Atef Khedher

Ben Kamel

Othman

Mohamed Benrejeb
email: mohamed.benrejeb@enit.rnu.tn

Didier Othman

Mohamed Maquin

Benrejeb

Design of an adaptive faults tolerant control: case of sensor faults

Keywords: multiple model, estimation, proportional integral observer, sensor faults, faults tolerant control

This paper presents a method of design of a sensor faults tolerant control. The method is presented for the case of linear systems and then for the case of non linear systems described by Takagi-Sugeno models. The faults are initially estimated using a proportional integral observer. A mathematical transformation is used to conceive an augmented system in which the sensor fault appear as an unknown inputs. The synthesized control depends on the estimated faults and the error between the state of a reference reference and the faulty system state. The fault tolearnt control is conceived using the augmented state. The conditions of the observer convergence and of the control existence are formulated in terms of Linear Matrix Inequalities (LMI). The formulation in LMI shows that the synthesis of the control and the observer can be independently made. For both cases (linear and non linear) The theoretical results are validated by their application to a noisy system affected by sensor faults.

Introduction

Physical processes are generally subjected to disturbances affecting their inputs or their outputs. The evolution in time of these disturbances is unknown and can damage the smooth running of the system. The consideration of the disturbances during the modelling and the state estimation becomes necessary to establish diagnosis procedures allowing faults detection and localization.

Faults estimation can be made using a proportional integral state observer in the case of non linear systems represented by multiple models [START_REF] Khedher | State and unknown input estimation via a proportional integral observer with unkbown inputs. 9th international conference on Sciences and Techniques of Automatic control and computer engineering[END_REF][START_REF] Orjuela | On the simultaneous state and unknown inputs estimation of complex systems via a multiple model strategy[END_REF]. PMI Observers can also be used for faults estimation [START_REF] Ichalal | Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi-Sugeno model with unmeasurable premise variables[END_REF]. That kind of observers gives some robustness property of the state estimation with respect to the system uncertainties and perturbations [START_REF] Beale | Robust control system design with a proportional integral observer[END_REF][START_REF] Ichalal | Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi-Sugeno model with unmeasurable premise variables[END_REF][START_REF] Linder | Rejecting disturbances to flexible structures using PI Kalman filters[END_REF][START_REF] Orjuela | Proportional-Integral observer design for nonlinear uncertain systems modelled by a multiple model approach[END_REF][START_REF] Weinmann | Uncertain models and robust control[END_REF]. Once the fault is estimated, its effect can be limited or eliminated using a fault tolerant control strategy.

The objective of a fault tolerant control is to find a control strategy which can limit or cancel, the fault effects on the system performances [START_REF]Commande multi-modèles tolérante aux défauts : Application au contrôle de la dynamique d'un véhicule automobile[END_REF]. There are two approaches of faults tolerant control synthesis: the passive approach and the active approach.

In the passive approach, the faults are taken into account during the design of the control. The method considers faults as disturbances which the control has to consider from its initial conception [START_REF] Chen | Fault-tolerant control systems design using the linear matrix inequality approach[END_REF][START_REF] Niemann | Passive fault tolerant control of double inverted penduluma case study example[END_REF].

The active approach reacts "actively" on the faults in on-line reconfiguration of the control so as to keep the stability and the nominal performances of the system [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF][START_REF] Weng | Active fault tolerant control of a double inverted pendulum. 6th IFAC Symposium on Fault Detection[END_REF]. This approach allows then to treat unforeseen faults but requires an effective method of faults detection and isolation allowing giving exactly information about the faults.

Our contribution in this paper lies in the synthesis of an active sensor fault tolerant control. For faults estimation, a mathematical transformation is used. It allows conceiving an augmented system in which the initial sensor fault appears as an actuator fault. By considering the augmented system, a proportional integral observer is conceived to estimate the faults [START_REF] Khedher | State and sensor faults estimation via a proportional integral observer[END_REF]. It is possible to estimate simultaneously actuator and sensor faults using this approach [START_REF] Khedher | An approach of faults estimation in Takagi-Sugeno fuzzy systems Accepted in the 8 th ACS/IEEE International Confer[END_REF][START_REF] Khedher | Sensor fault estimation for nonlinear systems described by Takagi-Sugeno models Accepted for publication in the International[END_REF]. The fault tolerant control is then synthesized. By being inspired of the works of Witczak and al. [START_REF] Witczak | A fault-tolerant control strategy for Takagi-Sugeno fuzzy systems[END_REF] made in the context of the discrete systems affected by actuator faults, and works presented in [START_REF] Edwards | A comparison of sliding mode and unknown input observers for fault reconstruction[END_REF][START_REF] Zhang | Coquempot V. Adaptive Observer-based Fast Fault Estimation[END_REF] treating linear systems, Khedher and al. [START_REF] Khedher | Fault tolerant control for nonliner system described by Takagi-Sugeno models[END_REF] have proposed an approach to conceive a fault tolerant control for actuator faults. this work is interested to the case of nonlinear systems described by Takagi-Sugeno fuzzy models affected by sensor faults.

The paper is structured in the following way. The section II presents the proposed method for the case of linear systems, the obtained results is applied to a numerical example. The section III recalls the Takagi-Sugeno models and details the proposed method for the multiple models case. An example of application of the proposed method to the Takagi-Sugeno models showing the efficiency of the method is presented in the section IV.

2 Proposed method for the linear system case

Problem formulation

The main objective of this part is to synthesize an active sensor fault tolerant control for the case of linear systems. The application of this control to a linear system presenting sensors faults allows to restore its original behaviour. A linear system can be described by the following state equation:

ẋ(t)=Ax(t)+Bu(t) y(t)=Cx(t) (1)
where x(t) ∈ IR n represents the system state, y(t) ∈ IR m is the measured output, u(t) ∈ IR r is the system input. A, B and C are known constant matrices with appropriate dimensions. Let consider a linear system with sensor fault described by the following state equation:

ẋf (t)=Ax f (t)+Bu f (t) y f (t)=Cx f (t)+Ef(t)+Dw(t) (2)
where x f (t) ∈ R n represents the system state, y f (t) ∈ R m is the measured output, u f (t) ∈ R r is the system input, f (t) represents the fault which is assumed to be bounded and w(t) is the measurement noise. E and D are respectively the fault and the noise distribution matrices which are assumed to be known.

Let us define the following states [START_REF] Edwards | A comparison of sliding mode and unknown input observers for fault reconstruction[END_REF] :

ż(t)= ĀCx(t) -Āz(t) żf (t)= ĀCx f (t) -Āz f (t)+ ĀEf (t)+ ĀDw(t) (3
) where -Ā is a stable matrix. Defining X and X f as: X = x T z T T and

X f = x T f z T f
T , these two state vectors can be written:

Ẋ(t)=A a X(t)+B a u(t) Y (t)=C a X(t) (4)
and:

Ẋf (t)=A a X f (t)+B a u f (t)+E a f (t)+F a w(t) Y f (t)=C a X f (t) (5) with:
A a = A 0 ĀC -Ā ,B a = B 0 ,C a = 0 I F a = 0 ĀD and E a = 0 ĀE ( 6
)
where I is the identity matrix with appropriate dimensions.

A proportional integral observer which permits the estimation of X f and f is considered:

     Ẋf (t)=A a Xf (t)+B a u f (t)+E a f (t)+K Ỹ (t) ḟ (t)=L Ỹ (t) Ŷf (t)=C a Xf (t) (7
) where Xf (t) is the estimated state, f (t) represents the estimated fault, Ŷf (t) is the estimated output, K is the proportional observer gain and L is its integral gain which must be computed and Ỹ (t)=Y f (t) -Ŷf (t). The control u f (t) is given by [START_REF] Witczak | A fault-tolerant control strategy for Takagi-Sugeno fuzzy systems[END_REF]:

u f (t)=-S f (t)+N (X(t) -Xf (t)) + u(t) (8)
where S and N are two constant matrices with appropriate dimensions. The objective is to find the matrices S and N which permit to the state X f to converge to X. Let define X(t) the error between X(t) and X f (t), Xf (t) the estimation error of the state X f (t) and f (t) the fault estimation error.

X(t)=X(t) -X f (t) (9) Xf (t)=X f (t) -Xf (t) (10) f (t)=f (t) -f (t) (11)
The dynamics of X(t) can be written: S is chosen so that E a = B a S, The dynamics of X(t) becomes:

Ẋ(t)= Ẋ(t) -Ẋf (t)=(A a -B a N ) X(t) +B a S f (t) -B a N Xf (t) -E a f (t) -F a w(
Ẋ(t)=( A a -B a N ) X(t) -B a N Xf (t) -E a f (t) -F a w(t) (13)
The dynamics of Xf (t) can be written:

Ẋf (t)= Ẋf (t) -Ẋf (t) =(A a -KC a ) Xf (t)+E a f (t)+F a w(t) (14)
The dynamics of f is written:

ḟ (t)= ḟ (t) -ḟ (t) = ḟ (t) -LC a Xf (t) (15)
The following vectors are introduced:

ϕ(t)=   X(t) Xf (t) f (t)   and ψ(t)= w(t) ḟ (t) (16)
(13), ( 14) and ( 15) can be written as:

φ(t)=A 0 ϕ(t)+B 0 ψ(t) (17)
with:

A 0 =   A a -B a N -B a N -E a 0 A a -KC a E a 0 -LC a 0   and B 0 =   -F a 0 F a 0 0 I   (18)
In order to analyse the convergence of the generalized error ϕ(t), let us consider the following quadratic Lyapunov candidate function V (t):

V (t)=ϕ(t) T P ϕ(t) (19)
where P denotes a symmetric positive matrix. The problem of robust state and fault estimation is to find the gains K and L of the observer to ensure an asymptotic convergence of ϕ toward zero when ψ(t) = 0 and to ensure a bounded error when ψ(t) = 0. This problem is reduced to find P verifying V<0, ie. A T 0 P + PA 0 < 0. The matrix A 0 can be expressed as:

A 0 = A a -B a NE 1 0 Ã -K C ( 20
)
where:

à = A a E a 00 , K = K L E 1 = -B a N -E a , C = C a 0 (21)
Assuming that P has the block diagonal form P = diag(P 1 ,P 2 ), it can be observed from the structure of A 0 that the eigenvalues of the matrix A 0 are the union of those of A a -B a N and à -K C. This indicates that the design of the control v(t) and the observer can be carried out independently (separation principle). Thus, it is clear from the expression of P that ϕ converges to zero if there exist matrices P 1 > 0 and P 2 > 0 such that these inequalities are satisfied:

(A a -B a N ) T P 1 + P 1 (A a -B a N ) < 0 (22) ( Ã -K C) T P 2 + P 2 ( Ã -K C) < 0 (23)
By multiplying [START_REF] Witczak | A fault-tolerant control strategy for Takagi-Sugeno fuzzy systems[END_REF] from left and right by

W = P -1 1 one obtains: W (A a -B a N ) T +(A a -B a N )W<0 (24)
The inequalities ( 23) and ( 24) are not linear. Substituting U = NW, and G = P 2 K, they become:

WA T a + A a W -U T B T a -B a U<0 (25) ÃT P 2 + P 2 Ã -G C -CT G T < 0 (26)
After the resolution of the linear matrices inequalities (LMI) (25) and (26), N and K are computed using the equations:

N = UW -1 (27) K = P -1 2 G (28)

Example

Consider the linear systems described by the equations ( 1) and ( 2) with C = I and:

A =     -0.2 -3 -0.60 .3 -0.6 -41 -0.6 3 -0.9 -7 -0.2 -0.5 -1 -2 -0.8     ,B     12 51 4 -3 12     D =     0.50 .5 0.20 .2 0.10 .1 00 .1     ,E=     2.54 2.50 .5 0 -0.5 44    
The system input u(t)= u 1 (t) T u 2 (t) T T with: u 1 (t) is a telegraph type signal varying between zero and one, u 2 (t)=0.3+0.1 sin(πt)

The fault f (t)= f 1 (t) T f 2 (t) T T with : The input u f (t) is computed using the equation [START_REF] Khedher | An approach of faults estimation in Takagi-Sugeno fuzzy systems Accepted in the 8 th ACS/IEEE International Confer[END_REF], this input permits to the system (2) to have the same behaviour with the system (1). This input is shown in figure [START_REF] Chen | Fault-tolerant control systems design using the linear matrix inequality approach[END_REF].

f 1 = 0,t ≤ 4sec 0.
The conceived observer allows to estimate the state x f and the control u f (t) is a fault tolerant

Conclusion

A method which permits simultaneously the fault estimation and the conception of the fault tolerant control is proposed in this section. This control is computed using the fault estimate and the error between the faulty system state and a reference model state. In the next section the proposed method will be extended to nonlinear systems described with multiple models.

On the multiple model representation

Multiple model approach is considered by many researches as an approach giving a simple structure for control or diagnosis of nonlinear systems [START_REF] Akhenak | Design of observers for Takagi-Sugeno fuzzy models for Fault Detection and Isolation[END_REF][START_REF] Murray-Smith | Multiple model approaches to modeling and control[END_REF]. This approach has been initially proposed, in a fuzzy modelling framework, by Takagi and Sugeno [START_REF] Takagi | Fuzzy identification of systems and its applications to model and control[END_REF] and in a multiple model modelling framework by Johansen and Foss [START_REF] Johansen | Non linear local model representation for adaptive systems[END_REF]. This model has been largely considered for analysis, modelling, control and state estimation of nonlinear systems.

The multiple model approach constitutes a tool which is largely used in the modelling of nonlinear systems [START_REF] Murray-Smith | Multiple model approaches to modeling and control[END_REF]. The basis of the multiple model approach is the decomposition of the operating space of the system into a finite number of operating zones. Hence, the dynamic behaviour of the system inside each operating zone can be modelled using a simple submodel, for example a linear model. The relative contribution of each submodel is quantified with the help of a weighting function. Finally, the approximation of the system behaviour is performed by associating the submodels and by taking into consideration their respective contributions. The choice of the structure used to associate the submodels constitutes a key point in the multiple modelling frameworks. This model has been largely considered for analysis, modelling, control and state estimation of nonlinear systems. The structure of a Takagi-Sugeno model is:

       ẋ(t)= M i=1 µ i (ξ(t)) (A i x(t)+B i u(t)) y(t)= M i=1 µ i (ξ(t))C i x(t) (29)
where x(t) ∈ R n is the state vector, u(t) ∈ R r control vector, y(t) ∈ R m vector of measures and A i , B i and C i are known constant matrices with appropriate dimensions.

The membership functions µ i (ξ(t)) assure a progressive passage between the local models. These have the following proprieties:

M i=1 µ i (ξ(t)) = 1,∀t (30) and 0 ≤ µ i (ξ(t)) ≤ 1, ∀i =1...M, ∀t (31)
The variable of decision ξ(t) is accessible in real time and it depends of measurable variables like system inputs or outputs.

Let's remark that state matrix of this kind of multiple model are built by the made of a levelheaded sum, with variable weight of different matrix of local models. One can also make a similarity between multiple model and system with variables parameters in time.

If, in the equation which defines the output, we impose that C 1 = C 2 = ... = C M = C, the output of the multiple model ( 29) is reduced to : y(t)=Cx(t) and the multiple model becomes:

   ẋ(t)= M i=1 µ i (ξ(t))(A i x(t)+B i u(t)) y(t)=Cx(t) (32)

Problem formulation

In this section the method proposed for linear system will be extended to nonlinear system described by multiple models. Suppose that the matrices B i are equals. Consider the nonlinear system described by the following multiple model structure:

   ẋ(t)= M i=1 µ i (ξ(t))A i x(t)+Bu(t) y(t)=Cx(t) (33)
where x(t) ∈ R n is the state vector, u(t) ∈ R r control vector, y(t) ∈ R m vector of measures and A i , B and C are known constant matrices with appropriate dimensions. The scalar M represents WSEAS TRANSACTIONS on SYSTEMS Atef Khedher, Kamel Ben Othman, Didier Maquin, Mohamed Benrejeb ISSN: 1109-2777 the number of local models. ξ(t) is the variable of decision which can depend on the input and/or the output and/or the system state. Consider the following Takagi-Sugeno model affected by sensor faults and measurement noise:

   ẋf (t)= M i=1 µ i (ξ(t))A i x f (t)+Bu f (t) y f (t)=Cx f (t)+Ef(t)+Dw(t) (34)
where x f (t) ∈ R n is the state vector, u f (t) ∈ R r is the input vector, y f (t) ∈ R m the output vector. f (t) represents the fault which is assumed to be bounded and w(t) is the measurement noise. E and D are respectively the fault and the noise distribution matrices which are assumed to be known.

The weighting functions must verify M i=1 µ i (ξ(t)) = 1, so, the states z and z f defined in (3) can be written :

ż(t)= M i=1 µ i (ξ(t))(-Āz(t)+ ĀCx(t)) żf (t)= M i=1 µ i (ξ(t))(-Āz f (t)+ ĀCx f (t) + ĀEf (t)+ ĀDw(t)) (35)
The two augmented state vectors X and X f are:

   Ẋ(t)= M i=1 µ i (ξ(t))A ai X(t)+B a u(t) Y (t)=C a X(t) (36)
and :

       Ẋf (t)= M i=1 µ i (ξ(t))A ai X f (t)+B a u f (t) +E a f (t)+F a w(t) Y f (t)=C a X f (t) (37) with :
A ai = A i 0 ĀC -Ā (38)
The Other matrices are given in (2.1). The structure of the proportional integral observer is chosen as follows:

                 Ẋf (t)= M i=1 µ i (ξ(t))(A ai Xf (t)+K i ( Ỹ (t))) +B a u f (t)+E a f (t) ḟ (t)= M i=1 µ i (ξ(t))L i ( Ỹ (t)) Ŷf (t)=C a Xf (t) (39)
where Xf (t) is the estimated state, f (t) represents the estimated fault, Ŷf (t) is the estimated output, K is the proportional observer gain, L is its integral gain which must be computed and Ỹ (t)=Y f (t) -Ŷf (t). K i are the local model proportional observer gains and L i are the local model integral gains to be computed. The control strategy is conceived using the following form of u f (t):

u f (t)=-S f (t)+u(t) (40)
Using the same notations given in ( 9), ( 10) and ( 11), the following is obtained:

Ẋ(t)= M i=1 µ i (ξ(t))A ai X(t)+B a S f (t) -E a f (t) -F a w(t) (41) If S verify E a = B a S, Ẋ becomes : Ẋ(t)= M i=1 µ i (ξ(t))A ai X(t) -E a f (t) -F a w(t) (42)
The dynamics of the error Xf (t) described by the equation ( 10) is written in multiple model case:

Ẋf (t)= M i=1 µ i (ξ(t))((A ai -K i C a ) Xf (t)) +E a f (t)+F a w(t) (43)
The dynamics of the fault estimation error is:

ḟ = ḟ (t) - M i=1 µ i (ξ(t))L i C a Xf (44)
The equations (42), ( 43) and ( 44) can be written:

φ(t)=A m ϕ(t)+B m ψ(t) (45)
where ϕ and ψ are given in ( 16) and:

A m = M i=1 µ i (ξ(t))A mi (46)
where:

A mi =   A ai 0 -E a 0 A ai -K i C a E a 0 -L i C a 0   and B m =   -F a 0 F a 0 0 I   (47)
Considering the Lyapunov fonction given in [START_REF] Takagi | Fuzzy identification of systems and its applications to model and control[END_REF], the errors converge to zero if V<0. V<0 if A T mi P +PA mi < 0, ∀i ∈ {1, ..., M }. The matrix A mi can be written:

A mi = A ai E 1 0 Ãi -Ki C (48)
with:

Ãi = A ai E a 00 , Ki = K i L i E 1 = 0 -E a , C = C a 0 ( 49
)
Assuming that P has the block diagonal form P = diag(P 1 ,P 2 ), ϕ converges to zero if there exist matrices P 1 > 0 and P 2 > 0 such that following inequality is satisfied:

A T ai P 1 + P 1 A ai E 1 P 2 + P 1 E 1 P 2 E T 1 + E T 1 P 1 Ξ < 0 (50)
with :

Ξ =( Ãi -Ki C) T P 2 + P 2 ( Ãi -Ki C) (51)
Substituing G i = P 2 Ki , (51) becomes:

Ξ = ÃT P 2 + P 2 Ã -G i C -CT G T i ( 52
)
The resolution of the linear matrix inequality (LMI) (50), which is now linear, permits to find the matrices P 1 , P 2 and G i . The matrices Ki are computed using Ki = P -1 2 G i .

