Kacper Pluta

Pascal Romon

Yukiko Kenmochi

Nicolas Passat

Bijective rigid motions of the 2D Cartesian grid

  L'archive ouverte pluridisciplinaire

Introduction

Rigid motions (i.e., rotations, translations and their compositions) defined on Z 2 are simple yet crucial operations in many image processing applications involving 2D data. One way to design rigid motions on Z 2 is to combine continuous rigid motions defined on R 2 with a digitization operator that maps the results back into Z 2 . However, the digitized rigid motion, though uniformly "close" to its continuous origin, often no longer satisfies the same properties. In particular, bijectivity is lost in general. In this context, it is useful to understand the combinatorial, geometrical and topological alterations associated with digitized rigid motions. More precisely, we observe the impact of rigid motions on the structure of Z 2 at a local scale. Few efforts were already devoted to such topic, in particular for digitized rotations. Especially, pioneering works by Nouvel and Rémila [START_REF] Nouvel | On colorations induced by discrete rotations[END_REF] led to an approach for characterizing bijective digitized rotations [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF], and more generally studying non-bijective ones [START_REF] Nouvel | Configurations induced by discrete rotations: Periodicity and quasiperiodicity properties[END_REF].

Our contribution is threefold. We first show the usefulness of a combinatorial model of the local behavior of rigid motions on Z 2 , proposed initially for rotations on Z 2 [START_REF] Nouvel | On colorations induced by discrete rotations[END_REF], [START_REF] Nouvel | Configurations induced by discrete rotations: Periodicity and quasiperiodicity properties[END_REF], called neighborhood motion maps. By using this model, we characterize bijective rigid motions on Z 2 , similarly to the characterization of bijective digitized rotations [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF]. As such characterization is made locally, we also show that the local bijectivity of rigid motions on Z 2 , i.e. bijectivity of rigid motions of finite sets on Z 2 , can be verified by using neighborhood motion maps.

This article is organized as follows. In Section 2 we recall basic definitions and generalize to digitized rigid motions on Z 2 the combinatorial model proposed previously by Nouvel and Rémila for rotations on Z 2 [START_REF] Nouvel | On colorations induced by discrete rotations[END_REF], [START_REF] Nouvel | Configurations induced by discrete rotations: Periodicity and quasiperiodicity properties[END_REF]. Section 3 provides a characterization of bijective rigid motions on Z 2 ; this characterization is an extension of the one proposed in [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF]. In Section 4 we provide new algorithms for the verification whether a given rigid motion is bijective or not when restricted to a finite subset of Z 2 . Finally, in Section 5 we conclude this article and provide some perspectives.

Basic notions 2.1 Digitized rigid motions

Rigid motions on R 2 are bijective isometric maps defined as

U : R 2 → R 2 x → Rx + t (1)
where t = (t x , t y ) t ∈ R 2 is a translation vector and R is a rotation matrix with θ ∈ [0, 2π) its rotation angle. This leads to the representation of rigid motions by a triple of parameters (θ,

t x , t y ) ∈ [0, 2π) × R 2 .
According to Equation (1), we generally have U(Z 2 ) Z 2 . As a consequence, in order to define digitized rigid motions as maps from Z 2 to Z 2 , we commonly apply rigid motions on Z 2 as a part of R 2 , and then combine the real results with a digitization operator

D : R 2 → Z 2 (x, y) → x + 1 2 , y + 1 2 (2)
where z denotes the largest integer not greater than z. Digitized rigid motions are then defined as

U = D • U |Z 2 .
Due to the behavior of D that maps R 2 onto Z 2 , digitized rigid motions are, most of the time, non-bijective. This leads us to define a notion of point status with respect to digitized rigid motions [START_REF] Ngo | Topology-preserving conditions for 2D digital images under rigid transformations[END_REF].

Definition 1. Let y ∈ Z 2 be an integer point. The set of preimages of y with respect to U is defined as S U (y) = {x ∈ Z 2 | U(x) = y}, and y is referred to as a s-point, where s = |S U (y)| is called the status of y.

Remark 1. In Z 2 , |S U (y)| ∈ {0, 1, 2} and only points p and q such that |p -q| = 1 can be preimages of a 2-point [START_REF] Jacob | On discrete rotations[END_REF].

The non-injective and non-surjective behaviors of a digitized rigid motion result in the existence of 2-and 0-points.

Neighborhood motion map

Let us consider the notion of neighborhood in Z 2 .

Definition 2. The neighborhood of p ∈ Z 2 (of squared radius r ∈ R + ), denoted N r (p), is defined as N r (p) = p + d ∈ Z 2 | d 2 ≤ r .
Remark 2. N 1 and N 2 correspond to the 4-and 8-neighborhoods, widely used in digital geometry [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF].

In order to track local alterations of a neighborhood during rigid motions, we introduce the notion of a neighborhood motion map. Definition 3. Given a digitized rigid motion U, the neighborhood motion map of p ∈ Z 2 for r ∈ R + is the function In other words, G U r (p) associates to each relative position of an integer point p + d in the neighborhood of p, the relative position of the image U(p + d) in the neighborhood of U(p). The squared radius r of N r (U(p)) is slightly larger than r. For instance, we have r = 2 (resp. 5) for r = 1 (resp. 2). Figure 1 presents a visual presentation of G U r (p), where r = 1, 2. Note that a similar notion for digitized rotations was previously proposed by Nouvel and Rémila [START_REF] Nouvel | On colorations induced by discrete rotations[END_REF].

G U r (p) : N r (0) → N r (0) (with r ≥ r) defined as G U r (p) : d → U(p + d) -U(p).

Remainder range partitioning

Digitized rigid motions U = D • U are piecewise constant, which is a consequence of the nature of D. In other words, the neighborhood motion map G U r (p) evolves noncontinuously according to the parameters of U that underlies U. Our purpose is now to express how G U r (p) evolves. Let us consider an integer point p + d in the neighborhood N r (p) of p. From Equation (1), we have

U(p + d) = Rd + U(p). (3)
We know that U(p) lies in the unit square centered at the integer point U(p), which implies that there exists a value ρ(p

) = U(p) -U(p) ∈ -1 2 , 1 2
2 . The coordinates of ρ(p), called the remainder of p under U, are the fractional parts of the coordinates of U(p); and ρ is called the remainder map under U. As ρ(p) ∈ -1 2 , 1 2 2 , this range, denoted by

P = -1 2 , 1 2
2 , is called the remainder range. Using ρ, we can rewrite Equation (3) by

U(p + d) = Rd + ρ(p) + U(p).
Without loss of generality, we can consider that U(p) is the origin of a local coordinate frame of the image space, i.e. U(p) ∈ P. In such local coordinate frame, the former equation rewrites as

U(p + d) = Rd + ρ(p). (4)
Still under this assumption, studying the non-continuous evolution of the neighborhood motion map G U r (p) is equivalent to studying the behavior of U(p+d) = D•U(p+d) for d ∈ N r (0) and p ∈ Z 2 , with respect to the rotation parameter θ defining R and the translation parameters embedded in ρ(p), that deterministically depend on (t x , t y , θ). The discontinuities of U(p + d) occur when U(p + d) is on the boundary of a digitization cell. Setting ρ(p) = (x, y) t ∈ P and d = (u, v) t ∈ N r (0), this is formulated by one of the following two formulas

x + u cos θ -v sin θ = k x + 1/2
(5)

y + u sin θ + v cos θ = k y + 1/2 (6)
where k x , k y ∈ Z. For a given d = (u, v) t and k x (resp. k y ), Equation (5) (resp. ( 6)) defines a vertical (resp. horizontal) line in the remainder range P, called a vertical (resp. horizontal) critical line. These critical lines with different d, k x and k y , divide the remainder range P into rectangular regions called frames. Note that for r = 1 (resp. r = 2) there are 4 (resp. 8) vertical and horizontal critical lines, respectively. As long as coordinates of ρ(p) belong to a same frame, the associated neighborhood motion map

G U r (p) remains constant. Proposition 4. For any p, q ∈ Z 2 , G U r (p) = G U r (q) iff ρ(p)
and ρ(q) are in the same frame.

A similar proposition was already shown in [START_REF] Nouvel | On colorations induced by discrete rotations[END_REF] for the case r = 1 and digitized rotations. The above result is then an extension for general cases, such that r ≥ 1 and digitized rigid motions. An example of the remainder range partitioning is presented in Figure 2.

Remark 3. Equations ( 5) and ( 6) of critical lines are similar to those for digitized rotations, since the translation part is embedded only in ρ(p) = (x, y) t , as seen in Equation (4).

Non-surjective and non-injective frames

Some frames correspond to neighborhood motion maps that exhibit 0-or 2-points, implying non-surjectivity or non-injectivity [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF].

Lemma 5. U(p) + d * is a 0-point if and only if ρ(p) is in one of the zones f 0 * (union of frames themselves) defined as follows: * defined as follows: We can characterize the non-surjectivity and non-injectivity of a digitized rigid motion by the presence of ρ(p) in these specific zones. Both types of the zones are presented in Figure 2.

f 0 ↑ = (1/2 -cos θ, sin θ -1/2) × (3/2 -cos θ -sin θ, 1/2), f 0 → = (3/2 -cos θ -sin θ, 1/2) × (1/2 -sin θ, cos θ -1/2), f 0 ↓ = (1/2 -sin θ, cos θ -1/2) × (-1/2, cos θ + sin θ -3/2), f 0 ← = (-1/2, cos θ + sin θ -3/2) × (1/2 -cos θ, sin θ -1/2), where * ∈ {↑, →, ↓, ←} and d ↑ = (0, 1) t , d → = (1, 0) t , d ↓ = (0, -1) t , d ← = (-1, 0) t .
f 2 ↑ = (sin θ -1/2, 1/2) × (-1/2, 1/2 -cos θ), f 2 → = (-1/2, 1/2 -cos θ) × (-1/2, 1/2 -sin θ), f 2 ↓ = (-1/2, 1/2 -sin θ) × (cos θ -1/2, 1/2), f 2 ← = (cos θ -1/2, 1/2) × (sin θ -1/2, 1/2). (a) f 2 ↓ f 2 ↑ f 2 → f 2 ← f 0 ↑ f 0 → f 0 ↓ f 0 ← (b)

Globally bijective digitized rigid motions

A digitized rigid motion is bijective if and only if there is no ρ(p), for all p ∈ Z 2 , in non-surjective and non-injective zones of P. In this section, we characterize bijective rigid motions on Z 2 while investigating such local conditions.

Let us start with the rotational part of the motion. We know from [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF] that rotations with any angle of irrational sine or cosine are non-bijective; indeed such rotations have a dense image by ρ (there exists p ∈ Z 2 such that ρ(p) lies in a non-surjective or/and non-injective zone of P). This result is also applied to U, whatever translation part is added.

