Laurent Daniel
email: laurent.daniel@lgep.supelec.fr

Olivier Hubert
email: olivier.hubert@lmt.ens-cachan.fr

An analytical model for the ∆E effect in magnetic materials

Keywords:

The ∆E effect is often presented as the dependency of the Young's modulus of a material on its state of magnetization. Nevertheless, the elastic properties of a magnetic material do not depend on the magnetization state. Actually, the sensitivity of the magnetostriction strain to the application of a stress explains the ∆E effect. According to this statement, a semi-analytical model for the ∆E effect is proposed, in which magnetization rotation is not considered. An experimental procedure to measure the ∆E effect in magnetic materials is then built-up. Experimental and modeling results are finally compared, with satisfying agreement.

Introduction

The Young's modulus E of a material is the ratio between the stress σ and the elastic strain ε el -measured in the direction parallel to the applied stress -in the case of a tension or compression test (equation [START_REF] Cullity | Introduction to magnetic Materials[END_REF]).

E = σ ε el (1)
When a stress is applied to a magnetic material, stressstrain response appears to be non-linear (figure 1). This effect is called the ∆E effect [START_REF] Cullity | Introduction to magnetic Materials[END_REF][START_REF] Du | Magnetostriction -Theory and applications of magnetoelasticity[END_REF]. It is often presented as a dependency of the Young's modulus E to the stress level. On the other hand, the ∆E effect depends on the state of magnetization of the material as illustrated in figure 1: the Young's modulus of a demagnetized speci-men appears to be lower than the Young's modulus of the same specimen magnetized at saturation. Thus, the ∆E effect could be seen as an apparent loss of linearity in the elastic behavior of demagnetized specimens. But it can also be interpreted as a consequence of the effect of stress on the magnetostriction strain (magnetostriction is the spontaneous deformation associated to magnetic domain structure evolution). The ∆E effect can consequently be dissociated from the elastic behavior. Indeed, the application of stress modifies the magnetization state of magnetic materials and generates a magnetostriction strain. This magnetostriction strain ε µ is superimposed to the elastic strain ε el , so that the total measured strain ε is defined by equation ( 2), all the strains being measured in the direction parallel to the applied stress.

ε = ε µ + ε el (2)
The apparent Young's modulus E a is defined by equation [START_REF] Squire | [END_REF].

E a = σ ε µ + ε el (3)
In the case of a saturated material, the magnetic domain structure has reached a saturated configuration and the magnetostriction strain cannot evolve anymore. The apparent Young's modulus E s a is then defined by equation [START_REF] Hubert | Magnetic domains[END_REF], corresponding to the original definition of the Young's modulus given by equation [START_REF] Cullity | Introduction to magnetic Materials[END_REF].

E s a = σ ε el = E (4)
For a given initial magnetic configuration, the ∆E effect can be quantitatively defined as a function of the applied stress σ following equation [START_REF] Daniel | [END_REF]:

∆E E = E -E a (σ) E a (σ) = ε µ (σ) ε el (σ) (5)
The value of the Young's modulus E can be easily identified thanks to an usual tensile test 1 , for a stress level such that the stress-strain curve is linear. In the linearity area of the curve, we define:

E = dσ dε (6)
A predictive model for the ∆E effect should then rely on the description of the effect of stress on the magnetostriction strain. Very few models are available in the literature. Squire treated the case of amorphous ribbons [START_REF] Squire | [END_REF],

but did not address the case of crystalline materials. This latter point is the purpose of that paper. After a brief presentation of the energetic terms involved in the magnetic equilibrium of a ferro-or ferri-magnetic body volume element, a simplified approach for the ∆E effect in cubic single crystals is presented, in which magnetization rota-1 whatever the magnetic state of the specimen.

tion is not considered 2 . It is applied both for materials with positive and negative anisotropy constants. An extension to the behavior of polycrystals is then proposed and results are compared to original experimental ones.

Magneto-elastic equilibrium

The magneto-elastic equilibrium of a ferro-or ferri-magnetic body can be seen as the result of a competition between several energetic contributions [START_REF] Hubert | Magnetic domains[END_REF].

-The exchange energy W ex is related to the ferromagnetic coupling effect between neighboring atoms, tending to favor an uniform magnetization in a volume element.

-The magneto-crystalline energy W K tends to align the magnetization along particular directions, called "easy axes". These easy magnetization directions are mostly connected to crystallographic structure. In the case of iron, whose crystallographic structure is body cubic centered, the anisotropy constant K 1 is positive and magnetization is aligned along <100> axes (six directions 3 ). In the case of Nickel, whose crystallographic structure is face cubic centered, the easy axes are the eight <111> directions (figure 2).

-The magneto-static energy W mag tends to align the magnetization direction with the magnetic field direc-2 Rotation is the mechanism considered in [START_REF] Squire | [END_REF]. In that sense, our proposal is complementary to this previous one and will not apply to amorphous materials. 3 the notation used for the crystallographic directions refers to the Miller indices.

[100]

[011]

[111] tion, or, at least, to energetically favor domains for which the magnetization direction is close to the magnetic field direction.

-The elastic energy W el introduces the magneto-elastic interactions in a ferromagnetic crystal. It is often called "magneto-elastic" energy.

The competition between these energetic contributions explains the existence of the typical magnetic domain microstructure of magnetic materials. Each magnetic domain is uniformly magnetized at saturation. For low magnetic field level, the magnetization of a magnetic domain is aligned along an easy axis.

The magnetization process is the result of two concomitant processes. On one hand, the magnetic walls, separating one domain from another, are moving, modifying the mean magnetization in the material. On the other hand, the magnetization direction can rotate out of its initial easy axis. This rotation mechanism is encountered when the energy given by the applied magnetic field is high enough to compensate the magneto-crystalline anisotropy energy. This situation is usually reached for medium to high magnetic fields. The application of a stress significantly modifies the magnetization of the material, through the contribution of the elastic energy.

Finally, the elastic energy strongly depends on the local magnetostriction strain, through the mechanical incompatibilities. This magneto-elastic coupling is at the origin of the ∆E effect.

3 A simplified approach for the ∆E effect in single crystals

We develop hereafter a simplified approach for the description of the ∆E effect in single crystals. This approach is inspired by a multiscale model for the prediction of magneto-elastic reversible behavior of ferromagnetic materials presented in [START_REF] Daniel | [END_REF]. The restriction to the case of no applied magnetic field allows an analytical derivation of the ∆E effect.

The approach is limited to the case when no magnetic field is applied, so that the magneto-static energy does not appear in the definition of the magnetic equilibrium (W mag = 0).

We suppose that no rotation mechanism occurs in the magnetic domains. The magneto-crystalline anisotropy energy is then uniform within a single crystal and does not participate to the evolution of the magnetostriction strain (W K = constant). The magnetization in a domain is always aligned along an easy crystallographic direction.

We choose a simplified description of the single crystal mi-crostructure. The crystal is seen as an aggregate of magnetic domains. Considering that only easy directions can be encountered for the magnetization in the domains, we divide the single crystal into domain families (num. α), each family being associated to the corresponding easy axis. In the case of <100> easy axes, only six domain families are possible (α = {1, . . . , 6}), eight in the case of <111> easy axes (α = {1, . . . , 8}).

The exchange energy W ex is responsible for the local coupling between magnetic moments. It does not participate anymore in the energetic description of such an aggregate (wall energy is not considered, exchange energy is hidden in the concept of domain family).

In such conditions, the elastic energy will be the only energetic term explicitly considered in the description of the magnetic equilibrium of the single crystal, because this term is not identical from one domain family to another.

The elastic energy W el α of a domain α can be written [START_REF] Daniel | [END_REF]:

W el α = -σ c : ε µ α (7)
where σ c is the mean stress -second order -tensor within the single crystal and ε µ α is the magnetostriction strain -second order -tensor in the domain family α. The latter, assumed to be homogeneous within a domain family, is written, in the crystallographic coordinate system of the cubic crystal (see for instance [START_REF] Bozorth | Ferromagnetism[END_REF]):

ε µ α = 3 2        λ 100 (γ 2 1 -1 3 ) λ 111 γ 1 γ 2 λ 111 γ 1 γ 3 λ 111 γ 1 γ 2 λ 100 (γ 2 2 -1 3 ) λ 111 γ 2 γ 3 λ 111 γ 1 γ 3 λ 111 γ 2 γ 3 λ 100 (γ 2 3 -1 3 )        (8)
(γ 1 , γ 2 , γ 3 ) are the direction cosines of magnetization in the domain family α, λ 100 and λ 111 are the magne-tostrictive constants of the material. If we consider a multiaxial applied stress state σ c , written in the crystallographic coordinate system (equation ( 9)), the elastic energy (equation ( 7)) can be written in the form of equation (10).

σ c =        σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33        (9)
W el α = -3 2 λ 100 σ 11 (γ 2 1 -1 3 ) + σ 22 (γ 2 2 -1 3 ) + σ 33 (γ 2 3 -1 3 ) -3 λ 111 (σ 12 γ 1 γ 2 + σ 13 γ 1 γ 3 + σ 23 γ 2 γ 3 ) (10)
It has to be emphasized that the condition on the magneto-crystalline energy (W K = constant) supposes that no magnetization rotation occurs. In particular, it means that the level of stress is not high enough to generate magnetization rotation in the domains. This condition can be expressed as

|σ c : ε µ α | ≪ |K 1 | in each domain.
The equilibrium configuration can be defined through the relative proportion of each domain family in the crystal. The volumetric fraction of a domain family is obtained using an explicit relation proposed by [START_REF] Buiron | [END_REF]:

f α = exp(-A s .W α ) α exp(-A s .W α ) = exp(-A s .W el α ) α exp(-A s .W el α ) (11)
A s being a material parameter linked to the initial anhysteretic susceptibility χ o and to the saturation magnetization M s [START_REF] Daniel | [END_REF]:

A s = 3χ o µ o M 2 s ( 12
)
For further simplification, we introduce the quantity

S: S = α exp(-A s .W el α ) (13)
An analytical model for the ∆E effect can then be derived from equation (11). Two cases are successively considered: material with <100> easy magnetization axes (positive anisotropy constant) and material with <111> easy magnetization axes (negative anisotropy constant).

3.1 Material with <100> easy magnetization directions

Definition of variables

Six domain families have to be considered: they will be noted abc. The subscripts abc can take the value 100, 100, 010, 010, 001 and 001. The magnetization rotation mechanism being neglected, the magnetostriction strain tensor in each domain family is greatly simplified:

ε µ abc = 1 2 λ 100        3a 2 -1 0 0 0 3b 2 -1 0 0 0 3c 2 -1        (14)
The elastic energy for each domain family is then 4 :

W abc = - 1 2 λ 100 (3a 2 -1)σ 11 + (3b 2 -1)σ 22 + (3c 2 -1)σ 33 (15)
4 It can be noticed that the shear terms of the stress tensor, expressed in the crystal coordinate system, do not appear in the definition of the elastic energy.