Summarizing the following theorem can be proposed: Theorem 1 The system (45) describing the evolution of the errors X(t), Xf (t) and f (t) is stable if there exist symmetric definite positive matrices P 1 et P 2 and matrices G i , i ∈ {1...M }, so that the following LMI are verified :

A T ai P 1 + P 1 A ai E 1 P 2 + P 1 E 1 P 2 E T 1 + E T 1 P 1 Ξ < 0 (53)
where

Ξ = ÃT P 2 + P 2 Ã -G i C -CT G T i ( 54
)
The observer gains (proportional and integral) are obtained by : Ki = P -1 2 G i .

Illustrative example

Consider the non linear systems described by ( 33) and (34) where: C = I, ξ(t)=u(t) and : The system input is: u = u 1 T u 2 T T with : u 1 (t) is a telegraph type signal varying in [0, 0.5] and u 2 (t)=0 .4+0.25 sin(πt). It is shown in the figure [START_REF] Edwards | A comparison of sliding mode and unknown input observers for fault reconstruction[END_REF]. The fault: f = f 1 T f 2 T T with: The chosen weighting functions depends on the two inputs of the system. They have been created on the basis of Gaussian membership functions. Figure 6 shows their time-evolution showing that the system is clearly nonlinear since µ 1 and µ 2 are not constant functions.

A 1 =     -0.
f 1 = sin(0.5πt), 15
The computation of the matrices K 1 , L

            K 2 =             3.
           
Simulation results are shown in figures [START_REF] Johansen | Non linear local model representation for adaptive systems[END_REF] to [START_REF] Khedher | Fault tolerant control for nonliner system described by Takagi-Sugeno models[END_REF].

The figure 7 shows the sensor faults and their estimated. It is clear that the proposed method allows the faults estimation even in the case of the faults varying in the time. The figure 9 shows the evolution of the fault tolerant control, if a fault appears the control changes in a way that the system guards its original behaviour. This result is

Conclusion

This work has presented a method of synthesis of active sensor fault tolerant control. The proposed method uses the fault estimation and the error between the reference state and the faulty system state to synthesize the fault tolerant control strategy. To estimate the sensor fault, an augmented system is conceived, this system has the advantage to let the sensor fault affecting the initial system appears as an unknown input which make its estimation simple. The advantage of this method

Figure 1 :

 1 Figure 1: Error between x and x f

Figure 2 :

 2 Figure 2: Estimation error of x f

Figure 3 :

 3 Figure 3: Faults and their estimations

Figure 4 :

 4 Figure 4: fault tolerant control u f

Figure 5 :

 5 Figure 5: System input

Figure 7 :

 7 Figure 7: Faults and their estimations

Figure 8 :

 8 Figure 8: Error between x and z

Figure 9 :

 9 Figure 9: fault tolerant control

  1 , K 2

												Atef Khedher, Kamel Ben Othman,
	WSEAS TRANSACTIONS on SYSTEMS				Didier Maquin, Mohamed Benrejeb
		1
											mu1
											mu2
	0.9
	0.8
	0.7
	0.6
	0.5
	0.4
	0.3
	0.2
	0.1
		0
		0	10	20	30	40	50	60	70	80	90	100
			Figure 6: Weighting functions
	and L 2 gives:
	L 1 =	1.0794 8.1861 6.6330 0.7608 4.3498 1.5344 -3.7962 8.9428
	L 2 =	-2.1645 8.1289 7.0558 -0.1195 4.2296 -5.7418 10.1167 2.2015
			3.8612		0.2280 -1.1161 1.1617
			-0.6495 0.7253		2.4040		0.4932
		 	0.8769		2.6781 -3.4879 -0.6686
	K 1 =	   	2.5354 0.3170		3.3790 2.0343 -9.7468 0.2042 1.0535 5.9432
		   	1.2085 -0.0015 0.0186 17.0517 0.0178 0.3383 -0.8709 -9.0551
			0.0808 -0.0122 0.0454 16.8713
	ISSN: 1109-2777								800	Issue 7, Volume 9, July 2010

Issue 7, Volume 9, July 2010

Atef Khedher, Kamel Ben Othman, Didier Maquin, Mohamed Benrejeb ISSN: 1109-2777


Bernard Dussardier

Wilfried Blanc

Pavel Peterka

Tailoring of the local environment of active ions in rare-earth-and transition-metal-doped optical fibres, and potential applications

Introduction

During the last two decades, the development of optical fibre-based sophisticated devices have benefited from the development of very performant optical fibre components. In particular, rare-earth (RE)-doped optical fibres have allowed the extremely fast development of fibre amplifiers for optical telecommunications [START_REF] Desurvire | Erbium Doped Fiber Amplifiers: principles and applications[END_REF][START_REF] Desurvire | Erbium Doped Fiber Amplifiers: device and system developments[END_REF], lasers [START_REF] Digonnet | Rare-Earth-Doped Fiber Lasers and Amplifiers[END_REF] and temperature sensors [START_REF] Grattan | Fiber optic sensor technology: an overview[END_REF]. The most frequently used RE ions are Nd 3+ , Yb 3+ , Er 3+ , Tm 3+ for their optical transitions in the near infrared (NIR) around 1, 1.5 and 2 µm. A great variety of RE-doped fibres design have been proposed for specific applications: depending of the RE concentration and nature of the fibre glass, various schemes (in terms of electronic transition within the RE populations) have been implemented. For example, Er 3+ -doped fibre amplifiers (EDFA) for long haul telecommunications use the very efficient 1.55 µm optical transition, whereas high concentrations of Yb 3+ and Er 3+ codoped in the same fibre allowed efficient non-radiative energy transfers from the 'sensitizer' (Yb 3+ ) to the 'acceptor' (Er 3+ ) in order to increase the power yield of the system, applied in power amplifiers and lasers. All the developed applications of amplifying optical fibres are the result of time consuming and careful optimization of the material properties, particularly in terms of dopant incorporation in the glass matrix, transparency and quantum efficiency. RE-doped fibres are made of a choice of glasses: silica is the most widely used, sometimes as the result of some compromises. Alternative glasses, including low maximum phonon energy (MPE) ones, are also used because they provide better quantum efficiency or emission bandwidth to some optical transitions of particular RE ions. The icon example is the Tm 3+ -doped fibre amplifier (TDFA) for telecommunications in the S-band (1.48-1.53 µm) [START_REF] Komukai | Upconversion pumped Thulium-doped fluoride fiber amplifier and laser operating at 1.47 µm[END_REF], for which low MPE glasses have been developed: oxides [START_REF] Minelly | Applications of antimony-silicate glasses for fiber optic amplifiers[END_REF][START_REF] Lin | Near-infrared emissions with widely different widths in two kinds of Er3+doped oxide glasses with high refractive indices and low phonon energies[END_REF], fluorides [START_REF] Durteste | Amplification and lasing at 1.3 µm in praseodymium-doped fluoridezirconate fibers[END_REF], chalcogenides [START_REF] Hewak | Low phonon-energy glasses for efficient 1.3 µm optical fiber amplifiers[END_REF], etc. Although some of these glasses have a better transparency than silica in the infrared spectral region (above 2.3 µm), for applications in the near infrared (NIR) these glasses have some drawbacks not acceptable at a commercial point of view: high fabrication cost, low reliability, difficult connection to silica components and, in the case of fibre lasers, low optical damage threshold and resistance to heat. To our knowledge, silica glass is the only material able to meet most of applications requirements, and therefore the choice of vitreous silica for the active fibre material is of critical importance. However a pure silica TDFA would suffer strong non-radiative de-excitation (NRD) caused by multiphonon coupling from Tm 3+ ions to the matrix. Successful insulation of Tm 3+ -ions from matrix vibrations by appropriate ion-site 'engineering' would allow the development of a practical silica-based TDFA. However, note that Tm 3+ -doped silica fibre are very good for 2 µm laser sources [START_REF] Jackson | Theoretical modelling of Tm-doped silica fiber lasers[END_REF]. This shows that the choice of optimal host glass is also dictated by the seeked application.

Other dopants have recently been proposed to explore amplification over new wavelength ranges. Bi-doped glasses with optical gain [START_REF] Murata | Bi-doped SiO2 as a new laser material for an intense laser[END_REF] and fibre lasers operating around 1100-1200 nm have been developed [START_REF] Dianov | CW bismuth fiber laser[END_REF][START_REF] Razdobreev | Efficient all-fiber bismuth-doped laser[END_REF], although the identification of the emitting center is still not clear, and optimization of the efficiency is not yet achieved. Transition metal (TM) ions of the Ti-Cu series would also have interesting applications as broad band amplifiers, super-fluorescent or tunable laser sources, because they have in principle ten-fold spectrally larger and stronger emission crosssections than RE ions. However, important NRD strongly reduces the emission quantum efficiency of TM ions in silica. Bi-and TM-doped fibres optical properties are extremely sensitive to the glass composition and/or structure to a very local scale. As we have shown, Cr ions in silica using standard fabrication methods provide strong and ultrabroad NIR luminescent, but only at low temperature [START_REF] Dussardier | Cr 4+ -doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF]. Interestingly, other application of TM-doped fibre have been proposed in lasers.

