Nathaël Gozlan

Ronan Herry

MULTIPLE SETS EXPONENTIAL CONCENTRATION AND HIGHER ORDER EIGENVALUES

.

Introduction

Let (M, g) be a smooth compact connected Riemannian manifold with its normalized volume measure µ and its geodesic distance d. The Laplace-Beltrami operator ∆ is then a non-positive operator whose spectrum is discrete. Let us denote by λ (k) , k = 0, 1, 2 . . ., the eigenvalues of -∆ written in increasing order. With these notations λ (0) = 0 (achieved for constant functions) and (by connectedness) λ (1) > 0 is the socalled spectral gap of M .

The study of the spectral gap of Riemannian manifolds is, by now, a very classical topic which has found important connections with numerous geometrical and analytical questions and properties. The spectral gap constant λ (1) is for instance related to Poincaré type inequalities and governs the speed of convergence of the heat flow to equilibrium. It is also related to Ricci curvature via the classical Lichnerowicz theorem [START_REF] Lichnerowicz | Géométrie des groupes de transformations[END_REF] and to Cheeger isoperimetric constant via Buser's theorem [START_REF] Buser | A note on the isoperimetric constant[END_REF]. We refer to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF][START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF] and the references therein for a complete picture.

Another important property of the spectral gap constant, first observed by Gromov and Milman [START_REF] Gromov | A topological application of the isoperimetric inequality[END_REF], is that it controls exponential concentration of measure phenomenon for the reference measure µ. The result states as follows. Define for all Borel sets A ⊂ M , its r-enlargement A r as the (open) set of all x ∈ E such that there exists y ∈ A with d(x, y) < r. Then, for any A ⊂ M such that µ(A) ≥ 1/2 it holds

µ(A r ) ≥ 1 -be -a √ λ (1) r , ∀r > 0,
where a, b > 0 are some universal constants (according to [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]Theorem 3.1], one can take b = 1 and a = 1/3). Note that this implication is very general and holds on any metric space supporting a Poincaré inequality (see [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]Corollary 3.2]). See also [6,[START_REF] Schmuckenschläger | Martingales, Poincaré type inequalities, and deviation inequalities[END_REF][START_REF] Aida | Moment estimates derived from Poincaré and logarithmic Sobolev inequalities[END_REF][START_REF] Nathael Gozlan | From dimension free concentration to the Poincaré inequality[END_REF] for alternative derivations, generalizations or refinements of this result. This note is devoted to a multiple sets extension of the above result. Roughly speaking, we will see that if A 1 , . . . , A k are sets which are pairwise separated in the sense that d(A i , A j ) := inf{d(x, y) : x ∈ A i , y ∈ A j } > 0 for any i = j and A is their union then the probability of A r goes exponentially fast to 1 at a rate given by √ λ (k) as soon as r is such that the sets A i,r , i = 1, . . . , k remain separated. More precisely, it follows from Theorem 1.1 (whose setting is actually more general) that, if A 1 , . . . , A k are such that µ(A i ) ≥ 1 k+1 and d(A i,r , A j,r ) > 0 for all i = j, then, denoting

A = A 1 ∪ . . . ∪ A k , it holds (0.1) µ(A r ) ≥ 1 - 1 k + 1
exp -c min(r 2 λ (k) ; r λ (k) ) ,

for some universal constant c. This kind of probability estimates first appeared, in a slightly different but essentially equivalent formulation in the work of Chung, Grigor'yan and Yau [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF][START_REF] Chung | Eigenvalues and diameters for manifolds and graphs[END_REF] (see also the related paper [START_REF] Friedman | Laplacian eigenvalues and distances between subsets of a manifold[END_REF] by Friedman and Tillich). Nevertheless, the method of proof we use to arrive at (0.1) (based on the Courant-Fischer min-max formula for the λ (k) 's) is quite different from the one of [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF][START_REF] Chung | Eigenvalues and diameters for manifolds and graphs[END_REF] and seems more elementary and general. This is discussed in details in Section 1.5. The paper is organized as follows. In Section 1, we prove (0.1) in an abstract metric space framework. This framework contains, in particular, the compact Riemannian case equipped with the Laplace operator presented above. The Section 1.5 contains a detailed discussion of our result with the one of Chung, Grigor'yan & Yau. In Section 2, we recall various bounds on eigenvalues on several non-negatively curved manifolds. Section 3 gives an extension of (0.1) to discrete Markov chains on graphs. In Section 4, we give a functional formulation of the results of Sections 1 and 3. As a corollary of this functional formulation, we obtain a deviation inequality as well as an estimate for difference of two Lipschitz extensions of a Lipschitz function given on k subsets. Finally, Section 5 discusses open questions related to this type of concentration of measure phenomenon.

Multiple sets exponential concentration in abstract spaces

1.1. Courant-Fischer formula and generalized eigenvalues in metric spaces. Let us recall the classical Courant-Fischer min-max formula for the k-th eigenvalue (k ∈ N) of -∆, noted λ (k) , on a compact Riemannian manifold (M, g) equipped with its (normalized) volume measure µ:

(1.1) λ (k) = inf V ⊂C ∞ (M ) dim V =k+1 sup f ∈V \{0} ´|∇f | 2 dµ ´f 2 dµ ,
where ∇f is the Riemannian gradient, defined through the Riemannian metric g (see e.g [START_REF] Chavel | Eigenvalues in Riemannian geometry[END_REF]) and |∇f | 2 = g(∇f, ∇f ). The formula (1.1) above does not make explicitly reference to the differential operator ∆. It can be therefore easily generalized to a more abstract setting, as we shall see below.

In all what follows, (E, d) is a complete, separable metric space and µ a reference Borel probability measure on E. Following [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF], for any function f : E → R and x ∈ E, we denote by |∇f |(x) the local Lipschitz constant of f at x, defined by

|∇f |(x) = 0 if x is isolated lim sup y→x |f (x)-f (y)| d(x,y)
otherwise.

Note that when E is a smooth Riemannian manifold, equipped with its geodesic distance d, then, the local Lipschitz constant of a differentiable function f at x coincides with the norm of ∇f (x) in the tangent space T x E. With this notion in hand, a natural generalization of (1.1) is as follows (we follow [23, Definition 3.1]):

(1.2) λ (k) d,µ := inf V ⊂H 1 (µ) dim V =k+1 sup f ∈V \{0} ´|∇f | 2 dµ ´f 2 dµ , k ≥ 0,
where H 1 (µ) denotes the space of functions f ∈ L 2 (µ) such that ´|∇f | 2 dµ < +∞. In order to avoid heavy notations, we drop the subscript and we simply write

λ (k) instead of λ (k)
d,µ within this section.

a i + k j=1 a j ≥ 1, ∀i ∈ {1, . . . , k}.

Recall the classical notation

d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B} of the distance between two sets A, B ⊂ E.
The following theorem is the main result of the paper and is proved in Section 1.3.

Theorem 1.1.

There exists a universal constant c > 0 such that, for any k ≥ 1 and for all sets

A 1 , . . . , A k ⊂ E such that min i =j d(A i , A j ) > 0 and (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k , the set A = A 1 ∪ A 2 ∪ • • • ∪ A k satisfies µ(A r ) ≥ 1 -(1 -µ(A)) exp -c min(r 2 λ (k) ; r λ (k) ) , for all 0 < r ≤ 1 2 min i =j d(A i , A j )
, where λ (k) ≥ 0 is defined by (1.2). Note that, since (1/(k + 1), . . . , 1/(k + 1)) ∈ ∆ k , Theorem 1.1 immediately implies Inequality (0.1).

Inverting our concentration estimate, we obtain the following statement that provides a bound on the λ (k) 's. (E,d,µ) be a metric measured space and λ (k) be defined as in (1.2). Let A 1 , . . . , A k be measurable sets such that (µ(A 1 ), . .

Proposition 1.2. Let

. , µ(A

k )) ∈ ∆ k , then, with r = 1 2 min i =j d(A i , A j ) and A 0 = E \ (∪A i ) r , λ (k) ≤ 1 r 2 ψ 1 c min i ln µ(A i ) µ(A 0 ) ,
where ψ(x) = max(x, x 2 ).

Proof. Let A = ∪ i A i . Inverting the formula in Theorem 1.1, we obtain

λ (k) ≤ 1 r 2 ψ 1 c ln 1 -µ(A) 1 -µ(A r ) , where ψ(x) = max(x, x 2 ). By definition of ∆ k , 1 -µ(A) = 1 - i µ(A i ) ≤ min i µ(A i ).
Therefore, letting A 0 = E \ A r , we obtain the announced inequality by non-decreasing monotonicity of ψ and ln.

The collection of sets ∆ k , k ≥ 1 has the following useful stability property:

Lemma 1.3. Let I 1 , I 2 , . . . , I n be a partition of {1, . . . , k}, k ≥ 1. Let a = (a 1 , . . . , a k ) ∈ R k and define b = (b 1 , . . . , b n ) ∈ R n by setting b i = j∈I i a j , i ∈ {1, . . . , n}. If a ∈ ∆ k then b ∈ ∆ n .
Proof. The proof is obvious and left to the reader.

Thanks to this lemma it is possible to iterate Theorem 1.1 and to obtain a general bound for µ(A r ) for all values of r > 0. This bound will depend on the way the sets A 1,r , . . . , A k,r coalesce as r increases. This is made precise in the following definition.

Definition 1.1 (Coalescence graph of a family of sets). Let A 1 , . . . , A k be subsets of E. The coalescence graph of this family of sets is the family of graphs G r = (V, E r ), r > 0, where V = {1, 2, . . . , k} and the set of edges E r is defined as follows: {i, j}

∈ E r if d(A i,r , A j,r ) = 0. Corollary 1.4. Let A 1 , . . . , A k be subsets of E such that min i =j d(A i , A j ) > 0 and (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k .
For any r > 0, let N (r) be the number of connected components in the coalescence graph G r associated to A 1 , . . . , A k . The function (0, ∞) → {1, . . . , k} : r → N (r) is non-increasing and right-continuous. Define r i = sup{r > 0 : N (r) ≥ k -i + 1}, i = 1, . . . , k and r 0 = 0 then it holds

(1.3) µ(A r ) ≥ 1 -(1 -µ(A)) exp -c k i=1 φ [r ∧ r i -r i-1 ] + λ (k-i+1) , ∀r > 0,
where φ(x) = min(x; x 2 ), x ≥ 0 and c is the universal constant appearing in Theorem 1.1.

Observe that, contrary to usual concentration results, the bound given above depends on the geometry of the set A.

µ(A r ) ≥ 1 -(1 -µ(A)) exp -cφ(r λ (k) ) , for all 0 < r ≤ 1 2 min i =j d(A i , A j ). Let k 1 = N ( 1 2 min i =j d(A i , A j )) and let i 1 = k -k 1 .
Then, for all i ∈ {1, . . . , i 1 }, r i = 1 2 min i =j d(A i , A j ). So that, for all 0 < r ≤ r i 1 , the preceding bound can be rewritten as follows (note that only the term of index i = 1 gives a non zero contribution)

µ(A r ) ≥ 1 -(1 -µ(A)) exp -c i 1 i=1 φ [r ∧ r i -r i-1 ] + λ (k-i+1) = 1 -(1 -µ(A)) exp -c k i=1 φ [r ∧ r i -r i-1 ] + λ (k-i+1) (1.4)
which shows that (1.3) is true for 0 < r ≤ r i 1 . Now let I 1 , . . . , I k 1 be the connected components of G r 1 and define, for all i ∈ {1, . . . , k 1 },

B i = ∪ j∈I i A j,r 1 . It follows eas- ily from Lemma 1.3 that (µ(B 1 ), . . . , µ(B k 1 )) ∈ ∆ k 1 . Since min i =j d(B i , B j ) > 0, the induction hypothesis implies that µ(B s ) ≥ 1 -(1 -µ(B)) exp   -c k 1 i=1 φ [s ∧ s i -s i-1 ] + λ (k 1 -i+1)   , ∀s > 0,
where

B = B 1 ∪ • • • ∪ B k 1 = A r 1 and s i = sup{s > 0 : N ′ (s) ≥ k 1 -i + 1}, i ∈ {1, . . . , k 1 } (s 0 = 0) with N ′ (s) the number of connected components in the graph G ′ s associated to B 1 , . . . , B k 1 . It is easily seen that r i 1 +i = r i 1 + s i , for all i ∈ {0, 1 . . . , k 1 }. Therefore, we have that, for r > r i 1 , µ(A r ) ≥ µ(B r-r i 1 ) ≥ 1 -(1 -µ(A r i 1 )) exp   -c k i=i 1 +1 φ [r ∧ r i -r i-1 ] + λ (k-i+1)   ≥ 1 -(1 -µ(A)) exp -c k i=1 φ [r ∧ r i -r i-1 ] + λ (k-i+1) ,
where the last line is true by (1.4).

