Philippe Bich

Rida Laraki

Externalities in Economies with Endogenous Sharing Rules

Keywords: Abstract Economies, Generalized Games, Endogenous Sharing Rules, Walrasian Equilibrium, Incomplete and Discontinuous Preferences, Better Reply Security JEL Classification: C02, C62, C72, D50

 where, by definition, players are restricted to pure strategies. We also provide a new interpretation of payoff indeterminacy in Simon and Zame's model in terms of preference incompleteness.

Introduction

The model of strategic games with endogenous sharing rules was introduced by Simon and Zame [START_REF] Leo | Discontinuous Games and Endogenous Sharing Rules[END_REF]. Formally, it is a (N + 1)-tuple G = ((X i ) i∈N , U ), where N is the set 1 of players, X i is the strategy set of player i ∈ N , and U is a multivalued function from X := i∈N X i to R N with nonempty values. The set U (x) ⊂ R N can be interpreted as the universe of payoff possibilities, given the strategy profile x ∈ X. When U (x) = {(u i (x)) i∈N } is a singleton for every x ∈ X, G reduces to a usual strategic game, u i being the payoff function of player i.

Simon and Zame [START_REF] Leo | Discontinuous Games and Endogenous Sharing Rules[END_REF] provide conditions that guarantee the existence of a solution for G, i.e., existence of a selection q = (q i ) i∈N of U (a sharing rule of U ), together with a mixed Nash equilibrium m * = (m * i ) i∈N of the game G = ((X i ) i∈N , (q i ) i∈N ).

1

The concept of sharing rule gives rise to many interpretations. Imagine a designer who must determine who wins an indivisible object in some auction including tie-breaking rules. In that case, selections of U represent admissible auction rules, and a solution can be seen as a mechanism and a Nash equilibrium of the induced game. Another motivation comes from the payoff indeterminacy that many economic models exhibit: for example, several producers have to choose, each, a location in an area where a continuum of consumers are uniformly distributed. Assume each consumer goes to the closest location. Then payoffs are not well defined when some producers choose the same location: indeed, any division of consumers between the producers is plausible. Simon and Zame's result guarantees the existence of a market sharing rule under which the discontinuous game played by the producers admits a mixed Nash equilibrium.

In this note, we prove existence of a Simon and Zame "solution" in economies with externalities (also called generalized games). This is a general equilibrium model, introduced by Arrow and Debreu [START_REF] Arrow | Existence of an Equilibrium for a Competitive Economy[END_REF], in which players play in pure strategies and each player admissible set of strategies is constrained by the strategies chosen by the opponents. For example, in exchange economies, consumers are limited by their budget constraint, which depends on the price vector, itself depending on consumers' demands.

Our second contribution is an interpretation of sharing rules indeterminacy in terms of preference incompleteness. As Aumann [START_REF] Aumann | Utility theory without the completeness axiom[END_REF] argues: "of all axioms of utility theory, the completeness axiom is perhaps the most questionable". Following this seminal paper, many extensions of equilibrium models to incomplete preferences have been investigated, either for continuous preferences [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF][START_REF] Shafer | Equilibrium in abstract economies without ordered preferences[END_REF], or discontinuous ones [START_REF] Carmona | An Existence Result for Discontinuous Games[END_REF]12,[START_REF] Yannelis | Equilibrium Theory with Discontinuous and Non-Ordered Preferences[END_REF]. In this note, we will assume that the ambiguity generated by the indeterminacy of payoffs creates incompleteness in the preferences. This permits to associate to every economy with externalities and endogenous sharing rules an economy with externalities and incomplete and discontinuous preferences. We prove that, in general, this economy does not possess a Nash equilibrium, but it is possible to complete the preferences in a weak sense to restore the existence of an equilibrium.

Economies with Endogenous Sharing Rules

An Economy with externalities and endogenous sharing rules E is a pair E = (G, B = (B i ) i∈N ) where G = ((X i ) i∈N , U ) is a game with endogenous sharing rules, and B i is a multivalued mapping from X -i to X i with a closed graph and nonempty convex values (i.e., a Kakutani-type mapping).

Definition 1 A solution of E is a pair (x * , q), where q = (q i ) i∈N is a selection of U , and x * ∈ X is a generalized Nash equilibrium of ((X i ) i∈N , (q i ) i∈N , B), i.e.: (i) For every i ∈ N ,

x * i ∈ B i (x * -i ). (ii) For every x i ∈ B i (x * -i ), q i (x i , x * -i ) ≤ q i (x * ).
Consider the following assumptions: A1: X is a convex and compact subset of a Hausdorff and locally convex topological vector space; A2: U is bounded; A3: The graph of E, defined by Γ := {(x, v) : v ∈ U(x) and x i ∈ B i (x -i ) for every i ∈ N }, is closed; A4: U admits a selection u = (u i ) i∈N , such that each u i is quasiconcave in player i's strategy.

Theorem 2 Any economy with externalities and endogenous sharing rules satisfying A1 to A4 admits a solution.

This can be related to Simon and Zame existence result [START_REF] Leo | Discontinuous Games and Endogenous Sharing Rules[END_REF]: they prove the existence of a solution in mixed strategies in every strategic games under A1, A2, A3 and convexity of U (x) for every x ∈ X. Theorem 2 proves the existence of a solution in pure strategies, even when the strategies of each player are constrained by the strategies of his opponent. In strategic games where B i (x -i ) = X i for every x -i ∈ X -i and every i ∈ I, we get the existence of a solution à la Simon-Zame in pure strategies. This was an open question in Jackson et al. [START_REF] Jackson | Communication and Equilibrium in Discontinuous Games of Incomplete Information[END_REF] and was solved recently in Bich and Laraki [START_REF] Bich | On the existence of approximate equilibria and sharing rule solutions in discontinuous games[END_REF] by using as a tool Reny 's [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF] better-reply security condition. But we shall see that adding externalities makes the proof more complex. Indeed, the result relies on a recent condition for Nash equilibrium existence in discontinuous games by Barelli and Meneghel [3].

Applications

Incomplete Preferences

Let us give an interpretation of Theorem 2 in terms of incomplete preferences. If G = ((X i ) i∈N , U ) is a game with endogenous sharing rules, then we can define the following preorders2 on X.

Definition 3 We say that y ∈ X is U -preferable to x ∈ X for player i, denoted x i y, if and only if u i (x) ≤ u i (y) for every selection3 u of U .

When x and y are distinct, x i y is equivalent to sup U i (x) ≤ inf U i (y), where U i (x) denotes the projection of U (x) ⊂ R N on the i-th component. In short, x i y if and only if y is at least as good as x, whatever the indeterminacy of payoffs modeled by U . Formally, to every economy with externalities (G, (B i ) i∈N ), one can associate an economy with externalities and incomplete preferences E = ((X i ) i∈N , ( i ) i∈N , (B i ) i∈N ), where the preorders i are derived from U as described above.

It is then standard to define a generalized Nash equilibrium of E as a profile x ∈ Π i∈N B i (x -i ) such that there is no player i ∈ N and no deviation y i ∈ B i (x -i ) with 4x i (y i , x -i ). The following example proves that, in general, E fails to have a generalized Nash equilibrium, even if the initial game G satisfies assumptions A1 to A4.

Example 4 Consider a strategic game with endogenous sharing rules and two players. The strategy spaces are [START_REF] Arrow | Existence of an Equilibrium for a Competitive Economy[END_REF][START_REF] Arrow | Existence of an Equilibrium for a Competitive Economy[END_REF]}. This satisfies assumption A1 to A4. In particular, any selection u of U satisfies the quasiconcavity requirement A4. As described above, this defines a game with incomplete preferences E = ((X i ) i=1,2 , ( i ) i=1,2 ). Clearly, for player 2, the unique bestresponse to x 1 is x 2 = 1x 1 . Thus, for every x 1 > 0, (x 1 , x 2 ) is not a Nash equilibrium of E, since it would imply x 2 = 1x 1 < 1, but then the only best-response of player 1 is x 1 = 0, a contradiction. Thus, the only candidate to be a Nash equilibrium is (0, 1), but it is not, since (0, 1)

X 1 = X 2 = [0, 1]. The endogenous sharing rules are defined by U (x 1 , x 2 ) = (1 -x 1 (1 -x 2 ), 1 -(1 -x 1 -x 2 ) 2 ) if (x 1 , x 2 ) = (0, 1) and U (0, 1) = {(-1, 1),
1 (ε, 1) for every ε ∈]0, 1]. Indeed, 1 = sup U 1 (0, 1) ≤ inf U 1 (ε, 1) = 1 and 1 = sup U 1 (ε, 1) > inf U 1 (0, 1) = -1.
Hence, E has no Nash equilibria. In particular, it is not generalized correspondence secure (see [START_REF] Carmona | Existence of Nash Equilibrium in Ordinal Games with Discontinuous Preferences[END_REF]), a condition that would imply the existence of a Nash equilibrium of E.

Thus, one cannot apply recent generalized Nash existence results to E (e.g., Yannelis, He [START_REF] Yannelis | Equilibrium Theory with Discontinuous and Non-Ordered Preferences[END_REF] or Carmona and Podzeck5 [START_REF] Carmona | Existence of Nash Equilibrium in Ordinal Games with Discontinuous Preferences[END_REF]) simply because the game may fail to have a Nash equilibrium. We now study the possibility of restoring existence after some completion of the preferences. Recall that a completion of the preorder i defined on X is a complete preorder ′ i on X such that:

(i) ∀(x, y) ∈ X 2 , x i y ⇒ x ′ i y; (ii) ∀(x, y) ∈ X 2 , x i y ⇒ x ′
i y. When the preorders i , i ∈ N , are defined from U as above, then for every selection q of U , one can define a q-completion of i as the complete preorder q i on X such that: x q i y ⇔ q i (x) ≤ q i (y). This is a weak completion of i , in the sense that it satisfies property (i) but not property (ii). This is because x i y is defined by: v i (x) ≤ v i (y) for every selection v of U , and w i (x) < w i (y) for at least one selection of U , and this may not imply q i (x) < q i (y).

Corollary 5 Consider an economy with externalities and endogenous sharing rule which satisfies assumptions A1 to A4, and let i be the preorders associated to U as described above. Then there exists a q-completions q i of the preorders i (i ∈ N ) for some selection q of U , such that ((X i ) i∈N , ( q i ) i∈N , (B i ) i∈N ) has a generalized Nash equilibrium x * ∈ X.

