be unable to deal with fair use situations and suitable only to express access, copy and reuse and associated time or fee restrictions. In order to balance the influence of the market (DRM technology, data and document s transaction contracts standard clauses), we studied public interest requirement s related to access to copyrighted works.

Legal and Social Norms: Open Access Usages and Creative Commons Metadata

Creative Commons [START_REF]Creative Common s[END_REF] proposes rights information measures allowing the licensor to grant to the public additional freedoms (legal prerogatives, or grants) to traditional copyright law and usages in order to reflect public interest. The licensor may choose her licensing conditions through a cognitive user interface generating a license under three formats:

1. a license in legal language, 2. a human -readable version sum marizing the conditions of usages granted by the licensor, each optional condition (Non Commercial Use, No Modification, ShareAlike or Copyleft…) is represented by a standardized semiotics, 3. legal metadata in RDF standard to tag works website or data files, this machine readable code can be further integrated in search engines, tracking, information retrieval or aggregating software applications.

Terminological Material and Requirements Toward an Ontolog y

Action -Based Dynamic Rights Expression

We claim that attaching usage conditions to a document at the time of its creation by embedding metadata to a file format is not satisfying. Indeed, access and usage rights are defined for a given moment and a given user. Some usage conditions can be expressed by the original creator and / o r the right holder at the time of the work's online divulgation; nevertheless, such a usage condition should be updated and adaptated for each single situation, defined at the crossroad of an Action, a User, a Document and Conditions. Rights expressions are to be built all along the life-cycle of the digital data embedde d in a document and re-configured at each action (transaction, modification…). An accurate rights expression is determined dynamically by the place and time of actions occurring on document s resulting from prior creations, modifications and transactions. RELs shall not preclude unspecified usages express access conditions to works in a descriptive ontology of technical actions.

A Network of Five Terminological Resources

Developing several ontological resources is a necessary step to qualify digital documents usages within an open access scheme, a commercial or private transaction, or a compulsory license mechanism. Five terminological pre -ontological ressources have been identified as a prerequisite material to build an ontological model to express online transaction contracts on digital documents and data. The resulting model is close to a meta -ontology as its high-level classes are generic and could potentially be mapped to upper -level ontologies (such as Dolce and LRI Core). The figure below shows the intrication of the five resources of our model, described in the following sections.

Rights Expression Languages State of the Art ( resource 1)

Instead of starting from scratch, we reused some existing RELs concepts. Indeed, some of their classes and categories reflect the implementation of license grants by a computer system (right of reproduction -> action of printing). We gathered material from several RELs, mostly MPEG-21 REL [START_REF]ISO/IEC FDIS 21000 -5, MPEG-21 Rights Expression Language (REL)[END_REF] and ODRL [START_REF]Open Digital Rights Language[END_REF]. These RELs commercial transaction terms are overrepresented and were synthesized toward a clearer model, where legal and unregulated usages factors have been taken into account.

Core Fields in Contracts for Documents and Data Usage (resource 2)

It is important to deal with practical usages coming from both social and market norms. Identifying core fields usually represented in both paper and electronic contracts has been achieved by compiling manually a selected corpus of various contracts from different background s (scientific publishing, music industry…), legal systems (civil and common law) and legal complexity (multimedia works). Clauses categorization answers requirement s for copyright contracts legal validity in French legislation ("rights" extend and destination, contract grant location and duration).

Legal Categories for the Use of Protected Documents and Data (resource 3)

Legal texts (legislation and doctrine) have been studied to understa nd copyright decision -making process. Privacy and secret data are managed through simpler but similar authorization patterns. IPROnto [START_REF]Intellectual Property Rights Ontology[END_REF] is an ontology of the REL domain which already includes positive law (WIPO international convention) and is build on actions [START_REF] Delgado | IPROnto: An Ontology for Digital Rights Managemen t[END_REF][START_REF] Gil | An interoperable framework for Intellectual Property Rights using web ontologies[END_REF]. However, implicit law has not been included in this ontology.