Therefore, we focus on rigid motions for which both cosine and sine of the angle θ are rational. Such angles are called Pythagorean angles [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF] and are defined by primitive Pythagorean triples (a, b, c) = (p 2q 2 , 2pq, p 2 + q 2 ) with p, q ∈ Z, p > q and pq is odd, such that (a, b, c) are pairwise coprime, and cosine and sine of such angles are a c and b c , respectively. The image of Z 2 by ρ when U is a digitized rational rotation corresponds to a cyclic group G on the remainder range P, which is generated by ψ ψ ψ = p c , q c and ω ω ω = -q c , p c and whose order is equal to c = p 2 + q 2 [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF]. When U contains a translation part, the image of ρ in P, which we will denote by G , is obtained by translating G (modulo Z 2 ) and |G | is equal to the order of G, its underlying group. Note also that a digitized rational rotation is bijective (the intersection of G with non-injective and non-surjective regions is empty) iff its angle comes from a twin Pythagorean triple-a primitive Pythagorean triple with the additional condition p = q + 1-see Nouvel and Rémila [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF] and, more recently, Roussillon and Coeurjolly [START_REF] Roussillon | Characterization of bijective discretized rotations by Gaussian integers[END_REF].

Our question is then whether a digitized rigid motion can be bijective, even when the corresponding rotation is not. In order to answer this question, we use the following equivalence property: digitized rational rotations are bijective if they are surjective or injective [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF]. Indeed, this allows us to focus only on non-surjective zones. Proposition 7. A digitized rigid motion whose rotational part is given by a non-twin Pythagorean primitive triple is always non-surjective.

Proof. We show that no translation factor can prevent the existence of an element of G in a non-surjective zone. We consider the length of a side of f 0 * , given by L 1 = 2q(p-q) c , and the side of the bounding box of a fundamental square in G, given by L 2 = p+q c . Note that any non-surjective zone f 0 * also forms a square. As p > q + 1, L 2 < L 1 , and thus G ∩ f 0 * ∅ (see Figure 3(a)).

If, on the contrary, the rotational part is given by a twin Pythagorean triple, i.e. is bijective, the rigid motion is also bijective, under the following condition. Proposition 8. A digitized rigid motion is bijective if and only if it is composed of a rotation by an angle defined by a twin Pythagorean triple and a translation t = t + Zψ ψ ψ + Zω ω ω, where t ∈ -1 2c ,

Proof. Let us first consider the case t = 0. Since L 2 > L 1 , there exists a fundamental square in G, i.e. whose vertices are (nω ω ω, mψ ψ ψ), ((n+1)ω ω ω, mψ ψ ψ), ((n+1)ω ω ω, (m+1)ψ ψ ψ), (nψ ψ ψ, (m+ 1)ψ ψ ψ), where n, m ∈ Z, and the vertices lie outside of f 0 ↓ , at N ∞ distance 1/2c (see Figure 3(b)). Now let us consider the case t 0. The above four vertices are the elements of G closest to f 0 ↓ , therefore if N ∞ ({t}) < 1/2c, where {.} stands for the fractional part function, then G ∩ f 0 ↓ = ∅. Moreover, if N ∞ ({t}) is slightly above 1/2c, then it is plain that some point of G will enter the frame f 0 ↓ . But G is periodic with periods ω ω ω and ψ ψ ψ, so that the set of admissible vectors t has the same periods. Then we see that the admissible vectors form a square (i.e. a N ∞ ball of radius 1/2c) modulo Zψ ψ ψ + Zω ω ω (see Figure 3(c)).

Locally bijective digitized rigid motions

As seen above, the bijective digitized rigid motions, though numerous, are not dense in the set of all digitized rigid motions. We may thus generally expect defects such as 2-points. However, in practical applications, the bijectivity of a given U on the whole Z 2 is not the main issue; rather, one usually works on a finite subset of the plane (e.g., a rectangular digital image). The relevant question is then: "given a finite subset S ⊂ Z 2 , is U restricted to S bijective?". Actually, the notion of bijectivity in this question can be replaced by the notion of injectivity since the surjectivity is trivial, due to the definition of U that maps S to U(S ).

The basic idea for such local bijectivity verification is quite natural. Because of its quasi-isometric property, a digitized rigid motion U can send at most two 4-neighbors to a same point (Remark 1). Thus, the lack of injectivity is a purely local matter, suitably handled by the neighborhood motion maps via the remainder map. Indeed, in accordance with Lemma 6, U is non-injective, with respect to S iff there exists p ∈ S such that ρ(p) lies in the union

f 0 ↑ f 0 → f 0 ↓ f 0 ← f 2 ↓ f 2 ↑ f 2 → f 2 ← L 1 L 2 L 1 L 2 (a) f 0 ↑ f 0 → f 0 ↓ f 0 ← f 2 ↓ f 2 ↑ f 2 → f 2 ← (b) f 0 ↓ (c)
F = f 2 ↓ ∪ f 2 ↑ ∪ f 2 ← ∪ f 2
→ of all non-injective zones. We propose two algorithms making use of the remainder map information, as an alternative to a brute force verification.

The first-forward-algorithm, verifies for each point p ∈ S the inclusion of ρ(p) in one of the non-injective regions of F . The second-backward-algorithm first finds all points w in G ∩ F , called the non-injective remainder set, and then verifies if their preimages ρ -1 (w) are in S .

Both algorithms apply to rational motions, i.e., with a Pythagorean angle given by a primitive Pythagorean triple (a, b, c) = (p 2q 2 , 2pq, p 2 + q 2 ) and a rational translation vector t = (t x , t y ) t . We capture essentially the behavior for all angles and translation vectors, since rational motions are dense. Methods for angle approximation by Pythagorean triples up to a given precision may be found in [START_REF] Anglin | Using Pythagorean triangles to approximate angles[END_REF]. These assumptions guarantee the exact computations of the algorithms, which are based on integer numbers.

Forward algorithm

The strategy consists of checking whether the remainder map ρ(p) of each p ∈ S belongs to one of non-injective zones f 2 * defined in Lemma 6; if this is the case, we check additionally whether p + d * ∈ N 1 (p) belongs to S ; otherwise, there is no 2-point with p under U |S .

This leads to the forward algorithm, which returns the set B of all pairs of points having the same image. We can then conclude that U |S is bijective iff B = ∅; in other words, U is injective on S \B. The break statement on line 7 comes from the fact that there is no 3-point. Also from Remark 1, we restrict the internal loop to the set {→, ↓}. The main advantage of the forward algorithm lies in its simplicity. In particular, we can directly check which neighbor p + d * of p constitutes a 2-point with p. Because rational rigid motions are exactly represented by integers, it can be verified without numerical error and in a constant time if ρ(p) ∈ F . The time complexity of this algorithm is O(|S |). Figure 4 illustrates the forward algorithm.

Remark 4. The forward algorithm can be used with non-rational rotations, at the cost of a numerical error.

Backward algorithm

In this section, we consider a rectangular finite set S as the input; this setting is not abnormal as we can find a bounding box for any finite set. The strategy of the proposed ρ(p) for all p ∈ S , under U-given by parameters arccos 6 37 , 0, 2 25 . Since no point ρ(p) lies within the non-injective zone F , we have a visual proof that U restricted to S is injective. backward algorithm consists of: Step 1: for a given U, i.e. a Pythagorean triple and rational translation vector, listing all the points w in the non-injective remainder set; Step 2: finding all their preimages ρ -1 (w); Step 3: finding among them those in S .

Step 1. As explained in Section 3, the cyclic group G is generated by ψ ψ ψ = p c , q c and ω ω ω = -q c , p c , and G is its translation (modulo Z 2 ). Therefore, all the points in G can be expressed as Zψ ψ ψ + Zω ω ω + {t}. To find these points of G in the non-injective zones, let us focus on f 2 ↓ given in Lemma 6. Note that the similar discussion is valid for other non-injective zones given by Lemma 6. The set of remainder points Zψ ψ ψ + Zω ω ω + {t} lying in f 2 ↓ is then formulated by the following four linear inequalities, and we define the non-injective remainder index set C ↓ such that

C ↓ =          (i, j) ∈ Z 2 -1 2 < p c i -q c j + {t x } < 1 2 -2pq p 2 +q 2 , p 2 -q 2 p 2 +q 2 -1 2 < q c i + p c j + {t y } < 1 2          . (7)
Solving the system of inequalities in [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF] consists of finding all pairs (i, j) ∈ Z 2 inside the given rectangle. This is carried out by mapping Zψ ψ ψ + Zω ω ω + {t} to Z 2 using a similarity; denoting by f 2 ↓ the image of f 2 ↓ under this transformation (Figure 5). To list all the integer points in (i, j) ∈ C ↓ , we first find the upper and lower corners of the rectangular region f 2 ↓ given by Equation ( 7): p-3q 2 , p-q 2 and q-p 2 , p+q 2 . We then find all the horizontal lines j = k where k ∈ Z ∩ p-q 2 , p+q 2 . For each line j = k, we obtain the two intersections with the boundary of f 2 ↓ as the maximal and minimal integers for i. The complexity of this step depends on the number of integer lines crossing f 2 ↓ , which is q, and thus it is O(q). This completes Step 1.

Step 2. We seek for the set of all preimages of iψ ψ ψ + jω ω ω + {t} for each (i, j) ∈ C ↓ , or equivalently, preimages of iψ ψ ψ + jω ω ω by the translationless remainder map. (The fact that this point is in f 2 ↓ plays no role in this step.) This is a Diophantine system (modulo Z 2 ) and the set of preimages of a point iψ ψ ψ + jω ω ω + {t} is given by a sublattice of Z 2 :

T(i, j) = p µ -v 2 i j + Z a -b + Z cσ cτ (8)
where µ, v and σ, τ are the Bézout coefficients satisfying µp 2 + vq 2 = 1, and σa + τb = 1, respectively.

To find these Bézout coefficients, we use the extended Euclidean Algorithm. The time complexity of finding µ and v (resp. σ and τ) is O(log q) (resp. O(log min(a, b)) [START_REF] Hunter | Essentials of Discrete Mathematics[END_REF]. As the Bézout coefficients are computed once for all (i, j) ∈ C ↓ , the time complexity of Step 2 is O(log q) + O(log min(a, b)) = O(log min(a, b)).