Remark 6 As said above, the completion above may not preserve the strict order. More precisely, it is possible (though not automatic) that the generalized Nash equilibrium x * ∈ X and the selection q in Corollary 5 satisfy u i (x) ≤ u i (y) for every selection u of U with at least one strict inequality, although q i (x) = q i (y). It is not surprising, since the endogenous sharing rule q can be seen as "summarizing actions taken by unseen agents whose behavior is not modelled explicitely" (see [START_REF] Leo | Discontinuous Games and Endogenous Sharing Rules[END_REF]). This unmodeled behaviour implies that the new sharing rule q can really change the preferences of the players (at least at indeterminacy points), and there is no reason why the associated q-completion would preserve the strict order. Anyway, a possibility to preserve the strict order is to strengthen the definition of the preorders i as follows. Say that y ∈ X is U -strongly preferable to x ∈ X for player i, denoted x ≪ i y, if and only if u i (x) < u i (y) for every selection u of U . This leads to a new notion of profitable deviation for player i (more restrictive than this defined by i ), thus to a weaker notion of Nash equilibrium. Under assumptions A1 to A4, the existence of a Nash equilibrium for such preorders ≪ i is a consequence of Shafer-Sonnenschein's Theorem (see [START_REF] Shafer | Equilibrium in abstract economies without ordered preferences[END_REF]). Indeed, if y i ∈ P i (x) := {y i ∈ X i : (x i , x -i ) ≪ i (y i , x -i )}, then, by definition of ≪ i , and because U has a closed graph, we get x ′ ≪ (y ′ i , x ′ -i ) for every (x ′ , y ′ i ) in some neighborhood of (x, y i ). This proves that P i has an open graph. Moreover, x i / ∈ coP i (x) for every i ∈ I (from assumption A4). Consequently, we can apply Shafer-Sonnenschein's Theorem to get the existence of x * ∈ X such that P i (x * ) = ∅ for every i ∈ N , i.e. x * is a Nash equilibrium in the game defined by the preorders ≪ i . This can be extended to the case of an economy with endogenous sharing rules.

Generalized Games with Discontinuous Payoffs

For every generalized game G = ((X i ) i∈N , (u i ) i∈N ), B) where each utility function u i is assumed to be bounded and quasiconcave with respect to x i , we can restore existence of a generalized Nash equilibrium by changing the payoff functions at discontinuity points under the constraint that the graph of the new game remains in the closure of the graph of the original game. More precisely, we can construct new payoff functions q = (q i ) i∈N such that:

(a) the economy with externalities G ′ = ((X i ) i∈N , (q i ) i∈N ), B) admits a generalized Nash equilibrium x * .

(b) for every y ∈ X with y i ∈ B i (y -i ) for every i ∈ N , there is a sequence (y n ) n∈IN converging to y such that y n i ∈ B i (y n -i ) for every i ∈ N and q(y) = lim n→+∞ u(y n );

This extends the sharing rule existence result in [START_REF] Bich | On the existence of approximate equilibria and sharing rule solutions in discontinuous games[END_REF] (Theorem 2). The proof is a direct consequence of Theorem 2. Indeed, for every profile y ∈ X, define U (y) to be the set of limit points of (u(y n )) n∈IN for all possible sequences (y n ) n∈IN converging to y and such that y n i ∈ B i (y n -i ) for all i ∈ N . Clearly, U satisfies all the assumptions A1 to A4. Consequently, from Theorem 2, there is a solution (x, q), which satisfies conditions (a) and (b) above.

Exchange Economies

Consider n consumers and m commodities. The initial endowment e i of consumer i is assumed to be an interior point in R m + . Consumer i's consumption set is equal to

X i = {x i ∈ R m + :
x i ≤ j∈N e j + (1, ..., 1)}. Following the interpretation of subsection 3.1, consumer's incomplete preferences are assumed to be represented by a multivalued function6 U i from X i to R + with a closed graph, nonempty bounded values on every compact set and admitting at least one quasiconcave selection u i . An example, for m = 2, could be:

(1) U 1 (x 1 , x 2 ) =    x 1 + x 2 if x 1 + x 2 < 2, x 1 + x 2 + 1 if x 1 + x 2 > 2, [x 1 + x 2 , x 1 + x 2 + 1] if x 1 + x 2 = 2.
In this economy, there is a bonus of 1 unit if the consumer has a sufficient quantity of goods, because he may have a substantial benefit if he has more than some minimal level. Moreover, consumer 1's payoff is indetermined when x 1 + x 2 = 2, and there are many ways to complete the preferences .

Under the above assumptions, there exists a selection (q i ) i∈N of (U i ) i∈N satisfying (a), (b) and (c) below:

(a) the economy {X i , q i , e i } i∈N admits a walrasian equilibrium (x * , p * ) ∈ i∈N X i × ∆(R m + ), that is:7 (1) i∈N x * i ≤ i∈N e i , and (2) x * i maximizes the utility function q i of agent i on his budget set

B i (p * ) = {x i ∈ X i : p * • (x i -e i ) ≤ 0}. (b) for every x i ∈ B i (p * ), there is a sequence (x n i , p n ) n∈IN converging to (x i , p * ), with x n i ∈ B i (p n
), and such that lim n→+∞ u i (x n i ) = q i (x i ).

(c) for every x i ∈ X i and

x i / ∈ B i (p * ), q i (x i ) = u i (x i ).
Conditions (b) and (c) guarantee that the payoff function q i is not too far from u i (modifications occur only at discontinuity points that are inside the budget set). In particular, u i (x i ) = q i (x i ) if u i is continuous at x i . The proof can be found in appendix B. Let us illustrate the result with an example.

Example 7 Consider the following walrasian economy with externalities and discontinuous payoffs: m = 2, e 1 = e 2 = (1, 1) and U 1 = U 2 defined as above in [START_REF] Arrow | Existence of an Equilibrium for a Competitive Economy[END_REF]. The following payoff functions u 1 and u 2 are quasiconcave selections of U 1 = U 2 :

u 1 (x 1 , x 2 ) = u 2 (x 1 , x 2 ) =    x 1 + x 2 if x 1 + x 2 < 2, x 1 + x 2 + 1 if x 1 + x 2 > 2, 3x 1 2 + x 2 if x 1 + x 2 = 2
However, the exchange economy defined by u 1 and u 2 has no walrasian equilibrium. Indeed, suppose p = (p 1 , p 2 ) is an equilibrium price vector. If p 1 ≤ p 2 , then no consumer demands x 1 , if p 1 > p 2 no consumer demands x 2 . Now, if we consider the selection:

q 1 (x 1 , x 2 ) = q 2 (x 1 , x 2 ) = x 1 + x 2 if x 1 + x 2 ≤ 2, x 1 + x 2 + 1 if x 1 + x 2 > 2 Then x * = (1, 1), p * = (1, 1
) is a walrasian equilibrium.

Appendix A: Proof of Theorem 2

The proof consists in several steps: first, we turns G into an auxiliary discontinuous strategic game G ′ . Second (steps 2 and 3), we prove the existence of a relaxed Nash equilibrium of G ′ . This is used to construct in step 4 a sharing rule solution of G ′ that satisfies some desirable properties. Finally, such a sharing rule solution is used to build a solution of G. This methodology follows the one developed in [START_REF] Bich | On the existence of approximate equilibria and sharing rule solutions in discontinuous games[END_REF] and [START_REF] Bich | A unified approach to equilibrium existence in discontinuous strategic games[END_REF] to prove existence of Nash, approximate and sharing rule equilibria in discontinuous games. But it is more complicated because of the externalities. Importantly, the existence results contained in steps 3 and 4 are valid for any quasiconcave compact discontinuous strategic game ((X i ) i∈N , (u i ) i∈N ).

By assumption, U admits a single-valued selection φ = (φ i ) i∈I where each φ i is quasiconcave in player i's strategy.

Step 1. Associate to G a discontinuous game G ′ .

Following an idea of Reny [START_REF] Reny | On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games[END_REF], we associate to the economy with externalities G = ((X i ) i∈N , (φ i ) i∈N ), B) a strategic game G ′ as follows. Because U is bounded, there exists Λ ∈ R such that φ i (x) ≥ Λ + 1 for every i ∈ N and every profile x ∈ X. The game G ′ has N players. For every i ∈ N , strategy set of player i is X i , and his payoff is

u i (x) = φ i (x) if x i ∈ B i (x -i ), Λ otherwise.
These new payoff functions are also quasiconcave.

Step 2. Generalized regularization of payoff functions of G ′ . Throughout this proof, for every i ∈ N , x ∈ X, and U in V(x -i ) (the set of open subsets of X -i ), denote by W U (x i , x -i ) the set of Kakutani-type8 multivalued mappings d i from U to X i such that x i ∈ d i (x -i ) for every x -i ∈ U . Let u i : X → R be the following regularization9 of the utility function u

i (2) u i (x) := sup U ∈V(x -i ) sup d i ∈W U (x) inf x ′ -i ∈U,x ′ i ∈d i (x ′ -i ) u i (x ′ ).
Remark that u i (x) ≤ u i (x) for every x ∈ X, since in the infimum above one can take

x ′ = x.
Step 3. Existence of a refined Reny solution of G ′ . Let us prove that there exists a pair (x * , v * ) ∈ Γ (where Γ := {(x, u(x)) : x ∈ X}) such that:

(3) ∀i ∈ N, sup

x i ∈X i u i (x i , x * -i ) ≤ v * i .
Such pair (x * , v * ) refines the Reny solution concept introduced in [START_REF] Bich | On the existence of approximate equilibria and sharing rule solutions in discontinuous games[END_REF]. When u i is continuous for every i ∈ N , x * is a Nash equilibrium and v * = u(x * ) is the associated payoff vector.

By contradiction, assume that there is no such pair, and let us prove that G ′ is generalized better-reply secure. Recall that G ′ is generalized better-reply secure (Barelli and Meneghel [START_REF] Barelli | A Note on the Equilibrium Existence Problem in Discontinuous Games[END_REF]) if whenever (x, v) ∈ Γ and x is not a Nash equilibrium, there exists a player i and a triple (d i , V x -i , α i ), where V x -i is an open neighborhood of x -i , d i is a Kakutani-type multivalued function from V x -i to X i and α i > v i is a real number such that for every x ′ -i in V x -i and x ′ i ∈ d i (x ′ -i ), one has u i (x ′ i , x ′ -i ) ≥ α i . For, consider (x, v) ∈ Γ such that x is not a Nash equilibrium. By assumption, (x, v) does not satisfy inequality (3), thus there exists some player i ∈ N such that sup

y i ∈X i u i (y i , x -i ) > v i . From the definition of u i , there is ε > 0, U ∈ V(x -i ), d i ∈ W U (x)
such that for every x ′ -i ∈ U and every

x ′ i ∈ d i (x ′ -i ), u i (x ′ i , x ′ -i ) ≥ v i + ε :
this implies generalized better-reply security. Consequently, from Barelli and Meneghel [START_REF] Barelli | A Note on the Equilibrium Existence Problem in Discontinuous Games[END_REF], since G ′ is generalized better-reply secure, it admits a Nash equilibrium. But this is a contradiction, since if x ∈ X is a Nash equilibrium, (x, u(x)) satisfies inequality (3) (because u i (x) ≤ u i (x) for every x ∈ X). By contradiction, this proves the existence of (x * , v * ) ∈ Γ satisfying inequality (3).

Step 4. Existence of a sharing rule solution of G ′ . We now prove that there exists some new payoff functions (q i ) i∈I and a pure Nash equilibrium x * ∈ X of G ′′ = ((X i ) i∈N , (q i ) i∈N ), with the additional properties: (i) for every i and

d i ∈ X i , q i (d i , x * -i ) ≥ u i (d i , x * -i ).
(ii) For every y ∈ X , there exists some sequence (y n ) converging to y such that u(y n ) converges to q(y).

For every i ∈ N , denote by S i (y) the space of sequences (y n ) n∈N of X converging to y such that lim n→+∞ u i (y n ) = u i (y). Then, define q : X → R N by

q(y) =      v * if y = x * , any limit point of u(x n ) n∈N if y = (d i , x * -i ) for some i ∈ N, d i = x * i , (x n ) n∈N ∈ S i (d i , x * -i ), q(y) = u(y)
otherwise.