To prove Theorem 1.1, we need some preparatory lemmas. Given a subset A ⊂ E, and x ∈ E, the minimal distance from x to A is denoted by

d(x, A) = inf y∈A d(x, y). Lemma 1.5. Let A ⊂ E and ǫ > 0, then (E \ A ǫ ) ǫ ⊂ E \ A. Proof. Let x ∈ (E \ A ǫ ) ǫ . Then, there exists y ∈ E \ A ǫ (in particular d(y, A) ≥ ǫ) such that d(x, y) < ǫ. Since the function z → d(z, A) is 1-Lipschitz, one has d(x, A) ≥ d(y, A) -d(x, y) > 0 and so x ∈ E \ A. Remark 1. In fact, we proved that (E \ A ǫ ) ǫ ⊂ E \ Ā. The converse is, in general, not true. Lemma 1.6. Let A 1 , . . . , A k be a family of sets such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k and r := 1 2 min i =j d(A i , A j ) > 0. Let 0 < ǫ ≤ r and set A = ∪ 1≤i≤k A i and A 0 = E \ (A ǫ ). Then, (1.5) max i=0,...,k µ(A i,ǫ ) µ(A i ) ≤ 1 -µ(A) 1 -µ(A ǫ ) .
Proof. First, this is true for i = 0. Indeed, by definition A 0 = E \ (A ǫ ) and, according to Lemma 1.5, (A 0 ) ǫ ⊂ A c (the equality is not always true), which proves (1.5) in this case. Now, let us show (1.5) for the other values of i. Since ǫ ≤ r, the A j,ǫ 's are disjoint sets. Thence, (1.5) is equivalent to

  1 - k j=1 µ(A j,ǫ )   µ(A i,ǫ ) ≤   1 - k j=1 µ(A j )   µ(A i ).
This inequality is true as soon as

(1 -µ(A i,ǫ ) -m i ) µ(A i,ǫ ) ≤ (1 -µ(A i ) -m i ) µ(A i ), denoting m i = k j =i µ(A j ). The function f i (u) = (1 -u -m i )u, u ∈ [0, 1], is decreasing on the interval [(1 -m i )/2, 1]. We conclude from this that (1.5) is true for all i ∈ {1, . . . , k}, as soon as µ(A i ) ≥ (1 -m i )/2 for all i ∈ {1, . . . , k} which amounts to (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k . For p > 1, we define the function χ p : [0, ∞[→ [0, 1] by χ p (x) = (1 -x p ) p , for x ∈ [0, 1] and χ p (x) = 0 for x > 1. It is easily seen that χ p (0) = 1, χ ′ p (0) = χ p (1) = χ ′ p (1) = 0, that χ p takes values in [0, 1]
and that χ p is continuously differentiable on [0, ∞[. We use the function χ p to construct smooth approximations of indicator functions on E, as explained in the next statement.

Lemma 1.7. Let A ⊂ E and consider the function

f (x) = χ p (d(x, A)/ǫ), x ∈ E, where ǫ > 0 and p > 1. For all x ∈ E, it holds |∇f |(x) ≤ p 2 ǫ -1 1 Aǫ\A
Proof. Thanks to the chain rule for the local Lipschitz constant (see e.g. [2, Proposition 2.1]),

∇χ p d(•, A) ǫ (x) ≤ ǫ -1 χ ′ p d(•, A) ǫ |∇d(•, A)|(x).
The function d(•, A) being Lipschitz, its local Lipschitz constant is ≤ 1 and, thereby,

|∇f |(x) ≤ χ ′ p d(x, A) ǫ .
In particular, thanks to the aforementioned properties of χ, |∇f | vanishes on A (and even on A) and on {x ∈ E : d(x, A) ≥ ǫ} = E \ A ǫ . On the other hand, a simple calculation shows that |χ ′ p | ≤ p 2 which proves the claim.

Proof of Theorem 1.1. Take Borel sets A 1 , . . . , A k with 1 2 min i =j d(A i , A j ) ≥ r > 0 and (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k and consider A = A 1 ∪ • • • ∪ A k . Let us show that, for any 0 < ǫ ≤ r, it holds (1.6) 1 + λ (k) ǫ 2 (1 -µ(A ǫ )) ≤ (1 -µ(A)). Let A 0 = E \ (A ǫ ) and set f i (x) = χ p (d(x, A i )/ǫ), x ∈ E, i ∈ {0, . . . , k}, where p > 1.
According to Lemma 1.7 and the fact that

f i = 1 on A i , we obtain (1.7) ˆ|∇f i | 2 dµ = p 4 ǫ 2 µ(A i,ǫ \ A i ) and ˆf 2 i dµ ≥ µ(A i ).
Since the f i 's have disjoint supports they are orthogonal in L 2 (µ) and, in particular, they span a k + 1 dimensional subspace of H 1 (µ). Thus, by definition of λ (k) ,

λ (k) ≤ sup a∈R k+1 ´|∇ k i=0 a i f i | 2 dµ ´ k i=0 a i f i 2 dµ ≤ sup a∈R k+1 ´ k i=0 |a i ||∇f i | 2 dµ ´ k i=0 a i f i 2 dµ ,
where the second inequality comes from the following easy to check sub-linearity property of the local Lipschitz constant:

|∇ (af + bg) | ≤ |a||∇f | + |b||∇g|.
Since the f ′ i s and the |∇f i | ′ s are two orthogonal families, we conclude using (1.7), that

λ (k) ǫ 2 p 4 ≤ sup a∈R k+1 k i=0 a 2 i (µ(A i,ǫ ) -µ(A i )) k i=0 a 2 i µ(A i )
, which amounts to

(1.8) 1 + λ (k) ǫ 2 p 4 ≤ max i=0,...,k µ(A i,ǫ ) µ(A i ) .
Applying Lemma 1.6 and sending p to 1 gives (1.6). Now, if n ∈ N and 0 < ǫ are such that nǫ ≤ r, then iterating (1.6) immediately gives

1 + λ (k) ǫ 2 n (1 -µ(A nǫ )) ≤ 1 -µ(A).
Optimizing this bound over n for a fixed ε gives

(1 -µ(A r )) ≤ (1 -µ(A)) exp -sup ⌊r/ǫ⌋ log 1 + λ (k) ǫ 2 : ǫ ≤ r .
Thus, letting

(1.9) Ψ(x) = sup ⌊t⌋ log 1 + x t 2 : t ≥ 1 , x ≥ 0, it holds (1 -µ(A r )) ≤ (1 -µ(A)) exp -Ψ λ (k) r 2 .
Using Lemma 1.8 below, we deduce that Ψ λ (k) r 2 ≥ c min(r 2 λ (k) ; r √ λ (k) ), with c = log(5)/4, which completes the proof.

Lemma 1.8. The function Ψ defined by (1.9) satisfies

Ψ(x) ≥ log(5) 4 min(x; √ x), ∀x ≥ 0.
Proof. Taking t = 1, one concludes that Ψ(x) ≥ log(1 + x), for all x ≥ 0. The function x → log(1 + x) being concave, the function x → log(1+x)

x is non-increasing. Therefore, log(1 + x) ≥ log (5) 4 x for all x ∈ [0, 4]. Now, let us consider the case where x ≥ 4. Observe that ⌊t⌋ ≥ t/2 for all t ≥ 1 and so, for x ≥ 4,

Ψ(x) ≥ 1 2 sup t≥1 t log 1 + x t 2 ≥ log(5) 4 √ x, by choosing t = √ x/2 ≥ 1. Thereby, Ψ(x) ≥ log(5) 4 x1 [0,4] (x) + √ x1 [4,∞) (x) ≥ log(5) 4 min(x; √ x),
which completes the proof.

Remark 2. The conclusion of Lemma Lemma 1.8 can be improved. Namely, it can be shown that

Ψ(x) = max    1 + ⌊ √ x a ⌋ log   1 + x 1 + ⌊ √ x a ⌋ 2    ; ⌊ √ x a ⌋ log   1 + x ⌊ √ x a ⌋ 2       ,
(the second term in the maximum being treated as 0 when √ x < a) where 0 < a < 2 is the unique point where the function (0, ∞) → R : u → log(1 + u 2 )/u achieves its supremum. Therefore,

Ψ(x) ∼ log(1 + a 2 ) a √ x
when x → ∞. The reader can easily check that log(1+a 2 ) a ≃ 0.8. In particular, it does not seem possible to reach the constant c = 1 in Theorem 1.1 using this method of proof. 1.4. Two more multi-set concentration bounds. The condition (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k can be seen as the multi-set generalization of the condition, standard in concentration of measure, that the size of the enlarged set has to be bigger than 1/2. Indeed, the reader can easily verify that ( 1 k+1 , . . . , 1 k+1 ) ∈ ∆ k . However, in practice, this condition can be difficult to check. We provide two more multi-set concentration inequalities that hold in full generality. The method of proof is the same as for Theorem 1.1 and is based on (1.8). Proposition 1.9. Let (E, d, µ) be a metric measured space and λ (k) be defined as in (1.2). Let (A 1 , . . . , A k ) be k Borel sets, A = ∪ i A i and A 0 = E \ A r . Then, with a (1) = min 1≤i≤k µ(A i ), the following two bounds hold:

1 -µ(A r ) ≤ (1 -µ(A)) 1 k i=1 µ(A i ) exp -c min r 2 λ (k) , r λ (k) ; 1 -µ(A r ) ≤ (1 -µ(A)) 1 µ(A) µ(A)/a (1) exp -c min r 2 λ (k) , r λ (k) .
Proof. Fix N ∈ N and ǫ > 0 such that N ǫ ≤ r. For i = 1, . . . , k and n ≤ N , we define

α i (n) = µ(A i,nǫ ) µ(A i,(n-1)ǫ )
;

M n = max 1≤i≤k α i (n) ∨ 1 -µ(A (n-1)ǫ ) 1 -µ(A nǫ ) ; L n = {i ∈ {1, . . . , k}|M n = α i (n)}; N i = ♯{n ∈ {1, . . . , N }|i = inf L n }; N 0 = N - k i=1 N i .
Roughly speaking, the number N i (0 ≤ i ≤ k) counts the number of time where the set A i growths in iterating (1.8). Lemma 1.6 asserts that in the case where (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k , then N 0 = N . However, we still obtain from (1.8), for 1

≤ i ≤ k, (1.10) 1 µ(A i ) ≥ N n=1 α i (n) ≥ 1 + λ (k) ǫ 2 N i .
The first inequality is true because µ(A i,N ǫ ) ≤ 1 and a telescoping argument. The second inequality is true because, as n ranges from 1 to N , by definition of the number N i and (1.8), there are, at least N i terms appearing in the product that can be bounded by (1 + λ (k) ǫ 2 ). The other terms are bounded above by 1. The case of i = 0 is handled in a similar fashion and we obtain:

1 -µ(A N ǫ ) ≤ (1 -µ(A)) 1 + λ (k) ǫ 2 -N 0 = (1 -µ(A)) 1 + λ (k) ǫ 2 -N k i=1 1 + λ (k) ǫ 2 N i .
(1.11)

The announced bounds will be obtain by bounding the product appearing in the righthand side and an argument similar to the end of the proof of Theorem 1.1. From (1.10), we have that,

(1.12)

k i=1 1 + λ (k) ǫ 2 N i ≤ 1 k i=1 µ(A i )
.