Step 3. We now consider the union of lattices T(i, j) for all couples (i, j) in C ↓ obtained in Step 1. To find their intersection with S , we apply to each an algorithm similar to Step 1, with an affine transformation mapping the basis ( a -b ), ( cσ cτ ) to ( 1 0 ), ( 0 1 ) and p µ-v 2 ( i j ) to ( 0 0 ). Thus, a rectangular S maps to a quadrangular Ŝ after such an affine transformation, and we find the set of integer points in Ŝ . Note that the involved transformation is the same for all the lattices, up to a translation. The complexity of listing all the preimages is given by |C ↓ | times the number of horizontal lines j = k, k ∈ Z, passing Ŝ , denoted by K. The cardinality of C ↓ is related to the area of f 2 ↓ given by 2q 2 (p-q) 2 (p 2 +q 2 ) 2 which cannot be larger than (thus avoiding the case when p and p + d * are mapped to the same point but p + d * is not in S ). This can be achieved during Step 3.

|C ↓ | = |G ∩ f 2 ↓ |, |C ↓ | ≤ ( 3 2 - √ 2
All the steps together allow us to state that the backward algorithm, whose time complexity is O(q + log min(a, b) + √ |S |), identifies non-injective points in finite rectangular sets. Figure 6 presents an experimental result of the backward algorithm applied to a computer tomography image. Remark 6. Even though the backward algorithm works with rectangles, one can approximate any set S by a union of rectangles and run the backward algorithm on each of them.

Conclusion

In this article, we have extended the neighborhood motion maps-which was previously proposed by Nouvel and Rémila [START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF] for digitized rotations-and we have shown that they are useful to characterize the bijectivity of rigid motions on Z 2 .

We first proved some necessary sufficient conditions of bijective rigid motions on Z 2 ; i.e., rigid motions such that no point p ∈ Z 2 has the image ρ(p) in either noninjective zones or non-surjective zones. Then, from a more practical point of view, we focused on finite sets of Z 2 rather than the whole Z 2 . In particular, we proposed two efficient algorithms for verifying whether a given digitized rigid motion is bijective when restricted to a finite set S . The forward algorithm consists of checking if points of S have preimages in non-injective zones. On the other hand, we used the reverse strategy to propose the backward algorithm consisting of the identification of points in G ∩ F and their preimages in S . The complexities of the forward and backward algorithms are O(|S |) and O(q + log min(a, b) + √ |S |), respectively. One of our perspectives is to extend the proposed framework to 3D digitized rigid motions.

Fig. 1 .

 1 Fig. 1. Visual representation of neighborhood motion maps G U r (p) with colors: reference neighborhoods N 1 (p) (a) and N 2 (p) (c), and examples of G U 1 (p) (b) and G U 2 (p) (d).

Lemma 6 .

 6 U(p) is a 2-point whose preimages are p and p + d * if and only if ρ(p) is in one of the zones f 2

Fig. 2 .

 2 Fig. 2. Examples of remainder range partitioning for: r = 1 (a), and r = 2 (b). Non-injective zones f 2 * and non-surjective zones f 0 * are illustrated by red and brown rectangles, respectively.

Fig. 3 .

 3 Fig. 3. Examples of remainder range partitioning together with G obtained for rotations by Pythagorean angles. (a) The non-bijective digitized rotation defined by the primitive Pythagorean triple (12, 35, 37) and (b) the bijective digitized rotation defined by the twin Pythagorean triple (7, 24, 25).The non-surjective and non-injective zones are illustrated by brown and red rectangles, respectively. (c) A fundamental square in G whose vertices are (nω ω ω, mψ ψ ψ), ((n + 1)ω ω ω, mψ ψ ψ), ((n + 1)ω ω ω, (m + 1)ψ ψ ψ), (nψ ψ ψ, (m + 1)ψ ψ ψ), represented by black circles, and f 0 ↓ in brown. The union of the areas filled with a line pattern form a square (i.e. a N ∞ ball of radius 1 2c ) of the admissible translation vectors modulo Zψ ψ ψ + Zω ω ω.

5 B 6 S

 56 Forward algorithm: A point-wise injectivity verification of U |S . Data: A finite set S ⊂ Z 2 ; a digitized rigid motion U. Result: The subset B ⊆ S whose points are not injective under U. 1 B ← ∅ 2 foreach p ∈ S do 3 foreach * ∈ {→, ↓} do 4 if p + d * ∈ S and ρ(p) ∈ f 2 * then ← B ∪ {{p, p + d * }} ← S \ {p, p + d * }

Fig. 4 .

 4 Fig. 4. (a) An initial finite set S ⊂ Z 2 , colored in black. (b) The remainder map image of S , i.e.ρ(p) for all p ∈ S , under U-given by parameters arccos6 37 , 0, 2 25 . Since no point ρ(p) lies within the non-injective zone F , we have a visual proof that U restricted to S is injective.

Fig. 5 .

 5 Fig.5. (a) Geometrical interpretation of the system of linear inequalities in Equation[START_REF] Nouvel | Characterization of bijective discretized rotations[END_REF], in the (i, j)-plane for (p, q) = (7, 2). The region surrounded by the four lines is f 2 ↓ , and the integer points within are marked by black circles. (b) The remainder range, G and f 2 ↓ illustrated by a red square, which corresponds to f 2 ↓ in (a).

  )c. On the other hand, K is bounded by d S /c, where d S stands for the diagonal of S . As the complexity of d S is given by O( √ |S |), the final complexity of Step 3 is O( √ |S |). Remark 5. A possible refinement consists of ruling out false positives at border points p of S , by checking whether p + d * belongs to S , where d * is given by the above procedure

Fig. 6 .

 6 Fig. 6. Non-injective pixels obtained for a computer tomography image of a human chest by the backward algorithm (preimages of points in G ∩ f 2 ↓ ). Each non-injective pixel is marked by a circle. Pixels of the same color are positioned periodically and are preimages of the same point in G ∩ f 2 ↓ . The result was obtained for θ = arcsin 591 19409 ≈ 1.7 • and t = 0. Note that from the point of view of backward algorithm the content of the image does not matter.

Acknowledgments. The authors express their thanks to Laurent Najman of ESIEE Paris for his remarks concerning the backward algorithm, and to Mariusz J ędrzejczyk of Norbert Barlicki Memorial Teaching Hospital No. 1, Department of Radiology and Diagnostic Imaging, for providing the computer tomography image of a human chest.

The research leading to these results received funding from the Programme d'Investissements d'Avenir (LabEx Bézout, ANR-10-LABX-58).


Ahmed Boubrima

Walid Bechkit

Hervé Rivano

Optimal Deployment of Dense WSN for Error Bounded Air Pollution Mapping

Keywords: Air quality monitoring, Wireless sensor networks deployment, error bounded mapping

Air pollution has become a major issue of modern megalopolis because of industrial emissions and increasing urbanization along with traffic jams and heating/cooling of buildings. Monitoring urban air quality is therefore required by municipalities and by the civil society. Current monitoring systems rely on reference sensing stations that are precise but massive, costly and therefore seldom. In this ongoing work, we focus on an alternative or complementary approach, using a network of low cost and autonomic wireless sensors, allowing for a finer spatiotemporal granularity of air quality sensing. We tackle the optimization problem of sensor deployment and propose an integer programming model, which allows to find the optimal network topology while ensuring air quality monitoring with a high precision and the minimum financial cost. Most of existing deployment models of wireless sensor networks are generic and assume that sensors have a given detection range. This assumption does not fit pollutant concentrations sensing. Our model takes into account interpolation methods to place sensors in such a way that pollution concentration is estimated with a bounded error at locations where no sensor is deployed.

I. INTRODUCTION

Air pollution affects human health dramatically. According to the World Health Organization (WHO), exposure to air pollution is accountable to seven million casualties in 2012. In 2013, the International Agency for Research on Cancer (IARC) classified particulate matter, the main component of outdoor pollution, as carcinogenic for humans. Air pollution has become a major issue of modern megalopolis, where the majority of world population lives, adding industrial emissions to the consequences of an ever denser urbanization with traffic jams and heating/cooling of buildings. As a consequence, the reduction of pollutant emissions is at the heart of many sustainable development efforts, in particular those of smart cities. Monitoring urban air pollution is therefore required by both municipalities and the civil society.

Current air quality monitoring is mostly operated by independent authorities. Conventional measuring stations are equipped with multiple lab quality sensors. These systems are however massive, inflexible and expensive. An alternativeor complementary -solution would be to use wireless sensor networks (WSN) [START_REF] Kumar | Environmental monitoring systems: a review[END_REF] which consist of a set of lower cost nodes that can measure information from the environment, process and relay them to some base stations, denoted sinks. The progress of electrochemical sensors, that are smaller and cheaper while keeping a reasonable measurement quality, makes the use of WSN for air quality monitoring viable [START_REF] Mead | The use of electrochemical sensors for monitoring urban air quality in low-cost, highdensity networks[END_REF].

The main advantage of the use of WSN for air pollution monitoring is to obtain a finer spatiotemporal granularity of measurements, thanks to the resulting lighter installation and operational costs. Although some WSN-based air quality monitoring systems are already operating, the deployment issue of these tiny nodes while taking into account the precision of the resulting network has not yet been investigated.

Minimizing the deployment cost is a major challenge in WSN design. The problem consists in determining the optimal positions of sensors and sinks so as to cover the environment and ensure network connectivity while minimizing the deployment cost [START_REF] Zhu | A survey on coverage and connectivity issues in wireless sensor networks[END_REF]. The deployment is constrained by the cost of the nodes and sinks, but also by operational costs such as the energy spent by the nodes. The network is said connected if each sensor can communicate information to at least one sink [START_REF] Younis | Strategies and techniques for node placement in wireless sensor networks: A survey[END_REF]. The coverage issue has often been modeled as a kcoverage problem in which at least k sensors should monitor each point of interest. Most research work on coverage uses a simple detection model which assumes that a sensor is able to cover a point in the environment if the distance between them is less than a radius called the detection range. This can be true for some applications like presence sensors but is not suitable for pollution monitoring. Indeed, a pollution sensor detects pollutants that are brought in contact by the wind. The notion of detection range is thus irrelevant in this context. Therefore, a deployment model is still needed for the air quality monitoring application.

In this ongoing work, we propose an integer programming model (ILP) of WSN deployment for error-bounded air quality mapping. We formulate the constraint of air pollution coverage based on interpolation methods in order to determine the optimal positions of sensors allowing to better estimate pollution concentrations at positions where no sensor is deployed. We base on the flow problem to formulate the connectivity constraint that ensures that the deployed sensors are able to send pollution data to at least one sink.

II. OPTIMIZATION MODEL A. Objective function

We consider as input of our model the map of a given urban area that we call the deployment region. We start by discretizing the deployment region in order to get a set of points P approximating the urban area at a high-scale (|P| = N ). Our goal is to be able to determine with a high precision the concentration value at each point p ∈ P . We ensure that for each point p ∈ P , either a sensor is deployed or the pollution concentration can be estimated with a high precision based on the data gathered by the neighboring deployed sensors.