The quantity S is given by:

S = 2 exp A s λ 100 (σ 11 -1 2 (σ 22 + σ 33 )) +exp A s λ 100 (σ 22 -1 2 (σ 11 + σ 33 )) +exp A s λ 100 (σ 33 -1 2 (σ 11 + σ 22 )) (16)
We deduce the associated volumetric fractions for each domain family:

                 f 100 = f 100 = 1 S exp A s λ 100 (σ 11 -1 2 (σ 22 + σ 33 )) f 010 = f 010 = 1 S exp A s λ 100 (σ 22 -1 2 (σ 11 + σ 33 )) f 001 = f 001 = 1 S exp A s λ 100 (σ 33 -1 2 (σ 11 + σ 22 )) (17)
We can verify that, in accordance with experimental observation, no magnetization is created in the single crystal by application of a stress:

-→ M c = α f α -→ M α = M s f 100 -f 100 f 010 -f 010 f 001 -f 001 = -→ 0 ( 18
)
But a magnetostriction strain ε µ c is created by application of a stress:

ε µ c = α f α ε µ α = 0 (19)

Uniaxial loadings

The case of a multiaxial applied stress can be first reduced to the simplified case of uniaxial loadings (tensile or compressive stress).

An uniaxial stress of amplitude σ along the [100] direction 5 leads to the strain ε µ 100 measured in the direction

5 σij = 0 except σ11 = σ.
parallel to the applied stress 6 :

ε µ 100 = λ 100 1 -exp -3 2 A s λ 100 σ 1 + 2 exp -3 2 A s λ 100 σ (20)
An uniaxial stress of amplitude σ along the [110] direction 7 leads to the strain ε µ 110 measured in the direction parallel to the applied stress:

ε µ 110 = λ 100 1 -exp -3 4 A s λ 100 σ 2 2 + exp -3 4 A s λ 100 σ (21)
If an uniaxial stress of amplitude σ is applied along the [111] direction 8 , we get:

f 100 = f 100 = f 010 = f 010 = f 001 = f 001 = 1 6 ( 22
)
so that:

ε µ 111 = 0 (23)
These results are plotted in figure 3 in the case of iron for which λ 100 = 21 10 -6 [START_REF] Jiles | Introduction to Magnetism and Magnetic Materials[END_REF]. The value for A s is 5 10 -3 m 3 .J -1 .

-50 0 50 6 if ε µ n is the projection of the tensor ε µ in the direction n, we have:

ε µ n = t n ε µ n. 7 σij = 0 except σ11 = σ22 = σ12 = 1 2 σ. 8 σij = 1 3 σ.
We can notice the dissymmetry between the tension and compression behaviors, visible for example in the values of the strain when -mechanical-saturation is reached:

       lim (σ→+∞) ε µ 100 = λ 100 lim (σ→-∞) ε µ 100 = - 1 2 λ 100 (24)        lim (σ→+∞) ε µ 110 = 1 4 λ 100 lim (σ→-∞) ε µ 110 = - 1 2 λ 100 (25)

Multiaxial loadings

More general and more complicated mechanical loadings can also be considered. We can study the particular cases of equi-bitension and hydrostatic pressure and compare them to uniaxial stress.

For example, under the hypotheses made, the magnetostriction strain in a <100> direction is defined, for any stress state, as:

ε µ 100 = λ 100 (2f 100 -f 010 -f 001 ) = λ 100 S 2 exp A s λ 100 (σ 11 -1 2 (σ 22 + σ 33 )) -exp A s λ 100 (σ 22 -1 2 (σ 11 + σ 33 )) -exp A s λ 100 (σ 33 -1 2 (σ 11 + σ 22 )) (26)
The magnetostriction strain in a <111> direction is defined, for any stress state, as:

ε µ 111 = 0 (27)
Figure 4 shows the response of a single crystal under uniaxial 9 , equibiaxial 10 and hydrostatic 11 loading along <100> directions.

9 σij = 0 except σ11 = σ.

10 σij = 0 except σ11 = σ22 = σ. We observe that a hydrostatic stress state has no effect on the magnetostriction strain.

11 σij = 0 except σ11 = σ22 = σ33 = σ.
3.2 Material with <111> easy magnetization directions

Definition of variables

In that case, eight domain families have to be considered: they will be noted abc. The subscripts abc take the values 111, 111, 111, 111, 111, 111, 111 and 111. The magnetization rotation mechanism being neglected, the magnetostriction strain tensor in each domain family is also greatly simplified:

ε µ abc = 1 2 λ 111        0 ab ac ab 0 bc ac bc 0        (28)
The elastic energy for each domain family is then 12 :

W abc = -λ 111 (ab σ 12 + ac σ 13 + bc σ 23 ) (29)
The quantity S is given by:

S = 2 [ exp (A s λ 111 (σ 12 + σ 13 + σ 23 )) +exp (A s λ 111 (-σ 12 -σ 13 + σ 23 )) +exp (A s λ 111 (-σ 12 + σ 13 -σ 23 )) +exp (A s λ 111 (σ 12 -σ 13 -σ 23 ))] (30)
We deduce the associated volumetric fractions for each domain family :

                           f 111 = f 111 = 1 S exp (A s λ 111 (σ 12 + σ 13 + σ 23 )) f 111 = f 111 = 1 S exp (A s λ 111 (-σ 12 -σ 13 + σ 23 )) f 111 = f 111 = 1 S exp (A s λ 111 (-σ 12 + σ 13 -σ 23 )) f 111 = f 111 = 1 S exp (A s λ 111 (σ 12 -σ 13 -σ 23 )) (31)
Here again, we verify that no magnetization can be created by application of a stress, but a magnetostriction strain ε µ c appears:

ε µ c = α f α ε µ α = 0 (32)

Uniaxial loadings

We study first the uniaxial case. An uniaxial stress of amplitude σ along the [111] direction leads to the strain ε µ 111 measured in the direction parallel to the applied stress:

ε µ 111 = λ 111 1 -exp -4 3 A s λ 111 σ 1 + 3 exp -4 3 A s λ 111 σ (33)
An uniaxial stress of amplitude σ along the [110] direction leads to the strain ε µ 110 measured in the direction parallel to the applied stress:

ε µ 110 = 1 2 λ 111 tanh 1 2 A s λ 111 σ (34)
If an uniaxial stress of amplitude σ is applied along the [100] direction, we get:

f 111 = f 111 = f 111 = f 111 = f 111 = f 111 = f 111 = f 111 = 1 8 ( 35
)
so that:

ε µ 100 = 0 (36)
These results are reported in figure 5 in the case of nickel for which λ 111 = -24 10 -6 [START_REF] Jiles | Introduction to Magnetism and Magnetic Materials[END_REF]. The value for A s is 5 10 -3 m 3 .J -1 (the same than for the iron single crystal). The dissymmetry between the tension and compression behaviors can also be noticed:

       lim (σ→+∞) ε µ 111 = - 1 3 λ 111 lim (σ→-∞) ε µ 111 = λ 111 (37)        lim (σ→+∞) ε µ 110 = - 1 2 λ 111 lim (σ→-∞) ε µ 110 = 1 2 λ 111 (38)

Multiaxial loadings

As previously said, the proposed modeling allows to consider more general and more complicated loadings.

The magnetostriction strain in a <100> direction is defined, for any stress state, as:

ε µ 100 = 0 (39)
The magnetostriction strain in a <111> direction is defined, for any stress state, as:

ε µ 111 = 2 3 λ 111 (3f 111 -f 111 -f 111 -f 111 ) = 2λ 111 3S [3 exp (A s λ 111 (σ 12 + σ 13 + σ 23 )) -exp (A s λ 111 (-σ 12 -σ 13 + σ 23 )) -exp (A s λ 111 (-σ 12 + σ 13 -σ 23 )) -exp (A s λ 111 (σ 12 -σ 13 -σ 23 ))] (40)
We developed a fully analytical model of the effect of -uniaxial and multiaxial -stress on the magnetostriction strain of cubic single crystals. This modeling allows a description of the ∆E effect consistent with the independence of the elastic properties of materials on their magnetization. The same principles can be applied to the prediction of the behavior of polycrystals.

Extension to the behavior of polycrystals

The magnetostrictive behavior of a polycrystal is supposed, as a first approximation, to be isotropic. The contrast of behavior along different directions, exhibited for example on figure 3 for the single crystal should not appear. The isotropic polycrystal can be seen as an aggregate of single crystals with random orientation. It can be defined as a single crystal for which all directions would be easy directions. In one domain of such a single crystal, the magnetostriction strain tensor can be written 13 (in the appropriate coordinate system):

ε µ m = 1 2 λ m        2 0 0 0 -1 0 0 0 -1        (41)
λ m denotes, for the polycrystal, the maximum magnetostriction strain that can be reached during a mechanical loading. The definition of its value requires a discussion.

Definition of λ m

λ m is the value of the maximum magnetostriction strain.

This parameter can be identified from experimental measurements on unstrained specimen, but it can also be defined from the value of the single crystal magnetostriction coefficient λ 100 or λ 111 . It is shown in [START_REF] Daniel | [END_REF] that the maximum magnetostriction strain λ m of a polycrystal, in the case when no magnetization rotation occurs can be written in the form: where k a and k b depend on the elastic properties of the single crystal and on the hypotheses chosen for the 13 in accordance with the usual isochoric hypothesis for the magnetostriction strain [START_REF] Du | Magnetostriction -Theory and applications of magnetoelasticity[END_REF].

λ m = 2

Multiaxial stress state

A general stress tensor is considered, with 6 independent components (see equation ( 9)). We choose to work in the principal coordinate system for the stress: in that particular framework, the stress tensor is diagonal and its components are called the principal stresses:

σ =        σ i 0 0 0 σ ii 0 0 0 σ iii        (43)
The definition of the magnetostriction strain of the polycrystal then follows the same strategy used for single crystals. Since a finite number of easy magnetization directions has been replaced by an infinite number, the symbol sum has to be replaced by an integral over the possible directions α.

The transformation matrix from the domain coordinate system to the principal coordinate system is noted P and defined by equation ( 44) where θ varies from 0 to 2π and ϕ from 0 to π.

P =        cosθ sinϕ sinθ cosθ cosϕ sinθ sinϕ -cosθ sinθ sinϕ cosϕ 0 -sinϕ        (44)
The magnetostriction strain in a domain α(θ, ϕ) can be expressed in the principal coordinate system according to equation (45).

