Nicolas Huin

Sébastien Martin

Jérémie Leguay

Virtual Multi-Topology Routing for QoS Constraints

Keywords: Multi-Topology Routing, Traffic Engineering, Quality of Service, Segment Routing

Multi-topology routing (MTR) provides an attractive alternative to segment routing for traffic engineering when network devices cannot be upgraded. However, due to a high overhead in terms of link state messages exchanged by topologies and the need to frequently update link weights to follow evolving network conditions, MTR is often limited to a small number of topologies and the satisfaction of loose QoS constraints. To overcome these limitations we propose vMTR, an MTR extension where demands are routed over virtual topologies that are silent, i.e., they do not exchange LSA messages, and that are continuously derived from a very limited set of real topologies, optimizing each QoS parameter. In this context, we present a polynomial and exact algorithm for vMTR and, as a benchmark, a local search algorithm for MTR. We show that vMTR helps reducing drastically the number of real topologies and that it is more robust to QoS changes.

I. INTRODUCTION

Traffic Engineering (TE) policies [START_REF] Wang | An overview of routing optimization for internet traffic engineering[END_REF] are key to controlling how bandwidth is shared in IP (Internet Protocol) networks. These policies can be tuned to satisfy application requirements in terms of end-to-end delay, packet loss, and jitter, for instance. They can be optimized by a centralized controller and applied by routers to steer traffic with data plane mechanisms such as Segment Routing (SR) [START_REF] Ventre | Segment routing: a comprehensive survey of research activities, standardization efforts, and implementation results[END_REF], [START_REF] Wu | A comprehensive survey on segment routing traffic engineering[END_REF] or MPLS (Multi-Protocol Label Switching). However, most IP networks today still rely on shortest path and hop-by-hop routing from Interior Gateway Protocols (IGP).

Extensions of IGP protocols such as Multi-Topology Routing (MTR) [START_REF] Psenak | Multi-topology (MT) routing in OSPF[END_REF], [START_REF] Przygienda | M-ISIS: multi topology (MT) routing in Intermediate System to Intermediate Systems (IS-ISs)[END_REF] or FlexAlgo [START_REF] Psenak | IGP Flexible Algorithm[END_REF] provide simple solutions to create and operate multiple routing planes, or "soft" slices [START_REF] Huin | Network slicing with multi-topology routing[END_REF], to support various QoS and operational requirements. Each routing plane minimizes a different additive link metric (e.g., IGP cost, TE metric, QoS) and can exclude some links when needed (i.e., constraints in FlexAlgo or exclusions in MTR). In theory, MTR and FlexAlgo only require a software upgrade of devices to update the control plane, and they can still rely on standard IP forwarding in the data plane. However, in practice, this is only the case for MTR, as FlexAlgo is always shipped along with SRv6 or SR-MPLS. Indeed, FlexAlgo's implementations rely on SR labels to identify routing planes and forward traffic consistently on top of them. When SR is available, TE policies can be configured at ingress nodes to provide source-based routing on top of existing IGP routing planes (or "algos" when FlexAlgo is rolled out). SR policies consist of a label stack that is injected in packet headers to encode the sequence of shortest-path segments that need to be traversed by packets. In this case, a data plane upgrade, i.e., device replacement or microcode update whenever possible, is necessary to support SR forwarding. Indeed, many legacy devices cannot support SR, as mentioned in [START_REF] Tian | Traffic engineering in partially deployed segment routing over ipv6 network with deep reinforcement learning[END_REF].

In this paper, we consider deployment scenarios where most devices still operate with legacy data planes (i.e., without SR support). In this context, IGP extensions like MTR can be used to create multiple routing planes, but most routers have not been replaced by SR-capable devices. The most popular IGP protocol is Open Shortest Path First (OSPF) [START_REF] Moy | OSPF: anatomy of an Internet routing protocol[END_REF]. In this protocol, routers periodically broadcast Link State Advertisement (LSA) messages that contain metrics related to their adjacent links. Upon reception of LSA messages, routers store them in an LSDB (Link State DataBase) and maintain a Shortest Path Tree (SPT) towards all the other nodes, which decides the next hop to use for all possible destinations. This information is stored in the data plane in a Forwarding Information Base (FIB) and used to forward packets. When only one link metric is used (e.g., IGP cost), a single routing plane is available. The set of link metrics is generally designed [START_REF] Fortz | Internet traffic engineering by optimizing OSPF weights[END_REF] to satisfy some traffic scenarios. However, with a single routing plane for all applications, it is, in general, not possible to satisfy a heterogeneous set of QoS requirements.

To increase routing options, MTR offers the possibility of executing multiple IGP instances in parallel, each working over a different metric. The SPT maintained for each instance, or topology, generally aims at satisfying a set of QoS requirements (e.g., cost, hop count, delay) and administrative constraints (e.g., exclusion of links or nodes). FlexAlgo [START_REF] Psenak | IGP Flexible Algorithm[END_REF] can be seen as an extension of MTR and adds the possibility to build a routing plane over 1) a given metric, 2) considering some constraints (i.e., exclusions of nodes/links), and 3) a particular shortest path algorithm. Each algo is defined by the tuple (metric, constraints, SPT algorithm). Each MTR topology, or "algo" in FlexAlgo, has its own IP prefix so that traffic can be forwarded consistently in a hop-by-hop fashion. In both cases, link metrics also need to be periodically advertised using LSAs so that routers can update their TE database. To meet a variety of QoS constraints from applications, a large number of MTR topologies may be required to satisfy all application requirements. However, multiplying the sets of TE metrics, i.e., MTR topologies, yields a significant network overhead as LSA messages are sent for each metric. On top of that, as network conditions fluctuate over time, topologies may require frequent updates from the management system to follow the evolution of traffic and network performance. However, link weights can only be updated at a slow pace, and they cannot follow the evolution of network quality over time.

To overcome this issue, we propose a new scalable and versatile IGP extension, called virtual MTR (vMTR), to dynamically maintain a set of virtual routing topologies. This solution aims at increasing routing flexibility in multi-topology routing to follow evolving network conditions and support heterogeneous QoS requirements while reducing the overhead from LSAs and weight updates. The main idea is to route flows over virtual topologies that are silent, i.e., they do not exchange LSA messages, and that are continuously derived from a very limited set of real MTR topologies, i.e., exchanging LSAs. Each real topologies computes shortest paths for a basic QoS parameter, e.g. delay or packet loss, while virtual ones are designed to generate paths meeting multiple QoS constraints.

Searching for a shortest path satisfying one or multiple end-to-end constraints such as latency or hop count usually involves algorithms relying on Lagrangian relaxation. These algorithms aim at finding feasible Lagrangian multipliers to modify link costs so that a shortest path provides a feasible solution. Well-known heuristic algorithms are LARAC [START_REF] Juttner | Lagrange relaxation based method for the qos routing problem[END_REF], for one constraint, or GEN-LARAC [START_REF] Xiao | Gen-larac: A generalized approach to the constrained shortest path problem under multiple additive constraints[END_REF] for multiple constraints. Inspired by this, we propose to derive virtual topologies from real ones applying a linear combination of their link metrics with designed multipliers. Compared to vanilla MTR, our vMTR solution reduces the overhead as each virtual topology is only defined by a set of multipliers, one per real IGP instance, and virtual topologies do not exchange LSAs. Furthermore, routing is automatically adjusted when basic QoS metrics vary and specific link metrics do not need to be updated when traffic conditions evolve.

In this paper, we first formulate the optimization problem to find optimal multipliers so as to minimize the number of virtual topologies in vMTR. We propose a polynomial and exact algorithm that finds the minimum set of Lagrangian multipliers to meet two QoS constraints for a set of demands. Then, as a benchmark, we revisit the IGP weight design problem [START_REF] Fortz | Internet traffic engineering by optimizing OSPF weights[END_REF] to minimize the number of topologies in MTR and derive a heuristic algorithm based on local search to solve the problem. Finally, we compare MTR and vMTR with numerical results and show that the number of real MTR topologies can be drastically reduced with vMTR when accepting all traffic demands. We also observe that vMTR topologies are more robust to QoS changes than MTR ones.

The rest of this paper is structured as follows. Related work is discussed in Sec. II. The system architecture and the problem formulation are introduced in Sec. III and IV. Sec. V presents algorithmic solutions. Sec. VI describes our numerical results and Sec. VII concludes this paper.

II. RELATED WORK

The design of link weights so that traffic requirements can be satisfied is known as the IGP weight design problem. This problem is a well-studied NP-hard problem [START_REF] Bley | Routing and capacity optimization for IP networks[END_REF].

Single topology routing. Fortz and Thorup [START_REF] Fortz | Internet traffic engineering by optimizing OSPF weights[END_REF], [START_REF] Fortz | Optimizing OSPF/IS-IS weights in a changing world[END_REF] [START_REF] Ericsson | A genetic algorithm for the weight setting problem in OSPF routing[END_REF] and later combined [START_REF] Buriol | A hybrid genetic algorithm for the weight setting problem in OSPF/IS-IS routing[END_REF] with the original local search. Bley et al. [START_REF] Bley | Routing and capacity optimization for IP networks[END_REF] proposed an improved version of the local search algorithm.

A math-heuristic based on column generation has also been proposed recently [START_REF] Benhamiche | Unsplittable shortest path routing: Extended model and matheuristic[END_REF].

Other variants related to energy-minimization have been considered [START_REF] Capone | OSPF optimization via dynamic network management for green IP networks[END_REF], [START_REF] Amaldi | Energy-aware IP traffic engineering with shortest path routing[END_REF]. Robust optimization variants have been studied to consider (weighted) link failures [START_REF] Fortz | Robust optimization of OSPF/IS-IS weights[END_REF] or worst case variations of demands [START_REF] Vallet | Online OSPF weights optimization in IP networks[END_REF] (i.e., minimum / maximum values for each demand) and hoses [START_REF] Ranaweera | Preventive start-time optimization of OSPF link weights against link failure for hose model[END_REF] at ingress routers to bound aggregate traffic. Oblivious routing extensions where traffic can be load balanced over multiple paths have also been considered [START_REF] Altin | Oblivious OSPF routing with weight optimization under polyhedral demand uncertainty[END_REF], [START_REF] Duffield | A flexible model for resource management in virtual private networks[END_REF] using the hose model. Other traffic polytopes have been used to model traffic uncertainty [START_REF] Moulierac | Optimizing IGP link weights for energyefficiency in multi-period traffic matrices[END_REF].

Beyond traditional optimization techniques, machine learning [START_REF] Kodialam | Network link weight setting: A machine learning based approach[END_REF] have recently been used to solve the deterministic version of the problem. It is based on a smoothed formulation of the problem that is suitable for gradient descent.

