Philippe Biane

Patrick Dehornoy
email: dehornoy@math.unicaen.fr

DUAL GARSIDE STRUCTURE OF BRAIDS AND FREE CUMULANTS OF PRODUCTS

Keywords: 1991 Mathematics Subject Classification. 20F36, 05A17, 46L54 braid group, dual braid monoid, Garside normal form, generating function, free cumulants, independent random variables, noncrossing partitions

We count the n-strand braids whose normal decomposition has length at most 2 in the dual braid monoid B + * n by reducing the question to a computation of free cumulants for a product of independent variables, for which we establish a general formula.

Introduction

A Garside structure on a group G is a generating family S that gives rise to distinguished decompositions of a particular type, leading in good cases to an automatic structure on G and, from there, to solutions of the word and conjugacy problems of G [START_REF] Dehornoy | Foundations of Garside Theory[END_REF]. In the case of the n-strand Artin braid group B n , two Garside structures are known: the so-called classical Garside structure, in which the distinguished generating family is a copy of the symmetric group S n [8, Chapter 9], and the so-called dual Garside structure, in which the distinguished generating family is a copy of the family NC(n) of all size n noncrossing partitions [START_REF] Bessis | Springer theory in braid groups and the Birman-Ko-Lee monoid[END_REF].

Whenever a finite Garside structure S is given on a group G, natural counting problems arise, namely counting how many elements of the group G or of the submonoid generated by S have length (at most) ℓ with respect to S. Call S-normal the distinguished decompositions associated with a Garside structure S. As S-normal sequences happen to be geodesic, the above question amounts to counting S-normal sequences of length ℓ. And, because S-normality is a purely local property, the central question is to determine S-normal sequences of length two, the general case then corresponding to taking the ℓth power of the incidence matrix associated with length two.

Initially motivated by the investigation of the logical strength of certain statements involving the standard braid ordering [START_REF] Carlucci | Unprovability statements involving braids[END_REF], the above mentioned counting questions in the case of the classical Garside structure of braids have been addressed in [START_REF] Dehornoy | Combinatorics of normal sequences of braids[END_REF], leading to nontrivial results involving Solomon's descent algebra and to natural conjectures, like the one established in [START_REF] Hivert | Sur une conjecture de Dehornoy[END_REF] using the theory of quasi-symmetric functions.

The aim of this paper is to address similar questions in the case of the dual Garside structure of braids and to obtain an explicit determination of the generating function for the number of normal sequences of length two (here Cat n denotes the nth Catalan number): Proposition 1.1. Let b * n,2 be the number of braids of length at most 2 in the dual braid monoid B + * n . Then the function R(z

) = 1 + n 1 b * n,2 z n is connected with M (z) = 1 + n 1 Cat 2 n z n by the equality (1.1) R(zM (z)) = M (z).
This formula, which inductively determines the numbers b * n,2 , will be deduced from a general formula for computing the free cumulants of a product of independent random variables.

Free cumulants where invented by Roland Speicher in order to compute with free random variables [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF]. In particular, free cumulants give a simple way of computing the free additive convolution of two probability measures on the real line, namely the free cumulants of the sum of two free random variables are the sums of the free cumulants of the variables. Free cumulants also appear when one tries to compute the free cumulants of a product of free random variables, although in a more complicated way.

Here is the general formula we establish (the notations are standard and explained in section 3): Theorem 1.2. Let X 1 , ..., X k be a family of commuting independent random variables, and let R (i) n be the free cumulants of X i . Then the free cumulants of the product X 1 X 2 •••X k are given by

(1.2) R n = π 1 ∨ ••• ∨π k =1n i R (i) π i ,
the sum being over all k-tuples of noncrosssing partitions in NC(n) whose join is the largest partition.

Free cumulants are specially designed to deal with highly noncommuting objects, therefore it may come as a surprise that there is also a simple formula for computing free cumulants of a product of independent commuting random variables, in terms of the free cumulants of the factors. Actually the formula also holds, with appropriate modifications, for classical and Boolean cumulants.

The organization of the paper is as follows. In Section 2, we recall the description of the dual Garside structure of braid groups and raise the induced counting questions. In Section 3, we review basic definitions about free cumulants and establish Theorem 1.2. This part can be read independently of the rest of the paper. In Section 4, we apply the result of Section 3 to braids and conclude with further questions and one additional result about the determinant of the involved incidence matrix.

The dual Garside structure of braids

Braid groups. For n 1, the n-strand braid group B n is the group defined by the presentation (2.1)

B n = σ 1 , ..., σ n-1 σ i σ j = σ j σ i for |i -j| 2 σ i σ j σ i = σ j σ i σ j for |i -j| = 1 .
The

G if g = s 1 •••s d (resp. g = t -1 e •••t -1 1 s 1 •••s d ) holds in G.
Every group G is trivially a Garside structure in itself, and the notion is interesting only when S is small, typically when G is infinite and S is finite or, at least, is properly included in G. Under mild assumptions, the existence of a finite Garside structure implies good properties for the group G such as the existence of an automatic structure or the decidability of the word and conjugacy problems.

Whenever S is a finite generating family in a group G, it is natural to consider the numbers

N S d = #{g ∈ G | g S = d}
, where g S refers to the minimal length of an S-decomposition. In the case of a Garside structure, symmetric S-normal decompositions are (essentially) unique, and they are geodesic. Therefore, N S d identifies with the number of length d symmetric S-normal sequences. In such a context, the submonoid S of G generated by S is the family of all elements of G whose symmetric Snormal decomposition has an empty denominator, that is, of all elements that admit an S-normal decomposition. Then, it is also natural to introduce the number

N S d,+ = #{g ∈ S | g S = d},
which, by the above remark, is the number of S-normal sequences of length d.

It then follows from the definition that a sequence (s 1 , ..., s d ) is S-normal if and only if every length two subsequence is S-normal, and the basic question is therefore to investigate the numbers

(2.2) N S 2,+ = #{(s 1 , s 2 ) ∈ S 2 | (s 1 , s 2 ) is S-normal}.
The classical Garside structure of B n . In the case of the braid group B n , two Garside structures are known. The first one, often called classical, involves permutations. By (2.1), mapping σ i to the transposition (i, i+1) induces a surjective homomorphism pr n : B n → S n . Considering the positive braid diagrams in which any two strands cross at most once ("simple braids") provides a set-theoretic section for pr n , whose image is a copy S n of S n inside B n . The family S n is a Garside family in B n [8, Chapter 9], the submonoid of B n generated by S n being the submonoid B + n generated by σ 1 , ..., σ n-1 [START_REF] Garside | The braid group and other groups[END_REF]. The associated numbers N Sn d,+ have been investigated in [START_REF] Dehornoy | Combinatorics of normal sequences of braids[END_REF]. In particular, writing b n,d for N Sn d,+ , it is shown that the numbers b n,2 are determined by the induction

(2.3) b n,2 = n-1 i=0 (-1) n+i+1 n i 2 b i,2
and that the double exponential series b n,2 z n /(n!) 2 is the inverse of the Bessel function J 0 ( √ z).

The dual Garside structure of B n . It is known since [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF] that, for every n, there exists an alternative Garside structure on B n , namely the dual Garside structure S * n , whose elements are in one-to-one correspondence with the noncrossing partitions of {1, 2, ..., n}. The question then naturally arises of determining the derived numbers N 2), we have

(2.4) b * n,d = #{(s 1 , ..., s d ) ∈ (S * n ) d | ∀i < n ((s i , s i+1 ) is S * n -normal)}.
In order to compute the numbers b * n,d , we shall describe the correspondence between S * n and noncrossing partitions and interpret the S * n -normality condition in terms of the latter.

We recall that a set partition of the set {1, ..., n} is called noncrossing if there is no quadruple 1 i < j < k < l n such that i and k belong to some part of the partition, whereas j and l belong to another part. The set of noncrossing partitions of {1, ..., n}, denoted by NC(n), is a poset for the reverse refinement order: for two partitions π and π ′ one has π π ′ if and only if each part of π is included in some part of π ′ . With this order, NC(n) is a lattice, with largest element 1 n the partition with only one part, and smallest element 0 n , the partition into n singletons. We will denote by

π 1 ∨ ••• ∨ π d the join of a family of noncrossing partitions. Definition 2.2. [4] For 1 i < j, put a i,j = σ i ...σ j-2 σ j-1 σ -1 j-2 ...σ -1 i in B n .
The dual braid monoid B + * n is the submonoid of B n generated by all elements a i,j , and S * n is the family of all left-divisors of ∆ * n in B + * n , with

∆ * n = σ 1 σ 2 •••σ n-1 .
Note that σ i = a i,i+1 holds for every i, hence B + * n includes B + n , a proper inclusion for n 3.

Proposition 2.3. [4]

The family S * n is a Garside structure on the group B n .

It is convenient to associate with the n-strand braid a i,j a graphical representation as the chord (i, j) in a disk with n marked vertices on the border as shown on the right.

1 i j n
Then the correspondence between the elements of S * n and noncrossing partitions stems from the following observation: Lemma 2.4. [START_REF] Bessis | Springer theory in braid groups and the Birman-Ko-Lee monoid[END_REF] For P a union of disjoint polygons in the n-marked disk, say

P = P 1 ∪ ••• ∪ P d , let a P = a P 1 •••a P d , with a P k = a i 1 ,i 2 a i 2 ,i 3 •••a i n k -1 ,in k where (i 1 , ..., i n k ) is a clockwise enumeration of the vertices of P k .
(i) The braid a P only depends on P and not on the order of enumeration.

(ii) Mapping P to a P establishes a bijection between unions of disjoint polygons in the n-marked disk and elements of S * n . It is standard to define a bijection between noncrossing partitions of {1, ..., n} and unions of disjoint polygons in the n-marked disk, yielding the announced correspondence. Noncrossing partitions can be embedded into the symmetric group, using geodesics in the Cayley graph [START_REF] Biane | Some properties of crossings and partitions[END_REF]: this amounts to mapping a union of polygons to the product of the cycles obtained by enumerating their vertices in clockwise order. Note that, although noncrossing partitions may be viewed as particular permutations, the associated braids need not coincide: for instance, the cycle [START_REF] Sloane | On-Line Encyclopedia of Integer Sequences[END_REF], which corresponds to the partition {{1, 3}, {2}}, is associated in S * 3 with the braid a 1,3 , that is, σ 1 σ 2 σ -1 1 , whereas it associated with σ 1 σ 2 σ 1 in S 3 .

For π a noncrossing partition of {1, ..., n}, we shall denote by a π the associated element of S * n . For instance, for π = {{1}, {2, 8}, {3, 5, 6}, {4}, {7}}, we find a π = a 2,8 a 3,5 a 5,6 , as shown on the right.