Also, from (1.10),

µ(A i,N ǫ ) ≥ 1 + λ (k) ǫ 2 N i µ(A i ).
Because N ǫ ≤ r, the sets A 1,N ǫ , . . . , A k,N ǫ are pairwise disjoint and, thereby,

1 ≥ µ(A i,N ǫ ) ≥ k i=1 1 + λ (k) ǫ 2 N i µ(A i ).
Fix θ > 0 to be chosen later. By convexity of exp,

1 + (1 -µ(A)) 1 + λ (k) ǫ 2 θ ≥ exp k i=1 µ(A i )N i + (1 -µ(A))θ log 1 + λ (k) ǫ 2 ≥ exp a (1) k i=1 N i + (1 -µ(A))θ log 1 + λ (k) ǫ 2 .
Finally, with p = 1 -µ(A) and t = θ log(1 + λ (k) ǫ 2 ), we obtain

k i=1 1 + λ (k) ǫ 2 N i ≤ e -pt
+p e (1-p)t 1/a (1) .

We easily check that, the quantity in the right-hand side is minimal for t = log 1 1-p at which it takes the value (1 -p) p-1 = µ(A) -µ(A)/a (1) . Thus, (1.13) (1) .

k i=1 (1 + λ (k) ǫ 2 ) N i ≤ 1 µ(A) µ(A)/a
Combining (1.12) and (1.13) with (1.11) and the same argument as for (1.9), we obtain the two announced bounds.

From Proposition 1.9, we can derive bounds on the λ (k) 's. The proof is the same as the one of Proposition 1.2 and is omitted. Proposition 1.10. Let (E, d, µ) be a metric measured space and λ (k) be defined as in (1.2). Let A 1 , . . . , A k be measurable sets, then, with r = 1 2 min i =j d(A i , A j ) and

A 0 = E \ (∪A i ) r , λ (k) ≤ 1 r 2 ψ 1 c ln a (1) µ(A 0 ) + 1 c k ln 1 a (1) ; λ (k) ≤ 1 r 2 ψ 1 c ln a (1) µ(A 0 ) + 1 c µ(A) a (1) ln 1 µ(A) ,
where ψ(x) = max(x, x 2 ) and a (1) = min 1≤i≤k µ(A i ).

1.5. Comparison with the result of Chung-Grigor'yan-Yau. In [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF], the authors obtained the following result:

Theorem 1.11 (Chung-Grigoryan-Yau [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF]). Let M be a compact connected smooth Riemannian manifold equipped with its geodesic distance d and normalized Riemannian volume µ. For any k ≥ 1 and any family of sets A 0 , . . . , A k , it holds

(1.14) λ (k) ≤ 1 min i =j d 2 (A i , A j ) max i =j log ( 4 µ(A i )µ(A j ) ) 2
,

where 1 = λ (0) ≤ λ (1) ≤ • • • λ (k) ≤ • • • denotes the discrete spectrum of -∆.
Let us translate this result in terms of concentration of measure. Let A 1 , . . . , A k be sets such that r = 1 2 min 1≤i<j≤k d(A i , A j ) > 0 and define , so that (1.15) is equivalent to the following statement:

A = A 1 ∪ • • • ∪ A k and A 0 = M \A s , for some 0 < s ≤ r. Then,
(1.16) µ(A s ) ≥ 1 - 4 a (1) exp(-λ (k) s), ∀s ∈ [min(s o , r); r].
We note that (1.16) holds for any family of sets, whereas the inequality given in Theorem 1.1 is only true when (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k . Also due to the fact that the constant c appearing in Theorem 1.1 is less than 1, (1.16) is asymptotically better than ours (see also Remark 2 above). On the other hand, one sees that (1.16) is only valid for s large enough (and its domain of validity can thus be empty when s o > r) whereas our inequality is true on the whole interval (0, r]. It does not seem also possible to iterate (1.16) as we did in Corollary 1.4. Finally, observe that the method of proof used in [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF] and [START_REF] Chung | Eigenvalues and diameters for manifolds and graphs[END_REF] is based on heat kernel bounds and is very different from ours. Let us translate Theorem 1.11 in a form closer to our Proposition 1.2. Fix k sets A 1 , . . . , A k such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k . Let 2r = min d(A i , A j ), where the infimum runs on i, j = 1, . . . , k with i = j. We have to choose a (k + 1)-th set. In view of Theorem 1.11, the most optimal choice is to choose A 0 = E \ (∪A i ) r . Indeed, it is the biggest set (in the sense of inclusion) such that min d(A i , A j ) = r where this time the infimum runs on i, j = 0, . . . , k and i = j. We let a (0) = µ(A 0 ) and we remark that if (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k then a (0) ≤ a [START_REF] Aida | Moment estimates derived from Poincaré and logarithmic Sobolev inequalities[END_REF] . The bound (1.14) can be read: for all r > 0,

λ (k) ≤ 1 r 2 log 4 a (1) a (0) 2 .
Therefore, to compare it to our bound, we need to solve

φ -1 1 c log a (1) a (0) 2 ≤ log 4 a (1) a (0) 2 .
Because the right-hand side is always ≥ 1, taking the square root and composing with the non-decreasing function φ yields

1 c log a (1) a (0) ≤ log 4 a (1) a (0)
.

That is a 1+c

(1) ≤ 4 c a 1-c (0) . In other words, on some range our bound is better and in some other range their bound is better. However, if the constant c = 1 could be attained in Theorem 1.1, this would show that our bound is always better. Note that comparing the bounds obtained in Proposition 1.10 and the one of [START_REF] Chung | Upper bounds for eigenvalues of the discrete and continuous Laplace operators[END_REF] is not so clear as, without the assumption that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k it is not necessary that a (0) ≤ a [START_REF] Aida | Moment estimates derived from Poincaré and logarithmic Sobolev inequalities[END_REF] and in that case we would have to compare different sets.

Eigenvalue estimates for non-negatively curved spaces

We recall the values of the λ (k) 's that appear in Theorem 1.1 in the case of two important models of positively curved spaces in geometry. Namely:

(i) The n-dimensional sphere of radius n-1 ρ , S n,ρ endowed with the natural geodesic distance d n,ρ arising from its canonical Riemannian metric and its normalized volume measure µ n,ρ which has constant Ricci curvature equals to ρ and dimension n.

(ii) The n-dimensional Euclidean space R n endowed with the n-dimensional Gaussian measure of covariance ρ -1 Id,

γ n,ρ (dx) = ρ n/2 e -ρ|x| 2 /2 (2π) n/2 dx.
This space has dimension ∞ and curvature bounded below by ρ in the sense of [START_REF] Bakry | Diffusions hypercontractives[END_REF]. These models arise as weighted Riemannian manifolds without boundary having a purely discrete spectrum. In that case, it was proved in [START_REF] Milman | Spectral Estimates, Contractions and Hypercontractivity[END_REF]Proposition 3.2] that the λ k 's of (1.2) are exactly the eigenvalues (counted with multiplicity) of a self-adjoint operator that we give explicitly in the following. Using a comparison between eigenvalues of [START_REF] Milman | Spectral Estimates, Contractions and Hypercontractivity[END_REF], we obtain an estimates for eigenvalues in the case of log-concave probability measure over the Euclidean R n .

Example 1 (Spheres). On S n,ρ , the eigenvalues of minus the Laplace-Beltrami operator (see for instance [START_REF] Atkinson | Spherical harmonics and approximations on the unit sphere: an introduction[END_REF]Chapter 3]) are of the form ρ -2 (n -1) 2 l(l + n -1) for l ∈ N and the dimension of the corresponding eigenspace

H l,n is dim H l,n = 2l + n -1 l l + n -2 l -1 , if l > 0and dim H l,n = 1, if l = 0.
Consequently,

D l,n := dim l l ′ =0 H l ′ ,n = n + l l + n + l -1 l -1 ,
and

λ (k) = ρ -2 (n -1) 2 l(l + n -1) if and only if D l-1,n < k ≤ D l,n where λ (k)
is the k-th eigenvalues of -∆ S n,ρ and coincides with the variational definition given in (1.2).

Example 2 (Gaussian spaces). On the Euclidean space R n , equipped with the Gaussian measure γ n,ρ , the corresponding weighted Laplacian is ∆ γn,ρ = ∆ R n -ρx • ∇. The eigenvalues of -∆ γn,ρ are exactly of the form ρ 2 q and the dimension of the associated eigenspace H q,n is dim H q,n = n + q -1 q .

Consequently,

D q,n := dim q q ′ =0 H q ′ ,n = n + q q ,
and λ (k) = ρ -2 q if and only if D q-1,n < k ≤ D q,n where λ (k) is the k-th eigenvalues of -∆ γn,ρ and coincides with the variational definition given in (1.2).

Example 3 (Log-concave Euclidean spaces). We study the case where E = R n , d is the Euclidean distance and µ is a strictly log-concave probability measure. By this we mean that µ(dx) = e -V (x) dx, where Proposition 4] that such a condition on V implies that the semigroup generated by the solution of the stochastic differential equation dX t = √ 2dB t -∇V (X t )dt, where B is a Brownian motion on R n , satisfies the curvature-dimension CD(∞, K) of Bakry-Emery and, therefore, holds the log-Sobolev inequality, for all

V : R n → R such that V is C 2 and satisfying ∇ 2 V ≥ K for some K > 0. It is a consequence of [4,
f ∈ C ∞ c (R n ), Ent µ f 2 ≤ 2 K ˆ|∇f (x)| 2 µ(dx).
Such an inequality implies the super-Poincaré of [27, Theorem 2.1] that in turns implies that the self-adjoint operator L = -∆ + ∇V • ∇ has a purely discrete spectrum. In that case, the λ (k) of (1.2) corresponds to these eigenvalues and [START_REF] Milman | Spectral Estimates, Contractions and Hypercontractivity[END_REF] showed that

λ (k) ≥ λ (k) γn,ρ , where λ (k)
γn,ρ is the eigenvalues of -∆ γn,ρ of the previous example.

Extension to Markov chains

As in the classical case (see [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]Theorem 3.3]), our continuous result admits a generalization on finite graphs or more broadly in the setting of Markov chains on a finite state space. We consider a finite set E and X = (X n ) n∈N be a irreducible time-homogeneous Markov chain with state space E. We write p(x, y) = P(X 1 = y|X 0 = x) and we regard p as a matrix. We assume that p admits a reversible probability measure µ on E : p(x, y)µ(x) = p(y, x)µ(y) for all x, y ∈ E (which implies in particular that µ is invariant). The Markov kernel p induces a graph structure on E by the following procedure. Set the elements of E as the vertex of the graph and for x, y ∈ E connect them with an edge if p(x, y) > 0. As the chain is irreducible, this graph is connected. We equip E with the induced graph distance d. We write L = p -I, where I stands for the identity matrix. The operator -L is a symmetric positive operator on L 2 (µ). We let λ (k) be the eigenvalues of this operator. Then, our Theorem 1.1 extends as follows:

Theorem 3.1. For any k ≥ 1 and for all sets A 1 , . . . , A k ⊂ E such that min i =j d(A i , A j ) ≥ 1 and (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k the set B = A 1 ∪ A 2 ∪ • • • ∪ A k satisfies µ(B n ) ≥ 1 -(1 -µ(B)) 1 + λ (k) -n , for all 1 ≤ n ≤ 1 2 min i =j d(A i , A j ) where λ (k)
is the k-th eigenvalue of the operator -L acting on L 2 (µ).

Proof. We let Π(x, y) = p(x, y)µ(x) and

E (f, g) = 1 2 (f (y) -f (x))(g(y) -g(x))Π(x, y) = f, -Lg µ .
For any set A, we define the discrete boundary of A as

∂A = A 1 \ A ∪ (A C ) 1 \ A C .
Let (X n ) be the Markov chain with transition kernel p and initial distribution µ. By reversibility of µ, (X 0 , X 1 ) is an exchangeable pair of law Π whose the marginals are given by µ. Then, for a set U , we have

E (1 U ) = E1 U (X 0 )(1 U (X 0 ) -1 U (X 1 )) = P(X 0 ∈ U, X 1 ∈ U ) ≤ P(X 1 ∈ ∂U ) = µ(∂U ).
Observe that if d(U, V ) ≥ 1, U and V are disjoint and U × V ∈ supp Π so that E (1 U , 1 V ) = 0. By Courant-Fischer's min-max theorem

λ (k) = min dim V =k+1 max f ∈V E (f, f ) µ(f 2 ) . Choose sets A 1 , . . . , A k with d(A i , A j ) ≥ 2n (i = j) and (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k . Set f i = 1 A i .
The f i 's have disjoint support and so they are orthogonal in L 2 (µ). By the previous variational representation of λ (k) , we have

λ (k) ≤ sup a i E k i=0 a i f i ´ k i=0 a i f i 2 dµ = sup a i a i a i ′ E (f i , f i ′ ) a i a i ′ ´fi f i ′ dµ = sup a i k i=0 a 2 i E (f i ) k i=0 a i ´f 2 i dµ .
In other words,

λ (k) ≤ max i=0,...,k µ((A i ) 1 ) + µ((A C i ) 1 ) -1 µ(A i ) ≤ µ((A i ) 1 ) -µ(A i ) µ(A i ) ,
where the last inequality comes from the fact that, by Lemma 1.5,

µ(E \ (E \ A) 1 ) ≥ µ(A). Consider the set B = ∪ k i=1 A i and choose A 0 = E \B 1 .
In that case, by Lemma 1.6 with ǫ = 1, we have

max i=0,...,k µ((A i ) 1 ) µ(A i ) ≤ 1 -µ(B) 1 -µ(B 1
) .

