Christian Commault
email: christian.commault@gipsa-lab.grenoble-inp.fr

Jacob Van Der Woude
email: j.w.vanderwoude@tudelft.nl

A classification of nodes for structural controllability

Keywords: Controllability, structured system theory, input connection condition, rank condition, steering node

A classification of nodes for structural controllability

Introduction

The study and analysis of large complex networks has become extremely important as nowadays society becomes more and more interconnected. Examples of such complex systems appear in biology, genetics, social networks, large communication or energy networks, etc. After the analysis of the behavior of a particular complex network, one of the most relevant questions is whether its behavior can be controlled. Therefore, the controllability of complex networks has received a lot of attention in the recent years. Especially, the question of interest is where to put so-called driver nodes by which the behavior of the network can be controlled, see [START_REF] Cowan | Nodal degree, not degree distributions, determine the structural controllability of complex networks[END_REF][START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Ruths | Control profiles of complex networks[END_REF].

Here we call a state/node of the network a steering node if it can be influenced from the outside of the network (externally) by a control. Various types of control can be distinguished, like controls that influence possibly all nodes simultaneously, controls that each influence an individual node, or a mix of the previous two, see [START_REF] Commault | Input addition and leader selection for the controllability of graph-based systems[END_REF].

Next to the question of where to put the steering nodes, also their minimal number is a relevant issue. Clearly, having as few as possible steering nodes may be important in applications.

In this paper, we focus on the situation that each steering node is supposed to be controlled by precisely one control. In principle, all nodes of a network may act as steering nodes. However, it is clear that occasionally for some nodes it is useless to be a steering node because they can also be controlled indirectly by other nodes of the network. Therefore, a classification seems appropriate of which nodes, seen as steering nodes, are always useless, which nodes may play a role in certain steering node configurations, and which nodes are essential in the sense that they always have to be present in any appropriate steering node configuration.

The above classification can be compared with the component classification in terms of the impact in case of their failure. This is a classical field of reliability theory, see [START_REF] Barlow | Importance of system components and fault tree events, stochastic processes and their applications[END_REF][START_REF] Beeson | Importance measures for non-coherentsystem analysis[END_REF]. Also in engineering, the classification of sensors in terms of their importance for preserving some property (such as observability) is an active research field, see [START_REF] Commault | Observability preservation under sensor failure[END_REF]. In this paper we study the classification of steering nodes in terms of their importance for controllability. It should be noted that, although we use here some concepts and tools of [START_REF] Commault | Observability preservation under sensor failure[END_REF], the present paper is much more than a dualisation of the results of [START_REF] Commault | Observability preservation under sensor failure[END_REF]. The main difference is that our classification does not rely on a given set of inputs and corresponding actuators. Here the inputs have to be chosen in a given set and they impact only one state.

The steering node classification in this paper resembles a similar classification (in critical, redundant and intermittent nodes) in [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF], see also [START_REF] Ding | Control backbone: An index for quantifying a node's importance for the network controllability[END_REF] for a quantitative approach of the importance of nodes. Notice that an alternative way to classify the nodes is to compute for each of them the control centrality [START_REF] Liu | Control centrality and hierarchical structure in complex networks[END_REF]. However, opposite to [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF][START_REF] Liu | Controllability of complex networks[END_REF], and as already indicated, we suppose that a control can only influence one steering node. The latter also seems to be much more realistic in applications. For this reason, we have to also take into account the input connection condition, apart from the rank condition that needs to be considered only when allowing controls that may influence several (or even all) steering nodes simultaneously.

The input connection condition can most easily be studied by decomposing the digraph associated to our networks into strongly connected components that are ordered themselves in an acyclic way. The rank condition that has to be checked can best be investigated by the so-called DMdecomposition. By using this decomposition the results in this paper can be presented in a more transparent and refined way than the results in [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF], where such decomposition is not considered.

We extend our classification to the case where some nodes cannot be influenced by inputs. This situation, which is often met in practice, has also been considered in [START_REF] Olshevsky | Minimum input selection for structural controllability[END_REF].

Our results are related with the recent literature about the Minimal Controllability Problem [START_REF] Commault | Input addition and leader selection for the controllability of graph-based systems[END_REF][START_REF] Commault | The single-input minimal controllability problem for structured systems[END_REF][START_REF] Olshevsky | Minimal controllability problems[END_REF][START_REF] Olshevsky | Minimum input selection for structural controllability[END_REF][START_REF] Assadi | Complexity of the minimum input selection problem for structural controllability[END_REF][START_REF] Pequito | A framework for structural input/output and control configuration selection in large-scale systems[END_REF].

The outline of this paper is as follows. In section 2 we introduce the type of systems that we consider and formulate the problem of steering node selection such that a system becomes controllable. We also introduce the notions of useless, useful and essential nodes for a specific property that we want the system to have. In section 3 we introduce structured systems, introduce their graph representation and recall some well-known conditions for structural controllability. The conditions consist of an input connection condition and a rank condition. In section 4 the two conditions are further analysed using connectivity aspects of the directed graph introduced in section 3, and for the rank condition by means of a DM-decomposition of the bipartite graph associated to the structured system. In section 5 we present conditions for a steering node to be useless, useful or essential for each of the two conditions for structural controllability. Our main result is the combination of all criteria to get characterizations of steering nodes that are useless, useful or essential for structural controllability as a whole. In section 6 we comment on the obtained results and illustrate them by means of two examples. In section 7 we extend the classification to the case where some nodes cannot be used as a steering node. In section 8 we pay some attention to the computational aspects of verifying the conditions obtained in this paper. We conclude by section 9 with a summary of the results of this paper and with some topics for future research.

Problem formulation

The controllability problem

In this paper, we consider a large scale system composed of n states interacting together with linear dynamics. We assume that we can represent the behavior of the whole system by the simple equation

ẋ(t) = Ax(t), (1)
where x(t) ∈ X = R n is the state vector and A is a real constant n × n matrix.

We will distinguish m states, called the steering nodes S = {x i 1 , . . . , x im }, with i j ∈ {1, . . . , n} and i 1 < i 2 < • • • < i m . To each steering node x i j we associate a control input u j that acts only on this state node. In this way we obtain a system that can be represented as

Σ : ẋ(t) = Ax(t) + Bu(t), (2)
where matrix B has m columns and its j-th column has all its entries equal to 0 except for b i j ( = 0), being the i j -th component of column j of B. Hence, the node set S is in 1-1 correspondence with the (structure of) matrix B.

In the following we will be looking for a set of steering nodes such that the pair (A, B), as introduced above, is controllable. Here we understand by controllability of system Σ, see [START_REF] Barlow | Importance of system components and fault tree events, stochastic processes and their applications[END_REF], the ability to steer Σ, in final time τ > 0, from an arbitrary initial condition x 0 = x(0) to an arbitrary final condition x 1 = x(τ ), by the use of an appropriate control function u(t) over the time interval [0, τ ).

It is well-known that controllability of system Σ is equivalent, for instance, to rank [B, AB, • • • , A n-1 B] = n. For this reason, the controllability of Σ is also referred to as the controllability of the pair (A, B).

A challenging problem is to find the minimal number of steering nodes such that (A, B) is controllable. This problem was tackled in [START_REF] Olshevsky | Minimal controllability problems[END_REF], where it was proved to be NP-hard, see [START_REF] Garey | Computers and intractability: A guide to the theory of NP-Completeness[END_REF] for more on this notion. Moreover, it is proved in [START_REF] Olshevsky | Minimal controllability problems[END_REF] that the minimum number of inputs needed for controllability in our context (one input acting on one state) is also the minimum number of states which are impacted by inputs when no structure is imposed to the B matrix. Minimizing the number of states that are impacted by inputs is certainly a more convincing way to approach the controllability of large scale systems. It is more appropriate than minimizing the number of inputs, without taking into account the number states impacted by these inputs as in [START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Ruths | Control profiles of complex networks[END_REF].