In general case, the set P is thus considered as the set of potential positions of WSN nodes. However, in smart cities applications, some restrictions on node positions may apply because of authorization or practical issues. When this is the case, we do not consider as potential positions the points p ∈ P where sensors cannot be deployed. We use decision variables x p resp. y p to specify if a sensor resp. a sink is deployed at point p or not. Sensors and sinks may have different costs, thus we denote by c sensor p resp. c sink p the sensor resp. the sink deployment cost at position p. The deployment cost function to minimize is thus given as follows:

F = p∈P c sensor p * xp + p∈P c sink p * yp (1)

B. Air quality coverage

Using numerical atmospheric dispersion models, we first get simulated pollution concentrations that may be considered as reference pollution concentrations. This does not mean that these reference concentrations are real but they reflect the best today's pollution knowledge. Let Z p denote the reference concentration value at point p. Given the set of selected points where sensors will be deployed {p where x p = 1}, we evaluate the estimated pollution concentrations Z p at points {p where x p = 0} based on reference values corresponding to the selected points, i.e. based on Z p where p ∈ {p where x p = 1}, as follows:

           Zp = q∈P-
We ensure that the denominator of Z p is never equal to zero using the second part of formula 2. The W pq parameter is the correlation coefficient between points p and q and is calculated using formula 3 based on the distance between the two points. D(p, q) is the distance function. α is the attenuation coefficient of the correlation distance, this means that for greater values of α, very low correlation coefficients are assigned to far points. The last parameter of formula 3 is the maximum correlation distance, which defines the range of correlated neighboring points of a given point.

In order to take into account the impact of the urban topography on the dispersion of pollutants, let D be the shortest distance along the roads network. This allows to assign small correlation values to points that are separated by buildings, even if they are close.

Wpq =    1 D(p, q) α if q ∈ Disc(p, d) -{p} 0 if q / ∈ Disc(p, d) (3)
In order to ensure that the concentration is estimated with high precision at points where no sensor is deployed, we define constraint 4. The E p parameter corresponds to the estimation error that is tolerated at point p. The choice of different values of E p in function of p allows to assign low tolerated estimation errors to locations that are sensitive to air quality such as hospitals, primary schools, etc.

Zp -Zp ≤ Ep, p ∈ P & xp = 0 (4)
By replacing Z p by its expression given in formula 2, we obtain the coverage constraints 5 and 6. These two constraints should be linearized in order to get an ILP formulation. 1) Linearization of constraint 5: The first step is to linearize the fraction part; this allows to get constraint 7. We now add x p * q∈P-{p} W pq * |Z q -Z p | to the right member of constraint 7 to relax it when x p = 1. Hence, we obtain constraint 8. Finally, we have to linearize the absolute-value function. Hence, we get the linear form of constraint 5 in constraints 9 and 10. 2) Linearization of constraint 6: The only thing to do to linearize constraint 6 is to relax the constraint when x p = 1. This can be obtained by replacing the right member of the constraint by -x p , which allows to get the constraint 11.

C. Network connectivity

We formulate the connectivity constraint as a network flow problem. We consider the same potential positions set P for sensors and sinks. We first denote by Γ(p), p ∈ P, the set of neighbors of a node deployed at the potential position p, Γ(p) = {q ∈ P where q ∈ Disc(p, R)} where R is the communication range of sensors. Then, we define the decision variables g pq as the flow quantity transmitted from a node located at potential position p to another node located at potential position q. We suppose that each sensor generates a flow unit in the network, and verify if these units can be recovered by sinks. The following constraints ensure that the deployed sensors and sinks form a connected WSN; i.e. each sensor can communicate with at least one sink. Constraints 12 and 13 are designed to ensure that each deployed sensor, i.e. such that x p = 1, generates a flow unit in the network. Constraint 14 ensures that non deployed nodes do not participate in communication. The flow sent by deployed sensors has then to be received by deployed sinks, thanks to constraint 15. At the end, our optimization model can be written as follows:

M inimize (1) Subject to. (9), (10), (11), ( 12), ( 13), ( 14) and (15)

III. SIMULATION RESULTS

A. Dataset

We perform our simulations on a pollution map generated by an enhanced atmospheric dispersion simulator called SIR-ANE. The dataset has been provided by Air-Rhone-Alpes, which is an observatory for air pollution monitoring within the Lyon region of France. We evaluate our ILP model on the La-Part-Dieu district, which is the heart of the Lyon City. Pollution map granularity is around 5 meters and concentrations correspond to the year 2012. We discretize the deployment region which is of around 850mX850m using a resolution of 50 meters, thus we get 306 discrete points. We consider all these points as potential positions of nodes.

B. Results

We study the dependency between the deployment precision and the needed number of sensors under different configurations of the correlation distance. Since we are studying the cost of the monitoring precision, we execute only the coverage constraint. We depict in Figure 1 the optimal deployment cost depending on the tolerated estimation error while considering two different functions of the correlation distance: the euclidean distance and the distance along roads. We notice that the number of sensors decreases as expected with the estimation error that is tolerated. Indeed, the interpolation method assumes a linear spatial evolution of pollution concentration between points where sensors are deployed (cf eq. ( 2)). When the tolerated error is small, it is required to deploy sensors at the locations where there is a high variability of pollution concentrations.

Regarding the impact of the distance metric, the distance along roads is obviously larger than the euclidean distance and takes into account the buildings that influence both pollution dispersion and radio propagation. Correlation coefficients (cf eq. ( 3)) are therefore smaller when using the distance along roads, hence more sensors are required in order to get a better estimation of pollution concentrations. IV. CONCLUSION In this ongoing work, we tackled the optimization problem of sensor deployment and proposed an integer programming model computing a cost-optimal network topology while ensuring the mapping of air quality with bounded error. Our idea is to constraint the deployment of sensors by the quality of the pollution estimation that can be interpolated between the sensors.

  {p} Wpq * Zq * xq q∈P-{p} Wpq * xq , p ∈ P & xp = 0 q∈P-{p} Wpq * xq > 0, p ∈ P & xp = 0

  q∈P-{p} Wpq * Zq * xq q∈P-{p} Wpq * xq -Zp ≤ Ep, p ∈ P & xp = 0 (5) q∈P-{p} Wpq * xq > 0, p ∈ P & xp = 0 (6)

  q∈P-{p} Wpq * xq * (Zq -Zp) ≤ Ep * q∈P-{p} Wpq * xq, p ∈ P, xp = 0 (7) q∈P-{p} Wpq * xq * (Zq -Zp) ≤ Ep * q∈P-{p} Wpq * xq +xp * q∈P-{p} Wpq * |Zq -Zp|, p ∈ P (8) q∈P-{p} Wpq * xq * (Zq -Zp) ≤ Ep * q∈P-{p} Wpq * xq +xp * q∈P-{p} Wpq * |Zq -Zp|, p ∈ P (9) q∈P-{p} -Wpq * xq * (Zq -Zp) ≤ Ep * q∈P-{p} Wpq * xq +xp * q∈P-{p} Wpq * |Zq -Zp|, p ∈ P (10)

  q∈P-{p}Wpq * xq > -xp, p ∈ P (11)

  gqp ≥ xp -(N + 1) * yp, p ∈ P (12) q∈Γ(p) gpq -q∈Γ(p) gqp ≤ xp, p ∈ P (13) q∈Γ(p) gpq ≤ N * xp, p ∈ P (14)

Fig. 1 :

 1 Fig. 1: Optimal coverage cost vs tolerated estimation error.IV. CONCLUSION In this ongoing work, we tackled the optimization problem of sensor deployment and proposed an integer programming model computing a cost-optimal network topology while ensuring the mapping of air quality with bounded error. Our idea is to constraint the deployment of sensors by the quality of the pollution estimation that can be interpolated between the sensors. ACKNOWLEDGMENT This work has been supported by the "LABEX IMU" (ANR-10-LABX-0088) and the "Programme Avenir Lyon Saint-Etienne" of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

ACKNOWLEDGMENT This work has been supported by the "LABEX IMU" (ANR-10-LABX-0088) and the "Programme Avenir Lyon Saint-Etienne" of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


F Kaiser

L A Ngah

A Issautier

T Delord

D Aktas

V D'auria

M P De Micheli

A Kastberg

L Labonté

O Alibart

A Martin

S Tanzilli

Polarization entangled photon-pair source based on quantum nonlinear photonics and interferometry

Keywords: Quantum Communication, Polarization Entanglement, Nonlinear Integrated Photonics 2008 MSC: 81P40, 81P45

We present a versatile, high-brightness, guided-wave source of polarization entangled photons, emitted at a telecom wavelength. Photon-pairs are generated using an integrated type-0 nonlinear waveguide, and subsequently prepared in a polarization entangled state via a stabilized fiber interferometer. We show that the single photon emission wavelength can be tuned over more than 50 nm, whereas the single photon spectral bandwidth can be chosen at will over more than five orders of magnitude (from 25 MHz to 4 THz). Moreover, by performing entanglement analysis, we demonstrate a high degree of control of the quantum state via the violation of the Bell inequalities by more than 40 standard deviations. This makes this scheme suitable for a wide range of quantum optics experiments, ranging from fundamental research to quantum information applications. We report on details of the setup, as well as on the characterization of all included components, previously outlined in F.

Introduction

Quantum communication science has become a very broad and active field of research. On one hand, quantum key distribution (QKD) [START_REF] Scarani | The security of practical quantum key distribution[END_REF], allowing secure distribution of cryptography ciphers between distant partners, has reached the commercial market as well as high-speed system capabablities [START_REF] Yoshino | High-speed wavelength-division multiplexing quantum key distribution system[END_REF]. Related networking protocols, such as entanglement based quantum relay operations, are employed as a means for extending the distance of quantum communication links [START_REF] Aboussouan | High-visibility two-photon interference at a telecom wavelength using picosecond-regime separated sources[END_REF][START_REF] Mcmillan | Two-photon interference between disparate sources for quantum networking[END_REF]. On the other hand, the study of light/matter interaction is a promising approach for implementing quantum storage devices [START_REF] Lvovsky | Optical quantum memory[END_REF]. Those devices are essential elements to achieve quantum repeater scenarios in which entanglement is distributed, stored, and distilled, all in a heralded fashion, making it possible to increase the overall link efficiency [START_REF] Sangouard | Quantum repeaters based on atomic ensembles and linear optics[END_REF].

