Brunda Kattekola

Marc Fivel

Jean-Pierre Franc

Cavitation erosion in UHMWPE: A three-dimensional FEM study

ABSTRACT: In high velocity liquid flows, bubbles nucleate in regions of low pressure. These bubbles then grow to large size (often their final radius is many times greater than nucleation radius) and subsequently collapse in the high pressure region. Repeated implosions of these bubbles near a solid surface lead to significant surface damage and mass erosion. It has been experimentally observed that polymers such as Ultra-High Molecular Weight Polyethylene (UHMWPE) possesses a better resistance to cavitation erosion than Aluminium alloys and Bronze alloys and even stainless steels. In this numerical study, we perform a finite element study on the cavitation erosion in UHMWPE. In the first phase, a rate-dependent elastic-viscoelastic-viscoplastic constitutive model is calibrated to simulate the response of UHMWPE under different loading conditions and within a rather large range of strain rates (10 -4 to 2000s -1 ). In the second phase, the effect of a single bubble collapse on a UHMWPE specimen investigated.

Cavitation damage has numerous implications in industrial context, hence there is a drive to develop materials that can minimize if not completely eliminate cavitation erosion. Experimental investigations, on the sintered samples of UHMWPE have shown that the resistance to cavitation erosion of UHMWPE is higher than some of its metallic counterparts for e.g.: Aluminum and Bronze alloys and Stainless steels A2205 [START_REF] Deplancke | Outstanding cavitation erosion resistance of ultra high molecular weight polyethylene (uhmwpe) coatings[END_REF]. Furthermore, it has been observed that the resistance of UHMWPE to cavitation improves with increasing the molecular weight [START_REF] Deplancke | Outstanding cavitation erosion resistance of ultra high molecular weight polyethylene (uhmwpe) coatings[END_REF]. This forms the thrust of the current numerical investigation where we simulate the cavitation erosion in UHMWPE of varying molecular weights.

Ultra High Molecular Weight Polyethylene (UHMWPE) belongs to commercially important class of semi-crystalline polymers. Its microstructure consists of two phases namely: amorphous and crystalline. When subjected to deformation, the two phases and their interaction with each other, gives the polymer a complex mechanical response. The response consists of several simultaneous deformation modes such as elastic, viscoelastic and viscoplastic. Further, when subjected to uniaxial tension, UHMWPE is known to exhibit post-yield strain softening followed by a strain hardening response. The stress-strain response further depends on loading rate, temperature etc. Given its complex response, numerous models (Hasan and Boyce 1993, 1 INTRODUCTION Cavitation is a major cause of damage in solid parts that are subjected to high velocity flow conditions. During the high velocity flow, bubbles nucleate in regions where the pressure of the fluid is less than its vapor pressure. The bubbles grow and collapse in the high pressure region causing damage to the solid surface. It should be remarked that only the bubbles that collapse in the vicinity of the solid surface contribute to material damage. The collapse of bubbles may generate a shock wave or a re-entrant jet that is directed towards the solid surface. The manner of bubble collapse is governed by the distance between the bubble and the solid surface. At a distance greater than the radius of fully formed bubble, the bubble implosion generates a shock wave with an amplitude in the order of Gigapascals. A bubble much closer to the solid surface (due to an asymmetry in the form of presence of solid surface) would form a reentrant jet (or microjet) directed at the solid surface [START_REF] Ewunkem | Cavitation erosion of engineering materials[END_REF][START_REF] Brennen | Cavitation and bubble dynamics[END_REF]. Nevertheless, irrespective of method of bubble collapse, it should be noted, that a single bubble is capable of creating a pit i.e. a small plastic deformation on the material surface whereas repeated bubble implosions cause critical wear and mass loss at the solid surface. The surface damage is quantified by measuring the depth of the accumulated pits. [START_REF] Bergstrm | Constitutive modeling of ultra-high molecular weight polyethylene under large-deformation and cyclic loading conditions[END_REF][START_REF] Dommelen | Micromechanical modeling of the elasto-viscoplastic behavior of semicrystalline polymers[END_REF][START_REF] Holmes | Constitutive model for large strain deformation of semicrystalline polymers[END_REF][START_REF] Pouriayevali | A constitutive description of the rate-sensitive response of semi-crystalline polymers[END_REF] have been developed to capture the response of semi-crystalline polymers. Despite this fact, there are few models that can comprehensively characterize the polymer strainstrain response. In this study, we adopt a model that has an ability to simulate UHMWPEs response both in quasi-static and dynamic modes corresponding to isothermal and adiabatic conditions.

This paper is divided into four sections. In Section 2, we present the essential features of the rate-dependent consitutive model [START_REF] Pouriayevali | A constitutive description of the rate-sensitive response of semi-crystalline polymers[END_REF]) and calibrate the model to simulate the mechanical response of UHMWPE. The experiments and numerical simulations of cavitation phenomenon are presented in Section 3. Further, in this section, we present the deformation induced in UHMWPE by a single dynamic bubble collapse. And finally, we conclude the paper with ungoing work and future plans.

CONSTITUTIVE MODEL

Model description

In this section, we briefly summarize the essential equations of the consitutive model, referring for a detailed description to [START_REF] Pouriayevali | A constitutive description of the rate-sensitive response of semi-crystalline polymers[END_REF].

The rheological representation of the constitutive model is shown in Figure 1. The model consists of two clusters arranged in series. The first cluster formed by a parallel combination of hyperelastic spring element 1 and Maxwell element governs the ratedependent reversible elastic-viscoelastic deformation. The second cluster of friction slider-dashpot governs the rate-dependent irreversible deformation. The friction slider is activated at stress greater than the yield stress of the material. Further, the viscosities associated with the viscous flow are rate-dependent and there are few model parameters that account for the decrease in stiffness with temperature.

The free energy associated with the model is given by:

ψ ψ ψ = + ψ e v ψ ψ + ψ e e ψ ψ
(1)

where ψ e ψ and ψ ve ψ ψ e ψ ψ correspond to the free energy of hyperelastic spring elements 1 and 2.

In the model, it is assumed that the free energies take a polynomial expression of the form: Here F is the distortional part of the deformation gradient tensor.

In the absence of heat flux and source, the free energy satisfies the Clausius-Duhem inequality, i.e.:

S C : - C - ≥ 1 2 0 ψ η -θ (4)
where S is the second Piola-Kirchoff stress, C = F T F is right Cauchy Green strain tensor, η is entropy and θ is absolute temperature. All quantities are defined per unit reference volume.

The expressions for stress and flow rules are derived by making an assumption that under all conditions, the rate of entropy change is positive. Closed form expression of stress can be found in [START_REF] Pouriayevali | A constitutive description of the rate-sensitive response of semi-crystalline polymers[END_REF]. For the sake of completion, flow rules are included below.

The viscoelastic and viscoplastic flow rules are given by: where the yield surface is defined as:

f y y T 0 2 3 , + y ( ) ) + y ε ) ) ) - = 0 ( ) y + ε y + σ (8)
where, σ y is the yield stress of the material and ε is the hardening function of exponential function of the form

ε = | | || H a Q ( ( p ) ) - E
. The Almansi plastic strain is further computed as E a = (I -B p )/2. B p is the plastic part of the left Cauchy Green strain tensor (B = FF T ). The model parameters H and Q govern the strain softening and hardening regimes.

Model calibration

Uniaxial tensile and compression experiments were conducted on dogbone and cylindrical specimens respectively. During the experiments, the loading rate as well as the molecular weights were varied. The constitutive model has 17 parameters whose values were determined from the uniaxial experimental stress-strain data. The parameters corresponding to each element are demonstrated in Figure 2.

Three dimensional Finite-Element simulations were performed on ABAQUS v6.11. The constitutive model described in the previous section was implemented as user subroutined VUMAT. For the purpose of consistency, the geometry and boundary conditions from experiments was replicated in simulations.

The parameters G eij and D ei associated with spring element 1 are related to shear G and bulk modulus K of the material according to the relationship:

i j n ij G G ij = j ∑ = / G 1 2 e G ( 9
)
and

i n i D K i = ∑ = / K 1 2 e D ( 10
)
Further, it is also assumed that the constants associated with hyperelastic element 2 (ref: Fig. 1) are given by G

YG Y Y j e ij v e G Y G G G Y ij
To compute the yield stress σ y σ , tangents are drawn to elastic-viscoelastic and yield regimes. The stress at the intersection of the tangents is chosen as yield stress parameter.

The remainder of the parameters are determined from iterations such that the error between experiments and simulated curves is minimal. In Figures 3 and4, the uniaxial response of UHMWPE from simulations (solid lines) is overlaid on experimental response (dashed lines). It can be seen from Figure 3 that with increasing molecular weight, UHMWPE yields at lower stress. Further, the necking in the specimen which manifests itself as strong peak in the yield is sharper at the lower molecular weight.

CAVITATION

Experiments

In Figure 5, the experimental setup for performing the cavitation tests is demonstrated. And in Figure 6, a picture of the circular specimen holder of 10 cm diameter is shown. The specimen holder contains six slots, that are used to place cylindrical specimens that can be simultaneously tested for cavitation erosion. More details on specimen preparation can be found in [START_REF] Deplancke | Outstanding cavitation erosion resistance of ultra high molecular weight polyethylene (uhmwpe) coatings[END_REF]. The specimen holder with six specimens is placed in a hydrodynamic tunnel where all the specimens are subjected to a jet of water at 90 m/s flow velocity. As mentioned before, only the bubbles collapsing in the vicinity of solid surface contribute to damage.

The pictorial evidence of cavitation erosion can be presented in Figure 6, where six UHMWPE cylindrical plate specimens of varying thicknesses and molecular weights have been for cavitation erosion for 100 hours in water jet with flow velocity 90 m/s. The specimens of lower molecular weight i.e. 0.6 Mg/mol experience much larger erosion compared to those with 10.5 Mg/mol molecular weight. Further, no evidence of thickness was observed either from images or profilometry measurements [START_REF] Deplancke | Outstanding cavitation erosion resistance of ultra high molecular weight polyethylene (uhmwpe) coatings[END_REF].

