Mahadi Ddamulira
email: mddamulira@tugraz.atmahadi@aims.edu.gh

On the Problem of Pillai with Tribonacci Numbers and Powers of 3

Let (T n ) n≥0 be the sequence of tribonacci numbers defined by T 0 = 0, T 1 = T 2 = 1, and T n+3 = T n+2 +T n+1 +T n for all n ≥ 0. In this note, we find all integers c admitting at least two representations as a difference between a tribonacci number and a power of 3.

Introduction

We consider the sequence (T n ) n≥0 of tribonacci numbers defined by T 0 = 0, T 1 = 1, T 2 = 1, and T n+3 = T n+2 + T n+1 + T n for all n ≥ 0.

The tribonacci sequence is sequence A000073 on the Online Encyclopedia of Integer Sequences (OEIS) [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]. The first few terms of the tribonacci sequence are (T n ) n≥0 = 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, . . . .

In this paper, we study the Diophantine equation

T n -3 m = c, (1)
for a fixed integer c and variable integers n and m. In particular, we are interested in finding those integers c admitting at least two representations as a difference between a tribonacci number and a power of 3. This equation is a variation of the Pillai equation

a x -b y = c, (2)
where x, y are non-negative integers and a, b, c are fixed positive integers.

In the 1930's, Pillai [START_REF] Pillai | On a x -b y = c[END_REF][START_REF] Pillai | A correction to the paper On a x -b y = c[END_REF] conjectured that for any given integer c ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2 to the equation ( 2) is finite. This conjecture is still open for all c = 1. The case c = 1 is the conjecture of Catalan which was proved by Mihȃilescu [START_REF] Mihȃilescu | Primary cyclotomic units and a proof of Catalan's conjecture[END_REF]. The work of Pillai work was an extension of the work of Herschfeld [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF][START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF], who had already studied a particular case of the problem with (a, b) = (2, 3). Since then, different variations of the Pillai equation have been studied. Several recent results for the different variations of the Pillai problem involving Fibonacci numbers, tribonacci numbers, Pell numbers, k-generalized Fibonacci numbers, and other linearly recurrent sequences, with powers of 2, have been completely studied. For example, see [START_REF] Bravo | On Pillai's problem with tribonacci numbers and powers of 2[END_REF][START_REF] Chim | On a variant of Pillai's problem[END_REF][START_REF] Chim | On a variant of Pillai's problem II[END_REF][START_REF] Ddamulira | On the problem of Pillai with Fibonacci numbers and powers of 3, To appear in[END_REF][START_REF] Ddamulira | On a problem of Pillai with Fibonacci numbers and powers of 2[END_REF][START_REF] Ddamulira | On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2[END_REF][START_REF] Hernane | On Pillai's problem with Pell numbers and powers of 2[END_REF][START_REF] Hernández Hernández | On Pillai's problem with the Fibonacci and Pell sequences, To appear in[END_REF].

We discard the situation when n = 1 and just count the solutions for n = 2 since T 1 = T 2 = 1. The reason for the above convention is to avoid trivial parametric families such as 1 -3 m = T 1 -3 m = T 2 -3 m . Thus, we always assume that n ≥ 2. The main aim of this paper is to prove the following result.

Theorem 1. The only integers c having at least two representations of the form T n -3 m with n ≥ 2 and m ≥ 0, are c ∈ {-2, 0, 1, 4}. Furthermore, all the representations of the above integers as T n -3 m with integers n ≥ 2 and m ≥ 0 are given by

-2 = T 5 -3 2 = T 2 -3 1 , 0 = T 9 -3 4 = T 2 -3 0 , (3) 1 = T 4 -3 1 = T 3 -3 0 , 4 = F 6 -3 2 = T 5 -3 1 .

Preliminary results

The tribonacci sequence

The characteristic polynomial of the tribonacci sequence (T n ) n≥0 is given by

Ψ(x) := x 3 -x 2 -x -1. Ψ(x) is irreducible in Q[x]
, and has a positive real zero

α = 1 3 1 + (19 + 3 √ 33) 1/3 + (19 -3 √ 33) 1/3 ,
lying strictly outside the unit circle and two complex conjugate zeros β and γ lying strictly inside the unit circle. Furthermore, |β| = |γ| = α -1/2 . According to Dresden and Zu [START_REF] Dresden | A simplified Binet formula for r-generalized Fibonacci numbers[END_REF], a Binet-like formula for the k-generalized Fibonacci sequences is established. For the tribonacci sequence, it states that

T n = C α α n-1 + C β β n-1 + C γ γ n-1 for all n ≥ 1, (4)
where C X = (X -1)/(4X -6). Dresden and Zu [START_REF] Dresden | A simplified Binet formula for r-generalized Fibonacci numbers[END_REF], also showed that the contribution of the complex conjugate zeros β and γ to the right-hand side of ( 4) is very small. More precisely,

T n -C α α n-1 < 1 2 for all n ≥ 1. ( 5
)
The minimal polynomial of C α over the integers is given by

44X 3 -44X 2 + 12X -1, has zeros C α , C β , C γ with |C α |, |C β |, |C γ | < 1. Numerically, 1.83 < α < 1.84, 0.73 < |β| = |γ| = α -1/2 < 0.74, 0.61 < |C α | < 0.62, 0.19 < |C β | = |C γ | < 0.20.
It is also a well known fact (see [START_REF] Bravo | On Pillai's problem with tribonacci numbers and powers of 2[END_REF][START_REF] Ddamulira | On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2[END_REF]) that

α n-2 ≤ T n ≤ α n-1 holds for all n ≥ 1. ( 6
)
Let K := Q(α, β) be the splitting field of the polynomial Ψ over Q. Then, [K, Q] = 6. Furthermore, [Q(α) : Q] = 3. The Galois group of K over Q is given by 1), (αβ), (αγ), (βγ), (αβγ), (αγβ)} ∼ = S 3 .

G := Gal(K/Q) ∼ = {(
Thus, we identify the automorphisms of G with the permutations of the zeros of the polynomial Ψ. For example, the permutation (αγ) corresponds to the automorphism σ : α → γ, γ → α, β → β.

Linear forms in logarithms

In order to prove Theorem 1, we need to use several times a Baker-type lower bound for a nonzero linear form in logarithms of algebraic numbers. There are many such bounds in the literature like that of Baker and Wüstholz [2]. We use the one of Matveev [START_REF] Matveev | An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II[END_REF]. Matveev [START_REF] Matveev | An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II[END_REF] proved the following theorem, which is one of our main tools in this paper.

Let γ be an algebraic number of degree d with minimal primitive polynomial over the integers

a 0 x d + a 1 x d-1 + • • • + a d = a 0 d i=1 (x -γ (i) ),
where the leading coefficient a 0 is positive and the η (i) 's are the conjugates of γ. Then the logarithmic height of γ is given by

h(γ) := 1 d log a 0 + d i=1 log max{|γ (i) |, 1} .
In particular, if γ = p/q is a rational number with gcd(p, q) = 1 and q > 0, then h(γ) = log max{|p|, q}. Some of the properties of the logarithmic height function h(•), that will be used in the next sections of this paper without reference are as follows:

h(η ± γ) ≤ h(η) + h(γ) + log 2, h(ηγ ±1 ) ≤ h(η) + h(γ), (7) h
(η s ) = |s|h(η) (s ∈ Z).
Theorem 2 (Matveev). Let γ 1 , . . . , γ t be positive real numbers in a number field K ⊆ R of degree D, and let b 1 , . . . , b t be nonzero integers. Assume that

Λ := γ b 1 1 • • • γ bt t -1, (8)
is nonzero. Then,

log |Λ| > -1.4 × 30 t+3 × t 4.5 × D 2 (1 + log D)(1 + log B)A 1 • • • A t ,
where

B ≥ max{|b 1 |, . . . , |b t |} and
A i ≥ max{Dh(γ i ), | log γ i |, 0.
16}, for all i = 1, . . . , t.

Baker-Davenport reduction procedure

During the course of our calculations, we get some upper bounds on our variables which are too large. Thus, we need to reduce them. To do so, we use some results from the theory of continued fractions. Specifically, for a nonhomogeneous linear form in two integer variables, we use a slight variation of a result due to Dujella and Pethő [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF], which itself is a generalization of a result of Baker and Davenport [START_REF] Baker | The equations 3x 2 -2 = y 2 and 8x 2 -7 = z[END_REF].

For a real number X, we write X := min{|X -n| : n ∈ Z} for the distance from X to the nearest integer.

Lemma 3 (Dujella, Pethő). Let M be a positive integer, p/q be a convergent of the continued fraction of the irrational number τ such that q > 6M , and let A, B, and µ be some real numbers with A > 0 and B > 1. Let further ε := µq -M τ q . If ε > 0, then there is no solution to the inequality Finally, the following lemma is also useful. It is Lemma 7 in [START_REF] Gúzman | Linear combinations of factorials and s-units in a binary recurrence sequence[END_REF].

0 < |uτ -v + µ| < AB -w ,
Lemma 4 (Gúzman, Luca). Let m 1, Y > (4m 2 ) m , and Y > x/(log x) m . Then

x < 2 m Y (log Y ) m .
3 Proof of Theorem 1

Let n, m, n 1 , and m 1 be non-negative integers such that (n, m) = (n 1 , m 1 ) and

T n -3 m = T n 1 -3 m 1 .
Without loss of generality, we assume that m

≥ m 1 . If m = m 1 , then T n = T n 1 , so (n, m) = (n 1 , m 1 )
, which gives a contradiction to our assumption. Thus, m > m 1 . Since

T n -T n 1 = 3 m -3 m 1 , (9)
and the right-hand side is positive, we get that the left-hand side is also positive and so n > n 1 . Thus, n ≥ 3 and n 1 ≥ 2.

Using the equation ( 9) and the inequality (6), we get

α n-4 ≤ T n-2 ≤ T n -T n 1 = 3 m -3 m 1 < 3 m , ( 10
) α n-1 ≥ T n ≥ T n -T n 1 = 3 m -3 m 1 ≥ 3 m-1 , (11)
from which we get that

1 + log 3 log α (m -1) < n < log 3 log α m + 4. ( 12
)
If n ≤ 300, then m ≤ 200. We ran a Mathematica program for 2 ≤ n 1 < n ≤ 300 and 0 ≤ m 1 < m ≤ 200 and found only the solutions from the list [START_REF] Bravo | On Pillai's problem with tribonacci numbers and powers of 2[END_REF]. From now, we assume that n > 300. Note that the inequality [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF] implies that m < 0.6n + 0.4. Therefore, to solve the Diophatine equation ( 1), it suffices to find an upper bound for n.