Over the past three decades, entanglement has been widely exploited as a resource in fundamental tests of quantum physics [START_REF] Aspect | Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: A new violation of Bell's inequalities[END_REF][START_REF] Tittel | Photonic entanglement for fundamental tests and quantum communication[END_REF]. We find, among others, nonlocality tests involving spacelike separated paired photons [START_REF] Tittel | Violation of Bell inequalities by photons more than 10 km apart[END_REF][START_REF] Weihs | Violation of bell's inequality under strict einstein locality conditions[END_REF], quantum delayed-choice experiments [START_REF] Ma | Experimental delayedchoice entanglement swapping[END_REF][START_REF] Peruzzo | A quantum delayed choice experiment[END_REF][START_REF] Kaiser | Entanglement-enabled delayed-choice experiment[END_REF], and demonstrations of micro-macro entangled states of light [START_REF] Lvovsky | Observation of micro?macro entanglement of light[END_REF][START_REF] Bruno | Displacement of entanglement back and forth between the micro and macro domains[END_REF]. Moreover, with the emergence of long-distance quantum communication links working at telecom wavelengths [START_REF] Lloyd | Long distance, unconditional teleportation of atomic states via complete Bell state measurements[END_REF][START_REF] Landry | Quantum teleportation over the Swisscom telecommunication network[END_REF][START_REF] Ursin | Entanglement-based quantum communication over 144 km[END_REF][START_REF] Hübel | Highfidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber[END_REF], new generation sources have been developed, featuring higher brightness, better stability, compactness, and near perfect entanglement fidelities. Photons can now be generated over narrow enough bandwidths (≤100 GHz) to avoid both chromatic and polarization mode dispersion along the distribution fibers [START_REF] Fasel | Quantum key distribution over 30 km of standard fiber using energy-time entangled photon pairs: a comparison of two chromatic dispersion reduction methods[END_REF]. More importantly, current light/matter interaction based quantum memories have, depending on both the physical system and the applied storage protocol, acceptance bandwidths ranging from some MHz to several GHz [START_REF] Lvovsky | Optical quantum memory[END_REF]. Consequently, to push long-distance quantum communication one step further, there is a need for implementing versatile solutions so as to benefit from the advantages of different quantum technologies. In this framework, sources based on quantum integrated photonics [START_REF] Tanzilli | On the genesis and evolution of quantum integrated optics[END_REF] appear to be natural and very promising candidates, offering the possibility to efficiently create polarization entanglement at wavelengths compatible with standard fiber components [START_REF] Kaiser | High-quality polarization entanglement state preparation and manipulation in standard telecommunication channels[END_REF][START_REF] Herrmann | Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs[END_REF].

In the following, we present the details of a versatile, high-brightness, source of polarization entangled photons, whose main results were first presented in Ref. [START_REF] Kaiser | A versatile source of polarization entanglement for quantum network applications[END_REF]. Its key features, i.e., the central emission wavelength, the spectral bandwidth, and the quantum state, can be tuned at will and adapted to a broad range of quantum network applications. This is enabled by a pertinent combination of an integrated nonlinear optics photon-pair generator, standard telecommunication components, and an entangled state preparation stage based on a stabilized Mach-Zehnder interferometer (MZI). After the presentation of the overall setup and the principle of the source, we will detail all the key elements, namely the integrated nonlinear generator, employed filters, and the quantum state preparation stage. We will then present the entanglement characterization proving the relevance of our approach. Related stabilization schemes and performances in terms of brightness and internal losses will also be outlined. Finally, we will summarize the obtained results, in the perspective of overall performances, and discuss potential improvements.

Specifications, setup, and principle of the source

Generation of polarization entanglement has been demonstrated using various strategies based on nonlinear media, such as microstructured fibers [START_REF] Fulconis | Nonclassical interference and entanglement generation using a photonic crystal fiber pair photon source[END_REF][START_REF] Fang | Polarization-entangled photon-pair generation in commercial-grade polarization-maintaining fiber[END_REF], single pass bulk crystals [START_REF] Fiorentino | Compact sources of polarization-entangled photons[END_REF][START_REF] Piro | An entangled photon source for resonant single-photonsingle-atom interaction[END_REF], crystals surrounded by a cavity referred to as optical parametric oscillators below the threshold [START_REF] Wang | Polarization-entangled mode-locked photons from cavity-enhanced spontaneous parametric down-conversion[END_REF][START_REF] Bao | Generation of Narrow-Band Polarization-Entangled photon pairs for atomic quantum memories[END_REF], or type-II waveguide crystals [START_REF] Kaiser | High-quality polarization entanglement state preparation and manipulation in standard telecommunication channels[END_REF][START_REF] Herrmann | Post-selection free, integrated optical source of non-degenerate, polarization entangled photon pairs[END_REF]. However, these strategies have all shown relatively low brightness, which becomes an issue when (ultra-)narrowband photons are needed. Other approaches, capable of generating narrowband polarization entangled photons based on quantum dots [START_REF] Dousse | Ultrabright source of entangled photon pairs[END_REF] or cold atomic ensembles [START_REF] Dudin | Entanglement of light-shift compensated atomic spin waves with telecom light[END_REF][START_REF] Yan | Generation of narrow-band hyperentangled nondegenerate paired photons[END_REF] have recently been demonstrated, albeit showing limited entanglement fidelities.

Our source specifications are outlined in the following. First, the paired photons are emitted at a wavelength lying in the telecom C-band (1530 -1565 nm), and further collected using a single mode telecom fiber in order to benefit from both standard components for routing and filtering purposes and low propagation losses in case of distribution over a long distance. Second, the photon bandwidth is made readily adaptable so as to be compatible with a broad variety of applications, ranging from QKD in telecommunication channels to quantum storage device implementations. Third, the coding of quantum information relies on the polarization observable since entanglement correlations can be measured using simple analyzers, being free of interferometric devices as opposed to the case of the time-bin observable [START_REF] Martin | Cross time-bin photonic entanglement for quantum key distribution[END_REF]. In addition, polarization entanglement can now be distributed over long distances thanks to active compensation systems of fiber birefringence fluctuations [START_REF] Xavier | Experimental polarization encoded quantum key distribution over optical fibres with real-time continuous birefringence compensation[END_REF]. Eventually, and importantly, the key figures of merit are a high rate of available photon-pairs, and a fidelity to the desired entangled state as close to unity as possible.

Based on the experimental setup depicted in Figure 1, we outline in the following the principles of the main building blocks.

Photon-pair generation via type-0 spontaneous parametric down-conversion

Light from a continuous-wave (CW) pump laser at 780.24 nm is sent through a type-0 periodically poled lithium niobate waveguide (PPLN/W) in order to generate, via spontaneous parametric down-conversion (SPDC), paired photons around the degenerate wavelength of 1560.48 nm. Using a vertically polarized (V) pump beam, the type-0 interaction permits to create vertically polarized twin photons, i.e., |V p type-0 -→ |V s |V i , where the indices p, s, and i represent the pump, signal and idler modes, respectively. Utilizing the type-0 process presents two main advantages compared to the type-II interaction which produces cross-polarized photons, i.e., in the state |H p type-II -→ |H s |V i , as detailed in [START_REF] Kaiser | High-quality polarization entanglement state preparation and manipulation in standard telecommunication channels[END_REF] and references therein. On one hand, the associated generation efficiency is at least 2 orders of magnitude higher. On the other hand, single photon bandwidth narrow filtering can readily be achieved using fiber filters, without problems with fiber birefringence. However, in contrast to the type-II interaction, where polarization entanglement can be formed by simply splitting the paired photons, the price to pay for using the type-0 process is a more complex experimental arrangement [START_REF] Kaiser | A versatile source of polarization entanglement for quantum network applications[END_REF].

Bandwidth filtering stage

In principle, the bandwidth of the paired photons can adapted at will depending on the desired quantum application. In our case, we work with fiber solutions only, compatible with all other elements of the source. By doing so, we avoid the losses due to in and out fiber coupling, yielding a higher long-term stability. After collection of the paired photons using a single mode fiber, we therefore take advantage of three filtering solutions based on telecom compatible components in order to demonstrate the versatility of our approach:

(a) a standard 100 GHz-spacing dense wavelength division multiplexing (DWDM) filter (AC photonics) compatible with DWDM-QKD protocols [START_REF] Yoshino | High-speed wavelength-division multiplexing quantum key distribution system[END_REF][START_REF] Kaiser | High-quality polarization entanglement state preparation and manipulation in standard telecommunication channels[END_REF];

(b) a 540 MHz phase shifted fiber Bragg grating filter (PSFBG, from AOS GmbH) compatible with broad acceptance bandwidth quantum memories, based on, e.g., room temperature atomic vapors or ion doped crystals [START_REF] Reim | Towards high-speed optical quantum memories[END_REF][START_REF] Clausen | Quantum storage of photonic entanglement in a crystal[END_REF][START_REF] Saglamyurek | Broadband waveguide quantum memory for entangled photons[END_REF];

(c) a 25 MHz PSFBG (Teraxion) compatible with the acceptance bandwidth of cold atom, and trapped ion based quantum memories [START_REF] Tanji | Heralded single-magnon quantum memory for photon polarization states[END_REF][START_REF] Piro | Heralded single-photon absorption by a single atom[END_REF].

Moreover, to avoid any polarization dependent transmission, and therefore degradation of entanglement, with the narrowband PSFBG filters, those have to placed in front of the state preparation stage. The polarization state of the photons is adjusted beforehand using a fiber polarization controller (PC 1 ) (see more details below).

Polarization entanglement preparation stage

To prepare the polarization entangled state, we employ an unbalanced MZI made of two fiber polarizing beam-splitters (f-PBS) connected by polarization maintaining fibers. This interferometer introduces a delay δt between the two polarization modes H (short arm) and V (long arm). By sending the paired photons prepared in the diagonal (D) state |D s |D i into this device, the exit states are of the form:

|D s |D i P rep. → 1 2 |H s,e |H i,e + e iφ/2 |H s,e |V i,l + e iφ/2 |V s,l |H i,e + e iφ |V s,l |V i,l , (1)
where e and l refer to "early" and "late" time bins, respectively. Here, φ/2 represents the phase difference between the short and long paths. Post-selecting only the cases where the two photons exit the interferometer simultaneously reduces the quantum state to |ψ post = 1 √ 2 |H s,e |H i,e + e iφ |V s,l |V i,l . This time-bin type post-selection constraints δt to be greater than the coherence time of the photon (τ p ) and the detector timing jitters (τ d ). It is important to note that the labels e and l have physical meanings only if the photon-pair creation time is known. In the case of CW SPDC, the creation time uncertainty is directly given by the coherence time of the pump laser (τ L ). In this way, using a laser with a coherence time much greater than δt ensures a constant phase for the two contributions of interest, |H s |H i and |V s |V i , and therefore allows obtaining polarization entangled states of the form:

|ψ post = 1 √ 2 |H s |H i + e iφ |V s |V i , (2)
where φ represents the phase experienced by the two-photon contribution |V s |V i in the long arm of the interferometer. To summarize the operation principle described above, the source can produce, together with a proper post-selection, the maximally polarization entangled state of Eq. 2 only if the conditions

τ L δt τ p + τ d (3)
are satisfied, as is the case in time-bin type experiments [START_REF] Tittel | Photonic entanglement for fundamental tests and quantum communication[END_REF][START_REF] Martin | Cross time-bin photonic entanglement for quantum key distribution[END_REF][START_REF] Franson | Bell inequality for position and time[END_REF]. By controlling the phase difference between the arms of the MZI, and by adjusting the input polarization state of the two photons, it is possible to prepare any arbitrary superposition of the maximally entangled Bell states |Φ + and |Φ -1 . Suppose that the input of the preparation stage is the two-photon state (α|H + β|V )(α|H + β|V ), with the normalization |α| 2 +|β| 2 = 1. In this case we obtain, after appropriate post-selection of events in the time domain, non-maximally entangled states of the form

|ψ post = 1 N α 2 |H s |H i + e iφ β 2 |V s |V i , where N = α 4 + β 4
is the normalization. This can be further generalized by simply adding a half-wave plate in the path of one of the two photons. This way, one can prepare non-maximally entangled states as a superposition of the four maximally entangled Bell states. 1 There are four maximally entangled states referred to as Bell states [START_REF] Tittel | Photonic entanglement for fundamental tests and quantum communication[END_REF]. They are defined as

|Φ ± = 1 √ 2 |H s |H i ± |V s |V i and |Ψ ± = 1 √ 2 |H s |V i ± |H s |V i ,
and form a complete orthonormal basis.