Numerical simulations

Geometry and boundary conditions

Utilizing the symmetry of the cylindrical specimens, the numerical simulations were performed on an axisymmetric plate. The finite element mesh in the θ = 0 plane is shown in Figure 7. The mesh is coarser at the far ends and much refined at the north-west corner where load boundary conditions are applied. The 4 noded axisymmetric element CAX4R were used to discretize the geometry. A Gaussian form of pressure distribution is applied at the top surface [START_REF] Kim | Advanced Experimental and Numerical Techniques for Cavitation Erosion Prediction[END_REF]). On the top surface, the applied pressure pulse is of the form:

σ σ - σ ⎛ ⎝ ⎜ ⎛ ⎛ ⎝ ⎝ ⎞ ⎠ ⎟ ⎞ ⎞ ⎠ ⎠ ⎛ ⎝ ⎜ ⎛ ⎛ ⎜ ⎝ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎞ ⎞ ⎟ ⎠ ⎠ ⎟ ⎟ H σ σ σ H exp 2 r r H ( 11
)
where r is the radius of the cylindrical plate, r H and σ H loading parameters. In the current investigation, the value of r H has been held constant at 50 μm and σ H is varied from 20 MPa to 200 MPa. At the center line (r = 0, z), the displacement in the radial direction u r is constrained, i.e.:

u r = 0 (12)
and at the bottom of the plate (r, z = 0), the displacement in the axial direction, u z is constrained, i.e.:

u z = 0 (13)

Primary results

In Figures 8 and9, the axial displacement u z along a line P -P′ (shown in the inset of Figs. 8 and9) is plotted for σ H = 20 MPa and 200 MPa respectively. Further, the dashed line in both the figures corresponds to the instance when the applied pressure at point P attains a maximum value of σ = σ H . The applied pressure is then gradually reduced to zero and the inelastic displacement at that instant is plotted in solid lines with label 'unload'. At σ σ H y σ σ σ σ < , the specimen is restored to undeformed configuration. It can be seen that the elastic part of the deformation is quickly recovered, however, the recovery of the viscoelastic deformation takes longer time. The time taken for restoration strongly depends on parameters governing the Maxwell element (see Fig. 2).

The contours of viscoelastic and plastic parts of deformation gradient tensor are plotted in Figures 10 and11 at the instant when the applied pressure σ is gradually reduced from σ = σ H to zero. It can be seen from Figure 10 that the viscoelastic deformation has not been fully recovered even after a complete unloading of the specimen. From the contours of the plastic deformation gradient tensor (Fig. 11), it is clear that the magnitude of plastic deformation is greatest at a region below the point suffering maximum displacement. So, if a failure were to initiate, it would most likely be in the interior of the specimen.

CONCLUSIONS

In this work, we have numerically simulated a plastic pit analysis with no mass loss in UHMWPE with molecular weight 0.6 Mg/mol at different magnitudes of pressure pulse. It is seen that with increasing load, the pit depth increases. Although not reported in the paper, it has been observed, during unloading, the time required to recover elasticviscoelastic deformation is very closely related to the viscosity parameters.

The magnitude of the applied pressure pulse is 200 MPa which is much lower than what is predicted from experiments. However, it should be observed that UHMWPE is a compliant material (as opposed to metals that are almost perfectly rigid). As a result, the amplitude of cavitation impact loads on UHMWPE is expected to be smaller than that on metals. This is due to the fluidstructure interaction, not modeled here. [START_REF] Roy | Cavitation induced damage: Fem inverse modeling of the flow aggressiveness[END_REF] used a finite element based inverse algorithm to determine the magnitudes and distribution of the pressure load that would be required to cause a pre-defined pit depth. It is expected that polymers would be subjected to pressure of the order of 1GPa.

Excessive distortion of mesh prevents us from increasing the magnitude σ H the of pressure pulse. We are currently exploring, Coupled Eulerian Lagrangian Analysis capability in ABAQUS v6.11 to avoid severe mesh distortion.

Finally, the investigation will include the effect of molecular weight on resistance to cavitation erosion.

  whereG e10 , G e01 , G e11 , G ve G G e 10 , D e1 , G ve G G e 01 , G ve G G e11 and D ve D D e 1 are material parameters associated with the model. And the invariants e1 I , e2 I , J J e are associated with the elastic and viscoelastic parts of right Cauchy Green strain tensor C = F F T .

Figure 1 .

 1 Figure 1. Rheological representation of elasticviscoelastic-viscoplastic consitutive model.

Figure 2 .

 2 Figure 2. Rheological representation of elasticviscoelastic-viscoplastic constitutive model with the list of parameters concerning each of the elements.

Figure 3 .

 3 Figure 3. Uniaxial response of UHMWPE in tension from numerical simulations (solid lines) are superposed on experimental data (dashed line).

Figure 4 .

 4 Figure 4. Uniaxial response of UHMWPE in compression from numerical simulations (solid lines) are superposed on experimental data (dashed line).

Figure 5 .

 5 Figure 5. Schematic of experimental setup for performing cavitation tests. The image is taken from Reference 3.

Figure 6 .

 6 Figure 6. Image showing the surface erosion in six UHMWPE cylindrical plate specimens of varying thicknesses and molecular weights when subjected to water jet at 90 m/s flow velocity for 100 hours. The image is taken from Reference 3.

Figure 7 .

 7 Figure 7. Finite element mesh on the axisymmetric plate. The mesh in the north-west corner is much refined.

Figure 8 .

 8 Figure 8. Axial displacement of a line P -P′ for σ H = 20 MPa. The 'load' and 'unload' curves correspond to displacement at σ = σ H and to σ = 0 MPa.

Figure 9 .

 9 Figure 9. Axial displacement of a line P -P′ for σ H = 200 MPa. The 'load' and 'unload' curves correspond to displacement at σ = σ H and σ = 0 MPa.

Figure 10 .

 10 Figure 10. Contours of viscoelastic part of deformation gradient tensor at an instant when applied pressure σ is reduced to zero for the case σ H = 200 MPa.

Figure 11 .

 11 Figure 11. Contours of plastic part of deformation gradient tensor at an instant when applied pressure σ is reduced to zero for the case σ H = 200 MPa.

ACKNOWLEDGEMENTS

Authors would like to thank Office of Naval Research, USA for providing the necessary financial assistance.


1/ p  1  2
A simple model: A two-component medium, each component is described by a two-level model

1/ p  1  2
We assume that the transparency range is very large: ω 1 (1/τ p ) ω 2 (τp: FCP duration).

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides 6 / 47

1/ p  1  2
A simple model: A two-component medium, each component is described by a two-level model

1/  p  1  2
We assume that the transparency range is very large: A simple model: A two-component medium, each component is described by a two-level model

1/  p  1  2
We assume that the transparency range is very large: ω 1 (1/τ p ) ω 2 (τp: FCP duration).

In a first stage, the two components are treated separately UV transition only, with (1/τ p ) ω 2 =⇒ Long-wave approximation A transparent medium A simple model: A two-component medium, each component is described by a two-level model

1/  p  1  2
We assume that the transparency range is very large: ω 1 (1/τ p ) ω 2 (τp: FCP duration).

In a first stage, the two components are treated separately UV transition only, with (1/τ p ) ω 2 =⇒ Long-wave approximation A transparent medium A simple model: A two-component medium, each component is described by a two-level model

1/  p  1  2
We assume that the transparency range is very large: ω 1 (1/τ p ) ω 2 (τp: FCP duration).

In a first stage, the two components are treated separately UV transition only, with (1/τ p ) ω 2 1/ p  2

A transparent medium

In a first stage, the two components are treated separately UV transition only, with (1/τ p ) ω 2 1/ p  2 =⇒ Long-wave approximation modified Korteweg-de Vries (mKdV) equation

∂E ∂ζ = 1 6 d 3 k dω 3 ω=0 ∂ 3 E ∂τ 3 - 6π nc χ (

A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

∂ 2 ψ ∂z∂t + c 1 sin ψ + c 2 ∂ ∂t ∂ψ ∂t 3 + c 3 ∂ 4 ψ ∂t 4 = 0 ∂u ∂z + c 1 sin t u + c 2 ∂u 3 ∂t + c 3 ∂ 3 u ∂t 3 = 0 with u = ∂ψ ∂t = E E r : normalized electric field
Integrable by inverse scattering transform in some cases: ,

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

∂ 2 ψ ∂z∂t + c 1 sin ψ + c 2 ∂ ∂t ∂ψ ∂t 3 + c 3 ∂ 4 ψ ∂t 4 = 0 Or ∂u ∂z + c 1 sin t u + c 2 ∂u 3 ∂t + c 3 ∂ 3 u ∂t 3 = 0 with u = ∂ψ ∂t = E E r : normalized electric field

A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

∂u ∂z + c 2 ∂u 3 ∂t + c 3 ∂ 3 u ∂t 3 = 0 with u = ∂ψ ∂t = E E r : normalized electric field
Integrable by inverse scattering transform in some cases: mKdV,

A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

A transparent medium

Then the two approximations are brought together to yield a general model:

The mKdV-sG equation

∂u ∂z + c 1 sin t u + c 2 ∂u 3 ∂t + 2c 2 ∂ 3 u ∂t 3 = 0 with u = ∂ψ ∂t = E E r : normalized electric field
Integrable by inverse scattering transform in some cases: mKdV, sG, and c 3 = 2c 2 .