ε µ p = t P ε µ m P (45)
In such conditions:

                                               ε µ p11 = λ m 2 (3 cos 2 θ sin 2 ϕ -1) ε µ p22 = λ m 2 (3 sin 2 θ sin 2 ϕ -1) ε µ p33 = λ m 2 (3 cos 2 ϕ -1) ε µ p12 = ε µ 21 = 3 λ m 2 cosθ sinθ sin 2 ϕ ε µ p13 = ε µ 31 = 3 λ m 2 cosθ cosϕ sinϕ ε µ p23 = ε µ 32 = 3 λ m 2 sinθ cosϕ sinϕ (46)
The elastic energy in a domain α, defined by equation (47), can be developed according to equation (48).

W el α = -σ : ε µ p (47) W el α = -1 2 λ m σ i 3 cos 2 θ sin 2 ϕ -1 +σ ii 3 sin 2 θ sin 2 ϕ -1 +σ iii 3 cos 2 ϕ -1 (48)
Parameter S of equation ( 13) is now defined by equation (49).

S = 2π 0 π 0 exp(-A s .W el α ) sinϕ dϕ dθ (49)
The magnetostrictive response E µ of the polycrystal can be defined in a similar way to the one obtained in a direction <100> of a single crystal with <100> easy magnetization directions (since all directions are easy axes).

E µ = α f α ε µ p dα (50)
with:

f α = 1 S exp(-A s .W el α ) (51)
The magnetostriction strain tensor components, defined in the principal coordinate system, are then written:

E µ ij = 1 S 2π 0 π 0 ε µ pij exp(-A s .W el α ) sinϕ dϕ dθ (52)
In accordance with experimental observation, the predicted magnetostriction strain is isochoric:

E µ 11 + E µ 22 + E µ 33 = 0 (53)
Moreover, we observe that the principal coordinate system for the magnetostriction strain tensor is the principal coordinate system for the stress tensor, so that:

E µ =        E µ 11 0 0 0 E µ 22 0 0 0 E µ 33        =        E µ i 0 0 0 E µ ii 0 0 0 E µ iii        (54)

Uniaxial tension-compression

The case of uniaxial tension-compression 14 of amplitude σ brings significant simplifications. The elastic energy (equation (48)) reduces to:

W el α = - 1 2 λ m σ 3 cos 2 ϕ -1 (
55) 14 For example σi = σii = 0 and σiii = σ.

Parameter S (equation ( 49)) is re-written:

S = 2π e (-1 2 Asλmσ) π 0 exp( 3 2 A s λ m σcos 2 ϕ) sinϕ dϕ (56)
The magnetostriction strain in the direction parallel to the applied stress is then defined by equation ( 57).

E µ iii = πλ m S exp(- 1 2 A s λ m σ) I 1 (57)
with:

I 1 = π 0 (3 cos 2 ϕ -1) exp( 3 2 A s λ m σcos 2 ϕ) sinϕ dϕ (58)
The calculation of the other terms of the tensor allows to verify the following expression for the magnetostriction strain tensor:

E µ = π λ m I 1 2 S exp( 1 2 A s λ m σ)        -1 0 0 0 -1 0 0 0 2        (59)

Experimental characterization of ∆E effect

The measurement of ∆E effect usually consists in the evaluation of the stress-strain response of a demagnetized specimen (Figure 1) thanks to a tensile-compressive machine. The ε µ component of the total deformation is then extracted according to equation ( 2). This procedure is nevertheless very difficult to apply since amplitude of magnetostriction is most of the time much lower than the total deformation ε. Polycrystalline iron is a classical example: the amplitude of longitudinal magnetostriction is about 10 -5 ; considering a Young's modulus of about 200 GPa, a 2 MPa tensile stress produces the same elastic amplitude of deformation than magnetostriction. When stress overcomes 20 MPa, the deformation of magnetostriction only accounts for 10% of the total deformation. This way of measurement is consequently not accurate. Other methods can be used [START_REF] Squire | [END_REF]. An alternative procedure, based on the hypothesis of magnetic saturation of the magnetostriction, is proposed in the next section.

Principle

The procedure is based on anhysteretic magnetostriction measurements under different levels of applied stress ie ε µ (M, σ) with σ constant. Accurate measurements require first to proceed to an efficient demagnetization under stress.

This step leads to an initial deformation ε i (equation( 60))

that is practically not possible to measure.

ε i (0, σ) = ε el (σ) + ε µ (0, σ) (60)
The deformation is then arbitrarily put to zero. Next step is to proceed to the anhysteretic magnetostriction measurement. Measurement is now corresponding to ε ′

given by equation (61).

ε ′ (M, σ) = ε(M, σ) -ε i (0, σ) = ε µ (M, σ) -ε µ (0, σ) (61)
The value of ε ′ (M = 0, σ) is artificially zero whatever the stress level. The extraction of magnetostriction behavior ε µ (M, σ) requires consequently to evaluate ε µ (0, σ).

Figure 6 gives a schematic view of ε ′ (M, σ) for σ = 0 (a) and σ = 0 (b). A very simple 2D scheme of the domain configuration is associated. If we make the hypothesis that the magnetization reaches M s at high magnetic field, the domain configurations and thus the values of magnetostriction are strictly identical whatever the stress level.

M ε' M s M ε' M s M ε' M s ε' (M,0) ε' (M,σ) S (σ) (a) (b) (c)
The ultimate value ε ′ (M s , 0)=ε µ (M s , 0) becomes a reference value that all the ε ′ (M, σ) curves must reach. We finally proceed to a shift S(σ) of the ε ′ (M, σ) curves (Figure 6c). S(σ) is intrinsically corresponding to the magnetostriction at zero applied field ie S(σ)=ε µ (0, σ), that is a direct observation of the ∆E effect. We note S(σ)=ε µ (σ).

Considering several stress levels σ, figure 6c is consequently corresponding to the complete magnetostriction behavior ε µ (M, σ) 15 .

Experimental procedure

The benchmark for magneto-mechanical measurements is based on a non-standard experimental frame [10]. It is constituted of two face-to-face positioned ferrimagnetic U-yokes (figure 7). Samples are placed between the two yokes. Their shape and length depend on the nature of the investigated material. In order to measure magnetostriction, samples have been instrumented with longitudinal and transverse strain gages. A half Wheatstone bridge configuration with temperature compensation has been chosen for strain measurement (low-pass second order Butterworth filtering). A primary winding is placed on the specimen. B-coil and H-coil ensure the measurement of magnetic quantities.

We restrict the experiment to reversible behavior with usual methods (so-called anhysteretic measurement). The anhysteretic curves are measured point by point by applying a sinusoidal magnetic field of mean value H, and of exponentially decreasing amplitude. 15 It is not rigorously "pure" magnetostriction because the parasitic elastic deformation due to the magnetic forces still remains (ie form effect). This deformation is sometimes of same order of magnitude and has the same dynamic (frequency, even function) than magnetostriction. A second correction procedure should be applied especially with sheet specimen. But, because ε µ (σ) is corresponding to a zero magnetization level, the correction is not necessary for this figure. Two solutions are possible to get uniaxial stress: the first solution is to suspend loads to the specimen, which is previously equipped with specific articulated heads (figure 7) 16 . This technique creates a pure uniaxial stress state and avoid vibrations, but compression is not possible; it is used for sheet format specimen (iron-silicon, iron-cobalt).

The second solution is to use a hydraulic machine. This solution leads to noisy deformation measurements but enables compression. It is used for bulk materials (pure iron, Ni-Zn ferrite). The procedure detailed in section 5.1 is finally applied. Figure 8 gives an example of measurement carried out with the experimental set-up [11]. It shows the longitudinal magnetostrictive behavior of pure iron under ten- 16 The maximal load is about 50 kg leading to a maximal stress from 16 MPa to 100 MPa depending on the section of the specimen.

sile and compressive stress 17 . ε µ (σ) is extracted from this measurement and plotted in figure 9 for longitudinal and transverse directions. Experimental results have been carried out with other materials: they are compared to the previsions of the model in the next section. 17 The specimen is a 10mm diameter plain cylinder of iron; form effect is negligible and so not withdrawn to the results. 42 700 [START_REF] Bozorth | Ferromagnetism[END_REF][START_REF] Jiles | Introduction to Magnetism and Magnetic Materials[END_REF] Ni-Zn ferrite -26 10 -6 -5 10 -6 -1 700 [12] FeSi alloy 25 10 -6 -5 10 -6 38 000 [START_REF] Bozorth | Ferromagnetism[END_REF][START_REF] Jiles | Introduction to Magnetism and Magnetic Materials[END_REF] FeCo alloy 100 10 -6 10 10 -6 35 000 [START_REF] Brissonneau | Magnétisme et matériaux magnétiques[END_REF] Table 1. Magnetostriction constants of the materials used for experiments.

As (m 3 /J) λm

Pure iron 5 10 -3 8.4 10 -6

Ni-Zn ferrite 3 10 -2 -3.0 10 -6

FeSi alloy 3 10 -2 10 10 -6

FeCo alloy 5 10 -3 40 10 -6

Table 2. Modeling parameters.

Bulk specimens

The results for pure iron and Ni-Zn bulk specimens are respectively presented in figures 10 and 11. Figure 10 exhibits a very good agreement between modeling and experimental results. Some significant discrepancies are observed for Ni-Zn ferrite. A higher value for λ m would be necessary in the modeling to get a better agreement. Considering the elastic constants of the single crystal [12],

another mechanical hypothesis than uniform stress state would not lead to a significant change of λ m . The relatively low value of K 1 explains these discrepancies: indeed for such a low magnetocrystalline anisotropy, the hypothesis of no magnetization rotation under stress is not verified. The maximum value for the magnetostriction strain cannot be defined as simply as in equation ( 42). This results points out a limitation of the proposed approach. In such a case, where the behavior results from the combinaison of domain wall motion and magnetization rotation mechanisms, the full multiscale [START_REF] Daniel | [END_REF] should be used. The corresponding results (presented in reference [12]) have been added on figure 11.

Sheet specimens

The results for iron-silicon and iron-cobalt sheet specimens 18 are respectively presented in figures 12 and 13.

The agreement between modeling (plain lines) and experimental (points) results is good considering the longi- 18 In both cases, the experimental data have been collected

with a tensile stress applied in the direction TD perpendicular to the rolling direction of the sheet. tudinal direction, but the comparison is not satisfactory in the transverse direction. The explanation of these discrepancies may be found in the initial distribution of the magnetic domains in the material. Indeed the model is designed for bulk materials so that the initial distribution of the domains is assumed to be random into a uni- form distribution (every domain directions have the same probability of existence). As a consequence, the behavior is isotropic and the transverse magnetostriction is equivalent in any direction perpendicular to the longitudinal direction. Considering an isochoric strain, it comes:

E µ Trans = - 1 2 E µ Long ( 62
)
This hypothesis obviously does not apply to sheet specimen. For such shapes, demagnetizing phenomena lead to a non-uniform distribution of domains [START_REF] Daniel | [END_REF]15]. This initial configuration is such that the domains with magnetization along the rolling direction of the sheet are in higher proportion than the others 19 . As a consequence, the behavior 19 This change (compared to the uniform distribution) depends on the material composition, on the crystallographic Plain line for isotropic strain and dot line (transverse direction) for configuration energy effect. Dashed lines: Results obtained with a full multiscale model with configuration energy effect [START_REF] Daniel | [END_REF]15,16].

is anisotropic. The simplifying hypotheses used do not apply. However, the ∆E effect can be predicted using the full -numerical -multiscale model [START_REF] Daniel | [END_REF]. The initial configuration is taken into account in the model using configuration energy [16,15]. This procedure has been applied to ironsilicon and iron-cobalt alloys. Results are plotted in figures 12 and 13 (dashed lines) showing a better agreement with experimental results.

