New insights in integral representation theory for the solution of complex canonical diffraction problems

J.M.L.

Chapter 1

New insights in integral representation theory for the solution of complex canonical diffraction problems J.M.L. Bernard 1 The problem of scattering by 2D and 3D canonical objects with imperfectly conducting surfaces requires some particular efforts on the representation of scattered and incident fields, and we present here some remarkable aspects of them in complex situations and their applications.

We begin with the study of 2D problems. The Sommerfeld-Maliuzhinets integral and its inversion in the spectral domain of complex angles has opened a new way of investigation on diffraction by a wedge-shaped domain. In this frame, we present this spectral method in a new perspective, by giving some novel exact general expressions and properties of the associated spectral function attached to the total field : novel spectral expression of free space Green function, single face representation and its consequences, uniqueness, existence, reciprocity, spectral causal representation of field in time domain. We then analyze in a distinct section the diffraction by an impedance wedge with curved faces. The diffraction coefficient, when limited to its principal order in curvature, leads us to some miscalculation when the wedge approaches the discontinuity of curvature case for example. Thus, we give here a general asymptotic representation taking into account a development at several orders for arbitrary wedge angle, which is uniformly valid from the discontinuity of curvature to the curved half-plane.

Concerning 3D problems, we begin with a section devoted to an efficient exact solution for the radiation of a point source above an impedance plane in electromagnetism. In classic approaches, we need to consider vertical and horizontal dipole cases separately, and we present here a method, consisting in filling a gap in representation of incident fields by potential theory, to finally permit an efficient global representation of the diffracted field for arbitrary primary sources in a direct manner. We conclude this chapter with the representation of fields for the diffraction by an imperfectly conducting cone. We present a general exact expression of fields with Debye potentials and Kontorovich-Lebedev (KL) integrals, and new general properties of spectral functions attached to them. We then insist on a novel compact representation of plane waves, which does not need any use of spherical harmonics series.

Representations of Spectral Function in

Basic integral representation

Let us consider the case of diffraction in free space of an incident plane wave

u i (ρ, ϕ) = U i (ω)e ikρ cos(ϕ-ϕ • ) , (1.1)
by a scatterer enclosed in a wedge-shaped region (fig. 1.1), defined in cylindrical coordinates (ρ, ϕ, z) as the domain outside the free space angular sector with origin O, -Φ r ≤ ϕ ≤ Φ l . An implicit harmonic dependence on time e iωt is understood and henceforth suppressed, and k = ω c denotes the wave number of the exterior medium with | arg(ik)| < π 2 . Physically, | arg(ik)| < π 2 means that there are some losses in free space, and | arg(ik)| = π 2 is considered as a limit case. The function U i (ω) is assumed O(ω -b ) for |ω| → ∞, b > 1 so that the scattered field u is O(ω -v ) with v > 1 as |Reω| → ∞ for Imω ≤ 0, and is analytic at any ω with Imω ≤ 0. The characteristics of the scatterer are supposed to be independent of z coordinate. We assume that the total field in the free space region, u = u s +u i , satisfies the Helmholtz equation,

(∆ + k 2 )u(ρ, ϕ) = 0, (1.2)
and that u is analytic with respect to ρ, ϕ and ϕ • , except possibly at the origin, and that there exists a constant s • such that ∞ 0 |u(ρ, ϕ)e -s • ρ |dρ < ∞. The total field u for -Φ r ≤ ϕ ≤ Φ l is then represented as a Sommerfeld-Maliuzhinets integral [START_REF] Maliuzhinets | Inversion formula for the Sommerfeld integral[END_REF],

u(ρ, ϕ) = 1 2πi γ f (α + ϕ)e ikρ cos α dα, (1.3)
which satisfies the Helmholtz equation. In this representation, f is an analytic function and the path γ consists of two branches: one, named γ + , going from (i∞+arg(ik)+ (a 1 + π 2 )) to (i∞+arg(ik) -(a 2 + π 2 )) with 0 < a 1,2 < π, as Imα ≥ d > 0, above all the singularities of the integrand, and the other, named γ -, obtained by inversion of γ + with respect to α = 0. Stationary phase methods [START_REF] Jones | The theory of electromagnetism[END_REF], [START_REF] Felsen | Radiation and scattering of waves[END_REF] can be applied to (1.3) to find the far field diffraction coefficient F(α, ϕ • ) = f (π + α)f (-π + α) [START_REF] Bernard | A spectral approach for scattering by impedance polygons[END_REF].

1.1.1.2 Basic properties of the total field and its spectral function f Some elementary properties can be assumed to hold for the field :

-(a') the only incoming plane wave, from the free space sector with origin O, -Φ r ≤ ϕ ≤ Φ l , is the incident field; -(b') the limit of the field u as ρ → 0 is finite and does not depend on ϕ, while the derivatives ∂ ρ u and ∂ ϕ u/ρ are locally summable with respect to ρ in the vicinity -(c') the field, except possibly its geometrical optics part when Im(k) = 0, does not grow at infinity. In addition, some bounds on the far field are assumed. We consider here that the field is O(e ikρ cos(ϕ-ϕ • ) ) for large ρ, | arg(ik)| < π 2 , which is a standard assumption in scattering theory.

These properties lead us to the following conditions on f [1], [START_REF] Bernard | A spectral approach for scattering by impedance polygons[END_REF], [START_REF] Bernard | Diffraction at skew incidence by an anisotropic impedance wedge in electromagnetism theory : a new class of canonical cases[END_REF] :

-(a) ( f (α) -u i (O)/(α -ϕ • )
) is regular at points with Re(α) belonging to the free space angular sector with origin O, -Φ r ≤Re(α) ≤ Φ l , which ensures (a').

-(b) there exist some constants g ± , some analytic function h, and some Maliuzhinets contour γ such that | f (α + ϕ) ∓ f (-α + ϕ)g ± | <| h(α) | on and inside the loop formed by the upper branch γ + of γ, when -Φ r ≤ ϕ ≤ Φ l , the function h being summable on γ + , regular on and within it. In this respect, we notice that

( f (i| ln ρ|) -f (-i| ln ρ|)) = -iu(0, ϕ) + O( ρ∂ u ∂ ρ ), as ρ → 0, with ρ∂ u ∂ ρ = o((ln ρ) -1
) and u(0, ϕ) = ig + . Since γ is odd, we can add a constant to f without changing u, which implies that we can define f with f (i∞) =f (-i∞). This ensures (b').

-(c) f (α + ϕ) has no singularity, except possibly those associated to incident, reflected or transmitted plane waves not vanishing at infinity, in the zone defined by Re(ikcosα

) > 0 as |Re(α)| < π, -Φ r ≤ ϕ ≤ Φ l , | arg(ik)| < π 2 .
Considering that the far field is O(e ikρ cos(ϕ-ϕ • ) ), f (α + ϕ) has no singularity in this region when Re(ik cos(ϕϕ

• )) < 0, i.e. π 2 < |ϕ -ϕ • | < 3π 2 .
This ensures (c').

1.1.2 Spectral functions f ± attached to the radiation of a single face and simple relation to f

1.1.2.

Radiation of a single face of a wedge-shaped region

To simplify the notation without losing generality, we take Φ r = Φ l = Φ with 0 < Φ < π. From the properties (b'), (c') on u, the scattered field in free space u s = uu i for |ϕ| < Φ can be written as the sum u + + u -of the radiations of equivalent surface currents carried by the faces ϕ = +Φ and -Φ,

u ± (ρ, ϕ) = -i 4 lim ρ 0 →0 + ∞ ρ 0 (u(ρ , ϕ ) ∂ H (2) 0 (kR) ∂ n - ∂ u(ρ , ϕ ) ∂ n H (2) 0 (kR))| ϕ =±Φ dρ (1.4) with R = ρ 2 + ρ 2 -2ρρ cos(ϕ -ϕ ), ∂ (.)/∂ n = n∇(.) = ∓∂ (.)/ρ ∂ ϕ , n the outward normal to the face ϕ = ±Φ, |ϕ| < Φ, | arg(ik)| < π 2 .
We show that it is possible to simply express the spectral function f ± attached to u ± with f as 0 < Φ < π. For this, we use an original representation of H

(2) 0 (kR).

Spectral function attached to Sommerfeld-Maliuzhinets representation of H

(2)

(kR)

As shown in [START_REF] Bernard | Progresses on the diffraction by a wedge: transient solution for line source illumination, single face contribution to scattered field, and new consequence of reciprocity on the spectral function[END_REF], the spectral function for H

(2)

0 (kR), where |ϕ | > π 2 , |ϕ| < |ϕ |, is given by f H (2) 0 (α) = -1 2π S e ikρ cos(α -ϕ ) (tan( 1 2 (α -α )) -g • (α ))dα , (1.5)
for α ∈]Sπ, S + π[ (i.e. the domain limited by Sπ and S + π), where S is the path from -i∞-arg(ik) to i∞+arg(ik) with Im(k sin α) = 0. By analyticity, the path S can be deformed continuously, as long as the integrand remains bounded, without changing f H

(2) 0

, and this expression can be continued, for |ϕ | ≤ π 2 by deforming S , or by f

H (2) 0 (π + α) -f H (2) 0 (-π + α) = -2ie ikρ cos(α-ϕ ) , for α outside ]S -π, S + π[. The terms g • , normally unnecessary because γ is odd, is chosen as g • (α) = -tan( α
2 ) in order to ameliorate the convergence of the integral.