Thus, we proved that

(1 + λ (k) )(1 -µ(B 1 )) ≤ (1 -µ(B)).
We derive the announced result by an immediate recursion.

Functional forms of the multiple sets concentration property

We investigate the functional form of the multi-sets concentration of measure phenomenon results obtained in Sections 1 and 3. (1) For all Borel sets A 1 , . . . ,

A k ⊂ E such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k , the set A = A 1 ∪ • • • ∪ A k satisfies (4.1) µ(A r ) ≥ 1 -(1 -µ(A))α k (r), ∀0 < r ≤ 1 2 min i =j d(A i , A j ).
(2) For all 1-Lipschitz functions f 1 , . . . , f k : E → R such that the sublevel sets

A i = {f i ≤ 0} are such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k , the function f * = min(f 1 , . . . , f k ) satisfies µ(f * < r) ≥ 1 -µ(f * ≤ 0)α k (r), ∀0 < r ≤ 1 2 min i =j d(A i , A j ).
Together with Theorem 1.1 or Theorem 3.1, one thus sees that the presence of multiple wells can improve the concentration properties of a Lipschitz function.

Proof. It is clear that (2) implies (1) when applied to f i (x) = d(x, A i ), in which case A i = {f i ≤ 0} and f * (x) = d(x, A). The converse is also very classical. First, observe that {f * < r} = ∪ k i=1 {f i < r}. Then, since f i is 1-Lipschitz, it holds A i,r ⊂ {f i < r} with A i = {f i ≤ 0} and so letting A = A 1 ∪ • • • ∪ A k , it holds A r ⊂ {f * < r}. Therefore, applying [START_REF] Aida | Moment estimates derived from Poincaré and logarithmic Sobolev inequalities[END_REF] to this set A gives (2). When (4.1) holds, we will say that the probability metric space (E, d, µ) satisfies the multi-set concentration of measure property of order k with the concentration profile α k .

In the usual setting (k = 1), the concentration of measure phenomenon implies deviation inequalities for Lipschitz functions around their median. The next result generalizes this well known fact to k > 1.

Proposition 4.2. Let (E, d, µ) be a probability metric space satisfying the multi-set concentration of measure property of order k with the concentration profile α k and f :

E → R be a 1-Lipschitz function. If I 1 , . . . , I k ⊂ R are k disjoint Borel sets such that (µ(f ∈ I 1 ), . . . , µ(f ∈ I k )) ∈ ∆ k , then it holds µ f ∈ ∪ k i=1 I i,r ≥ 1 -(1 -µ(f ∈ ∪ k i=1 I i ))α k (r), ∀0 < r ≤ 1 2 min i =j d(I i , I j )
Proof. Let ν be the image of µ under the map f . Since f is 1-Lipschitz, the metric space (R, | • |, ν) satisfies the multi-set concentration of measure property of order k with the same concentration profile α k as µ. Details are left to the reader.

Let us conclude this section by detailing an application of potential interest in approximation theory. Suppose that f : E → R is some 1-Lipschitz function and A 1 , . . . , A k are (pairwise disjoint) subsets of E such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k . Let us assume that the restrictions f |A i , i ∈ {1, . . . , k} are known and that one wishes to estimate or reconstruct f outside A = ∪ k i=1 A i . To that aim, one can consider an explicit 1-Lipschitz extension of f |A , that is to say a 1-Lipschitz function g : E → R (constructed based on our knowledge of f on A exclusively) such that f = g on A. There are several canonical ways to perform the extension of a Lipschitz function defined on a sub domain (known as Kirszbraun-McShane-Whitney extensions [START_REF] Kirszbraun | Uber die zusammenziehende und lipschitzsche transformationen[END_REF][START_REF] Mcshane | Extension of range of functions[END_REF][START_REF] Whitney | Analytic extensions of differentiable functions defined in closed sets[END_REF]). One can consider for instance the functions

g + (x) = inf y∈A {f (y) + d(x, y)} or g -(x) = sup y∈A {f (y) -d(x, y)}, x ∈ E.
It is a very classical fact that functions g -and g + are 1-Lipschitz extensions of f |A and moreover that any extension g of f |A satisfies g -≤ g ≤ g + (see e.g [START_REF] Heinonen | Lectures on Lipschitz analysis[END_REF]).

The following simple result shows that, for any 1-Lipschitz extension g of f |A , the probability of error µ(|f -g| > r) is controlled by the multi-set concentration profile α k . In particular, in the framework of our Theorem 1.1, this probability of error is expressed in terms of λ (k) . Proposition 4.3. Let (E, d, µ) be a probability metric space satisfying the multi-set concentration of measure property of order k with the concentration profile α k and f :

E → R be a 1-Lipschitz function. Let A 1 , . . . A k be subsets of E such that (µ(A 1 ), . . . , µ(A k )) ∈ ∆ k ; then for any 1-Lipschitz extension g of f |A , it holds µ(|f -g| ≥ r) ≤ (1 -µ(A))α k (r/2), ∀0 < r ≤ min i =j d(A i , A j ).
Proof.

: [0, ∞) → [0, ∞) and β k : [0, ∞) k → [0, ∞] such that for all Borel sets A 1 , . . . , A k ⊂ E, the set A = A 1 ∪ • • • ∪ A k satisfies µ(A r ) ≥ 1 -β k (µ(A 1 ), • • • , µ(A k ))α k (r), ∀0 < r ≤ 1 2 min i =j d(A i , A j ).
This framework contains the preceding one, by choosing β k (a) = 1 -k i=1 a i if a = (a 1 , . . . , a k ) ∈ ∆ k and +∞ otherwise. It also contains the concentration bounds obtained in Proposition 1.9, corresponding respectively to

β k (a) = 1 -k i=1 a i k i=1 a i , and β k (a) = 1 -k i=1 a i k i=1 a i k i=1 a i / min(a 1 ,••• ,a k )
, a = (a 1 , . . . , a k ).

Open questions

We list open questions related to the multi-set concentration of measure phenomenon. γn,ρ denotes the kth eigenvalue of the n-dimensional centered Gaussian measure with covariance matrix ρ -1 Id. Since the measure µ satisfies the log-Sobolev inequality, it is well known that it satisfies a (classical) Gaussian concentration of measure inequality. Therefore, it is natural to conjecture that µ satisfies a multi-set concentration of measure property of order k ≥ 1 with a profile of the form β k (r) = exp -C k,ρ,n r 2 , r ≥ 0, for some constant C k,ρ,n depending solely on its arguments. In addition, it would be interesting to see how usual functional inequalities (Log-Sobolev, transport-entropy, . . . ) can be modified to catch such a concentration of measure phenomenon.

Equivalence between multi-set concentration and lower bounds on eigenvalues in non-negative curvature.

Let us quickly recall the main finding of E. Milman [START_REF] Milman | On the role of convexity in isoperimetry, spectral gap and concentration[END_REF][START_REF] Milman | Isoperimetric and concentration inequalities: equivalence under curvature lower bound[END_REF], that is, under non-negative curvature assumptions, a concentration of measure estimate implies a bound on the spectral gap. Let µ be a probability measure with a density of the form e -V on a smooth connected Riemannian manifold M with V a smooth function such that (5.1) Ric + Hess V ≥ 0.

Assume that µ satisfies a concentration inequality of the form: for all A ⊂ M such that µ(A) ≥ 1/2 µ(A r ) ≥ 1 -α(r), r ≥ 0, where α is a function such that α(r o ) < 1/2 for at least one value r o > 0. Then, letting λ (1) be the first non zero eigenvalue of the operator -∆ + ∇V • ∇, it holds λ (1) ≥ 1 . It would be very interesting to extend Milman's result to a multiset concentration setting. More precisely, if µ satisfies the curvature condition (5.1) and the multi-set concentration of measure property of order k with a profile of the form α k (r) = exp(-min(ar 2 , √ ar)), r ≥ 0, can we find a universal function ϕ k such that λ (k) ≥ ϕ k (a)? This question already received some attention in recent works by Funano and Shioya [START_REF] Funano | Estimates of eigenvalues of the Laplacian by a reduced number of subsets[END_REF][START_REF] Funano | Concentration, Ricci curvature, and eigenvalues of Laplacian[END_REF]. In particular, let us mention the following improvement of the Chung-Grigor'yan-Yau inequality obtained in [START_REF] Funano | Estimates of eigenvalues of the Laplacian by a reduced number of subsets[END_REF]. There exists a universal constant c > 1 such that if µ is a probability measure satisfying the non-negative curvature assumption (5.1), it holds: for any family of sets A 0 , A 1 , . . . , A l with 1 ≤ l ≤ k (5.2)

λ (k) ≤ c k-l+1 1 min i =j d 2 (A i , A j ) max i =j log ( 4 µ(A i )µ(A j ) ) 2 .
Note that the difference with (1.14) is that λ (k) is estimated by a reduced number of sets. Using (5.2) (with l = 1) together with Milman's result recalled above, Funano showed that there exists some constant C k depending only on k such that under the curvature condition (5.1), it holds λ (k) ≤ C k λ (1) (recovering the main result of [START_REF] Funano | Concentration, Ricci curvature, and eigenvalues of Laplacian[END_REF]). The constant C k is explicit (contrary to the constant of [START_REF] Funano | Concentration, Ricci curvature, and eigenvalues of Laplacian[END_REF]) and grows exponentially when k → ∞. This result has been then improved by Liu [START_REF] Liu | An optimal dimension-free upper bound for eigenvalue ratios[END_REF], where a constant C k = O(k 2 ) has been obtained. As observed by Funano [START_REF] Funano | Estimates of eigenvalues of the Laplacian by a reduced number of subsets[END_REF], a positive answer to the open question stated above would yield that under (5.1) the ratios λ (k+1) /λ (k) are bounded from above by a universal constant.

Proposition 4 . 1 .

 41 Let (E, d) be a metric space equipped with a Borel probability measure µ. Let α k : [0, ∞) → [0, ∞). The following properties are equivalent:

5. 1 .

 1 Gaussian multi-set concentration. Using the terminology introduced in Section 4, Theorem 1.1 and the material exposed in Section 2 tell us that, if µ has a density of the form e -V with respect to Lebesgue measure on R n with a smooth function V such that Hess V ≥ ρ > 0, then the probability metric space (R n , | • |, µ) satisfies the multi-set concentration of measure property of order k with the concentration profile α k (r) = exp -c min(r 2 λ (k) γn,ρ ; r λ

4 1 -

 1 2α(ro) ro 2

  Optimizing over y∈ A gives that h(x) ≤ 2d(x, A). Therefore {h ≥ r} ⊂ {x : d(x, A) ≥ r/2} = A r/2 c and so, if 0 < r ≤ min i =j d(A i , A j ), it holds µ(|f -g| ≥ r) ≤ (1 -µ(A))α k (r/2).

	Remark 3. Let us remark that Propositions 4.1 to 4.3 can be immediately extended
	under the following more general (but notationally heavier) multi-set concentration of
	measure assumption: there exists functions α k

The function h : E → R defined by h(x) = |f -g|(x), x ∈ E, is 2-Lipschitz and vanishes on A. Therefore, for any x ∈ E and y ∈ A, it holds h(x) ≤ h(y) + 2d(x, y) = 2d(x, y).