The analytical breather solution

Waveguide description

The evolution of the electric field E : In (1+1) dimensions:

-→ The modified Korteweg-de Vries (mKdV) equation

∂ ζ E + β∂ 3 τ E + γ∂ τ E 3 = 0 Nonlinear coefficient γ = 1 2nc χ (3) , Dispersion parameter β = (-n ) 2c ,
Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

Waveguide description

The evolution of the electric field E : We generalize to (2+1) dimensions: -→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation

∂ ζ E + β∂ 3 τ E + γ∂ τ E 3 - V 2 τ ∂ 2 ξ Edτ = 0 Nonlinear coefficient γ = 1 2nc χ (3) , Dispersion parameter β = (-n ) 2c , Linear group velocity: V = c n . g c c x core cladding cladding -→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation ∂ ζ E + β α ∂ 3 τ E + γ α ∂ τ E 3 - V α 2 τ ∂ 2 ξ Edτ = 0
with α = g in the core and α = c in the cladding.

Nonlinear coefficient γ α = 1 2n α c χ (3) α , Dispersion parameter β α = (-n α ) 2c ,
Linear group velocity:

V α = c n α . g c c x core cladding cladding -→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation ∂ ζ E + β α ∂ 3 τ E + γ α ∂ τ E 3 + 1 V α ∂ τ E - V α 2 τ ∂ 2 ξ Edτ = 0
with α = g in the core and α = c in the cladding. Velocities :

V g < V c Nonlinear coefficient γ α = 1 2n α c χ (3) α , Dispersion parameter β α = (-n α ) 2c ,
Linear group velocity:

V α = c n α . g c c x core cladding cladding -→ The cubic generalized Kadomtsev-Petviashvili (CGKP) equation
In dimensionless form:

∂ z u = A α ∂ 3 t u + B α ∂ t u 3 + v α ∂ t u + W α 2 t ∂ 2 x udt
with α = g in the core and α = c in the cladding.

Relative velocities :

v g > v c
Nonlinear coefficient

γ α = 1 2n α c χ (3) α , Dispersion parameter β α = (-n α ) 2c ,
Linear group velocity:

V α = c n α .

Linear guided modes

The dimensionless CGKP equation

∂ z u = A α ∂ 3 t u + B α ∂ t u 3 + v α ∂ t u + W α 2 t ∂ 2 x udt
We linearize and seek for solutions of the form

u = f (x) exp[i (ωt -k z z)]. Waveguide dispersion relation: tan(k x a) = κ k x Mode profiles of the form f = R cos(k x x) in the core f = Ce -κ|x| in the cladding

Linear guided modes

The dimensionless CGKP equation

∂ z u = A α ∂ 3 t u + v α ∂ t u + W α 2 t ∂ 2 x udt
We linearize and seek for solutions of the form

u = f (x) exp[i (ωt -k z z)].
Material dispersion relation:

k 2 x,α = 2ω V α A α ω 3 -k z -ωv α Waveguide dispersion relation: tan(k x a) = κ k x Mode profiles of the form f = R cos(k x x) in the core f = Ce -κ|x| in the cladding

Linear guided modes

The dimensionless CGKP equation

∂ z u = A α ∂ 3 t u + v α ∂ t u + W α 2 t ∂ 2 x udt
We linearize and seek for solutions of the form

u = f (x) exp[i (ωt -k z z)]. Waveguide dispersion relation: tan(k x a) = κ k x Mode profiles of the form f = R cos(k x x) in the core f = Ce -κ|x| in the cladding Guiding condition: A 2 ω 3 -ωv 2 < k z < A 1 ω 3
As in the usual symmetric dielectric planar waveguide.

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

Nonlinear propagation in linear guide

We solve the CGKP equation starting from

u(x,t,z = 0) = A cos(ωt)f (x)e -t 2 /w 2 , f (x) = cos(k x x), for |x| ≤ a, Ce -κ|x| , for |x| > a, is a linear mode profile Normalized coefficients A 1 = A 2 = B 1 = B 2 = W 1 = W 2 = 1,
-→ we assume that -Temporal compression occurs -Spatial defocusing occurs, (else it collapses!) -Dispersion and nonlinearity are identical in core and cladding.

Guided wave profiles

(Normalized so that the total power is 1.

v 2 = 3, w = 2.)
The pulse is less confined in nonlinear (blue, and red) than in linear (pink and cyan) regime.

Nonlineary may reduce confinement. Indeed:

Waveguiding is due to total internal reflection.

-→ when the wave propagates too fast in the cladding, it cannot match the field oscillations in the core.

In the few-cycle regime, not only linear, but also nonlinear velocity (large and positive here).

The nonlinear velocity is larger in the core and creates a nonlinear variation of the index. It may compensate the linear part, and reduce the confinement. Important in very narrow guides, where linear confinement itself is low.

Nonlinear waveguide

Wave guided and confined by using nonlinear velocity: a higher nonlinear coefficient in the cladding than that in the core.

Two-cycle soliton of the nonlinear waveguide The field is u = Rf j (x)e i(ωt-βz) = Rf j (x)e iϕ , f j , (j = 1, 2): guided mode profile, R: amplitude. We seek for a solution as

∂ z u = A α ∂ 3 t u + B α ∂ t u 3 + V α ∂ t u + w α 2 t ∂ 2 x udt, α = g in
u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes. We report it into the GKP equation and get -in the cladding

∂ z Rf 1 + ∂ z Sf 2 = 0 -in guide 1 ∂ z Rf 1 + ∂ z Sf 2 = iw g 2ω (K c -K g )
Sf 2 and so on. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) , Involve overlap integrals I 1 = ∞ -∞ f 1 f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx
(" g j •dx" is the integral over the core j = 1 or 2.)

Individual waveguides, in the linearized model (B α = 0):

The field is u = Rf j (x)e i(ωt-βz) = Rf j (x)e iϕ , f j , (j = 1, 2): guided mode profile, R: amplitude. We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes.

We report it into the GKP equation and get -in the cladding

∂ z Rf 1 + ∂ z Sf 2 = 0 -in guide 1 ∂ z Rf 1 + ∂ z Sf 2 = iw g 2ω (K c -K g )
Sf 2 and so on. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) , Involve overlap integrals I 1 = ∞ -∞ f 1 f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx
(" g j •dx" is the integral over the core j = 1 or 2.)

Individual waveguides, in the linearized model (B α = 0):

The field is u = Rf j (x)e i(ωt-βz) = Rf j (x)e iϕ , f j , (j = 1, 2): guided mode profile, R: amplitude. We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes.

We report it into the GKP equation and get -in the cladding

∂ z Rf 1 + ∂ z Sf 2 = 0 -in guide 1 ∂ z Rf 1 + ∂ z Sf 2 = iw g 2ω (K c -K g )
Sf 2 and so on. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) , Involve overlap integrals I 1 = ∞ -∞ f 1 f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx
(" g j •dx" is the integral over the core j = 1 or 2.)

Individual waveguides, in the linearized model (B α = 0):

The field is u = Rf j (x)e i(ωt-βz) = Rf j (x)e iϕ , f j , (j = 1, 2): guided mode profile, R: amplitude. We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes.

We report it into the GKP equation and get -in the cladding

∂ z Rf 1 + ∂ z Sf 2 = 0 -in guide 1 ∂ z Rf 1 + ∂ z Sf 2 = iw g 2ω (K c -K g )
Sf 2 and so on. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) , Involve overlap integrals I 1 = ∞ -∞ f 1 f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx
(" g j •dx" is the integral over the core j = 1 or 2.)

Individual waveguides, in the linearized model (B α = 0):

The field is u = Rf j (x)e i(ωt-βz) = Rf j (x)e iϕ , f j , (j = 1, 2): guided mode profile, R: amplitude. We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes.

We report it into the GKP equation and get -in the cladding

∂ z Rf 1 + ∂ z Sf 2 = 0 -in guide 1 ∂ z Rf 1 + ∂ z Sf 2 = iw g 2ω (K c -K g )
Sf 2 and so on. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) , Involve overlap integrals I 1 = ∞ -∞ f 1 f 2 dx, I 2 = g 1 f 1 f 2 dx = g 2 f 1 f 2 dx
(" g j •dx" is the integral over the core j = 1 or 2.)

We for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) ,
The few-cycle pulse is expanded as a Fourier integral of such modes, u 1 = Re iϕ dω, u 2 = Se iϕ dω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms. Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 + C ∂ t u 2 + D∂ 3 t u 2 , ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 + C ∂ t u 1 + D∂ 3 t u 1 ,
We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) ,
The few-cycle pulse is expanded as a Fourier integral of such modes, u 1 = Re iϕ dω, u 2 = Se iϕ dω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms. Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 + C ∂ t u 2 + D∂ 3 t u 2 , ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 + C ∂ t u 1 + D∂ 3 t u 1 ,
We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) ,
The few-cycle pulse is expanded as a Fourier integral of such modes, u 1 = Re iϕ dω, u 2 = Se iϕ dω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms. Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 + C ∂ t u 2 + D∂ 3 t u 2 , ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 + C ∂ t u 1 + D∂ 3 t u 1 ,
We seek for a solution as

u = R(z)f 1 (x)e iϕ + S(z)f 2 (x)e iϕ ,
i.e., two interacting modes. Multiplying this by f 1 and integrating over all x removes the x-dependency of f 1 , f 2 and we get

∂ z R = ∂ z S = iw g 2ω (K c -K g ) I 2 1 + I 1 (R + S) ,
The few-cycle pulse is expanded as a Fourier integral of such modes, u 1 = Re iϕ dω, u 2 = Se iϕ dω.