Multi-topology routing. In the context of network slicing where end-to-end latency constraints must be met, a global optimization model along with results obtained with a genetic algorithm has been presented [START_REF] Huin | Network slicing with multi-topology routing[END_REF] to design multiple topologies (iteratively) and assign demands to them. Some papers have also proposed solution where topologies are designed offline to maximize edge-disjointness [START_REF] Wang | Adaptive multi-topology IGP based traffic engineering with near-optimal network performance[END_REF], [START_REF] Wang | Ample: an adaptive traffic engineering system based on virtual routing topologies[END_REF] or path diversity [START_REF] Mirzamany | An efficient traffic engineering based on multi-topology routing for future internet[END_REF]. These topologies are then used to route traffic based on actual QoS conditions. Other variants for MTR have been studied to consider robustness to traffic variations [START_REF] Wang | Robust traffic engineering using multitopology routing[END_REF] and failure scenarios [START_REF] Menth | Network resilience through multi-topology routing[END_REF].

Our contributions. In this paper, we extend MTR with virtual topologies that are silent, i.e., they do not exchange LSA messages, and that are continuously derived from a very limited set of real IGP topologies, i.e., exchanging LSAs.

III. MTR AND VMTR FOR QOS CONSTRAINTS

This section introduces MTR, and the vMTR extension we propose, from a system perspective. As a preliminary step, we also first formulate the QoS routing problem they address.

A. QoS satisfaction problem

Let's represent a network as a directed graph G = (V, A) where V is the set of nodes (i.e., routers) and A is the set of arcs (i.e., links). A set of basic IGP topologies T Q is associated with different QoS metrics such as delay, jitter or losses. On top of them, additional topologies, real for MTR or virtual for vMTR, can be designed to route demands that cannot be accepted on basic topologies. In this case, these topologies defined by the set of their link weights, also called resources, denoted r t a for arc a ∈ A and topology t ∈ T Q , need to be designed along with their mapping to demands. The set of traffic demands K is given as input. Each demand k goes from source node s k to destination node d k , and it has a set of QoS requirements that are defined by a maximum resource consumption r t k over basic (additive) QoS metrics from topology t ∈ T Q . Note that packet loss can be made additive using the log function. The problem is to find, for every demand k, a routing path p k satisfying a set of QoS constraints, as expressed as follows:

a∈p k r t a ≤ r t k ∀t ∈ T Q (1)
We consider a practical use case scenario where the total amount of traffic with QoS constraints is much smaller than best effort traffic. QoS constrained traffic is prioritized as high priority traffic in the data plane and the sum of high priority traffic (i.e., QoS constrained traffic) is not enough to saturate the bottleneck capacity. For this reason, we omit capacity constraints in the design of topologies and we focus only on QoS constraints satisfaction.

B. MTR: Multi-Topology Routing

When relying on multi-topology routing, the goal is to design the minimum number of topologies so that all demands have their QoS requirements satisfied. As mentioned in Sec. II, this is typically realized by solving a series of inverse shortest path trees problems. When link weights are designed and deployed, traffic is routed over a set of T D topologies, each one corresponding to a new IGP instance. Routers exchange periodically LSA messages to propagate weights and they maintain a minimum spanning tree SP T t for every t ∈ T D . Whenever network conditions evolve and the characteristic of links in terms of QoS changes, a significant update of the designed topologies, i.e. weights, may be required.

C. vMTR: virtual MTR

In the case of vMTR, additional virtual topologies need to be designed from the set of existing real topologies. The central idea behind vMTR is rooted in Lagrangian relaxation, similarly to the LARAC [START_REF] Juttner | Lagrange relaxation based method for the qos routing problem[END_REF] or GEN-LARAC [START_REF] Xiao | Gen-larac: A generalized approach to the constrained shortest path problem under multiple additive constraints[END_REF] algorithms that find a path satisfying one or multiple QoS constraints, respectively. Indeed, virtual topologies are derived from real ones using a set of Lagrangian multipliers. A set of multipliers, each one associated with a real topology, is used to calculate and maintain link weights in a virtual topology. In this way, virtual topologies are automatically updated when QoS metrics on real instances change. In addition, virtual topologies are completely silent and do not exchange LSAs. For instance, a virtual topology can be created from two real topologies on loss and delay so that (virtual) link weights are defined by λ l . Loss + λ d . Delay with a vector λ of multipliers associated to the topology. Virtual link weights are used by the shortest path tree algorithm inside routers to find constrained paths with end-to-end loss and delay constraints. Recall that, in practice, packet loss is made additive using the log function. In the general case, metrics from real topologies can be combined with any algebraic operation. However, without loss of generality, we will consider in the rest of the paper a linear combination. Considering other operations may help to further increase routing diversity and constraint satisfaction, but at the cost of some extra complexity. Indeed, Lagrangian algorithms are actually heuristic to solve the constrained shortest path problem and may not find feasible solutions when they exist (see Sec. VI). However, by design, vMTR based on a linear combination is guaranteed to route all demands for which paths can be found with LARAC.

To create and manage virtual topologies, a virtual weight designer module sits in the network controller, as depicted in Fig. 1. It takes as inputs the network topology, link metrics related to "real" IGP instances (e.g., cost, capacity, delay), a traffic matrix with a set of demands and their attributes (e.g., source and, destination, QoS requirements). It produces a set of |T D | vectors of multipliers, one for every virtual topologies, along with the mapping of demands to each topology. In practice it sends to all routers |T D | vectors to create |T D | virtual topologies, each vector being of size |T Q |, i.e., the number of existing / real IGP topologies. Every router continuously maintains a LSDB with the set of virtual link weights derived from real metrics. As virtual topologies are based on real ones, no specific LSAs need to be advertised. These topologies are almost immediately available to route traffic as routers do not need to wait for LSA exchanges. When metrics in real topologies are updated, all the virtual weights and their associated shortest path trees are updated locally by devices. Traditional Dynamic SPT [START_REF] Garg | Dynamizing dijkstra: A solution to dynamic shortest path problem through retroactive priority queue[END_REF] mechanisms implemented in IGPs can be used to stabilize the system in case of frequent updates. No routing loops can occur as multipliers and virtual metrics are positive. Unlike link weights in real topologies that need to be exchanged with LSAs, multipliers and the resulting virtual weights do not need to be integer. As for MTR, packets can be routed over a specific topology using QoS header fields like DSCP or ToS.

In terms of benefits and impact, one advantage of vMRT is the capability to explain a virtual topology as it is a combination of basic topologies related to metrics that users understand. This is impossible with MTR when weights are optimized by an algorithm. Actually, vMTR offers the possibility of manually or automatically setting multipliers to define a good tradeoff among basic topologies. An expert or customer can select the best multipliers according to his requirements. With regards to FlexAlgo, vMTR can be seen as a way to create new topologies (i.e., new metrics) over which "algos" are applied. We could even imagine an extension to the standard where multipliers are part of the algorithm definition and can be use to tune "algos" to meet QoS constraints. However, in this case, SR has to be rolled out.

Our proposition is in this paper two-fold: vMTR provides a way to create virtual topologies using multipliers and, on top of that, we propose in Sec. V-A an algorithm to automatically generate the said multipliers.

IV. TOPOLOGY DESIGN PROBLEMS

The optimization problem associated with the design of topologies for MTR, defined by link weights, consists in creating the minimum number of topologies such that all demands are accepted. This problem is similar in vMTR but differs in the way weights are computed. Indeed, weights of virtual topologies are derived from real topologies. This section presents the ILP (Integer Linear Programming) models for both cases.

In the rest of the paper, we will assume that MTR and vMTR are used to route demands that cannot be accepted over basic topologies |T Q | (each one maintaining a shortest path tree over a single QoS metric). Also, to ease the presentation, we will omit demands for which a constrained path cannot be found.

A. MTR

Let us introduce the variables to solve the topology design problem for MTR:

• x a,k,t ∈ {0, 1}: equals 1 if the demand k is routed along a path using link a in the topology t, and 0 otherwise. • π u,v,t ∈ R + : is the potential of the node u, i.e., the distance between node u and v on the topology t. • u a,d,t ∈ {0, 1}: equals 1 if the link a belongs to a shortest path towards destination d on topology t, and 0 otherwise. • w a,t ∈ R + : the weight assigned to link a ∈ A on the topology t. • y k,t ∈ {0, 1}: equals 1 if the demand k is assigned to the topology t (thus feasible for this topology), and 0 otherwise. • z t ∈ {0, 1}: equals 1 if the topology t is deployed.

Let us consider d K the set of nodes that are destinations of at least one demand. We denote by ω -(v) (resp. ω + (v)) the set of ingoing (resp. outgoing) links of node v.

As we do not know the needed number of topologies, we can consider an upper bound to create the right number of variables. This upper bound can be set by the number of designed topologies returned by the local search algorithm presented in the next section.

The ILP model is given by Fig. 2. This model is an extension of the well-known model from the literature (e.g., [START_REF] Wang | Robust traffic engineering using multitopology routing[END_REF], [START_REF] Huin | Network slicing with multi-topology routing[END_REF]) to jointly create multiple topologies. The main goal is to minimize the number of designed topologies. Constraints (3) link the activation of topologies and demands routed on top of them. Inequalities (4) ensure that each demand is assigned to one designed topology. Equalities ( 5) are the well-known flow conservation constraints. Constraints ( 6) and ( 8) link routing variables and shortest path variables. Inequalities [START_REF] Huin | Network slicing with multi-topology routing[END_REF] guarantee that the induced shortest path tree is a tree, i.e., there is at most one outgoing link at each node. Constraints ( 9) and [START_REF] Fortz | Internet traffic engineering by optimizing OSPF weights[END_REF] ensure that the shortest path tree follows the weight w. Inequalities (11) are resource constraints.

B. vMTR

For vMTR we need to add the following variables:

• λ t, t ∈ R: the multiplier associated with the basic topology t ∈ T Q and the virtual topology t ∈ T D . and the following constraints:

w a, t = t∈T Q λ t, tw a,t ∀a ∈ A, ∀ t ∈ T D (12)
Remark that in the case of vMTR, we complete the designed virtual topologies with real MTR topologies when all demands cannot be accepted with vMTR. To manage the two kinds of designed topologies, we penalize in the objective function the activation of MTR topologies.

V. ALGORITHMIC SOLUTIONS

This section presents algorithms for MTR and vMTR. While multiple algorithms exist for MTR (see Sec. II), we build upon one of them [START_REF] Bley | Routing and capacity optimization for IP networks[END_REF] to focus on QoS constraint satisfaction and minimize the number of designed topologies.

