Franck Montmessin
email: franck.montmessin@latmos.ipsl.fr

Anna Fedorova

Oleg Korablev

Alexander Trokhimovskiy

Kevin Olsen

L Rossi

T Fouchet

T Encrenaz

E Lellouch

Jean-Loup Bertaux

Exploring HDO and H 2 O on Mars with the ACS instrument onboard TGO

Introduction

The Trace Gas Orbiter (TGO) of the ESA-Roscosmos ExoMars mission has ended its trip to Mars, reaching the planet in October 2016. After more than a year-long aerobraking phase, its scientific mission has begun on April 22 nd 2018 with the execution of the first solar occultation. The primary objective of TGO is to detect, map and locate trace gas sources, possibly revealing a residual geophysical (or even biological) activity on Mars. The instrument of interest here is the infrared spectrometer Atmospheric Chemistry Suite (ACS). ACS covers a wavelength range from 0.7 to 17 μm at very high spectral resolution (λ / Δλ from 5,000 to 50,000). ACS operates in nadir and in solar occultation. Its performance and scientific objectives make it complementary to NOMAD, the other spectrometer dedicated to trace gas characterization.

The objectives of ACS [START_REF] Korablev | ACS Team: The Atmospheric Chemistry Suite (ACS) of three tpectrometers for the ExoMars 2016 Trace Gas Orbiter[END_REF] lie at the core of TGO mission goals (TGO will eventually serve as a telemetry relay for the ExoMars 2020 rover). However, the versatility of ACS makes it possible to contribute, beyond the sole topic of trace gases, to the more general knowledge of the Martian atmosphere, by characterizing in particular the Martian water cycle and that of its isotope HDO.

The water and heavy water cycles

Several members of our team have long been involved in the study of Martian HDO, having described their theoretical approach with several original works [2,3,4].

The D / H ratio, determined via the isotope ratio of the water vapor, is more than five times higher than the reference ratio of the terrestrial oceans. This relative enrichment of D appears as the result of a differentiated escape suggesting that a quantity of water at least five times higher was once present on Mars (the Earth managed how to preserve all of its water). What we know today about Martian D / H has been developed essentially from observations made from Earth. The latter have substantially increased in number thanks to several groups [5,6,7,8]. During condensation, a fractionation process takes place between H2O and HDO, the latter tends to concentrate in the ice phase, distilling an air depleted in deuterium. In fact, an isotopic gradient must theoretically settle between the cold regions subject to condensation and the warmer regions (see Figure ). The comparison of predictions of the 3D model published in [4] and ground observations agree well on the existence of a meridional gradient of D/H between the hot and humid summer hemisphere and the cold, dry winter hemisphere.

However, the observations also revealed a D/H contrast in longitude that contradicts model predictions [9]. No viable hypothesis can yet explain the presence of such a gradient. A fractionation action by adsorption / desorption of water in the regolith is theoretically possible, but such a phenomenon requires an exchange flux with the atmosphere that is too high. Many puzzles remain, and the field of HDO needs to be explored by observation.

Heavy water vapor: a major goal for ACS

The ACS measurements will produce the first vertical profiles obtained simultaneously for HDO and H2O in an altitude range limited in its lower part by the thick layer of aerosols (around 10 km typically) and in its upper part (above 80 km) by the gradual decrease in the concentration of these two species and that of the air mass factor. Nevertheless, these profiles will make it possible to identify the altitudes where it a sudden collapse of the two species is expected as a consequence of their condensation (hygropause), the associated formation of clouds (observed by ACS instrument too) and consequently the resulting fractionation. In fact, ACS shall be able to characterize in detail the space-time variability of H2O and HDO.

FFigure:

  Map of the D/H ratio on Mars at Ls 50° as predicted by the GCM presented in[4].

Acknowledgements

ExoMars is a space mission of ESA and Roscosmos. The ACS experiment is led by IKI Space Research Institute in Moscow. The project acknowledges funding by Roscosmos and CNES. Science operations of ACS are funded by Roscosmos and ESA. FM, KO, TF and EL activity within ACS is supported by CNRS and CNES.


Changgui Zhang
email: changgui.zhang@univ-lille.fr

On the positive powers of q-analogs of Euler series

). By considering a family of linear q-difference operators associated with a given first order non-homogenous q-difference equation, it will be shown that the summability order of q-analoguous counterparties of Euler series depends upon of the degree of power under consideration.

Introduction

It is well-known that, given any direction starting from the origin of the complex plane, the set of all Borel-summable series in this direction forms a differential algebra; see [START_REF] Malgrange | Sommation des séries divergentes[END_REF]Proposition 1.3.4.2] or [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF]Theorem 36]. This is not the same for the summability with respect to the power series solutions of q-difference equations. Indeed, if q > 1 and

(1. [START_REF] Andrews | Special Functions[END_REF] Êq (x) = n≥0 (-1) n q n(n-1)/2 x n , the divergent power series Êq (x) is Gq-summable of only one level but its square Êq (x) 2 is Gq-summable of two levels; see [11, §4.3.8] and [8, Théorème 2.2.1].