Since (x * , v * ) ∈ Γ, and by definition of q, condition (ii) above is satisfied at x * . Clearly, by definition, it is also satisfied at every y different from x * for at least two components, and finally also at every (d i , x * -i ) with d i = x * i (for some i ∈ N ), from the definition of q(d i , x * -i ) in this case. Condition (i) is true at every y different from x * for at least two components (from u i ≤ u i ), is true at every (d i , x * -i ) with d i = x * i by definition, and is finally true at x * from inequality (3). This ends the proof of Step 4.

Step 5. Existence of a solution of G. Now, we finish the proof of Theorem 2. Take

d i ∈ B i (x * -i ) = ∅. For every x ′ -i in some neighborhood of x * -i and every x ′ i ∈ B i (x ′ -i ), we have, by definition, u i (x ′ i , x ′ -i ) = φ i (x ′ i , x ′ -i ) ≥ Λ + 1.
Since B i is a Kakutani-type mapping, this implies, by definition, u i (d i , x * -i ) ≥ Λ + 1 (where u i is the regularization of u i , defined in the beginning of this proof). Thus, from condition (i) in step 4 above, we get [START_REF] Bich | A unified approach to equilibrium existence in discontinuous strategic games[END_REF]

∀d i ∈ B i (x * -i ), q i (d i , x * -i ) ≥ u i (d i , x * -i ) ≥ Λ + 1.
Since x * is a Nash equilibrium of G ′′ , we have:

∀i ∈ N, q i (x * ) ≥ sup d i ∈X i q i (d i , x * -i ) ≥ Λ + 1.
From condition (ii) in step 4 above, there is a sequence (x n ) converging to x * such that u(x n ) converges to q(x * ). Since q i (x * ) ≥ Λ + 1 for every i ∈ N , we cannot have u i (x n ) = Λ for n large enough. Consequently, from the definition of u i , we get u i (x n ) = φ i (x n ) and x n i ∈ B i (x n -i ) for n large enough. Passing to the limit, we get x * i ∈ B i (x * -i ) for every i ∈ I (because B i has a closed graph). A similar argument can be applied to any (y i , x * -i ) ∈ X for which y i ∈ B i (x * -i ): there is a sequence (x n ) converging to (y i , x * -i ) such that u(x n ) converges to q(y i , x * -i ). Since q i (y i , x * -i ) ≥ Λ + 1 (from inequality (4)), we cannot have u i (x n ) = Λ for n large enough. Consequently, u i (x n ) = φ i (x n ) and x n i ∈ B i (x n -i ) for n large enough. In particular, since φ is a selection of U and since U has a closed graph, we get (5) ∀y i ∈ B i (x * -i ), q(y i , x * -i ) ∈ U (y i , x * -i ). Now, define q(y i , x * -i ) = q(y i , x * -i ) whenever y i ∈ B i (x * -i ) for some i ∈ N , and q(y) = φ(y) elsewhere. The proof that x * is a equilibrium of ((X i ) i∈N , (q i ) i∈N , B) is a straightforward First, assume that we do not have i∈N (x * ie i ) ≤ 0. Then, let us define p = (p(1), ..., p(k), ..., p(m)) ∈ ∆(R m + ) with p(k) = 0 when i∈N (x * ie i )(k) ≤ 0 (where i∈N (x * ie i )(k) denotes k-component of i∈N (x * ie i )), and p(k) = λ. i∈N (x * ie i )(k) otherwise (where λ > 0 is a normalization coefficient that insures that p ∈ ∆(R m + )). By definition, we get p. i∈N (x * ie i ) > 0, thus from (iii) above, p * . i∈N (x * ie i ) > 0. But from condition (i) above, the budget constraint yields p * (x * ie i ) ≤ 0 for every i ∈ N , and summing these inequalities, we get p * . i∈N (x * ie i ) ≤ 0, a contradiction. This proves i∈N (x * ie i ) ≤ 0. From (ii) above, for every x i ∈ B i (p * ), we have q i (x i ) = qi (x i , x * -i , p * ) ≤ qi (x * , p * ) = q i (x * i ). Thus, for every i ∈ N , x * i maximizes q i in B i (p * ), which ends the proof.

A preorder is a reflexive and transitive binary relation.

Every preorder on X admits a multi-utility representation (see[START_REF] Evren | On the multi-utility representation of preference relations[END_REF]), that is there exists a family (v j ) j∈J of real-valued functions defined on X such that: x y ⇔ for every j ∈ J, v j (x) ≤ v j (y). Thus, there is no loss of generality in working with a cardinal multi-representation.

Here, i denotes the strict preorder associated to i , that is: for every (x, y) ∈ X 2 , x i y if and only if x i y and not (y i x).

Remark that conversely, our paper does not generalize the existence results of these two papers.

Here, to simplify the exposition, we do not allow externalities, that is U i depends only of player i's strategies.

The set ∆(R m + ) denotes the unit simplex of R m + .

A Kakutani-type multivalued mapping is a multivalued mapping with nonempty convex values and a closed graph.

This function was introduced by Carmona (see[START_REF] Carmona | Understanding some recent existence results for discontinuous games[END_REF]).

Laraki's work was supported by grants administered by the French National Research Agency as part of the Investissements d'Avenir program (Idex [Grant Agreement No. ANR-11-IDEX-0003-02/Labex ECODEC No. ANR11-LABEX-0047] and ANR-14-CE24-0007-01 CoCoRICo-CoDec).

1 For simplicity, we use the same letter N for the set of players or the number of players.

consequence of x * being a Nash equilibrium of ((X i ) i∈N , (q i ) i∈N ). Last, we have to prove that q(y) ∈ U (y) for every y ∈ X. This is clear at y = (y i , x * -i ) for y i ∈ B i (x * -i ), from (5) above. For others y, we have q(y) = φ(y) ∈ U (y) by definition. This ends the proof of Theorem 2.

5 Appendix B: proof of the statements in Section 3.3

From the exchange economy, define an economy with externalities and discontinuous payoffs (G, B) as follows:

1. There are (N + 1) players.

2. For i = 1, ..., N , player i's convex compact strategy space is

Last, define the strategy correspondences as follows: for every

Following section 3.2, this economy has a solution (x * , p * , q). This means that:

1. For every (x, p) ∈ i X i × ∆(R m + ) such that x i ∈ B i (p) for every i ∈ N , there is a sequence (x n , p n ) n∈IN converging to (x, p) such that x n i ∈ B i (p n ) for every i ∈ N and qi (x, p) = lim n→+∞ u i (x n i ). In particular, from the continuity of v N +1 , qN+1 (x, p) = v N +1 (x, p) = p. i∈N (x ie i ).

(i) For every

Let us now define q i (x i ) := qi (x i , x * -i , p * ) for every x i ∈ B i (p * ), and q i (x i ) = u i (x i ) otherwise. From 1 and 2 above, there is a sequence (x n , p n ) n∈IN converging to (

Thus condition (b) and (c) in section 3.3 hold. Let us prove that condition (a) also holds, that is, (x * , p * ) is a walrasian equilibrium of the economy with payoff functions (q i ) i∈N .


On the Effect of Low Temperatures on the Maximum Output Power of a Coherent Erbium Doped Fiber Amplifier Julien Le Gouët, Jérémy Oudin, Philippe Perault, Alaeddine Abbes, Alice Odier, and Alizée Dubois

Abstract-The influence of low temperatures on the performance of a high-power single-frequency fiber laser amplifier is evaluated with a numerical simulation. Cooling the fiber can allow to take advantage of both higher damping of the acoustic waves in the silica glass, and higher laser efficiency. We first report on the measurement of the stimulated Brillouin scattering (SBS) threshold in a silica fiber as a function of the temperature from 300 K down to 77 K. We then present the measurements of small-signal absorption and gain spectra of an erbium doped alumino-phosphosilicate fiber, at room temperature and liquid nitrogen temperature. Based on these data, we derive a numerical study of the combined effects of cooling on the SBS threshold and the amplifier efficiency, and conclude on the interest of this technique for SBS limited high power Er doped fiber amplifiers (EDFA). The temperature increase caused by the pump laser in the fiber core is also addressed.

Index Terms-Stimulated Brillouin scattering, erbium, fiber amplifier, liquid nitrogen temperature

I. INTRODUCTION

A MONG the countless types of lasers, high power sources with single frequency and single spatial mode emission have numerous applications. The amplification of coherent pulses (longer than 100 ns) up to the kW peak power range allow the long distance optical measurement of wind velocity [START_REF] Dolfi-Bouteyre | Pulsed 1.5µm lidar for axial aircraft wake vortex detection based on high brightness large-core fiber amplifier[END_REF], and continuous wave (cw) sources at 100 W or more [START_REF] Creeden | Single frequency 1560 nm Er:Yb fiber amplifier with 207 W output power and 50.5% slope efficiency[END_REF] are interesting for free-space telecommunications or defense applications. The single spatial mode offers the best atmospheric propagation, while the spectral linewidth defines the velocity measurement resolution or the capacity limit of coherent combining.

For the best compactness and robustness, these systems are preferably based on fiber coupled laser sources, which are amplified in rare-earth doped optical fibers. As it is wellknown, for coherent sources with spectral linewidth narrower than 10 MHz, the maximum power in fibers is limited by the stimulated Brillouin scattering (SBS) [START_REF] Kobyakov | Stimulated Brillouin scattering in optical fibers[END_REF]. In order to increase the power delivered by these fiber amplifiers, several approaches have been developped to hinder the SBS effect. In essence, they come down to either increase the effective mode Julien Le Gouët, Jérémy Oudin, Philippe Perault are with Office National d'Etudes et de Recherches Aérospatiales (ONERA), Palaiseau, France, e-mail: julien.le_gouet@onera.fr Alaeddine Abbes was with ONERA and is now at IES, Univ Montpellier, UMR CNRS 5214, Montpellier, France Alice Odier was with ONERA and is now at Amplitude Systemes, Pessac, France Alizée Dubois was with ONERA and is now at Service de Physique de l'Etat Condensé, CEA, Gif/Yvette, France area [START_REF] Ahmad | Polarizationmaintaining, large-effective-area, higher-order-mode fiber[END_REF], or reduce the interaction length [START_REF] Shi | Kilowatt-level stimulated-Brillouin-scattering-threshold monolithic transform-limited 100 ns pulsed fiber laser at 1530 nm[END_REF], or reduce the SBS gain. This last solution can be obtained for example by reducing the overlap between acoustic and optical waves [START_REF] Li | Al/Ge codoped large mode area fiber with high SBS threshold[END_REF], by introducing a thermal [START_REF] Imai | Dependence of stimulated Brillouin scattering on temperature distribution in polarization-maintaining fibers[END_REF] or strain gradients [START_REF] Yoshizawa | Stimulated Brillouin scattering suppression by means of applying strain distribution to fiber with cabling[END_REF] along the fiber to generate an inhomogeneous broadening of the SBS gain, or by changing the very nature of the core glass [START_REF] Ballato | Rethinking Optical Fiber: New Demands, Old Glasses[END_REF].