(2.5) b * n,d = #{(π 1 , ..., π d ) ∈ NC(n) d | ∀i < n ((a π i , a π i+1 ) is S * n -normal)
}, and we have to recognize when the braids associated with two partitions make an S * n -normal sequence.

By construction, the Garside structure S * n is what is called bounded by the element ∆ * n , that is, it exactly consists of the left-divisors of ∆ * n in B + * n . In this case, the normality condition takes a simple form.

Lemma 2.5. [7, Chapter VI] Assume that S is a Garside structure that is bounded by an element ∆ in a group G. Then, for s, t in S, the pair (s, t) is S-normal if and only if the only common left-divisor of ∂s and t in the monoid S is 1, where ∂s is the element satisfying s • ∂s = ∆.

We are thus left with the question of recognizing in terms of noncrossing partitions (or of the corresponding unions of polygons) when a braid a i,j left-divides the braid a π , and what is the partition π ′ satisfying a π ′ = ∂a π .

Lemma 2.6. [START_REF] Bessis | Springer theory in braid groups and the Birman-Ko-Lee monoid[END_REF] Assume that π lies in NC(n).

(i) For 1 i < j n, the braid a i,j left-divides a π in B + * n if and only if the chord (i, j) lies inside the union of the convex hulls of the polygons associated with π.

(ii) We have ∂a π = a π , where π is the Kreweras complement of π. Putting pieces together, we obtain:

Proposition 2.7. For all n and d, the number b * n,d of n-strand braids that have length at most d with respect to the Garside structure S * n is given by

(2.6) b * n,d = #{(π 1 , ..., π d ) ∈ NC(n) d | ∀i < n (π i ∧ π i+1 = 0 n )}.
Proof. By Lemma 2.6(i), saying that 1 is the only common left-divisor of a π and a π ′ in B + * n amounts to saying that the convex hulls of the polygons associated with π and π ′ have no chord in common, hence are disjoint, so, in other words, that the meet of π and π ′ in the lattice NC(n) is the minimal partition 0 n . By Lemma 2.6(ii), this condition has to be applied to π i and π i+1 for every i.

Free cumulants

In view of Proposition 2.7, we have to count sequences of noncrossing partitions satisfying lattice constraints involving adjacent entries. We shall derive partial results from a general formula expressing the free cumulants of a product of independent random variables.

Noncrossing partitions and free cumulants. We recall some basic facts, more details can be found in [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF].

Given a sequence of indeterminates T 1 , ..., T l , ... and a noncrossing partition π in NC(n), one defines

(3.1) T π = p∈π T |p| ,
where the product ranges over all parts p of π and |p| is the number of elements of p in {1, ..., n}. Two sequences of indeterminates M 1 , ..., M l , ... and R 1 , ..., R l , ... are related by the moment-cumulant formula if, for every n, one has

(3.2) M n = π∈NC(n) R π .
It is easy to see that this equation can be inverted and the R l can be expressed as polynomials in the M l . In fact, introducing the generating series

M (z) = 1 + ∞ l=1 z l M l , R(z) = 1 + ∞ l=1 z l R l ,
we can recast the relation (3.2) in the form

(3.3) R(zM (z)) = M (z).
If the M l are the moments of a probability measure (or a random variable), then the quantities R l are called the free cumulants of the probability measure (or random variable).

Free cumulants of a product of independent random variables.

We establish now the general formula for free cumulants of products of independent random variables stated as Theorem 1.2 in the introduction.

Proof of Theorem 1.2. Let M (i)
n be the moments of X i . Let us write the moment-cumulant formula for each of the variables in the product

X 1 X 2 •••X k . As the nth moment of X 1 X 2 •••X k is M n = M (1) n •••M (k) n , one has M n = M (1) n •••M (k) n = π 1 ,... ,π k ∈NC(n) k i=1 R (i) π i .
Let us now decompose the sum in the right hand side according to the value of π = π 1 ∨ •••∨π k . Since π i π holds, each part of π i is included in some part of π. Let p be a part of π. The intersections π i,p := π i ∩ p form a partition of the set p. If we identify p with {1, ..., |p|} by the only increasing bijection, then the sets π i,p form noncrossing partitions of {1, ..., |p|}. Furthermore one has

π 1,p ∨ ••• ∨ π n,p = 1 |p| , whence M n = π∈NC(n) p∈π   π 1,p ∨ ••• ∨πn,p=1 |p| i R (i) π i,p   .
Defining the sequence Q n by

Q n = π 1 ∨ ••• ∨πn=1n i R (i) π i ,
we obtain

M n = π∈NC(n) p∈π Q |p| = π∈NC(n) Q π .
Then it follows from (3.2) that the quantities Q n are the free cumulants of the sequence of moments M 1 , ..., M n , ... . The above argument is closely related to an argument in [START_REF] Speicher | A conceptual proof of a basic result in the combinatorial approach to freeness[END_REF]. The first author would like to thank Roland Speicher for pointing out this reference.

Corollary 3.1. The number of k-tuples (π 1 , ..., π k ) in NC(n) k satisfying π 1 ∨ ••• ∨ π k = 1 n is the nth free cumulant of the variable X 2 1 •••X 2
k where X 1 , ..., X n , ... are independent centered semi-circular variables of variance 1.

Classical and Boolean cumulants. Cumulants can also be defined using the lattice of all set partitions of {1, ..., n} (this is the classical case, studied by Rota, Schützenberger...), or the lattice of interval partitions (these are the Boolean cumulants, see [START_REF] Speicher | Boolean convolution, Free probability theory[END_REF]). In both cases it is immediate to check that the proof of Theorem 1.2 goes through and gives a formula for computing the corresponding cumulants of a product.

Back to braids

We now apply the result of Section 3 to braids.

Incidence matrices. For every n, there exists a binary relation on NC(n) that encodes S * n -normality. We introduce the associated incidence matrix.

Definition 4.1. For n 2, we let A * n be the Cat n × Cat n -matrix whose entries are indexed by noncrossing partitions, and such that (

A * n ) π,π ′ is 1 (resp. 0) if π ∧ π ′ = 0 n holds (resp. fails).
For instance, if the partitions of {1, 2, 3} are enumerated in the order , , , , , the matrix

A * 3 is       1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1       . It
follows from the properties of 0 n and 1 n that the column of 0 n and the row of 1 n in A * n contain only ones, whereas the row of 0 n , its 0 n -entry excepted, and the column of 1 n , its 1 n -entry excepted, contain only zeroes.

if (π 1 , ..., π d-1 ) contributes to b * n,d-1 (π d-1 ) and (π d-1 , π) contributes to b * n,2 (π), that is, (A * n ) π d-1 ,π = 1 holds. We deduce b * n,d (π) = π ′ ∈NC(n) b * n,d-1 (π ′ ) • (A * n ) π ′ ,π .
From there, an obvious induction shows that b * n,d (π) is the πth entry in (1, 1, ..., 1) (A * n ) d-1 , and the result follows by summing over all π.

(Note that, for d = 1, Proposition 4.2 gives b * n,1 = Cat n , which is indeed the sum of all entries in the size Cat n identity-matrix.)

Using Theorem 1.2, we can now complete the proof of Proposition 1.1.

Proof of Proposition 1.1. By Proposition 4.2, b * n,2 is the number of positive entries in the matrix A * n , that is, the number of pairs (π, π ′ ) in NC(n) 2 satisfying π ∧π ′ = 0 n . As the Kreweras complement is bijective, this number is also the number of pairs (π, π ′ ) in NC(n) 2 satisfying π ∧ π ′ = 0 n . By complementation, the latter is also the number of pairs satisfying π∨π ′ = 1 n . By Corollary 3.1, this number is the nth free cumulant of X 2 1 X 2 2 where X 1 , X 2 are independent centered semi-circular variables of variance 1. The moments of the latter are the squares of the Catalan numbers, so (3.3) gives the expected result.

Further questions. It is easy to compute the numbers b * n,d for small values of n and d, see Table 1.

A natural question is to ask for a description of the columns in Table 1 beyond the first two ones. The characterization of Proposition 1.1 does not extend to d 3: for instance, we have n : the first column (d = 1) contains the Catalan numbers, whereas the second column contains the sequence specified in Proposition 1.1, which is A168344 in [START_REF] Sloane | On-Line Encyclopedia of Integer Sequences[END_REF].

b * n,3 = {(π 1 , π 2 , π 3 ) ∈ NC(n) 3 | π 1 ∧ π 2 = 0 n and π 2 ∧ π 3 = 0 n } : d 1 2 3 4 5 6 7 b * 1,d 1 1 1 1 1 1 1 b * 2,d 2
replacing ∧ and 0 n with ∨ and 1 n is easy, but the Kreweras complement cannot be forgotten in this case.

On the other hand, describing the rows in

| det(A * n )| = n k=2 Cat ( 2n-k-1 n-1 ) k-1
.

Before proving this formula, we need an auxiliary result involving Möbius matrices. By definition, if X is a finite poset, the associated Möbius matrix µ is the inverse of the order matrix ζ indexed by the elements of X and given by ζ(x, y) = 1 for x ≥ y, 0 otherwise.

If the elements of X are ordered according to a linear extension of the partial order of X, then ζ is a lower triangular matrix, with ones on the diagonal. In particular, its determinant is 1. The same is true for µ.

Lemma 4.4. Assume that X is a lattice, and ϕ is a complex valued function on X. Let Φ be the matrix defined by Φ(x, y) = ϕ(x ∧ y). Then we have

det(Φ) = x∈X φ(x),
where φ is given by φ(x) = y≤x µ(x, y)ϕ(y).

Note that, under the above assumptions, we have ϕ(x) = y≤x φ(y) by definition of the Möbius matrix. In particular, for y = 1, we recover It follows that the generating series h k := f y (z, 1, k) satisfy the inductive rules h k+1 = h k -zh k-1 . By Pascal's relation for binomial coefficients, the series g k := n≥k z n 2n-k-1 n-1 satisfy the same induction rules. It remains to check g k = h k for k = 0 and 1, which is an easy exercise left to the reader, to conclude that g k = h k holds for every k.

  . Here we shall concentrate on the latter, hereafter denoted b * n,d , and specifically on b * n,2 . By (2.

8

  Under the above correspondence, ∆ * n corresponds to the (unique) n-gon in the n-marked disk, hence to the (noncrossing) partition 1 n with one part. Owing to (2.4), the numbers b * n,d are determined by

Figure 1 .

 1 Figure 1. The polygons associated with a partition π and with its image under the complement map ∂ and ∂ 2 : the red partition is the Kreweras complement of π, and repeating the operation leads to the image of the initial partition under a rotation by 2π/n, corresponding to conjugating under ∆ * n in B + * n .

