Ténan Yeo
email: yeo.tenan@yahoo.fr

DYNAMICAL SYSTEMS SERIES B STOCHASTIC AND DETERMINISTIC SIS PATCH MODEL

Keywords: 2020 Mathematics Subject Classification. Primary: 60J75, 34D23; Secondary: 60J20 epidemic patch model, law of large numbers, endemic equilibrium

Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. We next study the equilibria of the deterministic model. Our main contribution is a stability result of the endemic equilibrium in the case R 0 > 1. Finally we compare the equilibria with those of the homogeneous model, and with those of isolated patches.

1. Introduction. Early epidemic models were formulated assuming that individuals in the population mix homogeneously [START_REF] Andersson | Stochastic Epidemic Models and their Statistical Analysis[END_REF][START_REF] Bailey | The Mathematical Theory of Infectious Diseases and its Applications[END_REF][START_REF] Britton | Stochastic epidemic models with inference[END_REF][START_REF] Kermack | A contribution to the mathematical theory of epidemics[END_REF][START_REF] Weiss | On the asymptotic behavior of the stochastic and deterministic models of an epidemic[END_REF]. In this consideration, all pairs of individuals in the population have the same probability of coming into contact with each other. But, it is well known that in a large population several groups can be formed due to heterogeneity arising, for example, from social and economic factors. Some people may live in cities while others may live in rural areas. Consequently, demographic and disease parameters may vary for each group, and then the persistence and extinction of infectious diseases in those communities can be different. Furthermore, people may travel between the groups, which leads to the spread of the disease between groups. Then it is clear that spatial heterogeneity, habitat connectivity and rates of movement of individuals play an important role in the outbreak of an infectious disease. Those considerations motivate the development of multipatch epidemic models.

In this paper, we consider a deterministic multipatch SIS model, similar to the one studied by Allen et al. [START_REF] Allen | Asymptotic profiles of the steady states for an SIS epidemic patch model[END_REF]. We first show that it is the law of large numbers limit, as the size N of the population tends to infinity, of a continuous time finite population stochastic model. We next study the stability of the equilibria. Our main result is a stability result of the endemic equilibrium in the case R 0 > 1. We also compare the equilibria with those of the homogeneous model, and with those of isolated patches.

Several authors have exploited Lyapunov functions in order to study the stability of the endemic equilibrium in various types of models, see e.g. [START_REF] Fall | Epidemiological models and Lyapunov functions[END_REF][START_REF] Rebaza | Global stability of a multipatch disease epidemics model[END_REF][START_REF] Michael | Global stability of an epidemic model in a patchy environment[END_REF][START_REF] Zhisheng | Global stability of infectious disease models using Lyapunov functions[END_REF]. The same method does not seem to be applicable in our case.

The paper is organised as follows. In section 2, we introduce both deterministic and stochastic model on a finite number of patches. Section 3 is devoted to the law of large numbers. In section 4, we prove that the endemic equilibrium (EE) of the deterministic model is gloably asymptotically stable. Finally we compare this endemic equilibrium with the one of the homogeneous model and with the one when patches are isolated, in section 5.

• infectious individuals

• susceptible individuals Figure 1: Metapopulation 2. The models.

2.1.

The deterministic model. Consider a population consisting of N individuals, where each individual is located at one of j geographically distinct patches. Sites (or vertices) represent human communities in which the disease can diffuse and grow. The edges represent links between communities (see Figure 1 above). Individuals in that population can be classified according to their ability to transmit the disease to others. Susceptible individuals are those who do not have the disease and who can become infected. Infectious individuals are those who are infected by the disease and can transmit it to susceptible individuals. In this work, attention is given to the SIS model, but the same approach can be developed in the case of the SIRS model and of the SIR model with demography. For any patch j, the transmission of the disease depends on three factors: the rate of contacts, the probability that a contact is made with a susceptible individual, and the probability that a contact between an infectious and a susceptible individual leads to a successful transmission (see e.g [START_REF] Begon | A clarification of transmission terms in host-microparasite models: Numbers, densities and areas[END_REF][START_REF] Britton | Stochastic epidemic models with inference[END_REF]). S j (t) (resp. I j (t) ) denotes the number of susceptible (resp. infectious) individuals in patch j at time t. Here, we assume that • infections are local;

• when an infectious individual cures, he immediately becomes susceptible again;

• each infectious individual meets other individuals at some rate α j . The encounter results in a new infection with probability p j if the patner of the encounter is susceptible, which happens with probability S j (t) S j (t) + I j (t) , since we assume that individuals in each patch mix homogeneously. Letting λ j = α j p j and summing over the infectious individuals at time t gives the rate λ j S j (t) S j (t) + I j (t) I j (t) at time t;

• each infectious on site j recovers at rate γ j , so the total recovery rate at time t is γ j I j (t); • each susceptible migrates from j to k at rate ν S a jk , and similarly for the compartment I.

ν S and ν I are the diffusion coefficients for susceptible and infectious individuals, respectively. a ij represents the degree of movement from patch i into patch j.

Disease spreads between individuals in the same patch by contact between infectious and susceptible individuals, and spread from one patch to another by migration of infectious individuals. Hence the system of ordinary differential equations (ODEs) for the evolution of the number of susceptible and infectious individuals at each patch j is as follows

                           d S j dt (t) = - λ j S j (t)I j (t) S j (t) + I j (t) + γ j I j (t) + ν S k=1 k =j a kj S k (t) -a jk S j (t) d I j dt (t) = λ j S j (t)I j (t) S j (t) + I j (t) -γ j I j (t) + ν I k=1 k =j a kj I k (t) -a jk I j (t)
S j (0) ≥ 0 and I j (0) ≥ 0

j = 1, • • • , . (1)
By an obvious homogeneity property of the system of ODEs (1), both the numbers of individuals in each compartment and in each patch, and the proportions of individuals in each compartment and in each patch obey the system (1). In the sequel, we shall use the notation

z(t) = S 1 (t), I 1 (t), S 2 (t), I 2 (t), • • • , S (t), I (t)
T to denote the proportions.

2.2. The stochastic model. The deterministic models are based on the hypothesis of a population of large size. When it is not the case, the interactions between the individuals are not uniform but possess an intrinsic random character. We are going to expose now a probabilistic version of the previous model. For each given patch, Poisson processes count the number of new infections, removal and migrations between the patches during the time. So the propagation of the illness can be modeled by the following system of stochastic equations

                                           S j (t) = S j (0) -P inf j t 0 λ j S j (r)I j (r) S j (r) + I j (r) dr + P rec j t 0 γ j I j (r)dr - k=1 k =j P mig S,j,k t 0 ν S a jk S j (r)dr + k=1 k =j P mig S,k,j t 0 ν S a kj S k (r)dr I j (t) = I j (0) + P inf j t 0 λ j S j (r)I j (r) S j (r) + I j (r) dr -P rec j t 0 γ j I j (r)dr - k=1 k =j P mig I,j,k t 0 ν I a jk I j (r)dr + k=1 k =j P mig I,k,j t 0 ν I a kj I k (r)dr t ∈ [0, T ], j = 1, • • • , . (2)
where all the P j 's are mutually independent standard Poisson processes. Here infections, healings and migrations of individuals happen according to Poisson processes whose rates are given by the respective integrands. In system (2)

• P inf j t 0 λ j S j (r)I j (r) S j (r) + I j (r)
dr counts the number of transitions of type S -→ I on the patch j between time 0 and time t;

• recovery of an infectious happens at rate γ j , so P rec j t 0 γ j I j (r)dr counts the number of transitions of type I -→ S on the patch j between time 0 and time t.

• The term P mig S,j,k t 0 ν S a jk S j (r)dr counts the number of migrations of susceptible individuals from patch j to k, if we assume that each susceptible migrates from j to k at rate ν S a jk , and similarly for the compartment I.

Let mention that such Markov process has been used by Ethier & Kurtz [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF] to model both the logistic growth of a population, epidemics and chemical reactions.

In the next section we show that this stochastic model converges to the deterministic patch model as the total size of population tends to infinity.

3. Law of large numbers. We introduce the martingales M j (t) = P j (t) -t and we look instead at the renormalized model by dividing the size of the population in each compartment by N. Hence, by setting

S N j (t) = S j (t) N , I N j (t) = I j (t) N , S N (t) =    S N 1 (t) . . . S N (t)   , I N (t) =    I N 1 (t) . . . I N (t)   , and
Z N (t) = S N (t) I N (t)
, then the stochastic model takes the aggregated form

Z N (t) = Z N (0) + t 0 b r, Z N (r) dr + k j=1 h j N M j N t 0 β j Z N (r) dr , ( 3
)
where k is the total number of P j 's in the system, and b r,

Z N (r) = k j=1 h j β j Z N (r) ; (4)
the vectors h j ∈ {-1, 0, 1} 2 denote the respective jump directions with jump rates β j . The rates

β . Z N (t) ∈ λ j S N j (t)I N j (t) S N j (t) + I N j (t) , γ j I N j (t), ν S a ij S N j (t), (5)
ν I a ij I N j (t), i, j ∈ {1, • • • , } .
Concerning the initial condition, we assume that Z N (0) = z N = [Nx]/N, for some x ∈ [0, 1] , where [Nx] is a vector of integers.