As for Tm 3+ ions, practical applications based on silica doped with alternative dopants along 'low efficiency' optical transitions would be possible when the 'ion site engineering' will be performed in a systematic approach. This approach is proposed via 'encapsulation' of dopants inside glassy or crystalline nanoparticles (NP) embedded in the fibre glass, like reported for oxyfluoride fibres [START_REF] Samson | Efficient neodymium-doped glass-ceramic fiber laser and amplifier[END_REF] and multicomponent silicate fibres [START_REF] Samson | Nickel-doped nanocrystalline glass-ceramic fiber[END_REF]. In NP-doped-silica fibres, silica would act as support giving optical and mechanical properties to the fibre, whereas the dopant spectroscopic properties would be controlled by the NP nature. The NP density, mean diameter and diameter distribution must be optimized for transparency [START_REF] Tick | Transparent glass ceramics for 1300 nm amplifier applications[END_REF]. In this context, our group has made contributions in various aspects introduced above. Our motivations are usually application oriented, but we address fundamental issues. First, the selected dopants act as probes of the local matrix environment, via their spectroscopic variations versus ligand field intensity, site structure, phonon energy, statistical proximity to other dopants, etc. The studies are always dedicated to problems or limitation in applications, such as for EDFA and TDFA, or high temperature sensors. It is also important to use a commercially derived fabrication technique, here the Modified Chemical Vapor Deposition (MCVD), to assess the potential of active fibre components for further development.

The aim of this chapter is reviewing of our contributions towards the comprehension and improvement of the spectroscopic properties of some RE and TM ions doped into silica. Its outline is as follow : Section 2 will describe the MCVD fabrication method of preform and fibre samples, and the characterization techniques applied to all samples. In Section 3, we report on the spectroscopic investigations of Tm 3+ -doped fibres versus the material composition, including phonon interactions and non-radiative relaxations. It includes the proposal of potential applications by numerical simulations. Section 4 summarizes our original investigations on transition metals, focusing on chromium ions (Cr 3+ and Cr 4+ ) in silica-based fibres. Applications of Cr 4+ -doped optical fibres as integrated saturable absorbers for passively Q-switched lasers are investigated. In section 5 are reported our recent discoveries in RE-doped dielectric nanoparticles, grown by phase separation. Finally, perspectives and conclusions are drawn in section 6.

Experimental

Preforms and fibres fabrication

All fibres investigated in this article were drawn from preforms prepared by the MCVD technique [START_REF] Nagel | Modified chemical vapor deposition[END_REF] at the Laboratoire de Physique de la Matière Condensée (Nice). In this process, chemicals (such as O2, SiCl4) are mixed inside a silica tube that is rotating on a lathe. The flame from a burner translating along the lathe axis heats the tube and locally produces a chemical oxidizing reaction that transforms SiCl4 and O2 into SiO2 molecules and gaseous chlorine. Extremely fine silica particles are deposited as soot on the inner side of the tube. This soot is transformed into a glass layer when the burner is passing over. The cladding layers are synthesized first, followed by the core layer(s). Germanium and phosphorus can also be incorporated directly through the MCVD process. They are added to raise the refractive index. Moreover, the latter serves also as a melting agent, decreasing the melting temperature of the glass. All the other elements like RE, TM, Al (a common glass modifier) are incorporated through the so-called solution doping technique [START_REF] Townsend | Solution doping technique for fabrication of rare earth doped optical fibers[END_REF]. The core layer is deposited at lower temperature than the preceding cladding layers, so that they are not fully sintered and left porous. Then the substrate tube is filled with an alcoholic ionic solution and allowed to impregnate the porous layers. After soaking the solution is removed, the porous layer is dried and sintered. The tube is then collapsed at 2000 °C into a cylindrical preform. The preform is drawn into a fibre using a vertical tube furnace on a drawing tower. The preform tip is heated above 2000°C and as the glass softens, a thin drop falls by gravity and pulls a thin glass fibre. The diameter of the fibre is adjusted by varying the pulling capstan speed, under controlled tension. The fibre is then coated with a UV-curable polymer and is finally taken up on a rotating drum.

Material characterizations

Refractive index profiles (RIP) of the preforms and fibres were measured using dedicated commercial refractive index profilers. The oxide core compositions of the samples were deduced from RIP measurements on preforms, knowing the correspondence between index rising and AlO3/2, GeO2, PO5/2 concentration in silica glass from the literature. The composition was also directly measured on some preforms using electron probe microanalysis technique in order to compare results. The concentration of these elements is generally around few mol%. Luminescent ions concentrations are too low to be measured through the RIP. They were measured through absorption spectra. For example, the Tm 3+ ion concentration has been deduced from the 785 nm ( 3 H6=> 3 H4) absorption peak measured in fibres and using absorption cross-section reported in the literature [START_REF] Jackson | Theoretical modelling of Tm-doped silica fiber lasers[END_REF] : abs(785 nm) = 8.7x10 -25 m 2 .

Thulium-doped fibres

3.1

Improvement of the 800 and 1470 nm Tm 3+ emission efficiencies Thulium-doped fibres have been widely studied in the past few years. Because of Tm 3+ ion rich energy diagram (Fig. 1), lasing action and amplification at multiple infrared and visible wavelengths are allowed. In this paragraph, we will focus on the 3 H4 manifold. Thanks to the possible stimulated emission peaking at 1.47 µm ( 3 H4 => 3 F4, see Fig. 1), discovered by [START_REF] Antipenko | New lasing transition in the Tm 3+ ion[END_REF], one of the most exciting possibilities of Tm 3+ ion is amplifying optical signal in the S-band (1.47-1.52 µm), in order to increase the available bandwidth for future optical communications. The 3 H4 level can also decay radiatively by emitting 800 nm radiation which is of primary importance for high power laser and medical applications.

Unfortunately, the upper 3 H4 level of this transition is very close to the next lower 3 H5 level so NRD are likely to happen in high phonon energy glass hosts, causing detrimental gain quenching.Improvement of the Tm3+ spectroscopy was proposed through Tm-Tm [START_REF] Simpson | Energy transfer up-conversion in Tm 3+ -doped silica fiber[END_REF] and Yb-Tm [START_REF] Simpson | Visible and near infra-red up-conversion in Tm 3+ /Yb 3+ co-doped silica fibers under 980 nm excitation[END_REF] energy transfers mechanims. In this paragraph, we discuss on the modification of the local phonon energy around Tm 3+ ions to reduce NRD. Then, we present the resulting improvement in emission efficiencies for specific applications through numerical modeling.

Local Phonon interactions

The relevant NRD transition of interest here is from the 3 H4 level downto the 3 H5 level. The NRD rate is expressed as [START_REF] Van Dijk | On the nonradiative and radiative decay rates and a modified exponential energy gap law for 4f-4f transitions in rare-earth ions[END_REF]:

 W 3 nr  W 0  exp  VD E  2E p     (1)
where W0 and VD are constants depending on the material, E is the energy difference between the both levels and Ep is the phonon energy of the glass. To improve the quantum efficiency (which depends on the fluorescence lifetime) of the 800 and 1470 nm emission bands, modifications of Tm 3+ ion local environment were investigated by co-doping with selected modifying oxides [START_REF] Faure | Improvement of the Tm 3+ : 3 H4 level lifetime in silica optical fibers by lowering the local phonon energy[END_REF]. GeO2 and AlO3/2 have a lower maximum phonon energy than silica. As opposite demonstration, high phonon energy PO5/2 was also used. To investigate the modification of the local environment, decay curves of the 810 nm fluorescence from the 3 H4 level were recorded. All the measured decay curves are nonexponential. This can be attributed to several phenomena and will be discussed in the next paragraph. Here, we study the variations of 1/e lifetimes () versus concentration of oxides of network modifiers (Al or P) and formers (Ge). The lifetime strongly changes with the composition of the glass host. The most striking results are observed within the Tm(Al)

sample series:  linearly increases with increasing AlO3/2 content, from 14 µs in pure silica to 50 µs in sample Tm(Al) containing 17.4 mol% of AlO3/2. The lifetime was increased about 3.6 times. The lifetime of the 20 mol% GeO2 doped fibre Tm(Ge) was increased up to 28 µs whereas that of the 8 mol% PO5/2 doped fibre Tm(P) was reduced down to 9 µs. Aluminum codoping seems the most interesting route among the three tested codopants.

Non-exponential shape of the 810-nm emission decay curves

In aluminium-doped fibres, fluorescence decay curves from the 3 H4 level were found to be non-exponential. It is thought that Tm 3+ ions are inserted in a glass which is characterized by a multitude of different sites available for the RE ion, leading to a multitude of decay constants. This phenomenological model was first proposed by Grinberg et al. and applied to Cr 3+ in glasses [START_REF] Grinberg | Continuous function decay analysis of a multisite impurity activated solid[END_REF]. This model was applied for the first time to Tm 3+doped glass fibres [START_REF] Blanc | Thulium environment in a silica doped optical fibre[END_REF]. In this method, the luminescence decay is given by :

 I t    A i i  exp t /  i   (2)
where Ai and i are discrete distributions of amplitudes and decay constants (lifetimes), respectively. For the fitting procedure on the Ai series, 125 fixed values for i were considered, logarithmically spaced from 1 to 1000 µs. For a given composition (Fig. 2a), we can notice two main distributions of the decay constant. With the aluminium concentration, they increase from 6 to 15 µs and from 20 to 100 µs, respectively (Fig. 2b). From the histograms of the amplitude distributions obtained from the fittings, characteristic lifetimes were correlated with those expected for thulium located either in a pure silica or pure Al2O3 environment. The 3 H4 lifetime is calculating by using this equation:

 1 /  = 1 /  rad + W 3 nr (3)
where rad corresponds to the radiative lifetime which is given to be 670 µs in silica [START_REF] Walsh | Comparison of Tm:ZBLAN and Tm:silica fiber lasers: Spectroscopy and tunable pulsed laser operation around 1.9 µm[END_REF]. W0 and  were estimated for different oxide glasses [START_REF] Van Dijk | On the nonradiative and radiative decay rates and a modified exponential energy gap law for 4f-4f transitions in rare-earth ions[END_REF][START_REF] Layne | Multiphonon relaxation of rare-earth ions in oxide glasses[END_REF]. The energy difference E was estimated by measuring the absorption spectrum of the fibres. When Al concentration varies, this value is almost constant around 3700 cm -1 [START_REF] Faure | Improvement of the Tm 3+ : 3 H4 level lifetime in silica optical fibers by lowering the local phonon energy[END_REF]. With these considerations, the 3 H4 expected lifetime can be calculated. In the case of silica glass, silica = 6 µs and for a pure Al2O3 environment, alumina = 110 µs. Both values agree with those obtained from the fitting procedure. The distribution of decay constant around 10 µs corresponds to Tm 3+ ions located in almost pure silica environment whereas the second distribution is attributed to Tm 3+ located in Al2O3 -rich sites. This result shows that the global efficiency is increased by increasing the Al concentation in TDFAs. However concentrations above 20 at% would cause excess loss and glass stability problems in silica.