A. MTR Algorithm

We developed a greedy algorithm to design the sets of weights, i.e., topologies, as shown in Algo. 1. It iteratively generates random weight vectors, fine-tuned to maximize the number of accepted demands using a local search algorithm. MTR topologies are created one by one, each time trying to accept a maximum number of demands. If the local search fails to generate a weight vector accepting any demands, it falls back to computing a valid resource-constrained shortest path for one of the remaining demands (using the TAMCRA algorithm [START_REF] De Neve | Tamcra: a tunable accuracy multiple constraints routing algorithm[END_REF]), then uses that path for building a valid weight vector from a new random one.

min t∈T D z t (2) k∈K y k,t ≤ z t ∀t ∈ T D (3) t∈T D y k,t ≥ 1 ∀k ∈ K (4) a∈ω + (v)
x a,k,t -

a∈ω -(v) x a,k,t =      y k,t if v = s k , -y k,t if v = d k , 0 otherwise; ∀v ∈ V, ∀k ∈ K, ∀t ∈ T D (5) x a,k,t ≤ u a,d k ,t ∀a ∈ A, ∀k ∈ K, ∀t ∈ T D (6) a∈ω + (v) u a,d,t ≤ 1 ∀v ∈ V, ∀d ∈ d K , ∀t ∈ T D (7) u a,d,t ≤ k∈K:d=d k x a,d k ,t ∀a ∈ A, ∀d ∈ d K , ∀t ∈ T D (8)
w (u,v),t -π u,d,t + π v,d,t ≥ 1 -u (u,v),d,t ∀(u, v) ∈ A, ∀d ∈ d K , ∀t ∈ T D (9) w (u,v),t -π u,d,t + π v,d,t ≤ M (1 -u (u,v),d,t ) ∀(u, v) ∈ A, ∀d ∈ d K , ∀t ∈ T D ( 10
)
a∈A t∈T D r t a x a,k,t ≤ r t k ∀k ∈ K, ∀t ∈ T Q (11)
Fig. 2: Integer linear problem for the design of topologies in MTR.

The greedy algorithm uses a run-of-the-mill local search that encodes a solution by its weight vector, and the neighborhood of a solution is the set of weight vectors that differ from the solution by a single element. The fitness function used to evaluate a solution computes the shortest path tree according to the weight vector and counts the number of demands with a path respecting their required QoS.

Bley et al. [START_REF] Bley | Routing and capacity optimization for IP networks[END_REF] already proposed a local search algorithm for the IGP weight design problem, but we consider different assumptions. Firstly, while they consider an extended neighborhood (compared to ours) that takes into account load balancing, we only focus on the weight vector variation. Secondly, while they consider weights in the range [0, 20], we consider the whole range of possible OSPF weights ([0, 2 16 ]) The last point is particularly important since the definition of their neighborhood considers atomic changes in the weight (±1), which is quite effective considering the range of the weights. However, since we allow a much larger range of weights, we need to define a neighborhood that guarantees changes in the shortest path trees induced by the weights.

We thus generate meaningful neighbors (see line 25) computing the minimum increase or decrease in weight needed to change the shortest path trees. We call these values the up and the down delta-weight of a link, respectively. We denote by π(u, v) the distance of the shortest path from u to v with the current link weights.

First, we define the down delta-weight δ - v (a) of a link a = (s a , d a ) as the weight decrease needed to insert the link a in the current shortest path tree T v , rooted at v, in the place of the link a ′ = (s ′ a , d a ). Formally, we are looking for a value

δ - v (a) such that π(v, s a ) + w (a) -δ - v (a) < π(v, t a ), δ - v (a) = π(v, s a ) + w (a) -π(v, t a )
Second, we define the up delta-weight δ + v (a) of a link a = (s a , d a ) as the weight increase needed to remove the link from the current shortest path tree SP T v , rooted at v. Formally, we are looking for a value δ + v (a) such that π(v, s a ) + w (a) + δ + v (a) > min

a ′ ∈ω -(da)\{a} (π(v, s a ′ ) + w (a ′ ) ).
Thus, the up delta weight of a link a is given by

δ + v (a) = min a ′ ∈ω -(da)\{a} (π(v, s a ′ ) + w (a ′ ) ) -π(v, s a ) -w (a) (13) = min a ′ ∈ω -(da)\{a} δ - v (a ′ ) (14)
Once we computed the delta-weight of each link, we can propagate them (see line 40) up the tree. Indeed, any weight change occurring in the higher depth of the tree will impact the distance of the descendants. Given a link a 1 = (s 1 , d a1 ) and its swapped link a 2 = (s 2 , d a1 ), we can propagate their up and down delta-weight up to their lowest common ancestor in the tree. Formally, if u is the lowest common ancestor of s a1 and s a2 in SP T , and p T (v 1 , v 2 ) is the path from v 1 to v 2 in SP T , we have the following properties:

∀a ∈ p T (u, s 1 ), δ + (a) = δ + (a 1 ), ∀a ∈ p T (u, s 2 ), δ -(a) = δ -(a 2 )
Finally, we need to consider all possible shortest path trees in the graph. We can combine their up and down delta-weight by taking the minimum over all trees. w ← LOCALSEARCH(G, w, d, l, K)

6:

K ′ ← GETACCEPTEDDEMANDS(G, w, d, l, K) 7: if K ′ = ∅ then 8:
p ← TAMCRA(G, d, l, K) for all a ∈ p do 11:

wa ← 1

12:

K ′ ← GETACCEPTEDDEMANDS(G, w, d, l, K) 13:
T ∈ T ∪ (w, K ′ ) 14:

K ← K \ K ′ 15: function LOCALSEARCH(G = (V, A), w, d, l, K) 16: K ← GETACCEPTEDDEMANDS(G, w, d, l, K) 17: (w * , K * ) ← (w, GETACCEPTEDDEMANDS(G, w, d, l, K)) 18:
for i ← 0 to maxIte do

for all a ′ ∈ p T (v A ,sa) do 43: δ -(a ′ ) ← min(δ -(a ′ ), d(sa) + w(a) -d(ta)) 44: for all a ′ ∈ p T (v A ,ta) do 45: δ + (a ′ ) ← min(δ + (a ′ ), d(sa) + w(a) -d(ta))

B. vMTR Algorithm

The weight design problem for vMTR consists in maximizing the number of accepted demands (first objective) while minimizing the number of virtual topologies (second objective). To create a virtual topology with vMTR over 2 basic topologies (e.g., delay and loss), we consider a single multiplier λ that provides the following virtual weight r t a = r 1 a + λr 2 a for each link a ∈ A. A multiplier λ is said to be feasible for demand k if the shortest path p over link metrics r t a satisfies the maximum constraint on the end-to-end consumption for both resources, i.e., on r 1 k and r 2 k . In the following, we first present some properties for this particular case with 2 resources and then, we propose an exact algorithm that solves the problem in polynomial time.

Lagrangian properties. As one multiplier can cover several demands, a natural question is: does there exist one or several intervals of feasible multipliers for a given demand? In this section, we provide an answer to this question that allows solving the vMTR problem in polynomial time afterward.

Let us introduce the Constraint Shortest Path problem (CSP) where the goal is to find a path minimizing a resource under an additive end-to-end resource constraint. In [START_REF] Ying | The constrained shortest path problem[END_REF], the authors propose and analyze the LARAC algorithm, a polynomial time heuristic solution for the CSP problem based on Lagrangian relaxation. This algorithm computes a multiplier λ * and the solution it returns is the shortest path p λ * in the graph where the cost of each link a ∈ A is given by r 1 a + λ * r 2 a . The most interesting claim proved in [START_REF] Ying | The constrained shortest path problem[END_REF] for us is given below.

Claim 1. [34] If λ < λ * , then a∈p λ r 2 a ≥ r 2 k and if λ > λ * , then a∈p λ r 2 a ≤ r 2
This claim is the core of our vMTR algorithm. We denote by λ 1 * k (resp. λ 2 * k ) the multiplier provided by LARAC algorithm when resource 2 (resp. resource 1) is minimized and resource 1 (resp. resource 2) is constrained. As proved in [START_REF] Ying | The constrained shortest path problem[END_REF] this multiplier is optimal.

The following proposition ensures that, for a given demand, only one range of multiplier is feasible for the two resources.

Proposition 1. For a given demand k ∈ K, a multiplier λ k is feasible for k, if λ 2 * k ≤ λ k ≤ 1 λ 1 * k .
Proof. Let us consider the shortest path p where link costs r 2 a + λ 1 * k r 1 a , ∀a ∈ A are considered. The path p is also the shortest path in the graph where link costs are r 1 a + 1

λ 1 * k r 2 a , ∀a ∈ A. Indeed, for each a ∈ A, dividing r 2 a +λ 1 * k r 1 a by λ 1 * k induced r 1 a + 1 λ 1 * k r 2
a . Thus, thanks to the claim 1, all λ 2 k ≥ λ 2 * k induced a shortest path where resource 2 is respected and all 1

λ 1 k ≤ 1 λ 1 * k induced a shortest path where resource 1 is respected. Thus all λ k such that λ 2 * k ≤ λ k ≤ 1 λ 1 *
k are feasible for the two resources.

For a given multiplier λ, we may have several paths that are the shortest, but one of them can be unfeasible for resource 1 or 2. To address this issue, we modified the Dijkstra algorithm to find all shortest paths between two nodes in a graph and we slightly changed the λ value by adding an ϵ until all shortest paths are valid or only one path is the shortest.

In the following, for a given demand k, we denote λ min is discarded as it cannot be routed with vMTR. At the end of the procedure, the pool of discarded demand is processed with the MTR algorithm (Sec. V-A). The overall goal is to minimize the number of MTR topologies, as they generate LSAs and do not adapt to evolving network conditions.

Exact algorithm. The goal of this algorithm is to find a set of multipliers Λ such that all demands are covered. A multiplier λ covers a demand k if λ is feasible for the demand (see Proposition 1). At each step, we find a new multiplier to add in Λ such that some uncovered demands become covered with the new multiplier. More formally. Let K be the set of demands covered at the current step, and respectively, K \ K the set of uncovered demands.

Algo. 2 describes the whole algorithm and an example is given in Fig. 4. First, the algorithm orders the λ max k in increasing order. Then, at each step, it selects a λ max k * to add in Λ such that all uncovered commodities have an associated λ max k greater or equal to the selected λ max k * . It continues until all demands are covered.

Complexity and generalization. To summarize the approach above, the problem is solved in two steps. First, we define the space of valid multipliers for each demand. Second, we search the minimum number of multiplier vectors such that return Λ all demands are covered. The dimension of the space of valid multipliers is equal to |T Q | -1. For the case with 2 basic topologies, we have only 1 dimension, which can be represented by an interval graph and the problem is polynomial as presented above. If we consider 3 basic topologies, we have 2 dimensions that can be represented by a plane. In this case, we can consider that the area of valid multipliers, vector of size 2, represents a rectangle, which is a particular case of our problem. This kind of graph is called boxicity 2-graph. For the interval graph or a boxicity 2-graph, the selection of the minimum number of multiplier vectors consists in solving a clique cover problem in the graph H = (K, I) where K is the set of demands and I is the set of links. Two demands are linked if and only if they share a common valid multiplier vector. This problem is NP-hard for a boxicity 2-graph which is a particular case of the problem when 3 basic topologies are considered and polynomial for an interval graph [START_REF] Imai | Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane[END_REF].

VI. PERFORMANCE EVALUATION

We now compare MTR and vMTR over a set of instances generated from SNDlib [START_REF] Orlowski | SNDlib 1.0-Survivable network design library[END_REF].