Bounding n

From ( 4) and ( 5), we get

C α α n-1 -3 m = (C α α n-1 -T n ) + (T n 1 -3 m 1 ) = (C α α n-1 -T n ) + (T n 1 -C α α n 1 -1 ) + (C α α n 1 -1 -3 m 1 ) < 1 + 7 10 α n 1 -1 + 3 m 1 < α n 1 + 3 m 1 < 2 max{α n 1 , 3 m 1 }.
In the above we have used the fact that |C α | < 0.62 < 0.7. Multiplying through by 3 -m , using the relation [START_REF] Hernane | On Pillai's problem with Pell numbers and powers of 2[END_REF] and using the fact that α < 3, we get

C α α n-1 3 -m -1 < 2 max α n 1 3 m , 3 m 1 -m < max{α n 1 -n+6 , 3 m 1 -m+1 }. ( 13
)
For the left-hand side, we apply the result of Matveev, Theorem 2 with the following data:

t := 3, γ 1 := C α , γ 2 := α, γ 3 := 3, b 1 := 1, b 2 := n -1, and b 3 := -m.
Through out we work with the field K := Q(α) with D = 3. Since max{1, n -1, m} ≤ n, we take B := n. The minimal polynomial of C α over the integers is given by

44x 3 -44x 2 + 12x -1. Since |C α |, |C β |, |C γ | < 1, we get that h(C α ) = 1
3 log 44. So we can take A 1 := 3h(γ 1 ) = log 44. We can also take A 2 := 3h(γ 2 ) = log α and A 3 := 3h(γ 3 ) = 3 log 3. We put

Λ := C α α n-1 3 -m -1.
First we check that Λ = 0, if it were, then C α α n-1 = 3 m ∈ Z. Conjugating this relation by the automorphism (αβ), we obtain that C β β n-1 = 3 m , which is a contradiction because |C β β n-1 | < 1 while 3 m ≥ 3 for all m ≥ 1. Thus, Λ = 0. Hence, by Theorem 2, the left-hand side of ( 13) is bounded as follows:

log |Λ| > -1.4 • 30 6 • 3 4.5 • 3 2 (1 + log 3)(1 + log n)(log 44)(log α)(3 log 3).
By comparing with (13), we get min{(n -n 1 -5) log α, (m -m 1 -1) log 3} < 2.06 × 10 13 (1 + log n), which gives min{(n -n 1 ) log α, (m -m 1 ) log 3} < 2.12 × 10 13 (1 + log n).

Now we split the argument into two cases.

Case 1. min{(n -n 1 ) log α, (m -m 1 ) log 3} = (n -n 1 ) log α.

In this case, we rewrite (9) as

C α α n-1 -C α α n 1 -1 -3 m = C α α n-1 -T n ) + (T n 1 -C α α n 1 -1 ) -3 m 1 < 1 + 3 m 1 ≤ 3 m 1 +1 , which implies C α (α n-n 1 -1)α n 1 -1 3 -m -1 < 3 m 1 -m+1 . ( 14
)
We put

Λ 1 := C α (α n-n 1 -1)α n 1 -1 3 -m -1.
As before, we take K := Q(α), so we have D = 3. We have Λ 1 = 0, for if Λ 1 = 0, then

C α (α n-n 1 -1)α n 1 -1 = 3 m .
By conjugating the above relation by the Galois automorphism (αβ), we get that

C β (β n-n 1 -1)β n 1 -1 = 3 m .
The absolute value of the left-hand side is at most

|C β (β n-n 1 -1)β n 1 -1 | ≤ |C β β n-1 | + |C β β n 1 -1 | < 2,
while the absolute value of the right-hand side is at least 3 m ≥ 3 for all m ≥ 1, which is a contradiction. We apply Theorem 2 on the left-hand side of ( 14) with the following data: Since

t := 3, γ 1 := C α (α n-n 1 -
h(γ 1 ) ≤ h(C α ) + h(α n-n 1 -1) < 1 3 log 44 + 1 3 (n -n 1 ) log α + log 2 < 1 3 (log 11 + log 32) + 1 3 × 2.12 × 10 13 (1 + log n) < 1 3 × 2.50 × 10 13 (1 + log n). (15)
So, we can take A 1 := 2.50 × 10 13 (1 + log n). Furthermore, as before, we take A 2 := log α and A 3 := 3 log 3. Finally, since max{1, n 1 -1, m} ≤ n, we can take B := n. Then we get

log |Λ 1 | > -1.4 • 30 6 • 3 4.5 • 3 2 (1 + log 3)(1 + log n)(2.50 × 10 13 (1 + log n))(log α)(3 log 3). Then, log |Λ 1 | > -1.36 × 10 25 (1 + log n) 2 .
By comparing the above relation with ( 14), we get that

(m -m 1 ) log 3 < 1.40 × 10 26 (1 + log n) 2 . ( 16
)
Case 2. min{(n -n 1 ) log α, (m -m 1 ) log 3} = (m -m 1 ) log 3.

In this case, we rewrite [START_REF] Dresden | A simplified Binet formula for r-generalized Fibonacci numbers[END_REF] as

C α α n -(3 m-m 1 -1) • 3 m 1 = (C α α n-1 -T n ) + (T n 1 -C α α n 1 -1 ) + C α α n 1 -1 < 1 + 7 10 α n-1 < α n-1 (beacause n ≥ 3),
which implies that

|C α (3 m-m 1 -1) -1 α n-1 3 -m 1 -1| < α n 1 3 m -3 m 1 ≤ 3α n 1 3 m < 3α n 1 -n+4 < α n 1 -n+6 . ( 17
)
We put

Λ 2 := C α (3 m-m 1 -1) -1 α n-1 3 -m 1 -1. Clearly, Λ 2 = 0, for if Λ 2 = 0, then C α = (α -1 ) n-1 (3 m -3 m 1 )
implying that C α is an algebraic integer, a contradiction. We again apply Theorem 2 with the following data: We note that

t := 3, γ 1 := C α (3 m-m 1 -1) -1 ,
h(γ 1 ) = h(C α (3 m-m 1 -1) -1 ) ≤ h(C α ) + h(3 m-m 1 -1) = 1 3 log 44 + h(3 m-m 1 -1) < log(3 m-m 1 +2 ) = (m -m 1 + 2) log 3 < 2.50 × 10 13 (1 + log n).
So, we can take A 1 := 7.5 × 10 13 (1 + log n). Further, as in the previous applications, we take A 2 := log α and A 3 := 3 log 3. Finally, since max{1, n -1, m 1 } ≤ n, we can take B := n. Then, we get

log |Λ 2 | > -1.4 • 30 6 • 3 4.5 • 3 2 (1 + log 3)(1 + log n)(7.5 × 10 13 (1 + log n))(log α)(3 log 3). Thus, log |A 2 | > -4.08 × 10 26 (1 + log n) 2 .
Now, by comparing with [START_REF] Mihȃilescu | Primary cyclotomic units and a proof of Catalan's conjecture[END_REF], we get that

(n -n 1 ) log α < 4.10 × 10 26 (1 + log n) 2 . (18)
Therefore, in both Case 1 and Case 2, we have min{(n -n 1 ) log α, (m -m 1 ) log 3} < 2.12 × 10 13 (1 + log n), max{(n -n 1 ) log α, (m -m 1 ) log 3} < 4.10

× 10 26 (1 + log n) 2 . (19)
Finally, we rewrite the equation ( 9) as

C α α n-1 -C α α n 1 -1 -3 m + 3 m 1 = (C α α n-1 -T n ) + (T n 1 -C α α n 1 -1 ) < 1.
Dividing through by 3 m -3 m 1 , we get

C α (α n-n 1 -1) 3 m-m 1 -1 α n 1 -1 3 -m 1 -1 < 1 3 m -3 m 1 ≤ 3 3 m ≤ 3α -(n-4) ≤ α 6-n , (20)
since 3 < α ≤ α n 1 . We again apply Theorem 2 on the left-hand side of ( 20) with the following data:

t := 3, γ 1 := C α (α n-n 1 -1) 3 m-m 1 -1 , γ 2 := α, γ 3 := 3, b 1 := 1, b 2 := n 1 -1, and b 3 := -m 1 .
By using the algebraic properties of the logarithmic height function, we get

3h(γ 1 ) = 3h C α (α n-n 1 -1) 3 m-m 1 -1 ≤ h C α (α n-n 1 -1) + h(3 m-m 1 -1) < log 352 + (n -n 1 ) log α + 3(m -m 1 ) log 3 < 6.80 × 10 26 (1 + log n) 2 ,
where in the above inequalities, we used the argument from [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF] as well as the bounds [START_REF] Pillai | On a x -b y = c[END_REF]. Thus, we can take A 1 := 6.80 × 10 26 (1 + log n), and again as before A 2 := log α and A 3 := 3 log 3. If we put

Λ 3 := C α (α n-n 1 -1) 3 m-m 1 -1 α n 1 -1 3 -m 1 -1,
we need to show that Λ 3 = 0. If not, Λ 3 = 0 leads to

C α (α n-1 -α n 1 -1 ) = 3 m -3 m 1 .
A contradiction is reached upon a conjugation by the automorphism (αβ) in K and by taking absolute values on both sides. Thus, Λ

Reducing the bound for n

We need to reduce the above bound for n and to do so we make use of Lemma 3 several times. To begin, we return to [START_REF] Gúzman | Linear combinations of factorials and s-units in a binary recurrence sequence[END_REF] and put Γ := (n -1) log α -m log 3 + log C α .

For technical reasons we assume that min{n -n 1 , m -m 1 } ≥ 20. We go back to the inequalities for Λ, Λ 1 , and Λ 2 , Since we assume that min{n -n 1 , m -m 1 } ≥ 20 we get |e Γ -1| = |Λ| < 1 4 . Hence, |Λ| < 1 2 and since the inequality |y| < 2|e y -1| holds for all y ∈ -1 2 , 1 2 , we get 0

< |Γ| < 2 max{α n 1 -n+6 , 3 m 1 -m+1 } ≤ max{α n 1 -n+8 , 3 m 1 -m+2 }.
By substituting for Γ in the above inequality and dividing through by log 3, we get the inequality

0 < (n -1) log α log 3 -m + log C α log 3 < max α 8 (log 3)α n-n 1 , 9 (log 3)3 m-m 1 .
We apply Lemma 3 is such that q = q 88 > 6M . Furthermore, it yields ε > 0.0428119, and therefore either n -n 1 ≤ log ((α 8 / log 3)q/ε) log α < 193, or m -m 1 ≤ log ((9/ log 3)q/ε) log 3 < 105.

Thus, we have that either n -n 1 ≤ 193 or m -m 1 ≤ 105. Now we distinguish between the cases n -n 1 ≤ 193 and m -m 1 ≤ 105. First, we assume that n -n 1 ≤ 193. In this case we consider the inequality for Λ 1 , ( 14) and also assume that m -m 1 ≥ 20. We put

Γ 1 := (n 1 -1) log α -m log 3 + log C α (α n-n 1 -1) .
Then, inequality [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF] implies that

|Γ 1 | < 6 3 m-m 1 .
If we substitute for Γ 1 in the above inequality and divide through by log 3, we then get

0 < (n 1 -1) log α log 3 -m + log(C α (α n-n 1 -1)) log 3 < 6 (log 3)3 m-m 1 .
Again we apply Lemma 3 with the same τ as in the case of Γ. We use the 88-th convergent p/q = p 88 /q 88 of τ as before. But in this case we choose (A, B) := 6 log 3 , 3 and use 193]. For all values of ℓ, we get ε > 0.0000420218. Hence, by Lemma 3, we get m -m 1 < log ((6/ log 3)q/ε) log 3 < 110.

µ ℓ := log(C α (α ℓ -1)) log 3 , instead of µ for each possible value of ℓ := n -n 1 ∈ [1, 2, . . . ,
Thus, n -n 1 ≤ 193 implies that m -m 1 ≤ 110. Now let us turn to the case m -m 1 ≤ 105 and we consider the inequlity for Λ 2 , (17). We put

Γ 2 := (n -1) log α -m 1 log 3 + log C α 3 m-m 1 -1 ,
and we also assume that n -n 1 ≥ 20. We then have

|Γ 2 | < α 8 α n-n 1 .
If we substitute for Γ 2 in the above inequality and divide through by log 3, we then get 0 < (n -1) log α log 3 -m 1 + log(C α /(3 m-m 1 -1)) log 3 < α 8 (log 3)α n-n 1 .

We apply again Lemma 3 with the same τ, q, M, (A, B) := α 8 log 3 , α , and µ ℓ := log(C α /(3 ℓ -1)) log 3 for ℓ = 1, 2, . . . , 105.