Modeling of Tm-doped silica-fibre devices

Numerical models are useful tools for investigating thulium-doped fibre devices, predicting their performance and optimization of their parameters. We have developed comprehensive, spectrally and spatially resolved numerical model that is based on simultaneous solution of the laser rate equations and propagation equations that describe evolution of the optical power along the fibre. The rate equations for the relevant energy levels can be written according to the energy level diagram in Fig. 1 as follows:

), ( ) ( ) ( ) (
A A n W W W W n W W W W W n W W n dt dn               (4) , ) ( ) ( ) ( 2 0 3 3 30 31 35 3 53 54 5 5 14 13 1 04 03 0 3                        j r j nr r r nr W W W W W n W W W n W W n W W n dt dn (5) ), ( 4 0 5 5 50 5 35 3 05 0 5        j r j nr W W W n W n W n dt dn (6)
while it holds that the sum of population ni on each respective level is equal to the total thulium ion concentration nt in the core. It is assumed that the thulium ions are homogenously distributed and excited in the doped section of radius b in the fibre core of radius a and it holds that b ≤a. The transition rates Wij accounts for the stimulated absorption and emission between the respective levels. Spontaneous decay processes are described by Wij r and Wi nr , the radiative and nonradiative decay rates, respectively. The formulae for the transition rates can be found in [START_REF] Peterka | Theoretical modelling of S-band thulium-doped silica fibre amplifiers[END_REF]. In steady state the rate equations become a set of four linear algebraic equations. The propagation of optical power at respective wavelength is governed by the following propagation equation:

    ). ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( } 50 , 31 , 30 , 10 { 35 3 14 1 04 0 02 0 } 50 , 31 , 30 , 10 {                                           P n h n n n n P n n P dz dP ij ij i ij ij ji j ij i   (7)
where h is the Planck constant, P ± are the spectral power densities of the radiation propagating in both directions along the fibre axis and ij is the respective transition cross section. The cross sections are shown in Fig. 3. The spectral dependence of the cross sections are approximated by a set of Gaussian functions, the respective coefficients can be found in [START_REF] Peterka | Theoretical modeling of fiber laser at 810 nm based on thulium-doped silica fibers with enhanced 3 H4 level lifetime[END_REF]. The overlap factor  represents the fraction of the transversal field distribution that interacts with the RE ions. The first term in equation ( 7) describes the amplification and reabsorption of optical signals, the second term represents ground-state absorption and excited state absorption (ESA) in spectral bands with no significant emission, the third term accounts for spontaneous emission and the fourth term stands for background loss  of the fibre. Evolution of optical power in each ASE spectral slot around a wavelength  (typical 1 nm wide slot is used) is governed by its respective propagation equation. The propagation equations together with the set of the rate equations under steady state conditions are solved simultaneously along the fibre using Runge-Kutta-Gill method of the fourth order. Since the boundary conditions for the counterpropagating partial waves P -are not known at the beginning of the fibre, an iterative solution is applied. The above described numerical model was verified by experimental comparisons (Blanc et al., 2006;[START_REF] Lüthi | Distributed gain in a Tm-doped silica fiber: experiment and modeling[END_REF].

3.3

Applications

Thulium-doped fibres are renowned for their applications in high power fibre lasers at around 2 m. It is despite the low quantum conversion efficiency of 3 F4 level in thuliumdoped silica fibres, which is about 10% compared to ~100% quantum conversion efficiency of the ytterbium-and erbium-doped fibres, at around 1 m and 1.5 m, respectively. The lower quantum conversion efficiency increases the 2 m laser threshold but has almost no effect on the laser slope efficiency. Indeed, kW-class thulium-doped fibre lasers have been demonstrated recently [START_REF] Moulton | High power Tm:silica fiber lasers: current status, prospects and challenges[END_REF]. Quantum conversion efficiency of the 3 H4 level in non-modified silica fibres is much lower than that of 3 F4 level, only about 2% and therefore most of the applications of laser transitions originating from 3 H4 level are hindered by the lack of reliable low-phonon fibre host as discussed above. In the following two paragraphs we will show potential of the developed thulium-doped fibres with enhanced 3 H4 level lifetime for applications in fibre amplifiers for communication S-band and for fibre lasers around 810 nm. Two main obstacles have to be solved in order to use the 3 H4 → 3 F4 transition in the TDFA. Firstly, the quantum conversion efficiency of the 3 H4 thulium level should be increased, or in other words, the 3 H4 fluorescence lifetime should be made longer. One possible solution of the first problem is described above. Secondly, the lifetime of the upper laser level 3 H4 is shorter than that of the lower level 3 F4. Therefore, direct pumping of the 3 H4 level will hardly constitute the necessary population inversion. The second problem can be solved by using more complex pumping scheme employing gradual upconversion of the thulium ion to the upper laser level. It can be done by a single laser source as shown in Fig. 5a. The disadvantage is the possibility of the ESA 3 H4 → 1 G4 that results in loosing pump photons. Many dual-wavelength pumping scheme have been also reported as reviewed in [START_REF] Peterka | Theoretical modelling of S-band thulium-doped silica fibre amplifiers[END_REF]. Several results of TDFA performance simulation and of optimization of the TDFA parameters are shown in Fig. 5 and Fig. 6. The TDFA spectral gain is shown in Fig. 5b for different silica fibre host represented by the fluorescence lifetime of the 3 H4 level: 14 s of the typical non-modified silica fibre and 45 s of the alumina-doped fibre developed by the authors and 55 s of the highly germanium-oxide-doped fibre [START_REF] Cole | S-band amplification in a thulium doped silicate fiber[END_REF]. Similar value of 58 s was also measured in highly alumina-doped fibre [START_REF] Peterka | Thulium-doped silica-based optical fibers for cladding-pumped fiber amplifiers[END_REF]. Codirectional pump of 1 W at 1064 nm is assumed. Unless otherwise stated, we consider the thulium concentration nt=1.56×10 25 m -3 , the core diameter 2.6 m, numerical aperture NA=0.3 and the radiative lifetimes of the levels 3 F4, 3 H4 and 1 G4 are 3500, 650 and 860 s, respectively. The fluorescence lifetimes of the levels 3 F4, 3 H4 and 1 G4 are 430, 45 and 784 s, respectively. For sake of comparisons we set the cross sections of the 3 F4 and 1 G4 levels the same for all fibre host types, although in real fibres they also depend on the host material.

We have checked that their values have little effect on the calculated S-band gain compared to the effect of 3 H4 lifetime. The relevant branching ratios were estimated using Judd-Ofelt theory and were evaluated as follows: 54=0.03, 53=0.11, 52=0.30, 51=0.06, 50=0.50, 32=0.03, 31=0.09, and30=0.88. Zero background loss is assumed. Signal power evolution along the thulium-doped fibre is shown in Fig. 5c for three input pump powers at 1064 nm. The available gain is significantly reduced by the presence of ASE, mainly around 800 nm. Suppression of the ASE, e.g., by cascaded inscription of long-period fibre gratings into the doped fibre or by using thulium-doped photonic crystal fibre with tailored band-gaps, would ameliorate the gain as shown also in Fig. 5. Optimization of the fibre waveguide parameters is shown in Fig. 6a. Similarly as in the case of the EDFA, the gain gets higher with increasing numerical aperture and narrower core diameter. The effect of distributed filtering out of the ASE around 800 nm is also shown in the graph. Optimization of the pump wavelength is calculated in Fig. 6b. The optimum pump wavelength shifts towards shorter wavelength with increasing power levels. Shorter wavelength's pump sees lower GSA but at the same time higher first step ESA and consequently the integral value of the population inversion along the fibre is higher.

Thulium-doped fibre laser at 800 nm

The silica-based fibre lasers around 800 nm would extend the spectral range covered by high-power fibre lasers. The single-transversal mode, high-power laser source in the 800 nm spectral band is of interest for a variety of applications. The laser can be used for fibre sensors, instrument testing and for pumping of special types of lasers and amplifiers, e.g. the bismuth-doped lasers. Bismuth-doped fibres pumped around 800 nm may shift their gain to 1300 nm telecommunication band, where highly reliable silica-based fibre amplifiers are still unavailable. An efficient fibre laser in 800 nm spectral region could potentially be used as a replacement for titanium sapphire laser in some applications. A high-power amplifier in the 800 nm band would be useful in the optical fibre communications and shorthaul free-space communications. Although the laser diodes at this wavelength have been available for a long time, to our knowledge, commercially available single-mode laser diodes are limited to about 200 mW of output power in diffraction limited beam. It should not be confused with, e.g., laser diode stacks of ~kW output power, that are highly multimode with very high M 2 >1000 factor. The amplification and lasing at 800 nm band has been already investigated using fluoride-based TDFs and output power of up to 2 W and 37% slope efficiency was achieved [START_REF] Dennis | High power upconversion lasing at 810 nm in Tm:ZBLAN fibre[END_REF]. The output power was limited by the pump damage threshold of the fluoride fibre. Fibre-host reliability problems might be solved by using of silica-based fibres with enhanced 3 H4 lifetime. We proposed compact allsilica-fibre setup, example of which is shown in Fig. 7a. The proposed laser utilizes upconversion pumping scheme according to Fig. 7b.

We have performed optimization of the waveguide parameters, namely the core radius and NA [START_REF] Peterka | Theoretical modeling of fiber laser at 810 nm based on thulium-doped silica fibers with enhanced 3 H4 level lifetime[END_REF]. The optimum length of the TDF can be determined from the calculated dependence of the laser output on the TDF length as it is shown in Fig. 7c for the three laser hosts. The laser output power vs. pump wavelength is shown in Fig. 7d. In comparison with the TDFA for the S-band telecommunications with optimal pump wavelength around 1020 nm (Fig. 6), the region of optimal pump wavelengths is shifted towards longer wavelength because in this case high inversion between 3 H4 and 3 H6 levels and high pump absorption from 3 H6 level is desirable. The background loss of 0.1 dB/m was considered. It can be seen in Fig. 7c that lasing at 810 nm is hard to achieve with silica-based Tm-doped fibre in contrast to the fluoride host materials. However, the lasing might be possible even for silica-based fibre for specific short range of the fibre lengths. Especially when the 3 H4 lifetime is enhanced, the laser output is very close to the one of ZBLAN host. The waveguide parameters used for the parts (c) and (d) are: the core diameter of 3.4 m and numerical aperture of 0.2.