Description of instances. These instances contain the geographical coordinates for each node and the link capacities between them. We selected 15 large topologies. For each network, we have considered two resources, the loss and the delay. For each link, the packet loss is set to be inversely proportional to the link capacity and the delay is considered to be proportional to the distance. To avoid trivial instances, we generated all demands between each pair of routers. We set the maximum end-to-end loss (resp. delay) to one of the shortest path on the loss (resp. delay) minus ϵ. This ensures that the demands cannot be routed on basic topologies, i.e., loss or delay. For each demand, we solved optimally, i.e., not using LARAC, the constraint shortest path problem to ensure that a solution exists. If no solution exists for a demand then we did not consider it. All instances are available at [START_REF] Huin | Loss/delay instances for vIGP[END_REF].

Numerical results. Fig. 5 presents violin plots to summarize results over all 15 instances. We can see, in Fig. 5a, that vMTR reduces the number of real topologies compared to MTR, from an average of 8.41 to 5.41. It also reduces the maximum number of needed topologies from 20 to 12, and it also avoid the need for any real topology in some cases. vMTR requires an average of 4.08 virtual topologies. The decrease in the number of real topologies can be easily explained by looking at the number of average demands per topologies in Fig. 5b. vMTR virtual topologies can accept more demands than MTR real ones: in average, a vMTR topology holds 32.7 demands while, in average, MTR topologies hold 18.7 demands. In some cases, we can even have virtual topologies with 131.4 demands. However, real topologies in vMTR hold less demands because the remaining demands not covered by a virtual topology are harder to cover with the greedy MTR algorithm. Recall that as our implementation of vMTR with a linear combination of real topologies is based on the same principle as LARAC, which is a heuristic solution to the CSP problem, some demands cannot be satisfied with virtual topologies and real ones need to be added to ensure 100% traffic acceptance. Fig. 5c shows the execution times for both solutions. For vMTR, we present the time required to compute lambda multipliers for virtual topologies as well as the time needed by the greedy algorithm to provision the remaining demands over real topologies. vMTR, once again, shows an improvement over MTR with an average time of 86.2s over 141.3s, and it also decreases the maximum time from 740.5s to 411.1s. The computational time to compute only the virtual topologies is really small. It is due to the polynomiality of the algorithm.

Finally, we study the robustness of the two approaches Fig. 6: Violin plots of the QoS robustness which is, for a demand and a metric, the ratio between the end-to-end QoS of the provided paths and the requested QoS. A value of 1 for a given metric means that the path of a demand is at its maximum resource consumption. against changes of QoS metrics. To this end, we define the QoS robustness of a demand, for a given QoS metric, as the ratio of its provisioned path QoS over its end-to-end QoS requirement. A QoS robustness of 1 means that the provided path QoS equals the required QoS, and any increase of this QoS metric in the path's link would invalidate the path. While we can see in Fig. 6 that both solutions provision demands on strict paths (ratio of 1), on average, vMTR provides paths that are more robust to QoS changes than MTR (0.75 and 0.79 for vMTR vs 0.85 and 0.88 for MTR, for delay and losses, respectively). Remark that the robustness is slightly better from the delay point of view if we compare with the Loss. This is due to the distribution of metric values in our instances. For the two solutions, we have a better maximum robustness for the delay than for the loss. We then have more flexibility to improve the robustness of the delay.

VII. CONCLUSION

We introduced a new IGP extension, called virtual MTR (vMTR), to dynamically maintain a set of virtual routing topologies. This solution aims at increasing routing flexibility in multi-topology routing to follow evolving network conditions and support heterogeneous QoS requirements while reducing the overhead from LSAs and weight updates. We presented a polynomial and exact algorithm for vMTR when virtual topologies are derived from two real topologies and a local search algorithm for MTR as benchmark. We showed that vMTR can help reducing the number of real topologies and is more robust to QoS changes.

Future work along these lines include several directions. First, we may propose an efficient algorithm for the design of MTR topologies jointly with the creation of virtual ones in vMTR. We can also design lambda multipliers considering traffic scenarios or uncertainty sets on traffic. Also of interest, we may consider non additive and non linear ways to combine real topologies. Finally, we may develop efficient algorithms for the hard case where more than two real topologies are used to derive virtual ones.

Fig. 1 :

 1 Fig. 1: System overview for vMTR.

Algorithm 1

 1 Greedy algorithm for MTR 1: function GREEDY(G = (V, A), d, l, K)

19 : 35 :

 1935 W ← GENERATENEIGHBORHOOD(w, G, d, l) 20: w ← arg maxw∈W GETACCEPTEDDEMANDS(G, w, d, l) 21: K ←GETACCEPTEDDEMANDS(G, w, d, l, K) 22: if | K| > |K * | then 23: (w * , K * ) ← (w, K) 24: return w * 25: function GENERATENEIGHBORHOOD(w, G = (V, A)) 26: (δ + , δ -) ←COMPUTEDELTAWEIGHTS(G, w) function COMPUTEDELTAWEIGHTS(G = (V, A), w) 36: δ + ← {∞|a ∈ A} 37: δ -← {∞|a ∈ A} 38: for all v ∈ V do 39: T, d ← DIJKSTRASHORTESTPATHTREE(G, v, w) ▷ Get shortest path tree with distances 40: for all (sa, ta) ∈ A \ T do 41: vA ← LOWESTCOMMONANCESTOR(T, sa, ta) 42:

.Fig. 3 :

 3 Fig. 3: Example of non shortest paths uniqueness with λ = 13.

𝜆

Fig. 4 :

 4 Fig. 4: Example with 6 demands. Intervals between λ min k and λ max k are represented for each demand k. The first multiplier λ 1 covers the green demands, the second multiplier λ 2 covers the blue demands and λ 3 covers the orange demand. and λ max k for each demand k ∈ K. Each demand where λ min k

Algorithm 2 4 :

 24 Weight design algorithm for vMTR1: function (G = (V, A), K) 2: K = ∅, Λ = ∅ 3: σ ordering of demands k ∈ K in increasing order of λ max k while K \ K ̸ = ∅ do 5: λ * the smallest λ max k such that k ∈ K \ K 6:add in K all demands covered by λ * not in K 7:add λ * in Λ 8:

Fig. 5 :

 5 Fig. 5: Comparison of MTR and vMTR using violin plots.

  have initially proposed a local search algorithm where the main

	Symbol	Meaning
	K	Set of demands
	s k , d k r t a	Source and destination for demand k ∈ K Resource on arc a ∈ A for topology t
	T Q	Set of basic topologies (each related to a different QoS metric)
	T D r t k SP T t	Set of designed topologies Max resource consumption for demand k over topology t Shortest path tree on topology t
	λt	Vector of |T Q | multipliers to design vMTR topology t
		TABLE I: List of key notations.
	idea is to change a few link weights at each time step.
	They minimize a piece-wise linear cost function related to
	MLU (Maximum Link Utilization). Right after, a genetic
	algorithm has been proposed by Ericsson et al.


Alexis Wartelle
email: alexis.wartelle@ens-lyon.org

Marisel Di

Pietro Martínez

Olivier Fruchart

Philippe David

Guillaume Beutier

Small-angle X-ray resonant magnetic scattering at the Co M 2,3 and L 3 edges observed with photoemission electron microscopy

come

I. Introduction

Among the properties of matter which may be probed with X-rays, magnetism holds a somewhat peculiar place. In many, if not most situations, the interactions between such radiation and magnetic moments are several orders of magnitude weaker than the interactions originating from the material's electronic density [START_REF] Als-Nielsen | Elements of Modern X-Ray Physics[END_REF]. Yet, a variety of approaches have been developed over the years to explore magnetic orders using Xrays, from spectroscopy [START_REF] Baudelet | Magnetic properties of neodymium atoms in Nd-Fe multilayers studied by magnetic X-ray dichroism[END_REF][START_REF] Chen | Experimental Confirmation of the X-Ray Magnetic Circular Dichroism Sum Rules for Iron and Cobalt[END_REF] to diffraction [START_REF] Johnson | Cu3Nb2O8: A Multiferroic with Chiral Coupling to the Crystal Structure[END_REF][START_REF] Biffin | Unconventional magnetic order on the hyperhoneycomb kitaev lattice in βli2iro3: Full solution via magnetic resonant x-ray diffraction[END_REF][START_REF] Beutier | Band Filling Control of the Dzyaloshinskii-Moriya Interaction in Weakly Ferromagnetic Insulators[END_REF][START_REF] Misawa | Chirality and magnetic quadrupole order in pb(tio)cu4(po4)4 probed by interference scattering in resonant x-ray diffraction[END_REF] through imaging [START_REF] Kimling | Photoemission electron microscopy of threedimensional magnetization configurations in core-shell nanostructures[END_REF][START_REF] Streubel | Retrieving spin textures on curved magnetic thin films with full-field soft X-ray microscopies[END_REF][START_REF] Hierro-Rodriguez | Observation of asymmetric distributions of magnetic singularities across magnetic multilayers[END_REF][START_REF] Donnelly | Three-dimensional magnetization structures revealed with X-ray vector nanotomography[END_REF]. The latter field exploits the chemical selectivity and strongly enhanced scattering cross-sections [START_REF] Stöhr | Magnetism: From Fundamentals to Nanoscale Dynamics[END_REF][START_REF] Van Der Laan | Soft X-ray resonant magnetic scattering of magnetic nanostructures[END_REF] available at photon energies close to certain absorption edges of magnetic elements, where X-ray Resonant Magnetic Scattering (XRMS) occurs. Of particular interest are the L 2,3 absorption edges of 3d transition metals such as Fe, Co, and Ni, as well as the M 4,5 edges of rare-earth elements, such as Gd, Dy or Ho. These have been of crucial importance in the development of modern X-ray magnetic microscopy techniques [START_REF] Cheng | Studies of nanomagnetism using synchrotron-based x-ray photoemission electron microscopy (X-PEEM)[END_REF][START_REF] Donnelly | Imaging three-dimensional magnetic systems with X-rays[END_REF], which use XRMS as a contrast mechanism.