The first purpose of this paper is to obtain a linear q-difference equation for every positive power of the power series Êq (x). This will allow one to see the different level of summability for each of these corresponding power series. In what follows, let q be a given nonzero complex number, and let σ q be the associated q-difference operator defined by σ q f (x) = f (qx). By direct computation, one finds the series defined by (1.1) satisfies term by term the following linear qdifference equation :

(1.2) (xσ q + 1)y = 1 . Now, write (1.2) into the form xσ q y = 1 -y and then square both sides. As (σ q y) 2 = σ q (y 2 ), one obtains that x 2 σ q (y 2 ) -y 2 = 1 -2y. By using again (1.2), one gets the following equation:

(xσ q + 1)(x 2 σ q -1)(y 2 ) = x + 1 -2(xσ q + 1)y , what implies that

(1.3) (xσ q + 1)(x 2 σ q -1) Êq (x) 2 = x -1 .
We shall explain how to obtain a linear q-difference equation for Êq (x) n . For doing that, we shall start by recalling some non-commutative rings of q-difference operators and then arrive at a n-th order such operator associated with any given nth power of a power series solution of some first order non-homogenous q-difference equation. See Theorem 2.1 in § 2 and its generalisation stated in Theorem 4.3 in § 4. Section 3 is reserved to the proofs of Theorem 2.1 and two related Lemmas. In the last section, we shall apply, respectively, both Theorems 2.1 and 4.3 to the abovementioned power series Êq (x) and another q-analog Ê(x; q) of the classic Euler series. At the end of the paper will be outlined some results about the summability of both Êq (x) and Ê(x; q) when q > 1; see Theorem 5.3.

Throughout the present paper, we shall limit ourself to the q-summation method studied in our previous works [START_REF] Zhang | Développements asymptotiques q-Gevrey et séries Gq-sommables[END_REF] and [START_REF] Marotte | Multisommabilité des séries entières solutions formelles d'une équation aux q-différences linéaire analytique[END_REF], by means of a q-Laplace integral involving the multivalued function e log 2 x/2 ln q . There exists another summation, using a Jacobi theta function; see [START_REF] Ramis | Local analytic classification of q-difference equations[END_REF]. Comparing both summations can be seen as part of Stokes analysis, and this would be interesting to find identities on the associated special functions.

Notation and statements

As usual, we will denote by C{x} or C [[x]] the integral domain constituting of the germs of holomorphic functions at x = 0 in the complex plane or that of the power series of indeterminate x, respectively. Their respective corresponding field of fractions will be denoted by

C[x -1 ]{x} or C[x -1 ][[x]]. One has C{x} ⊂ C[[x]], as well as C[x -1 ]{x} ⊂ C[x -1 ][[x]]
, in a natural way via the classic Taylor or Laurent series.

Let R be one of the above-mentioned C-vector spaces C{x}, C

[[x]], C[x -1 ]{x} and C[x -1 ][[x]]
. By replacing x with qx, the q-difference operator σ q acts as an automorphism over R. Let σ k q = σ q k for any integer k; in particular, σ 0 q is simply the identity map on R. We will denote by R[σ q ] the set of the (linear) q-difference operators whose coefficients belong to R, i.e. L ∈ R[σ q ] if L = n k=0 a k σ k q , where a k ∈ R. By identifying each element of R with the corresponding multiplication map in R, the set R[σ q ] constitutes itself a non-commutative ring for the composition of operators. Given L 1 , L 2 ∈ R[σ q ], we will write

L 1 L 2 in stead of L 1 • L 2 .
Furthermore, we will denote by ν 0 (f ) ∈ Z ∪ {∞} the valuation of any given element f of the field C[x -1 ][[x]] "at x = 0". This means that (i) ν 0 (f ) = ∞ iff f is the identically vanishing series, and (ii)

ν 0 (f ) = ν ∈ Z iff f ≡ c x ν mod x ν+1 C[[x]]
for some nonzero complex number c.

Throughout the whole section, P will be some given nonzero power series belonging to the field C[x -1 ][[x]]. Define the associated family of power series {P n } n≥0 as follows: P 0 = 0, P 1 = P , and (2.1)

P n = n-1 k=0 (-xσ q ) k P for n ≥ 2.
By noticing that (-xσ q ) k P = (-x) k q k(k-1)/2 σ k q P , it follows from (2.1) that, for any pair (n, ) ∈ Z ≥0 × Z >0 ,

P n+ -P n = n+ -1 k=n (-x) k q k(k-1)/2 σ k q P ≡ (-1) n c q nν+n(n-1)/2 x ν+n mod x ν+n+1 C[[x]] ,
where c = 0 and ν = ν 0 (P ) ∈ Z. This implies that (2.2)

P m = P n if m = n.
Given any positive integer n ≥ 1, let

{L P n,k } 1≤k≤n ⊂ C[x -1 ][[x]] [σ q ]
be the associated family of q-difference operators defined in the following manner:

L P n,1 = 1 P 1 (x n σ q -(-1) n ) = 1 P (x n σ q -(-1) n ) ; (2.3) L P n,k+1 = 1 P k+1 x n-k σ q -(-1) n-k L P n,k , 1 ≤ k < n . (2.4)
Specifically, letting k = n, one gets the following form of the n-th order q-difference operator

L P n,n ∈ C[x -1 ][[x]] [σ q ]: (2.5) L P n,n = 1 P n (xσ q + 1) 1 P n-1 x 2 σ q -1 • • • 1 P 1 (x n σ q -(-1) n ) . Theorem 2.1. Let n be an integer ≥ 1, let P , f ∈ C[x -1 ][[x]
] with P = 0, and suppose that the following q-difference equation is satisfied:

(2.6) xσ q f + f = P .
Then:

(2.7)

L P n,n (f n ) = (-1) n(n-1)/2
, where L P n,n is the q-difference operator defined by (2.5). Theorem 2.1 will be proved with the help of the following lemmas.

Lemma 2.2. Let (n, k) ∈ Z 2 such that 1 ≤ k ≤ n, and let P , f be as in Theorem 2.1. For 0 ≤ j ≤ k, define A k;j ∈ C[x -1 ][[x]] by (2.8) A k;j = =j 0≤ ≤k (P j -P ) .

One has:

(2.9)

L P n,k (f n ) = (-1) k(2n-k+1)/2 k j=0 1 A k;j (f -P j ) n .
Lemma 2.3. Let n be a nonnegative integer, C an extension field of C, and let (α 0 , α 1 , ..., α n ) ∈ C n+1 such that α j = α for j = . Define the (n + 1)-tuple (a 0 , a 1 , ..., a n ) ∈ C n+1 as follows:

a j = =j 0≤ ≤n (α j -α ) (a 0 = 1 if n = 0).
The following identity holds in the ring C[T ] of the polynomial functions over C:

(2.10) n j=0 1 a j (T -α j ) n = (-1) n .

Proofs

Proof of Lemma 2.2. To simplify the presentation, we will write L n,k instead of L P n,k , and set n,k = (-1) k(2n-k+1)/2 . One has: (3. [START_REF] Andrews | Special Functions[END_REF] n;k+1 = (-1) n-k n,k . Let us proceed by induction on k. For k = 1, since P 1 = P , one has:

n,1 = (-1) n ; A 1;0 = -P , A 1;1 = P . Thus, (2.3) implies that L n,1 (f n ) = 1 P (x n σ q (f n ) -(-1) n f n ) .
At the same time, by considering (2.6), one gets that

x n σ q (f n ) = (xσ q f ) n = (P -f ) n ;
In this way, it follows that

L n,1 (f n ) = n,1 P ((f -P ) n -f n ) ,
that is exactly what wanted in (2.9) for k = 1. Now, suppose that equality (2.9) holds for some integer k between 1 and n -1. By using relation (2.4), one can express L n,k+1 (f n ) as follows:

(3.2) L n,k+1 (f n ) = n,k P k+1 k j=0 (B n,k;j -C n,k;j ) , where (3.3) B n,k;j = x n-k σ q 1 A k;j (f -P j ) n and (3.4) C n,k;j = (-1) n-k A k;j (f -P j ) n .
In view of (2.1), one finds that xσ q P j = P -P j+1 , hence:

xσ q (f -P j ) = xσ q f -xσ q P j = (P -f ) -(P -P j+1 ) = -(f -P j+1 ).
Thus, it follows that

B n,k;j = (xσ q (f -P j )) n x k σ q A k;j = (-1) n x k σ q A k;j (f -P j+1 ) n .
By using the expression of A k;j given in (2.8), one has :

x k σ q A k;j = =j 0≤ ≤k (xσ q (P j -P )) = (-1) k =j 0≤ ≤k (P j+1 -P +1 ) .
This is to say that

x k σ q A k;j = (-1) k P j+1 A k+1,j+1 .
So, one can write (3.3) as follows:

(3.5) B n,k;j = (-1) n-k P j+1 A k+1;j+1 (f -P j+1 ) n .
Rewrite (3.2) into the following form:

(3.6) L n,k+1 (f n ) = n,k P k+1   k-1 j=0 (B n,k;j -C n,k;j+1 ) + B n,k;k -C n,k;0   .
Let j < k; one has: A k+1,j+1 = (P j+1 -P k+1 ) A k,j+1 . Replace j by j + 1 in (3.4), and make use of (3.5). One gets that

B n,k;j -C n,k;j+1 = (-1) n-k P k+1 A k+1;j+1 (f -P j+1 ) n .
Furthermore, one sees that A k+1;0 = -P k+1 A k;0 . By letting j = k and j = 0 in (3.5) and (3.4) respectively, one has:

B n,k;k = (-1) n-k P k+1 A k+1;k+1 (f -P k+1 ) n and C n,k;0 = - (-1) n-k P k+1 A k+1;0 (f -P 0 ) n .
Thus, in view of (3.1), equality (3.6) implies that

L n,k+1 (f n ) = n,k+1 k+1 j=0 1 A k+1;j (f -P j ) n ,
which corresponds to (2.9) in which k was replaced by k + 1.