Robert E Tench
email: robert.tench@cybel-llc.com

Clément Romano

Jean-Marc Delavaux

Optimized Design and Performance of a Shared Pump Single Clad 2 µm TDFA

We report the design, experimental performance, and simulation of a single stage, co-and counter-pumped Tmdoped fiber amplifier (TDFA) in the 2 μm signal wavelength band with an optimized 1567 nm shared pump source. We investigate the dependence of output power, gain, and efficiency on pump coupling ratio and signal wavelength. Small signal gains of >50 dB, an output power of 2 W, and small signal noise figures of <3.5 dB are demonstrated. Simulations of TDFA performance agree well with the experimental data. We also discuss performance tradeoffs with respect to amplifier topology for this simple and efficient TDFA.

Introduction

Simplicity and optimization of design are critical for the practical realization of wide bandwidth, high power single clad Thulium-doped fiber amplifiers (TDFAs) for 2 µm telecommunications applications. Recent TDFAs [START_REF] Romano | Simulation and design of a multistage 10 W thulium-doped double clad silica fiber amplifier at 2050 nm[END_REF][START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF] have reported 2 µm band amplifiers with output power > 2W, gain > 55 dB, noise figure < 4 dB, and optical bandwidth greater than 120 nm. While these designs achieve high optical performance, they employ two or more optical stages and multiple pump sources. Therefore, it is desirable to investigate designs using one amplifier stage and one pump source. In this paper we report on the design, simulation, and experimental performance of a one-stage single clad TDFA using an L-band (1567 nm) shared fiber laser pump source, as a function of pump coupling ratio, active fiber length, pump power, and signal wavelength. Our one-stage TDFA data compare well with recently reported performance of multi-stage, multi-pump amplifiers [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. In addition, the simplicity of the single clad pump shared design and its potential for cost reduction offer a broad selection of performance for different applications.

The paper is organized as follows: Section 2 presents our experimental setup, a single stage TDFA with variable coupling in the pump ratio between co-pumping and counter-pumping the active fiber. Section 3 covers the dependence of simulated amplifier performance on active fiber length, pump coupling ratio, slope efficiency, and signal wavelength. Section 4 compares measurement and simulation of the TDFA performance. Section 5 contrasts our simple TDFA design with performance of a two-stage, three-pump amplifier as reported previously in [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. Finally, Section 6 discusses design parameter tradeoffs for different TDFA applications.

Experimental Setup for Shared Pump Amplifier

The optical design of our one-stage single pump TDFA is shown in Figure 1. A single frequency 2 µm DML source (Eblana Photonics) is coupled through attenuator A and into the active fiber F1. Pump light from a multiwatt fiber laser P1 at 1567 nm is split by coupler C1 with a variable coupling ratio k (%).The two pump signals WDM = Wavelength Division Multiplexer. co-pump and counter-pump fiber F1, with k = 100% and k = 0% corresponding to all counter-pumping and all co-pumping, respectively. The value of k was changed in the simulations and experiments to optimize amplifier performance. Isolators I1 and I2 ensure unidirectional operation and suppress spurious lasing. Input and output signal powers, and co-pump and counter-pump powers, are respectively referenced to the input and output of Tm-doped fiber F1 (7 meters of OFS TmDF200).

Simulated Amplifier Performance

We begin the design of a high performance optical amplifier by studying the critically important variations of fiber signal gain (G) and output power (Pout) as a function of active fiber length (L) and input signal power (Ps).

To do this we turn to the simulated amplifier performance [START_REF] Romano | Characterization of the 3F4 -3H6 Transition in Thulium-doped Silica Fibres and Simulation of a 2µm Single Clad Amplifier[END_REF][START_REF] Jackson | Theoretical modeling of Tm-doped silica fiber lasers[END_REF] shown in Figure 2, where G is plotted vs. L for four input signal power levels. Here the total 1567 nm pump power (co-+ counter-) (Pp) is 2.5 W, the signal wavelength λs is 1952 nm, and the coupling ratio k = 50%. We note that a similar set of gain curves can be generated for different wavelength bands of the TDFA, and this behavior will be investigated later in the section.

In Figure 2 we have measured the dependence of G vs. L for a 32 dB input dynamic range in signal power. The different Ps values illustrate the amplifier operating from a linear/unsaturated regime (Ps= -30 dBm) to a highly saturated regime (Ps = + 2 dBm). The equally important dependence of noise figure on these parameters will be dealt with later.

The first observation drawn from Figure 2 is that for low input signals (e.g. for Ps = -30 dBm) G is maximized for long fiber lengths of 12 meters or greater, while for saturating input powers (e.g. Ps = +2 dBm) G reaches a maximum value for lengths of about 2 meters. It is also clear that for small signal or unsaturated gain, most of the gain (i.e. more than 80%) is achieved in the first 5 meters of the fiber, while for saturated gain most of the gain occurs within the first 1.5 meters. The second observation is that saturated gain varies only slightly with active fiber length for values greater than 3 meters, indicating that a wide range of fiber lengths can be chosen for design of a power booster amplifier. However, later we will see that the choice of the fiber length affects the useful amplifier bandwidth. The next design study for the shared pump amplifier is to examine the dependence of the saturated output power on active fiber length L and coupling ratio k. To study this issue, we plot the output signal power with pump coupling ratio k for four active fiber lengths (i.e. L = 3, 5, 7 and 9 m) as shown in Figure 3. In this simulation Ps is set to +2 dBm at 1952 nm to saturate the amplifier, with the total pump set at Pp = 2.5 W at 1567 nm.

We first note that for a given fiber length, Pout increases linearly when moving from co-pumping (k = 0%) to nearly all counter-pumping (k = 95%) and then drops down for full counter-pumping (k = 100%). For all fiber lengths, the maximum output power is achieved for k = 95%. This behavior is not surprising because counterpumping maximizes the pump power available at the output of the fiber where the amplified signal power is the largest. We next observe that the maximum output power is achieved for a ratio of 95% counter-pumping to 5% co-pumping. This indicates that a small amount of co-pumping provides signal gain that offsets fiber absorption loss. Therefore full counter-pumping is not the most efficient way to pump this fiber.

For k = 50% in Figure 3, the relatively small variation in output signal power Pout with fiber length is consistent with the small variation in gain seen in Figure 2 as a function of fiber length for Ps = +2 dBm. We further note that as the fiber length is decreased from 9 m to 3 m, the output power Pout consistently increases. For very short fibers the difference between co-and counter-pumping will become negligible. However, as we will illustrate later, this comes at the expense of the amplifier operating bandwidth shifting from higher to shorter wavelengths. The amplifier performance illustrated in Figure 3 shows that we may consider three cases for design: k = 0%, k= 50%, and k = 95%. For k = 0%, Pout variation with fiber length is 18%. For k = 50%, it is as much as 10.9%, and for k = 95% it is about 2%. This indicates that a mostly counter-pumped amplifier will be less sensitive to changes in active fiber length than a co-pumped amplifier. Now let's consider the important design consideration of the dependence of saturated output power as a function of active fiber length L and pump power Pp. This behavior is illustrated in Figure 4 for a signal wavelength of 1952 nm, a coupling ratio k = 50%, and 1567 nm pump powers of 0.83 W, 1.7 W and 2.55 W, respectively.

In this plot we see that the maximum saturated output power is obtained for L = 2 m, relatively independent of fiber length and the pump power. It is apparent that above L = 2 m, Pout decreases slightly with increases in fiber length. This behavior is consistent with the simulation in Figure 2, and it illustrates that the optimum Pout for a saturated amplifier is not greatly dependent on L. The curves in Figure 4 lead to the important observation that the output power scales linearly with increases in the pump power. Therefore saturated output powers much higher than the 2.6 W already demonstrated [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF] can be achieved with this type of Thulium-doped fiber up to the stimulated Brillouin scattering (SBS) threshold which is estimated to be 10-20 W for a fiber length of 7 m [START_REF] Sincore | SBS Threshold Dependence on Pulse Duration in a 2053 nm Single-Mode Fiber Amplifier[END_REF].

So far our simulations have been carried out for the signal wavelength of 1952 nm. To more fully study performance of the amplifier, we now look at the amplifier slope efficiency η vs. signal wavelength λs and active fiber length L. Slope efficiency η = ΔPsat/ΔPp is defined as: the ratio of the change in saturated output signal power Psat to a change in pump power, for a given fiber length L and signal wavelength λs. It measures the efficiency of conversion of pump light into signal light and is an important figure of merit for the amplifier. The saturated output power Psat in our experiments and simulations is measured for a high input signal power Ps = +2 dBm.

Figure 5 shows simulations of η over the wavelength region of 1900 nm to 2050 nm, for fiber lengths ranging from 1.5 to 9 meters. Clearly the bandwidth of the amplifier shifts toward longer signal wavelengths for the longer fiber such as 9 meters. Shorter fibers such as 1.5 and 2 meters shift the operating bandwidth region toward shorter wavelengths.

Figure 5 indicates that for short fibers of 1.5 and 2 meters, η is optimum below 1950 nm, then diminishes rapidly with wavelength around 2000 nm and is negligible above 2020 nm. For longer fibers of 5 to 9 meters, η decreases more gradually with increasing wavelength and allows for a modest efficiency (i.e. 35%) up to 2050 nm. The simulated slope efficiencies in Figure 5 give a value at 1952 nm of 73% which is fully consistent with the value of 73% determined from Figure 4. Based on the simulation results, for this single stage configuration we can draw four conclusions. First, the most significant gain occurs in the first couple of meters of the active fiber. Second, the saturated output power scales with pump power and is not significantly affected by the fiber length. Third, the optimum coupling ratio k for a combination of large dynamic range and saturated output power is achieved for medium fiber lengths of 6-8 meters and a k value around 50%. Fourth, the choice of the fiber length affects the operating bandwidth of the TDFA and the slope efficiency η: shorter lengths yield shorter operating wavelengths, while longer lengths give longer wavelength operating regions. This last point will be discussed further in Section 4.

Comparison of Simulation and Experiment

We now turn to comparisons of simulation and experiment for the single stage amplifier of Figure 1. In all these comparisons, the experimental fiber length is 7 meters.

We start by looking at the signal output power Pout as a function of 1952 nm signal input power Ps over a range of -30 dBm to + 2 dBm. Pump powers Pp at 1567 nm range from 0.89 W to 3.09 W and the coupling ratio k is 50%. As illustrated in Figure 6, the simulations (in solid lines) agree well with the experimental data (points) with an average difference between simulation and experiment of 0.6 dB.

For 0 dBm input power, the measured output powers are 1.11 W and 1.86 W for pump powers of 1.93 W and 3.09 W, respectively. This corresponds to optical power conversion efficiencies of 58% and 60%, respectively. Figure 8 shows the dependence of G and NF on coupling ratio k for Ps = -30 dBm at 1952 nm and Pp= 1.70 W at 1567 nm. Experimental data are shown in points and the simulations in solid lines. The optimum operating setpoint for small signal gain is different from the optimum for noise figure, with the largest small signal gain occurring for k = 50% and the lowest noise figures for k = 0%. The noise figure increases slowly at first with k, and then rapidly to 6.1 dB as k reaches 100% which corresponds to counter-pump only. The agreement between simulation and experiment is good, validating the performance of our simulator over the full range of k values. From this graph, we observe that a good balance between optimum gain and optimum noise figure is achieved for a coupling ratio of k = 50%. In Figure 9 we plot the dependence of G and NF on signal wavelength λs over the range of 1900 -2050 nm for a coupling ratio of k = 50%, Pp = 1.12 W, and input signal power Ps = -30 dBm. The highest measured unsaturated gain is achieved at 1910 nm , with a small decrease with λ at 1952 nm and then a steady decrease up to 2050 nm. For G > 30 dB, the amplifier bandwidth is >120 nm. By extending the investigation to values of λ lower than 1900 nm, we can expect even larger bandwidths.

The smallest measured NF of 3.5 dB is at 1952 nm, and NF variation with wavelength is small (i.e.< 1.4 dB).