The most frequently used manifestation of XRMS is X-ray Magnetic Circular Dichroism (XMCD) [START_REF] Stöhr | Magnetism: From Fundamentals to Nanoscale Dynamics[END_REF], which is wellsuited to several X-ray microscopy techniques e.g. (Scanning) Transmission X-ray Microscopy [(S)TXM] or PhotoEmission Electron Microscopy (PEEM) [START_REF] Donnelly | Imaging three-dimensional magnetic systems with X-rays[END_REF]. However, the phase contrast counterpart of XMCD, namely X-ray Magnetic Circular Birefringence (XMCB), may also be used in experiments sensitive to sample refraction. This is relevant notably for Fourier Transform Holography (FTH) [START_REF] Stroke | Attainment of high resolutions in wavefront-reconstruction imaging[END_REF][START_REF] Guizar-Sicairos | Holography with extended reference by autocorrelation linear differential operation[END_REF] and ptychography [START_REF] Donnelly | Imaging three-dimensional magnetic systems with X-rays[END_REF][START_REF] Rodenburg | Hard-X-Ray Lensless Imaging of Extended Objects[END_REF]. As was demonstrated both in magnetic FTH [START_REF] Scherz | Phase imaging of magnetic nanostructures using resonant soft x-ray holography[END_REF][START_REF] Di Pietro Martínez | Three-dimensional tomographic imaging of the magnetization vector field using Fourier transform holography[END_REF] and in ptychography [START_REF] Donnelly | High-resolution hard x-ray magnetic imaging with dichroic ptychography[END_REF][START_REF] Neethirajan | Soft X-ray phase nano-microscopy of micrometre-thick magnets[END_REF], XMCB may not only yield substantial magnetic contrast, but also allow to work at photon energies significantly below the relevant peak of X-ray absorption. This in turn enables transmission imaging of relatively thick samples [START_REF] Di Pietro Martínez | Three-dimensional tomographic imaging of the magnetization vector field using Fourier transform holography[END_REF][START_REF] Neethirajan | Soft X-ray phase nano-microscopy of micrometre-thick magnets[END_REF].

While the typical spatial resolutions available with magnetic X-ray microscopy are not better than in electronbased techniques like electron holography and Lorentz microscopy [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF] or even (depending on the case) in Magnetic Force Microscopy (MFM) [START_REF] Abelmann | Magnetic Force Microscopy -Towards Higher Resolution[END_REF], their primary advantage lies with the direct sensitivity to magnetization, as opposed to induction along the electron path or to magnetostatic charges. Furthermore, as the X-ray attenuation lengths vary (for absorption edges relevant for XRMS) from ca. a dozen nanometres to several hundreds of nanometres in the soft X-ray regime, there is ample freedom to adjust the probed depth of material experimentally.

In the specific case of X-ray PEEM, this large variability provides a unique versatility when three-dimensional samples are studied. Indeed, XMCD-PEEM can usually yield a (mostly surface-related) image of the sample viewed from the top, but the transmission-type magnetic information in the object's elongated shadow may or may not be retrieved depending on signal-over-noise ratio. If the sample's absorption is too strong, there is no shadow contrast [START_REF] Hertel | Three-dimensional magneticflux-closure patterns in mesoscopic Fe islands[END_REF], whereas larger X-ray transmission allows for valuable, volume-related magnetic signal [START_REF] Kimling | Photoemission electron microscopy of threedimensional magnetization configurations in core-shell nanostructures[END_REF][START_REF] Da Col | Observation of Bloch-point domain walls in cylindrical magnetic nanowires[END_REF][START_REF] Bran | Spin configuration of cylindrical bamboo-like magnetic nanowires[END_REF][START_REF] Wartelle | Bloch-point-mediated topological transformations of magnetic domain walls in cylindrical nanowires[END_REF]. However, strictly speaking, shadow XMCD-PEEM images are not simply an elongated transmission view of a 3D object. As was noted by Jamet et al. [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF], sub-micron objects do create Fresnel diffraction patterns in the close vicinity of their shadow's rim. There appeared to be magnetic scattering in the corresponding XMCD-PEEM images at the Fe L 3 edge, however the confinement of the interference pattern is an obstacle to investigating small-angle XRMS from 3D objects with PEEM.

Considering the above, it is rather natural to look for a possibility to (i) decrease the Fresnel number [30] to suppress this confinement (ii) while preserving magnetic sensitivity (iii) as well as coherence. One route, when considering 3d transition metals, consists in moving from their L 2,3 edges down to M 2,3 edge. Indeed, the wavelength increases by an order of magnitude, XRMS is still sizeable [START_REF] Valencia | Faraday rotation spectra at shallow core levels: 3p of Fe, Co, and Ni[END_REF], and the coherence of synchrotron radiation is only expected to increase significantly [START_REF] Als-Nielsen | Elements of Modern X-Ray Physics[END_REF].

We point out that the relevance of XRMS at the Co M 2,3 edges for imaging has already been demonstrated by the work of Kfir et al. [START_REF] Kfir | Nanoscale magnetic imaging using circularly polarized high-harmonic radiation[END_REF], who employed extreme-ultraviolet FTH to recover the domain pattern in a Co/Pd multilayer. It was shown [START_REF] Kfir | Nanoscale magnetic imaging using circularly polarized high-harmonic radiation[END_REF] that the application of Coherent Diffraction Imaging (CDI [START_REF] Chapman | High-resolution ab initio three-dimensional x-ray diffraction microscopy[END_REF]) algorithms to the collected diffraction data (with the FTH images as starting point) enabled substantial increase both in signal-over noise ratio and spatial resolution. With respect to this approach and similar reports [START_REF] Tadesse | High resolution XUV Fourier transform holography on a table top[END_REF][START_REF] Eschen | Towards attosecond imaging at the nanoscale using broadband holography-assisted coherent imaging in the extreme ultraviolet[END_REF] utilizing high-order harmonic generation [START_REF] Mcpherson | Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases[END_REF], synchrotronbased PEEM at the same photon energy is expected to deliver a three orders of magnitude better energy resolution and a much larger photon flux. Last but not least, phase retrieval from small-angle scattering patterns acquired in a reflection geometry has been demonstrated by Menteş et al. [START_REF] Menteş | Coherent x-ray scattering in an XPEEM setup[END_REF]. There, the authors needed PEEM images of their artificial spin ice systems as input for the CDI algorithms. However, for a given sample-to-detector distance, the oversampling criterion [START_REF] Chapman | High-resolution ab initio three-dimensional x-ray diffraction microscopy[END_REF] is harder to fulfil at the Fe L 3 edge than at its M edge. Therefore, it seems that CDI in the above-mentioned, PEEM-based reflection geometry may become relevant for magnetic imaging at such energies. Two strong arguments to pursue this are (i) the fact that this modus operandi allows imaging under field since this is a photon-in, photon-out technique [START_REF] Menteş | Coherent x-ray scattering in an XPEEM setup[END_REF] as opposed to regular XMCD-PEEM, and (ii) the absence of need for alignment of many sample images before computing a difference image (to reveal magnetic contrast) with good stastistics.

With the above in mind, we explore in this work X-ray Resonant Magnetic Scattering (XRMS) from 3D objects at the Co M 2,3 edges, using a PEEM for imaging. To the best of our knowledge, this the first experimental application of extreme UV light to magnetic Fresnel diffraction in the specific configuration imposed by X-ray PEEM. The manuscript is organized as follows. To begin with, we describe the Co 1-x Gd x bead samples which are used in this work, as well as the laband synchrotron-based imaging methods that have been used. We then move on to the results that have been obtained, starting with a pre-characterization, then presenting the magnetic PEEM imaging. In this respect, we establish a solid overview of our samples' magnetic states with XMCD-PEEM at the Co L 3 edge, and as a second step perform imaging at the Co M 2,3 edges. There, we present our interpretation of the obtained scattering patterns.

II. Methods

A. Samples

Our sample fabrication began with Si substrates onto which a Ta\Pt bilayer was deposited for the sake of surface electrical conductivity. Based on prior calibrations of deposition rates in the used magnetron sputtering, the nominal stacking should be Ta(6 nm)\Pt(9 nm). Then, a pattern consisting of alignment marks was etched into the bilayer until the underlying Si became exposed, thanks to optical lithography and subsequent ion beam etching. During this second step, mass spectrometry allowed to interrupt the etching when all of the Ta had been removed inside the resist's patterned grooves.

Substrate pieces of adequate size were then introduced into a Pulsed Laser Deposition (PLD) system. There, large-fluence laser shots on a target with composition Co 80 Gd 20 led to the formation of Co 1-x Gd x beads on the substrate. Although we expect their stoichiometry to not differ too much from the target's, we do not have quantitative composition measurements on these samples [START_REF]X-ray ptychographic imaging performed on one such Co1-xGdx bead at the Gd L3 edge (not shown here) suggests a homogeneous composition in the volume[END_REF]. After several dozens of laser impacts on this first target, a Nb coating of a few nanometres was deposited with PLD so as to obtain a passivation layer and partially protect the beads from oxidation.

The PLD approach leads to a wide variety in terms of shapes and sizes. A small glimpse into the latter is provided in Fig. 1, whose Scanning Electron Microscopy (SEM) images display objects with lateral dimensions ranging from 200 nm to 3 µm. Since these images do not convey a full sense of shapes, we add that the beads may be close to spherical, flattened and globular, or (much) more distorted. Furthermore, a small proportion of them is somewhat polyhedral in shape, as is exemplified by the bead on the top left of Fig. 1. Though interesting in its own right, this stark contrast with the smoother and more round majority of objects goes beyond the scope of this manuscript. We point out that this collection is representative of objects on which Energy-Dispersive X-ray analysis was performed and confirmed the presence of Co 1-x Gd x ; however the specific beads shown in Fig. 1 have not been checked in this fashion themselves. There is a possibility that some of these are actually pure Nb, as we have determined that such beads may also be present. Yet, we insist that observations of dozens of Co 1-x Gd x beads lead us to consider these SEM image as typical.

B. Magnetic Force Microscopy imaging

Due to the aforementioned diversity in Co 1-x Gd x bead shapes and sizes, it would not be very informative to try to perform magnetometry measurements e.g. with a Vibrating Sample Magnetometer. Since magneto-optical measurements would be hindered by the curved bead surfaces as well as the typical size being at most a couple of times the light's wavelength, we turned to Magnetic Force Microscopy (MFM). Our goal was gain some insight into the magnetic configurations that can be present in our samples. All of our MFM acquisitions followed the common twopass scheme. The Atomic Force Microscopy (AFM) and MFM images presented in Section III were obtained with a custom-coated probe. A 25 nm-thin Co 80 Cr 20 layer was sputter-deposited on an AC240TS AFM tip from Asylum Research before the addition of a few nanometres of SiO 2 for protection against oxidation. The direction of the tip's magnetization was controlled by setting it in close proximity to a permanent magnet (with unambiguously identified poles) before imaging. The tip's oscillation amplitude was 25 nm, the second-pass tip lift was set to 11 nm, and the recorded MFM signal was the tip's oscillation phase with respect to the phase of its AC excitation. Therefore, a negative phase corresponds in our convention to attractive forces [START_REF] Thiaville | Magnetic force microscopy: Images of nanostructures and contrast modeling[END_REF]. We point out that the MFM data presented in this manuscript was acquired after removing the electrostatic contribution to the oscillation phase thanks to a suitably chosen applied bias voltage between tip and sample [START_REF] Jaafar | Distinguishing magnetic and electrostatic interactions by a Kelvin probe force microscopy-magnetic force microscopy combination[END_REF].

In addition to this dataset, MFM images of Co 1-x Gd x beads acquired with a different instrument are presented in the Supplemental Material.