0 (kR ε ), with R ε = ρ 2 + ρ 2 -2ρρ cos(ϕ -εΦ), ε ≡ +or-, in a more general form, f ε H (2) 0 (α) = -1 2π S ε e ikρ cos(α -εΦ) (tan( 1 2 (α -α )) -g • (α ))dα , (1.6)
where

S ε = S + a ε , with a ε a constant satisfying Φ -3π 2 < εa ε < Φ -π 2 for 0 < Φ < π.
The expression (1.6) is used for H

(2)

0 (kR ε ) and ∂ n H (2)
0 (kR ε ) in (1.4), and we then obtain a Sommerfeld-Maliuzhinets representation of u ± with f [4]- [START_REF] Bernard | Progresses on the diffraction by a wedge: transient solution for line source illumination, single face contribution to scattered field, and new consequence of reciprocity on the spectral function[END_REF] :

u ± (ρ, ϕ) = 1 2πi γ f ± (α + ϕ)e ikρ cos α dα, (1.7)
where

f ε (α) = 1 4πi S ε ε f (επ + α )(tan( 1 2 (α -α )) -g • (α ))dα , (1.8)
for α between S επ and

S ε + π, provided π 2 < Φ -εa ε < 3π 2 , π 2 < Φ -εϕ • < 3π 2 and g -= f (i∞) + f (-i∞) = 0, ε ≡ +or-, 0 < Φ < π.
Let us notice that the conditions (b) and (b'), which concern u and f , don't apply to u ε and f ε . However, we remark that

f ε (α + ϕ) -f ε (-α + ϕ) is bounded for large α on γ, even if f ε (α)
for Imα → ∞ (resp. u ε for ρ → 0) diverge when f (±i∞) = 0 (resp. u = 0 at ρ = 0). Since f is an analytic function, (1.8) can be analytically continued in the whole complex plane. We note in particular, that, taking into account of the poles of tan( 12 (αα )) which can be captured by S ε as α varies, f ± satisfies

f ± (π + α) -f ± (-π + α) = ± f (±π + α).
(1.9)

Concerning the dependence on ϕ • (or Φ), the expression (1.8) has been determined for π 2 < Φεϕ • < 3π 2 , but we can consider f ε (α) outside this domain by analytical continuation on ϕ • (or Φ), which corresponds to taking account of the contribution of any singularity that would go through S ε as ϕ • (or Φ) goes into these regions.

Exact expression of f from diffraction coefficient F and consequences

We have

f (α) = f + (α) + f -(α) + f i (α), with f i (α) = Res f | α=ϕ• 2 cot( 1 2 (α -ϕ • )
) and f ± verifying (1.8) and (1.9). Thus, we can write, when both conditions π 2 < Φ∓ϕ • < 3π 2 (which implies Φ > π/2) are satisfied, Stationary phase methods [START_REF] Jones | The theory of electromagnetism[END_REF], [START_REF] Felsen | Radiation and scattering of waves[END_REF] can be applied to (1.7) to find the far field radiation of the face ϕ = ±Φ, also denoted ϕ = εΦ. From the regularity of f ε , we can deform γ to stationary phase points α = +π and -π, when π 2 < Φεϕ • < 3π 2 and π 2 < Φεϕ < 3π 2 , and thus, out of the reflected and shadowed regions 2Φε(ϕ + ϕ • ) < π and |ϕϕ • | > π, we can write, for large kρ and using (1.9), (1.11) which can be considered as an important generalization of the result obtained by Michaeli for Dirichlet boundary conditions in [START_REF] Michaeli | Contribution of a single face to the wedge diffracted field[END_REF]. In other respects, we can use in (1.4) the formulas [START_REF] Gradshteyn | Table of integrals, series and products[END_REF],

f (α) = 1 4πi S F(α , ϕ • )(tan( 1 2 (α -α )) -g • (α ))dα + f i (α), (1.10
-2πkρe i(kρ+ π 4 ) u ± (ρ, ϕ) ∼ f ± (π + ϕ) -f ± (-π + ϕ) = ± f (±π + ϕ),
H (2) 0 (kR) ∼ 2 πkR e -ikR+iπ/4 , ∂ n H (2) 0 (kR) ∼ 2 πkR e -ikR-iπ/4 ∂ n (kR), (1.12) with R = ρ 2 + ρ 2 -2ρρ cos(ϕ -ϕ ), ∂ n (kR) = ∓∂ (kR)/ρ ∂ ϕ = (±kρ sin(ϕ - ϕ )/R
) at ϕ = ±Φ, and cos(ϕ ∓ Φ) < 0. Taking into account that

R = ρ -ρ cos(ϕ ∓ Φ) + (ρ sin(ϕ ∓ Φ)) 2 R + ρ -ρ cos(ϕ ∓ Φ) , (1.13)
and considering the properties (b')-(c') on u, we then obtain another expression for the far field that, compared with (1.11), gives us :

f (±π + ϕ) = 1 2 ∞ 0 (iku(ρ , ±Φ) sin(ϕ ∓ Φ) ± ∂ u ∂ n (ρ , ±Φ))e ikρ cos(ϕ∓Φ) dρ , (1.14) as π 2 < Φ ∓ ϕ • < 3π 2 and π 2 < Φ ∓ ϕ < 3π 2 , | arg(ik)| < π 2 , 0 < Φ < π. By changing Φ for Φ ±
e with 0 < Φ ± e < π and letting ±π + ϕ = α ± Φ ± e , we then notice that,

f (α ± Φ ± e ) = 1 2 ∞ 0 (-iku(ρ , ±Φ ± e ) sin α ± ∂ u ∂ n (ρ , ±Φ ± e ))e -ikρ cos α dρ , (1.15)
when π 2 < Φ ± e ∓ ϕ • < 3π 2 and π 2 < ∓α + π < 3π 2 , which is also valid, by analytic continuation, as Re(ik(cos αcos(Φ ± e ∓ ϕ • ))) > 0, |Reα| < π (note: on the straight semi-line ϕ = ±Φ ± e , we have

± ∂ u(ρ ,ϕ ) ∂ n = -∂ u(ρ ,ϕ ) ρ ∂ ϕ
). Thus, a knowledge of the field and of its normal derivative on one single face (a semi-line) gives a unique definition of the analytical spectral function f attached to the 'total' radiated field (uniqueness and existence). By the way, we also notice that, in accordance with the property (b), we have from (1.15)

, ( f (ix ± Φ ± e ) -f (-ix ± Φ ± e )) = -iu(0, ±Φ ± e ) + O( ∂ u ∂ ln ρ e | | ln ρ e |=x ) as x → ∞.
1.1.3.2 Deformation and simple exact expression of the spectral function f from fields on a piecewise smooth single face Using Green's theorem, we note that the contour of integration along ϕ = ±Φ in (1.4) can be deformed into a new path L ± 0,∞ from the origin to infinity without changing the field u ± (except at points captured by the path during its deformation), provided that the integral remains bounded and no source passes through the path during the deformation. Thus, we can write (1.14) in a more general form, with integration along a piecewise smooth path L ± 0,∞ , following

f (±π + ϕ) = 1 2 L ± 0,∞ (iku(ρ , ϕ ) sin(ϕ -ϕ t ) ± ∂ u ∂ n (ρ , ϕ ))e ikρ cos(ϕ-ϕ ) dl , (1.16)
where dl and ϕ t , both depending on (ρ , ϕ ), are respectively the element of length and the tangent angle along the piecewise smooth semi-line L ± 0,∞ .

If we divide the semi-infinite paths L ± 0,∞ into L ± 0,∆ ± (i.e. 0 < l < ∆ ± ) and L ± ∆,∞ (i.e. l > ∆ ± ), we have

f (±π + ϕ) = 1 2 L ± 0,∆ ± (iku(ρ , ϕ ) sin(ϕ -ϕ t ) ± ∂ u ∂ n (ρ , ϕ ))e ikρ cos(ϕ-ϕ ) dl (ρ , ϕ ) + f L ± ∆ ± ,∞ (±π + ϕ), (1.17)
where

f L ± ∆ ± ,∞ (α) = e -ikρ ∆ ± cos(α-ϕ ∆ ± ) f ± e (α), f ± e (α)
is the spectral function related to a shift of the origin at l = ∆ ± . This implies, by analytic continuation,

f (α) = 1 2 L ± 0,∆ ± (-iku(ρ , ϕ ) sin(α -ϕ t ) ± ∂ u ∂ n (ρ , ϕ ))e -ikρ cos(α-ϕ ) dl (ρ , ϕ ) + f L ± ∆ ± ,∞ (α)
.

(1.18)

Our expression can then lead us to a suitable solution for complex geometries and boundary conditions, in particular for the scattering by an impedance polygon [START_REF] Bernard | A spectral approach for scattering by impedance polygons[END_REF].

Exact causal time domain representation of a field above a dispersive wedge-shaped region

According to the behaviour of f (in particular for ω with large Im(ω) = -σ ≤ 0), we can assume that the path γ can be deformed into a contour D which consists of two branches D ± : D + with Imα > 0, given by ](i∞ + π), π

+ i0 + ] ∪ [π + i0 + , -π + i0 + ] ∪ [-π + i0 + , i∞ -π[,
F c (α + ϕ, τ) = 1 2π +∞-iσ -∞-iσ f (α + ϕ, ω)e iωτ dω, (1.19)
vanishes as τ < 0 from field causality, so that f (α + ϕ) is regular and O(ω -v ) on D with v > 1 as Imω ≤ 0. The time domain field u(t) deriving from Fourier transform of u, will be then simplified and expressed in an efficient explicitely causal expression on a finite subset H of D [START_REF] Bernard | Progresses on the diffraction by a wedge: transient solution for line source illumination, single face contribution to scattered field, and new consequence of reciprocity on the spectral function[END_REF], [START_REF] Bernard | On the time-domain scattering by a passive classical frequency dependent wedge-shaped region in a lossy dispersive medium[END_REF].