Another degree of freedom is the influence of low temperatures on the attenuation of the acoustic waves in silica glass. This effect has been analyzed since the 50's [START_REF] Anderson | Ultrasonic Absorption in Fused Silica at Low Temperatures and High Frequencies[END_REF], [START_REF] Pine | Brillouin scattering study of acoustic attenuation in fused quartz[END_REF], but its various physical origins have been understood in more detail only recently [START_REF] Courtens | Elasticity and anelasticity of oxide glasses[END_REF]. For hypersonic waves generated by optical radiations at a wavelength of 1.5 µm, the attenuation presents a maximum at about 100 K. This higher damping of the hypersound reduces the negative impact of the Brillouin backscattering, and allows the propagation of higher power in the fiber.

Cooling the fiber has the other well-known effect of improving the efficiency for lasers based on rare earth doped crystals [START_REF] Ter-Gabrielyan | Temperature dependence of a diode-pumped cryogenic Er:YAG laser[END_REF] or fibers [START_REF] Yao | Cryogenically cooled erbium-doped fiber lasers[END_REF]. This effect is due to the Boltzmann population distribution in the lower Stark levels, increasing the absorption cross-section in the vicinity of the zero-phonon transition.

The aim of this article is to study the influence of low temperatures on the maximum output power of a typical single-frequency EDFA. This study is based on two measurements between room and liquid nitrogen temperatures: the SBS linewidth and threshold power in a passive fiber, and the spectra of absorption and gain of an erbium (Er) doped singlemode fiber. To our knowledge, this is the first study about the combined effects of SBS linewidth and amplifier efficiency on the maximum reachable power at low temperature.

As a brief state of the art of the cryogenic cooling of optical fibers, we can mention several applications. In passive single-mode fibers, the spectral shift and width of the SBS spectrum are considered to monitor the thermal stability of superconductor magnets cooled down to a few Kelvins, such as those used at the LHC in CERN or in ITER [START_REF] Thévenaz | Brillouin optical fiber sensor for cryogenic thermometry[END_REF], [START_REF] Mahar | Real-time simultaneous temperature and strain measurements at cryogenic temperatures in an optical fiber[END_REF]. In the case of Er doped fibers, the spectral properties of the laser amplification have been analyzed down to 4 K, in order to analyze the influence of the Er gain homogeneous linewidth in wavelength multiplexing telecommunications [START_REF] Desurvire | Spectral gain hole-burning at 1.53 µm in erbium-doped fiber amplifiers[END_REF]. Singlemode Er doped fibers are exposed to even lower temperatures in demonstrations of quantum memories compatible with fiber telecommunications [START_REF] Veissier | Optical decoherence and spectral diffusion in an erbium-doped silica glass fiber featuring long-lived spin sublevels[END_REF]. As for high power lasers at cryogenic temperatures, numerous works have been dedicated to Ytterbium doped large mode area (LMA) fibers, either to increase the laser efficiency [START_REF] Jelger | Highly efficient temporally stable narrow linewidth cryogenically cooled Yb-fiber laser[END_REF], or to improve the gain in the low wavelength domain [START_REF] Steinborn | A continuous wave 10 W cryogenic fiber amplifier at 1015 nm and frequency quadrupling to 254 nm[END_REF]. It should be highlighted that throughout this literature, no particular issue was reported about the resistance of the fibers (core or clad pumped) to the low temperatures.

In the following, we first present the measurements of the SBS threshold in a passive single mode fiber as a function of the temperature. Then we report our measurements of the small-signal interaction spectra (absorption and emission) on a single-mode Er:Yb doped alumino-phosphosilicate fiber [START_REF] Bubnov | Optical properties of fibres with aluminophosphosilicate glass cores[END_REF]. We use these data as inputs for a numerical simulation of a single frequency high power amplifier (>100 W range) in an Er doped LMA fiber pumped in the same transition as the laser (so-called in-band pumping, here in the 4 I 15/2 -4 I 13/2 transition). Finally we conclude on the improvement of efficiency and maximum power that can be obtained by cooling the active fibers.

II. TEMPERATURE DEPENDENCE OF SBS THRESHOLD

POWER

The basic idea of this work stems from a well-referenced variation of the hypersound attenuation in amorphous media as a function of the temperature [START_REF] Anderson | Ultrasonic Absorption in Fused Silica at Low Temperatures and High Frequencies[END_REF]. In a crystal host, the hypersonic damping is dominated on a large temperature range by the phonon-phonon interactions, which involve the contribution of the thermal bath phonons. In that range (typically 20-300 K), the SBS linewidth (inverse of the hypersound lifetime) varies linearly with the temperature. However in noncrystalline solids, the thermal variation of the Brillouin linewidth is not monotonous [START_REF] Pine | Brillouin scattering study of acoustic attenuation in fused quartz[END_REF]. Because of the combination of several mechanisms, this variation exhibits a broad maximum at a temperature T that increases with increasing acoustic frequency, following an Arrhenius law [START_REF] Hertling | Internal friction and hypersonic velocity in vitreous germania under high pressure[END_REF], [START_REF] Floch | Experimental and theoretical study of the Brillouin linewidth and frequency at low temperature in standard single-mode optical fibres[END_REF].

In the case of standard silica optical fibers, the acoustic wave is generated at approximately 11 GHz for a 1.5 µm signal wavelength [START_REF]Nonlinear Fiber Optics[END_REF]. For that acoustic frequency, the SBS linewidth ∆ν B has been measured to increase from 50 MHz to 95 MHz when cooling from 300 K to 130 K [START_REF] Floch | Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4-370 K) and high hydrostatic pressures (1-250 bars)[END_REF]. Across the same temperatures, the Brillouin frequency shift varied by only 2% [START_REF] Floch | Experimental and theoretical study of the Brillouin linewidth and frequency at low temperature in standard single-mode optical fibres[END_REF].

For a single-frequency signal, with a spectral linewidth small compared to the SBS linewidth, the SBS small-signal gain writes as:

g B = 2πγ 2 e nv ac cλ 2 s ρ 0 ∆ν B ( 1
)
where n is the core refractive index, c the velocity of light in vacuum, λ s the wavelength of the optical signal field, v ac the acoustic waves velocity in the glass core, γ e is the electrostrictive coupling constant and ρ 0 the mean value of the glass density [START_REF] Boyd | Nonlinear Optics[END_REF]. If the ratio γ 2 e /ρ 0 does not vary significantly with T , an increase of ∆ν B translates into a decrease of g B . This finally corresponds to an increase of the incident SBS threshold power, according to its usual expression:

P th = 21 A ao g B L eff (2)
where A ao is the acousto-optic area [START_REF] Kobyakov | Stimulated Brillouin scattering in optical fibers[END_REF], and L eff is the interaction effective length, defined as :

L eff = 1 P sig (L) L 0 P sig (z)dz (3)

A. SBS linewidth measurements

We first measure the SBS linewidth as a function of the temperature, as reported in previous works [START_REF] Floch | Study of Brillouin gain spectrum in standard single-mode optical fiber at low temperatures (1.4-370 K) and high hydrostatic pressures (1-250 bars)[END_REF], for a 10 m long single-mode PM980 fiber (SMF). The method consists in analyzing the heterodyne beat-note between the injected signal and the SBS generated Stokes wave (Figure 1). For the signal, we use a laser diode (LD) at 1545 nm with linear polarization. The signal is split between a local oscillator and a probe arm. The probe is amplified by an EDFA before injection in the SMF. The power after the EDFA is high enough to provide a measurable SBS spectrum, and low enough to avoid spectral narrowing by the gain in a stimulated regime of Brillouin scattering [START_REF] Boyd | Noise initiation of stimulated Brillouin scattering[END_REF]. The SMF output is cleaved at an angle of 10 • to prevent any reflection that could seed the SBS process. A PM circulator is used to collect the Stokes wave, which has the same polarization as the input signal. The interference with the local oscillator is finally measured on a 12 GHz band photoreceiver, and analyzed on a 26 GHz band electric spectrum analyzer. The SMF is coiled on a 12 cm diameter aluminum support, and the temperature of the SMF can be varied between 77 K and 300 K by adjusting the height of the coil above a liquid nitrogen bath. Three thermistors are placed on the circumference of the fiber support to check the homogeneity of the SMF temperature. We find that in our nitrogen container, a vertical displacement of 1 cm corresponds to a temperature shift ranging from 5 K close to the top, to 15 K close to the nitrogen bath.

The measurement of the SBS linewidth as a function of T is displayed on Figure 2 (gray squares). It shows that the SBS linewidth finds a local maximum of about 75 MHz for T ≈ 100 K, like in previously reported measurements. At room temperature, the SBS linewidth is about 40 MHz, so we expect that the SBS threshold can be improved by a factor of 1.9.

B. Empirical definition of SBS

If the linewidth ∆ν B is the only parameter that varies significantly with temperature in the SBS gain expression (see eq. ( 1)), then the influence of T on the SBS threshold (eq. ( 2)) can be directly obtained from the variation of ∆ν B (see Figure 2, gray squares). In order to verify this correlation, we proceed to an empirical measurement that consists in monitoring the shape of high power pulses at the fiber output. Once the Brillouin scattering reaches the amplifying regime, close to the SBS threshold, a noticeable fraction of the input signal is backscattered. For a given input pulse shape, this translates into an alteration of the output pulse shape close to the peak.

The method for this measurement is straightforward: an acousto-optic modulator generates laser pulses from the 1545 nm laser diode, with a pulse duration of 400 ns and a repetition rate of typically 20 kHz. A two-stage EDFA can raise the injected peak power at the level of the SBS threshold in the 10 m SMF. The input peak power can be modified finely by tuning the repetition rate. The output pulse shape is monitored on a photodiode with a rise time of 10 ns. We define the empirical SBS threshold by the input power required to generate a noticeable dip of about 10% of the peak power. The relative error of this measurement is lower than 10%. From Figure 2 (black circles), we observe that the SBS threshold increases from 8 W at 300 K to 15 W at 100 K, which corresponds to an improvement by a factor of 1.9. Between 300 K and 77 K, the SBS threshold increases by a factor 1.8. We also observe a very good correlation between the SBS threshold and the SBS linewidth, as suggested by eq. ( 1) and ( 2) with the approximation that γ 2 e /ρ 0 does not vary significantly with T. Therefore this approximation is confirmed with a relative error lower than 10%.

C. Relative variation of the threshold

The SBS threshold variation with temperature can be confirmed by measuring the backscattered power as a function of the input power. The setup here is a direct measurement of the Stokes wave power, at the output of the optical circulator in the previous setup (Fig. 1). A direct time-resolved measurement cannot differentiate the contributions of the elastic Rayleigh and inelastic Brillouin backscattered waves, so we use an optical spectrum analyzer (OSA), with a spectral resolution of 0.05 nm. The corresponding frequency resolution of 6 GHz at 1545 nm allows the discrimination of the two contributions, typically separated by 11 GHz in silica fibers. On Figure 3 the peak power of the SBS Stokes wave is plotted as a function of the input peak power, for the fiber immersed in the liquid nitrogen bath (77 K) and outside (300 K). We did not perform the measurement at intermediary temperatures, because the acquisition of one curve is long enough to let some nitrogen evaporate, which would modify the height of the coil relative to the bath, and finally modify the temperature during the measurement.