It appears that the magnetostriction strain in the direction perpendicular to the sheet is close to zero (for these materials). The transverse magnetostriction strain satistexture, on the dimensions and on the forming process of the sheet.

fies the relation (63).

Fig. 1 .

 1 Fig.1. Illustration of ∆E effect on a tensile stress-strain curve (ε is the total strain).

Fig. 2 .

 2 Fig. 2. Crystallographic directions in the cubic symmetry (Miller indices).

Fig. 3 .

 3 Fig. 3. ∆E effect in the case of iron single crystal.

Fig. 4 .

 4 Fig. 4. ∆E effect in the case of iron single crystal for a uniaxial, equibiaxial and hydrostatic loading along <100> directions.

Fig. 5 .

 5 Fig. 5. ∆E effect in the case of nickel single crystal.

5 λ 100 k a for materials with <100> easy directions λ m = 3 5 λ

 5 111 k b for materials with <111> easy directions (42)

  description of the material. For instance, if we choose uniform stress (Reuss) hypotheses, we have k a = k b = 1, and if we choose uniform strain (Voigt) hypotheses, we have k a = 5µ a /(2µ a + 3µ b ) and k b = 5µ b /(2µ a + 3µ b ), µ a and µ b being the cubic shear modulus of the single crystal. For the sake of simplicity, we will chose k a = k b = 1 in further numerical applications.

Fig. 6 .

 6 Fig. 6. Schematic view of the measured deformation ε ′ and associated domain structure; (a) zero stress; (b) with σ applied stress; (c) shift of the σ applied stress curve to get the same saturation value.

Fig. 7 .

 7 Fig. 7. Apparatus for measurement of magnetostriction under applied stress -with articulated heads.

Fig. 8 .Fig. 9 .

 89 Fig.8. Influence of uniaxial stress on the anhysteretic longitudinal magnetostrictive behavior of pure iron[11].

Fig. 10 .

 10 Fig. 10. ∆E effect for pure iron: longitudinal and transverse magnetostriction strain as a function of the applied stress σ, modeling (line) and experimental results.

Fig. 11 .

 11 Fig. 11. ∆E effect for Ni-Zn ferrite: longitudinal and transverse magnetostriction strain as a function of the applied stress σ, modeling (line) and experimental results. Dashed lines: Results obtained with a full multiscale model [12].

Fig. 12 .

 12 Fig. 12. ∆E effect for an iron-silicon steel: longitudinal and transverse magnetostriction strain as a function of the applied stress σ, modeling (line) and experimental (points) results.Plain line for isotropic strain and dot line (transverse direction) for configuration energy effect. Dashed lines: Results obtained with a full multiscale model with configuration energy effect[START_REF] Daniel | [END_REF]15,16].

Fig. 13 .

 13 Fig.[START_REF] Brissonneau | Magnétisme et matériaux magnétiques[END_REF]. ∆E effect for an iron-cobalt alloy: longitudinal and transverse magnetostriction strain as a function of the applied stress σ, modeling (line) and experimental (points) results.

  as an hypothesis for the analytical model, the agreement between experimental and model data (dot line in figures 13 and 12) becomes better.7 ConclusionA model for the ∆E effect in magnetic materials has been proposed. This model is based on the description of the physical mechanisms responsible for magneto-elastic couplings at the single crystal scale. The proposed analytical approach does not include magnetization rotation as a source of magnetostriction strain. It is limited to material exhibiting high magneto-crystalline constants. A specific procedure for the experimental characterization of the ∆E effect has been proposed. Modeling and experimental results have been compared for bulk and sheet polycrystalline specimen. The results on Ni-Zn ferrite have allowed to illustrate the limitations and conditions of use of the model. In the case of sheet samples, an initial domain configuration has to be accounted for. This model provides a simple tool to describe the effect of stress on the magnetostriction strain. It could be used in electrical engineering to improve the macroscopic models for magnetoelastic coupling, that often neglect the effect of stress on magnetostriction.

It can be noticed that the diagonal terms of the stress tensor, expressed in the crystal coordinate system, do not appear in the definition of the elastic energy.


Anastasios Petropoulos

Dimitrios Goustouridis

Thanassis Speliotes

Grigoris Kaltsas
email: g.kaltsas@ee.teiath.gr

DEMONSTRATION OF A NEW TECHNOLOGY WHICH ALLOWS DIRECT SENSOR INTEGRATION ON FLEXIBLE SUBSTRATES

In this work we present a fabrication method for developing thermal sensors on flexible organic substrates. The constructed devices consist of Pt thermistors which are directly integrated to the copper tracks of a flexible copper-clad laminate. They reside on top of a 12μm thick SU-8 planarization layer, while a sacrificial layer utilized by the negative photoresist ma-N was used in order to define the thermistor pattern. The thermistors can act as both heating and temperature sensing elements, while due to small thickness and the low thermal conductivity of the Kapton substrate a very effective thermal isolation is achieved. The minimum radius of curvature of the fabricated devices was found to be 5mm. As the device is in direct communication to the macrowolrd, the need for wire bonding is eliminated, while the final surface of the produced sensor is relatively planar. The overall process is simple and cost-effective with minimal requirements in fabrication time. The potential application field of the presented devices is considered quite extensive as they can be directly expanded into flexible sensors able to measure quantities such as fluid flow rate, displacement or vacuum.

Introduction

The commercial availability of metalized flexible organic films introduced an increasing effort in using these substrates for the fabrication of various flexible devices [1]. Typically, the desired patterns are formed by chemical etching of the metal on selected areas. However, the resulting surface topology induces major restrictions on further process flow steps, thus limiting the fabrication of additional device structures. The technology proposed here provides the ability for microstructure formation on top of macro-structured flexible substrates. Thus a micro-device can be integrated using standard microsystem techniques on top of a flexible platform that implements the corresponding control and read out electronics.

Fabrication method 2.1 Overview

Recently, a novel fabrication method which concerns the formation of thermal sensors on printed circuit boards was presented [2]. The method which is described in this paper extends the specific technology to the formation of micro-devices in flexible substrates. The main problem regarding the integration of microstructures using lithography on commercial copper clad composites is the remaining topography on the substrate surface after the definition of the copper tracks, which is of the order of 10μm. The proposed method involves the formation of an SU-8 planarization layer on the substrate surface which aims to provide a smooth path for a subsequently deposited thin second metal film. The micro-device pattern is then defined by performing lithography on a sacrificial layer composed by the negative tone phototresist ma-N. The final device is produced after the second metal film deposition and the subsequent lift-off process. A layout of the process flow Regarding the involved lithography steps, high quality laser-printed transparencies were used in order to define the desired structures instead of the standard quartz photomasks. The minimum reproducible feature size obtained was 100μm. The use of transparencies allows a significant reduction of the involved cost, while the required time to implement a new design is minimal. These characteristics are highly beneficial since the proposed fabrication technology aims to provide a platform for the production of varying types of thermal sensors.

Copper pattern formation

The initial device substrate is a Pyralux® copper-clad laminated composite, which is constructed of DuPont Kapton® polyimide film with a 12.5μm thick copper foil on one side. In order to perform the necessary lithography steps, the flexible substrate is mounted onto a carrier wafer. The positive photoresist AZ was used for the pattern definition of the copper structures, while subsequent chemical etching removed the copper from the unprotected areas.

The Planarization layer

Direct metal deposition in order to establish electrical connections between distinct copper tracks is not applicable, due to the size and nature of the sidewalls of the copper structures. Furthermore, existing residuals of the adhesive sheet which bonds the copper foil to the polymide substrate introduce further anomalies on the surface as shown in Fig 2a. In order to provide a smooth path for the subsequently deposited metal lines, a planarization layer utilized by the thick negative photoresist SU-8 is spin coated on the patterned substrate. The planarization layer is extended so as to cover the edges of the copper tracks. The contact of the subsequently deposited thin Pt film to the copper tracks is performed through vias that are formed in the SU-8 layer on top of the copper tracks. The selection of the specific photoresist as the planarization layer material was based on the fact that it exhibits excellent adhesion to both the copper and the polymide surface, while it presents high resistivity to etchants. The exposure dose and the corresponding layer thickness are also significant parameters regarding the formation of a functional planarization layer. The proper tuning of these parameters requires the consideration of factors such as the difference in the reflectivity between the copper and the polymide substrate as well as the SU-8 layer non-uniformity in terms of thickness. The exposure dose window is quite narrow, as a potential overexposure of the photoresist leads to poor definition of the vias. Furthermore, underexposed SU-8 cannot survive the subsequent lift-off stage due to the uncompleted cross-linking process. The layer thickness used in the formation of the devices presented here was 12μm, with a corresponding exposure energy of 240 mJ/cm 2 .

Sacrificial layer and lift off

The perquisite regarding the photoresist used as sacrificial layer is to span a thickness range which is compatible to the rest of the process. Photoresists used for lift-off purposes are typically of the order of 1μm thick. Since the structures on the device surface impose thickness variations exceeding 10μm, the use of such photoresists does not provide a sacrificial layer of adequate homogeneity. The photoresist used in this case, was the negative tone ma-N 400 which exhibits a thickness range of 5-9μm, which assures that the spin coating over the existing structures is of adequate quality (Fig. 3). Furthermore, the fact that the photoresist is of negative tone allows the formation of negative sidewalls, so that the thermistor structures can be accurately defined. The factors affecting the wall gradient are the exposure dose and primarily the development time. For a specified exposure dose, a larger developing time leads to the removal of the less exposed part of the photoresist near the bottom of the structures, leading to the formation of an undercut. This can be seen in the SEM images of fig 4, where two structures of identical thickness are shown. The structure of fig 4b however has remained in the developer solution longer; this way the desired undercut is achieved. Actually a rather accurate estimation regarding the size of the undercut can be maintained during the developing process. In the bright field optical microscopy image shown in Fig. 3, the undercut is depicted as the thin pale line at the border of the ma-N structure. The line width is a function of the developing time. The latter is a significant point regarding the yield provided in the subsequent lift-off stage, as it allows the correction of effects that might be induced on the structures by certain critical process parameters affecting the thickness uniformity of the sacrificial layer. Such parameters are the exposure dose which may not be spatially uniform or the surface of the flexible substrate which may not be totally flat throughout the spinning process.