1.1.4.1 Causality of F c (α + ϕ, τ)
Let us use in (1.19) the single face expression (1.15) of f (α ± Φ ± e ) for Reα = 0. Changing the order of the integration (permitted), we can write

F c (α ± Φ ± e , τ) = 1 4π ∞ 0 dρ +∞-iσ -∞-iσ dω (-iku(ρ , ±Φ ± e ) sin α ± ∂ u ∂ n (ρ , ±Φ ± e ))e iωt , (1.20)
for Reα = 0 with t = -ρcosα

/c + τ, π 2 < Φ ± e ∓ ϕ • < 3π 2 , Φ ± e ≤ Φ.
The field u is non singular and O(ω -v ), with v > 1, as Imω < 0 and u(t) is causal with respect to the front of the incident plane wave, so that the contour of integration in ω can be closed at infinity for Im(ω) < 0 when t < -ρ/c, which implies that F c (α + ϕ, τ) = 0 for Reα = 0 as τ < 0. Considering the analyticity of f , F c (α + ϕ, τ) is assumed to be an analytic function of α as τ is fixed, when |Re(α + ϕ)| ≤ π + Φ and |Imα | > 0 + , and we then have,

F c (α + ϕ, τ) = 0, τ < 0, |Re(α + ϕ)| ≤ π + Φ.
(1.21)

Spectral causal expression of the field in time domain

From (1.3) and the causality of F c , the field in time domain is then [START_REF] Bernard | Progresses on the diffraction by a wedge: transient solution for line source illumination, single face contribution to scattered field, and new consequence of reciprocity on the spectral function[END_REF], [START_REF] Bernard | On the time-domain scattering by a passive classical frequency dependent wedge-shaped region in a lossy dispersive medium[END_REF],

u(t) = 1 2πi H F c (α + ϕ, τ c (α))dα, (1.22)
where

H = H + ∪ H -is a finite subset of D with τ c (α) = ρcosα/c + t > 0. Let iv(ρ,t) = | i ln(ct/ρ+ √ (ct/ρ) 2 -1), ct≥ρ -arccos(ct/ρ), -ρ<ct<ρ , H + is then given by [π + iv + i0 + , -π -iv + i0 + ] as |ct| < ρ, and ]π + iv(ρ,t), π + i0 + ] ∪ [π + i0 + , -π + i0 + ] ∪ [-π + i0 + , iv(ρ,t) -π[ as ct ≥ ρ,
above all the real singularities of the integrand as |Reα| < π, and H -is its inverse relatively to the origin. Considering the generality of (1.21), this expression can be used for a large class of causal dipersive wedge-shaped region, in particular for a wedge with frequency-dependent surface impedances [START_REF] Bernard | Progresses on the diffraction by a wedge: transient solution for line source illumination, single face contribution to scattered field, and new consequence of reciprocity on the spectral function[END_REF], [START_REF] Bernard | On the time-domain scattering by a passive classical frequency dependent wedge-shaped region in a lossy dispersive medium[END_REF].

Several Orders Asymptotic Representation for Scattering by

a Curved Impedance Wedge

Asymptotic representation in a region without creeping waves

Maliuzhinets type representation and asymptotic boundary conditions on curved faces

We consider a wedge with tangent planes at the edge defined by ϕ = ±Φ, in cylindrical coordinates (ρ, ϕ), and curved faces of radius a ± (fig. 1.2), with Φ > π/2 (note: Φ = π/2 is obtained by letting Φ → π/2). The electric and magnetic potentials E z and H z , indifferently denoted u, verify outside the wedge (∆ + k 2 )u = 0, with k = ω c the wave number of the free space, | arg(ik)| < π/2, and e iωt time dependence. As in previous section, we consider a plane wave illumination, with ϕ • the angle of incidence, and the total field u is represented as an integral of Sommerfeld-Maliuzhinets (1.3) with a contour γ composed of two infinite loops in the complex plane [10], [START_REF] Bernard | The diffraction by a curved impedance wedge : diffracted and creeping waves[END_REF], [START_REF] Bernard | [END_REF], [START_REF] Maliuzhinets | [END_REF], for |ϕ| < Φ, with f to be defined. After deformation of γ to the steepest descent path SDP, we can write

u(ρ,ϕ) = u i + ∑ u ± r + ∑ u ± s + e -ikρ 2πi SDP f (α + ϕ) e ikρ(cosα+1) dα, (1.23)
where

•the term u i is the incident field in the illuminated zone and zero in the shadow zone; •the terms u ± r correspond to geometrical optics field reflected by each faces ϕ = ±Φ (note: some secular terms are present in the asymptotic development of this term at high order).

•the terms u ± s are terms of guided waves excited by the edge, which are the contribution of complex pôles α ± s of f (α + ϕ); •the last term, named u d , is principally radiated cylindrically when ρ → ∞. Approximating f (α + ϕ) on SDP ± by its value at the stationnary phase points α = ±π, we obtain

u d ∼ -e -iπ/4-ikρ 2πkρ [ f (π + ϕ) -f (-π + ϕ)], (1.24)
where F(ϕ) = f (π + ϕ)f (-π + ϕ) is the diffraction (or far field) coefficient.

We assume impedance boundary conditions on the surface of the wedge,

( ∂ ∂ n -ik sin θ ± )u| curved face± = 0, (1.25)
with n the outward normal, and sin θ ± some constants with Reθ ± > 0.

We approximate (1.25) by asymptotic conditions in entire powers of the curvature 1/a ± of each face, B ± u = 0 on tangent plane at the edge ϕ = ±Φ, which gives [10], [START_REF] Bernard | [END_REF], [14], [15] :

(∓ ∂ ρ∂ ϕ -ik sin θ ± - 1 2a ± ( ∂ 2 ∂ ϕ 2 - ρ∂ ∂ ρ ± ik sin θ ± ρ∂ ∂ ϕ ) + O( ρ (a ± ) 2 ))u = 0. (1.26)
From the Maliuzhinets inversion theorem applied to B ± u = 0, we can write,

B ± α∓ π 2 f (α ± Φ) -B ± -α∓ π 2 f (-α ± Φ) = 0, (1.27)
where

B ± α+2π = B ± α (which implies B ± α± π 2 F(α ± Φ) + B ± -α± π 2 F(-α ± Φ) = 0)
, and,

B ± α∓ π 2 = (sin α ± sin θ ± ± D ± α ( . ) ika ± + O( 1 (ka ± ) 2 )), D ± α (.) = 1 2 [ ∂ 2 (.) ∂ α 2 - ∂ ∂ α (cotgα(.)) ± sin θ ± ∂ ∂ α ( 1 sin α ∂ ∂ α (.))].
(1.28)

We then let f = ∑ n≥0 f n /k n in the functional equation. Considering the terms of same powers in k, we then obtain a succession of functional equations of order m on the f n≤m , that we can solve, considering f • as the spectral function for a wedge with plane faces [START_REF] Maliuzhinets | [END_REF] (note: the constants sin θ ± are generally different from the ones for plane faces).

The first term of f influenced by the curvatures

Let us detail the integral expression given initially in [10], [START_REF] Bernard | The diffraction by a curved impedance wedge : diffracted and creeping waves[END_REF] for f 1 , which is the first term influenced by the curvatures. For this, we let f

• (α) = Ψ(α)σ (α)/Ψ(ϕ • )
where σ (α) = µ cos(µϕ • )/(sin(µα)sin(µϕ • )), µ = π/2Φ, ϕ • the angle of incidence, and f 1 (α)/k = Ψ(α)χ(α)/k. Taking the initial expression of χ in [10], we can shift the integration path with arbitrary real quantities ±d ± along the real axis, for each term concerning the faces ±, and, considering poles captured during the deformation, we obtain :

χ(α ) = i 8ΦΨ(ϕ • ) ∑ ± 1 ia ± ( i∞±d ± -i∞±d ± dα[(-W ± ∂ α ln((-sin α ± sin θ ± )Ψ(-α ± Φ)) + ∂ α W ± )(-σ (α ± Φ)∂ α ln( Ψ(-α ± Φ) (sin α ± sin θ ± ) ) -∂ α σ (α ± Φ)) 1 sin α ] ∓ 2πi sgnd ± ∑ 0<±sgnd ± Reα s <|d ± | Residue[...]| α=α s ), (1.29)
where

W ± =tg( π 4Φ (α ± Φ -α))
, Ψ is defined in [START_REF] Maliuzhinets | [END_REF] for wedge with plane faces. We note that this expression is the analytical continuation of the one given in [10] for Re(α

) ∈] -3Φ + d + , Φ + d + [∩] -Φ -d -, 3Φ -d -[,
which allows (also completed by (1.27)) a definition for any α .

Several orders asymptotic expressions in a region with

creeping waves terms

1.2.2.

Plane wave illumination and observation at infinity

We note, from (1.27) and (1.29), that f 1 (±(π + Φ)) and then the expression (1.24) of u d is singular for ϕ = ±Φ. In fact, near the tangents of the faces ±Φ, it is preferable to consider the radiation function F as the radiation of equivalent surface currents on a cylinder of radius a ± , that we characterize from the previous asymptotics of F, thereafter named F < . Therefore, we write [10], [START_REF] Bernard | The diffraction by a curved impedance wedge : diffracted and creeping waves[END_REF], [START_REF] Bernard | [END_REF], when

| ± ϕ -Φ| < π/2, | ± ϕ • -Φ| > π, F(ϕ) ≡ -e ika ± sin(±ϕ-Φ) iπka ± E e -iν(±ϕ-Φ) M ± (±ν) H (2) ν (ka ± ) + Q ± H (2) ν (ka ± ) dν, (1.30)
with Q ± = -i sin θ ± , and E a path from -∞iε to +∞iε with Imν < 0. Letting ν = ka ± sin α with |Reα | ≤ π/2 and | arg(ik)| < π/2, and α going from -i∞arg(ik) + 0 + to i∞ + arg(ik) + 0 + on the integration path, an asymptotic expansion of M ± is given by

M ± (±ka ± sin α ) = iπka ± 2cosα O ± ±α (F < (±(α -π/2 + Φ))), (1.31)
where

O ± α = ± 2 iπka ± ((1 + ∑ p≥1 P ± p (.) (a ± ) p+2 )B ± α )
, and P ± p are differential operators of sin α [START_REF] Bernard | [END_REF]. We have, letting α = απ/2 in (1.31) and using (1.28),

M ± (±ka ± cos α) = 1 sin α [ -iπka ± 2 O ± ±(α+π/2) ( . )]F < (±(α + Φ)) = 1 sin α [(sin α -sin θ ± ).( . ) - D ± ±α+π ( . ) ika ± + 0( 1 (ka ± ) 2 )] F < (±(α + Φ)).(1.32)
where M ± (±ka ± cos α) has no singularity at α = 0, from (1.27) and

O ± α± π 2 F < (α ± Φ) + O ± -α± π 2 F < (-α ± Φ) = 0 [12].
The path E is such that the singularities of the integrand in (1.30), due to M ± and to zeros of

H (2) ν (ka ± )+Q ± H (2)
ν (ka ± ), are respectively above and below E . Deforming the path E so that no singularity crosses it, we can analytically continue the edge diffraction coefficient for arbitrary ϕ • . Besides, by application of reciprocity principle, our expression remains valid if we exchange the roles of ϕ • and ϕ. Furthermore, we will derive an uniform transition expression at finite distance with all radiation terms.