C. Synchrotron-based imaging and spectroscopy

All the synchrotron experiments relevant to this work were performed at the undulator beamline Nanospectroscopy of Elettra (Sincrotrone Trieste). Our first step consisted in identifying Co 1-x Gd x beads with certainty; to that end, we performed X-ray Absorption Spectroscopy around the Co L 3 edge. Based on the obtained absorption spectra recorded on the top of the beads, we performed XMCD-PEEM imaging at energies up to the absorption peak, and as far down as 3.4 eV below that peak. The reason for this is the sharply-varying Xray penetration depth in this photon energy range. To set the scales, let us recall that for pure Co, taking magnetism into account, the X-ray attenuation length varies from 0.51 µm at the foot of the L 3 edge to as low as 12 nm to 21 nm at the absorption peak [START_REF] Nakajima | Electron-yield saturation effects in L-edge X-Ray Magnetic Circular Dichroism spectra of Fe, Co, and Ni[END_REF] i.e. over less than 6 eV.

As a result, we go from sensitivity to only the sample volume (in shadow XMCD-PEEM) well below the absorption peak to pure surface sensitivity at the top of the L 3 edge's white line. We note that the most suitable shift in photon energy (with respect to the peak of absorption) for shadow XMCD-PEEM imaging depends on both the bead's geometry and its composition. In any case, this optimization of magnetic contrast as a function of energy allows to probe relatively large depths of material, as has been reported by several groups [START_REF] Di Pietro Martínez | Three-dimensional tomographic imaging of the magnetization vector field using Fourier transform holography[END_REF][START_REF] Neethirajan | Soft X-ray phase nano-microscopy of micrometre-thick magnets[END_REF]. For each acquisition, tens of individual snapshots with exposure times on the order of a second were first recorded, the first half with a given sign of photon helicity, the second one with opposite photon helicity. The polarization state available on the Nanospectroscopy beamline is elliptical, with a degree of circular polarization of about 0.77 around the Co L 3 edge. In order to compensate for sample drift during exposure, each set of images was aligned using sharp features such as bead edges as references. Averaging the aligned series resulted in two micrographs I + and I -corresponding to opposite helicities; the latter two were then aligned before the asymmetry ratio A = (I + -I -)/(I + + I -) was computed for each pixel. In the following, we will refer to A as "magnetic image". The methodology for the final alignment consisted in minimizing the typical antisymmetric contrast in A that arises from under-/overcompensation of drift in both the horizontal and vertical directions.

Due to the strong absorption of Nb in the vicinity of the Co M 2,3 edges (at about 60 eV), we were initially not able to detect the latter with XAS. Therefore, we first performed three 30 min-long rounds of etching via Ar plasma, at operating pressures ∼3 mbar and accelerating voltage 1.5 kV, and a final 30 min round at an accelerating voltage of 1.2 kV. Thereafter, the observation of the Co M 2,3 edges indicated that magnetic imaging at these energies had become possible.

The Co M-edge magnetic images were acquired and processed similarly to the L 3 edge data. However, the total exposure times were twice larger for the sake of signal-overnoise ratio. Concerning image processing, we point out that the final alignment of M-edge data was determined not only by considering the above-mentioned criterion regarding driftrelated asymmetric contrast but also the behaviour of A on the substrate, away from beads, their shadow or interference patterns. For a set P of pixels corresponding to a (reasonably small) patch on the substrate, if the sample drift is nearly compensated, then we may expect the following behaviour for the summed values of A 2 over P:

(xi,yi)∈P A 2 (x i , y i ) ≃ A 2 0 +c X •(X d -X 0 ) 2 +c Y •(Y d -Y 0 ) 2 (1)
where c X and c Y are positive constants that are related to the substrate's roughness, and (X d -X 0 ) and (Y d -Y 0 ) are respectively the difference (in the horizontal/vertical direction) between the shift applied to align the I + and I -images and the optimal shift to fully compensate the sample drift. Finally, A 0 is a background level that is not necessarily null. Indeed, any slight imbalance in photon helicity and any change in illumination profile may lead to a small background. However, whatever this (non-physical) contrast value, minimizing the left-hand side in Eq. ( 1) for a given acquisition should suppress any artificial contrast. We stress that this refinement becomes necessary for Medge data for several reasons. First of all, the small attenuation length at these photon energies prohibited any shadow XMCD-PEEM: in these magnetic images, we sought XRMS contrast in the interference pattern behind a bead's shadow, corresponding to much lower signal-over-noise ratio than for L 3 edge (shadow) XMCD-PEEM. In addition, we encountered instrumental difficulties during acquisition that were due to the strongly enhanced beamline flux (and the resulting larger current in the imaging column) in the extreme ultraviolet range [START_REF] Locatelli | Image blur and energy broadening effects in XPEEM[END_REF]. With respect to this report by Locatelli et al., we expect the incident flux to be comparable in our experiments, and indeed we found a degraded image quality at the Co M 2,3 edges that could not be improved upon through the objective lens. Moreover, we experienced a lower-than-usual positional stability of the microscope's angle-limiting aperture (also known as contrast aperture), which we ascribe to the greatly-enhanced electron currents in these conditions. It must be noted that this aperture is of crucial importance in our magnetic imaging experiments on 3D objects since we rely on sample edges as well as substrate roughness (cf. previous paragraph) to refine the image alignment, and the aspect of such topographic features in PEEM images strongly depends on this aperture [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF][START_REF] Schneider | Investigating surface magnetism by means of photoexcitation electron emission microscopy[END_REF].

III. Results and discussion

A. Magnetic Force Microscopy

Several Co 1-x Gd x beads that have been imaged with MFM yield similar maps, featuring a background all over the object (which to a good approximation, appears proportional to the local bead height) and a clear central bump. To illustrate this, we present in Fig. 2.(a) an AFM image of a bead on which MFM has been performed twice, with opposite tip magnetizations. The corresponding magnetic images in Fig. 2.(b-c) clearly show that the background's sign is independent on the tip's magnetization, whereas the central bump's is not. Therefore, the former can be ascribed to a susceptibility contribution, which is always attractive and is indeed negative [START_REF] Thiaville | Magnetic force microscopy: Images of nanostructures and contrast modeling[END_REF], while the latter unambiguously indicates stray fields from the bead's magnetization. At this stage, bearing the similarity to published MFM images made on vortices [START_REF] Shinjo | Magnetic Vortex Core Observation in Circular Dots of Permalloy[END_REF][START_REF] Skidmore | Micromagnetic domain structures in cylindrical nickel dots[END_REF] as well as the bead's platelet shape in mind, it seems quite likely that this sample hosts a magnetic configuration comprising a vortex pointing perpendicular to the substrate.

We note that the lack of quantitative knowledge regarding this specific bead's composition and magnetic properties prevents us from backing this interpretation with direct arguments. However, we point out that the vortex state has been shown to persist in a wide variety of geometries and even in the presence of bulk and/or surface anisotropies [START_REF] Streubel | Equilibrium magnetic states in individual hemispherical permalloy caps[END_REF][START_REF] Pylypovskyi | Vortex polarity switching in magnets with surface anisotropy[END_REF][START_REF] Berganza | 3D quasi-skyrmions in thick cylindrical and dome-shape soft nanodots[END_REF][START_REF] Finizio | Three-Dimensional Vortex Gyration Dynamics Unraveled by Time-Resolved Soft X-ray Laminography with Freely Selectable Excitation Frequencies[END_REF] at the cost of more or less pronounced distortions. Thus, even if the Co 1-x Gd x alloy is not a magnetically soft material, it does not necessarily prevent the existence of a vortex or vortex-like configuration. Finally, such a state has the advantage of significant flux-closure (which is compatible with the absence of magnetostatic charges over most of the bead) and it is the simplest pattern that is compatible with our MFM images. As a matter of fact, departures from the textbook vortex [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF] are not important to us, since we only need non-uniform magnetic configurations to look for XRMS in PEEM.

B. Spectroscopy

The first result from our L 3 XAS measurements is the absence of Co oxidation near the beads' surface in most (if not all) cases. This is assessed from the peak shape of the photoemission recorded from the top of the objects: we usually do not observe the easily-recognizable structure that is characteristic of cobalt oxide [START_REF] Regan | Chemical effects at metal/oxide interfaces studied by x-ray-absorption spectroscopy[END_REF]. This confirms the efficiency of our Nb capping. As an example, Fig. Initially, our XAS measurements around the Co M 2,3 edges (expected around 60 eV [START_REF] Thompson | X-ray Data Booklet[END_REF]) did not feature any peak, but instead a rather constant trend which is illustrated in Fig. 4.(a). This came as a surprise considering the good signal-overnoise ratio at the L 3 edge. Since the same observation was made on several Co 1-x Gd x beads, notably some for which L 3 edge XAS had been acquired, the only explanation for the aspect of our low-energy XAS is that the Co M 2,3 edges are hidden from the X-ray point of view, not the point of view of pho-toemission. In this respect, we find that a first hint of our Nb layer's influence is the ratio of the PEEM intensities recorded (on the bead) at the absorption peak and at the edge's foot. Indeed, it has been shown for PEEM-recorded XAS on 3D objects that this ratio is usually in excess of 1.3 for 3d ferromagnetic metals [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF][START_REF] Wartelle | Motion of magnetic domain walls in engineered cylindrical nanowires[END_REF], reaching values of 2 to 3 at the Co L 3 even in samples where Co is a minority element [START_REF] Wartelle | Motion of magnetic domain walls in engineered cylindrical nanowires[END_REF]. In our Co 1-x Gd x beads, where we expect Co to be predominant, this intensity ratio is never higher than 1.5, which would be consistent with a capping thickness large enough to suppress photoemission from the underlying Co 1-x Gd x alloy [53]. In other words: the L 3 edge data already suggests a non-negligible masking by the Nb capping layer.

At this stage, we point out that Nb's attenuation length below 65 eV is below 40 nm [START_REF] Henke | X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92[END_REF]. By contrast, it is in excess of 130 nm around the Co L 3 edge. Also, it must be kept in mind that for the top, close to horizontal surface of the bead, the path length of X-rays through the capping is the latter's thickness enhanced by 1/ sin(Ψ) ≃ 3.6 because of the beam's incidence angle Ψ =16°. This means that effective Nb thickness is on the order of 10 nm or more, rather than the few nominally deposited nanometres. Concerning that thickness, we noted that the total Nb thickness estimated to be etched by our Ar plasma sputtering is 15 nm. Even if we assume the presence of a harder-to-etch, ca. 1.6 nm oxide layer [START_REF] Marks | Laser-Pulsed Plasma Chemistry: Surface Oxidation of Niobium[END_REF] with a 4:1 ratio of sputtering yields in our conditions, we are led to consider a total NbO x /Nb capping thickness of about 10 nm.