Setup and characterization of the key elements

Frequency stabilization of the pump laser

As laser source, we use a commercial tapered amplifier system (Toptica Photonics TApro) operating at a wavelength around 780.24 nm in the CW regime. A small fraction of the laser light (∼ 1 mW) is sent to a standard saturated absorption spectroscopy setup for frequency stabilization purpose. The stabilization is made on the |F = 2 → |F = 2 × 3 hyperfine crossover transition of the D 2 line of rubidium atoms ( 87 Rb). The Zeeman shifted resonance is modulated by ∼20 kHz using a small electromagnet. The saturation absorption signal is then demodulated using a lock-in amplifier (EG&G 5210), and fed forward to the laser diode current through a home-made PID controller. The speed of the stabilization system is sufficient in order to achieve a laser linewidth < 150 kHz, over all the relevant time scales, which guarantees a coherence time τ L > 3 µs. We can therefore fulfill the remaining requirement of inequality (3) even for long coherence time photons, i.e., up to τ p =20 ns, as will be detailed below.

Phase matching and optical characterization of the nonlinear integrated waveguide

The photon-pair generator is a 4.5 cm long, home-made, waveguide, integrated at the surface of a periodically poled lithium niobate substrate using the so-called soft proton exchange (SPE) technique [START_REF] Chanvillard | Soft proton exchange on periodically poled LiNbO 3 : A simple waveguide fabrication process for highly efficient nonlinear interactions[END_REF]. This permits obtaining low-loss (∼0.2 dB/cm) single-mode optical waveguide structures at telecom wavelengths, showing reasonably high mode confinement, via index variations on the order of 2×10 -2 . It also allows to maintain the integrity of the second-order non-linearity ([χ] 2 ) of the lithium niobate substrate. The desired quasi phasematching condition (780.24 nm → 1560.48 nm) for the type-0 SPDC process is obtained for a 7 µm-width waveguide, a poling period of the substrate of 16.3 µm, and for a temperature of 387 K. Note that heating the sample also permits canceling the photorefractive effect induced by the coupled pump intensity in the waveguide.

The PPLN/W internal down-conversion probability was characterized in the single photon counting regime, as described in Ref. [START_REF] Tanzilli | PPLN waveguides for quantum communication[END_REF], and found to be ∼4.8×10 -6 generated pairs per injected pump photon. This outperforms photon-pair generators based on optical parametric oscillators [START_REF] Wang | Polarization-entangled mode-locked photons from cavity-enhanced spontaneous parametric down-conversion[END_REF][START_REF] Bao | Generation of Narrow-Band Polarization-Entangled photon pairs for atomic quantum memories[END_REF] and bulk crystals [START_REF] Fiorentino | Compact sources of polarization-entangled photons[END_REF][START_REF] Piro | An entangled photon source for resonant single-photonsingle-atom interaction[END_REF] by a few orders of magnitude, and is comparable to the best values reported to date for SPDC in type-0 PPLN/Ws [START_REF] Tanzilli | PPLN waveguides for quantum communication[END_REF][START_REF] Halder | High coherence photon pair source for quantum communication[END_REF]. Such a high SPDC conversion probability can mainly be ascribed to the tight light confinement in the waveguide structure, since the efficiency grows quadratically with the optical power density [START_REF] Tanzilli | On the genesis and evolution of quantum integrated optics[END_REF].

The temperature dependent emission spectrum measured at the output of the PPLN/W is shown in Figure 2(a). Signal and idler wavelengths can each be tuned over more than 50 nm for a temperature change of quantum memory protocols typically demand bandwidths reduced down to 10 MHz-5 GHz. However, 4 THz is a broad enough bandwidth that can be further reduced so as to match a desired quantum application.

Characterization of the transmission of the filters

As mentioned previously, we employ three different fiber filters, i.e., a standard 100 GHz DWDM, a 540 MHz PSFBG, and a 25 MHz PSFBG, to reduce the photonic spectral bandwidth from the 4 THz initially generated by the PPLN/W. These filter bandwidths, together with the initial one, span more than 5 orders of magnitude.

As depicted in Figure 3(a), we properly characterize and set the three filters at the desired wavelength (1560.48 nm). Notably, the two PSFBG filters are configured using laser spectroscopy. Fort this, we inject light from a tunable diode laser (Toptica Photonics DL100 pro design), followed by a polarization controller (PC), and we monitor the transmitted power at the output. To obtain an accurate frequency reference, a fraction of the laser power is amplified using an erbium doped fiber amplifier (EDFA) and then frequency doubled in another home-made PPLN/W. For ∼ 100 mW of power at 1560.48 nm sent in the dedicated PPLN/W, we obtain ∼ 3 mW at 780.24 nm at its output. This is enough to perform saturated absorption spectroscopy in a Rb cell. The obtained hyperfine structure absorption lines here serve as accurate absolute frequency references when scanning the laser. As depicted in Figure 3(b), the 100 GHz DWDM filter shows a flat-top transmission bandwidth of 80 GHz. In Figure 3(c), we show the transmission of the 540 MHz PSFBG, which has lorentzian transmission bandwidths of 540 MHz and 580 MHz for vertically and horizontally polarized light, respectively. Due to fiber birefringence, the modes are separated by 480 MHz. The measurements for the 25 MHz PSFBG are given in Figure 3(d), with lorentzian transmission bandwidths of 25 MHz and 28 MHz for vertically and horizontally polarized light, respectively, and a mode separation of 80 MHz. Note that a consequence of the polarization dependent transmission behavior is that these PSFBGs cannot be directly applied to polarization entangled paired photons, as the frequency separation between horizontally and vertically polarized photons would reduce the entangled state to a product state.

The central frequency (wavelength) of both PSFBG filters can be finely tuned by adjusting the temperature. Experimentally, the 540 MHz and the 25 MHz filters present respectively a tunability of ∼1 GHz/K and ∼200 MHz/K, such that we can achieve about 50 GHz (↔ 400 pm) and 1 GHz (↔ 80 pm) over a temperature range between 30 • C and 80 • C. As a side effect, the temperature of the filters needs to be stabilized in order to avoid any central wavelength drift. In our case, this is done by isolating the filters hermetically from ambient air and by employing standard PID-type temperature controllers. Using such systems, we can set the transmission peak of both filters for vertically polarized light at exactly twice the wavelength of the Rb |F = 2 → |F = 2 × 3 hyperfine transition, against which the pump laser is stabilized (indicated by a star in the graphs of Figure 3(b) and(c)). The achieved temperature stability is 10 mK, corresponding to frequency stabilities of ∼ 10 MHz and ∼ 2 MHz for the 540 MHz and the 25 MHz filters, respectively.

Note that these PSFBG filters represent simple solutions compared to standard optical cavities, concerning both practicality and reliability. For instance, stabilizing a cavity to better than 2 MHz requires a ∼1 mK temperature stability, as well as a stringent isolation to ensure a constant pressure and a limited sensibility to acoustic vibrations. This is actually unnecessary with our fiber filters. Moreover, tuning the central frequency of a cavity necessitates the use of a piezo-electric transducer place on one of the cavity mirrors. This, in our case, can simply be achieved using basic temperature control. A good example of a narrow filtering made with an optical cavity can be found in reference [START_REF] Piro | An entangled photon source for resonant single-photonsingle-atom interaction[END_REF].

Polarization entanglement preparation stage and coincidence histogram

As described above, after paired photons have been generated in the PPLN/W and subsequently filtered in bandwidth (see Figure 1), we take advantage of an unbalanced MZI to engineer the polarization bi-photon state. When two diagonally polarized photons are injected in the device, they emerge in the state of Eq. 1, which can be further reduced to the maximally polarization entangled state of Eq. 2 by post-selection in the time domain. A proper post-selection can only be achieved provided the different contributions to the state of Eq. 1 are made temporally distinguishable. In other words, the delay δt between the two arms of the interferometer has to be much greater than the coherence time τ p of the photons, as stated by inequality 3. For this reason, we have chosen a fiber path length difference of 18 m, corresponding to a propagation time difference of 76 ns, enough to obtain a fidelity greater than 99% with spectral bandwidths as narrow as 19 MHz (coherence time of 20.5 ns).

We show in Figure 4(a) the experimental configuration employed for obtaining arrival time coincidence histograms at the output of the interferometer. A beam-splitter (BS) is used to separate and distribute the photons to Alice and Bob. Each user employs a free running indium-gallium-arsenide (InGaAs) avalanche photodiode (IDQ-220) as single photon detector (SPD). Both devices feature 20% detection efficiencies and dark count probabilities on the order of 10 -6 /ns. The SPD outputs are connected to a coincidence measurement apparatus (&).

As shown in Figure 4(b), we observe three distinct coincidence peaks for all three filters. The side peaks contain the cross-polarized photon-pair contributions, while the central peak is twice as high and contains the desired contributions |H |H and |V |V . Selecting only the events at zero relative delay generates the desired entangled state |ψ post of Eq. 2. For the 100 GHz DWDM filter, we expect a coherence time ∼ 5 ps smaller than the timing jitters of the detectors. Consequently the width of the corresponding coincidence peaks (dark curve in Figure 4) is mainly given by the convolution of the detectors' timing jitters (230 ps for each detector). Having the 540 MHz or the 25 MHz filter in place leads to a broadening of the coincidence peaks of 800±20 ps (red curve in Figure 4) and 15.6±0.7 ns (blue curve in Figure 4), respectively, which reflects the increased coherence time of the photons. These values are in good agreement with the corresponding filter bandwidths shown in Figure 3(b) and (c).

Phase stabilization of the polarization entanglement preparation stage

In order to demonstrate high quality polarization entanglement via the violation of the Bell inequalities (see below), the phase relation between the contributions |H |H and |V |V to the desired entangled state needs to be stable throughout the measurement time [START_REF] Martin | Analysis of elliptically polarized maximally entangled states for Bell inequality tests[END_REF]. Our calculations showed that phase fluctuations on the order of ∆φ ≈ 2 π 50 cause a degradation of the entanglement state fidelity of about 0.5%. In our MZI fiber configuration, phase variations are mainly due to refractive index fluctuations induced by temperature instabilities [START_REF] Leviton | Temperature-dependent absolute refractive index measurements of synthetic fused silica[END_REF]. The temperature phase dependency is approximately given by ∆φ ∆T ≈ 10 3 rad K -1 . This means that achieving a phase stability better than 2 π 50 would require a temperature stability of about 30 µK, which is technically demanding as the MZI would have to be placed in a vacuum chamber.