Proposition 4 . 2 .

 42 For all n and d 1, the number b * n,d is the sum of all entries in the matrix (A * n ) d-1 ; in particular, b * n,2 is the number of positive entries in A * n . Proof. For π in NC(n) let b * n,d (π) be the number of S * n -normal sequences of length d whose last entry is π. By Proposition 2.6, a length d sequence (π 1 , ..., π d-1 , π) contributes to b * n,d (π) if and only

Proof of Lemma 4 . 4 .

 44 First observe that ζ(x∧y, z) = ζ(x, z)ζ(y, z), therefore Φ(x, y) = ϕ(x ∧ = z∈X ζ(x ∧ y, z) φ(z) = z∈X ζ(x, z)ζ(y, z) φ(z).Multiplying the matrix Φ on the left by µ, one obtainsµ * Φ(x, y) = u µ(x, u)Φ(u, y) = u z µ(x, u)ζ(u, z)ζ(y, z) φ(z) = ζ(y, x) φ(x),the last equality coming from u µ(x, u)ζ(u, z) = δ xz . It follows that the matrix µ * Φ is upper diagonal, with the numbers φ(x) on the diagonal.Proof of Theorem 4.3. We apply Lemma 4.4 to X = NC(n) and ϕ defined by ϕ(π) = 1 for π = 0 n and ϕ(π) = 0 for π = 0 n . We obtain φ(π) = µ(π, 0 n ). It is known that the Möbius function for the lattice of noncrossing partitions is multiplicative: if π is made of p k parts of size k for k = 1, 2, ..., then one has µ(π,0 n ) = k ((-1) k-1 Cat k-1 ) p k .The matrix A * n coincides with the corresponding matrix Φ up to a permutation of the columns (given by taking the Kreweras complement). We deduce | det(A * n compute a k,n , we introduce the generating functionsf n,k (y) = π∈NC(n) y p k and f (z, y, k) = n≥k z n f n,k (y).Then one hasa k,n = f ′ n,k(1), whereas f n (1) = Cat n andφ(z) := f (z, 1, k)Let π be a noncrossing partition, such that the size of the part containing 1 is l. The complement of this part consists of l (possibly empty) noncrossing partitions π 1 , ..., π l , whose sizes add up to n -l. Moreover, we findp k (π) = p k (π 1 ) + ••• + p k (π l ) for l = k, = p k (π 1 ) + ••• + p k (π l ) + 1 for l = k.It follows from these considerations that the generating function f (z, y, k) satisfies the equation f (z, y, k) = 1/(1 -zf (z, y, k)) + (y -1)z k f k (z, y, k).

(4. 1 )

 1 φ(z) = 1/(1 -zφ(z)).Differentiating with respect to y and evaluating at y = 1, we findf y (z, 1, k) = zf y (z, 1, k)/(1 -zφ(z)) 2 + z k φ k (z). By (4.1), one has 1 -z/(1 -zφ(z)) 2 = 2 -φ(z), whence n≥k a n,k z n = f y (z, 1, k) = z k φ k (z)/(2 -φ(z)).

  group B n is both the group of isotopy classes of n-strand geometric braids, the mapping class group of an n-punctured disk, and the fundamental group of the configuration space obtained by letting the symmetric group act on the complement of the diagonal hyperplanes in C n[START_REF] Birman | Braids, Links, and Mapping Class Groups[END_REF]. Assume that G is a group and S is included in G.(i) A finite S-sequence (s 1 , ..., s d ) is called S-normal if, for i < d, every element of S left-dividing s i s i+1 left-divides s i , where "f left-divides g" means "f -1 g lies in the submonoid S of G generated by S".(ii) A pair of finite S-sequences ((s 1 , ..., s d ), (t 1 , ..., t e )) is called symmetric S-normal if (s 1 , ..., s d ) and (t 1 , ..., t e ) are S-normal and, in addition, the only element of S left-dividing s 1 and t 1 is 1.(iii) We say that (s 1 , ..., s d ) (resp. ((s 1 , ..., s d ), (t 1 , ..., t e ))) is a decomposition for an element g of

	Garside structures. A Garside structure in a group G is a subset S of G
	such that every element of G admits a decomposition of a certain syntactic
	form in terms of the elements of S, namely a symmetric S-normal decom-
	position in the following sense.
	Definition 2.1. [7]

Table 1 .

 1 The number b * n,d of n-strand braids of length at most d in the dual braid monoid B + *

	3	4	5	6	7	8
	b * 3,d b * 4,d b * 5,d b * 6,d 132 6 743 266 944 9 845 829 356 470 124 5 15 83 177 367 14 99 556 2 856 14 122 42 773 11 124 147 855 1 917 046 24 672 817 749 68 927 334 632 1 515

Table 2 .

 2 Table 1 leads to further natural questions. By Proposition 4.2, the generating series of the numbers b * n,d is rational for every n, and b * n,d can be expressed in terms of the d-power of the eigenvalues of the matrix A * n . For instance, one easily finds b * 3,d = 6•2 d -2d-5 for every d, as well as b * 3,d (π) = 2 d+1 -1 for π = 0 n , 1 n (as above, we write b * n,d (π) for the number of braids with a normal form finishing with π). Very little is known for n 4. It would be of interest to compute, or at at least approximate, the spectral radius of the matrix A * n . Here are the first values. Spectral radius of the incidence matrix A * Although we are not able to compute explicitly the eigenvalues of the matrix A * n , we can give a closed formula for its determinant. Theorem 4.3. For every n, we have

	n	1 2 3	4	5	6	7
	ρ(A * n ) 1 1 2 4.83... 12.83... 35.98... 104.87...

n

A determinant.


Discrimination power of measures of comparison


Maria Rifqi
email: maria.rifqi@lip6.frberger@thomson-lcr.fr

Vincent Berger

Bernadette Bouchon-Meunier
email: bernadette.bouchon-meunier@lip6.fr

The choice of a measure of comparison in granular learning methods

Introduction

The comparison of two objects is a usual task for many classification and learning methods.

Comparisons are usually evaluated through a measure of comparison. The used measure is often a distance. But, more and more, a similarity or a dissimilarity measure is chosen. But the choice of an appropriate measure among all avalaible measures in literature is not an easy task. It is linked to the problem of the characterization of relevant properties for the considered task. We have proposed [START_REF] Bouchon-Meunier | Towards general measures of comparison of objects[END_REF] a framework in order to deal with measures of comparison. This framework displays the main families of measures of comparison according to the properties they satisfy. Hence, the existing measures of comparison can be classified. Fyrther, it is known that a classification simplifies a problem.

However, the problem of the choice of a measure of comparison within a same family is still present. This paper gives an explicit method to choose a measure of comparison.

The proposed solution lies in a new representation of measures of comparison obtained by normalization of arguments. This normalization leads to a desirable property: measures of comparison do not depend on the scale of the system. Another consequence of this normalization is the fact that measures of satisfiability can be described by a unique argument and measures of resemblance by two arguments. The analysis of behaviours of measures of comparison is easy thanks to a geometrical interpretation and to a natural definition of the power of discrimination of a given measure of comparison.

We focus on two families of measures of similitude: measures of satisfiability and measures of resemblance. For each family, we discuss the role of the new parameter we point out and we propose new measures satisfying particular properties. For measures of satisfiability, the parameter controls the severity towards difference between sets whereas the parameter penalizes the differences between two sets for a measure of resemblance.

Measures of comparison

In [START_REF] Bouchon-Meunier | Towards general measures of comparison of objects[END_REF], we have proposed to formalize a measure of comparison between two fuzzy sets as a function of the common elements and the distinctive elements.

Formally, for any set of elements, let denote the set of fuzzy subsets of .

Definition 1 A fuzzy set measure

is supposed to be given, that is to say a mapping defined on and taking values in IR such that, for all and for all in :

MI1 : .

MI2 : if , then .

where denotes the classical inclusion of fuzzy sets.

A fuzzy set measure is close to the definition of an existential evaluator given by [START_REF] Dubois | A unifying view of comparison indices in a fuzzy set-theoretic framework[END_REF].

Definition 2 An -measure of comparison on is a mapping such that , for a given mapping IR IR IR and a fuzzy set measure on .

We denote:

where denotes the classical intersection of fuzzy sets and is a difference of fuzzy sets defined by:

D1 : if , then .
D2 :

is monotonous with regard to : entails We are interested in measures of comparison which evaluate the likeliness of two descriptions. We have called them measures of similitude.

Definition 3 An -measure of similitude on is anmeasure of comparison such that is non decreasing with respect to , non increasing with respect to and .

-measures of similitude can be distinguished more subtly in three types: measures of satisfiability, measures of resemblance and measures of inclusion. In this paper, we focus on two types of measures of similitude.

Measures of satisfiability

A measure of satisfiability corresponds to a situation in which we consider a reference object or a class and we need to decide if a new object is compatible with it or satisfies it. More particularly, measures of satisfiability are appropriate for rule base systems. For example, in [START_REF] Bouchon-Meunier | Learning from imperfect data[END_REF] objects are classified by means of a decision tree. In [START_REF] Bouchon-Meunier | Learning from imperfect data[END_REF], the comparison between the value of an attribute of the new example with test-values associated with each edge is realized by means of a measure of satisfiability.

The satisfiability of a reference description of by a new description defined as a fuzzy subset of has been defined as follows: With this definition, the value of the satisfiability of for can a priori be different with two pairs of fuzzy sets distinctive only because of the scale. It is desirable that a satisfiability measure depends only on the relative weights of and and not on the scale of the system. In order to obtain an objective measure, we propose to normalize the satisfiability measure.

We consider:

the reduced intersection the reduced distinctive feature As , the domain of definition of the measure of satisfiability is a quarter of circle. It can be described by a unique argument , with . We denote the measure of satisfiability . The conditions of definition 4 become: is decreasing with respect to . This new form of a measure of satisfiablity, expressed by a unique variable, has the advantage not to be dependent upon the size of the system. Furthermore, this normalization makes the definition of a measure of satisfiability more simple insofar as the argument is a segment and not a quarter of plan.

There are of course many possible choices for the satisfiability measure satisfying these three conditions. Among them, let us distinguish the two following forms:

. It is a linear satisfiability function.

. This function has the advantage of presenting a meaningful physical insight. If we represent the reference set by the vector in figure 1. If we describe each set and its related point by a vector from the origin, is the scalar product . When the two vectors are orthogonal, then the satisfiability vanishes:

. This is a good signification of orthogonality. More generally, the satisfiability appears as a projection, and the lack of satisfiability is represented as a deviation in figure 1: this is an intuitive notion of satisfiability.