Proof of Lemma 2.3. We give here two proofs.

Proof by induction -If n = 0, as a 0 = 1, equality (2.10) becomes evident. Suppose that (2.10) holds for some index n ≥ 0, and consider an (n + 2)-uplet (α 0 , ..., α n+1 ) where, as before, α j = α if j = . Set

P(T ) = n+1 j=0 1 ãj (T -α j ) n+1 , ãj = =j 0≤ ≤n+1 (α j -α ).
Let k be an integer such that 0 ≤ k ≤ n + 1. If one writes

a k;j = =k, =j 0≤ ≤n+1 (α j -α ) , one has a k;j = ãj /(α j -α k ) for j = k, what implies that (3.7) P(α k ) = - j =k 0≤j≤n+1 1 a k;j (α k -α j ) n .
By applying the induction hypothesis to the (n+1)-uplet (α 0 , ..., α k-1 , α k+1 , ...α n+1 ) appeared in the right hand side of (3.7), one obtains that P(α k ) = (-1) n+1 . As P is a polynomial with deg P ≤ n + 1, one finds that P(T ) = (-1) n+1 identically.

Proof by Lagrange polynomials 1 -Let K = C(T ) be the field of the rational functions over C. Let X be a new indeterminate, and set

F (X) = (T -X) n , Λ j (X) = =j 0≤ ≤n (X -α ) .
By noticing that F (X) ∈ K[X] be such that deg F = n < n + 1, the Lagrange interpolation formula [3, Chap. IV, §2, p. 26] implies that

F (X) = n j=0 F (α j ) a j Λ j (X) .
And now one compares the coefficients of X n for both sides of the above equality.

As that of Λ j is equal to 1, one finds that

(-1) n = n j=0 F (α j ) a j = n j=0 (T -α j ) n a j ,
which is exactly the expected identity (2.10).

End of the proof of Theorem 2.1. In view of Lemma 2.2, letting k = n in (2.9) yields that

L P n,n (f n ) = (-1) n(n+1)/2 n j=0 1 A n;j (f -P j ) n ,
where A n;j = =j 0≤ ≤n (P j -P ) . By (2.2), one knows that P j = P for j = . Thus, by applying Lemma 2.3 with

C = C[x -1 ][[x]
], X = f and α j = P j , one gets that

n j=0 1 A n;j (f -P j ) n = (-1) n .
This permits to finish the proof of Theorem 2.1.

One generalisation of Theorem 2.1

Instead of (2.6), let us consider the following slightly more general q-difference equation:

(4.1) α σ q y + y = β ,
where both α and β are nonzero power series belonging to the field

C[x -1 ][[x]].
In the same spirit as in (2.1), define the sequence

{β n } n≥0 in C[x -1 ][[x]
] as follows:

(4.2) We would like to thank our friend and collaborator J. Sauloy for this elegant proof.

β 0 = 0; β n = n-1 k=0 (-α σ q ) k β , n ≥ 1 .
Proposition 4.1. For any given positive integer n, the following conditions are equivalent.

(1) β j = 0 for any positive integer j such that 0 < j ≤ n.

(2) β j = β for any couple of integers (j, ) such that 0 ≤ < j ≤ n.

Proof. Indeed, by (4.2), it follows that (4.3)

β j -β = j-1 k= (-α σ q ) k β = (-α σ q ) β j-.
As α = 0, l'operator (-α σ q ) is an automorphism on the C-vector space

C[x -1 ][[x]].
So, this is the same for its -th power or iteration (-α σ q ) . This implies the equivalence between the conditions stated in Proposition 4.1.

Given a nonzero f ∈ C[x -1 ][[x]
], one remembers that ν 0 (f ) denotes the valuation of f at x = 0, that is the lowest degree of the terms of f . By using (4.2), one obtains that, for n > 0:

ν 0 (β n ) = ν 0 (β) if ν 0 (α) > 0; (n -1)ν 0 (α) + ν 0 (β) if ν 0 (α) < 0.
This gives the following statement for α, β

∈ C[x -1 ][[x]] \ {0}.
∈ C[x -1 ][[x]
]. Suppose that f satisfies the q-difference equation in (4.1) and that β j = 0 for any positive integer j such that 0 < j ≤ n. One has:

(4.4) L α,β n (f n ) = (-1) n(n-1)/2
, where L α,β n is the n-th order q-difference operator defined by

(4.5) L α,β n = 1 β n (α σ q + 1) 1 β n-1 α 2 σ q -1 • • • 1 β 1 (α n σ q -(-1) n ) .
Proof. Replace (k, P ) with (n, β) in (2.8), and define:

A n;j = =j 0≤ ≤n (β j -β ) , 0 ≤ j ≤ n.
By taking into account Proposition 4.1, it follows that A n;j = 0. Thus, one might proceed in the same way as for the proof of Theorem 2.1. We omit the details.