For each curve, two regimes can be identified, corresponding to the spontaneous and undepleted stimulated Brillouin scattering. For 300 K, the spontaneous regime does not appear clearly, as the backscattered power is too close to the noise floor of the OSA. For 77 K it appears as the first section, for P peak input < 10 W. The regime of undepleted stimulated Brillouin scattering appears for P peak input 5 W at 300 K, and for P peak input 10 W at 77 K. In this regime, the curves can be fitted with a power function:

P Stokes = a.P b input (4)
As indicated by the parallelism of the curves at 77 and 300 K, the exponent b is identical for both curves (b = 13.5 ± 0.5). For the amplitude coefficients, the fits yield a 300 = (1.8 ± 0.1).10 -16 and a 77 = (5.1 ± 0.4).10 -20 .

The SBS threshold power corresponds to the incident power P th that induces the backscattering of some arbitrary fraction η th = P Stokes /P th of this input power. Hence, the criterion η th can be chosen differently, for example if the the setup under study is more or less sensitive to the backscattered power. Given the expression (4) used to fit the data, the fraction η th can be expressed as:

η th = a 300 • P b-1 th 300 = a 77 • P b-1 th 77
which yields for the ratio of the SBS threshold powers at 77 and 300 K:

P th 77 P th 300 = a 300 a 77 1 b -1 = 1.8 ± 0.1 (5)
Given the relation 2 between the threshold power and the SBS gain, we deduce that g B is 80% lower at 77 K than at room temperature.

The result of this quantitative measurement is identical to the ratio obtained with the empirical determination of the SBS threshold (see Fig. 2). Therefore it confirms the relevance of the first method to evaluate the influence of the temperature on P th .

III. INFLUENCE OF THE TEMPERATURE ON THE AMPLIFICATION EFFICIENCY

The efficiency of a rare-earth based laser is known to increase at low temperatures, thanks to the redistribution of the excited state populations in the Stark sub-levels that are closer to the zero-phonon line (about 1.53 µm for Erbium in usual silica glasses). For signals at longer wavelengths, the absorption cross-section is strongly reduced, and the effective electronic structure corresponds to a four-level system [START_REF] Kagi | Temperature dependence of the gain in erbium doped fibers[END_REF], [START_REF] Millar | Thermal properties of an erbium-doped fiber amplifier[END_REF]. We present here our measurements of small-signal absorption and gain spectra at room and liquid nitrogen temperatures on a particular Er:Yb doped alumino-phosphosilicate (Al:P) fiber, and we investigate numerically the interest of cooling the active fiber in a high power single-frequency amplifier.

A. Brief spectroscopy of Er in Yb codoped aluminophosphosilicate fiber

The Al:P host has attracted a lot of interest in view of Erbium:Ytterbium codoping in silica fibers [START_REF] Unger | Codoped materials for high power fiber lasers: discussion behaviour and optical properties[END_REF], [START_REF] Vienne | Role of aluminum in ytterbium:erbium codoped phosphoaluminosilicate optical fibers[END_REF]. When Yb is used to increase the pumping efficiency of Er, a phosphorus doping is required to prevent the energy backtransfer from Er to Yb : the P 2 O 5 glass formers increase the maximum phonon energy and favor the multi-phonon decay, thus reducing the 4 I 11/2 lifetime in Er [START_REF] Vienne | Fabrication and Characterization of Yb3+ : Er3+ Phosphosilicate Fibers for Lasers[END_REF]. On the other hand, Al oxide codoping is required to improve the dilution of Er 3+ ions. Otherwise, for high doping concentrations, the ions can form clusters that share the laser pump energy and reduce its efficiency [START_REF] Myslinski | Effects of concentration on the performance of erbium-doped fiber amplifiers[END_REF]. By doping silica glass with both Al and P oxides, it was soon discovered that the local formation of AlPO 4 units in the silica network could reduce the refractive index of the core, leading to even lower index than the value of the pure silica for equimolar concentrations [START_REF] Digiovanni | Structure and properties of silica containing aluminum and phosphorus near the AlPO 4 join[END_REF]. This possiblity of tailoring the refractive index profile is particularly interesting to design large mode area (LMA) fibers with single spatial mode. In addition, several groups have reported the interest of adding Yb or Yb:P codopants in fibers with high Er concentrations to help the solubility of Er 3+ ions in the silica network [START_REF] Kiritchenko | Effect of ytterbium co-doping on erbium clustering in silica-doped glass[END_REF], [START_REF] Lim | Optimizing the pumping configuration for the power scaling of in-band pumped erbium doped fiber amplifiers[END_REF]. Therefore the combination of Yb codoping and Al:P host appears as an interesting candidate for Er doped fibers for high efficiency amplifiers.

As a first confirmation of this guess, we evaluated an important parameter in a higly doped fiber: the fraction 2k of Er ions in pairs among the total population. A common method consists in measuring the ratio of non-saturable and smallsignal absorptions at the peak wavelength λ peak for absorption and gain, using the expression derived in [START_REF] Myslinski | Effects of concentration on the performance of erbium-doped fiber amplifiers[END_REF]:

α ns (λ, T ) = 2k • α Er (λ, T ) • 1 - α Er + g * Er 2α Er + g * Er (λ, T ) (6)
where α Er (λ, T ) and g * Er (λ, T ) are respectively the absorption and gain spectra at a given temperature (see Fig. 4). Our Al:P fiber is a sample of IXF-2CF-EY-PM-12-130 (iXblue): it has a core diameter of 12.6 µm, a numerical aperture of 0.14, and a clad diameter of 130 µm. When carefully spliced, the fiber is essentially single-mode, and the overlap between doped core and optical mode is about 90%. From the peak absorption (see paragraph below) and the typical cross-section of 7.10 -25 m 2 for this type of host [START_REF] Laroche | Accurate efficiency evaluation of energy-transfer processes in phosphosilicate Er3+-Yb3+-codoped fibers[END_REF], we deduce an erbium concentration [Er]≈ 2.10 25 m -3 . To saturate the fiber absorption, we used a fiber laser delivering up to 5 W at 1535 nm. We obtained a fraction of paired Er 3+ ions 2k = 1.6 ± 0.5%, which is about 6 times lower than the value obtained with standard fibers for that concentration (see k values on Fig. 4-9 in [START_REF] Lim | Pump conditioning and optimisation for erbium doped fibre applications[END_REF]).

The absorption and gain spectra of this fiber were then measured at 300 K and 77 K (see Fig. 4). The usual cutback method was applied by coupling the broadband spectrum of a halogen lamp in the sample fiber, and measuring the transmission with an optical spectrum analyzer. At λ peak = 1535 nm, the small-signal absorption is 46.5 dB/m at 300 K, and 75 dB/m at 77 K, hence a ratio of 1.6 between the Er absorptions at 77 and 300 K. The same way, a ratio of 1.2 was found between the Yb absorptions at 77 and 300 K, for a small signal at 975 nm. This value confirms a previous measurement on Yb-doped fibers at these temperatures [START_REF] Alimov | Investigations of Yb-doped optical fiber using selective laser excitation[END_REF].

The gain spectra were obtained by measuring first the backpropagating fluorescence from a sample pumped in the core at 980 nm through a wavelength division multiplexer (WDM), and correcting from the WDM transmission. The sample was short enough (about 1 cm) to guarantee a full population inversion over its length. This was confirmed by comparing with the spectra obtained for shorter lengths. Then to convert the relative emission spectrum into an absolute gain spectrum, we measured the small-signal gain g * Er at λ peak = 1535 nm. This was done by injecting a signal seed of about 10 µW and measuring the output signal power for various fiber lengths, with the same requirement of full population inversion. The values obtained are g * Er (λ peak ) = 45 ± 1 dB/m at 300 K, and g * Er (λ peak ) = 76 ± 4 dB/m at 77 K. The resulting gain spectra are displayed on Fig. 4.

As expected, according to the Boltzmann statistics, at 77 K the populations of the various sub-levels thermalize to lower energy levels. The number of ions in the lower sublevels of the ground state 4 I 15/2 increases, corresponding to an increase of the absorption close to the zero-phonon line and a decrease at longer wavelengths. Similarly, for an optically pumped ion, the population of the highest sublevels of the excited state decreases, hence reducing the gain at shorter wavelengths.

B. Modeling of a high power EDFA

The interaction spectra measured above, together with the SBS threshold, are the only parameter that are expected to change notably with the temperature. The laser transition lifetime is another important model parameter, but its value does not vary significantly between 300 and 77 K [START_REF] Barnes | Absorption and emission cross section of Er 3+ doped silica fibers[END_REF]. Therefore, we can use the results of the previous measurements (Fig. 4 and 3) to investigate the interest of cooling the Er-doped fiber of an amplifier. We focus our study on the case of a LMA high power EDFA pumped in-band. This type of amplifier is generally interesting to deliver cw or average power in the range of 100 W or higher, with single spatial mode and single frequency, typically for lidar applications or free-space telecommunications [START_REF] Shi | Fiber lasers and their applications[END_REF]. Given the high powers involved, eye-safe radiation and low thermal load are required. These requirements can be met by using respectively 1.5 µm lasers and in-band pumping [START_REF] Jebali | 264 W output power at 1585 nm in Er-Yb codoped fiber laser using in-band pumping[END_REF].

Our simulation program uses the common 4th order Runge-Kutta algorithm to solve the propagation and rate equations along the fiber [START_REF] Desurvire | Erbium-Doped Fiber Amplifiers: Principles and Applications[END_REF], with the common approximation of the effective overlap integral between the active region and the propagating mode to suppress the radial dependence [START_REF] Giles | Modeling erbium doped fiber amplifiers[END_REF].

The erbium laser transition is considered as a homogeneously broadened two-level system. This program has been validated by numerous comparisons to experimental results, in various configurations (cw or pulsed regime, core pumped Er or clad pumped Er:Yb fibers...), and yields relative errors lower than 20% with respect to the output signal and residual pump powers [START_REF] Canat | Evidence of thermal effects in a high-power Er3+-Yb3+ fiber laser[END_REF], [START_REF] Canat | Performance and limitations of high brightness Er-Yb fiber sources[END_REF].

The fiber considered here is an LMA fiber with Er:Yb codoping in the same Al:P host as described above, with an absorption of 80 dB/m at λ peak and 300 K, injected with a cw signal P sig = 1 W at λ sig = 1560 nm and pumped at λ pump = 1535 nm. The fraction of ions in pairs can be scaled from the value reported above (2k = 1.6% for 46 dB/m), so we consider 2k = 3%. The effect of the pairs on the efficiency is modeled in a simple way: the powers involved are much higher than the saturation power at both λ pump and λ sig (about 3 mW and 20 mW respectively), so the attenuation induced by the ion pairs is essentially constant along the fiber and can be identified to the non-saturable absorption (see eq. ( 6)).

The values of the non-saturable absorption calculated at λ pump and λ sig , at 300 and 77 K, are listed in Table I, and are added to the background loss l glass at the corresponding wavelengths. The other model parameters derive from direct measurements or comparisons with results of amplifiers at room temperature with phosphosilicate fibers.

The model is first used to calculate the longitudinal evolution of P pump (z) and P sig (z). We then use eq. ( 2) to calculate the SBS threshold power along the fiber. This way, we determine for a given pump power the maximum signal output power P max sig , which is limited either by the pump power (for low pump and corresponding long fiber), or by SBS (for high pump and shorter fiber).