We report ∂ z R and ∂ z S into ∂ z u 1 , and get the linear coupling terms. Finally, we get the system of two coupled modified Korteweg-de Vries (mKdV) equations

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 + C ∂ t u 2 + D∂ 3 t u 2 , ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 + C ∂ t u 1 + D∂ 3 t u 1 ,

Nonlinear coupling

An analogous procedure, treating the nonlinear term as a perturbation, yields a evolution equation

∂ z u 1 = L(u 1 ) + ∂ t I 3 u 3 1 + I 4 3u 2 1 u 2 + u 3 2 .
The integrals involved are

I 3 = ∞ -∞ B α f 4 1 dx, I 4 = ∞ -∞ B α f 3 1 f 2 dx. The complete final system is ∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 +C ∂ t u 2 + D∂ 3 t u 2 + E ∂ t 3u 2 1 u 2 + u 3 2 ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 +C ∂ t u 1 + D∂ 3 t u 1 + E ∂ t 3u 1 u 2 + u 3

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 +C ∂ t u 2 + D∂ 3 t u 2 + E ∂ t 3u 2 1 u 2 + u 3 2 ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 +C ∂ t u 1 + D∂ 3 t u 1 + E ∂ t 3u 1 u 2 + u 3 1
We evidence a standard linear coupling term, a linear coupling term based on dispersion, a nonlinear coupling term

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 +C ∂ t u 2 + D∂ 3 t u 2 + E ∂ t 3u 2 1 u 2 + u 3 2 ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 +C ∂ t u 1 + D∂ 3 t u 1 + E ∂ t 3u 1 u 2 + u 3 1
We evidence a standard linear coupling term, a linear coupling term based on dispersion, a nonlinear coupling term

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 +C ∂ t u 2 + D∂ 3 t u 2 + E ∂ t 3u 2 1 u 2 + u 3 2 ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 +C ∂ t u 1 + D∂ 3 t u 1 + E ∂ t 3u 1 u 2 + u 3 1
We evidence a standard linear coupling term, a linear coupling term based on dispersion, a nonlinear coupling term

Nonlinear coupling

The complete final system is

∂ z u 1 = A∂ 3 t u 1 + B∂ t u 3 1 + V ∂ t u 1 +C ∂ t u 2 + D∂ 3 t u 2 + E ∂ t 3u 2 1 u 2 + u 3 2 ∂ z u 2 = A∂ 3 t u 2 + B∂ t u 3 2 + V ∂ t u 2 +C ∂ t u 1 + D∂ 3 t u 1 + E ∂ t 3u 1 u 2 + u 3 1
We evidence a standard linear coupling term, a linear coupling term based on dispersion, a nonlinear coupling term Two solitons with different frequencies can lock together. Initial pulses:

Output, uncoupled:

Output, coupled: We look for stationary states (vector solitons) in this model The "stationary" states oscillate with t and z: . few-cycle solitons are breathers. A typical example of few-cycle vector soliton (Dotted lines: u, solid lines: v . Left: at z = 0, right: at z = 60. < Au >= 1.837).

Evolution of soliton's maximum amplitude during propagation. Two types of oscillations:

• Fast: phase -group velocity mismatch • Slower: periodic energy exchange, as in linear regime.

Oscillations of the few-cycle vector solitons

The energies

E u = u 2 dt and E v = v 2 dt oscillate almost harmonically, as E u =< E u > +∆E u sin(2πσ a z + φ E ,u ), The same for A u = max t (|u|) and A v = max t (|v |) Spatial frequency σ a ∈ [1.06,1
.17], increasing with < A u >.

(linear: σ 0 = 1.326).

Amplitudes of oscillations vs amplitude of field u Well fitted with ∆E R A -< A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies

E u = u 2 dt and E v = v 2 dt oscillate almost harmonically, as E u =< E u > +∆E u sin(2πσ a z + φ E ,u ), The same for A u = max t (|u|) and A v = max t (|v |) Spatial frequency σ a ∈ [1.06,1
.17], increasing with < A u >.

(linear: σ 0 = 1.326).

Amplitudes of oscillations vs amplitude of field u Well fitted with ∆E R A -< A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies

E u = u 2 dt and E v = v 2 dt oscillate almost harmonically, as E u =< E u > +∆E u sin(2πσ a z + φ E ,u ), The same for A u = max t (|u|) and A v = max t (|v |) Spatial frequency σ a ∈ [1.06,1
.17], increasing with < A u >.

(linear: σ 0 = 1.326).

Amplitudes of oscillations vs amplitude of field u Well fitted with ∆E R A -< A >, etc., with A = 1.854.

Oscillations of the few-cycle vector solitons

The energies E u = u 2 dt and E v = v 2 dt oscillate almost harmonically, as E u =< E u > +∆E u sin(2πσ a z + φ E ,u ), Amplitudes of oscillations vs amplitude of field u Well fitted with ∆E u R A 0 -< A u >, etc., with A 0 = 1.854.

Evolution of the ratio v /u

Almost constant vs t

  FCP duration). eblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides 6 / 47

  3) (ω; ω,ω, -ω) ω=0 ∂E 3 ∂τ A transparent medium In a first stage, the two components are treated separately IR transition only, with ω 1 (1/τ p ) =⇒ Short-wave approximation sine-Gordon (sG) equation: and F. Sanchez, Phys. Rev. A 67, 013804 (2003) Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides A transparent medium In a first stage, the two components are treated separately IR transition only, with ω 1 (1/τ p ) field H. Leblond and F. Sanchez, Phys. Rev. A 67, 013804 (2003) Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides A transparent medium In a first stage, the two components are treated separately IR transition only, with ω 1 (1/τ p ) field H. Leblond and F. Sanchez, Phys. Rev. A 67, 013804 (2003) Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

  scattering transform in some cases: mKdV, sG, Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

  , S.V. Sazonov, I.V. Mel'nikov, D. Mihalache, and F. Sanchez, Phys. Rev. A 74, 063815 (2006) 1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

47 1

 47 and D. Mihalache, Phys. Rev. A 88, 023840 (2013) Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides 19 / Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons 1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons 2D waveguiding structure: two cores 1 and 2 and dielectric cladding The generalized Kadomtsev-Petviashvili (GKP) equation (dimensionless)

  the cores 1 and 2, α = c in the cladding. Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides Individual waveguides, in the linearized model (B α = 0):

1H.

  Leblond, and S. Terniche, Phys. Rev. A 93, 043839 (2016) Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

1

  Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons A soliton is launched in channel u and a smaller input in channel v . Initial pulses: u forms a soliton, v is spread out. Output, uncoupled: Output, coupled: A vector soliton forms. (Blue line: u, red line: v . Output at z = 2,coupling C = -1 or 0. Au = 2, Av = 0.2, λu = λv = 1, FWHMu = FWHMv = 3, ϕu = ϕv = 0, tu = tv = 0. Mutual trapping of solitons: It can occur if their center don't coincide initially Initial pulses: Output, uncoupled: Output, coupled: Blue line: u, red line: v . Output at z = 2, coupling C = -1 or 0. Au = Av = 3, λu = λv = 1, FWHMu = FWHMv = 2, ϕu = ϕv = 0, tu = 1, tv = -1).

  Blue line: u, red line: v . Output at z = 2, coupling C = -1 or 0. Au = Av = 1.8, λu = 1, λv = 0.8, FWHMu = FWHMv = 3, ϕu = ϕv = 0, tu = tv = 0). 1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

z

  Soliton with < Au >= 1.789.

  1.8 1.81 1.82 1.83 1.84 1.85 1.86 black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av .

  1.8 1.81 1.82 1.83 1.84 1.85 1.86 black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av .

  1.8 1.81 1.82 1.83 1.84 1.85 1.86 black saltires: ∆Eu; blue stars: ∆Au; red crosses: ∆Av .

  ∆Eu; blue stars: ∆Au; red crosses: ∆Av .

v

  Soliton with < Au >= 1.855. Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides 36 / 47Evolution of the ratio v /uOr θ = arctan v u . Oscillates almost harmonically with z. Amplitudes of oscillations vs field u amplitude: Black line: mean value < θ >; green line: ∆θ. Crosses: raw numerical data; solid lines: linear or parabolic fits. S. Terniche, H. Leblond, D. Mihalache, and A. Kellou, submitted to Phys.Rev. A Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides 1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons 1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

eblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides

1 Models for few-cycle solitons The mKdV-sG equation Nonlinear widening of the linear guided modes 2 Waveguide coupling in the few-cycle regime Derivation of the coupling terms Examples of behavior of the linear non-dispersive coupling Few-cycle optical solitons in linearly coupled waveguides 3 Few cycle spatiotemporal solitons in waveguide arrays Formation of a solitons from a Gaussian pulse Two kind of solitons

We assume a purely linear and non-dispersive coupling

The initial data is u = A u sin (ω u t + ϕ u ) e -(t-tu) 2 /τ 2 u , v = A v sin (ω v t + ϕ v ) e -(t-tv ) 2 /τv .

Leblond, Mihalache, Kremer, Terniche ( Laboratoire de Photonique d'Angers LϕA EA 4464, Université d'Angers., Horia Hulubei Nation Few-cycle optical solitons in coupled waveguides Consider now the coupled equations in the linearized case. The monochromatic solutions are

With, due to coupling,

The maximum amplitude and the power density of the wave oscillate with spatial frequency cω/π = σ 0 = 1.326.

Consider now the coupled equations in the linearized case. The monochromatic solutions are

With, due to coupling, A = u 0 cos cωz + iv 0 sin cωz, B = v 0 cos cωz + iu 0 sin cωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequency cω/π = σ 0 = 1.326.

Consider now the coupled equations in the linearized case. The monochromatic solutions are

With, due to coupling, A = u 0 cos cωz + iv 0 sin cωz, B = v 0 cos cωz + iu 0 sin cωz.

The maximum amplitude and the power density of the wave oscillate with spatial frequency cω/π = σ 0 = 1.326.

Consider now the coupled equations in the linearized case.

The monochromatic solutions are

With, due to coupling,

The maximum amplitude and the power density of the wave oscillate with spatial frequency cω/π = σ 0 = 1.326.

Oscillations of the few-cycle vector solitons

The energies

The same for A u = max t (|u|) and

.17], increasing with < A u >.

(linear: σ 0 = 1.326).

Amplitudes of oscillations vs amplitude of field u Well fitted with ∆E R A -< A >, etc., with A = 1.854.

A set of coupled waveguides within the same model, as:

We fix ϕ 0 = 0, x = 1, λ = 1, and we vary A 0 and τ . Two kind of solitons: breathing and fundamental.