We can also focus on the following measures which are known in the literature:

, which is usually defined with the sigma-count, can be also written:

. is discriminant for low satisfiability: a small difference between a set and the reference is tolerated. is discriminant for high satisfiablity: a small difference between a set and the reference is not tolerated. is discriminant for high and low satisfiability at the same time. For high satisfiability, this measure is between the linearly discriminant measure and the low satisfiability discriminant measure. For low satisfiability, it is between the linearly discriminant measure and the high satisfiability discriminant measure. But it is low discriminant for .

Remark

and are symmetrical relatively to : . has a low discrimination power for high satisfiability, and a high discrimination power for poor satisfiability, whereas does exactly the reverse.

We can consider that the discrimination power of a measure of satisfiability is given by the derivative of . For instance, for small and large differences between a set and the reference that is to say for or :

, for all . This means that the total discrimination power has to be distributed on the interval, but a high discrimination power somewhere implies a low discrimination power elsewhere, the integral being constant.

Accordingly, it is necessary to choose a measure with a discrimination power suitable for the considered application. This suggests a method of construction of a measure of satisfiability. The choice of the discrimination power is the first step. Then is obtained by integration of this function . For instance, a function with a high discrimination power for but a low discrimination for and is needed. This kind of measures means that if a description is not far from the reference, then the satisfiability is near from 1 because the difference is not significative. If a description is very far from the reference, we can consider that the satisfiability is null.

Once the behaviour of the wanted measure is known, the measure can be computed thanks to the discrimination power .

We would like to give the example of an interesting function for the measure of satisfiability using the Fermi-Dirac function. The analytic form is: and reflects essentially the Fermi-Dirac function. The above expression is used to ensure that and . The interest of this function lies on its physical meaning: in a physical system of temperature , containing a statistical set of states, the Fermi-Dirac function describes the statistical probability that a state of energy is filled or not, with

, where is the Boltzmann constant. Hence, it describes the probability of a state to belong to the Fermi sea [START_REF] Course | [END_REF] which is a good illustration of satisfiability. controls the decrease of the curve. The choice of enables to define a measure of satisfiability more or less severe, as shown on figure 3.

Measures of resemblance

We are now interested in resemblance measures. A measure of resemblance is used for a comparison between the descriptions of two objects, of the same level of Measures of resemblance are appropriate for a casebased reasoning or an instance-based learning. In clustering methods, distances can be replaced by a measure of resemblance. More generally, similarity-based classification methods [START_REF] Rifqi | Constructing prototypes from large databases[END_REF], [START_REF] Rifqi | Similarity and prototype based approach for classification of microcalcifications[END_REF] have to use resemblance measures as soon as all objects have the same level of generality.

Definition 5 An -measure of resemblance on is a measure such that:

is increasing with and decreasing with and for all .

-measures of resemblance which satisfy an additional property of -transitivity, for a triangular norm , are extensions of indistinguishability relations [START_REF] Trillas | On implication and indistinguishability in the setting of fuzzy logic[END_REF], [START_REF] Valverde | On the structure of t-indistinguishability operators[END_REF] to fuzzy sets. In the case where is the minimum, we obtain extensions of measures of similarity.

-measures of resemblance satisfying the property of exclusiveness: for all are called exclusive -measures of resemblance. We focus on them in the sequel.

Following our normalization procedure, we define: for . Similarly to the case of measures of satisfiability, this ensures that an exclusive measure of resemblance is not dependent on the scale of the problem.

The domain of study is now restricted to a piece of the unity sphere since . We have now to find the exclusive resemblance which satisfies the symmetry property . Geometrically, the sphere is simply obtained by a rotation of the satisfiability circle around the -axis (see fig- with . This means that can be described by any monotonous symmetrical function with respect to and and such that . Let us look for an expression of . We have: if These conditions show that the problem has been reduced to a satisfiability measure. We can therefore use again the solution described in the preceding section dealing with satisfiability. With this definition of , an exclusive resemblance appears as a satisfiability where a global distinctive feature is defined by , from the two individual distinctive features and (see Figure 5).

We can also consider different exclusive measures of resemblance as we have already done with measures of satis- . This measure corresponds to the measure of satisfiability . Furthermore, can be also written as: with such that: . This measure was introduced in [START_REF] Dubois | Fuzzy Sets and Systems, Theory and Applications[END_REF].

Other definitions of can be envisaged, for instance: or The choice of a particular form of has an effect on the measure of resemblance because this parameter represents distinctive elements. We can notice that:

(1)

As has a decreasing effect on an exclusive measure of resemblance, relation [START_REF] Bouchon-Meunier | Learning from imperfect data[END_REF] implies that, for a given and for all and , (2) Relation (2) means that penalizes more the differences between two sets than and that penalizes more the differences than . Furthermore, a particular is sensitive to the symmetry between and . Indeed, if differences are unbalanced, it means that or , or inversely, if differences are balanced, it means that , the behaviours of a given are not the same.

Conclusion

This paper enables to better analyse the behaviour of a measure of similitude thanks its discrimination power. This analysis is very useful in learning methods in general because a lot of them are based on comparisons of attribute values of objects in the learning database. Furthermore, the classification step is also often based on comparisons.

Each time a method needs a similitude measure, the questions are : when has the measure to be severe ? when has it to be tolerant ? The answers are given by the discrimination power described in this paper. We have given an example of a measure of satisfiability, the Fermi-Dirac function, with a parameter that controls the severity towards the difference with respect to the reference.

Definition 4

 4 An -measure of satisfiability on is a measure such that: is increasing with respect to , decreasing with respect to and independent of . Let us denote . Then, for all for all

Figure 1 .

 1 Figure 1. New representation of a measure of satisfiability

Figure 2 .

 2 Figure 2. Satisfiability measures Figure 2 displays the behaviour of various mentioned measures of satisfiability. We can see that: is linearly discriminant: satisfiability decreases linearly with the deviation.

  for every possible , we have:

Figure 3 .

 3 Figure 3. The Fermi-Dirac measure of satisfiability

Figure 4 .

 4 Figure 4. New representation of an exclusive measure of resemblance

Figure 5 .

 5 Figure 5. Correspondance between satisfiability measures and exclusive resemblance measures


Jessica Benedicto

Rémi Pollès

Cristian Ciracì

Emmanuel Centeno

David R Smith

Antoine Moreau

Nonlocality in metallo-dielectric multilayers: numerical tools and physical analysis

We provide theoretical and numerical tools to quantitatively study the impact of nonlocality arising from free electrons in metals on the optical properties of metallo-dielectric multilayers. Though effects due to nonlocality are in general quite small, they nevertheless can be important for very thin (typically below 10 nm) metallic layers -as are used in structures characterized by relatively flat dispersion curves. Such structures include those with negative refractive index; hyperbolic metamaterials; and materials with index near zero. We find in all cases that the inclusion of nonlocal effects through application of the hydrodynamic model to the electron response leads to a higher transmission through the considered medium. Finally, we examine the excitation of gap-plasmon resonances, where nonlocality plays a much greater role, and suggest possible routes for experimental investigation.

Metallo-dielectric multilayers 1 are a class of metamaterials that have attracted considerable attention because they are relatively easy to fabricate at infrared and visible wavelengths, and can support a wide scope of exotic behavior, including negative refraction and hyperbolic dispersion [2][3][4][5][6][7][8] . The unique properties of these multilayer structures can potentially play a role in applications such as, thermal radiation control [9][10][START_REF] Narimanov | Quantum Electronics and Laser Science Conference[END_REF] to subwavelength imaging 2,5,[START_REF] Fang | [END_REF][13][14][15][16][17][18][19] . Most of the analyses to date assume the classical, local model for the electron response of the metal layers. However, as the metals become very thin and are spaced close together, it can be expected that the classical model will no longer be applicable and must be substituted with a semiclassical or full quantum mechanical model. Recent experiments 20 have indicated there may be a length scale where the classical model fails, but where the semiclassical, hydrodynamic model for the free electron response is valid. The hydrodynamic model, while approximate, nevertheless provides a more tractable model for analytic and semi-analytic calculations, especially for larger structures, and is thus an attractive approach for estimating the impact of nonlocality in various scenarios.

Over the years, there have been diverging opinions as to whether structures composed of very thin metallic and dielectric layers are sensitive to the intrinsic nonlocality of metals [21][22][23] , that results from the interaction between free electrons. Here we provide a set of theoretical and numerical tools to take this nonlocality into account when simulating the behavior of any kind of metallo-dielectric multilayer. Our description relies on the hydrodynamic model [START_REF] Boardman | Electromagnetic surface modes[END_REF][START_REF] Frostmann | Metal optics near the plasma frequency[END_REF] , which has been shown to provide quantitative agreement for the plasmon resonance shifts observed on a system of gold nanospheres spaced sub-nanometer distances from a gold film 20 . This model was subsequently modified to take into account interband transitions [START_REF] Moreau | [END_REF] . In this paper, we present a dispersion relation for an infinitely periodic metallo-dielectric multilayer, as well as a scattering matrix formalism for finite layers, both of which incorporate the effects of nonlocality through the use of the hydrodynamic model. Our formalism makes it very easy to vary the boundary conditions [START_REF] Moreau | [END_REF] and is thus more general than previous works 23,27,28 . Using these tools, we show that the impact of nonlocality on a recently published optical negative index lens (n= -1) index lens design based on metallo-dielectric multilayers 8 can be observed but is very moderate. By contrast, we find nonlocal effects cannot be neglected for structures in the canalization regime 13 ; moreover, for very thin metallic layers, the bulk plasmon acts as a supplementary energy canal allowing light to tunnel through metallic layers. Finally, we use our tools to study the prism-coupling of a gap-plasmon [START_REF] Moreau | [END_REF] , where the impact of nonlocality is found to be appreciable. Based on this observation, we are led to suggest an experimental configuration that could be used to quantitatively probe nonlocality in metals.

I. HYDRODYNAMICAL MODEL FRAMEWORK

Within the framework of the hydrodynamical model, the conduction electrons of a metal are treated as a gas trapped within a volume bounded by the surface of the metal structure. Interactions between electrons are taken into account in an approximate manner through the introduction of a pressure term that includes both the electrostatic and quantum pressure. The hydrodynamical equations are linearized 29 to yield a relation between the electric field and the polarization of the metal linked to the free electrons displacements. The inclusion of the pressure terms in the hydrodynamic equations introduces a spatial derivative into the linearized equation of motion, and the response of the electron gas is thus nonlocal 30 .