We shall say that a vector u is nonnegative (resp. positive) if all its elements are nonnegative (resp. positive), in with case we will write u ≥ 0 (resp. u > 0). Note that the system (1) can be written in the following form

dz dt (t) = b(t, z(t)), x := z(0) ≥ 0, where b t, z(t) =                         - λ1 S1(t)I1(t) S1(t) + I1(t) + γ1 I1(t) + νS k=1 k =j a k1 S k (t) -a 1k S1(t) λ1 S1(t)I1(t) S1(t) + I1(t) -γ1 I1(t) + νI k=1 k =j a k1 I k (t) -a 1k I1(t) . . . - λ S (t)I (t) S (t) + I (t) + γ I (t) + νS k=1 k =j a k S k (t) -a k S (t) λ S (t)S (t) S (t) + I (t) -γ I (t) + νI k=1 k =j a k I k (t) -a k I (t)                         .
We set Theorem 3.1 ensures that, as the population size N becomes large, the proportion of susceptible and infectious individuals at each patch is well approximated, on any bounded interval [0, T ] by the solution of the ODEs (1), provided the scaled process starts close to an initial value of the ODEs.

F N t = σ{ Z N j (r), 0 ≤ r ≤ t, j = 1, • • • , }
Theorem 3.1 is a special case of Theorem 2.2.7 of Britton & Pardoux [START_REF] Britton | Stochastic epidemic models with inference[END_REF], where the proof written for the homogeneous model covers our situation as well. One of the earliest references on this convergence result is Ethier & Kurtz [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF] (chapter 11, Theorem 2.1). Thus, we do not give details and refer the reader to those papers for a complete proof. We briefly sketch the idea of the proof. First note that 0 ≤ Z N (t) ≤ 1, for all t ∈ [0, T ]. By using the law of large numbers for Poisson processes and the second Dini Theorem, it follows that sup 0≤t≤T j=1

h j N M j N t 0 β j (Z N (r))dr a.s.
-→ 0.

Next, it is not hard to see that b(t, z) is a globally Lipchitz function of z, locally uniformly in t. From this fact, it follows that, for all t ∈ [0, T ],

Z N (t) -z(t) ≤ z N -x + j=1 h j N M j N t 0 β j (Z N (r))dr (6)
+ C t 0 Z N (r) -z(r) dr,
where C is the Lipschitz constant of b. Finally, the result follows from Gronwall's Lemma and the fact that the two first terms in the right-hand side of (6) tend to zero as N → ∞.

In the following section, we study the equilibria of the ODEs (1).

4. Equilibria of the ODEs and their stability. In this section, we consider properties of the disease free equilibrium (DFE) and the endemic equilibrium (EE), including its existence, uniqueness and stability. Throughout this section, we assume that the connectivity matrix A = a ij 1≤i≤ 1≤j≤ is irreducible and symmetric.

The irreducibility assumption implies that the patches cannot be separated into two disjoint subsets such that there is no migration of individuals from one subset to the other. That is, for any

j, k ∈ {1, • • • , }, j = k, there exists s ≥ 2, a sequence j 1 , j 2 , • • • , j s ∈ {1, • • • , } such that j 1 = j, j s = k and a jiji+1 = 0, ∀i ∈ {1, • • • , s -1}.
We shall say that a matrix M = m ij 1≤i≤ 1≤j≤ is nonnegative (resp. positive) if all its elements are nonnegative (resp. positive), in with case we will write M ≥ 0 (resp. M > 0). In what follows, N j denotes the total population size in patch j, i.e. N j = S j + I j . Let us mention that the system of ODEs (1) has been studied by Allen et al. [START_REF] Allen | Asymptotic profiles of the steady states for an SIS epidemic patch model[END_REF], where the authors studied the asymptotic profiles of the steady states. First, using the irreducibility of the connectivity matrix, they show Lemma 4.1 (Allen [START_REF] Allen | Asymptotic profiles of the steady states for an SIS epidemic patch model[END_REF] ). [Existence and uniqueness of the DFE] The system (1) has a unique disease-free equilibrium which is given by

ẑ := Ŝ1 , Î1 , Ŝ2 , Î2 , • • • , Ŝ , Î = 1 , 0, 1 , 0, • • • , 1 , 0 .
The DFE always exists, an important question is whether an outbreak of the disease can occur when the population initially contains a small number of infected individuals. This question may be addressed using stability analysis of the DFE. In fact if R 0 , the basic reproduction number (the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual) is less than 1, the DFE is globally asymptotically stable. That is, the trajectory of the ODEs which starts close to the DFE will be attracted towards the DFE. That is the content of the next lemma. Before, we use the next-generation matrix approach of Van den Driessche & Watmough [START_REF] Driessche | Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF] to compute R 0 . Define the matrices

A = diag(γ j ) 1≤j≤ , B = diag(λ j ) 1≤j≤ and D = (d ij ) 1≤i, j≤ with d ij =        - k=1 k =i a ik if i = j, a ij if i = j.
A direct application of the result of the above reference yields the following Proposition.

Proposition 1. The basic reproduction number for the system (1) is the spectral radius of the next-generation matrix:

R 0 = ρ(-BV -1 ), where V = ν I D -A.
We have also the Lemma 4.2 (Allen et al. [START_REF] Allen | Asymptotic profiles of the steady states for an SIS epidemic patch model[END_REF] ). [Stability of the DFE] If R 0 < 1, then the disease-free equilibrium ẑ is globally asymptotically stable, that is z(t) -→ ẑ, as t → ∞.

We will show at the end of this section that, under a specific condition, the disease-free equilibrium is globally asymptotically stable if R 0 = 1. The existence and uniqueness of the EE for the system (1) is shown in Allen et al. [START_REF] Allen | Asymptotic profiles of the steady states for an SIS epidemic patch model[END_REF] under the assumption that R 0 > 1. In that work, the authors were not able to prove the stability of the EE, but conjectured that this EE attracts all solutions whose initial conditions have a nonzero proportion of infectious (and numerical simulations suggest that this is indeed the case). Here, we employ the approach in Bichara et al. [START_REF] Bichara | SIS and SIR epidemic models under virtual dispersal[END_REF] to prove the globally stability of the EE, under the assumption that infectious and susceptible individuals have the same diffusion rate ν S = ν I := ν. Assuming that ν S = ν I := ν, then the sytem given by ( 1) is equivalent to

                       d N j (t) dt = ν k=1 k =j a kj N k (t) -a jk N j (t) d I j (t) dt = λ j 1 - I j (t) N j (t) I j (t) -γ j I j (t) + ν k=1 k =j a kj I k (t) -a jk I j (t) j = 1, • • • , , (7)
which can be written in the form

     d N(t) dt = νDN(t) d I(t) dt = νD I(t) -A I(t) + I -diag(N -1 j (t))diag(I(t)) B I(t), (8)
where

N = (N 1 , • • • , N ) T , I = (I 1 , • • • , I )
T and I is the identity matrix with dimension × . Note that I -diag(N -1 j (t))diag(I(t)) B I(t) is the vector of new infections, νDI(t) is the vector of migrations of infectious individuals and AI(t) is the vector of transitions of individuals from the compartment I to the compartment S. Proof. Let Q be the matrix such that q ij = a ji for i = j and q jj = -k=1 a kj .

Hence the system d N(t) dt = νDN(t) is equivalent to d N T (t) dt = νN T (t)Q.
Notice that Q is the infinitesimal generator of an irreducible Markov process with state space n , n = 0, 1, • • • , . By using Theorem 5.1 of Pardoux [START_REF] Pardoux | Markov Processes and Applications: Algorithms, Networks, Genome and Finance[END_REF], it follows that there exists a unique strictly positive equilibrium N * which solves the equation (N * ) T Q = 0. Since the state space is finite, the Markov process associed to the infinitesimal generator Q is reccurent, and then the global asymptotic stability of N * is garanteed by the Theorem 6.5 of the above reference.

Next we treat the existence and stablity of the endemic equilibrium for the system given by the system [START_REF] Driessche | Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]. Notice that the system ( 8) is of triangular form, and hence the theory of asymptotically autonomous systems for triangular systems (Vidyasagar [START_REF] Vidyasagar | Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability[END_REF] ) guarantees that the asymptotic stability of its equilibrium is equivalent to that of the system

d I(t) dt = νD I(t) -A I(t) + I -diag(1/N * j )diag(I(t)) B I(t),
where (N * 1 , • • • , N * ) is the unique asymptotically stable equilibrium defined in Lemma 4.3. The Jacobian matrices of I -diag(N -1 j (t))diag(I(t)) B I(t) and νD I(t) -A I(t), respectively, at the disease free equilibrium are B and V. For the convenience of the reader, we recall the following result. Theorem 4.4 (Vidyasagar [START_REF] Vidyasagar | Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability[END_REF], Theorem 3.1 and 3.4).

Let f and g be two functions of class C 1 . Consider the following system

         ẋ = f (x) ẏ = g(x, y) x ∈ R n , y ∈ R m
with an equilibrium point (x * , y * ), i.e. , f (x * ) = 0 and g(x * , y * ) = 0.

If x * is globally asymptotically stable (GAS) in R n for the system ẋ = f (x), and if y * is GAS in R m for the system ẏ = g(x * , y), then (x * , y * ) is locally asymptotically stable for [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF]. Moreover, if all the trajectories of ( 9) are forward bounded, then (x * , y * ) is GAS for [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF].