We get ε > 0.00218297, therefore n -n 1 < log ((α 8 / log 3)q/ε) log α < 198.

To conclude, we first get that either n - Since n > 300, the inequality (20) implies that

|Γ 3 | < 2 α n-6 = α 8 α n .
Substituting for Γ 3 in the above inequality and dividing through by log 3, we get 0 < (n 1 -1) log α log 3 -m 1 + log(C α (α k -1)/(3 ℓ -1)) log 3 < α 8 (log 3)α n , where (k, ℓ) := (n-n 1 , m-m 1 ). We again apply Lemma 3 with the same τ, q, M, (A, B) := For the cases, we get ε > 0.0000115272, so we obtain n ≤ log ((α 8 / log 3)q/ε) log α < 207.

Hence, n ≤ 207. However, this contradicts our working assumption that n > 300. This completes the proof of Theorem 1.

  in positive integers u, v, and w with u ≤ M and w ≥ log(Aq/ε) log B .

1 ), γ 2 :

 12 = α, γ 3 := 3, b 1 := 1, b 2 := n 1 -1, and b 3 := -m.

γ 2 :

 2 = α, γ 3 := α, b 1 := 1, b 2 := n, and b 3 := -m 1 .

  3 = 0. Applying Theorem 2 gives log |Λ 3 | > -1.4 • 30 6 • 3 4.5 • 3 2 (1 + log 3)(1 + log n)(6.80 × 10 26 (1 + log n) 2 )(log α)(3 log 3), a comparison with (20) gives (n -6) < 3.70 × 10 39 (1 + log n) 3 , or n < 3.8 × 10 39 (1 + log n) 3 . (21) Now, by applying Lemma 4 on (21) with the data m := 3, Y := 3.8 × 10 39 , and x := n, leads to n < 3 × 10 46 .

n 1 ≤

 1 193 or m -m 1 ≤ 105. If n -n 1 ≤ 193, then m -m 1 ≤ 110, and if m -m 1 ≤ 105, then n -n 1 ≤ 198. Thus, we conclude that we always have n -n 1 ≤ 198 and m -m 1 ≤ 110.Finally, we go to the inequality of Λ 3 , (20). We putΓ 3 := (n 1 -1) log α -m 1 log 3 + log C α (α n-n 1 -1) 3 m-m 1 -1 .

α 8 log 3

 83 , α , and µ k,l := log(C α (α k -1)/(3 ℓ -1)) log 3 for 1 ≤ k ≤ 198 and 1 ≤ ℓ ≤ 110.

  with the following data Let τ = [a 0 ; a 1 , a 2 , . . .] = [0; 1, 1, 4, 13, 1, 6, 1, 4, 1, 10, 7, 1, 24, 3, 3, 2, 12, 4, 4, . . .] be the continued fraction expansion of τ . We choose M := 3 × 10 46 which is the upper bound on n. With the help of Mathematica, we find out that the convergent

	τ :=	log α log 3	, µ :=	log C α log 3	, and (A, B) :=	α 8 log 3	, α or	9 log 3	, 3 .
	p q	=	p 88 q 88	=	383979914200993729068715782793592146551951600940 692255294546383107303758900444711151890883197059

Acknowledgments

The author would like to thank the referee for the careful reading of the manuscript, the useful comments, and suggestions that greatly improved on the presentation of this paper. The author was supported by the Austrian Science Fund (FWF) grants: F5510-N26 -Part of the special research program (SFB), "Quasi-Monte Carlo Methods: Theory and Applications", P26114-N26 -"Diophantine Problems: Analytic, geometric and computational aspects", and W1230 -"Doctoral Program Discrete Mathematics".


Jeanne Deleforterie

Yael Kolasa

Levy Batista

Julie Hutin

Thierry Bastogne
email: thierry.bastogne@univ-lorraine.fr

Quality-by-Design for the safe development of medical devices containing nanomaterials. A study case in photodynamic therapy

est destinée au dépôt

(re-)assessment of risks. To that aim, the Quality-by-Design approach, as defined in ICH Q8-Q11 (QbD), is indisputably accepted and strongly recommended by the FDA and EMA for risk assessment during drug development. Some papers have emphasized the possible implementation of QbD in the medical device industry [1]. Nevertheless, to date no real and effective adaptation of this riskbased quality management approach has been adapted to biomedical devices manufacturing [2].

Objectives. Our goal is to develop both a new QbD paradigm and a web-based tool devoted to the safe development of class-III medical devices containing nanomaterials. This objective is pursued in the context of the European H2020 project TBMED (An Open Innovation test bed for the development of high-risk medical devices).

Methods.

A six-step QbD approach is proposed. The first four stages are devoted to the preclinical development while the next two steps concern the industrial implementation. Three categories of risk-assessment methods are used at different development steps: failure mode and effects analysis based on prior knowledge, statistical designs of experiments and Bayesian inference. To assess its applicability, we applied the integrated QbD approach to the development of a new medical device devoted to the realtime control of light during photodynamic therapy using nanoparticle-based photosensitizers.

Results. The new SaaS platform, entitled "Nanologic", is available at: (www.i-nano.eu). Four key documents for regulatory agencies are established during the preclinical study: the target product profile, the list of critical quality attributes (quality/safety descriptors), the list of critical material attributes and process parameters (risk factors associated with the design and production phases) and the design space: a key concept of risk assessment in QbD.

Conclusion.

We show how the QbD best practices can be adapted to the development of medical devices containing nanomaterials. Moreover, new questions still have to be investigated such as the solutions to be developed to better predict risks associated with the clinical proof of concept.

Your abstract may additionally contain one important figure and references. Use the space below if you want to include one figure.


Valentine Roulette

Guillaume Delplanque

Jeanne Deleforterie

G Delplanques

T Bastogne
email: thierry.bastogne@univ-lorraine.fr

A contribution in nanoinformatics

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

experimental data for regulatory purposes that demand sound scientific answers [1]. A major challenge is the establishment of common languages, standards and harmonised infrastructures with applicability to the needs of the different stakeholders. Nanoinformatics is the science and practice of determining which information is relevant to compare characterization of nanomaterials and to design optimized and safe nanodevices.

Objectives. Our goal is to bring a new contribution in nanoinformatics by proposing a text mining solution for the quasi-automatic collection of nanomaterials descriptors suited to the preparation of Quality-by-Design studies [2].

Methods. The first step relies on the construction of a non-structured database composed of selected scientific articles in PDF format. Secondly, the Quality-by-Design (ICH Q8-Q11) terminology is used to represent the main descriptors of nanomaterials with three main categories: (i) Critical Quality Attributes to describe the key physical, chemical and biological properties, (ii) Critical Material Attributes and (iii) Critical Process Parameters to characterize both design and manufacturing key variables. In a third step, a text mining algorithm, implemented in the Python language, is used to automatically detect CQA, CMA and CPP in articles and to build up a SQL database. A data curation step is then performed to complete and clean up the QbD database. The proposed approach was applied to a set of 30 scientific articles in Nanomedicine and performances in terms of sensitivity and specificity were finally assessed.

Results. In total, 1740 words were automatically analyzed, and we obtained an average response of 83,9% (1459/1740) correct identifications, decomposed as follows: 82,2% (235/286) of accuracy for the CQA descriptors, 92,9% (604/648) for CPP and 76,9% (620/806) for CMA.

Conclusion.

The proposed nanoinformatic solution has shown promising performances and allows to automate a very time-consuming task related to the collection and analysis of relevant scientific data for risk assessment in Quality-by-Design studies. Short-term perspectives will be focused to the automatic extraction of more complex descriptors.

Your abstract may additionally contain one important figure and references. Use the space below if you want to include one figure.

Fig. 1

 1 Fig. 1 Nanoinformatic process for the automatic analysis of scientific publications related to Qualityby-Design in Nanomedicine


Short-term effects of ocular 2% dorzolamide, 0.5% timolol or 0.005% latanoprost on the anterior segment architecture in healthy cats: a prospective study

Keywords: Cats, dorzolamide, latanoprost, timolol, UBM. Running

Dorzolamide 2%, timolol 0.5% and latanoprost 0.005% are widely used in the medical management of glaucoma in humans and pets. In this study, we wanted to evaluate the effect of these three molecules belonging to different hypotensive families on intraocular pressure (IOP) and anterior segment architecture in clinically healthy cats and compare these data with those obtained in a previous study under the same conditions in healthy dogs. In this prospective study 45 cats were included and divided into 3 homogeneous groups. For 7 days eye drops were instilled in the right eye (treated eye), dorzolamide 2% BID for the first group, timolol 0.5% BID for the second group and latanoprost 0.005% SID for the third group. The left (untreated) eye of all these cats constitute a control group useful for the statistical analysis of the data. On D0 and D7, the IOP was measured with Tonovet® and then, a high frequency ultrasound (ultrasound biomicroscopy UBM) was undertaken under general anesthesia. A biostatistical study of the 1440 data was then performed. Statistical test validated the use of the controlateral eye as a control, and the representativeness of a sample of 15 animals per group. At D7, the IOP of healthy cats treated with dorzolamide 2% BID decreased significantly whereas no variation was obtained with instillation of timolol 0.5% and latanoprost 0.005%. According to our protocol, these three hypotensive molecules do not significantly affect the architecture of the anterior segment of the clinically healthy feline eye as in dogs.

Introduction

A common feature of all glaucoma in animals is the neurodegenerative disorder of retinal ganglion cells (RGCs) and their axons, an abnormal increase in intraocular pressure (IOP) represents a constant risk factor [START_REF] Stiles | Feline ophthalmology -Glaucoma[END_REF]. The IOP is maintained at a relatively constant level and the rate of aqueous humor (AH) formation equals the outflow. Increase in IOP results from impairment of the aqueous humor outflow through the trabecular conventional pathway and/or through the uveoscleral pathway. It is estimated that glaucoma can affect about 1% of cats over the age of seven [Kroll et al., 2001] and most cases of feline glaucoma are secondary [START_REF] Stiles | Feline ophthalmology -Glaucoma[END_REF].

Current therapy for feline glaucoma follows the same guidelines used in other species. Medical therapeutic option includes many topical hypotensive agents used in humans and dogs. Among hypotensive drugs, 3 families are commonly used: carbonic anhydrase inhibitors (such as dorzolamide), beta-blockers (such as timolol) and prostaglandin analogs (such as latanoprost). Dorzolamide reduces the active formation of HA by inhibiting the action of carbonic anhydrase, which is found abundantly in the ciliary body epithelial cells. Timolol as a beta-blocker, lowers aqueous humor flow by altering the adrenergic neuronal control of aqueous humor formation by blockade of the beta-receptors in the ciliary body processes, it may also inhibit active transport and ultrafiltration related to sodium transfer system. Latanoprost reduces IOP by increasing the uveoscleral outflow. The remodeling of the extracellular matrix between the ciliary muscle fibers by local metalloproteases contributes to this pharmacological effect [START_REF] Plummer | The canine glaucomas -Topical and systemic medications for treatment of the glaucoma[END_REF].

In a recent study by [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF], these three hypotensive drugs lowered significantly IOP in dogs without modification of the biometry of the iridocorneal angle (ICA). This result is consistent with the hypotensive effect of dorzolamide and timolol on ciliary body epithelium but it is more surprising with latanoprost which increases the uveoscleral outflow of aqueous humor, all the more so as a previous study shows that the biometry of the anterior segment and the ciliary cleft is modified two hours after the application of a drop of latanoprost 0,005% in dogs [Gelatt and Mackay, 2001].