Steering node classification

A property P of the pair (A, B) can be seen as a mapping from the set R n×n × R n×m into the set {0, 1} where the property P is true precisely when

P(A, B) = 1.
When a set of steerings nodes S ⊆ X , defining matrix B, and therefore also the pair (A, B), is such that a given property P is true, we call S an admissible steering node set for property P.

Definition 1. For a given property P, a node x i seen as steering node, can be classified as follows, see for instance [START_REF] Commault | Observability preservation under sensor failure[END_REF][START_REF] Commault | Sensor classification for the fault detection and isolation, a structural approach[END_REF][START_REF] Staroswiecki | System component usefulness measures for fault tolerance evaluation[END_REF]].

1. Node x i is called a useless steering node if for any admissible steering node set S containing x i , S\{x i } is still an admissible steering node set for P, where S\{x i } is the set S minus the node x i . 2. A steering node which is not useless is called a useful steering node.

Hence, node x i is useful if there is an admissible steering node set S for P such that x i ∈ S, while S\{x i } is not admissible for P. 3. Node x i is called an essential steering node if x i belongs to any admissible steering node set S for P. Hence, x i is an essential node if S\{x i } is not admissible for any admissible steering node set S for P. The set of essential steering nodes is a subset of the set of useful steering nodes.

In this paper we will focus our attention on the search and classification of steering nodes for the controllability in the context of structured systems.

Linear structured systems and structural controllability

In the remainder we assume that system (1) is structured, meaning that we assume that only the zero/nonzero of the entries in matrix A is known.

A structured system of type (1) can be associated with a directed graph G(A) = (X , W) as follows:

• the node set is X , being the set of state nodes {x 1 , x 2 , . . . , x n },

• the edge set is W = {(x i , x j )|a ji = 0}, where a ji denotes the (j, i)th entry of matrix A and a ji = 0 means that the (j, i)-th entry of A is a structural nonzero.

We define a path in G(A) from a node x i 0 to a node x iq to be a sequence of edges, (x i 0 , x i 1 ), (x i 1 , x i 2 ) , . . . , (x i q-1 , x iq ), such that x it ∈ X for t = 0, 1, . . . , q, and (x i t-1 , x it ) ∈ W for t = 1, 2, . . . , q.

Let A be a structured matrix and let the total number of nonzero entries in A be given by l, then the structured system can be parameterized by a parameter vector λ ∈ R l , where each component of λ corresponds to precisely one nonzero. Hence, for each vector λ, a completely numerically specified system is obtained. The system is thus defined as

ẋ(t) = A λ x(t), (3)
where x(t) ∈ R n is the state vector, and A λ is the parameterized version of the structured matrix A.

The previous systems are called linear structured systems if only the zero/nonzero structure of the matrix A is known, or, equivalently, if the entries of the matrix A λ are either fixed zeros or independent parameters (not related by algebraic equations), see [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF][START_REF] Murota | Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics[END_REF]. For such systems, we can study generic properties, i.e., properties which are true for almost any value of the parameters.

One such property is, for instance, the generic controllability of a structured system. Another such property is the generic rank of a structured matrix. Given a structured matrix Q with k nonzeros and a parametrized version Q µ , the generic rank of Q will be defined as the rank of Q µ for almost all µ ∈ R k and will be denoted by g-rank Q. It can be shown that g-rank

Q = max µ∈R k rank Q µ .
Later in this paper the generic rank of matrix A will be used.

Example 1. Consider the system defined by the structured matrix

A =             0 0 * 0 0 0 0 0 0 0 * 0 0 0 0 0 * 0 0 0 0 0 0 0 0 * 0 0 0 0 * * 0 0 0 0 * 0 0 0 * 0 0 * 0 0 0 0 0 0 0 0 * * 0 0 0 0 0 0 0 0 0 0             ,
or by a parametrized version with λ ∈ R 11 given by

A Λ =             0 0 λ 1 0 0 0 0 0 0 0 λ 2 0 0 0 0 0 λ 3 0 0 0 0 0 0 0 0 λ 4 0 0 0 0 λ 5 λ 6 0 0 0 0 λ 7 0 0 0 λ 8 0 0 λ 9 0 0 0 0 0 0 0 0 λ 10 λ 11 0 0 0 0 0 0 0 0 0 0            
.

The associated digraph G(A) is depicted in Figure 1. In the figure, for later use, also the strongly connected components are already indicated.

x 5

x 6

x 7

x 4 x 8 x 3

x 1

x 8

x 2 As in Section 2, we can select a set of m steering nodes in X , to which we associate m control inputs. This induces an n × m matrix B with only m nonzero entries, one in each column and at most one in each row.

We can also parametrize the nonzeros in matrix B yielding a parameterized version B λ where each nonzero entry in B is replaced by a parameter and in which the parameter vector λ is updated to also include the parameters in B λ . Then λ includes the parameters of all nonzero entries in A and B.

We get then a controlled system Σ λ , which is the parametrized counterpart of Σ defined in (2), described by

Σ λ : ẋ(t) = A λ x(t) + B λ u(t). (4)
The graph G(Σ) can be obtained from G(A) by adding the m input nodes through the set U = {u 1 , . . . , u m } and m edges, one from each input node to the corresponding steering node. Hence, G(Σ) has node set X ∪ U, and the edge set is updated as

W := W ∪ {(u j , x i j )|j = 1, 2, . . . , m}. In G(Σ), a path (v i 0 , v i 1 ), (v i 1 , v i 2 ), . . . , (v i q-1 , v iq )
, where v i 0 ∈ U and v iq ∈ X , is called an input-state path. The system Σ is said to be input-connected if for any state node x i , i = 1, . . . , n, there exists an input-state path with end node x i .

The notion of structural controllability was introduced and studied by Lin, who proved a necessary and sufficient condition for structural controllability in terms of graph theoretic objects called cacti, see [START_REF] Lin | Structural controllability[END_REF]. The following result can proved to be equivalent to Lin's result, see for instance [START_REF] Reinschke | Multivariable control : A graph-theoretic approach[END_REF][START_REF] Shields | Structural controllability of multi-input linear systems[END_REF].

Theorem 1. Let Σ be the linear structured system defined by (2) with associated graph G(Σ). The system is structurally controllable if and only if

1. the system Σ is input-connected, 2. g-rank[A, B]=n.
In the following, the conditions 1 and 2 of Theorem 1 will be referred to as the input connection condition and the rank condition, respectively. Given a structured system of type (1), with associated graph G(A), the steering node selection problem then amounts to extend G(A) with edges (input-steering node) in such a way that the conditions of Theorem 1 are fulfilled in the extended graph G(Σ).

Structural controllability via steering node selection

We will first look at the two conditions for structural controllability individually.

Input connection condition

Consider the linear structured system defined by ( 1) with its associated graph G(A). A strongly connected component C is defined to be a maximum set of nodes of G(A) such that there exists a path, possibly of length zero, between any two nodes of C. The graph can be partitioned into a set of strongly connected components and this set can be endowed with a partial order. We are then interested in strongly connected components which have no incoming edges from other strongly connected components. From [START_REF] Commault | Input addition and leader selection for the controllability of graph-based systems[END_REF][START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF], we can deduce a condition for a set of steering nodes to be admissible for the input connection condition. Proposition 1. Consider the linear structured system defined by ( 1) with associated graph G(A). A steering node set S is admissible for the input connection condition if and only if there exists a node of S in any Critical Connection Component of G(A).