The origin of the hydrodynamic model can be traced to the physics of plasmas. It has been used to describe metallic structures since the 1960s and onwards 31,32 , raising many questions along the way as to the proper treatment of a metal-dielectric interface. The model thus suffered from many uncertainties, but was nonetheless widely used [33][34][35][36] and discussed [START_REF] Boardman | Electromagnetic surface modes[END_REF][START_REF] Frostmann | Metal optics near the plasma frequency[END_REF]37 , then somewhat abandoned -likely due to the lack of any clear experimental evidence for nonlocal effects. It is clear now for instance, that nonlocality has almost no impact on the surface plasmons of a thin metal film [START_REF] Moreau | [END_REF] , although the shift in the plasmon resonance wavelength of prismcoupled films was often considered as a test for nonlocal theories [START_REF] Frostmann | Metal optics near the plasma frequency[END_REF] .

A variety of other approaches have been proposed to take nonlocality into account, including the classical Random Phase Approximation 31,34,38,39 or Feibelman's model 40,41 . It is interesting to note that an essential improvement to the latter approach concerns the manner in which the contribution from bound electrons is taken into account 42 . However, this model is know for not having issues to take bulk absorption into account 39 .

More modern approaches to assessing electron response rely on density functional theory (DFT), considered one of the most accurate tools to incorporate the effects of quantum mechanical interactions on resonances and other properties of metallic nanoparticles and nanoclusters 43,44 . These methods also have limitations for example, the propagation of waves is usually not taken directly into account 45 , which is a reasonable approximation for tightly confined structures such as spherical dimers, but probably not descriptive for structures supporting gap-plasmons and similarly less confined excitations.

The hydrodynamical model has attracted increasing attention due to advances in nanofabrication and measurement techniques that allow structures to be designed and studied in the regime where nonlocal response is expected to dominate [46][47][48] . In general, the hydrodynamic model is attractive because (i) it seems that its predictions are in good agreement with the first experiments for which nonlocality clearly plays a role 20 , and well before other quantum effects kick in 49,50 ; (ii) it yields analytical results [START_REF] Moreau | [END_REF]51 and provides deeper insight into the physics of nonlocality; and (iii) it is easy to implement in numerical simulations 52 . In addition, the uncertainties about the boundary conditions are lifted when the contribution of the bound and free electrons are clearly distinguished [START_REF] Moreau | [END_REF] and its well-known tendency to overestimate the impact of nonlocality is lessened. This is especially relevant for very short wavelengths, when the permittivity is small and when the Drude term is of the order of the interband transition contribution. For the reasons above, the model has thus been used to study the enhancement by plasmonics tips 53 and dimers 54 , hyperbolic metamaterials 23,28 , subwavelength imaging by a silver slab 21,48 and gap-plasmon propagation [START_REF] Moreau | [END_REF]48,55 .

It should be finally stressed even if the model was found to be quite accurate 22 when compared to other approaches, the model still has to be backed by more fundamental studies 44 , comparison to experiments 56 or even more sophisticated models 30 .

We will in this section briefly remind the reader the fundamental physics of nonlocality within the framework of the hydrodynamical model.

A. Longitudinal waves

We consider here a multilayer as represented in Fig. 1. Nonlocal effects are expected to occur for p polarization only (sometimes referred to as TM), so that we will from now on assume we are considering this polarization only. We assume the structure is illuminated with a plane wave, characterized by frequency ω, a time dependency e -iωt and a wavevector whose component along the x axis is denoted k x .

0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 h h m d D z x FIG. 1.
Diagram of a periodic metallo-dielectric structure. The gray areas represent the metallic layers.

In the framework of the hydrodynamic model 57,58 , the electric and magnetic fields satisfy Maxwell's equations

∇ × E = iωµ 0 H (1)
∇ × H = -iωǫ 0 (1 + χ b )E + P f (2)
where the effective polarization of the medium is linked to the electric field by the fundamental relation

P f = ǫ 0 .ω 2 p ω 2 + iγω E -(1 + χ b ) β 2 ω 2 p ∇ (∇.E) . (3)
These equations can be easily solved to yield an analytical form for the fields [START_REF] Moreau | [END_REF] . In the jth dielectric layer, having a relative permittivity ǫ d , the magnetic and electric fields can be written

H y d = (A j e ik d z + B j e -ik d z )e i(kxx-ωt) (4)
E x = k d ωε 0 ε d (A j e ik d z -B j e -ik d z )e i(kxx-ωt) (5)
E z = -k x ωε 0 ε d (A j e ik d z + B j e -ik d z )e i(kxx-ωt) , (6)
with

k d = ε d k 2 0 -k 2
x and k 0 = ω c . Inside the jth metallic layer of permittivity ε m , in the framework of the hydrodynamical model, two types of waves are supported: the transverse and longitudinal waves. The transverse wave results can be found from consideration of the two Maxwell curl equations. The associated transverse fields thus satisfy

∂ z E x -∂ x E z = iω µ 0 H y (7) E x = 1 iωε 0 ε m ∂ z H y (8)
E z = - 1 iωε 0 ε m ∂ x H y . (9)
so that, inside the metal, they can be written

H ym = (A j e -κtz + B j e κtz )e i(kxx-ωt) (10)
E t x = iκ t ωε 0 ε m (A j e -κtz -B j e κtz )e i(kxx-ωt) (11) E t z = -k x ωε 0 ε m (A j e -κtz + B j e κtz )e i(kxx-ωt) , ( 12
)
where

κ t = k 2 x -ε m k 2 0 .
The longitudinal wave corresponds to a bulk plasmon supported by the free electron gas, with no accompanying magnetic field. Because the electric field corresponding to the longitudinal mode is thus curl free, it satisfies

∂ z E x = ∂ x E z . (13)
and can finally be written (the first equation below being the definition of C j and D j )

E ℓ x = 1 ωε 0 (C j e -κ ℓ z + D j e κ ℓ z ) e i(kxx-ωt) (14)
E ℓ z = -κ ℓ ik x ωε 0 (C j e -κ ℓ z -D j e κ ℓ z )e i(kxx-ωt) (15)
with

κ ℓ = k 2 x + ω 2 p β 2 1 χ f + 1 1 + χ b ( 16
)
where ω p is the plasma frequency of the considered metal, and χ f and χ b are the susceptibilities associated with the free and bound electrons, respectively. These three parameters are determined through careful fits of the metal local permittivity 59 . The parameter β is central, but not so easy to estimate. This constant can account for both coulomb interaction and quantum pressure through which free electrons interact strongly in the metal 29,30 . The recent experimental results on film-coupled nanoparticles show consistency with the theoretically calculated value of β = 5 3 E F m (with E F being the Fermi level and m the effective mass of free electrons in the metal 30 ) 20 . For gold and silver these values are very close, and we take here β = 1.35 × 10 6 m/s for both metals.

B. Additional boundary conditions

At the boundary between a metal and a dielectric, an additional boundary condition is needed to determine to what extent the longitudinal wave is excited. Since we consider here the response of the free electrons as nonlocal and the response of the bound electrons as purely local, there is only one physically sound boundary condition: We assume that the free electrons are confined within the metal volume, and thus the normal component of the polarization P f associated with the free electron density vanishes at the interface [START_REF] Moreau | [END_REF] . We emphasize that this condition, although it may seem the most logical, has rarely been applied in prior work 27,60 .

For the fields that are inside the metal, the boundary condition on the polarization can be written on the interface

P fz = - 1 iω ∂ x H y -ǫ 0 (1 + χ b )E z = 0 ( 17
)
where, of course, E = E t + E ℓ . Using equation 9, the previous condition can also be written

E ℓ z = κ ℓ ωǫ 0 k x ΩH y (18)
with

Ω = k 2 x κ l 1 ǫ - 1 1 + χ b . (19)

II. DISPERSION RELATION IN AN INFINITE METALLO-DIELECTRIC MULTILAYER

Metallo-dielectric multilayers are of heightened interest in many metamaterial configurations for their unique dispersion characteristics. While fully isotropic negative index and other novel metamaterial medium are difficult to realize at visible and infrared wavelengths, anisotropic media comprising alternating metal/dielectric layers can be readily fabricated and can often approximate the desired wave propagation effects. In particular, one striking property that can be achieved in metallodielectric multilayers is hyperbolic dispersion 61,62 . Hyperbolic metamaterials are compelling because the hyperbolic dispersion relation allows evanescent waves emitted by a source to be converted to propagating waves. This effect can be used to achieve sub-wavelength imaging 13,15,18,19,63,64 , projection of near-field information to far-field 5,14 and super-planckian thermal emission 9,[START_REF] Narimanov | Quantum Electronics and Laser Science Conference[END_REF] . Analytically, the dispersion relation when metals are assumed to be local is extremely similar to the dispersion relation of Bragg mirrors.

Several dispersion relations for nonlocal metallodielectric structures have already been published 23,27 , but do not consider the contribution of the interband transitions, which is expected to be quantitatively much more accurate when the correct boundary conditions are used.

Moreover, all the nonlocal parameters are united here in a single constant Ω making it very easy to change the characteristics of the model and thus allowing the retrieval of all the other dispersion relations. The demonstration leading to the dispersion relation is quite informative too. It shows not all the different transfer matrices algorithms 65 can handle nonlocality easily.

A. Dispersion relation

We consider an infinite structure made of alternating metallic and dielectric layers with thicknesses of respectively h m and h d . We will denote by a subscript m the fields inside the metal, and by d the fields inside the dielectric layer.

The transfer matrix (Abelès matrices 66,[START_REF] Markos | Wave propagation: from electrons to photonic crystals and left-handed materials[END_REF] ) can be written

H y d E x z+h d = cos(k d h d ) iωε0ε d k d sin(k d h d ) ik d ωε0ε d sin(k d h d ) cos(k d h d ) H y d E x z .
(20) which relates the transverse fields on either side of the dielectric layer.