Second metal deposition

Following the formation of the sacrificial layer, a bilayer of Ti/Pt of 30/300nm thickness is sputtered on the surface. Finally the lift-off stage is taking place, where the sacrificial layer and the metal residing on it are being removed by inserting the structure into acetone. There are two factors associated to the success of the lift off stage. The first one concerns the SU-8 layer, which has to be fully crosslinked in order to avoid the compromise of its adhesion to the substrate, while the second one is the need for negative sidewalls on the patterned ma-N structures.

The final device 3.1. Device Characteristics

In order to demonstrate the validity of the proposed technology an array of temperature sensors were fabricated and evaluated on top of a flexible organic substrate. A photograph of the fabricated device is shown in Fig 5 . The sensors' lateral dimensions are 1.5x0.1mm 2 . The fabricated thermistors can act both as heating or temperature sensing elements with a corresponding TCR of 0.0025/ o C. As it can be seen from fig. 5, the device can bend with regard to two axes of curvature. The sensing elements of the device are directly integrated to the circuit electronics; therefore there is no need for wire bonding which makes the device suitable for use in harsh environments. The final surface is relatively planar which allows for further lithography steps to be performed, but it can also provide minimally invasive measurements when the device is used as a flow sensor. The overall fabrication procedure is very cost-effective, with reduced equipment requirements as it can be carried out outside the clean room. Finally, the simplicity of the process implies that the device fabrication time is kept to a very low level.

The implementation of an alternative version of the presented technology [2] on rigid organic substrates has already produced a family of PCB-based sensors with extremely satisfying operational characteristics. All sensors employ the monitoring of heat transfer to the thermistors in order to measure quantities such as the gas flow rate, the liquid flow rate, vacuum, position and microchannel flow [3,4]. Flexible thermal sensors measuring the aforementioned quantities can be directly fabricated, with the potential of an expanded application field, for instance the measuring the gas and liquid flow on curved surfaces.

Mechanical operating limits

The mechanical strength of the devices is mainly defined by the SU-8 layer. The mechanical properties of SU-8 are dependent on process parameters such as the baking temperature, baking time and the UV dosage [5]. Typical values are a Young's modulus of 4.02 GPa [6] and a tensile strength of 40-60MPa [5]. Since the corresponding tensile strength for copper is 220MPa and 234Mpa for Kapton and alongside the fact that the thickness of the SU-8 layer is 12μm which can be considered a relatively large value, the planarization layer constitutes the less flexible part of the device. The minimum radius of curvature was experimentally established as 5mm. There is ongoing effort regarding the potential to minimize the layer thickness and subsequently the minimum radius of curvature. It has to be noted that, although the SU-8 layer thickness is 12μm, on top of the copper structures it reduces down to 5μm which makes it more flexible and therefore prevents potential detachment.

Thermal properties

Thermal sensors in general aim to measure a large number of quantities. Typically, they contain a heating element which gives rise to an increased temperature field and sensing elements that monitor the change in the temperature field as induced by the quantity that is measured. Heat conduction from the heating element to the substrate has a negative impact on the device resolution, sensitivity and measurement range. MEMS thermal sensors are typically fabricated on Si substrate, which exhibits a high thermal conductivity value. In order to address this issue, a number of isolation schemes have been presented in the literature such as the formation of free standing structures, vacuum cavities, the use of porous Si [7] etc. However these are quite often demanding process steps which might also introduce limitation to further lithography steps.

In this case however the materials that comprise the fabricated devices exhibit very low thermal conductivity values, which are 0.2 W/m•K for SU-8 and 0.12 W/m•K for Kapton®. Furthermore the overall thickness of the device is less than 100μm signifying a very effective thermal isolation of the heating element. In order to quantify the specific effect, the heating element temperature increase caused by the application of a specific amount of power was calculated and compared to corresponding temperature increase induced to a similar Pt thermistor fabricated on a rigid PCB. The latter is composed by FR4 which exhibits a thermal conductivity value of 0.2 W/m•K, but is 2mm thick. It was found that the temperature increase of the thermistor fabricated on the flexible substrate was 77% higher than that of a corresponding thermistor fabricated on a regular PCB, referring to the same power values (Fig 6). This fact implies that the effective thermal isolation of the fabricated thermistors is very high, with potential gain in the operational characteristics of the devices.

Conclusions

A new technology which allows the fabrication of micro-devices directly on flexible substrates was demonstrated. The proposed technique combines microsystem technology and standard PCB methods. The demonstration of the proposed technology was made through the fabrication of specific devices which are directly connected to macroscale copper structures of commercial flexible copper-clad laminated composites. The constructed Pt thermistors can act as heating or temperature sensing elements, fully integrated to the substrate electronics. A very high degree of thermal isolation of the thermistors is achieved, which makes them suitable for a variety of thermal sensing applications.

Fig 1 .

 1 Fig 1. The fabrication sequence: a) The copper clad laminated Kapton composite b) Chemical etching of copper c) Formation of the SU-8 planarization layer d) Formation of the ma-N sacrificial layer e) Pt sputtering and lift-off. The final form of the device

  Fig 2b shows the outcome of direct Pt deposition on the pre-defined flexible substrate, whereby the electrical continuity of the Pt film is destroyed at the edge of the copper track.

Fig 3 .

 3 Fig 3. An optical microscope image of the fabricated structures prior to the Pt deposition

Fig 2 .

 2 Fig 2. a) SEM image of the edge of the copper structure. b) Direct deposition of Pt on the pre-patterned organic substrate. The discontinuity of the Pt film at the edge of the copper track is clearly indicated.

Fig 5 .

 5 Fig 5. A photograph of the final device.

Fig 4 .

 4 Fig 4. SEM images of ma-N structures of 7.5μm thickness. The effect of development time in the formation of the undercut is shown. a) Development time 140 sec, b) Development time 170sec

Fig 6 .

 6 Fig 6. The temperature increase as a function of power regarding a thermistor fabricated on a flexible substrate and a similar one of the same dimensions fabricated on a rigid PCB

Acknowledgements

This work is co-funded by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%) under the framework of the research program PENED.


Chantal Khan Malek
email: chantal.khan-malek@femto-st.fr

Laurent Robert

Roland Salut

Femtosecond laser machining and lamination for large-area flexible organic microfluidic chips

Keywords: Reel-to-reel processing, roll-to-roll processing, microfluidics, femtosecond laser ablation, polymer, lamination, SU8, epoxy, heterogeneous material structuration, heterogeneous material integration, hybrid assembly

A hybrid process compatible with reel-to-reel manufacturing is developed for ultra low-cost large-scale manufacture of disposable microfluidic chips. It combines ultra-short laser microstructuring and lamination technology. Microchannels in polyester foils were formed using focused, high-intensity femtosecond (fs) laser pulses. Lamination using a commercial SU8-epoxy resist layer was used to seal the microchannel layer and cover foil. This hybrid process also enables heterogeneous material structuration and integration.

Introduction

The use of microfabrication techniques allows the creation of miniaturized analytical devices that are inherently smaller, enabling a reduction of sample and reagent volume, giving lower cost of operation for analysis mainly through decreased consumption of reagents, and allowing parallelisation of analysis (screening). Moreover, it may lead to high volume collective manufacturing.

Polymer-based microfabrication technologies are gaining momentum as they enable low cost fabrication of a variety of microsystems, in particular microfluidic-based systems. A wide variety of manufacturing approaches have been developed, in particular methods based on the replication of a master or mould structure (the geometrical inverse of the desired structure) that enable cost-effective high-volume manufacture [START_REF] Becker | Polymer microfabrication technologies for microfluidic systems[END_REF]. Polymeric materials like cyclic olefin copolymer (COC), polycarbonate (PC), polydimethylsiloxane (PDMS), polymethylmethacrylate (PMMA), polyethyleneterephtalate (PET), polyimide (PI), and epoxy (SU8) have been used to fabricate bioanalytical devices. However the demand for low-priced, single-use disposable analytic tests has steadily increased and there is a need for further reduction of cost of microfluidic devices in a number of high-volume, price-sensitive markets like point-ofapplication sensing in the health care industry (point-of-care diagnostic) as well as for food and water pathogen testing. The most promising approach nowadays for meeting this cost requirement is to switch from batch processing to continuous in-line large area reel-to-reel (or roll-to-roll) manufacturing as used in the press or packaging industry which corresponds to a new disruptive manufacturing paradigm allowing for high throughput cost-efficient manufacture of microfabricated products. This approach is now being applied in emerging fields such as organic electronics, lighting and displays, power sources (e.g. photovoltaics), RFIDs, whereas some other applications such as sensors are just starting.

A reel-to-reel manufacturing approach also requires specific developments in techniques compatible with large area, thin, mechanically flexible substrates [START_REF] Velten | Microfluidics on foil: state of the art and new developments[END_REF]. Here we employed two techniques, laser ablation using femtosecond laser and lamination that have the potential to meet these criteria.

Laser-based micromachining has many advantages as it is a precise, fast, flexible and contactless technique that can be used to accurately structure a variety of materials in a substractive mode by ablation without the need for the multistep processes of mask preparation, resist coating, exposure, developing involved in photolithography. A laser tool can also be easily integrated into reel-to-reel manufacturing processes. Lasers provide a flexible technology platform for creating microfluidic pathways as well as back-end processing of devices [START_REF] Khan Malek | Laser processing for bio-microfluidics applications (Part I)[END_REF][START_REF] Khan Malek | Laser processing for bio-microfluidics applications (Part II)[END_REF]. Laser ablation of polymers is commonly accomplished using UV light [START_REF] Roberts | UV laser machined polymer substrates for the development of microdiagnostic systems[END_REF], [START_REF] Waddell | UV Laser Micromachining of Polymers for Microfluidic Applications[END_REF]. Laser micromachining at longer wavelengths including CO2 lasers in the infrared (10µm) has been reported [START_REF] Klank | CO 2 laser micromachining and back-end processing for rapid production of PMMA-based microfluidic systems[END_REF], [START_REF] Cheng | Direct-write laser micromachining and universal surface modification of PMMA for device development[END_REF]. Ultra-short laser pulses, and in particular the use of femtosecond lasers, have opened up new opportunities in micromachining all types of materials. Transparent materials can also be processed with ultra-short laser pulses (multiphoton absorption induced by high peak power). This has been exploited for microfluidic applications, including in polymer bulk substrates [START_REF] Gomez | Femtosecond laser ablation for microfluidics[END_REF]. We use here a femtosecond laser to machine thin, flexible polymer foils which can be used in reel-to-reel processing scheme.