Plane wave illumination and observation at finite distance

To consider an observation point (ρ,ϕ) at finite distance, we can then use that,

u(ρ, ϕ) = - 1 2πi S • F(α ± (Φ -π/2))e -ikρ cos(α±(Φ-π/2)-ϕ) dα, (1.33)
is valid for 0 < ±ϕ < Φ when no source in the zone Φπ < ±ϕ < Φ, with correctly chosen S • fromarg(ik) -i∞ to arg(ik) + i∞. We then use (1.30) in (1.33) with

L ± < ≡ M ± iπka ± , following, u(ρ, ϕ) = 1 2πi E dν L ± < (±ν) H (2) ν (ka ± ) + Q ± H (2) ν (ka ± ) × S • dαe -ikρ cos(α±(Φ-π/2)-ϕ)-iν(±α-π/2)-ika ± cos α = E dν L ± < (±ν) H (2) ν (ka ± ) + Q ± H (2) ν (ka ± ) × e ∓iνΩ 2i ( e iνπ/2 π S • e -ikR cos(α-Ω)-iν(±(α-Ω)) dα), (1.34)
and we obtain,

u(ρ, ϕ) = 1 2i E L ± < (±ν)H (2) ν (kR) e ∓iνΩ H (2) ν (ka ± ) + Q ± H (2) ν (ka ± ) dν, (1.35)
where, relatively to the curvature center of each face, Ω is the angle from the edge to the point (ρ, ϕ), and

R = ρ 2 + (a ± ) 2 + 2ρa ± cos(Φ -π/2 ∓ ϕ) is the distance to (ρ, ϕ). The domain of validity of (1.35) is | ± ϕ -Φ| < π/2.
When we go beyond the frontier ϕ = ±Φ, there is no more stationnary phase point for this integral, and considering the residues terms at zeros ν

± p = ka ± cos α ± p of [H (2) ν (ka ± ) + Q ± H (2)
ν (ka ± )] in the zone Imν < 0, then closing the contour E at infinity, we obtain, when ±ϕ > Φ,

u(ρ, ϕ) = ∑ p -πL ± < (±ν ± p )H (2) ν ± p (kR)e ∓iν ± p Ω ∂ ∂ ν (H (2)
ν (ka ± ) + Q ± H (2) ν (ka ± ))| ν ± p .
(1.36)

The expressions of F and u are then uniform in the region ϕ ∼ ±Φ. It permits the continuous transformation of the field into creeping waves (a development in entire powers then changes continuously to one in fractional powers as ϕ varies), for any wedge angle. This result has been taked up and confirmed for first terms by Borovikov in his paper on impedance wedge with curved sides [14].

We remark that, contrary to expressions of Albertsen and Christiansen [START_REF] Albertsen | hybrid diffraction coefficients for first and second order discontinuities of two dimensional scatterers[END_REF], the operators go beyond first order. We can then consider a second order development of f without making diverge the expression (1.35), and we can study, for example, at low contrasts, the case of a curved plane with a discontinuity of material 'and' of curvature.

Remark : Noticing that the diffraction coefficient F < for a cylinder C a ± ,∞ of large radius a ± without discontinuity is null, we can analytically continue the solution for Φπ < ±ϕ • < Φ, by adding the exact contribution of C a ± ,∞ in (1.36).

Remark : For ν ≈ ka ± , H

ν (ka ± )+Q ± H

(2)

ν (ka ± ) ∼ i (m ± ) 2 √ π (w 2 (τ)-q ± w 2 (τ)), where m ± = (ka ± /2) 1/3 , q ± = -im ± sin θ ± , ν = ka ± + m ± τ, w 2 = √ π(Bi -iAi).

Point source illumination and observation at finite distance

We can then derive an expression at finite distance for observation point (ρ , ϕ ) and point source (ρ, ϕ) radiating u i = i 2 H

(2)

• (k ρ 2 + (ρ ) 2 + 2ρρ cos(ϕ -ϕ ))
. For this, we consider reciprocity, exchange the directions of observation and of incidence in

F < , let L ± (±ka ± cos α ) = A α ( 1 2πi γ f (α + ϕ,±(α + Φ))e ikρ cos α dα), with A α = -O ± ±(α+π/2) 2 sin α
, and obtain, when

| ± ϕ -Φ| > π and ϕ ∼ ±Φ, u(ρ , ϕ , ρ, ϕ) = 1 2i E L ± (±ν ) H (2) ν (kR ) e ∓iν Ω H (2) ν (ka ± ) + Q ± H (2) ν (ka ± ) dν , (1.37)
where

R = ρ 2 + (a ± ) 2 + 2ρ a ± cos(Φ -π/2 ∓ ϕ ),
and Ω is the angle, relatively to the cylinder center, from the edge to the observation point (ρ , ϕ ). This expression remains valid, by analytical continuation, for |ϕ| < Φ, and by reciprocity, when the roles of (ρ, ϕ) and (ρ , ϕ ), as source and observation points, are exchanged.

When we go beyond the frontier ϕ = ±Φ, there is no more stationnary phase point for this integral, and considering the residues terms at zeros ν

± p of [H (2) ν (ka ± )+ Q ± H (2)
ν (ka ± )] in the zone Imν < 0, we obtain, when ±ϕ > Φ,

u(ρ , ϕ , ρ, ϕ) = ∑ p -πL ± (±ν ± p )H (2) ν ± p (kR )e ∓iν ± p Ω ∂ ∂ ν (H (2)
ν (ka ± ) + Q ± H (2) ν (ka ± ))| ν ± p , (1.38)
where

L ± (±ka ± cos α p ) = A α p ( 1 2πi γ f (α + ϕ,±(α p + Φ))e ikρ cos α dα) can be de- veloped as A α p (u i,p + ∑ u ± r,p + ∑ u ± s,ε,p + e -ikρ 2πi SDP f (α + ϕ, ±(α p + Φ)) e ikρ(cosα+1) dα).
For lϕ > Φ and l ϕ > Φ (l,l are equal to +or-1), the problem is more complex and we need to express L ± (±ν ) as an integral of previous form. We then obtain for the diffracted part of the field :

u da = π 2 ∑ p,q A l α l p A l α l q ( f (π + l(α l p + Φ), l (α l q + Φ)) -f (-π + l(α l p + Φ), l (α l q + Φ))) H (2) ν l p (kR) e -liν l p Ω ∂ ∂ ν (H (2)
ν (ka l ) + Q l H (2)
ν (ka l ))| ν=ν l p × H (2)
ν l q (kR )e -l iν l q Ω ∂ ∂ ν (H (2)
ν (ka l ) + Q l H (2) ν (ka l ))| ν=ν l q , (1.39)
Let us notice that this expression has a form similar to the one defined by Thompson for the particular case of a discontinuity of materials on a circular cylinder.

Remark : If we denote f a ± ,∞ the asymptotic spectral function for a cylinder C a ± ,∞ of radius a ± (resp. for the PO contribution of face ±), we can analytically continue (1.37)-

(1.38) for Φ -π < ±ϕ < Φ by letting f (α) -f a ± ,∞ (α ∓ (Φ -π 2 )
) in place of f (α) and add the exact contribution of C a ± ,∞ (resp. of PO currents on face ±) in (1.37)-(1.38).

Some validations concerning the expression of

f = ∑ n≥0 f n /k n for arbitrary wedge angle

Perfectly conducting case: curved half plane to discontinuity of curvature

We now compare the results obtained from (1.29) with the expressions known for some particular wedge angles, with Φ ≥ π/2, in the limit case sin θ ± → ∞ so that Ψ = 1. Letting ε =sgn(Reα ), we choose to take d + = (1 + ε)Φ and d -= (1ε)Φ.

Denoting a δ = a + or a -when δ = +1 or -1, we can write

χ(α ) = ( 1 8Φ ∂ α ∂ ϕ • i∞ -i∞ dα[tg( π 4Φ (α -εΦ -α)) π 2Φ sin π 2Φ α (cos π 2Φ α + ε sin π 2Φ ϕ • ) × ( 1 a -ε sin α + 1 a ε sin(α + 2εΦ) )]) + (i ε a ε π 4Φ ∂ α ∂ ϕ • [ 1 sin(εΦ -ϕ • ) × tg( π 4Φ (α + ϕ • )) + tg( π 4Φ (α + ε(Φ -π))) επ 2Φ sin π 2Φ π (cos π 2Φ π -ε sin π 2Φ ϕ • ) ]), (1.40)
for Re(α ) ∈] -3Φ, 3Φ[ with ε =sgn(Reα ).

In the case of a curved half-plane, Φ = π and εa ε = a, and so we obtain

χ(α ) = (0) + i 1 4a ∂ α ∂ ϕ • [ 1 sin(ϕ • ) tg( 1 4 (α + ϕ • )) - 1 2 tg( 1 4 α ) 1 sin( 1 2 ϕ • ) ]. (1.41)
This gives us the second order term for far field function

( f 1 (π + ϕ) -f 1 (-π + ϕ))/k = i 4ka ∂ ϕ ∂ ϕ • ( tg((ϕ + ϕ • )/2) cos(ϕ/2) cos(ϕ • /2) ), (1.42)
which is conform to the result of Borovikov [17], first developed by V.B. Filippov.

For the discontinuity of curvature, Φ = π 2 and then the integral term for χ is 1 4π

∂ α ∂ ϕ • i∞ -i∞ dα[ cos α (cos α + ε sin α ) 1 (cos α + ε sin ϕ • ) ( 1 a + - 1 a -)] = i 2π ∂ α ∂ ϕ • ε cos α (sin ϕ • -sin α ) ( π/2 -εα cos α - π/2 -εϕ • cos ϕ • )( 1 a + - 1 a -), (1.43)
for -π/2 < εReα < 3π/2. The expression for χ then gives us

( f 1 (π + ϕ) -f 1 (-π + ϕ))/k = -2i cos ϕ cos ϕ • (sin ϕ + sin ϕ • ) 3 ( 1 ka + - 1 ka -), (1.44)
which is conform to the diffraction coefficient of Kaminetzky and Keller [15].