The goal of this study was to compare the effect of these three hypotensive molecules on IOP and biometry of the anterior segment of healthy cats, as well as to compare these data with those obtained by [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF] in healthy dogs with the same protocol under the same conditions. To evaluate the biometry of the anterior segment, we used high-frequency immersion ultrasound with high definition (20-50µm) for structures up to 5mm deep [Leibmann, 1998], while being non-invasive, and for which most of the parameters of the anterior segment have already been objectified in healthy cats [Aubin, 2003].

Materials and Methods

Inclusion criteria

This prospective study with a control group was performed using 45 healthy cats without any current local or systemic treatment. These 45 cats belonged to staff or customers of the Lorrainevet veterinary clinic. All the dogs were included with the consent of their owners. A complete general and ophthalmic examination including slit-lamp biomicroscopy and indirect ophthalmoscopy without pupillary dilatation was performed, followed by an evaluation of IOP using a rebound tonometer (Tonovetâ, Icare, Vantaa, Finland). All animals with an ophthalmologic history or a difference greater than 2 mmHg IOP (on an average of three IOP measurements) on both eyes were excluded from the study.

Experimental protocol

Commenté [MOU1]: cats

Cats were divided into 3 homogeneous groups (weight, age, sex, race) of 15 cats (Table I). The study was done in two steps, spaced 7 days apart. The first step (D0) was performed in a mesopic environment, a complete general and ophthalmic examination, without dilation and without any instillation, was carried out between 10 am and 1 pm for each animal. IOP was an average of three measures on each eye. Ultrasound biomicroscopy (UBM) examination (Avisoâ, Quantel Medical, Cournon d'Auvergne, France) was then performed under isoflurane gas anesthesia after medetomidine premedication and ketamine induction. At the awakening of each animal, eye drops were instilled into the right eye. For group 1, one drop of dorzolamide 2% (Trusopt 20mg / ml eye drops ND, MSD, Courbevoie, France) was instilled twice daily, at 7am and 7pm, from D0 at 7pm until D7 at 7am. For group 2, one drop of 0.5% timolol (Timoptol 0.5% eye drops, MSD, Courbevoie, France) was instilled twice daily according to the same protocol as group 1. For group 3, one drop of latanoprost 0.005% (Xalatan 0.005% eye drops ND, Pfizer, Paris, France) was instilled once a day at 10pm in the fornix of the right eye for seven days, from D0 at 10 pm until D6 at 10 pm included.

In the second step, on day 7, all animals were re-examined 12 hours after the last drop of latanoprost and 6 hours after the last drop of dorzolamide or timolol, with the same procedure under the same conditions as on D0. All examinations and UBM ultrasound were performed by the same operator on D0 and D7.

UBM high frequency ocular ultrasound

The ultrasound biomicroscopy for this study was a 50Mhz monotransducer with immersion probe with a geometric focalization and line scanning (UBM Avisoâ, Quantel Medical, Cournon d'Auvergne, France). The patient was positioned in dorsal recumbency with its head stabilized in a vacuum pillow. Two video sequences were recorded for each eye at D0 and D7.

The first sequence was recorded by placing the probe both centrally and perpendicular to the corneal surface to determine the distance between the apex of the corneal endothelium and the apex of the anterior capsule of the lens, at the 12 o'clock position. In the second sequence, the outflow pathways were evaluated in the dorsal quadrant with the probe placed perpendicular to the corneoscleral limbus at the 12 o'clock position. From each video sequence, one image was chosen for analysis (Figure 1).

The anterior segment parameters

Nine parameters were evaluated (Figure 2): (a) the ICA , (b) the width of the entrance of the ciliary cleft (CC), (c) the width of the mid-CC, (d) the length of CC, (e) the depth of the anterior chamber (AC), (f) the thickness of cornea at the corneoscleral limbus note, (g) the distance between Schwalbe's line (the borderline between the cornea and sclera) and the anterior lens capsule according to Kawata and Hasegawa., 2013 and (h) the area of the ciliary cleft. These was the distance between the inner sclera and the ciliary process in the central portion of the CC. The anterior chamber depth (e) was determined as the distance between the corneal endothelium in the apical region of the cornea and the anterior pole of the lens. The thickness of the cornea at the limbus (f) was the measure between the epithelium of the cornea and the endothelium in its most peripheral transparent part. The distance between the anterior capsule of the lens and the Schwalbe's line (g) was described by Kawata and Hasegawa., 2013. The area of the ciliary cleft (h) was calculated by the area formula (CC) = (d / 2 x (b + c) / 2) + ((cxd) / 4). All these measurements were performed by the UBM software caliper.

Statistical evaluation

For this study, 45 cats were divided into 3 homogeneous groups (weight, racial type, sex, age) of 15 individuals. Sample size n = 15 was determined using Student's t-test. Only five breeds of cats with a majority of European cats was represented in the three groups, the distribution was made in such a way as to obtain ages between 1 year and 10 years, an average weight of 4.1 kg and a male / female ratio close to 50% (Table I). Each animal was submitted to only one experiment (one type of eye drops) and two general anesthesia 7 days apart. The 1440 data obtained, including measurements of IOP and anterior segment biometry on the left (untreated) and right (treated) eyes, on D0 and D7 were integrated into a statistical model, in which the experimental unit was a cat, and the variable factor was the type of eyedrops instilled for 7 days.

The objective of the statistical study was to determine the effects of these three molecules on anterior segment biometry and IOP. For this, two statistical methods were used: the nonparametric Wilcoxon test to evaluate the significant or non-significant differences between two independent groups and a one-way analysis of variance (ANOVA) to compare the sample means. The level of significance was set at p<.05 for all statistical analyses.

Results

The IOP and biometric measurements of the anterior segment of the untreated eye (left eye) did not significantly differ between D0 and D7. And no significant difference in the biometric data of the anterior segment was found on day 0 between the untreated eye and the treated eye. After one week of instillation of dorzolamide 2%, the IOP of the treated eye was significantly 1mmHg lower (Figure 3). This decrease was relatively moderate (5.3%). After one week of instillation of latanosprost or timolol, the IOP of the treated eye did not significantly decrease (Figure 3). About biometric data, ICA values (a), the width of the entrance of the CC (b), the width of the mid-CC (c), the length of the CC (d), the anterior chamber depth (e), the distance between the anterior capsule of the lens and the Schwalbe's line (g) and the area of the ciliary cleft (h) were not significantly altered after 7 days of instillation whatever the eye drops. Only a significant decrease of 1.4% in corneal thickness (f) was observed with the instillation of dorzolamide (Figure 4), whereas there was a significant increase of 1.6% in thickness of the cornea (f) with the instillation of timolol (Figure 4).

Discussion

In this study, the parameters measured on the untreated eye (IOP and biometric data) did not vary between D0 and D7. It was therefore admitted that eye drops instilled on the treated eye (OD) for 7 days had no significant effect on the parameters measured on the untreated eye (OS) and this despite a possible systemic diffusion of eye drops. In addition, the absence of a significant difference in the data measured between the treated (OD) and untreated (OS) eye on D0 allowed to validate the use of the untreated eye (OS) as a control at D0 and D7. The study could also have been performed by comparing the data of the treated eye (OD) to D0 and D7.

Both analyses were performed with the same statistical tests and we observed the same variations.

In contrast with healthy dogs for which these three types of hypotensive agents cause a significant decrease in IOP [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF], only instillation of dorzolamide at 2% twice daily for 7 days resulted in a moderate but significant decrease in IOP in healthy cats in this study. This result is similar to previous studies evaluating the effects of 2% dorzolamide instilled in the eyes of healthy cats [START_REF] Rainbow | Effects of twice daily application of 2% dorzolamide on intraocular pressure in normal cats[END_REF], [Dietrich et al., 2007] and [START_REF] Rankin | Effects of ocular administration of ophthalmic 2% dorzolamide hydrochloride solution on aqueous humor flow rate and intraocular pressure in clinically normal cats[END_REF]. This moderate decrease in IOP of 1 mmHg (5.5%) is close to the decrease (1.6 mmHg) in the Dietrich study (2007) when 2% dorzolamide is instilled twice daily.

Increasing the frequency of instillation of dorzolamide 2% to 3 times daily does not appear to significantly increase the hypotensive effect of dorzolamide in healthy cats [Dietrich et al., 2007]. This modest effect of topical carbonic anhydrase inhibitors on the feline eye is also viewed in an earlier study of 20 healthy cats, where IOP did not significantly decrease after instillation of 1% brinzolamide for 7 days, twice a day [Gray and al., 2003]. This moderate response in healthy cats does not question the interest of dorzolamide in the medical management of glaucoma in cats, since the Sigle study in 2011 [START_REF] Sigle | The effect of dorzolamide 2% on circadian intraocular pressure in cats with primary congenital glaucoma[END_REF]] reveals an interesting decrease in IOP of 7 cats with primary congenital glaucoma with dorzolamide instilled 3 times a day.

The instillation twice daily of timolol for 7 days on the eyes of healthy cats did not lead to a significant reduction in IOP between day 0 and day 7 in this study. This result is similar to that obtained in the Kiland study [START_REF] Mcdonald | Effect of topical latanoprost 0,005% on intraocular pressure and pupil diameter in normal and glaucomatous cats[END_REF] Finally, instillation of latanoprost once daily for 7 days in healthy cats' eyes did not decrease the IOP between day 0 and day 7 in this study, which is consistent with previous studies on the effects of latanoprost on the IOP of healthy cats [Mc Donald and al., 2016], [START_REF] Studer | Effects of 0,005% latanoprost solution on intraocular pressure in healthy dogs and cats[END_REF]. This lack of efficacy of PGF2 alpha receptor analogs is attributed to species differences in prostaglandin receptors in the ciliary bodies of the cat's eye that express EP and DP receptors rather than FP receptors, found in particular in humans [Regnier and al., 2006].

The absence of significant modification of the anterior segment biometry in this study during instillation of timolol and latanoprost in healthy cats confirms the lack of response of these molecules to IOP after 7 days of treatment.

Regarding dorzolamide 2%, the absence of significant modification on the biometry of the anterior segment confirms that dorzolamide acts on the IOP by lowering the production of HA without interacting on the outflow pathway of HA. The lack of effect on AIC biometrics of these three molecules in healthy cats is similar to the results obtained in healthy dogs using the same protocol and the same operator [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF]. A decrease of 1.4% in the thickness of the peripheral cornea in cats treated with dorzolamide was found, this decrease similar to treated dogs [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF] can be related to a transient dehydration of the cornea. Conversely, a 1.6% increase in the thickness of the peripheral cornea in cats treated with timolol was noted: this can be related to a transient edema of the cornea.

Differences in IOP results with some studies during instillation of timolol may be related to the type of tonometer used. Nevertheless, Tonovet® provides accurate and reproducible IOP measurements in cats [START_REF] Mclellan | Validation of the TonoVet rebound tonometer in normal and glaucomatous cats[END_REF]. Although all the qualitative measurements on the anterior chamber angle and sclerociliary cleft are perfectly described by Dulaurent et al., 2012, the precision of the UBM measurements remains a variable dependent on the experience of the operator. It is important to remember that the comparison of the data between healthy dogs and healthy cats is reinforced by the fact that these two studies were carried out by the same operator and under the same conditions. This study has the same limitations as the study by [START_REF] Poinsard | Short-term effects of ocular 2% dorzolamide, 0,5% timolol or 0,005% latanoprost on the anterior segment architecture in healthy cats: a prospective study[END_REF], limitations related to the recruitment mode (private structure) causing a double selection bias, especially for breeds that may have specificities, and the number of cats presented. Even if these three molecules showed little or no influence on IOP and anterior segment biometry in healthy cats, it would be interesting to carry out the same study in a context of glaucomatous cats.