We observe that the agreement between simulation and experiment is good, and this shows our simulation predicts well the small signal gain G and noise figure NF as a function of λs. We now investigate the slope efficiency η as a function of total pump power Pp for a saturating input power of Ps ≈ + 2 dBm, Figure 10 shows experimental and simulated values for saturated output power as a function of pump power, for four different values of λs across the transmission band with k = 50%. The agreement between simulation (solid lines) and experiment (points) is good, illustrating the accuracy of our simulator over a wide region of λs and over pump powers from 0.3 to 3.2 W. Notice that at λ = 2050 nm the number of data points is limited by the onset of lasing due to the large ASE produced as the pump power increases. The experimental variation in signal output power with pump power is linear in all cases as expected from theory. The maximum measured output power is 2.00 W for λ = 1910 nm and pump power of 3.09 W, corresponding to an optical power conversion efficiency of 65%. In Figure 11 we compare the slope efficiencies measured in Figure 10 with the simulation of Figure 5, with an expanded span for λ of 1760 nm -2060 nm. The experimental slope efficiencies (points) agree reasonably well with the theory and this demonstrates that our simulation is valid over a wide range of values for λs for a saturated amplifier. The maximum measured slope efficiency is 68.2% at 1910 nm. This can be compared with the simulated value at this signal wavelength of 76.0%. Using a slope efficiency of greater than 50% as a criterion, Figure 11 shows that the simulated operating bandwidth BW and center operating wavelength λc of the amplifier vary significantly with fiber length. For a short fiber (3 m) the operating bandwidth BW at 50% slope efficiency is 198 nm as indicated by the horizontal arrows in the figure, and λc is 1896 nm. For the longest fiber simulated (9 m) BW is reduced to 160 nm, and λc is shifted up in wavelength to 1940 nm. Results for all the fiber lengths Table 2. Operating Bandwidth BW and Center Wavelength λc as a Function of Fiber Length L.

studied are summarized in Table 2. It is evident that shorter fiber lengths give greater operating bandwidths and lower center wavelengths. This behavior is consistent with previously reported results [START_REF] Li | Exploiting the short wavelength gain of silicabased thulium-doped fiber amplifiers[END_REF]. We note that Figure 11 and Table 2 are the first detailed comparisons of TDFA simulation and theory, since previous work on spectral performance has been either wholly experimental [START_REF] Li | Diodepumped wideband thulium-doped fiber amplifiers for optical communications in the 1800-2050 nm window[END_REF][START_REF] Li | Exploiting the short wavelength gain of silicabased thulium-doped fiber amplifiers[END_REF] or theoretical [START_REF] Gorjan | Model of the amplified spontaneous emission generation in thulium-doped silica fibers[END_REF][START_REF] Khamis | Theoretical Model of a Thulium-doped Fiber Amplifier Pumped at 1570 nm and 793 nm in the Presence of Cross Relaxation[END_REF].

Comparison of Multistage Amplifier Performance

In Sections 3 and 4, we have shown that the shared pump topology can deliver high performance that is fully in agreement with simulation results. Here we will compare the shared pump amplifier with a two stage-three pump TDFA [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. A summary comparison of the two amplifiers is given below in Table 3. The table reveals that there is no major difference in performance between the two TDFAs. Comparing the maximum saturated output powers, we see that the shared pump TDFA achieves 1.9 W output for 3.2 W available pump, while the 2 stage amplifier achieves 2.6 W for 3.6 W of available pump. The output power performance of the two amplifiers is seen to be comparable when the maximum pump power available is accounted for. NF values for the two amplifiers are similar, as are the operating dynamic ranges (measured over an input power span of -30 dBm to +2 dBm). The two-stage amplifier has a slightly higher small signal gain with 56 dB compared to 51 dB for the shared pump single stage TDFA.

Fiber
The difference in slope efficiencies, with 66% for the 1 stage shared pump configuration and 82% for the 2 stage, 3 pump configuration, can be explained by referring to the architecture of the 3 pump configuration [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF].

Here we recall the definition of slope efficiency η from Section 3: η = ΔPsat/ΔPp. Remembering that Psat is the output power for a highly saturated amplifier, we observe that in the 2 stage TDFA the first stage boosts the input signal power of +2 dBm to an intermediate level of about +20 dBm which is then input to the second fiber stage. This boost in power to +20 dBm increases the conversion efficiency for available pump power in the second stage and so increases η. Indeed the two stage amplifier brings the measured efficiency closer to the simulated value as shown in Figure 11.

In comparing the amplifier bandwidth for the two configurations, we see that the 167 nm simulated bandwidth for the one stage, shared pump amplifier is consistent with the estimated value of >120 nm for the two stage, three pump amplifier. The >120 nm value was obtained by measuring the 10 dB width of the ASE noise background in the saturated output spectrum of the two stage TDFA. We believe that a simulation of the slope efficiency for the two stage amplifier (currently in progress) will result in a more precise value for its bandwidth.

The comparisons in Table 3 illustrate that our single stage shared pump TDFA can match the performance targets of a complex two stage three pump amplifier. The simplicity of the architecture of the shared pump TDFA is a considerable advantage in the design simulation of TDFAs for broadband telecommunications systems.

Discussion of Parameter Optimization for TDFA Architecture

The data reported in Figures 2 -11 illustrate several salient points about the operation of the shared pump TDFA.

From our experimental and theoretical studies, it is evident that input power levels, saturated output power targets, noise figure specifications, small signal gain specifications, and operating signal bandwidths all depend in an interrelated way on the amplifier architecture. Design of an optimized amplifier requires a careful balancing of all these performance targets as a function of fiber length L and coupling ratio k.

For gain amplifiers, Figure 2 shows that it is very important to consider the input signal power when choosing an optimum fiber length. For example, at the coupling ratio of 50%, the fiber gain for -30 dBm input is highest for a fiber length of 14 meters. For -15 dBm input, the optimum gain occurs for lengths of 7-8 meters. Clearly the design specifications of the TDFA must be carefully considered when choosing an optimum fiber length for a preamplifier designed to operate at low signal input powers. For these low input powers the NF value remains close to the quantum limit of 3 dB.

For power amplifiers, maximum simulated output power occurs for a coupling ratio of k = 95% and an optimized fiber length of about 3.5 meters for a signal wavelength of 1952 nm. This optimized fiber length agrees well with the values obtained in Figures 2 and4 where the optimum length for maximum output power is between 3 and 4 meters for a pump coupling ratio of 50 %. We conclude that for maximizing output power at 1952 nm, coupling ratios anywhere between 50 and 95 % can be employed. Figure 4 demonstrates that the saturated output power Pout scales linearly with pump power up to the maximum simulated Pp of 2.55W. No Brillouin scattering or other nonlinear effects were observed in our experiments. This means that we can improve the output power of the amplifier simply by increasing the pump power, up to the limit where nonlinear effects start to be observed. The threshold for nonlinear effects in our shared pump amplifier is currently under study. For the parameters in the current experiments, the one stage shared pump design yields an attractive power amplifier that is simple to build and has high signal output power For generic or multipurpose amplifiers, Figure 5 and 11 illustrate that the operating bandwidth BW and center wavelength λc of the amplifier are strongly dependent on the active fiber length, with maximum long wavelength response above 2000 nm occurring for fiber lengths L of 9 meters and longer. Short wavelength response is maximized for short fiber lengths of 1.5 and 2 meters. The desired operating bandwidth and center wavelength can therefore be selected by choosing an appropriate active fiber length. The noise figure NF as shown in Figure 9 is slowly varying with signal wavelength λs for a coupling ration of k = 50 %, indicating that the noise performance of the multipurpose amplifier is highly tolerant of variations in signal wavelength λs. This is an attractive feature for the many applications of this type of TDFA.

Figure 1 .

 1 Figure 1. Optical Design of Single Stage Single Pump TDFA with a Shared Pump Arrangement.

Figure 2 .

 2 Figure 2. Signal Gain (G) as a Function of Fiber Length (L) for Four Different Levels of Ps.

Figure 3 .

 3 Figure 3. Simulated Output Signal Power (Pout) as a Function of Fiber Length (L) and Pump Coupling Ratio (k)

Figure 4 .

 4 Figure 4. Simulated Output Power Pout as a Function of Fiber Length L and Pump Power Pp for k = 50%

Figure 5 .

 5 Figure 5. Simulated Slope Efficiencies η vs. Signal Wavelength λs and Active Fiber Length L

Figure 6 .

 6 Figure 6. Output Signal Power Pout vs. Input Signal power Ps for k=50%, for Three Different Total Pump Powers Pp.

Figure 7 .

 7 Figure 7. Gain G and Noise Figure NF at 1952 nm as a Function of Input Signal Power Ps.

Figure 8 .

 8 Figure 8. Gain and Noise Figure as a Function of Coupling Ratio k.

Figure 9 .

 9 Figure NF, dB

Figure 10 .Figure 11 .

 1011 Figure 10. Saturated Output power Pout vs. Total Pump Power Pp as a Function of λs

Table 1 .

 1 Table 1 contrasts the measured and simulated values of slope efficiency η as a function of signal wavelength λs for a fiber length of 7 m. Comparison of Simulated and Measured Slope Efficiency η as a Function of λs .

		η, %
	λ, nm	Exp.	Sim.
	1910	68.2	76.0
	1952	65.9	72.9
	2004	52.1	55.0
	2050	13.5	9.6

Table 3 .

 3 Comparison of Single Stage, Shared Pump TDFA with Two Stage, Three Pump TDFA

	Length L = 7 m			TDFA Configurations, 1952 nm
	Parameter	Symbol Units	1 Stage, Shared Pump	2 Stage, 3 Pumps
	Pump Power (1567 nm)	Pp	W	3.2	3.6
	Saturated Output Power	Pout	W	1.9	2.6
	Small Signal Noise Figure	NF	dB	3.4	3.2
	Signal Dynamic Range	Pin	dB	32	32
	Small Signal Gain	G	dB	51	56
	Slope Efficiency (Saturated)	η	%	65.9	82
	Operating Bandwidth	BW	nm	167 (simulated)	> 120 (est. from ASE)

Summary

We have reported the experimental and simulated performance of a single stage TDFA with a shared in-band pump at 1567 nm. In particular we considered the dependence of amplifier performance on pump coupling ratio and signal wavelength. We determined that the optimum fiber length L and optimum coupling ratio k depend strongly on the design performance specifications for the TDFA such as signal wavelength band, saturated output power, noise figure, small signal gain, and dynamic range. Our simulations show that the operating bandwidth of the amplifier can be as high as 198 nm. Due to the broad Thulium emission bandwidth, this amplifier configuration can be tailored to meet a variety of performance needs. We achieved saturated output powers of 2 W, small signal gains as high as 51 dB, noise figures as low as 3.5 dB, and a dynamic range of 32 dB for a noise figure of less than 4.7 dB. In all cases we found good agreement between our simulation tool and the experiments. No Brillouin scattering or other nonlinear effects were observed in any of our measurements. Our experiments and simulations show that the shared pump TDFA can match the performance of more complex multistage, multi-pump TDFAs, and illustrate the simplicity and usefulness of our design. This opens the possibility for new and efficient TDFAs for lightwave transmission systems as preamplifiers, as in-line amplifiers, and as power booster amplifiers.

Acknowledgments

We gratefully acknowledge Eblana Photonics for the single frequency distributed mode 2 µm laser sources, and OFS for the single clad Tm-doped fiber.


Robert E Tench
email: robert.tench@cybel-llc.com

Clément Romano

Jean-Marc Delavaux

Optimized Design and Performance of a Shared Pump Single Clad 2 µm TDFA

Keywords: Fiber Amplifier, Thulium, 2000 nm, Silica Fiber, Single Clad

We report the design, experimental performance, and simulation of a single stage, co-and counter-pumped Tmdoped fiber amplifier (TDFA) in the 2 μm signal wavelength band with an optimized 1567 nm shared pump source. We investigate the dependence of output power, gain, and efficiency on pump coupling ratio and signal wavelength. Small signal gains of >50 dB, an output power of 2 W, and small signal noise figures of <3.5 dB are demonstrated. Simulations of TDFA performance agree well with the experimental data. We also discuss performance tradeoffs with respect to amplifier topology for this simple and efficient TDFA.