We shall need below the Theorem 4.5 (Hirsch [11], Theorem 6.1). Let F be a C 1 vector field in R q , whose flow φ preserves R q + for t ≥ 0 and is strongly monotone in R q + . Assume that the origin is an equilibrium and that all trajectories in R q + are bounded. If the matrixvalued map DF : R q -→ R q × R q is strictly decreasing, in the sense that if x < y then DF (x) > DF (y), then either all trajectories in R q + \{0} tend to the origin, or there is a unique equilibrium p * , (p * 0) in the interior of R q + and all trajectories in R q + \{0} tend to p * . Now, we are in a position to prove the main result of this section.

K = (u 1 , • • • , u ), (v 1 , • • • , v ) ∈ R + ×R + : 0 ≤ v i ≤ u i , 1 ≤ i ≤ and i=1 u i = 1
is a compact positively invariant for the system [START_REF] Britton | Stochastic epidemic models with inference[END_REF]. Define Notice that DL(I) is an irreducible Metzler matrix. Since B ≥ 0 and B = 0, DL is clearly stricly decreasing with respect of I. Applying Theorem 4.5, we deduce that either all trajectories in K tend to the origin, or there is a unique equlibrium in the interior of K and all trajectories in K\([0, ∞) × {0} ) tend to this equilibrium.

We introduce the stability modulus α(M ) of a matrix M , which is the largest real part of the elements of the spectrum Spec(M) of M :

α(M ) = max δ∈Spec(M ) Re(δ).
From Theorem 3.13 of Varga [START_REF] Varga | Matrix Iterative Analysis[END_REF] (chapter 3), R 0 > 1 is equivalent to α(B+V) > 0, and the disease free equilibrium is unstable in this case. It then follows from Theorem 4.5 that there exists a unique attracting endemic equilibrium I * = 0, which satisfies

(νD -A + B)I * -diag(1/N * j )diag(I * )B I * = 0. ( 10
)
Since B is a non-negative matrix and I * = 0, by using [START_REF] Fall | Epidemiological models and Lyapunov functions[END_REF], it follows that

DL(I * )I * = -diag(1/N * j )diag(BI * ) I * < 0. ( 11
)
Using the fact that DL(I * ) is a Metzler matrix, [START_REF] Hirsch | The dynamical systems approach to differential equations[END_REF] implies that it is stable (Berman & Plemmons [START_REF] Berman | Nonnegative Matrices in the Mathematical Sciences[END_REF]: criteron I 28 of Theorem 6.2.3). The stability modulus then satisfies α DL(I * ) < 0. This proves the local asymptotic stability of I * . Since the attractivity of I * is guaranteed by Hirsh's Theorem 4.5, we conclude that the endemic equilibrium I * is globally asymptotically stable if R 0 > 1.

Let us mention that, under the assumption ν I = ν S := ν, the DFE is globally assymptotically stable when R 0 = 1. Indeed, R 0 = 1 is equivalent to α(B + V) = 0. But, since B + V is an irreducible Metzler matrix, there exists a positive vector v such that (B + V) T v = 0. Let us consider the Lyapunov function L(I) = v | I . The derivative of this function is

L(I) = v | İ = v | B + V -diag(1/N * j )diag(I(t))B I(t) = -v | diag(1/N * j )diag(I(t))B I(t) ≤ 0.
Moreover, L(I) = 0 only at the DFE. Hence the DFE is GAS if R 0 = 1.

5.

Comparison of the equilibria: connected patches model, homogeneous model, isolated patches. We now consider the deterministic model in an homogeneous community:

       d S dt (t) = -λS(t)I(t) + γ I(t) d I dt (t) = λS(t)I(t) -γ I(t), (12)
where λ (resp. γ) is the rate of the disease transmission (resp. recovery).

The endemic equilibrium of the system of ODEs ( 12) is z * = γ λ , 1 -γ λ . We wish to compare this EE with the one of the deterministic heterogeneous model.

• First, if the disease transmission and recovery rates are the same on all patches, that is, for all j = 1, • • • , , λ j = λ and γ j = γ, then the EE of the ODEs (1) is

z * = γ λ , 1 1 - γ λ , • • • , γ λ , 1 1 - γ λ .
In this case, we note that the proportion of the infectious subpopulation in the hemogeneous model is equally distributed between all patches.

• We now look at the case where the patches have different disease transmission and recovery rates. In this case it is difficult to obtain an explicit expression for the EE, even for a small number of patches. But it can be found relatively simply using any numerical solver when the state space is small. Here, we consider the case of two patches and use the solver "Wolfram Alpha" to compute the EE.

In the Table 1, we give the proportion of the infectious subpopulation in each patch at the equilibrium for several values of the parameters. We take γ 1 = γ 2 = 1 and consider three cases. First λ 1 = 1.5, λ 2 = 2. In this case when, the patches are isolated, the value of the infectious subpopulation in patch 1 and patch 2 are I * 1 ≈ 0.333 and I * 2 ≈ 0.5, respectively. Secondly λ 1 = 3, λ 2 = 2.5, and I * 1 ≈ 0.666 and I * 2 ≈ 0.600 in isolated patches. Finally, in the case λ 1 = 1.5, λ 2 = 1.2, we have I * 1 ≈ 0.333 and I * 2 ≈ 0.166. In the above table, we have the proportions of the infectious subpopulation in each patch at the equilibrium, for some values of the diffusion coefficients. We observe that those proportions are very close to those when patches are isolated.

We have shown that the stochastic model is well approximated by a deterministic patch model. If R 0 > 1, the system of ODE (1) has a unique endemic equilibrium which is globally asymptotically stable. Moreover, considering the case of two patches, it appears that in the heterogeneity case, the final size of the epidemic in each patch is close to that of isolated patches, when the diffusion coefficients are small. In a future work, we will study the fluctuations of the stochastic model around its deterministic law of large numbers limit.

  and we shall assume that the process {Z N (t), t ≥ 0} is defined on the filtered probability space Ω, F, F N t , P . In what follows, u denotes the L 1 norm of an -dimensional vector u. More precisely, u = j=1 |u j |. The following theorem shows that the solution of the stochastic model (3) converges a.s. locally uniformly in t to the solution of a deterministic model, as the total population size N tends to infinity. Theorem 3.1. [Law of Large Numbers] Let Z N denote the solution of the SDEs (3) and z the unique solution of the ODEs (1). Let us fix an arbitrary T > 0. Then sup 0≤t≤T Z N (t) -z(t) -→ 0 a.s. , as N → ∞.

Lemma 4 . 3 .

 43 The system d N(t) dt = νDN(t) has a unique global asymptotically stable equilibrium.

Theorem 4 . 6 .

 46 [Existence and stability of the EE] Assume that ν I = ν S := ν and R 0 > 1. Then the system (1) has a unique endemicequilibrium z * = S * 1 , I * 1 , S * 2 , I * 2 • • • , S * , I * ,which is globally asymptotically stable. Proof. It follows from Theorem 4.4 that it is sufficient to study the stability of the reduced system d I(t) dt = νD I(t) -A I(t) + I -diag(1/N * j )diag(I(t)) B I(t). Note that the set defined by

L

  (I) = (B + V)I -diag(1/N * j )diag(I)B I. The derivative DL(I) is DL(I) = (B + V) -diag(1/N * j )diag(I)B -diag(1/N * j )diag(B I) = νD -A + B -diag(1/N * j )diag(I)B -diag(1/N * j )diag(B I).

Table 1 .

 1 proportion of I * 1 and I * 2 when patches are connected

	λ 1	λ 2	γ 1	γ 2	ν I	ν S	I * 1 1 + I * S * 1	,	I * 2 2 + I * S * 2
	1.5	2	1	1	0.0001	0.0001	(0.332 , 0.507)
	1.5	2	1	1	0.0001	0.0005	(0.334, 0.497)
	1.5	2	1	1	0.001	0.0001	(0.333, 0.497)
	1.5	2	1	1	0.0001	0.001	(0.332, 0.497)
	3	2.5	1	1	0.0001	0.0001	(0.667 , 0.598)
	3	2.5	1	1	0.0007	0.0001	(0.666 , 0.599)
	3	2.5	1	1	0.001	0.0001	(0.666 , 0.598)
	3	2.5	1	1	0.0001	0.001	(0.666 , 0.598)
	1.5	1.2	01	1	0.0001	0.0001	(0.332 , 0.165)
	1.5	1.2	1	1	0.0001	0.0009	(0.332 , 0.165)
	1.5	1.2	1	1	0.001	0.0001	(0.333 , 0.165)
	1.5	1.2	1	1	0.0001	0.008	(0.332 , 0.165)

Acknowledgments. I wish to thank the referee for his valuable advices and suggestions which helped me greatly in the writing of the revised version. I also would like to thank Prof. Etienne Pardoux for careful reading and plentiful suggestions that greatly improved the paper.

Funding. This research was supported by a thesis scholarship from the government of Ivory Coast, and a salary as instructor at University of Aix-Marseille.


Edlira Nano

Benoit Valot

Olivier Langella

Ludovic Bonhomme

Mélisande Blein

Michel Zivy

MassChroQ : A versatile tool for mass spectrometry quantification

MassChroQ : A versatile tool for spectrometry quantification B. Valot, O. Langella, E. Nano, M. Zivy, Proteomics, 2011, 11:3572-3577.