Discontinuity of curvature in an impedance surface

In this case, we have Φ = π/2 et sin θ ± = sin θ , and F(ϕ) = ( f 1 (π + ϕ)f 1 (-π + ϕ))/k. Instead of directly using (1.29), we prefer here to rewrite the functional equation in f 1 , letting α = ±π/2 + ϕ,

f 1 (±π + ϕ) cos ϕ + sin θ -cos ϕ + sin θ -f 1 (-ϕ) = = D ± ±π/2+ϕ ( f • (±π + ϕ)) -D ± ∓π/2-ϕ ( f • (-ϕ)) ia ± (cos ϕ -sin θ ) . (1.45)
From sin θ ± = sin θ and f • (π + ϕ) = f • (-π + ϕ), we have,

D + π/2+ϕ ( f • (π + ϕ)) -D + -π/2-ϕ ( f • (-ϕ)) = D - -π/2+ϕ ( f • (π + ϕ)) -D - π/2-ϕ ( f • (-ϕ)) (1.46)
and thus,

f 1 (π + ϕ) -f 1 (-π + ϕ) = i( 1 a + - 1 a -)× D + π/2+ϕ ( f • (π + ϕ)) -D + -π/2-ϕ ( f • (-ϕ)) (cos ϕ + sin θ ) . (1.47) Using f • (α) = cos α+sin θ cos ϕ • +sin θ cos ϕ •
sin α-sin ϕ • , and expanding D, we obtain

f 1 (π + ϕ) -f 1 (-π + ϕ) = ( 1 a + - 1 a -)× 2i cos ϕ cos ϕ • (1 + sin ϕ sin ϕ • -sin 2 θ ) (cos ϕ • + sin θ )(cos ϕ + sin θ )(sin ϕ + sin ϕ • ) 3 , (1.48)
which is perfectly in agreement with the result of Kaminetzky and Keller [15].

A Novel Expression of the Field for Arbitrary Bounded

Sources above a Passive or Active Impedance Plane

Formulation of the problem

We consider the scattering by an imperfectly reflective plane when it is illuminated by the field radiated by a bounded source, composed of arbitrary electrical and magnetic currents J and M (see fig. 1.3). The plane is defined by z = 0 in Cartesian coordinates (x, y, z). A harmonic time dependence e iωt , from now on assumed, is suppressed throughout. Each component of the scattered field is assumed to be regular in the domain z > 0, and O(e -γ|OP| ) at P(x, y, z), γ > 0, as z or ρ → ∞, when | arg(ik)| < π/2. Following Harrington [18, p.131] (see also Jones [START_REF] Jones | The theory of electromagnetism[END_REF]), we can write the electric field E and the magnetic field H satisfying the Maxwell equations, with two scalar potentials E and H , following

E = -ikcurl(H z) + (grad(div(.)) + k 2 )(E z) µ 0 ε 0 H = ikcurl(E z) + (grad(div(.)) + k 2 )(H z) (1.49)
where (∆+k 2 )E = 0 and (∆+k 2 )H = 0 outside the sources of radiation (i.e. outside J, M, and the scatterer), with k = ω √ µ 0 ε 0 , the constants ε 0 and µ 0 being respectively the permittivity and the permeability of the medium above the plane, | arg(ik)| ≤ π/2. Following the theory of this representation, the constant vector z can be chosen regardless of the sources, and E (or H )≡ e ±ikz has no influence on (E, H). Thereafter, we denote (E inc , H inc ) and (E s ,H s ) the potentials corresponding to the incident field (incoming wave) and the scattered field (outgoing wave), and write (1.49) in the compact form (E, µ 0 ε 0 H) = L ( zE , zH ). In [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF], we considered multimode boundary conditions on an isotropic plane, where β is the angle of incidence with the normal z [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF], g (e,h) j are complex constants, N and P are two positive numbers. This class of problem corresponds to the reflection by a multilayer [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF][START_REF] Brekhovskikh | Waves in layered media[END_REF][START_REF] Bernard | [END_REF], composed of isotropic media, or more generally, of uniaxial anisotropic media with the principal axis along z, backed with a perfectly reflective plane. From the symmetry at normal incidence, we notice that the condition R T E (0) = -R T M (0) has also to be satisfied. This implies that g e 1 = 1/g h 1 for monomode conditions, when N = P = 1. We now restrict ourselves to this latter case, commonly named the impedance case, and take an arbitrary complex number g e 1 = g e , corresponding to a passive (Reg e > 0) or active (Reg e < 0) plane. Using

N ∏ j=1 ( ∂ ∂ z -ikg e j )E z,tot | z=0 = 0, P ∏ j=1 ( ∂ ∂ z -ikg h j )H z,tot | z=0 = 0, (1.50)
E z = ∂ 2 E ∂ z 2 + k 2 E , µ 0 ε 0 H z = ∂ 2 H ∂ z 2 + k 2 H , (1.52)
we will search the potentials E s and H s , satisfying the Helmholtz equation as z > 0, regular and vanishing as z → ∞ when | arg(ik)| < π/2, that verify

( ∂ ∂ z -ikg e )E s (z) = ( ∂ ∂ z + ikg e )E inc (-z), ( ∂ ∂ z -ikg h )H s (z) = ( ∂ ∂ z + ikg h )H inc (-z) (1.53)
as z ≥ 0, with g h = 1/g e . Therefore, we need first a definition of (E inc , H inc ) as z ≤ 0.

Remark :

The boundary conditions on normal components can be simply deduced from boundary conditions on the tangential components after differentiation of them.

An expression of potentials (E inc , H inc ) for bounded sources J and M

Let us consider the incident field (E, H) at r of coordinates (x, y, z), radiated by arbitrary electric and magnetic bounded sources J and M [START_REF] Jones | The theory of electromagnetism[END_REF],

E = curl(G * M) + i ωε 0 (grad(div(.)) + k 2 )(G * J) µ 0 ε 0 H = - µ 0 ε 0 curl(G * J) + i k (grad(div(.)) + k 2 )(G * M) (1.54)
where G(r) = -e -ik|r(x,y,z)| 4π|r(x,y,z)| with |r| = x 2 + y 2 + z 2 , and * is the convolution product. The potentials (E inc , H inc ) for this field, satisfying the Helmholtz equation outside the sources, and vanishing at infinity when | arg(ik)| < π/2 as ±z → ∞, have a particularly compact expression, that we develop in [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF] for arbitrary sources. It is given by,

(E inc , H inc ) = z 8πk 2 ( µ 0 ε 0 (grad(div(J)) + k 2 J, ikcurl(J))+ + (-ik curl(M), grad(div(M)) + k 2 M)) * W = z 8πk 2 L ( µ 0 ε 0 J, M) * W ,(1.55)
where we can take, to solve (1.53) as z > 0,

W (r) = (e ik|z| E 1 (ik(|r|+|z|)) + e -ik|z| (E 1 (ik(|r|-|z|)) + 2 ln ρ)) (1.56)
with ρ = x 2 + y 2 , E 1 being the exponential integral function [START_REF] Abramowitz | Handbook of mathematical functions[END_REF], and the notation (A, B) * C ≡ (A * C, B * C). The reader can verify by inspection our expression, considering that the following conditions are satisfied when ±z > 0,

( ∂ 2 ∂ z 2 + k 2 ) W (r) 8πik = G(r), (∆ + k 2 )W (r) = 0 (1.57)
and that all derivatives of W are regular in these domains.

Remark :

It is worth noticing that we have ∆ xy ln(ρ) = 0 for ρ = 0, which implies that ln ρ in (1.56) has no influence on the field for z = 0, except by its singularity at ρ = 0. So, another way to proceed would be to suppress from the sources J and M a vertical tube enclosing the singularity, with a radius that we let tend to 0 in the field expression.

Expression of the potentials (E s , H s ) for an impedance plane

Using our expression of (E inc , H inc ) for the radiation of J and M, we can express the potentials E s and H s which satisfy the impedance boundary conditions (1.53), from the method developed in [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF]. So, letting N = P = 1 in [19, prop.(5.2)], we obtain, as z ≥ 0,

E s (x, y, z) = E inc (x, y, -z) + (( z ωε 0 grad(div(J)) + k 2 J 8πk + + z k (-ik curl(M)) 8πk
) * ∑ ε =-1,1 a e (g e -ε ) (V ε + ε K g e ))(x, y, -z) = E inc (x, y, -z) + (( z ωε 0 grad(div(J)) + k 2 J 8πk + z k (-ik curl(M)) 8πk ) * * ∑ ε =-1,1 (( ε + g e ε -g e -1)V ε + ε a e K g e (g e -ε ) ))(x, y, -z) (1.58)
and

H s (x, y, z) = H inc (x, y, -z) + (( z ωε 0 (ik curl(J)) 8πk + z k (grad(div(M)) + k 2 M) 8πk
) * ∑ ε =-1,1 a h (g h -ε ) (V ε + ε K g h ))(x, y, -z) = -H inc (x, y, -z) + (( z ωε 0 (ik curl(J)) 8πk + z k (grad(div(M)) + k 2 M) 8πk ) * * ∑ ε =-1,1 (( ε + g h ε -g h + 1)V ε + ε a h K g h (g h -ε ) ))(x, y, -z) (1.59)
where g e = 1/g h , a e,h = -2g e,h . In these expressions, the functions V ε , K g satisfy

V ε (x, y, -z) = e ε ikz (E 1 (ik(|r|+ε z)) + (1 -ε ) ln ρ), K g (x, y, -z) = e ikgz J g (ρ, -z) (1.60)
where ρ = x 2 + y 2 , g = g e or g = g h , and J g (ρ, -z) is given by the integral

J g (ρ, -z) = e -ikgz 2 D H (2)
0 (kρ sin β )e -ikz cos β cos β + g sin β dβ , (1.61)
with Re(ik sin β ) = 0 on D from -i∞arg(ik) to i∞ + arg(ik), which is a Fourier-Bessel integral commonly encountered in scattering theory [21, p.234], also called a Sommerfeld-type integral. Letting g = sin θ 1 with |Re(θ 1 )| ≤ π/2, we notice the presence of a cut with Re(ik cos θ 1 ) = 0 in active case (Reg < 0), which is due to poles of (cos β + g) -1 that can go through D. Thus, as discussed in [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF], any transformations that does not consider the presence of this branch cut can be wrong in active case.