Furthermore, in a first approximation, the XAS jump associated to an absorption edge is expected to scale with µ on -µ below where µ on and µ below are the X-ray linear attenuation coefficients at the absorption peak and at its foot, respectively. If we compare the M 2,3 and L 3 edges of Co, we find that (µ on -µ below )/µ below ∼ 1.6 at the M 2,3 edges [START_REF] Valencia | Faraday rotation spectra at shallow core levels: 3p of Fe, Co, and Ni[END_REF], whereas (µ on -µ below )/µ below ∼ 23 at the L 3 edge [START_REF] Nakajima | Electron-yield saturation effects in L-edge X-Ray Magnetic Circular Dichroism spectra of Fe, Co, and Ni[END_REF].

Therefore, we consider that the combination of weaker (in a relative sense) jump in edge absorption and much stronger Xray attenuation by the Nb capping are responsible for initially masking the Co M 2,3 edges. A crude estimate of the combination of both effects suggests a suppression of the edge jump's strength by a factor ∼ 33, which is very likely to bring the Co M 2,3 edges below the noise level.

C. X-ray magnetic imaging

Let us first review a few XMCD-PEEM images acquired at the Co L 3 edge. An overview of what we could observe at such photon energies is shown in Fig. 5. When imaged far enough below the absorption peak, and with the microscope focused on the substrate, the Co 1-x Gd x beads typically yield bipolar shadow XMCD-PEEM patterns, as can be seen in the left column of Fig. 5. Then, regular XMCD-PEEM imaging on the top surface of the beads performed with the photon energy at the maximum of absorption confirms the change of contrast from one side of the objects to the other, with the contrast inversions described by Jamet et al. [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF]. The several occurrences of these bipolar patterns allow to confirm the preliminary results from MFM: vortex or at least vortex-like . states seem to be predominant in our samples. Close to the absorption peak, the transmission XMCD-PEEM contrast is lost in the bulk of the shadow because of the sample thickness instead of saturating [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF], possibly because of harmonics contamination of the primary beam [56]. Thus, the shadow contrast in Fig. 5.(b,f) is not reliable; the notable exception in Fig. 5.(d) (with the correct contrast inversion between shadow and bead) is likely due to the slightly decreased absorption above the L 3 edge's white line peak.

However, we do see modulations in the shadow XMCD-PEEM contrast in Fig. 5.(a,c,e). Such modulations inside the shadow cannot result from errors in image alignment, unless (i) some variations in transmitted intensity occur near the shadow rim for the three beads, and (ii) alignment errors are every time along the same direction (roughly perpendicular to the X-ray beam). The combination of these two conditions seems unlikely. On the other hand, the modulation's asymmetry suggests a link to the asymmetry in average magnetization on both sides of the objects. Moreover, it must be kept in mind that for the images in Fig. 5.(a,c,e), the X-ray absorption is small enough to allow significant signal-over-noise ratio for transmission XMCD-PEEM over the whole shadow, contrary to the cases described by Jamet et al. [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF]. As a result, we can claim that these modulations are not artefacts related to saturation in absorption. Our conclusion is that these images are evidence for small-angle XRMS modulations on top of the usual "geometrical-optics" shadow XMCD-PEEM; in the following, we will refer to this effect as XRMS-PEEM. Since the visibility of the fringes is rather low and their pattern is strongly confined near the shadow's rim, we then tune the photon energy to the Co M 2,3 edges. As was explained in Sec. II C and Sec. III B, Co-sensitive image only became possible after significant sample sputtering, and with additional hindrances pertaining to the increased electron current in the PEEM's imaging column. This has an additional consequence on magnetic imaging, namely: the poorer spatial resolution impairs our accuracy in image alignment because all sharp features on the PEEM images (sample edges, substrate roughness etc.) are blurred. In practice, we find a poorer reproducibility than for L-edge XMCD-PEEM; however, certain repeated images are in good agreement with one another. To illustrate first the stark differences in PEEM imaging, we present in Fig. 6.(a,c,e 6(a) that the immediate vicinity of the bead appears distorted: there is a radial pattern around its perimeter that does not match the substrate roughness elsewhere. This enhanced optical aberration due to a 3D object is attributed to the instrumental complications described in Sec. II C. Beyond this, one can clearly see a much clearer and more extended interference pattern behind the bead, notably in Fig. 6(e), where the magnification is lower than in Figs. 6(a,c).

At this stage, let us examine the images that display the corresponding asymmetry ratio A in Figs. 6(b,d,f). While there are minute differences between them, the antisymmetry in the fringe pattern (perpendicular to the direction of Xray propagation) is the same across the three images, and it is plausible considering the previous magnetic images of that Co 1-x Gd x bead at the L 3 edge.

In order to support our hypothesis that this contrast is of magnetic origin, we present an additional pair of images of the same bead in Figs. 6(g,h), this time at a photon energy of 65 eV i.e. at a significant offset with respect to the previous images, as can be surmised from Fig. 4.(b). One would expect quite a strong reduction in XRMS strength upon moving the photon energy by an amount close to the edge jump's width, and the fringe contrast in Fig. 6(h) is clearly weaker than those in Figs. 6(b,d,f).

Using the magnetic contributions ∆δ and ∆β to the X-ray refractive index of Co around its M 2,3 edges determined by Valencia et al.

D. Analytical and numerical modelling

In the following, we will explore a very simplistic model for the interference pattern observed in PEEM beyond the shadow of the Co 1-x Gd x beads, at the Co M 2,3 edges. Our objective is merely to obtain physical insight into the relation between the XRMS-PEEM fringe patterns and the Fresnel diffraction from the sample. To that end, our coarse approach consists in approximating the complex wave propagation and scattering from a 3D object of unknown shape with a two-source interference problem. A schematic view of the situation of interest is presented in Fig. 8, with a spherical Co 1-x Gd x bead (in blue) on a substrate and an X-ray beam impinging on it with the experimental incidence angle Ψ. Let us now describe our model's features in more detail. First of all, we omit any three-dimensional aspect of the problem. Then, considering how opaque the scatterer is (with attenuation lengths of at most ≃12 nm), we neglect X-ray transmission for most of the bead and only keep contributions from parts with a low path length s traversed through the alloy. These contributions are gathered and considered as the first of our two sources.

} . . . ⇒ √ I 0 exp [i • kL(x)] ⇒ √ I 0 • t c • exp [±i • z m • ks] s Shadow -→ e x -→ e b Ψ x
The second source is of course the part of the beam that (in a geometrical-optics sense) does not impinge on the bead. For illustration purposes, a ray from the first source is shown in Fig. 8 (in red) intersecting a ray from the second one at a point on the substrate with abscissa x, behind the rim of the geometrical-optics shadow of the bead (in blue). As we are not aiming at a quantitative expression for the X-ray intensity as a function of abscissa x on the substrate, we will instead leave as unknown the path difference L(x) between the first source and that part of the beam which passes above it from here onwards referred to as the second source. In doing so, we absorb the integration over that second part of the incident wave into a beam amplitude √ I 0 and the path difference [START_REF]would only add the complexity of a spatially varying oscillation envelope with no further insight[END_REF] L(x). With respect to √ I 0 , we will use the modulus of a propagation prefactor t c for the first source's amplitude so as to take into account the sample's non-magnetic refractive index (we restrict ourselves to small-angle scattering)

n c = 1 -δ + i • β.
From this point on, we must determine the expression for the Fresnel diffraction that is caused only when charge scattering is considered. As a second step, we shall "turn on" XRMS by considering the magnetic contributions to our sample's refractive index, and recalculate the expression for the X-ray intensity on the substrate. Our goal is to assess whether XRMS modulates rather the oscillations' amplitude, resulting in a magnetic asymmetry A, or phase-shifts the patterns associated to each polarization depending on magnetization.

Let us denote as before n m = -∆δ For the sake of simplicity, we will consider that magnetization can only yield either +m b or -m b as projections. We may now write the amplitudes of our two sources a ± and a 0 as:

+ i • ∆β, m b = s -1 s 0 - → m• - → e b du
a ± = I 0 • t c • exp [±i • z m • ks], a 0 = I 0 • exp i • kL(x) . (2)
With Eq. ( 2) in mind, we can write down the X-ray intensities on the substrate I ± as: After a bit of algebra, we retrieve the expressions for the magnetic and non-magnetic fringe patterns. For the sake of normalization but also of computing an asymmetry ratio as we do in experiments, we use [I + (x) -I -(x)]/(2I 0 ) for the former, and [I + (x)+I -(x)]/(2I 0 ) for the latter. To first order in |t c |, these quantities read:

I ± (x) = |a 0 + a ± | 2 = I 0 • 1 + 2 • Re t c • e ±i•zmks e -i•kL(x) + |t c | 2 e ±2Im(zm
I + (x) -I -(x) 2I 0 =2m b • ks • Re(t c • n m ) • cos kL(x) + 2m b • ks • Im(t c • n m ) • sin kL(x) , (4)
and:

I + (x) + I -(x) 2I 0 =2Re(t c ) • cos kL(x) + 2Im(t c ) • sin kL(x) . (5)
Denoting φ t = arg(t c ) = -ks • δ and φ m = arg(n m ), we may rewrite the above equations as:

I + (x) -I -(x) 2I 0 = 2m b •ks•|t c |•|n m |•cos kL(x) -φ t -φ m (6)
and:

I + (x) + I -(x) 2I 0 = 2|t c | • cos kL(x) -φ t . (7)
With this, our model indicates that XRMS leads to intensity modulations ∝ |n m | • m b as expected, and phase-shifted with respect to the non-magnetic fringe pattern by -φ m . At 60 eV, -φ m ≃-240°, suggesting that we might expect close to a phase quadrature experimentally. A very accurate match would however require a very careful calibration of the Xray beam's energy beforehand since φ m varies by π/2 over ca. 3 eV according to Fig. 7. As an illustration, we present in the Supplemental Material a comparison between the values of ∆δ and ∆β obtained by Valencia et al. [START_REF] Valencia | Faraday rotation spectra at shallow core levels: 3p of Fe, Co, and Ni[END_REF], and those reported by Willems et al. [START_REF] Willems | Magneto-Optical Functions at the 3p Resonances of Fe, Co, and Ni: Ab initio Description and Experiment[END_REF].

It must be noted that the estimated magnetic phase shift does not depend on the precise values of s or m b (only the overall amplitude does), which is comforting. To conclude the discussion of our coarse analytical model, we propose the following interpretation: due to the larger contribution of XMCD with respect to XMCB (|∆β| ≃ 1.6•∆δ), the largest magnetic contrast are obtained close to when the non-magnetic fringe patterns reaches its average value (for kL(x)-φ t ≡ π/2 [π] ) because then the interference term is most sensitive to variation in amplitudes from the first source.