Abstract

Recently, many software tools have been developed to perform quantification in LC-MS analyses. However, most of them are specific to either a quantification strategy (e.g. label-free or isotopic labelling) or a mass-spectrometry system (e.g. high or low resolution). In this context, we have developed Mass-ChroQ, a versatile software that performs LC-MS data alignment and peptide quantification by peak area integration on extracted ion chromatograms. MassChroQ is suitable for quantification with or without labelling and is not limited to high resolution systems. Peptides of interest (for example all the identified peptides) can be determined automatically or manually by providing targeted m/z and retention time values. It can handle large experiments that include protein or peptide fractionation (as SDS-PAGE, 2D-LC). It is fully configurable. Every processing step is traceable, the produced data are in open standard format and its modularity allows easy integration into proteomic pipelines. The output results are ready for use in statistical analyses. Evaluation of MassChroQ on complex label-free data obtained from low and high resolution mass spectrometers showed low CVs for technical reproducibility (1.4%) and high coefficients of correlation to protein quantity (0.98). MassChroQ is freely available under the GNU General Public Licence v3.0 at http://pappso.inra.fr/bioinfo/masschroq/. This paper is freely available for download at http://pappso/downloads/masschroq/masschroq_article.pdf.


David J Moss

Keywords: Nonlinear optics, integrated waveguides, selfphase modulation, graphene oxide. 

We experimentally demonstrate enhanced self-phase modulation (SPM) in silicon nitride (Si3N4) waveguides integrated with 2D graphene oxide (GO) films. GO films are integrated onto Si3N4 waveguides using a solution-based, transfer-free coating method that enables precise control of the film thickness. Detailed SPM measurements are carried out using both picosecond and femtosecond optical pulses. Owing to the high Kerr nonlinearity of GO, the hybrid waveguides show significantly improved spectral broadening compared to the uncoated waveguide, achieving a broadening factor of up to ~3.4 for a device with 2 layers of GO. By fitting the experimental results with theory, we obtain an improvement in the waveguide nonlinear parameter by a factor of up to 18.4 and a Kerr coefficient (n2) of GO that is about 5 orders of magnitude higher than Si3N4. Finally, we provide a theoretical analysis for the influence of GO film length, coating position, and its saturable absorption on the SPM performance. These results verify the effectiveness of on-chip integrating 2D GO films to enhance the nonlinear optical performance of Si3N4 devices.

I. INTRODUCTION

elf-phase modulation (SPM) is a fundamental third-order nonlinear optical process that occurs when an optical pulse travelling in a nonlinear medium, where a varying refractive index of the medium is induced by the Kerr effect, thus produces a phase shift that leads to a change in the pulse's spectrum [START_REF] Stolen | Self-phase-modulation in silica optical fibers[END_REF][START_REF] Agrawal | Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers[END_REF][START_REF] Wu | Recent advances of spatial self-phase modulation in 2D materials and passive photonic device applications[END_REF]. It has been widely used as a relevant all-optical modulation technology for a variety of applications in broadband optical sources [START_REF] Guo | Mid-infrared frequency comb via coherent dispersive wave generation in silicon nitride nanophotonic waveguides[END_REF][START_REF] Yu | A broadband, quasi-continuous, mid-infrared supercontinuum generated in a chalcogenide glass waveguide[END_REF], optical spectroscopy [START_REF] Hult | High bandwidth absorption spectroscopy with a dispersed supercontinuum source[END_REF][START_REF] Kaltenecker | Near-infrared nanospectroscopy using a low-noise supercontinuum source[END_REF], pulse compression [START_REF] Krebs | Pulse compression of ultrashort UV pulses by self-phase modulation in bulk material[END_REF][START_REF] Peccianti | Subpicosecond optical pulse compression via an integrated nonlinear chirper[END_REF], optical logic gates [START_REF] Wu | 1D@ 0D hybrid dimensional heterojunction-based photonics logical gate and isolator[END_REF][START_REF] Singh | All-optical logic gates: designs, classification, and comparison[END_REF], optical modulators / switches [START_REF] Wu | 2D tellurium based high-performance all-optical nonlinear photonic devices[END_REF][START_REF] Wu | 2D MXene: MXene-based nonlinear optical information converter for all-optical modulator and switcher[END_REF], optical diodes [START_REF] Dong | Saturable absorption in 2D Ti3C2 MXene thin films for passive photonic diodes[END_REF][START_REF] Konorov | Experimental demonstration of a photonic-crystal-fiber optical diode[END_REF], and optical coherence tomography [START_REF] Moon | Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source[END_REF][START_REF] Sumimura | Quasisupercontinuum generation using 1.06 μmultrashort-pulse laser system for ultrahigh-resolution optical-coherence tomography[END_REF].

The ability to realize SPM based on-chip integrated photonic devices will reap attractive benefits of compact footprint, high stability, high scalability, and low-cost mass production [START_REF] Dulkeith | Self-phasemodulation in submicron silicon-on-insulator photonic wires[END_REF][START_REF] Boyraz | Self-phase-modulation induced spectral broadening in silicon waveguides[END_REF][START_REF] Tan | Group velocity dispersion and self phase modulation in silicon nitride waveguides[END_REF][START_REF] Duchesne | Efficient self-phase modulation in low loss, high index doped silica glass integrated waveguides[END_REF]. Although silicon (Si) has been a dominant device platform for integrated photonics [START_REF] Bogaerts | Silicon photonics circuit design: methods, tools and challenges[END_REF][START_REF] Feng | Silicon photonics: from a microresonator perspective[END_REF][START_REF] Reed | Silicon optical modulators[END_REF], its strong two-photon absorption (TPA) at near-infrared wavelengths leads to a low nonlinear figure-of-merit (FOM = n2 / (λβTPA), where n2 is the Kerr nonlinearity, βTPA is the two photon absorption coefficient, and λ the wavelength) of ~0.3 [START_REF] Leuthold | Nonlinear silicon photonics[END_REF], which significantly limits the SPM performance of Si devices in the telecom band. To address this, other complementary metal-oxide-semiconductor (CMOS) compatible integrated platforms such as silicon nitride compared to other 2D materials (over 2 orders of magnitude lower than graphene [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF]), facile synthesis processes [START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF][START_REF] Yang | Graphene-based multilayered metamaterials with phototunable architecture for on-chip photonic devices[END_REF], and high compatibility with CMOS fabrication [START_REF] Wu | 2D layered graphene oxide films integrated with micro-ring resonators for enhanced nonlinear optics[END_REF][START_REF] Jia | Fabrication technologies for the on-chip integration of 2D materials[END_REF]. In our previous work [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF], we demonstrated enhanced SPM of picosecond optical pulses in Si waveguides integrated with 2D GO films, achieving a maximum spectral broadening factor (BF) of ~4.3 and enhanced FOM by up to 20 times.

In this paper, we demonstrate significantly improved SPM performance for Si3N4 waveguides integrated with 2D GO films. By using a solution-based, transfer-free coating method, we achieve on-chip integration of GO films with precise control of their thicknesses. We perform SPM measurements using both picosecond and femtosecond optical pulses centered at telecom wavelengths. Compared to the uncoated Si3N4 waveguide, the GO-coated waveguides show more significant spectral broadening for both the picosecond and femtosecond optical pulses, achieving a maximum BF of ~3.4 for a device with 2 layers of GO. We also fit the SPM experimental results with theory and obtain a Kerr coefficient (n2) for GO that is about 5 orders of magnitude higher than Si3N4. Finally, we discuss the influence of GO film's length, coating position, and saturable absorption on the SPM performance. For Si waveguides the main challenge is to enhance the nonlinear FOM, whereas for Si3N4 waveguides the challenge is to enhance the nonlinear parameter  (= 2πn2 / (λAeff), where Aeff is the effective mode area) since their FOM is very large already.

We obtain an enhancement in  by a factor of up to ~18.4 for a Si3N4 waveguide coated with 2 layers of GO, compared to the uncoated waveguide, accompanied by only a modest increase in the linear loss of about 3 dB/cm per layer of GO and with no measurable decrease in the nonlinear FOM. These results confirm the high nonlinear optical performance of Si3N4 waveguides integrated with 2D GO films.