To avoid any restriction, we have developed in [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF], [START_REF] Bernard | On a novel expression of the field scattered by an arbitrary constant impedance plane[END_REF], [START_REF] Bernard | Propagation over a constant impedance plane: arbitrary primary sources and impedance, analysis of cut in active case, exact series, and complete asymptotics[END_REF], a novel expression for arbitrary g = sin θ 1 with |Re(θ

1 )| ≤ π/2 as | arg(ik)| ≤ π/2, J g (ρ, -z) = i i∞ b e -a cos α dα (1.62)
where the parameters a and b, with |Reb| ≤ π, Rea > 0, are defined following,

e ∓ib = ikR a (1 ± sin θ 1 )(1 ± cos ϕ), a = εikR sin ϕ cos θ 1 , -ia sin b = ikR(cos ϕ + sin θ 1 ), a cos b = ikR(1 + sin θ 1 cos ϕ) (1.63)
and z = R cos ϕ, ρ = R sin ϕ, R = ρ 2 + z 2 , ε =sgn(Re(ikR sin ϕ cos θ 1 )) (Re(a) = 0 being a limit case), 0 < ϕ < π/2. So defined, we have sgn(Reb) = -εsgn(Im(sin θ 1 )), and sgn(Ima) = -sgn(arg(ik)) when ε = -1.

We notice that, as g varies in the complex plane, this expression has a correct cut as ε changes of sign for Reg ≤ 0, is singular for g = -1, and is regular elsewhere (note: for Reg > 0, the change of sign of ε does not induce a cut as g varies).

Let us remark that this integral was also given in [START_REF] Koh | Exact closed-form expression of a Sommerfeld integral for the impedance plane problem[END_REF] for passive impedance case but it was with a definition of parameters which restricts its application (see details in section 2 of [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF]). It is intimately related to the incomplete cylindrical function in the Poisson form [START_REF] Agrest | Theory of incomplete cylindrical functions and their applications[END_REF] and to the leaky aquifer function [START_REF] Harris | Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions[END_REF]. The reader can refer to [START_REF] Bernard | On the expression of the field scattered by a multimode plane[END_REF], [START_REF] Bernard | On a novel expression of the field scattered by an arbitrary constant impedance plane[END_REF], [START_REF] Bernard | Propagation over a constant impedance plane: arbitrary primary sources and impedance, analysis of cut in active case, exact series, and complete asymptotics[END_REF] for its calculus in passive or active impedance cases.

Remark : The reader can verify by inspection that, when z > 0,

∂ J g (ρ, -z) ∂ z = e -ik(R+gz) R , (∆ + k 2 )(e ikgz J g (ρ, -z)) = 0. (1.64)
Remark : Let us notice that the solution for an impedance plane in acoustics given by Ingard [START_REF] Ingard | On the reflection of a spherical sound wave from an infinite plane[END_REF] presents, for some values of g with Re(g) > 0 (passive), non physical discontinuities as ϕ varies (due to errors in surface waves contributions reported by Wenzel [START_REF] Wenzel | Propagation of waves along an impedance boundary[END_REF]), that Thomasson [START_REF] Thomasson | Reflection of waves from a point source by an impedance boudary[END_REF] has corrected.

Spectral Representation of the Field for 3D Conical Scatterers

We describe here some new aspects of integral representations in problems with conical boundary conditions, encountered in electromagnetic scattering theory. We carefully analyse some new points in the general expression of Debye potentials in electromagnetism, and give an novel efficient representation of incident plane waves for solving the problems of scattering with conical geometries.

Formulation

We consider the diffraction of an electromagnetic plane wave by an imperfectly conducting cone C, in the domain Ω of the three-dimensional Euclidean space R 3 bounded by C. A dependence on time e iηt is assumed and suppressed throughout. The Maxwell's equations for the scattered wave field are given in Ω by

ik E = rot(Z 0 H), ikZ 0 H = -rot E, (1.65)
where k = η √ ε 0 µ 0 and Z 0 are respectively the wave number and the impedance of the free space, E and H are respectively the electric and the magnetic field, and k is complex with |arg(ik)| ≤ π/2. We introduce the representation of fields outside any sources with Debye potentials (E , H ) [START_REF] Jones | The theory of electromagnetism[END_REF],

E = rot rot( rE ) -ikrot( rH ), Z 0 H = rot rot( rH ) + ikrot( rE ), (1.66)
where rot rot = ∇div -, and the potentials E , H satisfy the Helmholtz equations.

We then have the expressions

E = e r (∂ 2 r (rE ) + k 2 rE ) + r -1 ∇ ω ∂ r (rE ) -ik∇ ω H ∧ e r , Z 0 H = e r (∂ 2 r (rH ) + k 2 rH ) + r -1 ∇ ω ∂ r (rH ) + ik∇ ω E ∧ e r , (1.67)
where

∇ ω = e θ ∂ θ + e ϕ
sin θ ∂ ϕ , (r, θ , ϕ) are the spherical coordinates, and ( e r , e θ , e ϕ ) are the unit vectors basis of the spherical coordinate system, which gives us,

E r = k 2 rE + ∂ 2 ∂ r 2 (rE ) , E θ = 1 r ∂ 2 ∂ r∂ θ (rE ) - ik sin θ ∂ H ∂ ϕ , E ϕ = 1 r sin θ ∂ ∂ r (r ∂ E ∂ ϕ ) + ik ∂ H ∂ θ , (1.68)
and

Z • H r = k 2 rH + ∂ 2 ∂ r 2 (rH ) , Z • H θ = 1 r ∂ 2 ∂ r∂ θ (rH ) + ik sin θ ∂ E ∂ ϕ , Z • H ϕ = 1 r sin θ ∂ ∂ r (r ∂ H ∂ ϕ ) -ik ∂ E ∂ θ . (1.69)
Let the two-dimensional unit sphere S 2 be centered at the vertex T of the convex cone C (fig. 1.4). We denote σ = S 2 ∩C the boundary of the domain Σ ⊂ S 2 which is cut out on the unit sphere by the cone C. We assume that the curve σ is smooth and that Σ is geodesically convex and belongs to a hemisphere of S 2 . The vector r = (r, ω) refers to the observation point, r = | r| is the distance from T to this point, and ω is the point of the unit sphere defined by ω = r/r, described by two angular coordinates (θ , ϕ) which have the traditional meaning as in the spherical coordinate system. In what follows, θ (ω, ω 0 ) = dist(ω, ω 0 ) = arccos < ω, ω 0 > R 3 , is the geodesic distance between the points ω and ω 0 on the unit sphere S 2 .

The conditions at infinity and at the vertex are those given in [33, sect.2]. To exhibit the properties of scattered fields for large kr, we rather choose to consider illumination by a point source at r 0 that we let tend to ∞, kr being large but fixed. Denoting θ (ω, ω 0 ) = inf ω ∈σ ( θ (ω, ω ) + θ (ω , ω 0 )) the length of the geodesics (broken for the reflected wave), we note that the reflection and shadow regions, in the sense of geometrical optics, are defined by the inequality θ (ω, ω 0 ) ≤ π, while θ (ω, ω 0 ) > π is the zone where the behaviour for large |kr| and |kr 0 | is dictated by the spherically diffracted wave excited by the tip of the cone. We can then assume that the scattered field in Ω, verifies for large kr as r 0 → ∞ , when |arg(ik)| < π/2 [START_REF] Bernard | Electromagnetic scattering by a smooth convex impedance cone[END_REF],

ikr 0 exp(ikr 0 )( E, Z 0 H) = O(exp(ikr cos(π/2 + ε(ω, ω 0 )))),
(1.70)

Integral and non-integral terms

Considering the general conditions at the apex given in [START_REF] Bernard | Electromagnetic scattering by a smooth convex impedance cone[END_REF], the wave field is O((kr) m ) with m > -3/2 as r tends to 0. Thus, assuming the Debye potentials and their angular derivatives are O(r l ) in vicinity of the tip of cone, we obtain, from the expressions of fields with potentials, that (l + 1)r l-1 = O((kr) m ), m > -3/2. The latter condition is satisfied if l > -1/2 or l = -1. We then look for the representations of the scattered wave field potentials in the form,

(E , H ) = 2 k 2 π ( i∞ -i∞ ν sin πν ν 2 -1/4 K ν (ikr) √ ikr g e,h (ω, ω 0 , ν)dν +iπ K 1/2 (ikr) √ ikr w e,h (ω, ω 0 )), (1.71)
where K ν is a modified Bessel function, and K 1/2 (ikr) = π 2ikr exp(-ikr). The integral terms are of Kontorovich-Lebedev type, and g e,h (ω, ω 0 , ν) are even analytical functions of ν that we assume definite at ν = ±1/2. We can derive, from the theory of KL transform [33, sect.3.3 and app.A], that these expressions are valid as |arg(ik)| < ε(ω, ω 0 ) when θ (ω, ω 0 ) ≤ π, and as |arg(ik)| ≤ ε(ω, ω 0 ) when θ (ω, ω 0 ) > π, which is equivalent to θ (ω, ω 0 ) > π/2 + |arg(ik)|.

We can write the Helmholtz equation for the potentials expressed with (1.71). Multiplying this equation by r 3 , and assuming that angular derivatives of the integral term in (1.71) remain O(r -1/2 ) as r tends to 0, we notice that ω w e,h (ω, ω 0 ) = O(r 1/2 ) as r → 0, where ω is the Laplacian on the unit sphere (also called the Laplace-Beltrami operator), and thus ω w e,h (ω, ω 0 ) = 0. In consequence, the KL integral term and the non-integral term in (1.71) satisfies the Helmholtz equation independently, and, we obtain that, ( ω + ν 2 -1/4)g e,h (ω, ω 0 , ν) = 0, ω w e,h (ω, ω 0 ) = 0.