Breathing soliton: localized in space and time oscillating wave packet


B Náfrádi

Á Antal

T Fehér

L F Kiss

C Mézière

P Batail

L Forró

A Jánossy

Frustration induced one-dimensionality in the isosceles triangular antiferromagnetic lattice of δ-(EDT-TTF-CONMe 2 ) 2 AsF 6

Keywords:

The 1/4-filled organic compound, δ-(EDT-TTF-CONMe2)2AsF6 is a frustrated two-dimensional triangular magnetic system as shown by high-frequency (111.2 and 222.4 GHz) electron spin resonance (ESR) and structural data in the literature. The material gradually orders antiferromagnetically below 40 K but some magnetically disordered domains persist down to 4 K. We propose that in defect free regions frustration prevents true magnetic order down to at least 4 K in spite of the large first-and second-neighbor exchange interactions along chains and between chains, respectively. The antiferromagnetic (AFM) order gradually developing below 40 K nucleates around structural defects that locally cancel frustration. Two antiferromagnetic resonance modes mapped in the principal planes at 4 K are assigned to the very weakly interacting one-dimensional molecular chains in antiferromagnetic regions.

I. INTRODUCTION

Materials with dense magnetic atomic or molecular sites usually have a magnetically ordered ground state. However, in some two-dimensional (2D) lattices quantum fluctuations and frustration of interactions between neighboring magnetic sites preclude long range order. In these the rotational symmetry is preserved and the ground state is an exotic spin liquid 1,2 . Kagome lattices with isotropic Heisenberg interactions between sites are prime examples 3 . However, it was noticed in the experimental realizations, vesignieite 4 and herbertsmithite, 5 that in the complex kagome geometry a strong Dzyaloshinskii-Moriya (DM) interaction freezes the quantum spin liquid at finite temperatures 6 . A simpler configuration is a triangular lattice of spin-1/2 moments with isotropic antiferromagnetic Heisenberg exchange interaction between sites. The equilateral triangular lattice with classical spins has an ordered ground state, but frustration in the quantum spin lattice leads to a spin liquid which is readily studied in organic materials like κ-(BEDT-TTF) 2 Cu 2 (CN) 3 and EtMe 3 Sb[Pb(dmit) 2 ] 2 7 .

A question of great interest is the robustness of the spin liquid state against deviations from the ideal structure. Namely, the nature of the magnetic ground state of the isosceles triangular antiferromagnetic lattice with different first-neighbor interactions, J and J 2 , on the base and legs respectively, is strongly debated [8][9][10] . It is believed that the system decouples into weakly interacting antiferromagnetic 1D chains if J is much larger than J 2 and a spin liquid state is formed with no magnetic order. The inorganic materials Cs 2 CuCl 4 and Cs 2 CuBr 4 were studied within this context 11 . In these systems the ground state is extremely sensitive to magnetic fields due to deviations from the isotropic exchange Heisenberg model, in particular the DM interaction 6,12,13 .

The organic magnet δ-(EDT-TTF-CONMe 2 ) 2 AsF 6 , hereafter EDT 2 AsF 6 , 24 is an excellent model system for studying frustration in an isosceles triangular lattice. The asymmetric EDT molecules (Fig. 1 (A), further on symbolized by a duck) form chains. The molecular separation is uniform along chains, the instability of the quasi-1D electronic system results in alternating "charge rich" and "charge poor" molecules with charges of 0.9 e + and 0.1 e + , respectively [14][15][16] (dark blue and light blue ducks in Fig. 1 (B)). The magnetic structure sketched in Fig. 1 (C) is approximated by a two-dimensional triangular system, with the exchange interactions J, J 2 and J 3 . The high-temperature magnetic properties follow a 1D spin-1/2 Heisenberg antiferromagnetic chain with an isotropic exchange of J = 298 K 17 . There is no direct measurement of J 2 ; we estimate it from the calculated overlap integrals along and perpendicular to the chains 14 to be about 30 K. Overlap is small in the c direction where AsF 6 ions separate the chains. The exchange parameters are somewhat modified below the orthorhombic to monoclinic transition 16 at 190 K. Anisotropies, deviations from the isotropic exchange Heisenberg model are weak 17 . An antiferromagnetic (AF) ordering has been observed 15 at T N = 8.5 K. The DM interaction between molecules along chains in the a direction is forbidden by symmetry 16 in the AF state and the first neighbor dipoledipole interaction is also canceled (see III E).

Here we report high-frequency electron spin resonance (ESR) measurements (complemented by static magnetization data) which elucidate the magnetic structure in EDT 2 AsF 6 . The magnetic resonances of the antiferro-magnetically ordered and the paramagnetic states are well separated in the ESR spectrum. Some paramagnetic regions persist to temperatures as low as 4 K. This is unexpected since the overlap integral between molecules on adjacent chains along b is quite large 14 and one expects intuitively a long range ordered AFM state to set in at much higher temperatures. We suggest that EDT 2 AsF 6 represents the remarkable case of an isosceles triangular lattice which is transformed by frustration into a system of weakly interacting antiferromagnetic chains. Without defects there would be no magnetic order down to the temperature range we investigated. However, defects locally lift the frustration in the imperfect crystals. As a result, at 4 K most of the crystal is antiferromagnetically ordered and depending on the crystal quality, some antiferromagnetic regions persist up to 40 K. The ESR in antiferromagnetically ordered regions shows that the chains (along a) interact surprisingly weakly in the b direction. This again is a result of frustration canceling the interaction between chains.

II. EXPERIMENTAL

EDT 2 AsF 6 single crystals were grown using electrochemical oxidation of EDT like in earlier studies of the same compound [14][15][16][17] . ESR was performed by homebuilt spectrometers operating at 111.2 and 222.4 GHz frequencies [18][19][20] . These setups are particularly suitable to detect the AFM resonance in organic magnets. The ESR spectra presented in this paper were recorded at fixed frequencies sweeping the magnetic field and measuring the derivative of the microwave intensity reflected from the sample.

The static magnetic susceptibility measured by SQUID and the paramagnetic susceptibility measured by the ESR intensity were compared in Refs. 14,17. The high temperature susceptibility data were used to calibrate the absolute value in our measurement.

III. RESULTS AND DISCUSSION

A. Paramagnetism and antferromagnetic ordering

Between 40 and 300 K, the static susceptibility follows well the ESR intensity measured at 9 GHz, as expected for a paramagnetic material where spins in the whole crystal contribute to the ESR at g ≈ 2. However, below 40 K, the 9 GHz ESR intensity decreases rapidly with decreasing temperature due to the decrease of the paramagnetic regions in the sample 14,17 . The static susceptibility remains large 14 , suggesting that below 40 K the material is inhomogeneous; it consists of paramagnetic and antiferromagnetically ordered regions. These latter ones cannot be detected at 9 GHz since this frequency is within the gap of the excitation spectrum. At low temperatures a small paramagnetic contribution proportional to the inverse temperature was found. At 1.8 K the large majority of the powder sample is antiferromagnetic. At high fields the small paramagnetic term is saturated and the magnetisation, M increases linearly with field. The effective antiferromagnetic exchange interaction determined from the slope of the magnetization vs. field curve at 1.8 K is roughly J = 410 K.

The ESR of the paramagnetic material was detected between 4 and 300 K at high excitation frequencies, ω/2π = 111.2 and 222.4 GHz. The paramagnetic resonance field is proportional to the exciting frequency and has the same g-factor anisotropy at all temperatures. The resonance splits at low temperatures into the ESR of the paramagnetic domains and the antiferromagnetic resonance (AFMR) of magnetically ordered domains (Fig. 2).

The onset temperature at which the AFMR lines appear resolved from the ESR varies from crystal to crystal between 15 and 40 K. The AFMR field is not proportional to the frequency and depends on the anisotropic coupling between sublattice magnetizations. The AFMR shift from the g = 2 position at fixed temperature is approximately inversely proportional to ω and increases with the increase of the sublattice magnetization at lower temperatures.

The ratio of the ESR and AFMR line intensities decreases gradually below the onset temperature showing the gradual expansion of antiferromagnetic domains at the expense of paramagnetic ones. However, some regions remain paramagnetic at 4 K since a weak, broadened ESR of uncorrelated EDT 2 AsF 6 chains is still observable. This ESR line is assigned to paramagnetic EDT 2 AsF 6 regions with few structural defects, embedded in antiferromagnetically ordered parts of the crystal. The ESR at 4 K appearing together with the AFMR of most of the sample does not arise from paramagnetic impurities since it has the same g-factor anisotropy as the ESR of the full sample at high temperatures (Fig. 3). The g factor depends on the orientation and type of the ESR active molecules. At the high fields of our experiment, the g-factor anisotropy of an impurity phase would be well resolved from that of paramagnetic EDT 2 AsF 6 .

B. Angular dependence of the AFMR modes

The angular dependence of the AFMR resonance fields was mapped at 4 K in the (a,b*) (Fig. 3), (a,c) (Fig. 4 A) and (b*,c) planes at 111.2 and 222.4 GHz. (b* is perpendicular to the (a,c) plane, and is close to b; it coincides with a well-defined edge of the crystal. See III C for details of the crystal structure). The accuracy of crystal alignment was better than 5 • . Two AFMR modes were resolved in general magnetic field directions. The two modes are degenerate in the b* and c directions but are split in a (Fig. 3). Each mode consists of several closely spaced lines.

In the (a,b*) plane, the modes depend on magnetic field history; the hysteresis is strongest near a (see Fig. 3). We note that line positions are different in subsequent 180 • rotations of the magnetic field. The nonsymmetric angular dependence around b* is also due to hysteresis. In our interpretation, it arises from pinning of magnetically ordered domains to a small concentration of defects. The angular dependence in the (b*,c) plane (not shown) is small and the measured AFMR felds depend sensitively on the precision of the sample orientation.

C. Assignment of AFMR modes

We draw our main conclusions from an analysis of the modes measured with field in the (a,c) plane (see Fig. 4).