As an example, for an injected pump power of 400 W, we show on Fig. 5 the different evolutions of P pump (z) and P sig (z), at 300 and 77 K. The SBS threshold powers are also indicated at the two temperatures, and are obtained from the expression (3) of the effective fiber length. From these curves, we extract the maximum signal power as the intersection between the signal P sig (z) and the SBS threshold P th (z). As expected, the 5. Evolutions of signal (solid) and pump (dashed) powers along the fiber, for an injected pump power of 400 W, at 300 K (red) and 77 K (blue). The SBS thresholds are indicated at 300 and 77 K, as well as the threshold at 300 K if the SBS gain could be reduced to the value of room temperature. efficiency of the amplifier is higher at low temperature: without taking into account the SBS limitation, the maximum power would be 60% higher at 77 K than at room temperature (350 W instead of 215 W), for a fiber length 30% shorter.

In addition, the SBS gain of a passive silica fiber is 80% lower at 77 K than at room temperature (see eq. ( 5)). Since the composition of the Er doped silica fiber is almost identical, we assume that the effect of cooling is identical on g B , even if the absolute values may differ. The SBS threshold in the amplifier is then higher at 77 K, with a ratio that depends on the effective lengths of the SBS interaction at the two temperatures. For the example on Fig. 5, the power is limited by the SBS threshold at both temperatures, and we find P max300 sig = 150 W at 300 K and P max77 sig = 315 W at 77 K. For an amplifier at room temperature, but with the same value of SBS gain as 77 K (reduced by 80%), the SBS limited power would increase to 185 W.

The figure 6 shows the variation with P pump of the maximum reachable signal power, which is first limited by the pump power, then by the SBS threshold. We compare three configurations: fiber amplifier at 300 K, at 77 K, and at 300 K with the SBS gain reduced by 80%. In the pump limited regime, i.e. for a pump power lower than 150 W, the optical-optical efficiency (slope of the dotted lines) is about 55% at 300 K and 90% at 77 K. The output power is thus superior by 60% at 77 K, thanks to the sole thermal population distribution. On the other hand in the SBS limited regime (Pump>250 W), the combination of higher acoustic damping and population redistribution at low temperature allow a power increase close to a factor of two.

We also consider the case of an EDFA at 300 K with the SBS gain value of 77 K, in order to compare the distinct influence of the sole SBS gain reduction only and of the fiber cooling. By reducing the SBS gain by 80%, the output power is improved by a small amount (20% for a 400 W pump power). This shows that the cold fiber efficiency is the most determining contribution in the improvement of the output power. In addition, let us note that the experimental techniques allowing the SBS gain reduction (high doping, fiber strain, signal broadening, etc) could also be applied at low temperature, increasing again the maximum output power.

As argued in the beginning, we find that cooling the fiber can bring together the two benefits of higher SBS threshold and higher amplifier efficiency. For all the points calculated, the amplified spontaneous emission (ASE) at the fiber input and output was negligible.

IV. ARGUMENT ABOUT THE LASER GENERATED HEAT

One could be concerned about the heat generated by this high power setup in the fiber core, which could increase the temperature. We show that there is no such limitation, thanks to the low quantum defect between pump and signal: most of the pump energy is converted into signal. We assume here that the main contribution to the heat would come from the non-radiative decay of the pump laser. Considering the same example presented previously (see Fig. 5), we calculate the temperature increment caused by the fraction of pump that is not converted into signal. At 77 K, the highest rate of pump absorption is about 90 W/m (between 3 and 5 m on Fig. 5). The quantum defect between pump and signal radiations is given by 1-λ pump /λ signal ≈ 2%, so the power that is converted into heat is lower than 2 W/m.

To convert this heat load into a quantitative temperature increase in the fiber core, the fiber is modeled with a finite element method. We consider that the fiber is cooled in a standard cryogenic setup, like the widespread Stirling machines used for infrared detectors. The system can be described as the succession of layers. The Er doped silica core is surrounded by a silica clad, with basically the same thermal properties. A cladding of acrylate polymer protects mechanically the silica fiber. Then the fiber is glued with epoxy between two copper plates. The successive stages of thermal contacts are supposed faultless. The thickness of each layer is reported in Table II, together with the thermal conductivity at 77 K. These parameters are used to calculate the temperature across the setup, for the region that experiences the highest thermal load (2 W/m, i.e. 6.37e9 W/m 3 in the core). To simplify, we assume that the fiber coils are not in contact with each other.

As can be seen on Fig. 7, the temperature in the fiber core increases by about 5 K in this section of highest absorption, for a pump power of 400 W. This calculation shows that despite the high pump power coupled into the fiber clad, the temperature does not increase significantly compared to the variation that would be required to affect the SBS threshold and the laser efficiency (see Fig. 2).

V. CONCLUSION

Taking advantage of the well-referenced effects of cryogenic temperatures on both phonon lifetime and on laser efficiency in silica fibers, we present the first study of their combined advantage for the performance of a single-mode and singlefrequency Er-doped fiber laser amplifier. The SBS threshold power is found 80% higher at 77 K than at 300 K. At the same temperatures, we report the measurement of the absorption and gain spectra of the 5 I 13/2 -5 I 15/2 laser transition of erbium, in a Er:Yb codoped alumino-phosphosilicate host.

These data are used in a numerical simulation of the signal amplification at 1560 nm in the same type of fiber, clad pumped at 1535 nm, with a pump power up to 400 W. In the pump limited regime, the amplifier at low temperature is 60% more efficient. In the SBS limited regime, given the higher SBS threshold power at 77 K, the amplifier can reach a power more than twice higher than at room temperature. Let us note that the cryogenic cooling can be combined with a number of other methods of SBS rejection, such as fibers with large mode area, low Brillouin gain (anti-guiding profile for acoustic waves), strain gradient, or signal frequency broadening... This paper demonstrates the interest of cooling the fiber, so the experiment should be tested in a configuration similar to that suggested here. In a longer term perspective, our work also illustrates the interest of understanding and using the physical properties of non-usual amorphous hosts for improving the performance of rare-earth doped fibers, notably in laser amplifiers. Historically, silica fibers have been used for EDFA because of their immediate compatibility (fabrication, splicing...) with the low absorption fibers used for longdistance telecommunications. Since this technological leap, many other glasses or doping methods were investigated [START_REF] Ballato | Rethinking Optical Fiber: New Demands, Old Glasses[END_REF], [START_REF] Dragic | Materials for optical fiber lasers: A review[END_REF], [START_REF] Hertling | Internal friction and hypersonic velocity in vitreous germania under high pressure[END_REF], [START_REF] Lambin-Iezzi | Stimulated Brillouin scattering in SM ZBLAN fiber[END_REF], [START_REF] Savelii | Benefit of Rare-Earth "Smart Doping" and Material Nanostructuring for the Next Generation of Er-Doped Fibers[END_REF], [START_REF] Shi | Kilowatt-level stimulated-Brillouin-scattering-threshold monolithic transform-limited 100 ns pulsed fiber laser at 1530 nm[END_REF] to reduce the SBS gain or improve the laser amplification efficiency, but few experiments have demonstrated a positive combination on these two parameters.

Alaeddine Abbes Alaeddine Abbes received the M.Sc. degree in Optoelectronics and Microwave Engineering in 2017, from the University of Montpellier. He is currently working toward the Ph.D. degree on the development of a compact, low-cost, highly tunable, highly coherent and powerful terahertz (THz) sources based on photomixing using VeCSEL (Vertical-external-Cavity Surface-Emitting-Laser) technology from the Institut d'Électronique et des Systèmes, Montpellier, France. He current research interests include THz generation and detection, THz optoelectronic generation using photoconductors and photodiodes, THz antenna, and semiconductor lasers.

Alice Odier Biography text here.

Alizée Dubois Biography text here.

Fig. 1 .

 1 Fig. 1. Setup for the measurement of the SBS spectrum linewidth.

Fig. 2 .

 2 Fig.2. Thermal variations of the SBS spectral linewidth (gray squares) and of the SBS threshold power (black circles), as defined empirically by the alteration of the output pulse shape.

Fig. 3 .

 3 Fig.3. Peak power of the backscattered Stokes wave as a function of the coupled peak power, at 77 (blue triangles) and 300 K (red circles). The dotted lines represent the corresponding exponential fits (blue and red dotted lines) in the regime of stimulated Brillouin scattering with undepleted pump.

Fig. 4 .

 4 Fig.4. Small-signal absorption (above) and gain (below) spectra, at 300 K (red) and 77 K (blue).

  Fig.5. Evolutions of signal (solid) and pump (dashed) powers along the fiber, for an injected pump power of 400 W, at 300 K (red) and 77 K (blue). The SBS thresholds are indicated at 300 and 77 K, as well as the threshold at 300 K if the SBS gain could be reduced to the value of room temperature.

Fig. 6 .

 6 Fig.6. Comparison of the maximum peak signal output as a function of the pump power, for various configurations: at 300 K (g B = 10 -11 m/W), at 77 K (g B = 0.56 10 -11 m/W), and at 300 K with the SBS gain reduced to the same value as 77 K. The dotted lines represent the signal power without SBS limitation.

Fig. 7 .

 7 Fig. 7. Temperature distribution across the section of the fiber, calculated with the parameters listed above.

TABLE I LIST

 I OF THE PARAMETERS FOR THE IN-BAND PUMPED ER FIBER

		AMPLIFIER MODEL
	Parameter	Value	Comment
	∅core	20 µm	Core diameter
	∅ clad	125 µm	Clad diameter
	N Acore	0.1	Numerical aperture
	l glass Cup	20 dB/km 5 10 -24 m 3 /s	Background loss Up-conversion factor
	g B (300 K) g B (77 K)	10 -11 m/W 0.56 10 -11 m/W	SBS gain at 300 K SBS gain at 77 K
	α Er (1535 nm)	80 dB/m	Absorption at 300 K
	α Er (1535 nm)	128 dB/m	Absorption at 77 K
	αns(λpump, 300 K)	20 dB/km
	αns(λ sig , 300 K)	150 dB/km
	αns(λpump, 77 K)	30 dB/km
	αns(λ sig , 77 K)	16 dB/km

TABLE II LIST

 II OF THE LAYERS USED TO CALCULATE THE TEMPERATURE INCREASE CAUSED IN THE FIBER CORE BY THE PUMP LASER.

ACKNOWLEDGMENT

We would like to thank Marie Foret from Laboratoire Charles Coulomb, Université Montpellier, for helpful discussions. We are also thankful to Agnès Dolfi-Bouteyre and François Gustave from ONERA, and Jean-Louis Le Gouët from Laboratoire Aimé Cotton (CNRS, UPR 3321) for careful reading.


David Tareste

Fre ´de ´ric Pincet

Eric Perez

Ste ´phane Rickling

Charles Mioskowski

Luc Lebeau

Energy of Hydrogen Bonds Probed by the Adhesion of Functionalized Lipid Layers

It is now well admitted that hydrophobic interactions and hydrogen bonds are the main forces driving protein folding and stability. However, because of the complex structure of a protein, it is still difficult to separate the different energetic contributions and have a reliable estimate of the hydrogen bond part. This energy can be quantified on simpler systems such as surfaces bearing hydrogen-bonding groups. Using the surface force apparatus, we have directly measured the interaction energy between monolayers of lipids whose headgroups can establish hydrogen bonds in water: nitrilotriacetate, adenosine, thymidine, and methylated thymidine lipids. From the adhesion energy between the surfaces, we have deduced the energy of a single hydrogen bond in water. We found in each case an energy of 0.5 kcal/mol. This result is in good agreement with recent experimental and theoretical studies made on protein systems showing that intramolecular hydrogen bonds make a positive contribution to protein stabilization.