In the metal, since both transverse as well as longitudinal waves propagate, two transfer matrices must be written, or

H ym E t x z+hm = cosh(κ t h m ) iωε0εm κt sinh(κ t h m ) κt iωε0εm sinh(κ t h m ) cosh(κ t h m ) H ym E t x z (21)
and

E ℓ x E ℓ z z+hm = cosh(κ ℓ h m ) ikx κ ℓ sinh(κ ℓ h m ) κ ℓ ikx sinh(κ ℓ h m ) cosh(κ ℓ h m ) E ℓ x E ℓ z z (22)
For a given period, writing down all the boundary conditions leads to the following equations:

The continuity of E x yields

E x (0) = E ℓ x (0) + E t x (0) (23) E x (h m ) = E ℓ x (h m ) + E t x (h m ), (24)
while the continuity of H y gives

H y d (0) = H ym (0) (25) H y d (h m ) = H ym (h m ). ( 26
)
Finally, if we apply the additional boundary conditions discussed above, we get the supplementary equations

E ℓ z (0) = κ ℓ ωǫ 0 k x ΩH ym (0) (27)
E ℓ z (h m ) = κ ℓ ωǫ 0 k x ΩH ym (h m ). ( 28
)
All the above equations (the transfer matrixes and the ones produced by the boundary conditions) can be used to yield

H y d E x hm+h d = C 11 C 12 C 21 C 22 H y d E x 0 (29)
where the coefficients of the matrix, denoted C, are

C 11 = cos(k d h d ) cosh(κ t h m ) + ε d k d κ t ε m + ε d k d ε m κ t Ω 2 sin(k d h d ) sinh(κ t h m ) - Ωε m κ t cos(k d h d ) cosh(κ ℓ h m ) sinh(κ t h m ) sinh(κ ℓ h m ) + Ωε d k d sin(k d h d ) sinh(κ ℓ h m ) (2 -2 cosh(κ t h m ) cosh(κ ℓ h m )) 1 - Ωε m κ t sinh(κ t h m ) sinh(κ ℓ h m ) -1 (30)
C 12 = cos(k d h d ) - Ωε d k d [cosh(κ ℓ h m ) -cosh(κ t h m )] sin(k d h d ) sinh(κ ℓ h m )   iωε0εm κt sinh(κ t h m ) 1 -εm κt Ω sinh(κthm) sinh(κ ℓ hm)   + iωε 0 ε d k d sin(k d h d ) cosh(κ t h m ) (31)
C 21 = ik d ωε 0 ε d sin(k d h d ) + iΩ ωε0 [cosh(κ ℓ h m ) -cosh(κ t h m )] cos(k d h d ) sinh(κ ℓ h m )   cosh(κ t h m ) -Ωεm κt sinh(κ t h m ) tanh(κ ℓ h m ) 1 -Ωεm κt sinh(κthm) sinh(κ ℓ hm)   + κ t iωε 0 ε m cos(k d h d ) sinh(κ t h m ) + iΩ ωε 0 cos(k d h d ) sinh(κ ℓ h m ) - iΩ ωε0 [cosh(κ ℓ h m ) -cosh(κ t h m )] cosh(κ ℓ h m ) cos(k d h d ) sinh(κ ℓ h m ) (32)
and finally

C 22 = ik d ωε 0 ε d sin(k d h d ) + iΩ ωε0 [cosh(κ ℓ h m ) -cosh(κ t h m )] cos(k d h d ) sinh(κ ℓ h m )   iωε0εm κt sinh(κ t h m ) 1 -εm κt Ω sinh(κthm) sinh(κ ℓ hm)   + cos(k d h d ) cosh(κ t h m ) (33)
We seek solutions that correspond to propagating modes in the structure, and that are expected to be pseudo-periodic, and thus satisfy the Bloch-Floquet condition

H y d E x hm+h d = e ±iKD H y d E x 0 (34)
The two solutions can be either propagating upward or downward, and e ±iKD appear as the two eigenvalues of the matrix C, and thus as the two solutions of the char-acteristic equation

λ 2 -T r(C)λ + Det(C) = 0 ( 35
)
which means we have

e ±iKD = 1 2 T r(C) ± T r(C) 2 -4Det(C) . (36)
Another way to write what is already a dispersion relation, is to sum up the two eigenvalues to get

cos(KD) = 1 2 (C 11 + C 22 ), (37)
which can be written, using the previous expressions of C 11 and C 22 ,

1 -Ωεm κt sinh(κthm) sinh(κ ℓ hm) cos(KD) = cos(k d h d ) cosh(κ t h m ) + 1 2 ε d k d κt εm -k d ε d εm κt + ε d k d εm κt Ω 2 sin(k d h d ) sinh(κ t h m ) + Ω sinh(κ ℓ hm) ε d k d sin(k d h d ) (1 -cosh(κ t h m ) cosh(κ ℓ h m )) -εm κt cos(k d h d ) cosh(κ ℓ h m ) sinh(κ t h m ) (38)
This expression can be checked for consistency in various limits. For instance, making Ω vanish reduces the expression to the local dispersion relation for a hyperbolic structure. When the metal thickness tends to infinity, the dispersion relation reduces to the gap-plasmon dispersion relation [START_REF] Moreau | [END_REF]55 . Furthermore, this expression recovers all previously published results with different boundary conditions 23,27 by just changing the expression of Ω and κ l , as done in [START_REF] Moreau | [END_REF] .

III. SCATTERING MATRIX ALGORITHM

Even if the previous dispersion relation is extremely informative and allows to retrieve very important parameters like the effective index 19 for metallo-dielectric structures, structures with a more complex pattern have recently attracted a lot of attention 8,[START_REF] Verhagen | [END_REF] . We underline that even for structures with simple patterns, the first and the last layer are usually made of the same material thus limiting the insight the dispersion relation can provide : it does not give access to the reflection coefficient 23 . For these reasons, a systematic way for calculating the reflection coefficient of a metallo-dielectric structure and the field inside any layer is required. This is all the more necessary that cavity resonances in metallo-dielectric layers 69 may complicate the global picture given by the dispersion curves.

A transfer matrix method has been proposed in the eighties 60 to study periodical multilayers above the plasma frequency. But transfer matrices are not numerically stable, especially below the plasma frequency when the waves are evanescent 65 , and are difficult to use systematically. As we will see in this section, the scattering matrices are adapted to taking nonlocality into account and they are numerically perfectly stable so that they constitute a natural choice. Our method has been validated by comparison with full COMSOL simulations, especially for the gap-plasmon resonance excitation.

A. Layer matrices

We consider a layer j, comprised between two interfaces located at z = z j for the upper interface and z = z j+1 for the lower one. The thickness of the layer is h j = z j -z j+1 . It is convenient to introduce here the coefficients A ± j and B ± j , that are defined in a dielectric layer by

H y = A + j e ik j z (z-zj ) + B + j e -ik j z (z-zj ) e i(kxx-ωt) (39)
= A - j e ik j z (z-zj+1) + B - j e -ik j z (z-zj+1) e i(kxx-ωt) (40) with

k j z = ε j k 2 0 -k 2
x , ε j being the relative permittivity of the dielectric medium. Using Maxwell's equations for p polarisation (9), the electric field (E x ,E z ) can be easily calculated.

This leads to introducing a scattering matrix for a dielectric layer that writes

A + j B - j = 0 e ik j z hj e ik j z hj 0 B + j A - j . (41)
Inside a metallic layer, coefficients can similarly be de-fined for the transversal wave as H y = A + j e -κt(z-zj ) + B + j e κt(z-zj ) e i(kxx-ωt) (42)

= A - j e -κt(z-zj+1) + B + j e κt(z-zj+1) e i(kxx-ωt) ,

and the corresponding electric field can be determined using (9). Taking the longitudinal wave into account leads to introducing coefficients C ± j and D ± j :

E ℓ x = 1 ωε 0 (C + j e -κ ℓ (z-zj ) + D + j e κ ℓ (z-zj ) )e i(kxx-ωt) (44) = 1 ωε 0 (C - j e -κ ℓ (z-zj+1) + D - j e κ ℓ (z-zj+1) )e i(kxx-ωt) . (45)
This leads to a scattering matrix for a metallic layer that writes

    A + j C + j B - j D - j     =     0 0 e -κthj 0 0 0 0 e -κ ℓ hj e -κthj 0 0 0 0 e -κ ℓ hj 0 0         B + j D + j A - j C - j     . (46)

B. Dielectric to metal scattering matrix

We assume here that medium j is dielectric, while medium j + 1 is metallic. At such an interface, the magnetic field H y is continuous, as is E x , that can be calculated using (8). A supplementary condition inside the metal is given by (18). A straightforward calculation shows that this leads to the following conditions on the coefficients

A - j + B - j = A + j+1 + B + j+1 ( 47
)
k j z ǫ j (A - j -B - j ) = iκ j+1 t ǫ j+1 (A + j+1 -B + j+1 ) (48)
+C + j+1 + D + j+1 D + j+1 -C + j+1 = iΩ(A + j+1 + B + j+1 ). (49)
Rearranging these equations, a scattering matrix for the dielectric-metal interface can be written

  A - j B + j+1 D + j+1   = 1 k j z ǫj + iκ j+1 t ǫj+1 -iΩ     k j z ǫj - iκ j+1 t ǫj+1 + iΩ 2 iκ j+1 t ǫj+1 2 2 k j z ǫj iκ j+1 t ǫj+1 - k j z ǫj + iΩ 2 2iΩ k j z ǫj 2iΩ iκ j+1 t ǫj+1 k j z ǫj + iκ j+1 t ǫj+1 + iΩ       B - j A + j+1 C + j+1   (50)
C. Metal to dielectric scattering matrix Similarly, at the interface between a metal (upper layer j) and a dielectric (lower layer j + 1), the boundary con-ditions lead to the following equations

A - j + B - j = A + j+1 + B + j+1 (51) i κ j t ǫ j (A - i -B - i ) + C - j + D - j = k j+1 z ǫ j+1 (A + j+1 -B + j+1 ) (52)
D - j -C - j = iΩ (A - j + B - j ), (53)
that become, once they have been re-arranged,

  A - j C - j B + j+1   = 1 i κ j t ǫj + k j+1 z ǫj+1 -i Ω     i κ j t ǫj - k j+1 z ǫj+1 + i Ω -2 2 k j+1 z ǫj+1 -2 i Ω i κ j t ǫj i κ j t ǫj + k j+1 z ǫj+1 + i Ω -2 i Ω k j+1 z ǫj+1 2 b - i -2 k j+1 z ǫj+1 - i κ j t ǫj + i Ω       B - j D - j A + j+1   (54)

D. Cascading method

Now that scattering matrixes have been defined for all kinds of interfaces, they have to be combined through a cascading method. Here the scattering matrixes are all square matrixes, but they may have more than 2 lines, so that the usual cascading algorithm cannot be used. Instead, one has to rely on the cascading algorithm that is usually employed for Fourier Modal Methods 70,71 . When trying to combine a scattering matrix S so that

A B = S 11 S 12 S 21 S 22 C D (55)
with a scattering matrix U such that

D E = U 11 U 12 U 21 U 22 B F (56)
then the resulting scattering matrix is

A E = S 11 + S 12 (1 -S 11 U 22 ) -1 U 11 S 21 S 12 (1 -S 11 U 22 ) -1 U 12 U 21 (1 -S 22 U 11 ) -1 S 21 U 22 + U 21 (1 -S 22 U 11 ) -1 S 22 U 12 C F . (57)
This method can be applied here, even if the U ij are in general not square. Here A, B, C, D, E and F may represent vectors as

A ± j C ± j , B ± j D ± j
, or simply A ± j or B ± j depending on the scattering matrix. Each time a cas-cade is needed, there is however no ambiguity on how to choose the vectors, given the size of the matrixes that have to be cascaded.