(1.72)

1.4.2.2
The equality w e,h (ω, ω 0 ) = g e,h (ω, ω 0 , 1/2), and the compatibility conditions on g e,h (ω, ω 0 , 1/2)

Considering that the energy flux through a small sphere surrounding the vertex vanishes as its surface collapses [START_REF] Jones | The theory of electromagnetism[END_REF], we can shown [33, sect.2] that we have no pure term of the form p(ω, ω 0 ) exp(-ikr)/ikr which satisfies the Maxwell equations in the expression of the field. To express this condition on g e,h , we begin to introduce the linear combinations C ± = E ± iZ 0 H and M ± = E ± iH which satisfy

C ± = rot rot( rM ± ) ∓ krot( rM ± ), (1.73)
or, developed in the spherical coordinates,

C ±,r = k 2 rM ± + ∂ 2 ∂ r 2 (rM ± ) , C ±,θ = 1 r ∂ 2 ∂ r∂ θ (rM ± ) ∓ k sin θ ∂ M ± ∂ ϕ , C ±,ϕ = 1 r sin θ ∂ ∂ r (r ∂ M ± ∂ ϕ ) ± k ∂ M ± ∂ θ .
(1.74)

Considering combinations of the terms in (1.74), we obtain

C ±,θ ± iC ±,ϕ = ( ∂ ∂ θ ± i ∂ sin θ ∂ ϕ )( ∂ r∂ r (rM ± ) + ik(M ± )), C ±,θ ∓ iC ±,ϕ = ( ∂ ∂ θ ∓ i ∂ sin θ ∂ ϕ )( ∂ r∂ r (rM ± ) -ik(M ± )).
(1.75)

Using (1.71), we can derive field terms with strict exp(-ikr)/ikr dependence in (1.75). Concerning integral terms, we exploit the relations 2νK ν (z)/z = K ν+1 (z) -K ν-1 (z) and 2K ν (z) = -K ν+1 (z)-K ν-1 (z). We then deform the contours of integration so that we capture residue terms attached to simple poles of K ν±1 (z)/(ν 2 -1/4) at ν = ∓1/2. We add to these terms the contribution of non integral terms of (1.71), and isolate the complete field term with strict exp(-ikr)/ikr dependence in (1.75).

Writing that it must vanish, we derive the compatibility conditions,

∂ g e (ω, ω 0 , 1/2) ∂ θ = ∂ g h (ω, ω 0 , 1/2) sin θ ∂ ϕ , ∂ g h (ω, ω 0 , 1/2) ∂ θ = - ∂ g e (ω, ω 0 , 1/2) sin θ ∂ ϕ , (1.76)
and

∂ (g e (ω, ω 0 , 1/2) -w e (ω, ω 0 )) ∂ θ = - ∂ (g h (ω, ω 0 , 1/2) -w h (ω, ω 0 )) sin θ ∂ ϕ , ∂ (g h (ω, ω 0 , 1/2) -w h (ω, ω 0 )) ∂ θ = ∂ (g e (ω, ω 0 , 1/2) -w e (ω, ω 0 )) sin θ ∂ ϕ .
(1.77)

All choices of w e,h satisfying this condition and Laplace-Beltrami equation gives the same fields E and H, and we can choose w e,h (ω, ω 0 ) = g e,h (ω, ω 0 , 1/2). We then obtain for the potentials,

(E , H ) = 2 k 2 π ( i∞ -i∞ ν sin πν ν 2 -1/4 K ν (ikr) √ ikr g e,h (ω, ω 0 , ν)dν +iπ K 1/2 (ikr) √ ikr g e,h (ω, ω 0 , 1/2)), (1.78)
with (1.76) which ensures that E = ( C + + C -)/2 and Z 0 H = ( C + -C -)/2i do not contain any non-integral terms with strict exp(-ikr)/r radial dependence.

It is useful to note that the equalities in (1.76) can be rewritten as

( e ϕ ∓ i e θ )∇ ω (g e (ω, ω 0 , 1/2) ± ig h (ω, ω 0 , 1/2)) = 0, (1.79) or ( ∂ ∂ ϕ ∓ i sin θ ∂ ∂ θ )(g e (ω, ω 0 , 1/2) ± ig h (ω, ω 0 , 1/2)) = 2 ∂ ∂ z ± (g e (ω, ω 0 , 1/2) ± ig h (ω, ω 0 , 1/2)) = 0, (1.80)
where z ± = ϕ ± i ln(tan(θ /2)), z + = z * -. We can generalize in a simple manner the conditions (1.79) and (1.80) on g e,h (ω, ω 0 , 1/2) for a new orthogonal vectors basis ( e 1 , e 2 = r ∧ e 1 ) on the unit sphere, either by noticing that ( e θ ± i e ϕ ) = ( e 1 ±i e 2 ) e 1 .( e θ ∓i e ϕ ) or by using a conformal transformation in (1.80), and we have in particular, ( e 2 ∓ i e 1 )∇ ω (g e (ω, ω 0 , 1/2) ± ig h (ω, ω 0 , 1/2)) = 0.

(1.81)

Because of the complexity of the expressions of the fields with potentials defined with (1.78), the properties of the spectral functions are more difficult to derive than in acoustics case [START_REF] Jones | Scattering by a cone[END_REF][START_REF] Bernard | Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section[END_REF]. For details, the reader can refer to [START_REF] Bernard | Electromagnetic scattering by a smooth convex impedance cone[END_REF], [START_REF] Bernard | Advanced theory of diffraction by a semi-infinite impedance cone[END_REF]. We now insist on an efficient expression of the potentials attached to an incident electromagnetic plane wave.

Remark : notice that ω g e,h (ω, ω 0 , 1/2) = 0 can be written, with the variables z

± , ∂ z + ∂ z -g e,h (ω, ω 0 , 1/2) = 0.
Remark : We easily deduce from (1.76) that the fields, respectively attached to the non-integral terms of E and H in (1.71), are equal. Another remarkable consequence of (1.76) is the fact that changing exp(-ikr) into exp(ikr) in the nonintegral terms of the potentials lead their contributions to vanish. This implies that we can take -2i sin(kr) in place of exp(-ikr) in the expressions of non-integral terms.

Remark : Functional equations on g e,h can be derived from (1.78) and boundary conditions on the cone, for some domain of complex k, then be used for arbitrary k from analyticity. Once g e,h determined, we can calculate (1.78), directly in the oasis region with θ (ω, ω 0 ) > π, or more generally after subtracting some approximation of g e,h as ν → i∞ due to geometrical optics contributions (with Physical Optics approximation for example) (see [34, p.63]).

An efficient expression

We here show how to obtain simple explicit expressions of potentials (E i , H i ) for an incident plane wave electromagnetic field [START_REF] Bernard | Electromagnetic scattering by a smooth convex impedance cone[END_REF], without usual series of spherical harmonics, and we detail some of their complex properties.

Let us consider a plane wave, described in spherical coordinates, following

E i = ( θ 0 sin β + ϕ 0 cos β )e ikr cos θ (ω,ω 0 ) , Z 0 H i = ( θ 0 cos β -ϕ 0 sin β )e ikr cos θ (ω,ω 0 ) , (1.82)
with ω 0 = (θ 0 , ϕ 0 ) the point on the unit sphere S 2 attached to the direction of incidence (i.e. the direction from which the wave comes), ( θ 0 , ϕ 0 ) the spherical vectors associated to ω 0 , and cos θ (ω, ω 0 ) = cos θ cos θ 0 + sin θ sin θ 0 cos(ϕϕ 0 ). Concerning the exponential term in (1.82), we notice [START_REF] Bernard | Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section[END_REF], that

u i := e ikr cos θ (ω,ω 0 ) = 4 i √ 2π +i∞ -i∞ ν sin(πν)u i ν (ω, ω 0 ) K ν (ikr) √ ikr dν, (1.83)
where u i ν (ω, ω 0 ) = -P ν-1/2 (cos θ (ω,ω 0 )) 4 cos(πν)

, with ( ω + ν 2 -1/4)u i ν (ω, ω 0 ) = δ (ωω 0 ) [START_REF] Bernard | Diffraction of acoustic waves by an impedance cone of an arbitrary cross-section[END_REF], [START_REF] Smyshlyaev | The high frequency diffraction of electromagnetic waves by cones of arbitrary cross section[END_REF], [START_REF] Babich | Numerical calculations of the diffraction coefficients for an arbitrary shaped perfectly conducting cone[END_REF] and the integral in (1.83) converging provided θ (ω, ω 0 ) > π/2 + |arg(ik)|. We then derive the radial components of the plane wave field from (1.82), and express them in the form,

E i r = D e ( e ikr cos θ (ω,ω 0 ) ikr ), Z 0 H i r = D h ( e ikr cos θ (ω,ω 0 ) ikr ), (1.84)
with ( term for the radial component is zero. We then apply an inverse Sommerfeld-Maliuzhinets transformation [START_REF] Maliuzhinets | Inversion formula for the Sommerfeld integral[END_REF] )(u i ν (ω, ω 0 )).