II. DEVICE FABRICATION AND CHARACTERIZATION

Fig. 1(a) shows a schematic of a GO-coated Si3N4 waveguide with a monolayer GO film. The bare Si3N4 waveguide has a cross-section of 1.6 μm × 0.66 μm, which was fabricated via a CMOS-compatible annealing-free and crack-free method [START_REF] Dirani | Annealing-free Si3N4 frequency combs for monolithic integration with Si photonics[END_REF][START_REF] Demongodin | Ultrafast saturable absorption dynamics in hybrid graphene/Si3N4 waveguides[END_REF]. First, two-step deposition of Si3N4 film (330-nm-thick layer in each step) was achieved via low-pressure chemical vapor deposition (LPCVD) for strain management and crack prevention. Next, 248-nm deep ultraviolet lithography and CF4/CH2F2/O2 fluorine-based dry etching were employed for patterning the low-loss Si3N4 waveguides. A silica upper cladding was then deposited using high-density plasmaenhanced chemical vapor deposition (HDP-PECVD), followed by opening a window on it down to the top surface of the Si3N4 waveguides via lithography and reactive ion etching (RIE). Finally, the 2D layered GO film was coated onto the Si3N4 waveguide by using a solution-based method that enabled transfer-free and layer-by-layer film coating, as reported previously [START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF][START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF][START_REF] Yang | Graphene-based multilayered metamaterials with phototunable architecture for on-chip photonic devices[END_REF][START_REF] Jia | Fabrication technologies for the on-chip integration of 2D materials[END_REF]. Compared to the sophisticated film transfer processes employed for on-chip integration of other 2D materials such as graphene and TMDCs [START_REF] Feng | Enhanced optical Kerr nonlinearity of graphene/Si hybrid waveguide[END_REF][START_REF] Liu | Enhanced optical Kerr nonlinearity of MoS2 on silicon waveguides[END_REF][START_REF] Yang | Bottom-up fabrication of graphene on silicon/silica substrate via a facile soft-hard template approach[END_REF], our GO coating method is highly scalable, enabling precise control of the GO layer number (i.e., film thickness), large-area film coating, and good film attachment on integrated chips [START_REF] Wu | Graphene oxide for integrated photonics and flat optics[END_REF][START_REF] Yang | Graphene-based multilayered metamaterials with phototunable architecture for on-chip photonic devices[END_REF].

Figs. 1(b-i) and (b-ii) show a schematic cross section and the transverse electric (TE) mode profile of the GO-coated Si3N4 waveguide in Fig. 1(a), respectively. The interaction between light and the GO film possessing an ultrahigh Kerr nonlinearity can be excited by the waveguide evanescent field, which underpins the enhancement of the SPM response in the hybrid waveguide.

Fig. 1(c) shows a microscope image of a Si3N4 integrated chip uniformly coated with a monolayer GO film, where the coated GO film exhibits good morphology, high transmittance, and high uniformity. The opened window on the silica upper cladding of the uncoated Si3N4 chip enables control of the film length and placement of the GO film that are in contact with the Si3N4 waveguide. Note that this can also be realized by patterning GO films on planarized Si3N4 waveguides (without silica upper cladding) via lithography and lift-off processes, as we did in our previous work [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF]. In this work, we used Si3N4 waveguides with opened windows mainly because they have lower coupling loss and propagation loss, which is beneficial for boosting the nonlinear response of SPM.

Figs. 1(d-i) and (d-ii) show the measured Raman spectra of a Si3N4 chip before and after coating 2 layers of GO, respectively, where the presence of the representative D and G peaks of GO in the latter one verifies the successful on-chip integration of the GO film [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF]. According to our previous measurements [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF][START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF], the GO film thickness shows a near linear relationship with layer number at small film thickness (i.e., layer numbers < 100), and the thickness for 1 layer of GO is ~2.0 nm. For the GO-coated Si3N4 waveguides used in the following SPM measurements, the measured film thicknesses for 1 and 2 layers of GO are ~2.1 nm and ~ 4.3 nm, respectively. III. LOSS MEASUREMENTS Fig. 2 shows the experimental setup used for measuring both loss and SPM of GO-coated Si3N4 waveguides. Three different laser sources were employed, including a tunable continuouswave (CW) laser and two different fiber pulsed lasers (FPLs) that can generate nearly Fourier-transform limited picosecond (pulse duration: ~1.9 ps) and femtosecond optical pulses (pulse duration: ~180 fs) centered at telecom wavelengths. An optical isolator was inserted after the laser source to prevent the reflected light from damaging it. A variable optical attenuator (VOA) and a polarization controller (PC) were used to tune the power and polarization of the input light, respectively. For both the loss and SPM measurements, TE polarization of input light injected into the device under test (DUT) was chosen because it supports in-plane interaction between the waveguide evanescent field and the 2D GO film, which is much stronger compared to the out-of-plane interaction given the significant optical anisotropy of 2D materials [START_REF] Gu | Regenerative oscillation and four-wave mixing in graphene optoelectronics[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF]. We used inversetaper couplers at both ends of the Si3N4 waveguide, which were butt coupled to lensed fibers to achieve light coupling into and out of the DUT.

For the loss measurements, the power of the light before and after passing the DUT was measured by two optical power meters, i.e., OPM 1 and OPM 2. All the three laser sources were used to measure the loss of bare and GO-coated Si3N4 waveguides. The corresponding results are compared in Fig. 3.

Fig. 3(a) shows the insertion loss (ILCW) of GO-coated Si3N4 waveguides versus input CW light power. Unless otherwise specified, the input power of CW light or optical pulses in this paper represents the power coupled into the waveguide after excluding the fiber-to-chip coupling loss. We measured the hybrid waveguides with 1 and 2 layers of GO (i.e., layer number N = 1, 2), and the corresponding results for the uncoated Si3N4 waveguide (N = 0) are also shown for comparison. The total length of the Si3N4 waveguide was 20 mm and the length of the opened window was 1.4 mm. The opened window started at 0.7 mm after the light input port.

As can be seen, the insertion loss does not show any obvious variation with the power of the input CW light, reflecting that the power-dependent loss induced by the photo-thermal changes in the GO films is negligible. This is because the photo-thermal changes are sensitive to the average light power coupling into the GO-coated waveguides [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Wu | 2D layered graphene oxide films integrated with micro-ring resonators for enhanced nonlinear optics[END_REF], and the average power of the input CW light here (< 7 mW) is much lower than those inducing significant photo-thermal changes (> 40 mW) in previous work [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Wu | 2D layered graphene oxide films integrated with micro-ring resonators for enhanced nonlinear optics[END_REF][START_REF] Zhang | Design and optimization of four-wave mixing in microring resonators integrated with 2D graphene oxide films[END_REF].

According to the results in Fig. 3(a), the excess propagation losses induced by the GO films are ~3.0 dB/cm and ~6.1 dB/cm for the hybrid waveguides with 1 and 2 layers of GO, respectively. These values are slightly higher than those of GOcoated dope silica waveguides [START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF][START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF] but lower than those of GO-coated Si waveguides [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF], mainly due to the moderate GO mode overlap in the GO-coated Si3N4 waveguides. It is also worth mentioning that the GO-induced excess propagation loss is about 100 times lower than graphene-induced excess propagation loss in graphene-coated Si3N4 waveguides [START_REF] Demongodin | Ultrafast saturable absorption dynamics in hybrid graphene/Si3N4 waveguides[END_REF][START_REF] Alexander | Electrically tunable optical nonlinearities in graphenecovered SiN waveguides characterized by four-wave mixing[END_REF], highlighting the low material absorption of GO compared to graphene and its advantage for implementing nonlinear photonic devices with relatively low loss. The low loss of GO is mainly induced by its large bandgap, which is typically between 2.1 eV -3.6 eV [START_REF] Wu | Graphene oxide for integrated photonics and flat optics[END_REF][START_REF] Lee | Graphene oxide/PEDOT: PSS composite hole transport layer for efficient and stable planar heterojunction perovskite solar cells[END_REF][START_REF] Rafique | Significantly improved photovoltaic performance in polymer bulk heterojunction solar cells with graphene oxide/PEDOT: PSS double decked hole transport layer[END_REF]. In principle, GO with a bandgap > 2 eV has negligible linear absorption below its bandgap, e.g., at near-infrared wavelengths (with a photon energy of ∼0.8 eV at 1550 nm). The light absorption of practical GO films is mainly caused by defects as well as scattering loss stemming from imperfect layer contact and film unevenness [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF][START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Wu | Graphene oxide waveguide and micro-ring resonator polarizers[END_REF].

Figs. 3(b-i) and 3(b-ii) show the insertion loss (ILpulse) of GOcoated Si3N4 waveguides versus input power of picosecond and femtosecond optical pulses, respectively. In contrast to CW light that has a peak power equaling to its average power, picosecond and femtosecond optical pulses have peak powers that are much higher than their average powers. Both the picosecond and femtosecond FPLs we used had the same repetition rate of ~60 MHz. For the picosecond pulses, the average input power ranged between 0.8 mW and 2.3 mW, which corresponded to a peak power range of 7 W -20 W. For the femtosecond pulses, the average input power ranged between 1.1 mW and 1.7 mW, which corresponded to a peak power range of 98 W -160 W. Given that the average powers of the picosecond and femtosecond optical pulses are in the same level as that of the CW light in Fig. 3(a), the photo-thermal changes in the GO films can be neglected when these optical pulses go through the hybrid waveguides.

In both Figs. 3(b-i) and 3(b-ii), the measured ILpulse of GOcoated Si3N4 waveguides decreases with the input power of optical pulses, and the waveguide with 2 layers of GO shows a more obvious decrease than the waveguide with 1 layer of GO.

In contrast, the result for the uncoated Si3N4 waveguide does not show such a trend. This indicates that there is saturable absorption (SA) induced by the GO films in the hybrid waveguides. Similar phenomenon has also been observed for the GO-coated Si waveguides [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF] and graphene-coated Si3N4 waveguides [START_REF] Demongodin | Ultrafast saturable absorption dynamics in hybrid graphene/Si3N4 waveguides[END_REF]. In our experiment, we also note that the change in the loss of the hybrid waveguides was not permanent, and the measured ILpulse in Fig. 3(b) is repeatable.