Conclusion

This prospective study revealed a significant decrease in IOP in healthy cats treated with dorzolamide at 2% BID for 7 days. Timolol 1% BID and latanoprost SID have no effect on IOP after 7 days of treatment in healthy cats. Finally, dorzolamide 2%, timolol 0.5% and latanoprost 0.005% have no effect on the biometry of AIC in healthy cats with the protocol used and the duration studied. -Leibmann, J.M., Ritch, R. and Esaki, K. 1998. Ultrasound biomicroscopy. Ophtalmol Clin Table III: Mean value of the corneal thickness (f) (mm) after one week of anti-glaucoma treatment on the treated eye.

  parameters were those described byDulaurent et al., 2012 and used by Poinsard et al., 2018. The ICA (a) was the angle formed by the base of the iris in the region of the CC entry and the inner corneoscleral junction. The length of the CC noted (d) corresponded to the distance between the pectinate ligament (or the most anterior visible portion of the uveal trabecular meshwork) and the anterior part of the ciliary body. The width of the CC entry, marked (b) was the distance between the corneoscleral limbus and the iris root. The width of the mid-CC (c)

  and contrary to the older studies of Wilkie and Latimer in 1991 and Colasanti and Trotter in 1981. In the Dietrich study (2007), concomitant administration of dorzolamide and timolol did not have a more intense hypotensive effect than dorzolamide alone.

  M.L., Powell, C.C., Gionfriddo, J.R. and Fails, A.D. 2003. Ultrasound biomicroscopy of the feline anterior segment. Vet. Ophthalmo. 6, 15-17. -Colasanti, B.K. and Trotter, R.R. 1981. Effects of selective beta 1-and beta 2-adrenoreceptor agonists and antagonist on intraocular pressure in the cat Invest Ophthalmol Vis Sci. 20, 69-76. -Dietrich, U.M., Chandler, M.J., Cooper, T., Vidyashankar, A. and Chen, G. 2007. Effect of topical 2% dorzolamide hydrochloride alone and in combination with 0,5% timolol maleate on intraocular pressure in normal feline eyes. Vet Ophthalmo. 10, 95-100. -Dulaurent, T., Goulle, F., Dulaurent, A., Mentek, M., Peiffer, R.L. and Isard PF. 2012. Effect of mydriasis induced by topical instillations of 0.5% tropicamide on the anterior segment in normotensive dogs using ultrasound biomicroscopy. Vet. Ophthalmol. 15, 8-13. -Gelatt, K.N. and MacKay, E.O. 2001. Effects of different dose schedules of latanoprost on intraocular pressure and pupil size in the glaucomatous Beagle. Vet. Ophthalmo. 4, 283-288. -Gray, H.E., Willis, A.M. and Morgan, R.V. 2003. Effects of topical administration of 1% brinzolamide on normal cat eyes. Vet. Ophthalmo. 6, 285-290.-Kawata, M. and Hasegawa, T. 2013. Evaluation of the distance between Schwalbe's line and the anterior lens capsule as a parameter for the correction of ultrasound biomicroscopic values of the canine iridocorneal angle. Vet.Ophthalmo. 16, 169-174. -Kiland, J.A., Voss, A.M., and McLellan, G.J. 2016. Effect of timolol maleat gel-forming solution on intraocular pressure, pupil diameter, and heart rate in normal and glaucomatous cats. Vet Ophthalmo. 19 (Suppl 1), 91-96.-Kroll, M.N., Miller, P.E. and Rodan, I. 2001. Intraocular pressure measurements obtained as part of a comprehensive geriatric health examination from cats seven years of age or older. J Am Vet MedAssoc. 219, 1406Assoc. 219, -1410. .

  Mean value of the width of the entrance of the ciliary cleft (b) (mm) after one week of anti-glaucoma treatment on the treated eye.

Acknowledgments:

The authors are grateful to Dr. Isabelle Raymond for the iconography and to Pr Thierry Bastogne for the statistical analysis.

Conflict of interest:

The Author(s) declare(s) that there is no conflict of interest. bimatoprost solution in normotensive feline eyes. Vet Ophthalmo. 9, 39-43.

- Wilkie, D.A. and Latimer, C.A. 1991. Effect of topical administration of timolol maleat on intraocular pressure and pupil size in cats. Am J Vet Res. 52, 436-440.

Iconography

Figure 1: Utrasound biomicroscopy (UBM) of the iridocorneal angle and ciliary cleft in a normal cat. Figure 2: Histological aspect of the anterior segment of a dog's eye showing the main biometric parameters studied in cats. Table I: Summary of race, age, gender and weight data for cats in the cohort. Figure 3: Box chart of IOP variations after one week of anti-glaucoma therapy on the treated eye (DOR: dorzolamide 2%, TI: timolol 0.5%, LAN: latanoprost 0.005%). Figure 4: Box chart of corneal thickness variations (f) after one week of anti-glaucoma treatment on the treated eye. Table II: Mean value of the IOP (mmHg) after one week of anti-glaucoma treatment on the treated eye. Table III: Mean value of the corneal thickness (f) (mm) after one week of anti-glaucoma treatment on the treated eye. Table IV: Mean value of the width of the entrance of the ciliary cleft (b) (mm) after one week of anti-glaucoma treatment on the treated eye. Table I


S Hertz

E Lecorné

O Szmygel

P Faucher

Béjot

F Billard

A Dubrouil

E Hertz

S Lecorné

E Szmygel

O Faucher

P Béjot

BOAR: Biprism based Optical Autocorrelation with Retrieval

Keywords:

  L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The apparition of mode-locked lasers in the mid-60s raised the problem of ultrashort laser pulse characterization. Since the characteristic time scale of electronic was overwhelmed, optical techniques had to be developed. Ultrafast lasers kept evolving and the need for characterization increased with the performances of the lasers. A breakthrough was made when the Chirp Pulse Amplification lasers appeared and over the past three decades, ultrashort laser pulse characterization has been a major concern for laser physicists. Substantial effort are still made to develop and improve multitudinous techniques of characterization [START_REF] Canhota | Inline self-diffraction dispersion-scan of over octave-spanning pulses in the single-cycle regime[END_REF][2][START_REF] Shen | Self-Referenced Spectral Interferometry for Femtosecond Pulse Characterization[END_REF] . Indeed, the optical pulses characterization is a rather complex problem that is highly dependent on the lasers characteristics. There are many different approaches and each of them having advantages and drawbacks. The choice of a techniques has to be made in consideration of the laser experimental conditions. As the research in the field has been very active, there is numerous different techniques of characterization and some of them are very atypical [START_REF] Prade | A simple method for the determination of the intensity and phase of ultrashort optical pulses[END_REF][START_REF] Prein | Complete characterization of femtosecond pulses using an all-electronic detector[END_REF][START_REF] Lai | Complete diagnostic of ultrashort pulses without nonlinear process[END_REF][START_REF] Diels | Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy[END_REF][START_REF] Ripoche | Determination of the duration of UV femtosecond pulses[END_REF][START_REF] Chilla | Frequency domain phase measurement of ultrashort light pulses. Effect of noise[END_REF][START_REF] Coello | Interference without an interferometer: a different approach to measuring, compressing, and shaping ultrashort laser pulses[END_REF][START_REF] Zeylikovich | Interferometric 2D imaging amplitude correlator for ultrashort pulses[END_REF][START_REF] Stibenz | Interferometric frequencyresolved optical gating[END_REF][START_REF] Langlois | Measurement of pulse asymmetry by three-photon-absorption autocorrelation in a GaAsP photodiode[END_REF][START_REF] Bender | Modified spectrum autointerferometric correlation (MOSAIC) for single-shot pulse characterization[END_REF][START_REF] Nicholson | Noise sensitivity and accuracy of femtosecond pulse retrieval by phase and intensity from correlation and spectrum only (PICASO)[END_REF][START_REF] Rhee | Realtime dispersion analyzer of femtosecond laser pulses with use of a spectrally and temporally resolved upconversion technique[END_REF][START_REF] Nibbering | Spectral determination of the amplitude and the phase of intense ultrashort optical pulses[END_REF][START_REF] Sharma | The effect of errors and detector noise on sensitive detection of chirp and pulse asymmetry of ultrashort laser pulses from interferometric autocorrelation signals[END_REF][START_REF] Bender | Ultrashort laser pulse characterization using modified spectrum autointerferometric correlation (MOSAIC)[END_REF][START_REF] Yellampalle | Unambiguous chirp characterization using modified-spectrum autointerferometric correlation and pulse spectrum[END_REF][START_REF] Nicholson | Unbalanced third-order correlations for full characterization of femtosecond pulses[END_REF] . We present here, a new technique for ultrashort pulses characterization based on single shot interferometric autocorrelation and two photon absorption. The devices and technique are so called, Biprism based Optical Autocorrelation with Retrieval (BOAR). It is a simple and compact single shot autocorrelator composed by only two elements: a Fresnel biprism and a camera. The latter is used to detect incident photon with energy lower than its absorption band gap, the only way to produce a signal is therefore to absorb two photons. This two-photon signal is a spatial interferogram that can be used to retrieve the pulse duration, the frequency chirp and the pulse spectrum. There is no non-linear crystal and no phase matching issues, the spectral working range is therefore very broad a) Electronic mail: pierre.bejot@u-bourgogne.fr (1200-2400 nm with a silicon based camera and 1800-3400 nm with an InGaAs camera). For the sake of brevity, we compare here our technique with the most common ones (Autocorrelation [START_REF] Langlois | Measurement of pulse asymmetry by three-photon-absorption autocorrelation in a GaAsP photodiode[END_REF][START_REF] Karkhanehchi | Autocorrelation measurements of modelocked Nd:YLF laser pulses using two-photon absorption waveguide autocorrelator[END_REF][START_REF] Spielmann | Measurement of interferometric autocorrelations: comment[END_REF][START_REF] Choi | Novel crosscorrelation technique for characterization of subpicosecond pulses from modelocked semiconductor lasers[END_REF] , FROGs [START_REF] Stibenz | Interferometric frequencyresolved optical gating[END_REF][START_REF] Kane | Characterization of arbitrary femtosecond pulses using frequencyresolved optical gating[END_REF][START_REF] Delong | Frequency-resolved optical gating with the use of secondharmonic generation[END_REF][START_REF] Delong | Improved ultrashort pulseretrieval algorithm for frequencyresolved optical gating[END_REF][START_REF] Paye | Measurement of the amplitude and phase of ultrashort light pulses from spectrally resolved autocorrelation[END_REF][START_REF] Kane | Singleshot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating[END_REF][START_REF] Kane | Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating[END_REF][START_REF] Peatross | Temporal decorrelation of short laser pulses[END_REF][START_REF] Amat-Roldan | Ultrashort pulse characterisation with SHG collinear-FROG[END_REF][START_REF] Bates | Ultrashort pulse characterization in the mid-infrared[END_REF][START_REF] Trebino | Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating[END_REF] and Spectral Interferometry [START_REF] Iaconis | Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses[END_REF][START_REF] Wong | Analysis of ultrashort pulse-shape measurement using linear interferometers[END_REF][START_REF] Wong | Linear filter analysis of methods for ultrashort-pulse-shape measurements[END_REF][START_REF] Lepetit | Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy[END_REF][START_REF] Ventalon | Time-domain interferometry for direct electric field reconstruction of mid-infrared femtosecond pulses[END_REF][START_REF] Fan | X-SEA-F-SPIDER characterization of over octave spanning pulses in the infrared range[END_REF][START_REF] Tokunaga | Induced phase modulation of chirped continuum pulses studied with a femtosecond frequency-domain interferometer[END_REF][START_REF] Geindre | Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma[END_REF][START_REF] Kovacs | Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry[END_REF][START_REF] Reynaud | Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses[END_REF] ), and more atypical ones that are related to our technique [START_REF] Sharma | The effect of errors and detector noise on sensitive detection of chirp and pulse asymmetry of ultrashort laser pulses from interferometric autocorrelation signals[END_REF][START_REF] Nicholson | Unbalanced third-order correlations for full characterization of femtosecond pulses[END_REF][START_REF] Peatross | Temporal decorrelation of short laser pulses[END_REF][START_REF] Kleimeier | Autocorrelation and phase retrieval in the UV using two-photon absorption in diamond pin photodiodes[END_REF][START_REF] Danielius | Characterization of phase modulated ultrashort pulses using single-shot autocorrelator[END_REF][START_REF] Goodberlet | Chirp-compensated autocorrelation of optical pulses[END_REF][START_REF] Delong | Chirped light and its characterization using the cross-correlation technique[END_REF][START_REF] Nicholson | Full-field characterization of femtosecond pulses by spectrum and cross-correlation measurements[END_REF][START_REF] Gebert | Michelson-type all-reflective interferometric autocorrelation in the VUV regime[END_REF][START_REF] Chen | Pulse retrieval from interferometric autocorrelation measurement by use of the population-split genetic algorithm[END_REF][START_REF] Yang | Spectral phase retrieval from interferometric autocorrelation by a combination of graduated optimization and genetic algorithms[END_REF] . Although autocorrelation needs an assumption on the temporal profile and it does not provide the exact pulse duration, it is the most simple and most robust technique, and therefore it is still widely used. Our technique is even more simple and more robust than an usual autocorrelator, although it allows to estimate the spectral phase and amplitude just like a FROG or a SPIDER. Spectrographic techniques like FROGs have a large working range and are suitable for complex and very chirped pulses 2, [START_REF] Kane | Singleshot measurement of the intensity and phase of a femtosecond UV laser pulse with frequency-resolved optical gating[END_REF][START_REF] Kane | Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating[END_REF] . They also have the advantage to be intuitive since an experienced user is able to estimate the chirp and its order just by looking at the frog trace. The disadvantage of FROGs is the heavy iterative algorithm that is needed to retrieve the phase. However it is still possible to retrieve the phase in real time [START_REF] Kane | Real-time measurement of ultrashort laser pulses using principal component generalized projections[END_REF] and some more powerful retrieval methods have recently appeared 2, [START_REF] Sidorenko | Ptychographic reconstruction algorithm for frequency-resolved optical gating: super-resolution and supreme robustness[END_REF][START_REF]On the Uniqueness of FROG Methods[END_REF] . Spectral phase interferometry based technique like SPIDER [START_REF] Iaconis | Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses[END_REF] or SRSI [START_REF] Oksenhendler | Self-referenced spectral interferometry[END_REF] have the advantage to retrieve the phase by direct Fourier Transform calculation. However, the working range is very narrow and measurements are possible only when the pulse duration are not too far from the Fourier limit. The BOAR is actually combining all advantages: as an autocorrelator, it is therefore extremely robust, it is suitable for rather chirped pulses and the retrieval is done directly by Fourier transformation of the interferogram in the time domain (i.e., an interferogram with encoded time). However the retrieval is direct and rapid if a Gaussian spectral profile is assumed, otherwise the calculations becomes heavier. Also BOAR relies on a single shot device and therefore it makes an important difference on the accessible information compared to multishot devices. The latter gives only an average measurement in time (on all the pulses required to obtain a measurement) and an average measurement in space [START_REF] Yasa | A rapid-scanning autocorrelation scheme for continuous monitoring of picosecond laser pulses[END_REF] (the beam is focused in multishot devices). On the contrary, single shot devices allow to access to pulse to pulse fluctuations and in this case the measurement is also spatially resolved [START_REF] Chuzavkov | Single-shot autocorrelator[END_REF][START_REF] Brun | Single-shot characterization of ultrashort light pulses[END_REF][START_REF] Liesfeld | Single-shot autocorrelation at relativistic intensity[END_REF] . It is therefore a pertinent information to evaluate the quality and stability of a laser system. Furthermore, it is known that the proper way to characterize stability performances is on a single shot basis [START_REF] Zhang | Single-shot, real-time carrier-envelope phase measurement and tagging based on stereographic abovethreshold ionization at short-wave infrared wavelengths[END_REF][START_REF] Manzoni | Single-shot detection and direct control of carrier phase drift of midinfrared pulses[END_REF][START_REF] Dubrouil | spectral structures in high-order harmonic beams generated with Terawatt 10-fs pulses[END_REF][START_REF] Feng | Complete analog control of the carrier-envelopephase of a high-power laser amplifier[END_REF] . In the following, we present the conception of the BOAR and the associated theory, experimental measurements, and numerical simulations. The latter are compared with the experimental data and then are used to determine the working range of the BOAR, the validity and accuracy of the results according to the experimental conditions.