About the summability of the powers of q-Euler series

Let us come back to the power series Êq (x) defined by (1.1), which satisfies the q-difference equation stated in (1.2). Letting P = 1, Theorem 2.1 implies immediately the following result.

Remark 5.1. Given any integer n ≥ 2, the n-th power Êq (x) n satisifies the following identity:

(5.1) (xσ q + 1) 1 P n-1

x 2 σ q -1 ...(x n σ q -(-1) n ) Êq (x) n = (-1) n(n-1)/2 P n ,

where

P k = 1 -x + ... + q (k-1)(k-2) (-x) k-1 for 1 ≤ k ≤ n.
In particular, when n = 2, as P 2 = 1 -x, (5.1) takes the form of (1.3).

Furthermore, it might be interesting to notice that, if q = 1, one has Ê1 (x) = 1 1 + x and, in this case, the above identity in (5.1) is simply equivalent to the following elementary relation:

n k=1 x k -(-1) k k-1 j=0 (-x) j = (-1) n(n-1)/2 (x + 1) n .
This observation related with the specific case of q = 1 may be also made for Theorem 4.3, by assuming, for example, both α and β to belong to the field C[x -1 ]{x} of the germs of meromorphic functions at x = 0 in C or to the sub-field C(x) of the rational functions over C.

Our next remark goes to another q-analog of the following so-called Euler series:

(5.2) Ê(x) = n≥0 (-1) n n! x n+1 .
This power series is divergent for all x ∈ C \ {0} but Borel-summable in every direction excepted R -. It satisfies the following first order ODE:

(5.3) (x δ + 1)y = x, δ = x∂ x = x d dx .
Letting Y = Ê(x) 2 , one can check that (5.4)

(x δ + 1 -x) (x δ + 2)Y = 2x 2 .
Indeed, if y = Ê(x), one deduces from (5.3) that xδ(y 2 ) = 2x y δy = 2y(x -y) = 2xy -2y 2 . This means that (xδ + 2)(y 2 ) = 2x y. Thus, applying again (5.3) yields that (x δ + 1) 1 2x (x δ + 2) y 2 = x . In this way, one gets (5.4), using the identity Besides, the correspondances n ↔ 1 -q n 1 -q and n! ↔ (1 -q)...(1 -q n ) (1 -q) n suggest one to consider the following q-analog of the Euler series:

(5.5) Ê(x;

q) = x + n≥1 (-1) n (1 -q)...(1 -q n ) (1 -q) n x n+1 .
This can be written in term of a basic hypergeometric series as follows:

Ê(x; q) = x 2 φ 1 (q, q; 0; q, -x 1 -q ) = x 2 φ 0 (q -1 , q -1 ; -; q -1 , x 1 -q -1 ).

See [6, (1.2.22), p. 4] for the general definition of r φ s (...; ...; q, z).

Let δ q = σ q -1 q -1 = x ∆ q , where ∆ q is defined in [1, p. 488, (10.2.3)]. By observing that n = δx n

x n and 1 -q n 1 -q = δ q x n x n , a direct computation shows that Ê(x; q) satisfies the q-analog of (5.3) as follows: (x δ q + 1)y = x . If one writes (5.6) α = x q -1 -x , β = (q -1) α , it follows from the above that (5.7) (α σ q + 1) Ê(x; q) = β .

Remark 5.2. Let α be as in (5.6). The following identity holds in the field

C[x -1 ][[x]]: (5.8) α δ q + 1 q -1 -x α δ q - 1 x + 1 q -1 -x Ê(x; q) 2 = α (σ q α -1) .
Moreover, when q → 1, (5.8) is reduced into the following equivalent form of (5.4):

(5.9)

δ + 1 x δ + 2 x Ê(x) 2 = 2 .
To obtain (5.8), one can apply (4.2) for n = 1 and n = 2, where α and β are defined by (5.6). This gives that

β 1 = (q -1) α, β 2 = β -α σ q β = (q -1) α (1 -σ q α) .
Thus, it follows from applying Theorem 4.3 to (5.7) that Ê(x; q) 2 satisfies the following q-difference equation:

(5.10) L Ê(x; q) 2 = α (σ q α -1) , where

L = 1 (q -1) 2 (α σ q + 1) α σ q - 1 α . (5.11)
By replacing σ q with (q -1)δ q + 1 in (5.11), one gets that

L = α δ q + 1 + α q -1 α δ q + α 2 -1 (q -1) α . As α + 1 = q -1 q -1 -x and α 2 -1 = (q -1)(2x -q + 1) (q -1 -x) 2
, one deduces immediately (5.8) from (5.10). The limit equation form given in (5.9) can be obtained from (5.8) by noticing both limits α → -1 and δ q → δ for q → 1.