Fig. 3(c) depicts the SA-induced excess propagation loss (∆SA, after excluding the linear propagation loss) versus the peak power of the input optical pulses, which is extracted from the results in Figs. 3 (a) and (b). The negative values of ∆SA indicate that the loss decreases with light power, showing an opposite trend to TPA where the loss increases with light power [START_REF] Leuthold | Nonlinear silicon photonics[END_REF][START_REF] Autere | Nonlinear optics with 2D layered materials[END_REF][START_REF] Wu | RF Photonics: An optical microcombs' perspective[END_REF]. The decreased loss induced by SA can facilitate more significant SPM driven by a high optical power. For femtosecond pulses, the decrease in loss is more significant than that for picosecond pulses, which can be attributed to their relatively high peak power that induces more significant SA in the GO.

IV. SPM MEASUREMENTS

In the SPM measurements, we used the same FPLs and the same fabricated devices as those employed for loss measurements in Section III to measure the SPM-induced spectral broadening. As shown in Fig. 2, picosecond or femtosecond optical pulses generated by these FPLs were coupled into the DUT, and the output signal was sent to an optical spectrum analyzer (OSA) for observation of spectral broadening. The corresponding results for the picosecond and femtosecond optical pulses are compared in Figs. 4 and5, respectively.

Fig. 4(a) shows the normalized spectra of picosecond optical pulses before and after propagation through the uncoated and GO-coated Si3N4 waveguides. The peak power of the input picosecond optical pulses was kept the same at ∼20 W. The output spectrum from the uncoated Si3N4 waveguide shows slight spectral broadening compared to the input pulse spectrum, which is mainly induced by the SPM in the Si3N4 waveguide. In contrast, the output spectra after propagating through the GO-coated Si3N4 waveguides show more significant spectral broadening, reflecting the enhanced SPM in these hybrid waveguides.

Fig. 4(b) shows the output spectra after propagation through the hybrid waveguide with 2 layers of GO measured using picosecond optical pulses with different peak powers. We chose 6 different input peak powers ranging from 7 W to 20 W -the same as those in Fig. 3(b-i). As expected, the spectral broadening of the output spectra becomes more significant as the peak power increases.

To quantitively compare the spectral broadening in these waveguides, we calculated the BFs for the measured output spectra. The BF is defined as [START_REF] Vermeulen | Graphene's nonlinear-optical physics revealed through exponentially growing self-phase modulation[END_REF][START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF][START_REF] Demongodin | Ultrafast saturable absorption dynamics in hybrid graphene/Si3N4 waveguides[END_REF]: where ∆ω0 and ∆ωrms are the root-mean-square (RMS) spectral widths of the input and output signals, respectively. Fig. 4(c) shows the BFs for the uncoated and GO-coated Si3N4 waveguides versus the peak power of input picosecond optical pulses. As can be seen, the BFs for the GO-coated Si3N4 waveguides are higher than that of the uncoated waveguide, and the BF for the hybrid waveguide with 2 layers of GO is higher than that for the device with 1 layer of GO, showing agreement with the results in Fig. 4(a). The BF increases with the peak power of the optical pulses, which is consistent with the results in Fig. 4(b). At a peak power of ~20 W, a maximum BF of ~1.3 is achieved for the hybrid waveguide with 2 layers of GO.

BF= ∆ω rms ∆ω 0 (1)
Fig. 5(a) shows the normalized spectra of femtosecond optical pulses before and after propagation through the uncoated and GO-coated Si3N4 waveguides, which were measured at the same input peak power of ∼160 W. Similar to Fig. 4(a), the output spectra after passing through the hybrid waveguides show more significant spectral broadening compared to the uncoated waveguide. achieved at a peak power of 160 W for the hybrid waveguide with 2 layers of GO, which is ~2.6 times higher than the maximum BF achieved for the picosecond optical pulses. This mainly results from the relatively high peak power of the femtosecond optical pulses that drives more significant SPM in the hybrid waveguide.

V. THEORETICAL ANALYSIS AND DISCUSSION

Based on the theory in Refs. [START_REF] Duchesne | Efficient self-phase modulation in low loss, high index doped silica glass integrated waveguides[END_REF][START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF][START_REF] Yin | Impact of two-photon absorption on self-phase modulation in silicon waveguides[END_REF], we simulated the evolution of optical pulses traveling along the GO-coated Si3N4 waveguides using the nonlinear Schrodinger equation as follows:

∂A ∂z = - iβ 2 2 ∂ 2 A ∂t 2 + iγ |A| 2 A - 1 2 αA ( 2
)
where i = √1, A(z, t) is the slowly varying temporal pulse envelope along the propagation direction z, β2 is the secondorder dispersion coefficient, and γ is the waveguide nonlinear parameter. The overall loss factor α includes both the linear propagation loss and the SA-induced excess propagation loss that are discussed in Fig. 3. Unlike in Refs. [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF][START_REF] Yin | Impact of two-photon absorption on self-phase modulation in silicon waveguides[END_REF], there are no free carrier absorption (FCA) and free carrier dispersion (FCD) items in Eq. ( 2) since the TPA in both Si3N4 and GO (with bandgaps > 2 eV) is negligible at near-infrared wavelengths. We retain only the second-order dispersion item in Eq. ( 2) because the physical length of the waveguides (20 mm) is much smaller than the dispersion length (> 1 m) [START_REF]Nonlinear fiber optics[END_REF]. In our simulation, we divided the GO-coated Si3N4 waveguides into uncoated (with silica cladding) and hybrid segments (coated with 1.4-mm-long GO films). Numerically solving Eq. ( 2) was performed for each segment, and the output from the previous segment was set as the input for the subsequent one.

Figs. 6(a) and (b-i) show the measured and fit spectra of the input picosecond pulses and the output signal after propagation through the uncoated Si3N4 waveguide, respectively. The simulated spectrum evolution of the input optical pulses propagating along the uncoated waveguide is shown in Fig. 6(bii). The peak power of the input picosecond pulses is ~20 W. The fit spectra and spectrum evolution were calculated based on Eq. ( 2), which show good agreement with the experimental results. The slight discrepancies between the measured and fit spectra mainly result from imperfections of the input pulse spectrum. The fit γ for the uncoated Si3N4 waveguide is ~1.5 W - 1 m -1in agreement with the reported values in previous literature [START_REF] Tan | Group velocity dispersion and self phase modulation in silicon nitride waveguides[END_REF][START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Dirani | Crack-free silicon-nitride-on-insulator nonlinear circuits for continuum generation in the C-Band[END_REF].

Figs. 6(c-i) and (d-i) show the measured and fit spectra for the output signals after transmission through the hybrid waveguides with 1 and 2 layers of GO, respectively. The input peak power is the same as that in Fig. 6(b). The corresponding spectrum evolutions along the hybrid waveguides are shown in Figs. 6(c-ii) and(d-ii). As can be seen, the theoretical simulations also agree well with the experimental results. The fit γ's for the hybrid waveguides with 1 and 2 layers of GO are ~11.5 and ~27.6, respectively, which are ~7.7 and ~18.4 times that of the uncoated Si3N4 waveguide, reflecting the significantly improved Kerr nonlinearity for the hybrid waveguides. The significant Kerr nonlinearity is also confirmed by the dramatical spectral broadening within the GO-coated region, as shown in the insets of Figs. 6(c-ii) and(d-ii).

Similar to Fig. 6, we also performed simulations based on Eq. ( 2) to fit the experimental results of femtosecond optical pulses. The measured and fit spectra of the input femtosecond pulses and the output signal after propagation through the uncoated Si3N4 waveguide are shown in Figs. 7(a) and(b-i) respectively. Figs. 7(c-i) and(d-i) show the measured and fit spectra for the output femtosecond signals after transmission through the hybrid waveguides with 1 and 2 layers of GO, respectively. The simulated spectrum evolution of the input pulses corresponding to Figs. 7(b-i), 7(c-i), and 7(d-i) are shown in Figs. 7(b-ii), 7(c-ii), and 7(d-ii), respectively. In all of these figures, the peak power of the input femtosecond pulses is ~160 W. As can be seen, all the theoretical curves show good agreement with the experimental ones. The fit γ for the uncoated Si3N4 waveguide and the hybrid waveguides with 1 and 2 layers of GO are the same as those obtained by fitting the experimental results of picosecond optical pulses in Fig. 6, highlighting the high consistency and further confirming the enhanced Kerr nonlinearity for the hybrid waveguides. Similar to the of Figs. 6(c-ii) and (d-ii), there is also dramatical spectral broadening within the GO-coated region in the insets of Figs.

7(c-ii) and (d-ii).

Based on the fit γ's of the hybrid waveguides, we further extract the Kerr coefficient (n2) of the layered GO films using [START_REF] Yang | Invited article: Enhanced four-wave mixing in waveguides integrated with graphene oxide[END_REF][START_REF] Donnelly | Ultra-large nonlinear parameter in graphenesilicon waveguide structures[END_REF][START_REF] Ji | Enhanced parametric frequency conversion in a compact silicon-graphene microring resonator[END_REF] :

γ = 2π λ c ∬ n 0 2 (x, y)n 2 (x, y)S z 2 D dxdy [∬ n 0 (x, y)S z D dxdy] 2 (3)
where λc is the pulse central wavelength, D is the integral of the optical fields over the material regions, Sz is the time-averaged Poynting vector calculated using mode solving software, n0 (x, y) is the refractive index profiles calculated over the waveguide cross section and n2 (x, y) is the Kerr coefficient of the different material regions. The values of n2 for silica and Si3N4 used in our calculation were 2.60 × 10 -20 m 2 W -1 [START_REF] Moss | New CMOScompatible platforms based on silicon nitride and Hydex for nonlinear optics[END_REF] and 2.59 × 10 -19 m 2 W -1 , respectively, with the latter obtained by fitting the experimental results for the uncoated Si3N4 waveguide.