Lamination has a quite long tradition in electronics in applying dry photoresist films for fabrication of printed circuit board (PCB) technology. However, use of laminated dry film resist in microsystem engineering applications is less common, despite the multiple advantages that it can offer. Structuring several layers of polymer films, then stacking, aligning and bonding them together using a lamination process to form a sealed unit has been used by several groups to manufacture microfluidic devices and miniaturized laboratories [START_REF] Weigl | Design and rapid prototyping of thin-film laminate-based microfluidic devices[END_REF], [START_REF] Yang | An integrated, stacked microlaboratory for biological agent detection with DNA and immunoassays[END_REF], [START_REF] Gast | Fabrication of multilayered microfluidic 3D polymer packages[END_REF]. Other notable example includes: lamination using Riston ® on top of SU8 or Riston ® layer patterned by photolithography [START_REF] Heuschkel | Buried microchannels in photopolymer for delivering of solutions to neurons in a network[END_REF]; various polymers micromachined by UV excimer laser and lamination using a two-layer polyethyleneterphtalate (PET) / polyethylene (PE) lamination tape for the manufacture of microanalytical devices [START_REF] Roberts | UV laser machined polymer substrates for the development of microdiagnostic systems[END_REF] or an adhesive-backed PET or polyimide for microdialysis unit [START_REF] Martin | Laser-micromachined and laminated microfluidic components for miniaturized thermal, chemical and biological systems[END_REF], or using Ordyl SY300 [START_REF] Vulto | Microfluidic channel fabrication in dry film resist for production and prototyping of hybrid chips[END_REF] or Etertec HQ-6100 [START_REF] Stephan | Fast prototyping using a dry film photoresist: microfabrication of soft-lithography masters for microfluidic structures[END_REF] dry resist and photolithography for prototyping fluidic circuits. Lamination with a transfer adhesive tape was also used as an adhesive bonding assembly method to manufacture polymeric piezoelectric micropumps [START_REF] Truongand | A polymeric piezoelectric micropump based on lamination technology[END_REF]. 3-D SU8 microfluidic networks with integrated electrodes were produced using lamination of SU8 dry resist films obtained via spin-coating of SU8 resist onto PET sheets and photolithography [START_REF] Abgrall | A novel fabrication method of flexible and monolithic 3D microfluidic structures using lamination of SU-8 films[END_REF]. Finally, commercial SU8 dry resist film were used first for packaging applications, then to construct microfluidic multilayer chips with embedded channels using a combination of lamination of SU8 commercial foils and photolithography [START_REF] Malek | Flexible microfluidics based on commercial SU8 foils[END_REF].

SU8 is a well-established epoxy-based material used within microsystem processing. Its unique properties as a thick photosensitive negative resin with its very high optical transparency for light above 360 nm makes it ideally suited for imaging structures of high aspect ratio with near-vertical sidewalls in very thick layers [START_REF] Lorentz | High-aspect-ratio, ultrathick, negative-tone near-UV photoresist and its applications for mems[END_REF]. Its optical transparency, biocompatibility and possibility to be fonctionalised with a number of biological protocols [START_REF] Marie | Immobilisation of DNA to polymerised SU-8 photoresist[END_REF] make SU8 a structural material that is often used for the construction of microfluidic devices using photolithography (SU8) and was used in the construction of a variety of sensing and lab-on-a-chip applications [START_REF] Heuschkel | Buried microchannels in photopolymer for delivering of solutions to neurons in a network[END_REF], [START_REF] Jackman | Microfluidic systems with on-line UV detection fabricated in photodefinable epoxy[END_REF], [START_REF] Zhang | Polymerization optimization of SU8 resist and its application in microfluidic systems and MEMS[END_REF], [START_REF] Svasek | Fabrication of miniaturized fluidic devices using SU-8 based lithography and low temperature wafer bonding[END_REF], [START_REF] Gadre | Fabrication of a fluid encapsulated dermal patch using multilayered SU-8[END_REF], [START_REF] Blanco | Novel three-dimensional embedded SU-8 microchannels fabricated using a low temperature full wafer adhesive bonding[END_REF], [START_REF] Carlier | Integrated microfluidics based on multi-layered SU-8 for mass spectroscopy analysis[END_REF], [START_REF] Ribeiro | A SU-8 fluidic microsystem for biological fluids analysis[END_REF], [START_REF] El-Ali | Simulation and experimental validation of a SU-8 based PCR thermocycler chip with integrated heaters and temperature sensor[END_REF]. SU8 epoxy layer is also used as a low temperature adhesive layer between structural layers of SU8 or to bond different materials [START_REF] Jackman | Microfluidic systems with on-line UV detection fabricated in photodefinable epoxy[END_REF], [START_REF] Blanco | Novel three-dimensional embedded SU-8 microchannels fabricated using a low temperature full wafer adhesive bonding[END_REF], [START_REF] Nijdam | Fluidic encapsulation in SU-8 μ-reservoirs with μ-fluidic through-chip channels[END_REF] as well as to integrate monolithically different functionalities [START_REF] Ruano-López | A new SU-8 process to integrate buried waveguides and sealed microchannels for a Lab-on-a-Chip[END_REF].

We report here on the manufacture of a low-cost disposable flexible polymer microfluidic passive device produced using a hybrid process combining two reel-to-reel compatible techniques, femtosecond laser ablation and lamination of three-layer polyester / SU8 epoxy / polyethylene commercially available foils.

Manufacturing process and discussion of results

The material used was a commercial SU8 dry epoxy photoresist film from Microchem (XP MicroformTM -DF series). The foils exist in 9, 14 and 20 µm nominal thickness. We worked with the 20 µm thick foils. The epoxy foil is sandwiched between two protective layers, a bottom one of 60 µm thick mylar (polyester) and a top one of 30 µm thick low density polyethylene. The three foils are optically transparent.

Ablation using femtosecond (fs) laser

Laser micromachining experiments were performed using a solid-state Ti:sapphire laser (Spectra Physics) which delivers a Gaussian beam at the wavelength λ = 800 nm. The pulse duration was 100 fs and its energy and maximum output power were respectively 0.44 mJ and 4.4 GW per impulsion. The pulse repetition rate was held constant at 5 kHz. Direct-write micromachining was carried out in a micromachining station (Astree 250 from Novalase; www.novalase.com). The power of the laser beam is controlled by a motorized attenuator, which consists of a half-wave plate. The laser was focused to a 10 µm spot size using an objective lens of 50 mm focal length. After beam shaping, the repetition rate was 5 kHz, the pulse duration 120fs, the average power 50mW, the energy per impulsion 0.01mJ, and the power per impulsion at the focal point 83 MW. Writing of the patterns can be performed by scanning of the laser beam on the sample placed in the focal plane and/or translation of the sample mounted on a computer-controlled 3D stage through the focal region of the laser beam. All of these parameters are controlled by the software. Note that the micromachining was performed at atmospheric pressure with a local aspiration.

In this experiment, the laser was scanned to laser-cut the fluidic circuit pattern. Only the contour of the circuit pattern was cut through the thickness of the material while the center portion of the circuit was mechanically removed with a tweezer and discarded. Multiple pass writing was used (50 in the case of the 20 µm thick SU8 film with its two protective foils) with a typical writing speed per pass of 20 mm/s. Channel widths down to 50 µm can be fabricated using this method. For smaller structures, the whole area in the pattern needs to be scanned and 10-20 µm wide structures can be written with a beam diameter of 10µm. Fig. 1 shows a polyester foil for which the pattern of a reservoir and channel were cut all the way through. We can see that the microchannels and reservoir were patterned with straight sidewalls. We can note that the thinnest the material sheet to be machined, the steepest the sidewalls. We can also observe that the ultrashort laser pulse does not lead to any deformation around the cut, indicating the absence of heat affected zone. The face on which the laser beam impacted presents some debris at the surface (as seen in Fig 1 -c) on the edges of the pattern, which can be easily removed but the other side of the foil leaves a very clean and smooth surface.

The great advantage of an ultra-short pulse laser is that it provides a structuring tool enabling highly precise and clean structuring using a simple, entirely dry process that is compatible with reel-to reel manufacturing.

a) b) c) Fig. 1: Polyester foil cut using femtosecond laser; a) photograph of the series of channels with reservoir in the flexible foil; b ) SEM close-up view of the 50 µm wide microchannel and 1 mm diameter reservoir patterns cut through the whole depth of the foil.

Most of the work to date on laser micromachining of polymers has been performed using ultraviolet lasers, mainly excimer lasers, which, although well established, has many limitations. Photo-induced and/or thermal decomposition lead to an ejection of fragments which are not completely removed and some material (debris) is redeposited on the surface. Ultra-short laser pulses, and in particular the use of femtosecond lasers, open up new opportunities in microfabrication. A main advantage of ultra-short pulse machining is minimum collateral effects, in particular a reduced heat-affected zone attributed to the absence of diffusion of heat during machining, as the shortness of the laser pulse precludes direct coupling of the laser energy into the thermal modes of the material during irradiation. Direct benefits of using femtosecond laser tool for manufacturing microfluidic devices include: a smooth surface of the microchannel walls, allowing for good fluid flow even in narrow channels; no edge effects on the sides of the channels enabling good sealing between microfluidic circuit and cover. Another advantage of using femtosecond laser is that in contrast to UV-based process, the femtosecond laser process does not require photo-absorbing material. The absorption edge for most polymers is in or near the UV. Consequently linear absorption between the near-infrared wavelength most commonly associated with femtosecond sources and transparent polymer is negligible. However the very high intensities associated with focused femtosecond pulses enable non-linear absorption through multiphoton processes, and efficient ablation of transparent materials. Other materials can also be patterned directly using femtosecond laser, for example thin metal sheets as electrodes, insulation layers and encapsulation layers. Complementary and post-processing concerning fluidic and electrical interconnection, access holes and vias can easily be formed; material modification (e.g. index refraction changes) or surface modification can be integrated in the process-flow. Also structures of sub-micrometric size can be produced using femtosecond laser thanks to an ablation rate well-controlled if we use a shorter focal length objective.

Laser-based techniques can be integrated easily in a chain of hybrid technologies for reel-to-reel manufacturing, therefore enabling precise patterning on large areas. More generally, laser-based processes meet the demands of great flexibility and rapid turn-around time in developing products. The direct-write fabrication does not require the generation of masks, like photolithographic techniques, or moulds, like the replication techniques, which adds an additional step to the chip fabrication process and reduces flexibility as minor modifications to the design then require the fabrication of a completely new mask or mould. They directly use digital data generated during the design of the chip, thereby allowing a design to be changed easily.