D e = sin β ∂ ∂ θ 0 + cos β ∂ sin θ 0 ∂ ϕ 0 and D h = cos β ∂ ∂ θ 0 -sin β ∂ sin θ 0 ∂ ϕ 0 . Denoting C i e,h = (E i r , Z 0 H i r )
(1.90)

It is worth noticing that the linear operator D e,h applies only on the variables θ 0 and ϕ 0 , which signifies a direct and easy work on u i ν for the integral representation of the incident field, and that the condition θ (ω, ω 0 ) > π/2 + |arg(ik)|, also applies for the convergence of integrals in the representation (1.86) of (E i , H i ). Remark: In the equations on g i e (ω, ω 0 , ν) ± ig i h (ω, ω 0 , ν), we have used that, D ± 0 (cos( θ (ω, ω 0 ))) = sin θ 0 (sin θ sin(ϕϕ 0 ) ± i(sin θ cos θ 0 cos(ϕϕ 0 )cos θ sin θ 0 )), D ∓ (cos( θ (ω, ω 0 ))) =sin θ (sin θ 0 sin(ϕϕ 0 )

± i(sin θ 0 cos θ cos(ϕϕ 0 )cos θ 0 sin θ )), D ∓ D ± 0 (cos( θ (ω, ω 0 ))) = sin θ sin θ 0 [cos(ϕϕ 0 )(1 + cos θ cos θ 0 ) + sin θ sin θ 0 ∓ i(cos θ 0 + cos θ ) sin(ϕϕ 0 ))], (1.95) with z = x ± i0 + , x =cos( θ (ω, ω 0 )).

Remark:

The function u i ν (ω, ω 0 ) satisfies the Helmholtz equation on the unit sphere with Dirac source [START_REF] Bernard | Electromagnetic scattering by a smooth convex impedance cone[END_REF]- [START_REF] Babich | Numerical calculations of the diffraction coefficients for an arbitrary shaped perfectly conducting cone[END_REF], and we can write that the spectral functions attached to the incident field satisfy ( ω + ν 2 -1/4)(g i e,h )(ω, ω 0 , ν) = -D e,h δ (ωω 0 ).

(1.96)

1 . 1 . 1

 111 Frequency and Time Domain, for the Scattering by a polygonal Region Basic elements in Sommerfeld-Maliuzhinets representation and properties 1.1.1.

Figure 1 . 1

 11 Figure 1.1 Geometry of a wedge-shaped region

  above all the singularities of the integrand as |Reα| < π, and D -, obtained by inversion of D + with respect to α = 0. Let us prove, from the single face expression of the function f , that

Figure 1 . 2

 12 Figure 1.2 Geometry of a curved impedance wedge

Figure 1 . 3

 13 Figure 1.3 Geometry : sources (J, M) and observation point above the plane z = 0

Figure 1 . 4 1 . 4 . 2

 14142 Figure 1.4 Geometry of a conical region

1. 4 . 3

 43 Potentials and properties for an incident plane wave 1.4.3.

1 4iν 2 ∞ 0 √

 120 and D e,h = (D e , D h ), we can exploit the representation for the modified Bessel function K ν (ikr) ikr = sin πν γ e ikr cos α sin α cos(να)dα, in (1.83) and express (1.84) with Sommerfeld integrals. By means of an inverse Sommerfeld-Maliuzhinets transformation [1], we then obtain -ik sin α ikrC i e,h e -ikr cos α dr = 2 i √ 2π +i∞ -i∞ D e,h (u i ν (ω, ω 0 ))π sin α cos ναdν.

  (ω, ω 0 , ν))π sin α cos ναdν. (1.88) Comparing the expressions (1.85) and (1.88), we have +i∞ -i∞ (-g i e,h (ω, ω 0 , ν)) cos ναdν = +i∞ -i∞ D e,h (u i ν (ω, ω 0 )) cos ναdν. (1.89)To reduce (1.89), we have to consider the elementary properties of functions involved in it. They are derived from (1.83)-(1.84) and (1.87) according to the behaviour of the field at the apex and at infinity : the even functions D e,h (u i ν (ω, ω 0 )) and g i e,h (ω, ω 0 , ν) are regular in the strip |Re(ν)| < 3/2 and are O(1/ cos(ν θ (ω, ω 0 ))) as Imν tends to infinity. These properties permit to use Fourier transformation for (1.89) and to obtain, g i e,h (ω, ω 0 , ν) = -D e,h (u i ν (ω, ω 0 )), which can be also written,g i e (ω, ω 0 , ν) = -(sin β ∂ ∂ θ 0 + cos β ∂ sin θ 0 ∂ ϕ 0 )(u i ν (ω, ω 0 )), g i h (ω, ω 0 , ν) = -(cos β ∂ ∂ θ 0 sin β ∂ sin θ 0 ∂ ϕ 0

± i sin θ 0 ∂ ∂ θ 0 ),( 1 -

 01 the functions g i e,h (ω, ω 0 , ν) verify,g i e (ω, ω 0 , ν) ± ig i h (ω, ω 0 , ν) =x 2 ) 1/2 P -1 ν-1/2 (x) (x 2 -1),g i e (ω, ω 0 , 1/2) ± ig i h (ω, ω 0 , 1/2) = e ∓iβ / sin θ 0 π(1cos( θ (ω, ω 0 ))) D ± 0 (cos θ (ω, ω 0 )),(1.91)with x =cos( θ (ω, ω 0 )), 0 ≤ θ (ω, ω 0 ) ≤ π.We can then consider the transform by the differential operatorD ± ≡ ( ∂ ∂ ϕ ± i sin θ ∂ ∂ θ ), following, sin θ 0 e ±iβ D ∓ (g i e (ω, ω 0 , ν) ± ig i h (ω, ω 0 , ν)) = D ∓ D ± 0 u i ν (ω, ω 0 ) = 1 4π (D ∓ D ± 0 (cos( θ (ω, ω 0 ))) ∂ P ν-1/2 (x) ∂ x -D ∓ (cos( θ (ω, ω 0 )))D ± 0 (cos( θ (ω, ω 0 ))) ∂ 2 P ν-1/2 (x) cos(πν)∂ x 2 ) = D ∓ D ± 0 (cos( θ (ω, ω 0 ))) 4π ( ∂ P ν-1/2 (x) ∂ x + (1cos( θ (ω, ω 0 ))) ∂ 2 P ν-1/2 (x) cos(πν)∂ x 2 ),(1.92)where we have used that,D ± 0 (cos( θ (ω, ω 0 )))D ∓ (cos( θ (ω, ω 0 ))) = -(1cos( θ (ω, ω 0 )))D ∓ D ± 0 (cos( θ (ω, ω 0 ))).(1.93) and recover the compatibility condition (1.80) at ν = 1/2.

2 ∂ 2 1 π

 221 P ν-1/2 (z) cos(πν)∂ z 2 =

  1.85) On the other hand, we consider the Debye potentials (E i , H i ), attached to a representation of the incident field with (1.68)-(1.69), in a form similar to (1.78), are even functions of ν. Exploiting the expressions (1.68-1.69) for E r , H r and substituting the expressions (1.86), we verify that

	(E i , H i ) =	k	4 √ 2π	( +i∞ -i∞	ν sin(πν) ν 2 -1/4	g i e,h (ω, ω 0 , ν)	K ν (ikr) √ ikr	dν	(1.86)
	+iπ	K 1/2 (ikr) √ ikr	g i e,h (ω, ω 0 , 1/2)),
	where the functions g i e,h C i e,h = (E i r , Z 0 H i r ) = (k 2 +	∂ 2 ∂ r 2 )(rE i , rH i )
	=	4i √ 2π	i∞ -i∞	ν sin πν g i e,h (ω, ω 0 , ν)	(ikr) 3/2 dν, K ν (ikr)	(1.87)
	noticing that the contribution of	K 1/2 (ikr) √ ikr

CEA-DIF, 91297 Arpajon, and LRC MESO, CMLA, ENS Cachan, 61, av. pres. Wilson, 94235, France, e-mail: jean-michel.bernard@cea.fr, jmbtrd@gmail.com.


Mikhayel Myriam

Benoît Doublet

Sébastien Leclercq

Dolla Sarkis

Molecular characterization of antibiotic resistant Enterobacteriaceae in poultry in Lebanon

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction & Objectives Methods

PC-18

We focused our study on 221 E.coli strains (being the most dominant strain with 83.6%), among the set of the representative isolates. They belonged to a great diversity of sequence-types, most being well-known avian associated (ST-117, ST-10, ST-48 and ST-93), and sometimes pathogenic. Poultry production is a main contributor of the global trend of antimicrobial resistance arising from food-producing animals worldwide. In Lebanon, abusive and inappropriate use of antibiotics is frequent in chickens for prophylactic reasons and to increase productivity. In absence of a surveillance system, the present objective is to decipher the spread of resistant Enterobacteriaceae in poultry in Lebanon.

Results

For the first time, this study illustrates the high prevalence of multidrug-resistant E. coli, especially to medically-important antibiotics, in broiler production in Lebanon. Further molecular studies are needed to understand the country-specific epidemiology of ESBL/AmpC and mcr-1 genes in Lebanese poultry. Our results also advocate the urge for surveillance programs and action plans in Lebanon to reduce the abusive use of antibiotics and to limit the spread of multidrug-resistance in food-producing animals.

Conclusions 280 rectal swabs from 56 farms across the country were screened for the presence of expanded-spectrum Cephalosporin (ESC)-resistant Enterobacteriaceae isolates (n=341). Antimicrobial susceptibility and expanded spectrum β-lactamase (ESBL)/AmpC production were determined by the disk-diffusion method. Whole genome sequencing of 263 representative isolates was performed to determine the phylogenetic diversity, acquired resistance genes and plasmids using the CGE tools suite.

A.

B1.

B3. B2. We can obviously notice a contrast of the proportions of ESBL gene's distribution, the number of strains carrying the mcr-1 genes, and the coexpression of ESBL genes and mcr-1 between all the regions. A pre-dominance of the bla CTX-M3 gene was visualized and a large bla CTX-M3 and mcr-1 gene co-expression link can be mentioned according to this map.

FIG 1 :

 1 FIG 1: Most relevant molecular results of the E.coli strains. A. The proportion of ESBL, pAmpC producers and ESBL/pAmpC coproducers. B1. The most prevalent ESBL genes (Only one isolate harboured the blaCTX-M-1 gene usually associated with livestock production). B2. The proportions of E.coli co-expressing ESBL/pAmpC genes. (100% of the pAmpC strains were carrying the gene blaCMY-2). B3. Resistant E. coli isolates co-expressing ESBL and/or AmpC genes and the mobile colistin resistance gene mcr-1.

FIG 2 :

 2 FIG 2: Distribution of main ESBL genes found in the 4 main regions of the Lebanese Territory and their co-expressions with the mcr-1 gene.