In order to look deeper into XRMS-PEEM, we now tackle the problem with a numerical model. Considering a rectangular simulation box containing a perfectly spherical magnetic bead on a substrate and with vacuum elsewhere, we compute the propagation of X-rays in an angular spectrum approach [START_REF] Schafer | Transducer characterization using the angular spectrum method[END_REF], with the incidence angle Ψ as in experiments. The diameter choice is a compromise between object size (as the angular spectrum approach would become computationally too heavy for micron-sized beads) and extent of the scattering pattern behind the bead shadow. Then, we obtain the expected PEEM image based on the absorption coefficient on the surface of the bead or on the substrate. In practice, we compute the X-ray propagation along the complete long axis of our simulation box, while the substrate is tilted by an angle Ψ, as illustrated in Fig. 9. We then obtain an image similar to experimental PEEM data based on the wave amplitudes on the topmost surfaces, weighted by the local absorption coefficient. As far as magnetism is concerned, the bead is made to host a simple configuration, with an analytical vortex ansatz whose core points perpendicular to the substrate.

In terms of X-ray physics, the relevant refractive indices for the substrate (Si) and for the magnetic material (considered to be Co 80 Gd 20 ) are computed based on the tabulated values by Henke et al. [START_REF] Henke | X-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92[END_REF] as well as resonant magnetic scattering factors from the Dyna software package [START_REF] Elzo | X-ray resonant magnetic reflectivity of stratified magnetic structures: Eigenwave formalism and application to a W/Fe/W trilayer[END_REF], and finally the First of all, as can be seen on Fig. 9.(b), we do retrieve fringes mostly outside of the shadow; the ones inside are made visible by the chosen logarithmic scale. We point out that the finite size of our box along the height (Y direction) is one of the main reasons for the artefacts present on the right-hand side of the simulated PEEM-like images. Then, the computed asymmetry ratio A in Fig. 9.(c) shows XRMS contrast both inside and outside the bead's shadow, as expected. The expected shadow XMCD-PEEM contrast is modulated, and beyond the shadow's rim, we see an asymmetric fringe pattern reminiscent of our experimental images. The latter can be investigated further with suitable line profiles along the beam direction; we therefore proceed in Fig. 10.(a), where two such profiles are drawn on the simulated XRMS image as red lines. They have been placed symmetrically around the middle of the simulation box. Both have been extracted on the XRMS image and on the non-magnetic image, and the four corresponding traces are plotted in Fig. 10.(b). As must be the case, the magnetic traces are opposite. Moreover, we can observe a significant phase shift between the latter and the profiles on the non-magnetic image. However, it is not found constant across the simulated oscillations; typical estimates are 60°to 80°.

At this stage, we perform similar simulations at a photon energy of 60 eV in a 1 µm-diameter bead. The vortex ansatz it hosts has been rescaled accordingly without further modifications. The results are shown in Fig. 11 similarly to the

IV. Conclusion

In the course of this work, we have performed magnetic imaging using X-ray PEEM and observed small-angle X-ray Resonant Magnetic Scattering at the Co M 2,3 edges. These experiments have been carried out on three-dimensional objects, namely Co 1-x Gd x beads, which we have taken care to characterize beforehand, including with more conventional L 3 -edge (shadow) XMCD-PEEM. Combined with the pristine metallic character of Co revealed by XAS, this first imaging scheme has confirmed the suitability of this sample for an investigation of M-edge XRMS. Indeed, the vortex-like configurations that have been evidenced both from direct XMCD-PEEM views and from the telltale shadow contrast ensure that the incoming X-rays probe opposite projections of magnetization. Our expectation was therefore to find an asymmetry in the pronounced Fresnel diffraction at the Co M 2,3 edges. After careful image analysis, a plausible contrast has been extracted; its magnetic origin is supported not only by its decrease above the absorption edge but also by our modelling. Coarse as the latter may be, it does hint at the same asymmetry as in our experiments. Unfortunately, our numerical simulations indicate that an accurate measurement and interpretation of the XRMS pattern's phase shift with respect to the non-magnetic fringe pattern requires more than the simple analytical model we presented. On the other hand, our L 3 -edge shadow XMCD-PEEM results suggest that smallangle XRMS can be sizeable and affect non-negligible parts of the shadow area [START_REF] Jamet | Quantitative analysis of shadow X-ray Magnetic Circular Dichroism Photo-Emission Electron Microscopy[END_REF], thus compromising efforts towards e.g. shadow XMCD-PEEM-based vector magnetic tomography using laminography [START_REF] Donnelly | Imaging three-dimensional magnetic systems with X-rays[END_REF].

However, it must be kept in mind that the photon-in, electron-out scheme is not the only one available in an Xray PEEM. Indeed, the reflection geometry and CDI phase retrieval approaches have recently been successfully demonstrated on artificial spin ice systems [START_REF] Menteş | Coherent x-ray scattering in an XPEEM setup[END_REF]. It stands to reason that the difficulty in reaching sufficient oversampling may be drastically suppressed by utilizing XRMS at M edges compared with L edges of 3d magnetic transition metals. In this respect, our investigation shows that magnetic contrast at this energies can already be accessed in the Fresnel diffraction of 3D samples, despite instrumental challenges which are specific to the electron imaging. Therefore, a longer-term prospect of this study is M-edge CDI in a reflection geometry. In addition to the possibilities of imaging under field [START_REF] Menteş | Coherent x-ray scattering in an XPEEM setup[END_REF], a very efficient use of photons can be foreseen thanks to the increased X-ray coherence as well as the absence of resolution degradation at large flux [START_REF] Locatelli | Image blur and energy broadening effects in XPEEM[END_REF]. Finally, the strong absorption is expected to be beneficial in the study of (ultra-)thin films, for which X-ray PEEM provides a valuable sensitivity to in-plane magnetization.

FIG. 1 .

 1 FIG. 1. Scanning electron micrographs of beads from PLD. All these images are on the same scale

FIG. 2 .

 2 FIG. 2. (a) AFM image on a Co1-xGdx bead. (b) MFM image corresponding to (a) and acquired with an upward-magnetized tip. (c) MFM image of the same bead acquired with a downward-magnetized tip.

FIG. 3 .

 3 FIG. 3. (a) Typical XAS traces obtained respectively from the top of a Co1-xGdx bead (blue) and from its shadow (red). Full black dots • highlight photon energies at which XMCD-PEEM was performed (cf. Fig. 5), while a cross × indicates the photon energy 778.6 eV at which the image from Fig. 3.(b) was acquired. (b) PEEM image at Co L3 edge's absorption peak of the bead (in the center) from which the XAS data from from Fig. 3.(a) was extracted. The adjacent wide patch on the top left of the bead corresponds to an alignment mark, and the smaller round item below the bead and slightly to the right is another Co1-xGdx object.

FIG. 4 .

 4 FIG. 4. (a) XAS traces around the Co M2,3 edges obtained on a Co1-xGdx bead before sputtering. It must be noted that several spectra qualitatively similar to this one were obtained on different beads. (b) XAS trace (blue curve) around the Co M2,3 edges obtained on a different Co1-xGdx bead after several rounds of Ar plasma sputtering. The significant (in terms of relative change) upturn in PEEM intensity starting a bit before 60 eV is attributed to the Co M2,3 edges, which were previously hidden by the Nb capping layer. The black curve corresponds to the X-ray's linear attenuation coefficient µ as derived from spectroscopic measurements published by Valencia and coworkers [31]..

FIG. 5 .

 5 FIG. 5. Examplary XMCD-PEEM images acquired close to the Co L3 edge, each row pertaining to one particular Co1-xGdx bead; (e-f) correspond to the one shown in Fig.3. In the left column, the photon energies are below (as far as 1.9 eV) the absorption peak, and tuned in order to favour shadow XMCD-PEEM contrast. In the right column, the photon energies are at or slightly below the absorption peak so as to obtain XMCD-PEEM contrast from the top of the beads. For all images, the colour scales correspond to the plotted asymmetry ratio A. The arrows indicate the direction of the X-ray beam, and the scale bars correspond to 1 µm. It must be noted that the focal plane was set to the substrate for shadow XMCD-PEEM and to the top of the beads for the "on-edge" measurements.

FIG. 6 .

 6 FIG.6. Regular PEEM images without magnetic contrast (left column) and their corresponding magnetic images with the same colour range for A (right column). All these images pertain to the same bead as in Fig.3.(b) Fig.5.(e-f), and have been acquired at the Co M2,3 edge, the three first rows at 60 eV and the last row (g-h) at 65 eV. These are independent measurements separated by several other acquisitions. While a different magnification was used in (e-f), all scale bars are 1 µm. The red arrows indicate the direction of X-ray propagation.

  ) several images acquired at a photon energy of 60 eV of the same bead as before [the one shown in Figs. 3.(b) and 5.(e-f)]. It can be noted for instance from Fig.

  FIG. 7. Norm (|nm|, blue dashed curve) and complex argument [arg(nm)] of the magnetic contribution to Co's refractive index near the M2,3 edges, computed from the results reported by Valencia et al. [31].

FIG. 8 .

 8 FIG. 8. Schematics for the two-source analytical model. The Co1-xGdx bead is represented in blue, and the relevant typical length s traversed by X-rays through it is shown between the yellow dashed lines. The black dashed line indicates the trajectory of X-rays at the geometrical-optics rim of the beads' shadow.

  the average projection of magnetization along the beam with propagation direction e b , and z m = n m • m b .

3 )

 3 )•ks . (If we use the values of ∆δ and ∆β from Valencia et al. [31], we have ∆δ = 8.9 • 10 -3 and ∆β = -1.4 • 10 -2 for a photon energy hν =60 eV; by contrast, δ = -2.0 • 10 -2 and β = 1.6 • 10 -1 . Since the attenuation length is lower than 12 nm, we may consider values of s of, at most, 24 nm. In that case, |t c | ≃ 0.31. With a wavelength λ =20.6 nm, we find n m • ks ≃ -0.065 -i • 0.10. Since |m b | ≤ 1 and |n m | • ks = 0.12, we may approximate exp [±i • z m ks] ≃ 1 ± i • z m ks.

  FIG. 9. (a) View from the side of the simulated sample, in terms of averaged linear attenuation coefficient. (b) computed PEEM-like image considering charge scattering only; a logarithmic scale is used for the sake of fringe visibility in the shadow. (b) XRMS pattern computed as normalized difference between PEEM-like images for opposite polarisations.

FIG. 10

 10 FIG. 10. (a) Zoom-in on Fig. 9.(c); the red lines indicate the profiles taken across this image. (b) Green dashed and purple full curves: profiles extracted from the lines drawn in (a). Full black and dashed blue curves: corresponding profiles extracted from the non-magnetic image.

FIG. 11

 11 FIG. 11. (a) View from the side of the simulated sample at 60 eV, in terms of averaged linear attenuation coefficient. (b) computed PEEM-like image considering charge scattering only; here the intensity scale is linear. (b) XRMS pattern computed as normalized difference between PEEM-like images for opposite polarisations.

V. Acknowledgements

We thank Andrea Locatelli, Tevfik Onur Menteş and Matteo Jugovac (ELETTRA) for assistance in the XMCD-PEEM measurements as well as fruitful discussions. We acknowledge the Agence Nationale de la Recherche for funding under project number ANR-19-CE42-0013-05.