INTRODUCTION

To recognize other proteins and achieve their biological functions, most proteins fold to globular conformations. For many years, it has been considered that the hydrophobic effect was the main driving force involved in protein folding [START_REF] Kauzmann | Some factors in the interpretation of protein denaturation[END_REF][START_REF] Tanford | Contribution of hydrophobic interactions to the stability of the globular conformation of proteins[END_REF]. Hydrogen bonds also play a role during the folding process, although it is not clear whether they make a favorable contribution to protein stability [START_REF] Baker | Hydrogen bonding in globular proteins[END_REF][START_REF] Mcdonald | Satisfying hydrogen bonding potential in proteins[END_REF]. When a protein folds, most of the intermolecular hydrogen bonds between polar groups and water molecules are broken and replaced by intramolecular hydrogen bonds (Fig. 1). A key question is whether it is energetically more favorable for the polar groups to make hydrogen bonds with water molecules in an unfolded protein or to make hydrogen bonds with each other in a folded one. The aim of this study is to bring new information on this question. Recent studies indicate that hydrogen bonds increase the stability of proteins and are as strong as the hydrophobic effect [START_REF] Pace | Forces contributing to the conformational stability of proteins[END_REF][START_REF] Myers | Hydrogen bonding stabilizes globular proteins[END_REF][START_REF] Pace | Tyrosine hydrogen bonds make a large contribution to protein stability[END_REF][START_REF] Pokkuluri | Increasing protein stability by polar surface residues: domain-wide consequences of interactions within a loop[END_REF]. It has been estimated that the average net stabilization is ϳ1 kcal/mol per intramolecular hydrogen bond. However, because of the complex structure of a protein, it appears that both experimentally and theoretically it is still difficult to obtain a reliable estimate of the contribution of the hydrogen bonds in the protein folding process. One way for probing hydrogen bonds is the measurement of interaction energies between surfaces bearing hydrogen-bonding groups. In such systems, the pure effect of the hydrogen bonds are measured, and the energy can be deduced from the adhesion energy between the surfaces. Very few studies have so far been reported on such measurements [START_REF] Pincet | Long-range attraction between nucleosides with short-range specificity: direct measurements[END_REF][START_REF] Berndt | Measurement of forces between surfaces composed of two-dimensionally organized, complementary and noncomplementary nucleobases[END_REF][START_REF] Schneider | Effect of substrate anchoring on the mechanical strength of Langmuir-Blodgett bilayers[END_REF]. This study presents forces measurements as a function of their separation distance between surfaces functionalized with groups that are able to establish hydrogen bonds in water: nitrilotriacetate (NTA), adenosine (A), thymidine (T), and methylated thymidine (MeT). The anchoring of these groups on the surfaces is obtained by coating mica sheets with monolayers of lipids having the appropriate headgroups. This method has the advantage of providing a controlled orientation of the headgroups, and it has proved so far that the strength of the anchoring is higher than the interactions between the studied groups. We will show that the headgroup/headgroup hydrogen bonds in our systems are energetically more favorable than the headgroup/water ones. The energetic contribution per hydrogen bond will be estimated for each type of group studied. It will also be shown that the adhesion is several orders of magnitude higher than the one of phospholipids that do not make hydrogen bonds.

MATERIALS AND METHODS

Surface force apparatus

The force measurements are made between two lipid bilayers in water by using a surface force apparatus (SFA). This technique gives the force between two surfaces (Ϯ0.1 N) versus the distance that separates them (Ϯ0.1 nm). The force is measured with a cantilever spring, and the distance is obtained interferometrically [START_REF] Israelachvili | Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0 -100 nm[END_REF]. Lipid bilayers are deposited on mica surfaces using the Langmuir-Blodgett technique [START_REF] Gaines | Insoluble Monolayers at Liquid-Gas Interfaces[END_REF]: a first monolayer of dimyristoyl-phosphatidylethanolamine (DMPE) with its headgroups on the mica surface and a second monolayer of the lipid to be studied with its headgroup facing the aqueous medium.

Submitted March 15, 2002, andaccepted for publication August 16, 2002 The two surfaces are in a crossed cylinders geometry (radius R). The ratio F/R is proportional to the interaction energy between two plane surfaces following the Derjaguin approximation: F/R ϭ 2E. Therefore, the SFA curves are shown as F/R as a function of the separation distance D between the two surfaces.

The forces are measured by cycles of approaching and then separating the surfaces. For each case, at least two different experiments and three cycles per experiment are performed.

Functionalized lipids

We have synthesized functionalized lipids made of two unsaturated C 18 alkyl chains covalently bound to the functional headgroup via a flexible spacer [START_REF] Lebeau | Rational design and synthesis of phospholipids for the two-dimensional crystallization of DNA gyrase, a key element in chromosome organization[END_REF]. By adding disorder, the unsaturations allow the lipids to be in a liquid expanded state even at high surface pressures. This configuration allows the functional headgroups to have translational and rotational freedom so they can adopt the more favorable configuration to be accessible to the ones of the other surface. The headgroups are NTA, A, T, and MeT (Fig. 2). The A and T lipids have been studied in a previous article [START_REF] Pincet | Long-range attraction between nucleosides with short-range specificity: direct measurements[END_REF]. Both A and T groups are able to form two hydrogen bonds. Besides the A-T link that matches perfectly, A-A and T-T links are also possible. In MeT, one hydrogen atom of the thymidine molecule is replaced by a methyl group therefore suppressing one hydrogen bond. The NTA molecule contains three carboxyl groups that can form hydrogen bonds. These carboxyl groups can chelate a metal ion such as nickel, in which case the hydrogen bonds are hindered. A, T, and NTA lipids, thus, offer the possibility of probing the strength of the hydrogen bonds they can form. As for the MeT and NTA-Ni lipids, they are used to measure the effect of reducing the number of potential hydrogen bonds on the forces.

RESULTS

Lipids isotherm and stability

Each lipid was first studied as a monolayer spread at the water/air interface in a Langmuir trough to obtain its com-FIGURE 1 Schematic illustration of the formation of intramolecular hydrogen bonds during the protein folding process: two protein/water hydrogen bonds in the unfolded state (state 1) break and one protein/ protein bond can form along with one water/water bond (state 2). The energetic balance is unclear because the number of hydrogen bonds remains constant. pression isotherm and verify its stability to desorption. They all presented a similar behavior. Their compression isotherm showed no phase transition plateau and thus was characteristic of a fluid monolayer (for the case of the NTA lipid, see Fig. 3). By keeping the monolayer under a constant surface pressure, the one at which the lipids will be deposited on the mica surfaces, and measuring the decrease with time of its area, we can estimate the quantity of lipid that leaves the water/air interface to go in solution. These pressures were 39.5 mN/m for the A and T lipids, 37.5 mN/m for the MeT lipid, and 35 mN/m for the NTA and NTA-Ni lipids. All the lipids revealed a very stable behavior with an average desorption of 1%/h.

When deposited as bilayers on a solid surface, the lipids may have different desorption kinetics. To verify the stability of the coated surfaces to desorption, we have deposited on a large mica sheet (ϳ10 cm 2 ), a first monolayer of DMPE, and a second monolayer of the lipid to be studied. The sample stayed under water (in a 1-l dish) up to 24 h, which is more than the time scale for an SFA experiment. Then the lipid was redeposited at the water/air interface, and the number of molecules were recounted. We found again that on average 1% of the lipid had desorbed per hour. So the lipids can be considered as being irreversibly bound to the DMPE surfaces, which ensures that the layers are closepacked on the time scale of the experiments.

SFA experiments

The NTA/NTA, MeT/T, and MeT/A systems display similar features. On approaching the surfaces, the force-distance profiles display no double layer repulsion. The resolution of the SFA being less than one charge per thousand lipids [START_REF] Pincet | Bilayers of neutral lipids bear a small but significant charge[END_REF], this establishes that the surfaces are electrostatically neutral. From a distance of ϳ100 nm, an attractive force sets in (see inset in Fig. 4 a for the case of the NTA/NTA system). Such an attraction has already been observed with the A and T lipids [START_REF] Pincet | Long-range attraction between nucleosides with short-range specificity: direct measurements[END_REF], but its origin is still unknown. When the force gradient is higher than the spring constant, the surfaces jump into contact. The resulting distance between the two mica surfaces right after the jump corresponds to the thickness of two close-packed monolayers (D 0 ). While the surfaces remain in contact (ϳ15 min), the distance decreases, indicating a loss of lipid in the outer monolayers and leading to a flat contact between the two surfaces. This is generally not observed with phospholipids layers, whereas it has been noticed with hydrogen bonding lipids [START_REF] Pincet | Specific forces between DNA bases[END_REF]. Upon pulling to separate the surfaces, the distance remains approximately constant until a sufficient force is applied. Then the distance increases back to D 0 (the contact is still flat), and the surfaces jump apart. In this article, we focus mainly on the pull off force F 0 corresponding to this detachment. The interaction profiles for the three systems are given in Fig. 4. In contrast to the NTA/NTA case, the NTA-Ni/NTA-Ni system presents a behavior similar to the one of neutral phospholipids: no double layer repulsion, no significant variation of the contact distance, and a weak pull off force (Fig. 4 a). One can also notice that the long-range attractive force is still present (inset in Fig. 4 a).

The macroscopic adhesion energy E 0 between the two surfaces can be deduced from the measured pull off force F 0 . Two main theories have been developed on this subject. First, the Johnson, Kendal, and Roberts theory (JKR theory; [START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF] considers that the adhesive forces between two surfaces cause an increase in contact area, so that the surfaces are flattened just before separation. It predicts that F 0 is related to E 0 by:

F 0 ϭ 3RE 0 /2 (1)
The second theory, the Derjaguin, Muller, and Toporov theory (DMT theory; [START_REF] Derjaguin | Effect of contact deformations on the adhesion of particles[END_REF], accounts for the effects of the forces between the two surfaces just outside the contact area, so that this contact area falls down to zero just before separation. F 0 is related to E 0 by:

F 0 ϭ 2RE 0 (2)
Maugis reconciled both theories by considering each of them as a special case [START_REF] Maugis | The JKR-DMT transition using a dugdale model[END_REF]. In the case of two surfaces having a strong adhesion energy in comparison to their elasticity, the contact area is flattened and the relation between F 0 and E 0 is given by Eq. 1 (JKR theory). This is the case for all the systems but the NTA-Ni/NTA-Ni system. In the opposite case, the contact area falls down to zero, and the relation between F 0 and E 0 is given by Eq. 2 (DMT theory). This is the case for the NTA-Ni/NTA-Ni system.