The AFMR fields of the two modes in this plane, B + AF and B - AF (red and blue lines, respectively) vary similarly but in opposite sense with the angle β measured from the a axis, i.e. B + AF (β) = B - AF (-β). Except for the relatively small splitting and hysteresis of lines near a, the angular dependence of the two modes fit well the expressions:

B + AF = B 0 + (1/2)b ac cos(2(β 0 + β)), (1)
B - AF = B 0 + (1/2)b ac cos(2(β 0 -β)). (2)
Each curve corresponds to a conventional twosublattice AFMR excited at frequencies much larger than the gap. This agrees with the observation that for the two frequencies, 111.2 and 222.4 GHz, the average B 0 is proportional to ω, and the amplitude, b ac is inversely proportional to ω.

We argue that the two modes described by Eqs. 1 and 2 have the symmetry in the angular dependence B + AF (β) = B - AF (-β) only if the chains are weakly coupled. The inevitable splitting of the modes near the apparent mode crossings is smaller than the experimental uncertainities. We assign the two AFMR modes to nearly independent modes of the chemically identical but structurally nonequivalent chains along a.

To explain, we first recall some details of the crystal structure [14][15][16] . The full structure including the CDW wavevectors has been determined at room temperatures; NMR shows that the CDW changes little at lower temperatures. Above 190 K the structure is orthorhombic. A monoclinic distortion arises below 190 K that gradually increases the angle γ from 90 to about 93 • at 100 K where it is close to saturation. This small distortion is assumed to be unimportant and we base the argument on the orthorhombic structure where not stated otherwise. Crystals are twinned, the twins are related by a reflection of all three principal axes above 190 K, thus they are magnetically equivalent in the sense that their ESR spectra are the same. Below 190 K the reflection symmetry between the b axes of the twins is broken, while it is unchanged for the a and c axes. Twins may split somewhat the observed AFMR spectra in general directions but not in magnetic fields in the (a,c) plane.

There are 4 chemically equivalent "a" chains that are structurally related by symmetry operations (Fig. 5 Clearly, if chains did not interact with their neighbors, the reflection symmetry b → -b would ensure that the resonance of chains I. and II. (III. and IV.) coincide, while the symmetry c → -c that I. and III. (II. and IV.) would rotate in the opposite sense for magnetic fields in the (a,c) plane. A significant magnetic interaction between first neighbor chains that couples the modes would inhibit mode crossings: for weakly interacting chains the modes split first near the crossing points.

To account for quasi-linear crossing of the counterrotating modes of the antiferromagnetic regions we suggest that the interaction between chains is small along both the b and c directions. It cannot be zero, otherwise there would be no magnetic order but it must be small to explain the lack of an observable splitting. The gradual development of the magnetic order in a large temperature range suggests that the coupling between chains is due to a small concentration of defects. A small part of the sample remains paramagnetic down to 4 K.

The electronic overlap in the c direction is very small as AsF 6 ions separate the first neighbor chains. On the other hand, the lack of magnetic interactions between the first neighbor chains in the b direction (e.g. between I. and II.) is unexpected. In view of the crystal and electronic structures it is surprising that the two AFMR modes are well described by Eqs. 1 and 2 even near modecrossing directions (Fig. 6) as AFMR in these high symmetry directions is rather sensitive to interactions between layers. The modes of interacting non-equivalent layers deviate strongly from the modes of isolated layers around regions of mode degeneracy. For example, the splitting of the AFMR modes of adjacent layers is well observable in the organic quasi-2D antiferromagnet, κ-(BEDT-TTF) 2 Cu(N[CN] 2 )Cl, despite the several orders of magnitude difference between inter-and intra-layer magnetic interactions 21,22 .

D. Justification of the isosceles triangular AF magnetism model

The triangular magnetic structure in Fig. 1 (C) explains the magnetic frustration causing the lack of interaction between the two AFMR modes. It is the simplest frustrated order compatible with the crystal and charge density wave structure and the static magnetic susceptibility. Here we summarize the arguments supporting the trianular magnetic structure.

The quasi 1D antiferromagnetic chain behavior of molecular chains along the a direction follows from the known crystal structure and the high temperature magnetic susceptibility. NMR shows that highly and poorly charged molecules alternate along a and that the charge on poor molecules is almost one electron smaller than on rich ones 14,15 . The full CDW structure was determined by XRD 16 . The magnetic susceptibility at high temperatures shows that along a the chains are quasi 1D antiferromagnetic with a large exchange interaction between molecules. It is a natural suggestion that the ordered magnetic structure consists of these antiferromagnetic chains. From the angular dependence of the two AFMR modes we find that magnetic interactions between a chains running parallel in the (a,b) plane are negligibly small in the antiferromagnetic state. This is explained by magnetic frustration. The negligible magnetic interaction between chains in the b direction does not follow from the crystal structure alone. According to band calculations the overlap between molecules in the b direction is not very small, the overlap integral between adjacent chains in the (a,b) plane is only an order of magnitude smaller than along the chain. (Overlap in the c direction is very small.) Thus we propose that frustration is the reason for the weak magnetic interaction. The simplest way to obtain frustration is evident from the crystal structure. Since neighboring chains of uniformly spaced molecules are shifted by a half lattice constant, the molecular lattice is a triangular network in the (a,b) plane. (Fig. 1 (B)). The uniform spacing between molecules along the chains that allows the triangular molecular structure is a unique feature of this compound 14 . To explain the magnetic frustration between neighboring a chains, we propose that the magnetic structure follows the triangular crystal structure. The main assumptions are that in the ordered state all a chains are simple two sublattice antiferromagnets and anisotropic exchange or other interactions between chains are small.

E. Comparison with the 2D magnetic lattice model

The two-dimensional magnetic lattice in Fig. 1 (C) serves as a model for the (a,b*) plane of EDT 2 AsF 6 . The isotropic antiferromagnetic interactions between first, second and third neighbor molecular pairs are characterized by J, J 2 and J 3 respectively. For J = J 2 and J 3 = 0 the model corresponds to the regular triangular frustrated spin system. On the other hand, if J J 2 , a two-sublattice antiferromagnetic order is established along the chains (the isosceles triangular case). Frustration is still present in this latter case since the molecular next neighbor interactions (J 2 ) cancel and the coupling between neighboring antiferromagnetic chains vanishes. A finite J 3 couples second neighbor chains and the system consists of two non-interacting 2D antiferromagnets.

The real material EDT 2 AsF 6 differs somewhat from the model. The structural transition at 190 K modifies in a subtle way the picture. Although in the orthorhombic structure isotropic antiferromagnetic interactions between neighboring chains fully cancel, the frustration is slightly weaker in the monoclinic structure below 190 K where molecules are inclined by a few degree and the J 2 interactions do not cancel completely on the neighboring molecules. The small anisotropic interactions determine the angular dependence of AFMR modes and have a profound effect on the magnetic order.

We assumed in the above argument that the chains are antiferromagnetic. From a general point of view, it is not evident whether a 1/4 filled chain is antiferromagnetic or ferromagnetic. The extended Hubbard model allows for both types 23 , depending on the parameters assumed in the calculations. However, in EDT 2 AsF 6 the AFMR mode diagram is incompatible with ferromagnetic chains since these would strongly interact in the (a,b*) plane. There would be no frustration in this case, independently of the sign of J 2 (i.e. whether the interaction between "a" chains is ferromagnetic or antiferromagnetic). Thus ferromagnetic chains along a are incompatible with the two independent AFMR modes in the (a,c) plane.

Finally, we suggest that the Néel temperature of the phase transition in a perfect crystal is below 4 K. We explain by a variation of defect centration that the onset temperature of an observable magnetic order varies from crystal to crystal. Defects magnetically connecting chains in the b direction locally break frustration and induce an incomplete static magnetic order in regions where the average defect distance is comparable to the in-chain correlation length. As a result, most but not all of the crystal is antiferromagnetically ordered at 4 K. The persistence of intrinsic paramagnetic regions signifies that at 4 K in some regions frustration overcomes the coupling between chains due to defects and residual interactions between chains. In these regions the same paramagnetic behaviour is observed in the ESR as at high temperatures, except for some line broadening. We suggest that the paramagnetic ESR corresponds to regions with small defect concentration where order is prevented by frustration at 4 K. The chains along a are quasi one-dimensional with a magnetic ordering temperature below 4 K. The continuous development of order and the persistance of paramagnetic regions show that the antiferromagnetic ordering temperature of the ideal, defect free system is lower than T NMR N = 8.5 K suggested in Ref. 15.

Hysteresis in high fields, which is unusual in antiferromagnets, suggests also that defects strongly modify the magnetic texture of the sample, creating antiferromagnetic and paramagnetic domains. Hysteresis due to disorder by structural defects is most important when anisotropic interactions between chains are particularly weak; this may be the case for magnetic fields in the (a,b*) plane. In organic spin-1/2 magnets, where there is no single ion anisotropy and anisotropic exchange between molecules composed of light elements is small, the anisotropy is mainly due to dipolar interactions. It is a simple matter to show that in EDT 2 AsF 6 the dipolar interactions between first neighbor chains cancel above the spin flop transition if the external field is in the (a,b*) plane. The fields of interest for the AFMR are well above the spin flop field; thus order is not established by dipolar interactions in the (a,b*) plane.

Symmetry arguments show that in EDT 2 AsF 6 the DM interaction is ineffective along "a" if chains are antiferromagnetic. On the other hand, the DM interaction is not zero between neighbor "a" chains in the (a,c) plane. A small, frequency dependent broadening of the ESR line has been attributed to the DM interaction between chains in the paramagnetic state 17 . A negligible ferrimagnetism is expected in the magnetically ordered regions, since the chains are antiferromagnetic with a large exchange, J in the ordered state. This is in accord with the absence of weak ferromagnetism in the static magnetization. However, the DM interaction can influence the magnetic order in a magnetic field dependent way. The relative orientation of sublattice magnetizations in neighboring chains can change abruptly when the external field direction is swept through the DM vector. The DM interaction may be at the origin of the hysteresis of the AFMR modes. Indeed, the DM vector of interchain interaction lies in the (a,b) plane and hysteresis effects are most pronounced when the magnetic field is rotated in this plane.