In order to compute the field inside the layers, beyond the reflection and transmission coefficients of the whole structure, it is necessary to compute the vectors that are eliminated during the cascading process. They can be obtained through the following relations

B D = (1 -S 22 U 11 ) -1 S 21 (1 -S 22 U 11 ) -1 S 22 U 12 (1 -S 11 U 22 ) -1 U 11 S 21 (1 -S 11 U 22 ) -1 U 12 C F . ( 58
)
Finally, once all the A ± j , B ± j have been obtained using the previous method, the most stable way to compute the magnetic field (but this is true for any other field) is to use the following hybrid expression inside a layer H y = A - j e ik j z (z-zj+1) + B + j e -ik j z (z-zj ) e i(kxx-ωt) (59) with k j z = iκ j t in the case of a metallic layer, to ensure the exponentials have a modulus as small as possible.

IV. IMPACT METAMATERIALS BASED ON METALLO-DIELECTRIC MULTILAYERS A. Hyperbolic dispersion relation

We now study the impact of nonlocality on the dispersion relation (38) in the case of a hyperbolic dispersion relation from a numerical point of view. One of the most promising experimental realizations of a medium with an engineered dispersion relation is the n=-1 effective index medium fabricated with alternating layers of silver and titanium oxide 8 , at an operating wavelength of 363.8 nm.

We choose here to consider the same kind of structures, especially the same media and properties, in order to be as realistic as possible.

Figure 2 shows the dispersion relation for an hyperbolic isofrequency curve presenting an effective index 19,64 close to -1. Obviously, the impact of nonlocality is negligible on this kind of structure. The change in the curvature of the dispersion relation induces a small change in the effective index of the structure, which may in turn cause a small change in the position of the image given by the lens equation 19 . But the change seems very small compared to what is suggested in previous works 23 Nonlocal effects are expected for much thinner metallic layers that those that are considered here with a thickness of 28.1 nm for metallic layers. In order for these effects to have an impact, the thickness of the layer needs not to be much larger than the penetration length of the longitudinal waves in the metal.

It turns out that, keeping the same period, the metallic layers are much thinner for a larger effective index (corresponding to a smaller curvature 19,64 ). In addition, when the dispersion curve is flattened it is much more sensitive to any perturbation. These are the reasons why nonlocality has a much larger impact on structures in the canalization regime 13,15 where metallic layers are much thinner than the dielectric ones. In order to illustrate this idea, we choose to study a periodic medium alterning a 5 nm metallic layer with a 22.6 nm dielectric layer.

Figure 3 shows that for such a structure, for which the isofrequency curve is expected to be flat or slightly negative, the nonlocal curve actually presents a clear positive curvature. We underline this means the structure, presenting a positive effective index, is unable to refocus light and cannot be used to build a lens at all. Nonlocality has actually a double impact on the optical behavior of the structure: (i) the plasmonic effect (linked to the fact that the Poynting vector is opposite in the metal and in the dielectric 72 ) is lowered by nonlocality, so that it generally takes thicker metallic layers to get the same effect and thus the same effective index and (ii) the longitudinal wave acts as a supplementary way for light to tunnel through metallic layers, so that the transmission is generally higher. The imaginary part of the Bloch wavevector, shown in Fig. 3, is always smaller when the longitudinal wave is taken into account. This is not straightforward, since for gap-plasmons the plasmonic effect is lowered too, leading to a lower effective index, but the losses are higher for the same effective index [START_REF] Moreau | [END_REF] .

We underline that there is no real homogeneization regime when nonlocality is taken into account, because when the thickness of the metallic layers tends to zero, nonlocal effects intervene more and more. It should be much more difficult in that case to design a structure that would work exactly as desired since no simple formulae are available in that case to guide the design. Numerical optimization procedures are likely to be required. Another consequence is that using thinner layers does have an impact on the behavior of the structure, contrarily to the local situation, where it is equivalent in the homogeneization regime to take two different periods as long as the dielectric-to-metal ratio is kept constant. Even if this ratio may have to be raised a little bit to get the desired properties, at least the transmission is higher for thinner metallic layers due to nonlocality. From that point of view the effects of nonlocality can thus be considered as beneficial and speak in favor of the use of thinner metallic layers. B. Impact of nonlocality on a slab of -1 index metamaterial

We have implemented the above scattering matrix algorithm, basing it on a code we have previously released 65 , and that has previsouly been used to simulate the propagation of light beams in metallo-dielectric layers 19,69 . Here we consider the structure known to behave as a -1 index medium [START_REF] Verhagen | [END_REF] and that have been recently fabricated 8 . This lens operation wavelength is of 363.8 nm, in the close UV range. In that range the plasmonic effects are actually much higher 72 . They are usually linked to the Poynting vector inside the metal, that is roughly proportionnal to 1 ǫ , a factor that becomes important when the permittivity is negative but small. It is however not possible to consider much shorter wavelength because the titanium oxide then becomes much more absorbent. This is what makes this close UV range so interesting to build flat lenses -and this is a range for which the nonlocal effects are much more likely to be noticeable too.

We have simulated what happens when the structure is illuminated with a Gaussian beam with a non-normal incidence, an experiment that has been made to study the negative refraction. A typical result is shown figure 5. In order to compare local and nonlocal simulations, we have plotted figure 4 different beam profiles for different incidence angles at the exit of the lens. There is actually very little difference between the two. This can be related to the fact that (i) we have seen above that for hyperbolic media with a -1 effective index nonlocality has not a huge effect on the dispersion relation and (ii) nonlocal effects can be seen for large wavevectors that are not concerned by this kind of experiment.

C. Impact in the canalization regime

Keeping the same materials and operation wavelength, it is possible to find a structure in the canalization regime, i.e. presenting a flat dispersion curve according to the local theory. In such a medium, all the waves (whether they are evanescent in the outside medium or not) propagate, and they do it in the same direction. As long as the ratio h d hm is equal to the ratio ǫ d ǫm ≃ 4.2, and the overall period stays small with respect to the wavelength, we can consider that we are in the canalization regime. Two cases are considered here (i) a 10 period structure with 10 nm metallic layer and a 42 nm dielectric layer and (ii) a 25 period structure with a 4 nm metallic layer and a 16.8 nm thick dielectric layer. Both structures begin and terminate with a metallic layer so that their respective thicknesses are 530 nm and 524 nm. The structure is illuminated with a Gaussian beam (normal incidence, wavelength of λ = 363.8 nm, waist of 0.1 nm and focused on the entrance of the structure). It should be underlined that what we call a Gaussian beam contains evanescents, so that it is actually almost a ponctual source. The results of the computation are shown Fig. 6 for the first case with 10 nm thick metallic layers and Fig. 7 for the second case with thinner layers.

The analysis of the dispersion curves is confirmed: the thinner the metallic layers, the more light manages to go through. Another point is that the predictions of the local theory are significantly different from the nonlocal one when it comes to the profile of the outgoing beam. This means that nonlocality can definitely not be ignored when the whole purpose of the structure is to make an image of a source with subwavelength resolution.

D. Gap-Plasmon Resonance

In many situations, nonlocality does not have a noticeable impact on the optical properties of a structure. Using the same structures as considered above, but for larger wavelength (outside of the close UV range mentioned above) makes that impact very often negligible.

In the eighties, the community was hoping that surface plasmons would be sensitive to nonlocality [START_REF] Frostmann | Metal optics near the plasma frequency[END_REF] , but this is not the case because the wavevector of the surface plasmon cannot reach high enough values. The gap-plasmon is the fundamental mode of a metallic waveguide 73 and it experiences a violent plasmonic slowdown when the size of the gap is decreased. Its wavevector can theoretically become arbitrarily large for a small enough gap. This makes this mode very sensitive to nonlocality [START_REF] Moreau | [END_REF] when the wavelength approaches the electrons mean free path 39 . A way to excite the gap-plasmon is to use gap-plasmon resonators [START_REF] Moreau | [END_REF]56 , but a more conventional way would be to couple the gap-plasmon using a prism. It is more difficult to reach extremely high wavevectors with such a setup. However this drawback would be compensated by a very high control of the experimental conditions so that the nonlocal effect should easily be measurable.

Here we consider the case of a gap-plasmon excited using a T iO 2 prism 74 . The prism is placed above, on top of a 18 nm thick silver layer, see Fig. 8. Two gap sizes are first considered here (10 and 12 nm respectively). The structure is illuminated by a Gaussian beam (wavelength of λ = 543 nm, waist of λ and focused on the entrance of the structure) and the incidence angle is made to vary. The results are shown in Fig. 9. The gapplasmon resonance (GPR) can clearly be seen for high incidence angle, and nonlocality has obviously a strong impact on the reflection coefficient of the structure. For a 12 nm gap, the main difference is a shift in the resonance : the GPR is excited for a smaller angle when nonlocality is taken into account (67.6 • instead of 68.9 • ). The corresponding field is shown in Fig. 10 For a 10 nm gap, the GPR is excited for very high incidence angles, so that the main difference there is more the depth of the resonance than its position.

Metal

Figure 11 shows the angle for which the GPR can be excited as a function of the gap width, for a T iO 2 prism covered with a 18 nm thick gold layer. For larger gaps, the difference between the local and the nonlocal theory can hardly be distinguished, but for gaps that are close to 10 nm the difference is quite obvious. Below 9 nm, the gap-plasmon has actually too high an effective index to be excited using a prism.

The difference between the local predictions and the nonlocal ones, however small, should be totally measurable.

We underline here that, for gaps that are of the order of 10 nm, any influence of the tunneling effect, and probably of the spill-out can totally be excluded. The effect relies purely on the fact that the gap-plasmon resonance is slowed down by plasmonic effects so that the mode becomes sensitive to nonlocality. This kind of very simple experiment may bring definitive answers regarding the validity of the different descriptions of nonlocality that are currently available.

V. CONCLUSION

We have presented here numerical tools that allow to take nonlocality in metals into account when simulating the propagation of a plane wave or of a beam in a metallo-dielectric multilayer. These tools, relying essentially on analytical calculations, are meant to be as accurate as possible -through the use of accurate material characteristics 59 and boundary conditions that can be considered conservative 26 compared to other implementation of the hydrodynamic model. The formula presented here are easy to adapt for different descriptions of the metal and different boundary conditions so that previous results can be retrieved and checked using the present work. Furthermore, if the hydrodynamic model needs further tuning to match future experiments, the present formalism should be very easy to adapt. Finally, we have made the codes we have written freely available 75 .