In the rest of this paper, we will suppose that q > 1 even if, in most cases, the hypothesis |q| > 1 may be really enough; see [START_REF] Roques | On classical irregular q-difference equations[END_REF] and [START_REF] Ramis | Local analytic classification of q-difference equations[END_REF]. Let us recall some results about the summability of power series solutions of q-difference equations. Given any linear q-difference operator L of the following form:

L = n j=0 a j σ j q ∈ C{x}[σ q ]
, a 0 a n = 0 , one definies its associated Newton polygon N P(L) as being the convex hull of the set {(j, m) : m ≥ ν 0 (a j )} in the strip [0, n] × [0, +∞). By following [8, (3.1.2)] and [START_REF] Zhang | Développements asymptotiques q-Gevrey et séries Gq-sommables[END_REF]Proposition 5.14], one knows that, if N P(L) admits only integer slopes, saying

κ 1 ≤ κ 2 ≤ ... ≤ κ n , then there exist ν ∈ Z, (h 1 , h 2 , ..., h n ) ∈ (1 + xC{x}) n and
(c 0 , c 1 , ..., c n ) ∈ (C \ {0}) n+1 such that (5.12) L = c 0 x ν h 1 (x κ1 σ q + c 1 ) h 2 (x κ2 σ q + c 2 ) ... h n (x κn σ q + c n ) .

Furthermore, let K + = Z >0 ∩ {κ j : 1 ≤ j ≤ n}, and define k as follows:

(5.13) k = ∅ if K + = ∅ (k 1 , ..., k m ) if K + = {k : 1 ≤ ≤ m} and k 1 < ... < k m .

By considering [8, § 3.3.5], it follows that any power series f ∈ C[[x]] such that L( f ) ∈ C{x} remains convergent or is Gq-summable of order k = (k 1 , ...k m ) ∈ Z m >0 depending on whether k = ∅ or not.

Theorem 5.3. Given any integer n ≥ 2, the n-th power Êq (x) n is Gq-summable of order (1, 2, ..., n). And this is the same for Ê(x; q) n .

Proof. This follows from taking into account (5.1) or applying Theorem 4.3 to (5.7) with the help of Remark 4.2.

Applying [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF]Theorem 36] or [START_REF] Malgrange | Sommation des séries divergentes[END_REF]Proposition 1.3.4.2] implies that every power Ê(x) n of the Euler series is Borel-summable at the same level, contrarily to what happens in the case of their respective q-analog Ê(x; q) n ; see Theorem 5.3. Furthermore, thanks to [START_REF] Vizio | On q-summation and confluence[END_REF]Theorem 3.15], one knows that the Gq-sum of Ê(x; q) tends toward the Borel-sum of Ê(x) when q → 1 + . In our coming work [START_REF] Dreyfus | A q-summation process that is a morphism of difference ring[END_REF], it would be shown that, given one generic direction d and two power series f1 and f2 such that L j ( fj ) ∈ C{x} for some L j ∈ C{x}[σ q ], where j = 1 or 2, the Gq-sum of f1 f2 along d is equal to the product of the Gq-sum of f1 along d with that of f2 . In this way, one would obtain that the Gq-sum of Êq (x) n or Ê(x; q) n can be really expressed as the n-th power of that of Êq (x) or Ê(x; q), respectively.

1991

  Mathematics Subject Classification. Primary : 39B32 ; 40G10 ; 30E15 . This work was partially supported by Labex CEMPI (Centre Européen pour les Mathématiques, la Physique et leurs Interaction).

1

 1

Remark 4 . 2 .Theorem 4 . 3 .

 4243 Both conditions considered in Proposition 4.1 are necessarily fulfilled if ν 0 (α) = 0. Let n ∈ Z >0 , and let α, β and f

(

  -x) in the non-commutative ring C(x)[δ] of the differential operators over C(x).


S Manakli

Y Trouiller

P Schiavone

Y Rody

P.-J Goirand

O ptimization of the depth of focus based on the analysis of the diffraction orders in the pupil plane

Keywords: Optical lithography, Depth of focus, Diffraction orders, Isofocal

The evolution of the depth of focus has been studied using a simple geometrical analysis of the diffraction orders in the pupil plane. This analysis is based on the observation that the diffraction orders participating to the interactions must be symmetrical with respect to the optical axis. The center of the gravity of the areas of the different diffraction orders captured by the pupil is evaluated as a function of pitch, numerical aperture and wavelength. Analytical expressions have been derived. They give which illumination settings (partial coherence) optimise the DOF for conventional, annular and quadrupole illuminations. The impact of the diffraction orders on the evolution of the DOF through pitch curve was also studied. The appearance of a peak in the DOF versus pitch curve, for annular and quadrupole cases is caused by the perfect symmetry between the diffracted waves interacting together.

. Introduction

As the size reduction of the elementary components of integrated circuits is outpacing the introduction of shorter exposure wavelengths and higher numerical aperture lenses, lithographers must look into other solutions such as resolution enhancement techniques (alternated PSM [1], attenuated PSM [2], scattering bars [START_REF] Trouiller | Proc SPIE[END_REF], double exposure [START_REF] Manakli | Proc SPIE[END_REF], off axis illumination) in order to extend the limits of the optical lithography.