The fit γ's of the hybrid waveguides with 1 and 2 layers of GO are ~11.5 W -1 m -1 and ~27.6 W -1 m -1 , respectively, which are ~7.7 and ~18.4 times that of the uncoated Si3N4 waveguide. The extracted n2 of 1 and 2 layers of GO are ~1.23 × 10 -14 m 2 W -1 and ~1.19 × 10 -14 m 2 W -1 , respectively. Both of the values are about 5 orders of magnitude higher than that of Si3N4 and agree reasonably well with our previous measurements [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF]. Note that the n2 of 1 layer of GO is higher than that of 2 layers of GO. We infer this may result from the increased inhomogeneous defects within the GO layers and imperfect contact between the multiple GO layers. Nonetheless, the higher GO mode overlap for the thicker 2-layer film, compared to the single-layer film, resulted in a more than doubling of the nonlinear parameter γ.

Based on the SPM modeling in Eq. ( 2) and the fit parameters obtained from Figs. 6 and7, we further investigate the influence of GO film length (Lc) and coating position (L0) on the SPM performance of GO-coated Si3N4 waveguides. being achieved for the picosecond and femtosecond optical pulses, respectively. This reflects that there is a large room for improvement in the SPM-induced spectral broadening by increasing the GO film length and the input peak power. The BF can also be improved by coating thicker GO films (i.e., N > 2), which was used for increasing the FWM conversion efficiency in Ref. [START_REF] Qu | Enhanced four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF]. The increased GO film thickness will also lead to loss increase for the hybrid waveguides and hence creates a need to balance the trade-off between the Kerr nonlinearity and loss [START_REF] Qu | Analysis of four-wave mixing in silicon nitride waveguides integrated with 2D layered graphene oxide films[END_REF][START_REF] Zhang | Optimizing the Kerr nonlinear optical performance of silicon waveguides integrated with 2D graphene oxide films[END_REF]. In contrast, it decreases with L0, with the maximum value being achieved at L0 = 0. This indicates that the largest spectral broadening can be achieved by coating GO films at the beginning, as expected since the light power is highest at the start of the waveguide.

As discussed in Section III, the decreased loss induced by the SA in the GO films affects the SPM performance. In Figs. 9(a) and (b), we show the influence of the SA on the spectral broadening of picosecond and femtosecond optical pulses after propagation through the hybrid waveguide with 2 layers of GO, respectively. In each figure, the solid curve shows the result when considering the SA that induces a slightly reduced loss, whereas the dashed curve shows the result that was calculated using a constant linear loss of GO measured at low CW powers (i.e., the loss in Fig. 3(a)). As can be seen, the SA of GO has a positive influence and yields more significant spectral broadening for both the picosecond and femtosecond optical pulses. The difference for the femtosecond pulses is more obvious, showing a similar trend to the results for the loss decrease in Fig. 3(b) and reflecting that there is a more significant influence of the SA on the spectral broadening for optical pulses with higher peak powers. In Table I, we provide comparisons for the parameters of the GO-Si3N4 waveguides in this work and the GO-Si waveguides in Ref. [START_REF] Zhang | Enhanced Kerr nonlinearity and nonlinear figure of merit in silicon nanowires integrated with 2D graphene oxide films[END_REF]. We note that the trade-offs and challenges involved with integrating GO films into these two very different platforms, are in turn very different. Compared to the GO-Si waveguides, GO-Si3N4 waveguides have a larger waveguide geometry, which results in lower mode overlap with GO films. Such a reduced GO mode overlap yields a lower GO-induced excess propagation loss, at the expense of a weaker light-GO interaction. Despite this, the nonlinear parameter γ of the GO-Si3N4 waveguide with 1 layer of GO is still ~7.7 times that of the uncoated waveguide. In contrast, there is only about a 2-fold improvement in the γ of the GO-Si waveguide with 1 layer of GO. This mainly due to the relatively low n2 of Si3N4 compared to Si, reflecting that integrating GO onto Si3N4 waveguides has a more dramatic impact on improving the nonlinear performance. In contrast to Si that has strong TPA at near infrared wavelengths, the TPA of Si3N4 in this wavelength range is absent, which yields much higher values of nonlinear FOM for both the uncoated and GO-coated Si3N4 waveguides. Hence, the motivation in integrating GO films onto Si waveguides lies very much in increasing the nonlinear FOM, whereas for Si3N4 waveguides, the main benefit of integrating GO films is to increase the nonlinearity (i.e., nonlinear parameter γ) without introducing additional nonlinear loss.

Finally, our observation of SA in the GO-coated Si3N4 waveguides effectively equates to having a negative nonlinear FOM, and so the very concept and utility of introducing a FOM, as first proposed [START_REF] Mizrahi | Twophoton absorption as a limitation to all-optical switching[END_REF], arguably does not apply here. For both the Si and Si3N4 platforms, reducing the GO film loss further through improved fabrication and integration methods (e.g., by using GO solutions with improved purity and optimized flake sizes) will directly benefit the nonlinear performance of all devices that incorporate GO films. This work demonstrates that GO, with its high nonlinearity and nonlinear FOM and relatively low loss, is an extremely interesting and attractive material to enhance the nonlinear performance of photonic chips for many applications, including Kerr microcombs and their use for microwave and RF photonics [64-218], integrated nonlinear optics [START_REF] Ferrera | CMOS compatible integrated all-optical RF spectrum analyzer[END_REF][START_REF] Pasquazi | Sub-picosecond phase-sensitive optical pulse characterization on a chip[END_REF][START_REF] Kues | Passively modelocked laser with an ultra-narrow spectral width[END_REF][START_REF] Bao | Turing patterns in a fiber laser with a nested microresonator: Robust and controllable microcomb generation[END_REF][START_REF] Little | Very high-order microring resonator filters for WDM applications[END_REF][START_REF] Bao | Direct soliton generation in microresonators[END_REF][START_REF] Ferrera | On-Chip ultra-fast 1st and 2nd order CMOS compatible all-optical integration[END_REF][START_REF] Bao | Turing patterns in a fiber laser with a nested microresonator: Robust and controllable microcomb generation[END_REF][START_REF] Ferrera | On-chip CMOS-compatible all-optical integrator[END_REF][START_REF] Ferrera | Low Power CW Parametric Mixing in a Low Dispersion High Index Doped Silica Glass Micro-Ring Resonator with Qfactor > 1 Million[END_REF][START_REF] Pasquazi | Self-locked optical parametric oscillation in a CMOS compatible microring resonator: a route to robust optical frequency comb generation on a chip[END_REF][START_REF] Pasquazi | Stable, dual mode, high repetition rate modelocked laser based on a microring resonator[END_REF][START_REF] Bao | Direct soliton generation in microresonators[END_REF][START_REF] Frigg | Optical frequency comb generation low temperature reactive sputtered silicon nitride waveguides[END_REF][START_REF] Ferrera | On-chip CMOS-compatible all-optical integrator[END_REF][START_REF] Pasquazi | Efficient wavelength conversion and net parametric gain via Four Wave Mixing in a high index doped silica waveguide[END_REF][START_REF] Pasquazi | Efficient wavelength conversion and net parametric gain via Four Wave Mixing in a high index doped silica waveguide[END_REF][START_REF] Pasquazi | All-optical wavelength conversion in an integrated ring resonator[END_REF][START_REF] Peccianti | Subpicosecond optical pulse compression via an integrated nonlinear chirper[END_REF][START_REF] Pasquazi | Sub-picosecond phase-sensitive optical pulse characterization on a chip[END_REF] and integrated quantum optics [START_REF] Kues | Passively modelocked laser with an ultra-narrow spectral width[END_REF][START_REF] Kues | Quantum optical microcombs[END_REF][START_REF] Roztocki | Complex quantum state generation and coherent control based on integrated frequency combs[END_REF][START_REF] Sciara | Generation and Processing of Complex Photon States with Quantum Frequency Combs[END_REF][START_REF] Caspani | Multifrequency sources of quantum correlated photon pairs on-chip: a path toward integrated Quantum Frequency Combs[END_REF][START_REF] Reimer | Integrated frequency comb source of heralded single photons[END_REF][START_REF] Reimer | Cross-polarized photon-pair generation and bichromatically pumped optical parametric oscillation on a chip[END_REF][START_REF] Reimer | Generation of multiphoton entangled quantum states by means of integrated frequency combs[END_REF][START_REF] Kues | On-chip generation of high-dimensional entangled quantum states and their coherent control[END_REF][START_REF] Zhang | Induced photon correlations through superposition of two four-wave mixing processes in integrated cavities[END_REF][START_REF] Roztocki | Practical system for the generation of pulsed quantum frequency combs[END_REF][START_REF] Zhang | Induced photon correlations through superposition of two four-wave mixing processes in integrated cavities[END_REF][START_REF] Reimer | High-dimensional one-way quantum processing implemented on d-level cluster states[END_REF].