3.4

Conclusions

Using a comprehensive numerical model we have shown the potential of the developed thulium-doped fibres for applications in fibre amplifiers for S-band telecommunication and for fibre lasers around 810 nm. The gain exceeding 20 dB of the S-band TDFA could be obtained with optimized fibre waveguiding parameters and optimized pump wavelength. Although the required pump power levels are relatively high, of the orders of Watts, thanks to the progress in ytterbium-doped fibre lasers even these pump powers can provide a cost effective solution. We have shown that efficient lasing at 810 nm can be achieved using silica-based Tm-doped fibre with enhanced 3 H4 lifetime for specific ranges of the fibre and laser cavity parameters. Such a fibre laser would broaden the spectral range currently covered by silica-based fibre lasers and it may replace the conventional laser systems in numerous applications.

Transition metal ions in silica-based optical fibres

Introduction

Many applications need ultra-broad band gain optical fibre materials. The well established tunability of RE-doped fibre devices is limited by shielding of the optically active electronic orbitals of RE ions, whereas unshielded orbitals are found in transition metal (TM) ions. One of the most promising is Cr 4+ because of its NIR ultra-broad band transitions. However incorporating of Cr 4+ ion in optical materials is difficult because it is less stable than the Cr 3+ ion. Some cristals such as Cr 4+ :YAG are very good broad-band gain media [START_REF] Sennaroglu | Efficient continuous-wave chromiumdoped YAG laser[END_REF], but little literature deals with Cr 4+ -doped bulk glasses [START_REF] Cerqua-Richardson | Spectroscopic Investigation of Cr 4+doped glasses[END_REF][START_REF] Hömmerich | Near infrared emission at 1.35 µm in Cr doped glass[END_REF], and even fewer on Cr-doped silica fibres [START_REF] Schultz | Optical Absorption of the Transition Elements in Vitreous Silica[END_REF][START_REF] Chen | 400-nmbandwidth emission from a Cr-doped glass fiber[END_REF]2007). Note that Ni 2+ in vitroceramic fibres is also a promising amplifying dopant [START_REF] Samson | Nickel-doped nanocrystalline glass-ceramic fiber[END_REF]. However, the proposed fibres lack of high transparency, reliability and cost-effectiveness of silica-based fibres, as produced by MCVD for instance. We have explored this field through basic studies on the optical properties of TM ions in silica optical fibres. In particular, the final TM oxidation states in the fibre core are strongly processdependent. Also, the optical properties of one particular TM oxidation state (say Cr 4+ ) is difficult to interprete because they depend (1) on the host composition and final structure, due to crystal-field (or ligand field, in glass) fluctuations [START_REF] Henderson | Optical Spectroscopy of Inorganic Solids[END_REF]) and

(2) on the experimental conditions (temperature, pressure, excitation wavelength,…). We describe the specific preparation details used for the Cr-doped fibres, we sum up spectroscopic results and interpretations, and finally we summarize our original studies on Cr-doped fibre saturable absorbers for all-fibre passively Q-switched (PQS) fibre lasers.

Fabrication and characterization of Chromium-doped samples

The fibres were prepared as in section 2.1, using Cr 3+ alcoholic doping solutions and oxidizing or neutral atmosphere for the drying-to-collapse stages. Samples containing Ge or/and Al were prepared, referred to as Cr(Ge), Cr(Ge-Al) and Cr(Al), respectively. We determined the absolute Cr content and relative concentrations of oxidation states using plasma emission and electron paramagnetic resonance spectroscopies, respectively. We assigned the optical transition of Cr 3+ and Cr 4+ using the Tanabe-Sugano (T.-S.) formalism [START_REF] Sugano | Multiplets of Transition-Metal Ions in Crystals[END_REF] and we qualitatively determined their strength, energy and bandwidth.

From composition and sharing out of valencies, we have determined the absorption crosssections. Absorption, emission and decay measurements were performed at room (RT) and low temperatures (LT, 12 or 77 K), using various pump wavelengths. Full details of the experimental procedures and analysis are given in [START_REF] Felice | Cr 4+ -doped silica optical fibres : absorption and fluorescence properties[END_REF][START_REF] Felice | Chromiumdoped silica optical fibres : influence of the core composition on the Cr oxidation states and crystal field[END_REF][START_REF] Dussardier | Cr 4+ -doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF].

Local structure, valency states and spectroscopy of transition metal ions

The main finding is that only Cr 3+ and Cr 4+ oxidation states were found. Cr 3+ is favoured by Ge co-doping in O-symmetry as in other oxide glasses [START_REF] Henderson | Optical Spectroscopy of Inorganic Solids[END_REF][START_REF] Rasheed | Disorder and the optical spectroscopy of Cr 3+ -doped glasses: I. Silicate glasses[END_REF], whereas Cr 4+ in distorded tetrahedral site symmetry (Cs) [START_REF] Anino | New Cr 4+ activated compounds in tetrahedral sites for tunable laser applications[END_REF] is present in all samples, and is promoted by Al or when [Cr] is high (Fig. 8). The most interesting samples are the low-doped Cr(Al) ones containing only Cr 4+ . The Cr 3+ absorption cross section abs 3+ (670 nm)= 43 x 10 -24 m 2 is consistent with reports in other materials like ruby [START_REF] Cronemeyer | Optical absorption characteristics of pink ruby[END_REF] and silica glass [START_REF] Schultz | Optical Absorption of the Transition Elements in Vitreous Silica[END_REF], while abs 4+ (1000 nm)~3.5 x 10 -24 m 2 is lower than in reference crystals for lasers [START_REF] Sennaroglu | Accurate determination of saturation parameters for Cr 4+ -doped solid-state saturable absorbers[END_REF] or saturable absorbers [START_REF] Lipavsky | Some optical properties of Cr 4+ -doped crystals[END_REF], but consistent with estimated values in alumino-silicate glass [START_REF] Hömmerich | Near infrared emission at 1.35 µm in Cr doped glass[END_REF]. The main normalized parameter from the T.-S. formalism (Dq/B=1.43) was lower than the so-called 'crossing value' (Dq/B = 1.6), and lower than those reported for Cr 4+ in laser materials like YAG and forsterite [START_REF] Anino | New Cr 4+ activated compounds in tetrahedral sites for tunable laser applications[END_REF]. As expected consequences, a broad featureless NIR emission band along the 3 T2 3 A2 transition is observed and no narrow emission line from the 1 E state is seen (Fig. 9, left). The LT fluorescence from Cr 4+ spreads from 850 to 1700 nm, and strongly varies depending on core chemical composition, [Cr] and p (pump wavelength). The observed bands were all attributed to Cr 4+ ions, in various sites.

Fig. 9 shows examples of fluorescence spectra of Cr 4+ in different samples and experimental conditions. Possible emission from Cr 3+ , Cr 5+ or Cr 6+ centers was rejected [START_REF] Dussardier | Cr 4+ -doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF]. The fluorescence sensitivity to [Cr] and p suggests that Cr-ions are located in various host sites, and that several sites are simultaneously selected by scanning of p. It is also suggested that although Al promotes Cr 4+ over Cr 3+ when [Cr] is low, Cr 4+ is also promoted at high [Cr] in Ge-modified fibres. The strong decrease of fluorescence from LT to RT (not shown) is attributed to temperature quenching caused by NRD through multiphonon relaxations, like in crystalline materials where the emission drops by typically an order of magnitude from 77 K to 293 K [START_REF] Sennaroglu | Efficient continuous-wave chromiumdoped YAG laser[END_REF].

The non-exponential LT fluorescence decays (Fig. 9 The lifetime of isolated ions (iso), measured on the exponential tail decay curves (not shown) reach high values: iso~200 to 300 µs at s~1100 nm, iso~70 µs at s~1400 nm. In heavily-doped Cr(Ge) samples, iso is an order of magnitude less. Hence, Cr 4+ ions are hosted in various sites: the lowest energy ones suffer more NRD than the higher energy ones. Also presence of Al improves the lifetime, even at high [Cr]. It is estimated that at RT, lifetime  would be much less than 1 µs.

As a summary of these spectroscopic studies, the main results are: [Cr]=4000 ppm.

4.4

Passively Q-switched fibre lasers using a Cr 4+ -doped fibre saturable absorber.

The development of Q-switched fibre lasers is an alternative to bulk pulsed solid-state lasers operating in the ns-to µs-range for applications such as material processing and metrology. Actively Q-switched (AQS) fibre lasers have been extensively studied [START_REF] Wang | Actively Q-switched fiber lasers: Switching dynamics and nonlinear processes (A review)[END_REF], as well as passively Q-switched (PQS) lasers [START_REF] Siegman | Lasers[END_REF]. Most AQS and PQS fibre lasers use externally driven bulk modulators or bulk passive components [START_REF] Hideur | Mode-lock, Q-switch and CW operation of an Yb-doped double-clad fiber ring laser[END_REF][START_REF] Paschotta | Passively Q-switched 0.1-mJ fiber laser system at 1.53 µm[END_REF] or intracavity TM-doped crystals [START_REF] Laroche | Compact diode-pumped passively Q-switched tunable Er-Yb double-clad fiber laser[END_REF], located in a free space section of the cavity, causing alignment and reliability critical problems. To avoid them, we are interest into all-fibre systems comprising a saturable absorber (SA) fibre. We had experimentally demonstrated the first all-fibre PQS laser using a Cr-doped optical fibre saturable absorber (SA) (Tordella et al., 2003). Before, only one numerical study had mentioned this possibility [START_REF] Luo | Passive Q-switched erbium-doped fibre laser with saturable absorber[END_REF], and 'auto' self-pulsing had been observed in heavily Er 3+ -doped fibres [START_REF] Sanchez | Effects of ion pairs on the dynamics of erbium-doped fiber lasers[END_REF]. However, to obtain stable PQS behaviour, the criterium of 'second laser threshold' must be met [START_REF] Siegman | Lasers[END_REF]: It imposes that the SA absorption cross-section be larger than the gain medium emission cross section. Also, the SA absorption must spectrally overlap the emission gain curve and the SA relaxation time must be short to allow for high repetition rates. RE absorption transitions have limited spectral width and sometimes very long decay times. For these reasons, TM ions are in principle better SA elements: they offer broad absorption and fast decay. For example, abs 4+ (1 µm) is at least 10 times larger than em Nd3+ [START_REF] Felice | Cr 4+ -doped silica optical fibres : absorption and fluorescence properties[END_REF] and Cr4+ <1 µs is much less than of Nd3+(500 µs) or Yb3+ (800 µs) [START_REF] Dussardier | Cr 4+ -doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF].