A mandatory step to further optimise the photolithographic process is the understanding of the *Corresponding author. E-mail address: serdar.manakli@st.com (S. Manakli).

optical phenomena in the pupil plane. Several papers have shown [START_REF] Terasawa | [END_REF][6][7] that the optimal depth of focus (DOF) is reached when the beams captured by the pupil are symmetrical to the optical axis. This means that the shape of the beams incident onto the mask impact the distribution of the diffraction orders in the pupil plane. This is why the illumination used in the circuit printing goes more and more to a symmetrical shape (from conventional to annular, quadrupole and dipole). The evolution of the depth of focus as a function of the pitch is studied in this paper. The measured DOF corresponds to the 0% energy latitude of 610% of targeted dimension which in this paper is 100 nm. This evolution of the DOF is studied using a simple geometrical analysis of the diffraction orders in the pupil plane. The barycentre of the areas of the different diffraction orders captured by the pupil is evaluated through the pitch, wavelength (l) and NA, and all steps of the analytical calculation are described. We compare the theoretical DOF optimisation using the center of the gravity (COG) analysis with a simulation of the aerial image at 193 nm wavelength and 0.63 NA and show that the analytical optimisation gives the same results and works faster. Finally, we study the impact of the diffraction orders positions on the evolution of the DOF curve versus feature pitch and partial coherence using the position of the barycentres.

. Area and barycentre of the diffracted orders through pitch

.1. Theoretical background

Fig. 1a shows a schematic sketch of a lithographic system [START_REF] Suzuki | Proc. SPIE[END_REF][START_REF] Williamson | Proc. SPIE[END_REF]. It consists of four parts: illumination optics, mask, projection optics and wafer. The light that falls onto the mask can be considered as a sum of plane waves with different angles of incidence. These plane waves are diffracted by the mask into the several directions [START_REF] Born | Principles of Optics[END_REF]. Here we will limit our study to the zero and first orders of diffraction (0, 1 and 2 1) because in the case of low k imaging most of the 1 contribution comes from these orders. We will consider only one-dimensional patterns.

A feature on a mask is reproduced on the wafer by a light interference process from the diffracted orders. Considering the coherence properties of lithographic light sources, we consider that two beams interfere if they come from the same source point. In the case where the feature pitch is small enough, as we can see in Fig. 2, from the source S1, P interacts with P . The first-order contribution P is not

21 0 1
captured by the pupil. We have here only an interaction between two beams (21 and 0). We have the same with another source point S2 where only P and P interact together because the pupil does not 0 1 capture P .

21

The shape and the position of these beams in the pupil plane have a great impact on the lithographic performance: if two beams are symmetrical to the optical axis, the DOF is at maximum [START_REF] Terasawa | [END_REF]6] (Fig. 1b). In the case of partially coherent illumination (conventional, annular or quadrupole illumination), as described by Abbe's theory [START_REF] Abbe | [END_REF], we can extend this to a set of source points. To evaluate the symmetry of a group of beams, we use the position of the COG of the diffracted order areas (21 and 1) captured by the lens. Considering that the interference of two symmetrical plane waves provides a theoretically infinite depth of focus, we can infer that if the positions of the COG of two groups of beams (0 and 2 1) are symmetrical, we optimise the DOF.

.2. Area and COG of the diffracted orders captured by the pupil plane

In order to evaluate the COG of the diffracted orders captured by the pupil plane, we need to derive an analytical expression representing the area of these diffracted orders. For that, we make use of the Hopkin's [7] treatment as follow: we consider the projection lens pupil as a circle C having NA (r ) 1 1

as radius and the diffracted first-order area as a circle (C ) having partial coherence s (r ) as radius. Fig. 3 shows an illustration of this for a conventional illumination. In the annular and quadrupole cases we are dealing with circular shapes as well and we only have to adapt the following expressions to them. All interaction surfaces can be computed easily as a superposition of portions of circles. Everything resume to the determination of the area and the center of gravity of an arc (e.g., hatched area of Fig. 3). The overlap area between C and C corresponds to the first diffracted orders area captured by the ]]]

SD S 5 r 3 Arctan 2 x 3 r 2 x (1) oe hatched area 1 1 1 1 x 1 3 3 r 3 sin u 2 x 3 tan u 2 1 1 ] ]]]]]]] x 5 (2)
g,hatched area 2 2

3 r 3 u 2 x 3 tan u C and C and u the semi-angle of the arc AB. Fig. 4 shows a drawing of the pupil plane for an annular illumination with 193 nm wavelength, 400 nm pitch, 0.63 NA and 0.50s / 0.80s . In dotted lines, we have the position of the first diffraction in out order COG captured by the pupil and the position of the COG of the 0 order which interfere with them. In solid, we find the position of the best symmetry which is located at 6l /(2p). More than the position of the COG, it also shows the number and areas of the interactions generated between the diffracted orders captured by the lens. These diffracted orders do not participate only to a two-wave interaction (between 1 / 0 and 21 / 0 orders). When the pitch becomes larger, we can have different scenarios as, for example, interaction between 1 / 0, 21/0, 1/0/21 (three-wave interaction) and presence of the 0 order illustrated in the Fig. 4.