To avoid the latter constraint, we opted for an active stabilization scheme. As depicted in Figure 5, light from a reference diode laser (Toptica Photonics DL100 pro design), diagonally polarized and emitted at 1558.6 nm, is Our phase stabilization scheme is based on a standard feedback loop, in which the reference wavelength is modulated via the current controller of the laser diode. To obtain the error signal, a lock-in amplifier demodulates the detector signal. The phase correction is applied to a home-made piezoelectric (PZT) fiber stretcher acting on the long arm of the MZI via an integrator circuit. Our system holds the phase term φ r + φ e constant at zero or π. The phase inside the interferometer, φ r , is then tuned by adjusting φ e . Our system is therefore capable of accurately setting the phase to any desired value over short time scales, i.e. on the order of 1 ms (mainly limited by the response time of the PZT).

Moreover, to ensure a long term stability, the 1558.6 nm reference laser is stabilized in a cavity, after frequency doubling to 779.3 nm. The cavity is itself stabilized upon that of the 780.24 nm pump laser (see above), which means that the reference laser is stabilized thanks to a transfer-cavity lock configuration. The overall system achieves an overall phase stability better than π 100 and works reliably over time scales of several

Entanglement analysis

In this section, we first demonstrate our capability to coherently tune the phase of the preparation interferometer. Thereafter, we set the phase to the desired value of π in order to create the entangled state |Φ -, and we perform entanglement analysis via the violation of the Bell inequalities.

Phase tuning demonstration and long-distance distribution of the paired photons

The experimental setup used to demonstrate the phase tunability is shown in Figure 6(a). The photon-pairs, at the output of the interferometric preparation stage, are expected to be prepared, together with a proper timebin post-selection, in the state |ψ post (see Eq. 2 and related discussion). The paired photons are then separated at a beam-splitter and sent to Alice and Bob. They both project their respective photons into the the phase sensitive diagonal/antidiagonal basis {|D , |A } using a polarization state analyzer made of a bulk polarizing beam-splitter (PBS) oriented at 45 • and an InGaAs avalanche SPD. In this situation, the coincidence rate, R c , recorded between the two users is given by

R c = 1 2 (1 -V cos φ) , (4)
where φ is the phase imprinted on the entangled state by the interferometric preparation stage, and V represents the fringe visibility of the interference pattern.

In Figure 6(b), we present the coincidence rate recorded as a function of the phase set in the preparation stage for all three filters. For these measurements, the phase is tuned between -π and 5π/2, using our stabilization system. The three curves follow the relation described by equation 4, and sinusoidal fits permit to infer the associated interference pattern visibilities. The net visibilities are 99.9 ± 1.2%, 99.4 ± 1.5%, and 97 ± 2%, for the 100 GHz (squares and dark line), 540 MHz (circles and red line), and 25 MHz (triangles and blue line) filters, respectively. Net visibilities mean that accidental coincidences induced by the detector dark counts have been subtracted. Without noise subtraction, the raw visibilities are 99.9±1.2%, 97.1±1.2%, and 88±2%, respectively. The last value is smaller than the others, owing to a degraded signal-to-noise ratio associated with the 25 MHz.

Moreover, Figure 6(c) shows the results obtained with the 540 MHz filter when the paired photons are distributed over 4 km of standard telecom optical fiber, and its direct comparison to a 20 m fiber distribution scenario. The obtained net and raw visibilities are 99 ± 2% and 96 ± 2%, respectively. The reductions in coincidence rate and raw visibility are ascribed to additional propagation losses of about 1.2 dB, which again causes a degradation of the signal-to-noise ratio.

To conclude this phase control and tunability study, it is worth mentioning that our preparation system works perfectly as it permits setting the phase of the entangled state to any desired value on short time scales (< 1 ms). The curves presented in Figure 6 therefore stand as a proof that any maximally entangled state of the form |ψ = 1 √ 2 |H |H + e iφ |V |V can be generated at will. For the following study, we now set φ = π in order to generate the state |Φ -, on which entanglement analysis is performed. , in order to project the entangled state on the phase sensitive {|D , |A } basis. The coincidence rate between the two detectors is recorded as a function of the phase set in the fiber preparation stage. (b) Results obtained for photon-pair distribution over 20 m of standard optical fiber: for all three filters, near perfect sinusoidal coincidence rate oscillations are observed, underlining the high phase stability achieved with our approach. (c) Results obtained for the 540 MHz filter when distributing the entangled photons over 20 m and 4 km of standard optical fiber, respectively. Again, high-visibility interference patterns are observed. Note that for all results presented here, noise contributions induced by accidental coincidences have been subtracted.

Entanglement analysis for the |Φ -state

In the following, we employ, among other possibilities [START_REF] Altepeter | Experimental methods for detecting entanglement[END_REF], the violation of the Bell inequalities as an entanglement [START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF][START_REF] Clauser | Proposed experiment to test local Hidden-Variable theories[END_REF]. To this end, we consider the standard experimental configuration shown in Figure 7(a), in which the users each employ a polarization state analyzer, comprising a half-wave plate (HWP), a PBS, and an SPD. As was the case for the phase control study, the two detectors are connected to a coincidence measurement apparatus (&).

The measurements are made in the following manner. Alice projects her photons consecutively along the horizontal (|H ), vertical (|V ), diagonal (|D ) and anti-diagonal (|A ) directions. For each of the four settings, we record the coincidence rates as a function of Bob's HWP angle. The experimental results corresponding to the three filters are shown in Figure 7(b), (c) and (d). For each filter and for all four settings chosen at Alice's side, sinusoidal coincidence rate oscillations are observed, underlining the rotational invariance of the generated entangled state. Sinusoidal fits permit to infer the associated interference pattern visibilities. We obtain average raw visibilities (including noise contributions) of 99.6 ± 1.3%, 97.1 ± 0.9% and 99 ± 3% for the 100 GHz, 540 MHz, and 25 MHz filters, respectively. These values are clearly above the non-local threshold of 71% for two-photon experiments [START_REF] Tittel | Long distance Bell-type tests using energy-time entangled photons[END_REF]. Note that in order to increase the visibility for the 25 MHz filter, two superconducting detectors (Scontel TCOPRS-001) have been used instead of the two InGaAs avalanche photodiodes that were in place for all the previous measurements. These superconducting devices both feature 7% detection efficiencies and less than 10 dark counts per second. This permits us to increase the signal-to-noise ratio at the price of lower coincidence counts.

Besides the obtained visibility, the Bell parameter is another quantitative measure for evaluating the quality of entanglement. We obtain raw Bell parameters S raw of 2.82 ± 0.01, 2.80 ± 0.02 and 2.82 ± 0.02 for the three employed filters, meaning that the Bell inequalities are violated by at least 40 standard deviations [START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF][START_REF] Clauser | Proposed experiment to test local Hidden-Variable theories[END_REF]. This underlines the high quality of the entangled states that can be generated with our scheme [START_REF] Kaiser | A versatile source of polarization entanglement for quantum network applications[END_REF].

Brightness and internal losses of the source

In addition to high entanglement qualities, the brightness of the photon-pair generator is one of its key features, since it provides the rate at which a quantum communication link can be operated. Typically, the brightness is defined as the number of paired photons available in one spatial mode, per of integration time, pump power, and spectral bandwidth. Achieving a high brightness requires the use of a high-efficiency generator associated with low propagation losses in the overall setup. In the following, we first detail the two latter figures, before giving the rate of entangled photon-pairs available at the output of the source.

On one hand, the internal down-conversion probability of the type-0 PPLN waveguide obtained over the full emission spectral bandwidth of 4 THz, shown in Figure 2(b), corresponds to a brightness B full ≈ 2400 pairs • (s • mW • MHz) -1 . However, all the three employed filters provide much narrower bandwidths than that of the full spectrum, such that the peak intensity within these bandwidths can be considered constant with an associated brightness B top ≈ 3600 pairs • (s • mW • MHz) -1 . As for all generators based on spontaneous processes, the main limitation comes from multiple pair emission issues, generally dominated, at weak pump powers, by double pair contributions. Those are due to the emission statistics that can be either poissonian or thermal, depending on both the coherence time of the photons and the time acquisition window [START_REF] Tapster | Photon statistics of pulsed parametric light[END_REF][START_REF] Monteiro | Narrowband photon pair source for quantum networks[END_REF]. In other words, operating the source at high pump powers would lead to non negligible double pair contributions and cause a severe degradation of the entangled state fidelity [START_REF] De Riedmatten | Two independent photon pairs versus four-photon parametric down conversion entangled states in parametric down conversion[END_REF]. For instance, working with a probability for single photon-pair emission equal to 0.01 per detection time window, generally defined as the maximum between the timing jitter of the detectors and the coherence time of the photons, leads to a decrease of the relative entangled state fidelity of 1%. Note that with our non-linear waveguide, a laser pump power of 7 mW is enough to achieve such a probability value per coherence time.

On the other hand, to compute the overall source brightness, we need to consider all losses from the output of the photon-pair generator to the output of the source itself, i.e., the two outputs of the fiber beam-splitter shown in Figure 1. We measure the single photon propagation losses between the output of the crystal and the input of the beam-splitter by replacing the pump laser by one emitting at 1560.48 nm. We obtain 4.5 dB, 5.2 dB, and 5.7 dB for the 100 GHz, 540 MHz and 25 MHz filters, respectively. However, when considering the propagation of the paired photons to the output of the source, additional losses have to be taken into account. We find the following values:

-3 dB due to the lorentzian transmission profiles of the PSFBG filters. Note that this is not the case for the DWDM filter, since it shows a flat-top transmission profile; -3 dB to the time-bin structure associated with the preparation stage, since only half of the generated photon-pairs are actually prepared in a maximally polarization entangled state; -3 dB introduced by a non-deterministic pair separation at the output beam-splitter.

The overall losses experienced by the paired photons are therefore 15 dB, 19.4 dB and 20.4 dB, respectively for the three filter configurations. Finally, combining the generator brightness and the internal losses of the full setup leads to available photonpair rates of about 9×10 6 , 11000, and ∼400 per second and mW of pump power, respectively for the three filter configurations. A summary of the relevant figures of merit for the source, in combination with the three different filters, is presented in table 1.