The previous notions and results can be illustrated on Example 1. The graph possesses five strongly connected components, namely {x 1 , x 3 }, {x 2 }, {x 5 }, {x 8 } and {x 4 , x 6 , x 7 }, where {x 1 , x 3 }, {x 5 } and {x 8 } are the Critical Connection Components, so that d c (A) = 3. It follows from Proposition 1, and it is clear from the graph, that input connection is verified if and only if x 5 , x 8 , and either x 1 or x 3 are steering nodes.

Rank condition

We next characterize the rank condition of Theorem 1, i.e., g-rank[A, B] = n. We do this by first computing the generic rank of matrix A. Once this rank is known, the rank condition of Theorem 1 can be fullfilled by an appropriate choice of the steering nodes. The previous implies that the rank condition of Theorem 1 can be studied by looking at the generic rank of matrix A. The rank defect n-g-rank (A), which is denoted d r (A), equals the number of steering nodes needed to make the rank condition of Theorem 1 become true.

Recall that in this paper matrix B is of special form as it consists of columns each having only one structural nonzero at distinct rows and therefore can be identified with a set of steering nodes.

The generic rank of A will be computed using a bipartite graph associated with the system (1). This bipartite graph H(A) can be introduced as follows. The bipartite graph associated with system (1) is

H(A) = (V + , V -; W )
where the sets V + and V -are two disjoint node sets, and W is the edge set.

• The node set V + is given by {x + 1 , . . . , x + n } and the node set V -is given by {x - 1 , . . . , x - n }. Notice that we have in fact split each state node x i of G(A) into two nodes x + i and x - i .

• The edge set W is described by W = {(x + i , x - j )|a ji = 0}. In the latter, as before, a ji = 0 means that the (j, i)-th entry of the matrix A is a structural nonzero.

A matching in the bipartite graph H(A) = (V + , V -; W ) is a set of edges M ⊆ W such that the edges in M have no common node. A node is covered by a matching if there exists an edge in the matching that is incident to the node. The cardinality of a matching, i.e., the number of edges it consists of, is also called its size. A matching M is called maximum if its cardinality is maximum. The maximum matching problem consists of finding a matching of maximum cardinality. The maximum matching problem can be solved by using efficient combinatorial algorithms, see for example [START_REF] Hopcroft | An n 5/2 algorithm for maximum matchings in bipartite graphs[END_REF]. We recall the following proposition, see [START_REF] Commault | Characterization of generic properties of linear structured systems for efficient computations[END_REF]. Proposition 2. Consider the linear structured system defined by (1) with associated bipartite graph H(A). The generic rank of A is equal to the size of a maximum matching in H(A). In particular, d r (A) = n-g-rank (A) is the minimal number of steering nodes needed to make the rank condition of Theorem 1 become true.

A useful tool to parameterize all maximum matchings in a bipartite graph is the Dulmage-Mendelsohn decomposition, see [START_REF] Dulmage | Coverings of bipartite graphs[END_REF], abbreviated as DM-decomposition, which will be presented now, see also [START_REF] Murota | Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics[END_REF]. The DMdecomposition of the bipartite graph H(A) = (V + , V -; W ) is the uniquely defined family of bipartite subgraphs

H i = (V + i , V - i ; W i ), called the DM- components, where {V + 0 , V + 1 , • • • , V + r , V + ∞ } is a partition of V +
, and likewise for V -and W . In the decomposition, the bipartite subgraph H 0 is called The DM-decomposition corresponding to the bipartite graph associated with Example 1 is given in Figure 2. The maximum size of a matching in H(A) is 6. Hence, the generic rank of A is equal to 6 and d r (A) = 2. From Proposition 5 it follows that a maximum matching of H ∞ can be (x + 3 , x - 1 ), which implies that a possible admissible steering node set for the rank condition is {x 2 , x 8 }, but also {x 1 , x 8 } can act an admissible steering node set when (x + 3 , x - 2 ) is choosen as maximum matching of H ∞ .

x 3 + x 2 - x 5 - x 1 - V ∞ - V ∞ + x 8 - x 4 + x 7 - V 1 + V 2 - x 7 + x 6 - V 2 + V 1 - x 5 + x - x 4 - x 1 + x 2 + x 8 + V 0 - V 0 + x 6 + V 1 V 1 - x 3 -

Steering node classification for structural controllability

We start now with a structured system of type (1), hence only defined by the matrix A. As before, we denote the associated directed graph by G(A) and the associated bipartite graph by H(A). In this section we will give a classification of steering nodes according to the definitions of Section 2. We will provide this classification first for each condition (input connection condition and rank condition) and then for controllability.

Classification of steering nodes for input connection

For the input connection condition we have the following.

Proposition 6. Consider a linear structured system of type (1) with associated graph G(A). For the input connection property, node x i , being element of an admissible steering node set S for the input connection condition, is Note that {x i } is a Critical Connection Component if and only if there is no edge (x j , x i ) in G(A) for j = i. Hence, node x i is essential for the input connection condition if and only if there is no edge (x j , x i ) in G(A) for j = i.

Classification of steering nodes for the rank condition

For the rank condition we have the following.

Proposition 7. Consider a linear structured system of type (1) with associated bipartite graph H(A) and the corresponding DM-decomposition. For the rank condition property, node x i , being element of an admissible steering node set S for the rank condition, is 1. useless if and only if x - i does not belong to the V - ∞ set of the DMdecomposition, 2. useful if and only if x - i belongs to the V - ∞ set of the DM-decomposition, 3. essential if and only if there exists no edge (x + j , x - i ) in H(A).

Proof:

1. ⇒ Assume that x - i ∈ V - ∞ .
Then according to point 4 of Proposition 3, there exists a maximum matching on the bipartite subgraph H ∞ that does not cover x - i . Denote such matching by M ∞ and denote the nodes of

M ∞ incident to V - ∞ by M - ∞ . Define S -= V - ∞ \M - ∞ and let S denote the corresponding nodes in X , i.e., x - k ∈ S -⇐⇒ x k ∈ S. Then, M - ∞ ∪ S -= V - ∞ , M - ∞ ∩ S -= ∅ and |S -| = |V - ∞ | -|V + ∞ |.
According to Proposition 5 the set S is admissible. Clearly, x i ∈ S -. Now leave x i out of S, or equivalently x - i out of S -, and consider S\{x i }. Then according to Proposition 5, the set S\{x i } is not admissible. Indeed, any maximum matching on H ∞ has size |V + ∞ |. The number of nodes not covered by such matching is |V - ∞ | -|V + ∞ |, implying that at least one of the uncovered nodes can also not be covered by S -\{x - i }, as

|S -\{x - i }| < |V - ∞ |-|V + ∞ |. With S\{x i } being not admissible, it follows that x i is useful, i.e., is not useless. ⇐ Assume that x - i ∈ V - ∞ . Then x - i ∈ V -\V - ∞ .
Since S is admissible, by Proposition 5, there exists a maximum matching on the bipartite subgraph H ∞ such that for every x - j ∈ V - ∞ not covered by this matching there holds that x j ∈ S. Now fix this matching. Recall that x i ∈ S and leave x i out of S. Then still for every x - j ∈ V - ∞ not covered by the matching there holds that x j ∈ S\{x i }. Hence, by Proposition 5, it follows that S\{x i } is also admissible and therefore x i is useless. 2. Obvious from point 1. 3. ⇒ Suppose that there exists an edge (x + j , x - i ). Since x i is at least useful, it follows from the ordering of the DM-components, see [START_REF] Murota | Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics[END_REF], that x + j ∈ V + ∞ . Then there always exists a maximum matching of H ∞ that covers x - i . Indeed, if a given maximum matching does not cover x - i , we can exchange the edge (x + j , x - k ), which belongs to the maximum matching, with (x + j , x - i ) to get a new maximum matching of H ∞ that does cover x - i . The nodes in V - ∞ not covered by this new matching induce an admissible steering node set that does not contain x i . Therefore, x i is not essential. ⇐ Assume that x - i ∈ V - ∞ . If H(A) has no edge (x + j , x - i ), then no (maximum) matching on H ∞ will ever cover x - i . Hence, by Proposition 5, node x i has to be in any admissible steering node set S. Hence, x i is contained in any admissible steering node set and is consequently essential.