Lamination

A lamination press (Rohm and Haas Manual Laminator 350) was used. The first foil of polyester with epoxy adhesive layer foil facing up, but without the polyethylene protective coating, was transferred to a 400 µm thick Si handle wafer. The second foil of polyester structured by laser ablation was stacked on it so that the epoxy film was sandwiched between both polyethelene foils. The assembly was fed into the press and laminated at 65°C and 1 bar applied on the silicon handle wafer reinforced with a 1.5 mm steel shim at its back. The feed rate was 0.4 m/min. The 20 µm thick SU8 photoresist acts as an adhesive layer that flows and ensures an easy joining of both polyester foils. The final device is composed of two polyester foils bonded together, one with the patterned microfluidic circuit and one cover foil (Fig. 2). The bonded composite foils are flat and do not show visible distortion at the macroscale, indicating low bow / warp and stress level in the final assembly. They also kept their optical transparency. However we can note that the adherence of the two polyester foils bonded using SU8 is not very strong as we used the protection layer of the SU8 commercial dry films which may have been treated for not sticking to the epoxy film, therefore modifying its surface properties. Other readily available polymer sheets will also be investigated as structural materials in future work. The lamination process presents several advantages: simple and very fast fabrication process not necessarily requiring a cleanroom environment when working with large enough microstructures and lending itself to upscaling; compatibility with reel-to-reel processes; low temperature and low pressure processing, thereby inducing low thermal stress, and fully compatible with substrates requiring low temperature budget; use of dry adhesive with no solvent or wet processes required; sufficient softness of the bonding layer for good and conformal contact as well as a range of thicknesses to cover surface non-uniformities or topographies; possibility to seal smaller microchannels without a risk of clogging using a thinner adhesive layer; low-cost and easy generation of 3-D structures via stacking; ability to bond different materials and incorporate hybrid functional elements into the design such as electrodes, membranes, sensors, actuators; compatibility with reel-to-reel processes. Challenges of this technique include adherence between layers and dimensional tolerances.

We would like to emphasize here that the lamination process offers a great degree of versatility in term of combining dissimilar materials (heterogeneous material integration) and employing different techniques for structuring the various layers. Using stacked lamina architecture is a fabrication strategy to build complexity and impart functionality in a device via each individual layer through seamless hybrid assembly. Moreover, reel-to-reel lamination allows a serial bonding approach of each layer without sacrificing manufacturing efficiency and throughput.

Conclusion

We demonstrated the manufacture of a simple flexible transparent passive closed microfluidic circuit in thin polyester film produced by direct-write micromachining with a femtosecond laser and lamination using a commercially available SU8-epoxy dry thin film. Ultra-short pulse laser patterning is a tool enabling precise and clean patterning in different materials and can be integrated easily in a chain of hybrid technologies for reel-to-reel manufacturing. The simple, generic, multilayer manufacturing method by stacking and bonding several layers of microstructured polymer films using a lamination process is a dry, low-cost, lowtemperature and low pressure technology to construct 3D microfluidic devices based on relatively simple two dimensional manufacturing processes compatible with reel-to-reel manufacturing. In addition, the lamination technique opens up the potential of creating mass-manufacturable disposable microfluidic devices in a variety of materials and combining different techniques. Therefore we implemented a powerful manufacturing method based on two generic and simple tools

Fig. 2 :

 2 Fig. 2: Flexible microfluidic passive device composed of two polyester foils, one foil with through-holes structured by femtosecond laser ablation and a cover foil laminated on top of the previous one and bonded with epoxy dry SU8 resist: a) photograph of the laminated assembly; b) SEM close-up picture of the bonded foils.

Acknowledgements

This work was performed within the framework of the 4M Network Of Excellence "Multi-Material Micro Manufacture: Technology and Applications (4M)" (EC funding FP6-500274-1; www.4m-net.org). We would like to thank the Novalase company (www.novalase.com) which is a company specialized in the fabrication of stations for laser micro-machining for their efficient support.


P Cailleu

G.-P Piau

S Haché

S N Burokur

A De Lustrac
email: andre.de-lustrac@u-psud.fr

F Gadot

Principles and Applications of a Controllable Electromagnetic Band Gap Material to a Conformable Spherical Radome

Keywords: 41.20, Jb, 42.70.Qs, 84.40.Ba

Principles and applications of a controllable electromagnetic band gap material to a conformable spherical radome

Introduction

Controllable Electromagnetic Band Gap (EBG) structures can find a variety of applications to antennas, RF screens, signature control and filters … [START_REF] Hook | Proc. 27th ESA Antenna Technology Workshop Innovative Periodic Antennas[END_REF][START_REF] Maci | [END_REF]. Several studies have been made on electronically active planar EBG structures in the microwave domain where their applications to electromagnetic windows and antennas have been demonstrated [3,4]. The main advantage of these structures resides in the large easiness of continuous directional control of transmission and of frequency adjustment, which depends on the structure's periodic parameters.

In this paper we use a controllable EBG to design an electronically controllable radome working around 10GHz. In the open literature, Frequency Selective Surfaces (FSS) have been proposed to design radomes [START_REF] Munk | Frequency selective surfaces, Theory and design[END_REF][START_REF] Chang | [END_REF][7][START_REF] Chang | IEE Proc. Microwaves Antennas and Propagation[END_REF][START_REF] Vardaxoglou | [END_REF][10][11]. Patents covering the use of active FSS to design controllable radomes can also be found [START_REF] Whelan | Ferro-electric frequency selective surface radome[END_REF][START_REF] Arceneaux | Absorptive/transmissive radome[END_REF]. However to the best of our knowledge it is the first time that a controllable radome is made of a conformed EBG material. The application of controllable EBG material to a radome implies that the classical properties of a radome will be satisfied, i.e. a transmission close to unity. Then we have optimized the radome structure to meet this criterion.

The EBG structures considered in this paper are composed of periodically organized copper strips on a conformal spherical shape. The final aim of this study is to achieve an electronic commutation of the transmission from an allowed state to a forbidden one at around 10 GHz. The structure must then consist of discontinuous parallel copper strips incorporating fast switching elements such as PIN diodes or photoconductors, necessary to obtain the commutation of the metallic wires from a discontinuous electrical state to a continuous one. In this present work, we consider the spherical radome only in a passive mode. This study on passive structures is the starting point to demonstrate the principle of the commutation before going towards electronically active structures. EBG structures are characterized by an alternation of forbidden frequency and allowed frequency bands depending on the geometrical and electromagnetic parameters of the structure, and also on the polarization of the incident wave. If the EBG structure is made of continuous metallic wires parallel to the electrical field of the incident wave, the first band at low frequencies is a forbidden band. In a pioneering work, J. B. Pendry has studied metallic electromagnetic band gap materials and has shown that a continuous metallic lattice may be considered like a diluted metal with a first forbidden frequency band from 0Hz to a plasma frequency ω p [START_REF] Pendry | [END_REF]. This plasma frequency depends on the geometrical parameters of the metallic lattice and on the used metal:

! " p 2 = ne 2 #r 2 $ 0 a 2 m eff with ! m eff = ln( a r )µ 0 e 2 r 2 n /2
where a is the period of the lattice, r is the radius of the wire, n is the electron density and m eff the effective electron mass.

When the continuous metallic wires are replaced by discontinuous ones with discontinuities periodically inserted along the wires with a period p, an allowed band is observed at low frequencies. The bandwidth of this allowed band depends on the capacitance and the period of the discontinuities [15]. If we can control the switching of the metallic wires from a discontinuous electrical state to a continuous one, then we can obtain the switching of the structure from a transmitting state to a reflecting one. This is the first control configuration investigated in this paper (Fig. 1a). However it is possible to use another mode. If we can control not only the electrical state of all the discontinuities, but also the periodicity of these discontinuities, we can then control the switching from a set of frequency bands, alternating forbidden and allowed bands, to a different set. If these periodicities are well chosen, an allowed band in the first set may be replaced by a forbidden band in the second set (Fig. 1b). This constitutes the second control configuration under study. We can see in Fig. 1 that at the frequency f 0 both control modes allow the switching from a transmitting state of the EBG material to a reflecting one. Several conformed prototypes have been simulated, fabricated and characterized to illustrate both control modes on a real scale spherical radome. The results obtained from both configurations show that the switching of the transmission is observed in both cases in simulations and measurements with a very good agreement. The first configuration is more suitable to the use of electronic switching components biased by DC current whereas the second one corresponds much better to the use of photoconductive components.

In this study we present calculated and measured results obtained on only passive prototypes. These prototypes have been fabricated and characterized to demonstrate the feasibility of the electrical state switching concept. To further validate this concept, the radome has been tested with a meteorological radar antenna actually used in airplanes.

Design and simulation of the EBG structures

The radome considered here is a conformable one on a spherical shape. This shape corresponds to the vertex of the real radome of an ATR 42 airplane, with a maximum radius R = 16.5 cm. The studied EBG structures are all composed of a one-dimensional grid of parallel copper strips of width w = 1 mm (Fig. 2a) regularly disposed on the spherical shape with a projection period a = 1 cm on the horizontal plane (xOy plane). This structure has two symmetry planes (xOz and yOz) that allow simplifying its simulation. The calculation is performed using Flomerics' Microstripes commercial software based on the Transmission Line Matrix method [16]. The software allows reducing the calculation time by simulating the wires using a wire model. The horn antenna used in the experimental characterizations of the radome is also included in the calculation to increase its accuracy. This type of excitation acts as a Transverse Electromagnetic (TEM) plane wave propagating towards the EBG structure in the +z direction with its electric field E oriented along the y-axis and its magnetic field H oriented along the x-axis. The two switching control configurations are tested using structures with either continuous wires or discontinuous wires:

Configuration 1: commutation between continuous and discontinuous wires

The first spherical EBG structure is composed of continuous metallic wires of width w = 1 mm. The continuous wires are supposed to present an allowed band at 10 GHz. For the second structure composed of discontinuous wires, the discontinuities are placed along each row of wires in such a manner that their projections on the y-axis are equally spaced with a period p. The width of the discontinuities in all the rows is 0.1 mm. For the simulation, the Efield is polarised parallel to the wires. Here, we have tested only this polarization, but it is also possible to design a structure operating for two perpendicular linear polarizations with the use of crossed grids at 90°. The wires are considered to be perfect conductors in the various simulations. The EBG structures are designed so as to obtain a switching between an allowed and a forbidden transmission band at around 10 GHz. In this first configuration, a commutation between the allowed band of the continuous wire strips structure and the forbidden band of discontinuous wire strips structure is desired. The position of the forbidden band depends on the discontinuities' periodicity and on the bend radius of the structure.