We have used these tools to assess the impact of nonlocality on realistic metallo-dielectric structures presenting a negative refractive index and in the canalization regime. Our conclusion are that for a negative index around -1 the impact of nonlocality should be expected to be negligible. For higher absolute values of the refractive index that are required to reach subwalength resolution, and especially in the canalization regime, the effect of nonlocality cannot be ignored. We underline that even small effects like the small change in the effective index due to nonlocality will have an impact on the operation of flat lenses, especially when they are able to reach super-resolution 19 . In that case, the propagation of high wavevector waves are actually responsible for the subwavelength resolution 13,15 . The structure has thus to be finely optimized 64 and there is little doubt that nonlocal effects should be taken into account. The tools we have provided here should help to finely simulate the optical behaviour of such structures.

Finally, we have shown that nonlocality has a small but definitely measurable impact on the excitation of a gap-plasmon using evancescent coupling. Such an experiment could bring clear and definitive answers to many questions regarding nonlocality, including the value of the β parameter or the right boundary condition. The gaps that are considered here are very large compared to previously studied situations 20,43,44,76 so that the measured effects can be attributed solely to nonlocality. At these scales, tunneling can be excluded, and the spill-out is not likely to have any noticeable impact [START_REF] Toscano | Self-consistent hydrodynamic approach to nanoplasmonics: Resonance shifts and spillout effects[END_REF] .

We hope that the present work will make it easy for the community to assess the impact of nonlocality thoroughly and to take it accurately into account, in order to design structures or experiments based on metallodielectric multilayers.

  .

FIG. 2 .

 2 FIG. 2. Dispersion curves for a metallo-dielectric structure (h d = 33.7 nm, hm = 28.1 nm, λ = 363.8 nm) presenting an effective index close to -1. Top: Real part. Bottom: Imaginary part. The local (black line) and nonlocal (red line) cases are shown. Very little difference can be seen, especially close to normal incidence.

FIG. 3 .

 3 FIG. 3. Dispersion curves for a metallo-dielectric structure (h d = 5 nm, hm = 22.6 nm, λ = 363.8 nm) in the canalization regime (i.e. a flat dispersion curve). Top: Real part. Bottom: Imaginary part. The local (black line) and nonlocal (red line) cases are shown. The difference is much more important here: nonlocality has a direct impact on the curvature of the dispersion curve.

FIG. 4 .FIG. 5 .

 45 FIG. 4. Transmitted beam for different incidence angles (dotted line: normal incidence, solid line:20 • , dashed line: 40 • ) for the local (black) and the nonlocal (red) computation in the case of the -1 lens 8 . The waist of the incident beam is equal to 1λ.

FIG. 6 .

 6 FIG. 6. Profile of the outgoing beam (magnetic field) for the local (black) and nonlocal (red) computation (case (i)). The field profile is computed at the very edge of the lens. Inset: local (left) and nonlocal (right) corresponding field maps for the magnetic field.

FIG. 7 .

 7 FIG.7. Profile of the outgoing beam for the local (black) and nonlocal (red) computation (case (ii)). The field profile is computed at the very edge of the lens. Inset: local (left) and nonlocal (right) corresponding field maps for the magnetic field.

FIG. 8 .

 8 FIG. 8. Diagram of the excitation of a gap-plasmon using a prism coupler. The thickness of the upper metallic layer is 18 nm.

FIG. 9 .

 9 FIG.9. Reflection coefficient of the structure described in Fig.8for a 12 nm gap (solid lines) in the local (black) and nonlocal (red) case and for a 10 nm (dashed lines) in the local (black) and nonlocal (red) case.

FIG. 11 .

 11 FIG. 11. Difference in degrees between the local predictions and the nonlocal ones, regarding the exact position of the gap-plasmon resonance. Inset: Angular position (in degrees) of the gap-plasmon resonance (GPR) as a function of the gap width (in nm).

ACKNOWLEDGMENTS

This work has been supported by the French National Research Agency, "Physics of Gap-Plasmons" project number ANR-13-JS10-0003.


domain: shs scco

text:
François Daniellou
email: francois.daniellou@ensc.fr

How do ergonomists influence work situations?

To begin with, I do not believe that there exists such a thing as "interventions in ergonomics". What exists are interventions "of ergonomists": it is not the discipline which intervenes, but persons. They do it in a diversity of frameworks, with different statuses, different origins of the requests, different objects, different resources and constraints. Now, what is an intervention? Intervening means "coming among others". The ergonomist is in the arena, among other people, not on the cliff ─ nor on the side of the field, advising them from a distance. He/she is there because he/she has been requested to help to solve a problem or to manage a project. He/she will endeavour to lead the company stakeholders to make decisions, to take actions or to develop practices that they would not have developed without his/her presence. He/she is the bearer of knowledge about human at work, but also of values and criteria, such as promoting health, the sustainable efficiency of the system, and the development of people and organisations. But how does he/she act ? By which mechanisms does he/she influence the situation?

 The first and most common answer brought to this question is the technological approach: the ergonomist acts by applying scientific knowledge about the human at work to solve the problems he/she addressing. This standpoint is widely shared worldwide. And it is true that the ergonomist brings interdisciplinary knowledge about the humans, their physiology, their cognition, their subjectivity. Unfortunately, we have been knowing for many years, notably thanks to Schön and Argyris, that practitioners do not apply scientific knowledge to solve a problem. Because what they come across in the workplace is not a problem, but, as Schön says, "a mess". The problem is not yet set when they arrive, or it has not been formulated in the right way. There is no one single way of setting the problem which has to be solved; the solution is not known in advance and is not unique; and the path from the problem to the solution is not defined in advance. From that point of view, the practitioner in ergonomics is in the same cognitive situation as a designer or an architect who will have to "converse with the situation" to "make moves" and anticipate their consequences. As in the case of the architect, different ergonomists would design different answers and different solutions ─ which is not a weakness but a strength.  Second possible proposal: the ergonomist acts by changing the stakeholders' representations about work and the workplace. This is one of the bases of "activity centred ergonomics", and I have contributed twenty years ago to promote this idea. Today, I think that it is a necessary, but not a sufficient description of our forms of action. Yes, ergonomists strive to modify the company stakeholders' representations of what is happening in the workplace, by making them discover the difference between prescribed work and real activity.

Because designers and organizers have set up the means for work. But, in spite of the attention paid to this anticipation, what will happen in real time always differs from what has been planned: the material drifts from its reference state, the tools wear out, incidents occur, customers have unexpected demands, etc. Any worker has to compensate for the limitations of design anticipations. To take care of the situation as it is, the worker mobilizes his/her body, his/her intelligence and experience, his/her networks, but also his/her subjectivity and values to strive to do "a job well done". He/she puts into play an adaptive activity to try to balance production goals, expectations of customers or colleagues, the actual state of the system and means available for work, the abilities, limitations and variations of his/her body, and the compliance with his/her values. He/she continuously elaborates trade-offs between those contradictory drives; in difficult situations, the inevitable imperfection of those compromises may result in problematic outcomes in terms of performance and/or health . We now know that the outcome mainly depends on the operational leeway the worker has to adapt his/her operating strategies to the variability of the context. Activity analysis is an attempt to understand the determinants of this adaptive activity, its favourable outcomes and its failures.

Changing the stakeholders' representations of what really happens in the workplace means inviting them to observe the workplace from the activity point of view, that is from a different viewpoint, which they have never adopted, from which it is possible to understand in different terms the problems of production, health, or safety that they are facing, and which allows them to imagine and discuss original ways out of these problems.

Adopting this viewpoint, that of work activity, it is also possible to anticipate the consequences of possible changes, through simulating possible future work. If such and such design decision is made, how will it be possible to work, not only in standard cases but also when greater variability occurs? Let us simulate it! Changing representations of work activity, and focusing on future possible activity at the design stage, is necessary. But the ergonomists' practices of activity analysis and simulation in the last two decades have revealed that it is not enough. It is not sufficient to analyse and circulate the determinants of work activity at the workplace to trigger relevant changes. There are two reasons for that:

-The first one is that one cannot reflect about the limitations of something when he/she is convinced that it cannot change. The limits to action are limits for thought. Therefore it is hopeless to convince stakeholders that a change is necessary when they are convinced it is impossible. The ergonomist not only has to change the representations of what exists, but also, and as a condition to this, to show that something else is possible.

-The second reason why it is not enough to change the representations of work activity and of its determinants is that it is also necessary to understand and influence the determinants of these determinants: How are decisions made? Why is the workers' knowledge not taken in account at the design stage? Why do workers have to cope daily with the same difficulties without any improvement occurring? Why is there such an unbalance between top-down and bottom-up information flows? Why do local managers have so little power of decision?

 The third proposal is, then, that ergonomists do not only contribute with recommendations to existing decisions processes, but that they try to influence the structure of the decision processes themselves:

-the ergonomist suggests a systemic approach: a given workplace cannot be improved independently of its relations with upstream and downstream workstations, nor independently of rethinking the work organisation and the local managers' power of decision; -the ergonomist promotes the need to confront different types of knowledge ─ that means he/she supports the "right to play" of stakeholders that were usually left by the wayside: operators, shop floor managers, etc. Debates between different logics, controversies between different standpoints are necessary to cope with all the aspects of any workplace. The ergonomist does not only recommend the participation of a number of stakeholders: he/she has methods to generate this participation at the right moments in a relevant and efficient way; -the ergonomist fuels these debates with knowledge about work and about humans, their physiology, their cognition, their subjectivity, knowledge which has been produced in an interdisciplinary framework by human factors and ergonomics for 70 years. But the ergonomist also brings new methods, new descriptions, new models of existing and future activity; -he/she tries to shift the focus of work organisation from "the search for the one best way" to "the development of relevant operational leeway everywhere", from centralism to subsidiarity, from topdown prescription to participation.

 This is the practice of many ergonomists, which cannot be confined to the application of scientific knowledge about the human at work. On the contrary, the discipline of ergonomics should recognize that practice is a relevant source of knowledge, if practice is conceptualized and modelled. Many excellent ergonomists worldwide have developed strategic skills and practices that have never been taught or even mentioned during their degree course. Ergonomics needs research about the ergonomists' craft.

 What is at stake is our responsibility, as teachers, to prepare future ergonomists for what they should really do : not only to apply scientific knowledge, but to be able to understand the preoccupations that are beyond expressed requests, to articulate different stakes from different stakeholders, to detect the operational leeway that exists in work situations and that which is missing, to make strategic analyses of organisations and decision processes to influence them, and to develop a reflective practice, which is the basis for the development of the discipline and the profession.