The previous results can be compared with those of [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF] (more precisely with the Supplementary Information associated with this paper). Point 3 of our Proposition 7 corresponds to the Supplementary Note 1, but our proof is purely a graph theoretic one, instead of using integer programming arguments as in [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF]. Point 1 of our Proposition 7 corresponds to the Supplementary Note 6, but our characterisation of useless nodes is much more explicit than in [START_REF] Jia | Emergence of bimodality in controlling complex networks[END_REF].

Classification of steering nodes for controllability

Next we combine the previous results to obtain a classification of steering nodes for structural controllability.

Theorem 2. Consider a linear structured system of type (1) with associated graph G(A), associated bipartite graph H(A) and the corresponding DMdecomposition. For controllability, the steering node x i , being an element of a steering node set S that is admissible for controllability, is 1. essential if and only if there exists no edge (x j , x i ) for j = i in G(A).

useless if x i belongs to no Critical Connection Component and x -

i does not belong to the V - ∞ set of the DM-decomposition.

Proof:

1. ⇒ Note that x i is a node in the steering node set S that is admissible for controllability. Then S is a steering node set that is also admissible for both the input connection condition and the rank condition. Now assume that there is an edge (x j , x i ) in G(A), or equivalently (x + j , x - i ) in H(A). Then, by Propositions 6 and 7, node x i is not essential for both the input connection condition and the rank condition. Hence, there exist steering node sets S c and S r that are admissible for the input connection and the rank condition, respectively, that both do not contain x i . Then the union S c ∪ S r does not contain x i too, and is an admissible steering node set for both the input connection condition and the rank condition, and consequently also for controllabiliy. Hence, node x i is not essential for controllability. ⇐ If there is no edge (x j , x i ) for i = j in G(A), then either there is no edge (x j , x i ) at all, or there is only an edge (x i , x i ) in G(A). In the latter case, {x i } is a strongly connected component without incoming edges and it follows from Proposition 6 that x i is essential for the input connection condition. In the former case, there is no edge (x + j , x - i ) at all in H ∞ , and it follows from Proposition 7 that x i is essential for the rank condition. Being essential for one of the two conditions, it follows that x i is essential for controllability. 2. Assume that x i is useless for both properties and consider any steering node set S containing x i that is admissible for controllability. The two properties remain satisfied with S\{x i }, therefore controllability is preserved too and consequently x i is useless for controllability.

Some remarks and examples

Some remarkable points follow from Theorem 2.

Remark 1. From the set of essential steering nodes for controllability, being the union of the essential steering nodes for the two subproperties, it follows that an essential steering node is a node with no incoming edge from another state (but may have a self-loop). The latter can be related with the interesting discussion in [START_REF] Cowan | Nodal degree, not degree distributions, determine the structural controllability of complex networks[END_REF][START_REF] Zhao | Intrinsic dynamics induce global symmetry in network controllability[END_REF] on the importance of self-loops in applications.

Remark 2. In Theorem 2, we only characterize a subset of the useless steering nodes for controllability (namely those which are useless for both subproperties). Indeed, as will be seen in a next example, some steering nodes may be useless for controllability, while being useful for one of the subproperties.

Example 1 (cont.) First we illustrate our main result on Example 1 whose graph G(A) is depicted in Figure 1, while the bipartite graph H(A) and its DM-decomposition are shown in Figure 2.

From Proposition 6 it follows that nodes x 2 , x 4 , x 6 , x 7 are useless for input connection, while nodes x 1 , x 3 , x 5 , x 8 are useful, with nodes x 5 , x 8 being essential, because they correspond to Critical Connection Components of cardinality one.

From Proposition 7 it follows that nodes x 3 , x 4 , x 5 , x 6 , x 7 are useless for the rank condition, nodes x 1 , x 2 , x 8 are useful, with node x 8 being essential, because it has no incoming edge. From Theorem 2 it follows that nodes x 5 , x 8 are essential steering nodes for controllability, while nodes x 4 , x 6 , x 7 are useless for controllability. It can be checked by inspection that besides nodes x 5 , x 8 also the nodes x 1 , x 2 , x 3 are useful for controllability. For example, node x 2 cannot be discarded from the admissible steering node set {x 2 , x 3 , x 5 , x 8 } because of the rank condition. We give now an example to illustrate Remark 2.

Example 2. Let A given below be a (3 × 3) structured matrix, whose digraph G(A) is depicted in Figure 3.

A =   0 * 0 * 0 0 * * 0   .
x 1 + x 2 + x 1 - x 3 - x 2 - V ∞ - V ∞ + x 3 + V 0 + Figure 4: DM-decomposition of Example 2
The graph G(A) possesses two strongly connected components, being {x 1 , x 2 } and {x 3 }, where {x 1 , x 2 } is the Critical Connection Component. Hence,

d c (A) = 1.
The DM-decomposition of H(A) is shown in Figure 4. Since the maximum size of a matching in H(A) is 2, the generic rank of A is equal to 2 and d r (A) = 1.

From Proposition 6 it follows that node x 3 is useless for input connection, while nodes x 1 and x 2 are useful.

From Proposition 7 it follows that there are neither useless nor essential nodes for the rank condition, so that nodes x 1 , x 2 and x 3 are just useful for the rank condition. From Theorem 2 we get no information about useless steering nodes for controllability because there are no useless steering nodes for both subproperties. However, it is clear that node x 3 is useless for controllability. Indeed, the input connection implies that nodes x 1 or x 2 should be steering nodes, which is also enough to ensure the rank condition.

Extension to the case of forbidden nodes

In this Section, as in [START_REF] Olshevsky | Minimum input selection for structural controllability[END_REF], we assume that there is a set of forbidden nodes, which cannot be used as steering nodes. This situation is frequently met in applications.

Let us denote by F ⊆ X the set of forbidden nodes, and the complementary subset by E = X \F. The nodes of E will be called effective nodes. These effective nodes E = {x i 1 , . . . , x ip }, with i j ∈ {1, . . . , n} and i 1 < i 2 < • • • < i p , can be associated with control inputs in order to define a steering node set S ⊆ E as in Section 2.

We extend the notions and definitions of Section 2, by restricting the possible sets of steering nodes S to be subsets of E. It is clear that, depending on E, an admissible steering node set for controllability does not necessarily exist. Let us first examine this question.

As was done for the steering node set in Section 2, we can associate a matrix B E with the effective node set E. The condition of existence of an admissible steering node set for controllability is stated in the following, rather obvious, lemma, which is given without proof.