Bandwidth [MHz]

80×10 1 for the three different filters. Note that the imperfections of the detectors provide limitations on data acquisition, since low efficiencies are associated with lower pair detection rates, and dark counts are one of the main limiting factor for achieving high raw visibility values [START_REF] Sekatski | Detector imperfections in photon-pair source characterization[END_REF]. Moreover, for both the 100 GHz DWDM and the 540 MHz PSFBG filters, the dead-times of the detectors induce an inherent reduction of the detected pair rate. † For these measurements, standard InGaAs avalanche photodiode SPDs have been used [START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF]. ‡ For this particular measurement, superconducting SPDs have been employed in order to reduce accidental coincidence events [START_REF] Hadfield | Single-photon detectors for optical quantum information applications[END_REF].

Potential improvements of the source

The pair rate could be improved considerably by reducing the internal propagation losses of the source, notably in the case of the narrowest employed filters. First, the lorentzian PSFBG filters (540 MHz and 25 MHz) could be replaced by flat-top ones, as for the 100 GHz DWDM filter. This would, in this case, lead to a pair loss reduction of 3 dB. Secondly, the propagation losses could also be reduced by splicing all the fiber patch cords instead of using fiber connectors. This would lead to approximately a 2 dB pair loss reduction. Thirdly, using a tapered waveguide structure at the output of the non-linear waveguide generator would yield another 1 dB of loss reduction, due to an adiabatic spatial mode adaptation between the waveguide and the collection fiber [START_REF] Castaldini | High performance mode adapters based on segmented SPE:LiNbO 3 waveguides[END_REF]. Finally, one could address the non-deterministic separation of the paired photons by utilizing a fiber coupled cavity able to transmit two peaks on both sides of the wavelength degeneracy (1560.48 nm). For instance, a free spectral range of 100 GHz combined with a transmission bandwidth of 25 MHz can be achieved by using a 1.2 mm-long cavity, finesse 4000. Despite the difficulty of building such a high finesse cavity, this would permit obtaining pairs of entangled photons easily separable using a simple DWDM placed after the interferometric preparation stage, providing a 3 dB pair loss reduction. Combining some of the above mentioned strategies could lead to an average pair loss reduction on the order of 6 dB, at the price of an increased setup complexity.

Conclusion

We have demonstrated a versatile and efficient scheme for the generating polarization entangled photon-pairs at telecom wavelengths. The versatility lies in both the spectral properties of the photons and in the generated quantum state. For instance, the wavelength of the emitted single photons can easily be adjusted over more than 50 nm, using basic temperature control of the photon-pair generator. We have also shown near perfect entanglement qualities over photonic spectral bandwidths ranging from 25 MHZ to 100 GHz, thus spanning more than four orders of magnitude. Our scheme is readily extendable to the 4 THz spectral bandwidth initially available at the output of the photon-pair generator, and therefore to five orders of magnitude. Moreover, the active stabilization system of the preparation stage allows us to set and to further tune the phase relation between the two contributions, |HH and |V V , to the entangled state. The weight of the contributions, and the contributions themselves, can be easily chosen by means of basic polarization controls in front of and after the preparation stage. We can therefore prepare any two-photon polarization state, ranging from product to maximally entangled states, as a superposition of two of the four Bell states.

All the presented results have been obtained with near perfect entanglement qualities, i.e., the Bell inequalities have been violated for all chosen bandwidths by more than 40 standard deviations, without noise subtraction. These results have also been made possible by the high brightness of our source, which is mainly due to a high-efficiency waveguide photon-pair generator, and optimized losses in the presented configuration. We have also outlined suggestions for future improvements that could lead to a potential brightness increase of almost one order of magnitude. These modifications are currently being implemented on our setup.

We believe that our scheme is a good candidate for a broad range of quantum experiments, ranging from fundamental quantum optics to quantum network applications. In the latter framework, our photon spectral bandwidths of 540 MHz and 25 MHz are already suitable for several quantum memory strategies based on hot atomic vapors and solid state devices. In order to adapt the wavelengths of the entangled photons to the absorption resonances of the corresponding memory devices, non-linear wavelength conversion processes such as sum frequency generation and difference-frequency generation have been proven to be very efficient strategies [START_REF] Tanzilli | A photonic quantum information interface[END_REF][START_REF] Curtz | Coherent frequency-down-conversion interface for quantum repeaters[END_REF].

Figure 1 :

 1 Figure 1: Experimental setup. (a) Light from a stabilized laser at 780.[START_REF] Kaiser | A versatile source of polarization entanglement for quantum network applications[END_REF] nm is sent through a PPLN/W for the generation of twin photons at the degenerate wavelength 1560.48 nm. After collection with a single mode telecom fiber, the paired photons pass through a polarization controller (PC 1 ) followed by a filter, adapting their bandwidth to the desired application. With commercially available components, the filtering transmission bandwidth can be chosen from some MHz to several hundreds of GHz. An unbalanced MZI made, of two fiber-polarizing beam-splitters (f-PBS) connected by polarization maintaining fibers, allows preparation of the desired polarization entangled state (via post-selection in the time domain). A fiber beam-splitter (BS) is used to separate the photons and to distribute them to Alice and Bob. At both locations, measurements are carried out using a polarization state analyzer, comprising a half-wave plate (HWP), a polarizing beam-splitter (PBS), and a single photon detector (SPD). (b) Principle of the entanglement preparation stage. Photon-pairs in the diagonal state |D |D enter the unbalanced MZI and exit in four possible states. If the photon-pair creation time remains unknown, the states |H |H and |V |V are indistinguishable in the time domain, and thus entanglement can be generated by post-selecting only when the two photons exit in the same time-bin [34].

Figure 2 :

 2 Figure 2: SPDC emission spectrum obtained from a PPLN/W having a poling period of 16.3 µm, a width of 7 µm, being pumped by a laser at 780.24 nm. (a) Spectral density over the full phase matching curve as a function of temperature. The colorbar on top indicates the relative intensity. Changing the crystal temperature by 3 K results in a wavelength tunability of more than 50 nm for either the signal or the idler photon. (b) Degenerate spectrum around 1560 nm, when the PPLN/W is heated to 387 K. The shape of the curve is a sinc function, as expected from the SPDC process, and its FWHM spectral bandwidth is of 32 nm, corresponding to 4 THz.

Figure 3 :

 3 Figure 3: Characterization of the employed filters. (a) Experimental setup. A tunable laser, emitting around the wavelength of 1560.48 nm, is sent through the filter under test and the polarization dependent transmission spectrum is recorded. In order to obtain a frequency reference in the case of the two PSFBG filters, the laser is amplified (EDFA) and frequency doubled in a PPLN/W, before being sent to a standard Rb saturated absorption spectroscopy setup. (b) Transmission spectrum of the 100 GHz DWDM filter showing a flat-top shaped profile. (c) Transmission spectrum of the 540 MHz filter. (d) Transmission spectrum of the 25 MHz filter. The two latter filters show lorentzian transmission profiles, which have a polarization dependence caused by fiber birefringence.

Figure 4 :

 4 Figure 4: Characterization of the polarization entanglement preparation stage, in combination with the three filters, for a diagonally polarized two-photon input state. (a) Experimental setup. At the output of the preparation stage, the photons are separated by a BS and subsequently detected. The arrival time correlations are recorded using a coincidence counter (&). (b) Coincidence histograms. Three well separated peaks are obtained for all the three filters. The two outer ones, located at ±76 ns, correspond to cross polarized contributions. The central peak at zero relative time delay contains the desired contributions |H |H and |V |V .

1 √ 2 2 |D + cos φr+φe 2 |A

 1222 injected in the interferometer via a telecom DWDM in the backward direction compared to that of the paired photons. Note that this component is a 200 GHz-spacing DWDM allowing multiplexing and demultiplexing of the standard ITU 2 telecommunication channels N • 21 and 23. At the output of the MZI device, the laser light is separated from the photons of interest via another DWDM and sent to an electro-optical phase modulator. The corresponding polarization state can therefore be written as |ψ r = |H + e i(φr+φe) |V , where φ r is the relative phase difference between the short (horizontal) and long (vertical) arms of the MZI, and φ e a controllable phase added by the modulator to the vertical contribution. Finally, a fiber PBS is used to project the light into the {|D , |A } basis, in which one obtains |ψ r = sin φr+φe . Consequently, measuring the light intensity at detector Det A allows us to infer the phase term (φ r + φ e ) mod 2π.

Figure 5 :

 5 Figure 5: Phase stabilization system of the interferometric entanglement preparation stage. The phase accumulated by a reference laser is measured and a correction signal is forwarded to a PZT. The phase can be accurately tuned to any desired value by adding an offset phase with a phase modulator.

Figure 6 :

 6 Figure 6: Demonstration of phase control capabilities. (a) Experimental setup. Alice and Bob both set their polarization state analyzers at 45• , in order to project the entangled state on the phase sensitive {|D , |A } basis. The coincidence rate between the two detectors is recorded as a function of the phase set in the fiber preparation stage. (b) Results obtained for photon-pair distribution over 20 m of standard optical fiber: for all three filters, near perfect sinusoidal coincidence rate oscillations are observed, underlining the high phase stability achieved with our approach. (c) Results obtained for the 540 MHz filter when distributing the entangled photons over 20 m and 4 km of standard optical fiber, respectively. Again, high-visibility interference patterns are observed. Note that for all results presented here, noise contributions induced by accidental coincidences have been subtracted.

Figure 7 :

 7 Figure 7: Demonstration of entanglement quality via the violation of the Bell inequalities. (a) Experimental setup. Both users a polarization state analyzer consisting of a half-wave plate (HWP), a PBS, and a SPD. The coincidence rate correlations for the standard polarization settings are recorded. For all three filters, DWDM (b), 540 MHz PSFBG (c) and 25 MHz PSFBG (d), near perfect visibility two-photon interference patterns are obtained for all polarization settings. This underlines the high quality entanglement generated with out approach.

Table 1 :

 1 Summary of the experimental results carried out with the experimental setup shown in Figure

	3 †	540 †	25 ‡

K. In the following of this paper, the PPLN/W is operated at 387 K, for which paired photons are emitted, as shown in Figure2(b), at the degenerate wavelength of 1560.48 nm, within a spectral bandwidth of

THz (↔ 32 nm). Note that such a broad bandwidth is in general unsuitable for many quantum communication applications, since, e.g., QKD in a DWDM environment requires

-200 GHz photonic bandwidths, while

International Telecommunication Union.

Acknowledgements

The authors would like to thank ID Quantique, Scontel, AOS GmbH, Draka Comteq, and Opton Laser International (Toptica Photonics) for technical support. The authors acknowledge financial support from the CNRS, the University Nice Sophia Antipolis, l'Agence Nationale de la Recherche for the 'e-QUANET' project (grant agreement ANR-09-BLAN-0333-01), the European ICT-2009.8.0 FET open program for the 'QUANTIP' project (grant agreement 244026), le Ministère de l'Enseignement Supérieur et de la Recherche (MESR), la Direction Générale de l'Armement (DGA), the Majlis Amanah Rakyat (MARA), and le Conseil Régional PACA.