The values of F 0 and E 0 for the different systems are displayed in Table 1. Except for the NTA-Ni/NTA-Ni system, the surface adhesion energies are much higher than the ones of phospholipids that do not make hydrogen bonds [START_REF] Marra | Direct measurement of forces between phosphatidylcholine and phosphatidylethanolamine bilayers in aqueous solutions[END_REF]. When the NTA group is functionalized with a nickel atom, the three OH endings are replaced by bonds between the oxygen atoms and the nickel ion and the measured energy E 0 decreases from 4 mJ/m 2 to 1 mJ/m 2 . This confirms that the adhesion energy measured in the NTA/NTA system is due to hydrogen bonds, which are blocked in the NTA-Ni/NTA-Ni case. From previous measurements, we also know than the bonds involved in the experiments with nucleosides are hydrogen bonds [START_REF] Pincet | Long-range attraction between nucleosides with short-range specificity: direct measurements[END_REF]. In the systems involving MeT groups for which one hydrogen atom has been replaced by a methyl group, therefore blocking one hydrogen bond, one can also notice that the adhesion energy is substantially decreased.

Therefore, it seems that the headgroup/headgroup hydrogen bonds are, in all the cases, energetically more favorable than the headgroup/water ones. This is coherent with previous results obtained on glycine lipids [START_REF] Schneider | Effect of substrate anchoring on the mechanical strength of Langmuir-Blodgett bilayers[END_REF].

DISCUSSION

Many of the force-distance profiles show characteristic features: an attraction from 100 nm, a strong adhesion, and a decrease of the distance when the bilayers are in contact. Such features have been previously observed on deliberately depleted lipid layers [START_REF] Helm | Molecular mechanisms and forces involved in the adhesion and fusion of amphiphilic bilayers[END_REF][START_REF] Helm | Role of hydrophobic forces in bilayer adhesion and fusion[END_REF][START_REF] Helm | Forces between phospholipid bilayers and their relationship to membrane fusion[END_REF] and were rightly interpreted as hydrophobic attractions and adhesion, which are now well documented. One could be tempted to give the same interpretation to the present data obtained on hydrogen bonding lipids. This interpretation would be right if the distance decrease was irreversible or if the layers were already depleted by lipid desorption before going into contact. However it was checked by desorption measurements from water/air interface and also from mica supported bilayers that no significant lipid desorption takes place on the time scale of the experiments. Moreover, the decreases seen in the contact distance were in our cases fully reversible: by pulling the surfaces apart the bilayers recovered their initial thickness. It was at this point, i.e., with full bilayers, that the adhesion energy was measured. These features have already been observed with A and T lipid layers for which hydrogen bonds accounted for the adhesion energies [START_REF] Pincet | Long-range attraction between nucleosides with short-range specificity: direct measurements[END_REF]. The molecular binding energies were indeed the ones known from the literature and these values were also obtained by two other methods [START_REF] Pincet | Electrostatic nanotitration of weak biochemical bonds[END_REF][START_REF] Pincet | From macroscopic adhesion energy to molecular bonds: a test of the theory[END_REF]. These adhesion energies can therefore be interpreted in terms of hydrogen bonds between close-packed lipid layers.

Force-distance profiles in contact

The unusual force-distance profiles when the surfaces are in contact in the NTA/NTA, MeT/T, and MeT/A cases have already been observed in similar experiments involving the A and T lipids (A/T, A/A, and T/T experiments). The decrease of the separation distance below D 0 indicates a rearrangement of the molecules in the contact area ("sticky fluid" behavior; [START_REF] Pincet | Specific forces between DNA bases[END_REF]. In the NTA-Ni/NTA-Ni case, where the hydrogen bonds are hindered, the "sticky fluid" phenomenon is not observed (Fig. 5). Therefore, the main new result about this behavior is that it appears to be very common between surfaces bearing close-packed hydrogen bonding groups.

Surface charges

All the interaction profiles presented here show no double layer repulsion and, thus, indicate that the surfaces are neutral. This is surprising because, given their pK a (see Fig. 6 for the cases of the A and T groups), all the systems but the T/T one are supposed to bear at least one charge per hundred lipids in pure water (up to one charge per lipid for the NTA-Ni/NTA-Ni system). This can be explained by the fact that, even in pure water, a considerable amount of counter-ions can bind to the surfaces [START_REF] Marra | Effects of counterion specificity on the interactions between quaternary ammonium surfactants in monolayers and bilayers[END_REF]. These ions may come from detachment from different parts of the SFA trough (stainless steel, teflon, glass) or from the dissolution of carbon dioxide. Marra found that more than 90 percent of the surface charges can be neutralized. In our case, this effect seems even stronger.

Hydrogen bonding energy

As the NTA group has the highest number of available hydrogen bonding sites, one should expect a higher adhesion energy in the NTA/NTA case. Each NTA group contains indeed three carboxyl groups corresponding to six possible hydrogen bonds. However, the relatively small adhesion observed between two NTA surfaces is understandable for two reasons. First, a high proportion of carboxyl groups have been ionized, and the hydrogen has probably been replaced by another cation from the solution.

Second, hydrogen bonds can be formed not only between carboxyl groups from opposite surfaces but also within the same molecule or within the same layer. From previous experiments made on weakly acidic and basic monolayers [START_REF] Payens | Ionized monolayers[END_REF][START_REF] Betts | The ionization characteristics of monolayers of weak acids and bases[END_REF][START_REF] Zhao | New method for determination of surface pK a using second harmonic generation[END_REF][START_REF] Vezenov | Force titrations and ionization state sensitive imaging of functional groups in aqueous solutions by chemical force microscopy[END_REF], one can estimate the surface pK a of the carboxylic groups of NTA to 5.8. Our experiments were performed in pure water whose pH is comprised between 5.3 and 5.7. Thus, we can consider that one-half of the carboxyl groups have been ionized. The layers being closepacked, it can be assumed that only one-third of the hydrogened carboxyl groups (which is the proportion for an hexagonal lattice) make interlayer bonds. Therefore, we estimate that only one-sixth of all the hydrogen bonding sites contribute to the adhesion energy, that is to say one hydrogen bonding site per NTA group.

In the case of A, T, and MeT lipids, hydrogen bonds within the same layer can be neglected. Indeed, it has been shown by x-ray diffraction performed on monolayers functionalized with nucleosides that the headgroups have a tendency to associate by setting their planes parallel to each other [START_REF] Perez | Translational order in liquid-expanded lipid monolayers functionalized with nucleosides[END_REF]. This phenomenon, called "stacking," strongly hinders the formation of hydrogen bonds within the same layer. The A and T groups can thus make two interlayer hydrogen bonds and the MeT group one interlayer hydrogen bond (Fig. 7).

All hydrogen bonds in the A/A system are identical (NHۙN) as well as for the T/T (NHۙO) and for the NTA/NTA (OHۙO) systems. Therefore, the molecular binding energy for each of these cases can be considered to be equal to the energy of one hydrogen bond times the number of such bonds between the molecules. In principle, one cannot do the same for the A/T case because it involves two different hydrogen bonds. However, as a first approximation, we will do it and discuss it further below.

From the adhesion energy E 0 and the number n H of hydrogen bonding sites involved in each single molecular bond, we can deduce the energy e H of a single hydrogen bond. Assuming that each molecule has n H hydrogen bonding sites that can be either bound or unbound, Boltzmann statistics leads to the relation: in which E 0 and e H are given in k B T units and in which is the average area occupied by a lipid inside the outer monolayer. The molecular areas are 0.56 nm 2 for NTA, 0.62 nm 2 for NTA-Ni, 0.63 nm 2 for A, 0.56 nm 2 for T, and 0.51 nm 2 for MeT.

n H and e H values are reported in Table 2. For each case, except the A/T one, we found a hydrogen bonding energy of the order of 0.5 kcal/mol. The unexpected similar values of the different hydrogen bonds justify a posteriori the approximation made in the case of A/T. However, one can notice that the hydrogen bond energy for A/T is larger than for the other cases. These stronger hydrogen bonds may have two origins: their geometry (better angle and length) gives them a more optimal strength, and a third hydrogen bond might partially be formed between the A and T groups (represented by the small dots in Fig. 7).

The values that we obtained for the different cases are in good agreement with previous studies on proteins stability through simulations on model compounds and site directed mutational techniques [START_REF] Pace | Forces contributing to the conformational stability of proteins[END_REF][START_REF] Pace | Tyrosine hydrogen bonds make a large contribution to protein stability[END_REF][START_REF] Myers | Hydrogen bonding stabilizes globular proteins[END_REF][START_REF] Pokkuluri | Increasing protein stability by polar surface residues: domain-wide consequences of interactions within a loop[END_REF].

CONCLUSION

Force measurements between surfaces functionalized with lipids having hydrogen bonding headgroups (NTA, A, T, and MeT lipids) lead to a reproducible value of the energy of a single hydrogen bond in pure water: ϳ0.5 kcal/mol. It shows that it is energetically more favorable for the headgroups to make hydrogen bonds with each other than to make hydrogen bonds with water molecules. This is coherent with past studies made on proteins stability, which showed that intramolecular hydrogen bonds in a folded protein are energetically more favorable than bonds with water molecules in an unfolded protein with an average stabilization of ϳ1 kcal/mol per intramolecular hydrogen bond. Moreover, in our systems, the hydrogen bonds can be perfectly isolated and probed, which is not the case in protein systems that are much more complex. These new results can thus give a more precise estimate of the energetic contribution of the hydrogen bonds in the protein folding process. This approach could be complementary to the usual FIGURE 7 Hydrogen bonds involved in each studied system. Each NTA group contains three carboxyl groups that can form two hydrogen bonds. This leads to six possible hydrogen bonds per NTA group. However, on average, only one hydrogen bond is formed between two NTA groups (see Discussion).

FIGURE 2

 2 FIGURE 2 Chemical structures of the functionalized lipids: A, T, MeT, NTA, and NTA-Ni lipids.

FIGURE 3

 3 FIGURE 3 Compression isotherm of the NTA lipid on pure water subphase.

FIGURE 5

 5 FIGURE 5 Force-distance profile in contact between two NTA surfaces (circles, "sticky fluid" behavior) and between two NTA-Ni surfaces (triangles, "classical" behavior).

FIGURE 6

 6 FIGURE 6 pK a values for the headgroups A and T.

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TABLE 1 Values of F 0 /R (F 0 is the pull off force, accuracy ϳ10%) and of the adhesion energy E 0 as deduced from Eq. 1 (Eq. 2 in the NTA-Ni/NTA-Ni case)

 1

						NTA-Ni/
	NTA	NTA/NTA MeT/T MeT/A A/A T/T A/T	NTA-Ni
	F 0 /R (mN/m)	19	28	22	50 43 110	6
	E 0 (mJ/m 2 )	4	5.9	4.7 10.6 9.1 23.3	1

TABLE 2 Values of the adhesion energy E 0 , of the estimated number n H of hydrogen bonds per lipid and of the hydrogen bonding energy e H as deduced from Eq. 3

 2 , i.e., site-directed mutational studies and simulations on model compounds. It could be used for any other systems that involve hydrogen bonds such as nucleic acids or colloidal systems.

		NTA/NTA	MeT/T	MeT/A	A/A	T/T	A/T
	E 0 (mJ/m 2 )	4	5.9	4.7	10.6	9.1	23.3
	n H	1	1	1	2	2	2
	e H
	(k B T/bond)	0.8	1.1	1	1.1	0.9	2.0
	(kcal/mol)	0.5	0.6	0.6	0.6	0.5	1.2

ones

Biophysical Journal Volume 83 December 2002 3675-3681

BiophysicalJournal 83(6) 3675-3681