Lemma 1. There exists an admissible effective steering node set for controllability for the system (1) if and only if the pair (A, B E ) is controllable.

If considering the problem in the structured system framework, the controllability of (A, B E ) can be tested using a version of Lin's theorem as in Theorem 1. First define the bipartite graph H(A, B E ) by adding to V + , on H(A), a set of inputs vertices u 1 , . . . , u p and edges (u j , x - i j ), for j = i, . . . , p.

Using refinements of the two conditions of Theorem 1, see for example [START_REF] Commault | Input addition and leader selection for the controllability of graph-based systems[END_REF], the result can be reformulated as follows.

Proposition 8. There exists an admissible steering node set for controllability of the structured system (1) if and only if

1. for any Critical Connection Component C i of G(A) we have C i ∩ E = ∅, 2.
the size of a maximum matching in H(A, B E ) is n.

Definition 1 can readily be extended to the case of forbidden nodes by simply adding that S ⊆ E, i.e., admissible steering nodes should be effective.

For a given property P, we denote by Es the set of essential nodes and by Us the set of useless nodes when there is no restriction on the set of possible steering nodes, i.e., when F = ∅. Denote by Es E the set of essential nodes and Us E the set of useless nodes when the set of possible steering nodes is E. Then we have the following simple general result. Lemma 2. With the previous definitions, we have:

1. Es ⊂ Es E , 2. Us ∩ E ⊂ Us E .

Proof:

Consider again the definitions of essential and useless nodes in Definition 1. It is obvious that the restriction of the set of possible nodes do not change their properties, and the result follows.

In simple terms, the previous means that the introduction of forbidden nodes can only increase the sets of essential and useless nodes.

Notice that in point 1 of Lemma 2, the intersection with E need not be explicited, since the essential nodes of the problem without forbidden nodes must belong to any set of effective nodes.

Classification of steering nodes for input connection

For the input connection condition we have the following.

Proposition 9. Consider a linear structured system of type (1) with associated graph G(A) and effective node set E. For the input connection property, node x i , being element of an admissible steering node set S ⊆ E for the input connection condition, is is n, the H ∞ part of H(A, B E ) is empty. Suppose now that u i j belongs to the V + 0 set in H(A, B E ). Then there exists a maximum matching in H(A, B E ) which does not cover u i j . Therefore one can build an admissible steering node set which does not contain x j , so x j is not essential. ⇐ Assume that u i j does not belong to the V + 0 set in H(A, B E ). Then, any maximum matching in H(A, B E ) contains u i j and discarding u i j (which is equivalent to discard the corresponding effective node x j from the set S), would decrease the size of a maximum matching. Then S\{x j } is not admissible and x j is essential.

Classification of steering nodes for controllability

The two previous propositions can be combined to get the general classification of nodes for controllability.

Theorem 3. Consider a linear structured system of type (1) with associated graph G(A), associated bipartite graph H(A), and the corresponding DM-decomposition. Consider a set of effective nodes E, B E the corresponding input matrix, and H(A, B E ) the associated bipartite graph with its DMdecomposition. For controllability, the steering node x j , being an element of a steering node set S ⊂ E that is admissible for controllability, is 1. essential if and only x j is the unique effective node in a Critical Connection Component or the corresponding input node u i j does not belong to the V + 0 set in H(A, B E ). 2. useless if x j belongs to no Critical Connection Component and x - j does not belong to the V - ∞ set of the DM-decomposition of H(A).

Proof: Follows the same lines as the proof of Theorem 2.

Examples

Example 1 (cont.) Consider again Example 1, assume that the set of forbidden nodes is F = {x 1 , x 4 , x 7 } and then that E = {x 2 , x 3 , x 5 , x 6 , x 8 }. As previously, nodes x 5 and x 8 are essential for input connection, but now x 3 is essential too, since it is the only effective node in the Critical Connection Component {x 1 , x 3 }. Concerning the rank condition, nodes x 3 , x 5 and x 6 in E are useless since x - 3 , x - 5 and x - 6 do not belong to the V - ∞ set of the DMdecomposition of H(A). The DM-Decomposition of the graph H(A, B E ), see Figure 5, shows that the input nodes u 2 and u 8 do not belong to the V + 0 set Notice that in this case, {x 2 , x 3 , x 5 , x 8 } is also a solution for the minimal controllability problem, see [START_REF] Olshevsky | Minimum input selection for structural controllability[END_REF][START_REF] Assadi | Complexity of the minimum input selection problem for structural controllability[END_REF][START_REF] Pequito | A framework for structural input/output and control configuration selection in large-scale systems[END_REF]. This solution contains 4 nodes, while when F = ∅, it can be checked that {x 1 , x 5 , x 8 } is a solution for the minimal controllability problem with only 3 nodes.

x 5 - u 2 x 4 + x 7 - u 5 x 5 + u 8 x 8 - V 1 + V 2 - x 3 + x 1 - V 2 + V 1 - x 2 - u 6 x 6 - V 0 - V 0 + x 1 + x
Example 3. In this example the use of forbidden and effective nodes will be illustrated in more details since the example will show that the inclusions in Lemma 2 can be strict. Therefore, consider the system described by the 4 × 4 structured matrix

A =     0 * 0 0 * 0 0 0 * 0 * 0 0 * 0 0     .
The digraph G(A) of the system is displayed in Figure 6. The DM- From the results of Section 4, we can compute the sets of useless and essential nodes. In summary, in the context of Lemma 2, we have the following sets Es and Us with the associated properties.

Es

Us input connection ∅ {x 3 , x 4 } rank ∅ {x 2 , x 3 } controllability ∅ {x 3 }

Now consider E = {x 1 , x 3 , x 4 } as a set of effective nodes, so that x 2 is a forbidden node.

From the results of this section, and using the DM-decomposition of the bipartite graph H(A, B E ), which is displayed Figure 8, one can compute the sets of useless and essential nodes. In summary, in terms of Lemma 2 and the effective node set E = {x 1 , x 3 , x 4 }, we have the following sets Es, Us, Es E and Us E with the associated properties.

Es

Us So, in the context of Lemma 2, we have a strict inclusion Es ⊂ Es E for the input connection condition and the controllability condition, and a strict inclusion Us ∩ E ⊂ Us E for the controllability condition.

Conversely, it can be seen from the comparison of Propositions 6 and 9, that for input connection, in point 2 of Lemma 2 (concerning useless steering nodes) we have indeed an equality. The same can be said for the rank condition by comparing Propositions 7 and 10.

Complexity analysis

The classification of steering nodes mainly implies the computation of the Critical Connection Components of G(A) and of the DM-decomposition of H(A).

The decomposition of a graph into strongly connected components is a standard combinatorial problem for which very efficient polynomial algorithms are available [START_REF] Tarjan | Depth-first search and linear graph algorithms[END_REF]. In particular, Kosaraju's algorithm gives the decomposition in strongly connected components with a complexity O(n + k), where k is the number of non-zero entries in A.

The DM-decomposition implies first the determination of a particular maximum matching, which is then completed by an alternate chain technique, see the details in [START_REF] Murota | Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics[END_REF]. Since the alternate chain technique is of a lower complexity than the maximum matching search, the complexity of the DM-decomposition is the same as, for example, in [START_REF] Hopcroft | An n 5/2 algorithm for maximum matchings in bipartite graphs[END_REF].

It follows that steering nodes classification for structured systems is a polynomial problem.