Our first all-fibre PQS laser, operating at 1084 nm, used a core-pumped Nd-doped fibre spliced to a Cr-doped fibre saturable absorber (CrSA) (Tordella et al., 2003). Although the pulse energy was low (~15 nJ), the characteristic behaviour agreed with theory, and the principle was demonstrated. Since, few PQS fibre lasers have implemented alternative SA ions in fibres, like RE (Ho 3+ , Tm 3+ , Sm 3+ ) or Bi, using sometimes rather complex cavity configurations, or heavily RE doped fibre SA [START_REF] Kurkov | Q-switched all-fiber lasers with saturable absorbers[END_REF]. More recently, double-clad ytterbium-doped fibre (DCYF) lasers have raised a great interest. In some cases, DCYF lasers self-pulse in a random manner [START_REF] Upadhyaya | Self-pulsing characteristics of a high-power single transverse mode Yb-doped CW fiber laser[END_REF], along two main operating regimes: the sustained self-pulsing (SSP: period longer than the cavity round-trip time), and the self mode-locking (SML: period equal to the cavity round-trip time) [START_REF] Brunet | A simple model describing both self-mode locking and sustained self-pulsing in ytterbium-doped ring fiber lasers[END_REF]. The setup of our second system is schematically shown in Fig. 10. It is based on a home-made Dshaped DCYF. All experimental details are found in [START_REF] Dussardier | Passively Q-switched ytterbium and chromium doped fibre laser[END_REF]. The cavity parameters were chosen so that it would provide favorable conditions for SSP. As a consequence, the laser slope efficiency (SE) was not optimized. That also increased the threshold of appearance of stimulated Brillouin scattering (SBS) far above the achieved output power, therefore SBS had not effect on the reported results, and all the observed dynamic could be attributed to PQS mode of operation. The DCYF laser without the CrSA had a chaotic behaviour, randomly switching between continuous-wave (CW) and Q-switched operation. The pulse envelopes (few µs long) were strongly modulated with a period of one cavity return-trip time (Fig. 11(a)), a sign of a combination of SSP and SML. The same was observed using a commercial Yb 3+ -heavily doped fibre. The CrSA fibre drastically changed the laser characteristics. The µs-long pulse trains were stabilized over the whole available pump range (peak RMS fluctuations < 10 % at 9.4 times the threshold) (Fig. 10). The PQS mode was present at any pump power. The smooth Q-switched pulses did not show the sub-modulation at the return trip period that was visible in the 'DCYF only' laser (Fig. 11(b)). The reasons of the stabilization are not fully understood yet, and are still under study. The PQS laser threshold and SE were 1 W and 6 %, respectively. Although the SE has not been optimized it compares well with other fibre lasers using RE-or Bi-doped fibre SA (up to ~10%) [START_REF] Kurkov | Q-switched all-fiber lasers with saturable absorbers[END_REF]. At 16 W of incident pump power the minimum pulse duration, highest repetition rate and peak power were 480 ns, 350 kHz and 20 W, respectively. Such a PQS laser source could be optimized to increase the intracavity optical density, and hence to saturate the CrSA more and faster, and produce more energetic pulses. It could also be a seed in a master-oscillator power amplifier configuration to be amplified by power Yb-doped fibre amplifiers.

(a) (b)

4.5

Conclusion on the study of Cr-doped optical fibres.

Some glass modifiers like Al in silica-based fibres induce major spectroscopic changes, even at low concentrations (~1-2 mol%). This would help engineering the Cr optical properties in silica-based fibres, using possibly alternative modifiers. An interesting application for PQS all-fibre lasers was found. However, increasing of the quantum efficiency would be necessary for amplifiers and light sources. One needs to control the TM local environment by some material engineering. Preliminary work was reported [START_REF] Dvoyrin | Effective room-temperature luminescence in annealed chromiumdoped silicate optical fibers[END_REF] on post-heat treatment of Cr 3+ -doped Ga-modified silica fibres. Other possible material engineering would be the nanoscale spontaneous phase separation.

Rare-earth-doped dielectric nanoparticles

Optical fibres are mostly based on silica glass for its interesting properties, as mentioned in section 1. However, some of its characteristics (high phonon energy, low solubility of luminescent ions, ...) induce drawbacks in spectroscopic properties of rare-earth (RE)-doped fibre lasers and amplifiers. Here we investigate the growing of RE-doped dielectric oxide nanoparticles (DNP) in silica-based optical fibres. Through this route, we keep the optical and mechanical advantages of silica glass, and we investigate the potential of engineering of RE spectroscopic properties through the choice of the DNP chemical composition.

Scarce reports on RE-doped transparent glass ceramics (TGC) singlemode fibres use low melting mixed oxides prepared by a rod-in-tube technique [START_REF] Samson | Nickel-doped nanocrystalline glass-ceramic fiber[END_REF], or mixed oxyfluorides using a double-crucible technique [START_REF] Samson | Efficient neodymium-doped glass-ceramic fiber laser and amplifier[END_REF], both with a subsequent ceramming stage. However the low melting point of these materials causes low compatibility with silica components. Transition metal-doped silica-based TGC fibres were prepared by MCVD (Modified Chemical Vapour Deposition) and using a slurry method [START_REF] Yoo | Development of a glass optical fiber containing ZnO-Al2O3-SiO2 glass-ceramics doped with Co 2+ and its optical absorption characteristics[END_REF], i.e. the particles were synthesized before insertion into the silica tubesubstrate.

In this section, we discuss a more straightforward and original technique to embed RE ions within in-situ grown oxide nanoparticles in silica-based preforms (Blanc et al., 2009a). The implemented principle is the spontaneous phase separation process [START_REF] Zarzycki | Glasses and the vitreous state[END_REF]. Silicate systems can exhibit strong and stable immiscibility when they contain divalent metals oxides (MO, where M= Mg, Ca or Sr) (Fig. 12). For example, if a silicate glass containing few mol% MO is heated it will decompose into two phases : one silica-rich and one MO-rich in shape of spherical particles. Two key advantages of this process are that (i) nanoparticles are grown in-situ during the course of the fabrication process and (ii) there is no need (and associated risks) of nanoparticles manipulation by an operator. Further, the process takes advantage of the high compositional control and purity typical of the MCVD technique. The so-called 'solution doping technique' was applied to incorporate alkaline-earth and erbium ions. To raise the core refractive index and ease the fabrication, germanium (~2 mol%) and small amounts of phosphorus (~1 mol%) were added. In the next paragraphs, we discuss the influence of the alkaline-earth ions on the nanoparticles formation. We also present the attenuation measurements as well as the nanoparticles composition. Finally, modifications of the erbium spectroscopy are described. The gray disk corresponds to the fibre core (~8 µm). The dark central part of the core is caused by the evaporation of germanium element; this is a common artifact of the MCVD technique that can be corrected through process optimization. Nanoparticles are visible as bright spots. They show an important compositional contrast compared to the silica background. The size distribution of the NP depends strongly on the nature of the alkaline earth ions. It is about 50 nm for Mg-doped fibre while it reaches 100 nm or more for Sr-and Ca-doped samples. The size of the nanoparticles is also dependent on the concentration of the alkaline-earth ions [START_REF] Blanc | Fabrication of rare-earth doped transparent glass ceramic optical fibers by modified chemical vapor deposition[END_REF]. For the fibre doped with the solution with 0.1 mol/l of MgCl2, the mean particle size is 48 nm and no particle bigger than 100 nm was observed in these MgO-doped fibres, unlike the case of CaO-doped fibres. Analysis of the SEM images also revealed that the inter-particle distance is in the 100 -500 nm range. When the solution concentration increases from 0.1 to 1 mol/l of MgCl2, the inter-particle distance remains nearly the same but the mean particle diameter almost doubles to reach 76 nm. With this concentration, NPs up to 160 nm were observed.

Sr Sr Ca

Ca

Mg

Mg

Fig. 13: SEM pictures of a MO-doped fibre (M = Sr, Ca, Mg). For each fibres, alkaline-earth concentration is 0,1 mol/l.

Transparency

The scattering loss in Mg-doped fibre was measured through the cut-back method. The fibre bending radius was kept > 20 cm to minimize bend loss. The 0.1 mol/l Mg-doped fibre attenuation spectrum is displayed in Fig. 14. At the wavelength of 1350 nm, losses were measured to be 0.4 dB/m only. This value is comparable with the attenuation measured in low-melting temperature transparent glass ceramics fibres [START_REF] Samson | Nickel-doped nanocrystalline glass-ceramic fiber[END_REF] and is compatible with amplifier applications. The loss variation vs wavelength is attributable to light scattering induced by the nanoparticles. Due to the small particle size we assume Rayleigh scattering to be the major source of scatter loss, which was estimated according to the formula:  (dB/m)=4.34.CRayleigh.N., where N is the DNP density (m -3 ),  is the overlap factor between the field and the core containing the DNP ( = 0.3 in fibre A) and CRayleigh (m 2 ) is the Rayleigh scattering coefficient [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]:

 C Rayleigh = ( 2 ) 5 48  d 6  4  n m 4  n n 2  n m 2 n n 2 + 2 n m 2       2 (8)
where d is the nanoparticle diameter, nm and nn the host material and particles refractive indices, respectively. The actual particle composition is not known but a high content of MgO is expected [START_REF] Stebbins | Characterization of phase separation and thermal history effects in magnesium silicate glass fibers by nuclear magnetic resonance spectroscopy[END_REF]. The nanoparticle refractive index is estimated at ~ 1.65, like that of Mg-based oxide such as Mg2SiO4 (Burshtein et al., 2003). Under these considerations, fitting of the experimental data with Eq. 8 yields a particle density N~ 0.4 x 10 20 m -3 , or equivalently a mean inter-particle distance ~300 nm. These values agree well with the results collected from the SEM pictures taken from cleaved fibres. At 1350 nm, the normalized frequency V was lower than 1.3 (LP11 mode cut-off wavelength is 700 nm). The difference between experimental data and Rayleigh scattering curve above 1350 nm is attributed to bending loss only. However, for practical application, the presented fabrication technique allows any necessary waveguide optimization without preventing the TGC growth. The attenuation of other fibres (1 mol/l of MgCl2 or Ca-and Sr-doped) was extremely high (several 100s dB/m), in agreement with the Rayleigh formula (120 dB/m) assuming large monodisperse 100 nm particles. This shows that the particle mean size must be less than 50 nm for potential applications, and that the technique presented here is able to produce fibres with acceptable scattering loss.