II. BASIC PRINCIPLES OF THE BOAR

The Biprism based Optical Autocorrelator with Retrieval (BOAR) relies on two basic principles: the creation of two identical temporally delayed sub-pulses from the pulse to be measured and a nonlinear optical effect. These two steps are respectively performed with a Fresnel biprism and a camera in which two-photon absorption occurs. The combination of these two elements makes the BOAR extremely compact, calibration-free and robust in alignment.

A. Geometrical considerations

Figure 1(a) illustrates the basic principle of the autocorrelator. The creation of the two replica is performed by means of a Fresnel biprism having an Apex angle A, a refractive index n, a thickness e 0 , and a height H (it will be supposed hereafter that the laser beam is larger than the biprism, i.e., that the biprism is the pupil of the optical system). The propagation of the laser pulse in the biprism leads to the creation of two identical beams crossing with an angle 2α in a plane parallel to the biprism input face, where α = asin [nsinβ] -β and β = π/2 -A/2. The optical path difference of the two beams imposes a temporal delay given in first approximation by ∆τ = 2xsinα c , where x is the distance relative to the propagation axis z. The temporal information is then directly transposed in the spatial domain, then allowing single-shot pulse duration measurements, provided that the integration times of the camera is shorter than the repetition period of the laser to be measured. The optimal distance d opt at which the two beams perfectly spatially overlap is given by:

d opt = H 4 ( 1 tanα -tanβ
) . At this optimal distance, the temporal window ∆τ max is given by: ∆τ max = H c sinα (1 -tanα tanβ). Note that these formula are obtained assuming that the laser pulse propagates at phase velocity in both the biprism and air, which leads to slightly underestimate the pulse duration. This can be corrected by applying a proportionality factor κ = 1 + ω0cosα ncosα-1 ∂n ∂ω to the retrieved pulse duration that takes into account the fact that the pulse envelope propagates at the group velocity.

B. Two-photon absorption 1. General formulas

The optical autocorrelation is performed by twophoton absorption in a camera having a pixel size dx and a number of pixels N along x placed at the distance d from the biprism output. The bandgap U g of the semiconductor material composing the camera pixels is chosen in the interval hc λ < U g < 2 hc λ , where h is the Planck constant, c is the light celerity, and λ is the central wavelength to be measured. For instance, a silicon (resp. InGaAs) camera (U g = 1.11 eV) can be used for measuring pulse duration in the [1200-2400 nm] (resp. [1800-3400 nm]) spectral range.

Considering the total complex electric field ε(t, τ ) = ε 1 (t) + ε 2 (t -τ ) hitting a pixel of the camera, where ε 1 (t) = A 1 (t)e iω0t (resp. ε 2 (t) = A 2 (t)e iω0t ) is the field at frequency ω 0 = 2πc/λ coming from the upward (resp. downward) face of the biprism, the photocurrent I phot induced by two-photon absorption in one pixel of the camera will is:

I phot (τ ) ∝ ∫ |ε 2 (t, τ )| 2 dt. ( 1
)
Assuming a perfect balance between the two replicas amplitudes (A 1 = A 2 = A), the photocurrent is given by:

I phot (τ ) ∝ G 2 (τ )+F 1 (τ )cos(ω 0 τ )+F 2 (τ )cos(2ω 0 τ ), (2)
where

G 2 = 2 ∫ I 2 (t)dt + 4 ∫ I(t)I(t -τ )dt, F 1 = 4 ∫ [I(t) + I(t -τ )] Re [A(t)A * (t -τ )] dt, F 2 = 2 ∫ Re ( A 2 (t)A * 2 (t -τ ) ) dt, (3)
and where I = |A| 2 is the field intensity. The contributions G 2 , F 1 , and F 2 oscillate at very different frequencies (ω=0, ω 0 , and 2 ω 0 , respectively). Therefore, their relative contributions to the two-photon signal I phot can be easily distinguished by an appropriate frequency filtering. One has to emphasize that the two-photon absorption signal then can be used to determine the central frequency of the laser to be measured provided that the optical characteristics of the biprism (Apex angle and refractive index) are well determined. Note that, since a part of the autocorrelation evolves with a frequency around 2ω 0 , this imposes an upper limit on the spatial resolution of the optical system. Indeed, for resolving F 2 , the camera pixel size has, at least, to be smaller than four times the interference period, i.e., dx ≤ λ 0 /(8sinα).

(4)

Autocorrelation measurement assuming chirped gaussian pulses

As every autocorrelator, an assumption about the temporal shape of the pulse has to be done so as to retrieve the pulse duration from the measurement. One of the widely used assumption is to consider a gaussian profile, possibly with a quadratic spectral phase. More particularly, if one imposes a quadratic spectral phase Φ(ω) = Kω 2 on a Gaussian electric field with an envelope A(t) = e -t 2 σ 2 t (and its associated spectral amplitude

A(ω) ∝ e -σ 2 t w 2 4
), the electric field complex envelope becomes:

A(t) = e -t 2 σ 2 tc e -iat 2 , ( 5
)
with

σ tc = √ σ 4 t +16K 2 σt and a = -4 K σ 4 t +16K 2 .
According to these definitions, σ t (resp. σ tc corresponds to the Fourier-Transformed limited (resp. effective) pulse duration, and a is the temporal phase. In this case, G 2 , F 1 , and F 2 reads respectively:

G 2 (τ ) = 1 + 2e -τ 2 σ 2 tc , F 1 (τ ) = 4 cos aτ 2 2 e -τ 2 2 ( 1 σ 2 tc + 1 2σ 2 t ) , F 2 (τ ) = e -τ 2 σ 2 t . (6)
A few remarks can then be done regarding the above expressions. First, the low-frequency part of the autocorrelation (G 2 ) gives access to the chirped pulse duration. However, a chirped pulse and a Fourier-Transform limited pulse with same duration produce the same signal for G 2 . Note that G 2 embeds an offset induced by the two-photon absorption signal produced by the two replicas taken individually. As discussed below, the presence of this offset can be detrimental for an accurate fit of the pulse duration. Secondly, the contribution oscillating around 2 ω 0 (F 2 ) is insensitive to the chirp and gives access to the unchirped pulse duration. More particularly, the amplitude of the Fourier Transform of F 2 corresponds to the field spectrum amplitude. Finally, the contribution F 1 oscillating around ω 0 is sensitive to both the pulse duration and the chirp. This function, however, is ill-suited in the case of highly chirped pulses. Indeed, this function tends to a distribution that is independent on the chirp applied to the field:

F 1 (τ ) ∼ K→∞ e -τ 2 4σ 2 t . ( 7
)
As a consequence, particular care must be taken if one uses this function for retrieving the effective pulse duration, in particular if the spatial profile of the beam is noisy.