On the sphere, each row of wires has a different bend radius. Since the transmission level through the EBG structure depends on this bend radius, a different discontinuities' periodicity p 1 must be applied to each row of wires in order to obtain a forbidden band at 10 GHz. The period p 1 for each row is determined separately from a cylindrical periodical structure, so as to obtain a commutation at the desired frequency. The cylindrical EBG structure has been previously presented and investigated in [START_REF] Haché | Proc. Loughborough Antennas and Propagation Conference (LAPC) 2007[END_REF]. The different parameters of the structure for a forbidden band at around 10 GHz are as follows: p 1 = 11 mm (discontinuities' periodicity) for the first six rows of wires, near of the summit of the radome, p 1 = 10 mm for the seventh (7 th ) to ninth (9 th ) rows and p 1 = 9 mm for the last row (10 th ). This set of wires will be referred to as having a discontinuities' periodicity p 1 in the rest of the text. Fig. 2b presents the transmission responses calculated for each row of wires separately on a cylindrical structure having a different specified radius. These rows of wires will then be applied on the spherical radome of variable radius. The responses show a very low transmission level for each row at 10 GHz ( ± 10%), indicating that the different optimized periods can be applied to the discontinuities in order to have a forbidden band with the spherical radome.

Configuration 2: commutation between two structures of discontinuous wires

In the second configuration a commutation between the allowed band of a structure composed of discontinuous wires with a projection period p 2 and the forbidden band of the structure composed of discontinuous wires with the projection period p 1 previously used in configuration 1. For this second structure composed of discontinuous wires, the same study to optimize the different values of p 2 has been done so to obtain an allowed band at 10 GHz. For the six first rows, the discontinuities periodicity is p 2 = 5.5 mm, for the 7 th to 9 th rows p 2 = 5 mm, and for the last row p 2 = 4.5 mm. This set of wires will be referred to as having a discontinuities' periodicity p 2 in the rest of the text. The simulated transmission responses are presented in Fig. 2c. We can observe that the transmission level is high and close to 0 dB for each row of wires, indicating an allowed band at 10 GHz. These rows of wires will then be applied on the spherical radome to exhibit an allowed band. The transmissions through the three EBG-based radomes normalized to that of the horn antenna are presented in Fig. 2d. For both configurations, a commutation of the transmitted signal can be observed at 10 GHz. This commutation reaches a level of more than 28 dB. In the first configuration this switching occurs between the EBG structure made of continuous wires (solid grey curve) and the structure made of discontinuous wires with periods p 1 (dashed black curve). In the second configuration, the commutation is observed between the EBG structure made of discontinuous wires with periods p 1 and an EBG structure made of discontinuous wires with periods p 2 (solid light grey curve). The main difference between these two commutation configurations is the transmission level when the radome is transparent to the horn antenna. In the first configuration, this level is weaker (-1.6 dB at 10 GHz) for the EBG structure made of continuous wires than for the second one made of discontinuous wires with periods p 2 (-0.6 dB).

Measurements

Gain versus frequency

Three different passive prototypes of the spherical EBG-based radome are fabricated according to the parameters mentioned above for the two switching control configurations. The first prototype is composed of continuous wire strips and the second one is composed of discontinuous wires with the periods of discontinuities p 1 . Finally, the last prototype has the periods of discontinuities p 2 for the rows of wires. The wires made of copper are printed on a flexible support and are disposed on the hollow conformable spherical object made of low loss foam having a relative permittivity close to 1 and a thickness h = 17 mm. The maximum radius of the sphere is 16.5 cm. The widths of the wires and of the discontinuities are respectively 1 mm and 0.1 mm. A first set of transmission measurements are done using a horn antenna working in the X band [8.2 GHz -12.4 GHz] and having a angular beam width of 30° at -3dB. It is placed at the centre inside the hollow foam sphere at a distance of 20 cm of the EBG structure. A picture of the EBG-based spherical radome with the emitting horn antenna inside is shown in Fig. 3a. A receiving horn antenna is placed at a distance of 8 m from the radome. The measurements are done in the EADS Innovation Works' anechoic chamber at Suresnes (France) as illustrated by the experimental setup shown in Fig. 3b. The gains of the emitting horn antenna alone and in the presence of the three different EBG-based radomes are measured versus frequency and the results are presented in Fig. 4a. The black curve represents the gain of the horn antenna alone in free space (without the use of the radome). For both configurations, a switching occurs at about 9.3 GHz. Compared to the simulation results, the measured commutation is shifted towards lower frequencies of about 0.7 GHz. This frequency shift between the simulated and the measured responses can be explained by the fabrication accuracy of the structures. The grey curve represents the gain response for the EBG structure with continuous wires. We can note that there is an average loss of 1.5 dB compared to the gain of the antenna alone, as it has been observed in simulations. In the first configuration, the switching between the EBG structures made of continuous wires and discontinuous wires with periods p 1 reaches 24.5 dB, slightly weaker than the simulated one. The light grey curve represents the gain response for the radome made of discontinuous wires with periods p 2 . Compared to the gain of the horn antenna alone, the transmission is quite lossless at 9.3 GHz. In the second configuration, the switching between the two EBG structures composed of discontinuous wires reaches a level of more than 26 dB.

A second set of measurements are done using as emitter a meteorological radar antenna array working at a centre frequency of 9.3 GHz with a bandwidth of about 10% [8.8 GHz -9.8 GHz] and having a angular beam width of 12° at -3 dB. This antenna is composed of an array of 256 elementary slot antennas. This type of antenna is used in the modern aircraft for weather forecasts and corresponds much better to the antenna associated really with this radome. In this case, the switching reaches a level of about 16 dB as shown in Fig. 4(b).

Compared to the gain of the meteorological antenna alone, there is an average loss of 2 dB when the EBG-based radome is transparent.

Radiation patterns

The E-plane radiation patterns are also measured at the frequency corresponding to the commutation (f = 9.3 GHz) for the horn antenna alone and in presence of the three EBGbased radomes. The diagrams are shown in Fig. 5a where the same color code of Fig. 4 holds. In these measurements, the radome is rotated from -120° to 120° with respect to the position of the horn antenna which remains fixed. From Fig. 5a, we can observe that the primary lobes of the continuous wires and the discontinuous wires with period p 2 have the same aperture angle at -3dB as that of the horn antenna alone. However the secondary lobes reaching a level of -12 dB are higher for the EBG structure with period p 2 . This may be due to the large number of discontinuities in this structure.

As for the horn antenna, measurements are done with the three EBG structures associated with the meteorological antenna array in order to show the commutation of the transmitted signal. The radiation patterns of the antenna alone and in presence of the EBG-based radomes at 9.3 GHz are presented in Fig. 5b. Concerning the first commutation configuration, i.e. between the continuous wires and the discontinuous ones with periods p 1 , we can observe a maximum level of commutation in transmission at 9.3 GHz of about 16 dB. For the second configuration between the EBG structure composed of discontinuous wires with periods p 2 and the one with p 1 , the commutation level is quite identical. We can also observe an increase in the level of the secondary lobes in the presence of the radome. But the level of these lobes lies under -20 dB, which is better than those observed in the measurements using the horn antenna as emitter.

Conclusion

This paper presents the principle of two types of conformable and controllable spherical radomes based on an Electromagnetic Band Gap (EBG) material operating at around 9.3 GHz.

The EBG structures are composed of a grid of metallic wires conformed on a hollow spherical object. Two configurations corresponding to two different operating modes have been tested: 1) an EBG structure made of electrically continuous wires and a structure made of discontinuous wires with periods p 1 , 2) two EBG structures made of discontinuous wires with different periods of discontinuity, p 1 and p 2 . Calculations and measurements performed on passive prototypes are in good agreement. A switching of the transmission is observed in simulations at 10 GHz and in measurements at 9.3 GHz. These EBG-based radomes have been tested with a horn antenna and a meteorological antenna array composed of 256 elementary slot antennas. The radiation patterns of both configurations of radomes have been measured and presented. A switching of about 26 dB is observed for the horn antenna, with an increase of the secondary lobes at a level of -12 dB. For the second antenna, which corresponds better to the real antenna used with this type of radome, we obtain a switching of 16 dB, with secondary lobes lying under -20 dB.

Figures captions

Fig. 1(a): First operating mode: switching between an EBG structure composed of continuous metallic wires (1:grey curve) and one made of discontinuous metallic wires with a period p 1 of discontinuities (2: black curve). (b) Second operating mode: switching between an EBG structure composed of discontinuous metallic wires with a period p 1 of discontinuities (1: black curve)and another one with a period p 2 of discontinuities (2: grey curve). At the frequency f 0 both operating modes allow the switching from a transmitting state of the EBG structure to a reflecting one. Calculated frequency response for both configurations of wires. The radome is transmitting when the wires are continuous (grey curve) and when they are discontinuous with a discontinuity period p 2 (solid light grey curve). The radome is reflecting when the wires are discontinuous with a period p 1 (dark dashed curve). The commutation reaches more than 28 dB at 10 GHz between the transmitting and the reflecting state. The transmission of the horn antenna alone is given by the black curve. with the meteorological antenna. The EBG radome is transmitting when the wires are continuous (solid grey curve) and when they are discontinuous with a discontinuity period p 2 (light grey curve). The radome is reflecting when the wires are discontinuous with a period p 1 (dashed black curve). The solid black curve illustrates the gain of the horn antenna alone.

Fig. 2 (

 2 Fig. 2(a): Schematic figure of the simulated structure with Microstripes. The inset shows half of the simulated radome in the xOz plane with the position of the different rows of wires. (b) Calculated transmission response for different row of wires with different periods p 1 . (c) Calculated transmission response for different row of wires with different periods p 2 . (d)Calculated frequency response for both configurations of wires. The radome is transmitting when the wires are continuous (grey curve) and when they are discontinuous with a discontinuity period p 2 (solid light grey curve). The radome is reflecting when the wires are discontinuous with a period p 1 (dark dashed curve). The commutation reaches more than 28 dB at 10 GHz between the transmitting and the reflecting state. The transmission of the horn antenna alone is given by the black curve.

Fig. 3 :

 3 Fig. 3: (a) EBG structure under test in the anechoic chamber of EADS (Suresnes, France). (b) Schematic view of the anechoic chamber with the radome under test.

Fig. 4 :

 4 Fig. 4: (a) Measured transmission responses for the two commutation configurations with the horn antenna. (b) Measured transmission responses for the two commutation configurationswith the meteorological antenna. The EBG radome is transmitting when the wires are continuous (solid grey curve) and when they are discontinuous with a discontinuity period p 2 (light grey curve). The radome is reflecting when the wires are discontinuous with a period p 1 (dashed black curve). The solid black curve illustrates the gain of the horn antenna alone.

Fig. 5 :

 5 Fig. 5: (a) Radiation patterns for the three EBG radomes and the horn antenna alone at 9.3 GHz. (b) Radiation patterns for the three EBG radomes and the meteorological antenna alone at 9.3 GHz. The color code is the same as in Fig. 4.

Figure 1 Figure

 1 Figure 1

Acknowledgement:

The authors would like to thank the EADS Foundation and DGA for its financial support.