VI. CONCLUSION

We experimentally demonstrate enhanced SPM in Si3N4 waveguides integrated with 2D GO films. The integration of GO films is achieved by using a solution-based, transfer-free 10 coating method with precise control of the film thickness. SPM measurements are performed using both picosecond and femtosecond optical pulses. The GO-coated Si3N4 waveguides show more significant spectral broadening than the uncoated waveguide, with a maximum BF of ~3.4 being achieved for a device with 2 layers of GO. The experimental results show good agreement with theory, achieving up to ~18.4 times improvement in the waveguide nonlinear parameter compared to uncoated waveguide and a fit n2 of GO that is about 5 orders of magnitude higher than Si3N4. Analysis for the influence of GO film's length, coating position, and SA on the SPM performance is also provided. This work demonstrates that the Si3N4 can be effectively transformed into a highly performing CMOS-compatible nonlinear photonic platform by integrating 2D GO films.

Competing interests: The authors declare no competing interests.

Fig. 1 .

 1 Fig. 1. (a) Schematic illustration of a GO-coated Si3N4 waveguide with a monolayer GO film. (b-i) Schematic illustration of cross section and (b-ii) corresponding TE mode profile of the GO-coated Si3N4 waveguide in (a). (c) Microscope image of a Si3N4 integrated chip uniformly coated with monolayer GO film. (d) Raman spectra of a Si3N4 chip (i) before and (ii) after coating 2 layers of GO. Insets show the corresponding microscope images.

Fig. 2 .

 2 Fig. 2. Experimental setup for measuring loss and SPM of GO-coated Si3N4 waveguides. CW laser: continuous-wave laser. FPL: fiber pulsed laser. PC: polarization controller. VOA: variable optical attenuator. OPM: optical power meter. DUT: device under test. CCD: charged-coupled device. OSA: optical spectrum analyzer.

4

 4

Fig. 3 .

 3 Fig. 3. (a) Measured insertion loss (ILCW) of GO-coated Si3N4 waveguides versus input power of continuous-wave (CW) light. (b) Measured insertion loss (ILpulse) of GO-coated Si3N4 waveguides versus input power of optical pulses. (c) Excess propagation loss induced by the SA (∆SA) versus peak power of input optical pulses. In (b) and (c), (i) and (ii) show the results for picosecond and femtosecond optical pulses, respectively. In (a) -(c), the results for uncoated (N = 0) and hybrid Si3N4 waveguides coated with 1 and 2 layers of GO (N = 1, 2) are shown for comparison. The data points depict the average of measurements on three samples and the error bars illustrate the variations among the different samples.

Fig. 5 (

 5 b) shows the output spectra measured at different input peak powers for the hybrid waveguide with 2 layers of GO, showing the similar trend as that in Fig. 4(b). The peak power of the input femtosecond optical pulses ranged from 98 W to 160 Wthe same as those in Fig. 3(b-ii). The calculated BFs for the uncoated and GO-coated Si3N4 waveguides versus the input peak power are shown in Fig. 5(c). A maximum BF of ~3.4 is

Fig. 4 .

 4 Fig. 4. SPM experimental results using picosecond optical pulses. (a) Normalized spectra of optical pulses before and after propagation through the GO-coated Si3N4 waveguides with 1 and 2 layers of GO at an input peak power of ∼20 W. (b) Optical spectra measured at different input peak powers for the hybrid waveguides with 2 layers of GO. (c) BFs of the measured output spectra versus input peak power for the hybrid waveguides with 1 and 2 layers of GO. In (a) and (c), the corresponding results for the uncoated Si3N4 waveguides are also shown for comparison.

Fig. 5 .

 5 Fig. 5. SPM experimental results using femtosecond optical pulses. (a) Normalized spectra of optical pulses before and after propagation through the GO-coated Si3N4 waveguides with 1 and 2 layers of GO at an input peak power of ∼160 W. (b) Optical spectra measured at different input peak powers for the hybrid waveguides with 2 layers of GO. (c) BFs of the measured output spectra versus input peak power for the hybrid waveguides with 1 and 2 layers of GO. In (a) and (c), the corresponding results for the uncoated Si3N4 waveguides are also shown for comparison

Fig. 6 .

 6 Fig. 6. (a) Measured and fit spectra of input picosecond optical pulses. (b-i) Measured and fit output spectra after propagation through the uncoated Si3N4 waveguides. (b-ii) Simulated spectra evolution along the uncoated Si3N4 waveguide. (c-i) Measured and fit output spectra after propagation through the hybrid waveguide with 1 layer of GO. (c-ii) Simulated spectra evolution along the hybrid waveguide with 1 layer of GO. (d-i) Measured and fit output spectra after propagation through the hybrid waveguide with 2 layers of GO. (d-ii) Simulated spectra evolution along the hybrid waveguide with 2 layers of GO. Insets in (c-ii) and (d-ii) show zoomin views for the GO-coated regions. In (b) -(d), the peak power of the input picosecond pulses is ~20 W.

Fig. 7 .

 7 Fig. 7. (a) Measured and fit spectra of input femtosecond optical pulses. (b-i) Measured and fit output spectra after propagation through the uncoated Si3N4 waveguides. (b-ii) Simulated spectra evolution along the uncoated Si3N4 waveguide. (c-i) Measured and fit output spectra after propagation through the hybrid waveguide with 1 layer of GO. (c-ii) Simulated spectra evolution along the hybrid waveguide with 1 layer of GO. (d-i) Measured and fit output spectra after propagation through the hybrid waveguide with 2 layers of GO. (d-ii) Simulated spectra evolution along the hybrid waveguide with 2 layers of GO. Insets in (c-ii) and (d-ii) show zoom-in views for the GO-coated regions. In (b) -(d), the peak power of the input femtosecond pulses is ~160 W.

Figs. 8

 8 (a) and (b) show the calculated BFs versus Lc and input peak power (Ppeak) for picosecond and femtosecond optical pulses after propagation through the hybrid waveguides, respectively. In each figure, (i) and (ii) show the results for the waveguides with 1 and 2 layers of GO, respectively. The coating position is fixed at L0 = 0.7 mm -the same as those of the fabricated devices in Sections III and IV. The black points mark the parameters corresponding to the SPM measurements in Section IV, where the calculated BFs are consistent with the experimental results in Figs. 4 and 5. The BF increases with both Lc and Ppeak, with maximum BFs of 4.2 (at Lc = 19.3 mm and Ppeak = 30 W) and 25.0 (at Lc = 19.3 mm and Ppeak = 240 W)

Figs. 8

 8 Figs.8(c) and (d) show the calculated BFs versus L0 and Ppeak for picosecond and femtosecond optical pulses after propagation through the hybrid waveguides, respectively, where the film length is fixed at Lc = 1.4 mm. The simulation results marked by the black points also agree well with the experimental results in Figs.4 and 5. The BF increases with Ppeak -a trend similar to that in Figs.8(a) and (b). In contrast, it decreases with L0, with the maximum value being achieved at L0 = 0. This indicates that the largest spectral broadening can be achieved by coating GO films at the beginning, as expected since the light power is highest at the start of the waveguide.As discussed in Section III, the decreased loss induced by the SA in the GO films affects the SPM performance. In Figs. 9(a) and (b), we show the influence of the SA on the spectral broadening of picosecond and femtosecond optical pulses after propagation through the hybrid waveguide with 2 layers of GO, respectively. In each figure, the solid curve shows the result when considering the SA that induces a slightly reduced loss, whereas the dashed curve shows the result that was calculated using a constant linear loss of GO measured at low CW powers (i.e., the loss in Fig.3(a)). As can be seen, the SA of GO has a positive influence and yields more significant spectral broadening for both the picosecond and femtosecond optical pulses. The difference for the femtosecond pulses is more obvious, showing a similar trend to the results for the loss decrease in Fig.3(b) and reflecting that there is a more significant influence of the SA on the spectral broadening for optical pulses with higher peak powers.

Fig. 8 .

 8 Fig. 8. (a) BFs versus GO film length (Lc) and input peak power (Ppeak) for picosecond optical pulses after propagation through the hybrid waveguides. (b) BFs versus Lc and Ppeak for femtosecond optical pulses after propagation through the hybrid waveguides. (c) BFs versus GO coating position (L0) and Ppeak for picosecond optical pulses after propagation through the hybrid waveguides. (d) BFs versus L0 and Ppeak for femtosecond optical pulses after propagation through the hybrid waveguides. In (a) -(d), (i) and (ii) show the corresponding results for the waveguides with 1 and 2 layers of GO, the black points mark the results corresponding to the device parameters and input powers in Figs. 4(a) and 5(a). In (a) and (b), L0 = 0.7 mm. In (c) and (d), Lc = 1.4 mm.

Fig. 9 .

 9 Fig. 9. (a) Comparison of spectral broadening of picosecond optical pulses after propagation through the GO-Si3N4 waveguide with and without considering the SA of GO. (b) Comparison of spectral broadening of femtosecond optical pulses after propagation through the GO-Si3N4 waveguide with and without considering the SA of GO. In (a) and (b), N = 2, Lc = 1.4 mm, and L0 = 0.7 mm. The peak power for the picosecond and femtosecond optical pulses are 20 W and 160 W, respectively.