To conclude, the pulse duration can be retrieved in two different ways. The first method consists to fit G 2 . The drawback of this measurement lies in the fact that this contribution is very sensitive to the quality of the beam profile. Indeed, since this contribution is a low-frequency oscillating function, almost all the noise coming from the beam shape imperfections is transferred on it. Moreover, the knowledge of G 2 does not provide any information on the frequency chirp. The second method is to use the combination of F 1 and F 2 . First, the Fourier-Transform limited pulse duration is retrieved by fitting F 2 , which also gives the pulse spectrum as described above. Then, the absolute value of the chirp parameter K, and consequently, the chirped pulse duration, can be determined by fitting F 1 . As mentioned above, the determination of K is limited to moderate chirps most currently encountered in real experiments. Note that the two methods can be performed simultaneously and independently, improving then the accuracy of the measurement.

III. EXPERIMENTAL SETUP AND RESULTS

The optical design of the autocorrelator is depicted in Fig. 1(b). Before the Fresnel biprism, a cylindrical reflective telescope, composed by two cylindrical mirrors (f = -25 mm and f = 125 mm, respectively) increases the beam size by a factor of 5 in the horizontal direction, direction on which the pulse duration measurement is performed. The telescope ensures that the beam size does not limit the range of pulse durations that can be measured by the autocorrelator. Then, a A=160 • 1.7 mm thick fused silica biprism (Newlight Photonics) is mounted in a manual rotation stage. Using this biprism and after correction coming from the difference between phase and group velocities, the total delay range ∆τ max is then ∆τ max ≃ 5.5 ps and the optimal distance is d opt = 6.3 cm. The camera used in the autocorrelator is a 12 bits silicon CMOS camera (Basler acA3800-14um). The minimal integration time is 35 µs, which makes the autocorrelator single-shot for lasers with a repetition rate lower than 28 kHz. The camera can be externally triggered with a 14 Hz maximal frame rate. The pixel size of the camera is dx=1.67 µm, which fulfills the condition given in Eq. 4 in the whole spectral range of interest. In the horizontal direction, the camera has 3840 pixels, which gives a 6.4 mm sensor size. Accordingly, the total delay range accessible with the present camera is only 3.5 ps, while the upper limit imposed by the biprism is 5.5 ps. As a consequence, since the camera field of view actually limits the delay range, the biprism was placed at a distance d = 4.7 cm, i.e. before d opt . The rotation of the biprism is adjusted by aligning the interference fringes along the vertical dimension of the camera. The signal is transferred by USB 3.0 to a computer for further processing. While the refractive index of the material composing the biprism (in the present case, fused silica) is well known, the calibration of the Apex angle has been accurately performed prior to any autocorrelation measurements by illuminating the optical system with a HeNe laser and by measuring the interferences fringes spacing on the camera. The calibration procedure finally led to A = 160.220 • ± 0.005 • . The laser characterized during the experiment is a commercial non-collinear optical parametric amplifier pumped by a 100 fs 800 nm chirped pulse amplified laser and delivers femtosecond laser pulses in the [1200-2400 nm] wavelength region. Figure 2(a) shows a typical signal recorded with the autocorrelator for a laser wavelength set at λ 0 =2020 nm. A fast Fourier transform along the horizontal dimension allows to easily separate the three oscillating functions G 2 , F 1 and F 2 that are shown in Figs. 2 (b-d). At the same time, knowing that F 1 and F 2 oscillates around ω 0 and 2 ω 0 respectively, the central wavelength of the laser is estimated to be λ e = 2022 nm, in very good agreement with the expectations. The retrieval procedure applied to G 2 gives a pulse duration (FWHM) ∆t ≃76 fs. Using F 1 and F 2 , one obtains ∆t ≃77 fs (in very good agreement with the result obtained with G 2 ), a Fourier-Transform limited pulse duration ∆t FTL ≃57 fs and a group delay dispersion (GDD) of ±1050 fs 2 . Then, the same procedure has been performed after having inserted a 1.4 cm thick BK7 plate in the laser path [Figs. 2(b-d)]. As expected, the contribution oscillating around 2ω 0 remains unchanged while both G 2 and F 1 gets broader than in the case without the plate. The fit of G 2 leads to ∆t ≃137 fs while those of F 1 and F 2 leads to ∆t ≃145 fs, a Fourier-Transform limited pulse duration ∆t FTL ≃56.7 fs and a group delay dispersion (GDD) of ±2720 fs 2 . Accordingly, one can estimate that the GDD introduced by the BK7 plate is about 1670 fs 2 , in good agreement with the theoretical expectation (1450 fs 2 ) calculated from a Sellmeier formula.

As discussed in the above section, F 2 can be used for estimating the laser spectrum. In order to confirm this, the spectrum retrieved with the autocorrelator has been compared with those actually measured with a commercial spectrometer (NIRQUEST from Ocean Optics) in the case of a laser pulse centered around λ 0 =1820 nm.

As shown in Fig. 3, the spectrum estimated by the autocorrelator agrees well with the spectrum measured with the commercial spectrometer. Usually, autocorrelators are sensitive to the input polarization. This is because almost autocorrelators are based on second-harmonic generation, which is a strongly polarization-dependent process. Accordingly, in such apparatus, the input pulse has to be linearly polarized in a particular direction. Since the presented autocorrelator is based on a different process (two-photon absorption), one can wonder which polarization has to be used during the measurement. The two-photon absorption process is a priori a polarization dependent process due to the tensor nature of the two-photon absorption cross-section. In order to check the polarization dependence, a Berek compensator has been inserted before the autocorrelator. This compensator allows to impose any polarization to the pulse to be measured. The comparison between linear and circular polarizations is depicted in Fig. 4. Apart a slightly less intense signal obtained in the circular case, the two measured autocorrelations are almost identical and the fitting procedure gives the same results in both case. Note that the autocorrelator has been also tested in the cases of elliptical polarizations and gives the same result.

IV. FULL-SPATIO TEMPORAL DYNAMICS

Since the temporal characteristics of the pulse is transposed in the spatial domain, the spatial beam shape can affect the autocorrelation measurement. It is then of prime importance to characterize the spatio-temporal shape of the laser pulse in particular at the plane at which the camera is placed. To this end, full spatiotemporal propagation simulations have been performed so as, first, to validate the experimental protocol and, second, in order to quantify potential limitations of the autocorrelator. In order to mimic as close as possible the experimental configuration, a H = 2 cm fused silica biprism with an 160 • Apex angle has been considered. Accordingly, the expected delay range accessible with this biprism is about ∆τ max =5 ps and the optimal distance is d opt ≃6.3 cm. The central wavelength of the laser has been set to λ 0 =2 µm. The spectrum E(x, ω) of the electric field E(x, t), chosen to be as close as possible to the experimental conditions presented here, reads in good approximation after the biprism:

E(x, ω, d = 0) = E 0 e -x 6 σ 6 p e -x 2 σ 2 x e -σ 2 t ω 2 4 e iKω 2
e iΦ(x,ω) cos (ω 0 t) .

(8) The super-gaussian function (σ p =0.98 cm) additionally superimposed to the spatial profile mimics the geometrical truncation induced by the biprism, the latter being smaller than the beam size (σ x =3.4 cm) after its magnification by the cylindrical telescope. Finally, Φ(x, ω) is the frequency-dependent phase introduced by propagation through the biprism. The latter reads:

Φ(x, ω) = n(ω)ωe(x)/c + ωl(x)/c, ( 9
)
where e(x) [resp. l(x)] is the thickness of the fused silica (resp. air) crossed by the ray arriving at a given height x at the output of the biprism:

e(x) = ( H 2 - |x| 1 -tan [α(ω)] tanβ ) tanβ, l(x) = |x| sinβ cos [α(ω) + β] . ( 10
)
The introduced phase then takes into account the fact that the central part of the laser pulse experiences a stronger dispersion than the outer region because of a longer propagation path through the biprism. It also takes into account, through the frequency dependence of α, that the frequencies within the pulse spectrum are not deviated exactly with the same angle because of the refractive index dispersion. These two parameters are those limiting the measurement of very short laser pulses as it will be described below. In the case of small angle, after a propagation distance d in the air, the electric field becomes in the Fourier space:

E(k x , ω, d) = E(k x , ω, 0)e i[kair(ω)-k 2 x /(2kair(ω))]d , ( 11
)
-1500 -1000 -500 0 1000 500 where k air (ω) = n air (ω)ω/c, k x is the conjugate variable of x in the reciprocal space and n air is the frequencydependent refractive index of the air. The spatiotemporal distribution of the electric field E(x, t, d) is then easily retrieved by an inverse Fourier-Transform in both space and time. Figure 5 shows the spatio-temporal distribution of the pulse intensity at d = d opt . The two parts of the beam coming from the two faces of the biprism intersect and interfere at the center because they temporally overlap. Note that the part of the energy located at positive times comes from the central edge of the biprism. Because of the discontinuity, it experiences diffraction so that oscillations can be noticed in the spatial profile. The photocurrent induced by two-photon absorption is then calculated by using Eq. 1 as exemplified in Fig. 6(a) for d = d opt = 6.3 cm in the case of a λ 0 = 2 µm 100 fs Fourier transform limited laser pulse. Then, the three oscillating functions F 2 , F 1 , and G 2 are isolated by a spectral filtering [Fig. 6(c)] and fitted using Eqs. 6. As explained earlier, working at d = d opt provides the largest temporal window. However, because of the gaussian spatial profile, the offset is not uniform at this distance, which can be detrimental to correctly fit G 2 using Eq. 6. For solving this issue and since the camera size actually limits the duration range that can be measured, a slightly shorter distance (d = 4.7 cm) was chosen experimentally so as to obtained a flat offset over the full temporal window. The numerical two-photon absorption signal obtained in this case and the associated oscillating functions are shown in Figs. 6(b) and (d), respectively. Finally, the output of the propagation code has been compared to the experiments (see Fig. 7). The pulse parameter chosen in the simulation are those retrieved by the fit procedure during the experiment (∆t FTL =57 fs and ∆t=77 fs). As shown, the simulation is in excellent agreement with the experiment, which then validates the propagation code.

A. of the autocorrelator 1. Pulse durations working range

Two distinct factors limit the pulse duration range that can be measured by means of the presented autocorrelator. The minimal FTL pulse duration that can be measured is limited by the group velocity dispersion occurring during the propagation through the biprism. In the case of a Fourier transform limited pulse, the propaga-tion within the biprism will make the pulse longer. More particularly, it can be shown that the quadratic phase introduced by the biprism Φ p is in good approximation

Φ p = k (2) (ω 0 ) 2 ω 2 e eff , ( 12
)
where k (2) (ω 0 ) = ∂ 2 k/∂ω 2 (ω 0 ) is the group velocity dispersion of the biprism material, e eff = e 0

( 1 -d 2dopt
) is the effective thickness of the biprism and e 0 = H 2 tanβ is the thickness of the biprism at the center. The propagation code described above was used to quantify the minimal and maximal pulse durations that can be measured with our autocorrelator. For a given and known initial set of parameters (FTL pulse duration and frequency chirp), the autocorrelation signal was numerically evaluated, giving in turn the three contributions G 2 , F 1 and F 2 after an appropriate spectral filtering. Then, these extracted contributions were fitted as it is done in experiments. The resulting fits then give access to the evaluated FTL pulse duration and frequency chirp, which can be compared to the initial set of parameters. The difference between the initial conditions and the fit of the parameters then gives the error introduced by the measurement apparatus itself. Figure 9(a) shows the relative error made by the pulse duration measurement as a function of the input pulse duration (full width at half maximum) in the conditions used during our experiment. The measurement of F 2 (FTL pulse duration) is almost not affected by the propagation through the biprism. This is because F 2 is insensitive to the chirp, and consequently, to the group velocity dispersion induced in the biprism. In fact, the only thing that introduces an error is the frequencydependence of the refraction angle at the biprism output leading to an angular chirp. However, this effect remains marginal for pulses longer than 15 fs if a fused silica biprism is used. On the contrary, the retrieved chirped pulse duration (using either G 2 or F 1 ) is longer than expected, in particular, for very short pulses and well fits the duration expected if one considers the dispersion induced by the propagation within the biprism. This means that the main effect introducing an error in the pulse measurement comes from the dispersion introduced by the propagation within the biprism and can be easily compensated during the fit algorithm. If the tolerance is set to 10 %, the minimal pulse duration that can be measured in this condition (i.e., with a fused silica biprism) is about 25 fs. More particularly, setting a 10 % tolerance in the pulse duration measurement and considering the biprism-induced groupe velocity dispersion only, one can show that the minimal pulse duration (FWHM) measurable with the present setup is given by:

∆ tmin (ω 0 ) = 2 √ log(2)k (2) (ω 0 )e eff ( 1.1 2 -1 ) 1/4 . ( 13
)
Since the group-velocity dispersion of the biprism is not constant over the full spectral window, the minimal pulse duration depends on the pulse central wavelength to be measured (Fig. 8). Note that, this limit can be pushed below by the use of a NaCl biprism instead of a fused silica biprism because the former has a smaller group velocity dispersion beyond 1.6 µm. Note also that these calculations do not take into account the finite spectral extension of the two-photon absorption band. In particular, close to the upper and lower limit of the band, the minimal pulse duration measurable with our autocorrelator will be limited by the fact that a part of the pulse spectrum goes beyond the spectral region of twophoton absorption. Figure 9(b) shows the error made when such a biprism is used (the apex angle of the latter has been chosen so as to keep the same deviation angle). In this case, the minimal pulse duration measurable is about 11 fs. As far as the maximal measurable FTL pulse duration is concerned, it is limited by the finite spatial expansion of the laser beam and does not depend on the biprism material as it can be noticed by comparing Fig. 9(a) and Fig. 9(b) in the long pulse region. In the chosen configuration, the maximal pulse duration that can be measured is about 1 ps (FWHM) when using G 2 and slightly more (≃ 1.5 ps) when using the combination of F 1 and F 2 . This difference is due to the fact that the quality of the fit of G 2 does depend on the offset flatness while these of F 1 and F 2 do not. In the case of a chirped laser pulse to be measured, the group delay dispersion induced by the propagation through the biprism impacts the retrieval of the real pulse duration while the FTL pulse duration retrieval by means of F 2 remains accurate. Indeed, the quadratic phase after propagation within the biprism Φ p is:

Φ p = ( K + k (2) (ω 0 ) 2 e eff ) ω 2 , ( 14
)
F 2 GDD FS GDD NaCl where K is the quadratic phase parameter of the pulse to be measured. Depending on the relative sign of K and the group delay dispersion introduced by the biprism, the chirped pulse duration effectively measured by the autocorrelator will be either longer or shorter than the pulse duration of the input pulse. Figures 9(c,d) show the relative error of the estimation as a function of K for a 60 fs (FWHM) 2µm laser pulse when using a fused silica or a NaCl biprism. For small K, the error is well fitted by the error introduced by the biprism dispersion.

In the highly chirped case, however, the limiting factor is not the group velocity dispersion of the biprism but the finite spatial extension of the laser beam. As it is the case for FTL laser pulse, the use of G 2 for estimating the laser pulse duration leads to underestimate the pulse duration in this case. Note also that the use of a NaCl biprism leads to a smaller error on the pulse estimation also in this case. It then seems that NaCl should be preferred for the design of the autocorrelator. However, apart from the case of very short pulses (<20 fs), fused silica biprism remains a good choice, in particular if one takes into account that the fact that NaCl biprisms are much more costly than silica one.

Impact of the spatial quality of the beam

Another experimental parameter affecting the accuracy of a single-shot autocorrelator is the spatial beam quality. Indeed, since the temporal information is transposed in the spatial domain, the presence of noise in the spatial domain in turn degrades the quality of the pulse duration retrieval. In order to estimate the error made when dealing with a noisy laser beam, simulations have been performed with spatial transverse profiles on which different noise amplitudes have been superimposed. The noise used in the simulation has not been chosen as a completely random function but behaves as 1/k x (where k x is the spatial frequency), i.e., a low frequency spatial noise. This choice reproduces somehow the behavior of the noise effectively affecting amplified laser pulses as it has also been verified with our own laser system. Figure 10 shows the relative error made by the fitting procedure for different noise amplitudes (keeping constant the noise spatial shape) as a function of the chirp amplitude for a 60 fs 2 µm laser pulse. The spatial profile of the electric field used in the three cases are depicted in the insets. As expected, the presence of noise is detrimental for the pulse duration retrieval. At low noise amplitude [Fig. 10(a)], the error remains limited by the dispersion and finite spatial extend of the laser pulse. This is not the case anymore for intermediate and strong noise amplitudes. It can be noticed that F 2 is almost unaffected by the presence of noise unlike F 1 and G 2 . There are two different reasons explaining the different impact of the noise on the fit accuracy. First, one has to remember that G 2 , F 1 and F 2 evolve around ω = 0, ω = ω 0 and ω = 2 ω 0 , respectively. This difference on the temporal domain is transposed in the spatial domain by the use of the biprism. Since the noise mostly affects the low spatial frequency, it impacts more G 2 than F 1 and leaves F 2 almost unaffected. The second reason explaining why F 2 is far less impacted than the two other functions is that G 2 and F 1 extend on a longer transverse dimension as the chirp parameter increases. As a consequence, they are more sensitive to the low-frequency spatial noise. Finally, as discussed above, the fit accuracy of F 1 decreases for long chirped laser pulse because this function tends to a chirp unsensitive function as the chirp increases. This is particularly true in presence of noise on the spatial profile. Note, however, that the impact of the noise can be decreased experimentally by a binning of the pixels over the vertical dimension, which is not used for pulse duration retrieval. This averaging can be directly performed in the hardware of the camera and/or a softwareperformed averaging over the vertical dimension of the two-photon absorption signal. As shown in Figs. 10, the presence of noise in the spatial profile mostly impacts the accuracy of the fit of G 2 , mainly because the fitting procedure of the latter is strongly dependent on the offset flatness.

V. EXTENSION OF THE PRINCIPLE AT LONGER WAVELENGTHS

As explained above, using a silicon camera allows to measure pulse durations for pulses located between 1200 nm and 2400 nm, which is the wavelength range of two-photons absorption of the silicon. Measuring pulse duration of pulses located at higher wavelengths needs to adapt the materials composing the detector. A proof-ofprinciple experiment was performed by using a InGaAs camera (Goldeye G-033 from Allied Vision). The camera sensor is composed by 640x512 15 µm pixels. The NaCl Fresnel biprism used in this experiment has a 175 • Apex angle.

VI. CONCLUSION

As a conclusion, a simple and compact single-shot autocorrelator based on two-photon absorption in a camera and working in the 1.2-2.4 µm spectral region has been presented and analyzed in details. The use of a Fresnel biprism for generating the two-replicas makes the autocorrelator extremely robust in alignment. Moreover, it has been shown that the autocorrelator is insensitive to the input pulse polarization contrary to almost all existing autocorrelators. The interferometric nature of the autocorrelator allows also to extract more information than a conventional intensimetric autocorrelator. In particular, it has been shown that the pulse spectrum, the frequency chirp and the pulse duration can be retrieved if ones assumes a particular temporal shape. Using a propagation code, the limitations of the autocorrelator have been analyzed. In particular, using a fused silica biprism, it has been shown that the pulse duration range accessible with the presented configuration extends from 25 fs to about 1 ps. Note that the lower limit can even be lowered by using a NaCl biprism instead of a fused silica one or by using a bimirror instead of a Fresnel biprism (i.e., by using an all-reflective geometry). Finally, the extension to higher wavelengths (up to 3.4 µm) has been demonstrated by substituting the silicon camera by a InGaAs one.

(Brice Gourier, Julien Lopez and Jean-Marc Muller) for the mechanical realization of the autocorrelator and the CRI-CCUB for CPU loan on the multiprocessor server.

Figure 1 .

 1 Figure 1. (a) Principle of the autocorrelator BOAR device. The black line is a typical experimental two-photon absorption signal recorded by the camera. (b) Experimental setup. The two cylindrical mirrors M1 and M2 (f = -25 mm and f = 125 mm, respectively) are used for increasing the beam size in the horizontal direction by a factor of 5 and FB is the Fresnel biprism.

Figure 2 .

 2 Figure 2. Autocorrelation obtained for a pulse at 2020 nm (a) and the associated oscillating function G2 (b), F1 (c), and F2. The measurement of the pulse duration performed at the output of the NOPA is shown in blue while those done after propagation in a 1.4 cm BK7 plate is depicted in dashed red.

Figure 3 .

 3 Figure 3. Laser spectrum measured with a commercial spectrometer (blue solid line) and evaluated with the BOAR (red dotted-dashed line). The black dashed line is a gaussian fit of the spectrum measured with the spectrometer. The central wavelength of the NOPA is set to 1820 nm.

Figure 4 .

 4 Figure 4. Comparison between experimental autocorrelations obtained for a linearly (dashed red) and circularly (solid blue) polarized pulse (a) and the associated oscillating functions G2 (b), F1 (c), and F2 (d) in the case of a 1400 nm laser pulse.

Figure 5 .

 5 Figure 5. Spatio-temporal distribution of the pulse intensity at d = dopt.

Figure 6 .Figure 7 .

 67 Figure 6. Typical theorical autocorrelation signal obtained by two-photon absorption with a 160 • fused silica biprism (a) placed at the optimal distance (dopt = 6.3 cm) and (b) where the biprism is placed experimentally (d = 4.7 cm) for a λ = 2µm, 100 fs pulse. The associated oscillating functions obtained after a spectral filtering are shown in (c) and (d) respectively.

Figure 8 .

 8 Figure 8. Theoretical minimal pulse duration measurable with the present autocorrelator as a function of the pulse central wavelength if one only considers the group velocity dispersion of the biprism (solid green line) and according to the full simulations (green squares).

Figure 9 .

 9 Figure 9. Relative error made by the fit as a function of the FTL pulse duration in the case of a fused silica (a) and NaCl (b) prism. Relative error made by the fit as a function of the initial chirp in the case of a fused silica (c) and NaCl (d) prism for a 60 fs laser pulse. The central wavelength of the pulse is 2 µm.

Figure 10 .

 10 Figure 10. Relative error made by the fit as a function of the initial chirp in presence of a low (a), intermediate (b) and strong (c) noise in the spatial profile for a fused silica 160 • biprism. The central wavelength of the pulse is 2 µm and the FTL pulse duration is 60 fs. The associated spatial electric field profile is shown in the subpanels.

  Figure 11(a) shows the two-photons absorption signal recorded by the camera in the case of a 3.1 µm femtosecond laser. As shown in Fig.11(b) that displays the Fourier-Transform of the signal, the three contributions G 2 , F 1 , and F 2 clearly appears, which confirms the two-photons nature of the process. By fitting the oscillating functions F 1 and F 2 [see Fig.11(c)], the pulse is found to be almost Fourier-Transform limited with a duration of approximatively 95 fs.

Figure 11 .

 11 Figure 11. Pulse measurement performed at λ0=3.1 µm with a InGaAs camera. (a) Two-photons absorption signal captured with the InGaAs camera. (b) Fourier-Transform of the two-photons signal showing the three different contributions oscillating at ω=0, ω0, and 2ω0, respectively. (c) Temporal distribution of F1 and F2. The retrieved pulse duration is about 95 fs.

ACKNOWLEDGMENTS

This work was supported by the Conseil Régional de Bourgogne (PARI program), the CNRS, the SATT Grand-Est, and Carnot-ARTS. We thank the CRM-ICB