Legal Categories for the Use of Unprotected Documents and Data (resource 4)

Unlike to the previous resource, ungoverned cases where no authorization is needed are left out by positive law analysis. The definition of a broad public domain (protection time expiration ie traditional public domain, document form and nature legal requirement s to reach copyright protection, fair use or exceptions to exclusive rights user and usage factors, open access licenses) have been kept separated from the previous resources for two reasons:

-Legal reasons: this notion of public domain is not founded with a specific statute in positive law, but can be deduced from the absence of protection by exclusive rights and is designated as "negative" or "implicit" law (as in a photography negative); -Architectural reasons: the statute of public domain implies that permission is already granted, there is no need to request an explicit authorization. Therefore, such situations do not require the implement ation of a negotiation interface or a technical protection measure. Distinction between permission and authorization was studied in [START_REF] Boella | Permissions and Authorizations in Normative Multiagent Systems[END_REF] and applied to normative multiagent comm unities. Following this model where permission is an exception to obligation, public domain works, fair use and exceptions to exclusive rights are permissions by nature of the work, user, usage. Open Access licenses could be handled as authorizations if one considers they add an autonomo us normative layer, but we think they are also permissions for multiple reasons. They are integrated in the legal system in addition of the law, and do not apply in an autonom ou s comm unity only. They are applicable a priori by avoiding request before the transaction happens, and they express that the sanction foreseen in the legal system will not occur.

Legal Categories in National Legislations

It matters to include legal categories and requirement s in RELs design so that management may respect the law and use all its combination possibilities, and not only the bricks provided by commercial business models.

Beyond national implementation differences, national legislations are using similar top -level categories and rules. However, we think our model shall not be based on national legislations texts linguistic parsing results, for both legal and ontology-building reasons:

-Legal argumenta tion: common law fair use criteria are more difficult to be represented in information systems than civil law exceptions factors. On the one hand, DRM systems can be considered as contractual private agreement s which do not have to respect all copyright fair uses, but only public order provisions. On the other hand, DRM systems have the same liberty to grant more generous provisions to the licensee than the minimal requirement provided by national exceptions. In the case of Open Access licenses, public interest principles are implemented on a voluntary basis, even if they do not have a status of public order.

-Ontology-building argument ation: manual vs automatic ontology building methodology. We are considering the hypothesis that automatic retrieval of legislative text through language processing analysis is more adequate for information retrieval on large corpus than for decision -making application building from small corpus. Further detailed metadata instances, to be contained in these four core classes makes it possible to express actions and, with simple relational and conditional connectors (if… then, and , or…), to build contracts which are dynamically updatable after each action: <If the requested docume nt is a private data (confidential information), then the licensee identity must be requested >; <If the requested action on the work is a legal exception to exclusive rights, then it is permitted according to usage and user conditions >

Rules Expressions and Relations Based on Logic

Automating Rights Expressions update at each action, the next step after category description, requires logical consistency. Creative Commons icons and licenses titles do not respect formal logic. Expressed conditions are based on the understa nding of a core prerogative1 (which is neither expressed in the icons nor in the license title: a royalty-free non -exclusive permission to reproduce and distribute the work), and not on a direct addend u m to basic copyright grant (all rights reserved). No explicit option reflects positive grants. An extension of the humanreadable system to the management of broader scope of usages (commercial usages) could hardly be logically supporte d. RDF Creative Commons machine -readable code presents a correct logic structure, such as the ODRL/Creative Commons profile [START_REF]ODRL Creative Common s Profile[END_REF] in XML, toward an ontology reflecting our model requirements and terminological plural component s. Positively granted actions, which are not expressed in the licenses icons and titles but implicitly asserted, are represented with the ODRL Permission /Prohibition /Require ment model [START_REF]Open Digital Rights Language[END_REF].

Conclusion: a Use -Case Methodolog y

The fifth resource of our model is a use case scenario (resource 5). We developed a a dozen of actions in order to compile and combine a wide range of technical, usage and legal possibilities to create and interact with multimedia document s: teaching, commerce, advertising, collective management… This use case is useful to check the proposed model consistency and completeness and to convey technology, usage and legal possibilities to a non -lawyer colleague2 , as a basis for further test to compare ontologies on a single complete example.

Some Creative Commons licenses which are out of the 6 licenses core suite and not available through the license generator interface (Developing Nations, Public Domain and Sampling licenses) grant less prerogative and may deserve a re-design of the options suite for the system's coherency sake.

Acknowledg men ts to Nadia Nadah for stimulating discussions on previous version of this model.