Composition of the nanoparticles

The composition was investigated by EDX analyses. When nanoparticles are analyzed, Ca or Mg, P and Si are found while only Si is detected outside of the particles. Germanium seems to be homogeneously distributed over the entire glass. When erbium is added to the composition, it is found to be inserted into the particles as it is presented on Fig. 15. Such conclusion was also drawn when emission intensity of Er 3+ was mapped on the end-face of a cleaved fibre under a confocal microscope [START_REF] Blanc | Erbium emission properties in nanostructured fibers[END_REF]. No erbium fluorescence is detected outside the DNP. This result was expected due to the low solubility of RE ions in silica (or even in germano-silicate), whereas strongly modified and amorphous silicates, such as in the DNP, have a high solubility for these ions. These results indicate that erbium ions are located inside or very close to the DNP.

Erbium spectroscopy

The emission spectra and lifetime from fibres doped with 0.1 (Fibre A) and 1 mol/l (Fibre B) were measured at room temperature around 1.55 µm under 980-nm pump excitation. The emission spectrum from Fibre A (Fig. 16) is similar to that in a silicate environment. Fibre B shows a distinct broadening of its spectrum (FWHM is 44 nm) by as much as ~ 50 % compared to Fibre A. In comparison, more than 10 at% of aluminum, as network modifier, would have been necessary to obtain the same FWHM in 'type III' Er-doped fibres in optical amplifiers for telecommunications [START_REF] Desurvire | Erbium Doped Fiber Amplifiers: principles and applications[END_REF], because Al and Er are evenly distributed across the core volume. Moreover, the shape of the fluorescence spectrum from Fibre B is quite unusual: it decreases monotonically between the peaks at 1.53 and 1.55 µm and the commonly observed dip at 1.54 µm is absent. These features would be attractive for realizing intrinsically gain flattened fibre amplifiers, provided sufficient minimization of the scattering loss is ensured through process optimization. Both fibres A and B produced single-exponential decaying fluorescence with 11.7 and 6.7 ms lifetimes, respectively.

Modifications of the Er 3+ spectroscopic properties in the TGC optical fibres are clearly evident from the above results when the Mg concentration increases. Such observations were previously reported in Ca-doped preforms. The broadening of the fluorescence curve and the lower lifetime obtained from Fibre B is tentatively interpreted as an effect of the modification of the erbium ions averaged local field. In other words, erbium ions in Fibre B are, on average, located in a medium with stronger local field compared to Fibre A. This induces stronger spontaneous transition probability, and hence a shorter lifetime [START_REF] Henderson | Optical Spectroscopy of Inorganic Solids[END_REF]. Although the exact composition of the NPs is not yet known, emission spectrum and fluorescence lifetime of erbium ions in Fibre B are closely related to the results reported in the literature in phosphate glasses [START_REF] Liu | Spectra and Laser Properties of Er 3+ , Yb 3+ :Phosphate Glasses[END_REF][START_REF] Jiang | Er 3+ Doped Phosphate Glasses and Lasers[END_REF]. When Mg concentration increases, erbium ions environment changes from silicate to phosphate.

Conclusion

A method to fabricate Er 3+ -doped TGC fibers entirely through the MCVD process is demonstrated. By adding magnesium to the silica-based composition, a low-loss fiber is obtained through in situ growth without requiring a separate process to realize NP (such as postprocess ceramming). An important result of this study is that the type of obtained nanoparticle by this technique could be as large as ~50 nm for applications such as in fiber amplifiers and lasers. Moreover, a broadening of the emission spectrum by as much as ~50% is observed with attractive features to realize gain flattened fiber amplifiers. More generally, this concept might have great potential as a possible solution to address various current issues in amplifying fibers, including (but not limited to) realization of intrinsically gain flattened amplifiers, etc.

General conclusion

The choice of a glass to develop new optical fibre component is most of the time a result of compromises. Silica glass is the most widely used for its many advantages (reliability, low cost fabrication, …). The overall composition suffers from different drawbacks, such as high phonon energy or low luminescent ions solubility, which affect quantum efficiency or emission bandwidth of luminescent ions, for example. However, spectroscopic properties of active ions are not directly related to the average properties, but to their local environment.

Then, new luminescent properties of dopants can be obtained by slightly modifying the silica composition.

In this chapter, we have shown that thulium emission efficiency can be improved by reducing the local phonon energy. Using a comprehensive numerical model we have shown that efficient S-band TDFA and lasing at 810 nm can be obtained. The chromium valence state was controlled by adding some glass modifiers like Al. This open the way to PQS allfibre lasers. Then, nanostructuration of doped fibre is proposed as a new promising route to 'engineer' the local dopant environment. All these results will benefit to optical fibre components such as lasers, amplifiers and sensors, which can now be realized with silica glass.

7.

Fig. 1 :

 1 Fig. 1 : Energy diagram of Tm 3+ ion. Parameters are described in the text.

Fig. 2

 2 Fig. 2 : a) Histogram of the recovered amplitude distributions obtained for silica-based Tm 3+ -doped fibres. b) variations of the short (circle) and long (triangle) characteristic lifetimes vs Al2O3 concentration.

Fig. 3 :

 3 Fig. 3 : Absorption and emission cross section spectra of thulium.

Fig. 4 :

 4 Fig. 4 : Typical layout of the TDFA.

Fig. 5 :

 5 Fig. 5 : Upconversion pumping of TDFA at around 1050 nm (a). The effect of glass host composition, mainly manifested by the 3 H4 level lifetime, on the gain in the S-band (b) and optimization of the TDFA length (c).

Fig. 6 :

 6 Fig. 6 : Optimization of the waveguide parameters, namely the NA and core radius (a) and optimiziation of the pump wavelength (b).

Fig. 7

 7 Fig. 7 Fibre laser setup in compact all-fibre arrangement (a), single wavelength upconversion pumping scheme (b), effect of the host material (c) and pump wavelength (d). The waveguide parameters used for the parts (c) and (d) are: the core diameter of 3.4 m and numerical aperture of 0.2.

Fig. 8 :

 8 Fig. 8:. Absorption from a Cr(Ge) preform ([Cr] = 1400 mol ppm) and a Cr(Al) fibre ([Cr] = 40 mol ppm). Solid lines are gaussian fittings to Cr 3+ (left) and Cr 4+ (right) transitions, respectively. Assignments are from the ground level Cr 3+ : 4 A2 or Cr 4+ : 3 A2 to the indicated level, respectively. The Cr 4+ energy states are referred to by their irreductible representation in the Td symmetry coordination. The Cr 4+ : 3 T2 level three-fold splitting is due to distorsion from perfect Td symmetry.

  , right) depend on [Cr] and s. The fast decay part is assigned to Cr clusters or Cr 4+ -rich phases within the glass. The 1/e-lifetimes ( at s = 1100 nm are all within the15-35 s range in Al-containing samples, whereas  ~ 3-11 s in Cr(Ge) samples, depending on [Cr].

  (i) the observed LT fluorescence of Cr 4+ is extremely sensitive to glass composition, total [Cr] and excitation wavelength; (ii) using Al as a glass network modifier has advantages: longer excited state lifetime and broader fluorescence bandwidth than in Ge-modified silica; (3) the RT emission is strongly limited by NRD in silica. Broadband light sources would need a better control of the local environment around TM ions. However this study has allowed to propose an alternative application to Cr-doped fibre for self pulsing lasers.

Fig. 9 :

 9 Fig. 9: Fluorescence spectra (a) : (1) fibre Cr(Al), p = 900 nm, T=77 K ; (2) preform Cr(Ge-Al), p = 673 nm, T=12K. Fluorescence decays (b) from Cr(Al) samples, p = 673 nm, T=12 K: (1) s~1100 nm and [Cr]=40 ppm, (2)s~1100 nm, [Cr]=4000 ppm, and (3) s~1400 nm,

Fig. 10 :

 10 Fig. 10: Yb 3+ :Cr 4+ PQS laser setup, and typical pulse train. DCYF: double-clad Yb-doped optical fibre, PC: polarization controler, CrSa: Cro-doped fibre saturable absorber, FBG: high reflectivity fibre Bragg grating mirror.

Fig. 11 :

 11 Fig. 11 : (a) zoom on a typical pulse envelope from the 'DCYF only' chaotic laser output. (b) zoom on few typical pulses from the stabilized 'DCYF+CrSA' laser.

Fig. 12 :

 12 Fig. 12: Immiscibility-gap in the phase-diagram of binary SiO2-MO glass (M = Mg, Ca, Sr)

Fig. 14 :

 14 Fig. 14: Transmission spectrum of a Mg-doped fibre (0.1 mol/l solution). Dots: experimental data, full line: calculated Rayleigh scattering curve.

Fig. 15 :

 15 Fig. 15: Energy dispersive x-ray analyses spectra of the preform sample doped with Ca and Er. The area analysed corresponds to the nanoparticle (a) and outside (b)

Fig. 16 :

 16 Fig. 16: Room temperature emission spectra from Mg-doped fibres. The Mg concentration in the doping solution is 0.1 (A) and 1.0 (B) mol/l, respectively. Excitation wavelength is 980 nm.