IV. CONCLUSION

In conclusion, EDT 2 AsF 6 is a quasi-one-dimensional compound which has a 1/4 filled electronic band with unusual electronic and magnetic properties. We find that the (a,b*) plane of EDT 2 AsF 6 is to a good approximation a 2D frustrated isosceles triangular magnetic lattice with a strong isotropic exchange interaction, J on the base and a much weaker interaction, J 2 on the legs. Frustration remains important in this triangular lattice; the ideal system decomposes into two networks of non-interacting one-dimensional antiferromagnetic chains preserving an overall paramagnetic response down to temperatures as low as 4 K. The AFMR modes show that the magnetic interaction between closely lying chains in the (a,b*) plane is unexpectedly weak due to frustration. Structural defects lift the frustration and nucleate antiferromagnetic regions up to temperatures as high as 40 K.

FIG. 1

 1 FIG. 1: (A) The EDT molecule and its schematic representation as a duck in further panels. (B) Schematic structure of the (a,b) plane of EDT2AsF6. Dark and light ducks represent charge rich and charge poor molecules. (C) The frustrated, isosceles triangular magnetic lattice. J, J2 and J3 are the first, second and third neighbor isotropic antiferromagnetic exchange interactions respectively. The exchange is large along chains while frustration cancels the isotropic interactions between chains.

FIG. 2 :

 2 FIG. 2: Temperature dependence of the ESR and AFMR spectra in an EDT2AsF6 crystal with B0 a, 222.4 GHz. The AFMR below 25 K signals an antiferromagnetic order in large regions. The ESR from paramagnetic regions persists to low temperatures. Reference at 7.941 T is KC60.

FIG. 3 :

 3 FIG. 3: Vertical lines: ESR and AFMR spectra in EDT2AsF6 at 4 K and 222.4 GHz with magnetic field in the (a,b*) plane. Red and blue lines: guides to the eye for the AFMR positions. Green line: ESR resonance fields at 200 K. The ESR resonance field oscillates sinusoidally with angle around 7.92 T.The ESR at 4 K, where most of the sample is antiferromagnetic, has the same g-factor anisotropy as the ESR at 200 K where all of the sample is paramagnetic. The AFMR modes oscillate with magnetic field angle around 7.88 T and depend on magnetic history. Resonance at 7.941 T is a KC60 field reference.

FIG. 4 :

 4 FIG. 4: AFMR modes in EDT2AsF6 with magnetic field in the (a,c) plane at 4 K. (A) resonance frquency 222.4 GHz. (Vertical lines are spectra. Red and blue lines are guides to the eye). For B in the (a,c) plane, one AFMR mode arises from chains in the b=0 plane, the other from chains in the b=b0/2 planes; (B) The (a,c) plane angular dependence of the AFMR resonance field at 111.2 GHz.

FIG. 5 :

 5 FIG. 5: Schematic projection of the (b,c) plane along the a (chain) axis showing charged molecules only. Molecules II. and IV. are in the a = a0/2 plane, molecules I. and III. are in the a = 0 plane. The AFMR modes are attributed to magnetically weakly interacting chains. For fields in the (a,c) plane, the angular dependence of the modes of chains I. and IV. rotate in opposite sense to the modes of chains II. and III. as expected from the glide plane symmetry relation c → -c (dashed lines) between the respective chains. The unexpectedly weak effective interaction along b is attributed to magnetic frustration. Interaction is weak in the c direction where AsF - 6 ions (red dots) separate the chains.

  ). Chains I. and III. are in the b = 0 plane while chains II. and IV. in the b = b 0 /2 plane. Chains I. and III. (similarly II. and IV.) are related by the rotation [x, y, z] → [x, -y, -z]. Chains I. and II. (III. and IV.) are related by the glide reflection [x, y, z] → [ 1 2 + x, 1 2 + y, 1 2 -z]. Finally, the glide reflection [x, y, z] → [ 1 2 + x, 1 2 -y, 1 2 + z] relates chains I. and IV. (II. and III.).

P. Fazekas and P. W. Anderson, Philosophical Magazine 30, 423 (1974).

P. W. Anderson, Science 235, 1196 (1987).

L. Balents, Nature

464, 199 (2010).

Acknowledgments

Support from the projects T ÁMOP-4.2.1/B-09/1/KMR-2010-0002, OTKA K107228. The work in Lausanne was supported by SCIEX by the Swiss National Science Foundation (Grant No. 200021 144419) and ERC advanced grant "PICOPROP" (Grant No. 670918). The work in Angers was supported by the ANR grant 3/4-Filled 2009-2011 (ANR-08-BLAN-0140-01) and the CNRS.


M Sacchi

N Jaouen

H Popescu

R Gaudemer

J M Tonnerre

S G Chiuzbaian

C F Hague

A Delmotte

J M Dubuisson

G Cauchon

B Lagarde

F Polack

The SEXTANTS beamline at SOLEIL: a new facility for elastic, inelastic and coherent scattering of soft X-rays

SEXTANTS is a new SOLEIL beamline dedicated to soft X-ray scattering techniques. The beamline, covering the 50-1700 eV energy range, features two Apple-II undulators for polarization control and a fixed-deviation monochromator. Two branch-lines host three end-stations for elastic, inelastic and coherent scattering experiments.

Introduction

SEXTANTS is a new beamline at SOLEIL synchrotron covering the 50-1700 eV energy range. The main scientific purpose is to investigate electronic and magnetic properties of solids using three scattering techniques, namely inelastic, magnetic, and coherent resonant x-ray scattering, including xray holographic imaging. These are all element selective photon-in-photon-out techniques with magnetic sensitivity and variable probing depths. In the following, we will present briefly the main characteristics of the SEXTANTS beamline, making reference to the schematic layout of Fig. 1.

Beamline layout 2.1. Source

The beamline features two Apple-II undulators of 44 mm and 80 mm period, providing full polarization control over the whole 50-1700 eV energy range [START_REF] Briquez | Proc. of the FEL-08 Conf[END_REF]. The two 1.6 m long modules are installed in the ID-14M medium-length section, one at its center (HU44), the other 1.8 m upstream (HU80) This configuration leaves the possibility of installing a third undulator module in the future.

M1-A/B mirrors

The first optical element of the beamline is a pair of horizontally deviating mirrors (M1-A/B) working at 4° deviation [START_REF]winlight-system[END_REF]. Both mirrors are water-cooled using a closed-circuit thermostatic bath, and are installed in an aluminium-alloy UHV vessel with all the motors placed outside the chamber [START_REF] Cinel | [END_REF]. The purpose is to minimize the strong and rapid carbon contamination of first optical elements that has previously been observed on similar SOLEIL beamlines [START_REF] Chauvet | [END_REF]. M1-A is a Pt-coated SiC plane mirror whose primary function is to absorb up to 90% of the thermal load from the undulator. At 15.5 mm undulator gap, linear polarization mode and full filling of the ring, maximum absorbed power is ~400 W, with a central power density of ~0.6 Wmm -2 . M1-A, placed 20 m from the center of the straight section, collects a maximum solid angle of 0.4×0.4 mrad 2 . M1-B is an Ir-coated Si spherical mirror (224 m radius), focusing the beam horizontally at the slit placed after M2 (see Fig. 1). The M1-A/B pair has two additional functions: i) fine adjustment of the deviation angle in order to account for different source positions (HU44 or HU80); ii) satisfy radiation safety requirements by displacing the beamline axis laterally with respect to the source axis.

Monochromator

The fixed-deviation (5°) monochromator [5] comprises five interchangeable VLS gratings (G1-G5) and one spherical mirror (M2, 116 m radius). The central line density of the gratings, in lines/mm, is 120(G1), 225(G2), 450(G3), 900(G4) and 1500(G5). With the exception of G5, gratings are of the variable-groove-depth (VGD) type [START_REF] Polack | 9 th SRI Conf.; AIP Conf. Proceedings[END_REF], allowing for a trade-off between spectral purity and efficiency. Calculated efficiency is >20% up to 700 eV and >15% up to 1500 eV. All gratings are mounted on the same base-plate for rotation (energy selection) and translation (grating and groove-depth selection). The rotation is measured using the graduations of a full-circle encoder with two read-heads. M2 focuses the radiation dispersed by the grating to the vertically occulting exit slit ES, placed 2.843 m downstream. For stability, the monochoromator hutch is thermalized to better than 0.1°C and all gratings and M2 are water-cooled via a closed-circuit thermostatic bath. In addition to the main elements G, M2 and ES, the monochromator ensemble features a 4-jaw assembly for defining the illuminated grating area and a vertical-blade-slit defining the horizontal source size for the optical elements downstream.

M3/M6 switching mirrors

The monochromatic beam is directed alternately along the two branches of the beamline by using two horizontally deviating toroidal mirrors [START_REF]winlight-system[END_REF]. M3 (r=122 mm; R=202 m) feeds the inelastic scattering branch, focusing the beam horizontally at working point A. M6 (r=144 mm; R=157 m) focuses the beam horizontally and vertically at working point B of the elastic scattering branch (see Fig. 1). Both M3/M6 mirrors are mounted on the same hexapod device [7]. When one mirror is selected as active in the control software, the hexapod movements refer automatically to its coordinate system, providing total control of its alignment through pure rotation and translation movements.

M4 refocusing mirror

Following the M3 switching mirror, vertical focusing at working point A is obtained via the upward deviating elliptical-cylindrical mirror M4 (R=26m, 4° deviation) [8]. Since a very small vertical size is required, the mirror produces a large demagnification (9:1) and features very low slope errors for its shape [START_REF] Polack | [END_REF]. The M4 mirror chamber is mounted on a hexapod device for alignment [7].

M7/M8 refocusing mirrors

After the intermediate focal point B, the beam reaches a pair of refocusing mirrors in Kirkpatrick-Baez configuration (M7, 4° horizontal deviation, and M8, 2° upward deviation). Both are plane silica mirrors [10] mounted on mechanical benders [11] driven by in-vacuum stepper motors. The range of bending radii makes it possible to displace the focal position of working point C between 2 m and 4 m from the M8 mirror position, allowing for a trade-off between spot-size and divergence.