Introduction

Simplicity and optimization of design are critical for the practical realization of wide bandwidth, high power single clad Thulium-doped fiber amplifiers (TDFAs) for 2 µm telecommunications applications. Recent TDFAs [START_REF] Romano | Simulation and design of a multistage 10 W thulium-doped double clad silica fiber amplifier at 2050 nm[END_REF][START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF] have reported 2 µm band amplifiers with output power > 2W, gain > 55 dB, noise figure < 4 dB, and optical bandwidth greater than 120 nm. While these designs achieve high optical performance, they employ two or more optical stages and multiple pump sources. Therefore, it is desirable to investigate designs using one amplifier stage and one pump source. In this paper we report on the design, simulation, and experimental performance of a one-stage single clad TDFA using an L-band (1567 nm) shared fiber laser pump source, as a function of pump coupling ratio, active fiber length, pump power, and signal wavelength. Our one-stage TDFA data compare well with recently reported performance of multi-stage, multi-pump amplifiers [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. In addition, the simplicity of the single clad pump shared design and its potential for cost reduction offer a broad selection of performance for different applications.

The paper is organized as follows: Section 2 presents our experimental setup, a single stage TDFA with variable coupling in the pump ratio between co-pumping and counter-pumping the active fiber. Section 3 covers the dependence of simulated amplifier performance on active fiber length, pump coupling ratio, slope efficiency, and signal wavelength. Section 4 compares measurement and simulation of the TDFA performance. Section 5 contrasts our simple TDFA design with performance of a two-stage, three-pump amplifier as reported previously in [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. Finally, Section 6 discusses design parameter tradeoffs for different TDFA applications.

Experimental Setup for Shared Pump Amplifier

The optical design of our one-stage single pump TDFA is shown in Figure 1. A single frequency 2 µm DML source (Eblana Photonics) is coupled through attenuator A and into the active fiber F1. Pump light from a multiwatt fiber laser P1 at 1567 nm is split by coupler C1 with a variable coupling ratio k (%).The two pump signals WDM = Wavelength Division Multiplexer. co-pump and counter-pump fiber F1, with k = 100% and k = 0% corresponding to all counter-pumping and all co-pumping, respectively. The value of k was changed in the simulations and experiments to optimize amplifier performance. Isolators I1 and I2 ensure unidirectional operation and suppress spurious lasing. Input and output signal powers, and co-pump and counter-pump powers, are respectively referenced to the input and output of Tm-doped fiber F1 (7 meters of OFS TmDF200). The input and output spectra of the TDFA were measured with an optical spectrum analyzer (Yokogawa AQ6375B).

Simulated Amplifier Performance

We begin the design of a high performance optical amplifier by studying the critically important variations of fiber signal gain (G) and output power (Pout) as a function of active fiber length (L) and input signal power (Ps).

The signal gain G is given by the following simple equation:

G(λs) = Pout (λs) / Ps (λs) ( 1
)
where λs is the signal wavelength, and Ps and Pout are signal powers measured at the input and output of the active Tm-doped fiber, respectively.

To study amplifier design we turn to the simulated TDFA performance [START_REF] Romano | Characterization of the 3F4 -3H6 Transition in Thulium-doped Silica Fibres and Simulation of a 2µm Single Clad Amplifier[END_REF][START_REF] Jackson | Theoretical modeling of Tm-doped silica fiber lasers[END_REF] shown in Figure 2, where G is plotted vs. L for four input signal power levels. Here the total 1567 nm pump power (co-+ counter-) (Pp) is 2.5 W, the signal wavelength λs is 1952 nm, and the coupling ratio k = 50%. We note that a similar set of gain curves can be generated for different wavelength bands of the TDFA, and this behavior will be investigated later in the section.

In Figure 2 we have measured the dependence of G vs. L for a 32 dB input dynamic range in signal power. The different Ps values illustrate the amplifier operating from a linear/unsaturated regime (Ps= -30 dBm) to a highly saturated regime (Ps = + 2 dBm). The equally important dependence of noise figure on these parameters will be dealt with later.

The first observation drawn from Figure 2 is that for low input signals (e.g. for Ps = -30 dBm) G is maximized for long fiber lengths of 12 meters or greater, while for saturating input powers (e.g. Ps = +2 dBm) G reaches a maximum value for lengths of about 2 meters. It is also clear that for small signal or unsaturated gain, most of the gain (i.e. more than 80%) is achieved in the first 5 meters of the fiber, while for saturated gain most of the gain occurs within the first 1.5 meters. The second observation is that saturated gain varies only slightly with active fiber length for values greater than 3 meters, indicating that a wide range of fiber lengths can be chosen for design of a power booster amplifier. However, later we will see that the choice of the fiber length affects the useful amplifier bandwidth. The next design study for the shared pump amplifier is to examine the dependence of the saturated output power on active fiber length L and coupling ratio k. To study this issue, we plot the output signal power with pump coupling ratio k for four active fiber lengths (i.e. L = 3, 5, 7 and 9 m) as shown in Figure 3. In this simulation Ps is set to +2 dBm at 1952 nm to saturate the amplifier, with the total pump set at Pp = 2.5 W at 1567 nm.

We first note that for a given fiber length, Pout increases linearly when moving from co-pumping (k = 0%) to nearly all counter-pumping (k = 95%) and then drops down for full counter-pumping (k = 100%). For all fiber lengths, the maximum output power is achieved for k = 95%. This behavior is not surprising because counterpumping maximizes the pump power available at the output of the fiber where the amplified signal power is the largest. We next observe that the maximum output power is achieved for a ratio of 95% counter-pumping to 5% co-pumping. For full counter-pumping, the pump is attenuated significantly within two meters after being launched, leaving the input end of the active fiber unpumped with no inversion achieved for the input Tm ion population. This indicates that a small amount of co-pumping provides signal gain that offsets fiber absorption loss. Therefore full counter-pumping is not the most efficient way to pump this amplifier.

For k = 50% in Figure 3, the relatively small variation in output signal power Pout with fiber length is consistent with the small variation in gain seen in Figure 2 as a function of fiber length for Ps = +2 dBm. We further note that as the fiber length is decreased from 9 m to 3 m, the output power Pout consistently increases. For very short fibers the difference between co-and counter-pumping will become negligible. However, as we will illustrate later, this comes at the expense of the amplifier operating bandwidth shifting from higher to shorter wavelengths. The amplifier performance illustrated in Figure 3 shows that we may consider three cases for design: k = 0%, k= 50%, and k = 95%. For k = 0%, Pout variation with fiber length is 18%. For k = 50%, it is as much as 10.9%, and for k = 95% it is about 2%. This indicates that a mostly counter-pumped amplifier will be less sensitive to changes in active fiber length than a co-pumped amplifier. Now let's consider the important design consideration of the dependence of saturated output power as a function of active fiber length L and pump power Pp. This behavior is illustrated in Figure 4 for a signal wavelength of 1952 nm, a coupling ratio k = 50%, and 1567 nm pump powers of 0.83 W, 1.7 W and 2.55 W, respectively.

In this plot we see that the maximum saturated output power is obtained for L = 2 m, relatively independent of fiber length and the pump power. It is apparent that above L = 2 m, Pout decreases slightly with increases in fiber length. This behavior is consistent with the simulation in Figure 2, and it illustrates that the optimum Pout for a saturated amplifier is not greatly dependent on L. The curves in Figure 4 lead to the important observation that the output power scales linearly with increases in the pump power. Therefore saturated output powers much higher than the 2.6 W already demonstrated [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF] can be achieved with this type of Thulium-doped fiber up to the stimulated Brillouin scattering (SBS) threshold which is estimated to be 10-20 W for a fiber length of 7 m [START_REF] Sincore | SBS Threshold Dependence on Pulse Duration in a 2053 nm Single-Mode Fiber Amplifier[END_REF].

So far our simulations have been carried out for the signal wavelength of 1952 nm. To more fully study performance of the amplifier, we now look at the amplifier slope efficiency η vs. signal wavelength λs and active fiber length L. Slope efficiency η = ΔPsat/ΔPp is defined as: the ratio of the change in saturated output signal power Psat to a change in pump power, for a given fiber length L and signal wavelength λs. It measures the efficiency of conversion of pump light into signal light and is an important figure of merit for the amplifier. The saturated output power Psat in our experiments and simulations is measured for a high input signal power Ps = +2 dBm.

Figure 5 shows simulations of η over the wavelength region of 1900 nm to 2050 nm, for fiber lengths ranging from 1.5 to 9 meters. Clearly the bandwidth of the amplifier shifts toward longer signal wavelengths for the longer fiber such as 9 meters. Shorter fibers such as 1.5 and 2 meters shift the operating bandwidth region toward shorter wavelengths.

Figure 5 indicates that for short fibers of 1.5 and 2 meters, η is optimum below 1950 nm, then diminishes rapidly with wavelength around 2000 nm and is negligible above 2020 nm. For longer fibers of 5 to 9 meters, η decreases more gradually with increasing wavelength and allows for a modest efficiency (i.e. 35%) up to 2050 nm. The simulated slope efficiencies in Figure 5 give a value at 1952 nm of 73% which is fully consistent with the value of 73% determined from Figure 4. Based on the simulation results, for this single stage configuration we can draw four conclusions. First, the most significant gain occurs in the first couple of meters of the active fiber. Second, the saturated output power scales with pump power and is not significantly affected by the fiber length. Third, the optimum coupling ratio k for a combination of large dynamic range and saturated output power is achieved for medium fiber lengths of 6-8 meters and a k value around 50%. Fourth, the choice of the fiber length affects the operating bandwidth of the TDFA and the slope efficiency η: shorter lengths yield shorter operating wavelengths, while longer lengths give longer wavelength operating regions. This last point will be discussed further in Section 4.

Comparison of Simulation and Experiment

We now turn to comparisons of simulation and experiment for the single stage amplifier of Figure 1. In all these comparisons, the experimental fiber length is 7 meters.

We start by looking at the signal output power Pout as a function of 1952 nm signal input power Ps over a range of -30 dBm to + 2 dBm. Pump powers Pp at 1567 nm range from 0.89 W to 3.09 W and the coupling ratio k is 50%. As illustrated in Figure 6, the simulations (in solid lines) agree well with the experimental data (points) with an average difference between simulation and experiment of 0.6 dB.

For 0 dBm input power, the measured output powers are 1.11 W and 1.86 W for pump powers of 1.93 W and 3.09 W, respectively. This corresponds to optical power conversion efficiencies of 58% and 60%, respectively.

In Equations ( 2) through (4), Δλ is the effective resolution bandwidth of the optical spectrum analyzer in m, and PASE is the measured internal forward spontaneous output power under the signal peak in Watts. h is Planck's constant, and c is the speed of light in vacuum. G(λ) is given by Equation (1). We measured the noise figure with a Δλ of 0.1 nm on the Yokogawa optical spectrum analyzer.

Using Equations ( 1) through ( 4), we now analyze the performance of the TDFA as shown in Figure 7. A maximum signal gain G (points) of 51 dB is measured at Ps = -30 dBm with an NF < 3.5 dB. Over the full range of input powers studied, the simulated gain values (solid lines) agree with the measured gain values to within 1 dB, validating the performance of our simulator over a wide range of input powers. Experimental values of noise figure are also plotted in points in Figure 7. The minimum measured noise figure is 3.5 dB, and the minimum simulated noise figure is 3.2 dB, close to the 3.0 dB quantum limit. Agreement between experiment and simulation for noise figure is good. The measured dynamic range for the amplifier is 32 dB for a noise figure of 4.7 dB or less.

Figure 8 shows the dependence of G and NF on coupling ratio k for Ps = -30 dBm at 1952 nm and Pp= 1.70 W at 1567 nm. Experimental data are shown in points and the simulations in solid lines. The optimum operating setpoint for small signal gain is different from the optimum for noise figure, with the largest small signal gain occurring for k = 50% and the lowest noise figures for k = 0%. The noise figure increases slowly at first with k, and then rapidly to 6.1 dB as k reaches 100% which corresponds to counter-pump only. The agreement between simulation and experiment is good, validating the performance of our simulator over the full range of k values. From this graph, we observe that a good balance between optimum gain and optimum noise figure is achieved for a coupling ratio of k = 50%. In Figure 9 we plot the dependence of G and NF on signal wavelength λs over the range of 1900 -2050 nm for a coupling ratio of k = 50%, Pp = 1.12 W, and input signal power Ps = -30 dBm. The highest measured unsaturated gain is achieved at 1910 nm , with a small decrease with λ at 1952 nm and then a steady decrease up to 2050 nm. For G > 30 dB, the amplifier bandwidth is >120 nm. By extending the investigation to values of λ lower than 1900 nm, we can expect even larger bandwidths.