Conclusions and outlook

In this paper we studied a classification of steering nodes into useless, usefull or essential ones in order for a large complex structured system to become structually controllable. For the individual conditions for structural controllability, being the input connection condition and the rank condition, this classification could be given completely. However, for their combination, culminating in a classification of steering nodes for structural controllabilty, this still is not settled completely as far as useless steering nodes are concerned. This will remain a topic for further research. The methods underlying the obtained classifications are based on well-understood algorithms coming from the theory of flows in networks. A generalization of the results is given in case the nodes of the system can be divided into forbidden and effective (=non-forbidden) nodes.

Figure 1 :

 1 Figure 1: Digraph of Example 1

Definition 2 .

 2 A strongly connected component of G(A) with no incoming edge from another strongly connected component is called a Critical Connection Component (CCC). The number of Critical Connection Components is called the connection defect and is denoted by d c (A).

Figure 2 :

 2 Figure 2: DM-decomposition of Example 1

Figure 3 :

 3 Figure 3: Digraph of Example 2 with strongly connected components

Figure 6 :

 6 Figure 6: Digraph of Example 3

Figure 7 :

 7 Figure 7: DM-decomposition of bipartite graph of Example 3

1 Figure 8 :

 18 Figure 8: DM-decomposition of bipartite graph of H(A, B E ) in Example 3

  1. useless if and only if it belongs to no Critical Connection Component, 2. useful if and only if it belongs to a Critical Connection Component, 3. essential if and only if it belongs to a Critical Connection Component of cardinality one. Assume that x i belongs to a Critical Connection Component C j and consider an admissible steering node set S composed of a steering node in each Critical Connection Component with x i the only steering node in C j . Then S\{x i } is no longer an admissible steering node set, and consequently x i is not useless. ⇐ Conversely, assume now that the steering node x i belongs to no Critical Connection Component, and consider any admissible steering node set S such that x i ∈ S. From Proposition 1, the set S must contain a node in each Critical Connection Component, therefore S\{x i } is still an admissible steering node set and x i is useless. 2. Obvious from point 1. 3. ⇒ Assume that x i belongs to a Critical Connection Component C j .

	Proof:
	1. ⇒

Hence, by the above point, x i is usefull. Now assume that C j of cardinality more than one. Denote by x k another node of C j . We can construct an admissible steering node set containing x k , but not x i . Therefore, x i is not essential. ⇐ Conversely, if x i belongs to a Critical Connection Component C j of cardinality one, it follows from Proposition 1, that any admissible steering node set must contain x i . Therefore x i is essential.

  1. useless if and only if it belongs to no Critical Connection Component, 2. useful if and only if it belongs to a Critical Connection Component, 3. essential if and only it is the unique effective node in a Critical Connection Component. Assume that x i is effective and belongs to a Critical Connection Component C j . Hence, by the above point, x i is usefull. Now assume that there exists another effective node x k ∈ C j . We can construct an admissible steering node set containing x k , but not x i . Therefore, x i is not essential. ⇐ Conversely, if x i is the unique effective node in a Critical Connection Component C j , it follows from Proposition 1, that any admissible steering node set must contain x i . Therefore x i is essential as effective node It is immediate to check that Proposition 9 reduces to Proposition 6 when F = ∅.7.2. Classification of steering nodes for the rank conditionProposition 10. Consider a linear structured system of type (1) with associated graph G(A), associated bipartite graph H(A), and the corresponding DM-decomposition. Consider a set of effective nodes E, B E the corresponding input matrix, and H(A, B E ) the associated bipartite graph with its DMdecomposition. For the rank condition property, node x j , being element of an admissible steering node set S ⊆ E for the rank condition, is We have at our disposal p effective basis vectors {vi 1 , . . . , v ip } such that rank [A, v i 1 , . . . , v ip ] = n. If v i k is such that rank [A, v i k ] = rankA, or v i k ∈SpanA, this node is clearly useless for the rank condition. In graph terms this corresponds to the fact that x - k does not belong to the V - ∞ set of H(A). ⇐ The arguments are the same as in the proof of Proposition 7. 2. Obvious from point 1. 3. ⇒ Notice first that, since the size of a maximum matching in H(A, B E )

	1. ⇒
	Proof:
	1. The arguments are exactly the same as those of point 1 in the proof of
	Proposition 6.
	2. Obvious from point 1.
	3. ⇒ 1. useless if and only if x -i does not belong to the V -∞ set of the DM-
	decomposition of H(A),
	2. useful if and only if x -i belongs to the V -∞ set of the DM-decomposition
	of H(A),
	3. essential if and only if the corresponding input node u i j does not belong to the V + 0 set in H(A, B E ).
	Proof:

  E ), so x 2 and x 8 are essential nodes for the rank condition. In summary, for controllability, as nodes in E nodes x 2 , x 3 , x 5 and x 8 are essential, and node x 6 is useless because it does not belong to a Critial Connection Component.

	u 6	x 6
	x 7 +	3 -
	u 3
	x 2 +	x 4 -
	x 6 +
	x 8

+ Figure 5: Digraph of bipartite graph of H(A, B E ) in Example 1 in H(A, B

  Es E Us E connection ∅ {x 3 , x 4 } {x 1 } {x 3 , x 4 } rank ∅ {x 2 , x 3 } ∅ {x 3 } controllability ∅ {x 3 } {x 1 } {x 3 , x 4 }

Acknowledgment

The authors wish to thank Jean-Michel Dion for his invaluable collaboration.

minimal inconsistent part, the bipartite subgraph H ∞ is called maximal inconsistent part, and the other subgraphs H i , i = 1, . . . , r, are called consistent parts. The used names come from a partial ordering ≺ that can be assigned to the bipartite subgraphs, implying that H 0 ≺ H i ≺ H ∞ , for i = 1, . . . , r.

The DM-decomposition and the above components have the following properties, for details see [START_REF] Murota | Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics[END_REF].

. . , r, ∞, as its DMcomponents. Then we have the following.

A maximum matching on H(A) is a union of maximum matchings on the DM-components

r) is covered by any maximum matching on H(A).

3. A node v + ∈ V + belongs to the minimal inconsistent part H 0 if and only if there exists a maximum matching on H(A) that does not cover node v + , implying that

only if there exists a maximum matching on H(A) that does not cover node v -, implying that

Note that for the consistent parts there holds

. . , r. The rank condition for controllability can then be expressed using only the maximal inconsistent part of the DM-decomposition as follows, see [START_REF] Commault | Observability recovering by additional sensor implementation in linear structured systems[END_REF]. Proposition 4. Consider the linear structured system defined by (1) with associated bipartite graph H(A) and the corresponding DM-decomposition. We have that g-rank

The result follows from Proposition 3, especially point 4. Indeed, the size of a maximum matching equals

Proposition 5. Consider the linear structured system defined by (1) with associated bipartite graph H(A) and the corresponding DM-decomposition. A steering node set S is admissible for the rank condition if and only if there exists a maximum matching of the bipartite subgraph H ∞ such that for every node x - i in V - ∞ that is not covered by the matching, there holds x i ∈ S.


est destinée au dépôt

Investigation of the coupling between pump amplitude noise and differential phase noise in an

We report experimental evidence of the coupling between the pump diode current and the output differential phase in a two-axis two polarization dual-frequency solid state laser. The effect has the same transfer function shape as that for pump current to output power one, suggesting a pump AM/laser PM coupling in the laser medium. We report that the pump amplitude noise is negligible in the laser differential phase noise assuming a linear coupling. © 2015 Optical Society of America