Monochromator performance

The beamline design was optimized to have a resolving power R>10000 up to 1000 eV, for a 10-20 µm opening of the ES. In general, optimization refers to the energy range 70-1300 eV. Below 70 eV, some 10-20% of the flux is lost due to increased source divergence. Above 1300 eV, the resolving power diminishes (R≈5000 up to 1700 eV) and the HU44 undulator must be used in third harmonic. Gas-phase absorption spectra confirmed the expected resolution performance. The highest energy that we tested up to now corresponds to the Ne-1s absorption at 867 eV. An example is shown in Fig. 2a, using the G5 grating and 5 µm exit slits. A Voigt fit, assuming a 252 meV Lorentzian broadening w L [START_REF] Kato | 9 th SRI Conf.; AIP Conf. Proceedings[END_REF], gives a Gaussian broadening w G of 31 meV and a resolving power R≈28000 (the large difference between w L and w G implies large uncertainties on the latter). Fig. 2b summarizes the R estimates obtained by N-1s and Ne-1s absorption measurements, using different gratings and diffraction orders. Although G4 (900 l/mm) seems to perform slightly worse than expected, the beamline attains its original goal of R=10000 up to 1000 eV. The behaviour of G5 at 867 eV suggests that R>10000 can be attained up to 1300 eV. Flux has been measured using a calibrated photodiode [START_REF]AXUV100 from International Radiation Detectors[END_REF]. To compare with calculations, data were reduced to take into account variable resolving power as a function of energy. The design goal of 10 13 ph.s -1 /0.1%BW is surpassed between 50 eV and 1000 eV, with a maximum of 10 14 ph.s - 1 /0.1%BW around 100 eV. 10 12 ph.s -1 /0.1%BW are obtained up to 1500 eV.

Working positions 4.1 Inelastic scattering branch, working position A

Working position A hosts permanently the high resolution spectrometer AERHA, whose characteristics and performance will be detailed elsewhere [START_REF] Chiuzbaian | [END_REF]. The vertical spot-size at A is a critical parameter for attaining nominal performance of the vertically-dispersing spectrometer. Using both knife-edge scans and measurements of the intensity through a narrow slit, the best value of the vertical FWHM at A was measured to be 2.9 µm.

Elastic scattering branch, working position B

Working position B is defined by the focal point of the M6 switching mirror. Using knife-edges driven by piezoelectric motors, the spot-size has been determined to be 80 µm×50 µm, h×v, in excellent agreement with design values. Position B can host either the IRMA-2 chamber [START_REF] Sacchi | IRMA-2 at SOLEIL: a set-up for magnetic and coherent scattering of polarized soft x-rays[END_REF], designed for reflectivity and coherent scattering experiments [START_REF] Sacchi | [END_REF], or external equipment brought-in by users.

Elastic scattering branch, working position C

Working position C is defined by the M7/M8 bendable mirrors. It hosts the reflectometer RESOXS [17], equipped for high-field (2 kOe) low-temperature (12K) experiments. External equipment can also be installed, if needed. The spot-size at 2.2 m from M8 has been characterized using a 2D imager (one pixel of the camera corresponds to 0.11 µm). Closing the beamline slits completely (Fig. 3

Conclusion

SEXTANTS is the Soleil beamline dedicated to polarized soft X-ray scattering experiments. It covers the 50 eV-1000 eV range with a resolving power that exceeds 10 4 and a flux at the sample that ranges from 1×10 14 ph./s/0.1%bw at 100 eV to 2×10 13 ph./s/0.1%bw at 1000 eV. The energy range up to 1700 eV can be covered with reduced performance. The SEXTANTS beamline was opened to external users in March 2011.

Figure 1 .

 1 Figure 1. SEXTANTS beamline layout. Main optical elements and the three working positions A, B and C are indicated.

Figure 2 .

 2 Figure 2. (a) Ne-1s absorption using G5 grating and 5 µm slits. (b) Summary of R values obtained by N-1s and Ne-1s absorption measurements.

  , left panel), we estimated the contribution to the spot-size coming from slope errors and aberrations of the three mirrors (M6, M7 and M8) forming the image at C. The measured h×v FWHM was 8.4 µm × 3.4 µm. When using 40 µm × 10 µm slits, we measured a FWHM of 17.3 µm × 7.4 µm, in excellent agreement with calculated values (right panel).

Figure 3 .

 3 Figure 3. Images of the focal spot at working position C (2.2m behind M8). Left panel: slit apertures <1 µm 2 . Right panel: slit apertures 40 µm × 10 µm (h×v); calculated values are given for comparison.


Oussama Meski

Paul François
email: paul.francois@univ-nantes.fr

Florent Laroche

An in virtuo system linking data corpus to 3D virtual model for industry 4.0

Keywords:

Introduction

In the fields of product design, reverse engineering or restitution, digital 3D modeling tools have become in many ways completely unavoidable. In recent years, virtual reality immersion has reinforced the use of 3D by allowing the virtual experimentation of study objects during their design phase [START_REF] Ashley | Exploring the Use of Virtual Environments in an Industrial Site Design Process[END_REF] or as an end in itself for all types of audiences [START_REF] Stuart | Challenging Heritage Visualisation: Beauty, Aura and Democratisation[END_REF]. In order to democratize the use of immersive 3D in areas where the data related to a three-dimensional model have a decisive influence on the very interest of the model, our main objective was to allow access to these metadata in virtuo. Immersive interfaces are used, as previously stated, in many fields where the data processed is highly spatial or geometric. Mechanical design or architecture can therefore easily make use of immersion without having to go through abstract representations of their data. In these fields too, geometric coherence or three-dimensional aspect are the data that take precedence over others: we try to test in virtual reality the coherence of a threedimensional model in itself. The approach we propose is hybrid since it focuses on the visualization of highly spatial data (a 3D model) -for which the immersion visit is naturally adapted -and data related to this virtual model, which can be of various formats and rather resemble a decision tree [START_REF] Buno | Decision Making Support Systems for the Archaeological Domain: a Natual Language Processing Proposal[END_REF]. The data manipulated are on the one hand of the text, image, sound, data, graphics, etc. type and are stored in a database with metadata; and on the other hand of the "link" type between several items in a database or between an item and an object in virtual reality. The main hypothesis we formulate is that the spatial positioning of the information and the link between it and the three-dimensional model makes it possible to judge more effectively the relevance of design choices or hypotheses. In addition, such a visualization would reveal logics related to the spatial distribution of information.

Development for industry 4.0

Virtual Reality allow the visualization and modelization of the business processes and products to be manufactured or maintained. Thus, it provides great operational benefits: on the one hand, virtual reality could facilitate and improve learning, and on the other hand, it could facilitate interactions between industrial production chains and their computerized management systems. Thanks to early modelling and immersion tests, all the production space can be optimally used, while the use of these technologies also makes it possible to predict maintenance or after-sales service needs [START_REF] Vincent | Développement de méthodes et outils basés sur la réalité augmentée et virtuelle pour l'assistance ou l'apprentissage d'opérations dans un contexte industriel[END_REF]. Two research projects are collaborating to integrate virtual reality technologies into the production industry. The development requires a global knowledge base structured in the form of ontology, gathering all the data and knowledge circulating in the industry [START_REF] Oussama | Towards a knowledge based framework for digital chain monitoring within the Industry 4.0 paradigm[END_REF]. Moreover, through the tools of machining monitoring, we capitalize on all the data necessary to detect the appearance of a malicious phenomenon during machining: chattering, tool breakage, etc. The aim of the proposal is to provide support for the various actors in the industry in order to better understand the design process and all its potential defects while remaining in a connected, accessible and easy to handle world: the immersion in virtual reality. Figure 2 details the principle of our scientific proposal to develop a framework that allows a new way of exploiting ontologies using virtual reality technologies. The challenge is being able to recover all the knowledge concerning a mechanical part in a given three-dimensional context (a part, a portion of a part, a defect) and which can be of a very different nature and type: product data, process, etc. The part in question may be a model or physical object identifiable by its serial number. By using virtual reality immersion, a 3D digital copy of the part (either from the source CAD file or from a digitization), a digital twin, can be analyzed and put into its manufacturing context at the same time. The user is then able to easily consult all the knowledge and KPIs (Key Performance Indicators) concerning a specific part. In addition, the user can also hierarchically access all the knowledge of the database related to the part (type of machine, details of a machine, etc.). In the example of a defect, it is therefore possible to access the details of the occurrence of a malicious phenomenon during the machining of the part (a chattering, a tool failure, a collision, etc.) and to know its causes and solutions. Another very important aspect that this system can cover is the possibility of commenting on your immersive experience, and communicating a diagnosis, some remarks or suggestions for improvement or even preventive solutions to deal with the appearance of a given phenomenon. These comments, in audio or text form after automatic transcription, could be reintegrated directly into the ontology and can therefore enrich the overall knowledge base. This immersion experience and its integration into the industrial environment would thus increase competitiveness and improve productivity from the design phase.

Fig. 1 .

 1 Fig. 1. Current situation regarding the different data and proposal.

Figure 2 :

 2 Figure 2: Framework concept to exploit ontologies using virtual reality technologies.

Conclusions and perspective

In this paper, we have presented a virtual reality tool allowing the in virtuo consultation of relational and spatialized data. This tool allows users to work more efficiently in virtual reality immersion by adding interactive levels of information to visible and manipulable objects. We have shown the interest of such a tool for two completely different use cases -heritage and industry 4.0 -although the same system must be adapted to meet the specific expectations of these domains and their existing professional tools. Of course, the question arises of access with this system to non-spatialized data. Indeed, we have insisted in this paper on the need to select a virtual object to display this information, but the design or reflection phase may require access to non-spatializable data (concepts, historical information, etc.). We are therefore convinced of the interest of developing a post-immersion experience in which other interactions would be possible.