The smallest measured NF of 3.5 dB is at 1952 nm, and NF variation with wavelength is small (i.e.< 1.4 dB).

We observe that the agreement between simulation and experiment is good, and this shows our simulation predicts well the small signal gain G and noise figure NF as a function of λs. We now investigate the slope efficiency η as a function of total pump power Pp for a saturating input power of Ps ≈ + 2 dBm, Figure 10 shows experimental and simulated values for saturated output power as a function of pump power, for four different values of λs across the transmission band with k = 50%. The agreement between simulation (solid lines) and experiment (points) is good, illustrating the accuracy of our simulator over a wide region of λs and over pump powers from 0.3 to 3.2 W. Notice that at λ = 2050 nm the number of data points is limited by the onset of lasing due to the large ASE produced as the pump power increases. The experimental variation in signal output power with pump power is linear in all cases as expected from theory. The maximum measured output power is 2.00 W for λ = 1910 nm and pump power of 3.09 W, corresponding to an optical power conversion efficiency of 65%. In Figure 11 we compare the slope efficiencies measured in Figure 10 with the simulation of Figure 5, with an expanded span for λ of 1760 nm -2060 nm. The experimental slope efficiencies (points) agree reasonably well with the theory and this demonstrates that our simulation is valid over a wide range of values for λs for a saturated amplifier. The maximum measured slope efficiency is 68.2% at 1910 nm. This can be compared with the simulated value at this signal wavelength of 76.0%. Using a slope efficiency of greater than 50% as a criterion, Figure 11 shows that the simulated operating bandwidth BW and center operating wavelength λc of the amplifier vary significantly with fiber length. For a short fiber (3 m) the operating bandwidth BW at 50% slope efficiency is 198 nm as indicated by the horizontal arrows in the figure, and λc is 1896 nm. For the longest fiber simulated (9 m) BW is reduced to 160 nm, and λc is shifted up in wavelength to 1940 nm. Results for all the fiber lengths studied are summarized in Table 2. It is evident that shorter fiber lengths give greater operating bandwidths and lower center wavelengths. This behavior is consistent with previously reported results [START_REF] Li | Exploiting the short wavelength gain of silicabased thulium-doped fiber amplifiers[END_REF]. We note that Figure 11 and Table 2 are the first detailed comparisons of TDFA simulation and theory, since previous work on spectral performance has been either wholly experimental [START_REF] Li | Diodepumped wideband thulium-doped fiber amplifiers for optical communications in the 1800-2050 nm window[END_REF][START_REF] Li | Exploiting the short wavelength gain of silicabased thulium-doped fiber amplifiers[END_REF] or theoretical [START_REF] Gorjan | Model of the amplified spontaneous emission generation in thulium-doped silica fibers[END_REF][START_REF] Khamis | Theoretical Model of a Thulium-doped Fiber Amplifier Pumped at 1570 nm and 793 nm in the Presence of Cross Relaxation[END_REF].

Comparison of Multistage Amplifier Performance

In Sections 3 and 4, we have shown that the shared pump topology can deliver high performance that is fully in agreement with simulation results. Here we will compare the shared pump amplifier with a two stage-three pump TDFA [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF]. A summary comparison of the two amplifiers is given below in Table 3. The table reveals that there is no major difference in performance between the two TDFAs. Comparing the maximum saturated output powers, we see that the shared pump TDFA achieves 1.9 W output for 3.2 W available pump, while the 2 stage amplifier achieves 2.6 W for 3.6 W of available pump. The output power performance of the two amplifiers is seen to be comparable when the maximum pump power available is accounted for. NF values for the two amplifiers are similar, as are the operating dynamic ranges (measured over an input power span of -30 dBm to +2 dBm). The two-stage amplifier has a slightly higher small signal gain with 56 dB compared to 51 dB for the shared pump single stage TDFA.

Fiber

The difference in slope efficiencies, with 66% for the 1 stage shared pump configuration and 82% for the 2 stage, 3 pump configuration, can be explained by referring to the architecture of the 3 pump configuration [START_REF] Tench | Broadband 2 W Output Power Tandem Thuliumdoped Single Clad Fibre Amplifier for Optical Transmission at 2µm[END_REF].

Here we recall the definition of slope efficiency η from Section 3: η = ΔPsat/ΔPp. Remembering that Psat is the output power for a highly saturated amplifier, we observe that in the 2 stage TDFA the first stage boosts the input signal power of +2 dBm to an intermediate level of about +20 dBm which is then input to the second fiber stage. This boost in power to +20 dBm increases the conversion efficiency for available pump power in the second stage and so increases η. Indeed the two stage amplifier brings the measured efficiency closer to the simulated value as shown in Figure 11.

In comparing the amplifier bandwidth for the two configurations, we see that the 167 nm simulated bandwidth for the one stage, shared pump amplifier is consistent with the estimated value of >120 nm for the two stage, three pump amplifier. The >120 nm value was obtained by measuring the 10 dB width of the ASE noise background in the saturated output spectrum of the two stage TDFA. We believe that a simulation of the slope efficiency for the two stage amplifier (currently in progress) will result in a more precise value for its bandwidth.

The comparisons in Table 3 illustrate that our single stage shared pump TDFA can match the performance targets of a complex two stage three pump amplifier. The simplicity of the architecture of the shared pump TDFA is a considerable advantage in the design simulation of TDFAs for broadband telecommunications systems.

Discussion of Parameter Optimization for TDFA Architecture

The data reported in Figures 2 -11 illustrate several salient points about the operation of the shared pump TDFA.

From our experimental and theoretical studies, it is evident that input power levels, saturated output power targets, noise figure specifications, small signal gain specifications, and operating signal bandwidths all depend in an interrelated way on the amplifier architecture. Design of an optimized amplifier requires a careful balancing of all these performance targets as a function of fiber length L and coupling ratio k.

For gain amplifiers, Figure 2 shows that it is very important to consider the input signal power when choosing an optimum fiber length. For example, at the coupling ratio of 50%, the fiber gain for -30 dBm input is highest for a fiber length of 14 meters. For -15 dBm input, the optimum gain occurs for lengths of 7-8 meters. Clearly the design specifications of the TDFA must be carefully considered when choosing an optimum fiber length for a preamplifier designed to operate at low signal input powers. For these low input powers the NF value remains close to the quantum limit of 3 dB.

For power amplifiers, maximum simulated output power occurs for a coupling ratio of k = 95% and an optimized fiber length of about 3.5 meters for a signal wavelength of 1952 nm. This optimized fiber length agrees well with the values obtained in Figures 2 and4 where the optimum length for maximum output power is between 3 and 4 meters for a pump coupling ratio of 50 %. We conclude that for maximizing output power at 1952 nm, coupling ratios anywhere between 50 and 95 % can be employed.

Figure 4 demonstrates that the saturated output power Pout scales linearly with pump power up to the maximum simulated Pp of 2.55W. No Brillouin scattering or other nonlinear effects were observed in our experiments. This means that we can improve the output power of the amplifier simply by increasing the pump power, up to the limit where nonlinear effects start to be observed. The threshold for nonlinear effects in our shared pump amplifier is currently under study. For the parameters in the current experiments, the one stage shared pump design yields an attractive power amplifier that is simple to build and has high signal output power For generic or multipurpose amplifiers, Figure 5 and 11 illustrate that the operating bandwidth BW and center wavelength λc of the amplifier are strongly dependent on the active fiber length, with maximum long wavelength response above 2000 nm occurring for fiber lengths L of 9 meters and longer. Short wavelength response is maximized for short fiber lengths of 1.5 and 2 meters. The desired operating bandwidth and center wavelength can therefore be selected by choosing an appropriate active fiber length. The noise figure NF as shown in Figure 9 is slowly varying with signal wavelength λs for a coupling ratio of k = 50 %, indicating that the noise performance of the multipurpose amplifier is highly tolerant of variations in signal wavelength λs. This is an attractive feature for the many applications of this type of TDFA.

To conclude, we have shown that an active fiber length L of 7 meters and a coupling ratio k = 50 % provide balanced performance over a wide range of operating parameters for the one stage, shared pump TDFA.

Figure 1 .

 1 Figure 1. Optical Design of Single Stage Single Pump TDFA with a Shared Pump Arrangement.

Figure 2 .

 2 Figure 2. Signal Gain (G) as a Function of Fiber Length (L) for Four Different Levels of Ps.

Figure 3 .

 3 Figure 3. Simulated Output Signal Power (Pout) as a Function of Fiber Length (L) and Pump Coupling Ratio (k)

Figure 4 .

 4 Figure 4. Simulated Output Power Pout as a Function of Fiber Length L and Pump Power Pp for k = 50%

Figure 5 .

 5 Figure 5. Simulated Slope Efficiencies η vs. Signal Wavelength λs and Active Fiber Length L

Figure 6 .

 6 Figure 6. Output Signal Power Pout vs. Input Signal power Ps for k=50%, for Three Different Total Pump Powers Pp.

Figure 7 .

 7 Figure 7. Gain G and Noise Figure NF at 1952 nm as a Function of Input Signal Power Ps.

Figure 8 .

 8 Figure 8. Gain and Noise Figure as a Function of Coupling Ratio k.

  Figure NF, dB

Figure 9 .

 9 Figure 9 . Small Signal Gain G and Noise Figure NF as a Function of λs

Figure 10 .

 10 Figure NF, dB

Table 1 .

 1 Table 1 contrasts the measured and simulated values of slope efficiency η as a function of signal wavelength λs for a fiber length of 7 m. Comparison of Simulated and Measured Slope Efficiency η as a Function of λs .

		η, %
	λ, nm	Exp.	Sim.
	1910	68.2	76.0
	1952	65.9	72.9
	2004	52.1	55.0
	2050	13.5	9.6

Table 2 .

 2 Operating Bandwidth BW and Center Wavelength λc as a Function of Fiber Length L.

	L, m	BW, nm	λc, nm
	3	198	1896
	5	182	1918
	7	167	1932
	9	160	1940

Table 3 .

 3 Comparison of Single Stage, Shared Pump TDFA with Two Stage, Three Pump TDFA

	Length L = 7 m			TDFA Configurations, 1952 nm
	Parameter	Symbol Units	1 Stage, Shared Pump	2 Stage, 3 Pumps
	Pump Power (1567 nm)	Pp	W	3.2	3.6
	Saturated Output Power	Pout	W	1.9	2.6
	Small Signal Noise Figure	NF	dB	3.4	3.2
	Signal Dynamic Range	Pin	dB	32	32
	Small Signal Gain	G	dB	51	56
	Slope Efficiency (Saturated)	η	%	65.9	82
	Operating Bandwidth	BW	nm	167 (simulated)	> 120 (est. from ASE)

Summary

We have reported the experimental and simulated performance of a single stage TDFA with a shared in-band pump at 1567 nm. In particular we considered the dependence of amplifier performance on pump coupling ratio and signal wavelength. We determined that the optimum fiber length L and optimum coupling ratio k depend strongly on the design performance specifications for the TDFA such as signal wavelength band, saturated output power, noise figure, small signal gain, and dynamic range. Our simulations show that the operating bandwidth of the amplifier can be as high as 198 nm. Due to the broad Thulium emission bandwidth, this amplifier configuration can be tailored to meet a variety of performance needs. We achieved saturated output powers of 2 W, small signal gains as high as 51 dB, noise figures as low as 3.5 dB, and a dynamic range of 32 dB for a noise figure of less than 4.7 dB. In all cases we found good agreement between our simulation tool and the experiments. No Brillouin scattering or other nonlinear effects were observed in any of our measurements. Our experiments and simulations show that the shared pump TDFA can match the performance of more complex multistage, multi-pump TDFAs, and illustrate the simplicity and usefulness of our design. This opens the possibility for new and efficient TDFAs for lightwave transmission systems as preamplifiers, as in-line amplifiers, and as power booster amplifiers.

Acknowledgments

We gratefully acknowledge Eblana Photonics for the single frequency distributed mode 2 µm laser sources, and OFS for the single clad Tm-doped fiber.