1.INTRODUCTION

Low phase-noise lasers are mandatory in many applications including telecommunications and lidar-radar [START_REF] Boyd | ʻʻA 1.55 µm solid-state laser source for DWDM applications''[END_REF][START_REF] Morvan | ʻʻ Dual-frequency laser at 1.53 µm for generating highpurity optically carried microwave signals up to 20 GHz[END_REF][START_REF] Pillet | ʻʻWideband dual-frequency lidar-radar for high-resolution ranging, profilometry, and Doppler measurement[END_REF][START_REF] Alouini | ʻʻOffset phase locking of Er,Yb:glass laser eigenstates for RF photonics applications[END_REF]. The laser linewidth of rare-earth doped solid state lasers is known to be smaller than in semiconductor lasers, where it is spoiled by amplitude-to-phase coupling (AM/PM) due to the Henry factor α [5,6]. Consequently most of the studies dealing with the Henry factor are related to semiconductor lasers. The few reported studies on amplitude to phase coupling in solid state lasers concern microchip lasers [START_REF] Szwaj | ʻʻLarge linewidth-enhancement factor in a microchip laser[END_REF][START_REF] Fordell | ʻʻModulation and the linewidth enhancement factor of a diode-pumped Nd:YVO4 laser[END_REF]. Yet, when ultrastable and narrow linewidth are required, the amplitude to phase conversion might become the limiting factor even in solid state lasers. We have been involved during the last years in developing a photonic adjustable ultra-stable sub-millimeter and millimeter wave reference based on a dual frequency laser [START_REF] Danion | ʻʻHigh spectral purity microwave and terahertz oscillator[END_REF] locked to a single ultralow expansion cavity. In this framework the knowledge of the amplitude to phase coupling coefficient is essential in order to ensure the best possible stability of the beat note between the two optical laser lines [START_REF] Rolland | ʻʻNarrow linewidth tunable terahertz radiation by photomixing without servo-locking[END_REF][START_REF] Pillet | ʻʻSelf-stabilization of the beatnote of a 1.5 µm dual-frequency laser. Applications to optoelectronic microwave oscillators[END_REF][START_REF] Alouini | Yb:glass laser for terahertz beat frequency generation[END_REF][START_REF] Danion | ʻʻDual frequency laser with two continuously and widely tunable frequencies for optical referencing of GHz to THz beatnotes[END_REF]. Given the extremely low coupling coefficients we are expecting in this kind of solid-sate laser, a specific measurement setup has to be designed.

In this paper, we first describe the experimental setup and the protocol used to investigate the AM/PM coupling between the pump and the dual frequency beatnote denoted in the following AM pump /PM beat . We then checked for possible measurement biases. Finally, a thorough analysis is conducted to recover the actual contribution of relative intensity noise of the pump on the differential phase noise of the dual frequency laser.

2.EXPERIMENTAL SETUP

The laser under consideration is an Er,Yb:glass laser oscillating at 1550 nm. The optical power of the two optical carriers is 1 mW after injection in the optical fiber [START_REF] Danion | ʻʻHigh spectral purity microwave and terahertz oscillator[END_REF][START_REF] Danion | ʻʻDual frequency laser with two continuously and widely tunable frequencies for optical referencing of GHz to THz beatnotes[END_REF]. The beat-note is tunable over a DC-900 GHz range by rotating the intra cavity etalons [START_REF] Danion | ʻʻDual frequency laser with two continuously and widely tunable frequencies for optical referencing of GHz to THz beatnotes[END_REF]. The diode pump is a fiber Bragg grating stabilized longitudinal monomode laser, providing 800 mW at 980 nm. The pump beam is split in two paths using a birefringent YVO4 crystal and orienting the input pump polarization at 45°. Hence, the two optical axes of the laser are pumped with the same pump, that is, with perfectly correlated pump noises, Fig. [1-a].

We seek to measure the AMpump/PMbeat eat coupling in the perturbation regime. As shown in Fig. 2, when the dual frequency laser is operated in the free running regime, the phase noise of its beatnote is relatively high which forbids the measurement of the AMpump/PMbeat coupling in the small signal regime. In order to detect small contributions, one has to reduce the beatnote drift by stabilizing the frequencies difference of the laser [START_REF] Alouini | Yb:glass laser for terahertz beat frequency generation[END_REF][START_REF] Gupta | ʻʻPhase-locked loops[END_REF][START_REF] Hsieh | ʻʻPhase-locked loop techniques[END_REF] as depicted in Fig. 1. This frequency difference is stabilized on a reference provided by a low phase noise electrical synthesizer and a homemade phase locked loop (PLL). The laser frequency difference excursion range being +/-15 MHz, a PLL is necessary to remove this drift and run the laser in constant conditions over the measurement time. In the experiments described in this paper, we use only one of the two internal LiTaO3 electro-optic crystals (2x2x8 mm3) to servo the frequency difference of the two optical carriers. The output beam is injected into a polarization maintaining 10/90 fibered coupler. Its first arm (10%) is used for optical monitoring purposes while the second arm (90%) is devoted to electrical beat note synthesis. This beatnote is obtained due to a fast InGaAs photodiode (Discovery 16 GHz) in front of which a polarizer oriented at 45° is placed to make the two laser polarizations interfere. The assembly starting from the laser intracavity electro-optic crystal to the photodiode output realizes a voltage controlled oscillator (VCO) synthesizer. In the experiment reported here the frequency beatnote, and thus the phase locked loop, is adjusted to 1 GHz. A directional coupler splits the microwave signal at the output of the photodiode. 10% of its power is directed on a phase noise measurement apparatus described in the following, whereas 90% of the power feeds the PLL. Basically, it is mixed with a microwave reference at 1 GHz provided by a low phase noise synthesizer. The

A C C E P T E D M A N U S C R I P T
correction filter has a proportional-integrator shape, so that the open loop unity gain bandwidth is 100 kHz. Finally, the correction signal is applied to the electro-optic crystal inside the laser cavity.

MEASUREMENT OF THE PUMP DIODE AMPLITUDE NOISE CONTRIBUTION

A careful characterization of the optical power modulation of the pump versus the voltage modulation of the driver from 10 to 10 5 Hz is performed using a network analyser (Stanford SR780) connected to a high bandwidth photodiode (1 GHz). This allows us to accurately adjust the modulation index of the pumping power.

The microwave phase noise of the beat note delivered by the laser is measured with a 26 GHz Aeroflex electrical spectrum analyser. The experiment is performed in a pertubative regime of low modulation order not to destabilize the laser operation. As shown in Fig. 2, to detect small signals, we need to stabilize the beat note to reduce the noise floor, hence the need for PLL. At 1 GHz, the noise floor is measured to be -80 dBc/Hz at 1 kHz offset frequency. In order to measure the AM pump /PM beat conversion efficiency, we applied a low modulation of the pump power. For example, when we modulate the current of the pump laser driver at 10 kHz in order to obtain 6 mW pump amplitude (0.75% of modulation index), we observe a modulation at the same offset frequency on the phase noise of the optically synthesized frequency. This phase modulation appears as a sharp peak in the phase noise spectrum (see Fig 2). It is worthwhile to notice that the action of the servo loop which reduces the phase noise excursion enabling such a measurement must be taken into account. Indeed, the measured phase noise must be multiplied by the gain of the open loop in order to unfold the effect of the PLL and then recover the actual phase modulation level. This correction factor being frequency dependent, a careful characterization of the open loop gain is performed. Moreover, we checked that the intensity modulation of the beatnote, associated to the intensity modulation of the pump laser, is efficiently filtered out by our I&Q demodulation apparatus. To this aim, we applied to the laser an external amplitude modulation of the same level and in the range 10-10 5 Hz and we did not notice after I&Q demodulation any residual peak in the phase noise spectrum. More importantly, we checked that the laser operates, as required, in perturbation regime. To this aim we increase the pump power modulation depth and check that the peak amplitude increases proportionally. We integrate the power fluctuation for frequencies between 900 Hz and 50 kHz and the power fluctuation between 1 mW and 6 mW. For each modulation frequency the slope of the line giving the phase modulation amplitude versus pump modulation amplitude is extracted using a linear regression as shown in Fig. 3. These slopes are then multiplied by the offset frequency to end up with the magnitude of the transfer function displayed with red dots in Fig. 5.