. DOF optimisation based on the symmetry of the diffracted waves

.1. DOF optimisation methods based on the position of the COG

The depth of focus can be optimised using the center of the gravity method. This method consists of optimising or symmetrizing the COG position of the diffracted orders interacting together. That means matching the COG positions with the best symmetrical COG positions, which are 6l /(2p) (Fig. 4). For a given wavelength, pitch, NA and for every illumination shape, the partial coherence is optimised until matching the COG positions of the captured 21 and 1 diffracted orders to the optimum COG positions. Fig. 5 shows the optimum s found using the COG method for varying c pitches. It is compared with the results of a full aerial image simulation (black squares). It can be seen that the agreement between the two methods is very good over the full range of pitch. This confirms the interest of our approach which provides much faster results. Another example is given in Fig. 6 where the depth of focus for a 120 / 200-nm grating is shown for different NAs. In both cases a clear optimum is observed. Using the optimisation method based on symmetry, the predicted optimum s c (0.63 and 0.74) are in very good agreement with those (0.63 and 0.73) determined by a full aerial image simulation study with Solid C.

On the other hand, the DOF can also be optimised for annular and quadrupole illumination. In the case of quadrupole, since increasing the pole radius s does not affect the position of the COG, the r classical formula deduced considering coherent illumination still holds (cf. Eq. ( 4)). In the case of annular illumination, the shape of the interfering orders is more complicated. However, we have remarked that we can, as a rule of thumb, apply the expression (3) for dense pitches and for an annular ring fixed at 0.3. Fig. 5 confirms the validity of this approximation for dense pitches. We can notice that today the usual s is between 0.4 and 0.7. From Fig. 5, we see that for a 0.63 NA c and annular illumination the optimum s is obtained only in the range between 0.6 and 0.7. In the c quadrupole case (0.63 NA), we verified that the range of the optimum s is 0.4 and 0.7. c 3 .2. Impact of the diffracted waves symmetry on the DOF curve versus pitch Fig. 7 shows the DOF variation through pitch and s for a quadrupole illumination. s is fixed at r c 0.55. We also add to this figure the simulated results of the COG positions around the peaks. As expected, we observe that the peaks correspond to the perfect symmetry between the 1 and 21 diffracted orders captured by the pupil plane and the 0 orders interacting with them. At these peaks, we have the situation that the COG positions and the best symmetry positions coincide (Fig. 4). We also see that the amplitude of the peak is larger for a more coherent illumination (for a smaller s ).

r Larger s tend to destroy the perfect two waves interference effect (which theoretically leads to an r infinite depth of focus). The case of annular illumination and fixed s is shown in Fig. 8. Here we can observe another out trend. The DOF decreases when the pitch increases. This can be attributed to the evolution of the isofocal CD [START_REF] Smith | Proc SPIE[END_REF]. It becomes smaller and move away from the target CD when the pitch increases thus decreasing the depth of focus. Peaks, corresponding to the symmetry of interfering diffraction orders discussed above, are also visible. These two causes of variation of the DOF (position of the isofocal CD and the symmetry of the diffraction orders) are also the explanation for the shape of the curves shown in Fig. 9. Here quadrupole illumination is concerned and both s and s vary. The decrease of DOF with increasing c r pitch as well as the symmetry peak previously mentioned can be observed on this graph. However, for certain illumination settings, a clear second peak can be observed. It is due to the incoming of the

Fig. 1 .

 1 Fig. 1. Schematic sketch of a lithographic system.

Fig. 2 .

 2 Fig. 2. Diffracted waves captured by the pupil plane.

Fig. 3 .

 3 Fig. 3. Schematic illustration of the projected lens (C ) and 1st diffracted order (C ).

1 2 Fig. 4 .

 124 Fig. 4. Illustration of the presence of several interactions at the pupil plane for an annular illumination and position of the COGs of the 21 and 1 diffracted orders captured by the pupil able to interfere with order 0.

  /(2a)?[r 2r 1a ], a the position of the centre of C , r and r the radii of the two circles

Fig. 5 .

 5 Fig. 5. Determination of the optimum s through the pitch using aerial image simulation, COG and approximated c (expression 3) methods for 0.63 NA and annular illumination.

Fig. 6 .

 6 Fig. 6. Simulation results of the DOF through s and NA for 320 nm pitch and 100 nm target feature for an annular c

Fig. 7 . 6 Fig. 8 .Fig. 9 .

 7689 Fig. 7. DOF through pitch and s for 100 nm target feature and quadrupole illumination (NA50.63). s is fixed at 0.55.r c

. Conclusions

In this paper, we have analysed the evolution of the depth of focus as function of the pitch. The partial coherence and the shape of the diffracted waves in the pupil plane have been confirmed to be of major importance. Our analysis is based on the position of the COG of the interfering waves that should be symmetrical. All the analytical calculations have been described. Finally, The shape of the DOF versus pitch curves has been completely explained. The influence of the position of the isofocal CD and the symmetry of the diffracted waves has been emphasized.