We then made systematic measurements of the transfer function with a network analyzer (Stanford SR780), Fig. 4. We measure the transfer function between the amplitude modulation and the conversion factor, the blue line on Fig. 5 Fig. 4. Scheme of experimental setup with a vectorial network: indirect method

A C C E P T E D M A N U S C

R I P T

To go further and confirm these results we now analyze the transfer function given by the ratio between the amplitude of the correction signal delivered by the PLL and the modulation amplitude of the pump diode current. Indeed, the correction signal, at frequencies smaller than the PLL unity gain bandwidth, is proportional to the laser free running noise weighted by the electro-optic voltage to frequency response. This calibration factor is measured to be constant and equal to 1.1 MHz/V up to a few MHz. The direct (red points) and indirect (blue full line) measurements are in good agreement except for the last point close to the laser relaxation oscillations. At this peculiar frequency, the laser is no more operated in the low signal regime because the output intensity modulation increases by more than 30 dB. Moreover, and because of this increase, the I&Q demodulation setup becomes not efficient enough to filter out the amplitude modulation. Thus, the spectral region around the oscillation relaxations must be avoided for such sensitive measurements.

Fig. 5. Conversion factor versus frequency modulation, with phase noise analyzer (dots) and PLL correction signal (plain curve).

It is worthwhile to notice that the slope of the transfer function does not correspond to the usual 1/f decrease but first to a 1/f decrease up to 1 kHz and then to a 1/f² decrease between 1 kHz and 30 kHz. This behavior is due to the energy exchange mechanism between Yb and Er in the active medium which has a millisecond time constant leading to an additional first order low-pass filter with 1 kHz pole [START_REF] Taccheo | ʻʻ Intensity noise reduction in a single-frequency ytterbium-codoped erbium laser[END_REF]. The frequency response of the energy exchange mechanism is extracted from low frequency RIN spectrum of the laser (see [START_REF] Danion | ʻʻDual frequency laser with two continuously and widely tunable frequencies for optical referencing of GHz to THz beatnotes[END_REF]). When this mechanism is taken into account, it turns out that the AM pump /PM beat coupling coefficient is independent of the modulation frequency as reported in Fig. 6. Moreover, we find that its magnitude is about 10 kHz per mW of pump power.

DEDUCTION OF AM/PM FACTOR

This result is of great importance since it will enable us to directly estimate the contribution of the pump noise on the differential phase noise in the dual frequency solid state laser. In particular, one wants to know if the pump diode choice has a detrimental effect on the phase noise of dual frequency laser beatnote. To this aim, we report in Fig. 7, different beatnote phase noise spectra as follows. The green spectrum represents the free running beatnote phase noise of the laser, measured using the correction signal of our PLL when open. The red spectrum represents the stabilized laser beatnote phase noise, i.e., when the PLL is closed. The two remaining dot spectra are deduced from the intensity noise of pump diode weighted by the measured AM pump /PM beat coupling coefficient. Hence, the blue dots spectrum represents the phase noise induced using a single mode fiber coupled pump diode whereas the orange dots spectrum corresponds to the phase noise induced by a multimode fiber coupled laser diode. In both cases, the dual frequency laser is operated in the same conditions, i.e., same pump power, same, threshold and same pump rate. Fig. 7 clearly evidences that AM pump /PM beat coupling is not the limiting factor for the two pumping schemes since both spectra are below that of the free running laser. The 20 dB phase noise offset between the multimode and monomode pumping schemes corresponds to the measured intensity noise offset between the two pump diodes. Consequently, the beatnote phase noise, although very low in dual frequency solid state lasers, is still limited by another noise source, and probably by intrinsic thermal noise in the active medium.

CONCLUSION

We highlighted the coupling factor between AM pump /PM beat between the amplitude pump fluctuation and the dual frequency beatnote for a two axis dual frequency solid state laser. We have described the experimental setup and the protocol used to investigate the coupling. To detect the small contribution of this factor, the laser is stabilized with a PLL. Two techniques of measurement, a direct method and an indirect method, allow to determine the coupling factor. We have shown that the slope of the transfer function AM pump /PM beat corresponds to a 1/f decrease up to 1 kHz and then to a 1/f² decrease between 1 kHz and 30 kHz, which corresponds at a low-pass filter due to energy exchange mechanism between Yb and Er in the active medium which has a millisecond time constant. Correcting the transfer function of this mechanism, the conversion factor is of 10 kHz per mW of pump power. This result, allows to estimate the fluctuation noise contribution of pump diode on the phase noise of the two-axis dual frequency solid state laser. We show that the phase noise induced by the RIN of the diode pump is not the limiting factor of this phase noise laser. One possibility of this limiting the phase noise of beat note is may be caused by intrinsic thermal noise in the active medium.

  Er, Yb: glass two-polarization dual-frequency solid-state laser Gwennaël Danion, Antoine Rolland, François Bondu, Goulc'Hen Loas, Mehdi Alouini To cite this version: Gwennaël Danion, Antoine Rolland, François Bondu, Goulc'Hen Loas, Mehdi Alouini. Investigation of the coupling between pump amplitude noise and differential phase noise in an Er, Yb: glass twopolarization dual-frequency solid-state laser. Journal of the Optical Society of America B, 2016, 33 (4), pp.589-593. 10.1364/JOSAB.33.000589. hal-01343398Investigation of the coupling between pump amplitude noise and differential phase noise in an Er,Yb:Glass two polarization dual frequency solid state laser Gwennaël Danion, * Antoine Rolland, François Bondu, Goulch'en Loas, and Mehdi Alouini Institut de Physique de Rennes, Département d'Optique et Photonique, UMR 6251 Université Rennes 1-CNRS Campus de Beaulieu, 35042 Rennes cedex, France *Corresponding author: gwennael.danion@univ-rennes1.fr

Fig. 1 .

 1 Fig. 1. (a) Schematic representation of the two-axis dual frequency laser. (b) Scheme of experimental setup stabilization and first measurement technique: direct measurement

Fig. 2 .

 2 Fig. 2.Phase noise of free running laser (green curve) and phase noise stabilized laser (purple curve) The sharp peak appearing on phase noise spectrum gives access to the AM pump /PM beat coupling induced by the pump power modulation.

Fig. 3 .

 3 Fig. 3. Measure of phase modulation versus amplitude modulation for different frequency of diode pump modulation.

Fig. 6 .

 6 Fig. 6. Conversion factor, between laser output amplitude and output phase modulation, corrected from oscillation relaxations

Fig. 7 .

 7 Fig. 7. Estimation of phase noise induced by a multimode pump (orange dotted line) and monomode pump (blue dotted line), and comparison with the on free running and stabilized laser phase noise.

Funding. This work is supported by Agence National de la Recherche (OSMOTUS ANR-2011-BS0301001) and by Région Bretagne.

Acknowledgment. The authors would like to thank L. Frein and S. Bouhier for their technical assistance.